Week Thirteen Economic Dynamics The Problem • Economy is dynamic – Exists in time – Changes over time • But economists analyse it “as if” static—ignore.
Download ReportTranscript Week Thirteen Economic Dynamics The Problem • Economy is dynamic – Exists in time – Changes over time • But economists analyse it “as if” static—ignore.
Slide 1
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 2
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 3
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 4
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 5
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 6
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 7
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 8
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 9
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 10
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 11
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 12
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 13
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 14
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 15
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 16
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 17
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 18
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 19
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 20
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 21
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 22
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 23
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 24
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 25
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 26
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 27
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 28
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 29
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 30
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 31
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 32
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 33
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 34
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 35
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 36
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 37
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 38
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 39
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 40
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 41
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 42
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 43
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 44
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 45
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 46
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 47
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 2
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 3
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 4
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 5
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 6
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 7
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 8
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 9
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 10
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 11
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 12
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 13
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 14
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 15
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 16
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 17
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 18
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 19
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 20
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 21
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 22
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 23
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 24
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 25
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 26
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 27
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 28
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 29
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 30
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 31
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 32
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 33
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 34
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 35
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 36
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 37
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 38
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 39
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 40
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 41
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 42
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 43
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 44
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 45
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 46
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes
Slide 47
Week Thirteen
Economic Dynamics
The Problem
• Economy is dynamic
– Exists in time
– Changes over time
• But economists analyse it “as if” static—ignore time
– Some mathematicians (e.g., Blatt) see this as
“immaturity”
• How to reconcile dynamic reality & static methods?
– Argue static determines long term
• Short-term cycles explained by external shocks to
stable economic system
• Don’t! Develop dynamic, nonequilibrium economics instead
• Both approaches compete in literature
An Analogy
• Riding a bicycle
– do you need to know how to balance a stationary
bicycle before you can ride it?
• Economist: “Yes!: you must learn statics before you
can do dynamics”
– (1) Learn how to balance bike while stationary
– (2) Ride in straight line, using skills acquired in (1)
– (3) Turn bike…? How? Try handlebars
– (4) Fall flat on face!
• Real world: “No!”
– Dynamic art of riding bike exploits centripetal
forces which don’t exist when bike is stationary
– Static art of balancing irrelevant to dynamic art
of riding!
• So why do economists “do” Statics?
The early days: Statics “because it was easy”
• Historically, the “KISS” principle:
– “If we wished to have a complete solution ... we should
have to treat it as a problem of dynamics. But it would
surely be absurd to attempt the more difficult
question when the more easy one is yet so imperfectly
within our power.” (Jevons 1871 [1911]: 93)
– “...dynamics includes statics... But the statical
solution… is simpler...; it may afford useful preparation
and training for the more difficult dynamical solution;
and it may be the first step towards a provisional and
partial solution in problems so complex that a complete
dynamical solution is beyond our attainment.”
(Marshall, 1907 in Groenewegen 1996: 432)
• Founding fathers expected their successors to develop
dynamic analysis, working from their static foundations:
20th Century as the Century of Dynamics?
• “A point on which opinions differ is the capacity of the
pure theory of Political Economy for progress. There
seems to be a growing impression that, as a mere
statement of principles, this science will sonn be fairly
complete… It is with this view that I take issue. The
great coming development of economic theory is to take
place, as I venture to assert, through the statement and
the solution of dynamic problems.” (J.B. Clark [father of
marginal productivity theory of distribution] 1898: 1)
• Why does dynamics matter (according to Clark)?:
– “A static state is imaginary. All actual societies are
dynamic… Heroically theoretical is the study that
creates, in the imagination, a static society.” (Clark:
1898: 9)
20th Century as the Century of Dynamics?
– “It [dynamic analysis] will bring the society that
figures in our theory into a condition that is like that
of the real world. It will supply what a static theory
openly and intentionally puts out of sight; namely,
changes that alter the mode of production, and act on
the very structure of society itself.” (Clark 1898: 1011)
• Great expectations… but little done until Great
Depression
– Frisch and exogenous explanation for economic cycles
– Harrod’s endogenous explanation
The beginnings of dynamics
• Frisch 1933: trade cycle explained “by the fact that
certain exterior impulses hit the economic mechanism and
thereby initiate more or less regular oscillations”
– Underlying highly stable “propagation mechanism” (like
rocking horse)
– Random shocks from outside (“impulses”)
– Each shock sets up single regular harmonic pattern
(like stone in pool of water)
– Overlay of many shocks gives irregular cycles (lots of
stones)
• Gave rise to econometrics
– Dominant method: fit “linear stochastic” model to
economic data
Harrod: Growth & cycle theory
• Harrod (1939)
– Criticised Frisch paradigm
• Divorces growth from cycles when “the trend of
growth may itself generate forces making for
oscillation” (OREF II 38)
• Has no explanation for growth or shocks
– Developed combined theory of growth/cycle
– Basic method: extension of Keynes’s GT into dynamics
– His dynamic equilibrium unstable: nonequilibrium model
– Derivation starts from static Keynesian equality of S
and I:
Harrod’s “knife edge”
S I Divide both sides by Y
S I
Y
Multiply LHS by
Y Y
Y
S I Y
Rearranging:
Y Y Y
S
I Y
Restating:
Y Y Y
s c g
Savings
Rate of growth
ratio
Incremental stock to
output ratio (“ICOR”)
Harrod’s “knife edge”
• Types of growth
– Actual growth:
• g.cp=s
• cp actual ICOR: actual accumulation of stocks in
given period
– “Warranted growth”: what fulfilled capitalist
expectations
• gw.c=s
• c desired ICOR: ratio of change in stocks to rate of
growth that capitalists want
– “Natural”: maximum sustainable rate of growth
• gn
Harrod’s “knife edge”
• Reciprocal relation between g & gw:
– If actual growth exceeds warranted, then actual ICOR
(accumulation of stocks) less than desired ICOR:
• If g > gw, then cp < c since both g.cp=gw.c=s
• Capitalists will increase orders to restore desired
ICOR
• Growth accelerates
– If actual growth below warranted, then actual ICOR
(accumulation of stocks) more than desired ICOR:
• If g < gw, then cp > c
• Capitalists decrease orders to restore desired
ICOR
• Growth declines
– Dynamic equilibrium unstable
Harrod’s “knife edge”
• Explains growth and cycles
– If g>gw
• economy booms
• eventually hits overfull employment constraints
• economy turns down
– If g
• hits rock bottom
• need to replace equipment (depreciation) forces
+ive investment
• restarts upward pattern
Hicks “interprets” Harrod
• Hicks could not accept that equilibrium unstable:
– “A mathematically unstable system does not fluctuate;
it just breaks down” (OREF II 56)
• Reworked Harrod’s model:
– Define growth as g Yt Yt 1
Yt 1
– Desired I function of change in Y: It c Yt 1 Yt 2
– Define actual consumption as
Ct 1 s Yt 1
– Therefore actual saving is
St Yt Ct Yt 1 s Yt 1
– Equate the two (Keynesian S=I):
c Yt 1 Yt 2 Yt 1 s Yt 1
– 2nd order difference equation:
Yt 1 s c Yt 1 cYt 2
Hicks “interprets” Harrod
Hicks's Difference Equation:
Cyclical Behaviour Through Lags
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
35
40
45
50 55
Periods
60
65
70
75
•Cycles alright, but whatever happened to
•Growth?
•“”Knife-edge” instability?
80
85
90
95
100
Hicks “interprets” Harrod
c<1, cycles peter out:
c>1, cycles explode:
For realistic values of
c, impossibly large
cycles & eventually
explosive negative
collapse of Y
Hicks “interprets” Harrod
• Problems
– Equation generates cycles, but not growth: Ye = zero!
– Cycles unstable for c > 1
• But c similar to v, the accelerator: ratio of capital stock
to output
K
v
Y
– v between 2 & 3 for most countries
•“Solutions”
–Assume exogenous growth at “natural” rate
–Assume c < 1
–Assume exogenous shocks to explain persistence of cycle
Hicks “interprets” Harrod
• Hicks interpretation dominates trade cycle theory
• Growth becomes separate topic, dominated by
neoclassical models
• Hicks approach extended/modified by Samuelson, Domar
• Led nowhere; interest in trade cycle declined over 6070s
• Revival in 80s with neoclassical “real business cycle”
models
• But Hicks’s model based on an economic error
– Equation results from equating desired investment to
actual savings
– Keynesian S=I applies ex-post: actual figures only
– Correcting this:
Hicks “interprets” Harrod
• Actual investment is defined as change in capital:
It Kt Kt 1 or Kt Kt 1 It 1
• We can relate this to output using the accelerator:
Kt v Yt
• Expanding, this is definitely not equal to Hicks’s relation
It Kt Kt 1
v Yt Yt 1
c Yt 1 Yt 2
It Kt 1 Kt
or
v Yt 1 Yt
c Yt 1 Yt 2
• Nor is his term for savings really actual savings:
• Savings a residual from this year’s income, not last year’s
Hicks “interprets” Harrod
• Defining savings:
St Yt Ct
Yt 1 s Yt
s Yt
s Yt 1
• Equate actual savings and actual investment:
It v Yt Yt 1 St s Yt
• A first order growth equation: growth but no cycles!
Yt
v
v s
Yt 1
Hicks “interprets” Harrod
• Hicks’s 2nd order difference equation model
Yt 1 s c Yt 1 cYt 2
• Therefore a complete waste of time
• Effectively asks the question “what level of output will
guarantee that desired investment equals actual
savings?”
• The answer? Zero output! So-called trade cycle is just
oscillations en route to what mathematicians call “the
trivial solution”
The rise & fall (& rise?) of dynamics
• Unfortunately, Hicks’s useless model dominated early
attempts at economic dynamics
• Failure to get useful results led to waning of initial 1950s
enthusiasm for dynamics
• Equilibrium analysis continued to dominate economics
• Pinnacle of this “general equilibrium”… but even this
raises dynamic issues:
– Is Walras’ general equilibrium dynamically stable?
• Walras himself abstracted from dynamic processes
by assumptions of “tatonnement” by an “auctioneer”
who did not allow any trades to take place until
equilibrium was achieved…
Dynamics & Stability of General Equilibrium
• “First, let us imagine a market in which only consumer
goods and services are bought and sold... Once the prices
or the ratios of exchange of all these goods and services
have been cried at random…, each party to the exchange
will offer at these prices those goods or services of
which he thinks he has relatively too much, and he will
demand those articles of which he thinks he has
relatively too little for his consumption during a certain
period of time. … the prices of those things for which the
demand exceeds the offer will rise, and the prices of
those things of which the offer exceeds the demand will
fall. New prices now having been cried, each party to the
exchange will offer and demand new quantities. And again
prices will rise or fall until the demand and the offer of
each good and each service are equal. Then the prices
will be current equilibrium prices and exchange will
effectively take place.” (Walras 1874)
Dynamics & Stability of General Equilibrium
• Presumed process would eventually converge because direct
effects would outweigh indirect:
• “This will appear probable if we remember that the change
from p’b to p’’b, which reduced the above inequality to an
equality, exerted a direct influence that was invariably in the
direction of equality at least so far as the demand for (B) was
concerned; while the [consequent] changes from p’c to p’’c, p’d
to p’’d, ..., which moved the foregoing inequality farther away
from equality, exerted indirect influences, some in the
direction of equality and some in the opposite direction, at
least so far as the demand for (B) was concerned, so that up
to a certain point they cancelled each other out. Hence, the
new system of prices (p’’b, p’’c, p’’d, ...) is closer to equilibrium
than the old system of prices (p’b, p’c, p’d, ...); and it is only
necessary to continue this process along the same lines for the
system to move closer and closer to equilibrium.” (Walras
1874; my emphasis)
Dynamics & Stability of General Equilibrium
• Modern mathematics shows Walras’ belief incorrect
– In model with production and growth
• All input & net output quantities are non-negative
– Can’t use negative quantities of an input
– Can’t produce negative quantities of any output and
have sustainable growth
• All prices must be non-negative
– Process can be modelled with matrix of input-output
quantities, all non-negative
– Matrix plays two roles in model: it determines output
dynamics; its inverse determines price dynamics
• Will “pull” of direct effects exceed sum of “push & pull”
of indirect effects, as Walras surmised?
– Both price & quantity dynamics must be stable for
stable outcome —will process converge to equilibrium
price & quantities, or diverge?
Dynamics & Stability of General Equilibrium
• Answer depends on key characteristic of matrix: its “dominant
eigenvalue” (German for “principal value”)
– tells how much the matrix “stretches” space.
– If greater than certain value, matrix stretches space—
instability
– If less, matrix contracts space—stability
• (key value 1 for discrete models, 0 for
continuous ones)
• Perron-Frobenius theorem proves that the dominant eigenvalue
of matrix with all non-negative entries is greater than zero.
• Matrix and its inverse have inverse eigenvalues: e.g., if dominant
eigenvalue of production matrix is 1/10, then dominant
eigenvalue of price matrix is 10. So either the quantity matrix
or its inverse must have eigenvalue>1.
• So…? Either price or quantity process will be unstable:
either quantities or prices (possibly both) will not converge
to equilibrium
Dynamics & Stability of General Equilibrium
• Mathematical results proved in early 1900s; considered in
economic literature in 1960s:
– Jorgenson, D.W., (1960) 'A dual stability theorem',
Econometrica 28: 892-899; (1961). 'Stability of a
dynamic input-output system', Review of Economic
Studies, 28: 105-116; (1963) 'Stability of a dynamic
input-output system: a reply', Review of Economic
Studies, 30: 148-149.
– McManus, M., (1963). 'Notes on Jorgenson’s model',
Review of Economic Studies, 30: 141-147.
– See Blatt, Dynamic Economic Systems, Ch. 6
• Response of GE modellers?
– Ignore issue of the time process by which a market
economy does or does not approach equilibrium:
Dynamics & Stability of General Equilibrium
• “For Walras, general equilibrium theory was an abstract but
nevertheless realistic description of the functioning of a
capitalist economy. He was therefore more concerned to show
that markets will clear automatically via price adjustments in
response to positive or negative excess demand … than to
prove that a unique set of prices and quantities is capable of
clearing all markets simultaneously. By the time we got to
Arrow and Debreu, however, general equilibrium theory had
ceased to make any descriptive claim about actual economic
systems and had become a purely formal apparatus about a
quasi economy. It had become a perfect example of what
Ronald Coase has called “blackboard economics”, a model that
can be written down on blackboards using economic terms like
“prices”, “quantities”, “factors of production”, and so on, but
that nevertheless is clearly and even scandalously
unrepresentative of any recognizable economic system.” (Blaug
1998)
Dynamics & Stability of General Equilibrium
• “For any economic agent a complete action plan (made
now for the whole future), or more briefly an action, is a
specification for each commodity of the quantity that he
will make available or that will be made available to him,
i.e., a complete listing of the quantities of his inputs and
of his outputs...
• For a producer, say the jth one, a production plan (made
now for the whole future) is a specification of the
quantities of all his inputs and all his outputs... The
certainty assumption implies that he knows now what
input-output combinations will be possible in the future
(although he may not know the details of technical
processes which will make them possible)…”
Dynamics & Stability of General Equilibrium
• “As in the case of a producer, the role of a consumer is
to choose a complete consumption plan... His role is to
choose (and carry out) a consumption plan made now for
the whole future, i.e., a specification of the quantities of
all his inputs and all his outputs.
• The analysis is extended in this chapter to the case
where uncertain events determine the consumption sets,
the production sets, and the resources of the economy. A
contract for the transfer of a commodity now specifies,
in addition to its physical properties, its location and its
date, an event on the occurrence of which the transfer is
conditional. This new definition of a commodity allows one
to obtain a theory of uncertainty free from any
probability concept and formally identical with the theory
of certainty developed in the preceding chapters.”
(Debreu 1953; my emphases)
Time matters…
• Have to treat time seriously
– Processes occur in time
– Dynamic process over time need not converge to
equilibrium
– Basic mathematical techniques for handling time-based
processes are differential and difference equations
– Essential aspect of mathematical models of real-world
processes is nonlinearity
• Simplest relationship between two variables (apart from
none at all) is linear: double independent, double the
dependent
• Real world relationships between variables in real world
dynamic systems are nonlinear (I.e., not that simple!)…
The importance of being nonlinear
• Economist attitudes garnered from understanding of
linear dynamic systems
– Stable linear systems do move from one equilibrium to
another
– Unstable linear dynamic systems do break down
– Statics is the end point of dynamics in linear systems
• So economics correct to ignore dynamics if economic
system is
– linear, or
– nonlinearities are minor;
– one equilibrium is an attractor; and
– system always within orbit of stable equilibrium
• Otherwise
– Nonlinearity necessary for proper dynamics
The importance of being nonlinear
• Behavior of modified Hicks-Harrod model a “quirk”:
– Linear model (just constants and variables, no powers,
etc.)
– But generates sustained cycles (for c>=v)
• Most linear models:
– Cycle to equilibrium (c<1 in Hicks)
– Rigid cycles (c=1 in Hicks)
– Unstable (c>1 in Hicks)
• Frisch/Hicks argument that “unstable system … just
breaks down” only true for linear systems
– Nonlinear systems can have unstable equilibria and not
break down
The importance of being nonlinear
• Kaldor (1940) first economist to realise importance of
nonlinearity
– Began with linear model
– Realised that this had only 2 states:
• either “dangerous instabilities” or
• “more stability than the real world appears, in fact,
to possess”
– Deduced that therefore, economic relations “cannot
be linear”
• Nonlinearity makes it possible for equilibria to be
unstable, and yet model to be determinate
– System therefore oscillates—never converges to
equilibrium, but never reaches unattainable values
either
The importance of being nonlinear
Linear models can be:
Ind e te m
r inan t
U
G
lob
al
ly
S
le
b
ta
G
lob
ly
al
ns
tab
le
Cycles in linear system require
S tab le equ ilib r ium
w ith sh o ck s
U n stab le equ ilib r ium
w ith ba rr ie rs (thu s
n on lin ea r )
Frisch/Hicks/Econometrics approach
Harrod’s initial model
The importance of being nonlinear
• Nonlinear systems can be:
L
a
oc
lly
U
ns
tab
le
G loba lly S tab le
• Cycles can occur because system is:
L o ca lly S tab lew ith S h o ck s
L o ca lly un stab le : F
" ar
from E qu ilib r ium "
Not so different from linear model
Completely unlike linear model
An example: Lorenz’s weather model
The importance of being nonlinear
• Any complex model can be simulated using a polynomial
– a + bx + cx2 + dx3 + …
• Near equilibrium, linear components of model dominate
– If equilibrium unstable, linear components push system
away from equilibrium
• For x<1, x>x2>x3>…
– Far from equilibrium, nonlinear components dominate &
push system back towards equilibrium
• For x>1, x3>x2>x> …
• Many non-mainstream nonlinear models developed
• One example: Goodwin’s “predator-prey” model (1967)
– Based on Marx’s model, Capital I Ch. 25 (see lecture
Week 10)
Sample nonlinear model
• Marx’s model
– High wages—>low investment—>low growth—>rising
unemployment—>falling wage demands—>increased
profit share—>rising investment—>high growth—
>high employment—>High wages: cycle continues
• Goodwin draws analogy with biology “predator-prey”
models
– Rate of growth of prey (fish—>capitalists!) depends
+ively on food supply and -ively interactions with
predator (shark—>workers)
– Rate of growth of predator depends -ively on number
of predators and +ively on interactions with prey:
Predator-Prey cycles
Rate of growth
of Fish
Food supply
1 dF
a b S
F dt
1 dS
c d F
S dt
Rate of growth
of Sharks
Interactions with Sharks
Interactions with Fish
Nonlinear bits
stabilise far from
equilibrium
Rate of death
dF
in absence of
a F b S F
Linear bits
dt
Fish to eat
unstable
dS
near equilibrium
c S d F S
dt
• Generates a cycle:
– Lots of fish—>lots of interactions with Sharks—>rapid growth
of Sharks—>Fall in Fish numbers—>less interactions with Sharks
—>Fall in Shark numbers—>Lots of fish again...
Predator-Prey cycles
Equilibrium here, but
system will never reach it
Fish
1100
1000
1100
1050
Sharks
0
200
400 600
Time
8001000
Fish
900
1000
102
950
101
900
98
100
99
98
0
200
400 600
Time
8001000
99
100
Sharks
101
102
Linear forces push away
Nonlinear forces push back in
System cycles indefinitely
Current state of dynamics in economics
• Undergoing revival since mid-80s
• 3 streams
– Neoclassical
• “real business cycle”
– Increasing returns to scale explanation
– Still using linear models
– No longer major element of modern theory
– Non-neoclassical
• Kaldor/Goodwin based nonlinear models
– “Complexity” analysis
• Inspired by “chaos theory” in physics, evolution in
biology
Complexity Theory
• Nonlinear dynamic systems can develop complicated
behaviour from interaction of simple rules
• Systems live on border between “chaos” and “order”
• Tiny changes can push system from order into chaos
• Undermines “rational expectations” (Week 7):
– Impossible to predict course of complex system
•Example: lemmings
–Rate of growth of lemmings
•+ive fn of current population Lt 1 1 a Lt
•-ive fn of overcrowding
•Combining:
Lt 1 b Lt 2
Lt 1 1 a Lt b Lt
2
Complexity Theory
For low values of a,
tapers to
stable equilibrium:
For a=2, a 2-valued
cycle: population
overshoots equilibrium,
then undershoots, etc.
For a > 2.7, apparently
random pattern…
But nothing random about it: deterministic nonlinear models
can generate sustained, unpredictable aperiodic cycles—like
those in real world systems
The story today
• Dynamics now “hot” area of economics
• Much interaction with other sciences
– Biology
– Computing
– Physics
• Non-neoclassical models now match neoclassicals in
mathematical sophistication
• Economics may finally “grow up”
– but most economists today still woefully ignorant of
dynamics:
Ignorance of dynamics the rule…
• Jevons/Marshall attitude still dominates most schools of
economic thought, from textbook to journal:
– Taslim & Chowdhury, Macroeconomic Analysis for
Australian Students: “the examination of the process of
moving from one equilibrium to another is important and
is known as dynamic analysis. Throughout this book we
will assume that the economic system is stable and most
of the analysis will be conducted in the comparative
static mode.” (1995: 28)
– Steedman, Questions for Kaleckians: “The general point
which is illustrated by the above examples is, of course,
that our previous 'static' analysis does not 'ignore'
time. To the contrary, that analysis allows enough time
for changes in prime costs, markups, etc., to have their
full effects.” (Steedman 1992: 146)
Conclusion: Mathematicians & Physicists on…
• “A baby is expected to first crawl, then walk, before
running. But what if a grown-up man is still crawling? At
present, the state of our dynamic economics is more akin
to a crawl than a walk, to say nothing of a run. Indeed,
some may think that capitalism as a social system may
disappear before its dynamics are understood by
economists.” (Blatt 1983: 5)
• Mirowski, More Heat than Light claims neoclassical
economics based on analogy to 19th century physics
• Modern physics based on completely different paradigm
today (quantum mechanics, thermodynamics)
– Many physicists quite disparaging about modern
economics
• “Unreal assumptions”, “absurd proposition/lemma
style of argument”…
Conclusion: The debate goes on…
• Some physicists now developing “econophysics” to apply
these new ideas to economics
– See for example http://www.unifr.ch/econophysics/
• Influences on economics also coming from evolutionary
theory
– See for example http://www.orgs.bucknell.edu/afee/
• Competing schools still alive and well
– Tiny fragmented minority still developing what they
call Marxian economics
• See for example http://www.marx.org
– Substantial, more cohesive minority of Post Keynesians
• http://csf.colorado.edu/pkt
– Quasi-neoclassical Austrian school accepts rejects
equilibrium analysis, focuses on uncertainty
• http://www.hayekcenter.org/
Conclusion: The end of history?
• These and other non-neoclassical approaches considered
in Political Economy (next semester)
• Many economists believe History of Economic Thought is
about the past of economics
• But current state of economics far from “cut & dried”
– Many competing paradigms (though one dominant)
– Many unresolved disputes