#### Transcript Slide 1

```Prerequisites
Almost essential
General equilibrium:
Basics
Frank Cowell: Microeconomics
Useful, but optional
General Equilibrium:
Price Taking
November 2006
General Equilibrium: Excess
Demand and the Rôle of Prices
MICROECONOMICS
Principles and Analysis
Frank Cowell
Some unsettled questions
Frank Cowell: Microeconomics

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


Under what circumstances can we be sure that an
equilibrium exists?
Will the economy somehow “tend” to this
equilibrium?
And will this determine the price system for us?
We will address these using the standard model of
a general-equilibrium system
To do this we need just one more new concept.
Overview...
Frank Cowell: Microeconomics
General Equilibrium:
Excess Demand+
Excess Demand
Functions
Definition and
properties
Equilibrium
Issues
Prices and
Decentralisation
Frank Cowell: Microeconomics
Ingredients of the excess demand
function
Aggregate demands (the sum of individual
households' demands)
 Aggregate net-outputs (the sum of
individual firms' net outputs).
 Resources
 Incomes determined by prices

check this out
Aggregate consumption, net output
Frank Cowell: Microeconomics
Consumer:
Aggregation
Firm and the
market
From household’s demand function  Because incomes
depend on prices
xih = Dih(p, yh)
= Dih(p, yh(p) )
So demands are just functions of p  xih(•) depends on holdings
xih = xih(p)
of resources and shares
If all goods are private (rival) then
aggregate demands can be written:
xi(p) = Sh xih(p)
From firm’s supply of net output
qif = qif(p)
Aggregate:
qi = Sf qif(p)
 “Rival”: extra consumers
Same as in “consumer:
aggregation”
 standard supply functions/
demand for inputs
graphical
 aggregation is valid
if there
summary
are no externalities. Just as
in “Firm and the market”)
Derivation of xi(p)
Frank Cowell: Microeconomics
 Alf’s demand curve for good 1.
 Bill’s demand curve for good 1.
 Pick any price
 Sum of consumers’ demand
 Repeat to get the market demand
curve
p1
p1
p1
Alf
x1a
Bill
x1b
The Market
x1
Derivation of qi(p)
Frank Cowell: Microeconomics
 Supply curve firm 1 (from MC).
 Supply curve firm 2.
 Pick any price
 Sum of individual firms’ supply
 Repeat…
 The market supply curve
p
p
p


q1
low-cost
firm
q1+q2
q2
high-cost
firm
both firms
Subtract q and R from x to get E:
p1
p1
Demand
Supply
q1
x1
p1
Resource
stock
Frank Cowell: Microeconomics
p1
R1
net output of i
aggregated over f
demand for i
aggregated over h
Ei(p) := xi(p) – qi(p) – Ri
1
E1
Resource
stock of i
Frank Cowell: Microeconomics
Equilibrium in terms of Excess
Demand
Equilibrium is characterised by a
price vector p*  0 such that:
• For every good i:
Ei(p*)  0
• For each good i that has a
positive price in equilibrium
(i.e. if pi* > 0):
Ei(p*) = 0
The materials balance condition
(dressed up a bit)
If this is violated, then somebody,
somewhere isn't maximising...
You can only have excess supply
of a good in equilibrium if the
price of that good is 0.
Using E to find the equilibrium
Frank Cowell: Microeconomics
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
Five steps to the equilibrium allocation
1.
From technology compute firms’ net output
functions and profits.
2.
From property rights compute household incomes
and thus household demands.
3.
Aggregate the xs and qs and use x, q, R to
compute E
4.
Find p* as a solution to the system of E functions
5.
Plug p* into demand functions and net output
functions to get the allocation
But this begs some questions about step 4
Issues in equilibrium analysis
Frank Cowell: Microeconomics
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Existence

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Uniqueness

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Is there only one p*?
Stability

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Is there any such p*?
Will p “tend to” p*?
For answers we use some fundamental properties
of E.
Two fundamental properties...
Frank Cowell: Microeconomics
consumer
demand
• Walras’ Law. For any price p:
You only have to work
with n-1 (rather than n) equations
n
S
i=1
pi Ei(p) = 0
demand functions...
• Homogeneity of degree 0. For any
price p and any t > 0 :
Ei(tp) = Ei(p)
You can normalise the prices by
any positive number
homogeneity property of
demand functions...
Can you explain why they are true?
Reminder : these hold for any competitive allocation, not just
equilibrium
Price normalisation
Frank Cowell: Microeconomics
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We may need to convert from n numbers p1,
p2,…pn to n1 relative prices.
The precise method is essentially arbitrary.
The choice of method depends on the
It can be done in a variety of ways:
You could divide by
n
labour
MarsBar
pi
S
i=1
a numéraire
pppn
to give a
neat
oftheory
n-1
standard
value
system
“Marxian”
Mars
bar
theory
of
ofvalue
value
set
of set
prices
thatprices
sum
to 1
 This method might seem weird
 But it has a nice property.
 The set of all normalised prices
is convex and compact.
Normalised prices, n=2
Frank Cowell: Microeconomics
 The set of normalised prices
p2

 The price vector (0,75, 0.25)
(0,1)
J={p: p0, p1+p2 = 1}
(0, 0.25)
•
(0.75, 0)

(1,0)
p1
Normalised prices, n=3
Frank Cowell: Microeconomics
p3
 The set of normalised prices
 The price vector (0,5, 0.25, 0.25)

(0,0,1)
J={p: p0, p1+p2+p3 = 1}
(0, 0, 0.25)
•

(0,1,0)

(1,0,0)
(0, 0.25 , 0)
0
(0.5, 0, 0)
p1
Overview...
Frank Cowell: Microeconomics
General Equilibrium:
Excess Demand+
Excess Demand
Functions
Is there any p*?
Equilibrium
Issues
Prices and
Decentralisation
•Existence
•Uniqueness
•Stability
Approach to the existence problem
Frank Cowell: Microeconomics

Imagine a rule that moves prices in the direction of excess
demand:





This rule uses the E-functions to map the set of prices into
itself.
An equilibrium exists if this map has a “fixed point.”


a p* that is mapped into itself?
To find the conditions for this, use normalised prices



“if Ei >0, increase pi”
“if Ei <0 and pi >0, decrease pi”
An example of this under “stability” below.
p  J.
J is a compact, convex set.
We can examine this in the special case n = 2.

In this case normalisation implies that p2  1  p1.
Why?
Existence of equilibrium?
Frank Cowell: Microeconomics
Why boundedness below?
 ED diagram, normalised prices
 Excess demand function with
well-defined equilibrium price
 Case with discontinuous E
 Case where excess demand
for good2 is unbounded below
As p2  0, by normalisation, p11
As p2  0 if E2 is bounded below
then p2E2  0.
By Walras’ Law, this implies p1E1 
0 as p11
So if E2 is bounded below then E1
can’t be everywhere positive
1
p1
Excess
supply
E-functions are:
 continuous,
 bounded below
good 2 is
free here

p1*
good 1 is
free here
0
Excess
demand
No equilibrium
price where E
crosses the axis

E1
E never
crosses the axis

Existence: a basic result
Frank Cowell: Microeconomics

An equilibrium price vector must exist if:
1.
2.


Boundedness is no big deal.


excess demand functions are continuous and
bounded from below.
(“continuity” can be weakened to “upper-hemicontinuity”).
Can you have infinite excess supply...?
However continuity might be tricky.


Let's put it on hold.
We examine it under “the rôle of prices”
Overview...
Frank Cowell: Microeconomics
General Equilibrium:
Excess Demand+
Excess Demand
Functions
Is there just one
p*?
Equilibrium
Issues
Prices and
Decentralisation
•Existence
•Uniqueness
•Stability
The uniqueness problem
Frank Cowell: Microeconomics
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Multiple equilibria imply multiple allocations, at
normalised prices...
...with reference to a given property distribution.
Will not arise if the E-functions satisfy WARP.
If WARP is not satisfied this can lead to some
startling behaviour...
let's see
Multiple equilibria
Frank Cowell: Microeconomics
 Three equilibrium prices
 Suppose there were more of
resource 1
 Now take some of resource 1
away
1
p1
single equilibrium
jumps to here!!


three equilibria
degenerate to one!


0
E1
Overview...
Frank Cowell: Microeconomics
General Equilibrium:
Excess Demand+
Excess Demand
Functions
Will the system
tend to p*?
Equilibrium
Issues
Prices and
Decentralisation
•Existence
•Uniqueness
•Stability
Stability analysis
Frank Cowell: Microeconomics


We can model stability similar to physical sciences
We need...




A definition of equilibrium
A process
Initial conditions
Main question is to identify these in economic
terms
Simple
example
A stable equilibrium
Frank Cowell: Microeconomics
Stable:
Equilibrium:
If we apply a small shock
Status
left
the
built-in
undisturbed
by gravity
process (gravity)
restores
the status quo
An unstable equilibrium
Frank Cowell: Microeconomics
Equilibrium:
Unstable:
This
If weactually
apply afulfils
small the
shock
definition.
process (gravity) moves us
But….
away from the status quo
“Gravity” in the CE model
Frank Cowell: Microeconomics




Imagine there is an auctioneer to announce prices,
If good i is in excess demand, increase its price.
If good i is in excess supply, decrease its price (if
Nobody trades till the auctioneer has finished.
Frank Cowell: Microeconomics
“Gravity” in the CE model: the
auctioneer using tâtonnement
individual
individual
dd & ss
Announce p
individual
dd & ssdd
individual
& ss
dd & ss
…once we’re at
equilibrium we
Equilibrium?
Equilibrium?
Equilibrium
?
Equilibrium?
Evaluate
Evaluate
excess dd
Evaluate
excess
dd
Evaluate
excess dd
excess dd
Frank Cowell: Microeconomics

Adjust prices according to sign of Ei:


If Ei > 0 then increase pi
If Ei < 0 and pi > 0 then decrease pi


Define distance d between p(t) and equilibrium p*

Two examples:
with/without
WARP
Given WARP, d falls with t under tâtonnement
Globally stable...
Frank Cowell: Microeconomics
1
 Yields excess supply
 Under tâtonnement price falls
p1(0)
Excess
supply
•p *
1
 Start instead with a low price
 Yields excess demand
Excess
demand
 Under tâtonnement price rises
 If E satisfies
p1
p1(0)
E1
0
E1(0)
E1(0)
WARP then
the system
must
converge...
Not globally stable...
Frank Cowell: Microeconomics
1
 Start again with very low price
p1
•
Excess
supply
Unstable
 …now try a (slightly) low price
Locally
Stable
•
•0
 …now try a (slightly) high price
 Check the “middle” crossing
Excess
demand
 Here WARP
does not
hold
 Two locally
Also locally
stable
stable
equilibria
E1
 One unstable
Overview...
Frank Cowell: Microeconomics
General Equilibrium:
Excess Demand+
Excess Demand
Functions
The separation
theorem and the
role of large
numbers
Equilibrium
Issues
Prices and
Decentralisation
Decentralisation
Frank Cowell: Microeconomics
Crusoe:
and market

Recall the important result on decentralisation



The counterpart is true for this multi-person world.
Requires assumptions about convexity of two sets,
defined at the aggregate level:



discussed in the case of Crusoe’s island
the “attainable set”: A := {x: x q+R, F(q) 
the “better-than” set: B(x*) := {Shxh: Uh(xh )Uh(x*h ) }
To see the power of the result here…


use an “averaging” argument
previously used in lectures on the firm
Decentralisation again
Frank Cowell: Microeconomics
 The attainable set
 The “Better-than-x* ” set
 The price line
 Decentralisation
x2
 x*
B
p1


A = {x: x  q+R, F(q)0}

B = {Shxh: Uh(xh)  Uh(x*h)}

x* maximises income
over A

x* minimises
expenditure over B
p2
A
0
x1
Problems with prices
Frank Cowell: Microeconomics
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Either non-convex technology (increasing returns
or other indivisibilities) for some firms, or...
...non-convexity of B-set (non-concave- contoured
preferences) for some households...
...may imply discontinuous excess demand
function and so...
...absence of equilibrium.
But if there are are large numbers of agents
everything may be OK.
two
examples
A non-convex technology
output
Frank Cowell: Microeconomics
One unit of input
produces exactly
one of output
B
 q'
 The case with 1 firm
 Rescaled case of 2 firms,
 … 4, 8 , 16
 Limit of the averaging process
 The “Better-than” set
• q*
 “separating” prices and equilibrium
 Limiting attainable
set is convex

A
q°
input
 Equilibrium q* is
sustained by a
mixture of firms at q°
and q' .
Non-convex preferences
Frank Cowell: Microeconomics
 The case with 1 person
 Rescaled case of 2 persons,
x2
 A continuum of consumers
 The attainable set
No equilibrium here
 “separating” prices and equilibrium
 x'
• x*
 Limiting better-than
set is convex
B
 Equilibrium x* is
 x°
A
x1
sustained by a
mixture of consumers
at x° and x' .
Summary
Frank Cowell: Microeconomics

Excess demand functions are handy tools for
getting results
Review

Continuity and boundedness ensure existence of
equilibrium
Review

WARP ensures uniqueness and stability
Review

But requirements of continuity may be demanding
Review
```