Transcript The optimal growth setup with exogenous technological change

Agenda
Monday
- Presentation of impacts in agriculture
- Discounting
Wednesday
- Presentation on extremes (hurricanes)
- Summary on impacts
- Pass out exam
- Hand in pset 4
Friday
- Review exam
- Go over any questions on psets 3 and 4
Discounting in economics:
The fundamental determinants
Economics 331b
2
Start with a consumption path
For this, assume that:
1. c is per capita consumption
2. Population is constant
3. c is a comprehensive measure
(includes all external and nonmarket effects.
c1(t)
time
Which is the preferable path, c1 or c2?
For this, assume that:
1. c is per capita consumption
2. Population is constant
3. c is a comprehensive measure
(includes all external and nonmarket effects.
c2(t)
c1(t)
time
Which is the preferable path?
c2(t)
c1(t)
time
Which is the preferable path?
c1(t)
c2(t)
time
Philosophical-ethical foundations
of environmental economics
Religious and Kantian views: Largely absent
Utilitarianism: Bentham, J.S. Mill (Utilitarianism), Peter Singer (on animals)
In modern bioethics: Maximize QALYs
Happiness research and hedonic psychology (among behavior psychologists
and some economists)
Welfarism (of the social welfare function): Bergson, Samuelson, Foundations
Paretianism: An incomplete ordering of social states based on welfarism used
throughout economics
Critiques of Utilitarianism, conequentialism, and welfarism:
Ordinalists: cannot measure utility
Positivism: ethics as esthetics
J. Rawls, Theory of Justice: emphasis on equality and justice
Amartya Sen, (Sen and Williams, Beyond Utilitarianism), emphasis on
capabilities and equity
Ethical fundamentals
Alternative universes: {c1(t), …, cn(t)}
Process values, rules, religions,…
Human values, non-human (intrinsic)
values, …
Dynamic one-sided game
among generations
Consequentialism
Human values
Welfarism and individualistic “social
welfare function”: W = V(U1, … Un)
Ethical fundamentals
States of the world: {c1(t), …, cn(t)}
.
.
.
Welfarism and individualistic
“social welfare function”
Pure 19th century
“add them up”
Complex welfare functions
(non-separable,
altruistic, dynastic,…)
W 
N
 U n (c n )
n1
Standard approach
in modern economics: DU
W 
T
 U t (c t ) e   t
t 1
Ethical fundamentals
States of the world: {c1(t), …, cn(t)}
.
.
.
Standard DU approach DU:
Alternative preference functions
Rawlsian: maximin or leximin
W=min[c1, … ,cn,… ]
Environmental (E) or
other constraints
(1) U=V(c); E > E*
(2) U=V(c) + Z(E)
Standard economic approach: can trade off
environmental and non-environmental values
U = U(c, E)
The optimal growth setup with
exogenous technological change
We assume that there is labor augmenting technological change, E ,
at rate h. This increases the "effective labor force," where L  LE.
(2b) Production function : Y  F ( K , EL)  F ( K , L )
y  Y / L  LF ( K / L, 1)  Lf ( k )
(4a) Growth in effective labor supply :
L / L  n  h = exogenous
So the new steady state is :
(6a)
sf ( k *)  ( n  h   ) k *
With long run growth rate of real output:
(7)
gY  n  h
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The solution
In optimal economic growth, we choose the path of K(t) to maximize the
utility of future consumption (alternatively, the savings rate).
Here is the semi-technical version (this is in the spirit of the calculus of
variations).
“Splish splash” optimal growth experiment: Suppose that we invest in period
t with a return in period (t+θ). An investment is a withdrawal from
consumption of Δ, with the return being an increase in consumption with
real rate of return r. The fundamentals are the following (with no
population growth):

(8)
W   U[c(t)]e t dt
0
(9)
U [c(t)] = c(t)1- / (1-  )
Splish-splash
Consumption
∆(1+r)θ
c(t)
-∆
|
t
|
t+θ
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So the solution for the change in total welfare is:
(10) change in W  0   e   t u'[c(t )]   e   ( t  )u'[c(t   )][  e r ]
or
u'[c(t )]  u'[c(t  1 )]e( r   )
The marginal utility for our assumed utility function is
u'[c(t )]  c(t )
e( r   )  [c(t  1 ) / c(t )]
Let h = growth rate of c(t), so [c(t   ) / c(t )]  e h .
This yields
e( r   )  e h
Take logs, so
(11) r    h
which is the basic result of the Ramsey model. Equation (11) is
called the "Ramsey equation."
More…
To add population growth, we change the objective function to
the following:

W   L(t)U[c(t)]e  t dt
0
If population growth is constant at rate n, we see that the new
Ramsey equation is:
(12)
r    h  n
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Key distinction, often confused
Utility discount rate (pure rate of social time preference), ρ:
This refers to the comparison of well-being or utilities over time, space,
or generations.
Goods discount rate (tradeoff in markets), r:
This refers to the return on private or social investments in goods,
services, etc.
More…
The debate on discounting generally will use the framework of the Ramsey
model. Begin with the Ramsey equation from (12) above:
r    h  n
This shows the relationship between the equilibrium real interest rate and
underlying parameters.
Observables:
r = real interest rate or real return on capital
g = rate of growth of real consumption per capita
Non-observables:
  rate of social time preference (time or utility discount rate)
  curvature of utility function (inequality aversion, risk aversion)
Two schools of discounting theory
What real interest rate on goods shall we actually use for discounting costs
and benefits of long-term investments (climate, radioactive wastes, dams,
technology,…)?
Descriptive: Argues that we can observe the rate of return and should make
sure that our decisions are consistent with opportunity cost on other
investments. This leads to a relatively high utility discount rate (Feldstein,
Eckstein, Lind, Nordhaus):
  r   h  .055  2  .02  .015
Prescriptive: Argues that we can know the normative parameters on
philosophical grounds and derive the interest rate from that (Cline, Stern):
r     h  .001  1  .013  .014
Why this is so important in climate-change analysis:
Damages are so late in the game.
Numerical example of effect over 200 years
1,000,000.0000
Present value (billions of 2010 $)
100,000.0000
10,000.0000
1,000.0000
100.0000
10.0000
1.0000
0
0.02
0.04
0.06
0.1000
0.0100
0.0010
Discount rate
0.08
0.1
0.12
Why this is so important in climate-change analysis:
Damages are so late in the game.
Numerical example of effect over 200 years
700,000
Present value (billions of 2010 $)
600,000
500,000
400,000
300,000
200,000
100,000
0
0
0.02
0.04
0.06
Discount rate
0.08
0.1
0.12
Problems with each
Major criticism of descriptive is that a positive ρ is unethical and violates
intergenerational fairness. (Good point)
Major criticism of prescriptive is that it leads to distorted investment
decisions because actual return on investment is much higher than the
prescriptive discount rate. (Good point)
Is there any way to reconcile all this?
Two polar cases
The standard descriptive model uses market rates of return (circa 5-6 percent
per year for goods and services). Assume g = 2 % per year
In the Ramsey framework, this can be interpreted as a solution of the
following equation:
5.5 = ρ + α 2
We take the solution of ρ = 1.5 and α = 2.
The prescriptive approach (Cline, Stern) argues that it is ethically indefensible
to have generational discounting. Stern also assumes that α = 1. With their
assumed g = 1.4 % per year, this yields:
r = 0.1 + 1 (1.4) = 1.5 % per year
Conclusions on discounting
Remember the ethical foundations.
A key question is whether you take the mixed-market solution (prices) as a
constraint on decisions; or whether you want to argue that the
preferences as revealed (by market prices) in the market are wrong.
In the ethical/prescriptive view, the goods discount rate will be determined by
both time discounting and view on income distribution over time.
These issues are particularly important for very long-run decisions (global
warming, radioactive wastes, …).
The arguments also involve questions such as market imperfections, taxes,
the equity premium – pretty horrible.