Transcript The optimal growth setup with exogenous technological change

Agenda
Monday
- Discounting
- Hand in pset (what was your calculation of SCC?)
Wednesday
- Complete discussion of Integrated Assessment Models and RICE-2010
model
Friday
- Open discussion
Next week:
- Uncertainty, learning, fat tails
Discounting in economics:
The fundamental determinants
Economics 331b
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The choice set
Consider a dynamic path for important variables. This should consider a
wide range of important variables for different people or generations.
Market consumption, non-market consumption, public goods,
environmental goods, …
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For example, work of John Roemer et al.*
A quality of life function with consumption, environment, etc.:
(i) Consumption of goods and services (c);
(ii) Educated leisure (x);
(iii) “Education, which modifies the value of leisure time to the
individual;
(iv) “Knowledge, in the form of society’s stock of culture and science,
which directly increases the value of life (in addition to any indirect
effects through productivity), via improvements in health and life
expectancy, and because an understanding of how the world works
and an appreciation of culture are intrinsic to human well-being (Sn),
(v) “An undegraded biosphere, which is valuable to humans for its direct
impact on physical and mental health (Sm).”
Welfare function takes the form QuoL =
*Humberto Llavador, John E. Roemer, Joaquim Silvestre, "A Dynamic Analysis of
Human Welfare in a Warming Planet“http://pantheon.yale.edu/~jer39/
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This leads to the problem of how to rank paths
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Start with a consumption (or QuoL) path
For this, assume that:
1. c is per capita consumption or Roemer’s QuoL
2. Population is constant
c1(t)
time
Which is the preferable path, c1 or c2?
For this, assume that:
1. c is per capita consumption or Roemer’s QuoL
2. Population is constant
c2(t)
c1(t)
time
Which is the preferable path?
For this, assume that:
1. c is per capita consumption or Roemer’s QuoL
2. Population is constant
c2(t)
c1(t)
time
Which is the preferable path?
c1(t)
c2(t)
time
Philosophical-ethical foundations
of environmental economics
Modern economics of welfarism (as in 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, emphasis on capabilities and equity
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) (alternatively, the
savings rate) to maximize the utility of future consumption.
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
(9)
U [c(t)] = c(t)1- / (1-  )
0
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
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
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
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 should 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 – very technical issues.
What does the jury say?
HUNG JURY.