SWF  E   e  t u( c(t ))dt 


1 Eu '(ct )
rt    ln
t
u '(c0 )
• What is the status of , u?
― Ethical, or preference-based?
― Concavity of u: aversion to risk, fluctuation, inequity?
•
•
•
•
Should we disentangle these three dimensions?
Functional form of u? limc0 u '(c)   ?
Link with market prices?
Calibration of ct?
1
The extended Ramsey rule
in the iid lognormal case
ct 1  ct e xt , with xt i.i.d.  N (  ,  )
g  ln( Ec1 / c0 )  Ee x    0.5 2
 
  x 
E c0  e 
t
Eu '(ct )



 x t
  (   0.5 2 ) 


  Ee   e
.



u '(c0 )
c0
1 Eu '(ct )
     0.5 2 2 .
rt    ln
t
u '(c0 )
r     g  0.5 (  1) 2 .
Impatience
wealth effect
precautionary effect
2
Interpersonal MRS approach
• Consider an economy with 2 social groups of equal
size, A and B. Each agent in group A is 2 times
wealthier than in group B.
• We can transfer wealth from A to B. What is the
maximum sacrifice of A that Society should accept
for B to get one more1€?
Certainty equivalent approach
• You are indifferent between
― 50-50 chance to live with a daily income of 80 or 120;
― A sure daily income of X.
• Risk aversion or aversion to inequity (veil of ignorance).
Standard time-series calibration of the
extended Ramsey rule
•
Kocherlakota (1996), using United States annual data over the period
1889-1978, estimated the standard deviation of the growth of
consumption per capita to 3.6% per year.
 2  (0.036) 2 and   2 implies 0.5 (  1) 2  0.4%.
•
Benchmark calibration
g







r     g  0.5 (  1) 2  3.6%
5
Calibration of the Ramsey rule
6
Calibration of the Ramsey rule (Ct’d)
7
Alternative ross-sectional)calibration of the
extended Ramsey rule
• 190 countries over the period 1969-2009:
400,00%
Growth rate 1969-2009
350,00%
300,00%
250,00%
200,00%
150,00%
100,00%
50,00%
0,00%
-50,00%
3
6
9
LN(GDP/cap 1969)








r     g  0.5 (  1) 2
 0%  3%  2.42%
 0.58%
8
Pricing the future
The economics of discounting
and sustainable development
Markov switches
Christian Gollier
Auteur
Two-regime Markov process
ct 1  ct e xt

s
 xt   t   t

g
b
 P  st 1  b st  g    ; P  st 1  g st  b   
E u '(c1 ) s 
s
s
2 2
s
s
 (1   s ) Ee  (   0 )   s Ee  (   0 )  e0.5  (1   s )e    s e   .


u '(c0 )
r1s     m1s  0.5 2 2 ,
s
e m1  (1   s )e   s e .
s
s
Precautionary equivalent
growth rate
m1g  m1b
10
Numerical sim I
• Link with the literature on extreme events (Rietz (1988),
Aase (1993), Barro (2006)).
• Cecchetti, Lam and Mark (2000) estimated a two-state
regime-switching process for the US economy using the
annual per capita consumption data covering the period
1890-1994.
• The unconditional expected growth rate is 1.89%.
g
b
g



b



11
r  3.26%
12
Persistent shocks on the growth rate
• Daily wage (in pounds of wheat):
― In Babylon (1880-1600 B.C.): around 15;
― In the golden age of Pericles in Athens: around 26;
― In England around 1780: 13.
• Malthus Law? Stable 0% growth of GDP/cap.
• Switch to a trend of 2% around 1800-1850.
13
Numerical sim II
• The calibration based on data covering the period 18901994 fails to recognize a crucial aspect of economic
history: Malthus’ trap.
g
b
g
b






14
Pricing the future:
The economics of discounting
and sustainable development
Parametric uncertainty and fat tails
Christian Gollier
Auteur
Uncertain growth
• Dynamic process on ct parametrized by .
• =1,…,n with probabilities q1,q2,…,qn.
• By the law of iterated expectations, we have that
n
Eu '(ct )   q E u '(ct )  .
 1
E u '(ct )  
1 n
1 n
rt    ln  q
  ln  q e  rt t
t  1
u '(c0 )
t  1
1 E u '(ct )  
rt    ln
t
u '(c0 )
16
Conditional to , the growth process is a
random walk
ct 1  ct e xt

 x0 , x1 ,...  i.i.d .  N (  ,   ) 
  (1, q ;...; n, q )
1
n

rt      0.5 2 2 .
2 2
1 n
rt    ln  q e(   0.5  ) t .
t  1
  (1%,1/ 2;3%,1/ 2)
  3.6%
 2
  0%
17
The case of an unknown trend of economic
growth
• Suppose that is known, but  is normally distributed with
mean  and std deviation .
2 2
2 2
1
rt    ln e( 0  0.5 t 0  0.5  ) t    0  0.5 2 ( 2   02t ).
t
ln
ct

 ,   N (  t ,  2t ) 
ct
2
2 2
c0
  ln  N ( 0t ,  t   0 t )
c0
2 2

 t  N ( 0t ,  0 t ) 
min r  
18
The case of an unknown volatility of
economic growth
• Weitzman (2007, 2009) : Suppose alternatively that  is
known, but  is not.
• We work with the precision p   2  (a, b).
• Unconditional distribution of xt:
x p  N (  ,   1/ p ) 
x
 Student (2a)

1/
ab
p   ( a, b)

• As is well-known also, this Student’s t-distribution has fatter
tails than the corresponding normal distribution with the same
mean and variance.
1 Eu '(ct )
 
rt    ln
t
u '(c0 )
19