Economic growth and economic development 265

Introduction to Modern Economic Growth
t = 1, i.e., solve the problem
max u (x (1)) + δ

{x(t)}T
t=1

T
X

β t−1 u (x (t))

t=2

subject to
x (t) ∈ [0, x¯]

G (x∗ (0) , ..., x (T )) ≤ 0.
Prove that the solution from t = 1 onwards, {x∗∗ (t)}Tt=1 is not necessarily

the same as {x∗ (t)}Tt=1 .

(4) Explain which standard axioms of preferences in basic general equilibrium
theory are violated by those in parts 2 and 3 of this exercise.
Exercise 5.2. This exercise asks you to work through an example that illustrates
the difference between the coefficient of relative risk aversion and the intertemporal elasticity of substitution. Consider a household with the following non-timeseparable preferences over consumption levels at two dates:

# 1
"à
à 1
ả 1

ả 1



1

1
c1
c
1
+ 2
,
V (c1 , c2 ) = E
1−θ
1−θ
where E is the expectations operator. The budget constraint of the household is
1
c2 ≤ W,
1+r
where r is the interest rate and W is its total wealth, which may be stochastic.
c1 +

(1) Let us first suppose that W is nonstochastic and equal to W0 > 0. Characterize the utility maximizing choice of c1 and c2 .
(2) Compute the intertemporal elasticity of substitution.

Ô
Ê
(3) Now suppose that W is distributed over the support W , W with some

distribution function G(W ), where 0 < W < W < ∞. Characterize the

utility maximizing choice of c1 and compute the coefficient of relative risk

aversion. Provide conditions under which the coefficient of relative risk
aversion is the same as the intertemporal elasticity of substitution. Explain
why the two differ and interpret the conditions under which they are the
same.
251