Salyer, ECN 200E, Spring 2010
1
Homework #1
1. Within the context of the Ramsey-Cass-Koopmans model of optimal growth, assume that agents
have instantaneous utility given by:
u
(
c
) =
°
e
°
°c
°
where
° >
0
:
In the economy, there is no technological progress, no labor growth, and assume that
capital does not depreciate. Agents discount future utility at the subjective rate of
± >
0
:
Technology
is represented by
Y
(
t
) =
F
(
K
(
t
)
; L
(
t
))
where
F
(
:
)
exhibits constant returns to scale. Agents are
endowed with one unit of labor.
(a) Relate
°
to the concavity of the utility function. Compute the elasticity of intertemporal sub-
stitution (between consumption at dates
t
1
and
t
2
) which is de°ned as:
²
=
d
ln (
c
(
t
2
)
=c
(
t
1
))
d
ln (
u
0
[
c
(
t
1
)]
=u
0
[
c
(
t
2
)])
(b) Express the social planner±s problem as a Hamiltonian and derive the associated necessary
conditions.
(c) Construct the phase diagram associated with this economy.
ANSWER: This utility function exhibits constant absolute risk aversion:
ARA
=
°
u
00
u
0
=
°:
Hence, greater values implies greater curvature but note that this functional form exhibits
increasing relative risk aversion
(
RRA
)
as a function of consumption. Since relative risk aversion
is the inverse of the elasticity of intertemporal elasticity of substitution, we know that the IES
is decreasing in consumption. To see this, assume that the ratio
c
2
c
1
is changing strictly through
a change in
c
2
(a convenience which we can do since utility is time separable). Also note that
d
ln (
u
0
[
c
(
t
1
)]
=u
0
[
c
(
t
2
)]) =
°
d
ln (
u
0
[
c
(
t
2
)]
=u
0
[
c
(
t
1
)])
:
So we have:
²
=
°
d
°
c
2
c
1
±
d
°
u
0
2
u
0
1
±
u
0
2
u
0
1
c
2
c
1
=
°
°
dc
2
c
1
±
°
du
0
2
u
0
1
±
u
0
2
u
0
1
c
2
c
1
=
°
u
0
2
du
0
2
dc
2
1
c
2
=
°
u
0
2
u
00
2
1
c
2
=
1
RRA
=
1
°c
2
The Hamiltonian associated with this problem is:
H
=
e
°
±t
²
°
e
°
°c
(
t
)
°
³
+
³
(
t
) [
f
(
k
(
t
))
°
c
(
t
)]
The associated necessary conditions are:
@H
@c
= 0
)
e
°
±t
e
°
°c
(
t
)
°
³
(
t
)
(1)
@H
@k
=
°
_
³
(
t
)
)
³
(
t
)
f
0
(
k
(
t
)) =
°
_
³
(
t
)
(2)
@H
@³
=
_
k
(
t
)
)
_
k
(
t
) =
f
(
k
(
t
))
°
c
(
t
)
(3)
Take the time derivative of eq.
(1)
, simplifying and using in eq.
(2)
yields the Keynes-Ramsey
condition:
_
c
(
t
) =
f
0
(
k
(
t
))
°
±
°
(4)