Graduate Macro II, Homework 2
Jonathan Heathcote
Due in class on Thursday February 16th 2006
Consider the neoclassical growth model in discrete time. In the two exam
ples below, you are asked to solve for e
ﬃ
cient allocations by considering social
planner’s problems.
1. Assume the planner’s preferences are given by
T
P
t
=0
β
t
u
(
c
t
)
where
u
(
c
t
) =
ln(
c
t
)
,
and the technology is
c
t
+
i
t
=
y
t
∀
t
≥
0
y
t
=
k
θ
t
∀
t
≥
0
k
t
+1
=
(1
−
δ
)
k
t
+
i
t
∀
t
≥
0
c
t
≥
0
, k
t
≥
0
∀
t
≥
0
k
0
given
(a) Describe a set of equations that implicitly de
fi
nes a solution to this
planning problem.
(b) Assume the following parameter values:
β
= 0
.
96
, T
= 100
, θ
= 0
.
36
,
δ
= 0
.
08
.
Assume
k
0
= 1
.
Solve for the e
ﬃ
cient allocation
{
c
∗
t
, k
∗
t
}
T
t
=0
.
I suggest you use the following “shooting method”:
i. Guess bounds such that
c
1
l
≤
c
∗
0
≤
c
1
h
,
and make an initial guess
for
c
∗
0
,
denoted
c
1
0
,
where
c
1
0
= 0
.
5
¡
c
1
l
+
c
1
h
¢
ii. Use the set of equations de
fi
ning a solution to the problem to
compute
k
T
+1
given
c
1
0
and
k
0
iii. Use “bisection" to update the guess for
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 '05
 would
 Dynamic Programming, Equations, Recursion, Bellman equation

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