Homework 1
Macroeconomic Theory I. Viktor Tsyrennikov
Question 1.
An individual seeks to maximize his utility
U
(
c
) =
∞
s
t
=0
β
t
u
(
d
t
)
,
β
∈
(0
,
1)
,
where
d
t
is a
nonnegative
consumption level of a
durable
good in period
t
.
Durable goods are accumulated according to:
d
t
+1
= (1

δ
)
d
t
+
c
t
, δ
∈
(0
,
1]
,
where
c
t
g
0 is date
t
spending on durable goods. Assume that borrowing
limits are such that they are never binding. Initial values
a
0
, d
0
are given.
a)
Derive the durablegoodsconsumption Euler equation. How does is it
compare with the case when goods are perishable?
Solution.
The Lagrangian for this problem is
L
=
∞
s
t
=0
b
β
t
u
(
d
t
)+
λ
t
(
R
t
+1
(
a
t
+
y
t

c
t
)

a
t
+1
)+
κ
t
((1

δ
)
d
t
+
c
t

d
t
+1
)+
μ
t
+1
(
a
t
+1
+
B
t
+1
)
B
,
(1)
where
λ
t
is the Lagrange multiplier on the agent’s date
t
budget constraint,
κ
t
is the lagrange multiplier on the evolution of durable goods equation and
μ
t
+1
is the Lagrange multiplier on the borrowing constraint on the date
t
+1
assets. The FOCs for this problem for each date
t
are:
a
t
+1
:

λ
t
+
R
t
+2
λ
t
+1
+
μ
t
+1
= 0
,
d
t
+1
:

κ
t
+
κ
t
+1
(1

δ
) +
β
t
+1
u
′
(
d
t
+1
) = 0
,
c
t
:

R
t
+1
λ
t
+
κ
t
= 0
.
Combining the FOCs gives:
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 Spring '11
 ViktorTsyrennikov
 Macroeconomics, Utility, Trigraph, Theory I. Viktor, I. Viktor Tsyrennikov, Macroeconomic Theory I.

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