Econ 387L: Macro II
Spring 2003, University of Texas
Instructor: Dean Corbae
Midterm Exam Answers
Consider the following investment problem faced by a
&
rm. The
&
rm&s revenues are
given by
R
(
K
t
,Z
t
)
and there are costs of adjusting its capital holdings
C
(
K
t
+1
,K
t
)
where
K
t
is the capital stock at the beginning of period
t
,
K
t
+1
≥
0
is capital chosen at the
beginning of period
t
to come on line in
t
+1
,
Z
t
is a technology innovation, and
K
0
>
0
is given.Gross investment is given by
I
t
=
K
t
+1
−
(1
−
δ
)
K
t
and the relative price of
investment goods in terms of consumption goods is
p
t
. Firms discount the future at rate
β
∈
(0
,
1)
.The
&
rm&s problem is
max
{
K
t
+1
}
∞
t
=0
E
"
∞
X
t
=0
β
t
{
R
(
K
t
t
)
−
C
(
K
t
+1
t
)
−
p
t
[
K
t
+1
−
(1
−
δ
)
K
t
]
}
#
.
We will assume the following:
R
(
K
t
t
)=
Z
t
K
t
,
C
(
K
t
+1
t
γ
2
&
I
t
K
t
¶
2
K
t
,and
p
t
=
p,
∀
t.
There are 4 parts to this question.
1. What are the
&
rst order conditions? Interpret them in terms of the marginal cost of
adding capital versus the marginal bene
&
t.
Answer:
The f.o.c. is given by
p
t
+
∂C
(
K
t
+1
t
)
∂K
t
+1
=
βE
t
•
∂R
(
K
t
+1
t
+1
)
t
+1
−
(
K
t
+2
t
+1
)
t
+1
+
p
t
+1
(1
−
δ
)
‚
or
γ
&
I
t
K
t
¶
+
p
=
t
•
Z
t
+1
+
γ
2
&
I
t
+1
K
t
+1
¶‰&
I
t
+1
K
t
+1
¶
+2(1
−
δ
)
±
+
p
(1
−
δ
)
‚
(1)
The left hand side is the marginal cost of adding capital (the pecuniary cost plus the
adjustment cost) in period
t
. The right hand side is the marginal bene
&
tinperiod
t
from adding the capital; the marginal product of capital less the costs of adjustment plus
the sales price.
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 Spring '07
 CORBAE
 Economics, zt

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