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Econ 387L: Macro II
Spring 2004, University of Texas
Instructor: Dean Corbae
Answers  Midterm Exam
(1) 15 points. Consider the following business cycle model. The only source
of uncertainty comes from exogenous government expenditure
G
t
on defense
that is
f
nanced by lump sum taxes
T
t
. Household preferences are given by
E
£P
∞
t
=0
β
t
ln(
C
t
)
¤
.
Production satis
f
es
Y
t
=
K
θ
t
.
Capital depreciates fully in a
period (i.e.
δ
= 1). The law of motion for government expenditure is given by
G
t
=(1
−
γ
)
G
+
γG
t
−
1
+
u
t
where
G
is steady state expenditure and
u
t
is drawn
fromazeromeanun
i
formd
istr
ibut
ionw
ithsupport[
u
min
,u
max
] such that
G
t
satis
f
es
Y
t
>G
t
>
0 around the steady state.
(a) Formulate the planner’s problem of allocating capital in the economy as
a nonlinear programming problem.
Answer:
The planner’s problem is
max
{
K
t
}
E
0
∞
X
t
=0
β
t
ln(
C
t
)(
1
)
subject to
C
t
+
K
t
+1
+
G
t
=
K
θ
t
(2)
(b) Solve for the steady state. How do changes in
G
a
f
ect the steady state
capital stock and implied interest rate?
Answer:
Plugging
C
t
from the resource constraint (2) into the objective
function and taking the FOC of (1) with respect to
K
t
+1
we have
1
K
θ
t
−
K
t
+1
−
G
t
=
βE
t
"
θK
θ
−
1
t
+1
K
θ
t
+1
−
K
t
+2
−
G
t
+1
#
.
(3)
At the steady state,
K
t
+1
=
K
t
=
K
and
G
t
+1
=
G
t
=
G
implies
K
=(
βθ
)
1
1
−
θ
.
(4)
Hence, the steadystate capital stock is independent of government expenditure
(th
isisduetothelumpsumnatureo
ftaxation in this example). This implies
that the steady state interest rate (i.e. the marginal product of capital) is
independent of government expenditure.
(c) Explain a process to generate linear decision rules. What are the state
variables in the decision rules? Explainhowyouwou
lddeterm
inewhetherthe
”reduced form” coe
ﬃ
cients of the decision rule is invariant to government policy.
Answer:
The process we have used in class is to linearize the necessary
conditions (speci
f
cally the second order di
f
erence equation in
K
t
given in (3)
andthenusethemethodo
fundeterm
inedcoe
ﬃ
cients to solve for the “reduced
form” coe
ﬃ
cients of the decision rule (speci
f
cally, conjecturing the decision
1
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View Full Document rule is of the form
b
k
t
+1
=
A
b
k
t
+
B
b
G
t
and plugging into the linearized version
of (3) yields two nonlinear equations as functions of the deep parameters (say
f
1
(
A, B, γ,
G, β, θ
)=0and
f
2
(
A, B, γ,
G, β, θ
) = 0). In general, the solution of
this nonlinear system of equations yields
A
(
γ,
G, β, θ
)and
B
(
G, β, θ
)
.
While we did not explicitly ask you to do it, here’s the solution. Writing (3)
as
V
(
K
t
+2
,K
t
+1
t
,G
t
+1
t
)=
1
C
t
−
βE
t
"
θK
θ
−
1
t
+1
C
t
+1
#
(5)
where
C
t
=
K
θ
t
−
K
t
+1
−
G
t
C
t
+1
=
K
θ
t
+1
−
K
t
+2
−
G
t
+1
.
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This note was uploaded on 08/06/2008 for the course ECON 387 taught by Professor Corbae during the Spring '07 term at University of Texas at Austin.
 Spring '07
 CORBAE

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