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
fi
nanced by lump sum taxes
T
t
.
Household preferences are given by
E
£P
∞
t
=0
β
t
ln(
C
t
)
¤
.
Production satis
fi
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
from a zero mean uniform distribution with support [
u
min
, u
max
] such that
G
t
satis
fi
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
ff
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
(this is due to the lump sum nature of taxation 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? Explain how you would determine whether the
”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
fi
cally the second order di
ff
erence equation in
K
t
given in (3)
and then use the method of undetermined coe
ﬃ
cients to solve for the “reduced
form” coe
ﬃ
cients of the decision rule (speci
fi
cally, conjecturing the decision
1
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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
) = 0 and
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
(
γ,
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
, K
t
, G
t
+1
, G
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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 CORBAE
 Thermodynamics, Kt Kt K

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