Econ 387L: Macro II
Spring 2006, University of Texas
Instructor: Dean Corbae
Midterm Exam
1. Consider the following inventory control problem faced by a
fi
rm. In any date
t,
the
fi
rm takes as given a stochastic process for its sales
S
t
and chooses inventories
I
t
+1
taking
as given its current inventories
I
t
in order to maximize its revenues given by sales minus
costs. The costs associated with inventory holdings
C
t
are given by
1
C
t
=
(
S
t
+
I
t
+1
−
I
t
)
2
2
+
αe
−
(
I
t
−
4
S
t
)
.
The
fi
rst term is associated with the costs of production and changes in inventory while the
second term is the cost of stockouts. The sales process is given by
S
t
+1
= (1
−
ρ
) +
ρS
t
+
ε
t
+1
(1)
where
ε
t
+1
is distributed
N
(0
, σ
2
)
and
S
−
1
= 1
.
Firms discount the future at rate
β
∈
(0
,
1)
.
a. (5 points) State the
fi
rm’s problem.
Answer
. The
fi
rm’s problem is
max
{
I
t
+1
≥
0
}
∞
t
=0
E
"
∞
X
t
=0
β
t
{
S
t
−
C
t
(
I
t
+1
, I
t
, S
t
)
}
#
subject to (
??
).
b. (5 points) State necessary conditions for the optimal inventory choice.
Answer.
The necessary F.O.C. with respect to
I
t
+1
is:
−
(
S
t
+
I
t
+1
−
I
t
)
−
βE
t
h
−
(
S
t
+1
+
I
t
+2
−
I
t
+1
)
−
αe
−
(
I
t
+1
−
4
S
t
+1
)
i
= 0
,
∀
t.
(2)
c. (5 points) What is the steady state? Under what condition on parameters are steady
state inventories nonnegative?
Answer
. With
ε
t
= 0
,
∀
t,
then (
??
) implies steady state sales are normalized to 1 (i.e.
S
= 1)
.
Furthermore (2) implies
−
1 +
β
(1 +
αe
−
(
I
−
4)
)
=
0
(3)
⇐⇒
I
= ln(
α
)
−
ln(1
−
β
) + ln(
β
) + 4
.
Note that
I
≥
0
⇔
ln(
α
)
−
ln(1
−
β
) + 4
≥
ln(
β
)
.
d. (10 points) What are the model parameters? Calibrate these numbers to the following
data (Hint: You needn’t actually calculate the numerical value of the parameters, just state
the system of equations in a way that makes it clear how you would calculate them):
(i). The long run inventory to sales ratio is 0.5
(ii). The interest rate that makes
fi
rms not want to borrow or save is given by
4
.
16%
(iii). The autocorrelation of sales
0
.
9
and the variance of sales is
1
.
Answer
. The set of parameters to calibrate is
(
β, α, ρ, σ
2
)
.
1
Recall that
e
−
x
>
0
is everywhere positive for all
x
∈
R
,
strictly convex,
lim
x
→−∞
=
∞
,
lim
x
→∞
= 0
.
Furthermore, recall that
ln(
e
g
(
x
)
) =
g
(
x
)
and
de
g
(
x
)
dx
=
g
0
(
x
)
e
g
(
x
)
for some function
g
(
x
)
.
1
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From (ii) we can obtain
β
. Speci
fi
cally,
−
1
invested today yields
β
(1 +
r
)
tomorrow.
Hence
−
1 +
β
(1 +
r
) = 0
→
β
=
1
(1+
r
)
= 0
.
96
.
Combining (i), equation (3), and the last result we pin down
α
. Speci
fi
cally,
βαe
0
.
5
−
4
= 1
−
β.
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 Spring '07
 CORBAE
 Game Theory, Necessary and sufficient condition, inventory control problem

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