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Department of Economics
The Ohio State University
Midterm Answers—Econ 805
Prof. Peck
February 7, 2008
1. (35 points)
Consider the following economy with two goods, two
f
rms, and one con
sumer. The consumer’s utility function is given by
u
(
x
1
,x
2
)=
A
log(
x
1
)+log(
x
2
)
,
where
A
is a strictly positive parameter. The consumer owns both
f
rms and
has the initial endowment vector,
ω
=(1
,
1)
. Firm 1 produces good 1 with good
2 as an input (that is,
y
1
1
≥
0
and
y
2
1
≤
0
)
,andhasaproduct
ionfunc
t
ionor
boundary of the production set given by
y
1
1
=
−
1
2
y
2
1
.
Firm 2 produces good 2 with good 1 as an input (that is,
y
2
2
≥
0
and
y
1
2
≤
0
),
and has a production function or boundary of the production set given by
y
2
2
=
−
1
2
y
1
2
.
(a) (10 points) De
f
ne a competitive equilibrium for this economy.
(b) (20 points) Normalize the price of good 2 to be one, so the price vector
is given by
(
p,
1)
. Compute the competitive equilibrium price and allocation, as
afunct
iono
ftheparameter
A
. (Hint: Depending on
A
,i
tisposs
ib
letha
tone
or both
f
rms do not produce.)
(c) (5 points) For what values of
A
is the competitive equilibrium allocation
Pareto optimal?
Answer:
(a) Note: because of strict monotonicity, we can write budget constraints
and market clearing as equalities.
A CE is a price vector
(
p
1
,p
2
)
and an
allocation,
(
x
1
2
,y
1
1
2
1
1
2
2
2
)
, such that
(i)
(
x
1
2
)
solves
max
A
log(
x
1
x
2
)
subject to
p
1
x
1
+
p
2
x
2
=
p
1
+
p
2
+
π
1
+
π
2
x
≥
0
,
1
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View Full Document(ii)
(
y
1
1
,y
2
1
)
solves
max
p
1
y
1
1
+
p
2
y
2
1
subject to
y
1
1
=
−
1
2
y
2
1
y
1
1
≥
0
2
1
≤
0
,
(iii)
(
y
1
2
2
2
)
solves
max
p
1
y
1
2
+
p
2
y
2
2
subject to
y
2
2
=
−
1
2
y
1
2
y
1
2
≤
0
2
2
≥
0
,
(iv) markets clear
x
1
=1+
y
1
1
+
y
1
2
x
2
y
2
1
+
y
2
2
.
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 Spring '08
 PECK
 Utility

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