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
fi
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
fi
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
), and has a production function or
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
fi
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
a function of the parameter
A
.
(Hint:
Depending on
A
, it is possible that one
or both
fi
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
, x
2
, y
1
1
, y
2
1
, y
1
2
, y
2
2
)
, such that
(i)
(
x
1
, x
2
)
solves
max
A
log(
x
1
) + log(
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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(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
, y
2
1
≤
0
,
(iii)
(
y
1
2
, y
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
, y
2
2
≥
0
,
(iv) markets clear
x
1
=
1 +
y
1
1
+
y
1
2
x
2
=
1 +
y
2
1
+
y
2
2
.
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 Spring '08
 PECK
 Economics, Utility, Economic equilibrium, Akerlof

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