Department of Economics
The Ohio State University
Final Exam Answers–Part II–Econ 805
Profs. Levin, Morelli, and Peck
March 13, 2001
3. (33 points)
The following economy has n consumers and k commodities.
Each con
sumer’s utility function satis…es strict monotonicity, strict quasiconcavity, and
continuity. Suppose that
(
p
¤
;x
¤
)
is a competitive equilibrium, and that
x
¤¤
is
a feasible allocation that is
not
Pareto optimal.
For each of the following statements, either prove the statement or provide
a counterexample.
Carefully explain.
[You can use the theorems proven in
class without proving them here.]
(a) For
all
i, we have
u
i
(
x
¤
i
)
¸
u
i
(
x
¤¤
i
)
:
(b) At least one consumer prefers her competitive equilibrium bundle to
any other consumer’s bundle. That is, for
some
i, we have
u
i
(
x
¤
i
)
¸
u
i
(
x
¤
h
)
for all h.
(c)
p
¤
¢
n
X
i
=1
x
¤
i
¸
p
¤
¢
n
X
i
=1
x
¤¤
i
:
ANSWER:
(a) This statement is false.
There must be some allocation that Pareto
dominates
x
¤¤
, but it does not have to be
x
¤
.
For example, in the following
diagram,
x
¤
is the allocation determined by the point (.5, .5), and
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 Spring '08
 PECK
 Economics, Utility, competitive equilibrium, Economic equilibrium, competitive equilibrium allocation, es strict monotonicity

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