Department of Economics
The Ohio State University
Prof. Peck
March 16, 2004
Directions:
Answer all questions, carefully label all diagrams, and show
all work.
1. (30 points)
Consider an exchange economy with three consumers and two goods,
n
= 3
and
k
= 2
.
For
i
= 1
;
2
;
3
, the utility function of consumer
i
,
u
i
(
x
i
)
is dif-
ferentiable, strictly concave, and strictly monotonic. For each of the following
statements, either prove the statement (if the statement is true) or ±nd a coun-
terexample (if the statement is false).
(a) If we have a feasible allocation,
x >>
0
, such that consumer 1²s marginal
rate of substitution is equal to consumer 2²s marginal rate of substitution, then
there cannot exist another feasible allocation,
x
0
, such that
u
1
(
x
0
1
)
>
u
1
(
x
1
)
u
2
(
x
0
2
)
>
u
2
(
x
2
)
u
3
(
x
0
3
)
u
3
(
x
3
)
:
(b) Given initial endowments,
!
= (
!
1
; !
2
; !
3
)
, it is impossible for any two
of the consumers to strictly be better o/ excluding the third consumer from
trade. That is, two of the consumers cannot each receive strictly higher utility
by trading among themselves, intead of having the three consumers trade to a
competitive equilibrium.
(c) If the initial endowments,
!
= (
!
1
; !
2
; !
3
)
, are strongly Pareto optimal,
then there is a competitive equilibrium in which no trade takes place.
Answer:
(a) False.
Although consumers 1 and 2 have the same marginal rate of
substitution, consumer 3²s MRS can be di/erent, so there can be potential gains
from trade.
For example, suppose that, for
i
= 1
;
2
;
3
, we have
u
i
(
x
1
i
; x
2
i
) =
log(
x
1
i
) + log(
x
2
i
)
, and let
x
1
= (100
;
1)
; x
2
= (100
;
1)
;
and
x
3
= (1
;
150)
. Now
let
x
0
1
= (50
;
50)
; x
2
= (50
;
50)
;
and
x
3
= (101
;
52)
. This is feasible and provides
everyone with higher utility.
A few of you noticed that we never required
x
to be nonwasteful, so counterexamples could be constructed based on
x
being
wasteful.
(b) True. Although one of the consumers might want to exclude one of the
others, it is impossible for two of the consumers each to prefer to exclude the
third. To prove it, let
(
p
; x
)
be the competitive equilibrium with 3 consumers,
and suppose without loss of generality that consumers 1 and 2 could each receive
1