Econ 51: Final Answer Key
1 Walrasian Equilibrium and Pareto E
ﬃ
ciency (23 points)
(a) (5 points) Apples and bananas are perfect substitutes for both Rob and Tom. Tom
likes more apples, Rob likes more bananas. Therefore, a Pareto e
ﬃ
cient allocation
assigns all the apples and some bananas to Tom, or all the bananas and some apples
to Rob. In math:
{
(
a
R
,b
R
):
b
R
=40
,
0
≤
a
R
≤
100
}
(the upper side of the Edgeworth
box) or
{
(
a
R
,b
R
):
a
R
=0
,
0
≤
b
R
≤
40
}
(the left side of the Edgeworth box).
(b) (7 points) Because the
f
rms can transform one-to-one, any Pareto e
ﬃ
cient allocation
should assign all bananas to Rob and all apples to Tom. To see why, suppose for
example that Rob has some apples. He can use the
f
rms to transform those apples to
bananas. This will make him happier, without a
f
ecting anyone else in the economy.
Given that Rob gets all bananas and Tom gets all apples, any production plan is Pareto
e
ﬃ
cient. In math:
{
(
a
R
,b
R
,a
T
,b
T
):
a
R
=0
,b
T
=0
,a
R
+
b
T
=140
}
.
(c) (7 points) Normalize
p
a
=1.If
p
b
6
=
p
a
one of the
f
rms can make in
f
nite pro
f
ts, so we
cannot have an equilibrium. Therefore, in equilibrium it must be that
p
b
=1
. W
ith
these prices, Rob would spend all his income on bananas, and Tom would spend all
his income on apples. Therefore, consumption will be
c
R
=(0
,
100) and
c
T
=(40
,
0).
To make this feasible,
f
rm 2 will transform 60 apples to 60 bananas.
(d) (4 points) The answers to parts (b) and (c) are unchanged. Firms 3 and 4 are less
e
ﬃ
cient than
f
rms 1 and 2. For part (b): it is easy to see that it is not e
ﬃ
cient to use
them for production; their inputs can produce more if used by the other
f
rms. For
part (c): if either
f
rm 3 or 4 produces, one of the other
f
rms will have in
f
nite demand,
so this cannot be an equilibrium.
2 Externalities (22 points)
(a) (8 points) Normalize
p
m
=1
.C
lear
ly
,the
f
rm produces in equilibrium, so we must have
p
c
= 1. Rob’s demand is given by
MRS
R
=
3
c
R
=
p
c
p
m
= 1, i.e.
c