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Problem Set 5 — Solutions
Econ 3102, Winter 2011
1 PE in an Exchange Economy
For the purpose of taking derivatives, it may be useful to write
u
1
as follows:
u
1
(
x
1
,x
2
) =
√
x
1
y
1
=
√
x
1
·
√
y
1
=
x
1
/
2
1
y
1
/
2
1
First, solve for the marginal utilities:
MU
x
1
=
∂u
1
∂x
1
=
y
1
/
2
1
·
1
2
·
x

1
/
2
1
=
√
y
1
2
√
x
1
x
2
=
2
2
= 1
y
1
=
1
∂y
1
=
x
1
/
2
1
·
1
2
·
y

1
/
2
1
=
√
x
1
2
√
y
1
y
2
=
2
2
= 1
We can now solve for the marginal rates of substitution:
MRS
1
=
x
1
y
1
=
√
y
1
2
√
x
1
√
x
1
2
√
y
1
=
√
y
1
2
√
x
1
·
2
√
y
1
√
x
1
=
y
1
x
1
2
=
x
2
y
2
= 1
The following equality identiﬁes the interior PE allocations:
1
=
y
1
x
1
= 1 =
2
This simpliﬁes to
y
1
=
x
1
. Therefore, an interior allocation is Pareto eﬃcient if and only if it lies
on the line
y
1
=
x
1
and satisﬁes the feasibility conditions 0
≤
x
1
≤
12 and 0
≤
y
1
≤
8. Note that
including both feasibility constraints is a bit redundant because
y
1
=
x
1
. It suﬃces to just put
bounds on either
x
1
or
y
1
. However, since
e
x
= 12
>
8 =
e
y
, the feasibility constraint for good
y
is binding before that of good
x
. Thus, the set of interior PE allocations are those for which
0
≤
x
1
≤
8,
y
1
=
x
1
,
x
2
= 12

x
1
, and
y
2
= 8

y
1
.
Graphically, the set of interior PE allocations is the line starting in the bottomleft corner of
the Edgeworth box and increasing with a slope of 1 until it reaches the upper boundary of the
box. That is, it is the line connecting the allocation ((0
,
0)
,
(12
,
8)) to the allocation ((8
,
8)
,
(4
,
0)).
Figure 1 illustrates the Edgeworth box and this part of the contract curve. It also includes some
indiﬀerence curves for each consumer.
There are several ways to solve for the boundary PE allocations. I think the simplest method
is to divide the Edgeworth box into two regions, (i) allocations with
y
1
> x
1
, and (ii) allocations
with
y
1
< x
1
. Let’s consider which, if any, allocation in these regions could be PE:
1
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View Full Document(a) Case (i) —
y
1
> x
1
: In this region, note that
MRS
1
=
y
1
x
1
>
1 =
2
. Therefore, at such
an allocation, both individuals would beneﬁt if consumer 1 trades some
y
to consumer 2 in
exchange for some
x
(at a rate of
E
units of
y
per unit of
x
for some
1
> E > MRS
2
).
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 Spring '08
 Witte
 Macroeconomics

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