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Shrinking Core Example
Consider two individuals,
A
and
B
, who have identical utility functions
x
1
x
2
.
A
starts oﬀ with
e
A
1
= 10 and
e
A
2
= 0 while
B
starts oﬀ with
e
B
1
= 0 and
e
B
2
= 10. Note that both
A
and
B
get zero
utility from their current endowments. Since both are getting zero utility from their endowments,
all Pareto eﬃcient points that give utility greater than or equal to zero for both
A
and
B
will be
in the core. Therefore,
x
A
1
= 0,
x
A
2
= 0,
x
B
1
= 10, and
x
B
2
= 10 is a point in the core:
A
gets zero
utility and this a PE point as there is no way of making
A
or
B
better oﬀ without making the
other worse oﬀ.
Suppose now that there are two
A
’s and two
B
’s. If we give both A’s zero of each good and both
B’s 10 of each good will this allocation be in the core? The answer is no, as the core requires both
Pareto eﬃciency
and
that no coalition of individuals could be made better oﬀ by trading among
themselves. The only coalition when there is one
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 Fall '09
 CUR
 Microeconomics, Utility

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