5. Trivially,
x
∗
is a feasible allocation with endowments
w
𝑖
=
x
∗
𝑖
and
𝑚
𝑖
=
p
∗
w
𝑖
=
p
∗
x
∗
𝑖
. To show that (
p
∗
,
x
∗
) is a competitive equilibrium, we have to show that
x
∗
𝑖
is the best allocation in the budget set
𝑋
𝑖
(
p
, 𝑚
𝑖
) for each consumer
𝑖
. Suppose
to the contrary there exists some consumer
𝑗
and allocation
y
𝑗
such that
y
𝑗
≻
x
𝑗
and
py
𝑗
≤
𝑚
𝑗
=
px
∗
𝑗
.
By continuity, there exists some
𝛼
∈
(0
,
1) such that
𝛼
y
𝑗
≻
𝑖
x
∗
𝑗
and
p
(
𝛼
y
𝑗
) =
𝛼
py
𝑗
<
py
𝑗
≤
px
∗
contradicting (3.66). We conclude that
x
∗
𝑖
≿
𝑖
x
𝑖
for every
x
∈
𝑋
(
p
∗
, 𝑚
𝑖
)
for every consumer
𝑖
. (
p
∗
,
x
∗
) is a competitive equilibrium.
3.229
By the previous exercise, there exists a price system
p
∗
such that
x
∗
𝑖
is optimal
for each consumer
𝑖
in the budget set
𝑋
(
p
∗
,
p
∗
x
∗
𝑖
), that is
x
∗
𝑖
≿
𝑖
x
𝑖
for every
x
𝑖
∈
𝑋
(
p
∗
,
p
∗
x
∗
𝑖
)
(3.67)
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 Fall '10
 Dr.DuMond
 Macroeconomics, All rights reserved, p∗, Duality, Foundations of Mathematical Economics

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