3. Assume
≿
is locally nonsatiated. Then, for every
x
∈
𝑋
, there exists some
y
∈
𝑋
such that
y
≻
x
. Therefore, there is no best element.
1.234
Assume otherwise, that is assume that
x
∗
≿
x
for every
x
∈
𝐵
(
p
, 𝑚
) but that
∑
𝑛
𝑖
=1
𝑝
𝑖
𝑥
𝑖
< 𝑚
. Let
𝑟
=
𝑚
−
∑
𝑛
𝑖
=1
𝑝
𝑖
𝑥
𝑖
be the unspent income. Spending the residual
on good 1, the commodity bundle
x
=
x
∗
+
𝑟
𝑝
1
e
1
is affordable
𝑛
∑
𝑖
=1
𝑝
𝑖
𝑥
𝑖
=
𝑛
∑
𝑖
=1
𝑝
𝑖
𝑥
∗
𝑖
+
𝑝
1
𝑟
𝑝
1
=
𝑚
Moreover, since
x
≩
x
∗
,
x
≻
x
∗
, which contradicts the assumption that
x
∗
is the best
element in
𝑋
(
p
, 𝑚
).
1.235
Assume otherwise, that is assume that
x
∗
≿
x
for every
x
∈
𝐵
(
p
, 𝑚
) but that
∑
𝑛
𝑖
=1
𝑝
𝑖
𝑥
∗
𝑖
< 𝑚
. This implies that
x
∗
∈
int
𝑋
(
p
, 𝑚
). Therefore, there exists a neigh
borhood
𝑁
of
x
∗
with
𝑁
⊆
𝑋
(
p
, 𝑚
).
Within this neighborhood, there exists some
x
∈
𝑁
⊆
𝑋
(
p
, 𝑚
) with
x
≻
x
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 Fall '10
 Dr.DuMond
 Macroeconomics, Open set, Z0, x∗, Michael Carter, open neighborhood, Foundations of Mathematical Economics

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