there exists some
x
∈
𝑋
such that
𝑛
∑
𝑖
=1
𝑝
𝑖
𝑥
𝑖
≤
𝑚
Therefore
x
∈
𝑋
(
p
, 𝑚
) which is nonempty.
Let ˇ
𝑝
= min
𝑖
𝑝
𝑖
be the lowest price of the
𝑛
goods. Then
𝑋
(
p
, 𝑚
)
⊆
𝐵
(
0
, 𝑚/
ˇ
𝑝
) and
so is bounded. (That is, no component of an affordable bundle can contain more than
𝑚/
ˇ
𝑝
units.)
To show that
𝑋
(
p
, 𝑚
) is closed, let (
x
𝑛
) be a sequence of consumption bundles in
𝑋
(
p
, 𝑚
). Since
𝑋
(
p
, 𝑚
) is bounded,
x
𝑛
→
x
∈
𝑋
. Furthermore
𝑝
1
𝑥
𝑛
1
+
𝑝
2
𝑥
𝑛
2
+
⋅ ⋅ ⋅
+
𝑝
𝑛
𝑥
𝑛
𝑛
≤
𝑚
for every
𝑛
This implies that
𝑝
1
𝑥
1
+
𝑝
2
𝑥
2
+
⋅ ⋅ ⋅
+
𝑝
𝑛
𝑥
𝑛
≤
𝑚
so that
x
𝑛
→
x
∈
𝑋
(
p
, 𝑚
). Therefore
𝑋
(
p
, 𝑚
) is closed.
We have shown that
𝑋
(
p
, 𝑚
) is a closed and bounded subset of
ℜ
𝑛
. Hence it is compact
(Proposition 1.4).
1.232
Let
x
,
y
∈
𝑋
(
p
, 𝑚
). That is
𝑛
∑
𝑖
=1
𝑝
𝑖
𝑥
𝑖
≤
𝑚
𝑛
∑
𝑖
=1
𝑝
𝑖
𝑦
𝑖
≤
𝑚
For any
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Dr.DuMond
 Macroeconomics, Weighted mean, All rights reserved, Order theory, Monotonic function, Convex function, Compact space

Click to edit the document details