3. Since
𝑆
is open,
𝑇
=
𝑓
−
1
(
𝑆
) is open. Therefore,
𝑇
=
𝑓
(
𝑆
) is a neighborhood of
𝑓
(
x
0
). Therefore,
𝑓
is locally onto.
4.62
Assume to the contrary that there exists
x
0
∕
=
x
1
∈
𝑆
with
𝑓
(
x
0
) =
𝑓
(
x
1
). Let
x
=
x
1
−
x
0
. Define
𝑔
: [0
,
1]
→
𝑆
by
𝑔
(
𝑡
) = (1
−
𝑡
)
x
0
+
𝑡
x
1
=
x
0
+
𝑡
x
. Then
𝑔
(0) =
x
0
𝑔
(1) =
x
1
𝑔
′
(
𝑡
) =
x
Define
ℎ
(
𝑡
) =
x
𝑇
(
𝑓
(
𝑔
(
𝑡
)
)
−
𝑓
(
x
0
)
)
Then
ℎ
(0) = 0 =
ℎ
(1)
By the mean value theorem (Mean value theorem), there exists 0
< 𝛼 <
1 such that
𝑔
(
𝛼
)
∈
𝑆
and
ℎ
′
(
𝛼
) =
x
𝑇
𝐷𝑓
[
𝑔
(
𝛼
)]
x
=
x
𝑇
𝐽
𝑓
(
𝑔
(
𝛼
))
x
= 0
which contradicts the definiteness of
𝐽
𝑓
.
4.63
Substituting the linear functions in (4.35) and (4.35), the ISLM model can be
expressed as
(1
−
𝐶
𝑦
)
𝑦
−
𝐼
𝑟
𝑟
=
𝐶
0
+
𝐼
0
+
𝐺
−
𝐶
𝑦
𝑇
𝐿
𝑦
𝑦
+
𝐿
𝑟
𝑟
=
𝑀/𝑃
which can be rewritten in matrix form as
(
1
−
𝐶
𝑦
𝐼
𝑟
𝐿
𝑦
𝐿
𝑟
) (
𝑦
𝑟
)
=
(
𝑍
−
𝐶
𝑦
𝑇
𝑀/𝑃
)
where
𝑍
=
𝐶
0
+
𝐼
0
+
𝐺
. Provided the system is nonsingular, that is
𝐷
=
1
−
𝐶
𝑦
𝐼
𝑟
𝐿
𝑦
𝐿
𝑟
∕
= 0
the system can be solved using Cramer’s rule (Exercise 3.103) to yield
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 Fall '10
 Dr.DuMond
 Macroeconomics, Mean Value Theorem, All rights reserved, Michael Carter, Foundations of Mathematical Economics

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