3.172
Continuity is a necessary and suﬃcient condition for the existence of a utility
function representing
≿
(Remark 2.9).
Suppose
𝑢
represents the homothetic preference relation
≿
. For any
x
1
,
x
2
∈
𝑆
𝑢
(
x
1
) =
𝑢
(
x
2
) =
⇒
x
1
∼
x
2
=
⇒
𝑡
x
1
∼
𝑡
x
2
=
⇒
𝑢
(
𝑡
x
1
) =
𝑢
(
𝑡
x
2
) for every
𝑡 >
0
Conversely, if
𝑢
is a homothetic functional,
x
1
∼
x
2
=
⇒
𝑢
(
x
1
) =
𝑢
(
x
2
) =
⇒
𝑢
(
𝑡
x
1
) =
𝑢
(
𝑡
x
2
) =
⇒
𝑡
x
1
∼
𝑡
x
2
for every
𝑡 >
0
3.173
Suppose that
𝑓
=
𝑔
∘
ℎ
where
𝑔
is strictly increasing and
ℎ
is homogeneous of
degree
𝑘
. Then
ˆ
ℎ
(
x
) =
(
ℎ
(
x
)
)
1
/𝑘
is homogeneous of degree one and
𝑓
= ˆ
𝑔
∘
ˆ
ℎ
where
ˆ
𝑔
(
𝑦
) =
𝑔
(
𝑦
𝑘
)
)
is increasing.
3.174
Assume
x
1
,
x
2
∈
𝑆
with
𝑓
(
x
1
) =
𝑔
(
ℎ
(
x
1
)) =
𝑔
(
ℎ
x
2
)) =
𝑓
(
x
2
)
Since
𝑔
is strictly increasing, this implies that
ℎ
(
x
1
) =
ℎ
(
x
2
)
Since
ℎ
is homogeneous
ℎ
(
𝑡
x
1
) =
𝑡
𝑘
ℎ
(
x
1
) =
𝑡
𝑘
ℎ
(
x
2
) =
ℎ
(
𝑡
x
2
)
for some
𝑘
. Therefore
𝑓
(
𝑡
x
1
) =
𝑔
(
ℎ
(
𝑡
x
1
)) =
𝑔
(
ℎ
(
𝑡
x
2
)) =
𝑓
(
𝑡
x
2
)
3.175
Let
x
0
∕
=
0
be any point in
𝑆
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 Fall '10
 Dr.DuMond
 Macroeconomics, Utility, Foundations of Mathematical Economics, homothetic preference relation

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