October 21, 2005 12:18
master
Sheet number 76 Page number 60
60
Lecture Five
Example 3:
Consider a consumer who spends
α
k
of his wealth on commodity
k
(where
α
k
≥
0 and
6
K
k
=
1
α
k
=
1). This rule of behavior is not formu
lated as a maximization of some preference relation. It can how
ever be fully rationalized by the preference relation represented by
the CobbDouglas utility function
u
(
x
)
=
5
K
k
=
1
x
α
k
k
. A solution
x
∗
to
the consumer’s problem
B
(
p
,
w
)
must satisfy
x
∗
k
>
0 for all
k
(notice
that
u
(
x
)
=
0 when
x
k
=
0 for some
k
). Given the differentiability
of the preferences, a necessary condition for the optimality of
x
∗
is
that
v
k
(
x
∗
)/
p
k
=
v
l
(
x
∗
)/
p
l
for all
k
and
l
where
v
k
(
x
∗
)
=
du
/
dx
k
(
x
∗
)
=
α
k
u
(
x
∗
)/
x
∗
k
for all
k
. It follows that
p
k
x
∗
k
/
p
l
x
∗
l
=
α
k
/α
l
for all
k
and
l
and thus
x
∗
k
=
α
k
w
/
p
k
for all
k
.
Example 4:
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This note was uploaded on 12/29/2011 for the course ECO 443 taught by Professor Aswa during the Fall '10 term at SUNY Stony Brook.
 Fall '10
 aswa

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