Jan Hagemejer
1
Solutions to selected problems from homework 1.
Question 1
Let
u
be a utility function which generates demand function
x
(
p, w
) and indirect utility function
v
(
p, w
). Let
F
:
R
→
R
be a strictly increasing function. If the utility function
u
*
is defined by
u
*
(
x
) =
F
(
u
(
x
)) what are the demand functions
generated by
u
*
(answer in terms of
x
(
p, w
) and
v
(
p, w
)).
In fact, by mistake I ommited the part of the question that I intended to include and this has mislead many of you.
The missing sentence should be “How are the indirect utility functions related”. Of course everybody who handed in the
homework gets full credit for the question 1. So here is the solution:
Step 1. You can show that the utility maximization of
u
leads to the same demand
x
(
p, w
) maximization of
u
*
.
You do
not have to do that since we said that in the lecture, but here it is:
L
1
=
u
(
x
)

λ
(
X
i
p
i
x
i

w
) and
L
2
=
F
(
u
(
x
))

λ
(
X
i
p
i
x
i

w
)
if we take the FOC’s with respect to
x
i
and
x
j
for
L
1
we get:
∂L
1
∂x
i
=
∂U
(
x
)
/∂x
i

λp
i
= 0
∂L
1
∂x
j
=
∂U
(
x
)
/∂x
j

λp
j
= 0
these
lead to:
∂U
(
x
)
/∂x
i
∂U
(
x
)
/∂x
j
=
p
i
p
j
(1)
for any pair of
i
and
j.
For
L
2
we get:
∂L
1
∂x
i
= (
∂F
(
·
)
/∂U
(
x
))(
∂U
(
x
)
/∂x
i
)

λp
i
= 0
∂L
1
∂x
j
= (
∂F
(
·
)
/∂U
(
x
))
∂U
(
x
)
/∂x
j

λp
j
= 0
And these lead to the same condition as in (1).
Since the budget constraint is the same for both problems,
x
(
p, w
) =
x
*
(
p, w
).
Step 2. What about theV*(p, w)? We know that
1. We know that
u
*
(
x
) =
F
(
u
(
x
)).
So we know with certainty that
h
*
(
p, u
*
) =
h
(
p, F

1
(
u
*
)) =
h
(
p, u
), since
F

1
(
u
*
) =
u
.
2. To calculate expenditure we take
e
*
(
p, u
*
) =
∑
l
p
l
h
*
l
(
p, u
*
) =
∑
l
p
l
h
l
(
p, F

1
(
u
*
)) =
∑
l
p
l
h
l
(
p, u
) =
e
(
p, u
)
.
Questions 3 and 4
Note that these questions are related.
You can do it 2 ways.
First of all, please note that the function is
constructed in such a way that for any positive level of utility, the zero consumption is impossible. Therefore we can rule
out corner solutions (
x
= 0),
One way:
Question 3.
Rederive CobbDouglas Walrasian and Hicksian demands for the general case:
U
(
x
) =
n
Y
i
=1
x
α
i
i
with
∑
n
i
=1
α
i
= 1.
You've reached the end of your free preview.
Want to read all 7 pages?
 Fall '15
 Utility