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Solutions to Problem Set 2: Consumer Theory
Compulsory Home Assignment. Deadline: 13 November
1. Consider the indirect utility function given by
v
(
p
1
,p
2
,m
)=
m
p
1
+
p
2
.
(a) What are the (Marshallian) demand functions?
Solution: Use Roy´s Identity:
x
1
(
p
1
2
−
∂v
(
p
1
,p
2
,m
)
∂p
1
(
p
1
,p
2
,m
)
∂m
=
−
−
m
(
p
1
+
p
2
)
2
1
p
1
+
p
2
=
m
p
1
+
p
2
,
x
2
(
p
1
2
−
(
p
1
,p
2
,m
)
2
(
p
1
,p
2
,m
)
=
−
−
m
(
p
1
+
p
2
)
2
1
p
1
+
p
2
=
m
p
1
+
p
2
.
(b) What is the expenditure function?
Solution: Use the identity
v
(
p, e
(
p, u
)) :=
u.
Replacing
m
with
e
in the indirect utility function and setting it equal
to
u
yields
e
p
1
+
p
2
=
u,
implying
e
(
p
1
2
,u
)=(
p
1
+
p
2
)
u.
(c) What is the direct utility function?
Solution: Since the consumer demands exactly the same quantity of
both goods, irrespective of relative prices, the utility function must be
of the Leontief type, where the two goods are perfect complements in
consumption. From the expression of the indirect utility function, we
know that the utility function must be
u
(
x
1
,x
2
)=min
{
x
1
2
}
.
2. Use the utility function
u
(
x
1
2
x
1
2
1
x
1
3
2
and the budget constraint
m
=
p
1
x
1
+
p
2
x
2
to calculate
1
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This note was uploaded on 02/01/2011 for the course ECONOMY 6 taught by Professor Fallahi during the Spring '10 term at Cambridge.
 Spring '10
 fallahi
 Utility

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