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Cornell University
Economics 3130
Problem Set 3 Solutions
1. Consider the utility function
U
(
x
1
,x
2
,x
3
) =
α
ln
x
1
+
β
ln
x
2
+
γ
ln
x
3
. Assume the
consumer has income
I
and faces prices
p
1
,
p
2
, and
p
3
. Find the Marshallian demands.
We can use the Lagrangian method.
L
=
α
ln
x
1
+
β
ln
x
2
+
γ
ln
x
3
+
λ
(
I

p
1
x
1

p
2
x
2

p
3
x
3
)
The ﬁrst order conditions (FOC’s) are as follows.
x
1
:
αx

1
1

λp
1
= 0
x
2
:
βx

1
2

λp
2
= 0
x
3
:
γx

1
3

λp
3
= 0
λ
:
I

p
1
x
1

p
2
x
2

p
3
x
3
= 0
Using the ﬁrst two FOC’s we ﬁnd that
x
2
=
βp
1
αp
2
x
1
. Using the ﬁrst and third FOC’s we
ﬁnd that
x
2
=
γp
1
αp
3
x
1
. Substituting into the budget constraint (the fourth FOC), we
ﬁnd that
x
1
=
α
α
+
β
+
γ
I
p
1
. It follows that
x
2
=
β
α
+
β
+
γ
I
p
2
and
x
3
=
γ
α
+
β
+
γ
I
p
3
.
2. Consider the twogood CES utility function
U
(
x
1
,x
2
) =
1
(1

α
)
(
x
1

α
1
+
x
1

α
2
). Suppose
that
α
can take any value from 0 to inﬁnity. As usual, assume that the consumer has
income
I
and faces prices
p
1
and
p
2
.
(a) What is the marginal rate of substitution?
MRS
= (
x
2
x
1
)
α
(b) What value of
α
corresponds to perfect substitutes? to CobbDouglas? to perfect
complements?
For perfect substitutes
α
= 0, for CobbDouglas
α
= 1, and for perfect comple
ments
α
goes to inﬁnity.
(c) In class we calculated the elasticity of substitution as follows:
σ
=
MRS
r
dr
dMRS
. We
deﬁned
r
=
x
2
x
1
. The answer from (a) above is a simple function of
r
, so you can
write the ﬁrst term using
r
and not
x
1
or
x
2
. Then, with the MRS written in
terms of
r
, take the derivative
dMRS
dr
. The inverse of that is the second term in the
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This note was uploaded on 08/29/2009 for the course ECON 3130 taught by Professor Masson during the Spring '06 term at Cornell University (Engineering School).
 Spring '06
 MASSON
 Microeconomics, Utility

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