Solution Homework Assignment 3
Econ 150b. Intermediate Microeconomics
Fabian Duarte
*
February 5, 2008
1.
Consider a consumer whose utility function is given by
u
(
x
1
,
x
2
) = (
α
x
ρ
1
+(
1

α
)
x
ρ
2
)
1
ρ
(1)
where
ρ ε
(
0
,
1
)
is a parameter. Using any of the three methods we’ve seen in class, find the
optimal consumption bundle as a function of prices
(
p
1
,
p
2
)
and income
m
. Plot the demand
and Engel curves.
(a)
First method: We use two equations:
•
MRS
=

p
1
p
2
•
p
1
x
1
+
p
2
x
2
=
m
So, first we calculate
MRS
, for that we need to calculate first partial derivatives
∂
u
∂
x
1
=
1
ρ
(
α
x
ρ
1
+(
1

α
)
x
ρ
2
)
1
ρ

1
(
αρ
x
ρ

1
1
)
and
∂
u
∂
x
2
=
1
ρ
(
α
x
ρ
1
+(
1

α
)
x
ρ
2
)
1
ρ

1
((
1

α
)
ρ
x
ρ

1
2
)
then
MRS
=

α
x
ρ

1
1
(
1

α
)
x
ρ

1
2
=

p
1
p
2
(2)
*
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then
x
1
=
‡
(
1

α
)
p
1
α
p
2
·
1
ρ

1
x
2
(3)
Now, we replace
x
1
in the second equation
p
1
x
1
+
p
2
x
2
=
m
and we get
m
=
p
1
‡
(
1

α
)
p
1
α
p
2
·
1
ρ

1
x
2
+
p
2
x
2
m
=
x
2
h‡
(
1

α
)
α
p
2
·
1
ρ

1
p
ρ
ρ

1
1
+
p
2
i
m
=
x
2
h
(
1

α
)
1
ρ

1
p
ρ
ρ

1
1
+
α
1
ρ

1
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 Spring '08
 EduardoFaingold
 Microeconomics, Utility, p1, Fabian Duarte, [email protected]

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