Microeconomics
ECON 100A
Problem Set 5 Solutions
Due October 26, 2010 (Solutions Posted October 29, 2010)
1.
Consider the expenditure minimization problem subject to achieving a given level of utility
5
.
1
5
.
0
8
z
x
U
=
a.
Illustrate the consumer’s problem
Here’s an example. The numbers are the optimal consumption point for taken from an
earlier utility maximization problem, using px = 2.50 and pz = 3.33 and Y = 40.
Z
9
U=432
4
X
b.
Write down the first order conditions for this constrained optimization problem
.
0
8
)
3
(
0
12
)
2
(
0
4
)
1
(
.
.
.
)
8
(
5
.
1
5
.
0
5
.
0
5
.
0
5
.
1
5
.
0
5
.
1
5
.
0
=

=
∂
∂
=

=
∂
∂
=

=
∂
∂


+
=

U
z
x
L
z
x
p
z
L
z
x
p
x
L
C
O
F
U
z
x
z
p
x
p
L
z
x
z
x
λ
c.
Solve for the optimal consumption bundle, X
h
and Z
h
, as a function of
x
p
,
z
p
, and
U
.
X
h
(Px, Pz,
U
) =
(
U
/8)
1/2
(p
z
/3p
x
)
3/4
Z
h
(Px, Pz,
U
) = (
U
/8)
1/2
(3p
x
/p
z
)
1/4
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Solve for minimum expenditures as a function of
x
p
,
z
p
, and
U
E (Px, Pz,
U
) =
p
x
(
U
/8)
1/2
(p
z
/3p
x
)
3/4
+ p
z
(
U
/8)
1/2
(3p
x
/p
z
)
1/4
(…can you simplify this further?)
e.
OK, now you will need to get a calculator.
Let Px=$2.50 and Pz=$3.33 and
U
=432.
What
are the numerical values of X
h
, Z
h
and E?
Put these values on your graph.
Z
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 Fall '08
 ZAMBRANO
 Microeconomics, Utility, Px, Utility maximization problem, pz

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