Microeconomics
ECON 100A
Problem Set 7
Due November 9, 2010 (Solutions Posted November 12, 2010)
1.
From problem set 4 question 3 you already have seen the following utility function:
z
x
z
x
U
2
)
,
(
+
=
a.
Solve for the optimal consumption bundle, x* and z*, as a function of
x
p
,
z
p
, and Y.
z*(p
x
, p
z
, Y)=
2
)
(
z
x
p
p
,
x*(p
x
, p
z
, Y)=
x
z
x
p
p
p
Y
2
)
(

[If
2
)
(
z
x
p
p
Y

<0
then x*=0, z*=
z
p
Y
]
b.
What are x*, and z* if Y=100,
2
=
x
p
1
=
z
p
x*=48, z*=4
c.
What is the level of utility given x* and z*?
U=52
d.
Solve for the Hicksian demand functions X
h
(p
x
, p
z
,
U
) and Z
h
(p
x
, p
z
,
U
).
)
(
2
pz
p
U
x
x
h

=
,
2
)
(
z
x
h
p
p
z
=
e.
NOW, Suppose P
x
falls to $1. At this new, lower price of X, what are the ordinary demands,
X* and Z*, and the consumer’s maximum achievable utility?
x*=99, z*=1, U=101
f.
What are the values of X
h
(p
x
, p
z
,
U
) and Z
h
(p
x
, p
z
,
U
) if U is equal to value you solved for
in part c, and
1
=
x
p
1
=
z
p
50
=
h
x
,
1
=
h
z