Microeconomics
ECON 100A
Problem Set 3 Solutions
Due October 13, 2010 (Solutions Posted October 15, 2010)
1.
Toni’s utility is
3
2
3
)
,
(
z
x
z
x
U
=
a.
Does Toni consider X and Z goods?
(“Goods” are items we like more of, ie. more is better.
“Bads” are items that we like to have less of, ie. less is better. For example, pizza is a good.
Garbage is a bad.) Prove your answer.
b.
What is Toni’s marginal rate of substitution between goods x and z (her MRS)?
c.
Are Toni’s indifference curves convex?
Prove you answer.
a.
X and Z are goods (assuming x and z>0) because MU is positive for x and z.
3
6
)
,
(
xz
x
z
x
U
=
∂
∂
>0 and
2
2
9
)
,
(
z
x
z
z
x
U
=
∂
∂
>0
b. The MRS is
x
z
z
x
xz
z
z
x
U
x
z
x
U
MRS
3
2
9
6
)
,
(
)
,
(
2
2
3

=

=
∂
∂
∂
∂

=
,
Yes, the indifference curves are convex since the absolute value of the MRS falls as z
decreases and x increases.
2.
Consider the constrained optimization problem. [This question is especially important because it
enables you to solve for consumer demand functions from scratch.]
Y
z
p
x
xz
z
x
U
z
z
x
=
+
=
x
3
,
p
subject to
3
)
,
(
max
a.
List the choice variables.
The choice variables are x and z
b.
List the parameters.
The parameters are P
x
, P
z
, Y.
c.
Are they goods or bads? Prove this.
Yes, they are goods.
One way to prove this is to show that U
x
and U
z
are positive for all
positive x, z (i.e. utility is increasing in x and z).
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
9
)
,
(
and
0
3
)
,
(
2
3
>
=
∂
∂
>
=
∂
∂
xz
z
z
x
U
z
x
z
x
U
d.
Are they convex?
Yes, they are convex.
You can use any of the four methods shown in class.
One way is
to show that MRS is decreasing as x increases and z decreases along an indifference
curve.
Here, the MRS=
x
z
z
x
U
z
x
U
z
x
3
)
,
(
)
,
(

=

which falls as z decreases and x increases.
e.
Draw an indifference curve diagram and budget line and show the optimal bundle of
goods for the consumer.
See class notes.
Graph with X on horizontal axis and Z on vertical axis.
Budget line has
X intercept of I/P
x
and Z intercept of I/P
z
.
Then draw indifference curves (level sets of
utility function) with goal of reaching indifference curve tangent to budget constraint.
f.
Set up the Lagrangian.
The Langragian is
))
,
(
(
)
,
(
z
x
h
z
x
f
L
λ
+
=
where f(.) is the objective function (what we
want to maximize) and h(.) is our constraint (something our solution must satisfy).
In
our case:
)
(
3
3
z
P
x
P
Y
xz
L
z
x


+
=
g.
Write down the first order conditions.
F.O.C (firstorder conditions) are that the firstpartial derivatives of L with respect to
each choice variable and
λ
equal 0 for a maximum, i.e.:
0
0
9
0
3
2
3
=


=
∂
∂
=

=
∂
∂
=

=
∂
∂
z
P
x
P
Y
L
P
xz
z
L
P
z
x
L
z
x
z
x
2
3
9
3
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 ZAMBRANO
 Microeconomics, Utility, Px, pz, Z. Toni

Click to edit the document details