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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).
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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
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This note was uploaded on 11/30/2011 for the course ECON 311 taught by Professor Zambrano during the Fall '08 term at Cal Poly.
 Fall '08
 ZAMBRANO
 Microeconomics, Utility

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