Microeconomics 100A
Problem Set 3 Solutions
Due February 3, 2010
Professor Michael Noel
University of California San Diego
1.
Draw the budget line for the following situations.
a.
You have an income of $40 per week to spend on two commodities.
Commodity X costs $10
per unit and commodity Z costs $5 per unit.
b.
You have an income of $50 per week to spend on two commodities.
Commodity X costs $10
per unit and commodity Z costs $5 per unit.
c.
You have an income of $50 per week to spend on two commodities.
Commodity X costs $20
per unit and commodity Z costs $5 per unit.
d.
Your Aunt Kristin gives you a $20 gift certificate for the purchase of commodity X.
You can
not use the gift certificate to purchase Z and you can not sell the gift certificate.
You also
have an income of $50 per week , commodity X costs $10 per unit, and commodity Z costs
$5 per unit.
e.
You have an income of $40 per week to spend on two commodities.
Commodity Z costs $5
per unit.
Commodity X costs $10 per unit.
If you buy more than two X, the price falls to $5
per unit.
Z
Budget line part A
10
8
Budget line part B
Budget Line Part C
Budget Line Part D
Budget Line Part E
2.5
4
5 7
6
X
a.
40=10X+5Z
b.
50=10X+5Z.
c.
50=20X+5Z.
d.
The most Z you can consume is $50/5=10 units.
You can still consume 10Z and 2X if you
wish because you can use the gift certificate to buy up to two X.
If you want more than two
X, then you will have to use your cash.
As before, you will have to give up 2Z to get 1X
since P1/P2=2.
Therefore, the X intercept is 7, (you can buy two X using the gift certificate
and 5 more X using your cash.)
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8
Slope =2
4
Slope =1
2
6
X
e. If she buys only Z, she can have 8Z.
To get the first X she must give up 2Z.
To get the second
X she must also give up 2Z (note that the slope of the budget line is 2=Px/Pz=$10/$5).
Now
that she has two X, the price falls to $5 and so, to get the third X, she only has to give up one
Z (note that the slope of the budget line is
1=Px/Pz=$5/$5).
2.
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 MRS falls as z decreases
and x increases.
3.
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 Winter '08
 staff
 Microeconomics, Utility, pz, Z. Toni

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