Econ 3070
Prof. Barham
Problem Set –Chapter 5 Solutions
1. Aunt Joyce purchases two goods, perfume and lipstick. Her preferences are
represented by the utility function
U (P , L ) = PL ,
where P denotes the ounces of perfume used and L denotes the quantity of
lipsticks used. Let PP denote the price of perfume, PL denote the price of lipstick,
and I denote Aunt Joyce’s income.
a. What is Aunt Joyce’s maximization problem?
Max U (P, L) = P.L
L,P
s.t. PL L + Pp P = I
b. Derive her demand for perfume. Your answer should be an equation that
gives P as a function of PP , PL , and I. Determine this by using calculus and
maximizing the objective function, do not use the tangency condition.
To find the demand for perfume we need to find the optimal amount of perfume, it will
be a function of income and prices.
Step 1: Utility is a function of two variables. Since we don’t know how to maximize
when utility is a function of two variables we need to substitute for one of them.
Since we are trying to find the demand for perfume, we will substitute for lipstick, so
we are utility will be just a function of perfume.
Rewriting the budget constraint:
L=
I − PP P
PL
Now we can substitute this into the utility function
"I −P P%
" I % "P
%
P
Max U(P)=P*$$
'' = P $$ '' − $$ P * P 2 ''
P
# PL &
# PL & # PL
&
Step 2: Now we need to find the value of P that maximizes utility. We know that we
need the value of P where the slope of the utility curve is zero.
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Econ 3070
Prof. Barham
∂U
= gives us the slope of the utility function with respect to P
∂P
∂U
I P
= − P 2P = 0
∂P PL PL
2
PP
I
P=
PL
PL
P=
I PL
PL 2PP
P=
I
2PP
Demand function for perfume
c. Derive her demand for lipstick. Your answer should be an equation that gives
L as a function of PP , PL , and I.
To find the demand function for lipstick, we can repeat a similar exercise that we did in
part A. Or we can substitute our value of P back into the budget constraint. (Or
recognize that the answer has to be symmetric).
! I $
PL L + PP ##
&& = I
" 2PP %
I
PL L = I −
2
I
PL L =
2
I
L=
demand function for lipstick
2PL
d. What can be said about her cross-price elasticity of demand of perfume with
respect to the price of lipstick?
In part a, we found that Aunt Joyce’s demand for perfume is given by
P=
I
.
2PP
Since her demand for perfume does not depend on PL, Aunt Joyce’s cross-price
elasticity of demand of perfume with respect to the price of lipstick is zero. That is,
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Econ 3070
Prof. Barham
a 1% change in the price of lipstick generates a 0% change in the demand for
perfume.
2. Ch 5, Problem 5.7
Karl’s preferences over hamburgers (H) and beer (B) are described by the utility
function U(H,B)=min(2H,3B). His monthly income is I dollars, and he only buys
these two goods out of his income. Denote the price of hamburgers by PH and of
beer by PB.
a. Write out the consumer’s maximization problem. Remember this is a case of
perfect complements so the indifference curves would be L shaped.
Max U (P, L) = min(2H ,3B)
L,P
s.t. PH H + PB B = I
b. Draw a graph in H and B space of where the budget constraint and
indifference curves must be for the utility maximization. Don’t use real
numbers just draw what the basic shape of the curves look like.
Optimum is at the corner, so 2H=3B
IC
BL
c. Derive Karl’s demand curve for beer as a function of the exogenous variables
(hint, you can’t maximize this function normally but you know that to be at
an optimum 2H=3B).
We are maximizing utility at the corner point of the L shaped indifference curve, so at
that point
Equation 1 from Utility function: 2H=3B of H=3/2B (I rewrote it this way as we are trying
to find the demand curve for B.
Equation 2: The budget constraint also has to hold PHH + PBB=I
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Econ 3070
Prof. Barham
We have 2 equations and 2 unknowns. We can substitute equation 1 into equation 2.
PH(3/2B) + PBB=I
Now I can try to solve for B as a function of price and income which give us our demand
curve.
3
B( PH + PB ) = I
2
3
B( PH + PB ) = I
2
I
B=
3
PH + PB
2
d. You can answer this just by looking at the demand curve. Because it has a larger
coefficient, the price of hamburgers affects the demand for beer more than the price
of beer. A one dollar increase in PH decreases demand for beer more than a one
dollar increase in PB .
3. Uncle Bob purchases two goods, tweed sport coats and bow ties. His preferences
are represented by the utility function
U (B, C ) = B 0.25 C 0.75 ,
where B denotes the number of bow ties purchased and C denotes the number of
sport coats purchased. Let $25 be the price of bow ties and $60 be the price of
sport coats. And finally, let I denote Uncle Bob’s income.
a. Derive Uncle Bob’s Engel curve for bow ties. Your answer should be an
equation that gives B as a function of I.
Uncle Bob’s maximization problem is:
Max U ( B, C ) = B1/ 4C 3/ 4
B ,C
s.t. PB B + PC C = I
I - PB B
Step 1: Rewrite budget constraint. C =
sub into utility function
Pc
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Econ 3070
Prof. Barham
3
⎛ I − P B ⎞4
B
Max U (B) = B * ⎜⎜
⎟⎟
B
P
⎝
⎠
c
1
4
3
−
1
⎛ I − P B ⎞4 3 1 ⎛ I − P B ⎞ 4 P
∂U 1
B
B
= B ⎜⎜
⎟⎟ + B 4 ⎜⎜
⎟⎟ (− B ) = 0
∂B 4
P
4
P
PC
⎝
⎠
⎝
⎠
c
c
−
3
4
3
1 ⎛ I − PB B ⎞ 4
⎜
⎟
4 ⎝ Pc ⎠
B
3
4
1
3 PB 4
B
4 PC
=
1
⎛ I − P B ⎞4
B
⎜
⎟
⎝ Pc ⎠
3
1
1 3
1 ⎛ I − PB B ⎞ 4 ⎛ I − PB B ⎞ 4 3 PB 4 4
B B
⎜⎜
⎟⎟ ⎜⎜
⎟⎟ =
4 ⎝ Pc ⎠ ⎝ Pc ⎠ 4 PC
1 ⎛ I − PB B ⎞ 3 PB
B
⎜
⎟=
4 ⎜⎝ Pc ⎟⎠ 4 PC
P
I PB B
−
=3 B B
Pc
Pc
PC
4
PB
I
B=
Pc
Pc
4B=
B=
I Pc
Pc PB
I
I
=
4*25 100
Therefore, Uncle Bob’s Engel curve for bow ties is given by
I
.
B=
100
c. Are bow ties a normal good?
Bow ties are a normal good because the demand for bow ties increases as income
increases. Since bow ties are a normal good, Uncle Bob’s income elasticity of
demand for bow ties is positive.
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Econ 3070
Prof. Barham
4. Ch 5, Problem 5.9
Rick purchases two goods: food and clothing. He has a diminishing marginal
rate of substitution of food for clothing. Let x denote the amount of food
consumed and y the amount of clothing. Suppose the price of food increases
from Px1 to Px 2 . On a clearly labeled graph, illustrate the income and substitution
effects of the price change on the consumption of food. Do so for each of the
following cases:
a. Case 1: Food is a normal good.
Given the increase in the price of x, we expect to see the following effects:
Substitution Effect
Income Effect
¯
¯
x
Because the price of x increased, x became relatively more expensive, and y became
relatively less expensive. As a result, Rick substitutes away from x in favor of y. This
is represented in the table by a down arrow for x and an up arrow for y in the
substitution effect column.
Moreover, the increase in the price of x reduced Rick’s purchasing power. Since x
and y are both normal goods (x being a normal good is given by the problem, y being
a normal good is assumed), the reduction in purchasing power causes Rick to
purchase less of both x and y. This is represented in the table by the down arrows in
the income effect column.
The following diagram gives us a graphical representation of the information
presented in the table:
y
C
•
B
•
A
•
BL2
BL1
x
The initial consumption bundle is represented by point A, which lies on the initial
budget line BL1. The increase in the price of x causes the budget line to shift
inwards to BL2. The new consumption bundle is represented by point C. We then
construct point B in order to separate the substitution effect from the income effect.
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Econ 3070
Prof. Barham
The movement from A to B represents the substitution effect. Note that as
suggested by the table, the movement from A to B shows x going down.
The movement from B to C represents the income effect. Once again, note that as
suggested by the table, the movement from B to C shows x going down.
5. Ch 5, Problem 5.18
The demand function for Kendamas is given by D(P)=16-2P (note that D(P) is just a
way of saying writing the demand function where the quantity demanded is a
function of P which you are used to seeing as QD. Compute the change in consumer
surplus when the price of a widget increases from $1 to $3. First show your results
graphically.
First lets graph this demand curve it is linear, so the slope is -2
If P=0, Q=16
If Q=0 then P= ? just write the inverse demand curve P=(16-QD)/2 so P=8
If P=1 then QD or D(1) =14
If P=3 then QD or D(1) =10
For price of a widget equal to $1 consumer surplus is
CS$1 = ½ · (8 – 1) · D(1) = ½ · 7 · 14 = 49.
When price is equal to $3 consumer surplus is
CS$3 = ½ · (8 – 3) · D(3) = ½ · 5 · 10 = 25.
So the change in consumer surplus is 49-25=24 or Area EBDC
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Econ 3070
Prof. Barham
P
$8
A
Area of ABE triangle
CS when P = $3 is 25
D(P) = 16 – 2P
$3
$1
E
D
B
Area of ACD triangle
CS when P = $1 is 49
C
D(P)
7.
Ch 5, Problem 5.26 ed. 5.
Suppose that Bart and Homer are the only people in Springfield who drink
7-UP. Moreover, their inverse demand curve for 7-UP are:
Bart: P=10-4QB
Homer: P=25-2QB
Neither one can consume a negative amount.
Write down the market demand curve for 7-UP in Springfield, as a function of
all possible prices.
Bart will only consume when the price is less than 10. To see this, see what the price has
10 - P
to be if QB is zero. Therefore his demand curve for 7-UP is QB =
, when P<10
4
and zero otherwise.
Homer will only consume if the price is less than 25 so his demand curve is
25 - P
QH =
, when P < 25 and zero otherwise.
2
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Econ 3070
Prof. Barham
Therefore the market demand curve for 7-UP as a function of all possible values of price
is:
Q M = 0, if P > 25
25 − P
, if 10 < P < 25
2
60 − 3P
QM =
, if P < 10
4
QM =
9