Consumer Behavior
4
1. Which assumption about consumer preferences does each of the following individuals violate?
a. Adam likes basketball more than football; football more than baseball; and baseball more than basketball.
b. Christina prefers prune juice to orange juice but cannot decide how she feels about grapefruit juice.
c. Blake likes superhero comic books but prefers 5 comic books to 10 comic books.
1. a. Suppose Adam’s preferences are transitive. If basketball is preferred to football and football is preferred to baseball, then by transitivity, basketball is preferred to baseball. But this contradicts the statement that Adam likes
baseball more than basketball. Adam’s preferences violate the assumption of transitivity.
b. Christina’s preferences violate the assumption of completeness since she is unable to assess her own utility or
well-being from a bundle of grapefruit juice.
c. Blake’s preferences violate the assumption that more is better than less.
2. By assumption, individual preferences must be transitive so that if A is preferred to B, and B is preferred to C, then
A is preferred to C. Suppose that Marsha, Jan, and Cindy individually have transitive preferences over three goods:
oranges, apples, and pears. If Marsha, Jan, and Cindy were to vote on whether to name oranges, apples, or pears the
“fruit of the month,” show that it is possible the preferences for the group might not be transitive.
2. Suppose the three consumers have the following preferences that satisfy transitivity:
Most preferred
Least preferred
Marsha
Jan
Cindy
Oranges
Apples
Pears
Apples
Pears
Oranges
Pears
Oranges
Apples
When voting on which fruit should be named “fruit of the month,” the following results will be obtained:
Oranges
vs. Apples
Oranges
2
Apples
1
Pears
Apples
vs. Pears
Oranges
vs. Pears
1
2
1
2
For the group preferences, we have oranges preferred to apples (2 votes to 1), apples preferred to pears, and pears preferred to oranges so the group preferences do not satisfy transitivity even though the preferences of each individual
do.
3. Draw two indifference curves for each of the following pairs of goods. Put the quantity of the first good on the horizontal axis and the quantity of the second good on the vertical axis.
a. Paul likes pencils and pens, but does not care which he writes with.
b. Rhonda likes carrots and dislikes broccoli.
c. Emily likes hip-hop iTunes downloads and doesn’t care about heavy metal downloads.
d. Michael only likes dress shirts and cufflinks in 1 to 2 proportions.
e. Carlene likes pizza and shoes.
27
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3. In the figures that follow, bundles along the indifference curve labeled U 2 are strictly preferred to bundles along the
indifference curve U 1.
a. Paul is equally as happy with a pen as with a pencil. Therefore, these two goods are perfect substitutes.
Pens
U1
U2
Pencils
b. Carrots are a good and broccoli is a bad for Rhonda.
Broccoli
U1
U2
Carrots
c. Hip-hop is a good, heavy metal is a neutral.
U2
U1
Heavy
metal
music
Hip-hop music
d. Dress shirts and cufflinks are perfect complements.
Pairs of
cufflinks
2
U2
1
U1
0
Goolsbee2e_Solutions_Manual_Ch04.indd 28
1
2
Dress shirts
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Consumer Behavior
Chapter 4
29
e. Both pizza and shoes are good for Carlene.
Shoes
U2
U1
Pizza
*4. Suppose that John is indifferent between consuming bundle A, which consists of 4 apples and 1 peach, and bundle
B, which consists of 4 peaches and 1 apple. If John were given the choice between bundle A and bundle C, which
contained 3 peaches and 2 apples, which should he pick? (Hint: Draw an indifference curve or two.)
4. A and B are two bundles on the same indifference curve U1, so any bundle that lies along
the straight line joining A and B will lie on a higher
indifference curve. Bundle C, consisting of
3 peaches and 2 apples, is such a bundle so John will
choose bundle C over bundle A.
Peaches
6
5
B
4
C
3
2
A
U2
1
U1
0
5. The table at right displays the total utility U(X ) that corresponds to the number of units of X consumed by three different
consumers (Auon, Barbara, and Camira), holding everything
else constant:
a. Compute the marginal utility of X for each of the three
consumers at each level of X.
b. Based on the data in the table, can you tell whether any
of these consumers are violating any of the standard
assumptions about preferences?
c. Is it possible that any of these three consumers have the
exact same preferences, and that columns for the three
consumers differ only because of the arbitrary units that
are used to measure utility? Explain.
Goolsbee2e_Solutions_Manual_Ch04.indd 29
1
2
3
Auon
4
5
6
Barbara
Apples
Camira
U (X )
X
U (X )
X
U (X )
X
10
2
10
2
10
2
14
3
10
3
12
3
16
4
10
4
15
4
17
5
9
5
19
5
17.5
6
8
6
24
6
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5. a.
MUxAuon
MUxBarbara
MUxCamira
2
—
—
—
3
4
0
2
4
2
0
3
5
1
–1
4
6
0.5
–1
5
X
b. Barbara’s marginal utility is negative for the 5th and 6th units consumed. This is a violation of the assumption that
more is better (or, at a minimum, more is not any worse than less).
c. No two consumers have the same preferences in this example. Auon experiences diminishing, but positive, marginal utility. Barbara’s marginal utility actually becomes negative. Camira’s marginal utility increases as more is
consumed.
*6. A consumer’s utility function is given by U = XY, where MUX = Y and MUY = X.
a. What is the utility derived from 1 unit of X and 2 units of Y ? What is the utility derived from 2 units of X and 1 unit of
Y ? What is the utility derived from 5 units of X and 2 units of Y ?
b. How does the consumer rank the following bundles?
Bundle
Quantity of X
Quantity of Y
A
2
2
B
10
0
C
1
5
D
3
2
E
2
3
c. Graph indifference curves that show the bundles of X and Y for which U = 6 and U = 8. Is the “more is better”
assumption satisfied for X and Y ?
6. a.
U(1, 2) = (1)(2) = 2
U(2, 1) = (2)(1) = 2
U(5, 2) = (5)(2) = 10
b.
Bundle
Quantity of X
Quantity of Y
Utility
A
2
2
4
B
10
0
0
C
1
5
5
D
3
2
6
E
2
3
6
From the table, U(D) = U(E) > U(C) > U(A) > U(B), and we have the ranking (> denotes strictly preferred,
~ denotes indifferent)
D~E>C>A>B
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Consumer Behavior
Chapter 4
31
c. Good Y
4
3
E
2
A
0
D
2
U=8
U=6
Good X
3
The “more is better” assumption is satisfied.
7. Kelly’s utility function is given by U = 5X + 2Y, where MUX = 5 and MUY = 2.
a. What is M RSXY ?
b. What is M RSXY when X = 1 and Y = 5? When X = 2 and Y = 2.5?
c. Draw a few sample indifference curves.
MUX
M RSXY = _
MUY
5 = 2.5
=_
2
b. M RSXY is constant and equal to 2.5 no matter what consumption bundle is being considered (X = 1, Y = 5 or
X = 2, Y = 2.5).
c. Good Y
7. a.
Slope = -2.5
U1
U2
Good X
8. Andrea loves to eat burritos with hot sauce. In fact, she cannot enjoy a burrito (B ) unless it has three servings of hot
sauce (H ). She gets no additional enjoyment from more than three servings per burrito. Thus, her utility function is
U = min{B, __13 H }. Graph Andrea’s indifference curves for U = 1 and U = 2.
H = 3 for the vertices of the indifference curves.
8. Perfect complements with _
B
Hot
sauce
U2
6
3
0
U1
1
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2
Burritos
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Consumption and Production
9. Suppose John’s utility function is 4XY, where X is consumption of beer and Y is consumption of pizza. For this utility
function, the marginal utility of X is given by MU x = 4Y; the marginal utility of Y is given by MU y = 4X.
a. Suppose Y = 3. Calculate John’s utility for X = 2, 3, 10, and 11. For a given level of Y, does good X display
diminishing marginal utility?
b. Suppose X = 3. Calculate John’s utility for Y = 2, 3, 10, and 11. For a given level of X, does good Y display
diminishing marginal utility?
c. Find three different bundles containing X and Y that give John 48 utils of satisfaction. Plot the three bundles and
connect them with an indifference curve. What happens to the marginal rate of substitution between X and Y as
consumption of X increases?
d. Does the principle of diminishing MRS depend on the diminishing marginal utility of X and Y?
9. a. When X = 2 and Y= 3, total utility = 4 × 2 × 3 = 24.
When X = 3 and Y = 3, total utility = 4 × 3 × 3 = 36.
When X = 10 and Y = 3, total utility = 4 × 10 × 3 = 120.
When X = 11 and Y = 3, total utility = 4 × 11 × 3 = 132.
No, X does not display diminishing marginal utility for a given level of Y. When Y = 3, every additional unit of
X increases utility by 12.
b. When X = 3 and Y = 2, total utility = 4 × 3 × 2 = 24.
When X = 3 and Y = 3, total utility = 4 × 3 × 3 = 36.
When X = 3 and Y =10, total utility = 4 × 3 × 10 = 120.
When X = 3 and Y =11, total utility = 4 × 3 × 11 = 132.
No, Y does not display diminishing marginal utility for a given level of X. When X = 3, every additional unit of
Y increases utility by 12.
c. There are a number of bundles that yield 48 units of utility. Some examples are: X = 2 and Y = 6; X = 3 and
Y = 4; X = 4 and Y = 3; X = 12 and Y = 1. The marginal rate of substitution diminishes as more of X is consumed,
because each additional unit of X yields less utility than the previous one.
Good Y
6
4
3
1
0
2 3 4
12
Good X
d. Yes, the reason why the MRS diminishes is because marginal utility is also diminishing.
10. José gets satisfaction from both music and fireworks. José’s income is $240 per week. Music costs $12 per CD, and
fireworks cost $8 per bag.
a. Graph the budget constraint José faces, with music on the vertical axis and fireworks on the horizontal axis.
b. If José spends all his income on music, how much music can he afford? Plot a point that illustrates this scenario.
c. If José spends all his income on fireworks, how many bags of fireworks can he afford? Plot a point that illustrates
this scenario.
d. If José spends half his income on fireworks and half his income on music, how much of each can he afford? Plot
a point that illustrates this scenario.
e. Connect the dots to create José’s budget constraint. What is the slope of the budget constraint?
f. Divide the price of fireworks by the price of music. Have you seen this number before and, if so, where?
g. Suppose that a holiday bonus raises José’s income temporarily to $360. Draw José’s new budget constraint.
h. Indicate the new bundles of music and fireworks that are feasible, given José’s new income.
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Consumer Behavior
Chapter 4
33
10. José has income of I = $240, the price of music is PM = $12, and the price of fireworks is PF = $8.
a. Music (M)
(CDs)
20
0
b.
Fireworks (F )
(Bags)
30
I = 20
_
PM
This is shown as the point labeled b in the figure below.
I = 30
c. _
PF
This is shown as the point labeled c in the figure below.
d.
_1 I
2 = 15
F= _
PF
_1 I
2 = 10
M= _
PM
This is shown as the point labeled d in the figure below.
e.
Music (M )
(CDs)
20
10
b
d
c
0
15
30
Fireworks (F )
(Bags)
The slope of the budget constraint is
20 = – _
ΔM = – _
2
_
ΔF
30
3
f.
PF _
2
_
= 8 =_
PM
12
3
This is –1 times the slope of the budget constraint.
g.
Music (M )
(CDs)
30
20
0
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BC 1
BC 2
30
45
Fireworks (F )
(Bags)
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Consumption and Production
11. Suppose that only one person in the world sells ice cream. She employs a strange pricing policy: You can buy 1 ice
cream cone for $1, but if you buy 2 cones, you have to pay $2 each. If you buy 3, you have to pay $3 each, and so on,
so that if you buy 10, you have to pay $10 each. You have $100 dollars to spend on ice cream cones and chocolate
milk, and chocolate milk costs $1 per unit. Draw your budget constraint. This strange ice cream pricing, where buying more costs you more, is called a quantity surcharge.
11. The equation for the budget constraint PI I + PC C = M
and given PC = $1.00, M = $100, and PI = $I gives I 2 +
C = 100.
Chocolate
milk (C )
100
0
10
Ice cream (I )
12. John enjoys ordering out for pizza and renting movies online. He makes $30 each week at a part-time job. If movies
cost $2 per rental, and pizza costs $7.50 per slice, graph John’s budget constraint. Then, illustrate the effects of each
of the following events:
a. John’s mother finds a coupon good for one free pizza, and gives it to John.
b. The company that John rents movies from sponsors a holiday week promotion: Rent the first five movies at regular price; and all movies after the fifth are half-off.
c. John’s favorite pizza place increases the price of pizza from $7.50 to $10.
12. a.
Movies
15
BC 1
BC 2
0
b.
1
4
5
Pizzas
Movies
25
BC 2
15
BC 1
5
0
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4
Pizzas
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Consumer Behavior
c.
Chapter 4
35
Movies
15
BC 1
BC 2
5
0
3
4
Pizzas
13. Good X sells for $4, and good Y sells for $2. At your current level of consumption, the marginal rate of substitution
between X and Y is 4.
a. Draw an indifference curve and budget constraint that reflects the facts given above. (Hint: You will have to
choose an initial income level and an initial bundle of X and Y.)
b. Are you maximizing your utility?
c. If you are not maximizing your utility, are you buying too much X or too much Y? Explain.
13. a.
Good Y
Current
consumption
bundle
Good X
b. You are not maximizing utility, because the indifference curve is steeper than the budget constraint at the point
you have selected.
c. You are buying too much Y. The number of units of Y that you would be willing to give up to get one more unit of
X is more than what you would actually have to give up.
14. André gets utility from playing laser tag and reading books. Each week, André spends his entire $100 paycheck on
both goods. One hour of laser tag costs $20; a book costs $10.
a. Graph André’s budget constraint. Put books on the horizontal axis and hours of laser tag on the vertical axis.
b. When André maximizes his utility, he buys 6 books. Carefully add an indifference curve to your graph to reflect
André’s utility-maximizing bundle of books and laser tag.
c. When André maximizes his utility, how many hours of laser tag does he play?
d. When André maximizes utility, the marginal utility he receives from playing an extra hour of laser tag is 12 utils.
How much utility must André get from reading another book?
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Consumption and Production
14. a.
Hours of
laser tag
5
b.
10
Books
10
Books
Hours of
laser tag
5
2
6
c. With the remaining $40 left after buying 6 books at $10 each, André can play 2 hours of laser tag.
d. The ratio of P B/P L is 10/20 = 1/2, so the ratio of the marginal utility of the last book purchased must be one-half
of the marginal utility of the last hour of laser tag. MU B = 1/2 × 12 = 6.
*15. For Maahir, shampoo and conditioner are perfect complements. He likes to use 1 squirt of shampoo and 1 squirt of
conditioner each time he washes his hair.
a. Draw a set of indifference curves for Maahir that illustrate the utility he derives from using shampoo and conditioner.
b. Assume that shampoo costs $4 and conditioner costs $2. Construct a budget constraint for Maahir and describe his
purchasing habits. What is his optimal bundle likely to look like? (Hint: Assume some level of income for Maahir.)
c. Suppose that prices change so that shampoo costs $2 and conditioner costs $4. What is likely to happen to
Maahir’s optimal bundle as a result? Explain.
d. How would your answer to (c) change if Maahir used 2 squirts of shampoo and 1 squirt of conditioner each time
he washed his hair?
15. a.
Conditioner
2
U2
1
U1
0
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2
Shampoo
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Consumer Behavior
b. Maahir’s budget constraint is
Chapter 4
37
–PS /PC = –2
Conditioner
4S + 2C = 12
Since Maahir likes to consume the two goods,
shampoo and conditioner, in a 1 to 1 ratio, he can
purchase together 1 unit of each for a total of $2
+ $4 = $6. With I = $12 of income, Maahir can
afford 2 units of each.
c. When there is a change in prices so that PS + $2 and
PC = $4, the new budget constraint for Maahir will
be 2S + 4C = 12. The slope of the budget constraint is
PS
1 , but the total cost of a unit of each
now – _
= –_
PC
2
good (Maahir’s ideal consumption ratio) is still
$6, so he can afford to purchase the same bundle
(2 units of each good) as before.
2
U2
1
U1
0
1
2
3
Shampoo
Conditioner
2
U2
1
0
–PS /PC = – 12
U1
1
2
3
4
5
6
Shampoo
d. If Maahir likes to use shampoo and conditioner in the ratio 2 to 1, then under the original prices, a bundle consisting of S = 2 and C = 1 would have cost 2(4) + 1(2) = $10. With the change in prices, the same bundle would
cost 2(2) + 1(4) = $8. Maahir would now be able to consume the same bundle but would have money left over to
consume more of other goods under the new prices.
16. Suppose that there are only two goods, books and coffee. Justine gets utility from both books and coffee, but her
indifference curves between them are concave rather than convex to the origin.
a. Draw a set of indifference curves for Justine.
b. What do these particular indifference curves tell you about Justine’s marginal rate of substitution between books
and coffee?
c. What will Justine’s utility-maximizing bundle look like? (Hint: Assume some level of income for Justine, and
some prices for books and coffee; then draw a budget constraint.)
d. Compare your answer to (b) to real-world behaviors. Does the comparison shed any light on why economists
generally assume convex preferences?
16. a.
Coffee
U1
U2
Books
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Consumption and Production
b. Justine’s marginal rate of substitution of coffee for books is increasing as the consumption level of books increases.
c. Justine will maximize utility by consuming only a single good.
Coffee
Optimal consumption
Budget constraint
U1
U2
Books
d. Concave indifference curves imply that a consumer’s marginal rate of substitution is increasing as consumption
of a good increases. This means that the marginal utility of a good increases (relatively) as consumption of that
good increases. If indifference curves are concave, consumers maximize utility by consuming only one good. The
fact that we observe consumers choosing a variety of goods would suggest that convex preferences are a realistic
assumption.
*17. Chrissy spends her income on fishing lures (L ) and guitar picks (G ). Lures are priced at $2, while a package of guitar
picks cost $1. Assume that Chrissy has $30 to spend and her utility function can be represented as U(L,G ) = L 0.5G 0.5.
For this utility function, MUL = 0.5L –0.5G 0.5 and MUG = 0.5L 0.5G –0.5.
a. What is the optimal number of lures and guitar picks for Chrissy to purchase? How much utility does this combination bring her?
b. If the price of guitar picks doubles to $2, how much income must Chrissy have to maintain the same level
of utility?
17. a. Chrissy’s marginal rate of substitution can be determined from her marginal utility for each good. The marginal
rate of substitution will be equal to the ratio of relative prices at the optimal consumption bundle:
–0.5 0.5
MUL _
M RSLG = _
= 0.5L G–0.5
MUG 0.5L 0.5G
G =_
2
=_
L
1
G* = 2L*
Chrissy’s optimal bundle must also lie on her budget constraint, 2L + G = 30. Substituting the relation from the
tangency condition gives
2L* + 2L* = 30
4L* = 30
L* = 7.5
and
G* = 15
The optimal consumption bundle is (L*, G*) = (7.5, 15), and this will give Chrissy utility U =
(7.5)0.5(15)0.5 = 10.6.
b. If there is a doubling in the price of guitar picks to PG = $2, then from the tangency condition:
PL
M RSLG = _
PG
G =_
2
_
L
2
G * = L*
Chrissy will want to consume the two goods in equal quantities. In order to maintain utility at U = 10.6,
U(L,G) = L 0.5G 0.5
10.6 = L 0.5G 0.5
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Consumer Behavior
Chapter 4
39
with G* = L* = k
10.6 = k 0.5k 0.5
10.6 = k
Chrissy will consume the bundle (L*, G*) = (10.6, 10.6), which at the new price for guitar picks will cost
(2)(10.6) + (2)(10.6) = $42.40
Chrissy will require income I ' = $42.40 in order to maintain the same level of utility.
18. A popular cellular telephone provider offers its customers the following pay-as-you-go data plan: The first 4 gigabytes of data cost $10 each; each gigabyte of data after that costs $20.
a. Suppose the typical data customer has $100 to spend on either phone use or chocolate bars, which cost $1 each.
Graph the typical customer’s budget constraint.
b. Add an ordinary-looking (downward-sloping, convex to the origin) set of indifference curves to the graph, to
represent the preferences of the typical customer. How many gigs of data is the typical customer likely to use?
c. Does your answer to (b) depend critically on the steepness or flatness of the typical customer’s indifference
curves? Explain.
18. a.
Chocolate
bars
100
60
0
4
7
Data (in gigabytes)
b. The typical consumer will use 4 gigs of data.
Chocolate
bars
100
60
0
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4
7
Data (in gigabytes)
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Consumption and Production
c. There are many different shapes of indifference curves that will have a tangency at the point of using 4 gigs
of data.
Chocolate
bars
100
60
0
4
7
Data (in gigabytes)
19. Suppose that doctors’ visits cost $20, and the typical consumer has an income of $100. Consumers spend all of their
incomes on doctors’ visits and a “composite good” that costs $1 per unit.
a. Draw a graph that illustrates the consumer’s budget constraint, putting doctor’s visits on the horizontal axis. Make
sure you indicate the horizontal and vertical intercepts.
Now, suppose the local government is considering two health plans. Under plan A, the government will give
out vouchers worth 2 free visits to the doctor. Under plan B, the government will give out four 50% coupons to
be used at the doctor’s office.
b. Draw the new budget constraint the consumer faces under plan A.
c. Draw the new budget constraint the consumer faces under plan B.
d. For whom is the choice of plan A or plan B not likely to matter — those who are quite well, or those who are quite
sick? (Hint: Superimpose some indifference curves on your budget constraints.)
e. Which plan would someone who is generally well be likely to choose, if offered a choice?
19. a.
Composite
good
100
0
5
Doctors’ visits
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Consumer Behavior
b.
Chapter 4
41
Composite
good
100
0
c.
2
5
7
Doctors’ visits
5
7
Doctors’ visits
Composite
good
100
60
0
4
d. Consumers who consume a high quantity of doctor’s visits will be equally as well off under either plan (utility U2 )
and are better off than without a plan (utility U1).
Composite
good
100
60
U2
U1
0
Goolsbee2e_Solutions_Manual_Ch04.indd 41
2
4
5
7
Doctors’ visits
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e. Consumers who consume low quantities of doctor’s visits will achieve the highest utility under plan A (utility U3),
second highest under plan B (utility U2), and lowest utility with no plan (utility U1).
Composite
good
100
U3
U2
U1
60
0
2
4
5
7
Doctors’ visits
20. Cherise considers movies and concerts to be perfect substitutes. When Cherise maximizes her utility, what will optimal bundle of movies and concerts probably look like? Explain your reasoning by drawing a graph that includes her
indifference curves and a hypothetical budget constraint.
20. Cherise’s indifference curves will be straight lines. Her optimal consumption bundle will be determined by the relative prices of movies and concerts, and will most likely involve consuming only one good.
Concerts
Optimal
bundle
Budget
constraint
Indifference
curves
Movies
21. Economist Joel Waldfogel may be America’s biggest Grinch. He bemoans what he calls the “deadweight loss
of Christmas” created when people give gifts (such as ugly sweaters) the recipients would rarely, if ever, buy for
themselves.
a. Draw a graph with a budget constraint showing affordable bundles of a composite good costing $1 and ugly
sweaters. (You may assume some level of income and a price for ugly sweaters.)
b. Cheryl might get some utility from an ugly sweater, but is currently spending all of her income on the composite
good. Add an indifference curve that reflects Cheryl’s situation. What must her indifference curves look like?
c. Suppose Cheryl receives a gift of one ugly sweater from a co-worker. Show the effect of the gift on Cheryl’s
budget constraint. Where on the constraint is Cheryl likely to maximize her utility?
d. Waldfogel suggests that the world might be a happier place if instead of giving ugly sweaters, people simply gave
an equivalent amount of cash. Draw the budget constraint Cheryl would face if this were the case. Add an indifference curve or two to show what happens when Cheryl maximizes her utility. Does the cash gift make her happier
than the sweater?
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Consumer Behavior
Chapter 4
43
21. a. The budget constraint should be a straight line with intercept points determined by the prices and income level
you have selected.
b. Her indifference curves would be flat.
Composite good
Optimal
bundle
Indifference curves
Budget constraint
Ugly sweaters
c. Cheryl will likely consume only one ugly sweater.
Composite
good
Budget constraint
Ugly sweaters
d. Yes, Cheryl can attain a higher level of utility with a cash gift rather than a sweater. When a cash gift is received,
the budget constraint pivots upward at the kink to become a straight line segment intersecting the vertical axis at
a higher point. Cheryl can attain a higher indifference curve.
Composite
good
Budget constraint
1
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Ugly sweaters
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Goolsbee2e_Solutions_Manual_Ch04.indd 44
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