Answers to the Problems and Applications
1.
Max enjoys windsurfing and
snorkeling. The table shows
the total utility he gets from
each activity.
a. Find Max’s marginal utility
from windsurfing at each
number of hours per day.
Hours
per
day
1
2
3
4
5
6
7
Total
Total
utility
utility
from
from
windsurfing snorkeling
120
40
220
76
300
106
360
128
396
140
412
150
422
158
Max’s marginal utility from
windsurfing 1 hour per day is
120; from windsurfing 2 hours
per day is 100; from
windsurfing 3 hours per day is
80; from windsurfing 4 hours per day is 60; from windsurfing 5
hours per day is 36; from windsurfing 6 hours per day is 16;
and, from windsurfing 7 hours per day is 10.
b. Find Max’s marginal utility from snorkeling at each number
of hours per day.
Max’s marginal utility from snorkeling 1 hour per day is 40;
from 2 hours per day is 36; from snorkeling 3 hours per day is
30; from snorkeling 4 hours per day is 22; from snorkeling 5
hours per day is 12; from snorkeling 6 hours per day is 10; and,
from snorkeling 7 hours per day is 8.
c. Do Max’s marginal utility from windsurfing and from
snorkeling obey the principle of diminishing marginal
utility?
Max’s marginal utility from windsurfing and from snorkeling both
obey the principle of diminishing marginal utility because both
decrease as the consumption increases.
d. Which does Max enjoy more: his 6th hour of windsurfing or
his 6th hour of snorkeling?
Max’s marginal utility from his 6th hour of windsurfing is 16
and his marginal utility from his 6th hour of snorkeling is 10.
Max enjoys his 6th hour of windsurfing more than his 6th hour of
snorkeling.
2.
Max in problem 1 has $35 a day to spend, and he can spend as
much time as he likes on his leisure pursuits. Windsurfing
equipment rents for $10 an hour, and snorkeling equipment
rents for $5 an hour.
a. Make a table that shows the various combinations of hours
spent
Marginal
Marginal
windsurfin
utility per
utility
g and
Hours
dollar from
Hours
per dollar
snorkeling
windsurfing windsurfing snorkeling
from
that Max
snorkeling
can
3
8.0
1
8.0
afford.
2
10.0
3
6.0
The table
1
12.0
5
2.4
is to the
0
7
1.6
right.
b. In your table, add two columns and list Max’s marginal
utility per dollar from windsurfing and from snorkeling.
The columns are in the table.
c. How long does Max spend windsurfing and how long does he
spend snorkeling to maximize his total utility?
To maximize his utility, Max windsurfs for 3 hours and snorkels
for 1 hour.
Max uses his $35 so that all of the $35 is spent and so that the
marginal utility per dollar from each activity is the same. When
Max windsurfs for 3 hours and snorkels for 1 hour, he spends $30
renting the windsurfing equipment and $5 renting the snorkeling
equipment—a total of $35.
The marginal utility from the third hour of windsurfing is 80
and the rent of the windsurfing equipment is $10 an hour, so the
marginal utility per dollar from windsurfing is 8. The marginal
utility from the first hour of snorkeling is 40 and the rent of
the snorkeling equipment is $5 an hour, so the marginal utility
per dollar from snorkeling is 8. The marginal utility per dollar
from windsurfing equals the marginal utility per dollar from
snorkeling.
d. If compared to c, Max spent a dollar more on windsurfing
and a dollar less on snorkeling, by how much would his
total utility change?
If Max windsurfs another hour, he pays $10 and gains 60 units of
utility (the marginal utility from the 4th hour), which is 6.0
units of utility per dollar. So if he spends a dollar more on
windsurfing, his utility from windsurfing increases by 6.0. If
he spends an hour less on snorkeling, he saves $5 and loses 40
units of utility (the marginal utility from the 1st hour of
snorkeling), which is 8.0 units of utility per dollar. So if he
spends a dollar less on snorkeling, he loses 8.0 units of
utility. Overall, spending a dollar more on windsurfing and a
dollar less on snorkeling lowers Max’s total utility by 2.0
units of utility.
e. If compared to c, Max spent a dollar less on windsurfing
and a dollar more on snorkeling, by how much would his
total utility change?
If Max snorkels another hour, he pays $5 and gains 36 units of
utility (the marginal utility from the 2nd hour), which is 7.2
units of utility per dollar. So if he spends a dollar more on
snorkeling, his utility from snorkeling increases by 7.2. If he
spends an hour less on windsurfing, he saves $10 and loses 80
units of utility (the marginal utility from the 3rd hour of
windsurfing), which is 8.0 units of utility per dollar. So if he
spends a dollar less on windsurfing, he loses 8.0 units of
utility. Overall, spending a dollar more on snorkeling and a
dollar less on windsurfing lowers Max’s total utility by 0.8
units of utility.
f. Explain why, if Max equalized the marginal utility per hour
from windsurfing and from snorkeling, he would not maximize
his utility.
Snorkeling costs half as much per hour as does windsurfing. If
Max equalized the marginal utility from windsurfing and
snorkeling the marginal utility per dollar from snorkeling would
be twice that from windsurfing. Max can therefore increase his
utility by spending less on windsurfing and more on snorkeling.
For example, by decreasing his expenditure on windsurfing by a
dollar Max loses utility but by spending that dollar on
snorkeling Max gains almost twice the amount of utility he lost.
3.
Max in problems 1 and 2 is offered a special deal on
windsurfing equipment: a rental rate of $5 an hour. His
income remains at $35 a day and the rental price of
snorkeling equipment remains at $5 an hour.
a. Make a
Marginal
Marginal
table that
utility per
utility
shows the
Hours
dollar from
Hours
per dollar
new
windsurfing windsurfing snorkeling
from
combinatio
snorkeling
ns of
7
2.0
0
hours
6
3.2
1
8.0
spent
5
7.2
2
7.2
windsurfin
4
12.0
3
6.0
g and
3
16.0
4
4.4
snorkeling
2
20.0
5
2.4
that Max
1
24.0
6
2.0
can
0
7
1.6
afford.
The table is to the right.
b. In your table, list Max’s marginal utility per dollar from
windsurfing and snorkeling.
The columns are added in the table to the right.
c. How many hours does Max now spend windsurfing and how many
hours does he spend snorkeling?
Max will now maximize his total utility by spending 5 hours
windsurfing and 2 hours snorkeling. This combination of
windsurfing and snorkeling uses all of Max’s income and sets the
marginal utility per dollar from windsurfing equal to the
marginal utility per dollar from snorkeling.
4.
Given the information about Max in problems 1, 2, and 3,
a. Find two points on Max’s demand curve for rented
windsurfing equipment.
From problem 2 part c, when the price of renting windsurfing
equipment is $10 per hour, Max rents windsurfing equipment for 3
hours. From problem 3 part c, when the price of renting
windsurfing equipment is $5 per hour, Max rents windsurfing
equipment for 5 hours.
b. Draw Max’s demand curve for rented windsurfing equipment.
The demand curve is in Figure
8.1.
c. Is Max’s demand for renting
windsurfing equipment elastic
or inelastic?
Max’s elasticity of demand for
renting windsurfing equipment is
inelastic because a fall in the
price decreases Max’s total
expenditure on renting
windsurfing equipment.
5.
Max, with the utility schedules
in problem 1, gets an increase
in income from $35 to $55 a day.
Windsurfing equipment rents for
$10 an hour, and snorkeling
equipment rents for $5 an hour.
Show the effect of the increase
in Max’s income on Max’s demand curve for
a. Rented windsurfing equipment, and explain whether, for Max,
windsurfing equipment is a normal good or an inferior good.
To maximize his utility, Max windsurfs for 4 hours and snorkels
for 3 hours. Max uses his $55
such that all of the $55 is spent
and marginal utility per dollar
for each activity is the same.
When Max windsurfs for 4 hours
and snorkels for 3 hours, he
spends $40 renting the
windsurfing equipment and $15
renting the snorkeling equipment—
a total of $55. The marginal
utility from the fourth hour of
windsurfing is 60 and the rent of
the windsurfing equipment is $10
an hour, so the marginal utility
per dollar from windsurfing is 6.
The marginal utility from the
third hour of snorkeling is 30
and the rent of the snorkeling
equipment is $5 an hour, so the
marginal utility per dollar from
snorkeling is 6. The marginal utility per dollar from
windsurfing equals the marginal utility per dollar from
snorkeling.
Max’s demand for rented windsurfing equipment increases. The
quantity of windsurfing equipment rented at a price of $10 per
hour increases from 3 hours (problem 2c) to 4 hours (this
problem). As a result Max’s demand curve for rented windsurfing
equipment shifts rightward as illustrated in Figure 8.2 (on the
previous page) by the shift from D1 to D2. Windsurfing equipment
is a normal good.
b. Rented snorkeling equipment, and explain whether, for Max,
snorkeling equipment is a normal good or an inferior good.
Max’s demand for rented
snorkeling equipment increases.
The quantity of snorkeling
equipment demanded at a price of
$5 per hour increases from 1
hour (problem 2c ) to 3 hours
(this problem). As a result
Max’s demand curve for rented
snorkeling equipment shifts
rightward as illustrated in
Figure 8.3 by the shift from D1
to D2. Snorkeling equipment is a
normal good.
6.
Schools Get a Lesson in Lunch
Line Economics
Sharp rises in the cost of milk,
grain, and fresh fruits and
vegetables are hitting
cafeterias across the country, forcing cash-strapped schools
to raise prices or pinch pennies by serving more economical
dishes. … Fairfax schools, for instance, serve oranges—14
cents each—instead of grapes, which are a quarter a serving.
The Washington Post, April 14, 2008
Assume that a Fairfax school has a $14 daily fruit budget.
a. How many oranges a day can the school afford to serve if it
serves no grapes?
If the school serves no grapes, it can afford to serve 100
oranges.
b. How many servings of grapes can the school afford each day
if it serves no oranges?
If the school serves no oranges, it can afford to serve to serve
56 servings of grapes.
c. If the school provides 50 oranges a day and maximizes
utility, how many servings of grapes does it provide?
If the school maximizes utility, it will spend all of its
budget. If the school is providing 50 oranges, it is spending $7
on oranges, leaving it $7 to spend on grapes. With $7 to spend
on grapes, the school buys 28 servings of grapes.
d. If the marginal utility from an orange is 14 units of
utility, what is the marginal utility from a serving of
grapes?
The (MUO/PO) = (MUG/PG). Rearranging gives (MUO/PO) × PG = MUG, so
(14 units of utility/14 cents) × 25 cents = MUG, so the marginal
utility of a serving of grapes is 25 units of utility.
7.
Can Money Buy Happiness?
“Whoever said money can’t buy happiness isn’t spending it
right.”... You know that there must be some connection
between money and happiness. If there weren’t, you’d be less
likely to stay late at work (or even come in at all). … “Once
you get basic human needs met, a lot more money doesn’t make
a lot more happiness.” … Going from earning less than $20,000
a year to making more than $50,000 makes you twice as likely
to be happy, yet the payoff for then surpassing $90,000 is
slight.
CNN, July 18, 2006
a. What does the fundamental assumption of marginal utility
theory suggest about the connection between money and
happiness?
Marginal utility theory assumes that total utility increases
with increases in the consumption of a good or service, so a
person’s total happiness (that is, their total utility) is
assumed to increase with income (so that their consumption
increases).
b. Explain why this article is consistent with marginal
utility theory.
Marginal utility theory assumes that total utility increases
with increases in the consumption of a good or service but
marginal utility decreases with increases in the consumption of
a good or service. Hence a person’s total happiness increases
when income increases but the marginal happiness (that is, the
change in total happiness) decreases when income increases.
These results are precisely what the story reports: Total
happiness increases as income increases from $20,000 to $50,000
to $90,000 but the change in happiness diminishes.
8.
Eating Away the Innings in Baseball’s Cheap Seats
Baseball and gluttony, two of America’s favorite pastimes,
are merging in a controversial trend taking hold at Major
League Baseball stadiums across the nation: all-you-can-eat
seats. … Some fans try to “set personal records” during their
first game in the section. By their second or third time in
such seats … they eat like they normally would at a game.
USA Today, March 6, 2008
a. What conflict might exist between utility maximization and
setting “personal records” for eating?
Utility maximization means that the person will eat until the
marginal utility per dollar of food equals the marginal utility
per dollar of all other goods and services. Setting a personal
record, however, implies that the person’s objective is to eat
until he or she has eaten more than at past events and not to
maximize his or her utility.
b. What does the fact that fans eat less at subsequent games
indicate about what happens to the marginal utility from
ballpark food as the quantity consumed increases?
The fact they eat less implies that the marginal utility from
ballpark food decreases as more is consumed.
c. How can setting personal records for eating be reconciled
with marginal utility theory?
The marginal utility of food consumption includes not only the
“usual utility” from food but also the utility from setting a
food-eating record.
d. Which ideas of behavioral economics are consistent with the
information in the news clip?
Bounded willpower seems very consistent with the information.
Undoubtedly the people who “set personal records” in the stadium
regret their decisions at later dates when they either have less
income to spend than they desire and/or need to lose weight.
9.
Compared to Other Liquids, Gasoline is Cheap
Think a $4 gallon of gas is expensive? Consider the prices of
these other fluids that people buy every day without
complaint…
Gatorade, 20 oz @ $1.59 = $10.17 per gallon …
Wite-Out, 7 oz @ $1.39 = $25.42 per gallon …
HP Ink Cartridge, 16 ml $18 = $4,294.58 per gallon
The New York Times, May 27, 2008
a. What does marginal utility theory predict about the
marginal utility per dollar from gasoline, Gatorade, WiteOut, and printer ink?
Marginal utility theory predicts that the marginal utility per
dollar of each of these liquids is the same.
b. What do the prices per gallon reported in this news clip
tell you about the marginal utility from a gallon of
gasoline, Gatorade, Wite-Out, and printer ink?
Marginal utility theory concludes that the higher the price, the
higher the marginal utility. Thus the marginal utility of a
gallon of gasoline is less than the marginal utility of gallon
of Gatorade, which is less than the marginal utility of a gallon
of Wite-Out, which, in turn, is less than the marginal utility
of a gallon of HP ink.
c. What do the prices per unit reported in this news clip tell
you about the marginal utility from a gallon of gasoline, a
20 oz bottle of Gatorade, a 7 oz bottle of Wite-Out, and a
cartridge of printer ink?
Marginal utility theory concludes that the higher the price, the
higher the marginal utility. Thus the marginal utility of 7 oz.
of Wite-Out is less than the marginal utility of 20 oz. of
Gatorade, which is less than the marginal utility of a gallon of
gasoline, which, in turn, is less than the marginal utility of
an HP Ink Cartridge.
d. How can the paradox of value be used to explain why the
fluids listed in the news clip might be less valuable than
gasoline, yet far more expensive.
Gasoline presumably is more valuable than Gatorade, Wite-Out, or
HP Ink Cartridges because gasoline is really essential to our
modern life. But gasoline is much more common than these other
three products. As a result the marginal utility and hence the
price of gasoline per gallon is less than that of the other
products even though the consumer surplus from gasoline vastly
exceeds that from the other products.
10. Exclusive Status: It’s in The Bag; $52,500 Purses. 24
Worldwide. 1 in Washington.
Forget your Coach purse. Put away your Kate Spade. Even
Hermes’s famous Birkin bag seems positively discount. The
Louis Vuitton Tribute Patchwork is this summer’s ultimate
status bag, ringing in at $52,500. And it is arriving in
Washington. ... The company ... [is] offering only five for
sale in North America and 24 worldwide....
The Washington Post, August 21, 2007
a. Use marginal utility theory to explain the facts reported
in the news clip.
The number of purses is extremely limited, so the marginal
utility from these purses will be very high. When the potential
consumers equate the marginal utility per dollar from this purse
to the rest of the goods and services they buy, the price of the
purse will be extremely high due to the very high marginal
utility.
b. If Louis Vuitton offered 500 Tribute Patchwork bags in
North America and 2,400 worldwide, what do you predict
would happen to the price that buyers would be willing to
pay and what would happen to the consumer surplus?
If Louise Vuitton increased the number of these purses, the
marginal utility from the purse would fall and the price would
fall. Assuming that the demand for the purse did not change, the
consumer surplus from the purse would increase.
c. If the Tribute Patchwork bag is copied and thousands are
sold illegally, what do you predict would happen to the
price that buyers would be willing to pay for a genuine bag
and what would happen to the consumer surplus?
The marginal utility and the demand for the purse decrease
because the counterfeits are substitutes for the real article.
As a result the price buyers are willing to pay for the genuine
bag falls. The presence of the counterfeits increases the number
of substitutes for the real item and the demand becomes more
elastic. If the supply is perfectly inelastic, then the decrease
in demand combined with the increased elasticity of demand means
that the consumer surplus decreases.