In this chapter you will study ways to measure a consumer’s valuation of
a good given the consumer’s demand curve for it. The basic logic is as
follows: The height of the demand curve measures how much the consumer
is willing to pay for the last unit of the good purchased—the willingness to
pay for the marginal unit. Therefore the sum of the willingnesses-to-pay
for each unit gives us the total willingness to pay for the consumption of
the good.
In geometric terms, the total willingness to pay to consume some
amount of the good is just the area under the demand curve up to that
amount. This area is called gross consumer’s surplus or total benefit
of the consumption of the good. If the consumer has to pay some amount
in order to purchase the good, then we must subtract this expenditure in
order to calculate the (net) consumer’s surplus.
When the utility function takes the quasilinear form, u(x) + m, the
area under the demand curve measures u(x), and the area under the
demand curve minus the expenditure on the other good measures u(x) +
m. Thus in this case, consumer’s surplus serves as an exact measure of
utility, and the change in consumer’s surplus is a monetary measure of a
change in utility.
If the utility function has a different form, consumer’s surplus will not
be an exact measure of utility, but it will often be a good approximation.
However, if we want more exact measures, we can use the ideas of the
compensating variation and the equivalent variation.
Recall that the compensating variation is the amount of extra income
that the consumer would need at the new prices to be as well off as she
was facing the old prices; the equivalent variation is the amount of money
that it would be necessary to take away from the consumer at the old
prices to make her as well off as she would be, facing the new prices.
Although different in general, the change in consumer’s surplus and the
compensating and equivalent variations will be the same if preferences are
quasilinear.
In this chapter you will practice:
• Calculating consumer’s surplus and the change in consumer’s surplus
• Calculating compensating and equivalent variations
Suppose that the inverse demand curve is given by P(q) = 100 − 10q
and that the consumer currently has 5 units of the good. How much
money would you have to pay him to compensate him for reducing his
consumption of the good to zero?
Answer: The inverse demand curve has a height of 100 when q = 0
and a height of 50 when q = 5. The area under the demand curve is a
trapezoid with a base of 5 and heights of 100 and 50. We can calculate
the area of this trapezoid by applying the formula
Area of a trapezoid = base ×
1
2
(height1 + height2).
In this case we have A = 5 × 1
2 (100 + 50) = $375.
Suppose now that the consumer is purchasing the 5 units at a price of $50
per unit. If you require him to reduce his purchases to zero, how much
money would be necessary to compensate him?
In this case, we saw above that his gross benefits decline by $375.
On the other hand, he has to spend 5 × 50 = $250 less. The decline in
net surplus is therefore $125.
Suppose that a consumer has a utility function u(x1, x2) = x1 + x2. Initially
the consumer faces prices (1, 2) and has income 10. If the prices
change to (4, 2), calculate the compensating and equivalent variations.
Answer: Since the two goods are perfect substitutes, the consumer
will initially consume the bundle (10, 0) and get a utility of 10. After the
prices change, she will consume the bundle (0, 5) and get a utility of 5.
After the price change she would need $20 to get a utility of 10; therefore
the compensating variation is 20 − 10 = 10. Before the price change, she
would need an income of 5 to get a utility of 5. Therefore the equivalent
variation is 10 − 5 = 5.
14.1 (0) Sir Plus consumes mead, and his demand function for tankards
of mead is given by D(p) = 100 − p, where p is the price of mead in
shillings.
(a) If the price of mead is 50 shillings per tankard, how many tankards of
mead will he consume? .
(b) How much gross consumer’s surplus does he get from this consumption?
.
(c) How much money does he spend on mead? .
(d) What is his net consumer’s surplus from mead consumption?
.
14.2 (0) Here is the table of reservation prices for apartments taken
from Chapter 1:
Person = A B C D E F G H
Price = 40 25 30 35 10 18 15 5
(a) If the equilibrium rent for an apartment turns out to be $20, which
consumers will get apartments? .
(b) If the equilibrium rent for an apartment turns out to be $20, what
is the consumer’s (net) surplus generated in this market for person A?
For person B? .
(c) If the equilibrium rent is $20, what is the total net consumers’ surplus
generated in the market? .
(d) If the equilibrium rent is $20, what is the total gross consumers’
surplus in the market? .
(e) If the rent declines to $19, how much does the gross surplus increase?
(f) If the rent declines to $19, how much does the net surplus increase?
14.3 (0) Quasimodo consumes earplugs and other things. His utility
function for earplugs x and money to spend on other goods y is given by
u(x, y) = 100x −
x2
2
+ y.
(a) What kind of utility function does Quasimodo have?
.
(b) What is his inverse demand curve for earplugs? .
(c) If the price of earplugs is $50, how many earplugs will he consume?
.
(d) If the price of earplugs is $80, how many earplugs will he consume?
.
(e) Suppose that Quasimodo has $4,000 in total to spend a month. What
is his total utility for earplugs and money to spend on other things if the
price of earplugs is $50? .
(f) What is his total utility for earplugs and other things if the price of
earplugs is $80? .
(g) Utility decreases by when the price changes from $50 to
$80.
(h) What is the change in (net) consumer’s surplus when the price changes
from $50 to $80? .
14.4 (2) In the graph below, you see a representation of Sarah Gamp’s
indifference curves between cucumbers and other goods. Suppose that
the reference price of cucumbers and the reference price of “other goods”
are both 1.
Cucumbers
Other goods
0
4040
3030
2020
10
10 2020 3030 40
B
A
(a) What is the minimum amount of money that Sarah would need in
order to purchase a bundle that is indifferent to A? .
(b) What is the minimum amount of money that Sarah would need in
order to purchase a bundle that is indifferent to B? .
(c) Suppose that the reference price for cucumbers is 2 and the reference
price for other goods is 1. How much money does she need in order to
purchase a bundle that is indifferent to bundle A? .
(d) What is the minimum amount of money that Sarah would need to
purchase a bundle that is indifferent to B using these new prices? .
(e) No matter what prices Sarah faces, the amount of money she needs
to purchase a bundle indifferent to A must be (higher, lower) than the
amount she needs to purchase a bundle indifferent to B. .
14.5 (2) Bernice’s preferences can be represented by u(x, y) = min{x, y},
where x is pairs of earrings and y is dollars to spend on other things. She
faces prices (px, py) = (2, 1) and her income is 12.
(a) Draw in pencil on the graph below some of Bernice’s indifference
curves and her budget constraint. Her optimal bundle is pairs
of earrings and dollars to spend on other things.
Dollars for other things
16
12
8
4
0 4 8 12 16
Pairs of earrings
(b) The price of a pair of earrings rises to $3 and Bernice’s income stays the
same. Using blue ink, draw her new budget constraint on the graph above.
Her new optimal bundle is pairs of earrings and dollars
to spend on other things.
(c) What bundle would Bernice choose if she faced the original prices and
had just enough income to reach the new indifference curve?
Draw with red ink the budget line that passes through this bundle at
the original prices. How much income would Bernice need at the original
prices to have this (red) budget line? .
(d) The maximum amount that Bernice would pay to avoid the price
increase is This is the (compensating, equivalent) variation in
income. .
(e) What bundle would Bernice choose if she faced the new prices and had
just enough income to reach her original indifference curve?
Draw with black ink the budget line that passes through this bundle at
the new prices. How much income would Bernice have with this budget?
.
(f) In order to be as well-off as she was with her original bundle, Bernice’s
original income would have to rise by This is the (compensating,
equivalent) variation in income. .
14.6 (0) Ulrich likes video games and sausages. In fact, his preferences
can be represented by u(x, y) = ln(x + 1) + y where x is the number of
video games he plays and y is the number of dollars that he spends on
sausages. Let px be the price of a video game and m be his income.
(a) Write an expression that says that Ulrich’s marginal rate of substitution
equals the price ratio. ( Hint: Remember Donald Fribble from
Chapter 6?) .
(b) Since Ulrich has preferences, you can solve this equation
alone to get his demand function for video games, which is
His demand function for the dollars to spend on sausages is
.
(c) Video games cost $.25 and Ulrich’s income is $10. Then Ulrich demands
video games and dollars’ worth of sausages.
His utility from this bundle is (Round off to two decimal
places.)
(d) If we took away all of Ulrich’s video games, how much money would
he need to have to spend on sausages to be just as well-off as before?
.
(e) Now an amusement tax of $.25 is put on video games and is passed
on in full to consumers. With the tax in place, Ulrich demands
video game and dollars’ worth of sausages. His utility from this
bundle is (Round off to two decimal places.)
(f) Now if we took away all of Ulrich’s video games, how much money
would he have to have to spend on sausages to be just as well-off as with
the bundle he purchased after the tax was in place? .
(g) What is the change in Ulrich’s consumer surplus due to the tax?
How much money did the government collect from Ulrich by
means of the tax? .
14.7 (1) Lolita, an intelligent and charming Holstein cow, consumes
only two goods, cow feed (made of ground corn and oats) and hay. Her
preferences are represented by the utility function U(x, y) = x−x2
/2+y,
where x is her consumption of cow feed and y is her consumption of hay.
Lolita has been instructed in the mysteries of budgets and optimization
and always maximizes her utility subject to her budget constraint. Lolita
has an income of $m that she is allowed to spend as she wishes on cow
feed and hay. The price of hay is always $1, and the price of cow feed will
be denoted by p, where 0 < p ≤ 1.
(a) Write Lolita’s inverse demand function for cow feed. (Hint: Lolita’s
utility function is quasilinear. When y is the numeraire and the price of
x is p, the inverse demand function for someone with quasilinear utility
f(x) + y is found by simply setting p = f′
(x).) .
(b) If the price of cow feed is p and her income is m, how much hay does
Lolita choose? (Hint: The money that she doesn’t spend on feed is used
to buy hay.) .
(c) Plug these numbers into her utility function to find out the utility
level that she enjoys at this price and this income. .
(d) Suppose that Lolita’s daily income is $3 and that the price of feed is
$.50. What bundle does she buy? What bundle would she
buy if the price of cow feed rose to $1? .
(e) How much money would Lolita be willing to pay to avoid having the
price of cow feed rise to $1? This amount is known as the
variation.
(f) Suppose that the price of cow feed rose to $1. How much extra money
would you have to pay Lolita to make her as well-off as she was at the
old prices? This amount is known as the
variation. Which is bigger, the compensating or the equivalent variation,
or are they the same? .
(g) At the price $.50 and income $3, how much (net) consumer’s surplus
is Lolita getting? .
14.8 (2) F. Flintstone has quasilinear preferences and his inverse demand
function for Brontosaurus Burgers is P(b) = 30 − 2b. Mr. Flintstone is
currently consuming 10 burgers at a price of 10 dollars.
(a) How much money would he be willing to pay to have this amount
rather than no burgers at all? What is his level of (net)
consumer’s surplus? .
(b) The town of Bedrock, the only supplier of Brontosaurus Burgers, decides
to raise the price from $10 a burger to $14 a burger. What is Mr.
Flintstone’s change in consumer’s surplus?
.
14.9 (1) Karl Kapitalist is willing to produce p/2 − 20 chairs at every
price, p > 40. At prices below 40, he will produce nothing. If the price
of chairs is $100, Karl will produce chairs. At this price, how
much is his producer’s surplus? .
14.10 (2) Ms. Q. Moto loves to ring the church bells for up to 10
hours a day. Where m is expenditure on other goods, and x is hours of
bell ringing, her utility is u(m, x) = m + 3x for x ≤ 10. If x > 10, she
develops painful blisters and is worse off than if she didn’t ring the bells.
Her income is equal to $100 and the sexton allows her to ring the bell for
10 hours.
(a) Due to complaints from the villagers, the sexton has decided to restrict
Ms. Moto to 5 hours of bell ringing per day. This is bad news for Ms.
Moto. In fact she regards it as just as bad as losing dollars of
income.
(b) The sexton relents and offers to let her ring the bells as much as she
likes so long as she pays $2 per hour for the privilege. How much ringing
does she do now? This tax on her activities is as bad as
a loss of how much income? .
(c) The villagers continue to complain. The sexton raises the price of bell
ringing to $4 an hour. How much ringing does she do now?
This tax, as compared to the situation in which she could ring
the bells for free, is as bad as a loss of how much income? .
Some problems in this chapter will ask you to construct the market demand
curve from individual demand curves. The market demand at any
given price is simply the sum of the individual demands at that price.
The key thing to remember in going from individual demands to the market
demand is to add quantities. Graphically, you sum the individual
demands horizontally to get the market demand. The market demand
curve will have a kink in it whenever the market price is high enough that
some individual demand becomes zero.
Sometimes you will need to find a consumer’s reservation price for
a good. Recall that the reservation price is the price that makes the
consumer indifferent between having the good at that price and not having
the good at all. Mathematically, the reservation price p∗
satisfies
u(0, m) = u(1, m − p∗
), where m is income and the quantity of the other
good is measured in dollars.
Finally, some of the problems ask you to calculate price and/or income
elasticities of demand. These problems are especially easy if you
know a little calculus. If the demand function is D(p), and you want to
calculate the price elasticity of demand when the price is p, you only need
to calculate dD(p)/dp and multiply it by p/q.
15.0 Warm Up Exercise. (Calculating elasticities.) Here are
some drills on price elasticities. For each demand function, find an expression
for the price elasticity of demand. The answer will typically be
a function of the price, p. As an example, consider the linear demand
curve, D(p) = 30 − 6p. Then dD(p)/dp = −6 and p/q = p/(30 − 6p), so
the price elasticity of demand is −6p/(30 − 6p).
(a) D(p) = 60 − p. .
(b) D(p) = a − bp. .
(c) D(p) = 40p−2
. .
(d) D(p) = Ap−b
. .
(e) D(p) = (p + 3)−2
. .
(f) D(p) = (p + a)−b
. .
15.1 (0) In Gas Pump, South Dakota, there are two kinds of consumers,
Buick owners and Dodge owners. Every Buick owner has a demand function
for gasoline DB(p) = 20 − 5p for p ≤ 4 and DB(p) = 0 if p > 4.
Every Dodge owner has a demand function DD(p) = 15 − 3p for p ≤ 5
and DD(p) = 0 for p > 5. (Quantities are measured in gallons per week
and price is measured in dollars.) Suppose that Gas Pump has 150 consumers,
100 Buick owners, and 50 Dodge owners.
(a) If the price is $3, what is the total amount demanded by each individual
Buick Owner? And by each individual Dodge owner?
.
(b) What is the total amount demanded by all Buick owners?
What is the total amount demanded by all Dodge owners? .
(c) What is the total amount demanded by all consumers in Gas Pump
at a price of 3? .
(d) On the graph below, use blue ink to draw the demand curve representing
the total demand by Buick owners. Use black ink to draw the
demand curve representing total demand by Dodge owners. Use red ink
to draw the market demand curve for the whole town.
(e) At what prices does the market demand curve have kinks?
.
(f) When the price of gasoline is $1 per gallon, how much does weekly
demand fall when price rises by 10 cents? .
(g) When the price of gasoline is $4.50 per gallon, how much does weekly
demand fall when price rises by 10 cents? .
(h) When the price of gasoline is $10 per gallon, how much does weekly
demand fall when price rises by 10 cents? .
Dollars per gallon
6
5
4
3
2
1
0 500 1000 1500 2000 2500 3000 3500 4000
Gallons per week
15.2 (0) For each of the following demand curves, compute the inverse
demand curve.
(a) D(p) = max{10 − 2p, 0}. .
(b) D(p) = 100/
√
p. .
(c) ln D(p) = 10 − 4p. .
(d) ln D(p) = ln 20 − 2 ln p. .
15.3 (0) The demand function of dog breeders for electric dog polishers
is qb = max{200−p, 0}, and the demand function of pet owners for electric
dog polishers is qo = max{90 − 4p, 0}.
(a) At price p, what is the price elasticity of dog breeders’ demand for
electric dog polishers? What is the price elasticity of
pet owners’ demand? .
(b) At what price is the dog breeders’ elasticity equal to −1?
At what price is the pet owners’ elasticity equal to −1? .
(c) On the graph below, draw the dog breeders’ demand curve in blue
ink, the pet owners’ demand curve in red ink, and the market demand
curve in pencil.
(d) Find a nonzero price at which there is positive total demand for dog
polishers and at which there is a kink in the demand curve.
What is the market demand function for prices below the kink?
What is the market demand function for prices above the
kink? .
(e) Where on the market demand curve is the price elasticity equal to
−1? At what price will the revenue from the sale of electric
dog polishers be maximized? If the goal of the sellers is to
maximize revenue, will electric dog polishers be sold to breeders only, to
pet owners only, or to both? .
Price
300
250
200
150
100
50
0 50 100 150 200 250 300
Quantity
15.4 (0) The demand for kitty litter, in pounds, is ln D(p) = 1, 000 −
p + ln m, where p is the price of kitty litter and m is income.
(a) What is the price elasticity of demand for kitty litter when p = 2 and
m = 500? When p = 3 and m = 500? When
p = 4 and m = 1, 500? .
(b) What is the income elasticity of demand for kitty litter when p = 2
and m = 500? When p = 2 and m = 1, 000? When
p = 3 and m = 1, 500? .
(c) What is the price elasticity of demand when price is p and income is
m? The income elasticity of demand? .
15.5 (0) The demand function for drangles is q(p) = (p + 1)−2
.
(a) What is the price elasticity of demand at price p? .
(b) At what price is the price elasticity of demand for drangles equal to
−1? .
(c) Write an expression for total revenue from the sale of drangles as a
function of their price. Use calculus to find
the revenue-maximizing price. Don’t forget to check the second-order
condition. .
(d) Suppose that the demand function for drangles takes the more general
form q(p) = (p + a)−b
where a > 0 and b > 1. Calculate an expression for
the price elasticity of demand at price p. At what price
is the price elasticity of demand equal to −1? .
15.6 (0) Ken’s utility function is uK(x1, x2) = x1 + x2 and Barbie’s
utility function is uB(x1, x2) = (x1 + 1)(x2 + 1). A person can buy 1
unit of good 1 or 0 units of good 1. It is impossible for anybody to buy
fractional units or to buy more than 1 unit. Either person can buy any
quantity of good 2 that he or she can afford at a price of $1 per unit.
(a) Where m is Barbie’s wealth and p1 is the price of good 1, write an
equation that can be solved to find Barbie’s reservation price for good 1.
What is Barbie’s reservation price for good 1?
What is Ken’s reservation price for good 1? .
(b) If Ken and Barbie each have a wealth of 3, plot the market demand
curve for good 1.
Price
4
3
2
1
0 1 2 3 4
Quantity
15.7 (0) The demand function for yo-yos is D(p, M) = 4 − 2p + 1
100 M,
where p is the price of yo-yos and M is income. If M is 100 and p is 1,
(a) What is the income elasticity of demand for yo-yos? .
(b) What is the price elasticity of demand for yo-yos? .
15.8 (0) If the demand function for zarfs is P = 10 − Q,
(a) At what price will total revenue realized from their sale be at a maximum?
.
(b) How many zarfs will be sold at that price? .
15.9 (0) The demand function for football tickets for a typical game at a
large midwestern university is D(p) = 200, 000 − 10, 000p. The university
has a clever and avaricious athletic director who sets his ticket prices so
as to maximize revenue. The university’s football stadium holds 100,000
spectators.
(a) Write down the inverse demand function. .
(b) Write expressions for total revenue and marginal
revenue as a function of the number of tickets
sold.
(c) On the graph below, use blue ink to draw the inverse demand function
and use red ink to draw the marginal revenue function. On your graph,
also draw a vertical blue line representing the capacity of the stadium.
Price
30
25
20
15
10
5
0 20 40 60 80 100 120 140 160
Quantity × 1,000
(d) What price will generate the maximum revenue? What
quantity will be sold at this price? .
(e) At this quantity, what is marginal revenue? At this quantity,
what is the price elasticity of demand? Will the stadium be
full? .
(f) A series of winning seasons caused the demand curve for football
tickets to shift upward. The new demand function is q(p) = 300, 000 −
10, 000p. What is the new inverse demand function?
.
(g) Write an expression for marginal revenue as a function of output.
MR(q) = Use red ink to draw the new demand function
and use black ink to draw the new marginal revenue function.
(h) Ignoring stadium capacity, what price would generate maximum revenue?
What quantity would be sold at this price?
.
(i) As you noticed above, the quantity that would maximize total revenue
given the new higher demand curve is greater than the capacity of the
stadium. Clever though the athletic director is, he cannot sell seats he
hasn’t got. He notices that his marginal revenue is positive for any number
of seats that he sells up to the capacity of the stadium. Therefore, in order
to maximize his revenue, he should sell tickets at a price of
.
(j) When he does this, his marginal revenue from selling an extra seat
is The elasticity of demand for tickets at this price quantity
combination is .
15.10 (0) The athletic director discussed in the last problem is considering
the extra revenue he would gain from three proposals to expand the
size of the football stadium. Recall that the demand function he is now
facing is given by q(p) = 300, 000 − 10, 000p.
(a) How much could the athletic director increase the total revenue per
game from ticket sales if he added 1,000 new seats to the stadium’s capacity
and adjusted the ticket price to maximize his revenue? .
(b) How much could he increase the revenue per game by adding 50,000
new seats? 60,000 new seats? (Hint: The athletic director
still wants to maximize revenue.) .
(c) A zealous alumnus offers to build as large a stadium as the athletic
director would like and donate it to the university. There is only one hitch.
The athletic director must price his tickets so as to keep the stadium full.
If the athletic director wants to maximize his revenue from ticket sales,
how large a stadium should he choose? .