Answers to Chapter 6 Exercises
Review and practice exercises
6.1. Perfect price discrimination. Consider a monopolist with demand D = 120 − 2 p
and marginal cost MC = 40. Determine profit, consumer surplus, and social welfare in the
following two cases: (a) single-price monopolist; (b) perfect price discrimination.
Solution: As we have seen before, in these problems it is useful to work with the inverse
demand curve. In the present case, this is given by p = 60 − .5 q. Marginal revenue is given
by MR = 60−q. Equating to marginal cost and solving with respect to q, we get 60−q = 40,
or q = 20. This implies p = 60 − 20/2 = 50. Monopoly profits are (50 − 40) × 20 = 200,
whereas consumer surplus is 21 (60 − 50) × 20 = 100. It follows that social surplus is equal
to 300.
Consider now the case of a perfect discriminating monopolist. Price for the qth unit
is now given p = 60 − .5 q, so long as the resulting price is greater than marginal cost. If
follows that the monopolist’s optimal output is given by solving p = MC , or 60 − .5 q = 40,
which yields q = 40. Under perfect discrimination, consumer surplus is zero (each consumer
pays a price equal to its willingness to pay, or else it doesn’t buy). Seller’s profit is given
by the integral of the difference between price and marginal cost. Since demand is linear,
this is the area of the triangle formed by the demand curve and the MC curve, from 0 to
optimal q: 21 (60 − 40) × 40 = 400.
6.2. The Economist. First-time subscribers to the Economist pay a lower rate than
repeat subscribers. Is this price discrimination? Of what type?
Answer: This is an example of price discrimination by indicators (also known as thirddegree price discrimination). The market is segmented into new subscribers and repeat
subscribers. New subscribers, know the product less well and are thus likely to be more
price sensitive. Moreover, the fact that they have not subscribed in the past indicates that
they are likely to be willing to pay less than current subscribers. It is therefore optimal to
set a lower price for new subscribers.
6.3. Cement.
Cement in Belgium is sold at a uniform delivered price throughout the
country, that is, the same price is set for each customer, including transportation costs,
regardless of where the customer is located. The same is practice is also found in the sale
of plasterboard in the United Kingdom.15 Are these cases of price discrimination?
Answer: Yes, these are cases of price discrimination. Consider the total price being paid by
each customer, P , as being composed of the price actually charged and the transportation
cost; P = pi + ti . Since locations are different, transportation costs are different, thus, each
consumer is charged a price pi that depends on his or her location. This is a clear example
of geographic price discrimination, one instance of discrimination by indicators.
6.4. Fulton fish market. A study of the New York fish market (when it was the Fulton
fish market) suggests that the average price paid for whiting by Asian buyers is significantly
lower than the price paid by White buyers.16 What type of price discrimination does this
correspond to, if any? What additional information would you need in order to answer the
question?
Answer: This appears to be a case of price discrimination by indicators (also known as
third-degree price discrimination), whereby a group of buyers (a market segment) pays
a different price than another group. Theory predicts that in a non-competitive market
(monopoly, oligopoly) buyers with higher price elasticity should be charged a lower price;
as a result, we can conclude that Asian buyers have higher price elasticity than white buyers.
In order to have a more accurate picture, however, more information is needed. Different
prices could could simply result from quantity discounts and the possible fact that different
quantities are bought by the different groups. If that were the case, we would have price
discrimination by self-selection, not by indicators. In other words, it may be that Asian
buyers pay a lower price not because they are Asian but because they purchase larger
quantities. Similarly, the time of purchase (e.g., before 5am or after 5am) could be correlated
with race, so that it is not race that determines the price difference.
A similar reasoning applies to the type of establishment does the buyer represents (store,
fry shop, etc.). Finally, it could also be the case that different groups use different types of
payment type (cash or credit), so that different prices reflect different costs.
For a more complete discussion, see the cited reference.
6.5. Coupons.
Supermarkets frequently issue coupons that entitle consumers to a
discount in selected products. Is this a promotional strategy, or simply a form of price
discrimination? Empirical evidence suggests that paper towels are significantly more expensive in markets offering coupons than in markets without coupons.17 Is this consistent
with your interpretation?
Answer: This may be interpreted as a case of price discrimination by self selection. By
offering coupons (hence a lower price), supermarkets can serve the buyers with a higher price
elasticity at a different price. In order for this strategy to improve revenues with respect to
single price, supermarkets should then set a higher regular price. Hence, empirical evidence
is consistent with the explanation that this is a form of price discrimination.
6.6. Coca-Cola. In 1999, Coca-Cola announced that it was developing a “smart” vending
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machine. Such machines are able to change prices according to the outside temperature.18
Suppose, for the purposes of this problem, that the temperature can be either “High” or
“Low.” On days of “High” temperature, demand is given by Q = 280 − 2 p, where Q is
number of cans of Coke sold during the day and p is the price per can measured in cents.
On days of “Low” temperature, demand is only Q = 160 − 2 p. There is an equal number
days with “High” and “Low” temperature. The marginal cost of a can of Coke is 20 cents.
(a) Suppose that Coca-Cola indeed installs a “smart” vending machine,
and thus is able to charge different prices for Coke on “Hot” and
“Cold” days. What price should Coca-Cola charge on a “Hot” day?
What price should Coca-Cola charge on a “Cold” day?
Answer: On a Hot day, Q = 280−2 p, or p = 140−Q/2. Marginal revenue is MR = 140−Q.
Equating to marginal cost (20) and solving, we get Q∗ = 120 and p∗ = 80. On a Cold day,
Q = 160 − 2 p, or p = 80 − Q/2. Marginal revenue is MR = 80 − Q. Equating to marginal
cost (20) and solving, we get Q∗ = 60 and p∗ = 50.
(b) Alternatively, suppose that Coca-Cola continues to use its normal
vending machines, which must be programmed with a fixed price,
independent of the weather. Assuming that Coca-Cola is risk neutral,
what is the optimal price for a can of Coke?
Answer: Observe from part (a) that even on a Hot day the optimal price is no greater than
80 cents. So, we can restrict our attention to prices of 80 cents or less. In this price range,
the expected demand is given by Q = 12 (280 − 2p) + 21 (160 − 2 p) = 220 − 2 p.q Solving for p
gives p = 110 − Q/2. The marginal revenue associated with this expected demand curve is
given by MR = 110 − Q. Equating this marginal revenue to marginal cost, we get Q∗ = 90
and p∗ = 65.
(c) What are Coca-Cola’s profits under constant and weather-variable
prices? How much would Coca-Cola be willing to pay to enable
its vending machine to vary prices with the weather, i.e., to have
a “smart” vending machine?
Answer: Under price discrimination, from part (a), profits on a Hot day are (80 − 20) ×
120 = $72, and profits on a Cold day are (50 − 20) × 60 = $18. Expected profits per day
are therefore ($72 + $18)/2 = $45. Under uniform pricing, expected profits per day are
(65 − 20) × 90 = $40.50. It follows that Coca-Cola should be willing to pay up to an extra
$4.50 per day for a “smart” vending machine.
However, one might add that there are other important considerations. The fact that
Coca-Cola did not pursue the idea of a “smart” vending machine is probably less due to
the economics of it as it is due to the consumer backlash that the idea created.
6.7. Sal’s satellite. Sal’s satellite company broadcasts TV to subscribers in LA and NY.
q . If p > 80, then we must consider the possibility of zero consumption by at least one of the market
segments.
3
The demand functions are given by
QNY = 50 −
QLA = 80 −
1
3
2
3
PNY
PLA
where Q is in thousands of subscriptions per year and P is the subscription price per year.
The cost of providing Q units of service is given by
TC = 1, 000 + 30 Q
where Q = QNY + QLA .
(a) What are the profit-maximizing prices and quantities for the NY and
LA markets?
Answer: PNY = 90, QNY = 20, PLA = 75 and QLA = 30.
(b) As a consequence of a new satellite that the Pentagon developed,
subscribers in LA are now able to get the NY broadcast and vice
versa, so Sal can charge only a single price. What price should he
charge?
Answer: Note that demand in NY is positive if and only if price is lower than 150; whereas
demand in LA is positive if and only if price is lower than 120. It follows that total demand
is given by


if P < 120
 QNY + QLA = 130 − P
1
Q=
QNY = 50 − 3 P
if 120 < P < 150


0
if P > 150
Suppose that P < 120, so that the first formula applies. Then the inverse demand curve
is given by P = 130 − Q. Optimal price is P = (130 + 30)/2 = 80, which confirms the
hypothesis that P < 120. Quantity is given by Q = 130 − P = 50.
(c) In which situation is Sal better off? In terms of consumers’ surplus,
which situation do people in LA prefer? What about people in NY?
Why?
Answer: Sal is better off without the new satellite (that is, when he can price discriminate).
People in NY prefer the new satellite, since they pay a lower price. People in LA preferred
the situation before the introduction of the satellite, because the price increased for them.
As often is the case, price discrimination makes some people better off, some people worse
off.
6.8. Stadium pricing. Stanford Stadium has a capacity of 50k and is used for exactly
seven football games a year. Three of these are OK games, with a demand for tickets
given by D = 150k − 3 p per game, where p is ticket price. (For simplicity, assume there
is only one type of ticket.) Three of the season games are not so important, the demand
being D = 90k − 3 p per game. Finally, one of the games is really big, the demand being
D = 240k − 3 p. The costs of operating the Stadium are essentially independent of the
number of tickets sold.
(a) Determine the optimal ticket price for each game, assuming the objective of profit maximization.
4
Answer: Demand for OK games is given by D = 150 − 3 p, where number of tickets is
measured in thousands. Inverse demand is p = 50−Q/3. Marginal revenue is MR = 50− 32 Q.
Marginal cost is zero, since costs do not depend on the number of tickets sold. Equating
marginal cost to marginal revenue, we get Q = 75. This is greater than capacity. Therefore,
the optimal solution is simply to set price such that demand equals capacity: 150−3 p = 50,
which implies p = $33.3
Demand for not-so-important games is given by D = 90 − 3 p. Inverse demand is
p = 30 − Q/3. Marginal revenue is MR = 50 − 23 Q. Equating marginal revenue to marginal
cost, we get Q = 45. Substituting back in the inverse demand curve we get p = $15.
Since demand for the Big Game is greater than for the OK games, it will surely be the
case that MR = MC implies a demand level greater than capacity. The optimal price is
therefore determined by equating demand to capacity: 240 − 3 p = 50, or simply p = $63.3
Given that the Stadium is frequently full, the idea of expanding the Stadium has arisen.r
A preliminary study suggests that the cost of capacity expansion would be $100 per seat
per year.
(b) Would you recommend that Stanford go ahead with the project of
capacity expansion?
Answer: The marginal gross profit of an additional seat is the sum of the difference between
marginal revenue and marginal cost for all games where capacity was a constraint. For
OK games, marginal revenue is given by MR = 50 − 32 50 = 16.7. For the Big Game,
MR = 80 − 23 50 = 46.7. Adding these up (three times the first plus the second) we get
$96.7. Since this is less than the marginal cost of capacity expansion, it is not worth it to
pursue the project.
6.9. Spoken Word.
Your software company has just completed the first version of
SpokenWord, a voice-activated word processor. As marketing manager, you have to decide
on the pricing of the new software. You commissioned a study to determine the potential
demand for SpokenWord. From this study, you know that there are essentially two market
segments of equal size, professionals and students (one million each). Professionals would
be willing to pay up to $400 and students up to $100 for the full version of the software.
A substantially scaled-down version of the software would be worth $50 to consumers and
worthless to professionals. It is equally costly to sell any version. In fact, other than the
initial development costs, production costs are zero. Although you know there are two
market segments, you cannot directly identify a consumer as belonging to a specific market
segment.
(a) What are the optimal prices for each version of the software?
Answer: It is optimal to price the full version at 400 and the scaled-down version at 50.
Total profits are 450.
Suppose that, instead of the scaled-down version, the firm sells an intermediate version that
r . Ignore the fact that Stanford Stadium used to hold 90,000 seats and was thought to be too big.
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is valued at $200 by professionals and $75 by students.
(b) What are the optimal prices for each version of the software? Is
the firm better off by selling the intermediate version instead of the
scaled-down version?
Answer: One first possibility would be to price the intermediate version at 75 and the full
version at 400. However, this would lead professionals to choose the intermediate version
since the difference between willingness to pay and price is greater for the intermediate
version. In order to induce professionals to buy the full version, the full version’s price
would need to be 75 + (400 − 200) = 275, where the value in parentheses is the professionals’
difference in willingness to pay between the two versions. This would lead to a total profit
of 275 + 75 = 350, which is lower than initially. Still another possibility would be to price
the full version at 400 and the intermediate version at 400 − (400 − 200) = 200. In this
case, professionals would buy the full version but students would not buy the intermediate
version. Profits would then be 400: better than 350 but still less than the 450 the firm
would get with the truly scaled-down version.
6.10. SoS. SoS (Sounds of Silence, Inc) prepares to launch a revolutionary system of
bluetooth-enabled noise-cancellation headphones. It is estimated that about 800,000 consumers would be willing to pay $450 for the headphones; an additional 1,500,000 consumers
would be willing to pay $250 for the headphones. Though SoS knows this marketing information, it cannot identify a consumer as belonging to one group or the other.
SoS is considering the launch of a stripped-down version of the headphones (the strippeddown version uses wires instead of bluetooth). The 800,000 high-valuation consumers would
only be willing to pay $325 for the stripped-down version. The remaining 1,500,000 consumers don’t particularly care about bluetooth vs. wire connections; they are willing to pay
the same $250 for either version.
Both the bluetooth version and the stripped-down version cost the same to produce:
$100 per unit.
(a) Determine the optimal pricing policy assuming that SoS only sells
bluetooth-enabled headphones.
Answer: There are two candidate price levels: 450 and 250. At 450, profit is given by
(450 − 100) × 800, 000 = $280m. At 250, profit is given by (250 − 100) × (800, 000 +
1, 500, 000) = $345m. It follows the optimal price is $250.
(b) Determine the optimal pricing policy assuming that SoS offers the two
versions.
Answer: The new alternative to consider is offering the stripped-down version at a lower
price and the full version at a higher price. The relevant constraint is that high-end users
have no incentive to choose the stripped down version. This implies that
450 − pH ≥ 325 − pL
Since low-valuation buyers will not have an incentive to buy the more expensive version (a
fact that we can confirm later on), the best we can do is to charge their valuation: 250. It
6
follows from the above inequality that the highest price we can charge the full version is
pH = 450 − 325 + 250 = 375. Under this strategy, total profit is given by
π = (375 − 100) × 800, 000 + (250 − 100) × 1, 500, 000 = $445m
Since this is better than $345m, the optimal solution determined in (a), versioning is the
optimal strategy.
(c) Suppose that SoS finds out that the estimate regarding the number of
low-valuation users is overly optimistic. In fact, there are only 300,000
consumers who would be willing to pay $250. How would you change
your answer to (a) and (b)?
Answer: Given a different number of low-valuation consumers, we must recompute both the
values in (a) and the values in (b). As to (a), we still have $280m from targeting high-end
users only. A price of $250 now leads to a profit of (250−100)×(800, 000+300, 000) = $165m.
It follows that, if the seller were to sell one version only, then the optimal price is now $450.
Consider now the case of versioning. The no-deviation constraint is still the same and
so pH would still be equal to $375. Total profit is now given by
π = (375 − 100) × 800, 000 + (250 − 100) × 300, 000 = $265m
This is less than $280m. It follows that the overall optimal policy is now to forget about
versioning and simply target high-end users with a price of $450.
6.11. RawDeal. RawDeal is the new sushi bar in the neighborhood. Their estimated
marginal cost is 10 cents per sushi unit. RawDeal estimates that each consumer has a
demand for sushi given by q = 20 − 10 p, where q is number of sushi units and p is price in
dollars per unit.
(a) Determine the optimal price per sushi unit.
Answer: Inverse demand is given by
p = 2 − 0.1 q
If demand is linear and given by p = a − b Q, then optimal price is given by
p=
a+c
2
It follows that, in the present case,
p=
2 + .1
= 1.05
2
(that is, one dollar and 5 cents).
(b) RawDeal is considering switching to an all-you-can-eat-sushi policy.
Determine the optimal price per customer. How does profit compare
to pricing per unit?
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Answer: All you can eat means the price per unit is zero. It follows that each consumer
will eat 20 units of sushi. The consumer surplus from 20 units at zero price is given by
CS =
1
2 × 20 = 20
2
It follows that revenue per customer is $20. Since each customer will eat 20 units, cost
per customer will be 20 × .1 = $2. It follows that profit per customer is given by $18. By
contrast, charging 1.05 per piece leads to a demand of
q = 20 − 10 p = 20 − 10 × 1.05 = 9.5
and a profit of
π = (1.05 − .10) × 9.5 = .95 × 9.5 = 9.025
It follows that switching to all you can eat nearly doubles RawDeal’s profit.
(c) Discuss other advantages and disadvantages of each pricing option.
Answer: Implementation costs play an important role here. One of the advantages of allyou-can eat is that you don’t need to monitor how much each patron eats: all you need to
do is to charge the entrance fee. One potential problem is that all patrons are not equal: if
you get stuck with people who eat a lot, then the deal may be better for the customers than
for the seller. In fact, one problem when you have all-you-can eat restaurants competing
against “conventional” ones is that consumers naturally self select, with the big eaters
disproportionately visiting the all-you-can-eat venues more often.
(d) Ignoring implementation costs, what is the optimal two-part tariff for
sushi (i.e., a fee at the door plus a price per sushi piece).
Answer: The optimal two-part tariff is to set price equal to marginal cost and charge a
fixed fee equal to the consumer surplus at that price. If unit price is 10 cents, then each
consumer will order
q = 20 − 10 p = 19
units of sushi. This implies that consumer surplus is given by the area of the triangle
between the demand curve and price level, that is,
1
× (2 − .1) × 19 = 18.05
2
Profit per consumer is then given by
π = F + pq − cq
where F is the fixed fee, p is price per unit, c is cost per unit, and q quantity consumed.
However, since we set p = c, we simply get π = F . In other words, under the optimal
two-part tariff profit per consumer is given by 18.05. As expected, this is better than all
you can eat (18 per consumer), though by very little (less than 1%). In fact, given the
transactions costs of charging on a per unit basis, all-you-can eat may well be the overall
optimal solution.
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Challenging exercises
6.12. Pricing with limited capacity. Consider the model of a monopolist with two
markets presented earlier in the chapter. Suppose that the seller has a limited capacity and
low marginal cost up to capacity. An example of this would be an airline with two types of
passengers or a football stadium with two types of attendees.
Derive the conditions for optimal pricing. How do they relate to the case when there
are no capacity constraints?
Answer: Let K denote capacity and p1 (q1 ), p2 (q2 ) the inverse demand functions. The
monopolist’s problem becomes:
max q1 p1 (q1 ) + q2 p2 (q2 ) − c (q1 + q2 )
q1 ,q2
subject to
q1 + q2 ≤ K
Suppose that the constraint q1 + q2 ≤ K is not binding. Then the elasticity rule applies
and we have
MR 1 = MR 2 = c
Suppose that the constraint q1 + q2 ≤ K is binding. Then the cost term in the objective
function is simply c K, a constant; and the constraint can be solve to yield q2 = K − q1 .
Since cost is constant, the objective is to maximize the revenue in market 1 plus the revenue
in market 2. Since q2 = K − q1 , each time we increase q1 by one unit we must decrease q2
by 1 unit as well, and vice-versa. This implies that the derivative of the objective function
with respect to q1 is given by
MR 1 + (−1) MR 2
Since this must equal zero (first-order condition) we again obtain MR 1 = MR 2 .
The same result can be obtained intuitively. Suppose that the seller is capacity constrained. Is the current set of prices optimal? One alternative is to take one unit from one
market and sell it the other market, changing prices accordingly. Would the seller want to
do this? By taking one unit away from Market 1, the seller would lose MR 1 . By selling it
in Market 2, the seller would get MR 2 . Optimality then requires that MR 1 = MR 2 .
The difference with respect to the case when capacity is not a constraint is that, if
before MR 1 = MR 2 = c, now MR 1 = MR 2 takes on a higher value, whichever value solves
q1 + q2 = K.
For math aficionados. From a mathematical point of view, the best way to solve a
maximization problem with constraints is to write down the Lagrangean for this problem.
In this case we have
L = q1 p1 (q1 ) + q2 p2 (q2 ) − c (q1 + q2 ) + λ (K − q1 − q2 )
The first-order conditions are:
MR 1
= MC + λ
MR 2
= MC + λ
9
Depending on whether capacity constraints are binding or not, we will have λ positive or
zero. Whichever is the case, the above equations show that optimality implies that marginal
revenue be equated across markets. Notice that, if demand elasticity differs across markets,
then this implies different prices for the different markets.
6.13. BlackInk. Printing Solutions, the maker of the printer BlackInk, faces an important
product design dilemma: deciding the speed of its popular laser printer. There are two
market segments: Professionals are willing to pay up to $800 (a − .5) for the printer, where
a is printer speed. Students, in turn, are willing to pay up to $100 a. Maximum printer
speed corresponds to a = 1, whereas a = 0 corresponds to a worthless printer. There are
one million professionals and one million students. It is equally costly to produce a printer
with any level of a. In fact, other than the initial development costs, production costs are
zero.
How many versions of the BlackInk should Printing Solutions sell? Which versions?
What are the optimal prices of each version?
Answer: Suppose first that the firm sells one version only. If that is the case, then it might
as well choose a = 1. There are then two candidates for optimal price: $400 and $100.
Profits are given by $400m in the first case and $200m in the second case (recall that there
are one million professionals and one million students). It follows that a = 1, p = 400 is the
optimal solution (conditional on there being only one version).
Since there are only two types of consumers, it will not be necessary to offer more than
two different versions. Since it is equally costly to produce any version and willingness
to pay is increasing in a, it follows that one of the versions should have maximum speed
(a = 1).
If there is to be self-selection between two different versions, it will be the case that
professionals choose the faster printer and students the slower one. We thus have four
possible constraints: incentive and participation; for type H and for type L (professionals
and students, respectively). I next argue that, if the value of a is optimally chosen, then all
constraints except the low type incentive constraint must be binding:
1. The type H incentive constraint must be binding. If that were not the case, then we
could increase the value of a a little bit without violating the incentive constraint.
This would imply a higher value for the L type, which would then allow me to charge
a higher price.
2. The type L participation constraint must be binding. Were that not the case, I
could increase the price of the slower printer and increase profit without violating any
constraint.
3. The type H participation constraint must be binding. If that were not the case, then I
could decrease the slow printer’s speed a bit, decrease the price of the slower printer by
the decline in valuation to type L buyers, and then increase the price of the fast printer
such that the type H incentive constraint remains valid. At the margin, this would
increase seller’s profit, thus contradicting the possibility that the type H participation
constraint is not binding.
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Given the above binding constraints, it must be that p1 (the price of the full version) equals
$400 (type H participation constraint), whereas pa (the price of the slow version, with speed
a) equals 100 a (type L participation constraint). Finally, the type H incentive constraint
implies that type H gets zero surplus from buying a slow printer:
800 (a − .5) − pa = 0
which implies a = 4/7, which in turn implies and pa = 100 × (4/7) ≈ 57.14.
Profits under one version are $400m. Under two versions, profits are $457.14m, an
increase of $57.14m. Basically, the increase corresponds to student sales.
Note for aficionados: the fact that three of the four constraints are binding need not
always be the case (see the baby iMac example in the main text). Suppose for example
that students have no use for printers that are slower than a = .5 but otherwise are not
interested in speed (that is, they are willing to pay 50 for any printer with a ≥ .5). Then it
is optimal to offer a printer with a = .5 and none of the incentive constraints are binding.
Suppose instead that students have no use for printers that are slower than a = .6 but
otherwise are not interested in speed (that is, they are willing to pay 60 for any printer
with a ≥ .6). Then it is optimal to offer a printer with a = .6 and the type H participation
constraint is not binding, for the optimal p1 is then 380 = 400 − 80 + 60.
6.14. Multiple two-part tariffs. Consider the model of non-linear pricing introduced
in Section 6.2. Suppose there are two types of consumers, in equal number: type 1 have
demand D1 (p) = 1 − p, and type 2 have demand D2 (p) = 2 (1 − p). Marginal cost is zero.
(a) Show that if the seller is precluded from using non-linear pricing, then
the optimal price is p = 12 and profit (per consumer) 83 .
Answer: Total demand from a consumer of Type 1 and a consumer of Type 2 is given by
D(p) = D1 (p) + D2 (p) = 1 − p + 2 (1 − p) = 3 (1 − p). The monopolist’s problem is:
max 3 p (1 − p)
p
The solution to this problem is given by the first order condition, 1 − 2 p = 0, so that we
get p = 21 and the profit is 43 . Social welfare is given by the sum of the firm’s profit and the
consumer surplus and is equal to: Wa = 3 p (1 − p) + (1 − p)2 = 1.
(b) Show that if the seller must set a single two-part tariff, then the
9
9
and p = 41 , for a profit of 16
.
optimal values are f = 32
Answer: In this case the monopolist’s demand is the same. However, the monopolist now
can also charge a fixed fee, f , from both consumers. The problem becomes:
max 3 p (1 − p) + 2 f
p
s.t.
(1 − p)2
≥f
2
where the constraint comes from the fact that the consumer of Type 1 must have a positive
surplus, otherwise it will not buy. Once the constraint for the Type 1 consumer is satisfied,
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the constraint for Type 2 is also satisfied; we can therefore ignore it. The monopolist is
better off when it extracts as much surplus as possible from consumers. Thus, its optimal
policy requires that the fixed fee be equal to the Type 1 consumer surplus, that is, the
constraint should be binding. The monopolist’s problem becomes:
max 3 p (1 − p) + (1 − p)2
p
and the solution is given by the first order condition, 3 − 6 p − 2 + 2 p = 0, so that we get
2
9
p = 41 , f = 32
and the profit is 98 . Welfare is given by Wb = 3 p (1−p)+(1−p)2 +0+ (1−p)
=
2
45
>
W
.
a
32
(c) Show that if the seller can set multiple two-part tariffs, then the optimal values are f1 = 81 , p1 = 12 , f2 = 87 , p2 = 0, for a profit of
5
8.
Answer: In this case the monopolist’s problem is more complex:
max p1 (1 − p1 ) + f1 + 2 p2 (1 − p2 ) + f2
p1 ,p2
s.t.
CS 1 (p1 ) ≥ f1
(PC1)
CS 2 (p2 ) ≥ f2
(PC2)
CS 1 (p2 ) − f2 ≤ CS 1 (p1 ) − f1
(IC1)
CS 2 (p1 ) − f1 ≤ CS 2 (p2 ) − f2
(IC2)
where the participation constraints assure that the consumer will prefer to consume and the
incentive compatibility constraints assure that each plan is chosen by the targeted type of
consumers, that is, Type 1 consumers will prefer plan 1 to plan 2 while Type 2 consumers
will prefer plan 2 to plan 1.
One can show that PC1 and IC2 are binding, while PC2 and IC1 are not. Suppose that
PC1 and IC2 are satisfied. We have:
CS 2 (p2 ) − f2 ≥ CS 2 (p1 ) − f1 ≥ CS 2 (p1 ) − CS 1 (p1 ) ≥ 0
where the last inequality comes from the fact that, at any price, the surplus of the Type 2
consumers is higher, since they consume more. Therefore, PC2 is automatically satisfied.
PC2 will not be binding unless consumers of Type 1 are not served. To see this, suppose
PC2 is binding. From IC2 and PC1 we get
CS 2 (p1 ) ≤ f1 ≤ CS 1 (p1 )
which is obviously impossible. In contrast, PC1 must be binding: if PC1 and PC2 would
not bind the monopolist could increase its profits by increasing both f1 and f2 with the
same small amount without violating the ICs. If IC2 is not binding the monopolist could
increase f2 with a small amount and keep all other constraints satisfied, while increasing
her profits.
2
1)
and
Therefore, we have f1 = CS 1 (p1 ) = (1−p
2
f2 = CS 2 (p2 ) − CS 2 (p1 ) + f1 = (1 − p2 )2 −
12
(1 − p1 )2
2
The monopolist’s problem becomes:
max
p1 ,p2
p1 (1 − p1 ) + 2 p2 (1 − p2 ) + (1 − p2 )2
The first order conditions are: 1 − 2 p1 = 0 and 2 − 4 p2 − 2 + 2 p2 = 0, and the solutions
are: p1 = 21 , f1 = 18 , p2 = 0, f2 = 87 , and profit is equal to 54 . Social welfare in turn is given
by
(1 − p1 )2
11
Wc = p1 (1 − p1 ) +
+ 2 p2 (1 − p2 ) + (1 − p2 )2 =
< Wb
2
8
(d) Show that, like profits, total surplus increases from (a) to (b) and
from (b) to (c).
Answer: The proof is already contained in the previous points.
6.15. Sales. Many retail stores set lower-than-usual prices during a fraction of the time
(sale). One interpretation of this practice is that it allows for price discrimination between
patient and impatient buyers.
Suppose that each buyer wants to purchase one unit per period. Each period is divided
into two subperiods, the first and the second part of the period. Suppose there are two
types of buyers, i = 1, 2. Each type of buyer is subdivided according to the part of the
period they would ideally like to make their purchase. One half the buyers would prefer to
purchase during the first part of the period, one half during the second part. A buyer of
type i is willing to pay v i for a purchase during his or her preferred part of the period; and
v i for a purchase at another time.
Buyers of type 1, which constitute a fraction α of the population, are high-valuation,
impatient buyers; that is, v h is very high and v h very low. High valuation implies that v h
is very high; impatience implies that v h is very low: buyers of type 1 are not willing to
buy at any time other than their preferred time. Buyers of type 2, by contrast, are very
patient: v l ≈ v l . Assume that α is relatively low; specifically, α < v l /v h . To summarize:
v h > v l ≈ v l > α v h > v h ≈ 0.
(a) Show that, under a constant-price strategy, the seller optimally sets
p = vl .
Answer: If p > v 1 , then there is no sale. If v 2 < p < v 1 , then the only purchasers are the
impatient, high-valuation buyers, and the seller’s profit is π = α p, with maximum value
α v 1 . If p < v 2 , then all buyers make a purchase and the seller’s profit is π = p, with
maximum value v 2 . Since α v 1 < v 2 , it is clear that the best constant-price strategy is to
set p = v 2 .
(b) Determine firm profits when it sets prices p = v h and p = v l in the
first and second parts of the period, respectively. Show that profits
are greater under the “sales” strategy.
Answer: Under this strategy the seller’s profit is
π=
1
1
1
α, v 1 + (1 − α) v 2 + α v 2 = v 2 + α (v 1 − v 2 ) > v 2
2
2
2
where the last inequality is based on the fact that v 2 ≈ v 2 .
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6.16. Optimal bidding strategy. Consider a first-price auction with two bidders. Suppose
Bidder 1 believes that Bidder 2’s bid is some number between 0 and 21 , with all numbers
equally likely. Show that Bidder 1’s optimal bid is given by b1 = v1 /2.
Answer: By bidding b1 , Bidder 1’s expected profit is given by
π1 = (v1 − b1 ) P(b1 > b2 )
where vi and bi is bidder i’s value and bid, respectively.
The higher b1 , the lower the net gain from winning the auction, v1 −b1 ; but the higher the
probability of winning the auction, P(b1 > b2 ). Specifically, if b1 = 0, then P(b1 > b2 ) = 0;
whereas, if b1 = 21 , then P(b1 > b2 ) = 1. More generally, for b1 ∈ [0, 21 ],
P(b1 > b2 ) = 2 b1
It follows that
π1 = (v1 − b1 ) 2 b1
Taking the derivative with respect to b1 and equating to zero, we get the first-order condition
for profit maximization (see Section 3.2):
2 ( − b1 + v 1 − b1 ) = 0
which leads to the desired expression.
This exercise begs the question of what should we expect Bidder 1’s belief about Bidder
2’s bid to be. This gets into the realm of game theory, which I introduce in Chapter 7. See
specifically Exercise 7.11.
Applied exercises
6.17. Selling mechanisms field experiment. Set up a seller identity in a online trading
site (eBay, Taobao, Alibaba, etc). Obtain a series of objects of uniform quality (e.g., sports
trading cards, USB memory drives, etc). Sell different units of the object using different
selling mechanisms: fixed price, auction, negotiation. Compare the price obtained with
each method and discuss the extent to which the differences can be explained by economic
theory.
14