MGEC41H3
Industrial Organization
Fall 2021
Problem set 3 – ANSWER KEY
Question 1 (30 points)
Consider a 2-firm (firm i and j) differentiated product environment where firms choose the price of
their good but not the product type. The product is differentiated in one dimension; in particular,
the good can take on types ranging from 0 to 1. Consumers derive an inherent utility from the
good v̄ but face an adjustment cost t = 1 when deviating from their preferred type. Suppose both
firms only face marginal costs such that ci = α and cj = 23 α and a constant fixed costs F . (Hint:
This problem set up is the same as the one in the Hotelling model.)
1. (15 points) Suppose we force firm i to locate at 0 and firm j to locate at 1. What is the
optimal price that each firm i and j will charge and what is the corresponding profit for each
firm?
Answer: Note that this problem set up is the same as the one in the Hotelling line. This
means that in order to solve for the optimal prices for each company, we must first determine
their demand. This is given by the location of the indifferent consumer, xI , which must
satisfy:
(2 points)
v̄ − pi − txI = v̄ − pj − t(1 − xI )
Since t = 1,
v̄ − pi − xI = v̄ − pj − (1 − xI )
pj − pi + 1
xI =
2
The profit maximization for firm i is:
max(pi − ci )xI − F = max(pi − α) ×
(1.5 points)
pi
pi
pj − pi + 1
2
−F
And for firm j:
(1.5 points)
pi − pj + 1
2
max(pj − cj )(1 − xI ) − F = max pj − α ×
−F
pj
pj
3
2
(2 points) The FOCs for each firm are:
[i] :
[j] :
pi − α pj − pi + 1
pj + α + 1
+
= 0 ⇒ pi =
2
2
2
pj − (2/3)α pi − pj + 1
pi + (2/3)α + 1
−
+
= 0 ⇒ pj =
2
2
2
−
1
(2 points, 1 each best response) Plugging the best responses into each other, we get:
pi =
pj + α + 1
=
2
pi +(2/3)α+1
2
+α+1
2
⇒ 2pi =
pi + (2/3)α + 1
+α+1
2
2
4pi = pi + 1 + α + 2 + 2α ⇒
3
9 + 8α
(1.5 point)
pi =
9
(1.5 point) And pj is
pj =
9+8α
+ 1 + (2/3)α
pi + 1 + (2/3)α
= 9
2
2
18 + 14α
9 + 7α
pj =
=
18
9
(3 points, 1.5 for each firm) The profits for each firm are then:
9 + 8α
−α/9 + 1
(9 − α)2
Firm i:
−α ×
−F =
−F
9
2
162
9 + 7α 2
α/9 + 1
(9 + α)2
Firm j:
− α ×
−F =
−F
9
3
2
162
2. (15 points) Suppose firm i is considering entry into a differentiated products market where
1
there are N firms each with a cost of ci = 12 and a fixed cost F = 16
. The consumers in the
market are the same as before. Set up the profit maximization problem for a firm. Where is
the indifferent consumer located? How many firms will enter? (Hint: This effectively a Salop
circular city model).
Answer: To set up the profit maximization problem for a firm i, we must first determine
their demand. It is given by the location of the indifferent consumer between firm i and the
closest rivals, which must satisfy:
1
(2 points)
pi + txI = pj + t
− xI
N
where firm j is one of firm i’s competitor. Since t = 1,
pj − pi +
1
p i + xI = p j +
− xI ⇒ xI =
N
2
1
N
In the Salop’s circular city model, each firm i has rivals j and k on both sides of the circle.
Therefore, firm i’s profits are:
pj − pi + N1
pk − pi + N1
+
− 1/16
(3 points)
max(pi − ci )2xI − F = max(pi − 1/2) ×
2
2
(2 points) The FOC for firm i is
−(pi − 1/2) +
pj − pi +
2
1
N
+
pk − pi +
2
1
N
=0
(2 points) In a symmetric equilibrium, pi = pj = pk = p and hence p =
of the indifference consumers is
(3 points)
xI =
2
pj − pi +
2
1
N
=
1
2N
1
2
+
1
N.
The location
The number of firms in equilibrium is such that profits are zero:
1
− 1/16 = 0 ⇒
N2
√
N = 16 = 4
(1.5 point)
(1.5 point)
Question 2 (30 points)
A cellphone carrier wants to increase her profits by targeting different consumer segments. The
company was able to identify two types of consumers: budget and heavy-use consumers. Budget
consumers value low-cost plans and their inverse demand curve is PL = 200 − qL . Heavy-use
consumers, on the other hand, are willing to pay more for the service and their demand curve for
PH = 320 − qH . For simplicity, assume the cellphone carrier is the only service provider in both
markets.
(a) (15 points) Suppose the firm collects information on each consumer and knows which type
of buyer they are (budget or heavy use). Moreover, the firm has perfect information on
their willingness to pay. Suppose the firm faces the same costs for each market segment,
C(qL ) = C(qH ) = C(q) = 23 q 2 + 10. What is the firm’s total profits? Hint: Firm profits are
equal to the producer surplus minus fixed costs.
Answer: From the information given (collections of information on each consumer and perfect
information on their willingness to pay), we know this is a first degree price discrimination
problem.
(1 point) For the budget market, we know the firm produces until PL = M CL , and therefore
we must have:
200 − qL = PL = M CL = 3qL ⇒ qL = 50
(2 points)
PLc = 200 − 50 = 150
Similarly for the heavy-users market, (1 point) PH = M CH , and we get
320 − qH = PH = M CH = 3qH ⇒ qH = 80
(2 points)
c
PH
= 320 − 80 = 240
Students can calculate profits for each market in two ways: using quantities and prices in
each market directly or calculating total surpluses. For either way, 4 points for individual
profits in each market. We know that in the first degree price discrimination, the monopolist
extracts the entire surplus in the market. In the budget market, the monopolist’s surplus is
M SL =
(200 − 150) × 50
2
|
{z
}
150 × 50
2 }
| {z
+
CSL in perfect competition
= 5000
PS in perfect competition
Since profits equal to producer surplus minus fixed costs, monopolist’s profits from the budget
market are πL = 5000 − 10 = 4990.
In the heavy use market, the monopolist’s surplus is
M SH =
(320 − 240) × 80
2
|
{z
}
+
CSH in perfect competition
3
240 × 80
2 }
| {z
PS in perfect competition
= 12800
Since profits equal to producer surplus minus fixed costs, monopolist’s profits from the heavy
use market are πH = 12800 − 10 = 12790.
(2 points for total profits)Thus, the total profits from both markets are π = πL + πH =
4990 + 12790 = 17780
(b) (15 points) Design 2 two-part tariffs, one for each market, that allow the cellphone carrier to
obtain the profits in (a). Assume that the firm is able to keep the markets separate ( i.e.,
there is no incentive compatibility problem) and there are 100 identical consumers of each
type.
Answer: (6 points, 3 for CS in each market) Since CSL = 1250 and CSH = 3200 and there
are 100 identical consumers in each market, we can set the two-part tariff for each market as
(2.5 points)
(2.5 points)
CSL
+ PLc qL = 12.50 + 150qL
100
CSH
c
+ PH
TH (qH ) =
qH = 32 + 240qL
100
TL (qL ) =
(4 points) Note that the monopolist’s total profits from these two-part tariffs are:
3
3
2
2
12.5 × 100 + 150 × 50 − (50) − 10 + 32 × 100 + 240 × 80 − (80) − 10 =
2
2
4990 + 12790 = 17780
Students do not need to show consumer surplus again if they calculated it in part (a) for each
market. However, they must show it in part (b) if they had not done so. Students must also
show that those two-part tariffs yield the combined profits obtained in (a).
Question 3 (15 points)
Consider the same monopoly markets as in Question 2. Suppose now that the firm only knows
each consumer’s market type and no longer knows the consumers’ individual demands. Moreover,
suppose the costs are non-separable by market, i.e., there is a cost structure that accounts for costs
in both markets. This cost structure is C(Q) = 12 Q2 + 2Q. What is the profit and consumer surplus
in each market?
Answer: If the firm only knows the market type, then this is a 3rd degree price discrimination
problem. The monopolist’s will charge different prices for the budget and heavy-use markets, but
all consumers in the same market will pay the same, uniform price. The monopolist’s total profits
are then:
(3 points)
1
max (200 − qL )qL + (320 − qH )qH − (qL + qH )2 − 2(qL + qH )
2
qL ,qH
(2 points, 1 each) The FOCs are:
198 − qH
3
318 − qL
=
3
[qL ] :
200 − 2qL − (qH + qL ) − 2 = 0 ⇒ qL =
[qH ] :
320 − 2qH − (qH + qL ) − 2 = 0 ⇒ qH
4
(2 points) Plugging the best responses into each other, we get:
318 − qL
3
8qL = 276 ⇒ qL = 34.5
318 − 34.5
qH =
= 94.5
3
3qL = 198 −
(3 points) The monopolist’s profits are then:
1
π = (200 − 34.5) × 34.5 + (320 − 94.5) × 94.5 − (34.5 + 94.5)2 − 2(34.5 + 94.5) = 18441
2
(5 points, 2.5 each) The consumer surplus in each market is:
[200 − (200 − 34.5)] × 34.5
= 595.125
2
[320 − (320 − 94.5)] × 94.5
=
= 4465.125
2
CSL =
CSH
Question 4 (25 points)
Suppose that there is a firm selling chips and salsa. Moreover, there are three consumers in this
marketplace. Consumer 1 values chips and salsa at $10 and $15, respectively, consumer 2 values
them at $15 each, and consumer 3 at $14 and $17, respectively.
(a) (8 points) Suppose the firm could charge individual prices by selling chips at $15 and salsa at
$16, a bundle for $30 or mixed bundling. Which is more profitable for the firm, assuming no
costs for the goods?
Answer: The firm’s profits from each type of strategy are:
- Individual pricing: Only consumer 2 buys chips and only consumer 3 buys salsa. Profits
are then π = 15 + 16 = 31;
- Pure bundling: Consumer 2 and consumer 3 buy the bundle: π = 30 + 30 = 60;
- Mixed bundling: Both consumer 2 and consumer 3 are indifferent between buying the
bundle and buying either chips or salsa individually, respectively. In either case, π = 60.
Grading: 2 point for individual pricing profits, 3 points for pure bundling profits and 3 points
for mixed bundling profits.
(b) (5 points) Suppose a resale market was introduced. Would there be any effects of the resale
market on the firm’s profit? Explain.
Answer: If there was a resale market, it would push the price of chips down to $14 and the
price of salsa down to $15. This would hurt the firm’s bundling prospects since people could
just buy the good they prefer at a lower price.
Grading: 3 points for saying that the price of each good would be lower in the resale market.
1 point for each price (need to say 14 and 15)
2
(c) (12 points) Now suppose the firm has a cost structure for chips of C(qC ) = 4qC
and a cost
2
structure for salsa of C(qS ) = 2qS . Does this change which option is more profitable? If so,
why?
Answer:The firm’s profits from each type of strategy are:
5
- Individual Pricing: Only consumer 2 buys chips and only consumer 3 buys salsa ⇒ π =
31 − 4 − 2 = 25
- Pure Bundling: Consumer 2 and consumer 3 buy the bundle ⇒ π = 60 − 4(2)2 − 2(2)2 =
36
- Mixed Bundling: Both consumer 2 and consumer 3 are indifferent between buying the
bundle and buying either chips or salsa individually, respectively. Assume that they
purchase the bundle. Profits are then π = 60 − 4(2)2 − 2(2)2 = 36
Grading: 3.5 points for individual pricing, 3.5 points for pure bundling and 5 points for mixed
bundling.
6