Trinity College Dublin Lecturer: Martin P aredes
Department of Economics Hilary Term 2007
Industrial Organization
Answer Key to A ssignm ent # 3
1. In a linear Hotelling town ther e are 100 potential costumers that are uniformly distributed on a unit
mile. Each consumer has a willingness to pay for pizza of $30, andwouldbuyonlyonepizzaper
week. Itcostaresident$10 per mile to travel. Assume two pizza stores are considering opening
shops on opposite ends of the street. After opening, each store would have a marginal cost of $5 per
pizza, an there is no fixed cost for opening a store.
(a) What are the equilibrium prices each store will charge for pizza? What would their profits be?
Each pizza store will charge p = t + c =10+5=15, where t is the transportation cost
and c is the marginal cost. At these prices, the stores would split the market. All customers
would purchase because the farthest any customer has to travel is 0.5 miles, incurring a $5
transportation cost. So, the highest effective price is $20 (because it is p + tx = 15 + 10(0.5)),
w ell below the consumers willingness to pay. Thus q
1
= q
2
=50. So the profitperweekforeach
firm would be π
1
= π
2
=(15− 5) ∗ 50 = $500.
(b) Intuitively, would both stores be happy with their price and location choice, or would one of
them want to change their price/location? In fact, what would happen to the firms’ prices if
they were located right next to each other?
Both firms have an incentive to change their location. Keeping prices fixed, if one firm moves
towards the other, it would expand its market share (simply draw the two firms effective prices
before and aft er one of the firms moves to see this). Thus, when both firms change $15 for their
pizzas, neither one is happy with their location. However, the closer and closer firms locate,
the less their products differentiated, there will be fierce price competition. In the limit, if they
are located right next to each other, prices will go do wn to marginal cost (p = c).
(c) Suppose the firms decide to merge (i.e., they become a monopolist). What would the firms
incentive to merge be (assume that the firms would serve the entire market after the merger)?
We need to compare the profits before and after the merger. Before the merger (part (a)),
both firms get π
1
+ π
2
=2∗ 500 = $1000. Let’s calculate now the profits post merger. First,
notice that, if the firms merge, they no longer have to w orry about price competition. Since the
maximum a consumer is willing to pay is $30 and the maximum transportation cost a consumer
will pay is $5 (this is the transportation cost paid by the marginal consumer, located at the
z =
1
2
), then the merged firm can charge $25 and still sell to all the consumers. In this case the
profits per week are Π
M
=(25− 5) ∗ 100 = $2000
Therefore, before the merger both firms together got π
1
+ π
2
= $1000, and after the merger the
new firm gets Π
M
= $2000. The difference ($1000) is the total incentive to merge.
2. Cabral, problem 10.1.: First-time subscribers to the Economist pay a lower rate than repeat sub-
scribers. Is this price discrimination? Of what type?
This is an example of third-degree price discrimination. The market is segmen ted in to new subscribers
and repeat subscribers. New subscribers, know the product less well and are thus likely to be more
price sensistive. Moreover, the fact that they have not subscribed in the past indicates that they are
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likely to be willing to pay less than current subscribers. It is therefore optimal to set a lower price
for new subscribers.
3. Cabral, problem 10.3.: Cement in Belgium is sold at a uniform delivered price thro ughout 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. Are these cases of price discrimination?
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 = p
i
+ t
i
.Since
locations are different, transportation costs are different, thus, each consumer is charged a price p
i
that depends on his or her location. This is a clear example of geographic price discrimination.
4. Cabral, problem 10.4.: A restaurant in London has recently removed prices from its menu: each
consumer is asked to pay what he or she thinks the meal was worth. Is this a case of price discrimi-
nation?
It is likely that each consumer will pay a price that reflects his or her willingness to pay. In that
sense, this is a situation of close to perfect price discrimination.
5. Cabral, problem 10.8.: Coca-Cola recently announced that it is developing a ”smart” vending ma-
chine. Such machines are able to change prices according to the outside temperature. 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 − 2p,whereQ 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 − 2p. There is an equal number days with ”High” and ”Low” temperature. The marginal
cost of a c an of Coke is 20 cents.
(a) Suppose that Co ca-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?
On a Hot day, Q = 280 −2p,orp = 140−
1
2
Q. 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 −2p,or
p =80−
1
2
Q. Marginal revenue is MR =80− Q. Equating to marginal cost (20) and solving,
we get Q
∗
=60and 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?
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 =0.5(280−2p)+0.5(160−2p) = 220−2p.Solvingforp gives p = 110−
1
2
Q.
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?
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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 da y for a ”smart” vending
machine.
6. A monopolist faces the inverse demand curve P = z (36 − Q),whereP is price, Q is total output
and z is the quality og product sold, which can take on only two values. The monopolist can choose to
market a low-quality product for which z =1. Alternatively, the monop olist can choose to market a
high-quality product for which z =2. Marginal cost is independent of quality and is constant at zero.
Fixed, cost, however, depends on the product design and increases with the quality chose. Specifically,
fixed cost is equal to 65z
2
.
(a) Find the monopolist’s profits if it maximises profits and chooses a low-quality design.
For z =1, profits for this firms are given by:
Π = PQ− VC(Q) − FC
=(1)(36− Q)(Q) − 0 − 65(1)
2
=36Q − Q
2
− 65
The first order condition yields:
dΠ
dQ
=36− 2Q =0
→ Q =18
→ P =36− (18) = 18
Profitisthengiveby:
Π = PQ− VC(Q) − FC
= (18)(18) − 65
= 324 − 65 = 259
(b) Find the monopolist’s profits if it maximises profits and chooses a high-quality design.
For z =2, profits for this firms are given by:
Π = PQ− VC(Q) − FC
=(2)(36− Q)(Q) − 0 − 65(2)
2
=72Q − 2Q
2
− 260
The first order condition yields:
dΠ
dQ
=72− 4Q =0
→ Q =18
→ P =(2)(36− (18)) = 36
3
Profitisthengiveby:
Π = PQ− VC(Q) − FC
= (36)(18) − 260
= 648 − 260 = 388
(c) Comparing your answers to (a) and (b), what quality choice should the monopolist make?
The monopolist will go with high quality.
7. General Foods is a monopolist and knows that its market for Bran Flakes contains two types of
consumers. Type A consumers have indirect utility functions V
A
=20z, whiletypeBconsumers
have indirect utility functions V
B
=10z. In each case, z is a measure of product quality, which can
be chosen from the interval [1, 2]. Ther e are N consumers in the market, of which General Foods
knows that a fraction λ is of type A, and the remainder from typ e B. For simplicity, assume that all
costs are zero.
(a) Suppose that General Foods can tell the differentconsumertypesapartandsocanchargethem
different pric es for the same quality of breakfast cereal. What is the pr ofit-maximising strategy
for General Foods?
Since both types have increasing willingness to pay as quality rises, the firm will sell maximum
quality z
A
= z
B
=2to each type. Price to type A consumers is 40 while the price to type B
consumers is 20.
(b) Suppose now that General Foods does not know what type of consumer is which. Show how its
profit-maximising strategy is determined by λ.
From the participation constraint of type B consumers, we get p
B
=10z
B
. From the incentive
compatibility constraint of type A consumers, we get p
A
=20z
A
− 10z
B
. Then the firm’s total
profitisΠ = N [λp
A
+(1− λ) p
B
]=N [20λz
A
+10(1− 2λ) z
B
] . It is easy to see that the firm
will set z
A
as high as possible, so z
A
=2. The quality for type B consumers will depend on the
value of λ.
• If λ ≤
1
2
(so that there more low-type consumers), then profitisincreasinginz
B
. As such,
the firm will set z
B
=2. In other words, General Foods will offer a unique product, of the
highest quality, at price p =20
• If λ ≥
1
2
(so that there more high-type consumers), then profitisdecreasinginz
B
. As such,
the firm would set z
b
=1.Atthispointthefirm even has to decide whether to offer a
low-qualit y good. If the firm only sells the high-quality product, it can set a price as high
as p =40, getting profits Π =40Nλ.Ifthefirm sells both products, then it will charge
apricep
B
=10for the low-quality good, but the highest possible price they can charge
for the high quality good is p
A
=30(in order to satisfy type A’s incentive compatibility
constraint). As a result, profits will be Π = N [30λ +10(1− λ)] = (10 + 20λ) N. Hence,
they will only offer the high quality good.
8. In a two-period economy, one consumer wishes to buy a TV set in period 1. The copnsumer lives for
two periods, and is willing to pay a maximum price of 100 euros per period of TV usage. In period
2, two consumers (who live in period 2 only) are born. each of the newly-born consumers is willing
to pay a maximum of 50 euros for using a TV in period 2. Suppose that in this market there is only
one firm producing TV sets, that TV sets are durable, and that production is costless.
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(a) Calculate the prices the monopoly charges for TV sets in perios 1 and 2.
We solve for the monopolys profit maximising prices starting from the second period. The
second period outcome may depend on two cases:
• First-period consumer does not buy in period 1: Clearly, in this case, the second
period profit maximising price is P
2
=50, yielding a profit level of Π
2
=3× 50 = 150.
• First-period consumer buys in period 1: In this case the second period profitmax-
imising price is again P
2
=50, yielding a profitlevelofΠ
2
=2× 50 = 100.
Altogether, the second-period price is independent of the action of the first period buyer in the
first period. Therefore, the maxim um price the monopoly can c harge the first-period buyer in
the first period is P
1
= 150.
(b) An swer the previous question assuming that in the first period a consumer who lives two periods
is willing to pay no more than 20 euros per period for TV usage.
The second period outcome may depend on two cases:
• First-period consumer does not buy in period 1: In this case, the monopoly has two
choices: (i) charging P
2
=20, and sell to all three consumers, thereby earning a second
period profitofΠ
2
=3× 20 = 60; or, (ii) charging P
2
=50, and selling only to the second
period consumers, thereby earning a second period profitofΠ
2
=2× 50 = 100.
• First-period consumer buys in period 1: In this case the second period profitmax-
imising price is again P
2
=50, yielding a profitlevelofΠ
2
=2× 50 = 100.
Altogether, the second period price is independent of the actions of the first period buyer.
Now, in order to attract the first-period buy er to purchase in period 1, the monopoly should
set P
1
=40, thereby extracting all surplus from all consumers.
9. Cabral, problem 15.2.: In less than one year after the deregulation of the German telecommunications
market at the start of 1998, domestic long-distance rates have fallen by more than 70%. Deutsche
Telekom, the former monopolist, accompanied some of these rate drops by increases in monthly fees
and local calls. MobilCom, one of the main competitors, fears it may be unable to match the price
reductions. Following the announcement of a price reduction by Deutche Telekom at the end of 1998,
shares of MobilCom fell by 7%. Two other competitors, O.tel.o and Mannesmann Arcor, said they
would match the price cuts. VIAG Interkom, however, accused Telekom of ”competition-distorting
behavior,” claiming the company is exploiting its (still remaining) monopoly power in the local market
to subsidize its long-distance business. Is this a case of predatory pricing? Present arguments in favor
and against such assertion.
One could indeed argue that this is a case of predatory pricing. If Deutsche Telekon has monopoly
in local markets, it likely has financial resouces strong enough to afford losing money in the long
distance market by pricing below marginal cost. However, since there are two other competitors that
matched Deutsche Telekom’s prices, one can argue that there exists technology with marginal cost
less than the low-price charged. Evidently, other explanations can also invoked, namely low-cost
signaling and reputation for toughness.
10. Cabral, problem 15.3.: ”The combined output of two merging firms decreases as a result of the
merger.” True or false?
If the merger implies little or no cost efficiencies (namely at the level of marginal cost), we w ould
expect the combined output of the merging firms to decline. If however the merger reduces the
5
marginal cost of the combined firm significantly, then it is possible that the combined output increases
as a result of the merger.
11. Kikkoman is the dominant supplier in the market of say sauce, but it faces continuous entry threats.
Suppose that Kikkoman incurs a cost C(q
1
)=6q
1
. Kikkoman faces potential entry by Red Zen, which
produces a perfect substitute for Kikkoman’s product. However, Red Zen’s production costs are given
by C(q
2
) = 100 + 12q
2
, so that MC
2
=12. The demand for soy sauce is given by D(P ) = 120 − P.
(a) Suppose initially that the incumbent, Kikkoman, can credibly commit to produce some output,
after which Red Zen will choose its own output
i. Find Red Zen’s best response function
Red Zen’s profit function is π
2
= P · q
2
−C(q
2
) = (120 −q
1
−q
2
)q
2
−100 −12q
2
. Marginal
revenue is MR
2
= 120 − q
1
− 2q
2
, while marginal cost is MC
2
=12. Since MR
2
= MC
2
,
we obtain the entrant’s best response function:
q
∗
2
= BR(q
1
)=54−
1
2
q
1
.
ii. If Kikkoman accommodates entry, find the incumbent’s profit-maximizing quantity and its
resulting profits.
Kikkoman will act as a Stackelberg leader and will ch oose its quantity knowing what Red
Zen’s best response will be. Thus, Kikkoman’s effective demand is P = 120 − q
1
− q
∗
2
=
120 − q
1
−
¡
54 −
1
2
q
1
¢
=66−
1
2
q
1
, so marginal revenue is MR
1
=66− q
1
, while marginal
cost is MC
1
=6. This gives q
1
=60,makingprofits π
1
= 1800
(b) Alternatively, the incumbent can attempt to deter entry by engaging in “limit pricing”. In fact,
it would set a quantity so that the entrant will not be able to make a profit.
i. Show that q
1
=88is the quantity that results in the limit price, and find that price and the
incumbent’s associated profit.
Since Kikkoman knows Red Zen’s best response function, it can find the quantity that will
make Red Zen achieve zero profits. Red Zen’s profits are π
2
= (120−q
1
−q
2
)q
2
−100−12q
2
.
Substituing q
2
=54−
1
2
q
1
into π
2
, we get:
π
2
=
µ
108 − q
1
−
µ
54 −
1
2
q
1
¶¶µ
54 −
1
2
q
1
¶
− 100 = 0
This reduces to
¡
54 −
1
2
q
1
¢
2
= 100 ⇒ q
1
=88.
When q
1
=88, Red Zen is indifferent between entry and non-entry, so q
2
=0. Then,
P = 120 − 88 = 32 and π
1
=(32− 6)88 = 2288.
ii. Will the incumbent prefer to deter entry or accommodate entry?. Explain.
The incumbent will prefer to deter entry. By limit pricing, Kikkoman receives higher profits
with detering entry (π
1
= 2288) than by acommodating entry (π
1
= 1800).
12. Consider a Cournot industry c omposed of 3 firms, facing a demand D(P ) = 150 − P. Initially,
the three firms are identical and have the same marginal cost $ 30. As such, the Cournot-Nash
equilibrium is for each firm to produce 30 units at a price of $60. Suppose that two of those firms
decide to merge, and that, as a result, the merged firm will realize a savings in its variable cost. In
other words, post-merger marginal cost would be equal to $ c<30.
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(a) Calculate the post-merger equilibrium output level of the merged firm and the non-merged firm,
and compute the corresp onding price and profits.
Now we have an asymmetric Cournot duopoly. Without loss of generality, suppose firm 1 & 2
merge. The best response function for the merged firm is:
q
m
=
150 − c
2
−
q
3
2
The other firm’s best response function is:
q
3
=
150 − 30
2
−
q
m
2
=60−
q
m
2
Solving we get:
q
m
=
150 − c
2
−
1
2
³
60 −
q
m
2
´
⇒ q
m
=60−
2
3
c
and
q
3
=30+
1
3
c
Then, P = 150 −
¡
60 −
2
3
c
¢
−
¡
30 +
1
3
c
¢
=60+
1
3
c.Profits for the merged firm are π
m
=
¡
60 +
1
3
c − c
¢¡
60 −
2
3
c
¢
=
¡
60 −
2
3
c
¢
2
= 3600 − 80c +
4
9
c
2
and profits for the nonmerged firm
are π
3
=
¡
60 +
1
3
c − 30
¢¡
30 +
1
3
c
¢
=
¡
30 +
1
3
c
¢
2
= 900 + 20c +
1
9
c
2
.
(b) Compare your results in p art (a) with the (pre-merger) Cournot-Nash equilibrium and deter-
mine:
i. How large should the savings be for the merger to be profitable?
The merger is profitable if
π
m
≥ π
1
+ π
2
.
3600 − 80c +
4
9
c
2
≥ 2(900)
1800 − 80c +
4
9
c
2
≥ 0
Solving, we get c ≤ 90 −45
√
2 ≈ 26.3. In other words, the new marginal cost has to reduce
up to aproximately c =26in order for the merger to be profitableforthemergerfirms.
ii. How large should the savings be for the merger to benefitconsumers?
The merger will benefit consumers if
P
m
≤ P
60 +
1
3
c ≤ 60
It is very easy to see that the merger will never benefitconsumers
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