Boğaziçi University
Department of Economics
Fall 2016
EC 301 ECONOMICS of INDUSTRIAL ORGANIZATION
Problem Set 1 - Answer Key
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1. For the production function f (L, K) = L + K, state and prove whether it has IRTS, DRTS or CRTS.
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Answer: Let λ > 0. Then f (λL, λK) = λL + λK < λL + λ K = λ(L + K). Thus, f (λL, λK) <
λf (L, K), that is, f has DRTS.
2. Suppose that a perfectly competitive firm has a long-run supply function LT C(Q) = kQ2 . Suppose
also that the prevailing market price is p. For what values of k and p does this firm exit the market?
Answer: Note that AC(Q) = kQ and M C(Q) = 2kQ. Then, M C(Q∗ ) = p implies 2kQ∗ = p. Thus,
kp
Q∗ = p/2k. The firm exits if AC(Q∗ ) > p, that is
> p which can never happen. Thus, there is no
2k
p > 0 and k such that this firm exits the market.
3. Suppose that the supply of the competitive fringe is given by Qf (p) = 10 + 2p. Suppose that there is
a dominant firm with a marginal cost M C = 2 and no fixed costs. The market demand function is
given by QM (p) = 100 − p.
(a) Find the profit maximizing price for the dominant firm, and its profit.
Answer: Note that QD (p) = QM (p) − Qf (p) = 100 − p − 10 − 2p = 90 − 3p. Thus the profit is
π D (p) = (p − 2)(90 − 3p). Then, the first order condition is
dπ D
dp
= 0 which implies pD = 16 and
π D = 588.
(b) Find the monopoly price and its profits (that is, suppose the firm faces the whole market demand
and there is no fringe) and compare it to the dominant firm’s profit you get above. Which market
structure is more socially desirable, dominant firm with a fringe or a monopoly? Why?
Answer: The profit is now π M (p) = (p − 2)(100 − p). Then the first order condition is
dπ M
dp
=0
implying pM = 51 and π M = 2401. Thus, π M > π D . The dominant firm with a fringe is socially
more desirable because the price is closer to the competitive price.
(c) What happens to the dominant firm’s profit when the demand is more price elastic? Why?
Answer: If the demand is more price elastic, then the profit of the dominant firm decreases. To
see this, note LD =
as |εd | increases.
sD
εfs sf +|εd |
is a measure of market power of the dominant firm and it decreases
4. In a two-period lived economy, one consumer wishes to buy a TV set in period 1. The consumer lives
for two periods, and is willing to pay a maximum price of 100TL per period of TV usage. In period
2, two new consumers join the market, so they live in period 2 only. Each of the two new consumers
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is willing to pay a maximum of 50TL for using a TV in period 2. Suppose that there is only one firm
producing TV sets, that TV sets are durable for two periods, and that the production is costless. Find
the prices the monopoly charges for TV sets in periods 1 and 2.
Answer: Solve for profit maximizing prices starting from the second period. In the 2nd period we
have the following two cases.
Case1: First period consumer does not buy in period 1. Clearly second period price should be p2 = 50
yielding π2 = 3.50 = 150.
Case2: First period consumer buys in period 1. In this case, again, p2 = 50, yielding π2 = 2.50 = 100.
Note that second period price is independent of the action of the first-period buyer in the first period.
Therefore maximum price the monopoly can charge the first-period buyer in the first period is p1 =
100 + 50 = 150, where 100 is the maximum value for the usage in the 1st period and 50 is the
value(price) of the usage of TV in the 2nd period.
5. What is the effect of used units on a durable good monopolist’s new unit price? What would happen
to this effect if the monopolist could reduce the durability of the new units? And if the monopolist
does so, what would happen to the availability of used units? Give an example of a real world practice
along these lines.
Answer: According to Swan (1980), because a good’s secondhand market price is reflected in the
good’s initial price, the availability of used goods would not reduce the durable good monopolist’s
profit. However other papers (Waldman 1996, Fundeberg and Tirole 1998) indicate that there may be
substitutability between different used units, and the availability of used units may lower the price of
the durable good monopolist. But then, the durable good monopolist might react to this by reducing
its new units’ durability, in order to decrease the substitutability between new and used units. Thus
monopoly can increase the new unit price. A closely related real-life practice is leasing instead of
selling the good, which we discussed in class in detail. (For more on this issue please see Waldman
(2003) which is the second paper listed on our reading list on the course webpage.)
6. Suppose there is a single firm, producing washing machines. The washing machines are durable for
only two periods. The marginal cost is 30 with no fixed cost. The firm does not discount. There are
100 potential buyers in the first period. These buyers live for two periods and are willing to pay a
maximum price of 100 TL for one period of watching machine usage. They have a discount factor of
δ = 1/2. In the second period, there are 300 many new washing machine buyers joining the market,
and they live in the second period only (at the end of the second period everyone dies!). 100 of these
new buyers are willing to pay 80TL and the rest are willing to pay 120 TL, for one period usage of
washing machines. Also assume that the indifferences are resolved in favor of buying in the first period
and that the consumers are strategic. Find the maximum overall profit the firm can achieve and the
optimal prices it charges each period.
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Answer: Suppose the buyers in the first period choose to wait. Then, in the second period, if the
price is 80, then 400 buyers will buy and the profit will be 400 × (80 − 30) = 20000. If the second
period price is 100, then 300 buyers will buy, and the profit is then 300 × (100 − 30) = 21000. If the
price is 120, only 200 will buy, and the profit will be 200 × (120 − 30) = 18000. Thus, if each buyer in
the first period waits for the second period to buy, then the price must be 100 in the second period.
Thus, if a buyer in the first waits to buy in the second period, then she will get zero surplus since
she will buy in the second period at price 100, leaving her zero surplus. However, if she buys in the
first period, say at price P1 , then she gets 100 + (1/2)100 − P1 = 150 − P1 . Thus, if P1 = 150, each
buyer in the first period will actually buy in the first period. Then the second period price will be
120, since 200 × (120 − 30) = 18000 is larger than 300 × (80 − 30) = 15000. Thus, overall profit will
be 100 × (150 − 30) + 15000 = 12000 + 18000 = 30000.
Alternatively, the firm could pick P1 > 150 and make the buyers in the first period wait. Then the firm
would charge 100 in the second period and get 21000 overall profit as we calculated above. Thus, since
30000 > 21000, the optimal prices are P1 = 150 and P2 = 120 with an overall total profit π = 30000.
7. Suppose there is one single firm, Zenheizer, producing headphones. The headphones are durable for
only two periods and the production is costless. Zenheizer does not discount. There are 200 many
headphone buyers in period 1. These buyers live for two periods and are willing to pay a maximum
price of 60TL per period of headphone usage. they have a discount factor of δ = 2/3. In period 2,
there are 500 many new headphone buyers join the market, so they live in period 2 only. Each of
these 500 new buyers is willing to pay 90TL for one period usage of headphones. Also assume that the
indifferences are resolved in favor of buying in the 1st period and that the consumers are strategic.
(a) Find the maximum overall profit Zenheizer can achieve.
Answer: In the 2nd period if the price is 90TL, then only the new users will buy. So the 2nd period
profit would then be 45000TL. But if the 2nd period price is 60TL, then all users would buy it, and
the profit would be 60 × 700 = 42000TL. Thus the 2nd period price would be 90TL. So, the users in
the 1st period will face a price of 90 in the 2nd period, and end up not buying, getting zero surplus. In
the 1st period, with a price P1 , a user receives 60+(2/3)60−P1 = 100−P1 if he buys in the 1st period.
So, users in the 1st period will buy it in the 1st period at any price P1 ≤ 100. Pick P1 = 100TL. Then
the overall profit of the firm is 100 × 200 + 90 × 500 = 65000TL.
(b) What if instead of 500 new buyers, only 300 new buyers join the market in the 2nd period?
Answer: In the 2nd period if the price is 90TL, then only the new users will buy. So the 2nd period
profit would then be 27000TL. But if the 2nd period price is 60TL, then all users would buy it, and
the profit would be 60 × 500 = 30000TL. Thus the 2nd period price would be 60TL if everyone buys
in the 2nd period. The users in the 1st period will face a price of 60 in the 2nd period which gives
them zero surplus if they buy in the 2nd period. If, only new users are buying in the 2nd period then
the price would be 90. Again, in the 1st period, with a price P1 , a user receives 100 − P1 if he buys
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in the 1st period. Then, again, P1 = 100TL. In the 2nd period only the new users are buying, so P2
must be 90. Then the overall profit of the firm is 100 × 200 + 90 × 300 = 47000TL.
8. Suppose there is one single firm producing refrigerators which are durable for only three periods and
the production is costless. The firm does not discount. There are only 2 buyers. No new buyers join
at any period. Both buyers live for three periods. One of the buyers has a willingness to pay of 12TL
and the other buyer has a willingness to pay of 8TL, per period of refrigerator usage. Both buyers
have a discount factor of δ = 1/2. Also assume that the indifferences between buying and waiting are
resolved in favor of buying and that the consumers are strategic. Each period the monopoly sets a
price Pt for t = 1, 2, 3 and then each buyer decides whether to buy or not (if hasn’t bought yet). Find
the optimal prices for the monopoly.
Answer: Suppose that the prices, P1 , P2 and P3 are such that,
Case1: they both buy in t = 1. Then, the optimal P1 for this case would be 14(=8+4+2) together
with large enough P2 and P3 so that they both buy at t = 1. The profit is then 14 · 2 = 28.
Case2: they both buy in t = 2. Then, the optimal P2 for this case would be 12(=8+4) together with
large enough P1 and P3 so that they both buy at t = 2. The profit is then 12 · 2 = 24.
Case3: they both buy in t = 3. Then, the optimal P3 for this case would be 8 together with large
enough P1 and P2 so that they both buy at t = 2. The profit is then 8 · 2 = 16.
Case4: high type buys at t = 1 and low type buys at t = 2. Then, the net benefit high type gets is
21 − P1 and net benefit that the low type gets is 8 + 4 − P2 = 12 − P2 (discounted to t = 2). Setting a
high enough P3 , P2 must be 12 and then P1 must solve 21 − P1 = 1/2(12 + 6 − P2 ), that is, P1 = 18.
Then, the optimal prices are P1 = 18 and P2 = 12 and a large enough P3 . In fact, we need to make
sure both types do not wait and buy at t = 3. So, we need 21 − 18 = 3 ≥ (1/4)(12 − P3 ) for the high
type, and 12 − 12 = 0 > (1/2)(8 − P3 ). Any P3 > 8 works. With these prices the profit is 30.
Case5: high type buys at t = 1 and low type buys at t = 3.Then, the net benefit high type gets is
21 − P1 and net benefit that the low type gets is 8 − P3 = 8 − P3 (discounted to t = 3). So clearly
P3 = 8. Now, we need to make sure high type prefers buying at t = 1 rather than buying at t = 3.
To do this, we need 21 − P1 = (1/4)(12 − 8). Thus, P1 = 20. Note that, for P2 > 16, no buyer buys
at t = 2 when P1 = 20 and P3 = 8. So with such prices high type buys at t = 1 and low type buys at
t = 3. With these prices the profit is 28.
Case6: high type buys at t = 2 and low type buys at t = 3. Then, the net benefit high type gets is
(12 + 6) − P2 = 18 − P2 (discounted to t = 2) and net benefit that the low type gets is 8 − P3 = 8 − P3
(discounted to t = 3). So clearly P3 = 8. Now, we need to make sure high type prefers buying at t = 2
rather than buying at t = 3. For this, we need 18 − P2 = (1/2)(12 − 8) = 2. Thus, P2 = 16. And, if
P1 > 20, then the high type buyer will buy at t = 2. With these prices the profit will be 24.
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Thus, the highest profit is when the prices are such that P1 = 18, P2 = 12 and P3 > 8 with a profit of
30.
(In the other cases, where the low type buys first, the prices will be lower than the ones above. Hence
the profit will be lower. So it is enough to consider only the cases above.)
9. Suppose there is a single firm, producing a durable good X, which is durable for only two periods.
The marginal cost is zero and there are no fixed costs. The firm does not discount, that is, the firm’s
discount factor is δf = 1. There are 3 potential buyers, buyer A, buyer B and buyer C. Buyer A’s
valuation for one period usage of good X is vA = 16, and she has a discount factor δA = 0. Buyer B’s
valuation for one period usage of good X is vB = 10, and she has a discount factor δB = 1/2. Buyer
C’s valuation for one period usage of good X is vC = 6, and she has a discount factor δC = 1/2. All
buyers live for two periods and are demanding at most one unit of good X in their entire life-time.
Assume that the indifferences between buying now and waiting are resolved in favor of buying now and
that the consumers are strategic. Suppose, at the beginning of period first period, the firm announces
prices for each period, P1 and P2 , and credibly commits to these prices. Find the maximum overall
profit the firm can achieve and the optimal prices it announces for each period.
Answer: First note that buyer A has an overall valuation for one unit of good X, if she buys it at
t = 1, is 16(1 + δA ) = 16(1 + 0) = 16. If she waits she always gets a present value 16δA = 0. Thus,
buyer A will buy in the first period for all P1 ≤ 16, and for any other price P1 she will never buy.
Thus there is no way this buyer waits for t = 2.
Also note that buyer B’s overall valuation for one unit of good X, if she buys it in t = 1, is 10(1+δB ) =
10(1 + (1/2)) = 15. And, buyer C’s overall valuation for one unit of good X, if she buys it in t = 1,
is 6(1 + δC ) = 6(1 + (1/2)) = 9. Note also that, if B chooses to wait, then C will also choose to wait
since they have same discount factors and B has a higher valuation than C.
Thus, we have five cases to consider:
case1: both B and C buy at t = 1: All buyers buy at t = 1. If P1 > 9, then buyer C waits. Thus, for
this case to be the outcome P1 = 9 and P2 must large enough. The profit of the firm then would be
π1 = 3 · 9 = 27.
case2: B buys at t = 1 and C buys at t = 2: A and B buy at t = 1 and C buys at t = 2. For this
case, the second period price must be P2 = 6, since there will be only buyer C at t = 2. Then, for
the first period price, we know for A to buy we must have P1 ≤ 16. For buyer B to be indifferent
between buying now at t = 1 and waiting to buy at t = 2, we need 15 − P1 = δB (10 − P2 ), that is,
15 − P1 = (1/2)(10 − 6) = 2. Thus, firm sets P1 = 13 and P2 = 6. With these prices the profit is
π2 = 2 · 13 + δf · 6 = 26 + 6 = 32.
case3: both B and C buy at t = 2: In the second period t = 2, there are buyers B and C available.
For both to buy we need P2 = 6. Then the profit of the firm is π3 = 16 + δf · (2 · 6) = 15 + 12 = 28.
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case4: C never buys: If prices are such that both A and B buy at t = 1 and C never buys, setting
P2 large enough (P2 > 10 for instance), with P1 = 15, both A and B buy at t = 1. At any P1 > 15, B
does not buy at all, and at any price P1 < 15 profit is not maximized within this case. Thus, P1 = 15
and P2 > 10 is optimal for this case, bringing a profit of π4 = 30.
case5: B and C never buy: Then clearly the optimal first period price is P1 = 16, bringing a profit
of only π5 = 16.
Therefore the highest profit is achieved in case2, where A and B buy at t = 1 and C buys at t = 2, at
prices P1 = 13 and P2 = 6, bringing a maximum profit of 32.
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