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Answer Key: Problem Set 2
February 8, 2016
Problem 1
See also chapter 4 in the textbook.
a. Under leasing, the monopolist chooses monopoly pricing each period. The profit of the monopolist in
each period is:
⇡t= Pt Qt
Demand in each period is:
Pt = 1000
Qt
where t = 1, 2. The first order conditions w.r.t Qt is:
1000
2Qt = 0
where t = 1, 2. Solving them gives us Q1 = Q2 = 500, the profit-maximizing rental rate P1 = P2 =
500, profit per period $250,000, and total profits $ 250, 000(1 + ).
b. A way to think about this problem is start with the second period. In the second period everything
that happened in the first period (whatever that was) can be taken as given. So we can infer the
optimal behavior of the monopolist as a function of the first period choices: solve for q2 as a function
of q1 . Once we know this optimal behavior we can go back to the first period and look at the total
profits (⇡1 + ⇡2 ) as a function of q1 for which we solve. We follow this procedure to solve the
problem:
The monopolist knows that her first-period sales are irreversible, so in the second-period, she faces
a residual demand:
1
Q2 (P2 ) = Q(P2 )
Q1 = 1000
P2
Q1
So the inverse demand function for the second period is:
P2 = 1000
Q1
Q2
The monopolist’s profit in the second period (given Q1 units sold in the first period) is:
⇡2 = Q2 (1000
Q1
Q2 )
1000 Q1
= Q2 . Price in the second period is: P2 = 10002 Q1 , and the profit
2
2
(1000 Q1 )
.In the first period, consumers who buy the durable good this
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The first order condition is:
in the second period is ⇡2 =
period can consume it for two periods. Therefore, their valuation should incorporate the benefit they get
in the second period (what if they could sell the durable good in the second period at price p2 ). They are
willing to pay:
P1 = 1000
Q1 + P2 = 1000
Q1 + (
1000 Q1
) = (1 + )(1000
2
2
Q1 )
The monopoly chooses the optimal quantity Q1 to maximize her total profit in the two periods
(remember, her profit in the second period is a function of the quantity she sells in the first period, since
she can not commit to a second-period price (or quantity) beforehand. In other words, the quantity she
sells in the second period is contingent on the quantity she sells in the first period.):
⇡ = (1 + )(1000
2
Q1 )Q1 +
This is a function of Q1 only. From the F.O.C, we have Q1 =
and P2 =
500(2+ )
.
4+
Q1 ) 2
(1000
4
2000
4+ ,
and Q2 =
500(2+ )
4+
and P1 =
500(2+ )2
4+
Notice here there is some intertemporal price discrimination. Aggregate profits are:
⇡ = (1, 000, 000 + 250, 000)(
2+ 2
)
4+
c. Suppose the monopolist can commit to P1, P2 (P1 > P2 ). The residual demand in the second period
is: Q2 = 1000 Q1 P2 , the same as before. In period one, consumers are willing to pay: P1 =
1000 Q1 + P2 . So Q1 = 1000 P1 + P2 . The monopolist chooses P1, P2 to maximize her profit
in both periods:
max P1 (1000
P1 ,P2
P 1 + P2 ) + P2 (1000
2
Q1
P2 )
Plug in Q1 = 1000
P 1 + P2 in the second term, we have:
max P1 (1000
P1 ,P2
P 1 + P2 ) + P2 (P1
P2 (1 + ))
P1
From the first order conditions with respect to P1 and P2 , we get: P1 = 500 + P2 , and P2 = (1+
).
Solving these two equations gives us the profit maximizing prices in both periods: P1 = 500(1 + ), and
P2 = 500. All sales occur in period one. The quantity sold in period one is Q1 = 500. (You can check
that quantity sold in period 2 is zero.) The monopolist’s profit is: ⇡ = 250, 000(1 + ). It is the same as
the profit she gets under leasing.
d. We know from part b. that if the monopolist produces the durable good, she gets ⇡ = ( 35 )2 ⇤
1, 250, 000 = 450, 000. If she produces the nondurable good, her profit in each period is:
⇡t = (1000
Qt
c)Qt
1000 c
2
(1000 c)2
.
2
where t = 1, 2. From the first order condition, we know the optimal quantity is Qt =
price is Pt =
1000+c
2
for both periods. When
= 1, the total profit is ⇡ = 2 ⇤
monopolist will choose to produce the nondurable good only when:
(1000 c)2
2
c 2
( 1000
)
2
=
> 450, 000,
and
The
or c < 51.
e. Under leasing, the monopolist’s profit is 500,000. Since the marginal cost c can’t be negative, the
profit she gets by producing a nondurable good at cost c will not be bigger than her leasing profit.
So she will not choose to produce the nondurable good if she can produce the durable good and
lease it (when c = 0, she is indi↵erent).
f. First, leasing does not involve intertemporal price discrimination, so for that purpose the policy (forbidding leasing) is poorly designed. When c < 51, the monopolist will produce and sell the nondurable
good. Here there is no intertemporal price discrimination, but it involves efficiency loss: less quanc
tity is provided for both periods (Qt = 1000
< 500) at a higher cost (c instead of zero). If c 51,
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the monopolist will produce the durable good. In period one, she sells 400 units at P1 = 900. Her
profit is 360,000, and the total consumer surplus from two periods’ consumption of these 400 units
is: 280,000 (the area under the demand curve times 2 minus the total payment). In the second
period, the monopolists sells 300 units at P2 = 300. Her profit is 90,000, and consumer surplus is
45,000. The total profit is 450,000, and the total consumer surplus is 325,000. The social surplus
is 775,000. Under leasing, the total social surplus is 750,000. Here forbidding leasing improves
efficiency, but it involves intertemporal price discrimination.
Problem 2
a. (D,R) is the unique pure strategy Nash Equilibrium. A possible interpretation is about two firms
choosing whether to advertise (R,D) or not advertise (L,U). Advertising is costly, but “steals” customers from the other firm. However, if both advertise, the stealing cancels. Another interpretation
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is that the two firms are Samsung and Apple, deciding whether to innovate substantially in their
next release of their flagship smartphone (R,D), or simply release an update (L,U). Innovation can
steal customers from the competitor firm, but it’s costly. Notice that in the NE both innovate, paying the corresponding cost, and consumers switch from one firm to another symmetrically, leaving
them with a lower payo↵.
b. The two pure strategy Nash Equilibria are (U,L) and (D,R). This game corresponds to a location game
between two stores selling the same product; they can choose on of two location, which di↵er in
profitability: (D,R) corresponds to the more profitable, while (U,L) correspond to the less profitable.
When the two firms choose the same location, however, they get zero profit because competition
at that location pushes prices down. Similarly, one might think of a game in which firms choose
whether to produce a high quality or low quality product, and cannot di↵erentiate their product in
any other way. The market for high quality products is potentially more profitable, since it allows
for higher markups, but if both firms produce a high quality product all profits will be competed
away.
c. The two pure Nash Equilibria are (U,R) and (D,L). Suppose that the two player are two entrepreneur
who want to establish a restaurant in the same market. L-M-R and U-M-D correspond to French
Italian and Chinese. Italian is favored by consumers (as shown by the fact that they can still both
turn a positive profit if they choose (M,M)), but is a closer substitute for French and Chinese than
those two are for each other. Hence, in equilibrium none will establish an Italian restaurant.
d. The two pure strategy Nash Equilibria are (U,R) and (D,L). This game can be interpreted as an entry
game in a market that can only support one firm. For instance, suppose that Walmart and Target
are simultaneously and independently deciding whether to establish a store in a medium-sized rural
county; U and L correspond to entry, while D and R correspond to staying out. If they both enter,
neither will be able to make positive profits, but both would like to be the ones operating alone in
this market.
Problem 3
a. Suppose there is a MXNE in which player 1 randomizes between U and D; say it does U with probability
↵ 2 [0, 1] . Say moreover that in this equilibrium player 2 enters (i.e. does L) with probability
2 [0, 1]. The expected payo↵ of player 1 will then be:
⇡1 (↵, ) =
5↵ + 10↵ (1
= 10↵
)
15↵ .
This function is linear in player 1’s decision variable, ↵, so that its maximum will be reached at one
of the corners of the feasible range of ↵, unless 10 = 15 , that is = 23 . This means that the only
case in which ↵ is interior, i.e. di↵erent from both 0 and 1, is when = 23 , and thus the expected
payo↵ from doing U (↵ = 1) is 10/3, the same as the expected payo↵ from doing L (↵ = 0) . This
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means that player 1 in this case is indi↵erent between the two actions. This discussion proves that
indi↵erence between pure strategies is a necessary (but not sufficient) condition for randomization.
Intuitively, if you prefer one action to another, it makes no sense to toss a coin and play the action
you dislike with positive probability.
b. The argument above also implies that player 2 must be randomizing with probability
1 to randomize in equilibrium.
=
2
3
for player
c. Since ⇡2 (↵, ) = 10
15↵ , it must be the case that ↵ = 23 for player 2 to randomize. We have found
in this way the unique MXNE of the game, in which both players enter with probability 23 . Notice
that this pair of probabilities satisfies the definition of Nash Equilibrium: given the other player’s
randomization, each player is indi↵erent between the two actions, and thus chooses optimally to
randomize. A peculiar feature of any MXNE which you can observe from the previous discussion is
that each player picks a precise randomization strategy not because of his own optimizing behavior
(in fact, given his opponent’s behavior, he’s indi↵erent between any mixture and any pure strategy!)
but rather because that strategy makes his opponent indi↵erent and thus also willing to randomize.
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