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The problems in this chapter provide the student with some practice with many of the different
models of imperfect competition introduced in the text. Space considerations forced us to omit
problems on search, advertising, and innovation. The instructor may wish to supplement the
problem set with exercises from these areas depending on interest.
Comments on Problems
15.1 This problem compares the monopoly outcome to the outcome under Cournot and
Bertrand competition in a simple example with perfect substitutes and linear demand.
15.2 This problem generalizes the previous problem to general linear demand functions and
arbitrary numbers of firms.
15.3 This problem analyzes Cournot competition when firms have different marginal costs.
This departure from identical firms allows the student to shift around firm’s best
responses independently on a diagram.
15.4 This problem analyzes Bertrand competition when firms have different marginal costs. If
we adopt the usual assumption that demand is allocated evenly to equal-priced firms, then
we encounter a technical problem, called the open-set problem, in this setting. It would
not be an equilibrium for firms to charge 10 (the high marginal cost). The firm 2 would
profit from undercutting slightly and capturing all demand. The problem is that firm 2 has
no “best” undercutting price. For any price just below 10, say 9.999, firm 2 could earn
more by increasing price slightly but keeping it below 10, say 9.9999. One way to avoid
this problem is to assume prices are denominated in discrete units, say pennies, and
fractions of pennies are not allowed. The solution to the open-set problem suggested here
is to assume that the low-cost firm gets all the demand at equal prices.
15.5 This problem analyzes Bertrand competition with differentiated products. The problem
gives students practice in drawing diagrams with upward-sloping best responses (as
opposed to downward-sloping with Cournot).
15.6 This problem exercises in collusion in infinitely repeated games.
15.7 This problem analyzes the Stackelberg game both with and without the possibility of
entry-deterring investment.
CHAPTER 15:
Imperfect Competition
15.8 This problem analyzes entry deterrence in a Hotelling model.
Analytical Problems
15.9 Herfindahl index of market concentration. Many economists subscribe to the
conventional wisdom that increases in concentration are bad for social welfare. This
problem leads students through a series of calculations showing that the relationship
between welfare and concentration is not this straightforward.
15.10 Inverse elasticity rule. This problem derives alternatives form of the inverse elasticity
rule for a Cournot firm that are related to the one derived for a monopoly.
15.11 Competition on a circle. This problem is a useful twist on the Hotelling model that has
been used in a wide variety of applications because the symmetry of a circle makes the
analysis of entry easier.
15.12 Signaling with entry accommodation. In this problem, the incumbent must
accommodate entry because the fixed cost is low enough that entry cannot be deterred.
Given that best responses are upward-sloping (strategic complements), the incumbent
pursues a “puppy dog” strategy of trying to convince its rival it is a weak (high-cost)
competitor, with the hope of inducing its rival to charge a high price.
Behavioral Problem
15.13 Can competition unshroud prices? This new behavioral problem shows that market
forces (competition, advertising) may not be sufficient to overcome consumers’
behavioral biases.
Solutions
15.1 a. The monopolist maximizes profit
)150(QQ −
yielding first-order
b. Cournot firm 1 maximizes profit
)150(211 qqq −−
yielding first-order
Chapter 15: Imperfect Competition
174
d.
15.2 a. A monopolist maximizes
( ),Q a bQ c−−
yielding first-order condition
b. Cournot firm 1 maximizes
1 1 2
[ ( ) ],q a b q q c− + −
yielding first-order condition
P
D
Chapter 15: Imperfect Competition
175
c. The Nash equilibrium of the Bertrand game involves marginal-cost pricing:
d. Cournot firm
i
maximizes
[ ( ) ],
i i i
q a b Q q c
−
− + −
yielding first-order condition
15.3 a. Skipping preliminary calculations, firm 1’s best-response function is
3
Chapter 15: Imperfect Competition
176
Further,
b. The reduction in firm 1’s marginal cost shifts its best response out and shifts the
equilibrium from
E
to
.E
15.4 a. The most reasonable Nash equilibrium is for both firms to charge the high
q2
BR1(q2)
Chapter 15: Imperfect Competition
177
15.5 a. Firm 1 maximizes
)1( 211 bppp +−
with respect to
1,p
yielding the first-
c. An increase in
b
pivots the best-response functions, shifting the equilibrium from
E
to
.E
15.6 a. The present discounted value of the stream of payoffs from colluding is
Chapter 15: Imperfect Competition
178
b. From Example 15.7, we know that
11n
−
for collusion to be sustainable, or
rearranging to get a condition on
n
, we have
15.7 a. Solve the game using backward induction starting with firm 2’s action. We saw
from Problem 15.1 part (b) that firm 2’s best-response function is
b. If firm 1 accommodates 2’s entry, the outcome in part (a) arises, and 1 earns
Chapter 15: Imperfect Competition
179
15.8 a. The two firms engage in Bertrand competition in homogeneous products at the
right end of the beach, leading to prices equal to marginal cost (here zero). Firm
c. A’s entry-deterring strategy is not credible. From its strategy in part (a), it earns
Chapter 15: Imperfect Competition
180
Analytical Problems
15.9 Herfindahl index of market concentration
a. Reprising the analysis from Problem 15.2, firm i’s profit is
)( cbQbqaq iii −−− −
b. We can obtain a rough idea of the effect of merger by seeing how the variables in
Chapter 15: Imperfect Competition
181
e. Comparing part (a) with part (b) suggests that increases in the Herfindahl index
15.10 Inverse elasticity rule
Equation 15.2 can be rearranged as follows:
15.11 Competition on a circle
a. This is the indifference condition for a consumer located distance
x
from firm i:
Chapter 15: Imperfect Competition
182
f. Total transportation costs equal the number of half-segments between firms,
2,n
15.12 Signaling with entry accommodation
a. The high type’s best-response function is
b. This is similar to part (a) except the relevant best-response function for firm 1 is
that for the low type:
c. The best response for the high type is given in part (a) and for the low type in part
(b). The best response for firm 2 comes from maximizing its expected profit:
Chapter 15: Imperfect Competition
183
d. Firm 2 earns an expected payoff of about 0.542 under complete information and
e. We need to check that the low type would prefer its equilibrium profit to the
profit from charging the high type’s price in the first period and then having firm
Behavioral Problem
15.13 Can competition unshroud prices?
a. Sophisticated consumers will only pay
i
s
if it is less than
,e
their personal
avoidance cost. If
e
is very small, firms will find it more profitable to try to
exploit myopic consumers instead, charging the highest value that keeps them
from voiding the purchase. They do not void if
.
ii
p s v+
Hence,
.
ii
s v p=−
b. The posted prices of color laser printers is close to their monochrome counterparts
even though they must be considerably more complicated to manufacture (and
come with four or five separate toner cartridges, which must be costlier than the
Chapter 15: Imperfect Competition
184
c. If firm 1 advertises, the net surplus consumers (who are all sophisticated now)
obtain from firm 2 is
d. By “de-biasing” myopic consumers, advertising educates them about how to buy
at the posted price from the rival firm, which is very difficult for the advertising
e. Let
a
p
be the equilibrium posted price from part (a):
