CHAPTER 15:
Imperfect Competition
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.
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
)150(QQ
5,625.
m
)150( 211 qqq
simultaneously,
cc
iPq 50
and
2,500.
c
i
d.
P
MC = 0 Bertrand
D
Q
Cournot
Monopoly
( ),Q a bQ c
2
()
.
4
mac
b
b. Cournot firm 1 maximizes
1 1 2
[ ( ) ],q a b q q c
yielding first-order condition
02 21 cbqbqa
and best-response function
2
1.
2
a bq c
qb
Symmetrically for firm 2,
1
2.
2
a bq c
qb
The Nash equilibrium outcome is
,
3
c
i
ac
qb
2,
33
cac
P
2
()
.
9
c
i
ac
b
c. The Nash equilibrium of the Bertrand game involves marginal-cost pricing:
,
b
Pc
d. Cournot firm
i
maximizes
[ ( ) ],
i i i
q a b Q q c
yielding first-order condition
2 0.
ii
a bQ bq c
Once the first-order condition has been taken, we can
apply the fact that firms are symmetric and so the equilibrium will be symmetric.
Substituting
c
i
ciqnQ )1(
into the first-order condition and solving for
c
i
q
yields
.
( 1)
c
i
ac
qbn
Therefore,
,
1
cn a c
Qnb
,
1
ca nc
Pn
2
2
()
,
( 1)
c
i
ac
bn
2
2
()
.
( 1)
cn a c
bn
e. It is easy to verify the answers to parts (a)–(c) by making the indicated
substitutions for
.n
15.3 a. Skipping preliminary calculations, firm 1’s best-response function is
21
1
1.
qc
q
equilibrium from
E
to
.E
BR2(q1)
q1
q2
BR1(q2)
E
E’
15.4 a. The most reasonable Nash equilibrium is for both firms to charge the high
marginal cost:
**
12
10.pp
(The are other Nash equilibria in which both firms
500 20 10 300
15.5 a. Firm 1 maximizes
)1( 211 bppp
with respect to
1,p
yielding the first-
12
1 2 0.p bp
b.
**
2
11
,.
2 (2 )
ii
qbb
c. An increase in
b
pivots the best-response functions, shifting the equilibrium from
E
to
.E
BR2(p1)
p1
p2
BR1(p2)
E
E’
15.6 a. The present discounted value of the stream of payoffs from colluding is
mmd
V