NI_11th Edtion (1) (1)-75

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PROBLEMS
15.1
Assume for simplicity that a monopolist has no costs of production and faces a demand curve given by Q ¼ 150 $ P.
a. Calculate the profit-maximizing price–quantity combination for this monopolist. Also calculate the monopolist’s profit.
b. Suppose instead that there are two firms in the market facing the demand and cost conditions just described for their iden-
tical products. Firms choose quantities simultaneously as in the Cournot model. Compute the outputs in the Nash equilib-
rium. Also compute market output, price, and firm profits.
c. Suppose the two firms choose prices simultaneously as in the Bertrand model. Compute the prices in the Nash equilibrium.
Also compute firm output and profit as well as market output.
d. Graph the demand curve and indicate where the market price–quantity combinations from parts (a)–(c) appear on the curve.
15.2
Suppose that firms’ marginal and average costs are constant and equal to c and that inverse market demand is given by P¼ a$ bQ,
where a, b> 0.
a. Calculate the profit-maximizing price–quantity combination for a monopolist. Also calculate the monopolist’s profit.
b. Calculate the Nash equilibrium quantities for Cournot duopolists, which choose quantities for their identical products
simultaneously. Also compute market output, market price, and firm and industry profits.
c. Calculate the Nash equilibrium prices for Bertrand duopolists, which choose prices for their identical products simultane-
ously. Also compute firm and market output as well as firm and industry profits.
d. Suppose now that there are n identical firms in a Cournot model. Compute the Nash equilibrium quantities as functions
of n. Also compute market output, market price, and firm and industry profits.
e. Show that the monopoly outcome from part (a) can be reproduced in part (d) by setting n ¼ 1, that the Cournot duopoly
outcome from part (b) can be reproduced in part (d) by setting n ¼ 2 in part (d), and that letting n approach infinity yields
the same market price, output, and industry profit as in part (c).
15.3
Let ci be the constant marginal and average cost for firm i (so that firms may have different marginal costs). Suppose demand is
given by P ¼ 1 $ Q.
a. Calculate the Nash equilibrium quantities assuming there are two firms in a Cournot market. Also compute market output,
market price, firm profits, industry profits, consumer surplus, and total welfare.
b. Represent the Nash equilibrium on a best-response function diagram. Show how a reduction in firm 1’s cost would change
the equilibrium. Draw a representative isoprofit for firm 1.
15.4
Suppose that firms 1 and 2 operate under conditions of constant average and marginal cost but that firm 1’s marginal cost is
c1 ¼ 10 and firm 2’s is c2 ¼ 8. Market demand is Q ¼ 500 $ 20P.
a. Suppose firms practice Bertrand competition, that is, setting prices for their identical products simultaneously. Compute
the Nash equilibrium prices. (To avoid technical problems in this question, assume that if firms charge equal prices, then
the low-cost firm makes all the sales.)
b. Compute firm output, firm profit, and market output.
c. Is total welfare maximized in the Nash equilibrium? If not, suggest an outcome that would maximize total welfare, and
compute the deadweight loss in the Nash equilibrium compared with your outcome.
15.5
Consider the following Bertrand game involving two firms producing differentiated products. Firms have no costs of
production. Firm 1’s demand is
q1 ¼ 1$ p1 þ bp2,
where b > 0. A symmetric equation holds for firm 2’s demand.
Chapter 15: Imperfect Competition 569
a. Solve for the Nash equilibrium of the simultaneous price-choice game.
b. Compute the firms’ outputs and profits.
c. Represent the equilibrium on a best-response function diagram. Show how an increase in b would change the equilibrium.
Draw a representative isoprofit curve for firm 1.
15.6
Recall Example 15.6, which covers tacit collusion. Suppose (as in the example) that a medical device is produced at constant
average and marginal cost of $10 and that the demand for the device is given by
Q ¼ 5,000$ 100P:
The market meets each period for an infinite number of periods. The discount factor is d.
a. Suppose that n firms engage in Bertrand competition each period. Suppose it takes two periods to discover a deviation
because it takes two periods to observe rivals’ prices. Compute the discount factor needed to sustain collusion in a
subgame-perfect equilibrium using grim strategies.
b. Now restore the assumption that, as in Example 15.7, deviations are detected after just one period. Next, assume that n is
not given but rather is determined by the number of firms that choose to enter the market in an initial stage in which
entrants must sink a one-time cost K to participate in the market. Find an upper bound on n. Hint: Two conditions are
involved.
15.7
Assume as in Problem 15.1 that two firms with no production costs, facing demand Q ¼ 150 $ P, choose quantities q1 and q2.
a. Compute the subgame-perfect equilibrium of the Stackelberg version of the game in which firm 1 chooses q1 first and then
firm 2 chooses q2.
b. Now add an entry stage after firm 1 chooses q1. In this stage, firm 2 decides whether to enter. If it enters, then it must sink
cost K2, after which it is allowed to choose q2. Compute the threshold value of K2 above which firm 1 prefers to deter firm
2’s entry.
c. Represent the Cournot, Stackelberg, and entry-deterrence outcomes on a best-response function diagram.
15.8
Recall the Hotelling model of competition on a linear beach from Example 15.5. Suppose for simplicity that ice cream stands
can locate only at the two ends of the line segment (zoning prohibits commercial development in the middle of the beach). This
question asks you to analyze an entry-deterring strategy involving product proliferation.
a. Consider the subgame in which firm A has two ice cream stands, one at each end of the beach, and B locates along with A at
the right endpoint. What is the Nash equilibrium of this subgame?Hint: Bertrand competition ensues at the right endpoint.
b. If B must sink an entry cost KB, would it choose to enter given that firm A is in both ends of the market and remains there
after entry?
c. Is A’s product proliferation strategy credible? Or would A exit the right end of the market after B enters? To answer these
questions, compare A’s profits for the case in which it has a stand on the left side and both it and B have stands on the
right to the case in which A has one stand on the left end and B has one stand on the right end (so B’s entry has driven A
out of the right side of the market).
Analytical Problems
15.9 Herfindahl index of market concentration
One way of measuring market concentration is through the use of the Herfindahl index, which is defined as
H ¼
Xn
i¼1
s2i ,
where st ¼ qi/Q is firm i’s market share. The higher is H, the more concentrated the industry is said to be. Intuitively, more
concentrated markets are thought to be less competitive because dominant firms in concentrated markets face little competitive
pressure. We will assess the validity of this intuition using several models.
570 Part 6: Market Power
a. If you have not already done so, answer Problem 15.2d by computing the Nash equilibrium of this n-firm Cournot game.
Also compute market output, market price, consumer surplus, industry profit, and total welfare. Compute the Herfindahl
index for this equilibrium.
b. Suppose two of the n firms merge, leaving the market with n $ 1 firms. Recalculate the Nash equilibrium and the rest of
the items requested in part (a). How does the merger affect price, output, profit, consumer surplus, total welfare, and the
Herfindahl index?
c. Put the model used in parts (a) and (b) aside and turn to a different setup: that of Problem 15.3, where Cournot duopolists
face different marginal costs. Use your answer to Problem 15.3a to compute equilibrium firm outputs, market output, price,consumer surplus, industry profit, and total welfare, substituting the particular cost parameters c1 ¼ c2 ¼ 1/4. Also com-
pute the Herfindahl index.
d. Repeat your calculations in part (c) while assuming that firm 1’s marginal cost c1 falls to 0 but c2 stays at 1/4. How does
the cost change affect price, output, profit, consumer surplus, total welfare, and the Herfindahl index?
e. Given your results from parts (a)–(d), can we draw any general conclusions about the relationship between market concen-
tration on the one hand and price, profit, or total welfare on the other?
15.10 Inverse elasticity rule
Use the first-order condition (Equation 15.2) for a Cournot firm to show that the usual inverse elasticity rule from Chapter 11
holds under Cournot competition (where the elasticity is associated with an individual firm’s residual demand, the demand left
after all rivals sell their output on the market). Manipulate Equation 15.2 in a different way to obtain an equivalent version of
the inverse elasticity rule:
P $MC
P
¼ $ si
eQ,P
,
where si ¼ qi/Q is firm i’s market share and eQ, P is the elasticity of market demand. Compare this version of the inverse
elasticity rule with that for a monopolist from the previous chapter.
15.11 Competition on a circle
Hotelling’s model of competition on a linear beach is used widely in many applications, but one application that is difficult to
study in the model is free entry. Free entry is easiest to study in a model with symmetric firms, but more than two firms on a
line cannot be symmetric because those located nearest the endpoints will have only one neighboring rival, whereas those
located nearer the middle will have two.
To avoid this problem, Steven Salop introduced competition on a circle.18 As in the Hotelling model, demanders are located
at each point, and each demands one unit of the good. A consumer’s surplus equals v (the value of consuming the good) minus
the price paid for the good as well as the cost of having to travel to buy from the firm. Let this travel cost be td, where t is a
parameter measuring how burdensome travel is and d is the distance traveled (note that we are here assuming a linear rather
than a quadratic travel-cost function, in contrast to Example 15.5).
Initially, we take as given that there are n firms in the market and that each has the same cost function Ci ¼ K þ cqi, where
K is the sunk cost required to enter the market [this will come into play in part (e) of the question, where we consider free
entry] and c is the constant marginal cost of production. For simplicity, assume that the circumference of the circle equals 1
and that the n firms are located evenly around the circle at intervals of 1/n. The n firms choose prices pi simultaneously.
a. Each firm i is free to choose its own price (pi) but is constrained by the price charged by its nearest neighbor to either side.
Let p" be the price these firms set in a symmetric equilibrium. Explain why the extent of any firm’s market on either side
(x) is given by the equation
p þ tx ¼ p" þ t[(1/n) $ x].
b. Given the pricing decision analyzed in part (a), firm i sells qi ¼ 2x because it has a market on both sides. Calculate the
profit-maximizing price for this firm as a function of p", c, t, and n.
c. Noting that in a symmetric equilibrium all firms’ prices will be equal to p", show that pi ¼ p" ¼ c þ t/n. Explain this result
intuitively.
d. Show that a firm’s profits are t/n2 $ K in equilibrium.
e. What will the number of firms n" be in long-run equilibrium in which firms can freely choose to enter?
18See S. Salop, ‘‘Monopolistic Competition with Outside Goods,’’ Bell Journal of Economics (Spring 1979): 141–56.
Chapter 15: Imperfect Competition 571
f. Calculate the socially optimal level of differentiation in this model, defined as the number of firms (and products) that min-
imizes the sum of production costs plus demander travel costs. Show that this number is precisely half the number calcu-
lated in part (e). Hence this model illustrates the possibility of overdifferentiation.
15.12 Signaling with entry accommodation
This question will explore signaling when entry deterrence is impossible; thus, the signaling firm accommodates its rival’s entry.
Assume deterrence is impossible because the two firms do not pay a sunk cost to enter or remain in the market. The setup of
the model will follow Example 15.4, so the calculations there will aid the solution of this problem. In particular, firm i’s demand
is given by
qi ¼ ai $ pi þ
pj
2
,
where ai is product i’s attribute (say, quality). Production is costless. Firm 1’s attribute can be one of two values: either a1 ¼ 1,
in which case we say firm 1 is the low type, or a1 ¼ 2, in which case we say it is the high type. Assume there is no discounting
across periods for simplicity.
a. Compute the Nash equilibrium of the game of complete information in which firm 1 is the high type and firm 2 knows that
firm 1 is the high type.
b. Compute the Nash equilibrium of the game in which firm 1 is the low type and firm 2 knows that firm 1 is the low type.
c. Solve for the Bayesian–Nash equilibrium of the game of incomplete information in which firm 1 can be either type with
equal probability. Firm 1 knows its type, but firm 2 only knows the probabilities. Because we did not spend time this chap-
ter on Bayesian games, you may want to consult Chapter 8 (especially Example 8.7).
d. Which of firm 1’s types gains from incomplete information? Which type would prefer complete information (and thus
would have an incentive to signal its type if possible)? Does firm 2 earn more profit on average under complete information
or under incomplete information?
e. Consider a signaling variant of the model chat has two periods. Firms 1 and 2 choose prices in the first period, when firm 2
has incomplete information about firm 1’s type. Firm 2 observes firm 1’s price in this period and uses the information to
update its beliefs about firm 1’s type. Then firms engage in another period of price competition. Show that there is a separat-
ing equilibrium in which each type of firm 1 charges the same prices as computed in part (d). You may assume that, if firm 1
chooses an out-of-equilibrium price in the first period, then firm 2 believes that firm 1 is the low type with probability 1. Hint:
To prove the existence of a separating equilibrium, show that the loss to the low type from trying to pool in the first period
exceeds the second-period gain from having convinced firm 2 that it is the high type. Use your answers from parts (a)–(d)
where possible to aid in your solution.
SUGGESTIONS FOR FURTHER READING
Carlton, D. W., and J. M. Perloff. Modern Industrial Organi-
zation, 4th ed. Boston: Addison-Wesley, 2005.
Classic undergraduate text on industrial organization that cov-
ers theoretical and empirical issues.
Kwoka, J. E., Jr., and L. J. White. The Antitrust Revolution,
4th ed. New York: Oxford University Press, 2004.
Summarizes economic arguments on both sides of a score of im-
portant recent antitrust cases. Demonstrates the policy rele-
vance of the theory developed in this chapter.
Pepall, L., D. J. Richards, and G. Norman. Industrial
Organization: Contemporary Theory and Practice, 2nd ed.
Cincinnati, OH: Thomson South-Western, 2002.
An undergraduate textbook providing a simple but thorough
treatment of oligopoly theory. Uses the Hotelling model in a va-
riety of additional applications including advertising.
Sutton, J. Sunk Costs and Market Structure. Cambridge, MA:
MIT Press, 1991.
Argues that the robust predictions of oligopoly theory regard the
size and nature of sunk costs. Second half provides detailed case
studies of competition in various manufacturing industries.
Tirole, J. The Theory of Industrial Organization. Cambridge,
MA: MIT Press, 1988.
A comprehensive survey of the topics discussed in this chapter
and a host of others. Standard text used in graduate courses,
but selected sections are accessible to advanced undergraduates.572 Part 6: Market Power
EXTENSIONSSTRATEGIC SUBSTITUTES AND COMPLEMENTS
We saw in the chapter that one can often understand the nature
of strategic interaction in a market simply from the slope of
firms’ best-response functions. For example, we argued that a
first mover that wished to accept rather than deter entry should
commit to a strategy that leads its rival to behave less aggres-
sively. What sort of strategy this is depends on the slope of
firms’ best responses. If best responses slope downward, as in a
Cournot model, then the first mover should play a ‘‘top dog’’
strategy and produce a large quantity, leading its rival to reduce
its production. If best responses slope upward, as in a Bertrand
model with price competition for differentiated products, then
the first mover should play a ‘‘puppy dog’’ strategy and charge a
high price, leading its rival to increase its price as well.
More generally, we have seen repeatedly that best-response
function diagrams are often helpful in understanding the na-
ture of Nash equilibrium, how the Nash equilibrium changes
with parameters of the model, how incomplete information
might affect the game, and so forth. Simply knowing the slope
of the best-response function is often all one needs to draw a
usable best-response function diagram.
By analogy to similar definitions from consumer and pro-
ducer theory, game theorists define firms’ actions to be strate-
gic substitutes if an increase in the level of the action (e.g.,
output, price, investment) by one firm is met by a decrease in
that action by its rival. On the other hand, actions are strategic
complements if an increase in an action by one firm is met by
an increase in that action by its rival.
E15.1 Nash equilibrium
To make these ideas precise, suppose that firm 1’s profit, p1(a1,
a2), is a function of its action a1 and its rival’s (firm 2’s) action
a2. (Here we have moved from subscripts to superscripts for
indicating the firm to which the profits belong tomake room for
subscripts that will denote partial derivatives.) Firm 2’s profit
function is denoted similarly. A Nash equilibrium is a profile of
actions for each firm, ða"1, a"2Þ, such that each firm’s equilibrium
action is a best response to the other’s. Let BR1(a2) be firm 1’s
best-response function, and let BR2(a1) be firm 2’s; then a Nash
equilibrium is given by a"1 ¼ BR1ða"2Þ and a"2 ¼ BR2ða"1Þ.
E15.2 Best-response functions
in more detail
The first-order condition for firm 1’s action choice is
p1
1ða1, a2Þ ¼ 0, (i)
where subscripts for p represent partial derivatives with
respect to its various arguments. A unique maximum, and
thus a unique best response, is guaranteed if we assume that
the profit function is concave:
p1
11ða1, a2Þ < 0: (ii)
Given a rival’s action a2, the solution to Equation i for a maxi-
mum is firm 1’s best-response function:
a1 ¼ BR1ða2Þ: (iii)
Since the best response is unique, BR1(a2) is indeed a function
rather than a correspondence (see Chapter 8 for more on cor-
respondences).
The strategic relationship between actions is determined
by the slope of the best-response functions. If best responses
are downward sloping [i.e., if BR01ða2Þ < 0 and BR02ða1Þ < 0],
then a1 and a2 are strategic substitutes. If best responses are
upward sloping [i.e., if BR01ða2Þ > 0 and BR02ða1Þ > 0], then
a1 and a2 are strategic complements.
E15.3 Inferences from the profit function
We just saw that a direct route for determining whether
actions are strategic substitutes or complements is first to
solve explicitly for best-response functions and then to differ-
entiate them. In some applications, however, it is difficult or
impossible to find an explicit solution to Equation i. We can
still determine whether actions are strategic substitutes or
complements by drawing inferences directly from the profit
function.
Substituting Equation iii into the first-order condition of
Equation i gives
p1
1ðBR1ða2Þ, a2Þ ¼ 0: (iv)
Totally differentiating Equation iv with respect to a2 yields, af-
ter dropping the arguments of the functions for brevity,
p1
11BR
0
1 þ p1
12 ¼ 0: (v)
Rearranging Equation v gives the derivative of the best-
response function:
BR01 ¼ $
p1
12
p1
11
: (vi)
In view of the second-order condition (Equation ii), the de-
nominator of Equation vi is negative. Thus, the sign of BR01 is
the same as the sign of the numerator, p1
12. That is, p1
12 > 0
implies BR01 > 0 and p1
12 < 0 implies BR01 < 0. The strategic
relationship between the actions can be inferred directly from
the cross-partial derivative of the profit function.
E15.4 Cournot model
In the Cournot model, profits are given as a function of the
two firms’ quantities:
p1ðq1, q2Þ ¼ q1Pðq1, q2Þ $ Cðq1Þ: (vii)
The first-order condition is
p1
1 ¼ q1P 0ðq1 þ q2Þ þ Pðq1 þ q2Þ $ C 0ðq1Þ, (viii)
as we have already seen (Equation 15.2). The derivative of
Equation viii with respect to q2 is, after dropping functions’
arguments for brevity,
p1
12 ¼ q1P 00 þ P 0: (ix)
Because P 0 < 0, the sign of p1
12 will depend on the sign of
P00—that is, the curvature of demand. With linear demand,
P00 ¼ 0 and so p1
12 is clearly negative. Quantities are strategic sub-
stitutes in the Cournot model with linear demand. Figure 15.2
illustrates this general principle. This figure is drawn for an
example involving linear demand, and indeed the best responses
are downward sloping.
More generally, quantities are strategic substitutes in the
Cournot model unless the demand curve is ‘‘very’’ convex
(i.e., unless P00 is positive and large enough to offset the last
term in Equation ix). For a more detailed discussion see
Bulow, Geanakoplous, and Klemperer (1985).
E15.5 Bertrand model with
differentiated products
In the Bertrand model with differentiated products, demand
can be written as
q1 ¼ D1ð p1, p2Þ: (x)
See Equation 15.24 for a related expression. Using this nota-
tion, profit can be written as
p1 ¼ p1q1 $ Cð q1Þ
¼ p1D1ð p1, p2Þ $ CðD1ð p1, p2ÞÞ: (xi)
The first-order condition with respect to p1 is
p1
1 ¼ p1D1
1ð p1, p2Þ þ D1ð p1, p2Þ
$ C 0ðD1ð p1, p2ÞÞD1
1ð p1, p2Þ: (xii)
The cross-partial derivative is, after dropping functions’ argu-
ments for brevity,
p1
12 ¼ p1D1
12 þ D1
2 $ C 0D1
12 $ C 00D1
2D
1
1: (xiii)
Interpreting this mass of symbols is no easy task. In the
special case of constant marginal cost (C00 ¼ 0) and linear
demand ðD1
12 ¼ 0Þ, the sign of p1
12 is given by the sign of D1
2
(i.e., how a firm’s demand is affected by changes in the rival’s
price). In the usual case when the two goods are themselves
substitutes, we have D1
2 > 0 and so p1
12 > 0. That is, prices are
strategic complements. The terminology here can seem con-
tradictory, so the result bears repeating: If the goods that the
firms sell are substitutes, then the variables the firms choose—
prices—are strategic complements. Firms in such a duopoly
would either raise or lower prices together (see Tirole, 1988).
We saw an example of this in Figure 15.4. The figure was
drawn for the case of linear demand and constant marginal
cost, and we saw that best responses are upward sloping.
E15.6 Entry accommodation in a
sequential game
Consider a sequential game in which firm 1 chooses a1 and
then firm 2 chooses a2. Suppose firm 1 finds it more profitable
to accommodate than to deter firm 2’s entry. Because firm 2
moves after firm 1, we can substitute firm 2’s best response
into firm 1’s profit function to obtain
p1ða1, BR2ða1ÞÞ: (xiv)
Firm 1’s first-order condition is
p1
1 þ p1
2BR
0
2|fflffl{zfflffl}
S
¼ 0: (xv)
By contrast, the first-order condition from the simultaneous
game (see Equation i) is simply p1
1 ¼ 0. The first-order condi-
tions from the sequential and simultaneous games differ in
the term S. This term captures the strategic effect of moving
first—that is, whether the first mover would choose a higher
or lower action in the sequential game than in the simultane-
ous game.
The sign of S is determined by the signs of the two factors
in S. We will argue inthe next paragraph that these two fac-
tors will typically have the same sign (both positive or both
negative), implying that S > 0 and hence that the first mover
will typically distort its action upward in the sequential game
compared with the simultaneous game. This result confirms
the findings from several of the examples in the text. In Figure
15.6, we see that the Stackelberg quantity is higher than the
Cournot quantity. In Figure 15.7, we see that the price leader
distorts its price upward in the sequential game compared
with the simultaneous one.
Section E15.3 showed that the sign of BR02 is the same as the
sign of p2
12. If there is some symmetry to the market, then the
sign of p2
12 will be the same as the sign of p1
12. Typically, p
1
2 and
p1
12 will have the same sign. For example, consider the case of
Cournot competition. By Equation 15.1, firm 1’s profit is
p1 ¼ Pðq1 þ q2Þq1 $ Cðq1Þ: (xvi)
Therefore,
p1
2 ¼ P 0ðq1 þ q2Þq1: (xvii)
574 Part 6: Market Power
Because demand is downward sloping, it follows that p1
2 < 0.
Differentiating Equation xvii with respect to q1 yields
p1
12 ¼ P 0 þ q1P 00: (xviii)
This expression is also negative if demand is linear (so P 00 ¼ 0)
or if demand is not too convex (so the last term in Equation
xviii does not swamp the term P 0).
E15.7 Extension to general investments
The model from the previous section can be extended to gen-
eral investments—that is, beyond a mere commitment to a
quantity or price. Let K1 be this general investment—(say)
advertising, investment in lower-cost manufacturing, or prod-
uct positioning—sunk at the outset of the game. The two
firms then choose their product-market actions a1 and a2
(representing prices or quantities) simultaneously in the
second period. Firms’ profits in this extended model are,
respectively,
p1ða1, a2, K1Þ and p2ða1, a2Þ: (xix)
The analysis is simplified by assuming that firm 2’s profit is
not directly a function of K1, although firm 2’s profit will indi-
rectly depend on K1 in equilibrium because equilibrium
actions will depend on K1. Let a"1ðK1Þ and a"2ðK1Þ be firms’
actions in a subgame-perfect equilibrium:
a"1ðK1Þ ¼ BR1ða"2ðK1Þ, K1Þ,
a"2ðK1Þ ¼ BR2ða"1ðK1ÞÞ:
(xx)
Because firm 2’s profit function does not depend directly on K1
in Equation xix, neither does its best response in Equation xx.
The analysis here draws on Fudenberg and Tirole (1984)
and Tirole (1988). Substituting from Equation xx into Equa-
tion xix, the firms’ Nash equilibrium profits in the subgame
following firm 1’s choice of K1 are
p1"ðK1Þ ¼ p1ða"1ðK1Þ, a"2ðK1Þ, K1Þ,
p2"ðK1Þ ¼ p2ða"1ðK1Þ, a"2ðK1ÞÞ:
(xxi)
Fold the game back to firm 1’s first-period choice of K1.
Because firm 1 wants to accommodate entry, it chooses K1 to
maximize p1"(K1). Totally differentiating p1"(K1), the first-
order condition is
dp1"
dK1
¼ p1
1
da"1
dK1
þ p1
2
da"2
dK1
þ @p
1
@K1
¼ p1
2
da"2
dK1|fflfflffl{zfflfflffl}
S
þ @p
1
@K1
:
(xxii)
The second equality in Equation xxii holds by the envelope
theorem. (The envelope theorem just says that p1
1 & da"1=dK1
disappears because a1 is chosen optimally in the second pe-
riod, so p1
1 ¼ 0 by the first-order condition for a1.) The first
of the remaining two terms in Equation xxii, S, is the strategic
effect of an increase in K1 on firm 1’s profit through firm 2’s
action. If firm 1 cannot make an observable commitment to
K1, then S disappears from Equation xxii and only the last
term, the direct effect of K1 on firm 1’s profit, will be present.
The sign of S determines whether firm 1 strategically over-
or underinvests in K1 when it can make a strategic commit-
ment. We have the following steps:
signðSÞ ¼ sign p2
1
da"2
dK1
" #
¼ sign p2
1BR
0
2
da"1
dK1
" #
¼ sign
dp2"
dK1
BR02
" #
: (xxiii)
The first line of Equation xxiii holds if there is some symme-
try to the market, so that the sign of p1
2 equals the sign of p2
1.
The second line follows from differentiating a"2ðK1Þ in Equa-
tion xx. The third line follows by totally differentiating p2" in
Equation xxi:
dp2"
dK1
¼ p2
1
da"1
dK1
þ p2
2
da"2
dK1
¼ p2
1
da"1
dK1
,
(xxiv)
where the second equality again follows from the envelope
theorem.
By Equation xxiii, the sign of the strategic effect S is deter-
mined by the sign of two factors. The first factor, dp2"/dK1,
indicates the effect of K1 on firm 2’s equilibrium profit in the
subgame. If dp2"/dK1 < 0, then an increase in K1 harms firm
2, and we say that investment makes firm 1 ‘‘tough.’’ If dp2"/
dK1 > 0, then an increase in K1 benefits firm 2, and we say
that investment makes firm 1 ‘‘soft.’’ The second factor, BR02,
is the slope of firm 2’s best response, which depends on
whether actions a1 and a2 are strategic substitutes or comple-
ments. Each of the two terms in S can have one of two signs
for a total of four possible combinations, displayed in Table
15.1. If investment makes firm 1 ‘‘tough,’’ then the strategic
effect S leads firm 1 to reduce K1 if actions are strategic com-
plements or to increase K1 if actions are strategic substitutes.
The opposite is true if investment makes firm 1 ‘‘soft.’’
For example, actions could be prices in a Bertrand model
with differentiated products and thus would be strategic com-
plements. Investment K1 could be advertising that steals mar-
ket share from firm 2. Table 15.1 indicates that, when K1 is
observable, firm 1 should strategically underinvest to induce
less aggressive price competition from firm 2.
E15.8 Most-favored customer program
The preceding analysis applies even if K1 is not a continuous
investment variable but instead a 0–1 choice. For example, con-
sider the decision by firm 1 of whether to start a most-favored
customer program (studied in Cooper, 1986). A most-favored
customer program rebates the price difference (sometimes in
addition to a premium) to past customers if the firm lowers its
Chapter 15: Imperfect Competition 575
price in the future. Such a program makes firm 1 ‘‘soft’’ by
reducing its incentive to cut price. If firms compete in strategic
complements (say, in a Bertrand model with differentiated
products), then Table 15.1 says that firm 1 should ‘‘overinvest’’
in the most-favored customer program, meaning that it should
be more willing to implement the program if doing so is observ-
able to its rival. The strategic effect leads to less aggressive price
competition and thus to higher prices and profits.
One’s first thought might have been that such a most-
favored customer program should be beneficial to consumers
and lead to lower prices because the clause promises payments
back to them. As we can see from this example, strategic con-
siderations sometimes prove one’s initial intuition wrong, sug-
gesting that caution is warranted when examining strategic
situations.
E15.9 Trade policy
The analysis in Section E15.7 applies even if K1 is not a choice
by firm 1 itself. For example, researchers in international trade
sometimes take K1 to be a government’s policy choice on
behalf of its domestic firms. Brander and Spencer (1985) stud-
ied a model of international trade in which exporting firms
from country 1 engage in Cournot competition with domestic
firms in country 2. The actions (quantities) are strategic sub-
stitutes. The authors ask whether the government of country
1 would want to implement an export subsidy program, a de-
cision that plays the role of K1 in their model. An export sub-
sidy makes exporting firms ‘‘tough’’ because it effectively
lowers their marginal costs, increasing their exports to coun-
try 2 and reducing market price there. According to Table
15.1, the government of country 1 should overinvest in the
subsidy policy, adopting the policy if it is observable to
domestic firms in country 2 but not otherwise. The model
explains why countries unilaterally adopt export subsidies and
other trade interventions when free trade would be globally
efficient (at least in this simple model).
Our analysis can be used to show that Brander and
Spencer’s rationalization ofexport subsidies may not hold up
under alternative assumptions about competition. If exporting
firms and domestic firms were to compete in strategic com-
plements (say, Bertrand competition in differentiated prod-
ucts rather than Cournot competition), then an export subsidy
would be a bad idea according to Table 15.1. Country 1 should
then underinvest in the export subsidy (i.e., not adopt it) to
avoid overly aggressive price competition.
E15.10 Entry deterrence
Continue with the model from Section E15.7, but now sup-
pose that firm 1 prefers to deter rather than accommodate
entry. Firm 1’s objective is then to choose K1 to reduce firm
2’s profit p2" to zero. Whether firm 1 should distort K1
upward or downward to accomplish this depends only on the
sign of dp2"/dK1—that is, on whether investment makes firm
1 ‘‘tough’’ or ‘‘soft’’—and not on whether actions are strategic
substitutes or complements. If investment makes firm 1
‘‘tough,’’ it should overinvest to deter entry relative to the case
in which it cannot observably commit to investment. On the
other hand, if investment makes firm 1 ‘‘soft,’’ it should
underinvest to deter entry.
For example, if K1 is an investment in marginal cost reduc-
tion, this likely makes firm 1 ‘‘tough’’ and so it should over-
invest to deter entry. If K1 is an advertisement that increases
demand for the whole product category more than its own
brand (advertisements for a particular battery brand involving
an unstoppable, battery-powered bunny may increase sales of
all battery brands if consumers have difficulty remembering
exactly which battery was in the bunny), then this will likely
make firm 1 ‘‘soft,’’ so it should underinvest to deter entry.
References
Brander, J. A., and B. J. Spencer. ‘‘Export Subsidies and Inter-
national Market Share Rivalry.’’ Journal of International
Economics 18 (February 1985): 83–100.
Bulow, J., G. Geanakoplous, and P. Klemperer. ‘‘Multimarket
Oligopoly: Strategic Substitutes and Complements.’’ Jour-
nal of Political Economy (June 1985): 488–511.
Cooper, T. ‘‘Most-Favored-Customer Pricing and Tacit Collu-
sion.’’ Rand Journal of Economics 17 (Autumn 1986):
377–88.
Fudenberg, D., and J. Tirole. ‘‘The Fat Cat Effect, the Puppy
Dog Ploy, and the Lean and Hungry Look.’’ American
Economic Review, Papers and Proceedings 74 (May 1984):
361–68.
Tirole, J. The Theory of Industrial Organization. Cambridge,
MA: MIT Press, 1988, chap. 8.
TABLE 15.1 STRATEGIC EFFECT WHEN ACCOMMODATING ENTRY
Firm 1’s Investment
‘‘Tough’’ (dp2"/dK1 < 0) ‘‘Soft’’ (dp2"/dK1 > 0)
Actions Strategic Complements (BR 0 > 0) Underinvest ($) Overinvest (þ)
Strategic Substitutes (BR 0 < 0) Overinvest (þ) Underinvest ($)
576 Part 6: Market Power
	PART SIX: Market Power
	CHAPTER 15 Imperfect Competition
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