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Chapter 14
Learning Objectives
After reading Chapter 14, you will
have learned:
Oligopoly
W How
In May 1999, two of the world’s largest pharmaceutical companies, the Swiss
firm Roche Holding and the German firm BASF, agreed to pay criminal fines
totaling $725 million to settle a U.S. Department of Justice price-fixing
antitrust case that charged the two with conspiring to raise the price of vitamins.
Between 1996 and 1999 the U.S. Department of Justice imposed its ten largest
fines for price fixing on firms in the following markets: vitamins, electricity conductors, feed supplements, food additives, offshore oil construction, and industrial cleaners. These firms all have one thing in common: They are all oligopolies,
or markets where a few large firms supply virtually all of the industry’s output.
Why did these firms conspire to fix prices, despite the risk of large fines and bad
publicity? And why were other firms in oligopolies able to set their prices without entering into illegal price-fixing agreements with their competitors? In this
chapter, we examine how firms behave in oligopolies.
Oligopoly market structures can result in a wide variety of firm behavior and
industry performance. Consider the following examples of oligopolistic behavior:
• In 1995, an upstart airline, Spirit Airlines, entered the Detroit market
and offered a one-way Detroit-Philadelphia fare of $49. At that time,
Northwest Airlines, the dominant carrier out of Detroit, was charging
an average Detroit-Philadelphia fare of $170. After Spirit entered,
Northwest reduced its Detroit-Philadelphia fare to $49 for most seats.
• In 1967, a British, French, and German government consortium provided subsidies to enable the Europeans to enter the commercial aircraft industry and compete with Boeing. In 1974, the first European
aircraft appeared on the market, but it was a commercial failure. The
three governments then decided it was necessary to produce a complete
line of aircraft to compete with Boeing. This goal was finally achieved
when a European-subsidized company, Airbus, emerged. Initially,
because of its small size relative to Boeing, Airbus faced a significant
375
to derive the Cournot-Nash
equilibrium using reaction functions
and how the Cournot-Nash equilibrium compares to the competitive
and monopoly equilibriums.
W How
to derive the Stackelberg equilibrium and how the Stackelberg
equilibrium differs from the
Cournot-Nash equilibrium.
W How
to derive the Bertrand equilibrium using reaction functions for a
homogeneous product and for a
heterogeneous product.
W The
dominant firm model for the
case of an oligopoly market dominated by one large firm that competes with a group of small fringe
firms.
W Why
the Prisoner’s Dilemma typically makes collusion difficult and
how firms try to solve the
Prisoner’s Dilemma by developing
methods of coordinated behavior.
Featured Applications
W 14.1
Application Experimental
Games and the Cournot-Nash
Model
W 14.2
Application Fuji’s Purchase of
Wal-Mart’s Photo-Processing Plants
W 14.3
Application Bertrand Pricing
in the Airline Industry
W 14.4
Application The Decline of
Dominant Firms
W 14.5A
Application Price Fixing in
the World Vitamin Market
W 14.5B
Application Price Leadership
in Cyberspace: The Airlines Case
376
Chapter 14
Oligopoly
cost disadvantage that resulted in a need for the European governments to supply ever-larger subsidies. Today, however, after decades of large subsidies, Airbus
competes effectively with Boeing.
• In the days when most gasoline contained lead, only a few firms produced leadbased anti-knock gasoline additives. From 1920 to 1948, Ethyl Corporation was
the only American producer. DuPont entered in 1948, followed by PPG
Industries and Nalco Chemical in the early 1960s. From 1974 to 1979, there
were 24 price changes in the market. In 20 of these cases, all four companies
changed price on the same day. In the other four cases, all four firms set identical prices within a day or two.
Each of these examples of an oligopoly market illustrates a different type of competitve
behavior. In the Northwest Airlines example, prices are highly competitive. In the case of
the Airbus consortium, several leading European nations spent billions of their taxpayers’
euros and pounds to compete with an American worldwide-dominant firm—Boeing. In
the lead-based anti-knock gasoline additives market, the firms found a method of eliminating price competion. These disparate competitive outcomes show that in oligopoly
markets, the possibilities range from successful agreements to keep prices high to competition that drives prices down toward marginal cost. This spectrum of possibilities makes
it impossible for economists to explain oligopoly behavior using one model.
In this chapter, we explore a number of models of oligopoly behavior: the CournotNash model, the Stackelberg model, the Bertrand model, the dominant firm model, and
models used to explain successful collusion. Of the chapter’s five sections, the first two
deal with quantity-based models, the third and fourth with price-based models, and the
fifth with models of successful collusion.
14.1
The Cournot-Nash Model
The 19th-century French philosopher, mathematician, and economist Augustin Cournot
made the first formal attempt to model oligopoly behavior in 1838.1 Cournot considered
the case of a duopoly market with two identical firms. The firms face identical costs and
there is no product differentiation. Under these conditions, price is a simple function of
the total quantity that the two firms produce. The driving force behind the Cournot
model is one additional assumption: Both firms assume that their opponent will maintain its output at the current level forever. This is a rather unrealistic assumption that is
inconsistent with most real-world situations. For example, suppose that your economics
professor is ten minutes late for the first class of the semester. According to Cournot’s
assumption, you should assume he will be ten minutes late for every class for the entire
semester. Furthermore, if he is five minutes late for the second class, you then assume he
will be five minutes late for all the remaining classes. Later in this chapter, we return to
the unrealistic nature of Cournot’s output maintenance assumption.
1 Augustin Cournot. Recherches sur les principes mathematiques de la theorie des richesses. Paris: M. Riviere & Cie.,
1838. Translated in Researches Into the Mathematical Principles of Wealth (New York: A.M. Kelly, 1960).
14.1
The Cournot-Nash Model
377
In his original paper, Cournot began by assuming that one firm has a monopoly in
a market in which a second firm is about to enter. Suppose, for example, that initially
Kodak is a monopolist in the American photographic film industry and Fuji is considering entering. The demand for film in the United States is:
p 10.00 Q
residual demand curve
A demand curve that is the leftover portion of an industry
demand after a group of firms
in the industry have chosen
their outputs.
Figure 14.1 Kodak’s
Initial Monopoly and
Fuji’s Equilibrium When
qK 4.5
In panel (a), Kodak is a
monopolist that maximizes
profit by producing 4.5 million rolls of film and charging $5.50 per roll. In panel
(b), Fuji assumes that
Kodak will produce 4.5 million rolls forever, and therefore faces a residual demand
p 5.50 qF and maximizes profit by producing
2.25 million rolls of film
and charging $3.25. Total
industry output is now 6.75
million rolls, so the new
equilibrium price is $3.25.
where p is price in dollars and Q is output in millions of rolls of film. Assume further that
Kodak’s marginal cost and average cost equal $1.00 per roll; that is, MC AC $1.00.
Figure 14.1(a) shows Kodak’s current monopoly equilibrium. Setting MR MC,
Kodak’s profit-maximizing output is 4.5 million rolls of film, and the monopoly price is
$5.50 per roll. Kodak’s monopoly profit is pq (AC )q $5.50(4,500,000) $1.00(4,500,000) $24,750,000 $4,500,000 $20,250,000, or $20.25 million.
Next Cournot turned his attention to the behavior of the firm entering the market—
in our example, Fuji. If the second firm, Fuji, assumes that Kodak will produce 4.5 million rolls of film per year forever, what output should Fuji produce? Recall that the
industry’s demand curve is p 10.00 Q, which can also be written as p (10.00 qK ) qF , where qK is Kodak’s output and qF is Fuji’s output. Given Fuji’s assumption
that Kodak produces 4.5 million rolls, then Fuji’s demand curve is p (10.00 4.5) qF 5.50 qF . Fuji’s demand curve is depicted in Figure 14.1(b), and is referred to as
a residual demand curve, because it is derived by assuming that Fuji faces a demand
curve that is left over, or residual, after Kodak has chosen its output.
To better understand the concept of a residual demand curve, consider what happens
to price if Fuji begins to produce any output greater than zero and Kodak produces qK 4.5. Note that for each 1 million rolls of film that Fuji sells, the price declines by one dollar below $5.50. If Fuji produces 1 million rolls, total industry output equals 5.5 million
rolls and p $4.50. If Fuji sells 1 million rolls, price equals $4.50, so one point on Fuji’s
residual demand curve is (qF 1 million, p $4.50). What if Fuji sells 2 million rolls?
Then Q qK qF 6.5 million rolls, and p $3.50, so a second point on Fuji’s residual
$
$
10.00
Kodak’s demand curve
p 10.00 Q
5.50
5.50
Fuji’s residual demand
p 5.50 qF
3.25
MC AC
1.00
4.5
9
MRK
10 Millions of
rolls
(a) Kodak’s Cournot equilibrium given qF 0
MC AC
1.00
2.25
4.5 5.5 Millions of
rolls
MR
F
(b) Fuji’s Cournot equilibrium
given qK 4.5 million rolls
378
Chapter 14
Oligopoly
demand curve is (qF 2 million, p $3.50). Fuji’s residual demand curve is, therefore,
p $5.50 qF .
Figure 14.1(b) shows how to derive Fuji’s profit-maximizing output. With identical
costs to Kodak’s, MC $1.00. Recalling that the MR curve bisects a linear demand
curve, Fuji maximizes profit by producing where MRF $5.50 $2qF $1.00 MC,
so qF 2.25 million rolls. Total industry output is Q qK qF 4.5 2.25 6.75
million rolls, so p $3.25.
Now it’s Kodak’s turn to respond to Fuji’s output. According to Cournot, Kodak
assumes that Fuji will maintain an output of 2.25 million rolls per year forever. Figure
14.2(a) depicts Kodak’s new equilibrium, given that Fuji is selling 2.25 million rolls of
film per year. With qF 2.25, Kodak’s residual demand curve is p (10.00 2.25) qK 7.75 qK , and its marginal revenue curve is MRK 7.75 2qK . Kodak maximizes profit by producing where MRK 7.75 2qK 1.00 MC, so qK 3.375 million rolls. Total industry output is 2.25 3.375 5.625 million rolls, and p $4.375
per roll.
Fuji now realizes that its strategy was wrong. After Fuji entered, Kodak did not maintain its output at 4.5 million rolls, but instead reduced its output to 3.375 million. Once
Fuji observes that Kodak has changed its output, should Fuji drop its assumption that
Kodak will maintain its output forever? Not according to Cournot. Instead, Cournot
argued that Fuji would now assume that Kodak would maintain its output at 3.375 million rolls forever. As shown in Figure 14.2(b), Fuji’s residual demand curve becomes p (10.00 3.375) qF 6.625 qF . Fuji equates its new MRF curve to MC 1.00
and produces 2.8125 million rolls. Industry output is now 2.8125 3.3750 6.1875
million rolls, and p $3.8125.
$
$
7.75
6.625
4.375
Kodak’s residual demand
p 7.75 qK
Fuji’s residual demand
p 6.625 qF
3.8125
MC AC $1.00 1.00
1.00
3.375
6.75 7.75 Millions of
rolls
MRK 7.75 2qK
(a) Kodak’s Cournot equilibrium given
qF 2.25 million rolls
MC AC $1.00
2.8125
5.625 6.625 Millions of
rolls
MRF 6.625 2qF
(b) Fuji’s Cournot equilibrium given
qK 3.375 million rolls
Figure 14.2 Kodak’s Equilibrium When qF 2.25 and Fuji’s Equilibrium When qK 3.375
In panel (a), Kodak assumes that Fuji will produce 2.25 million rolls per year forever, and therefore, Kodak faces a residual
demand p 7.75 qK and maximizes profit by producing 3.375 million rolls of film and charging a price of $4.375. In panel
(b), Fuji assumes Kodak will produce 3.375 million rolls forever, and therefore, Fuji faces a residual demand of p 6.625 qF ,
and maximizes profit by producing 2.8125 million rolls of film and charging a price of $3.8125.
14.1
The Cournot-Nash Model
379
Now Kodak’s assumption that Fuji will maintain its output at 2.25 million rolls has
been violated, because Fuji has increased its output to 2.8125 million rolls. Kodak
responds by changing its output yet again. This in turn results in yet another output
response by Fuji, which causes Kodak to change its output again, and so on. When does
all of this end? Because of our assumption of symmetric costs and demands, only when
both Kodak and Fuji face identical residual demand curves, and therefore both produce
the same output, will this back-and-forth series of responses end. Figures 14.1–14.3 illustrate how the residual demand curves converge until they are identical when Kodak and
Fuji face identical residual demand curves of p (10.00 3.00) qK 7.00 qK and
p (10.00 3.00) qF 7.00 qF , respectively. Figure 14.3 shows this equilibrium.
Only when each firm produces 3 million rolls will each firm’s assumption regarding the
other’s output be correct. In equilibrium, total industry output is 6 million rolls of film
and p $4.00 per roll. One of the important implications of the Cournot equilibrium
is that duopolists produce an equilibrium output between the profit-maximizing monopoly output of 4.5 million and the perfectly competitive output of 9 million, where p MC $1.00.
Reaction Functions and the Cournot-Nash Equilibrium
reaction function
A function identifying one
firm’s optimal output (price)
response to every possible output (price) produced (charged)
by competitors.
We can find the Cournot equilibrium using algebra; however, first we need to understand
an important new concept—a firm’s reaction function.2 In a duopoly Cournot game, a
firm’s reaction function identifies the profit-maximizing output that one firm produces
in response to its competitor’s output. More generally, in the context of other models, a
reaction function might identify the profit-maximizing price that one firm would charge
in response to any price that a competitor charged, or the profit-maximizing level of
advertising expenditures that one firm would make in response to the level of advertising
expenditures of its competitors.
To derive Kodak’s and Fuji’s reaction functions, we begin with Cournot’s key
assumption: Kodak assumes that Fuji’s level of output will remain constant, and Fuji
Figure 14.3 Final
Cournot Equilibrium
When qK qF 3.0
In the final Cournot equilibrium, Fuji and Kodak
each assume the other firm
will produce 3.0 million
rolls per year forever, and
therefore both firms face a
residual demand p 7.00
q, and maximize profit
by producing 3.0 million
rolls of film and charging a
price of $4.00 per roll.
$
7.00
Cournot equilibrium
residual demand
p 7.00 q
4.00
MC AC $1.00
1.00
3.00
MR
2
6.00 7.00 Millions of
rolls
The concept was introduced by Arthur L. Bowley, The Mathematical Groundwork of Economics (Oxford: Oxford
University Press, 1924).
380
Chapter 14
Oligopoly
assumes that Kodak’s level of output will remain constant. Given these assumptions, there
exists a profit-maximizing output response for Kodak associated with any given output
produced by Fuji, and vice versa. These functional relationships can be written as:
qK f (qF ) and qF g(qK )
(14.1)
To derive Kodak’s reaction function, qK f (qF ), we use the twice-as-steep rule from
Chapter 11, which states that for a linear demand curve, the marginal revenue curve has
the same price intercept and twice the slope of the demand curve. This rule permits us
to derive the reaction functions using algebra. (Appendix 14.1 uses calculus to derive
these reaction functions in a more precise mathematical way.) For industry demand curve
p 10.00 qK qF , Kodak’s residual demand curve can be identified by noting that
for any given level of Fuji’s output, qF , Kodak assumes that the quantity (100 qF ) is
constant. It follows that for any given qF , Kodak’s demand curve is:
pK (10.00 qF ) qK.
(14.2)
Equation 14.2 is a linear demand curve, with constant intercept $(10.00 qF ) and
slope 1, and therefore Kodak’s marginal revenue is:
MRK (10.00 qF ) 2qK
To obtain Kodak’s reaction function, set MRK MC for profit maximization:
MRK (10.00 qF ) 2qK 1.00 MC
(14.3)
2qK 9.00 qF
(14.4)
or:
Solving Equation 14.4 for qK , Kodak’s reaction function is:
qK 4.5 (1/2)qF
(14.5)
Because Fuji faces identical demand and cost conditions, Fuji’s reaction function is then:
qF 4.5 (1/2)qK
(14.6)
Figure 14.4 illustrates both reaction functions (Equations 14.5 and 14.6) on the
same graph. The Cournot equilibrium occurs at the intersection of the two reaction functions, because only at that point are both firms making the correct assumption concerning the output of the other. We can derive the intersection by substituting the value of qF
from Equation 14.6 into Equation 14.5 and solving for qK as follows:
qK 4.5 1
1
1
a 4.5 qK b 4.5 2.25 qK
2
2
4
412.25 2
9
3
qK 2.25 1 qK 3
4
3
3
14.1
Figure 14.4 The Cournot
Equilibrium Using
Reaction Functions
Kodak’s reaction function is
qK 4.5 (1/2)qF , and
Fuji’s reaction function is qF
4.5 (1/2)qK . Only at
the Cournot equilibrium
where the two reaction
functions intersect and qK
qF 3.0 million rolls is
there no incentive for either
firm to change its output.
The Cournot equilibrium is
not on the red collusion
curve where the firms produce a combined output of
4.5 million rolls and maximize joint profits.
The Cournot-Nash Model
381
qF
9.0
Kodak’s reaction function
4.5
4.0
3.25
3.0
A Cournot-Nash equilibrium
B
Fuji’s reaction function
Collusion
curve
2.5 3.0
4.5
9.0
qK
Substituting qK 3 into Equation 14.6:
qF 4.5 1
1
qK 4.5 132 4.5 1.5 3
2
2
When each firm produces 3 million rolls, neither firm has an incentive to change output;
therefore, Cournot’s assumption that the other will maintain output is correct.
THE COURNOT EQUILIBRIUM
AS A NASH EQUILIBRIUM In Figure 14.4, no
other output combination can be sustained. Suppose Fuji produces 4 million rolls; then
Kodak should produce at the point A on its reaction function, qK 4.5 (4/2) 2.5
million rolls. However, if Kodak produces 2.5 million rolls, then Fuji should operate at
point B on its reaction function and produce qF 4.5 (2.5/2) 3.25 million rolls.
However, that would result in a Kodak response, which would result in a Fuji response,
and so on. A stable equilibrium is achieved only when the two reaction functions intersect and each firm produces the Cournot equilibrium quantity of 3.0 million rolls.
The Cournot equilibrium in Figure 14.4 has the characteristic that both players are
doing the best they can, given the choice of their opponent. Recall from Chapter 13 that
this is precisely the definition of a Nash equilibrium. Figure 14.4 reveals that there is one,
and only one, Nash equilibrium, where qK qF 3 million. If both firms produce 3
million rolls, then a unilateral move to produce more or less than 3 million always reduces
profits for the firm making the move. For example, if Fuji unilaterally reduces its output
from 3 million to 2 million rolls, but Kodak maintains its output at 3 million rolls, then
Fuji’s profits decline from $9 million [ pqF AC (qF ) $(10.00 3 3)(3) ($1.00)(3) $4(3) $3 $9 million] to $8 million [ pqF AC(qF ) $(10.00 2 3)(2) ($1.00)(2) $5(2) $2 $8 million]. If Kodak unilaterally moves to
produce 5 million rolls, but Fuji maintains its output at 3 million, Kodak’s profit declines
from $9 million to $5 million ( pqK AC (qK ) $(10.00 5 3)(5) ($1.00)(5)
$2(5) $5 $5 million). Given any combination of outputs produced by the two
firms except qK qF 3 million, it is always possible for at least one of the firms to
382
Chapter 14
Oligopoly
Cournot-Nash
equilibrium
The equilibrium outcome in a
simultaneous move game where
all the firms assume that the outputs of all the other firms in the
industry will remain constant.
increase its profit through a unilateral move. Because the Cournot equilibrium is also a
Nash equilibrium, the equilibrium is called a Cournot-Nash equilibrium.
THE COLLUSION CURVE Does the Cournot-Nash equilibrium maximize the two
firms’ joint profits? If the firms arrive at the Cournot-Nash equilibrium, each firm earns
$9 million, so joint profits are $18 million. But there are many output combinations that
result in a larger joint profits. For example, if each firm produces 2 million rolls, then each
earns $10 million [ pq AC(q) $(10.00 2 2)(2) ($1.00)(2) $6(2) $2 $10 million], and combined profits are $20 million. Better yet for Kodak and Fuji,
if they each produce 2.25 million rolls, they combine to produce the monopoly output of
4.5 million rolls at p $5.50 per roll, and each earns profit of $10.125 million [ pq AC(q) $(10.00 2.25 2.25)(2.25) ($1.00)(2.25) $5.50(2.25) $2.25 $10.125 million] and joint profits are maximized at $20.25 million. In fact, if Kodak’s and
Fuji’s combined output equals the monopoly output of 4.5 million rolls of film, combined
profits equal $20.25 million, no matter how the output is divided between the two firms.
We have already established that the Cournot equilibrium is based on the questionable assumption that both firms assume the other firm will maintain its current level of
output forever; we have also just discovered that the Cournot equilibrium is also a Nash
equilibrium in a simple two-player game. The Cournot-Nash game is a simultaneous
move, quantity-choice game with homogeneous products. The information structure of
the Cournot-Nash game is imperfect (because the firms move simultaneously), complete,
14.1 APPLICATION Experimental Games and the Cournot-Nash Model
Empirical evidence on the likelihood of Cournot-Nash
behavior is difficult to obtain, because it requires showing that oligopolists select prices between the competitive
price and the monopoly price. Because of this difficulty,
economists have turned to the use of experimental games
to try to determine the likelihood of the Cournot-Nash
outcome. These games are played typically by college students who receive information concerning their profit
payoffs after making an output decision that is similar to
the information in a Prisoner’s Dilemma game. For
example, the students know that if they charge a lower
price than their opponent, they sell more units and are
likely to earn a larger profit (unless the price is too low),
and they know that if they charge a higher price than
their opponent, they sell fewer units and are likely to sustain an economic loss. The student participants are often
allowed to keep some of the profits they earn, so there is
a strong incentive to do well.
3
In one experiment run by economists Lawrence
Fouraker and Signey Siegel, 16 pairs of student duopolists played a game for 25 rounds.3 The results showed
that the Cournot-Nash solution was the most common
outcome occurring in 7 of the 16 games, compared to 5
competitive outcomes, 3 joint profit-maximizing outcomes, and 1 outcome between the Cournot-Nash and
joint profit-maximizing outcome. The mean outcome
was also the Cournot-Nash outcome. When three students instead of two played the game, the competitive
outcome became the most common. In another experiment, economist Charles Holt, Jr. found that the
Cournot-Nash result was the most common outcome
under a variety of different game structures.4 A survey of
experimental results by economist Charles Plott also
found evidence of the Cournot-Nash outcome to be
fairly common.5
Lawrence Fouraker and Signey Siegel. Bargaining Behavior. New York: McGraw-Hill, 1963.
Charles A. Holt, Jr., “An Experimental Test of the Consistent-Conjectures Hypothesis,” American Economic
Review, 1985, pp. 314–325.
5 Charles R. Plott, “Industrial Organization Theory and Experimental Economics,” Journal of Economic Literature,
1982, pp. 1485–1527.
4
14.2
collusion curve
A curve identifying all the output combinations that firms can
produce to maximize joint
industry profits.
The firm that moves second in
the Stackelberg game.
Stackelberg leader
The firm that moves first in the
Stackelberg game.
383
certain, and symmetric (see Chapter 13). In such a game, both firms realize that it makes
sense to produce the Nash equilibrium immediately without the back-and-forth jockeying for output position described by Cournot. Although the final outcome is the same,
the Nash equilibrium is based on a much more reasonable set of behavioral assumptions
than the Cournot equilibrium. The realization that Cournot’s original equilibrium is also
a Nash equilibrium resurrected Cournot’s model, which has become one of the most
widely used models of economic behavior.
We know that to maximize total industry profits at $20.25 million, Kodak and Fuji
must produce together 4.5 million rolls of film, regardless of how output is distributed
between the firms. In Figure 14.4, the red line where qK qF 4.5 million is called the
collusion curve. The collusion curve identifies all the combinations of qK and qF that maximize joint profits. If the two firms combine to produce 4.5 million rolls, however, the
combination will not be a Nash equilibrium. Each firm could, for example, produce 2.25
million rolls and earn a profit of $10.25 million, but then each firm would have an incentive to unilaterally increase output. For example, if Fuji unilaterally increased output to
3.5 million rolls, it would earn a profit of $11.375 million ( pqF AC(qF) $(10.00 2.25 3.5)(3.5) ($1.00)(3.5) ($4.25)(3.5) $3.5 $14.875 $3.5
$11.375 million). None of the points on the collusion curve are Nash equilibria. In
Section 14.5, we explore how Kodak and Fuji might attempt to reach a point on the collusion curve where joint profits are maximized.
14.2
Stackelberg follower
The Stackelberg Model
The Stackelberg Model
You just learned that the Cournot-Nash model is a one-period simultaneous move game
in which both players assume that their opponent’s output is constant even after their
opponent changes its output. In 1934, economist Heinrich von Stackelberg questioned
Cournot’s assumption that both firms take their opponent’s output as given, by considering what would happen if only one of the two firms, known as the Stackelberg follower, assumed its opponent would maintain its output, while the other firm, known as
the Stackelberg leader, used this knowledge about its opponent to select an output that
maximized its profit.6 In the Stackelberg model, the leader makes a more logical assumption than the follower, and as a result, the game’s equilibrium changes.
Firms with Identical Costs and Demand
The Stackelberg model is a sequential game where the Stackelberg leader moves first, and
the Stackelberg follower moves second. Returning to our Kodak-Fuji game, suppose
Kodak is the leader and moves first; then Fuji, the follower, moves; and then the game
ends. The demand and cost conditions are the same as they were before: demand is p 100 qK qF and costs are MC AC $1.00 for both firms. As an initial simplifying assumption, suppose Kodak can select only from eight possible quantities: 2.0, 2.5,
3.0, 3.5, 4.0, 4.5, 5.0, or 5.5 million rolls. Fuji observes Kodak’s quantity and then selects
its quantity. The game tree in Figure 14.5 identifies some of the possible outcomes of
such a game; it is only a partial depiction of the game, because it identifies only Fuji’s
profit-maximizing responses—as opposed to all of Fuji’s possible responses—to Kodak’s
move. Once Kodak selects an output, Fuji chooses its profit-maximizing output by
6
The original work was in Heinrich von Stackelberg, Marktform und Gleichgewicht (Vienna: Springer, 1934).
384
Chapter 14
Oligopoly
Figure 14.5 The
Extensive Form of the
Stackelberg Game ($millions Kodak profit, $millions Fuji profit)
If Kodak can produce only
eight possible outputs
between 2.0 and 5.5 million
rolls, Fuji responds to
Kodak’s output by producing an output that maximizes Fuji’s profit. Kodak
moves first and selects qK 4.5 million rolls, knowing
that Fuji will respond by
producing 2.25 million
rolls. Kodak produces 4.5
million rolls, because that
output maximizes Kodak’s
profit at $10.125 million.
qF 3.5
qK 2.0
F1
qK 2.5
F2
qK 3.0
F3
qK 3.5
F4
qK 4.0
F5
qF 2.5
(10.000, 6.250)
qK 4.5
F6
qF 2.25
(10.125, 5.063)
qK 5.0
F7
qF 2.0
(10.000, 4.000)
qK 5.5
F8
qF 1.75
(9.625, 3.063)
qF 3.25
qF 3.0
qF 2.75
(7.000, 12.250)
(8.125, 10.563)
(9.000, 9.000)
(9.625, 7.562)
K
substituting Kodak’s quantity for qK in Equation 14.6. If Kodak produces 2.0 million
rolls, Fuji produces qF 4.5 (1/2)qK 4.5 1.0 3.5 million rolls; if Kodak produces 2.5 million rolls, Fuji produces qF 4.5 (1/2)qK 4.5 1.25 3.25 million
rolls; and so on. In Figure 14.5, the numbers in parentheses show Kodak’s and Fuji’s profits in millions of dollars, respectively.
What is Kodak’s optimal move? Kodak knows that if it produces 2.0 million rolls, Fuji
produces 3.5 million rolls and Kodak’s profit equals $7 million; if Kodak produces 2.5 million rolls, Fuji produces 3.25 million rolls and Kodak’s profit equals $8.125 million, and
so on. Kodak selects the output that maximizes its profit, or 4.5 million rolls. Fuji
responds by producing 2.25 million rolls. Kodak earns profit of $10.125 million, and Fuji
earns $5.063 million. Then the game ends. Figure 14.5 shows the Cournot-Nash equilibrium at F3 where Kodak and Fuji both produce 3.0 million rolls, and Kodak’s profit is $9.0
million. By moving first, Kodak is able to “force” Fuji to its reaction function, and as a
result, Kodak earns a larger profit than in the Cournot-Nash equilibrium.
Next, we eliminate the simplifying assumption that Kodak can produce only eight
possible outputs, and consider the more realistic game where Kodak can produce any
output. While Kodak can select any output in this new version of the game, Fuji must
be on its reaction function when it takes the second move. This restriction is placed on
Fuji by substituting Fuji’s reaction function, qF 4.5 (1/2)qK , directly into Kodak’s
demand curve, which yields:
1
1
p 110.00 qF 2 qK a 10.00 a 4.5 qK b b qK 5.5 qK qK
2
2
p 5.5 (1/2)qK
(14.7)
14.2
The Stackelberg Model
385
Equation 14.7 is Kodak’s demand curve. Using the twice-as-steep rule, Kodak’s marginal
revenue is:
MRK 5.5 qK
(14.8)
To maximize profit, Kodak sets MRK MC:
MRK 5.5 qK 1.00
or:
qK 4.5
Kodak’s profit-maximizing output is 4.5 million rolls. (Appendix 14.2 derives this result
using calculus.)
Once Kodak produces 4.5 million rolls, we know from Figure 14.5 that Fuji’s optimal response is to substitute qK 4.5 into its reaction function and produce 2.25 million rolls (qF 4.5 (1/2)qK 4.5 2.25 2.25). Total industry output is Q qK qF 4.5 2.25 6.75 million rolls, and price is p 10.00 6.75 $3.25 per roll.
The Stackelberg equilibrium thus yields a greater total industry output and lower price
than the Cournot-Nash equilibrium.
Firms with Different Costs
Having identified the Stackelberg equilibrium in the case of completely symmetric duopolists, it is useful to address one common question: Why is Kodak, and not Fuji, the
Stackelberg leader? The truth is, if both firms are symmetric, there is no theoretical reason why one firm should be the leader and the other the follower. Complete symmetry,
however, is rare in real-world oligopolies. For example, General Motors and Toyota are
not completely symmetric in terms of production costs or product demands. The same
is true for Coca-Cola and Pepsi, Dell and Gateway, Boeing and Airbus, and Kodak and
COMMON ERROR 14.1
Failing to Correctly Substitute the Follower’s Reaction
Function into the Leader’s Demand Curve
To correctly determine the Stackelberg equilibrium, it
is necessary to substitute the follower’s reaction function into the leader’s demand curve. Many students
mistakenly substitute the follower’s reaction function
into the leader’s reaction function instead. However,
one of the key points about the Stackelberg equilibrium is that in equilibrium, the leader is not on its own
reaction function. Ironically, by constraining both
firms to their reaction functions, this error results in
finding the Cournot-Nash equilibrium. For example,
if you substitute the follower Fuji’s reaction function
into the leader Kodak’s reaction function, it yields:
qK 4.5 1
1
1
a 4.5 qK b 2.25 qK
2
2
4
3
q 2.25
4 K
qK 412.25 2
9
3
3
3
This is Kodak’s Cournot-Nash equilibrium output of
3 million rolls. As the Stackelberg leader, however,
Kodak’s equilibrium output is 4.5 million rolls, or 50
percent greater. Don’t make this common error!
386
Chapter 14
Oligopoly
Fuji. In each of these oligopolies, there is some variation between the firms in terms of
costs and demands.
Consider a case where one of the duopolists has an advantage because of lower production costs and/or greater demand. In this case, it is logical to assume that the firm
with the advantage is the Stackelberg leader. Suppose, for instance, that Kodak’s costs are
MCK ACK $1.00 per roll, but Fuji’s costs are 10 percent higher, so MCF ACF $1.10 per roll. Under these cost conditions, it makes sense to assume that Kodak, the
low-cost firm with the theoretical ability to drive Fuji from the market by charging a price
below $1.10, controls the game by moving first.
Fuji’s higher costs change its reaction function and the Stackelberg equilibrium.
Kodak’s reaction function, however, remains unchanged, because its costs and demand have
not changed. To calculate the new Stackelberg equilibrium, it is necessary to first identify
Fuji’s new reaction function and then substitute the new reaction function into Kodak’s
demand curve. To derive Fuji’s new reaction function, we start with Fuji’s demand curve:
pF (10.00 qK) qF
(14.9)
According to the twice-as-steep rule, Fuji’s marginal revenue is:
MRF (10.00 qK) 2qF
To obtain Fuji’s reaction function, set MRF MC $1.10:
MRF (10.00 qK ) 2qF 1.10 MCF
(14.10)
2qF 8.90 qK
(14.11)
or:
Solving Equation 14.11 for qF, Fuji’s new reaction function is:
qF 4.45 (1/2)qK
(14.12)
Substituting qF 4.45 (1/2)qK from Equation 14.12 into Kodak’s demand curve
yields:
1
1
p 110.00 qF 2 qK a 10.00 a 4.45 qK b b qK 5.55 qK qK
2
2
p 5.55 (1/2)qK
(14.13)
Equation 14.13 is Kodak’s new demand curve, so Kodak’s marginal revenue is:
MRK 5.55 qK
To maximize profit, Kodak equates marginal revenue to marginal cost:
MRK 5.55 qK 1.00
so:
qK 4.55
(14.14)
14.3
The Bertrand Model
387
14.2 APPLICATION Fuji’s Purchase of Wal-Mart’s Photo-Processing Plants
In the market for wholesale photofinishing, Kodak is by
far the leading firm, controlling more than 60 percent of
the market. Fuji is the second-largest wholesale photofinisher. In 1993 and 1994, Kodak began to expand its photofinishing capacity by purchasing two photofinishing
labs from Walgreen’s, which owns a large chain of drugstores. In 1995, Kodak also signed an agreement with
Kmart to run all 320 of Kmart’s in-store photofinishing
mini-labs. In 1996, Kodak continued expanding its output, when it gained control of all seven of Eckard Drugs’s
photofinishing labs. The Eckard acquisition pushed
Kodak’s market share beyond 80 percent.
In early 1996, Fuji controlled 15 photofinishing labs
in the United States and its market share was between
five and six percent. After observing Kodak’s output
expansion from 1993 to 1996, Fuji responded in July
1996 by purchasing all six of Wal-Mart’s large wholesale
photofinishing labs. Furthermore, Fuji gained control
over all of Wal-Mart’s out-of-store photofinishing services. The acquisition of Wal-Mart’s labs, which followed
Fuji’s building of a large photofinishing lab in South
Carolina earlier in the year, increased Fuji’s market share
to 16 percent in 1997.
In the context of the Stackelberg model, Kodak
moved first and expanded its photofinishing operations,
then Fuji, after observing Kodak’s expansion, expanded
its photofinishing capacity to a level it considered optimal, given Kodak’s capacity. As expected, after Fuji’s
capacity expansion, photofinishing prices declined dramatically, as did Kodak’s market share.
To obtain Fuji’s profit-maximizing output, substitute qK 4.55 into Equation 14.12:
1
1
qF 4.45 qK 4.45 14.552 4.45 2.275 2.175
2
2
If Fuji’s costs are 10 percent higher than Kodak’s, Kodak is the Stackelberg leader and
produces 4.55 million rolls of film, while Fuji is the follower and produces 2.175 million
rolls. Total output is 4.552.175 6.725 million rolls and p 10.00 6.725 $3.275. A 10 percent increase in Fuji’s costs results in the following:
• A 1.1 percent increase in Kodak’s output from 4.5 million rolls to 4.55 million
• A 3.3 percent decrease in Fuji’s output from 2.25 million rolls to 2.175 million
• A 0.37 percent decrease in total output from 6.75 million rolls to 6.725 million
• A 0.80 percent increase in price from $3.25 to $3.275
14.3
The Bertrand Model
In 1883, Joseph Bertrand criticized Cournot’s equilibrium by showing that if each duopolist assumes that its opponent holds constant its price (instead of its quantity), Cournot’s
logic results in an entirely different equilibrium.7 Betrand considered the case where the
two firms produce a homogeneous product. Later theorists expanded the model to
include cases where the firms produce differentiated products. We will consider both
cases, beginning with the simpler but less-realistic case of homogeneous products.
The Betrand Model with Homogeneous Products
Because most consumers consider a seat on any non-stop flight between the same two
cities a homogeneous product, consider a game that Southwest Airlines and US Airways
play on flights between Albany, New York and Baltimore, Maryland. Suppose the
7 Bertrand’s model was developed in a critical review of Cournot published 45 years after the original article in
Joseph Bertrand, book review of Recherches sur les principes mathematiques de la theorie des richesses, in Journal de
Savants, Vol. 67, 1883, pp. 499–508.
388
Chapter 14
Oligopoly
demand for seats is p 1,000 Q and the marginal cost of adding an additional passenger on a one-way flight is $70. If Southwest assumes that US Airways always maintains its price at its current level, pUS , then Southwest’s demand is dependent upon the
current relationship between Southwest’s price, pS , and the US Airways price, pUS .
Because seats on any flight are a homogeneous product, passengers always purchase a
ticket on the less-expensive airline. If Southwest’s price is greater than the US Airways
price, US Airways captures the entire market, and qS 0. If Southwest’s price is less than
the US Airways price, Southwest captures the entire demand, and qS 1,000 pS .
Finally, if pS pUS , the two firms split the market, in which case each obtains one-half
of the total industry demand, and Southwest’s demand curve is:
qS 1
11,000 pS 2
2
1
qS 500 pS for pS pUS
2
(14.15)
Southwest’s demand curve can therefore be identified as:
0 if pS 7 pUS
1
qs ≥ 500 pS if pS pUS
2
1,000 pS if pS 6 pUS
Bertrand model
A model developed by Joseph
Bertrand that criticized
Cournot’s model by arguing
that if firms assume that all
other firms hold their prices
constant, Cournot’s logic results
in an entirely different outcome, with price equal to marginal cost.
(14.16)
The Bertrand model with a homogeneous product is driven by the assumption that
one of the airlines can capture the entire market if it charges a lower fare than its competitor. Given this assumption, if Southwest charges a fare just slightly less than pUS , it
will virtually double its output and profit. This implies that if pS pUS , where represents a number that is infinitesimally greater than 0, Southwest captures the entire market. Of course, if pUS pS , then US Airways captures the entire market.
Figure 14.6 identifies Southwest’s demand curve if US Airways charges a fare of $400
for a one-way ticket. If Southwest charges any fare greater than $400, US Airways sells
all the tickets, Southwest sells no tickets, and Southwest’s demand curve is the vertical
blue line along the price axis. If Southwest matches the US Airways fare and charges
$400, the airlines divide the total demand of 600 tickets, and Southwest sells 300 tickets
at point A. Finally, if Southwest charges any fare less than $400, US Airways sells no tickets, Southwest sells all the tickets, and Southwest’s demand curve is identical to the industry’s demand curve p 1,000 qS .
What is Southwest’s optimal pricing strategy? If Southwest assumes US Airways
maintains its price at $400, then Southwest can charge a fare slightly less than $400 and
capture the entire market. To simplify the analysis, we assume that fares can be quoted
only in whole dollar amounts. If Southwest lowers its fare by $1.00 to $399, its output
will more than double, from 300 to 601 tickets. Southwest’s marginal revenue associated
with a fare reduction from $400 to $399 is calculated as follows. First, calculate the
change in total revenue associated with the fare reduction: At pS $400, TRS pqS $400(300) $120,000, and at pS $399, TRS pqS $399(601) $239,799. Next,
calculate Southwest’s marginal revenue:
¢TRS
$119,799
$239,799 $120,000
$398.00
MRS ¢qS
601 300
301
Because marginal revenue is $398 and marginal cost is $70, Southwest is better off if it
reduces its fare from $400 to $399.
14.3
Figure 14.6 The
Southwest Airlines
Demand Curve if US
Airways Charges $400
for a Ticket
If Southwest Airlines
assumes that US Airways
will maintain its fare at
$400, its demand curve
consists of three segments.
If Southwest charges a fare
higher than $400, it sells
zero tickets. If Southwest
charges a fare equal to
$400, it sells half the 600
tickets demanded or 300
tickets at point A. If
Southwest charges a fare
below $400, it sells all the
tickets demanded.
The Bertrand Model
389
$
1,100
1,000
900
800
700
600
500
A
400
300
200
100
70
MC AC
100 200 300 400 500 600 700 800 900 1,000 1,100 qS
Should Southwest reduce its fare further in order to sell more tickets? To sell one
additional ticket to the 602nd passenger, Southwest must reduce its fare to $398. The
marginal revenue of reducing its fare from $399 to $398 is:
¢TRS
$239,596 $239,799
$203
MRS $203
¢qs
602 601
1
Marginal revenue is negative (notice that Southwest is operating on the bottom half of a
linear demand curve, where demand is inelastic), and therefore, any further fare reductions reduce Southwest’s total revenues and profit. Southwest’s optimal policy is to lower
its fare to $399.
But surely, from US Airways’s perspective, the game cannot end there. After
Southwest lowers its fare to $399, what is US Airways’ optimal pricing strategy? If US
Airways assumes that Southwest will maintain its fare at $399, then US Airways can
charge a fare slightly less than $399 and capture the entire market. If US Airways lowers
its fare to $398, its output increases from zero at a fare of $400 to 602 at a fare of $398.
US Airways’ marginal revenue associated with this price reduction is:
MRUS ¢TRUS
¢qUS
$239,596 $0
$239,596
$398.00
602 0
602
Because US Airways’ marginal revenue is $398 and marginal cost is $70, US Airways is
better off if it reduces its fare to $398.
But now Southwest is left out of the market, unless it lowers its fare to $397, in
which case US Airways will respond by lowering its fare to $396. When does the game
end? Any fare greater than MC $70 will result in one airline reducing fares, which in
turn will result in a fare reduction by the other, and so on. Equilibrium occurs only when
both fares equal MC $70, because only then will neither firm have an incentive to
reduce its fare further.
Figure 14.7 identifies the reaction functions for Southwest and US Airways.
Southwest’s reaction function goes through the point ($70, $70), but is not identified for
390
Chapter 14
Oligopoly
Figure 14.7 A Bertrand
Equilibrium with
Homogeneous Products
If Southwest charges a fare
above MC $70, US
Airways responds by charging a fare $1.00 less than
Southwest’s fare, and the
US Airways reaction function lies at a distance $1.00
below the 45-degree line. By
the same reasoning,
Southwest’s reaction function lies at a distance $1.00
to the left of the 45-degree
line. The Bertrand equilibrium is the Nash equilibrium at the intersection
point ($70, $70).
pU S
Southwest’s reaction
function
US Airways’ reaction
function
70
pS
70
fares below $70, because a fare cut below $70 would result in economic losses; the fare
would be below average cost. For all pUS $70, Southwest charges a fare pS (pUS $1.00); therefore, for pUS $70, Southwest’s reaction function lies at a distance $1.00
to the left of the 45-degree line. Similarly, the US Airways reaction function lies at a distance $1.00 below the 45-degree line. For p $70, the two reaction functions will be
parallel to each other and also parallel to the 45-degree line where pS pUS. The Nash
equilibrium occurs at the intersection point ($70, $70), because this is the only point
where both firms are doing the best they can given the choice of their competitor.
We can interpret the Bertrand model as a simultaneous move game. Both firms reason: “If we set our fare at any p MC, then our opponent will set its fare at p , where
is a number infinitesimally greater than zero, and we will sell nothing. But, if we set
our fare at p MC $70, then we either capture the entire market or split the market
50-50. We should, therefore, set our fare equal to MC.”
The Bertrand Model with Product Differentiation
In the Bertrand duopoly model with a homogeneous product, price equals MC, the perfectly competitive price. In many oligopolies, however, firms produce differentiated products. Let’s analyze the Bertrand model in the case where duopolists Ben & Jerry’s and
Häagen Dazs produce differentiated premium ice cream.8
Suppose that because of consumer brand loyalty, Ben & Jerry’s and Häagen Dazs face
the following symmetric demand curves for pints of ice cream:
qB 9.60 2pBpH
(14.17)
qH 9.60 2pHpB
(14.18)
8 For a fairly simple treatment of the introduction of product differentiation into the Bertrand model, see Eric
Rasmusen, Games and Information: An Introduction to Game Theory (Oxford: Basil Blackwell, 1989), pp. 267–269.
14.3
The Bertrand Model
391
14.3 APPLICATION Bertrand Pricing in the Airline Industry
Pricing behavior in the airline industry has often been
consistent with Bertrand price behavior. Many airlines
have followed a policy of pricing at near-marginal cost on
routes where they face competition, especially from lowprice carriers such as Southwest Airlines or JetBlue. The
carriers’ rationale for this behavior is consistent with the
Bertrand model assumptions. Each carrier fears that if its
fares are even slightly higher than the competition, it will
lose nearly its entire market share.
Consider the following fares offered by US Airways
over the Internet on August 28, 2002. The distance from
Syracuse to Baltimore is almost identical to the distance
from Albany to Baltimore.
US Airways competed against Southwest Airlines in
the Albany market, but not in the Syracuse market. In
Route
the Albany market, US Airways essentially matched
Southwest’s low fares. In the Syracuse market, US
Airways was a virtual monopolist on flights to Baltimore
and Raleigh-Durham. US Airways’ lowest advancedpurchase fare from Syracuse to Baltimore was 50.4 percent higher than its lowest advanced-purchase fare from
Albany to Baltimore, and its lowest fare from Syracuse to
Raleigh-Durham was 22.7 percent higher than its lowest
fare from Albany to Raleigh-Durham. If Southwest
charged fares close to marginal cost, then the Albany fares
can be interpreted as Bertrand equilibrium fares. US
Airways reasoned that if its Albany fares were higher than
Southwest’s, it would fly mostly empty planes out of
Albany to Baltimore or Raleigh-Durham.
US Airways Lowest
Round-Trip Advanced Fare
Southwest Lowest
Round-Trip Advanced Fare
Syracuse, NY to Baltimore, MD
$173.00
No flights
Albany, NY to Baltimore, MD
$115.00
$120.00
Syracuse, NY to Raleigh-Durham, NC
$216.00
No flights
Albany, NY to Raleigh-Durham, NC
$167.00
$172.00
where qB , qH 0 and pB , pH $4.80. Quantities are measured in millions of pints and
price is measured in dollars. The restrictions that qB and qH 0 prevent quantities from
becoming negative, and the restrictions that pB and pH $4.80 prevent output from
becoming infinite as an opponent’s price increases. In Equation 14.17, Ben & Jerry’s
demand is a negative function of its own price pB , and a positive function of Häagen
Dazs’ price pH. Unlike the case of a homogeneous product, even if the price of Ben &
Jerry’s is greater than the price of Häagen Dazs—that is, pB pH —Ben & Jerry’s still
sells some ice cream. Some consumers buy Ben & Jerry’s ice cream because of its unique
taste or certain flavors it produces, even if it is more expensive than Häagen Dazs.
Furthermore, an increase in pH results in an increase in qB ; that is, an increase in the price
of Häagen Dazs increases the quantity demanded of Ben & Jerry’s.
Assume MCB MCH $1.20 per pint, so that in the absence of product differentiation, the Bertrand equilibrium price equals $1.20. In the following derivation of Ben
& Jerry’s reaction function, remember that the reaction function pB f (pH ) is in terms
of price, not quantity, and pH is assumed to be constant throughout. We first solve
Equation 14.17 for pB :
2pB 9.60pH qB
392
Chapter 14
Oligopoly
Dividing both sides of the equation by 2 yields:
1
1
pB a 4.80 pH b qB
2
2
(14.19)
Because pH is assumed to be constant, using the twice-as-steep rule, MRB is:
1
MRB a 4.80 pH b qB
2
(14.20)
Using Equations 14.19 and 14.20, it is possible to calculate all the points on Ben &
Jerry’s reaction function. First, from Equation 14.20, we compute Ben & Jerry’s profitmaximizing quantity for any given price of Häagen Dazs pH. Then from Equation 14.19,
we calculate Ben & Jerry’s price. For example, if the price of Häagen Dazs is pH $2.00
per pint, we compute Ben & Jerry’s profit-maximizing quantity by setting MRB MC
and substituting pH $2.00 into Equation 14.20, which yields MRB (4.80 (1/2)(2.00)) qB 1.20 MC. Solving for qB, qB (4.801.00) 1.20 4.60 million pints. Once we know that Ben & Jerry’s profit-maximizing quantity is 4.60 million
pints, we calculate Ben & Jerry’s price from Equation 14.19: pB (4.80 (1/2)(2.00))
(1/2)(4.60) (4.801.00) 2.30 $3.50 per pint. This identifies one point on
Ben & Jerry’s reaction function: When Häagen Dazs charges pH $2.00, Ben & Jerry’s
maximizes its profit by charging pB $3.50. If the price of Häagen Dazs is pH $4.00,
then MRB (4.80 (1/2)(4.00)) qB 1.20 MC; qB (4.802.00) 1.20 5.60 million pints; and pB (4.80 (1/2)(4.00)) (1/2)(5.60) (4.802.00) 2.80
$4.00. This identifies a second point on Ben & Jerry’s reaction function: When
Häagen Dazs charges pH $4.00, Ben & Jerry’s charges pB $4.00.
Generalizing these results for any price of Häagen Dazs, Ben & Jerry’s maximizes its
profit by equating MRB to MC from Equation 14.20:
1
MRB a 4.80 pH b qB 1.20 MC
2
(14.21)
Solving Equation 14.21 for qB :
1
qB 3.60 pH
2
(14.22)
To obtain Ben & Jerry’s reaction function pB f (pH ), substitute qB from Equation 14.22
into Equation 14.19:
1
1
1
1
1
pB a 4.80 pH b qB a 4.80 pH b a 3.60 pH b
2
2
2
2
2
1
1
pB a 4.80 pH b 1.80 pH
2
4
1
pB 3.00 pH
4
(14.23)
Equation 14.23 is Ben & Jerry’s reaction function. Because both firms face symmetric
demand and costs, Häagen Dazs’ reaction function is derived the same way:
1
pH 3.00 pB
4
(14.24)
Figure 14.8 shows the two reaction functions and the Nash equilibrium. To obtain
the Nash equilibrium, we find the intersection point of the two reaction functions by sub-
14.4
Figure 14.8 A Bertrand
Equilibrium with Product
Differentiation
With product differentiation, each firm’s profitmaximizing response to a
price increase by its opponent is to increase price. If
Häagen Dazs charges a price
of $2.00, Ben & Jerry’s
profit-maximizing response
is to charge a price of $3.50
at point A. If Häagen Dazs
charges a price of $4.00,
Ben & Jerry’s response is
also to charge a price of
$4.00 at the intersection of
the two reaction functions
at point B. Point B is the
Bertrand equilibrium.
pH
The Dominant Firm Model
393
Ben & Jerry’s reaction function
1
pB 3.00 pH
4
B
4.00
Häagen Dazs’ reaction function
1
pH 3.00 pB
4
3.00
A
2.00
3.00 3.50 4.00
pB
stituting Häagen Dazs’ reaction function pH 3.00(1/4)pB from Equation 14.24 into
Ben & Jerry’s reaction function from Equation 14.23, and then solving for pB as follows:
pB 3.00 or:
1
1
1
a 3.00 pB b 3.75 pB
4
4
16
1613.75 2
15
pB 3.75 1 pB 4.00
16
15
By analogous reasoning, pH $4.00.
The Bertrand equilibrium with product differentiation yields a price of $4.00 per
pint. In the absence of product differentiation, the Bertrand equilibrium is pB pH MC $1.20 per pint. The introduction of product differentiation has resulted in a dramatically increased price, and a price that is well above marginal cost.
14.4
The Dominant Firm Model
The Cournot-Nash, Stackelberg, and Bertrand games are played by large firms. These
models are therefore best applied to real-world markets in which large firms compete
against each other. Some markets, however, are characterized by one dominant firm that
controls a large percentage of total industry output and a significant number of relatively
small fringe firms that compete with the dominant firm.9 Such markets are structured
like the personal computer market in the 1980s, which was dominated by IBM, but also
included many smaller PC manufacturers such as Compaq, Hewlett-Packard, Digital
9 This model is sometimes referred to as the Forchheimer Model, because it was derived in Karl Forchheimer,
“Theoretisches zum unvollstandegen Monopole,” Schmollers Jahrbuch, 1908, pp. 1–12; A. J. Nichol, Partial
Monopoly and Price Leadership (Philadelphia: Smith-Edwards, 1930). See also Dean A. Worcester, “Why Dominant
Firms Decline,” Journal of Political Economy, August 1957, pp. 338–347.
394
Chapter 14
Oligopoly
Equipment, Epson, Dell, and Gateway. In markets dominated by one large firm, we can
assume that the dominant firm is able to set the industry price, and that the fringe firms
then take the dominant firm’s price as given and set their prices to maximize profits, given
the dominant firm’s price.
Figure 14.9 illustrates the dominant firm model. The industry demand curve, DI, is
the linear demand curve ABC and the fringe firms’ supply curve is Sf . The dominant
firm’s demand curve, DD , is a residual demand curve that is obtained by subtracting the
fringe’s supply curve Sf from the total industry demand curve DI. For example, if price
is p4, total industry quantity demanded equals q2 units, and the fringe would supply the
entire industry demand of q2 units. Therefore, if p p4, the residual quantity demanded
for the dominant firm would be zero at point E. If price is p1, total industry quantity
demanded equals q5 units, the fringe would supply zero, and the residual demand for the
dominant firm equals q5 units. If p p2, then total industry quantity demanded equals
q4 units, the fringe firms supply q1 units, and the residual demand for the dominant firm
equals q3 q4 q1 units. Similar calculations can be done to obtain the dominant firm’s
residual demand for any price between p1 and p4. Such calculations yield the dominant
firm’s demand curve DD, which is the dark-blue kinked line EGIBC. The dominant firm’s
marginal revenue curve for quantities less than q5 is MRD and bisects the segment EGIB
of the demand curve DD. Recall from Chapter 11 that a profit-maximizing monopolist
always prices where demand is elastic; therefore, the segment of the marginal revenue
curve for quantities greater than q5, that is, for the highly inelastic segment BC of the
kinked demand curve EGIBC, is not shown in Figure 14.9.
If the dominant firm’s marginal cost curve is MCD , the dominant firm maximizes
profit by setting MRD MCD at point F and by producing qD units of output, which it
sells at price p3 taken from its demand curve at point G. The fringe firms take the price
p3 as given, respond as perfectly competitive firms, and produce qf units at point K. Total
industry demand equals qT qD qf at point L.
A major implication of the dominant firm model is that the dominant firm’s market
share often declines over time. If the competitive fringe firms earn positive economic
Figure 14.9 Equilibrium
in the Dominant Firm
Model
Industry demand DI is the
straight line ALJBC. The
fringe firms’ supply curve is
Sf . To obtain the dominant
firm’s residual demand
curve, subtract Sf from DI ,
which yields the dark-blue
demand curve EGIBC. The
dominant firm equates its
marginal cost, MCD, to
marginal revenue, MRD,
and produces output qD and
sets the price at p3. The
fringe firms take the price
p3 as given and produce an
output of qf units. Total
output is qT qD qf .
$
p5 A
Sf
DI
p4 E
DD
K
p3
G
L
MCD
p2 H
p1
q1
J
I
F
qf
q2 qD
qT
q3 q4
B
q5
MRD
C
q6 Quantity
14.4
COMMON ERROR 14.2
The Dominant Firm Model
395
Incorrectly Identifying the Price in the Dominant Firm Model
Because of the complexity of Figure 14.9, students
often get confused and misidentify the dominant
firm’s equilibrium price. For example, in this figure,
the dominant firm’s residual demand curve DD , and
marginal cost curve MRD , are correctly identified;
however, the price p is taken from point B, where
MCD MRD. Students often fail to go up to the
demand curve to obtain the price at which the dominant firm would sell the qD units it produces. Given
price p, the fringe firms supply qf . Therefore, according
to these mistaken students, total output qT qf qD ,
which is wrong. By construction of the residual
demand curve, we know that qT qf q1, not qf qD .
Be careful not to make this common error when analyzing the dominant firm model.
$
Sf
MCD
DI
DD
This is wrong!
p
A
qf
B
C
E
qD
q1 qT
MRD
Quantity
Common Error Figure 14.2 The dominant firm’s
demand curve is DD and marginal revenue is MRD, but
the price is not p.
14.4 APPLICATION The Decline of Dominant Firms
There is considerable empirical evidence providing support for the dominant firm model. This evidence suggests
that dominant firms’ market shares often decline substantially over time. U.S. Steel is a case in point. A classic dominant firm after its formation in 1902, U.S. Steel’s
market share was 65 percent at the time of its creation.
U.S. Steel charged high prices and earned high profits
during the first twenty years of its existence. As a result,
new competitors entered and established competitors
expanded their outputs. By 1920, U.S. Steel’s market
share had declined to 50 percent.
In the low-volume segment of the copier industry,
Xerox behaved as a classic dominant firm price leader.
Xerox set high profit-maximizing prices in this sector and
conceded market share to its smaller rivals. Between
1961 and 1967, more than twenty-five firms entered the
low-volume segment of the market, and by the mid1970s, copier prices had declined and Xerox had lost its
dominant position.
Reynolds International Pen Corporation is one of
the most dramatic examples of a dominant firm’s decline.
Reynolds invented an improved ballpoint pen that
enabled the ink to flow much more smoothly to the pen’s
ball point. It began selling these pens in 1945. Initially,
the pens cost approximately $0.80 each to produce and
Gimbel’s department stores sold them for $12.50.
Gimbel’s sold 10,000 pens the first day they went on sale.
By early 1946, Reynolds was producing 30,000 pens a
day and earning large economic profit. Reynolds’s high
prices encouraged large-scale entry. By Christmas of
1946, many firms were in the industry and prices had
fallen to as low as $0.88. By 1948, competitors had managed to greatly reduce the average cost of producing pens,
and prices declined to $0.39. In 1951, prices declined
further and reached a low of $0.19. By then, Reynolds,
which had lowered its prices as entry occurred, was long
gone from the industry. Reynolds came and went quickly,
but for one brief moment it was a dominant firm price
(continued)
396
Chapter 14
Oligopoly
leader and earned very large economic profit. This example points out the risks associated with dominant firm
price leadership. By being completely passive toward its
competitors, Reynolds found itself knocked out of the
market in just a few years.
There are many more examples of dominant firm
price leaders—International Harvester in farm equip-
ment, Goodyear in tires, RCA in color televisions,
General Electric in appliances, and IBM in computers—
all of these firms lost significant market share to competitors as a result of a high price policy and passivity.
With too little concern for competitors’ entry, these companies stood by while they tumbled from positions of
complete prominence.
profit at price p3 in Figure 14.9, there will be an incentive for the fringe supply Sf to
increase over time as new firms enter and as existing fringe firms expand their outputs. An
increase in the fringe supply curve causes the market share of the fringe to increase.
Furthermore, as the fringe supply Sf increases, the residual demand for the dominant firm
DD shifts to the left (decreases), and the dominant firm’s relative share of output declines.
The dominant firm is remarkably passive in this model. This passivity is often cited
as a major weakness of the dominant firm model. As we will see in Chapter 15, there are
strategies that a dominant firm can adopt to try to maintain its market power in the face
of competition from the competitive fringe.
14.5
Collusion: The Great Prisoner’s Dilemma
Recall from Chapter 13 that the basic lesson of the Prisoner’s Dilemma is that defecting
is a dominant strategy, which makes it difficult for oligopolists to collude effectively. In
this section, we consider how firms attempt to solve the Prisoner’s Dilemma. We begin
by analyzing the case of firms with identical demand and cost curves and then proceeding to the case of differing demand and cost curves.
Collusion with Identical Demand and Cost Conditions
Table 14.1, which reproduces Table 13.2, shows Kodak and Fuji as completely symmetric firms that face identical demands and produce with identical costs. Regardless of
whether this game is played once or any finite number of times, defecting is the dominant strategy for Kodak and Fuji. The companies, however, would be better off if they
Table 14.1 A Prisoner’s Dilemma Game of Collusion ($millions Kodak’s profit, $millions
Fuji’s profit)
Fuji Action
Collude
high price
Kodak Action
Collude
high price
500
Defect
low price
600
500
100
Defect
low price
100
300
600
300
14.5
infinite game
A static game that is repeated
forever.
tit-for-tat
A strategy in a Prisoner’s
Dilemma game in which a
player starts off cooperating in
the first round, and in every
subsequent round adopts its
opponent’s strategy in the previous round.
Collusion: The Great Prisoner’s Dilemma
397
could successfully collude and charge a high price. There is a great incentive for Kodak
and Fuji to find a method of achieving the high price outcome. In the United States,
overt collusion is illegal because of the antitrust laws, and therefore firms often look for
alternative methods of coordination.
Suppose Kodak and Fuji believe that the game in Table 14.1 will go on forever; that
is, that the game is an infinite game. If the firms are going to successfully solve the
Prisoner’s Dilemma, they must establish an environment where each believes the other
will abide by a collusive agreement. To establish such a trusting environment, Kodak and
Fuji must first figure out a way to signal each other their willingness to abide by a collusive agreement. Second, they must develop a system for detecting any defections from the
agreement. And third, defections must be punished in order to deter future defections.
Signaling, detection, and punishment: In the absence of these three conditions, Kodak
and Fuji have such a strong incentive to defect that the outcome of the game will almost
certainly be the dominant strategy of: Defect, Defect.
One possible strategy that incorporates signaling, detection, and punishment is the
following:
1. Start in the first round by colluding and charging a high price.
2. In every subsequent round, adopt your opponent’s strategy in the previous
round: that is, in round N, adopt your opponent’s strategy in round N 1.
Suppose Kodak adopts this strategy. In the first round, Kodak charges a high price. If Fuji
also charges a high price in round 1, then Kodak charges a high price in round 2.
However, if Fuji charges a low price in round 1, Kodak punishes Fuji’s behavior in round 1
by charging a low price in round 2. If Kodak and Fuji both adopt this strategy, then both
charge a high price in round 1 and in every subsequent round. This simple two-part strategy incorporates signaling, detection, and punishment, and is known as tit-for-tat. By
setting a high price in round 1, Kodak and Fuji signal to each other their willingness to
collude. Furthermore, any future defection is quickly detected when prices are
announced, and all defections are punished in the next round.
Tit-for-tat became a famous solution to the Prisoner’s Dilemma when political scientist Robert Axelrod invited a group of world-renowned game theorists to enter a competition to try to solve the Prisoner’s Dilemma.10 In Axelrod’s competition, 14 invited
experts submitted computer programs to solve the Prisoner’s Dilemma. Each program
was run a total of 15 times: once in head-to-head competition against each of the other
13 programs, once against a random computer-generated program, and once against
itself. The entrants knew that the game would last 200 rounds. To the surprise of many,
the winner of the competition was Anatol Rapoport, who used the simple tit-for-tat strategy. At the end of the competition, Axelrod concluded that tit-for-tat won because it
exhibits three important characteristics: It is nice toward competitors; it punishes all
defections; and it forgives defectors who return to the fold. Tit-for-tat is “nice” in the
sense that it never initiates an aggressive action. Yet it is “aggressive” in the sense that it
punishes all aggressive moves, even first-time defections. It is “forgiving” in that it rewards
defectors who revert to collusion by also reverting to collusion. Virtually all of the highly
rated strategies submitted to Axelrod exhibited the characteristic of “niceness.” In fact,
each of the top eight ranking strategies was nice in the sense that a player using any of
these strategies was never the first player to defect, and never defected until very close to
the end of the game.
10
Robert Axelrod. The Evolution of Cooperation. New York, Basic Books, 1984.
398
Chapter 14
Oligopoly
14.5A APPLICATION Price Fixing in the World Vitamin Market
At the beginning of this chapter, we learned that Roche
Holding and BASF were fined a combined $725 million
for fixing the prices of vitamins in violation of the
Sherman Act (see Chapter 12). The vitamin price-fixing
conspiracy lasted from January 1990 through February
1999—almost the entire decade of the 1990s. It included
not only the Swiss firm Roche and the German firm
BASF but Aventis (France), Solvay Pharmaceuticals
(Netherlands), Merck (Germany), Daiichi (Japan), Esai
(Japan), and Takeda Chemical (Japan). In all, the conspiracy included eight large companies in five different
countries. If you used vitamin A, E, B1, B2, B5, B6, C,
D3, biotin, folic acid, or beta carotene during the 1990s,
you paid a price that was set by a world-wide price-fixing
conspiracy at artificially high levels well above the average
cost of producing these vitamins. Furthermore, even if
you never took a vitamin in the 1990s, if you ate beef,
chicken, or pork, you paid a price that was elevated,
because the vitamin additives given to cattle, chickens,
and pigs were sold at artificially high prices.
How did these eight companies solve the Prisoner’s
Dilemma? First, consistent with Axelrod’s conclusion,
they were extremely nice to each other. They held
numerous meetings to fix prices and divide the world
vitamin market. At these meetings, they allocated sales
volumes and market shares. They went so far as to allocate specific customers and rig bids to specific buyers.
The meetings signaled a strong desire to set a high collusive price. Second, they worked out a formalized method
of detecting defectors. Meetings and conversations were
held to ensure compliance with the agreements, and the
firms exchanged their sales and pricing data with each
other. Finally, a method of punishment was available for
any firm that violated the agreement by selling too many
units or by charging too low a price: that firm’s market
share could be reduced at a later meeting.
The vitamin conspiracy shows that it is possible for
firms to overcome the Prisoner’s Dilemma through direct
collusion. To emphasize this point, when Assistant U.S.
Attorney General Joel Kline announced that Roche and
BASF had agreed to plead guilty and pay the largest
antitrust fines in U.S. history, he stated:*
Let me be clear about this: contrary to what
some have suggested, these kinds of cartels are
by no means transient or unstable. They are
powerful and sophisticated and, without intervention by antitrust authorities, will often go on
indefinitely.
*Statement of Assistant Attorney General Joel I. Kline, May 20, 1999, http://www.usdoj.gov.
Collusion with Differing Costs or Demand Conditions
As difficult as it is for firms with identical costs and demands to successfully solve the
Prisoner’s Dilemma, it is even more difficult for firms with differing costs or demands to
successfully collude. Economists have developed many models of oligopoly behavior
under conditions of differing demands and costs. In this section, we explore briefly two
of those models: the kinked demand curve and price leadership models.
THE KINKED DEMAND CURVE MODEL In Figure 14.10(a), duopolists produce
a homogeneous product with different marginal costs. Assume that if the firms charge the
same price, they divide the demand equally, so demand curve D represents one-half of
the total industry demand curve. One firm has low marginal costs, MCl , and the other
firm has higher marginal costs, MCh . The low-marginal-cost firm prefers an industry
price of pl ; the high-cost firm prefers a higher price, ph .
Suppose the two firms play a sequential game. The low-cost firm moves first and sets
the price at pl . The high-cost firm moves next. Before moving, the high-cost firm makes
two assumptions: (1) If it charges a price greater than pl , the low-cost firm maintains its
price at pl , and the high-cost firm loses many sales; and (2) If it charges a price equal to
14.5
Figure 14.10 The Kinked
Demand Curve Model
In panel (a), two firms have
identical demands but different marginal costs. The
high-cost firm prefers a
price ph , and the low-cost
firm prefers a price pl . In
panel (b), the high-cost firm
faces a kinked demand
curve ABC. As a result, the
high-cost firm faces a gap in
its marginal revenue curve
between points E and F.
Even if the high-cost firm’s
marginal cost increased to
MC h2 , it maintains a price
of pl .
kinked demand curve
model
An oligopoly model in which a
firm’s demand curve is based on
the assumptions that if it independently raises price, its competitors maintain their prices,
whereas a price decrease will be
matched. As a consequence, the
firm’s demand curve is highly
elastic for price increases but
highly inelastic for price
decreases.
price leadership
A model of oligopoly behavior
in which the firms depend on
one firm to signal price changes
to the other firms, who then
decide whether to follow the
changes.
Price
Collusion: The Great Prisoner’s Dilemma
Price
MC 2h
A
ph
pl
399
X MCh
Y
MCl
B
pl
MCh
E
G
D
Dh
F
Z
Quantity
qh ql
MR
(a) Prefered prices for a low-cost
firm and a high-cost firm
C
Quantity
ql
MRh
(b) Kinked demand curve for
a high-cost firm
or less than pl , the low-cost firm matches its price. Figure 14.10(b) shows the demand
curve for the high-cost firm given these assumptions. If the high-cost firm increases its
price above pl , its demand is highly elastic, as indicated by the segment AB. If the highcost firm charges a price less than or equal to pl , the low-cost firm matches the price and
the two firms continue to share the demand curve along the less-elastic segment BC,
which is identical to the segment YZ in panel (a). The demand curve in panel (b) has a
kink at point B, and as a result, the marginal revenue curve has a gap between points E
and F. The elastic segment of the demand curve between points A and B has a marginal
revenue curve equal to the line segment AE, and the more inelastic segment of the
demand curve between points B and C has a marginal revenue curve equal to the line segment below point F. The dramatic decrease in elasticity at point B results in a large
decrease in marginal revenue if the high-cost firm reduces its price below pl . The highcost firm maximizes profit by equating marginal cost to marginal revenue at point G,
which is in the gap between points E and F, produces output ql , and charges a price pl .
In fact, if the high-cost firm’s marginal cost curve lies anywhere in the gap between points
E and F, it will charge a price pl . For example, if the high-cost firm’s marginal cost
increased to MC h2 , its profit-maximizing price is still pl .
The preceding model is the kinked demand curve model, which economist Paul
Sweezy developed in the 1930s to explain why prices are often inflexible or rigid in oligopolistic markets. Sweezy’s model explained price rigidity well, but provides no explanation for how the price, pl , was set in the first place. Instead, the model assumed that
once the price was set in an oligopoly, all the firms faced a kinked demand curve at that
price, and therefore the firms maintained the current price.
PRICE LEADERSHIP The preceding analysis of the kinked demand curve combined
Sweezy’s original theory with another theory, the theory of price leadership. Price leadership is a model of oligopoly behavior where the oligopolists depend on one firm to signal price changes to the other firms, who then decide whether to follow the changes.
Oligopolists often use price leadership to help solve the Prisoner’s Dilemma. When price
400
Chapter 14
Oligopoly
price leader
The firm in an oligopoly that is
first to change price in response
to a change in an industry’s
underlying demand or cost
conditions.
leadership works, all the firms rely on the price leader to signal the “correct” price to the
other firms. The price leader is expected to understand the industry’s underlying demand
and cost conditions, and if those conditions change, to change the price. For example, if
demand increases, the price leader increases the price and waits to see whether the other
firms follow suit. If they do, the price increase sticks. If they don’t, the leader rescinds the
price increase and the price returns to its former level. If costs decrease because of a new
technological advance, the price leader decreases price and waits to see whether the other
firms follow the price decrease. Once again, if they do, the price decrease sticks. If they
don’t, the leader rescinds the price decrease.
In some industries, the same firm plays the role of the price leader for years or
decades, while in other industries, different firms serve as the price leader from one pricechanging episode to the next. The system works well as long as all the firms recognize the
price leader’s role as a signaling agent rather than viewing the price leader as a breakaway
maverick trying to destroy the industry’s pricing discipline. Price leadership has been
common in many American industries, from automobiles and cigarettes to gasoline
retailing and electric turbines.
In the models presented in this chapter, the firms shared power in their respective
markets and there were no attempts to eliminate existing firms or to prevent new firms
from entering the market. The next chapter considers cases in which dominant firms
attempt to maintain their market power by taking strategic actions against their competitors or potential competitors.
14.5B APPLICATION Price Leadership in Cyberspace: The Airlines Case
Successful price leadership requires that firms signal price
changes to each other. Before the advent of the Internet,
this signaling typically took place in a public forum such
as a newspaper or a private forum such as a trade association. The signaling of price information through a
newspaper might still work, but it is very slow compared
to the instantaneous signaling of information that takes
place over the Internet. In December 1992, the U.S.
Department of Justice filed an antitrust case against eight
major airlines (Alaska, American, Continental, Delta,
Northwest, TWA, United, and USAirways) and the
Airline Tariff Publishing Company (ATPCO), charging
that the nine entities had used information exchanged
over the Internet through ATPCO to fix fares.
ATPCO is an information clearinghouse for the airlines. The airlines inform ATPCO of all fare changes:
new fares to be added, old fares to be deleted, and existing fares to be changed. Each day ATPCO produces an
electronic list of all the fares and distributes the list to all
the major airlines and the four major computer reservations systems, which make the list available to travel
agents and airlines throughout the world. Included in
this information is the fare, the first and last dates to sell
tickets at that fare, the first and last dates that travel is
permitted at that fare, and any restrictions on the fare
such as a seven-day advanced purchase requirement or a
Saturday night stay-over requirement. Because the fares
submitted to ATPCO always included a first-ticket sale
date, it was possible for an airline to announce a fare
increase but delay its implementation until a future date.
The ability to pre-announce a fare increase became a crucial issue in the antitrust case because it gave firms the
ability to rescind a pre-announced price increase if the
other airlines failed to follow the fare increase.
The government charged the eight airlines with
using information exchanged through ATPCO to fix
fares by linking fares on different routes. For example,
Northwest Airlines (NW) and American Airlines (AA)
compete on flights to and from Boston, Detroit, Chicago, and
Los Angeles. Northwest has a hub at Detroit and
American has a hub at Chicago. Northwest flies non-stop
from Detroit to LA, but American only flies from Detroit
(continued)
Summary
to LA with a change of planes in Chicago. American flies
non-stop from Chicago to Boston, but Northwest only
flies from Chicago to Boston with a change of planes in
Detroit. Consumers prefer non-stop flights and are often
willing to pay a price premium for a non-stop flight.
Suppose that before the price signaling began, American
competed on the Detroit–LA route by charging a lower
fare than Northwest, and Northwest competed on the
Chicago–Boston route by charging a lower fare than
American. One possible price-fixing agreement would be
for American to raise its fare from Detroit to LA in return
for an agreement from Northwest to raise its fare from
Chicago to Boston. But how can American signal its
intentions to Northwest? According to the Justice
Department, American might signal Northwest through
ATPCO as follows:
1. American sends a message to ATPCO lowering
its fare from Detroit to LA. The new fare has the
same set of restrictions as Northwest’s lowest fare
from Chicago to Boston. American is using this
fare change to signal Northwest of the linkage
between American’s Detroit–LA fare and
Northwest’s Chicago–Boston fare. The last-ticket
date (the last day to sell this type of ticket) for
this low fare is two weeks from today.
2. At the exact same time, American also sends a
message to ATPCO setting a Detroit–LA fare
401
matching Northwest’s current non-stop fare, but
with a first-ticket date (the first day to buy this
type ticket) of two weeks from today.
3. American then waits to see whether Northwest
receives the signal. If Northwest gets the message, it puts a last-ticket date of two weeks from
today on its current low fare from Chicago to
Boston, and at the same time, institutes a new
higher fare matching American’s fare from
Chicago to Boston, but with a first-ticket date of
two weeks from today.
4. Two weeks from today, American’s Detroit–LA
fare and Northwest’s Chicago–Boston fares will
increase to match their competitors’ non-stop
fares and the lower fares will be eliminated.
On May 17, 1994, the eight airlines and ATPCO
settled the antitrust case out of court by agreeing to eliminate the use of first-ticket dates, which effectively eliminated the pre-announcement of price increases. The
defendants also agreed to eliminate the use of last-ticket
dates except in public advertising campaigns. The idea
behind permitting last-ticket dates in public advertising
campaigns was to encourage discount sales in newspaper,
television, and radio advertising. Finally, the airlines
agreed to eliminate signaling each other through subtle
messages passed back and forth in the footnotes to their
listed fare restrictions.
Summary
The Cournot model is built upon the questionable assumption that firms’ competitors maintain their current outputs. The Cournot equilibrium is also a Nash
equilibrium, because all the firms are doing the best they can given their opponents’ choices; therefore, this equilibrium is called a Cournot-Nash equilibrium.
The Cournot-Nash equilibrium results in a greater output and lower price than
monopoly, but a lower output and higher price than perfect competition.
2. The Stackelberg model consists of a leader and follower. The leader moves first and
assumes that the follower will respond to the leader’s output choice by producing
on its reaction function. The leader selects a quantity that maximizes its profit,
given the follower’s decision to remain on its reaction function. The Stackelberg
leader produces a greater output than a Cournot-Nash equilibrium would have it
produce, while the Stackelberg follower produces less output than it would with a
Cournot-Nash equilibrium.
3. In the Bertrand model, firms take their competitors’ prices, not quantities, as fixed
forever. With a homogeneous product, this assumption results in a price equal to
1.
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Chapter 14
Oligopoly
marginal cost. The Bertrand outcome is therefore identical to the perfectly competitive outcome. When product differentiation is added to the Bertrand model, the
equilibrium price increases and is well above marginal cost.
4. In the dominant firm model, an industry is characterized by one large dominant
firm and a group of small fringe firms. The dominant firm sets the price and the
competitive firms take the price as given and behave as firms in a perfectly competitive industry. In the long run, if the fringe firms earn positive economic profits,
entry tends to increase the fringe supply and decrease the dominant firm’s market
share.
5. The Prisoner’s Dilemma makes it difficult for firms to collude successfully.
Solutions to the Prisoner’s Dilemma require that firms signal their intentions to
each other, detect any defections, and punish defectors. Successful solutions often
require all the firms to exhibit nice behavior toward their competitors. Tit-for-tat is
one possible solution to the Prisoner’s Dilemma, in which the firms are nice.
Differences in demand and cost conditions across firms make it more difficult to
solve the Prisoner’s Dilemma and may result in the industry using a price leader to
signal when it is time to change price. Firms other than the price leader may face a
kink in their demand curves that results in maintaining price at the current level.
Oligopolists may use price leadership to maintain industry discipline and prevent
price competition.
Self-Test Problems
QUIZ
Consider a market with two identical firms. The
market demand is P 100 4Q, where Q qa qb. MCa MCb ACa ACb 4.
a. Solve for the Cournot reaction functions of each firm.
b. Solve for the Cournot-Nash equilibrium. Calculate
the quantity, price, and profit for each firm.
c. Solve for output and price with collusion and
identify the collusion curve. Because costs are the
same, assume the firms evenly divide the collusive
output. Show that Firm A has an incentive to
increase its collusive output by 1 unit.
d. Now assume this market has a Stackelberg leader.
Assume Firm A is the Stackelberg leader. Solve
for each firm’s quantity and price and profit.
e. Compare the Stackelberg result to the Cournot
result. Which outcome is more allocatively efficient?
2. Harvey (h) and Kim (k) both sell home-made decorated T-shirts on a college campus. They each produce
a differentiated product, and because of brand loyalty,
they face the following symmetric demand curves:
1.
qh 5 (1/5)ph (1/10)pk and
qk 5 (1/5)pk (1/10)ph
In addition, MCh MCk 1.5.
a. Using the Bertrand model, if there were no product differentiation, what would be the price of
the T-shirts?
b. Assuming product differentiation, using the
Bertrand model, solve for Harvey’s reaction function ph f (pk ) and for Kim’s reaction function
pk f (ph ). What is the price in the market?
How many T-shirts will Harvey sell? How many
T-shirts will Kim sell?
3. Consider an industry that is characterized by a dominant firm (d ) and a fringe (f ). The market demand
curve is: P 208 4Q, where Q represents the
total industry output (qdqf ). The supply of the
fringe (Sf ) is Pf 10 2qf , and the marginal cost
of the dominant firm is MCd 4 (1/3)qd .
a. Solve for the residual demand curve for the dominant firm.
b. What output will the dominant firm produce?
What price will it charge?
c. What is the fringe’s output?
d. Show your results on a graph.
Questions and Problems for Review
403
Questions and Problems for Review
1.
2.
3.
4.
5.
6.
7.
Consider a market with two identical firms. The
market demand is P 26 2Q, where Q qa qb. MCa MCb ACa ACb 2.
a. Solve for the Cournot reaction functions of each
firm.
b. Solve for the Cournot-Nash equilibrium. Calculate
the quantity, price, and profit for each firm.
Consider a market with two identical firms. The
market demand is P 93 3Q, where Q qa qb. MCa MCb ACa ACb 3.
a. Solve for the Cournot reaction functions of each
firm.
b. Solve for the Cournot-Nash equilibrium. Calculate
the quantity, price, and profit for each firm.
Consider a market with two identical firms. The
market demand is P 26 2Q, where Q qa qb. MCa MCb ACa ACb 2. (Same equations as question 1.)
a. Solve for output and price with collusion and
identify the collusion curve.
b. Assume that the firms initially evenly split the
collusive output. Show that Firm A has an incentive to increase output by 1 unit. Does Firm B
also have an incentive to increase output?
Consider a market with two identical firms. The
market demand is P 93 3Q, where Q qa qb. MCa MCb ACa ACb 3. (Same equations as question 2.)
a. Solve for output and price with collusion and
identify the collusion curve.
b. Assume that the firms initially evenly split the
collusive output. Show that Firm A has an incentive to increase output by 1 unit.
Consider a market with two identical firms. The
market demand is P 26 2Q, where Q qa qb. MCa MCb ACa ACb 2. (Same equations as question 1.) Now assume this market has a
Stackelberg leader, Firm A. Solve for the quantity,
price, and profit for each firm.
Consider a market with two identical firms. The
market demand is P 93 3Q, where Q qa qb. MCa MCb ACa ACb 3. (Same equations as question 2.) Now assume this market has a
Stackelberg leader, Firm A. Solve for the quantity,
price, and profit for each firm.
Consider a market with two firms. The market
demand is P 26 – 2Q, where Q qa qb.
8.
9.
10.
11.
Assume Firm A has a lower MC than Firm B. MCa
ACa 2 and MCb ACb 2.5. This difference
in cost functions provides better justification for the
existence of a Stackelberg leader.
a. Which firm would you expect to be the
Stackelberg leader? Why?
b. Solve for Firm B’s Cournot reaction function.
c. Solve for Firm A’s demand curve.
d. Solve for the Stackelberg quantity, price, and
profit for each firm.
Consider a market with two firms. The market
demand is P 93 3Q, where Q qa qb.
Assume Firm A has a lower MC than Firm B. MCa
ACa 3 and MCb ACb 3.5. This difference
in cost functions provides better justification for the
existence of a Stackelberg leader.
a. Which firm would you expect to be the
Stackelberg leader? Why?
b. Solve for Firm B’s Cournot reaction function.
c. Solve for Firm A’s demand curve.
d. Solve for the Stackelberg quantity, price, and
profits for each firm.
Consider a market with two identical firms. The
market demand is P 26 2Q, where Q qa qb. MCa MCb ACa ACb 2. (Same equations as question 1) Assume there is no product differentiation in this case and the firms follow a
Bertrand pricing model. Solve for the Bertrand
equilibrium: Calculate output for each firm, and
price and profit as well.
Consider a market with two identical firms. The
market demand is P 93 3Q, where Q qa qb. MCa MCb ACa ACb 3. (Same equations as question 2.) Assume there is no product differentiation in this case and the firms follow a
Bertrand pricing model. Solve for the Bertrand
equilibrium: Calculate output, price, and profit for
each firm.
In questions 1, 3, 5, 7, and 9, we have used the same
demand function to solve for the Cournot, collusive,
Stackelberg, Stackelberg with different costs, and the
Bertrand outcomes. Compare the results of these five
different models in terms of price, output for each
firm, total market output, profit for each firm, and
total industry profits. Comment on the differences.
Which outcomes are best from an economic perspective? Which are worst?
404
Chapter 14
Oligopoly
In questions 2, 4, 6, 8, and 10, we have used the
same demand function to solve for the Cournot, collusive, Stackelberg, Stackelberg with different costs,
and the Bertrand outcomes. Compare the results of
these five different models in terms of price, output
for each firm, total market output, profit for each
firm, and total industry profits. Comment on the
differences. Which outcomes are best from an economic perspective? Which are worst?
13. Consider two firms (A and B) that price using the
Bertrand model and produce the same good, but
with product differentiation. Because of brand loyalty,
they face the following symmetric demand curves:
12.
16.
qa 21 3pa 2pb and qb 21 3pb 2pa
In addition, MCa MCb 1.
a. If there were no product differentiation, what
would be the Bertrand equilibrium price in this
market?
b. Assuming product differentiation, solve for Firm
A’s reaction function pa f (pb ) and for Firm B’s
reaction function pb f (pa ). What is the price
in the market?
c. Compare the price with product differentiation
in question 13(b) to the price without product
differentiation in question 13(a).
14. Consider two firms (A and B) that produce the
same good, but with product differentiation.
Because of brand loyalty, they face the following
symmetric demand curves:
qa 12 2pa pb and qb 12 2pb pa
In addition, MCa MCb 2.
a. If there were no product differentiation, what
would be the Bertrand equilibrium price in this
market?
b. Assuming product differentiation, solve for Firm
A’s reaction function pa f (pb ) and for Firm B’s
reaction function pb f (pa ). What is the price
in the market? Calculate output for each firm.
c. Compare the results with no product differentiation in question 14(a) to those with product differentiation in question 14(b).
15. Consider an industry that is characterized by a dominant firm (d ) and a fringe ( f ). The market demand
17.
18.
19.
20.
curve is: P 486 2Q, where Q represents the
total industry output (qd qf ). The supply of the
fringe (Sf ) is Pf 6 qf , and the marginal cost of
the dominant firm is MCd 4 (2/3)qd .
a. Solve for the residual demand curve for the dominant firm.
b. What output will the dominant firm produce?
What price will it charge?
c. What is the fringe’s output?
d. Show your results on a graph.
Consider an industry that is characterized by a dominant firm (d ) and a competitive fringe (f ). The market demand curve is: P 200 2Q where Q
represents the total industry output (qd qf ). The
supply of the fringe (Sf ) is Pf 11 qf , and the marginal cost of the dominant firm is MCd 2 (2/3)qd .
a. Solve for the residual demand curve for the dominant firm.
b. What output will the dominant firm produce?
What price will it charge?
c. What is the fringe’s output?
d. Show your results on a graph.
Consider a duopoly game of collusion with a different form of punishment where if one firm cheats,
the other firm is “triggered” into playing its Cournot
equilibrium outcome forever. Consider Axelrod’s
three important characteristics of successful punishment strategies. Compare the features of the punishment strategy described in this question to
tit-for-tat.
The kinked demand curve model has been highly
criticized as unrealistic. Describe its assumptions.
Do you think the kinked demand curve model is
realistic? Explain.
How is the price leadership model different from
the dominant firm model? Explain. How are the
two models similar?
Consider three of the models presented in this chapter: Cournot, Stackelberg, and Bertrand.
a. Discuss the assumptions of each model.
b. Discuss the realistic nature of each model.
c. How did the development of the concept of a
Nash equilibrium help to resurrect the Cournot
model?
Internet Exercises
Visit http://www.myeconlab.com/waldman for this chapter’s Web exercises. Answer real-world economics problems by
doing research on the Internet.
Appendix 14.1
405
Appendix 14.1
Deriving Reaction Functions Using Calculus
Using calculus, Kodak’s reaction function can be derived as follows. To maximize profit,
Kodak must set MR MC for any given level of Fuji’s output. Total revenue for Kodak,
TRK , is:
TRK qK p qK 1100 qF qK 2
or:
TRK 100qK qK2 qK qF
Marginal revenue, MRK, is simply the partial derivative of TRK with respect to qK ;
therefore:
MRK 0TRK
0qK
100 2qK qF
Because MC 10, we set MRK 10 and obtain:
100 2qK qF 10
or:
qK 90 qF
2
1
45 qF
2
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Chapter 14
Oligopoly
Appendix 14.2
Deriving the Stackelberg Equilibrium Using Calculus
Using calculus:
1
1
TRK pqK c 100 a 45 qK b qK d qK 55qK qK2
2
2
and:
MRK dTRK
dqK
55 qK
Setting MR MC:
55 qK 10 or qK 45