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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. 402 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 2 406 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