Oligopoly. Chapter 25

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1 Chapter 25 Oligopoly We have thus far covered two extreme market structures perfect competition where a large number of small firms produce identical products, and monopoly where a single firm is isolated from competition through some form of barrier to entry (and through a lack of close substitutes that could be produced by someone else). 1 The models that represent these polar opposites are incredibly useful because they allow us to develop intuition about important economic forces in the real world. At the same time, few markets in the real world really fall on either of these extreme poles, and so we now turn to some market structures that fall in between. The first of these is the case of oligopoly. An oligopoly is a market structure in which a small number of firms is collectively isolated from outside competition by some form of barrier to entry. Just as in the case of monopolies, this barrier to entry may be technological (as, for instance, when there are high fixed costs) or legal (as when the government regulates competition). We will assume in this chapter s analysis of oligopoly that the firms produce the same identical product and will leave the case where firms can differentiate their products to Chapter 26. Were the firms in the oligopoly to combine into a single firm, they would therefore become a monopoly just like the one we analyzed in Chapter 23. Were the barriers to entry to disappear, on the other hand, the oligopoly would turn to a competitive market as new firms would join so long as positive profits could be made. Since there are only a few firms in an oligopoly, my firm s decision about how much to produce will have an impact on the price the other firms can charge, or my decision about what price to set may determine what price others will set. Firms within an oligopoly therefore find themselves in a strategic setting a setting in which their decisions have a direct impact on the economic environment in which they operate. You can see this in how airlines behave as they watch each other to determine what fares to set or how many planes to devote to particular routes, or in how the small number of large car manufacturers set their financing packages for new car sales. Below, we will develop a few different ways of looking at the limited and strategic competition that such oligopolistic firms face. 1 This chapter builds primarily on Chapter 23 and Section A of Chapter 24. Only Section 25B.3 of this chapter requires knowledge of Section B from Chapter 24 and this section can be skipped if you only read Section A of Chapter 24. The chapter also presumes an understanding of the different types of costs covered in the earlier chapters on producer theory (as summarized in the first section of Chapter 13) as well as a basic understanding of demand and elasticity as covered in the first section of Chapter 18.

2 966 Chapter 25. Oligopoly 25A Competition and Collusion in Oligopolies While we could think of oligopolies with more than two firms, we will focus here primarily on the case where two firms operate within the oligopoly market structure (that is then sometimes called a duopoly ). The basic insights extend to cases where there are more than two firms in the oligopoly but as the number of firms gets large, the oligopoly becomes more and more like a perfectly competitive market structure. We will also simplify our analysis by assuming that the two firms are identical (in the sense of facing identical cost structures) and that the marginal cost of production is constant. In end-of-chapter exercises, we then explore how our results are affected by changing these baseline assumptions. To fix ideas, lets think of the following concrete situation: I am a producer of economist cards but I recently discovered that you are also producing identical cards. Suppose both of us applied for a copyright on this idea and, since we both applied at the same time, the government has granted both of us the copyright but will not grant it to anyone else. For some inexplicable reason that suggests a general lack of sophistication on the part of the general public, the only people who buy these cards are economists who attend the annual American Economic Association (AEA) meetings every January and you and I therefore have to determine our strategy for selling cards at these meetings. Each of our firms in this oligopoly then has, essentially, two choices to make: (1) how much to produce and (2) how much to charge. It might be that it s really easy to duplicate the cards at the AEA meetings, in which case we might decide to simply post a price at our booth and produce the cards as needed. In this case, price is the strategic variable that we are setting prior to getting to the meetings as we advertise to the attendees to try to get them to come to our booth. Alternatively, it might be that we have to produce the cards before we get to the AEA meetings because it s not possible to produce them on the spot as needed. In that case, quantity is the strategic variable since we have to decide how many cards to bring prior to getting to the meetings, leaving us free to vary the price depending on how many people actually want to buy cards when we get there. Whether price or quantity is the right strategic variable to think about then depends on the circumstances faced by the firms in an oligopoly on what we will call the economic setting in which the firms operate. We will therefore develop two types of models models of quantity competition and models of price competition. The other feature of oligopoly models is that they either assume that the firms in the oligopoly make their strategic decision simultaneously or sequentially. Maybe it takes me longer to get my advertising materials together and I therefore end up posting my price after you do, or maybe I work in a local market where I have to set the capacity for producing a certain quantity of cards before you do. As we have seen in our discussion of game theory, we can employ the concept of Nash equilibrium for the case of simultaneous decision making while we use the concept of subgame perfect (Nash) equilibrium in the case of sequential decisions. Sometimes, as we will see, it matters who moves first. We therefore have four different types of models we will discuss: (1) price competition where firms make strategic decisions about price simultaneously; (2) price competition where firms make strategic decisions about price sequentially; (3) quantity competition where firms make strategic decisions about quantity simultaneously; and (4) quantity competition where firms make strategic decisions about quantity sequentially. We will begin with price competition and then move to quantity competition, each time considering both the simultaneous and the sequential case, and we will see that firms could in principle do better by simply combining forces and behaving like

3 25A. Competition and Collusion in Oligopolies 967 a single monopoly. Following our discussion of oligopoly price and quantity competition, we will therefore consider the circumstances under which oligopoly firms might succeed in forming cartels that behave like monopolies by eliminating competition between the firms in the oligopoly. 25A.1 Oligopoly Price (or Bertrand ) Competition Competition between oligopoly firms that strategically set price (rather than quantity) is often referred to as Bertrand competition after the French mathematician Joseph Louis Francois Bertrand ( ). Bertrand took issue with another French mathematician, Antuine Augustin Cournot ( ) whose work on quantity competition (which we discuss in the next section) had suggested that oligopolies would price goods somewhere between where price would fall under perfect competition and perfect monopoly. Bertrand came up with a quite different and striking conclusion: he suggested that Cournot had focused on the wrong strategic variable quantity and that his result goes away when firms instead compete on price. In particular, Bertrand argued that such price competition will result in a price analogous to what we would expect to emerge under perfect competition (price equal to marginal cost) even if only two firms are competing with one another. 25A.1.1 Simultaneous Strategic Decisions about Price Bertrand s logic is easy to see in a model with two identical firms that make decisions simultaneously and face a constant marginal cost of production (with no recurring fixed cost). Suppose we face no real fixed costs and we can easily adjust the quantity of cards we produce on the spot at the AEA meetings. We therefore decide to advertise a price and produce whatever quantity is demanded by consumers at that price. But as we think about announcing a price, we have to think about what price the other might announce and how consumers might react to different price combinations. One conclusion is pretty immediate: If we announce different prices, then consumers will simply flock to the firm that announced the lower price and the other firm won t be able to sell anything. I will therefore want to avoid two scenarios: First, I don t want to set a price that is so low that it would result in negative profits if I managed to attract consumers at this price. Since we are assuming no recurring fixed costs and constant marginal costs, this means I don t want to set a price below marginal cost. Second, assuming your firm similarly won t set a price below marginal cost, I don t want to set a price higher than what you set because then I don t get any customers. Put differently, whatever price you set, it cannot be a best response for me to set a higher price or a price below marginal cost. The same is true for you which means that, in any Nash equilibrium in which we both do the best we can given the strategy played by the other, we will charge identical prices that do not fall below marginal cost. But we can say more than that. Suppose that the price announced by both of us is above marginal cost. Then I am not playing a best response because, given that you have announced a price above marginal cost, I can do better by charging a price just below that and getting all the customers. The only time this is not true is if both firms are announcing the price equal to marginal cost. Given that you are charging this price, I can do no better by charging a lower price (which would result in negative profits) or a higher price (which would result in me getting no customers). The same is true for your firm given that I am charging a price equal to marginal cost. Thus, by each announcing a price equal to marginal cost, we are both playing best response strategies to the other and the outcome is a Nash equilibrium.

4 968 Chapter 25. Oligopoly Exercise 25A.1 Can you see how this is the only possible Nash equilibrium? Is it a dominant strategy Nash equilibrium? Exercise 25A.2 Is there a single Nash equilibrium if more than two firms engage in Bertrand competition within an oligopoly? 25A.1.2 Using Best Response Functions to verify Bertrand s Logic While the logic behind Bertrand s conclusion that price competition leads oligopolistic firms to behave competitively is straightforward, this is a good time to develop a tool that will be useful throughout our discussion of oligopoly: best response functions. These functions are simply plots of the best response of one player to particular strategic choices by the other. They are useful when players have a continuum of possible actions they can take in a simultaneous move game rather than a discrete number of actions as in most of our game theory development in Chapter 24. When best response functions for both players are then plotted on the same graph, they can help us identify the Nash equilibria easily. Suppose I am firm 1 and you are firm 2. Consider panel (a) of Graph On the horizontal axis, we plot p 1 the price set by me, and on the vertical axis we plot p 2 the price charged by you. We then plot your best responses to different prices I might announce. We already know that you will never want to set a price below marginal cost (M C), and if I were to ever be stupid enough to set a price below MC, any p 2 > p 1 would be a best response for you (since it would simply result in you not selling anything and letting me get all the business.) For purposes of our graph, we can then simply let your best response to p 1 < MC be p 2 = MC. If I announce a price p 1 above MC, we know that you will want to charge a price just below p 1 to get all consumers away from my booth. Thus, for p 1 > MC, your best response is p 2 = p 1 ǫ (where ǫ is a small number close to zero). Since p 1 = p 2 on the 45-degree line in the graph, this means that your best response in panel (a) will lie just below the 45-degree line for p 1 > MC. Graph 25.1: Best Response Functions for Simultaneous Bertrand Competition In panel (b) of Graph 25.1, we do the same for my firm only now p 2 (on the vertical axis) is taken as given by firm 1, and firm 1 finds its best response to different levels of p 2. If you set your

5 25A. Competition and Collusion in Oligopolies 969 price below MC, my best response can then be taken to simply be p 1 = MC, and if you set your price p 2 above MC, my best response is p 1 = p 2 ǫ (which lies just above the 45-degree line). We defined a Nash equilibrium in Chapter 24 as a set of strategies for each player that are best responses to each other. In order for an equilibrium to emerge in our price setting model, my price therefore has to be a best response to your price, and your price has to be a best response to my price. Put differently, when we put the two best response functions onto the same graph in panel (c), the equilibrium happens where the two best response functions intersect. This happens at p 1 = p 2 = MC just as we derived intuitively above. 25A.1.3 Sequential Strategic Decisions about Price In the real world, it is often the case that one firm has to make a decision about its strategic variable before the other with the second firm being able to observe the first firm s decision when its turn to act comes. As we argued in our chapter on game theory, sometimes this makes a big difference with the first mover gaining an advantage (or disadvantage) from having to declare its intentions in advance of the second mover. It s easy to see that this is not, however, the case for our two firms engaging in Bertrand competition. Suppose I move first and you get to observe my advertised price before you advertise your own. Remember that in such sequential settings, subgame perfection requires that I will have to think through what you will do for any action I announce. But our discussion above already tells us the answer: you will choose a price just below p 1 whenever p 1 > MC, leaving me with no consumers. Since I will not choose a price below MC, this implies that I will set p 1 = MC and you will follow suit with our two firms splitting the market by charging prices exactly equal to MC. Exercise 25A.3 How would you think about subgame perfect equilibria under sequential Bertrand competition with 3 firms (where firm 1 moves first, firm 2 moves second and firm 3 moves third)? 25A.1.4 Real-World Caveats to Bertrand s Price Competition Result While Bertrand s logic is intuitive, few economists believe that his result is one that truly characterizes many real world oligopoly outcomes. There are several real-world considerations that considerably weaken the Bertrand prediction regarding price competition in oligopolies, and here we will briefly mention some of them. (In end-of-chapter exercises, we additionally explore how the Bertrand predictions change with different assumptions about firm costs.) First, the pure Bertrand model assumes that firms are able to produce any quantity demanded at the price that they announce. This might in fact be true in some markets but typically does not hold. As a result, real world firms have to set some capacity of production as they think about announcing a price, and this capacity choice, as we will again mention in Section 25A.2.2, then introduces quantity as a strategic variable. In cases where capacity choices are in fact binding on the Bertrand competitors, the model predicts that each firm will again announce the same price but that this price will be above marginal cost in much the way that it is under strict quantity competition (as we will demonstrate in the next section). 2 Second, we have assumed throughout that the two firms in our oligopoly interact only one time, whether simultaneously or sequentially. But in the real world, firms typically interact repeatedly which implies that price competition 2 This solution to the Bertrand Paradox of p = MC was first developed by Francis Edgeworth ( ) at the end of the 19th century and has since been formalized using modern economic tools.

6 970 Chapter 25. Oligopoly of the type envisioned by Bertrand occurs in a repeated game context. Again, we would expect an equilibrium in which the firms in the oligopoly announce the same price in each period. In the non-repeated game, we concluded that the only such equilibrium price has to be equal to marginal cost because, were this not the case, neither firm is best responding to the strategic choice of the other. But now suppose that firms are engaged in repeated price competition and consider whether p > MC could emerge in a given period. A strategy for each firm must then specify a price for any possible previous price history, which opens the possibility of trigger strategies of the following form: I will begin our repeated interactions by charging a price p > MC and will continue to do so in future periods as long as that price has been played by both of us in all previous periods; otherwise, I will charge p = MC forever. Suppose we both play this strategy. Then, in any given period, I have to weigh whether the short run gain from charging a price slightly below p (which results in me getting all the customers this period) outweighs the long-run cost of reverting to p = MC in all future periods. It is quite plausible that this short run benefit is smaller than the long run cost which would make my strategy a best response to yours (and yours a best response to mine). In infintiely repeated interactions, or in interactions where there is a good chance we will meet again, we can therefore see how p > MC can emerge as an equilibrium under price competition. Exercise 25A.4 Suppose our two firms know that we will encounter each other n times and never again thereafter. Can p > MC still be part of a subgame perfect equilibrium in this case assuming we engage in pure price competition? Finally, Bertrand assumed that firms are restricted to producing identical products. If we allow for the possibility that consumers differ somewhat in their tastes for how economist cards look and what exactly they say on the back, we might however decide to produce slightly different versions of economist cards and through such product differentiation become able to charge p > MC. This is because consumers that have a strong preference for my type of card will still buy from me at a somewhat higher price, and similarly those with a preference for your type of card will continue to buy yours at a somewhat higher price. Product differentiation therefore also introduces the possibility of p > MC emerging under price competition. We will develop this more in Chapter A.2 Oligopoly Quantity Competition The implicit assumption that underlies Bertrand competition is that firms can easily adjust quantity once they set price. In our example, we assumed that we can both just produce the required cards on the spot at the AEA meetings. But, as we just mentioned, many firms have to set capacity for their production and, once they have done so, cannot easily deviate from this in terms of how much they will produce. It might be hard for us to have our card factory at our booth at the AEA meetings, which means we will have to produce our cards ahead of time and bring them with us to our booths. In such circumstances, it is more reasonable to assume that firms choose capacity (or quantity ) first and then sell what they produce at the highest price they can get. This is the scenario that Cournot had in mind when he investigated competition between oligopolistic firms, and it is the scenario we turn to next. As we will see, this model, known as the Cournot Model, has very different implications regarding the equilibrium price at which oligopolistic firms produce. As in the previous section, we will continue by assuming that firms in our oligopoly are identical and face constant marginal cost.

7 25A. Competition and Collusion in Oligopolies A.2.1 Simultaneous Strategic Decisions about Quantity: Cournot Competition We can again use best response functions to see what Nash equilibrium will emerge when two firms in an oligopoly choose capacity simultaneously. In panel (a) of Graph 25.2, we begin by considering firm 2 s best response to different quantities x 1 set by firm 1. If I set x 1 = 0, then you would know that you will have a monopoly on economist cards at the AEA meetings. From our work in Chapter 23, we can then easily determine the optimal quantity for you by solving the monopoly problem. This is depicted in panel (b) of the graph where D is the market demand curve and MR is your monopoly s marginal revenue curve that has the same intercept (as D) but twice the slope. Your firm, firm 2, would then produce the monopoly quantity x M where MR = MC (and charge the monopoly price p M ). The quantity x M therefore becomes your best response to x 1 = 0 and determines the intercept of your best response function in panel (a). Graph 25.2: The Best Response Function for Firm 2 under Simultaneous Cournot Competition Now suppose I set x 1 = x 1 > 0. You then know that you no longer face the entire market demand curve because I have committed to filling x 1 of the market demand. Put differently, you now face a demand curve that is equal to the market demand curve D minus x 1. In panel (c) of Graph 25.2, we therefore shift the demand D by x 1 to get the new residual demand D r that remains given that I will satisfy a portion of market demand. From this, we can calculate the residual marginal revenue curve MR r that now applies to your firm. Once again, you will maximize profit where marginal revenue equals marginal cost; i.e. MR r = MC. This results in a new optimal quantity given x 1 denoted x 2 (x 1 ), which in turn becomes your best response to me having set x 1 = x 1. Note that x 2 (x 1 ) necessarily lies below x M i.e. your best response quantity decreases as x 1 increases. We can imagine doing this for all possible quantities of x 1 to get the full best response function for your firm 2 as depicted in panel (a). Exercise 25A.5 Can you identify in panel (b) of Graph 25.2 the quantity that corresponds to the horizontal intercept of firm 2 s best response function in panel (a)? Exercise 25A.6 What is the slope of the best response function in panel (a) of Graph 25.2? (Hint: Use your answer to exercise 25A.5 to arrive at your answer here.)

8 972 Chapter 25. Oligopoly We can then do what we did for Bertrand competition by putting the best response functions of the two firms together into one graph to see where they intersect. Since our two firms are identical, my best response function can be similarly derived. This is done in panel (a) of Graph 25.3, which is just the mirror image of the best response function for your firm that we derived in the previous graph. The two best response functions then intersect at x 1 = x 2 = x C in panel (b), with x C the Cournot-Nash equilibrium output for each of our firms in the oligopoly. Graph 25.3: Simultaneous Move Cournot-Nash Equilibrium 25A.2.2 Comparing and Reconciling Cournot, Bertrand and Monopoly Outcomes In panel (c) of Graph 25.3 we can then see how the quantities produced under monopoly, Cournot and Bertrand competition compare. As illustrated in panel (b), C represents each firm s output under Cournot (or quantity) competition. From constructing the best response functions, we know that the vertical intercept of firm 2 s best response function is the monopoly quantity, as is the horizontal intercept of firm 1 s best response function. When we connect these (with the dashed magenta line in panel (c)), we get all combinations of firm 1 and firm 2 production that sum to the monopoly quantity. Were the two firms to collude, for instance, and simply split the monopoly quantity, they would produce half of x M at the point labeled M. Thus, production is unambiguously higher under Cournot competition than it would be under monopoly production. We can also see how Cournot production compares to Bertrand production. From our work in the last section we know that Bertrand or price competition results in both firms charging a price equal to MC. At such a price, market demand will be equal to x in panel (b) of Graph Now suppose that, under Cournot competition, firm 2 determines its best response to firm 1 setting its quantity to x. This would imply that firm 2 s residual demand is equal to D shifted inward by x, leaving it with a residual demand curve that has a vertical intercept at MC. Thus any output that firm 2 would produce given that firm 1 is producing x would have to be sold at a price below MC which implies firm 2 s best response is to produce x 2 = 0. This implies that firm 2 s best response function reaches zero at x 1 = x = 2x M ; i.e. the horizontal intercept of firm 2 s best response function lies at x. (Note: This is the answer to within-chapter-exercise 25A.5.) Since the two firms are identical, the same is true for firm 1 s vertical intercept.

9 25A. Competition and Collusion in Oligopolies 973 If we connect the horizontal intercept of firm 1 s best response function with the vertical intercept of firm 2 s best response function (with the dashed blue line) in panel (c), we then get all the different ways in which the two firms could split production and produce x, the quantity that would be sold when p = M C as happens under Bertrand competition. If we assume that, when both firms charge the Bertrand price of p = MC, the two firms split overall output, each firm would produce half of x as indicated at point B in the graph. Thus, Bertrand competition leads to unambiguously higher output than Cournot competition. Exercise 25A.7 Which type of behavior under simultaneous decision making within an oligopoly results in greater social surplus: quantity or price competition? Exercise 25A.8 True or False: Under Bertrand competition, x B 1 = x B 2 = x M. As we will note again in Chapter 26, the dramatic difference between the Bertrand and Cournot competition seems quite strange, and it is not easy to choose between the two models on intuitive grounds: On the one hand, it seems that firms in the real world often set prices (when they are not in perfectly competitive settings), and this seems to speak in favor of the Bertrand model. (In Chapter 26, for instance, I give the example of Apple coming out with a new computer and immediately setting its price long before it finds out how much it will have to produce.) On the other hand, the Bertrand prediction of price being set equal to marginal cost even when only two firms are competing seems a stretch, which speaks in favor of the Cournot model which not only arrives at the intuitively reasonable prediction that price falls between the monopoly and the competitive level when there are only two firms but also predicts (as we will show in Section B) that oligopoly prices converge to competitive prices as the number of firms in the oligopoly becomes large. Much work has, as a result, been done by economists to reconcile these models of oligopoly competition. One of the most revealing results, which we already mentioned in our discussion of Bertrand competition, is the following: Suppose that firms really do set prices (as the Bertrand model assumes) but they set capacities for production (which sounds a lot like the quantity setting of the Cournot model) before announcing prices. Then under plausible conditions, it has been shown that this Bertrand equilibrium outcome of price competition results in Cournot quantities and prices. 3 Economists have therefore often come to view oligopoly competition as guided in the long run by production capacity competition (as envisioned by Cournot) equilibrated through price competition (as envisioned by Bertrand) in the short run when capacities are fixed. Both models appear to have their place, and both play important roles in how we think of oligopoly competition. 25A.2.3 Sequential Strategic Decisions about Quantity: The Stackelberg Model Under Bertrand competition, we concluded that it does not matter whether firms determine their price simultaneously or sequentially in either case, firms end up charging p = MC in equilibrium. The same is not true for quantity competition, as we will see now. The sequential quantity competition model is known as the Stackelberg model, 4 and the firm designated to move first is called the Stackelberg leader while the firm that moves second is called the Stackelberg follower. In sequential move games, we concluded in Chapter 24 that non-credible threats are eliminated by restricting ourselves to Nash equilibria that are subgame-perfect i.e. to equilibria in which early movers look forward and determine the best responses by their opponents 3 This was demonstrated by Kreps, D. and J. Scheinkman (1983), Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes, Rand Journal of Economics 14, The model is named after Heinrich Freiherr von Stackelberg ( ), a German economist.

10 974 Chapter 25. Oligopoly later on in the game. When she decides how much capacity to set, the Stackelberg leader will then take into account the entire best response function of the follower because that function tells the leader exactly how the follower will respond once she finds out how much the leader will be producing. Thus, rather than guessing about the quantity the opposing firm will set (as is the case under simultaneous quantity competition), the leader now has the luxury of inducing how much the follower will set by her own actions in the first stage. Suppose, then, that you firm 2 are the follower and I firm 1 am the leader. I already know your best response function for any quantity that I might set we derived this in Graph 25.2a which we now replicate in panel (a) of Graph In deciding how much capacity to set, I then simply have to determine my residual demand curve given your best response function. The grey demand curve D in panel (b) is simply the market demand curve. For any output level x 1 x, we know that your best response is simply not to produce, which implies that I know I will own the market demand curve if I chose to produce above x. Thus, my residual demand is equal to market demand for quantities greater than x. If I set capacity below x, however, I know that you will produce along your best response function once you find out how much capacity I set. To arrive at my residual demand, I therefore have to subtract the quantity that I know you will produce for any x 1 < x. If I set my capacity close to x, you will choose to produce relatively little, but as x 1 falls, your best response quantity rises and reaches x M, the monopoly quantity, when x 1 = 0. My residual demand curve D r therefore begins at the monopoly price p M (which would be charged by you if I set x 1 = 0) and reaches the market demand curve D when it crosses MC. Once we have figured out firm 1 s residual demand, we can now do what we always do to identify my firm s optimal capacity: simply plot out the MR r curve that corresponds to D r and find its intersection with M C. Because all the relationships are linear, this intersection occurs at half the distance between x and zero which happens to be the monopoly quantity x M. Thus, the Stackelberg leader, firm 1, will set x 1 = x M, and the Stackelberg follower will produce half this amount as read off its best response function. Given what I as the leader have done in the first stage, you as the follower are doing the best you can, and given your predictable output decisions in the second stage (as summarized in your best response function), I have done the best I can. We have reached a sub-game perfect equilibrium. Exercise 25A.9 Determine the Stackelberg price in terms of p M the price a monopolist would charge and MC. Adding this outcome to our predicted outputs for Bertrand, Cournot and monopoly settings from Graph 25.3, we can then see that the Stackelberg quantity competition results in greater overall output than simultaneous Cournot competition but less overall output than Bertrand price competition. Exercise 25A.10 Where is the predicted Stackelberg outcome in Graph 25.3c? 25A.2.4 The Difference between Sequential and Simultaneous Quantity Competition We can now step back a little and ask why the Stackelberg model differs fundamentally from the Cournot model. Why, for instance, don t I threaten to act like a Stackelberg leader when you and I are competing simultaneously? Suppose you and I set quantity simultaneously before we arrive at the AEA meetings, but I call you ahead of time and tell you that I will produce the Stackelberg leader quantity. Would you

11 25A. Competition and Collusion in Oligopolies 975 Graph 25.4: Stackelberg Equilibrium have any reason to believe me when I threaten to do this? The answer is that you should not take my threat seriously. After all, if you thought that I thought you would produce x M /2, my best response (according to my best response function in Graph 24.5) would be to produce less than x M! (You can see this in panel (c) of the graph where the horizontal (dashed) grey line that passes through M at an output level of x M /2 for you crosses my best response function to the left of x M.) Your best response to me producing less than x M would then be to produce more than x M /2. My threat to produce x M is therefore simply not credible when I try to bully you over the phone. When the game assumes a sequential structure, however, the threat becomes real because you know how much I have produced by the time that you have to decide how much to produce. It s no longer an idle threat for me to say I will produce the Stackelberg leader quantity I have just done so. Now it is indeed a best response for you to produce the Stackelberg follower quantity, and given that you will do so it is best for me to have produced the Stackelberg leader quantity. It is the sequential structure of the game that results in the difference in equilibrium behavior, and without that sequential structure, there is no way for me to credibly threaten to do anything other than produce the Cournot quantity.

12 976 Chapter 25. Oligopoly 25A.3 Incumbent Firms, Fixed Entry Costs and Entry Deterrence The insight that the sequential structure of the oligopoly quantity competition changes the outcome of that competition can then get us to think of other ways in which sequential decision-making might matter. An important case is the case in which one firm is the incumbent firm that currently has the whole market but is threatened by a second firm that might potentially enter the market and turn its structure from a monopoly to an oligopoly. Is there anything (aside from sending someone with a baseball bat) the incumbent firm can do to prevent the potential entrant from coming into the market? The answer depends on two factors: (1) how costly is it for the potential entrant to actually enter the market and begin production, and (2) to what extent can the incumbent firm credibly threaten the potential entrant. 25A.3.1 Case 1: Incumbent Quantity Choice follows Entrant Choice Suppose the potential entrant has to pay a one-time fixed entry cost FC in order to be able to begin production. Now consider the case in which the potential entrant makes her decision on whether to enter the market before either firm makes a choice about how much to produce. Panels (a) and (b) in Graph 25.5 picture two such scenarios. In both panels, firm 2 first decides whether or not to enter, and if she does not enter, firm 1 sets its quantity x 1. If firm 2 does enter, the firms are assumed to choose their production quantities simultaneously in panel (a) and sequentially in panel (b). Graph 25.5: Possible Sequences of Entry and Quantity Choices

13 25A. Competition and Collusion in Oligopolies 977 Recall that we solve games of this kind from the bottom up in order to find subgame perfect equilibria. If firm 2 does not enter, we know that firm 1 will optimize by simply producing the monopoly quantity and thus will make the monopoly profit π M while firm 2 will make zero profit. If firm 2 enters, on the other hand, the two firms will engage in simultaneous Cournot competition in panel (a), with each firm making the Cournot profit π C but with firm 2 paying the fixed entry cost FC. Firm 2 therefore looks ahead and makes its entry decision based on whether or not (π C FC) is greater than zero. Put differently, so long as the profit from producing the Cournot quantity at the Cournot price is greater than the fixed cost of entering, firm 2 will enter the market. Similarly, in panel (b), firm 2 knows that she will be a Stackelberg follower if she enters, and so she will enter so long as the profit π SF from producing the Stackelberg follower quantity at the Stackelberg price is greater than the fixed cost of entering. Exercise 25A.11 True or False: Once the entrant has paid the fixed entry cost, this cost becomes a sunk cost and is therefore irrelevant to the choice of how much to produce. Exercise 25A.12 Is the smallest fixed cost of entering that will prevent firm 2 from coming into the market greater in panel (a) or in panel (b)? Notice that in neither of these cases can the incumbent firm (firm 1) do anything to affect firm 2 s entry decision because the entry decision happens before quantities are set. This implies that firm 2 s entry decision is entirely dependent on the size of the fixed entry cost FC. The problem (from firm 1 s perspective) is once again that there is no way it can credibly threaten firm 2, a problem that can disappear if firm 1 gets to commit to an output quantity before firm 2 makes its entry decision (as we will see next). 25A.3.2 Case 2: Entry Choice follows Incumbent Quantity Choice Now consider the sequence pictured in panel (c) of Graph 25.5 where the incumbent (firm 1) chooses its quantity x 1 before the potential entrant (firm 2) makes its decision on whether to enter the market and produce. Again, we can solve the resulting game from the bottom up, beginning with the case in which firm 2 has decided to enter the market. Firm 2 s optimal quantity is then simply given by its best response function (derived in Graph 25.2) to the quantity set by firm 1 (which is known to firm 2 at the time it makes its quantity decision). Firm 1 knows firm 2 s best response function which implies that if firm 2 enters the market, firm 1 is simply a Stackelberg leader. Thus, if firm 2 enters, the equilibrium payoffs are the Stackelberg profits, π SL and π SF, minus the fixed entry cost for firm 2. The incumbent firm, however, would very much like to remain the only firm in the market. Short of sending in big guys with baseball bats to beat up firm 2, the only way to persuade firm 2 to stay out of the incumbent s (monopoly) market is for the incumbent to insure that firm 2 cannot make a positive profit by entering. And the only way to do that is to commit to producing a larger quantity in order to drive the price down sufficiently to keep firm 2 from wanting to come into the market. Whether it is possible for firm 1 to do this and thereby to make a profit higher than that of a Stackelberg leader depends on just how big the fixed entry cost FC is for firm 2. This is illustrated in the two panels of Graph In panel (a), we plot the profit that the incumbent can expect from different output levels if it remains the only firm in the market. The highest possible profit occurs at the monopoly quantity x M (which, as we have seen, is also the Stackelberg leader quantity x SL ). If the fixed entry cost is very high, the incumbent can simply produce x M and rest assured in its monopoly given that it is simply too costly for any potential

14 978 Chapter 25. Oligopoly entrant to enter the market. This is illustrated in panel (b) where, for FC FC, firm 1 produces x M while firm 2 stays out of the market (and thus produces zero). If the fixed entry cost is very low, on the other hand, there is little that firm 1 can do to keep the entrant out of the market and so firm 1 simply produces the Stackelberg leader quantity x SL and accepts firm 2 s production of the Stackelberg follower quantity x SF. This is illustrated in panel (b) for FC FC. Graph 25.6: Setting Quantity to Deter Entry The interesting case of entry deterrence arises for fixed entry costs between FC and FC. Suppose, for instance, that FC is just below FC i.e. suppose that firm 2 would make a slightly positive profit by entering if firm 1 behaved like a Stackelberg leader and produced x SL. If firm 1 then produces just a little more than x SL, this will insure that firm 2 can no longer make a positive profit by entering. The incumbent firm can therefore deter entry by producing above x SL. While this will mean that firm 1 s profit falls below the monopoly profit, it is preferable to engaging in Stackelberg competition with firm 2 (in which case firm 1 would only get π SL ). As the fixed entry cost falls, it becomes harder and harder for firm 1 to do this necessitating higher and higher levels of output to deter entry. But it s worth it as long as the incumbent s profit remains above the Stackelberg leader profit π SL. Thus, the highest quantity that firm 1 would ever be willing to produce to deter entry, x ED max, is the quantity that will insure πsl. When fixed entry costs fall below FC, it is too costly for the incumbent to deter entry and firm 1 reverts back to producing simply the Stackelberg leader quantity. This is, then, a more rigorous treatment of an idea that we raised in Chapter 23 when we discussed the possibility that a monopoly might be restrained in its behavior (and might produce more than the monopoly quantity) if it feels threatened by potential competitors. Notice that, if it could, the incumbent firm would like to reduce its output back to the monopoly quantity x M once it has successfully deterred an entrant, but the only way that deterrence could succeed is if the incumbent was able to commit to not doing so by setting output prior to firm 2 s entry decision. It is this commitment that made the threat to the entrant credible were it possible to then go back on the commitment, the threat would not be credible and entry could not be deterred. It is a little like the general that would like to strike fear into the opposing army on the battlefield by telling them that his army will fight to the death. Of course just saying We will fight to the

15 25A. Competition and Collusion in Oligopolies 979 death! is not credible anyone can say it. So the general might cross a bridge into the battlefield and then burn the bridge down thus cutting off any possibility of retreat. This would certainly make the threat to fight to the death more credible just as the incumbent firm s threat to increase production to prevent entry becomes credible when the firm actually does it and thus cuts off any possibility of retreat. 25A.4 Collusion, Cartels and Prisoner s Dilemmas So far, we have assumed that you and I will act as competitors within the oligopoly strategically competing on either price or quantity decisions. Now suppose instead, however, that I call you before the AEA meetings and say: Why don t we stop competing with each other and instead combine forces to see if we can t do better by coordinating what we do? Logically, we should be able to do better if we don t compete. After all, if we could act like one firm that has a monopoly, we would be able to do at least as well as we can do if we compete by simply producing the same quantity as we do under oligopoly competition. But we know from Graph 25.3c that as a monopoly we would produce less than we do under Cournot, Stackelberg or Bertrand competition. Our joint profit would therefore be higher if we could find a way of splitting monopoly production and charging a higher price than it would be under any competitive outcome that results in a price below the monopoly price. We therefore have an incentive to find a way to collude instead of compete. 25A.4.1 Collusion and Cartels A cartel is a collusive agreement (between firms in an oligopoly) to restrict output in order to raise price above what it would be under oligopoly competition. The most famous cartel in the world is OPEC the Organization of Petroleum Exporting Countries which is composed of countries that produce a large portion of the world s oil supply. Oil ministers from OPEC countries routinely meet to set production quotas for each of the countries. Their claim is to aim for a stable world price of oil, but what they really aim for is a high price for oil. There are many other examples of attempts by producers of certain goods to form cartels, some of which we will analyze in end-of-chapter exercises. Suppose our two little firms are currently engaged in Cournot competition, with each of us producing x C as depicted in Graph 25.3b. It s then easy to see how we can do better all we have to do is figure out what the monopoly output level x M would be and agree to each limit our own production to half of that. This would allow us to sell our economist cards at the AEA meetings at the monopoly price p M, with each of us making half the profit we would if our individual firm was the sole monopoly. The same cartel agreement would make each of us better off if we currently engaged in Bertrand competition. Exercise 25A.13 * How might the cartel agreement have to differ if we were currently engaged in Stackelberg competition? (Hint: Think about how the cartel profit compares to the Stackelberg profits for both firms, and use the Stackelberg price you determined in exercise 25A.9 along the way.) 25A.4.2 A Prisoner s Dilemma: The Incentive of Cartel Members to Cheat Suppose, then, that you and I enter a collusive cartel agreement and decide to each produce half of x M in order to maximize our joint profit. It is certainly in our interest to sign such an agreement.

16 980 Chapter 25. Oligopoly But is it optimal for us to stick by our agreement as we prepare to come to the AEA meetings with our economist cards? Suppose I believe you will stick by the agreement. We can then ask what I would have to gain from producing one additional set of economist cards above the quota we set in our cartel. In panel (a) of Graph 25.7, we assume that we we have agreed to behave as a single monopolist, jointly producing x M which allows us to sell all our cards at price p M. Were we, as a monopoly, to produce one more set of cards, we would have to drop the price in order to sell the larger quantity. This would result in a loss of profit equal to the magenta area since we can no longer sell the initial x M goods at the price p M. It would also result in an increase in profit equal to the blue area since we get to sell one more set of cards. For a monopoly, the quantity x M is profit maximizing because the magenta area is slightly larger than the blue area i.e. our monopoly profit would fall if we produced one more set of cards. Graph 25.7: The Incentive to Cheat on a Cartel Agreement But now think of the question of whether to produce one more set of cards from the perspective of one of the members of the cartel that has agreed to behave as a single monopolist. In our cartel agreement, we agreed that I would produce half of the monopoly output level x M and you would produce the other half. If you produce one more set of cards, you will therefore lose only half the magenta area in profit from having to accept a price slightly lower than p M for the half of x M you are producing under the cartel agreement, but you would get all of the blue area in additional profit from the additional unit you produce. Since the magenta area is only slightly larger than the blue area, half of the magenta area is certainly smaller than all of the blue area in the graph which means your profit will increase if you cheat and produce one more set of cards than you agreed to in the cartel. Panel (b) looks at this another way and asks not only whether it would be in your best interest to produce one unit of output beyond the cartel agreement but how much more you would in fact want to produce assuming you believe that I will be a sucker and stick by the agreement to produce only half of x M. The residual demand D r that you would face given that I produce x 1 = 0.5x M is equal to the market demand D minus 0.5x M which intersects MC at the quantity 1.5x M. The corresponding residual marginal revenue curve MR r has twice the slope and therefore intersects MC at 0.75x M implying that it would be optimal for you to produce 0.75x M rather than 0.5x M

17 25A. Competition and Collusion in Oligopolies 981 as called for in your cartel agreement. Put differently, if you believe I will produce 0.5x M, your best response is to produce 0.75x M. Exercise 25A.14 Can you verify the last sentence by just looking at the best response functions we derived earlier in Graph 25.2? Now, if you are smart enough to figure out that it is in your best interest to cheat on the cartel agreement, chances are that I am smart enough to figure this out as well. But that means that, unless we can find a way to enforce the cartel agreement, the cartel will unravel as each of us cheats. And if each of us knows that the other will cheat, we are right back to Cournot competition and will end up behaving as if there was no cartel agreement at all. Put in terms of the game theory language we developed earlier, we face a classic Prisoner s Dilemma: We would both be better off colluding and producing in accordance with the agreement than we would be by competing with one another (either in Bertrand or Cournot competition), but we also both have a strong incentive to cheat on the agreement (whether the other party cheats or not) and bring more economist cards to the AEA meetings than we had promised. As we noted in our discussion of Prisoner s Dilemmas, these types of games do not result in the optimal outcome for the two players unless the players can find a way to enforce the agreement. Inconveniently for us, cartel agreements are usually illegal. (Usually, but not always as we will see shortly.) Exercise 25A.15 The Prisoners Dilemma you and I face as we try to maintain a cartel agreement works toward making us worse off. How does it look from the perspective of society at large? While the incentives of cartel members therefore contain seeds that undermine cartel agreements, there are real world examples of cartel agreements that have lasted for long periods. They may not always be successful at maintaining exactly monopoly output, but they often do restrict output beyond what Cournot competition would predict. This raises the question of how firms can overcome the Prisoners Dilemma incentives that would, if unchecked, lead to a full unraveling of a cartel. We can think of two possible ways of accomplishing this: First, firms might find ways of hiring an outside party to enforce the cartel, just as our two prisoners in the classic Prisoners Dilemma might do by joining a mafia that enforces silence when the prisoners are interrogated by the prosecutor. Second, in our discussion of repeated Prisoners Dilemmas in Chapter 24, we found that, if the game is repeated an infinite number of times or, more realistically, if the players know that there is a decent chance that they will meet again each time that they meet, cooperation in the Prisoners Dilemma can emerge as part of a subgame perfect equilibrium strategy. We will now briefly discuss each of these paths that can lead to successful cartel cooperation among oligopolists. 25A.4.3 Enforcing Cartel Agreements through Government Protection In 1933, in the midst of the Great Depression, Congress passed the National Industrial Recovery Act (NIRA) at the urging of the newly inaugurated President Franklin D. Roosevelt who proclaimed it the most important and far-reaching legislation ever enacted by the American Congress. The act represented a stark departure from laissez faire attitudes toward industry, envisioning a more planned economy in which industrial leaders would coordinate production and prices to foster fair competition, with compliance enforced by the newly created National Recovery Administration (NRA). In essence, the act legalized cartels in major manufacturing sectors thus putting the force of law behind oligopolists efforts to set price and quantity within particular markets. It

18 982 Chapter 25. Oligopoly generally received strong support from large corporations but was opposed by smaller firms. 5 The NIRA has become the clearest example in the U.S. of how oligopolists can employ the government as an enforcer of cartel agreements to limit quantity and raise price. Less than two years after its enactment, the U.S. Supreme Court unanimously declared the portion of the NIRA that established cartels as unconstitutional. Exercise 25A.16 Why would oligopolists who cannot voluntarily sustain cartel agreements want to have such agreements enforced? While this large-scale establishment of cartels vanished in the U.S. with the demise of the NIRA, similar legislation often governs industry in other countries. And, there continue to be more modest attempts to establish cartels through government action, typically with the stated purpose of benefitting the general welfare but the actual consequence of restricting quantity and raising price. In the 1990 s, for instance, Congress authorized the Northeast Interstate Dairy Compact that permitted the setting of minimum wholesale prices of milk across six New England states (amending extensive federal price regulation of milk that predated the establishment of the Compact) and restrictions of competition from milk producers in other regions. Other regional milk cartels were similarly authorized in other regions. The stated intent of such legislation was to assure the continued viability of dairy farming in the Northeast and to assure consumers of an adequate, local supply of pure and wholesome milk at a fair and equitable price. The cooperative suggested that dramatic price fluctuations, with a pronounced downward trend, threaten the viability of the Northeast dairy region and that cooperative, rather than individual state action, may address more effectively the market disarray. But the ultimate aim of the cartel was the same as that of all cartels: to curtail competition and raise price. Predictably, such legislation tends to be fought vigorously by consumer groups and is advocated by firms producing the cartel good. 6 In some cases, it is generally recognized that the purpose of government sponsored cartels is to limit competition in order to raise price. Few, for instance, would argue that this is not the prime mission of OPEC the Organization of Petroleum Producing Countries that meets frequently to set production quotas for each of its 13 member countries. Yet one would not be able to tell this from the official mission statement by OPEC which states: OPEC s mission is to coordinate and unify the petroleum policies of Member Countries and to ensure the stabilization of oil prices in order to secure an efficient, economic and regular supply of petroleum to consumers, a steady income to producers and a fair return on capital to those investing in the petroleum industry. The words sound similar to those used to advocate for the NIRA in 1933 and continue to be similar to those articulated whenever government enforcement for cartel agreements is sought by firms. 25A.4.4 Self-Enforcing Cartel Agreements in Repeated Oligopoly Interactions Alternatively, we can turn to the case where oligopolists who seek to establish a cartel agreement know that they will meet repeatedly. From our game theory chapter, we know that this is not sufficient for cooperation to emerge: If the firms know they will interact repeatedly but that this interaction will end at some definitive point in the future, subgame perfection leads to an unraveling of cooperation from the bottom of the repeated game tree upwards. The firms know that, in their 5 The act also encouraged collective bargaining through unions, set maximum work hours and minimum wages and forbid child labor. 6 To the extent to which milk cartels are intended to support the viability of small, family-owned dairy farmers, they appear not to be very successful. Most of the economic benefits accrue to larger corporate dairy farms, with little evidence that cartels slow the disappearance of smaller, less efficient farms.

19 25B. The Mathematics of Oligopoly 983 final interaction, neither will have an incentive to stick by the cartel agreement. But that means that in the second to last period, there will also be no incentive to cooperate since there is no credible way to punish non-cooperation in the final interaction. But that then means that there is no way to enforce cooperation in the third-to-last interaction given that both firms know that non-cooperation will take place in the last two periods. And by the same logic, cooperation cannot emerge in any period. But the real world is rarely quite as definitive as setting up a finitely repeated set of interactions with a clear end-point. Rather, firms will know that they are likely to interact again each time that they meet, and for our purposes, we can therefore treat such interactions as infinitely repeated. Again, as we saw in our discussion of repeated Prisoners Dilemmas in Chapter 24, this removes the unraveling feature of finitely repeated games because there is no definitive final interaction. And it opens the possibility of simple trigger strategies under which firms begin by complying with the cartel agreement, continue to do so as long as everyone complied in previous interactions, and revert to oligopoly competition if someone deviates from the agreement. Such strategies can sustain cartel cooperation so long as the immediate payoff from violating the cartel agreement is not sufficiently large to overcome the long-run loss from the disappearance of the cartel and the reversion to oligopoly competition. Real world strategies of this type are complicated by the fact that firms might not in fact be able to tell for sure whether another firm has violated the agreement. For instance, suppose that oil producers cannot observe how much oil is produced by any given company but they can only see the price that oil sells for in the market. Suppose further that oil price in any given period depends on both the overall quantity of oil supplied by the oligopoly firms and unpredictable (and unobservable) demand shocks to the oil market. If a firm then observes an unexpectedly low price in a given period, it might be because a member of the cartel has cheated and has produced more oil than the agreement specified, but it might also be because of an adverse demand shock in the oil market. Firms in such markets may then find it difficult to be certain about whether cartel members are cheating and run the risk of mis-interpreting an unexpectedly low price as a sign of cheating. Economists have introduced such complicating factors into economic models of oligopolies and cartels, and it becomes plausible to observe equilibria in which cartel agreements break down and re-emerge in repeated oligopoly interactions. This corresponds well to observed cartel behaviors in some industries. Exercise 25A.17 In circumstances where firms are not certain about demand conditions in any given period, why might a more forgiving trigger strategy (like Tit-for-Tat) that allows for the re-emergence of cooperation be better than the extreme trigger strategy that forever punishes perceived non-cooperation in one period? 25B The Mathematics of Oligopoly Throughout most of this section, we will assume for simplicity that firms face a constant marginal cost M C = c (with no recurring fixed costs) and that the market demand for the oligopoly good x is linear and of the form x = A αp. (25.1) In some of our end-of-chapter exercises, we will explore how the various oligopoly models are affected by different assumptions, including different marginal costs and the presence of recurring

20 984 Chapter 25. Oligopoly fixed costs for the firms. For now, note that, under our current assumptions, were the oligopoly to function as a single monopoly, we know from our work in Chapter 23 that, assuming no price discrimination, the firm would produce the monopoly quantity x M and sell it at the monopoly price p M where x M = A αc 2 Exercise 25B.1 Verify x M and p M in equation (25.2). 25B.1 Bertrand Competition and p M = A + αc 2α. (25.2) From our work in part A, we know that Bertrand competition, whether simultaneous or sequential, will result in both firms setting price equal to marginal cost. It is therefore quite easy to determine the overall Bertrand oligopoly output level by simply substituting MC = c for price in the market demand function to get the joint output level x = A αc. Assuming that the consumers will come to our two firms in equal numbers when we charge the same price, this implies Bertrand output levels for our two firms of x B 1 = x B 2 = A αc (25.3) 2 sold at the Bertrand price of p B = c. Thus, for the linear demand and constant MC model we are using, the Bertrand model predicts that each of the two firms will produce the quantity that a single monopolist would choose to produce on her own, because the competitive quantity is twice the monopoly quantity. The Bertrand model becomes more interesting, as we will see in Chapter 26, when firms can differentiate their products, i.e. when firms are not producing identical products but are still part of an oligopoly. We will also demonstrate in end-of-chapter exercise 25.1 how the inclusion of recurring fixed costs and differences in marginal costs across firms can alter the stark Bertrand predictions. 25B.2 Quantity Competition: Cournot and Stackelberg Next we briefly describe the mathematics behind Cournot and Stackeberg competition as treated in Section A before covering some other aspects of quantity competition in Section 25B.3. 25B.2.1 Cournot Competition In order to calculate the best response functions for our two firms in the economist card oligopoly described in part A, we begin (as we did in Graph 25.2c) by calculating my residual demand given I assume you produce x 2. If the market demand is given by equation (25.1), then my residual demand if you produce x 2 is simply x r 1 = A αp x 2. (25.4) To make this analogous to the residual demand curve graphed in Graph 25.2c, we need to put it in the form of an inverse demand function; i.e. ( ) ( ) A p r 1 = x2 1 x 1. (25.5) α α

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