Chapter 11. T he economy that we. The World of Oligopoly: Preliminaries to Successful Entry Production in a Nonnatural Monopoly Situation


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1 Chapter T he economy that we are studying in this book is still extremely primitive. At the present time, it has only a few productive enterprises, all of which are monopolies. This economy is certainly far from the type of The World of Oligopoly: Preliminaries to Successful Entry highly competitive free enterprise system that we are familiar with in the United States. It is unlikely that such a system will grow out of the situation described in Chapter 0, where we saw a governmentregulated natural monopoly develop to supply pure water to the community. To find the origins of competitive markets, we will have to investigate other industries in which the technology is such that several firms, if not many firms, can survive simultaneously at the equilibrium. In this chapter, we will see a new industry that is not a sustainable or natural monopoly develop in our primitive society. This industry produces a recently discovered product called a gadget. The first entrepreneur in the industry quickly realizes that once she establishes her firm and begins to sell her product, other firms will attempt to imitate the product and enter the industry. Unless she can prevent the entry of such potential competitors, the industry will rapidly undergo a transformation from a monopoly to a duopoly to an oligopoly. As we investigate the events that occur in this industry, we will examine the theory of duopoly and oligopoly the theory of markets with two or a few competing firms. This theory will be of major importance in Chapter when we see our gadget maker plan a strategy to keep potential entrants out of her market.. Production in a Nonnatural Monopoly Situation Let us assume that an agent in our primitive society comes upon a technology to make a product that we will call a gadget. This technology is such that the marginal cost of producing gadgets rises as more are produced and rises at an increasing rate. Figure. 407
2 408 Chapter The World of Oligopoly: Preliminaries to Successful Entry Figure. Marginal and Average Cost Curves for a Firm That Produces Gadgets. Cost MC AC Marginal cost rises at an increasing rate as gadget output rises. 0 Quantity shows such a marginal cost curve along with the assumed average cost curve for the firm producing the gadgets. Remember that we usually depict marginal cost curves as being Ushaped, like the average cost curve shown in Figure.. We could easily have done so again, but, for the sake of simplicity, we will assume that in this case the marginal cost is always increasing. There is no contradiction between a Ushaped average cost curve and a constantly rising marginal cost curve because the firm presumably has fixed costs that make the average cost of production high at low levels of output. As these average fixed costs fall with the increase in output, so does the average cost until the rising marginal cost of production pulls the average cost up again. We will assume that the inverse demand for gadgets is a simple linear function p = A bq, where p is the market price of the good, A is a constant, b is the slope of the inverse demand function, and q is the total output placed on the market. As we saw in Chapter 9, this function tells us the maximum price attainable for any given quantity sold and is therefore called the inverse demand function. Our gadget maker sees immediately that if she can keep competitors out of her market, she will be able to make a substantial profit for herself. Figure. demonstrates this fact by showing her demand function superimposed on her cost function. Figure. also indicates the monopoly price that the gadget maker can set in the absence of any competitors in the market. If she uses this price, it will yield profits equal to the area dcba. Unfortunately, however, her technology does not give her a natural monopoly. She will have to battle to keep competitors from entering her market because, presumably, two or even more firms will be able to profitably coexist in this market.
3 . The Cournot Theory of Duopoly and Oligopoly 409 Figure. The ProfitMaximizing Output for the Gadget Monopoly. If there are no other market entrants, the entrepreneur can earn monopoly profits that are equal to the area dcba. Price, Cost MC AC d c a b 0 MR D Quantity. The Cournot Theory of Duopoly and Oligopoly As our gadget maker starts to plan a strategy for keeping other firms out of her market, she realizes that in order to know how to prevent entry, she needs an understanding of what the industry might be like with competition present. For example, what kind of market equilibrium might she face if the industry is a duopoly that is, if there are two firms selling the product? To obtain this information, our gadget maker hires a consulting firm, which bases its opinions on Cournot s theory of duopoly, a famous theory that was devised in
4 40 Chapter The World of Oligopoly: Preliminaries to Successful Entry the nineteenth century to analyze the behavior of quantities, prices, and profits in a twofirm market. The consultants issue a report that summarizes the essential features of the Cournot model. (Although this model is for a duopoly, it can be, and often is, extended to an oligopoly a market that is dominated by a few sellers of a product.) To understand the Cournot model better, let us assume that there are two firms in the gadgets market and that these firms face a linear inverse demand function of p = A b(q + q ). Clearly, the price that will prevail in the market will be a function of the outputs of both duopolists, q and q. We can express the cost function for firm as C(q ) and the cost function for firm as C(q ). Note that while the demand function and the price depend on the output levels of both duopolists, each duopolist s cost function is determined by its own output level. We will assume that both cost functions have marginal and average costs as depicted in Figure.. Marginal costs rise as output grows and rise at an increasing rate, while average costs are Ushaped. Given a twofirm industry with the linear demand function and cost functions shown in Figure., we want to use the Cournot model to find the answers to the following questions: What will be the equilibrium output levels of the two duopolists? What price will prevail in the market? What profit will each firm make? By equilibrium, we mean a pair of output levels, one for each firm, which are such that after they are chosen, neither firm has any incentive to change its output level. This type of equilibrium is called a Cournot equilibrium. It is simply the Nash equilibrium that we defined in Chapter 7 applied to a model in which duopolistic or oligopolistic firms compete with each other by choosing output levels. To understand how a Cournot equilibrium is reached, we must examine the concept of reaction functions. Reaction Functions In every market, there is a strategic interaction among firms. Each firm in the market will respond to the actions of the other firms in some manner. These responses are summarized by what are called reaction functions or bestresponse functions. The reaction function specifies a firm s optimal choice for some variable such as output, given the choices of its competitors. Using the Cournot model, let us assume that both firms in a duopolistic market want to make as much profit as they possibly can. However, each firm has a problem because its profit depends on the output level its competitor chooses as well as the output level it chooses, and it does not know what the choice of its competitor will be when it makes its own choice. We can summarize this situation by saying that both duopolists (firms and ) want to maximize their profits, as indicated by the following profit functions: π = (A b(q + q )) q C(q ) and π = (A b(q + q )) q C(q ). Antoine Augustin Cournot (80 877) was a French mathematician, economist, and philosopher. He was one of the first scholars to use mathematical techniques to analyze economic problems. In his most noted work, which was published in 838, Cournot examined problems of pricing in monopolistic, duopolistic, oligopolistic, and perfectly competitive markets. This work appeared in English as Researches into the Mathematical Principles of the Theory of Wealth, translated by Nathaniel Bacon (New York: Macmillan, 897).
5 . The Cournot Theory of Duopoly and Oligopoly 4 Consulting Report. Using the Cournot Model to Determine an Equilibrium for a Duopolistic Market The consultants explain that the Cournot model is based on two key concepts about the firms in a duopolistic market: that each will behave in a profitmaximizing manner and that each will assume that the other firm will keep its output constant at the existing level when it changes its own output. We can think of the Cournot model as one in which firms alternate making decisions about the quantity they wish to produce. First, one firm chooses what it considers to be a profitmaximizing level of output. Then, given that firm s choice of a quantity and assuming it will not change, the other firm sets its own profitmaximizing quantity. This process of adjustment continues through several stages of action and reaction until the two firms reach an equilibrium and have no further incentive to change their outputs. During the entire process of adjustment, each firm believes that the other firm s current level of output is fixed and uses this assumption in selecting its own level of output. In the Cournot model, the quantity, price, and profits produced at the equilibrium for a duopolistic market will be between those that occur in a monopolistic market and those that occur in a perfectly competitive market. Note that the price that either duopolist faces (A b(q + q )) and the profit it will earn depend on the output of both duopolists. Note also that each profit function is composed of two parts. The first part is the revenue component of profit, which is represented by the price of the good (A b(q + q )) times the output of the firm, q or q. The second part of the profit function is simply the duopolist s cost, C(q ) or C(q ). Let us now say that firm decides to produce q units of output. Under the Cournot model, firm assumes that no matter what output choice it makes, firm will not change its own output choice in response. Economists today call this assumption the Cournot conjecture. More generally, we will let the conjectural variation denote the change that a firm expects in its competitor s choice of an output level in response to a change the firm made in its own output level. Using this definition, we can say that the conjectural variation in the Cournot model is zero. (Later we will consider oligopoly models with nonzero conjectural variations.) Given firm s decision, firm s profit is now solely determined by its own output choice, which means that we can express its profit function as π = (A b(q + q )) q C(q ). In a sense, firm is now a monopolist because with firm s output fixed at q, the price of the good is only a function of the output choice of firm. Given firm s decision, firm should now choose an output level that equates its marginal revenue to its marginal cost. To see how the optimal output for firm is derived, consider Figure.3. Figure.3 shows the demand function and the associated marginal revenue function facing firm at different levels of output set by firm. To understand how these functions
6 4 Chapter The World of Oligopoly: Preliminaries to Successful Entry Figure.3 The Optimal Output for a Gadget Duopolist. Given firm s production of q, firm maximizes its profit by choosing output level q, which equates its marginal cost to its marginal revenue given that q units are already in the market. A bq * Price, Cost MC D (q, q ) A bq A bq MR 3 MR MR D (q, q ) 0 q q D 3 (q, q * ) Quantity are determined, consider what happens when firm chooses an output of q. If the inverse demand curve for the product is p = A b(q + q ), then with an output of q for firm, the inverse demand curve facing firm will be p = (A bq ) bq. In other words, if firm s output is fixed at q and firm sets an output of zero, then the price will be A bq. The output of firm reduces the price from A, which is what it would be if both firms set a zero output. As firm raises its output, the price will fall even further and the slope of the demand curve will be b. In Figure.3 the demand curve labeled D is the one that will result when firm sets its output at q.iffirm chooses a higher level of output, say q, then the demand curve for firm will shift toward the origin, as shown by D. The reason for this shift is that the price that will result now if firm sets an output of zero is A bq, which is less than A bq,because q > q. Note, however, that the slope of the demand curve remains the same. If firm chooses a level of output that is lower than q,sayq*, then the demand curve for firm will be further from the origin, as depicted by D 3. In this case, if firm sets an output of zero, the market price will be A bq*, which is greater than A bq, because q* < q. In Figure.3 we also see the marginal cost curve for firm. Depending on the output level chosen by firm, we can now define the output level that represents the
7 . The Cournot Theory of Duopoly and Oligopoly 43 Figure.4 The Reaction (BestResponse) Function for Gadget Duopolist. Output of Firm (q ) Given firm s choice of q, firm s optimal response is q = f (q ). q = f (q ) 0 Output of Firm (q ) best response the profitmaximizing choice for firm. Finding the best response is a simple matter. First we locate the demand curve for firm that is associated with the quantity chosen by firm, and then we find the output level at which firm s marginal revenue equals its marginal cost. For example, say that q is the quantity set by firm. We therefore look at the demand curve labeled D and the marginal revenue curve labeled MR in Figure.3 and see that q is the optimal level of output for firm. It is firm s best response because it is the output level that maximizes the firm s profit. Similarly, if firm sets a quantity of q, the relevant demand and marginal revenue curves for firm are D and MR, and its best response is to choose an output level of q. In this way, we can define a best response for firm to every hypothetical output level of firm. The reaction function for firm is formally presented in Figure.4. Note that in Figure.4 the output level of firm is inversely related to the output level of firm. The more firm produces, the less firm will produce. It is important to realize that each point on this reaction function represents the optimal (profitmaximizing) choice or best response of firm to a possible output level of firm. A similar analysis can be made for firm in order to derive its reaction function and find its best response to each level of output that firm might choose. Such a curve appears in Figure.5. In Figure.5, we again see that the optimal or profitmaximizing output level for one duopolist (firm ) is a decreasing function of the output level chosen by the other duopolist (firm ), given the Cournot conjecture. For convenience, we can express these reaction functions for the duopolists in the gadgets market as q = f (q ) and q = f (q ).
8 44 Chapter The World of Oligopoly: Preliminaries to Successful Entry Figure.5 The Reaction (BestResponse) Function for Gadget Duopolist. Given firm s choice of q, firm s optimal response is q = f (q ). Output of Firm (q ) q = f (q ) 0 Output of Firm (q ) Solved Problem. Question (Application and Extension: The Best Responses) Assume that there are two firms, each producing with a constant marginal cost function of MC =. Assume that the demand function for the product is defined as follows: p = 00.5(q + q ). If firm sets a quantity of q = 00, what is the best response quantity for firm. Answer If firm sets a quantity of q = 00, then the residual demand curve facing firm is p = 00.5(q + 00) = 00.5(00).5q = 50.5q. If the marginal cost of production is constant at, then the firm should set a quantity at which the marginal revenue associated with the residual demand curve is equal to the constant marginal cost. We know that the marginal revenue curve associated with this demand curve has the form MR = 50 q, since it has a slope twice the steepness of the demand curve. Setting MR = MC means setting 50 q =. Solving, we see that q = 48, and this is the best response. An Alternative Derivation of Reaction Functions There is an alternative, and perhaps simpler, way to derive the reaction function for a firm. Consider Figure.6. In Figure.6 each point, like point x, represents output levels for firm and firm. For example, at point x, firm produces q and firm produces q. At point q m, the output
9 . The Cournot Theory of Duopoly and Oligopoly 45 Figure.6 Reaction Functions. For each level of q, firm s reaction function gives the level of q that places firm on the lowest attainable isoprofit curve (that is, the level of q determined by the tangency of the isoprofit curve and a horizontal line). Given q = q, firm chooses q = q ; given q = q, firm chooses q = q. Output of Firm (q ) q q Firm s Reaction Function y x a 0 c q q b q m d Output of Firm (q ) of firm is q m, and the output of firm is zero. In other words, q m is firm s monopoly output. This output combination of (q m, 0), in which firm produces its monopoly output and firm produces zero output, yields profits for firm that are greater than the profits it can earn with any other output combination. Now look at the output combination at point a of Figure.6. At that point, firm continues to produce its monopoly output of q m, but now firm produces a positive output. Clearly, firm will receive lower profits at point a than it will at point q m, because the positive output of firm decreases the price that firm can obtain for its output. Let us now locate the isoprofit curves in this space the sets of points that yield the same profits to firm. We will start by looking for those output combinations that yield the same profits as point a. Let us examine point b, where again firm produces the monopoly output of q m, but now firm is producing less than it did at point a. Clearly, the reduction in firm s output raises firm s profits above what they were at point a. To bring the profits of firm back to what they were at point a, we have two options. We can either move to point c or point d. At point c, firm has lower output than it did at point b, which raises the price of gadgets but lowers the firm s profits because it is now selling a smaller
10 46 Chapter The World of Oligopoly: Preliminaries to Successful Entry quantity of the product. At point d, firm has higher output than it did at point b, which allows it to sell a greater quantity, but the additional output lowers the price and also increases the firm s costs. The net result is that points c, a, and d all have the same profit levels. In general, the isoprofit curves for firm have the shape shown in Figure.6. The curves closer to the horizontal axis (and closer to the monopoly output level) contain higher levels of profit. To derive the reaction functions for firms and, let us look first at firm. For any given output level chosen by firm, the reaction function should tell us the profitmaximizing output level for firm. Say that firm sets an output level of q. Given this choice by firm, firm will want to choose the output level that places it on the lowest possible isoprofit curve because profits increase as firm moves toward the horizontal axis. This output level will be characterized by the tangency of the isoprofit curve and the line drawn parallel to the horizontal axis at the height of q. Such a tangency occurs at point x in Figure.6, where the output level is q. When firm chooses a higher level of output, such as q, tangency occurs at point y and the optimal level of output for firm falls to q. By successively choosing different levels of output for firm and finding the tangency points for firm, we can trace firm s reaction function. A similar analysis can produce firm s reaction function. Deriving a Cournot Equilibrium To find the Cournot equilibrium for this duopolistic market, we can simply take the two reaction functions for firms and, place them on the same diagram, and see where they intersect. The point of intersection represents the equilibrium, as shown in Figure.7. Note that the two reaction functions intersect at point e in Figure.7. At this point, firm is producing an output of q e and firm is producing an output of q e. If firm produces q e, then the best response for firm is to produce q e. Similarly, if firm produces q e, then the best response for firm is to produce q e. In short, q e and q e are the best responses of the two firms to each other. If both firms choose these output levels, neither firm will have an incentive to change its choice. Put differently, if these output levels are set, they will remain unchanged the market will be in equilibrium. Stable and Unstable Cournot Duopolies At this point in our analysis, the following question arises. According to the Cournot theory of duopoly, the gadgets market will be in equilibrium if firms and choose output levels q e and q e, but what guarantee do we have that the two firms will actually choose these output levels? There are two responses to this question. The first response is that the Cournot theory does not claim that the duopolists will choose these output levels. All it says is that if they do choose these output levels, the market will be in equilibrium. The second response goes further and says that we can actually expect that the output levels in the market will eventually reach q e and q e. Let us examine the reasoning behind the second response. Say that we are not at the equilibrium in Figure.7. Also say that firm chooses an output level that is higher than its equilibrium output level of q e, such as q. From firm s reaction function, we see that it will then choose an output level of q, which is lower than its equilibrium output level of q e. However, when firm chooses q, firm s reaction function indicates that it will decrease its output from q to q*. With the output of firm at q*, the reaction function of
11 . The Cournot Theory of Duopoly and Oligopoly 47 Figure.7 The Reaction Function Equilibrium (Nash Equilibrium) for the Gadget Duopoly. The intersection of the two firms reaction functions at (q e, q e ) is the point at which each firm is responding optimally to the other s choice. Output of Firm (q ) q = f (q ) q e q * e q = f (q ) q 0 q e q * q Output of Firm (q ) firm shows that it will now increase its output from q to q*, and so on. This process is convergent. If allowed to continue, it will lead the firms to converge on q e and q e, which are the equilibrium output levels for the market. Does convergence on the equilibrium depend on how we draw the reaction functions? The answer to this question is yes. In Figure.7 the reaction functions are drawn in such a way that the one for firm is flatter than the one for firm. If the opposite is true, then we will have the situation depicted in Figure.8. If the reaction functions are shaped as shown in Figure.8, it will be difficult for the market to converge to the equilibrium. In this case, it is still true that if the market ever reaches the equilibrium, it will remain there, but we cannot rely on the process of convergence to bring about the equilibrium. To illustrate this point, let us say that the two firms are not at the equilibrium in Figure.8. We will assume that firm chooses an output that is higher than its equilibrium output of q e, such as q. From firm s reaction function, we can see that it will then choose q, which is lower than its equilibrium output of q e. When firm chooses q, firm s reaction function shows that it will increase its output from q to q*, moving it further away from the equilibrium. With the output of firm now at q*, the reaction function of firm indicates that it will decrease its output from q to q*, moving it further away from the equilibrium as well, and so on. This process is divergent. At each stage, it moves the two firms further away from the equilibrium.
12 48 Chapter The World of Oligopoly: Preliminaries to Successful Entry Figure.8 An Unstable Cournot Equilibrium. The reaction function of the firm whose output is measured on the vertical axis is steeper than the reaction function of the firm whose output is measured on the horizontal axis. In this case, the reactions of the two firms to a disequilibrium situation will take them further and further away from the equilibrium point. Output of firm (q ) q = f (q ) q e e q q* q = f (q ) 0 q e q q * Output of Firm (q ) Solved Problem. Question (Application and Extension: Nash Equilibrium Quantities) Assume that you have two firms competing in a market and that these firms have the following reaction functions: q q = 50 q = 50 q a) For what quantity produced by firm would firm prefer to shut down and produce nothing? b) What quantity would firm produce if firm never existed? c) Verify that q = 33.3 and q = 33.3 is a Nash equilibrium.
13 . The Cournot Theory of Duopoly and Oligopoly 49 Answers a) Obviously, if firm set a quantity of 00, then q = 50 (00) = 0 and firm would shut down. b) If firm never existed, then q = 0 and the best output for q would be 50. c) We can verify that 33.3 for each firm is a Nash equilibrium by seeing if those quantities constitute a best response for each firm. Plugging 33.3 into each reaction function we see that = 50 (33.33); thus, if firm is choosing 33.33, firm will also want to choose 33.33, and viceversa. That is the definition of an equilibrium. Using Game Theory to Reinterpret the Cournot Equilibrium Our previous analysis of the Cournot equilibrium can be restated in terms of game theory. We can think of the strategic interaction between firms in a duopolistic market as a game, which we might call the simultaneousmove quantitysetting duopoly game. In this game, there are two players, firms and ; and each player has a strategy set from which it can choose a feasible strategy whenever it must make a move. These strategy sets are equivalent to all the positive output levels in the Cournot model. However, to simplify the game, it might make sense to restrict the strategy sets to those output levels between 0 and A/b. Because A/b drives the price of the good to zero, we can presume that rational firms will never choose output levels above A/b. The payoff functions for this game are presented in two equations that we used previously: π = (A b(q + q )) q C(q )forfirm and π = (A b(q + q )) q C(q ) for firm. These are the components of the game s normal form. The game is played as follows: First, both firms choose their output levels simultaneously, with neither firm knowing what level the other firm has chosen. Once these quantities are placed on the market, the demand curve tells the players what the price will be, and each firm calculates its payoffs (profits) accordingly. The equilibrium defined by the Cournot model is nothing more than the Nash equilibrium in this simultaneousmove quantitysetting duopoly game. Solved Problem.3 Question (Application and Extension: Best Response Functions and Nash Equilibria) Bob and Art live next door to each other and share a driveway between their houses. Both men are botanists and love flowers. Each has decided to plant pansies on his side of the driveway. On one Saturday morning, each announces to the other that he is going to his own favorite gardening shop to purchase pansies. Since each man will plant the flowers so close to each other, both will get enjoyment from his own and his neighbor s pansies.
14 40 Chapter The World of Oligopoly: Preliminaries to Successful Entry Art s preferences for how many pansies to purchase depend on how many Bob has bought and can be described by the bestresponse function A where B is the number of pansies bought by Bob. Bob s preferences can be described by his bestresponse function B = 0 = 4 where A is the number of pansies Art buys. a) If Bob finds out that Art is going to buy pansies, how many should he buy? b) What is the equilibrium amount of pansies the men should buy if neither knows how many the other is going to purchase? 4 B A Answer a) If Art buys pansies, Bob s best response is B = 4 = 4 ( ) Thus, Bob should buy pansies. b) The equilibrium is found by solving the two bestresponse functions simultaneously that is, solving the following two equations: for A and B. This can easily be done by substituting Bob s function into Art s function and solving, A = 0 A 4 4 = 8 + A 8 7 A = 8 8 A = A = 0 B = B A
15 .3 Criticisms of the Cournot Theory: The Stackelberg Duopoly Model 4 and substituting back into Bob s bestresponse function B = 4 = Thus, assuming that pansies (like all other goods) are perfectly divisible, in equilibrium Art buys 7 pansies and Bob buys 7. Since Art s best response is to buy pansies if Bob is going to buy 7 pansies, and Bob s best response is to buy pansies if Art is going to buy 7, each pansy decision is a best response to the other. This is the definition of an equilibrium Criticisms of the Cournot Theory: The Stackelberg Duopoly Model An Asymmetric Model Our gadget maker thinks that she now understands the Cournot theory and its game theory interpretation quite well. However, she questions the relevance of the Cournot theory to her situation because she is worried about preventing potential competitors from entering the market in which she is already entrenched. As a result of this entrenchment, she believes that any entrant will view her as a kind of leader and will view himself, a relative upstart, as a follower. In other words, she feels that the Cournot theory treats firms or players symmetrically, but, in reality, the situation she faces is asymmetric, with her firm established as the leader and any firm that enters the market now taking the role of a follower. (Imagine a new firm starting to manufacture automobiles in the United States and having to compete with General Motors and Ford. Clearly, a theory that treats all firms in this type of market symmetrically would be unrealistic.) To obtain information about how an asymmetric market functions, our gadget maker turns to another consulting firm, one that bases its opinions on the work of Heinrich von Stackelberg, another economist who studied the problem of duopoly and developed a wellknown duopoly model. The new consulting firm issues the following report in which it describes the main features of the Stackelberg model. Let us now take a closer look at the Stackelberg model by applying it to the gadgets market. We will assume the same demand, cost, and profit functions as we did with the Cournot model. We will say that demand is linear, that marginal costs are strictly increasing, and that profits are represented by the following two equations: π = (A b(q + q )) q C(q ) for firm and π = (A b(q + q )) q C(q ) for firm. We will also assume Heinrich von Stackelberg was a German economist who examined market organization and the strategic interaction of firms. He proposed the leaderfollower concept for duopolistic markets in Marktform und Gleichgewicht (Vienna: Julius Springer, 934). This work appeared in English as The Theory of the Market Economy, translated by A.T. Peacock (New York: Oxford University Press, 95).
16 4 Chapter The World of Oligopoly: Preliminaries to Successful Entry Consulting Report. Using the Stackelberg Model to Determine an Equilibrium for a Duopolistic Market The consultants explain that the Stackelberg duopoly model is an extension of the Cournot model but allows for asymmetric behavior by the two firms in a duopolistic market. The Stackelberg model assumes that one firm will play an aggressive role in the market (be the leader) and the other firm will play a passive role (be the follower). The leader will choose its level of output first. It will set a profitmaximizing quantity, taking into consideration the quantity it expects the follower to set in reaction to its own choice. The leader assumes that the follower will also want to maximize its profits but that it will accept the leader s output choice as a given. This assumption permits the leader to predict the follower s output choice and take that choice into account when it makes its own output choice. that each firm has a Cournot reaction function defining its best response to any given output level chosen by its competitor. We will denote these reaction functions as q = f (q ) for firm and q = f (q ) for firm. To make our example asymmetric, assume that firm chooses its quantity first. Then, firm, knowing what firm has done, makes its choice. After both firms have sequentially chosen their outputs, the price of the good on the market and the profits of both firms are determined. Because firm moves first, it is the Stackelberg leader and can commit itself to a fixed output. Firm, the Stackelberg follower, then takes firm s output as a given and chooses a best response. In such a model, there is an advantage to moving first, as depicted by Figure.9. In Figure.9 we see the reaction function of firm, the Stackelberg follower in this market. Firm, the Stackelberg leader, knows that for any output level it might set, firm will set the output level that represents its best response to firm s choice. Therefore, firm can predict firm s choice of an output level from its reaction function. If firm is rational, it will choose the output level that maximizes its profits after taking into consideration firm s best response to that output level. The Stackelberg Equilibrium To understand what output level would be consistent with a Stackelberg equilibrium, let us look at Figure.9 again. Note that the isoprofit curves of firm are superimposed on firm s reaction function. The task for firm is to choose the output level that will place it on the lowest possible isoprofit curve consistent with firm s choice of an optimal (profitmaximizing) quantity. In short, by moving first, firm can actually choose at which point on firm s reaction function it wants the market to be the point where its own profits will be greatest. In Figure.9 this profit maximization occurs at point E, where firm s isoprofit curve is tangent to firm s reaction function. Such a point represents a Stackelberg equilibrium. To demonstrate that point E in Figure.9 must be a Stackelberg equilibrium, consider any other point such as B. Note that point B is on isoprofit curve π,whichinvolves
17 .3 Criticisms of the Cournot Theory: The Stackelberg Duopoly Model 43 Figure.9 The Stackelberg Solution. Firm (the leader) chooses the point on the reaction function of firm (the follower) that is on the lowest attainable isoprofit curve of firm : point E. Output of Firm (q ) Firm s Reaction Function Cournot Equilibrium E B Firm s Reaction Function Output of Firm (q ) lower profits than isoprofit curve π, the one where point E is located. No output combination on isoprofit curves like π 3 and π 4 can be the equilibrium because it is not on firm s reaction function and therefore does not fit our assumption that firm will act in a rational manner. Algebraically, we can describe the Stackelberg model as follows: Let q = f(q ) be firm s reaction function and let π = (A b(q + q )) q C(q ) be firm s profit function. However, because firm knows that firm will respond to its output choice by choosing a best response, we can replace q in firm s profit function by f (q ). Firm s profit function now reads π = (A b(q + f (q )) q C(q ). Firm s problem is simply to maximize this profit function by choosing q, knowing that firm will respond optimally to its choice.
18 44 Chapter The World of Oligopoly: Preliminaries to Successful Entry As we see in Figure.9, at the Stackelberg equilibrium, firm chooses a higher level of output than it previously did at the Cournot equilibrium and receives greater profits. This is the essence of the firstmover advantage that the leader has in the Stackelberg model. The Stackelberg model matches our gadget maker s view of her market as an asymmetric one, in which her firm will be the leader and an entrant will be the follower. Remember that we characterized the Cournot model as a simultaneousmove quantitysetting duopoly game. Similarly, we can think of the Stackelberg model as a sequentialmove quantitysetting duopoly game that results in a greater payoff for the leader and a smaller payoff for the follower than they would receive at the Cournot equilibrium of the same market..4 The Welfare Properties of Duopolistic Markets As our primitive society begins to develop markets where two or a few firms compete, a question naturally arises about the welfare aspects of the Cournot equilibrium. Do such markets produce a better welfare outcome than monopolistic markets? The answer to this question is as follows: The Cournot equilibrium outputs for firms in duopolistic markets yield better welfare results than those that occur in monopolistic markets (when welfare is measured in terms of consumer surplus plus producer surplus), but the welfare results in such markets are not optimal. They are in between the welfare levels produced in perfectly competitive markets and those that occur in monopolistic markets. To prove this statement, let us again turn our attention to the gadgets market a duopolistic market. As we did previously, we will assume that the inverse demand for gadgets is linear and is represented by p = A b(q + q ). For the sake of simplicity, we will also assume that the marginal cost of production is zero. (The results that we derive would not be different if we were to assume that the marginal cost is Ushaped or strictly rising, as in Figure..) Because each firm has zero marginal cost, it will set its marginal revenue equal to zero when the other firm chooses a level of output. Reformulating the problem, with the assumption of zero marginal cost, we can express the profit function for the duopolists as follows: π = (A b(q + q )) q for firm and π = (A b(q + q )) q for firm. Note that when the marginal cost is zero, maximizing profits is the same as maximizing revenue. We can derive the reaction functions for the two firms from these profit functions by equating the marginal revenue for each firm to zero after the other firm has set its level of output. Using partial derivatives, we find that the marginal revenue is MR = A bq bq for firm and MR = A bq bq for firm. Solving for q and q will give us the reaction functions of the two firms. These reaction functions will specify the profitmaximizing output that each firm should set for any given output of the other firm. We can express the reaction functions as follows: q = (A bq )/b for firm and q = (A bq )/b for firm. If a monopolist with the same cost structure were to provide all the gadgets for this market, then she would have to determine how much of output q to produce so as to maximize her profits. This problem can be stated as: Max π =(A bq)q. Note that because there is only one producer, the total output (q) for the market is the same as the sum of the
19 .4 The Welfare Properties of Duopolistic Markets 45 Figure.0 The Monopoly Solution with Zero Marginal Costs. The monopolist will choose output A/b, at which the marginal revenue equals the marginal cost of zero. At the welfareoptimal output level, x, the price equals zero. The deadweight loss is area (A/b)xy under the demand curve and between the monopoly and welfareoptimal output levels. Price p m y p= (A bq) Deadweight Loss MC = 0 0 A b MR Demand x Quantity firms output (q = q + q ). To maximize this function, we take the derivative and make it equal to zero. We then find that the optimal monopoly output is q = A/b. Figure.0 represents the monopolistic market by our familiar inverse demand and marginal revenue curves. However, note that in this example, the marginal cost curve is flat and moves along the horizontal axis. In Figure.0 we see the market or aggregate demand for the product along with the marginal cost curve of the monopolist (which is the same as the marginal cost curve of each duopolist). We also see that the monopolist will choose an output of A/b and a price of p m. This monopoly pricequantity combination will create a deadweight loss of consumer surplus equal to the area in the triangle (A/b)xy. The welfareoptimal price will be zero, and the welfareoptimal quantity will be x units. Note that with an inverse demand curve of p = A bq, a zero price results when A = bq so that q = A/b, which is the output at point x. In other words, because the marginal cost is zero, any consumer willing to pay more than a price of zero should be allowed to buy gadgets, which means that the optimal price must be zero. The monopoly and welfareoptimal pricequantity combinations therefore represent
20 46 Chapter The World of Oligopoly: Preliminaries to Successful Entry Figure. The Equilibrium Price and Quantity Compared. The Cournot equilibrium, with a total quantity of A/3b, is on an isooutput line strictly between the monopoly line (a total quantity of A/b) and the welfareoptimal line (a total quantity of A/b). Output of Firm (q ) A b q = f (q ) A b Welfareoptimal Output A q + q = b A 3b E Monopoly Output A q + q = b Cournot Output A q + q = 3b q = f (q ) 0 A A A 3b b b Output of Firm (q ) the two extremes between which prices and quantities can fall. Let us now demonstrate that in a duopolistic market, the price and the quantity will be between these two extremes. Consider Figure.. Solved Problem.4 Question (Application and Extension: The Equilibrium Price and Quantity Under Duopoly) a) The town of Deadeye has only one chicken farmer to supply eggs. Assume that the marginal cost of egg production is zero and that in Deadeye the inverse de
21 .4 The Welfare Properties of Duopolistic Markets 47 mand for eggs (measured in cents) is p = 00 q. At what price and quantity will the chicken farmer sell eggs? b) Now assume that a second chicken farmer moves to Deadeye who also has zero marginal cost. Taking demand into consideration, this means that chicken farmer has a reaction function q where q and q are the quantities of eggs supplied by chicken farmers and, respectively. Chicken farmer, likewise, has a reaction function q = 5 q = 5 q What will the quantities supplied and price for eggs be now? Answer a) The chicken farmer will set the quantity that equates marginal revenue to marginal cost. Since demand is linear, marginal revenue will be MR(q) = 00 4q. Thus, to find the optimal quantity Putting the quantity back into the demand curve, we find that the price of eggs is 00 (5) = 50 b) To find the quantities, simultaneously solve the two reaction functions. q 5 q 5 =. 75q =. 5 q 00 4q = 0 q = 5 = 6 3 Substituting back into farmer s reaction function, we find that q is also equal to 6 3. This is a result of the fact that the two chicken farmers are symmetric; they face identical demands and marginal costs. Whenever this occurs, both firms in the Cournot duopoly game will produce identical amounts. The price of eggs is, then, p = = 33 3 Notice that the price has gone down and the total quantity supplied of eggs has gone up as compared to the case when there was only one chicken farmer. In the
22 48 Chapter The World of Oligopoly: Preliminaries to Successful Entry case when there was only one chicken farmer, consumer surplus was $6.5. When the second chicken farmer entered the market, consumer surplus increased to $.. This shows that increased competition lowers the price for the consumer. In Figure. we see our familiar reaction functions, this time for firms that have a zero marginal cost. Note that the horizontal axis represents the output of firm, while the vertical axis represents the output of firm. From the reaction functions of these two firms, which we previously saw in equation form as q = (A bq )/b and q = (A bq )/b, we find that when firm produces a zero output, firm s best response is to set the monopoly output because, in effect, firm is a monopolist. The monopoly output for firm occurs at point A/b along the horizontal axis. If firm produces a zero output, firm s best response is to choose the monopoly output at point A/b along the vertical axis. Similarly, if either firm were ever to choose the welfareoptimal output of A/b, the other firm s best response would be to set an output of zero. For example, suppose that firm is at point A/b, on the horizontal axis. At this point the price would be zero, so firm s best response is the produce zero (see point x in Figure.0). Now consider the line drawn between the two monopoly outputs in Figure. (the line between points A/b and A/b). Any output combination along this line is such that the total output on the market will be the monopoly output. The points along this line differ only according to the amount that each firm individually supplies to the market. A line of this type is called an isooutput line. Because the market price depends on the sum of the outputs produced, the price at any point on the isooutput line will be the same. For example, the price at any point on the line between the monopoly outputs A/b will be the monopoly price. Similarly, consider the line between the welfareoptimal outputs A/b. At any point along this isooutput line, the total output is equal to the welfareoptimal quantity and the price is equal to the welfareoptimal price. The only thing that differs at any point on the line is the portion of the welfareoptimal quantity that each firm supplies. Finally, note that point E in Figure., the Cournot equilibrium quantity (A/3b), is on an isooutput line strictly between the monopoly line and the welfareoptimal line 3 Because price decreases as quantity produced increases, it must be true that at the Cournot equilibrium, the price and the quantity are between the monopoly and welfareoptimal levels. This proves our premise that in a duopolistic market, the outcome for society (as measured by the consumer surplus plus the producer surplus) will be between the monopoly outcome and the welfareoptimal outcome. Note that this result will occur for any technologies that produce reaction functions with the same general shape as the ones in Figure., not just for technologies with zero marginal costs. 3 Note that the Cournot output (/3) (A/b) can be written as (n/(n + )) A/b, where n is the number of firms in the market. As n becomes large and moves toward infinity, (n/(n + )) A/b approaches A/b, the competitive result. We will derive this result more formally in Chapter 4.
23 Solved Problem.5.4 The Welfare Properties of Duopolistic Markets 49 Question (Application and Extension: Monopoly, Duopoly, and Consumer Surplus) Consider a market with a demand curve for a product produced by two firms, firms and. The demand for the products can be summarized by the following demand function: p = 50 ( q + q) There is zero marginal cost. Given this demand function, the reaction function for each firm is q = 50 q and q = 50 q. As discussed in Solved Problem. earlier in the chapter, the Nash equilibrium for this market is one where each firm sets a quantity of The monopoly solution is one where the total quantity on the market is 50. What is the improvement in consumer surplus by having the market organized as a duopoly? Answer Let us answer this question with a diagram (see Figure.). In this diagram we have the market demand curve p = 50 ( q + q) and we have the two quantities of and 50. As we see, if the price for the good in the market is 50 or more, the quantity demanded is zero. No one is willing to pay more than 50 for the good. Using the demand curve, we see that if both firms produce 33.33, the total Figure. Welfare Comparisons of Monopoly and Duopoly. Price 50 5 c d 6.66 a b p = 50 (q + q ) Quantity
24 430 Chapter The World of Oligopoly: Preliminaries to Successful Entry quantity is and the market price is p = 50 (66.66) = At the monopoly quantity, the market price is p = 50 (50) = 5. Now in the diagram we see that the consumer surplus is represented by the triangle above the line ab for the duopoly situation (that is, the line representing a quantity of and a price of 6.66) and the triangle above the line cd in the monopoly situation (that is, the line representing a price of 5 and a quantity of 50). The areas of these triangles are as follows: CS CS monopoly duopoly = ( 50 5)( 50) = 65 = ( )( ) =,..5 Criticisms of the Cournot Theory: The Bertrand Duopoly Model One feature of the Cournot model that often strikes people as odd is the fact that it assumes that firms compete in a market by choosing the quantities of a good they will produce. Our usual perception is that firms compete through the prices they charge for their goods. For example, when we look at the advertisements in a newspaper, we see automobile dealers, consumer electronics stores, supermarkets, and many other types of firms competing on the basis of price. Clearly, the view that firms compete through the prices they choose is at variance with the assumption of the Cournot model that they compete through the quantities they decide to produce. Our gadget maker is one of the people who questions the validity of the Cournot model for just this reason. She therefore asks for advice about the nature of a duopolistic market from still another consulting firm. This firm bases its opinions on the work of Joseph Bertrand, who had the same reservations about the Cournot model and, as a result, developed a different kind of duopoly model. 4 In their report, the consultants summarize the basic features of the Bertrand model. Let us use a simple example to look at the Bertrand model more closely. Say that two firms, i and j, sell an identical product. According to the Bertrand model, these firms can think of themselves as facing a demand function with the following characteristics: If one firm charges a price that is above the price set by its competitor, the demand for its product will be zero. If it charges a price below the price set by its competitor, it will capture all the demand in the market. (In the Bertrand model, it is assumed that all firms have enough production capacity to supply the entire market. Later, we will see what happens when this assumption is violated.) If both firms set the same price for the product, they will split the demand in the market. This type of demand function can be depicted in algebraic terms as follows: 4 Joseph Bertrand was a nineteenthcentury French mathematician and economist. His critique of the Cournot model and presentation of an alternative model appears in Theorie Mathematique de la Richesse Sociale, Journal des Savants (September 883):
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