27 Oligopoly - CHAPTER 2 7 OLIGOPOLY We have now...

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Unformatted text preview: CHAPTER 2 7 OLIGOPOLY We have now investigated two important forms of market structure: pure competition, where there are typically many small competitors, and pure monopoly, where there is only one large firm in the market. However, much of the world lies between these two extremes. Often there are a number of competitors in the market, but not so many as to regard each of them as having a negligible effect on price. This is the situation known as oligopoly. The model of monopolistic competition described in Chapter 24 is a special form of oligopoiy that emphasizes issues of product differentiation and entry. However, the models of oligopoly that we will study in this chapter are more concerned with the strategic interactions that arise in an industry with a small number of firms. There are several models that are relevant since there are several different ways for firms to behave in an oligopolistic environment. It is unreason- able to expect one grand model since many different behavior patterns can be observed in the real world. What we want is a guide to some of the possible patterns of behavior and some indication of what factors might be important in deciding when the various models are applicable. QUANTlTY LEADERSHIP 481 For simplicity, we will usually restrict ourselves to the case of two firms; this is called a situation of duopoly. The duopoly case allows us to cap- ture many of the important features of firms engaged in strategic interaction without the notational complications involved in models with a larger num- ber of firms. Also, we will limit ourselves to investigation of cases in which each firm is producing an identical product. This allows us to avoid the problems of product differentiation and focus only on strategic interactions. 27.1 Choosing a Strategy If there are two firms in the market and they are producing a homogeneous product, then there are four variables of interest: the price that each firm charges and the quantities that each firm produces. When one firm decides about its choices for prices and quantities it may already know the choices made by the other firm. If one firm gets to set its price before the other firm, we call it the price leader and the other firm the price follower. Similarly, one firm may get to choose its quantity first, in which case it is a quantity leader and the other is a quantity follower. The strategic interactions in these cases form a sequential game.1 On the other hand, it may be that when one firm makes its choices it doesn’t know the choices made by the other firm. In this case, it has to guess about the other firm’s choice in order to make a sensible decision itself. This is a simultaneous game. Again there are two possibilities: the firms could each simultaneously choose prices or each simultaneously choose quantities. This classification scheme gives us four possibilities: quantity leadership, price leadership, simultaneous quantity setting, and simultaneous price set— ting. Each of these types of interaction gives rise to a different set of strategic issues. There is also another possible form of interaction that we will examine. Instead of the firms competing against each other in one form or another they may be able to collude. In this case the two firms can jointly agree to set prices and quantities that maximize the sum of their profits. This sort of collusion is called a cooperative game. 27.2 Quantity Leadership In the case of quantity leadership, one firm makes a choice before the other firm. This is sometimes called the Stackelberg model in honor of the 1 We will examine game theory in more detail in the next chapter, but it seems appro- priate to introduce these specific examples here. 482 OLIGOPOLY (Ch. 27) first economist who systematically studied leader-follower interactions.2 The Stackelberg model is often used to describe industries in which there is a dominant firm, or a natural leader. For example, IBM is often con— sidered to be a dominant firm in the computer industry. A commonly observed pattern of behavior is for smaller firms in the computer industry to wait for IBM’s announcements of new products and then adjust their own product decisions accordingly. In this case we might want to model the computer industry with IBM playing the role of a Stackelberg leader, and the other firms in the industry being Stackelberg followers. Let us turn now to the details of the theoretical model. Suppose that firm 1 is the leader and that it chooses to produce a quantity yl. Firm 2 responds by choosing a quantity yg. Each firm knows that the equilibrium price in the market depends on the total output produced. We use the inverse demand function p(Y) to indicate the equilibrium price as a function of industry output, Y = y1 + yg. What output should the leader choose to maximize its profits? The answer depends on how the leader thinks that the follower will react to its choice. Presumably the leader should expect that the follower will attempt to maximize profits as well, given the choice made by the leader. In order for the leader to make a sensible decision about its own production, it has to consider the follower’s profit-maximization problem. The Follower’s Problem We assume that the follower wants to maximize its profits 115% 19041 + 92ly2 — 62(92)- 2 The follower’s profit depends on the output choice of the leader, but from the viewpoint of the follower the leader’s output is predetermined-the production by the leader has already been made, and the follower simply views it as a constant. The follower wants to choose an output level such that marginal revenue equals marginal cost: A MR2 = p(y1+ 1/2) + Aim = M02. W The marginal revenue has the usual interpretation. When the follower increases its output, it increases its revenue by selling more output at the 2 Heinrich von Stackelberg was a German economist who published his influential work on market organization, Marktform and Gleichgewz‘cht, in 1934. QUANTITY LEADERSHIP 483 market price. But it also pushes the price down by A39, and this lowers its profits on all the units that were previously sold at the higher price. The important thing to observe is that the profit—maximizing choice of the follower will depend on the choice made by the leader. We write this relationship as 92 = film)- The function f2(y1) tells us the profit-maximizing output of the follower as a function of the leader’s choice. This function is called the reaction function since it tells us how the follower will react to the leader’s choice of output. Let’s derive a reaction curve in the simple case of linear demand. In this case the (inverse) demand function takes the form p(y1+112) = a—b(y1 +y2). For convenience we’ll take costs to be zero. Then the profit function for firm 2 is 7T2(y1,y2) = [a — 5(91 + y2lly2 OI' W2(y1,y2)= ay2 - b14192 — 531% We can use this expression to draw the isoprofit lines in Figure 27.1. These are lines depicting those combinations of y1 and y2 that yield a constant level of profit to firm 2. That is, the isoprofit lines are comprised of all points (y1,y2) that satisfy equations of the form (192 — ll3911/2 — by% = W2. Note that profits to firm 2 will increase as we move to isoprofit lines that are further to the left. This is true since if we fix the output of firm 2 at some level, firm 2’s profits will increase as firm 1’s output decreases. Firm 2 will make its maximum possible profits when it is a monopolist; that is, when firm 1 chooses to produce zero units of output. For each possible choice of firm 1’s output, firm 2 wants to choose its own output to make its profits as large as possible. This means that for each choice of y1, firm 2 will pick the value of 3,12 that puts it on the isoprofit line furthest to the left, as illustrated in Figure 27.1. This point will satisfy the usual sort of tangency condition: the slope of the isoprofit line must be vertical at the Optimal choice. The locus of these tangencies describes firm 2’s reaction curve, f2(y1). To see this result algebraically, we need an expression for the marginal revenue associated with the profit function for firm 2. It turns out that this expression is given by MR2(y1,yzl = a - byi — 2by2. 484 OLIGOPOLY (Ch. 27) y2 = OUTPUT OF FIRM 2 lsoprofit lines for firm 2 f2(Y1) Reaction curve l‘2(y1 ) y1 y1 = OUTPUT or: FIRM I Derivation of a reaction curve. This reaction curve gives the profit—maximizing output for the follower, firm 2, for each output choice of the leader, firm 1. For each choice of y1 the follower chooses the output level f2(y1) associated with the iso- profit line farthest to the left. (This is easy to derive using calculus. If you don’t know calculus, you’ll just have to take this statement on faith.) Setting the marginal revenue equal to marginal cost, which is zero in this example, we have a— byl - 2by2 = O, which we can solve to derive firm 2’s reaction curve: anbyl 2b ' 92: This reaction curve is the straight line depicted in Figure 27.1. The Leader’s Problem We have now examined how the follower will choose its output given the choice of the leader. We turn now to the leader’s profit—maximization problem. Presumably, the leader is also aware that its actions influence the output choice of the follower. This relationship is summarized by the reaction QUANTITY LEADERSHIP 485 function f2(y1). Hence when making its output choice it should recognize the influence that it exerts on the follower. The profit—maximization problem for the leader therefore becomes Iggx p(y1 + y2)y1 — 01(y1) such that 3/2 = f2(y1)- Substituting the second equation into the first gives us Iggx Plyl + f2(y1)ly1 — 01(91)- Note that the leader recognizes that when it chooses output yl, the to- tal output produced will be yl + f2(y1): its own output plus the output produced by the follower. When the leader contemplates changing its output it has to recognize the influence it exerts on the follower. Let’s examine this in the context of the linear demand curve described above. There we saw that the reaction function was given by (1 —~ byl 2b f2(y1) = y2 = . (27.1) Since we’ve assumed that marginal costs are zero, the leader’s profits are 7r1 ($11,312) = plyi + y2)yl = ayl — by? — byiyz- (27-2) But the output of the follower, yg, will depend on the leader’s choice via the reaction function y2 = f2(y1). Substituting from equation (27.1) into equation (27.2) we have «1(91, y2) = ayi * by? — by1f2(y1) a — byl =— ayl - by? — byl 2b . Simplifying this expression gives us a b 2 7T1(y17y2) — 51/1 “ 291- The marginal revenue for this function is O, M = R 2 byl . Setting this equal to marginal cost, which is zero in this example, and solving for yl gives us yl 2 2b' 486 OLIGOPOLY (Ch. 27) In order to find the follower’s output, we simply substitute yi‘ into the reaction function, ._ a—byi‘ y2__ 2b *1 “411' These two equations give a total industry output of yf + yfi = 3a / 4b. The Stackelberg solution can also be illustrated graphically using the isoprofit curves depicted in Figure 27.2. (This figure also illustrates the Cournot equilibrium which will be described in section 27.5.) Here we have illustrated the reaction curves for both firms and the isoprofit curves for firm 1. The isoprofit curves for firm 1 have the same general shape as the isoprofit curves for firm 2; they are simply rotated 90 degrees. Higher profits for firm 1 are associated with isoprofit curves that are lower down since firm 1’s profits will increase as firm 2’s output decreases. .)’2 Reaction curve. for firm 1 Reaction- curve fer firm 2 Cournot equilibrium Stackelberg equilibrium lsoprofit . curves for . firml Y1 Stackelberg equilibrium. Firm 1, the leader, chooses the point on firm 2’s reaction curve that touches firm 1’s lowest possible isoprofit line, thus yielding the highest possible profits for firm 1.; _ - Firm 2 is behaving as a follower, which means that it will choose an output along its reaction curve, f2(y1). Thus firm 1 wants to choose an PRICE LEADERSHIP 487 output combination on the reaction curve that gives it the highest possible profits. But the highest possible profits means picking that point on the reaction curve that touches the lowest isoprofit line, as illustrated in Figure 27.2. It follows by the usual logic of maximization that the reaction curve must be tangent to the isoprofit curve at this point. 27.3 Price Leadership Instead of setting quantity, the leader may instead set price. In order to make a sensible decision about how to set its price, the leader must forecast how the follower will behave. Accordingly, we must first investigate the profit-maximization problem facing the follower. The first thing we observe is that in equilibrium the follower must always set the same price as the leader. This follows from our assumption that the two firms are selling identical products. If one charged a different price from the other, all of the consumers would prefer the producer with the lower price, and we couldn’t have an equilibrium with both firms producing. Suppose that the leader has set a price 19. We will suppose that the follower takes this price as given and chooses its profit-maximizing output. This is essentially the same as the competitive behavior we investigated earlier. In the competitive model, each firm takes the price as being outside of its control because it is such a small part of the market; in the price- leadership model, the follower takes the price as being outside of its control since it has already been set by the leader. The follower wants to maximize profits: max pyz — 02(y2). 112 This leads to the familiar condition that the follower will want to choose an output level where price equals marginal cost. This determines a supply curve for the follower, S (p), which we have illustrated in Figure 27.3. Turn now to the problem facing the leader. It realizes that if it sets a price p, the follower will supply S (p) That means that the amount of output the leader will sell will be R(p) = D(p) — S (p) This is called the residual demand curve facing the leader. Suppose that the leader has a constant marginal cost of production c. Then the profits that it achieves for any price p are given by: mp) = (p — 0) [13(10) — 500)] = (p — C)R(p)- In order to maximize profits the leader wants to choose a price and output combination where marginal revenue equals marginal cost. However, the marginal revenue should be the marginal revenue for the residual demand curveithe curve that actually measures how much output it will be able to 488 OUGOPOLY (Ch. 27) Demand curve facing leader (residual demand) MR facing leader yZ‘ y: QUANTITY Price leader. The demand curve facing the leader is the market demand curve minus the follower’s supply curve. The leader equates marginal revenue and marginal cost to find the optimal quantity to supply, ’92. The total amount supplied to the market is y; and the equilibrium price is p*. sell at each given price. In Figure 27.3 the residual demand curve is linear; therefore the marginal revenue curve associated with it will have the same vertical intercept and be twice as steep. Let’s look at a simple algebraic example. Suppose that the inverse de- mand curve is D(p) : ae-bp. The follower has a cost function 02(y2) : 36/2, and the leader has a cost function c1(y1) 2 cyl. For any price p the follower wants to Operate where price equals marginal cost. If the cost function is 62(y2) : 313/2, it can be shown that the marginal cost curve is M Cg (y2) = y2. Setting price equal to marginal cost gives us 17:92- Solving for the follower’s supply curve gives yg 2 5(1)) : p. The demand curve facing the leader—the residual demand curve—is R(p)IBM-5(1))=a—bp—p=a~(b+1)p- From now on this is just like an ordinary monopoly problem. Solving for p as a function of the leader’s output yl, we have 91- (27.3) SIMULTANEOUS QUANTITY SETTING 489 This is the inverse demand function facing the leader. The associated marginal revenue curve has the same intercept and is twice as steep. This means that it is given by a 2 MR1=b+1—b+1 91- Setting marginal revenue equal to marginal cost gives us the equation a _ 2 b+1 b-i-l MR1: y1=C=M01. Solving for the leader’s profit-maximizing output, we have a—c(b+1) 111: 2 We could go on and substitute this into equation (27.3) to get the equilib- rium price, but the equation is not particularly interesting. 27.4 Comparing Price Leadership and Quantity Leadership We’ve seen how to calculate the equilibrium price and output in the case of quantity leadership and price leadership. Each model determines a different equilibrium price and output combination; each model is appropriate in different circumstances. One way to think about quantity setting is to think of the firm as making a capacity choice. When a firm sets a quantity it is in effect determining how much it is able to supply to the market. If one firm is able to make an investment in capacity first, then it is naturally modeled as a quantity leader. On the other hand, suppose that we look at a market where capacity choices are not important but one of the firms distributes a catalog of prices. It is natural to think of this firm as a price setter. It’s rivals may then take the catalog price as given and make their own pricing and supply decision accordingly. Whether the price—leadership or the quantity-leadership model is appro— priate is not a question that can be answered on the basis of pure theory. We have to look at how the firms actually make their decisions in order to choose the most appropriate model. 27.5 Simultaneous Quantity Setting One difficulty with the leader—follower model is that it is necessarily asym- metric: one firm is able to make its decision before the other firm. In some 490 OLIGOPOLY (Ch. 27) situations this is unreasonable. For example, suppose that two firms are simultaneously trying to decide what quantity to produce. Here each firm has to forecast what the other firm’s output will be in order to make a sensible decision itself. In this section we will examine a one-period model in which each firm has to forecast the other firm’s output choice Given its forecast, each firm then chooses a profit~maximizing output for itself. We then seek an equi- librium in forecasts#a situation where each firm finds its beliefs about the other firm to be confirmed. This model is known as the Cournot model, after the nineteenth-century French mathematician who first examined its implications.3 We begin by assuming that firm 1 expects that firm 2 will produce 3;; units of output. (The e stands for ewpected output.) If firm 1 decides to produce yl units of output, it expects that the total output produced will be Y = y1 + 11%, and output will yield a market price of p(Y) : p(y1 + y§). The profit—maximization problem of firm 1 is then Himmler + 31391 - C(91)- For any given belief about the output of firm 2, pg, there will be some optimal choice of output for firm 1, pl. Let us write this functional rela- tionship between the empected output of firm 2 and the optimal choice of firm 1 as 91 = 161(95)- This function is simply the reaction function that we investigated earlier in this chapter. In our original treatment the reaction function gave the follower’s output as a function of the leader’s choice. Here the reaction function gives one firm’s optimal choice as a function of its beliefs about the other firm’s choice. Although the interpretation of the reaction function is different in the two cases, the mathematical definition is exactly the same. Similarly, we can derive firm 2’s reaction curve: 1’2 : f2(yl3)i which gives firm 2’s optimal choice of output for a given expectation about firm 1’s output, yf. Now, recall that- each firm is choosing its output level assuming that the other firm’s output will be at y? or y§. For arbitrary values of pi and pg this won’t happen—in general firm 1’s optimal level of output, yl, will be different from what firm 2 expects the output to be, yf. Let us seek an output combination (yf, 143) such that the optimal output level for firm 1, assuming firm 2 produces 213...
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