Oligopoly. Chapter Overview


 Brent Cannon
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1 Chapter 10 Oligopoly 10.1 Overview Oligopoly is the study of interactions between multiple rms. Because the actions of any one rm may depend on the actions of others, oligopoly is the rst topic which requires us to consider interactive decision making. In our study of supply and demand, we assumed that rms and consumers took market conditions to be given and did not realize their decisions a ected the other side of the market. In our study of monopoly pricing, we allowed the monopolist to account for the interdependence of quantity and price, but still assumed that consumers took market prices as given. That is, monopoly amounted to a single person strategic decision problem where the rm had complete control of the market outcome. But oligopoly requires a serious assumption of some sort about the interactions between the decisions of the rms because each rm has partial e ect on the market price. The standard models of oligopoly fall partway between the models of monopoly and perfect competition. Each rm accounts for its own e ect on prices, and maximizes pro ts, while taking the actions of other rms to be xed according to some prevailing conjecture about what those rms will do. An equilibrium requires that the prevailing conjectures about the actions of all rms actually correspond to their pro tmaximizing decisions given these conjectures. This de nition of equilibrium matches the de nition of Nash equilibrium in game theory. 261
2 10.2 Classical Cournot Outcomes The most prominent model of competition among a small number of rms is Cournot competition, as studied originally by Auguste Cournot in Each rm decides on a quantity to produce and the total quantity of production sets the market price. The formal model of Cournot competition is based on rms 1; :::; m. Each rm is de ned by a cost function c j (q j ) which gives its cost for production a give quantity of goods. The market demand function p(q) gives the marketclearing price as a function of total quantity, Q. assume that production costs are strictly increasing and convex (c 0 j > 0,. c00 j We 0) and that the market price is decreasing in quantity sold (p 0 (Q) 0). Each rm faces the same type of problem: (F P j ) max q j q j p(q 1 ; q 2 ; :::; q m ) c j (q j ): De nition 1 A Cournot equilibrium consists of quantities (q 1 ; q 2 ; :::; q m ) such that each rm maximizes pro ts given the production of other rms. The assumption that each rm takes the quantities of the others as given may seem unrealistic, for a rm may have many competitors and is unlikely to know about the speci c production plans of each and every one of them. In fact, since we assume that the market price depends only on the total quantity Q and not on the quantities chosen by individual rms, the informational requirements for a Cournot equilibrium can be somewhat relaxed any rm needs only to know the aggregate production of others to determine its pro tmaximizing quantity in equilibrium. One way to proceed with equilibrium analysis is very mechanical. Crank out the rst order conditions for each rm, check that the combination of cost function c j (q j ) and price function p(q) satisfy the secondorder conditions so that the FOC s for each rm. Assuming that all secondorder conditions are satis ed, then the combination of rstorder conditions for all the rms produce a system of m equations with m unknowns (the quantities of the rms) and should generally produce an interior solution which is a Cournot equilibrium. 1 But a more 1 It is also possible that there could be a boundary solution. For example, if one or more rms had very 262
3 intuitive approach is to analyze the reaction curves of each rm, which describe each rm s optimal quantity as a function of the total quantity produced by others. We begin by studying reaction curves in the case of duopoly (two rms) and then extend our analysis to the case of many rms. Reaction Curves De ne q1 (q 2) as the optimal production for rm 1 as a function of rms 2 s production, q 2. De ne q2 (q 1) similarly. An equilibrium consists of quantities ^q 1 ; ^q 2 such that q1 (q 2) = ^q 1 ; q2 (q 1) = ^q 2. For xed and known q 2, the rstorder conditions for rm 1 s maximization problem are p(q 1 + q 2 ) + q 1 p 0 (q 1 + q 2 ) c 0 1(q 1 ) = 0: The secondorder conditions for a maximum are given by 2p 0 (q 1 + q 2 ) + q 1 p 00 (q 1 + q 2 ) c 00 1(q 1 ) 0: The rst and secondorder conditions for rm 2 are exactly parallel to the conditions identi ed above for rm 1. We have assumed that p 0 0, c 00 0, so the secondorder conditions always hold if p 00 is nonpositive, and will also hold in some cases where p 00 > 0, but is not su ciently positive to o set the two other negative terms in the sum. In the simplest example, which we discuss at length below, both p and c are linear functions, so that p 00 = c 00 j = 0 and the secondorder conditions are guaranteed to hold. Proposition 2 Suppose that the rstorder conditions characterize the reaction functions for each rm 2 and that p 00 (q) 0. Then q 1 (q 2) is decreasing in q 2 and q 2 (q 1) is decreasing in q 1. We illustrate this result by totally di erentiating the rstorder conditions for q 1. analysis for q 2 is similar.) (The high costs of production that they would only lose money by setting q j > 0, then these rms should not produce anything in equilibrium. 2 This is equivalent to assuming that there is a unique solution to (FOC) and that the secondorder conditions hold as well. 263
4 p 00 (q 1 +q 2 )dq 1 +p 00 (q 1 +q 2 )dq 2 +p 0 (q 1 +q 2 )dq 1 +q 1 p 00 (q 1 +q 2 )dq 1 +q 1 p 00 (q 1 +q 2 )dq 2 c 00 1(q 1 )dq 1 = 0: Rearranging terms to isolate the ratio dq 1 dq 2 ; dq 1 dq 2 = q 1 p 00 (q 1 + q 2 ) + p 0 (q 1 + q 2 ) q 1 p 00 (q 1 + q 2 ) + p 00 (q 1 + q 2 ) + p 0 (q 1 + q 2 ) c 00 (q 1 ) : By assumption, each term in the numerator is (weakly) negative and each term in the denominator is (weakly) negative, implying dq 1 dq 2 0. Example 3 Consider a simple linear example with linear demand and constant marginal cost for the rms: p(q) = A Q = A q 1 q 2, MC 1 = c 1, MC 2 = c 2 :Then the rms are distinguished only by their marginal costs, with 1 (q 1 ; q 2 ) = q 1 (A q 1 q 2 ) c 1 q 1 ; 2 (q 1 ; q 2 ) = q 1 (A q 1 q 2 ) c 2 q 2 The derivative of rm 1 s pro t with respect to its quantity is 1 1 = A c 1 q 2 2q 1. Given the parameters A, Q, and a conjectured value q 2, rm 1 s marginal pro ts are decreasing in its quantity q 1, and thus its secondorder conditions for a maximum are satis ed. This is analogous to a monopoly problem with rm 1 as monopolist facing demand function D 1 (q 1 ) = A 0 q 1, where A 0 = A q2 0 and q0 2 represents the conjectured value of rm 2 s production, which rm 1 takes as given. In this respect, we can view each rm in a Cournot duopoly as a residual monopolist for customers who are not served by the other rm. As in the monopoly case, each rm recognizes that as it increases its quantity, it cuts the price to its existing customers, thereby producing decreasing marginal pro ts and a unique solution to its maximization problem for a given quantity produced by the other rm. Thus, rm 1 s rstorder conditions are su cient to identify its optimal quantity (assuming that they identify a positive value for q 1 ), yielding the solution q 1(q 2 ) = (A q 2 c 1 )=2: 264
5 This equation identi es rm 1 s reaction function to rm 2 s production since it identi es an optimal quantity as a function of rm 2 s production. rm 1 s reaction function is given by the equation Similarly, rm 2 s reaction function to q 2(q 1 ) = (A q 1 c 2 )=2: Given the assumption of linear costs and prices, these reaction functions are linear as well. Figure 1 graphs both reaction functions simultaneously; note that rm 1 s production is represented as a function from the vertical axis ( rm 2 s quantity) to the horizontal axis ( rm 1 s quantity).for the speci c parameter values A = 10, c 1 = 3; c 2 = 2. A Cournot equilibrium is an intersection of the two reaction functions, for this is the only way that the conjectured quantities for the rms could match the solutions to their maximization problems. As seen in Figure 1, there is a unique intersection of the reaction curves in the linear example and thus a unique Cournot equilibrium with q 1 = A 2c 1 + c 2 3 ; q2 = A 2c 2 + c 1 : 3 In Figure 1, rm 2 has lower marginal costs of production than does rm 1, so naturally rm 2 produces a greater quantity in the resulting equilibrium at (2, 3). Note in addition that if the two reaction curves are linear, they can intersect at most once, so there is a unique Cournot equilibrium. As depicted in the gure below, Firm 1 s reaction curve q1 (q 2) is more steeply sloping than rm 2 s reaction curve q2 (q 1). This is a general property of a linear duopoly model 3 that guarantees a unique equilibrium, either at an intersection of the reaction curves or (if there is no intersection of the reaction curves) at a boundary with only one rm producing a positive quantity. 4 3 In a linear duopoly model, rm 1 s reaction function is given by q1(q 2) = A c 1 q 2, which can be represented 2 as the inverse function q 1 2 (q 1) = A c 1 2q1, which has a slope of 2. By comparison, rm 2 s reaction function has a slope of 1/2 (q2(q 1) = A c 2 q 1 ), so in graphical representation, rm 2 s reaction function is always atter 2 than rm 1 s reaction function. 4 A boundary equilibrium occurs, if for example q2(0) = q 2m is on rm 2 s reaction curve, but q1(q 2m) < 0. In this case, rm 2 simply produces its monopoly quantity and rm 1 produces nothing in equilibrium. The negative value for rm 1 s reaction function indicates that rm 1 s production costs are so high that it cannot compete pro tably with rm 2 in equilibrium. 265
6 Duopoly Reaction Curves Firm 2 quantity q1(q2) q2(q1) Firm 1 quantity 266
7 In a symmetric case with c 1 = c 2 = k, the equilibrium values for the two rms converge to the same value (A k)=3. Tatonnement One challenge is that the equilibrium analysis above does not provide a description of why rms would produce the equilibrium quantities in a Cournot duopoly. If each rm makes the right conjecture about what the other will produce, then they would certainly produce their equilibrium quantities, but this leaves the question of how they might come to these conjectures. One extreme answer, suggested by Cournot, is that sophisticated rms should come to expect equilibrium actions from each other. The basis for this answer in Cournot competition is based on what is known as a tatonnement (slight adjustment) process. Suppose that the rms compete over time in many consecutive periods, where rm i produces q i;t in period t. Further, the rms use last period s output as the best prediction of this period s output for its rival. That gives a dynamic process for the outputs over time based on the equations q 1;t+1 = q 1(q 2;t ); q 2;t+1 = q 2(q 1;t ): As shown in Figure 2 for a linear duopoly example, the tatonnement process leads to "cobweb" dynamics that converge towards the equilibrium point. For example, if we start at an arbitrary point B, where both rms are producing more than their equilibrium quanities, then they will move to point C in the second period, where both rms are producing less than their equilibrium quantities, to point D in the third period, and so on. Each period, the rms switch their production from one side of the equilibrium point to the other (alternating between producing relatively more and relatively less than the equilibrium quantities), while moving ever closer to equilibrium. If we allow this dynamic process to continue for many periods, the rms will move asymptotically closer and closer to the Cournot equilibrium over time for any nite number of periods t, they will not actually reach equilibrium, but for any distance from the equilibrium point, there is some t () so that after t or more periods, the rms are always within of their equilibrium quantities. 267
8 . The Cournot equilibrium is nice because it has this interpretation as the longrun dynamic outcome of some version of strategic interactions. At the same time, though, the tatonnement description is dubious. First, it only applies in repeated competition. But more worrying is the fact that the assumption of each rm that their competitor is acting as in the previous period is clearly false, yet they cling to that assumption over many periods of play. Furthermore, rms may wish to try to develop a reputation over time in order to in uence their rivals. Thus, we cannot really rely on the tatonnement story provided by Cournot and are left without a justi cation of the Cournot equilibrium as the result of a dynamic convergence process Comparison of Cournot and Monopoly Outcomes A monopolist in a simple linear demand economy p(q) = A Q would set p = Q = (A k)=2, where k is the constant marginal cost of production. By contrast, two Cournot competitors would each produce (A k)=3 for total production of 2(A k)=3 and price of (A k)=3. Since the monopoly output maximizes total pro ts for the industry, the Cournot rms would do better to reduce their production. They do not do so because their relationship to the product market mixes competition and monopoly. Taking the production of the other competitor as xed (competition), each acts as a monopolist in what it perceives as the residual market, where 2(A k)=3 consumers remain and the demand curve appears to be p(q i ) = 2A=3 k=3 q i : 268
9 As a result, each rm incorporates the e ect of its production on the prices of its own goods, but it ignores the e ect on the sales of the other rm. With more rms, the pattern continues as each has less and less of a residual market to monopolize. Then, q i = (A k)=(n 1); Q = (A k) n=(n + 1); p = A=(n + 1) + nk=(n + 1): As n increases, total quantity Q increases and p decreases. Ultimately, as n! 1, p! k and we reach the state of perfect competition. This example demonstrates the earlier statement that as the number of rms increases (from 1), the behavior of the rms moves further away from monopoly and towards perfect competition Incentives and the Market The reaction curves illustrate a simple point. As rm 1 produces more, rm 2 s optimal reaction is to produce less. If the rms recognize that point (and they will), they may try to in uence their rival s action. Note that the strategic interaction between the actions of the rms is dismissed in the de nition of Cournot equilibrium. If rm 1 can somehow commit to a more aggressive production policy, it could improve its lot from the original equilibrium because rm 2 would produce correspondingly less than before. Firm 1 can attempt to commit to more production in several ways Stackleberg Competition In the Stackelberg version of competition, one rm moves rst. This rm must be at least as well o as its rival because it can always choose the earlier level of production (e.g. (A k)=3 with two rms). Instead, it will produce more, based on the logic above, giving it a rstmover advantage. Formally, rm 1 can anticipate rm 2 s production as a function of its own decision. 5 q2(q 1 ) = A q 1 k : 2 5 Note that rm 1 s decision precedes rm 2 s decision in time, so only rm 1 can view the problem in this manner. 269
10 (F P 1 ) max q 1 q 1 p(q) = q 1 (A q 1 q 2(q 1 )): Substituting and taking the derivative gives q 1 = (A k)=2; q 2 = (A k)=4: Total output has increased from the Cournot equilibrium, but rm 1 is better o. Although we have only solved for a special case here, it is easy to demonstrate that both of these properties carry over when the demand function is more general (not necessarily so simple and linear) and marginal costs are increasing rather than constant: rm 1 increases output if it can move rst, increases pro ts from the symmetric equilibrium because rm 2 reduces its output in response. The Stackelberg equilibrium is not an equilibrium of the original Cournot simultaneousmoves game. Firm 2 is responding optimally to rm 1, but rm 1 s is producing too much to be optimal. Firm 1 increased its production to constrain rm 2, but it cannot do that and choose the best response to rm 2 simultaneously. What rm 1 would really like to do is 1) pretend to commit to (A k)=2, 2) induce rm 2 to produce (A k)=4, 3) actually respond optimally to q 2 = (A k)=4 by producing 3(A k)=8. Of course, then rm 2 would wish to increase its production a bit from (A k)=4 and so on. What happens if we allow this process to of thought to continue? We will inevitably end up back at the symmetric Cournot equilibrium because this is exactly the tatonnement process Fixed and Marginal Costs: Another way to become more aggressive is to shift costs from marginal costs to xed costs. With lower marginal costs of production, a rm hits the point MC = MR at higher production level than before (in its monopoly calculations for its residual market). Again, because of production in uence on rivals, it is possible to improve pro ts by shifting towards a xed cost production technology, e.g. building new, faster (but expensive) machinery. The natural extension of this result is that it might be desirable to choose a more costly technology which is weighted towards xed costs over a cheaper one weighted towards marginal costs. Example: Suppose that p(q) = 9 Q; k 1 = k 2 = 3: The rms are identical, each with marginal costs of 3 and no xed costs. The Cournot equilibrium is for each to produce (A k)=3 or 2 each, with a market price of 5. Each rm produces 3 units of the good at a total cost of 6, while incurring a cost of 3*2 =
11 How much would rm 1 pay to reduce its marginal cost to 0? You would not think that it would ever pay more than its current production costs of 6. But let s see what happens. With c 1 = 0, rm 1 s optimal reaction to rm 2 is to produce (9 9q 2 )=2, which is an increase from the earlier value of 3 (q 2 =2). Note that xed costs play no role in the market competition because they are taken to be sunk costs at that point. Then the Cournot equilibrium is q 1 = 4; q 2 = 1; p = 4. By shifting to a lower marginal cost technology, rm 1 in uences rm 2 to cut production by 50%! Now rm 1 s production costs are given by the xed cost, F C and its pro t is 16 F C. It is pro table to spend up to 12 to change technologies, even though that shift could increase the production costs in the relevant range of possibilities Managerial Contracts A nal method to become more aggressive is to pay your managers a bonus based on marketshare rather than total pro ts. That is quite unusual because most contracting issues involve attempts to align the incentives of employees with those of the rm. Paradoxically, oligopoly competition is one instance where it can improve pro ts of the rm to have its managers value something else. If the managers value market share, then they will always choose to produce more output than suggested by the reaction curve. That in turn causes other rms to be more conservative. It is the e ect on other rms that is paramount here. Once again, the commitment to aggressiveness can increase pro ts. Kreps problem 16.2 develops this idea further Price Competition It may seem that price rather than quantity competition is a more natural model. After all, rms cannot choose their market shares, which is at the core of Cournot competition. In fact, it is said that Cournot wished to study a situation in which the strategic choice of rms was the price rather than the quantities, but that he felt that it was too di cult that way. Price competition is known as Bertrand competition. In its extreme form, the rms have unlimited capacity of production, constant and identical marginal costs k, and undi erentiated goods. The rms choose prices simultaneously and the rm which charges a lower price will 271
12 capture the whole market. They split the market at equal prices. Unfortunately, this model produces an astonishing and somewhat unbelievable result: the only equilibrium is for both rms to choose a price equal to their marginal cost, yielding no pro ts whatsoever. The proof of this result is really quite simple: the best response for rm 1 to any price p 2 > MC = k is a price just less than p 2. Similarly, the best response for rm 2 to any price p 1 > k is a price just less than p 1. Price equals marginal cost equals k is the only way both can be simultaneously playing best responses to the other s action. In this case, tatonnement takes the form of continual undercutting of prices. If the initial price is above marginal cost, the rms will reduce those prices ruthlessly to attempt to take the whole market, and they will continue cutting prices until price reaches marginal cost. Comment: This actual story of price competition is an explanation of why it might be that rms o er minimum price guarantees. If you have to o er rebates to previous customers whenever you cut the price, then this reduces your incentives to cut the price, even if it allows you to capture the whole market. If all the rms o er minimum price guarantees, then they may be able to maintain a pro table price and escape from the cutthroat competition that drove the price down to marginal cost Linking Bertrand and Cournot Results Since Bertrand and Cournot each o er such di erent conclusions about the results of market pricing, we would like to resolve their seeming clash. With slightly more realistic assumptions, the Bertrand model adjusts to produce results close to Cournot outcomes. 1. Let the rms choose capacities in period 1 (building expenses are increasing in capacity) and then choose prices in period 2. If the capacity of the rms are not large enough to meet the full market demand individually, then they could charge di erent prices and yet each still have some market share. 2. Allow for a cost of congestion. As one rm gains more customers, it cannot o er such good service to them, either because of long lines or because it is straining its production limits. 3. The rms o er slightly di erentiated products. Even if Coke is considerably cheaper than Pepsi, some customers will still buy Pepsi. Any of these adjustments produces an equilibrium which resembles pure Cournot competi 272
13 tion more than it resembles pure Bertrand outcomes. 273
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