I. Introduction and definitions

Transcription

1 Economics 335 March 7, 1999 Notes 7: Noncooperative Oligopoly Models I. Introduction and definitions A. Definition A noncooperative oligopoly is a market where a small number of firms act independently, but are aware of each other s actions. B. Some Typical Assumptions for These Markets 1. Consumers are price takers. All firms produce homogeneous products 3. There is no entry into the industry 4. Firms collectively have market power: they can set price above marginal cost. 5. Each firm sets only its price or output (not other variables such as advertising). C. Some Tentative Conclusions 1. The equilibrium price lies between that of monopoly and perfect competition. Firms maximize profits based on their beliefs about actions of other firms 3. The firm s expected profits are maximized when expected marginal revenue equals marginal cost 4. Marginal revenue for a firm depends on its residual demand curve (market demand minus the output supplied by other firms) II. Oligopoly Models and Game Theory A. Idea of Game Theory A game is a formal representation of a situation in which a number of decision makers (players) interact in a setting of strategic interdependence. By that, we mean that the welfare of each decision maker depends not only on her own actions, but also on the actions of the other players. Moreover, the actions that are best for her to take may depend on what she expects the other players to do. We say that game theory analyzes interactions between rational, decision-making individuals who may not be able to predict fully the outcomes of their actions B. Assumptions About Behavior of Firms in Oligopolistic Games 1. Firms are rational. Firms reason strategically

2 C. Elements of Typical Oligopolistic Games 1. There are two or more firms (not a monopoly).. The number of firms is small enough that the output of an individual firm has a measurable impact on price (not perfect competition). We say each firm has a few, but only a few rivals. 3. Each firm attempts to maximize its expected profit (payoff). 4. Each firm is aware that other firm s actions can affect its profit. 5. Equilibrium payoffs are determined by the number of firms, the rules of the games and the length of the game. D. Comparison of Oligopoly with Competition In competition each firm does not take into account the actions of other firms. In effect they are playing a game against an impersonal market mechanism that gives them a price that is independent of their own actions. E. Single Period or Static Games In such games the rivals meet only once to compete but do not compete again. Such models are appropriate for markets that last only a brief period of time. F. Nash Equilibrium A set of strategies is called a Nash equilibrium if, holding the strategies of all other firms constant, no firm can obtain a higher payoff by choosing a different strategy. At equilibrium, no firm wants to change its strategy. III. Single Period Models of Oligopoly A. Definition of a Single Period Model Firms meet only once in a single period model. The market then clears one and for all. There is no repetition of the interaction and hence, no opportunity for the firms to learn about each other over time. B. Three Standard Duopoly Models 1. Cournot. Bertrand 3. Stackelberg

3 3 IV. The Cournot Model A. Historical Background Augustin Cournot was a French mathematician. His original model was published in He started with a duopoly model. His idea was that there was one incumbent firm producing at constant unit cost of production and there was one rival firm considering entering the market. Since the incumbent was a monopolist, p > MC, and there was potential for a rival to enter and to make a profit. Cournot postulated that the rival firm would take into account the output of the incumbent in choosing a level of production. Similarly he postulated that the monopolist would consider the potential output of the rival in choosing output. B. Cournot duopoly model and solution 1. Assumptions a. Two firms with no additional entry b. Homogeneous product such that q 1 + q = Q where Q is industry output and q i is the output of the ith firm. c. Single period of production and sales (consider a perishable crop such as cantaloupe or zucchini). d. Market and inverse market demand is a linear function of price. p ö A BQ ö A B(q 1 ø q ) Q ö A B p B ö A p B ö a bp (1) As an example assume market demand is given by Q(p) ö p () and inverse demand is given by p ö 1.001Q. ö 1.001(q 1 ø q ). (3) With this function when p = $1.00, Q = 0, and when p = 0, Q = e. Each firm has constant and equal marginal cost equal to c. With constant marginal cost, average cost is also constant and equal to c. For the example assume that AC = MC = $0.8. f. The decision variable in the quantity of output to produce and market

4 4. The residual demand curve for firm If firm believes that firm 1 will sell q 1 units of output, then its residual inverse demand curve is given by p ö A Bq 1 Bq ö (A Bq 1 ) Bq (4) In equation 4 we view (A - Bq 1 ) as a constant. We can also write this in quantity dependent form as q (p) ö Q(p) q 1 (5) We obtain equation 5 by shifting the market demand curve to the left by q 1 units. For example if q 1 = 300 then the residual demand curve will hit the horizontal axis at 700 units. Thus q (p) = p = p. Now if p = 0 then q (p) = 700. The residual inverse demand curve is given by substituting 300 in equation 3 to obtain p ö q. (6) We can also find residual marginal revenue by taking the derivative of the residual revenue function. With residual demand given by p = A - Bq 1 - Bq, revenue for the second firm is given by R ö (A Bq 1 Bq )q ö Aq Bq 1 q Bq (7) while residual marginal revenue is given by the derivative of residual revenue R ö Aq Bq 1 q Bq MR ö A Bq 1 Bq (8) For the example with residual demand given by p = q, revenue for the second firm is given by R = q ( q ) =.70 q - 001q.. Residual marginal revenue is given by the derivative of residual revenue R ö 0.70q.001q 1 < MR ö dr dq ö q (9)

5 5 3. Profit maximization for firm Setting marginal revenue equal to the constant marginal cost will allow us to solve for q as a function of q 1, c, and the demand parameters as follows MR ö dr dq öa Bq 1 Bq ö MC öc < A Bq 1 Bq ö MC öc < Bq ö A Bq 1 c < q ö A Bq 1 c B < q õ ö A c B q 1 (10) This is called the reaction function for firm. For any level of firm 1 output, it gives the optimal level of output for firm 1. For the example firm marginal cost is given by $0.8, so setting marginal revenue equal to marginal cost with q 1 = 300 will give MR ö dr dq ö.70.00q ö MC ö.8 <.70.00q ö MC ö.8 <.00q ö.4 < q õ ö.4.00 ö10 (11) Rather than specifying a level of q 1 in this example we can perform the same analysis for a generic value of q 1 as follows. In this case

6 6 q (p) ö p q 1 ö 1000 q p < p ö 1000 q q ö 1.001(q 1 øq ) (1) < R ö [1.001(q 1 øq )]q ö q.001q.001q 1 q < MR ö dr dq ö1.001q 1.00q Since marginal cost is given by $0.8 setting marginal revenue equal to marginal cost will give MR ö dr dq ö 1.001q 1.00q ö MCö.8 < 1.001q 1.00q ö MCö.8 <.00q ö.7.001q 1 < q õ ö q 1 ö 360.5q 1 (13)

7 7 For different values of q 1 we get different values of q õ as follows q 1 q * MR(q *) MC

8 8 Graphically we can depict the optimum as follows Demand and Residual Demand 1.00 $ Q Demand Res Demand MR MC We can also plot the reaction curve for firm given various levels of q 1. Reaction Curve for Firm Firm Output Firm 1 Output q When q 1 = 300 the optimum quantity of q is at 10. The firm will attempt to charge a price of p = (10+300) =.49. Plugging this in the demand function will give Q = (.49) = 510. Alternatively if firm acted competitively, equilibrium would be where the residual demand curve intersected marginal cost. This is at an output of 40. With q 1 at 300 this gives an industry output of 70. This then gives a market price of p = (40+300) =.8 which is equal to marginal cost.

9 9 4. Profit maximization for firm 1 Firm 1 is symmetric to firm so consider the profit maximization problem for firm 1. Œ 1 ö max [pq 1 C(q 1 )] q 1 ö max [(A Bq 1 Bq )q 1 cq 1 ] q 1 ö max [Aq 1 Bq 1 Bq 1 q cq 1 ] q 1 < dœ 1 ö A Bq dq 1 Bq c ö 0 1 < Bq 1 ö A Bq c < q õ 1 ö A c B q (14) This is the reaction function for firm 1. The parameters in the example problem are A = 1, B =.001 and c =.8. This then gives a response function of q õ 1 ö A c B q 1.8 ö ()(.001) 0.5q.7 ö q ö q (15) Now suppose firm 1 thinks that firm will produce 10 units. Firm 1 will then produce q õ 1 ö q ö 360 (0.5)(10) ö ö 55 (16) Unfortunately when firm 1 thinks firm will produce 10, it chooses to produce 55, not the 300 that would have induced firm to produce 10. Thus this set of beliefs and decisions is not a Nash equilibrium since firm would now prefer to produce not 10, but 3.5. But if firm produces 3.5 then firm 1 may also choose a different level of output etc.

10 10 5. Finding a Nash equilibrium In order to find an equilibrium we need to find a point where each firms beliefs about the other firm are fulfilled. This can be done using the reaction functions for each firm which are the same. Consider first the best response for firm 1 if firm produces nothing. This is given by q õ 1 ö A c B 0 ö A c B (17) A c which is just the monopoly solution. But what if firm 1 did produce. The best response of firm B would be to produce q õ ö A c B q 1 ö A c B (0.5) A c B ö A c 4B (18) But then firm 1 will no longer assume it is a monopolist but will take this output as given and produce q õ 1 ö A c B q ö A c B (0.5) A c 4B ö 3(A c) 8B (19) Cournot s insight was that for an outcome to be an equilibrium, it must be the case that each firm is responding optimally to the (optimal) choice of its rival. Each firm must choose a best response based upon a prediction about what the other firm will produce, and in equilibrium, each firm s prediction must be correct. An equilibrium requires that both firms be on their respective reaction curves. To find this point we can substitute the optimal response for firm 1 into the response function for firm as follows

11 11 q õ < 3q õ 4 ö A c B q 1 ö A c B (0.5) A c B q ö A c 4B ø q 4 ö A c 4B < q õ ö A c 3B (0) Similarly we find that q õ 1 ö A c 3B (1) We can also see this graphically by plotting the reaction functions as follows Cournot Equilibrium q q1 q* q1* Industry output is the sum of the output of the two firms Q õ ö q õ 1 ø q õ ö A c 3B ö (A c) 3B ø A c 3B ()

12 1 The equilibrium price is given by substituting Q* in the demand equation p ö (A BQ õ ) ö A B ö (A c) 3B A 3 Aø 3 c (3) ö 1 3 Aø 3 c which gives p = 1 3 (1) ø 3 (.8) ö.5 in the example. Each firm then has profit of Œ i ö pq õ i cq õ i ö 1 3 Aø 3 c qõ i ö 1 3 A 1 3 c qõ i cq õ i (4) For the numerical example we obtain q õ 1 ö ö 40 We can also get this using the numerical response functions directly q ö 360.5q 1 ö 360.5(360.5q ) ö ø.5q <.75q ö 180 < q õ ö 40 (5) In a similar fashion it is clear that q 1 * = 40. Thus the Cournot equilibrium for this market is q 1 * = q * = 40. Each firm believes the other will sell 40 units, each will sell 40 units. At this point industry output is 480 units and industry price will be p = (480) = 0.5. Each firm will have profits equal to Œ = (.5) (40) - (.8)40 = 57.6 and total industry profits will be The equilibrium presented is the only plausible solution to this problem since it is the only one where beliefs are consistent with results. The difficulty is how the firms arrive at these correct beliefs in a one period, one-shot model.

13 13 C. Comparison of Duopoly with the Cartel Equilibrium If the firms act as a cartel, they will consult and set output such that joint profits are maximized. The problem is set up as follows where only total output is of concern Œ ö max [p(q 1 øq ) C(q 1 ) C(q )] q 1 øq ö max [(A BQ)Q cq] Q ö max [AQ BQ cq] Q < dœ dq ö A BQ c ö 0 < BQ ö A c < Q ö A c B (6) This is the same as the monopoly solution. If the firms split the output and profit they obtain total profit of Œ m ö (A BQ õ )Q õ cq õ ö A B (A c) B Q õ cq õ ö A 1 Aø 1 c Qõ cq õ (7) ö 1 Aø 1 c Qõ cq õ ö 1 A 1 c Qõ For the numerical example we obtain

14 14 Œ ö max [p(q 1 øq ) C(q 1 ) C(q )] q 1 øq ö max [(1.001Q)Q.8Q] Q ö max [Q.001Q.8Q] Q (8) < dœ dq ö 1.00Q.8 ö 0 <.00Q ö.7 < Q ö360 At this quantity the market price will be p = (360) =.64 which is higher than the duopoly price of.5 and is higher than marginal cost of.8. Thus consumers are better off with the Cournot duopoly as compared to the cartel solution. The total profits in the cartel are given by Œ =.64(360) -.8(360) = $ How the firms divide up these profits is rather arbitrary ranging from an equal split to one firm having almost all of them. Since the cartel profits are higher than the Cournot ones, there is a definite incentive for the firms to collude. D. Comparison with the Competitive Equilibrium If the firms act competitively they will each take price as given and maximize profits ignoring the other firm. For firm 1 the supply function is derived as follows Œ 1 ö max q 1 [pq 1 C(q 1 )] ö max q 1 [pq 1 cq 1 ] < dœ dq 1 ö p c ö 0 < p ö c (9) With constant marginal cost the firm will supply an infinite amount at a price of c and none if price is less. Similarly for the second firm. Thus the equilibrium price will be c. For the numerical example this means that p =.8. Market demand and supply will be given by Q ö A p B ö A c B. For the example market demand is given by Q = p = (1000)(.8) = = 70. Neither firm will make a profit and will be indifferent about production levels. Thus the firms will clearly take account of each others actions as in the duopoly solution.

15 15 E. A Cournot Model with Many Firms 1. Inverse demand equations The market inverse demand equation is given by p ö A BQ ö A B(q 1 ø q ø à ø q N ) N ö A B( q j ) jö1 (30) For the ith firm we can write p ö A BQ ö A B(q 1 ø à ø q i 1 ø q iø1 ø àø q N ) ö A B( q j ) Bq i jgi (31) ö A BQ i Bq i. Profit maximization Profit for the ith firm is given by Œ i ö max [pq i C(q i )] q i ö max [(A BQ i Bq i )q i cq i ] q i ö max [Aq i BQ i q i Bq i cq i ] q i (3) Optimizing with respect to q i will give dœ i dq i ö A BQ i Bq i c ö 0 < Bq i ö A BQ i c < q õ i ö A c B Q i (33) Since all firms are identical, this holds for all other firms too.

16 16 3. Nash equilibrium In a Nash equilibrium, each firm i chooses a best response q i * that reflects a correct prediction of the other N-1 outputs in total. Denote this sum as Q -i *. A Nash equilibrium is then q õ i ö A c B Q õ i (34) Since all the firms are identical we can write for the identical firms. We can then write < q õ ø (N 1)q õ < (Nø1)q õ q õ ö A c B < q õ ö Q õ i ö q õ j ö (N 1)q õ igj ö A c B ö A c B A c (Nø1) B (N 1)q õ where q* is the optimal output (35) Industry output and price are then given by Q õ ö N(A c) (Nø1) B p õ ö A BQ õ N(A c) ö A B (Nø1) B ö (Nø1)A NA Nø 1 (Nø1) Nc Nø1 A ö Nø1 ø N Nø1 c (36) Notice that if N =1 we get the monopoly solution while if N = we get the duopoly solution, etc. Consider what happens as N 9 7. The first term in the expression for price goes to zero and the second term converges to 1. Thus price gets very close to marginal cost. Considering quantities, the N(A c) (Nø1)B 9 (A c) B ratio which is the competitive solution.

17 17 4. Numerical example It easily can be shown that the reaction function for a representative firm in a Cournot model with many firms will take the form q õ ö 360 (.5)(q õ )(N 1) (37) where q is the identical output of each of the other firms. The optimal q for each firm will be q õ ö 70 Nø1 (38) with equilibrium price given by p õ ö 1ø.8N Nø1 (39) Clearly the larger the number of firms the smaller is output per firm, the higher is industry output and the lower is price. F. A Cournot Model with Differing Costs 1. Individual firm optima with different marginal costs Consider first a two firm model where firm i has marginal cost c i. The optimal solution is obtained as before by defining profit and choosing q i to maximize profit given the level of output of the other firm. Consider the first firm as follows. Œ 1 ö max [pq 1 C(q 1 )] q 1 ö max [(A Bq 1 Bq )q 1 c 1 q 1 ] q 1 ö max [Aq 1 Bq 1 Bq 1 q c 1 q 1 ] q 1 < dœ 1 dq 1 ö A Bq 1 Bq c 1 ö 0 < Bq 1 ö A Bq c 1 (40) < q õ 1 ö A c 1 B q Similarly the best response function for the second firm is given by < q õ ö A c B q 1 (41)

18 18. Nash equilibrium We solve for the optimal q i * simultaneously as in the equivalent marginal cost case. q õ ö A c B q 1 ö A c B (0.5) A c 1 B q ö 0.5A c ø0.5c 1 B ö A c øc 1 4B ø q 4 ø 0.5q (4) < 3q õ 4 ö A c øc 1 4B < q õ ö A c øc 1 3B We can solve for q 1 * in a similar fashion to obtain q õ 1 ö A c 1 øc 3B (43) Graphically consider a case where c 1 = c and c > c. Cournot Equilibrium with Different Costs q q1 q* q1* q* High Notice that the response function of firm has moved inward so that it produces less for each value of firm 1. Its optimal output is less. For the numerical example and a marginal cost for the second firm of 0.43 we obtain

19 19 q õ 1 ö A c 1 øc 3B ö 1 (.8)ø.43 3(.001) ö 90 q õ ö A c øc 1 3B ö 1 (.43)ø.8 3(.001) ö 140 (44) p ö.57 At this price each firm has marginal revenue equal to marginal cost. Total production of 430 is less than before (480). V. The Bertrand Model A. Introduction In the Cournot model each firm independently chooses its output. The price then adjusts so that the market clears and the total output produced is bought. Yet, upon reflection, this phrase the price adjusts so that the market clears appears either vague or incomplete. What exactly does it mean? In the context of perfectly competitive markets, the issue of price adjustment is perhaps less pressing. A perfectly competitive firm is so small that its output has no effect on the industry price. From the standpoint of the individual competitive firm, prices are given i.e., it is a price-taker. Hence, for analyzing competitive firm behavior, the price adjustment issue does not arise. The issue of price adjustment does arise, however, from the perspective of an entire competitive industry. That is, we are obliged to say something about where the price, which each individual firm takes as given, comes from. We usually make some assumption about the invisible hand or the Walrasian auctioneer. This mechanism is assumed to work impersonally to insure that the price is set at its market clearing level. But in the Cournot model, especially when the number of firms is small, reliance upon the fictional auctioneer of competitive markets seems strained. After all, the firms clearly recognize their interdependence. Far from being a price-taker, each firm is keenly aware that the decisions it makes will affect the industry price. In such a setting, calling upon the auctioneer to set the price is a bit inconsistent with the development of the underlying model. Indeed, in many circumstances, it is more natural to cut to the chase directly and assume that firms compete by setting prices and not quantities. Consumers then decide how much to buy at those prices. The Cournot duopoly model, recast in terms of price strategies rather than quantity strategies, is referred to as the Bertrand model. Joseph Bertrand was a French mathematician who review and critiqued Cournot s work nearly fifty years after its publication in 1883, in an article in the Journal des Savants. A central point in Bertrand s review was that the change from quantity to price competition in the Cournot duopoly model led to dramatically different results.

20 0 B. Basic Bertrand Model 1. Assumptions a. Two firms with no additional entry b. Homogeneous product such that q 1 + q = Q where Q is industry output and q i is the output of the ith firm. c. Single period of production and sales d. Market and inverse market demand is a linear function of price. p ö A BQ ö A B(q 1 ø q ) ö 1.001Q Q ö A B p B ö A p B ö p (45) e. Each firm has constant and equal marginal cost equal to c. With constant marginal cost, average cost is also constant and equal to c. f. The decision variable in the price to charge for the product. The residual demand curve for firm If firm believes that firm 1 will sell q 1 units of output, then its residual inverse demand curve is given by p ö A Bq 1 Bq ö (A Bq 1 ) q (46) while residual demand is given by Q ö A B p B ö a bp where a ö A B and b ö 1 B (47) In order to determine its best price choice, firm must first work out the demand for its product conditional on both its own price, p, and Firm 1's price, p 1. Firm 's reasoning then proceeds as follows. If p > p 1, Firm will sell no output. The product is homogenous and consumers always buy from the cheapest source. Setting a price above that of Firm 1, means Firm serves no customers. The opposite is true if p < p 1. When Firm sets the lower price, it will supply the entire market, and Firm 1 will sell nothing. Finally, we assume that if p = p 1, the two firms split the market. The foregoing implies that the demand for Firm 's output, q, may be described as follows:

21 1 q ö 0 if p > p 1 q ö a bp if p < p 1 q ö (a bp ) if p ö p 1 (48) Consider a graphical representation of this demand function. Demand Function for Firm in Bertrand Model p p 1 Demand a bp q The demand structure is not continuous. For any p greater than p 1, demand for q is zero. But when p falls and becomes equal to p 1, demand jumps from zero to (a - bp )/. When p then falls still further so that it is below p 1, demand then jumps again to a - bp. This discontinuity in Firm 's demand curve is not present in the quantity version of the Cournot model. This discontinuity in demand carries over into a discontinuity in profits. Firm 's profit, ù, as a function of p 1 and p is given by:

22 ù (p 1,p ) ö 0 if p >p 1 ù (p 1,p ) ö (p c)(a bp ) if p <p 1 ù (p 1,p ) ö (p c)(a bp ) if p ö p 1 (49) 3. Best response functions To find Firm 's best response function, we need to find the price, p, that maximizes Firm 's profit, ù (p 1, p ), for any given choice of p 1. For example, suppose that Firm 1 chose a very high price higher even aøbc b ö A B ø c B B ö Aø c than the pure monopoly p M =. Since Firm can capture the entire market by selecting any price lower than p 1, its best bet would be to choose the pure monopoly price, p M, and thereby earn the pure monopoly profits. Conversely, what if Firm 1 set a very low price, say one below its unit cost, c? If p 1 < c, Firm is best setting its price at some level above p 1. This will mean, we know, that Firm will sell nothing and earn zero profits. What about the more likely case in which Firm 1 sets its price above marginal cost, c, but below the pure monopoly price, p M? How should Firm optimally respond in these circumstances? The simple answer is that it should set a price just a bit less than p 1. The intuition behind this strategy is illustrated in the next figure which shows Firm 's profits given a price, p 1, satisfying (a+bc)/b > p 1 > c. Observe that Firm 's profits rise continuously as p rises from c to just below p 1. Whenever p is less than p 1, Firm is the only company that any consumer buys from. However, in the case where p 1 is less than p M, the monopoly position that Firm obtains from undercutting p 1 is constrained. In particular, it cannot achieve the pure monopoly price, p M, and associated profits, because at that price, Firm would lose all its customers. Still, the firm will wish to get as close to that result as possible. It could, of course, just match Firm 1's price exactly. But whenever it does so, it shares the market equally with its rival. If, instead of setting p = p 1, Firm just slightly reduces its price below the p 1 level, it will double its sales while incurring only a infinitesimal decline in its profit margin per unit sold. This is a trade well worth the making as the figure makes clear. In turn, the implication is that for any p 1 such that c < p 1 < p M, Firm 's best response is to set p = p 1 - J, where J is an arbitrarily small amount. The last case to consider is the case in which Firm 1 prices at cost so that p 1 = c. Clearly, Firm has no incentive to undercut this value of p 1. To do so, would only involve Firm in losses. Instead, Firm will do best to set p either equal to or above p 1. If it prices above p 1, Firm will sell nothing and earn zero profits. If it matches p 1, it will enjoy positive sales but break even on every unit sold. Accordingly, Firm will earn zero profits in this latter case, too. Thus, when p 1 = c, Firm 's best response is to set p either greater than or equal to p 1. Let p õ denote Firm 's best response price for any given value of p 1. Our preceding discussion may be summarized as follows

23 3 p õ ö aøbc b if p 1 > aøbc b p õ ö p 1 J if c < p 1 & aøbc b (50) p õ ' p 1 if c ö p 1 p õ > p 1 if c > p 1 ' 0 By similar reasoning, Firm 1's best response, p õ 1, for any given value of p is given by: p õ 1 ö aøbc b if p 1 > aøbc b p õ 1 ö p J if c < p & aøbc b (51) p õ 1 ' p if c ö p p õ 1 > p 1 if c > p ' 0

24 4 π p1 p (, ) Profits for Firm in Bertrand Model a + bc > p > 1 c b c p 1 p 4. Equilibrium We are now in a position to determine the Nash equilibrium for the duopoly when played in prices. We know that a Nash equilibrium is one in which each firm s expectation regarding the action of its rival is precisely the rival s best response to the strategy chosen by the firm in question in anticipation of that response. For example, the strategy combination, [p 1 = (a+bc)/b, p = (a+bc)/b-j] cannot be an equilibrium. This is because in that combination, Firm is choosing to undercut Firm 1 on the expectation that Firm 1 chooses the monopoly price. But Firm 1 would only choose the monopoly price if it thought that Firm was going to price above that level. In other words, for the suggested candidate equilibrium, Firm 1's strategy is not a best response to Firm 's choice. Hence, this strategy combination cannot be a Nash equilibrium. There is, essentially, only one Nash equilibrium for the Bertrand duopoly game. It is the price pair, (p õ 1 ö c, p õ ö c). If Firm 1 sets this price in the expectation that Firm will do so, and if Firm acts in precisely the same manner, neither will be disappointed. Hence, the outcome of the Bertrand duopoly game is that the market price equals marginal cost. This is, of course, exactly what occurs under perfect competition. The only difference is that here, instead of many small firms, we have just two, large ones. C. Criticisms of the Bertrand model 1. Small changes in price lead to dramatic changes in quantity The chief criticism of the Bertrand model is its assumption that any price deviation between the two firms leads to an immediate and total loss of demand for the firm charging the higher price. It is this assumption that gives rise to the discontinuity in either firm s demand and profit functions. It is also this assumption that underlies the derivation of each firm s best response function.

25 5 There are two reasons why one firm s decision to charge a price somewhat higher than its rival may not cause it to lose all its customers. One of these factors is the existence of capacity constraints. The other is that the two goods many not be identical.. Capacity constraints Consider a small town with two feed mills. Suppose that both mills are currently charging a fee of $4.85 per cwt for mixed feed. to cut and style hair. According to the Bertrand model, one mill, say Firm should believe that any fee that it sets below $4.85 for the same service will enable it to steal instantly all of Firm 1's customers. But suppose, for example, both firms were initially doing a business of 0 customers per day and each had a maximum capacity of 30 customers per day, Firm would not be able to service all the demand implied by the Bertrand analysis at a price of $4.75. More generally, denote as Q C, the competitive output or the total demand when price is equal to marginal cost i.e., Q C = a - bc. If neither firm has the capacity to produce Q C (neither could individually meet the total market demand generated by competitive prices), but instead, each can produce only a smaller amount, then the Bertrand outcome with p 1 = p = c will not be the Nash equilibrium. In the Nash equilibrium, it must be the case that each firm s choice is a best response to strategy of the other. Consider then the original Bertrand solution with prices chosen to be equal to marginal cost, c, and profit at each firm equal to zero. Because there is now a capacity constraint, either firm, say Firm for instance, can contemplate raising its price. If Firm sets p above marginal cost, and hence, above p 1, it would surely lose some customers. But it would not lose all of its customers. Firm 1 does not have the capacity to serve them. Some customers would remain with Firm. Yet Firm is now earning some profit from each such customer (p > c). So, its total profits are now positive whereas before they were zero. It is evident therefore that p = c, is not a best response to p 1 = c. So, the strategy combination (p 1 = c, p = c) cannot be a Nash equilibrium, if there are binding capacity constraints. 3. Homogeneity of products The second reason that the Bertrand solution may fail to hold is that the two firms do not, as assumed, produce identical products. The two feed mills in the example may not produce exactly the same quality of feed. Indeed, as long as the two firms are not side-by-side, they differ in their location and customers may prefer one or the other based on distance from their own operation. VI. The Stackelberg Model A. Firms set output in this model but one firm acts first and the other firm follows. B. The leader assumes that the follower will follow his best response function. Therefore the leader will set output taking this into account. In particular, the leader will use the follower s response function to determine the residual demand curve. C. The profit maximization problem for firm 1 and resulting industry equilibrium In this case we will consider the numerical example only in working out the solution and assume that firm 1 is the leader. Firm 1 realizes that once it chooses its output, q 1, the other firm (firm ) will use its Cournot best response function to picks its optimal output q = R (q 1 ) Consider first the residual demand for the leader firm (firm 1). The response function for firm is given by q 1.

26 6 The residual demand facing firm 1 is given by q 1 (p) = p - q (p). This gives q 1 (p) ö p q (p) ö p [360.5q 1 ] ö pø.5q 1 <.5q 1 (p) ö p < q 1 (p) ö p < 000p ö 180 q 1 < p ö q 1 (5) where we substitute the firm response function for q (p). Thus the residual inverse demand curve is given by p = q 1. The profit maximization problem for firm 1 is then Œ ö max [pq 1 C(q 1 )] q 1 ö max [( q 1 ) q 1.8q 1 ] q 1 ö max [.64q q 1.8q 1 ] q 1 < dœ dq 1 ö q 1.8 ö 0 <.001q 1 ö.36 < q 1 ö 360 (53) The optimum for firm two is given by q (p) ö 360.5q 1 ö 360.5(360) ö 180 Total industry supply is given by Q = = 540 with an industry price of p = (540) = This is lower than the Cournot price but still higher than the competitive price. Consumers are thus better off under the Stackelberg model.

27 7 VII. Comparison of Models Firm 1 quantity Firm quantity Total quantity Price Competition?? 70.8 Cartel/Monopoly Cournot Duopoly Bertrand Duopoly?? 70.8 Stackelberg Duopoly VIII. Multiperiod Games and Oligopoly A. Idea The models developed to this point have assumed a one period structure where the game is only played once. In the real world most firms engage in competition over a long time period. For example two supermarkets in the same town compete over many products and over months and years. They will take into account how the other has behaved in the past in making decisions. The firms may also signal to each other that they are willing to implicitly cooperate or they can implicitly threaten to punish a firm which does not cooperate. B. A Single Period Game Consider a simple single period game based on the previous duopoly problem. Suppose instead of an infinite number of output levels each firm is constrained to choose either 180 or 40 units of output. The Cournot solution is to produce 40 while the cartel solution is to produce 180. Suppose the firms cannot explicitly collude. Based on the information in the table created previously we can compute the profits to each firm for each of its strategies for each of the other firms strategies. For example if the first firm produces 40 and the second firm produces 180, the first firm has profits of 7. This is computed using the residual demand equation q 1 = p - q which gives residual inverse demand of p = (q 1 + q ). The profits are computed from Œ 1 ö pq 1 C(q) ö (1.001(q 1 øq ))q 1.8q 1 (55) ö q 1.001q 1.001q 1 q.8q 1 For q 1 = 40 and q = 180 this will give

28 8 Œ 1 ö q 1.001q 1.001q 1 q.8q 1 ö (40).001(40)(180).8(40) ö 7 (56) Similarly if q 1 = 40 and q = 40 the first firm has profits of We can summarize is a table as follows where the first number in each cell is the payoff to firm 1. Firm Firm The question then is what is the optimal strategy for each firm. Since the firms do not know which strategy the other will play they must make a decision ex ante. The way to play the game is to consider the best strategy contingent on the other firm s action. Consider the case for firm 1. Suppose firm chooses to produce 180 then firm 1 will have profits of 7 by producing 40 and profits of 64.8 by producing 180. So firm 1 will prefer 40. Suppose instead that firm chooses to produce 40. Firm 1 will have profits of 57.6 by choosing 40 and 54 by choosing 180. Thus firm 1 will prefer 40. Since firm 1 will prefer 40 in wither case, 40 is a dominant strategy. Now consider firm. If firm 1 chooses 40, then firm is better off by choosing 40. And if firm 1 chooses 180, firm is better off with 40. So firm has a dominant strategy of 40. Thus the equilibrium of this game is for each firm to produce 40 units which is the duopoly solution. This has total profits of = This is not the best solution for the firms as the cartel solution of 360 total units of product will give profits of This single period game where the firms do not collude results in a lower profit outcome for the firms than is possible if they do collude. This game is a prisoner s s dilemma because both firms have dominant strategies that lead to a payoff that is inferior to what it would be if they cooperated. C. An Infinitely Repeated Single Period Game 1. Signaling a willingness to collude A firm could signal that it is willing to collude by choosing to produce 180 units for several periods even though it has a lower payoff. The other firm may notice this and voluntarily cut back on output and its profits knowing that the first firm will not keep this up forever. If the second firm doesn t respond, the first firm may return to the higher level of 40.. Threatening to punish If both firms produce 40, they each make $57.6. If they both produce 180, they each make $64.8. Suppose firm 1 makes it known that it will produce 180 as long as firm does the same. This could be done by announcing a list price of $.64 and selling no more than 180 units. Assuming the other firm offers only 180 units this price will hold. The first firm could offer discounts if the firm offers more than 180 and the market price starts to fall. Firm 1 could make it very clear that if it has to discount prices at all to remain competitive it will produce 40 units which is the Cournot solution with an equilibrium price of $.5. If firm two believes firm 1 will follow this strategy, it will produce 180 units since if it tries to produce 40, firm 1 will raise its output and lower the profits of firm.

29 9 D. Issues in Repeated Games 1. Credibility For a threat or signal to be meaningful, it must be credible. A credible threat is one that a firm s rivals believe is rational in the sense that it is in the best interest of the firm to continue to employ it. For example a threat to price below marginal cost forever is not a credible threat.. Interest rates If interest rates are very high, the impact of future periods income is reduced. So situations with lower interest rates may lead to more meaningful threats and signals. 3. Length of game The longer the game, the more chance that a threat or signal has meaning. 4. Perfect Nash equilibria Nash equilibria in which threats are credible as called perfect Nash equilibria. For example in a two period game, a threat by firm 1 to produce more than 40 in the second period if firm produces more than 180 in the first period is not credible since in the last period of a game the best solution is the one period solution of producing Subgame perfect Nash equilibria A subgame is a game that starts in period t and continues until the end of the game. If a proposed set of strategies is the best response in any subgame, then these strategies are called a subgame perfect Nash equilibrium. The problem with finite games is that threats are usually not meaningful in the last period and so they are not meaningful in previous period.

30 30 IX. Cartels A. Introduction and Definition 1. Definition An association of firms that explicitly agrees to coordinate its activities is called a cartel. A cartel that includes all the firms in an industry is, in effect, a monopoly. Cartels are more likely to occur in oligopolistic industries which have a small number of firms.. Cheating and cartel discipline Members of a cartel have an incentive to cheat on any agreements that restrict production. This temptation to cheat weakens the power of the cartel. The firms also have an incentive to modify output sharing rules in their favor. Thus, cartels may often break apart. 3. Legal standing of cartels. In most countries cartels that restrict output and thereby raise price are illegal and are prosecuted to some degree. B. Formation and Maintenance of Cartels 1. Why cartels form In a competitive industry, all firms will produce where price is equal to marginal cost and obtain no economic profits (otherwise more firms would enter). If these firms coordinate their output, they may be able to charge a price that is higher than the competitive price and obtain monopoly profits as a group. These profits can then be shared, leaving each firm with a positive profit. Consider the competitive industry equilibrium below. The industry supply curve is the sum of the supply curves of the L identical firms. Competitive Equilibrium Typical Firm in Equilibrium Industry Equilibrium $ MC AC $ S p e p s p e p s D(p) y e Output Q e = Ly e Output The equilibrium price will be p c with industry output Ly c where there are L identical firms in the industry. Each firm produces where price is equal to marginal cost. None of these firms will make any profit since the price just covers average cost at the equilibrium. Now consider a case where the firms coordinate output. In this case they can act as a monopolist and price using the marginal revenue curve. The diagram will now look as follows:

31 31 Cartel Equilibrium Typical Firm in Equilibrium Industry Equilibrium $ MC AC $ S p m p m p e p e MR D(p) y m y e y w Output Q m = Ly m Q e = Ly e Output The industry will produce where MR = S (sum of MC). This will give a lower output Q m and a higher price p m. The individual firm will now get 1/L of industry profit and produce y m. The price will be p m. Note that the firm would prefer to produce y w at the price p m. In the absence of the cartel the individual firms have no incentive to reduce output since the impact on price will be minimal unless others go along. C. Creating and Maintaining the Cartel 1. Why join a. If everyone joins and restricts output, then each member is better off. b. Probability of getting caught and fined if they are illegal is low. Why not join a. Moral grounds b. Might get caught if cartels are illegal c. You re better off letting others restrict output. You can be a free-rider. 3. Why cheat a. Cheating is profitable since you get the higher price on more output b. Cheating is hard to detect

32 3 4. Why cartels fail a. Cheating b. Getting caught 5. Factors that facilitate cartel formation a. Ability to raise price An inelastic demand curve will lead to increases in revenue with decreases in output. A demand curve is more elastic when more substitutes are. Thus, industries in which production is specialized are more likely to be able to form cartels. For example, OPEC controlled oil, for which there are few substitutes in motor vehicle transportation. When entry by non-members is more difficult, a cartel can raise its price more easily. b. Low expectation of severe punishment c. Low organizational costs (1) Few firms () Industry is highly concentrated (3) Nearly identical products (4) Existence of a trade association 6. Enforcing cartel agreements. Detecting cheating is easier if the following apply a. There are few firms b. Prices tend to be stable c. Prices are widely known d. The firms produce similar products at similar points in market channel (minimal vertical integration)

33 33 7. In some cartels the incentive to cheat is small a. Steep marginal costs reduce incentive to cheat Cartel Equilibrium Steep versus Flat MC Typical Firm in Equilibrium Steep MC Typical Firm in Equilibrium Flatter Marginal Cost $ MC AC $ $ MC AC p m p m p e p e y m y e y w Output y m y e y w Output b. Lower fixed costs give less incentive to cheat than high fixed costs with relatively constant marginal costs. In the case of high marginal costs the firm will tend to cheat less. Average and Marginal Cost Curves High Fixed Costs $ AFC AVC ATC MC Price Output c. If purchases are small and frequent information about price reductions does not lead to many more sales

34 34 Average and Marginal Cost Curves Low Fixed Costs High Marginal Costs $ Output AFC AVC ATC MC Price 8. Ways to discourage cheating a. Fix more than just price b. Divide the market c. Fix market shares d. Use most-favored-nation clauses which guarantee that the seller is not selling at a lower price to another buyer e. Use meeting-competition clauses which guarantee that if another firm offers a lower price, the original firm will match it or release the buyer from the contract f. Use trigger prices that effectively return output to the pre-cartel level if prices fall below the trigger price D. Consumers and Cartels Consumers will always benefit as cartels break apart. E. Laws on price-fixing 1. Sherman Anti-trust Act of 1890 to protect trade and commerce against unlawful restraints and monopolies. Federal Trade Commission Act of 1914 unfair methods of competition are hereby declared illegal. 3. Private lawsuits 4. These laws have generally only prevented overt collusion.

35 35 Appendix This appendix works out three examples. Duopoly Example I Consider two firms in a Cournot duopoly each with constant marginal cost of.35 and an industry demand curve of Q = p. If the first firm produces 500 units, we can determine many units the second firm want to produce. q ö p 500 ö p <800p ö 800 q <p ö q R ö (1.0015q )q ö q.0015q MR ö1.005q ö.35 ö MC <.65ö.005q.65 < q ö.005 ö60 or Œ ö (1.0015q )q.35q ö q.0015q.35q ö.65q.0015q dœ dq ö q ö 0 < q ö ö60

36 36 We can also find the second firm s best response function as follows. q ö p q 1 <800p ö 1300 (q 1 øq ) <p ö (q 1 øq ) Œ ö ( (q 1 øq ))q.35q ö 1.65q.0015q 1 q.0015q.35q ö 1.75q.0015q 1 q.0015q dœ ö q dq 1.005q ö 0 < q ö q 1 ö510.5q 1 We can also find the first firm s best response function. q 1 ö p q <800p ö 1300 (q 1 øq ) <p ö (q 1 øq ) Œ 1 ö ( (q 1 øq ))q 1.35q 1 ö 1.65q q q 1 q.35q 1 ö 1.75q q q 1 q dœ dq 1 ö q 1.015q ö 0 < q 1 ö q ö510.5q

37 37 The equilibrium for this market is found by substituting one firms best response function into the other firms response function. q ö510.5q 1 ö510.5(510.5q ) ö q ö55.5q <.75q ö55 <q ö 340 q 1 ö 510.5q ö 510.5(340) ö ö 340

38 38 Duopoly Problem I Consider two firms in a Cournot duopoly each with constant marginal cost of 1.6 and an industry demand curve of Q = p. If the first firm produces 500 units, how many will the second firm want to produce? q ö p 500 ö p <500p ö 100 q < p ö.4.00q R ö (.4.00q )q ö.4q.00q MR ö.4.004q ö 1.6 ö MC <.8ö.004q.8 < q ö.004 ö00 or Œ ö (.4.00q )q 1.60q ö.4q.00q 1.60q ö.8q.00q dœ dq ö q ö 0 < q ö ö00 Show the second firm s best response function. q ö p q 1 <500p ö 1700 (q 1 øq ) <p ö (q 1 øq ) Œ ö (3.4.00(q 1 øq ))q 1.60q ö 3.4q.00q 1 q.00q 1.60q ö 1.80q.00q 1 q.00q dœ ö q dq 1.004q ö 0 < q ö q 1 ö450.5q 1

39 39 Show the first firm s best response function. q 1 ö p q <500p ö 1700 (q 1 øq ) <p ö (q 1 øq ) Œ 1 ö (3.4.00(q 1 øq ))q 1 1.6q 1 ö 3.4q 1.00q 1.00q 1 q 1.6q 1 ö 1.8q 1.00q 1.00q 1 q dœ dq 1 ö q 1.00q ö 0 < q 1 ö q ö450.5q What is the equilibrium for this market? q ö450.5q 1 ö450.5(450.5q ) ö q ö5.5q <.75q ö5 < q ö 300 q 1 ö 450.5q ö 450.5(300) ö ö 300

40 40 Stackelberg Example I Consider two firms in a Stackelberg duopoly each with constant marginal cost of. and an industry demand curve of Q = p. If the first firm produces 00 units and the second firm is a follower we can find the output of the second firm by maximizing profits given the first firm s production of 00. q ö p 00 ö p <400p ö 900 q <p ö.5.005q Œ ö (.5.005q )q.q ö.5q.005q.q ö.05q.005q dœ dq ö q ö 0 < q ö ö410 In general we can find the second firm s best response function using residual demand as follows q ö p q 1 <400p ö 1100 (q 1 øq ) <p ö (q øq ) Œ ö ( (q 1 øq ))q.q ö.75q.005q 1 q.005q.q ö.55q.005q 1 q.005q dœ ö q dq 1.005q ö 0 < q ö q 1 ö510.5q 1

41 41 If the first firm is a Stackelberg leader and uses the second firm s best response function to obtain its residual demand, we can find its optimal output by maximizing profit subject to the response of firm as follows. The residual demand facing firm 1 is given by q 1 (p) = p - q (p). This then gives q 1 (p) ö p q (p) ö p [510.5q 1 ] ö pø.5q 1 <.5q 1 (p) ö p < q 1 (p) ö p < 800p ö 1180 q 1 < p ö q 1 (67) Thus the residual inverse demand curve is given by p = q 1. The profit maximization problem is given by Œ ö max [pq 1 C(q 1 )] q 1 ö max [( q 1 ) q 1.q 1 ] q 1 ö max [1.475q q 1.q 1 ] q 1 < dœ ö q dq 1. ö 0 1 <.005q 1 ö 1.75 < q 1 ö 510 (68) (69) The optimum for firm two is given by q (p) ö 510.5q 1 ö 510.5(510) ö 55