5. Oligopoly Oligopoly: ? Small number of firms: Firms depend on each other. ? Identical products: Firms jointly face a downward sloping industry demand. ? No entry: Long-run positive profits are possible. Duopoly: oligopoly with two firms. Game Theory analyzes strategic interaction. It is a tool for problems of a small number of economic agents with conflicts of interests. Nash equilibrium: no one wants to change, assuming others won’t. Players move simultaneously. Cournot equilibrium: a Nash equilibrium in which both firmsplayNashinquan- tities. Bertrand equilibrium: a Nash equilibrium in which both firms play Nash in prices. Stackelberg equilibrium: a leader and a follower maximize profits. Players move sequentially. 5.1. Bertrand Equilibrium Consider a duopoly, for which the two firms compete in prices. Let x(p 1 ,p 2 ) be the market demand. Assume a constant marginal cost c>0 for both firms. The two firms simultaneously o?er their prices p 1 and p 2 . Sales for firm i are then given by x i (p)= ? ? ? ? ? ? ? ? ? ? x(p) if p i <p j , 1 2 x(p) if p i = p j, 0 if p i >p j . The profitis π i =(p i ?c)x i (p). 2—6 Proposition 2.1. (Bertrand). There is a unique stable Bertrand equilibrium (p ? 1 ,p ? 2 ), in which p ? 1 = p ? 2 = c.  When there are n identical firms with constant marginal cost c, the Bertrand equi- librium is: p ? 1 = ···= p ? n = c. Thus, we have a competitive outcome even with only two firms. 5.2. Cournot Equilibrium Firm 1’s problem: max y 1 p(y 1 + y 2 )y 1 ?c 1 (y 1 ). (2.7) Firm 2’s problem: max y 2 p(y 1 + y 2 )y 2 ?c 2 (y 2 ). FOCs: p 0 (?y 1 + y 2 )?y 1 + p(?y 1 + y 2 )=c 0 1 (?y 1 ), p 0 (y 1 +?y 2 )?y 2 + p(y 1 +?y 2 )=c 0 2 (?y 2 ), (2.8) Cournot equilibrium (y ? 1 ,y ? 2 ): p 0 (y ? 1 + y ? 2 )y ? 1 + p(y ? 1 + y ? 2 )=c 0 1 (y ? 1 ), p 0 (y ? 1 + y ? 2 )y ? 2 + p(y ? 1 + y ? 2 )=c 0 2 (y ? 2 ). The FOC for firm 1 determines a reaction function of firm 1: ?y 1 = f 1 (y 2 ), and the FOC for firm 2 determines the other: ?y 2 = f 2 (y 1 ). The Cournot equilibrium is where the two reaction curves intersect. ? The stability con- dition is null null null null ?f 2 ?y 1 ?f 1 ?y 2 null null null null < 1. Proposition 2.2. (Cournot). In a Cournot equilibrium with constant marginal cost c for both firms,themarketpricep c is greater than the competitive price p ? and smaller than the monopoly price p m .  2—7 Most firms in reality seem to choose their prices, yet the reality tends to produce a Cournot-like outcome. That is, the Cournot model gives the right answer for the wrong reason. One explanation is capacity constraints. We can think of Cournot quantity competi- tion as capturing long-run competition through capacity choices, with price competition occurring in the short run, given the chosen levels of capacity. Example 2.6. Consider two identical firms with c i (y i ) ≡ cy i . The industry demand: p = a?y. Then, π i ≡ (a?y 1 ?y 2 )y i ?cy i ,i=1,2. If both play Nash in quantity, y N 1 = y N 2 = a?c 3 , π N 1 = π N 2 = 1 9 (a?c) 2 .  5.3. Stackelberg Equilibrium Let firm 1 be the leader and firm 2 be the follower. Firm 2’s reaction function y ? 2 = f 2 (y 1 ) in (2.8). Firm 1’s problem is max y 1 p[y 1 + f 2 (y 1 )]y 1 ?c 1 (y 1 ). (2.9) If (2.9) gives y ?? 1 , firm 2’s choice is y ?? 2 ≡ f 2 (y ?? 1 ), and the Stackelberg equilibrium is (y ?? 1 ,y ?? 2 ). Example 2.7. Re-consider the firms in Example 2.6. If firm1isleader,thesolutionis y S 1 = a?c 2 ,y S 2 = a?c 4 , π S 1 = (a?c) 2 8 , π S 2 = (a?c) 2 16 .  5.4. Cooperative Equilibrium Noncooperative game: each player acts in his own best interest. Cooperative game: all players work as a team for their total benefit. For a duopoly, if the two firms agree to cooperate, their problem is max y 1 ,y 2 p(y 1 + y 2 )(y 1 + y 2 )?c 1 (y 1 )?c 2 (y 2 ). FOC: p 0 (y ? 1 + y ? 2 )(y ? 1 + y ? 2 )+p(y ? 1 + y ? 2 )=c 0 1 (y ? 1 )=c 0 2 (y ? 2 ). 2—8 The cooperative equilibrium is (y ? 1 ,y ? 2 ). ? Players in a cooperative equilibrium may negotiate to divide the total benefit. The negotiation process may be modelled as a bargaining game. Example 2.8. Re-consider the firms in Example 2.6. If they share production equally, y C 1 = y C 2 = a?c 4 , π C 1 = π C 2 = 1 8 (a?c) 2 .  5.5. Competition vs Cooperation For a duopoly, three possible games: Nash, cooperation, and Stackelberg. Which game will they play? Will the firms compete or cooperate? The Prisoners’ Dilemma: Confess Deny Confess Deny Prisoner 1 Prisoner 2 -3, -3 -10, -1 -1, -10 -2, -2 Dominant strategy: the best strategy regardless of the others’ actions. Dominant strategy equilibrium: the strategy in the equilibrium is a dominant strategy for each player. The Nash equilibrium for the prisoner’s dilemma is a dominant strategy equilibrium. Consider the firms in Example 2.6. Two strategies: cooperate or compete. If firm 1 produces at the cooperative quantity and firm 2 cheats and plays a Nash strategy, the profits are π NC 1 = 3 32 (a?c) 2 , π NC 2 = 9 64 (a?c) 2 . The result is a prisoners’ dilemma: Nash Coop Nash Coop Firm 1 Firm 2 1/9, 1/9 9/64, 3/32 3/32, 9/64 1/8, 1/8 2—9 5.6. Cooperation in a Repeated Game How can the prisoners achieve Pareto optimal outcome? Repeated game: agameisplayedrepeatedly. In a repeated game, one player can penalize the other for a bad behavior. Trigger strategy: Once cheated, no more cooperation. For the above game, if interest rate r< 8 9 , the cooperative equilibrium is sustainable. 5.7. Transportation Cost Consumers with total size M are located uniformly on [0, 1]. Each consumer buys one unit of the good. There are two identical firms located at 0 and 1 selling the same product with a constant marginal cost c. 1 The total cost of buying the product from firm i is p i + td, where d is the distance of the consumer to the firm. The Bertrand equilibrium is: p ? 1 = p ? 2 = c + t. 5.8. Locational Equilibrium Firms may locate themselves in best locations. How will firmslocatethemselvesin equilibrium? 0 1 1 x 2 xz ? Assume M =1. Given locations (x 1 ,x 2 ) with condition x 1 <x 2 , the Bertrand equilibrium is p ? 1 = c + 2 3 t + 1 3 t(x 1 + x 2 ), p ? 2 = c + 4 3 t? 1 3 t(x 1 + x 2 ). 1 MWG (1995), 396—399. 2—10