香港科技大学：《微观经济学》（英文版） Lecture 7

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 conficts of interests Nash equilibrium: no one wants to change, assuming others wont. Players move simultaneously Cournot equilibrium: a Nash equilibrium in which both firms play Nash in quan- 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 a(p1, P2) be the market demand. Assume a constant marginal cost c>0 for both firms The two firms simultaneously offer their prices p1 and p2. Sales for firm i are then (p) if Pi The profit

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 firms play Nash in quan￾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(p1, p2) be the market demand. Assume a constant marginal cost c > 0 for both firms. The two firms simultaneously offer their prices p1 and p2. Sales for firm i are then given by xi(p) = ⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ x(p) if pi pj . The profit is πi = (pi − c)xi(p). 2—6

Proposition 2.1.(Bertrand). There is a unique stable Bertrand equilibrium (pi, p?) in which p=n=C.口 When there are n identical firms with constant marginal cost c, the Bertrand equi- librium is: Pi=..=Pn=C. Thus, we have a competitive outcome even with only two 5.2. Cournot Equilibrium Firm ls problen max P(y1+y2)g1-C1(y1) Firm 2s problem (1+y2)-e2(v) FOCs: p(1+y2)1+p(1+y)=C1(m), p(+2)2+p(1+2)=c2(2) Cournot equilibrium (yi, y%) p(1+2)+p(1+y2)=d1(), p(1+)2+p(+2)=2() The FoC for firm 1 determines a reaction function of firm 1 91=fi(y2), and the Foc for firm 2 determines the other v2=f2(y1) The Cournot equilibrium is where the two reaction curves intersect T The stability con- dition is 20f1 Proposition 2.2.( Cournot ) In a Cournot equilibrium with constant marginal cost c for both firms, the market price p is greater than the competitive price p* and smaller than the monopoly price p". L

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 y1 p(y1 + y2)y1 − c1(y1). (2.7) Firm 2’s problem: max y2 p(y1 + y2)y2 − c2(y2). FOCs: p0 (ˆy1 + y2)ˆy1 + p(ˆy1 + y2) = c0 1(ˆy1), p0 (y1 + ˆy2)ˆy2 + p(y1 + ˆy2) = c0 2(ˆy2), (2.8) Cournot equilibrium (y∗ 1, y∗ 2) : p0 (y∗ 1 + y∗ 2)y∗ 1 + p(y∗ 1 + y∗ 2) = c0 1(y∗ 1), p0 (y∗ 1 + y∗ 2)y∗ 2 + p(y∗ 1 + y∗ 2) = c0 2(y∗ 2). The FOC for firm 1 determines a reaction function of firm 1: yˆ1 = f1(y2), and the FOC for firm 2 determines the other: yˆ2 = f2(y1). The Cournot equilibrium is where the two reaction curves intersect. † The stability con￾dition is     ∂f2 ∂y1 ∂f1 ∂y2     < 1. Proposition 2.2. (Cournot). In a Cournot equilibrium with constant marginal cost c for both firms, the market price pc is greater than the competitive price p∗ and smaller than the monopoly price pm.  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 eason One explanation is capacity constraints. We can think of Cournot quantity competi tion as capturing long-run competition through capacity choices, with price competition ccurring in the short run, given the chosen levels of capacit Example 2.6. Consider two identical firms with ci(yi)= cyi. The industry demand y. The 丌:≡(a--v)-cv,i=1,2 If both play Nash in quantity, 3 5.3. Stackelberg Equilibrium Let firm 1 be the leader and firm 2 be the followe Firm 2s reaction function y2= f2(g)in(2.8) Firm 1s problem is max pl+f 2(yn)ly1-c1(yr If(2. 9 )gives yt*, firm 2's choice is 2*= f2(3i"), and the Stackelberg equilibrium is (3*,y*) Example 2.7. Re-consider the firms in Example 2.6. If firm 1 is leader, the solution is C-c 4 8 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 maxp(1+y2)(+v2)-c1(m)-c2(v) FOC. p(M1+)ⅵ+2)+p(1+2)=1()=以2()

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 ci(yi) ≡ cyi. The industry demand: p = a − y. Then, πi ≡ (a − y1 − y2)yi − cyi, i = 1, 2. If both play Nash in quantity, yN 1 = yN 2 = a − c 3 , πN 1 = πN 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 = f2(y1) in (2.8). Firm 1’s problem is max y1 p[y1 + f2(y1)]y1 − c1(y1). (2.9) If (2.9) gives y∗∗ 1 , firm 2’s choice is y∗∗ 2 ≡ f2(y∗∗ 1 ), and the Stackelberg equilibrium is (y∗∗ 1 , y∗∗ 2 ). Example 2.7. Re-consider the firms in Example 2.6. If firm 1 is leader, the solution is yS 1 = a − c 2 , yS 2 = a − c 4 , πS 1 = (a − c)2 8 , πS 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 y1, y2 p(y1 + y2)(y1 + y2) − c1(y1) − c2(y2). FOC: p0 (y∗ 1 + y∗ 2)(y∗ 1 + y∗ 2) + p(y∗ 1 + y∗ 2) = c0 1(y∗ 1) = c0 2(y∗ 2). 2—8

The cooperative equilibrium is (yi, g*).t Players in a cooperative equilibrium may negotiate to divide the total benefit. The negotiation process may be modelled as a bargaining game. Cxample 2.8. Re-consider the firms in Example 2.6. If they share production equally = 4 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 lIonel Confess Den 3,-3 1,-10 Prisoner l 10,-1 Dominant strategy: the best strategy regardless of the others'actions Dominant strategy equilibrium: the strategy in the equilibrium is a dominan 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 profit C 9 The result is a prisoners'dilemma: F ash Nash 1/9, 1/9 9/64,3/32 Irm Coop3/32,9/641/8,1/8

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, yC 1 = yC 2 = a − c 4 , πC 1 = πC 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: C onfess Deny C onfess Deny P risoner 1 P risoner 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 πNC 1 = 3 32(a − c) 2 , πNC 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: a game is played repeatedly 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<o, 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 I The total cost of buying the product from firm i is pi+ td, where d is the dist of the consumer to the firm. The Bertrand equilibrium is Pi=p2=c+ 5.8. Locational Equilibrium Firms may locate themselves in best locations. How will firms locate themselves in equilibrium Assume M=1. Given locations (1, 2)with condition 1 <J2, the Bertrand equilibrium is =c+t+t(x1+m2) 41 n2=c+;t-t(x1+x2) MWG(1995),396-

5.6. Cooperation in a Repeated Game How can the prisoners achieve Pareto optimal outcome? Repeated game: a game is played repeatedly. 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 pi + 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 firms locate themselves in equilibrium? 0 x1 zˆ x2 1 Assume M = 1. Given locations (x1, x2) with condition x1 < x2, the Bertrand equilibrium is p∗ 1 = c + 2 3 t + 1 3 t(x1 + x2), p∗ 2 = c + 4 3 t − 1 3 t(x1 + x2). 1MWG (1995), 396—399. 2 — 10