# Bertrand Stackelberg Cartels. Oligopoly (contd.) Chapter 27

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1 Oligopoly (contd.) Chapter 7

2 Midterm Bring pencil/pen, bluebook, pink scantron 8 MC questions, like last time Important skills: Find monopoly p, q Analyze effects of policy (e.g. tax, subsidy, price ceiling) on monopoly Max profits using price discrimination (lecture and especially workouts 5.5-8) Analyze duopoly (Cournot, Bertrand, Stackelberg, collusion)

3 Oligopoly Considerations: Do firms compete on price or quantity? Do firms act sequentially (leader/followers) or simultaneously (equilibrium) Stackelberg models: quantity leadership Cournot equilibrium models: simultaneous choice quantity competition Bertrand equilibrium models: simultaneous choice price competition

4 Today Price competition (Bertrand duopoly) More quantity competition Stackelberg Duopoly Cartels/collusion

5 Bertrand Duopoly: price competition Firms compete on price No clear leader, follower so firms effectively choose p simultaneously Take the other firm s price as given Market demand determines equilibrium output Both choose same price: divide demand evenly One sets lower price: that firm captures entire market

6 Bertrand Duopoly: price competition Suppose two firms have same MC What price to charge? Apply Nash equilibrium Each chooses optimal p i given p j No one has incentive to deviate If pricing above marginal cost, each has incentive to undercut competitor (p i > p j > MC is not an equilibrium) p i = p j = MC is the only possible equilibrium Zero profits for both, but no incentive to deviate: Higher price means no sales Lower price means losses

7 Stackelberg Duopoly: leader/follower Two firms compete in the same market Firm 1 chooses q 1 Firm observes q 1, chooses q This determines total Q which determines price

8 Stackelberg Duopoly: how to solve the model Analyze using backwards induction Start at the end: what does Firm do given q 1? Derive reaction function just like we did for Cournot Then find optimal q 1, given Firm 1 can deduce s reaction

9 Stackelberg Duopoly: back to our example Find s reaction function (recall: p = a Q, MC = c) Firm s profits, given q 1 : Π (q 1, q ) = (p c)q = (a q 1 q c)q implies q (q 1) = a q 1 c Realizing this, 1 factors in s response when computing profits: Π 1 (q 1, q (q 1 )) = (p c)q 1 = (a q 1 q(q 1 ) c)q 1 = (a q 1 a q 1 c c)q 1 = a q 1 c q 1

10 Stackelberg Duopoly: back to our example Π 1 (q 1, q(q 1 )) = a q 1 c q 1 Differentiate to find q 1 Solving yields q 1 = a c Plug into reaction function: Π 1 = a q 1 c q 1 = 0 q (q 1) = a q 1 c = a a c c = a c 4 Same per unit profit, so q 1 > q π 1 > π First-mover advantage

11 Stackelberg Duopoly: back to our example Fill in details: q1 = a c, q = a c 4 Q = 3 a+3c 4 (a c) so P = 4 p c = a c 4, so π 1 = (a c) 8, π = (a c) 16, and Π = 16 (a c) CS = 9 3 (a c) so W = Π + CS = 15 3 (a c)

12 Cartels Can firms make more profit by colluding and behaving as a monopoly? The cartel will behave as if it s a monopoly = MR = MC Q = 1/(a c) and q 1 = q = 1/4(a c), P = a+c Profits: Π = 1/4(a c) so π 1 = π = 1/8(a c) Compare to Cournot: lower Q, CS, W ; higher P, Π

13 Summary Table PC Mon. Stck. Cour. Cour. (n) cart. q 1 1/ 1/3 1/(n + 1) 1/4 q 1/4 1/3 1/4 Q 1 1/ 3/4 /3 n/(n + 1) 1/ a+c a+3c a+c a+nc a+c P c 4 3 n+1 π 1 1/8 1/9 1/(n + 1) 1/8 π 1/16 1/9 1/8 Π 0 1/4 3/16 /9 n/(n + 1) 1/4 CS 1/ 1/8 9/3 /9 1 n 1/8 (n+1) W 1/ 3/8 15/3 4/9 n +n (n+1) 3/8 Note: quantities (q 1, q, Q) in units of a c; welfare (Π, CS, W ) in units of (a c).

14 Cartels So what is stopping firms colluding and making more profits? Anti-trust regulation Ok, but even without regulation, e.g. what problem does OPEC face? Collusion is not individually rational: each firm has incentive to cheat and produce more E.g. increase q and earn more profit (foreshadowing: how is this like tragedy of the commons?) E.g. slightly decrease p and capture entire market Cartels can only succeed if they can effectively monitor and punish cheating, which is difficult, esp. if collusion is illegal

15 Cartels What are the incentives facing each firm? Suppose each duopolist can choose to cooperate (C) and produce the collusive quantity (q i = a c 4 ), or to cheat/defect (D) and produce the Cournot quantity (q i = a c 3 ). C C 1 8, 1 8 D 0 144, D , , 1 9 Duopolists Dilemma Note: 1 8 = and 1 9 =

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