Econ 37 Solution: Problem Set # Fall 00 Page Oligopoly Market demand is p a bq Q q + q.. Cournot General description of this game: Players: firm and firm. Firm and firm are identical. Firm s strategies: q Firm s strategies: q Solve for Cournot Nash Equilibrium Payoff (or profit function Firm : (q ;q q (a b(q + q cq. Firm : (q ;q q (a b(q + q cq. Firm s problem: Choose q as to maximize its profit by taking q as given. Max q (a b(q + q cq q FOC: a-b(q + q c -bq 0 a-bq c bq a-c q q - b BR (q * The solution q is the best response of firm to the quantity chosen by firm. Firm s problem: Max q (a b(q + q cq q a-c q BR (q - b Since both firms are identical (i.e. same demand and cost functions in NE *. q q q Substitute BR (q into BR (q a-c q - a-c b q - b 3 a-c q 4 * a-c q 3b * a+c p 3 p* a-bq* a-b(q * + q * a-b( a-c 3b
Econ 37 Solution: Problem Set # Fall 00 Page * q*(p*-c a-c a+c c 3b 3 * 9b Cartel (this is similar to the monopoly case Let q be the quantity produced by firm and firm under cartel. Joint profit (q +(q {q (a b(q c} Cartel chooses q so as to maximize oint profit. Max (q (a b(q c q FOC: (a b(q c q 0 (a-c q q a-c < q* p a-b(q a-b( a-c a+c > p* q (p -c a-c a+c c > * Firm cheat when firm sets q q. How will firm cheat? Set q that maximize firm s profit given q q. That is, firm produce at BR (q.
Econ 37 Solution: Problem Set # Fall 00 Page 3 BR (q a-c - b ( a-c (- a-c ( 4 b 3(a c q 8 b > q Q q +q q +q p a-b(q +q a-b( a-c 3(a c + 8 b p 3 a + 5c 8 < p q (p-c ( a c 3 a + 5c ( -c 8 3 < * 3b q (p-c 3(a c 3 a + 5c ( -c 8 9 > 6 We can summarize the results so far by using payoff matrix Firm, Firm Don't Coop Coop Don't Coop 9 (Cournot, 9b 9b 6 Coop 3 9 (Cartel,, 3b 6 3, 3b a c NE of this game is that both firms choose "Don't coop", set q. This is a 3b unique equilibrium, which is inferior to what would have been if they cooperate. But both firms decide to cheat because of the higher profit. This is similar to Prisoner's Dilemma case. To solve multi-period game, we use "Backward Induction" method.
Econ 37 Solution: Problem Set # Fall 00 Page 4 In the -period game, we solve nd period first. The best response of both firms is "Don't coop". Knowing that they will play Cournot strategy in the last period, they will also cheat in the first period. Thus, cartel will not survive. Analogically, the 0-period game will be played in the same way. Both firms will cheat in 0 th, 9 th,..through the first period. As long as the game has finite horizon, firms are always better off cheating in the last period. However, if they play forever, a cartel may survive. To see this, suppose firm cheats in period t then, in period t+, firm can retaliate by increasing its output. As a result, payoffs of both firms will be lowered. The threat of retaliation is likely to prevent firms from cheating. Survival of cartel also depends on firms' time preference. That is how much firms care about future punishments.. Bertrand Game Players: Firm and firm, both are identical. Strategies: p and p Inverse demand curve: Q a b b p Consider profit of firm. If p < p, all consumers will buy form firm. Firm 's demand is the market demand. If p > p, all consumers will buy form firm. Firm 's demand is zero. If p p, firm share the market with firm by / : /. Thus, a (p -c( p b b if p < p a (p -c( p b b if p p 0 if p > p. a (p -c( p b b if p > p a (p -c( p b b if p p 0 if p < p. The best response for firm As long as p is above marginal cost, firm can maximize profit by set price slightly lower than p and capture the entire market. If p is equal to marginal cost, the best that firm can do is to set price equal to marginal cost as well. It will not set price below marginal cost since it will incur a loss. Thus, BR (p p - ε if p - ε > c c otherwise BR (p p - ε if p - ε > c c otherwise. Nash Equilibrium
Econ 37 Solution: Problem Set # Fall 00 Page 5 Suppose firm set price at p where p > c. Firm 's best response is to set p p - ε. But this cannot be NE since, given p p - ε, the best response of firm is to set p p - ε not p. As long as p > c, both firms will undercut each other. They will not do so if p c. Thus, NE occurs when both firms choose price equal to MC. c q b a c b b 0 p b 3. Stakelberg Game Players : Firm (leader, firm (follower Strategy: q and q Period : firm chooses q Period : given q, firm chooses q to maximize its profit. Use "Backward Induction." Solve for period first then solve for period. Period Firm uses BR (q as we derived in question. a-c q BR (q - b Period Firm knows how much firm will produce through BR. Max q (a b(q + BR (q cq q q s a c b Substitute into BR (q BR (q s a c a c - b q s a c p s a-b(q s +q s a c a c a-b( + b p s a + 3c 4
Econ 37 Solution: Problem Set # Fall 00 Page 6 s q s (p s -c ( a c ( b a + 3c -c 4 s s q s (p s -c a c a + 3c ( -c 4 s 6b 4. Rankings At firms level q b q s > q* >q s q s > * > s b At industry level Q q + q Q b > Q s > Q* >Q p > p* > p s > p b > * > s > b 5. (a Player 's strategies are A, B, and C (b Player 's strategies are (L,L,L, (L,L,R, (L,R,R, (R,L,R, (R,R,R, (R,R,L, (L,R,L, (R,L,L (c To find NE, write down payoff matrix Player, Player A B C LLL +5,9* 4, 4, LLR 5,9* 4, +7,3* LRR 5,9* 6,3 +7,3* RLR, 4, +7,3* RRR, 6,3* +7,3* RRL, +6,3* 4, LRL 5,9* +6,3* 4, RLL, +4, +4, +: BR for player *: BR for player shaded area: NE
Econ 37 Solution: Problem Set # Fall 00 Page 7 NE are (A;LLL, (B;RRL, (B;LRL, (C;LLR, (C;LRR, (C;RLR, (C;RRR. (d To solve for SPE, use backward induction. Player A B C Player L R L R L R (5,9 (, (4, (6,3 (4, (7,3 SPE is (C;LRR : Optimal action in each subgame 6. Single period NE in Bertrand game NE: p p c 0 If firms cooperate, firms play p p p such that p maximize cartel's profit, a (p-c( p b b a FOC: ( p - (p-c 0 b b b a-p-p+c0 p a+c (notice that this is the same as question part 3 If firm plays p, firm can cheat by playing BR (p p - ε a+c - ε In the situation where one firm cheat wile the other cooperate, profit of the coop firm 0. Profit of the cheating firm, x a c a a+c ( - ε ( ( b b - ε a c ε a c + ε ( ( b
Econ 37 Solution: Problem Set # Fall 00 Page 8 4ε This is also a prisoner's dilemma problem. Payoff Matrix Firm, Firm Cheat Cartel Cheat 0,0 x,0 Cartel 0, x, Nash reversion strategy for this game is Firm plays p p if p,t- p plays p c otherwise. Similarly for firm. To show that these strategy constitute an SPE, we have to find conditions that prescribe best response behavior for firm given firm is following this strategy in each subgame. I. Consider the period after which cheating has occurred. Suppose firm cheated. If firm follows Nash reversion, it should play p c. Firm's best response is also to play pc. Thus, in these subgames, Nash reversion prescribes best response behavior. II. Consider period after which no cheating has occurred. Will firm play p given firm plays Nash reversion? Recall i 0 a i a lim0 ε - Let δ be discount factor where 0 δ. PDV (present discounted value of profit from playing p PDV of profit form cheating x δ + 0 x δ Nash reversion prescribes best response if > x δ 4ε > ( δ δ > - ( 4ε ( 4ε δ As ε 0, if δ >, then Nash reversion is SPE.