Question

Mathematical Question 3 (30pts) 3. Consider two firms are performing Cournot price competition in two differentiated goods markets. Firm 1 produces goods 1, and firm 2 produces goods 2, and two market demand functions are given by 91 (P1,P2) = 12-2p1 + P2 and 921,P2) = 12-2p2 + P 1. Furthermore, assume that the two firms have the same cost function such that fixed cost is $20 and variable cost is zero. a. (10pts) Calculate the equilibrium prices, quantities and profits for both firms. b. (10pts) Assume the two firms collude and form a cartel. That is, they maximize their total profits by setting prices. What will be the new equilibrium prices, quantities, and profits? c. (5pts) Assume two firms agree to set prices at the collusion equilibrium level given in (b), but one of the firm unilaterally deviates its price to the Cournot equilibrium level given in (a), what will be the quantities and profits for both firms in this case? Is this deviation profitable for this firm? d. (5pts) Based on the information given in (a), (b),(c), draw out the payoff matrix for two firms which have two potential strategies: Collude or Deviate. Explain why this game is a prisoner dillema. (Hint: the strategy is what price the firm chooses; the payoff is the corresponding profit)

Answer 1

91( P1, P2] = 12-2 pit P2 variable Fixed cost=0 cost = 20 92 (bi, P2) = 12-2p+bi (a) Best Responce function for both the firms. Profit function of from I Ti = TRI- TC. = P1.9 - 20 Ti = p. (12-2bit P2) - 20 F.O.C Ji = 12-upit b2 = 0 dpi = 12 + P2 - pi = 3 + 0.25 kz = k BRI (P2) Similarly z = TR2 - TC2 = P2.42 - 20 T2 = pz (12-2 b 2 + b )-20 F.o.c

JT12 - 12-4 Pzt bi = 0 Jp2 - 12 tbl - p2 u - 13+ 0.25 k, = R2 | BR2 (b) Solving Be, ana BR-2 Simul teously P = 3 + 0.25P2 = 3+0.25 (3+0.25",) pi = 3 + o. 75+ 0.0625 ki = b c - 0.0625 p = 3.75 - 0.9375 Pi = 3.75 h - 3075 0.9375 -4. from Bha 3+0.25 pi= P2 3+0.25 (4) = p ² = 4 Ibro pz CB 4 L

9 = 12-2 kit pe = 12-2 (4) + 4 = 12-8 +4 9, = 8 2. = 12-2p2 + bi = 12-2 (u) + 4 = 16-8 q2 = 8 Th,= TRI- TCI - br.9,- TC, = 4(8) - 20 = 32-20 =12 T2 = TR2 - TC2 = br. 22 - TC2 = 4(8) - 20 = 32-20 - 12 IT = TS=12] 12 = 12

(b) collusion Coucesive for Profit T12 = TRI2- TC12 Piz=p = P.2.912 - TC 12 9, +22=212=9 = p. 9 - TC TC 12 = 20+20 =40=To = b.q - 40 - Þ(12-2p+ p + 12-2p+p) -40 = b (24-46+ 2 b)-40 The = b (24-2b)-40 F.O.C dll = 24-up = 0 ap - 24 = p =) b = 6e pi=p2 = p = 6 VEX q = 12-2 bit pa - 12-2 (6) + 6 = 12-12 +6 2 =6 - qu (By Symmetry).

- TRI- TC, = p. 91-20 = (6) (6)-20 = 36-20 CT = 16 = 8 By Symmetry. be = P₂ C = 6 9 = = 2 2 = 6 T = Th 2 = 16 (C) Iy one of firm deviates 2et say firm- I deviates and set b = 4 By Setting Prices p = 4 all the market demand will be sarued by from I and firm-2 at b c = 6 will get 2080 market demand And so consequence of this Firm- 2 will get zero revenue and its Jixed cost wes &208 So, firm - 2 will earn - 20$ Profit. (loss of $20).

from-2 a = 12-2 bit pa Q2 = 12-2 Pzt bi q = 91 +92 - 12-2 bit b2 + 12-2 Pzt bi q = 24-bi-p2 when fim-I Charge P, = p= 4 Q = 24-p-p - 24-26 = 24-2 (4) = 24-8 - 16 TI, TRi - Tci = 6.9-20 = 64-20 = 44

d=deviation po = 4 ,P2 = 6 7,4 = 16,20 T = 6, 44, T = 0 -20. since I was the so it is pofitable Jos fiom-Z To deviate Player-2 deviates Collude) deviates (12, 12 ) (44,-20) Player - 7 (charge b=4) 16=4 6=6 | (-20, 44) (16, 16) Dont deviates (collude) Charge to a b=6 Nash equilibrium of the game is (deviate, deviatus] and change (bay, b=4) That gives Pay (12, 12) Payoff for To both of the firm, where as if They choose to collude They both will get

(16, 16) Payoyf> (12, 12). where as Prisoncas dillema game can be described as Conyess (-3,-8) sont conyess Co,-10) Confess pont (-10,0) (-1,-1) Confess (Hypothetical Parafss) Similarly hash cquilibrium of Prisoners dilema is conyess, confess which gives of of years of Jail To both where as Best Payoff was (-7, -1) which is Possible when they both will choose (Dont Conjess, Don't con Jess). in both of games, They don't Coolcrate or not Play the coordination Strategies and get dess Payoffs of Than what they can get wts if they play coordination Strategiels.