Question

Consider a Cournot game with 2 firms. Inverse demand function is given by P= 20 – (91 +92). The firm has MC=AC=5. Firms choose qi € (0,0) a) Find the Nash equilibrium (9*.*). Calculate profits in equilibrium. b) Now suppose that a firm also has to pay a fixed cost of 20 if it produces some output. Write down the cost function of the firm. c) Find the Nash equilibrium (9.97) fixed costs are 20. Calculate equilibrium profits. How does your answer relate to part a? d) Now suppose that a firm has to pay a fixed cost of 40 if it produces some output. Find the Nash equi- librium (91,9%). Calculate equilibrium profits. How does your answer relate to part a and c?

Answer 1

a) IT = P21 - 521 (20-21-2272, - 52, = 2081 - 93 – 222, – 52, = 152, - 22 - 922, = 15 - 29,- qz put T = 0 OT 091 02) 15- 29,- 22 = 0 15- 92 = 28, 9. = 15 - 22 T2 = = 2112 P22 - 522 (20 - 21-22) 22 - 522 = 2022 - 2,22 - 2 2 - 522 = 1522 - 2122 - 222 = 15 - 2- 222 put oll2 = 0 022 15 - 21- 222 = 0 15 - 2 = 222 022 from (1) 8 (1) 22= -3 (1-22)

92 = 9.2 518 x 4 / 4 Mua T = P" 20-5-5 = 10 P2,- 52 = (P-5721 - (10-5) 5 = 5x5 = 25 P22- 522 = (P-5 22 = (10- 5)(52 3 5* 5 = 25 [T2 = FC = 20 i TG = TC2 52, + 20 = 522 +20 c) IT = P2 - TC, PE, 52,- 20 10x5 - 5x5 - 20 = 50- 25-20 = = 5 T2 = = P22 - TC2 Paz - 522-20 = 10x5 – 5x5 - 20 = 5 when FC =0, profits are I = 25 12 = 25, but it res to 20 profits and have gone down to 5.

reads to clearly in an profits & in fixed cart of the firms. d) Now fixed cont = 40 in Total cast TC = 521 + 4o TC2 = 522+40 Since any change in fixed cont doen not read to change in eque librium quantities 2, = 5 2* = 5 P"= 10 T = P2 - TCI 10x5 - 5x5 - Yo = 50 - 25 - 40 = - 15 T12 = P22 - TC2 = P22- 542 - 40 = 10 X 5- 5x5 - 40 = - 15 In part (a) FC =0, T₂ = 25 T2 = 25 (0) FC = 20 Ti = 5, 12=5 (1) FC = 40, T, = -15, T2 = - 15 Thur, FC = 40 is not desirable because it maker negative profits negative which makes no sense in economics. Even though firms are producing Cournot equilibriem quantities but due to high fixed cost firms are making negative profits.