Question

In this problem you will show how 5 firms facing a market demand curve P = 260 ? 2Q, zero fixed costs, and constant marginal cost C = 20, can cooperate in every period of an infinitely-repeated game in which they employ the trigger strategies that we saw in class; a firm cooperates until someone fails to cooperate, which triggers noncooperation forever. More specifically, you need to derive the following:

a (5). Firm profits from collusion.

b (5). Firm Cournot profit.

c (5). A firm’s profit when it defects from the cartel.

d (5). The expected present value of a firm’s profit stream from colluding.

e (5). The expected present value of a firm’s profit stream from defecting.

f (5). The probability adjusted discount factor that allows the firms to sustain collusion in every period.

Answer 1

a) In collusion, all 5 firms act as a monopoly so they produce at MR = MC

260 - 4Q = 20

Q = 60 units and P = 260 - 120 = 140.

Profit to each firm = (Revenue - cost)/5

= (60*140 - 20*60)/5 = 1440

b) Cournot output = (260 - 20)/(2*6) = 20 unit and market quantity is 100. Price = 260 - 2*100 = $60

Profit to each firm = 20*60 - 20*20 = $800.

c) In case a firm deviates it charegs a slightly lower price and supplies the entire market so it earns monopoly profit of 60*140 - 20*60 = 7200

d) Let the discount rate be d. The expected present value of a firm’s profit stream from colluding is 1440.

e The expected present value of a firm’s profit stream from defecting is 7200in first period and 800 in every period thereafter = 7200(1 - d) + 800d = 7200 - 6400d.

f (5). The probability adjusted discount factor that allows the firms to sustain collusion in every period.

1440 > or =  7200 - 6400d

d > or = 5760/6400

d > or = 96%