Question

Suppose that demand is given by P = 130 ? Q and marginal cost equals 10. Firms are Cournot competitors and play a supergame. The collusive agreement being considered is for each to produce half of the monopoly output. What is the critical discount factor to sustain collusion using grim punishment strategies if detection of deviation requires two periods?

Answer 1

Market demand = 130 – Q. Profit maximization yields cartel output Qm = (130 – 10)/2 = 60

Profits for the cartel are (130 – 10)^2/4 = 3600

Each firm in the cartel produces qm = (130 – 10)/2*2 = 30 units

Profit of each firm = (130 – 10)^2/4*2 = 1800.

Deviating firm has an output qr = (130 – 10)(2 + 1)/4*2 = 45 units

Profits for the deviating firm = (130 – 10)^2(2 + 1)^2/16*(2^2) = 2025

Cournot firms are each producing qc = (130 – 10)/(2 + 1) = 40 units

Each of the firm is earing a profit of ? = (130– 10)^2/(2 + 1)^2 = 1600

There are two outcome possible for this subgame: Collusive output in for all periods including the current one or Cournot output in all periods as the punishment is given forever.

For the first case, a firm's payoff is 1800 for infinite period.

If he deviates in first period he will be able to secure 2025 in that period but will receive only 1600 for each period forever. Hence the payoff is 2025 + 1600? + 1600?² + ... = 2025(1??) + 1600? = 2025 - 425? . The firm has no incentive to deviate if the payoff from not deviating exceed the payoff from deviating:

1800 ? 2025 - 425?

? ? 225/425 or 0.53. This is the critical discount factor to sustain collusion