Question

Assume in a Cournot model, there are 3 identical firms in the market with the same constant marginal cost of 30. The demand function is P = 150 ? Q. Firms share the monopoly profit equally if they participate in the cartel. Suppose that the market game is now repeated indefinitely. Derive the condition under which the collusion/cartel will be successful. (Hint: show the range of the probability adjusted discount factor ? that makes the cartel sustainable.)

Answer 1

Market demand = A – Q or 150 - Q.

Total profit of cartel is ? = (A – Q)Q – cQ. = (150 - Q)Q - 30Q

Profit maximization yields cartel output Qm = (A – c)/2 = (150 - 30)/2 = 60 units (20 units to each member)

Profits for the cartel are (A – c)^2/4 = (150 - 30)^2/4 = 3600

Each firm in the cartel produces qm = (A – c)/2n = (150 - 30)/6 = 20 units

Profit of each firm = (A – c)^2/4n = (150 - 30)^2/12 = 1200

Deviating/Cheating firm has an output qr = (A – c)(n + 1)/4n = (150 - 30)*4/12 = 40 units

Profits for the deviating firm = (A – c)^2(n + 1)^2/16n^2 = 120^2*16/(16*9) = 1600

Cournot firms are each producing qc = (A – c)/(n + 1) = 120/4 = 30 units

Each of the firm is earing a profit of ? = (A – c)^2/(n + 1)^2 = 120^2/16 = 900

Let the discount factor be d that makes cartel sustainable. Hence there are two outcome possible for this subgame: (Cartel, Cartel) in for all periods including the current one or (Cpurnot, Cournot) in all periods as the punishment is given forever. Under cartel sustainability, a firm’s payoff is 1200 for infinite period.  

If a firm deviates in first period it will be able to secure a cheating profit of 1600 in that period but will receive only 900 for each period forever. Hence the payoff is 1600 + 900d + 900d² + ... = 1600(1?d) + 900d . The firm has no incentive to deviate if the payoff from not deviating exceed the payoff from deviating:

1200 ? 1600(1?d) + 900d

1200 ? 1600 ?1600d + 900d

1200 ? 1600 ? 700d

This gives d ? 4/7

A discount factor greater than or equal to this level will make the cartel sustainable.