Question

Assume in a Bertrand 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. With probability 0.3 the cartel is detected by the end of period t, in which case each cartel member has to pay the fine $1000. Cartel investigation takes one period. If detected by the end of period t, the cartel is disbanded at the end of the period and the firms return to the noncooperative game permanently. Derive the condition required for the cartel to succeed

Answer 1

ans.....
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.