Question

Q4. Now consider collusion between n price-setting firms which produce a homogenous good. The monopoly profit in this industry is 100. Marginal cost is 2 for each firm and there are no fixed costs. The n firms collude by setting the monopoly price and dividing the monopoly profit evenly between them. a. When the game is played once, show that there is a profitable unilateral deviation for each firm from this collusive agreement. b. Now suppose that the firms interact repeatedly, with the probability of the game continuing each period being δ. Consider the following grim trigger strategies: for each firm, collude until one player deviates, then price at marginal cost forever. Show that these strategies constitute a Nash equilibrium of the repeated game if δ ≥ 1 − 1/n. [Hint: recallthat: x+δx+δ^2x+...= x/1-δ ] (5marks) c. Intuitively, why is a higher δ required to support collusion in the repeated game when the number of firms n is larger? (Remember to show all working)

Answer 1

Ans....
a) Under collusive agreement, each firm charges a price and produces a quantity deteremined by MR = MC = 2. Now we know that monopoly price is greater than MC. so P* > 2. Under the agreement all produces the same quantity, share 100 as 100/n for each firm and charges P* > 2. Every firm has an incentive to slightly charge less than P* but greater than 2 so that it acquires all the market share and profit. Because the game is repeated once, there is an incentive to deviate as punishment is not provided.
b) Assume that all firms follow the grim trigger strategy. This implies punsihment is given forever once any of the firm deviates. Hence there are two outcome possible for this subgame: produce collusively for all periods including the current one or produce according to P = MC in all periods as the punishment is given forever. For the first case, a player’s payoff is 100/n for infinite period.  
If a firm deviates in first period it will be able to secure 100 in that period but will receive 0 for each period forever. Hence the payoff is 100 + 100δ + 100δ2 + ... = 100(1−δ) + 0δ . The player has no incentive to deviate if the payoff from not deviating exceed the payoff from deviating
100/n > or = 100(1−δ) + 0δ
1/n > or = (1 - δ)
δ > or = 1 - 1/n
Hence the discount rate depends on the number of firms.
c) You can observe that a higher n makes 1/n smaller and so 1 - 1/n larger. So the discount rate has to be higher to encourage firms to follow the agreement and discourage them to deviate from the collusion.