Question

Two firms are involved in an infinitely repeated game. If they cooperate on pricing, they can split the monopoly profit, each earning $500 per period. But if one firm undercuts slightly, then the firm that undercuts can collect the entire monopoly profit of $1000. The firms interact repeatedly and attempt to cooperate using the following strategy: Let’s start by splitting the monopoly profit. If anyone ever undercuts, we’ll revert to the competitive outcome in the following period, and both of us will earn a profit of 0 forever after.a.For what values of the interest rate will firms cooperate? For what values of the interest rate is cooperation impossible?b.Suppose, instead of the competitive outcome and a profit of zero, that the punishment for undercutting is to revert to a duopoly game where both firms earn a profit of $400 per period. For what values of the interest rate will firms cooperate now? Is cooperation easier or more difficult than in (a)? Provide an economic interpretation

Answer 1

A)

Both the firms will follow the grim trigger strategy. This implies punishment is given forever once any of the firm undercuts. Hence there are two outcomes possible for this sub game: split the monopoly profit for all periods including the current one or competitive profit in all periods as the punishment is given forever.

For the first case, each firm's payoff is 500 for infinite period.  

If a firm undercuts and deviates in first period it will be able to secure 1000 in that period but will receive 0 for each period forever. Hence the payoff is 1000 + 0d + 0d² + ... = 1000(1 -d) + 0d = 1000 - 1000d . Firm has no incentive to deviate if the payoff from not deviating exceeds the payoff from deviating:

500 > or = 1000 - 1000d

-500 > or -1000d

d >or = 1/2

Interest rate should be d > or = 1/2 so that cooperation is possible.

B)

Again both the firms will follow the grim trigger strategy. Punishment is given forever once any of the firm undercuts. Hence there are two outcomes possible for this sub game: split the monopoly profit for all periods including the current one or duopoly profit in all periods as the punishment is given forever.

Each firm's payoff is 500 for infinite period under cooperation  

If any firm deviates in first period it will be able to secure 1000 in that period but will receive only 400 for each period forever. Hence the payoff is 1000 + 400d + 400d² + ... = 1000(1 - d) + 400d = 1000 - 600d. Firm has no incentive to deviate if the payoff from not deviating exceeds the payoff from deviating:

500 > or = 1000 - 600d

-500 > or = – 600d

d > or = 5/6

Interest rate should be > or = 5/6 so that cooperation is possible. In part a it is easier to cooperate because the interest rate is lower and threat is more credible.