Question

Two firms are operating as a cartel. Firms engage in Bertrand pricing, each firm has a constant marginal cost equal to c = 4, and the firms sell a homogenous, perfectly substitutable output. Demand is given by D(p) = 12 − p, and the collusive arrangement has it that they split the market evenly. There are no demand shocks.

The firms decide that the grim trigger strategy punishes collusion too harshly. Instead, they decide that the punishment should only last for T periods. This means that if in any period t, one of the firms cheats on the agreement, then the firms will revert back to Bertrand pricing for the next T periods (i.e. in period t+1, t+2, all the way to period t+T). Once the T periods of punishment have elapsed, the firms will then go back to cooperating.

For example, if T = 2, that means that after a firm cheats, the two firms charge the Bertrand price for 2 periods, and go back to cooperating thereafter.

(a) Suppose T = 1. For what values of δ, the discount factor, will the above collusive strategy be a SPNE? (Remember to calculate the present value of profits for each firm if it cooperates and if it deviates from the agreement.)

Answer 1

To determine the values of δ for which the collusive strategy is a SPNE, we need to calculate the present value of profits for each firm if it cooperates and if it deviates from the agreement.

Step-by-step explanation:

Collusive Strategy in a Cartel with Punishment

In this scenario, the firms are operating as a cartel and using the Bertrand pricing model. The firms have a constant marginal cost of c = 4 and the demand function is given by D(p) = 12 - p. The collusive arrangement states that they split the market evenly. However, instead of using the grim trigger strategy for punishment, they decide to implement a punishment that lasts for T periods.

Calculation of Discount Factor, δ

To determine the values of δ for which the collusive strategy is a SPNE (subgame perfect Nash equilibrium), we need to calculate the present value of profits for each firm if it cooperates and if it deviates from the agreement.