Question

Suppose that two firms produce a pair of imperfectly substitutable goods at the same constant marginal cost c = 16 and compete à la Bertrand (firm 1 chooses p_{1} and firm 2 chooses p_{2} ). There are no fixed costs. The market demands that the firms face each period ( q_{1} for firm 1 and q_{2} for firm 2) are given by

q_{1} = 24 - 3p_{1} + 2p_{2}

q_{2} = 24 - 3p_{2} + 2p_{1}

The horizon is infinite (T = [infinity]) and all firms discount the future by the same discount factor delta in (0, 1) . Given these firms compete repeatedly, they may be able to earn higher profits by engaging in tacit collusion. Let delta ^ * be the minimum discount factor that sustains collusion. Answer the following questions.
d) Suppose that instead of updating quantities each period, the firms are only allowed to update them in odd-numbered periods (the choice made then persists for two periods). Find the new threshold discount factor, δ ∗∗ . Is collusion now easier or harder to sustain than in the baseline case?

please show all working/ reasoning.

Answer 1

To find the new threshold discount factor, δ∗∗, we need to consider the profit-maximizing strategies of the firms under the updated condition. In the baseline case, the firms update their quantities each period. However, in the new scenario, the firms are only allowed to update them in odd-numbered periods, and the choice made then persists for two periods.

Step-by-step explanation:

To find the new threshold discount factor, δ∗∗, we need to consider the profit-maximizing strategies of the firms under the updated condition. In the baseline case, the firms update their quantities each period. However, in the new scenario, the firms are only allowed to update them in odd-numbered periods, and the choice made then persists for two periods.

Under this new condition, the firms need to consider the potential strategies and their payoffs in the alternate periods. By analyzing the payoffs and potential gains from deviating from collusion, we can determine whether collusion is easier or harder to sustain than in the baseline case.