Two firms compete in price in a market for infinite periods. In this market, there are N consumers; each buys one unit per period if the price does not exceed $10 and nothing otherwise. Consumers buy from the firm selling at a lower price. In case both firms charge the same price, assume N/2 consumers buy from each firm. Assume zero production cost for both firms. A possible strategy that may support the collusive equilibrium is: Announce a price of $10 if the equilibrium price has always been $10; otherwise, announce the price as in Nash equilibrium of the one-shot Bertrand game. 1.a Let δ be the discount factor. Find the condition on δ such that the above strategy can indeed support the collusive equilibrium. Now suppose that Firm 2’s marginal cost is $4, but Firm 1’s marginal cost remains at zero. 1.b Find the condition on δ under which Firm 2 will not deviate from the collusive equilibrium. 1.c Find the condition on δ under which Firm 1 will not deviate from the collusive equilibrium. 1.d Knowing that both firms’ discount factor is 0.6, how should Firm 2 set its capacity constraint so that the collusive equilibrium can still be supported? (Hint: The idea here is that, by limiting its own output, Firm 2 lets Firm 1 have a greater market share. As a result, Firm 1’s gain of deviating from the collusive agreement would be smaller.)

Two firms compete in price in a market for infinite periods. In this market, there are N consumers; each buys one unit per period if the price does not exceed $10 and nothing otherwise. Consumers buy from the firm selling at a lower price. In case both firms charge the same price, assume N/2 consumers buy from each firm. Assume zero production cost for both firms.

A possible strategy that may support the collusive equilibrium is: Announce a price of $10 if the equilibrium price has always been $10; otherwise, announce the price as in Nash equilibrium of the one-shot Bertrand game.

1.a Let δ be the discount factor. Find the condition on δ such that the above strategy can indeed support the collusive equilibrium.

Now suppose that Firm 2’s marginal cost is $4, but Firm 1’s marginal cost remains at zero.

1.b Find the condition on δ under which Firm 2 will not deviate from the collusive equilibrium.

1.c Find the condition on δ under which Firm 1 will not deviate from the collusive equilibrium.

1.d Knowing that both firms’ discount factor is 0.6, how should Firm 2 set its capacity constraint so that the collusive equilibrium can still be supported? (Hint: The idea here is that, by limiting its own output, Firm 2 lets Firm 1 have a greater market share. As a result, Firm 1’s gain of deviating from the collusive agreement would be smaller.)