2. Cournot-Nash: Two Cournot duopolists produce in a market with demand p = 100 - Q. The marginal cost for firm 1 is constant and equals c = 20. The marginal cost for firm 2 is also constant and equals c2 = 30. Firm 1 produces q and firm 2 produces (2, and Q = 91+q2. (a) Find the best-response functions for each firm and plot these on a graph. Plot 02 on the horizontal axis. [1 mark] (b) Derive and plot the iso-profit curves for the Nash equilibrium, and shade the area indi- cating quantities by the two firms that would result in higher profits for both firms. [1 mark] (c) Replot the best-response functions for each firm with q2 on the horizontal axis. Suppose that the firms move in sequence, and that firm 1 chooses the value of q that it would choose were q2 = 0. Then find firm 2's best response to this quantity, then find firm l's best response to firm 2's best response, and so on. Show whether under this sequence, the firms converge to the Cournot-Nash equilibrium. (d) Find the Cournot-Nash equilibrium output by each firm, market price, and profits to each firm. [1 mark] (e) Graph demand and the marginal costs of the two firms and show profits, consumer's surplus, and the dead-weight-loss relative to the low-cost firm producing all output on a competitive basis. [1 mark] (f) The two firms want to merge. They argue for the merger on the grounds that marginal production costs would fall to ci for all units of output after the merger, since all pro- duction would be at the low marginal cost. Find the equilibrium quantity the merged firm will produce, the market price at which this will sell, and the profits to the merged firm. (1 mark] (g) Graph demand and the marginal cost of the merged firm and show profits, consumer's surplus, and the dead-weight-loss relative to the firm producing all output on a compet- itive basis. [1 mark] (h) Given this information, would you recommend the merger? Explain by calculating the benefits and costs from the merger. Hint: How will the merged firm behave? [1 mark] 2. Cournot-Nash: Two Cournot duopolists produce in a market with demand p = 100 - Q. The marginal cost for firm 1 is constant and equals c = 20. The marginal cost for firm 2 is also constant and equals c2 = 30. Firm 1 produces q and firm 2 produces (2, and Q = 91+q2. (a) Find the best-response functions for each firm and plot these on a graph. Plot 02 on the horizontal axis. [1 mark] (b) Derive and plot the iso-profit curves for the Nash equilibrium, and shade the area indi- cating quantities by the two firms that would result in higher profits for both firms. [1 mark] (c) Replot the best-response functions for each firm with q2 on the horizontal axis. Suppose that the firms move in sequence, and that firm 1 chooses the value of q that it would choose were q2 = 0. Then find firm 2's best response to this quantity, then find firm l's best response to firm 2's best response, and so on. Show whether under this sequence, the firms converge to the Cournot-Nash equilibrium. (d) Find the Cournot-Nash equilibrium output by each firm, market price, and profits to each firm. [1 mark] (e) Graph demand and the marginal costs of the two firms and show profits, consumer's surplus, and the dead-weight-loss relative to the low-cost firm producing all output on a competitive basis. [1 mark] (f) The two firms want to merge. They argue for the merger on the grounds that marginal production costs would fall to ci for all units of output after the merger, since all pro- duction would be at the low marginal cost. Find the equilibrium quantity the merged firm will produce, the market price at which this will sell, and the profits to the merged firm. (1 mark] (g) Graph demand and the marginal cost of the merged firm and show profits, consumer's surplus, and the dead-weight-loss relative to the firm producing all output on a compet- itive basis. [1 mark] (h) Given this information, would you recommend the merger? Explain by calculating the benefits and costs from the merger. Hint: How will the merged firm behave? [1 mark]
