For all problems consider a market containing four identical firms, each of which makes an identical product. The inverse demand for this product is P = 100?Q, where P is price and Q is aggregate output. The production costs for firms 1, 2, and 3 are identical and given by C(qi)= 20qi (i= 1,2,3), where qi is the output of firm i. This means that for each of these firms, variable costs are constant at $20 per unit. The production costs for firm 4 are C(q4)= (20+ ?)q4, where ? is some constant. Note that if ? > 0, then firm 4 is a high-cost firm, while if ? < 0, firm 4 is a low-cost firm (|?| < 20). Note also that Q is the total outputs in the market.
1. Assume that the firms each choose their outputs to maximize profits given that they each act as Cournot competitors.
a. Identify the Cournot equilibrium output for each firm, the product price, and the profits of the four firms. For this to be a true equilibrium, all of the firms must at least be covering their variable costs. Identify the constraint that? Must satisfy for this to be the case.
b. Assume that firms 1 and 2 merge and that all firms continue to act as Cournot competitors after the merger. Confirm that this merger is unprofitable.
c. Now assume that firms 1 and 4 merge. Can this merger be profitable if ? is positive so that firm 4 is a high- cost firm? What has happened to the profits of firm 2 as a result of this merger?