Consider two firms facing the demand curve P = 50 - 5Q, where Q = Q1 + Q2. The firms’ cost functions are C1(Q1) = 20 + 10Q1 and C2(Q2) = 10 + 12Q2.

a. Suppose both firms have entered the industry. What is the joint profit-maximizing level of output? How much will each firm produce? How would your answer change if the firms have not yet entered the industry?

b. What is each firm’s equilibrium output and profit if they behave noncooperatively? Use the Cournot model. Draw the firms’ best-responses (reaction curves) and show the equilibrium

Explanation:

If both firms enter the market, and they collude, they will face a marginal revenue curve with twice the slope of the demand curve: MR = 50 - 10 Q . Setting marginal revenue equal to marginal cost (the marginal cost of Firm 1, since it is lower than that of Firm 2) to determine the profit-maximizing quantity, Q : 50 - 10 Q = 10, or Q = 4. Substituting Q = 4 into the demand function to determine price: P = 50 – 5*4 = $30. The question now is how the firms will divide the total output of 4 among themselves. Since the two firms have different cost functions, it will not be optimal for them to split the output evenly between them. The profit maximizing solution is for firm 1 to produce all of the output so that the profit for Firm 1 will be: π1 = (30)(4) - (20 + (10)(4)) = $60. The profit for Firm 2 will be: π2 = (30)(0) - (10 + (12)(0)) = -$10. Total industry profit will be: π T = π1 + π2 = 60 - 10 = $50. If they split the output evenly between them then total profit would be $46 ($20 for firm 1 and $26 for firm 2). If firm 2 preferred to earn a profit of $26 as opposed to $25 then firm 1 could give $1 to firm 2 and it would still have profit of $24, which is higher than the $20 it would earn if they split output. Note that if firm 2 supplied all the output then it would set marginal revenue equal to its marginal cost or 12 and earn a profit of 62.2. In this case, firm 1 would earn a profit of –20, so that total industry profit would be 42.2