Assume that two companies (A and B) are duopolists who produce identical products. Demand for the products is given by the following linear demand function:
P = 200 - QA - QB
where QA and QB are the quantities sold by the respective firms and P is the selling price. Total Cost functions for the two companies are
TCA = 1,500 + 55QA+ Q2A
TCB= 1,200 + 20QB+ 2Q2B
Assume that the firms form a cartel to act as a monopolist and maximize total industry profits (sum of firm A and firm B profits).
a. Determine the optimum output and selling price for each firm.
b. Determine Firm A, Firm B, and total industry profits at the optimal solution found in Part (a).
c. Show that the marginal costs of the two firms are equal at the optimal solution found in Part (a).
First, we determine how many units should be produced to maximize their joint profit.
profit = price X total output - cost of firm A - cost of firm B
profit = (200 - Qa - Qb)(Qa + Qb) - 1,500 - 55Qa - Qa^2 - 1,200 - 20Qb - ...