Two firms produce differentiated products. Firm 1 faces the demand curve Q1 = 75 - P1 + .5P2. (A lower competing price robs the firm of some, but not all, sales. Thus, price competition is not as extreme as in the Bertrand model.) Firm 2 faces the analogous demand curve Q2 = 75 - P2 + .5P1. For each firm, AC = MC = 30.
a. Confirm that firm 1’s optimal price depends on P2 according to P1 =52.5 + .25P2. (Set up the profit expression π1 = (P1 - 30) Q1 = (P1 - 30)(75 - P1 + .5P2) and set Mπ = ∂π1/∂P1 = 0 to solve for P1 in terms of P2. Alternatively, set MR1 = MC and solve for Q1 and then P1 in terms of P2.
b. Explain why a lower price by its competitor should cause the firm to lower its own price.
c. In equilibrium, the firms set identical prices: P1 = P2. Find the firms’ equilibrium prices, quantities, and profits.