Consider the following Bertrand duopoly model involving two rms producing differentiated products. For simplicity, assume that both rms have no production cost. The demand functions for rm 1's and 2's products are q1 = 30 - 2p1 + p2 and q2 = 30 + p1 - 2p2: where p1 and p2 denote the prices of firm 1 and firm 2, respectively.

(a) (10) Find rms' prices and profits in the Nash equilibrium of this Bertrand model.

(b) (10) Suppose that the two rms collude to maximize the sum of their profits. What is the collusive outcome (rms' prices)? Show that every firm has a higher profit in this collusive outcome than in the Nash equilibrium of part (a), and show this collusive outcome is not an equilibrium outcome.

(c) (5) Suppose that the two rms engage in this Bertrand competition over infinitely many periods, and every rm discounts its payoff by discount factor 2 (0; 1) per period. Find the condition on the discount factor under which the collusive outcome from part (b) can be supported as a subgame perfect equilibrium outcome using the trigger strategy prole with the Nash equilibrium from part (a) as the punishment