(10 points) There are two firms (1 and 2) in a market that sell heterogeneous goods. Let q1 and q2 be the demand of the goods for firms 1 and 2 if the prices charged by them are p1 and p2 respectively. Suppose we are given the following relationship between the prices and the quantities: p1 = max{10 2q1 + q2, 0}, and p2 = max{10 2q2 + q1, 0}. Suppose the firms are engaged in price competition. That is, they each pick a price simultaneously and independently. This results in a demand for each of the firms, which determines profits for the firms. (a) (2 points) Given prices p1 and p2, determine the market-clearing quantities q1 and q2 of the two firms respectively. (b) (3 points) Derive the best-response functions of the firms and use these to compute the Nash equilibrium prices. Compute the associated profits of both firms. (c) (2 points) Compute the prices at which the total profit of the two firms is maximized. Compute the associated profits of both firms. (d) (3 points) Now suppose that this price competition game is played infinitely many times, and that the firms' payoffs are discounted by a common factor . Can you realize the outcome of part (c) as the outcome of a subgame perfect Nash equilibrium strategy for each firm? If so, what conditions should satisfy?