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4. Consider a duopoly with firms that offer homogeneous products where each has constant mar- ginal cost of c. Let D(P) denote market demand. Firms make simultaneous price decisions. Letting P, and p2 be the prices of firms 1 and 2, respectively, the demand function of firm 1 is specified to be if p, P2. If firm l's price is lower than firm 2's price, then all consumers buy from it, so that its demand equals market demand. If both firms charge the same price, then they equally split market demand. If firm l's price is higher than firm 2's price, then all consumers go to firm 2. Firm 2's demand function is similarly defined. Each firm chooses prices to maximize its profit. a. Show that both firms pricing at marginal cost is a Nash equilibrium. b. Show that any other pair of prices is not a Nash equilibrium. Suppose that we limit firms to choosing price equal to c, 2c, or 3c. c. Compute the payoff/profit matrix. d. Derive all of the Nash equilibrium price pairs