2. Two firms, X and Y, produce vertically differentiated goods. There are 400 consumers who might want to buy the firms' goods. Given their tastes, each consumer has a characteristic b, which tells us something about how much they like the goods. Specifically, a consumer with characteristic b is willing to pay up to dollars for one of firm X's goods, and up to dollars for one of firm Y's goods. Consumers' values of b are all between 10 and 50. More specifically, for any two values of b between 10 and 50, and , with , the number of consumers with values of b between and is . So, for example, the number of consumers with values of b between 18 and 31 is 10x(31-18)=130. The firms compete by simultaneously choosing prices for their goods, for firm X, for firm Y. Once the firms choose their prices, consumers decide which firm's good they want to buy (the goods are close substitutes, even if not identical, and no consumer gets any extra benefit from owning a second good). A consumer will make her purchase based on consumer surplus; her consumer surplus from buying a good from firm X would be ; her consumer surplus from buying a good from firm Y would be . (a) Suppose that the firms charge prices and , with (what a consumer with the lowest possible b of 10 would be willing to pay for a Y good); and such that the difference between and is between 100 and 500 (this assumption makes certain things about the problem work out more simply; you don't need to use the assumption in your work). Write down the condition for a consumer with characteristic b to prefer buying good X to good Y, a condition that will have and in it. Use that condition to find the number of consumers firm X will sell to, and the number of consumers firm Y will sell to, as a function of the prices. (8 points) (b) Each good that firm X sells costs it 60 dollars to produce; each good that firm Y sells costs it 30 dollars to produce. Using your last answer to part (a), write down firm X's profit, as a function of and , and firm Y's profit, as a function of and . Solve for the prices the firms will charge in equilibrium. Do this by finding each firm's profit- maximizing choice of price, given a guess about the other firm's price, followed by a solution for those prices when their guesses are correct. Solve for the number of consumers that buy from each firm, and for the firms' profits, in the equilibrium. (15 points) (c) Suppose that consumer tastes changed, so that all consumers like both goods more. Specifically, a consumer with characteristic b, who valued one of firm 2's goods at 25b dollars, and one of firm I's goods at 15b dollars, now values one of firm 2's goods at 100b dollars, and one of firm I's goods at 90b dollars. Explain how this would affect the predicted prices and firm profits. You can do this without recalculating everything from parts (a) and (b). What is the intuition for how prices and profits respond to this change in the values of the goods to consumers? (8 points)
