3. Two rms, X and Y, produce horizontally differentiated goods. There are 300 consumers who might want to buy the rms' goods. Given their tastes, each consumer has a characteristic x, which tells us something about how much they like the goods. Specically, a consumer with characteristic x is willing to pay up to 400 10x dollars for one of rm X's goods, and up to 400 10(1 - 1:) dollars for one of rm Y's goods. Consumers' values of x are all between 0 and 10. More specically, for any two values of x between 0 and 10, xmgh and xmw, with xhigh > xmw, the number of consumers with values of x between army\" and 2:10\2. Two rms, X and Y, produce vertically differentiated goods. There are 300 consumers who might want to buy the rms' goods. Given their tastes, each consumer has a characteristic b, which tells us something abOut how much they like the goods. Specically, a consumer with characteristic bis willing to pay up to 130!) dollars for one of rmX's goods, and up to 1001:! dollars for one of rm Y's goods. Consumers' values of b are all between S and 15. More specically, for any two values of b between S and 15, bhigh and blow, with bhigh > blow, the number of consumers with values of b between bhigh and btow is 30 x (bhigh - brow) So, for example, the number of consumers with values of b between 7 and 11 is 30X(l 1-7)=120. The rms compete by simultaneously choosing prices for their goods, PX for firm X, Py for rm 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 benet from owning a second good). A consumer will make her purchase based on consumer surplus; her conSumer surplus from buying a good from rm X would be 1301) PX; her conSumcr Surplus from buying a good from rm Y would be 100!) Py. (a) Suppose that the rms charge prices PX and Py, with P}: