This exercise will guide you through a proof of a version of Theorem 5.4 when the consumer's utility function is quasiconcave instead of strictly quasiconcave and strictly increasing instead of strongly increasing 2 This exercise will guide you through a proof of a version of Theorem 5.4 when the consumer's utility function is quasiconcave instead of strictly quasiconcave and strictly increasing instead of strongly increasing (a) If the utility function u: R. - R is continuous, quasiconcave and strictly increasing, show that for every & E (0. 1) the approximationg utility function v,: R" - IR defined by 1. ( xX ) - 1 1 + ( 1 - 8 7 +. ... . . + (1 - 5 2 is continuous, strictly quasiconcave and strongly increasing. Note that the approximation to u(.) becomes better and better as & - 1 because v, (x) - u(x) as z - 1. (b) Show that if in an exchange economy with a positive endowment of each good, each consumer's utility function is continuous, quasiconcave and strictly increasing on R4, there are approximate ing utility functions as in part (a) that define an exchange economy with the same endowments and possessing a Walrasian equilibrium. If, in addition, cach consumer's endowment gives him a positive amount of each good, show that any limit of such Walrasian equilibria, as the approxi- mations become better and better (e.g., as e - 1 in the approximations in part (a)) is a Walrasian equilibrium of the original exchange economy. (c) Show that such a limit of Walrasian equilibria as described in part (b) exists. You will then have proven the following result. If each consumer in an exchange economy is endowed with a positive amount of each good and has a continuous, quasiconcave and strictly increasing utility function, a Walrasian equilibrium exists