3. Consider two consumers with utility functions: UA = log(x) + x2 log(xB), UB = log(x ) + x2 2 - log (xA). Both consumers have income M, and the price of both goods is 1. a. Calculate the market equilibrium. b. Calculate the social optimum for a utilitarian social welfare function. c. Show that the optimum can be sustained by a neutral tax placed on good 1 (so that the after-tax price becomes 1 + t (where 1 is the initial price and t is the tax). d. Assume now that preferences are given by: UA = p4 log(xi) + x7 - - log(x), UB = p log(x5) + x2 log (xA). 2 NI Calculate the taxes necessary to decentralize the optimum