4. Consumer's Problem [35 points]. Ernie's preferences for apples (qa) and bananas (qb) are represented by the utility function U(qa,qb)=(qa)31(qb)32. The price of an apple is pa, the price of a banana is pb, and Ernie's income is Y. (a) Derive the marginal utilities of apples and bananas. Show that, for qa>0 and qb>0, Ernie's preferences satisfy "more is better." (5 points) (b) State the consumer's problem of Ernie. Then explain why the budget constraint in the problem, paqa+pbqbY, can be replaced by the budget line, paqa+pbqb=Y. (3 points) (c) Derive Ernie's demand functions of apples and bananas. Show your steps. (18 points) (d) Based on the demand function for bananas you derive in (c), (i) calculate the income elasticity using the "one-point" formula and argue accordingly that bananas are necessities; ( 6 points) (ii) explain how the substitution effect interacts with the income effect when there is a price change in banana (a diagram is helpful but not necessary). (3 points)