Consider an economy in which there are two goods, 1 and 2, whose prices are pi > 0 and p2 > 0, respectively. The two goods can only be consumed in non-negative amounts x1 and t2, respectively. A consumer has preferences over R2 which are represented by the utility function The consumer's income is I > 0. (a) Formulate the consumer's utility maximization problem, find the first- order conditions for utility maximization, and find the Marshallian (un compensated) demand functions ri(Pi, P2, I) and r2(P1,P2, I) for goods 1 and 2, respectively. (Note: Use the Lagrangian method. Assume that the budget constraint holds with equality and that the solution is in terior (i. e. x1 > 0 and r2 > 0), thus disregarding the non-negativity constraints on x1 and x2.) Check that the second order conditions are satisfied
(b) Now formulate the corresponding expenditure minimization problem and find the Hicksian (compensated) demand functions r(p, P2, U) and r£(Pi, p2, U) for the two goods, where U 0 is some arbitrary level of utility. (Note again: Use the Lagrangian method. Assume that the solu- tion is interior and that the other relevant constraint holds with equalitv. Do not check the second order conditions for this problem.)
(c) Compute the indirect utility function V(pi, P2, I) and the expenditure function E(P1,P2, U).