Solve the following questions

Problem 1 (150 marks). Consider a consumer with wealth to who consumes two goods, which we shall call goods 1, 2. Let the amount of good 6 that the consumer consumes be it's and the price of good E be pg. Suppose that the consumer's preferences are described by the utility function nets) =(s?1+ we)? (1) Is this utility function well behaved as we have dened it? Is it very well behaved? (2) Set up the utility maximisation problem and write down the Lagrangian. (3) Write down the rst order necessary conditions for an interior maximum. (4) Solve the rst order conditions to obtain the Marshallian (or uncompen sated) demand functions. (5) Substitute the Marshallian demands back into the utility function to obtain the indirect utility function. (6) State Roy's Theorem and verify that Roy's Theorem does indeed give the same Marshallian demands that you found above. (7) Invert the indirect utility function to nd the expenditure function. (8) Consider the expenditure minimisation problem min P1331 + 132352 3312322 subject to: (1/31 + 3:2 J2 = it Write down the Lagrangian for this problem. (9) Write down the rst order necessary conditions for an interior minimum. (15) Write the indirect utility function as a function of the normalised prices q1 = p1/w and q2 = p2/w rather than as functions of p1, p2, and w. Call this function v(q1, q2) = v(q1, 92, 1). (16) Consider the minimisation problem min v(q1, q2) 91 , 92 subject to: q121 + 922 = 1. with x1, 22 > 0. Write down the Lagrangian for this problem. (17) Write down the first order necessary conditions for an interior minimum. (18) Solve the first order conditions to obtain the inverse demand functions, giving normalised prices as a function of the consumption bundle. Call this function q : R3 - R3. (19) Take these inverse demand functions and substitute them into v to give a utility function u* giving utility as a function of the consumption bundle. Confirm that this is the same as the utility function that you started with. (20) Write the Marshallian demands you found in part 4 as functions of the normalised prices q1 and q2, rather than p1, p2, and w. Call these functions x1 and x2, so we have a : R2 - R2 . Show that x is the inverse function to q.(10) Solve the first order conditions to obtain the Hicksian (or compensated) demand functions. (11) Take the Hicksian demand functions that you just found, which we shall denote he(p1, p2, u), and substitute them back into the objective function, Pix1 + P242, to obtain the expenditure function e(P1, P2, u) = pihl(P1, P2, u) + p2h2(P1, P2, u). and confirm that it is the same as the expenditure function you found above. (12) Use Shephard's Lemma to find the Hicksian demand functions from the expenditure function and confirm that are the same functions that you found by solving the expenditure minimisation problem. (13) Substitute the indirect utility function v(p1, p2, w) you found in part (5) into the Hicksian demand functions you just found and confirm that you obtain the Marshallian demand functions that you found in part (4). (14) Solve the equation e(P1, p2, v(P1, P2, w)) = w to find the indirect utility function and confirm that it is the same as the function you found above