Question

A consumer has the following preferences u(11, 12) = log (11) + 12 Suppose the price of good 1 is pı and the price of good 2 is P2. The consumer has income m. (a) Find the optimal choices for the utility maximization problem in terms of P1, P2 and m. Denote the Lagrange multiplier by 1. (b) How do the optimal choices change as m increases? What does the income offer curve (also called the income expansion path) look like for this consumer? (You can show it on a diagram.) (c) What is the slope of the Marshallian demand curve for good 1? Use the Slutsky equation to find the slope of the Hicksian demand curve for good 1, without actually solving the expenditure minimization problem. (d) For a utility level ū, solve the expenditure minimization problem and find the optimal choices in terms of P1, P2 and ū. Denote the Lagrange multiplier by .. (e) Find the Hicksian demand curve for good 1. What is the slope of this curve? Does it match your answer in (c)? (f) Find the expenditure function. Find its partial derivative with respect to P2. Provide an interpretation of this derivative in terms of choice behavior. (g) Use the answer in (e) to find compensating variation for a change in p. from a to b, a < b. Hint: d 1 log(x) = d. and the fundamental theorem of calculus tell us $"<(t)dt = e(f(b) – f(a)) where c is a constant. (h) Use the answer in (f) to find compensating variation for a change in pı from a to b. Does this match your answer in (9)? (i) What is the consumer surplus lost as a result of the change in pı from a to b, a<b?

Answer 1

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Q a U ( XX2) = ln X, FX2 B C₂ m= pix + P2X2. at eam MRS 1, 2= Pilpa L= lnx, +X2+ Acm-Pixi-P2X2) Foci əlaxis /x- Piazo, & allax 2 = 1- P22=0 thus ta Pippa X=1/12 X 80 of m712 from B.Co Pall M= K (P218) + P2X2 1 X2M = (m-P2) | P2 Ang b ) m7P2, →gcc. as m↑ & m LP2, then Xozo x* nise ma, then xx remain unchanged, Xg* rises. gec → Xg c) slope of Marshallion dd of Good1 Pi= P2/x Slope +. DR /əx, = – P2 / x2 Ang from slusky Eq" ax h - api a Pi Now əxllom =0 $0 - P2 api api P12 of hicksian (fat fi) xi- х. Эх, = am әх aXL - = Selope

Its Mandatory to Answer first Four parts only