Question

Suppose a consumer has a utility function U(x1, x2) = Inxi + x2. The consumer takes prices (p1 and p2) and income (I) as given. > 1. What is special about P2 1) Find the demand functions for and x2 assuming these demand functions? Are both goods normal? Are these tastes homothetic? 2) Now find the demand functions for x1 and x2 assuming-<1. You probably P2 assumed the opposite above, so now will you find something different. Explain 3) Graph the income consumption curve for good 2 as a function of income for given prices > 1. Verify Roy's identity 4) Find the consumer's indirect utility function when P2

Answer 1

Q 1. UCX1, X2) = lnxi +X2 It Income Now at Eqm MRS, 2= Pilla - MRS= MUNU2 = /x, 80 at Eqma. I = Pilpa - Pg= Pixi - X, * = P2/P, I from B.C. | Pixt + P2 X2 = 1 - Pixelio X2 = I- Pi (P21P;) P2 Xg&= (1- P2) IP2 = Ill2-1. [X2 = I -11 If IlP2 > 1. 80 demand funch of I is independent of Income Level, which implies that I is a necessity. yes Both Goods are Normal, as aX2A >O. a1 No tastes are not homo Thetic, Boz for Momo theticity, the to are radial expansion of each other, along a ray through the origin Don Pout here for Quasi Linear Pref, Preferences) , ic are radial expansion along either Honizontal or vertical line.

Ma Now D/P2 L1, then x2=0 & X, *= Illy. Ang entire Income is spent on Good I only a comer soln exist 3.) - ICC. gcc for Good 2 After I SP2, till then X I <P2 , Then Xo =0, xanh ↑ linearly with Income 4) I (XM, X24) = In 1 P214) + 1/22 - 1 Income LP2 (DIP) X→ Roy Identity, x=-20/ap, 20/01 2 0 2 then OPIE | * -P2 0 0 ƏD = VP (P2IPJ (Pj2 -aulap = +P2 P & P2 = P21P 201 as Piz AP Je Boved