Question

3. There are two goods, Xi and X2 with prices pı > 0 and P2 = 1. Assume that a consumer has income I> 0 that she will allocate for the bundle (X1, X2), and has preferences represented by the utility function u(X1, X2) = a ln x1 + x2, for some a > 0. (a.) Derive the marginal utilities and bang-for-bucks for each good. (b.) Find the optimal bundle assuming an interior solution, i.e. x > 0 and x > 0. (c.) Find the optimal bundle assuming I < a. (d.) Suppose you're a seller of good X1. Show that your revenues (p1 x xi) do not depend on the price you set as long as I > a. (e.) Imagine you have data on sales, prices and income. Using this data, you want to estimate the parameter a of the utility function. Explain why we cannot estimate a if observed income is too low, i.e. I < a. (Hint: What is the optimal bundle when I < a?]

Answer 1

Income I Q) UCXI, X2) = Llnx, +X2 g P2=1 Piso a) Muy = 24 = x/x1, MU2 = 24/0x2= 1 bang-for bucks = ones Moneys worth .. 6.) at eam, MUL = MU2 M X = 1 [X2M = D-x 7 J Ang " (1) Now optimal Bundle, If Is a D-2 LO 80 X2=0, only xi is consumed, (X19X2) * = (P/P, , o) Ang d.) Now seller of Good x,, Revenue = Px* = Pia/p = x 2.) If IKA, then Xi = P1P, 80 No Information on a could be obtained. Ans (independent of Bice Pi).