【高级微观经济学】08年习题二.docxVIP

GS/ECON 5010 Answers to Assignment 2 October 2008 Q1. Could the following 3 equations be Hicksian demand functions (if the reference level of utility u were high enough so that u + ln p2 + ln p3 2 + 2 ln p1)? Explain briey. x1(p; u) = u 2 2 ln p1 + ln p2 + ln p3 p1 x2(p; u) = u + p1 x3(p; u) = u + A1 What would the expenditure function e(p; u) be, if these three functions were the Hicksian demand functions? Since e(p; u) = p1xH1 (p; u) + p2xH2 (p; u) + p3xH3 (p; u) it would have to be the case that e(p; u) = (p1 + p2 + p3)u + p1(ln p2 + ln p3 2 ln p1) (1 1) if the Hicksian demand functions xHi (p; u) were the ones listed in the question. The expenditure function e(p; u) de ned in (1 1) is homogeneous of degree 1 in prices : since ln(ka) = ln k + ln a, e(kp; u) = k(p1 + p2 + p3)u + kp1(ln p2 + ln k + ln p3 + ln k 2 ln p1 2 ln k) = ke(p; u) (1 2) The derivatives of the expenditure function (1 1) are e1(p; u) = u + ln p2 + ln p3 2 ln p1 2 = xH1 (p; u) e2(p; u) = u + p1 = xH (p; u) p2 2 e3(p; u) = u + p1 = xH (p; u) p3 3  (1 3) (1 4) (1 5) so that Shephard’s Lemma holds. If a function (such as e(p; u) de ned above) is homogeneous of degree t, then all its rst derivatives must be homogeneous of degree t 1. So equation (1 2) establishes that all of the functions xi(p; u) are homogeneous of degree 0. [This fact can be established more directly : tpi=tpj = pi=pj so that x2(p; u) and x3(p; u) are homogeneous of degree 0 in prices ; ln tp = ln t + ln p so that x1(p; u) is homogeneous of degree 0.] The matrix H of second derivatives of e(p; u) is 1 0 2 p1 H(p; u) = B p2 @ 1 p3  1 p2 p1 (p2)2 0  1 1 p3 C A p1 (p3)2 This matrix is symmetric. [This must be the case if xi(p; u) is the i{th derivative of some function e(p; u) as has already been shown.] The principal minors [the determinants of the 1{by{1, 2{by{2 and 3{by{3 matrices in the top left corner of H] are M1 = 2 0 p1 2 1 1 M2 = = 0 (p2)2 (p2)2 (p2)2 M3 = 2 p1 + p1 + p1 = 0 (p2)2(p3)2 (p2)2(p3)2 (p2)2(p3)2 so