10 Solved Problems on Utility Functions in Optimization - Homework 4 | ECN 505, Assignments of Econometrics and Mathematical Economics

Download 10 Solved Problems on Utility Functions in Optimization - Homework 4 | ECN 505 and more Assignments Econometrics and Mathematical Economics in PDF only on Docsity! ECN 505 (001) - ASSIGNMENT 4 - SPRING 2009 Constrained Optimization 1. H 8.4 2. H 9.1 3. In a two good world, a consumer faces two constraints: the usual income constraint and a coupon constraint. Let ci = the price of good i in coupons, C = number of coupons. Determine the algebraic form of a consumer’s coupon constraint. Then in each of the following cases, determine whether or not an optimum at x̄ for a consumer is represented. If so, derive the Kuhn Tucker conditions that “correspond” to the picture. If not, carefully explain why not. Assume that the consumer has positive marginal utilities. In the last case, prove that MRS12 (x ∗) = α · p1 p2 + (1− α) c1 c2 , where α ∈ (0, 1). Determine the specific value for α; interpret your result. 1 4. An individual has a utility function of the form U(x, y) = α log(x− γx) + (1− α) log(y − γy), where 0 < α < 1; γx ≥ 0, γy ≥ 0; and x > γx, y > γy. If x = γx or y = γy, then u(x, y) = −∞. This is a form of the Stone-Geary utility function. Assume throughout that M > pxγx + pyγy. (Why?) (a) Derive the individual’s Marshallian (usual) demand functions. Is it possible to have corner solu- tions? Discuss. Derive the Lagrange multiplier. (b) Derive the indirect utility function. (c) Verify that your Lagrange multiplier has the expected interpretation. (d) Now assume that γx < 0 and γy < 0; also, assume that x ≥ 0 and y ≥ 0. Is it possible to have corner solutions in the individual’s constrained utility maximization problem? You need not actually work the problem. Graph some indifference curves and discuss. (e) Verify Roy’s Identity for good x; that is, show that x(px, py,M) = − ∂V (px, py,M) ∂px ∂V (px, py,M) ∂M . 5. A firm’s production function is y = f(K,L) = KαLβ , where y = output, K = capital ≥ 0, L = labor ≥ 0, α > 0, and β > 0. (a) Show that this production function is homothetic. (b) Find the marginal rate of technical substitution of labor for capital associated with the production function. That is, find MRTSLK = MPL MPK ≡ fL fK . (c) How does the marginal rate of technical substitution of capital for labor change (if at all) as labor and capital are both multiplied by the same positive constant? Explain why your answer doesn’t surprise you given your finding in (a). mIn fact, show generally that if F (K,L) is homothetic, then the output expansion path is a straight line through the origin. (d) Find the firm’s long run conditional input demand functions K(w, v, y) and L(w, v, y); that is, solve the problem min K≥0,L≥0 wL+ vK st y = KαLβ . Also, find the Lagrange multiplier. (e) Given your answers in (d), determine the firm’s long run cost function; that is, find c(w, v, y) = wL(w, v, y) + vK(w, v, y). (f) Verify that your Lagrange multiplier has the expected interpretation. (g) Show that the long run total cost function is homogeneous of degree 1 in w and r. Why doesn’t this result surprise you? Explain. (h) For labor, verify Shephard’s Lemma, i.e., show that ∂c(w, v, y) ∂w = L(w, v, y). 6. H 8.8 2