Convex set, epigraph, continuous convex function, hyperplanes

Instructions: Liberal partial credit will be awarded IF you show your work. Unjustied answers are worth at most half-credit! 1. Let C = fx 2 Rn j kxk1 1g which is the unit ball in `1(Rn). (a) In the case that n = 2, draw a careful picture of the ball. Is it convex? Why or why not? (b) For a general dimension n, let ^x be a point on the boundary of C. Explicitly identify the supporting hyperplanes of C at ^x. [CAUTION ! Dierent points on the boundary may well have multiple supporting hyperplanes.] 2. Show that a set C Rn is convex if and only if its intersection with any line is convex. 3. Show that the epigraph of a continuous convex function f : Rn ??! R is a closed set. 4. In the notes on consumer behavior we have dened what it means for a preference relation to be non-satiated. A preference relation is called locally non-satiated provided for every > 0 there is a y 2 X with ky ?? xk < such that y x. (a) Show that local non-satiation implies non-satiation. (b) Typically, a consumer has limited resources and hence must meet a budget. If p is a price vector and x is a typical bundle, then the price of that bundle is given by the inner product hp; xi. Given a utility function, a rational consumer chooses a bundle which is a solution to the constrained optimization problem max x2X u(x) subject to hp; xi m: where m is the maximum buying power available to the given consumer. Show that under local non-satiation a utility maximizing bundle x? must be such that hp; x?i = m. (c) Show that if is strictly convex, i.e., if x0 x then, for all 2 (0; 1) ; x+(1??)x0 x, then, for each price vector p > 0 there is a unique solution to the constrained maximization problem. DUE: My oce by 5 p.m. Friday, December 18 together with your signature to the statement below. I certify that the work submitted for this nal examination is my own work and that I have not consulted with any other person concerning the examination problems or their solution. NAME: SIGNATURE: .