topology, application of mathematics in economics

answers are worth at most half-credit! 1. Let C = {x 2 Rn | kxk1 ? 1} 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 ˆx be a point on the boundary of C. Explicitly identify the supporting hyperplanes of C at ˆx. [CAUTION ! Di?erent 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 defined 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