Consider an agent with preferences over RE: representable by the CES utility function n in U(a:) : (Z owl") for some parameters r, a1, ..., an > 0. 1. For what values of T > 0 are the agent's preferences convex? When answering this question, follow the following instructions: a) If preferences are convex for some i", prove this fact. (You may nd the following facts useful: 1) monotone transformations of quasiconcave functions are quasicon cave; 2) concave functions are quasiconcave; and 3) sums of concave functions are concave.) b) If preferences are not convex for some 1*, provide an example of a pair of bundles for which some weighted average of the bundles is not preferred to either. 2. Let n : 2 and (051,052) : (1/5,4/5). Suppose that the agent has total wealth w > 0 and faces linear prices (pl, 1) for some p1 > 0. a) Suppose that r : 1/2. Find the agent's optimal consumption bundle as a function of w and p1. Plot 3:\"; and 3:; as functions of pi when w : 10. b) Repeat the previous part with r : 2. Be careful to specify the full set of optimal bundles whenever there are multiple