Consider an agent with preferences over R1 representable by the CES utility function 71 1/1' U(:L*) = (2 aim?) for some parameters 1", 041, ..., an > 0. 1. For what values of 'r > 0 are the agent's preferences convex? When answering this question, follow the following instructions: a) If preferences are convex for some 7', 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 r, 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 (ahag) = (1/5,4/5). Suppose that the agent has total wealth w > 0 and faces linear prices (101,1) for some 101 > 0. a) Suppose that "r = 1/2. Find the agent's optimal consumption bundle as a function of w and p1. Plot m'f and as; as functions of p1 when w = 10. b) Repeat the previous part with 7' = 2. Be careful to specify the full set of optimal bundles whenever there are multiple