The following problem describes properties of the Constant Elasticity of Substitution (CES) utility function. 1. Consider the utility function N U(CCN) = B1/1-1/ n=1 1/(1-1/8) where the parameters and all the , are positive. For simplicity, ignore the case = = 1. a. Given the vector of prices p = (p, ..., PN), define P(p) = min N PnCn: U(C,..., CN) (C1.....CN) n=1 (1) (i) Set up the Lagrangian. Write down the first-order conditions and use the constraint U(c1,..., CN) = 1 to simplify these conditions to p = (/cn)/, for n = 1,..., N, where is a positive Lagrange multiplier. (ii) Determine the consumption choices (c1,..., CN) that solve (1). (iii) Verify that N P(p) = ) n=1 1/(1-E) (2) b. Show that the consumption choices that solve V(p, x) = max {U (c1,...,CN) N U(c1,..., CN): Pnen1 n=1 can be written as Cn = for n = 1,..., N. c. Determine V(p, x). Pn -()) = P(p). P(p)'