The following problem describes properties of the Constant Elasticity of Substitution (CES) utility function. 1. Consider the utility function 1/(1-1/e) U(C1, . . ., CN) = E 31/6 1-1/8 n=1 where the parameters & and all the , are positive. For simplicity, ignore the case = = 1. a. Given the vector of prices p = (p1, . . ., PN), define P(p) = min [PIC : U ( G ... . CN ) 21) (1) (C1,...,CN) (i) Set up the Lagrangian. Use the first-order conditions and the constraint to show that pn = H(Bn/Cn)1/, for n = 1, ..., N, where u is a positive Lagrange multiplier. (ii) Determine the consumption choices (c1, . .., CN) that solve (1). (iii) Verify that N 1/(1-2) P(p) = pl-e (2) n=1 b. Show that the solution to V(p, x) = max (C1,....CN) U ( a , . .., CN ) : [ Pron SIS n=1 can be written as -E Cn = Bn Pn P(p) P(p) for n = 1, . .., N. c. Determine V(p, x)