1. a) Suppose an agent lives for three periods and maximizes a three-period utility function: The prices of goods in each period are p, =1, p, =2, and p, =2. i) If the consumer enters period 2 with an income of W, find the value function, I , for the final two periods. Use this value function to convert the problem into a 2- period problem and solve, assuming that the consumer enters period 1 with income Wi. ii) Determine the value function at the start of period 1, V(W,). b) Consider the equation V(x, ) = u(c,)+ BV(/(x,,6,)) where / is a constant. The control variable is c and exogenous variable is x. Let c, denote the optimal value of consumption; i.e., the value that maximizes the right-hand side. i) What is c a function of? Use the Envelope Theorem to find the derivative of each side with respect to x, . iii) How does this formulation of a problem relate to the one in part a)? Explain