1. The optimal growth model is an extension to the cake-eating problem. Consumers decide whether to consume or invest in capital. Capital is converted to income according to a production function y = Ake, where A > 0, and a E (0, 1). In other words, this problem assumes log utility, Cobb-Douglas production, and no stochastic shocks. The sequence problem is, max E Bt In(ct) { ct , kt+1) t= 0 Subject to: ko = ko > 0 (a) Write the Bellman equation for this problem. (b) Use the method of undetermined coefficients to solve for the value function of the form V (k) = A + BIn(K). (c) Show that the following guesses can not be forms for the value function: i. V (k) = A ii. V (k) = BIn(k) iii. V (k) = Bka