Exercise 6.8. Consider the following discrete time optimal growth model with full depreciation: max {c(t),k(t)}∞ t=0 X∞ t=0 βt ³ c (t) − a 2 [c (t)]2 ´

subject to k (t + 1) = Ak (t) − c (t) and k (0) = k0. Assume that k (t) ∈ £ 0, ¯ k ¤ and a < ¯ k−1, so that the utility function is always increasing in consumption. (1) Formulate this maximization problem as a dynamic programming problem. (2) Argue without solving this problem that there will exist a unique value function V (k) and a unique policy rule c = π (k) determining the level of consumption as a function of the level of capital stock. (3) Solve explicitly for V (k) and π (k) [Hint: guess the form of the value function V (k), and use this together with the Bellman and Euler equations; verify that this guess satisfies these equations, and argue that this must be the unique solution].