Consider the following discrete-time optimal growth model with full depreciation:
Assume that u(.) is strictly concave and increasing for c ≥ 0, and f (.) is concave and increasing.
(a) Formulate this maximization problem as a dynamic programming problem.
(b) Prove that there exists a unique value function V (k) and a unique policy rule c = π(k), and that V (k) is continuous and strictly concave and π(k) is continuous and increasing.
(c) When will V (k) be differentiable?
(d) Assuming that V (k) and all other functions are differentiable, characterize the Euler equation that determines the optimal path of consumption and capital accumulation.
(e) Is this Euler equation enough to determine the path of k and c? If not, what other condition do we need to impose? Write down this condition and explain intuitively why it makes sense.