Introduction to Modern Economic Growth (2) Prove that there exists 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 (3) When will V (k) be differentiable? (4) Assuming that V (k) and all the other functions are differentiable, characterize the Euler equation that determines the optimal path of consumption and capital accumulation (5) Is this Euler equation enough to determine the path of k and c? If not, what other condition we need to impose? Write down this condition and explain intuitively why it makes sense Exercise 6.13 Prove that, as claimed in Proposition 6.4, in the basic discretetime optimal growth model, the optimal consumption plan c (k) is nondecreasing, and when the economy starts with k0 < k∗ , the unique equilibrium involves c (t) ↑ c∗ Exercise 6.14 Prove that as claimed in the proof of Proposition 6.3, Assumption implies that s (ε) > ε for ε sufficiently small Provide an intuition for this result Exercise 6.15 * Provide a proof of Proposition 6.1 without the differentiability assumption on the utility function u (·) imposed in Assumption 3’ Exercise 6.16 Prove that the optimal growth path starting with capital-labor ratio k0 , which satisfies (6.37) is identical to the competitive equilibrium starting with capital-labor ratio and satisfying the same condition (or equivalently, equation (6.42) Exercise 6.17 Consider the model of searching for ideas introduced in Section 6.8 Modify the model so that the entrepreneur can use any of the techniques he is discovered in the past to produce at any point in time (1) Prove that if the entrepreneur has turned down production at some technique a0 at date t, he will never accept technique a0 at date t + s, for s > (i.e., he will not accept it for any possible realization of events between dates t and t + s) (2) Prove that if the entrepreneur accepts technique a0 at date t, he will continue to produce with this technique for all dates s ≥ t 311