Introduction to Modern Economic Growth (1) Using the transversality condition together with a (0) and {w (t)}∞ t=0 , find an expression implicitly determining the initial level of consumption, c (0) What happens to this level of consumption when a (0) increases? (2) Consider the special case where u (c) = ln c Provide a closed-form solution for c (0) (3) Next, returning to the general utility function u (·), consider a change in the earnings profile to a new sequence {w ˜ (t)}∞ t=0 such that for some T < ∞, P −t w (t) < w ˜ (t) for all t < T , w (t) ≥ w ˜ (t) for all t ≥ T , and ∞ t=0 R w (t) = P∞ −t ˜ (t) What is the effect of this on the initial consumption level t=0 R w and the consumption path? Provide a detailed economic intuition for this result Exercise 6.11 Consider the following discrete time optimal growth model ∞ X u (c (t)) max ∞ {c(t),k(t)}t=0 t=0 subject to k (t + 1) = k (t) − c (t) and k (0) = k0 < ∞ Assume that u (·) is a strictly increasing, strictly concave and bounded function Prove that there exists no optimal solution to this problem Explain why Exercise 6.12 Consider the following discrete time optimal growth model with full depreciation: max ∞ {c(t),k(t)}t=0 subject to ∞ X β t u (c (t)) t=0 k (t + 1) = f (k (t)) − c (t) and k (0) = k0 Assume that u (·) is strictly concave and increasing, and f (·) is concave and increasing (1) Formulate this maximization problem as a dynamic programming problem 310