Economic growth and economic development 264

Introduction to Modern Economic Growth 5.12 Exercises Exercise 5.1 Recall that a solution {x (t)}Tt=0 to a dynamic optimization problem is time-consistent if the following is true: whenever {x (t)}Tt=0 is an optimal solution starting at time t = 0, {x (t)}Tt=t0 is an optimal solution to the continuation dynamic optimization problem starting from time t = t0 ∈ [0, T ] (1) Consider the following optimization problem max {x(t)}T t=0 T X β t u (x (t)) t=0 subject to x (t) ∈ [0, x¯] G (x (0) , , x (T )) ≤ Although you not need to, you may assume that G is continuous and convex, and u is continuous and concave Prove that any solution {x∗ (t)}Tt=0 to this problem is time consistent (2) Now Consider the optimization problem max u (x (0)) + δ {x(t)}T t=0 subject to T X β t u (x (t)) t=1 x (t) ∈ [0, x¯] G (x (0) , , x (T )) ≤ Suppose that the objective function at time t = becomes u (x (1)) + P δ Tt=2 β t−1 u (x (t)) Interpret this objective function (sometimes referred to as “hyperbolic discounting”) (3) Let {x∗ (t)}Tt=0 be a solution to this maximization problem Assume that the individual chooses x∗ (0) at t = 0, and then is allowed to reoptimize at 250