Economic growth and economic development 381

Introduction to Modern Economic Growth x (0) = and x (1) = 1, where y (t) ∈ R and f is an arbitrary continuously differen- tiable function Show that the unique solution to this maximization problem does not satisfy the necessary conditions in Theorem 7.2 Explain why this is Exercise 7.16 * Consider the following maximization problem: Z x (t)2 dt max − x(t),y(t) subject to x˙ (t) = y (t)2 x (0) = and x (1) = 1, where y (t) ∈ R Show that there does not exist a continu- ously differentiable solution to this problem Exercise 7.17 Consider the following discounted infinite-horizon maximization problem max Z subject to ∞ ¸ ∙ 1/2 exp (−ρt) 2y (t) + x (t) dt x˙ (t) = −ρx (t) y (t) and x (0) = (1) Show that this problem satisfies all the assumptions of Theorem 7.14 (2) Set up at the current-value Hamiltonian and derive the necessary conditions, with the costate variable µ (t) (3) Show that the following is an optimal solution y (t) = 1, x (t) = exp (−ρt), and µ (t) = exp (ρt) for all t (4) Show that this optimal solution violates the condition that limt→∞ exp (−ρt) µ (t), but satisfies (7.55) Exercise 7.18 Consider the following optimal growth model without discounting: Z ∞ [u (c (t)) − u (c∗ )] dt max subject to k˙ (t) = f (k (t)) − c (t) − δk (t) with initial condition k (0) > 0, and c∗ defined as the golden rule consumption level c∗ = f (k∗ ) − δk∗ 367