Exercise 7.23. Consider the following continuous time discounted infinite horizon problem: max Z 0 exp (t)

Exercise 7.23. Consider the following continuous time discounted infinite horizon problem: max Z ∞ 0 exp (−ρt) u (c (t)) dt subject to x˙ (t) = g (x (t)) − c (t) with initial condition x (0) > 0.

Suppose that u (·) is strictly increasing and strictly concave, with limc→∞ u0 (c)=0 and limc→0 u0

(c) = ∞, and g (·) is increasing and strictly concave with limx→∞ g0 (x)=0 and limx→0 g0 (x) = ∞. (1) Set up the current value Hamiltonian and derive the Euler equations for an optimal path. (2) Show that the standard transversality condition and the Euler equations are necessary and sufficient for a solution. (3) Characterize the optimal path of solutions and their limiting behavior.