Introduction to Modern Economic Growth t ∈ R+ , then yˆ (t) and the corresponding xˆ (t) achieve the unique global maximum of (7.46) Theorem 7.16 (Arrow Sufficient Conditions for Discounted InfiniteHorizon Problems) Consider the problem of maximizing (7.46) subject to (7.47) and (7.48), with f and g continuously differentiable and weakly monotone Define ˆ (x, y, µ) as the current-value Hamiltonian as in (7.50), and suppose that a soluH tion yˆ (t) and the corresponding path of state variable x (t) satisfy (7.51)-(7.53) and which leads to (7.55) Given the resulting current-value costate variable µ (t), define ˆ (x, yˆ, µ) Suppose that limt→∞ V (t, xˆ (t)) exists and that M (t, x, µ) M (t, x, µ) ≡ H is concave in x Then yˆ (t) and the corresponding xˆ (t) achieve the unique global maximum of (7.46) The proofs of these two theorems are again omitted and left as exercises (see Exercise 7.12) We next provide a simple example of discounted infinite-horizon optimal control Example 7.3 One of the most common examples of this type of dynamic optimization problem is that of the optimal time path of consuming a non-renewable resource In particular, imagine the problem of an infinitely-lived individual that has access to a non-renewable or exhaustible resource of size The instantaneous utility of consuming a flow of resources y is u (y), where u : [0, 1] → R is a strictly increasing, continuously differentiable and strictly concave function The individual discounts the future exponentially with discount rate ρ > 0, so that his objective function at time t = is to maximize Z ∞ exp (−ρt) u (y (t)) dt The constraint is that the remaining size of the resource at time t, x (t) evolves according to x˙ (t) = −y (t) , which captures the fact that the resource is not renewable and becomes depleted as more of it is consumed Naturally, we also need that x (t) ≥ The current-value Hamiltonian takes the form ˆ (x (t) , y (t) , µ (t)) = u (y (t)) − µ (t) y (t) H 349