Economic growth and economic development 363

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 1. 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 = 0 is to maximize
Z ∞
exp (−ρt) u (y (t)) dt.
0

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) ≥ 0.
The current-value Hamiltonian takes the form

ˆ (x (t) , y (t) , µ (t)) = u (y (t)) − µ (t) y (t) .
H
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