Economic growth and economic development 362

Introduction to Modern Economic Growth
where the first inequality exploits the fact that limt→∞ x˙ (t) > gˆ
x (t) and the second,
the fact that λ (t) ≡ exp(−ρt)µ(t) → λ and that xˆ(t) is increasing. But from (7.56),

˙
= 0, so that all the inequalities in this expression must
limt→∞ exp(−ρt)µ(t)x(t)
hold as equality, and thus (7.55) must be satisfied, completing the proof of the
theorem.

Ô

The proof of Theorem 7.14 also clarifies the importance of discounting. Without
discounting the key equation, (7.56), is not necessarily true, and the rest of the proof
does not go through.
Theorem 7.14 is the most important result of this chapter and will be used
in almost all continuous time optimizations problems in this book. Throughout,
when we refer to a discounted infinite-horizon optimal control problem, we mean
a problem that satisfies all the assumptions in Theorem 7.14, including the weak
monotonicity assumptions on f and g. Consequently, for our canonical infinitehorizon optimal control problems the stronger transversality condition (7.55) will
be necessary. Notice that compared to the transversality condition in the finitehorizon case (e.g., Theorem 7.1), there is the additional term exp (−ρt). This is because the transversality condition applies to the original costate variable λ (t), i.e.,
limt→∞ [x (t) λ (t)] = 0, and as shown above, the current-value costate variable µ (t)
is given by µ (t) = exp (ρt) λ (t). Note also that the stronger transversality condition
takes the form limt→∞ [exp (−ρt) µ (t) xˆ (t)] = 0, not simply limt→∞ [exp (−ρt) µ (t)] =
0. Exercise 7.17 illustrates why this is.
The sufficiency theorems can also be strengthened now by incorporating the
transversality condition (7.55) and expressing the conditions in terms of the currentvalue Hamiltonian:
Theorem 7.15. (Mangasarian Sufficient Conditions for Discounted
Infinite-Horizon Problems) Consider the problem of maximizing (7.46) subject
to (7.47) and (7.48), with f and g continuously differentiable and weakly monotone.

ˆ (x, y, µ) as the current-value Hamiltonian as in (7.50), and suppose that a
Define H
solution yˆ (t) and the corresponding path of state variable x (t) satisfy (7.51)-(7.53)
and (7.55). Suppose also that limt→∞ V (t, xˆ (t)) exists and that for the resulting
ˆ (x, y, µ) is jointly concave in (x, y) for all
current-value costate variable µ (t), H
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