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)] = 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 348