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Introduction to Modern Economic Growth discussion, the decision-maker considered updating his or her plan, with the payoff function being potentially different after date t1 (at least because bygones were bygones) In contrast, here the payoff function remains constant The issue of time consistency is discussed further in Exercise 7.19 We now state one of the main results of this chapter Theorem 7.9 (Infinite-Horizon Maximum Principle) Suppose that problem of maximizing (7.28) subject to (7.29) and (7.30), with f and g continuously differentiable, has a piecewise continuous solution yˆ (t) with corresponding path of state variable xˆ (t) Let H (t, x, y, λ) be given by (7.12) Then the optimal control yˆ (t) and the corresponding path of the state variable xˆ (t) are such that the Hamiltonian H (t, x, y, λ) satisfies the Maximum Principle, that H (t, xˆ (t) , yˆ (t) , λ (t)) ≥ H (t, xˆ (t) , y, λ (t)) for all y (t) , for all t ∈ R Moreover, whenever yˆ (t) is continuous, the following necessary con- ditions are satisfied: (7.34) Hy (t, xˆ (t) , yˆ (t) , λ (t)) = 0, (7.35) λ˙ (t) = −Hx (t, xˆ (t) , yˆ (t) , λ (t)) , (7.36) x˙ (t) = Hλ (t, xˆ (t) , yˆ (t) , λ (t)) , with x (0) = x0 and lim x (t) ≥ x1 , t→∞ for all t ∈ R+ The proof of this theorem is relatively long and will be provided later in this section.5 Notice that the optimal solution always satisfies the Maximum Principle In addition, whenever the optimal control, yˆ (t), is a continuous function of time, the conditions (7.34)-(7.36) are also satisfied This qualification is necessary, since we now allow yˆ (t) to be a piecewise continuous function of time The fact that yˆ (t) 5The reader may also wonder when an optimal piecewise continuous solution will exist as hypothesized in the theorem Unfortunately, the conditions to ensure that a solution exists are rather involved The most straightforward approach is to look for Lebesgue integrable controls, and impose enough structure to ensure that the constraint set is compact and the objective function is continuous In most economic problems there will be enough structure to ensure the existence of an interior solution and this structure will also often guarantee that the solution is continuous 334