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Introduction to Modern Economic Growth is a piecewise continuous function implies that the optimal control may include discontinuities, but these will be relatively “rare”–in particular, it will be continuous “most of the time” More important, the added generality of allowing discontinuities is somewhat superfluous in most economic applications, because economic problems often have enough structure to ensure that yˆ (t) is indeed a continuous function of time Consequently, in most economic problems (and in all of the models studied in this book) it will be sufficient to focus on the necessary conditions (7.34)-(7.36) It is also useful to have a different version of the necessary conditions in Theorem 7.9, which are directly comparable to the necessary conditions generated by dynamic programming in the discrete time dynamic optimization problems studied in the previous chapter In particular, the necessary conditions can also be expressed in the form of the so-called Hamilton-Jacobi-Bellman (HJB) equation Theorem 7.10 (Hamilton-Jacobi-Bellman Equations) Let V (t, x) be as defined in (7.31) and suppose that the hypotheses in Theorem 7.9 hold Then whenever V (t, x) is differentiable in (t, x), the optimal pair (ˆ x (t) , yˆ (t)) satisfies the HJB equation: (7.37) f (t, xˆ (t) , yˆ (t)) + ∂V (t, xˆ (t)) ∂V (t, xˆ (t)) + g (t, xˆ (t) , yˆ (t)) = for all t ∈ R ∂t ∂x Proof From Lemma 7.1, we have that for the optimal pair (ˆ x (t) , yˆ (t)), Z t f (s, xˆ (s) , yˆ (s)) ds + V (t, xˆ (t)) for all t V (t0 , x0 ) = t0 Differentiating this with respect to t and using the differentiability of V and Leibniz’s rule, we obtain ∂V (t, xˆ (t)) ∂V (t, xˆ (t)) + x˙ (t) = for all t ∂t ∂x Setting x˙ (t) = g (t, xˆ (t) , yˆ (t)) gives (7.37) f (t, x (t) , y (t)) + Ô The HJB equation will be useful in providing an intuition for the Maximum Principle, in the proof of Theorem 7.9 and also in many of the endogenous technology models studied below For now it suffices to note a few important features First, given that the continuous differentiability of f and g, the assumption that V (t, x) is differentiable is not very restrictive, since the optimal control yˆ (t) is piecewise 335