Introduction to Modern Economic Growth Since the pair (ˆ x (t) , yˆ (t)) is optimal, we have that Z ∞ f (t, xˆ (t) , yˆ (t)) dt V (t0 , xˆ (t0 )) = t0 Z ∞ ≥ f (t, xδ (t) , yδ (t)) dt t0 t0 +∆t = Z f (t, xδ (t) , yδ (t)) dt + V (t0 + ∆t, xδ (t0 + ∆t)) , t0 where the last equality uses the fact that the admissible pair (xδ (t) , yδ (t)) is optimal starting with state variable xδ (t0 + ∆t) at time t0 + ∆t Rearranging terms and dividing by ∆t yields V (t0 + ∆t, xδ (t0 + ∆t)) − V (t0 , xˆ (t0 )) ≤− ∆t R t0 +∆t t0 f (t, xδ (t) , yδ (t)) dt ∆t for all ∆t ≥ Now take limits as ∆t → and note that xδ (t0 ) = xˆ (t0 ) and that R t0 +∆t f (t, xδ (t) , yδ (t)) dt = f (t, xδ (t) , yδ (t)) lim t0 ∆t→0 ∆t Moreover, let T ⊂ R+ be the set of points where the optimal control yˆ (t) is a continuous function of time Note that T is a dense subset of R+ since yˆ (t) is a piecewise continuous function Let us now take V to be a differentiable function of time at all t ∈ T , so that V (t0 + ∆t, xδ (t0 + ∆t)) − V (t0 , xˆ (t0 )) ∂V (t, xδ (t)) ∂V (t, xδ (t)) = + x˙ δ (t) , ∆t→0 ∆t ∂t ∂x ∂V (t, xδ (t)) ∂V (t, xδ (t)) = + g (t, xδ (t) , yδ (t)) , ∂t ∂x lim where x˙ δ (t) = g (t, xδ (t) , yδ (t)) is the law of motion of the state variable given by (7.29) together with the control yδ Putting all these together, we obtain that f (t0 , xδ (t0 ) , yδ (t0 )) + ∂V (t0 , xδ (t0 )) ∂V (t0 , xδ (t0 )) + g (t0 , xδ (t0 ) , yδ (t0 )) ≤ ∂t ∂x for all t0 ∈ T (which correspond to points of continuity of yˆ (t)) and for all admissible perturbation pairs (xδ (t) , yδ (t)) Moreover, from Theorem 7.10, which applies at all t0 ∈ T , (7.41) f (t0 , xˆ (t0 ) , yˆ (t0 )) + ∂V (t0 , xˆ (t0 )) ∂V (t0 , xˆ (t0 )) + g (t0 , xˆ (t0 ) , yˆ (t0 )) = ∂t ∂x 341