Introduction to Modern Economic Growth Integrating both sides over [0, t1 ] yields (7.17) Z t1 Z M (t, x (t) , λ (t)) dt ≤ 0 Moreover, we have (7.18) t1 Z M (t, xˆ (t) , λ (t)) dt+ t1 Mx (t, xˆ (t) , λ (t)) (x (t) − xˆ (t)) dt Mx (t, xˆ (t) , λ (t)) = Hx (t, xˆ (t) , yˆ (t) , λ (t)) = −λ˙ (t) , where the first line follows by an Envelope Theorem type reasoning (since Hy = from equation (7.13)), while the second line follows from (7.15) Next, exploiting the definition of the maximized Hamiltonian, we have Z t1 Z t1 M (t, x (t) , λ (t)) dt = W (x (t) , y (t)) + λ (t) g (t, x (t) , y (t)) dt, and Z t1 M (t, xˆ (t) , λ (t)) dt = W (ˆ x (t) , yˆ (t)) + Z t1 λ (t) g (t, xˆ (t) , yˆ (t)) dt Equation (7.17) together with (7.18) then implies (7.19) W (x (t) , y (t)) ≤ W (ˆ x (t) , yˆ (t)) Z t1 λ (t) [g (t, xˆ (t) , yˆ (t)) − g (t, x (t) , y (t))] dt + Z t1 − λ˙ (t) (x (t) − xˆ (t)) dt Integrating the last term by parts and using the fact that by feasibility x (0) = xˆ (0) = x0 and by the transversality condition (t1 ) = 0, we obtain ả Z t1 Z t1 · ˙λ (t) (x (t) − xˆ (t)) dt = − λ (t) x˙ (t) − xˆ (t) dt 0 Substituting this into (7.19), we obtain (7.20) W (x (t) , y (t)) ≤ W (ˆ x (t) , yˆ (t)) Z t1 + λ (t) [g (t, xˆ (t) , yˆ (t)) − g (t, x (t) , y (t))] dt ∙ ¸ Z t1 · λ (t) x˙ (t) − xˆ (t) dt + 327