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Introduction to Modern Economic Growth Once more using the fact that xδ (t0 ) = xˆ (t0 ), this implies that ∂V (t0 , xˆ (t0 )) g (t0 , xˆ (t0 ) , yˆ (t0 )) ≥ ∂x ∂V (t0 , xˆ (t0 )) g (t0 , xδ (t0 ) , yδ (t0 )) f (t0 , xδ (t0 ) , yδ (t0 )) + ∂x for all t0 ∈ T and for all admissible perturbation pairs (xδ (t) , yδ (t)) Now defining (7.42) (7.43) f (t0 , xˆ (t0 ) , yˆ (t0 )) + λ (t0 ) ≡ ∂V (t0 , xˆ (t0 )) , ∂x Inequality (7.42) can be written as f (t0 , xˆ (t0 ) , yˆ (t0 )) + λ (t0 ) g (t0 , xˆ (t0 ) , yˆ (t0 )) ≥ f (t0 , xδ (t0 ) , yδ (t0 )) +λ (t0 ) g (t0 , xδ (t0 ) , yδ (t0 )) H (t0 , xˆ (t0 ) , yˆ (t0 )) ≥ H (t0 , xδ (t0 ) , yδ (t0 )) for all admissible (xδ (t0 ) , yδ (t0 )) Therefore, H (t, xˆ (t) , yˆ (t)) ≥ max H (t, xˆ (t) , y) y This establishes the Maximum Principle The necessary condition (7.34) directly follows from the Maximum Principle together with the fact that H is differentiable in x and y (a consequence of the fact that f and g are differentiable in x and y) Condition (7.36) holds by definition Finally, (7.35) follows from differentiating (7.41) with respect to x at all points of continuity of yˆ (t), which gives ∂f (t, xˆ (t) , yˆ (t)) ∂ V (t, xˆ (t)) + ∂x ∂t∂x ∂ V (t, xˆ (t)) ∂V (t, xˆ (t)) ∂g (t, xˆ (t) , yˆ (t)) + g (t, xˆ (t) , yˆ (t)) + = 0, ∂x ∂x ∂x for all for all t ∈ T Using the definition of the Hamiltonian, this gives (7.35) Ô 7.4 More on Transversality Conditions We next turn to a study of the boundary conditions at infinity in infinite-horizon maximization problems As in the discrete time optimization problems, these limiting boundary conditions are referred to as “transversality conditions” As mentioned 342