Economic growth and economic development 356

This establishes the Maximum Principle.. Condition 7.36 holds by definition.. Using the definition of the Hamiltonian, this gives 7.35.. More on Transversality Conditions We next turn to

Introduction to Modern Economic Growth Once more using the fact that xδ(t0) = ˆx (t0), this implies that

f (t0, ˆx (t0) , ˆy (t0)) + ∂V (t0, ˆx (t0))

∂x g (t0, ˆx (t0) , ˆy (t0))≥ (7.42)

f (t0, xδ(t0) , yδ(t0)) + ∂V (t0, ˆx (t0))

∂x g (t0, xδ(t0) , yδ(t0)) for all t0 ∈ T and for all admissible perturbation pairs (xδ(t) , yδ(t)) Now defining (7.43) λ (t0)≡ ∂V (t0∂x, ˆx (t0)),

Inequality (7.42) can be written as

f (t0, ˆx (t0) , ˆy (t0)) + λ (t0) g (t0, ˆx (t0) , ˆy (t0)) ≥ f (t0, xδ(t0) , yδ(t0))

+λ (t0) g (t0, xδ(t0) , yδ(t0))

H (t0, ˆx (t0) , ˆy (t0)) ≥ H (t0, xδ(t0) , yδ(t0))

for all admissible (xδ(t0) , yδ(t0)) Therefore,

H (t, ˆx (t) , ˆy (t))≥ maxy H (t, ˆx (t) , 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 ˆy (t), which gives

∂f (t, ˆx (t) , ˆy (t))

∂2V (t, ˆx (t))

∂t∂x

2V (t, ˆx (t))

∂x2 g (t, ˆx (t) , ˆy (t)) +∂V (t, ˆx (t))

∂x

∂g (t, ˆx (t) , ˆy (t))

for all for all t∈ T Using the definition of the Hamiltonian, this gives (7.35) ¤

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 limit-ing boundary conditions are referred to as “transversality conditions” As mentioned

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