Economic growth and economic development 356

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)) ∂ 2 V (t, xˆ (t))
+
∂x
∂t∂x
2
∂ V (t, xˆ (t))
∂V (t, xˆ (t)) ∂g (t, xˆ (t) , yˆ (t))
+
g (t, xˆ (t) , yˆ (t)) +
= 0,
2
∂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
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