Moreover, let T ⊂ R+ be the set of points where the optimal control ˆy t is a continuous function of time.
Trang 1Introduction to Modern Economic Growth Since the pair (ˆx (t) , ˆy (t)) is optimal, we have that
V (t0, ˆx (t0)) =
Z ∞
t 0
f (t, ˆx (t) , ˆy (t)) dt
≥
Z ∞
t 0
f (t, xδ(t) , yδ(t)) dt
=
Z t 0 +∆t
t 0
f (t, xδ(t) , yδ(t)) dt + V (t0+ ∆t, xδ(t0+ ∆t)) , 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, ˆx (t0))
Rt 0 +∆t
t 0 f (t, xδ(t) , yδ(t)) dt
Now take limits as ∆t→ 0 and note that xδ(t0) = ˆx (t0) and that
lim
∆t→0
Rt 0 +∆t
t 0 f (t, xδ(t) , yδ(t)) dt
∆t = f (t, xδ(t) , yδ(t)) Moreover, let T ⊂ R+ be the set of points where the optimal control ˆy (t) is a
continuous function of time Note that T is a dense subset of R+ since ˆy (t) is a
piecewise continuous function Let us now take V to be a differentiable function of
time at all t∈ T , so that
lim
∆t→0
V (t0+ ∆t, xδ(t0+ ∆t))− V (t0, ˆx (t0))
∂V (t, xδ(t))
∂V (t, xδ(t))
∂x ˙xδ(t) ,
= ∂V (t, xδ(t))
∂V (t, xδ(t))
∂x g (t, xδ(t) , yδ(t)) , where ˙xδ(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))
∂x g (t0, xδ(t0) , yδ(t0))≤ 0 for all t0 ∈ T (which correspond to points of continuity of ˆy (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, ˆx (t0) , ˆy (t0)) + ∂V (t0, ˆx (t0))
∂V (t0, ˆx (t0))
∂x g (t0, ˆx (t0) , ˆy (t0)) = 0.
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