By Kenneth Hurlstone Jackson
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Additional resources for Aislinge Meic Con Glinne
The following proposition gives suﬃcient conditions for a feasible path from x0 to be optimal. Observe that one of the conditions is that X is a subset of the positive orthant Rn+ . 9 (Mangasarian Lemma). Assume H1, H’2, H’3, H4, X ⊂ Rn+ , X contains 0, F is diﬀerentiable in int(graph(Γ )) and F2 ≤ 0. If x is a feasible path from x0 which satisﬁes ∀t, (xt , xt+1 ) ∈ int(graph(Γ )) and (i) Euler equation: ∀t, F2 (xt , xt+1 ) + βF1 (xt+1 , xt+2 ) = 0 and (ii) Transversality Condition: lim β t F1 (xt , xt+1 ) · xt = 0 t→∞ then x is optimal.
One can remark that the proof of the diﬀerentiability of V is fairly simple. g. Araujo and Scheinkman , Araujo , Santos , Montrucchio , Blot and Crettez ). The One Dimension Case In the one-sector models, the optimal paths always converge to a steady state. In the introduction, we have given an example of a two-sector model that can be transformed into a one dimensional optimal growth model. However the optimal path may be non-monotonic. 11. Assume X = R+ and H1, H’2, H’3, H4. Assume also that the function F is strictly concave with respect to the second variable.
Now, since xt+1 ∈ Γ (xt ) and since V satisﬁes the Bellman equation, we have V (xt ) ≥ F (xt , xt+1 ) + βV (xt+1 ), ∀t ≥ 0. Now, let x ∈ Π(x0 ) satisfy ∀t ≥ 0, V (xt ) = F (xt , xt+1 ) + βV (xt+1 ). Then, by induction, we get: T β t F (xt , xt+1 ) + β T +1 V (xT +1 ). ∀T, V (x0 ) = t=0 2. 3(iii), limT →+∞ β T +1 V (xT +1 ) = 0. , the sequence x is optimal. We have proved statement (i). (ii) If x is optimal, then by statement (i) we have V (xt ) = F (xt , xt+1 ) + βV (xt+1 ). That means that xt+1 ∈ G(xt ).
Aislinge Meic Con Glinne by Kenneth Hurlstone Jackson