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Introduction to Modern Economic Growth However, in general the returns to the holding an asset come not only from dividends but also from capital gains or losses (appreciation or depreciation of the asset) In the current context, this is equal to V˙ (x) Therefore, instead of ρV (x) = d, we have ρV (x) = d + V˙ (x) Thus, at an intuitive level, the Maximum Principle amounts to requiring that the maximized value of dynamic maximization program, V (x), and its rate of change, V˙ (x), should be consistent with this no-arbitrage condition 7.3.3 Proof of Theorem 7.9* In this subsection, we provide a sketch of the proof of Theorems 7.9 A fully rigorous proof of Theorem 7.9 is quite long and involved It can be found in a number of sources mentioned in the references below The version provided here contains all the basic ideas, but is stated under the assumption that V (t, x) is twice differentiable in t and x As discussed above, the assumption that V (t, x) is differentiable in t and x is not particularly restrictive, though the additional assumption that it is twice differentiable is quite stringent The main idea of the proof is due to Pontryagin and co-authors Instead of smooth variations from the optimal pair (ˆ x (t) , yˆ (t)), the method of proof considers “needle-like” variations, that is, piecewise continuous paths for the control variable that can deviate from the optimal control path by an arbitrary amount for a small interval of time Sketch Proof of Theorem 7.9: Suppose that the admissible pair (ˆ x (t) , yˆ (t)) is a solution and attains the maximal value V (0, x0 ) Take an arbitrary t0 ∈ R+ Construct the following perturbation: yδ (t) = yˆ (t) for all t ∈ [0, t0 ) and for some sufficiently small ∆t and δ ∈ R, yδ (t) = δ for t ∈ [t0 , t0 + ∆t] for all t ∈ [t0 , t0 + ∆t] Moreover, let yδ (t) for t ≥ t0 +∆t be the optimal control for V (t0 + ∆t, xδ (t0 + ∆t)), where xδ (t) is the value of the state variable resulting from the perturbed con- trol yδ , with xδ (t0 + ∆t) being the value at time t0 + ∆t Note by construction xδ (t0 ) = xˆ (t0 ) (since yδ (t) = yˆ (t) for all t ∈ [0, t0 ]) 340