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Introduction to Modern Economic Growth maxy∈G(x) U (x, y) + βV (y) is strictly increasing This establishes that T V (y) Ô C00 (X) and completes the proof proof of Theorem 6.6 From Corollary 6.1, Π (x) is single-valued, thus a function that can be represented by π (x) By hypothesis, π(x (0)) ∈ IntG(x (0)) and from Assumption 6.2 G is continuous Therefore, there exists a neighborhood N (x (0)) of x (0) such that π(x (0)) ∈ IntG(x), for all x ∈ N (x (0)) Define W (·) on N (x (0)) by W (x) = U[x, π(x (0))] + βV [π(x (0))] In view of Assumptions 6.3 and 6.5, the fact that V [π(x (0))] is a number (independent of x), and the fact that U is concave and differentiable, W (·) is concave and differentiable Moreover, since π(x (0)) ∈ G(x) for all x ∈ N (x (0)), it follows that (6.17) W (x) ≤ max [U(x, y) + βV (y)] = V (x), y∈G(x) for all x ∈ N (x (0)) with equality at x (0) Since V (·) is concave, −V (·) is convex, and by a standard result in convex analysis, it possesses subgradients Moreover, any subgradient p of −V at x (0) must satisfy p · (x − x (0)) ≥ V (x) − V (x (0)) ≥ W (x) − W (x (0)), for all x ∈ N (x (0)) , where the first inequality uses the definition of a subgradient and the second uses the fact that W (x) ≤ V (x), with equality at x (0) as established in (6.17) This implies that every subgradient p of −V is also a subgradient of −W Since W is differentiable at x (0), its subgradient p must be unique, and another standard result in convex analysis implies that any convex function with a unique subgradient at an interior point x (0) is differentiable at x (0) This establishes that −V (·), thus V (·), is differentiable as desired The expression for the gradient (6.4) is derived in detail in the next section Ô 6.5 Fundamentals of Dynamic Programming In this section, we return to the fundamentals of dynamic programming and show how they can be applied in a range of problems 280