Economic growth and economic development 294

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.
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