Economic growth and economic development 291

Introduction to Modern Economic Growth
for this to show how this conclusion can be reached either by looking at Problem
A1 or at Problem A2, and then exploiting their equivalence. The first proof is more
abstract and works directly on the sequence problem, Problem A1.
Proof of Theorem 6.3. (Version 1) Consider Problem A1. The choice set of
this problem Φ (0) is a subset of X ∞ (infinite product of X). From Assumption 6.1,
X is compact. By Tychonof’s Theorem (see Mathematical Appendix), the infinite
product of a sequence of compact sets is compact in the product topology. Since
again by Assumption 6.1, G (x) is compact-valued, the set Φ (x (0)) is bounded. A
bounded subset of a compact set, here X ∞ , is compact. From Assumption 6.2 and
the fact that β < 1, the objective function is continuous in the product topology.
Then from Weierstrass’ Theorem, an optimal path x (0) exists.

Ô

Proof of Theorem 6.3. (Version 2) Consider Problem A2. In view of
Assumptions 6.1 and 6.2, there exists some M < ∞, such that |U(x, y)| < M for

all (x, y) ∈ XG . This immediately implies that |V ∗ (x)| ≤ M/(1 − β), all x ∈ X.

Consequently, V ∗ ∈ C (X), where C (X) denotes the set of continuous functions

defined on X, endowed with the sup norm, kf k = supx∈X |f (x)|. Moreover, all

functions in C (X) are bounded since they are continuous and X is compact.
Over this set, define the operator T
(6.15)

T V (x) = max U(x, y) + βV (y).
y∈G(x)

A fixed point of this operator, V = T V , will be a solution to Problem A2. We first
prove that such a fixed point (solution) exists. The maximization problem on the
right hand side of (6.15) is one of maximizing a continuous function over a compact
set, and by Weierstrass’s Theorem, it has a solution. Consequently, T is well defined.
It can be verified straightforwardly that it satisfies Blackwell’s sufficient conditions
for a contraction in Theorem 6.9 (see Exercise 6.6). Therefore, applying Theorem
6.7, a unique fixed point V ∈ C(X) to (6.15) exists and this is also the unique
solution to Problem A2. Now consider the maximization in Problem A2. Since

U and V are continuous and G (x) is compact-valued, we can apply Weierstrass’s
Theorem once more to conclude that y ∈ G (x) achieving the maximum exists. This
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