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 277