Economic growth and economic development 77

Introduction to Modern Economic Growth Theorem 2.3 Consider the following nonlinear autonomous system (2.25) x (t + 1) = G [x (t)] with initial value x (0), where G :Rn → Rn Let x∗ be a steady state of this system, i.e., G (x∗ ) = x∗ , and suppose that G is continuously differentiable at x∗ Define A ≡∇G (x∗ ) , and suppose that all of the eigenvalues of A are strictly inside the unit circle Then the steady state of the difference equation (2.25) x∗ is locally asymptotically stable, in the sense that there exists an open neighborhood of x∗ , B (x∗ ) ⊂ Rn such that starting from any x (0) ∈ B (x∗ ), we have x (t) → x∗ Proof See Luenberger (1979, Chapter 9) Ô An immediate corollary of Theorem 2.3 the following useful result: Corollary 2.1 Let x (t) , a, b ∈ R, then the unique steady state of the linear difference equation x (t + 1) = ax (t) + b is globally asymptotically stable (in the sense that x (t) → x∗ = b/ (1 − a)) if |a| < Let g : R → R be a continuous function, differentiable at the steady state x∗ , defined by g (x∗ ) = x∗ Then, the steady state of the nonlinear difference equation x (t + 1) = g (x (t)), x∗ , is locally asymptotically stable if |g0 (x∗ )| < Moreover, if |g (x)| < for all x ∈ R, then x∗ is globally asymptotically stable Proof The first part follows immediately from Theorem 2.2 The local stability of g in the second part follows from Theorem 2.3 Global stability follows since |x(t + 1) − x∗ | = |g(x(t)) − g(x∗ )| ¯Z ¯ ¯ x(t) ¯ ¯ ¯ = ¯ g0 (x)dx¯ ¯ x∗ ¯ < |x(t) − x∗ |, where the last inequality follows from the hypothesis that |g (x)| < for all x Ô R We can now apply Corollary 2.1 to the equilibrium difference equation of the Solow model, (2.16): 63