Economic growth and economic development 76

Introduction to Modern Economic Growth where x (t) ∈ Rn and G : Rn → Rn Let x∗ be a fixed point of the mapping G (·), i.e., x∗ = G (x∗ ) Such a x∗ is sometimes referred to as “an equilibrium point” of the difference equation (2.23) Since in economics, equilibrium has a different meaning, we will refer to x∗ as a stationary point or a steady state of (2.23) We will often make use of the stability properties of the steady states of systems of difference equations The relevant notion of stability is introduced in the next definition Definition 2.4 A steady state x∗ is (locally) asymptotically stable if there exists an open set B (x∗ ) x∗ such that for any solution {x (t)}∞ t=0 to (2.23) with x (0) ∈ B (x∗ ), we have x (t) → x∗ Moreover, x∗ is globally asymptotically stable ∗ if for all x (0) ∈ Rn , for any solution {x (t)}∞ t=0 , we have x (t) → x The next theorem provides the main results on the stability properties of systems of linear difference equations The Mathematical Appendix contains an overview of eigenvalues and some other properties of difference equations Theorem 2.2 Consider the following linear difference equation system (2.24) x (t + 1) = Ax (t) + b with initial value x (0), where x (t) ∈ Rn for all t, A is an n × n matrix and b is a n × column vector Let x∗ be the steady state of the difference equation given by Ax∗ + b = x∗ Suppose that all of the eigenvalues of A are strictly inside the unit circle in the complex plane Then the steady state of the difference equation (2.24), x∗ , is globally asymptotically stable, in the sense that starting from any x (0) ∈ Rn , ∗ the unique solution {x (t)}∞ t=0 satisfies x (t) → x Proof See Luenberger (1979, Chapter 5, Theorem 1) Ô Next let us return to the nonlinear autonomous system (2.23) Unfortunately, much less can be said about nonlinear systems, but the following is a standard local stability result 62