Economic growth and economic development 78

Introduction to Modern Economic Growth Proposition 2.5 Suppose that Assumptions and hold, then the steady-state equilibrium of the Solow growth model described by the difference equation (2.16) is globally asymptotically stable, and starting from any k (0) > 0, k (t) monotonically converges to k ∗ Proof Let g (k) ≡ sf (k) + (1 − δ) k First observe that g (k) exists and is always strictly positive, i.e., g0 (k) > for all k Next, from (2.16), we have (2.26) k (t + 1) = g (k (t)) , with a unique steady state at k∗ From (2.17), the steady-state capital k∗ satisfies δk∗ = sf (k∗ ), or (2.27) k∗ = g (k ∗ ) Now recall that f (·) is concave and differentiable from Assumption and satisfies f (0) ≥ from Assumption For any strictly concave differentiable function, we have (2.28) f (k) > f (0) + kf (k) ≥ kf (k) , where the second inequality uses the fact that f (0) ≥ Since (2.28) implies that δ = sf (k∗ ) /k ∗ > sf (k∗ ), we have g (k∗ ) = sf (k∗ ) + − δ < Therefore, g0 (k∗ ) ∈ (0, 1) Corollary 2.1 then establishes local asymptotic stability To prove global stability, note that for all k (t) ∈ (0, k ∗ ), k (t + 1) − k∗ = g (k (t)) − g (k∗ ) Z k∗ = − g0 (k) dk, k(t) < where the first line follows by subtracting (2.27) from (2.26), the second line uses the fundamental theorem of calculus, and the third line follows from the observation 64