Economic growth and economic development 370

Introduction to Modern Economic Growth i(t) k(t)=0 i’(0) δk* Stable arm i’’(0) i(t)=0 k(0) k* k(t) Figure 7.1 Dynamics of capital and investment in the q-theory Theorem 7.18 Consider the following nonlinear autonomous differential equation (7.64) x˙ (t) = F [x (t)] where F : Rn → Rn and suppose that F is continuously differentiable, with initial value x (0) Let x∗ be a steady-state of this system, given by F (x∗ ) = Define A =∇F (x∗ ) , and suppose that m ≤ n of the eigenvalues of A have negative real parts and the rest have positive real parts Then there exists an open neighborhood of x∗ , B (x∗ ) ⊂ Rn and an m-dimensional manifold M ⊂ B (x∗ ) such that starting from any x (0) ∈ M, the differential equation (7.64) has a unique solution with x (t) → x∗ Put differently, these two theorems state that when only a subset of the eigenvalues have negative real parts, a lower-dimensional subset of the original space leads to stable solutions Fortunately, in this context this is exactly what we require, since i (0) should adjust in order to place us on exactly such a lower-dimensional subset of the original space 356