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

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∗ ) = 0. 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.
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