Economic growth and economic development 369

Introduction to Modern Economic Growth
The analysis of dynamics in this case requires somewhat different ideas than
those used in the basic Solow growth model (cf., Theorems 2.4 and 2.5). In particular, instead of global stability in the k-i space, the correct concept is one of
saddle-path stability. The reason for this is that instead of an initial value constraint, i (0) is pinned down by a boundary condition at “infinity,” that is, to satisfy
the transversality condition,
lim exp (−rt) q (t) k (t) = 0.

t→∞

This implies that in the context of the current theory, with one state and one control
variable, we should have a one-dimensional manifold (a curve) along which capitalinvestment pairs tend towards the steady state. This manifold is also referred to
as the “stable arm”. The initial value of investment, i (0), will then be determined
so that the economy starts along this manifold. In fact, if any capital-investment
pair (rather than only pairs along this one dimensional manifold) were to lead to
the steady state, we would not know how to determine i (0); in other words, there
would be an “indeterminacy” of equilibria. Mathematically, rather than requiring
all eigenvalues of the linearized system to be negative, what we require now is saddlepath stability, which involves the number of negative eigenvalues to be the same as
the number of state variables.
This notion of saddle path stability will be central in most of growth models
we will study. Let this now make these notions more precise by considering the
following generalizations of Theorems 2.4 and 2.5:
Theorem 7.17. Consider the following linear differential equation system
(7.63)

x˙ (t) = Ax (t) +b

with initial value x (0), where x (t) ∈ Rn for all t and A is an n × n matrix. Let
x∗ be the steady state of the system given by Ax∗ + b = 0. Suppose that m ≤ n of
the eigenvalues of A have negative real parts. Then there exists an m-dimensional

manifold M of Rn such that starting from any x (0) ∈ M, the differential equation

(7.63) has a unique solution with x (t) → x∗ .
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