This paper establishes conditions for the asymptotic stability of balanced growth

paths in dynamic economic models as typical cases of homogeneous dynamical systems.

Results for common two-dimensional deterministic and stochastic models are

presented and further applications are discussed.

According to Solow & Samuelson (1953) balanced growth paths for deterministic

economies are induced by so-called Perron-Frobenius solutions defined by an eigenvalue

λ > 0 (the growth factor) and by an eigenvector $\bar{x}$ , a fixed point of the system

in intensive form. Contraction Lemma A.1 states for continuous deterministic systems

that convergence to a balanced path occurs whenever the product λ · M($\bar{x}$)

of the eigenvalue λ multiplied with the contractivity 0 < M($\bar{x}$) < 1 of the stable

eigenvector $\bar{x}$ of the intensive form is less than one. For λ·M($\bar{x}$) > 1 all unbalanced

orbits in the neighborhood of the balanced path diverge in spite of convergence

in intensive form. This confirms that convergence to a stable eigenvector of the

intensive form is only a necessary condition for convergence in state space.

In the stochastic case, the condition for asymptotic stability of balanced growth

paths (Theorem B.2) uses results from a stochastic analogue of the Perron-Frobenius

Theorem on eigenvalues and eigenvectors. Convergence (divergence) occurs if the

expectation of the product λ(ω) · M(ω) is less than (greater than) one, i.e. if the

product is mean contractive. This is equivalent to the condition that the sum of

the expectations of the logarithmic values of the stochastic growth rate and of the

contractivity factor of the intensive form are less than (greater than) zero.