Stable Balanced Expansion in Homogeneous Dynamic Models

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 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.