Global dynamics in a non-linear model of the equity ratio

A model for firms’ financial conditions is proposed, which ultimately reduces to a two-dimensional non-invertible map in the variables mean and variance of the equity ratio. The possible dynamics of the model and the global behaviour are investigated. We describe the mechanism of bifurcations leading to fractalization of the basins and/or fractalization of their boundaries, showing how a locally stable attractor may be almost globally unstable. Multistability is also investigated. Two, three or four co-existing attractors have been found and we describe the mechanism of bifurcations leading their basins to become chaotically intermingled, and thus to unpredictability of the asymptotic state in a wide region. The knowledge of such regimes, besides those associated with simple dynamics, may be of help for the operators. While the use of the technical tools we propose to study the global dynamics and bifurcations may be of help for further investigations. ” 2000 Elsevier Science Ltd. All rights reserved. In this paper we present a model of fluctuating growth in which firms’ financial conditions play a crucial role. Our analysis starts from the distribution of firms according to their equity ratio, that is the ratio of the equity base or net worth to the capital stock, a proxy of financial robustness. We identify two dynamic laws for the mean and the variance of this distribution. The motion over time of the average equity ratio is the engine of growth and fluctuations. The dynamic pattern of the dispersion of the distribution, captured by the variance, however, interacts with evolution of the average equity ratio. Given the non-linear nature of the map which describes the laws of motion of the mean and the variance of the equity ratio, a wide range of dynamic patterns are possible. Fixed points or periodic orbits, attracting closed invariant curves and thin annular chaotic areas wide chaotic areas or explosions may occur. For quite plausible values of the parameters which characterize the map, the dynamics of the equity ratio can be regular or chaotic, and the motion of capital and output can be characterized as a process of fluctuating growth, although, as we shall see, often very sensitive to small perturbations. The goal of the present paper is to show how the global properties (deriving from the structure of the basins, their boundaries and the critical curves of non-invertible two-dimensional maps) may be used to understand the dynamic behaviour of the model, especially when analytical results are not accessible, as in our case, where not only the equilibrium values, but also the number of existing fixed points, cannot be explicitly known.