Numerical Solution of Huge Sets of Nonlinear Differential Equations: The Coupling of Open Economic Systems

Kai Brandt
II. Institute of Theoretical Physics, University of Stuttgart
kai@theo2.physik.uni-stuttgart.de

Abstract

Maastricht is only one catchword which emphasizes the fast growing importance of interactions between economic areas. On the other hand synergetics is one catchword which represents the progress in the exploration of complex, nonlinear, dynamic systems. Here, we make use of one evolution in order to investigate the other. This is one of the first attempts to model spatial economic proceedings by a nonlinear, dynamic, descriptive, and behavioural decision model. It also illustrates the importance of time and flexibility in economic behaviour. Special emphasis is given to a modular construction and a big variability of the model frame. The master equation approach by WEIDLICH and HAAG and an extension of the frequency dependence which has been defined by LEIBENSTEIN and ARTHUR are used as tools. Eventually, model consistent exchange streams between the single regions are formulated for a multiregional application. Thereby we can investigate in particular the meaning of frequency dependence in a spatial context. We will find a variety of new manifestations of this dependence which do not occur in connection with single economies.

In the simulations, the model is specified to two or three regions with two commodities each. This yields a 16- or 30-dimensional set of differential equations. They are integrated numerically, comparing different integrators like Runge-Kutta, predictor-corrector, Burlisch-Stoer, etc. Different results concerning performance and robustness against inherent instabilities can be found.Particularly, the mutual influences regarding business cycles and long waves are investigated and the affiliated economic coupling is analysed.

For more detailed information, please refer to BRANDT 1995 and HAAG 1996.