Stability—independent of economic structure? A prototype analysis

Abstract

This paper investigates the price-quantity disequilibrium dynamics of production economies as sketched in the writings of Walras when production enters the scene. The resulting cross-dual dynamic structure closely resembles the classical process of equalizing profit rates when the profitability consequences of price movements are taken into account (and the tâtonnement procedure of Walrasian economics is modified in an appropriate way). The paper therefore indirectly considers some basic aspects of classical dynamics as well. It starts from a formal version of these dynamics recently analyzed by Mas-Colell in the framework of general equilibrium models. Modifications of cross-dual dynamics are then proposed which greatly improve its stability properties. This is done by assuming that the direction and the magnitude of changes in existing disequilibria will also influence future price and quantity movements. It is shown that such additional forces—if sufficiently pronounced—will make all equilibria of a given economy asymptotically stable. Furthermore, this integration of additional economic forces gives rise to a dynamic system which provides a new example for the abstract discussion of so-called Generalized Newton Methods, i.e. depending on the strength of the derivative terms the new dynamics will have fairly ‘universal stability’ properties. The final section of the paper then investigates the stability properties of this extended cross-dual dynamics when the above derivative forces are differentiated with respect to their strength (only changes in profitability are important or only changes in excess supply do matter). Such situations—though more plausible—are shown to be less ‘universally stable’ and thus demand further investigation.