Does an Unstable Keynesian Unemployment Equilibrium in a Non-Walrasian Dynamic Macroeconomic Model Imply Chaos?

Abstract

Simonovits (1982) introduced a non-Walrasian dynamic macromodel which is based on the disequilibrium model with inventory dynamics due to Honkapohja and Ito (1980). This model describes transactions on two markets (one market for labor and one market for goods) and contains five parameters. Obviously, there is an equilibrium which is the so-called Keynesian unemployment equilibrium point. For suitable choices of the parameters, this equilibrium point is locally stable and, according to Simonovits (1982), the Keynesian unemployment equilibrium point is also globally stable, i.e., the trajectories of all initial values converge to the equilibrium point. For many parameter values the equilibrium point is unstable; this is in fact the case when the product of the two eigenvalues at the equilibrium point is greater than 1. Simonovits' (1982) conjecture says: If the Keynesian unemployment equilibrium point is unstable, then there is First, we recall some definitions of chaos. We restrict our attention to systems whose trajectories are bounded. A first definition is the following. A system is chaotic if there exist infinitely many periodic points with different period and if there is an uncountable set of aperiodic points. This kind of chaos is nowadays called Li-Yorke chaos; see Li and Yorke (1975) and Diamond (1976). Before turning to a second definition, we need the notion of dependence on initial values. A system may be said to have sensitive dependence on initial values if we can find, with probability p (where 0 < p < 1), a point x such that for every open neighborhood U of x there is a point y in U such that the trajectories of x and y will not be close