Piecewise linear systems in economic models

Abstract

We construct a market where agents make decisions to buy or sell stocks and bonds. Agents make decisions based on a convex optimization problem with a risk-sensitive utility as the objective function. The solutions to each optimization problem are piecewise linear demands for the securities. We use a discrete time map for price dynamics of the stock such that price changes are proportional to excess demand. The sum of all agent stock demands is the stock excess demand and in our case also a piecewise linear function. The dynamics of the stock price are given by a discrete time piecewise linear system (PLS). We analyze stability of a one stock one bond market PLS, give sufficient conditions for global stability and characterize the origins of unstable complex price dynamics. The multi-security market problem is derived and the dependence on parameters of the basins of attraction for an equilibrium price and periodic orbits is discussed.