Modelling a Market Society with Stochastically Varying Money Exchange Frequencies

Abstract

We propose and examine a model expressed by stochastic differential equations for the evolution of a complex system. We refer in particular to a market society, in which the state of each individual is identified by the amount of money at his/her disposal. The evolution of such a system over time is described by suitable equations that link the instantaneous changes in the probability of each state with the probable outcomes of pairwise interactions between elements of the system. In the context at hand, these pairwise interactions simply represent money exchanges, due to the sales and purchases of goods and services. In this paper, unlike the usual method in the literature, the interaction frequencies and the consequent probabilities of passing from one state to another are not considered as assigned once and for all but are supposed to be randomly variable. This choice, as also shown by several numerical simulations, seems likely to have fruitful consequences, especially for a more realistic representation of economic issues and phenomena.