Non-conservative kinetic model of wealth exchange with saving of production

Abstract

In this paper we propose a non-conservative kinetic model of wealth exchange with saving of production as an extension of the Chakraborti–Chakrabarti model of money exchange. Using microeconomic arguments, we achieve rules of interaction between economic agents that depend on two exogenous parameters, the exchange aversion of the agents (λ) and the saving of production (s), such that in the limit s = 0, these rules can be reduced to the ones of the Chakraborti–Chakrabarti model. The non-conservative dynamics are approached analytically through a mean field approximation and the Boltzmann kinetic equation. Both approximations allow us to compute a theoretical rate of exponential growth (g) and to fit the emergent distributions of wealth to a gamma probability density function, in such a way that g, the fit parameters and the Gini index can be expressed analytically in terms of λ and s. In general, the emergent distributions do not reach a stationary state, however it is possible to study the emergence of self-similar distributions that hold the gamma pattern and maximize the Shannon entropy. With the purpose of addressing labor income, we explore additionally the effect of salary income in the model by defining a two-class structure where population is separated into workers and producers. This assumption leads to an emergent rate of economic growth g̃_e. The macroeconomic implications of this model are studied by means of the wealth/income ratio, which can be predicted as s∕g̃_e, in accordance with the Solow model of economic growth. The results in this paper allow to tie some of the important facts of the modern economic speech, as well as the microeconomic theory, with some methods and ideas developed in the context of non-conservative exchange models.

Graphical abstract