Kinetic models for the trading of goods

TL;DR

This paper develops kinetic equations to model the distribution and trading dynamics of two goods among many agents, incorporating economic principles and uncertainty, leading to insights on wealth distribution tails.

Contribution

It introduces a novel kinetic framework based on price theory and utility functions for modeling binary exchanges, including the derivation of linearized models and analysis of Pareto tails.

Findings

Solutions develop Pareto tails influenced by exchange uncertainty.

Linearized models exhibit Pareto index dependence on gain and variance ratios.

Fokker-Planck equations describe the limit of quasi-invariant trades.

Abstract

In this paper we introduce kinetic equations for the evolution of the probability distribution of two goods among a huge population of agents. The leading idea is to describe the trading of these goods by means of some fundamental rules in price theory, in particular by using Cobb-Douglas utility functions for the binary exchange, and the Edgeworth box for the description of the common exchange area in which utility is increasing for both agents. This leads to a Boltzmann-type equation in which the post-interaction variables depend in a nonlinear way from the pre-interaction ones. Other models will be derived, by suitably linearizing this Boltzmann equation. In presence of uncertainty in the exchanges, it is shown that the solution to some of the linearized kinetic equations develop Pareto tails, where the Pareto index depends on the ratio between the gain and the variance of the uncertainty. In particular, the result holds true for the solution of a drift-diffusion equation of Fokker-Planck type, obtained from the linear Boltzmann equation as the limit of quasi-invariant trades.