Kinetic modeling of knowledge and wealth dynamics in national and global markets

TL;DR

This work develops a kinetic theory for the co-evolution of knowledge and wealth in domestic and global markets by coupling binary wealth exchanges with knowledge dynamics, incorporating background learning and wealth-dependent risk. Through a Boltzmann formulation for a single population and a quasi-invariant limit to a Fokker–Planck equation, it reveals Pareto-tailed steady states for both wealth and knowledge. The framework is then extended to multiple populations with inter-country transfers, yielding FP systems with fast label-switching equilibria and macroscopic laws for population sizes, mean wealth, and mean knowledge in a two-country setting. The analysis highlights how migration and transfers shape long-time distributions and inequality, providing a mathematical bridge between micro-interactions and macroeconomic patterns. Potential applications include policy analysis and calibration against empirical data, with future work focusing on numerical simulations and more complex multi-country configurations.

Abstract

We propose a kinetic model to describe the dynamical evolution of wealth and knowledge in national and global markets, starting from a microscopic description of individual interactions. The model is built upon interaction rules that account for a strong interdependence between the microscopic variables, influencing agents' trading and saving propensities, knowledge acquisition, and the stochastic market effects. We begin with a domestic market scenario and extend the framework to international trade, incorporating the possibility of individual transfers between different countries. The dynamics of the system are described through Boltzmann-type equations, which allow for a detailed study of the evolution of the agent distribution in each country. In this context, we study the evolution of macroscopic quantities of the system, focusing on the number density of individuals, and the mean wealth and knowledge of each population, and we discuss these results in relation to existing models in the literature. Finally, under a quasi-invariant trading limit, we derive simplified Fokker-Planck type equations that reveal some emergent behaviors of the system, including the formation of Pareto tails in the long-term wealth and knowledge distributions.