Rigorous results for the distribution of money on connected graphs

TL;DR

This paper provides rigorous mathematical proofs for the equilibrium money distributions in spatially explicit models of monetary transactions on connected graphs, confirming prior physics-based conjectures about their asymptotic behavior.

Contribution

It extends existing models to include local interactions on connected graphs and offers rigorous proofs for the equilibrium distributions, advancing theoretical understanding.

Findings

Confirmed exponential and gamma distribution convergence at equilibrium

Extended models to include local interactions on connected graphs

Provided rigorous proofs for conjectured distributions

Abstract

This paper is concerned with general spatially explicit versions of three stochastic models for the dynamics of money that have been introduced and studied numerically by statistical physicists: the uniform reshuffling model, the immediate exchange model and the model with saving propensity. All three models consist of systems of economical agents that consecutively engage in pairwise monetary transactions. Computer simulations performed in the physics literature suggest that, when the number of agents and the average amount of money per agent are large, the distribution of money at equilibrium approaches the exponential distribution for the first model, the gamma distribution with shape parameter two for the second model and a gamma distribution whose shape parameter depends on the saving propensity for the third model. The main objective of this paper is to give rigorous proofs of and extend these conjectures to generalizations of the first two models and a variant of the third model that include local rather than global interactions, i.e., instead of choosing the two interacting agents uniformly at random from the system, the agents are located on the vertex set of a general connected graph and can only interact with their neighbors.