Journal of Statistical Physics

, Volume 171, Issue 4, pp 727–743 | Cite as

Rigorous Results for the Distribution of Money on Connected Graphs

  • Nicolas Lanchier
  • Stephanie Reed


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 limiting distribution of money as time goes to infinity approaches the exponential distribution for the first model, the gamma distribution with shape parameter two for the second model and a distribution similar but not exactly equal to 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 these conjectures and also 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.


Interacting particle systems Econophysics Distribution of money Uniform reshuffling Immediate exchange model Uniform saving 

Mathematics Subject Classification

Primary 60K35 91B72 



The authors would like to thank two anonymous referees for their useful comments on a preliminary version of this work.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA

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