Computer Animation of Money Exchange Models

For a detailed description of the money exchange models, see the paper A. A. Dragulescu and V. M. Yakovenko, "Statistical mechanics of money", The European Physical Journal B, v. 17, pp. 723-729 (2000)pdf

The computer animation videos presented below were produced by Justin Chen, an undergraduate student at Caltech, as a summer project in 2007 under guidance of Victor Yakovenko.

Description of computer simulations

Initially, all agents are given the same amount of money.  After simulation starts, certain amounts of money Dm are repeatedly transferred from one randomly selected agent to another.  If the selected agent does not have enough money to pay Dm , the transaction does not take place, and simulation continues with another pair of agents.  The transfers of money are supposed to represent payments from one agent to another for certain products and services.  However, we do not keep track of the goods offered in exchange for money and only keep track of money balances of all agents.  The random character of money transfers is supposed to reflect the wide variety of products and connections in modern economy.

The histogram on the upper plot shows time evolution of the money distribution between the agents.  As time goes on, the initial d-function distribution of money broadens.  The vertical scale is adjusted with time, so that the histogram fits into the screen.  Eventually, the money distribution stabilizes at the exponential shape, shown by the reference line in red, when the system reaches statistical equilibrium.

The bottom plot shows time evolution of the entropy of the money distribution.  The entropy increases in time from the initial value 0 to the maximal value achieved when the system reaches statistical equilibrium.

Models with different rules for the transferred amount Dm were considered.

We observe that a narrow initial distribution, where all agents have the same amount of money, is unstable and evolves in time into a broad and skewed distribution, where many agents have low money balances and few agents have high money balances.  Eventually, the distribution of money reaches statistical equilibrium at the exponential shape (the Boltzmann-Gibbs distribution), in agreement with general principles of statistical physics and the principle of maximal entropy. 

For more information on this subject, see the papers at  When the rules for money transfers have different symmetries, e.g. Dm  is proportional to the money balance of an agent, other distributions may be obtained.

Last updated August 22, 2007

Home page of Victor Yakovenko