# Computer Animation of Money Exchange Models

Computer simulation of money exchange models is based on the paper by

A. A. Dragulescu and V. M. Yakovenko, "Statistical Mechanics of Money"
The European Physical Journal B 17, 723 (2000),  PDF,  cond-mat/0001432

Computer animation videos presented below were produced by Justin Chen, then an undergraduate student at Caltech, as a summer project in 2007 guided by Victor Yakovenko in Maryland.

## 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, transaction does not take place, and simulation continues with another pair of agents. 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 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 a wide variety of products and connections in modern economy.

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

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

Two models with different rules for the transferred amount Dm are shown below.

## A model with a fixed transferred amount

In this model, Dm=\$1 has the same value for all transactions. The initial balance of each agent is \$10. Simulations are performed with 500 agents and repeated 1000 times. The histogram shown in the animation is obtained by adding the histograms of all runs to produce a smoother distribution. Since Dm is small, money distribution evolves in a diffusive manner. The initial distribution first broadens into a symmetric Gaussian curve. Then, probability density starts to accumulate around m=0, which acts as the impenetrable boundary, because money balances of agents cannot go below zero. As a result, probability distribution eventually acquires a skewed (asymmetric) exponential shape.

## A model with random transferred amount

In each transaction, Dm is selected to be a random fraction of \$1000, which is the average amount of money per agent in this simulation. The simulation is performed with 5000 agents. The histogram shown in the animation is obtained by averaging the histograms of 10 runs of simulations to produce a smoother distribution.

For both models, 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. However, if a rule for money transfers does not have time-reversal symmetry, e.g. Dm is proportional to the money balance of an agent, other distributions may be obtained.