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.

Initially, all agents are given the same amount of money. After simulation starts, certain amounts of money D*m*
are repeatedly transferred from one randomly selected agent to another. If
the selected agent does not have enough money to pay D*m*,
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 D*m* are shown below.

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. D*m* is proportional to the money balance of an agent, other distributions may be obtained.

For more information on this subject, see https://physics.umd.edu/~yakovenk/econophysics/.

*Last update
2023-6-18*

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