# Distribution of Monetary Wealth

J. C. Sprott
Department of Physics, University of Wisconsin, Madison, WI 53706, USA
August 6, 2012

Many processes in nature produce probability distributions that approximate a power law. One example is the distribution of monetary wealth. Here is are some simple computer experiments that demonstrates how that might arise.

In the simulations, we take a model society with one million individuals and give them all \$100,000 which they are free to invest or spend in such a way that their wealth changes by an amount in the range of -10% to +10% each year. We then look at the distribution of wealth within the society after twenty years. The code was written in PowerBASIC and is available for download.

In the first simulation, the rate is chosen uniform random over the allowed range with no memory of the past. Thus each individual executes a random walk, some years gaining wealth and other years losing it. The distribution of wealth after twenty years is shown on a double logarithmic scale of two decades in the graph at the right. The result is approximately Gaussian as expected.
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In the second simulation, the rate is chosen uniform random over the allowed range but with the same rate for each person throughout the twenty year period. As one would expect, the rich get richer and the poor get poorer so that after twenty years the distribution of wealth is as shown in the graph at the right. The result is quite accurately a power law with a slope close to -1. The cut-offs at the high and low ends each expand by 10% each year.
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In the third simulation, the rate each year is half determined by the past and half chosen randomly, with a result as shown in the graph at the right. Interestingly, over a good portion of the range, the distribution of wealth is still quite accurately a power law with the same slope close to -1. As before, the range over which the distribution is a power law expands in time.
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This fourth simulation is the same as the second simulation except that the rates are chosen randomly from a Gaussian (normal) distribution with mean zero and standard deviation 10%. The results at the right show that the distribution of wealth after twenty years is not quite a power law, but it is far from a normal distribution and decreases monotonically over the indicated range.
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This fifth simulation is the so-called "yard sale model" in which each individual starts with \$100,000 and executes 1000 trades with another randomly chosen individual. In each trade, between 0 and 10% (randomly chosen) of the wealth of the poorer individual is exchanged between the two, half the time favoring the richer and half the time favoring the poorer. The resulting distribution is still roughly a power law despite the fact that all individuals are treated as equals, other than the fact that each trade is a larger fraction of the poorer's wealth. The PowerBASIC code for the yard sale model is available for download.
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These results provide evidence that power laws are a robust feature of such dynamical systems whenever there is some degree of correlation of the events that give rise to the non-uniform distribution. Any process that spreads things (such as wealth) out in a more-or-less uniform way on a logarithmic scale will produce an approximate power law with a slope of -1. Note that progressive taxation in which the wealthy are taxed at a higher rate than the poor does not alter the power law, only making it grow more slowly from an initial distribution in which everyone has the same wealth. If the initial condition is something close to a power law, it will remain so in the presence of any reasonable taxation and redistribution of wealth. There will always be some segment of the population whose wealth consistently increases faster than another segment. Of course individuals can change where they lie on the curve through hard work, good money management, and luck. Furthermore, even if all the individuals are identical and equally wealthy at the start but exchange wealth through random trades with no influx of wealth, an approximate power law distribution will occur after many such trades.

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