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Conceived and designed the experiments: JEF CL SP. Performed the experiments: CL. Analyzed the data: JEF CL. Contributed reagents/materials/analysis tools: SP. Wrote the paper: JEF CL SP.

Current address: Department of Ecology, Evolution and Behavior, University of Minnesota, St. Paul, Minnesota, United States of America

The authors have declared that no competing interests exist.

In many economies, wealth is strikingly concentrated. Entrepreneurs–individuals with ownership in for-profit enterprises–comprise a large portion of the wealthiest individuals, and their behavior may help explain patterns in the national distribution of wealth. Entrepreneurs are less diversified and more heavily invested in their own companies than is commonly assumed in economic models. We present an intentionally simplified individual-based model of wealth generation among entrepreneurs to assess the role of chance and determinism in the distribution of wealth. We demonstrate that chance alone, combined with the deterministic effects of compounding returns, can lead to unlimited concentration of wealth, such that the percentage of all wealth owned by a few entrepreneurs eventually approaches 100%. Specifically, concentration of wealth results when the rate of return on investment varies by entrepreneur and by time. This result is robust to inclusion of realities such as differing skill among entrepreneurs. The most likely overall growth rate of the economy decreases as businesses become less diverse, suggesting that high concentrations of wealth may adversely affect a country's economic growth. We show that a tax on large inherited fortunes, applied to a small portion of the most fortunate in the population, can efficiently arrest the concentration of wealth at intermediate levels.

The distribution of wealth is a fundamental property of how society is structured and has myriad economic, political, and social implications. The right to keep a large part of what one earns is one of the basic tenets of democratic capitalism, which provides incentives to invest and contribute to the productivity of the economy. However, large concentrations of wealth raise equity issues and may be incompatible with democracy itself; as put bluntly by U.S. Supreme Court Justice Louis Brandeis: “We can either have democracy in this country or we can have great wealth concentrated in the hands of a few, but we cannot have both.”

Models of the wealth distribution

Recent work has identified the importance of entrepreneurship in generating high concentrations of wealth

We analyze whether a simple individual-based stochastic model that includes compounding returns can generate the highly concentrated wealth distribution observed among entrepreneurs in real populations. Before considering more complicated explanations, we believe it is useful to understand whether wealth concentration could occur due to the effects of chance alone. The effects of chance on wealth distribution may be revealed in models that track the wealth of individual entrepreneurs and include stochasticity, as opposed to more commonly used aggregate general equilibrium models, which do not allow for effects of stochastic variation among individuals. We isolate the role of chance by starting with assumptions that favor equality of wealth and exclude other factors that could lead to the concentration of wealth. We assume that all individuals have equal talent and begin with the same amount of capital. We also assume that business success in one year is not correlated with future business success. After exploring the implications of these assumptions, we test whether our conclusions are robust to variations in assumptions.

Among entrepreneurs, the dynamics of wealth concentration are determined largely by growth (or loss) of business worth. Therefore, we track capital wealth and assume that labor income does not factor into the growth of capital wealth. In economic terms, another way to arrive at this assumption is if all existing capital is invested, all capital income is reinvested, and consumption is equal to labor income. This allows us to track capital without the need to track labor income or consumption.

We assume all entrepreneurs begin with equal capital, set to 1 unit of wealth. In each time period (

This simple model demonstrates that, with passing time, the proportion of wealth held by an arbitrarily small fraction of entrepreneurs asymptotically approaches 1–that is, a small proportion of entrepreneurs come to possess essentially all of the wealth. Given a rate of return

The integral in the numerator can represent the wealth in any segment of the population. Parameter

In this simplest model, the concentration of wealth occurs merely because some individuals are lucky by randomly receiving a series of high growth rates, and once they are ahead with exponentially growing capital, they tend to stay ahead. Because the variance in the sum of return rates is additive, over time the individuals with interest rates at the right tail of the ever-widening normal distribution come to dominate the wealth. Recall that it is the exponents that are normally distributed, not the amount of wealth, so that individuals at the high end of the distribution achieve exponentially greater fortunes. Because of the law of large numbers, our results are robust to changes in the assumption that returns on investment are drawn from a normal distribution. Annual returns drawn from any distribution that obeys the central limit theorem will give exponents whose sum approaches a normal distribution. Note that wage income, because it does not grow exponentially, is not expected to have similar wealth-concentrating effects.

The analytical results can be illustrated by simulations of individual-based models (

All simulations start with an even distribution of wealth. Unless otherwise noted, all simulations were run with 100,000 individuals and a 5% yearly average return on investment. Red lines show the analytically expected trajectories (Eqn. 1); points show the results from individual-based simulations. Three replicate simulations were run for each high variance simulation. (A) Higher variance among individual rates of return increases the rate of wealth concentration. (B) Inequality as measured by the Gini coefficient also increases over time. (C) Wealth concentrates even when the mean growth rate varies over time, such that in some years the total economy grows and in others the economy shrinks. Average annual rates of return were randomly drawn from a normal distribution with

The concentration of wealth has consequences for the most likely growth rate of the sum of all the entrepreneurs' capital, hereafter referred to, for brevity, as “the economy.” While the average return for an individual is

In simulations with

Thus, the wealth of the economy as a whole grows faster than the wealth of most individuals who make up the economy. In large populations with a diverse distribution of wealth, the most likely growth rate of the economy will approach the mean of

If centrally planned economies are viewed as having only one line of capital, then our results suggest that a centrally planned economy will likely have lower economic growth than an economy with diverse entrepreneurial activity. Ironically, the benefits derived from diversity in capitalist economies can be destroyed by a property inherent in the economy itself–the tendency of compounding chance, left unchecked, to concentrate wealth and effectively reduce the diversity that led to the high rates of economic growth in the first place. However, real capitalist and real centrally planned economies have many other differences that are also likely to contribute to differences in growth.

The purpose of our model is to illustrate how concentration of wealth arises naturally under the simplest conditions, not to realistically describe all the features of a free market economy. However, it is important to consider whether the tendency towards concentration of wealth observed in our model is likely to be swamped by modifications that incorporate additional features of real economies. We find that our conclusions are robust to several such modifications.

(1) Real economies have periods of growth and recession, such that in some years the average rate of return is high and in other years it is low or negative. We simulated conditions in which the rate of return for the market varied normally across years with a mean annual increase of 8 percent per annum and a standard deviation of 19. These parameters reflect the distribution of real inflation-corrected returns for the S&P 500 between 1871 and 2009. Allowing for this economy-wide temporal variation in growth did not affect the concentration of wealth in our model simulations (

(2) Consider a model with population increase where entrepreneurs may divide an inheritance among multiple offspring. Assume that an individual dies and that his or her estate is split evenly between two offspring on average every 80 years. Individual-based simulations of these conditions show that such division of inherited wealth does not significantly affect the rate of wealth concentration (

(3) Immigration can bring new entrepreneurs to a society. Simulations show that immigrant entrepreneurs with little individual wealth speed concentration of wealth (not shown), whereas immigrant entrepreneurs with mean wealth (which is much higher than median wealth) slightly slow the rate of wealth concentration (

(4) Individuals who are relatively successful entrepreneurs today are more likely to be successful in the future, such that there is temporal autocorrelation in returns. Temporal autocorrelation acts to speed up the concentration of wealth, because individuals with initially high rates of return are likely to continue to receive high rates of return (not shown).

Thus, many of the modifications we have discussed to make the model more realistic produce an even faster rate of wealth concentration than that seen in the simplest, purely random models.

Because entrepreneurs drive patterns of wealth concentration among the richest citizens, and because one of nine Americans is self-employed, patterns of wealth concentration observed across the whole population could be predicted by our simple model. Our model predicts a log-normal distribution of wealth. In contrast, the Italian economist Pareto suggested that wealth in all societies is distributed according to what has become known as Pareto's law

Solid line represents the best fit for the Pareto distribution (

Historically, wealth concentrations have varied widely. For example, in the United States, the top 1% of the population has owned between 15 and 40% of the wealth during the 20th Century

We model an inherited fortune tax with the following assumptions: (1) life expectancy is 80 years, (2) a tax is applied only to the inherited fortunes of the wealthiest fraction of the entrepreneurs, with wealth above a designated cutoff [

The results show that an inherited fortune tax effectively halts the concentration of wealth in our models (

This analysis illustrates that limiting inter-generational transfer of wealth through an inherited fortune tax or equivalent mechanism can moderate the concentration of wealth, based on our model of entrepreneurs in industrialized societies. Recent empirical work in small-scale societies found that concentration of wealth also occurs there and is positively correlated with the degree of inter-generational wealth transmission

Empirical patterns of wealth distribution show greater concentration of wealth than is predicted by current economic models, and this wealth is disproportionately concentrated in the hands of wealthy entrepreneurs. Our analysis demonstrates that an inexorable effect of chance can lead to unlimited concentrations of wealth in the hands of a few. This occurs whenever different entrepreneurs invest in different businesses, experience different rates of return on their investments, and reinvest their capital income. Thus, inevitable random fluctuations may help explain the high concentrations of wealth that are commonly observed empirically. Indeed, the log-normal distribution of wealth predicted by our model is a better fit to recent observed wealth distribution data than is the Pareto function.

Concentrations of wealth reduce the diversity of independent capital lines that can meaningfully contribute to business growth, thus reducing the most likely aggregated business growth. Progressively deepening disparities between modal and mean wealth, as in

We thank Holly MacCormick, Richard McGehee, Benjamin Kerr, Adrienne Keen, and Eville Gorham for crucial insights and discussions of this material, and an anonymous reviewer for suggestions that led to significant improvements in the manuscript.