### A Simple Mathematical Model of Economic Inequality

#### by David Radcliffe

**Most people acknowledge that economic inequality has been increasing in the United States. Statistics bear this out. In 2007, the top 1% of families received 24% of the nation’s income, but their share during the 1960s and 1970s was about 9%.**

Liberals and conservatives have vastly different explanations for economic inequality. Liberals generally believe that the rules of the game favor the wealthy, while the poor face discrimination and diminished opportunities. Conservatives generally believe that people are wealthy due to superior ability or effort, and that inequality is the natural result of giving people the opportunity to achieve their potential.

Both of these explanations for inequality are deterministic; they discount the role of blind luck. But I claim that inequality can arise purely by chance, and I will explain how this occurs.

Let us begin with a simple thought experiment. Imagine a society in which all citizens begin with equal wealth of 100 units. Every year, half of the population enjoys a 20% increase in wealth, and the other half suffers a 20% decrease in wealth. We suppose that the process is completely random, and it does not favor the rich over the poor. In this imaginary society, the (expected) average wealth remains constant at 100 units, but the median wealth will gradually decline.

Years |
Median wealth |

0 | 100.0 |

10 | 81.5 |

20 | 66.5 |

30 | 54.2 |

40 | 44.2 |

50 | 36.0 |

The reason for this decline is that the median person’s wealth will experience the same number of up years and down years; but an increase of 20% followed by a decline of 20% results in a *decline* of 4%, since 1.20 × 0.80 = 0.96. Nevertheless, the society’s total wealth remains the same, because the increases and decreases in wealth cancel each other out. Consequently, the majority of the society’s wealth is concentrated in fewer hands each year.

Let’s generalize this model. We suppose that the distribution of wealth in a society is specified by a random variable X. (This does not mean that wealth is distributed randomly, only that it can vary between individuals.) We need a mathematical measure of the inequality of a distribution, and we choose to define it by the following formula.

I(X) = E[X^{2}]/(E[X])^{2}

What does this formula mean? E[X] denotes the expected value (or average) of X, so we are dividing the average value of X^{2} by the square of the average wealth. The quantity I(X) is equal to 1 in the case of perfect equality, and it is greater than 1 otherwise. Larger values of I(X) indicate greater income inequality. An equivalent formula is

I(X) = 1 + (σ/μ)^{2
}

where σ is the standard deviation and μ is the mean.

Now, we suppose that each person’s wealth increases or decreases by a random percentage. This is equivalent to multiplying X by a positive random variable Y. We assume that the percentage of increase in wealth is independent of a person’s current wealth; that is, X and Y are independent. But if X and Y are independent, then X^{2} and Y^{2} are also independent, which implies that

I(XY) = E[X^{2}Y^{2}]/(E[XY])^{2} = E[X^{2}]/(E[X])^{2} × E[Y^{2}]/(E[Y])^{2} = I(X) I(Y).

But I(Y) > 1, and so I(XY) > I(X), which means that social inequality has increased by a factor of I(Y).

There are many objections that could be raised to this analysis. The model is too simple to capture the complexity of a national economy, and I make no claims to the contrary. If the model were literally true, then we would expect the distribution of wealth to follow a log-normal distribution; but as Ben Goldacre observed, the true distribution more closely resembles a power law. But I do argue that this simple model demonstrates that, in the absence of other factors, random chance will inexorably lead to unequal distribution of wealth.

A more sophisticated treatment of this idea is discussed in the article Entrepreneurs, Chance, and the Deterministic Concentration of Wealth by Joseph E. Fargione, Clarence Lehman, and Stephen Polasky. See this article for a non-technical summary.

I re-read the article by Fargione et al, The authors found that the distribution of wealth in 1995 was much closer to a log-normal distribution (r^2 = 0.98) than to a power-law distribution (r^2 = 0.87). This agrees with the predictions of my model.

Very interesting. Not that it means anything, but I was curious how this would compare to the actual income distribution (which I have looked at in the past). So I split the population into income buckets using a binomial distribution (41 income buckets are created in 40 years) & looked at the distribution of the wealth.

IRS records show that for 2008 Adj. Gross Income, the top 5% had 35% of AGI, the top 10% had 46%, the top 25% had 67%, and the top 50% had 87%. If you run your scenario for 40 years, the top 5% will have about 35% of the income, the top 10% will have about 50%, the top 25% will have about 70%, and the top 50% will have about 90%. The ratio of median AGI to mean AGI is 33K to 60K or .55. Running your scenario for 40 years gives a ratio of about .44. Interesting that the distributions are so similar.