# Getting the units right for the Pareto distribution of wealth: x = people, dollars, dollars per person?

When people talk about the 80-20 rule in the context of wealth, it is usually expressed, verbally, by stating that the 20 percent of the people with the highest wealth get 80 percent of the wealth, and the top 20 percent of the richest 20 get 80 percent of that, or 64% of total wealth, and so forth. Then there is hand-waving about how the Pareto distribution represents this constant proportionality.

However, when you graph a Pareto distribution in income, the x value, income, has to be measured either in dollars or in dollars per person. This implies that the pdf is measured in 1/dollars or in persons per dollar, respectively, so that the area of the integral can be a unitless probability.

If we measure income in dollars, the interpretation is straightforward. The CDF F(x) gives the probability that an arbitrary dollar will be found in the pocket of someone with an income below x. But this relates dollars to dollars, or a unitless proportion to dollars of income per person. Suppose I want to know about the income of groups of people, as in the popular explanation with which I opened. I have not been able to make either the dollars to (1/dollars) units or the dollars per person to persons per dollar units come out in a way that gives me a percent of people, or of people in a particular income range, instead of a percent of dollars in all dollars.

Is there someone who could give a clear explanation of how you set the units up so as to get the answer in terms of a ratio between the percent of people and the percent of wealth? Assuming that the value of alpha is set at the level that give you the 80/20 rule exactly, I'd like to see what, if this were chemistry, I'd call the stoichiometry: an arrangement of the problem so that the units on the left hand side are equal to the units on the RHS, and those units come out relating a proportion of people to a proportion of wealth, e.g. showing that the top 1/25 of the people get 64% of the wealth.

We can regard the wealth of an individual chosen at random in the population as a r.v. $X$ having density $f(x)$ over $(x_{\text{min}}, \,\infty)$ and survival $S(x):= \Pr\{X>x\}$. Assume that $n$ independent individuals $X_i$ are chosen at random. Their total wealth $S := \sum_{i=1}^n X_i$ has expectation $n\,\mathbb{E}[X]$. If $u \geq x_{\text{min}}$ is fixed, the total wealth for those with wealth exceeding $u$, i.e. such that $X_i > u$ is $S_u := \sum_{i=1}^n X_i 1_{\{X_i >u\}}$ with expectation $n\,\mathbb{E}[X 1_{\{X >u\}}]$. The ratio of sums $S_u/S$ writes as well as a ratio of means, and by the strong law of large numbers it tends almost surely for large $n$ to the ratio of expectations: $$\frac{S_u}{S} \underset{\text{a.s.}}{\to} \frac{ \mathbb{E}\left[X 1_{\{X >u\}}\right] }{\mathbb{E}\left[X\right]} = \frac{\int_{u}^\infty x f(x)\,\text{d}x}{\int_{x_{\text{min}}}^\infty x f(x)\,\text{d}x }.$$ This result is valid as soon as $X$ has a finite expectation, i.e. $\mathbb{E}\left[X\right] < \infty$. By choosing $u$ as the quantile of probability $p$, we get at the right hand side $1-L(p)$ where $L(p)$ is the value of the Lorenz curve, which is unitless. Provided that $n$ is large enough, $L(p)$ (e.g. $L(0.8)$) thus gives the percent of the total wealth owned by the fraction $p$ (e.g. $80\%$) of individuals with lowest wealth.
Now consider the Pareto distribution with shape $\alpha$ and scale $x_{\text{min}}$; it has survival $S(x) = (x_{\text{min}}/x)^\alpha$ with $\alpha>0$. It has an interesting stability property: for any $u \geq x_{\text{min}}$ the conditional distribution $X \, \vert \{X >u\}$ is Pareto with shape $\alpha$ and scale $u$. So the same `"concentration" of the total wealth applies to the sub-population of individuals with wealth $>u$. The expectation is finite for $\alpha > 1$ and the Lorenz curve is then given by $L(p) = 1 - (1-p)^{1-1/\alpha}$. Only one shape parameter leads to the 80-20 rule. It is obtained with by solving $L(0.8) = 0.2$ in $\alpha$, leading to $\alpha \approx 1.16$.