Pareto Distribution

The Pareto distribution is a probability distribution for quantities where a small number of very large values make up a big share of the total. It is the best-known example of a power law: the chance of seeing a value above some level falls off slowly, in proportion to a power of that level, rather than collapsing quickly as in the normal distribution. Economists and statisticians use it to describe top incomes and wealth, the size of cities and firms, the returns of venture capital portfolios and the cost of large insurance claims.

It is named after the Italian economist and engineer Vilfredo Pareto, who studied income data from several countries in the 1890s and noticed a similar mathematical pattern at the top of each distribution. The popular "80/20 rule", the idea that 80% of effects come from 20% of causes, is named after him, although it was the quality-management writer Joseph Juran who coined that label decades later. The Pareto distribution is a classic case of fat tails, and is unrelated to the separate idea of Pareto efficiency, which comes from Pareto's work on welfare economics.

Think of a big music streaming service. Millions of songs are available, but a few hundred hits collect a huge share of all plays, while most tracks are hardly heard. Picking a song at random, you will usually find a modest number of plays, yet the overall total is dominated by the rare giants.

Many economic quantities look like that. Most people have moderate incomes and savings, but a small group at the top holds a large slice of the total. Most towns are small, a few cities are huge. The Pareto distribution is a mathematical way to describe this "a few get most" pattern and to measure how extreme it is.

For values above a minimum level, the share of cases larger than a given value $x$ is:

• $X$ is the random quantity, such as an income.
• $x_m$ is the minimum value from which the distribution applies (the scale).
• $\alpha$ is the shape parameter or tail index. A smaller $\alpha$ means a fatter tail and more concentration at the top.

Suppose top incomes start at $x_m = 20{,}000$ and $\alpha = 2$. The share of people earning above 80,000 is $(20{,}000 / 80{,}000)^2 = (1/4)^2 = 1/16$, or 6.25%. The average income in this group is $\alpha x_m / (\alpha - 1) = 2 \times 20{,}000 / 1 = 40{,}000$, twice the minimum. If $\alpha$ fell to 1.5, the share above 80,000 would rise to about 12.5% and the average to 60,000. When $\alpha$ is 1 or less, the average is infinite in theory, which shows how extreme heavy tails can be. The 80/20 split corresponds to an $\alpha$ of about 1.16.

• Income and wealth. Researchers measuring economic inequality, including the teams behind the World Inequality Database, use Pareto tails to estimate top income and wealth shares from income tax records. The tail index links directly to concentration measures such as the Gini coefficient and informs debates on the wealth tax.
• Cities and firms. City populations in many countries roughly follow Zipf's law, a Pareto pattern with $\alpha$ close to 1. Firm sizes are similarly skewed, with a few large companies accounting for much of output and market capitalisation.
• Insurance and reinsurance. Actuaries use Pareto models for large losses, such as major fires or catastrophes, to price excess-of-loss cover.
• Investment returns. Venture capital and start-up returns are dominated by a few big winners, so averages are driven by rare outcomes.
• Market risk. The tails of daily asset returns are often modelled with Pareto-type distributions in extreme value theory, used alongside measures such as value at risk.

Several mechanisms can produce Pareto tails. Proportional growth, where incomes, firms or cities grow by random percentages, tends to generate them, especially when combined with entry and exit. Returns on wealth that are higher for those who already have more add to concentration, as do "winner-takes-all" effects such as network effects in digital markets. Economists including Xavier Gabaix and Charles Jones have linked the shape of the income tail to differences in the growth of top incomes and to entrepreneurial returns.

The Pareto distribution usually fits only the upper tail; incomes in the middle and bottom are better described by other distributions, such as the lognormal. Estimates of $\alpha$ are sensitive to where the tail is assumed to start, and data on the very top can be incomplete because of tax havens and under-reporting. Pareto patterns can also appear by chance in limited samples, so statisticians test whether a power law is really the best fit before drawing conclusions.

• World Inequality Database