# Power laws and wealth

From Alison Griswold at Slate, reporting on the Wealth-X and UBS billionaire census (warning: obnoxious auto-playing music at the second link): “The typical billionaire has a net worth of $3.1 billion.” Does “typical” mean mean? or median? It appears that “mean” is intended, because the front page of this census says there are 2,325 billionaires globally, with a combined net worth of 7.3 trillion dollars; the quotient is just around 3.1 billion. Furthermore, wealth supposedly follows a Pareto distribution – the number of people wealthier than $x$ is proportional to $x^{-\alpha}$. (But note that this may not be true; in general identifying power laws is tricky. But let’s play along, and observe that: • the median of a Pareto distribution is $x_m 2^{1/\alpha}$. Let $x_m = 1$ (i. e. measure money in units of billions of dollars) and you get that $alpha = log(2)/log(3.1) \approx 0.61$ , if “typical” means median. • the mean of a Pareto distribution is $\x_m (\alpha/(\alpha-1))$, so you get $\alpha/(\alpha-1) = 3.1$, or$\alpha = 31/21 \approx 1.48\$, if “typical” means mean.

These two parameters are very different! In particular, with the parameter $\alpha = 1.48$ (derived from assuming the mean billionaire has a net worth of 3.1 billion), 81 percent of billionaires have less net worth than what the article calls the “typical” billionaire, and the median billionaire has a net worth of “only” 1.6 billion. In contrast, a Pareto distribution with $\alpha < 1$, such as any one where the median is at least twice the minimum, doesn’t even have a well-defined mean. (Of course the actual distribution of billionaire net worths has a well-defined mean, whatever it is, because there are a finite number of them.)

The original survey also mentions that there’s a “wealth ceiling” around 10 billion USD; see the plot at quartz. But I don’t see any really clear evidence for this. There could be such a ceiling, though, a function of the size and growth rate of the world economy, the typical length of human lives, tax rates on the income of the very wealthy, and so on.