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.

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