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.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s