So, there’s an article published in yesterday’s Guardian titled, “The mathematical law that shows why wealth flows to the 1%,” which is fine, except for the fact that the “law” is not really a law, nor does it necessarily show “why” wealth flows anywhere.

To be fair, it’s a perfectly reasonable article with a crap, misleading headline, so I blame the editor, not the author.

The point of the article is to introduce the idea of a power law distribution, or heavy-tailed distributions more generally. These pop up all over the place, but are something that many people are not familiar with. The critical feature of such distributions, if we are talking about, say, wealth, is that an enormous number of people have very little, while a small number of people have a ton. In these circumstances it can be misleading, or at least uninformative, to talk about “average” wealth.

The introduction is nicely done, and it represents an important part of the “how” of wealth is distributed, but what, if anything, does it tell us about the “why”?

To try to answer that, we’ll walk through three distributions with the same “average,” to see what a distribution’s shape might tell us about the process that gave rise to it: Normal, Log Normal, and Pareto.

The core of the issue, I think, is that there are three different technical definitions that we associate with the common-usage term “average,” the mean, the median, and the mode. This is probably familiar to most readers who have made their way here, but here’s a quick review:

**mean**is what you usually calculate when you are asked to find the average of something. For instance, you would determine the average wealth of a nation by taking its total wealth and dividing it by the number of people.

**median**is the point where half of the distribution lies to the right, and half lies to the left. So the median wealth would be the amount of money X where half of the people had more than X and half had less than X.

**mode**is the high point in the distribution, its most common value. In the picture above, the mode of the blue curve is at about 300, while the mode of the red curve is a little less than 50.

Image from Alex Pardee‘s 2009 exhibition “Hiding From The Normals” |

*can*lead to a particular distribution, observing that distribution does not prove that your particular mechanism was actually at work. It seems like that should be obvious, but you actually see a disturbing number of scientific papers that basically make that error. There will typically be whole families of mechanisms that can give rise to the same outcome. However, looking at the outcome (the distribution, in this case) and asking what mechanisms are consistent with it is an important first step.

Vilfredo Pareto, who grew a very long beard in order to illustrate the idea of a distribution with a very long tail. |

*should be*somewhat Normally distributed. They think that it

*is*more Log Normally distributed. They fail to recognize that,

*in reality*, it is more like Pareto distributed.

Clauset, A., Shalizi, C., & Newman, M. (2009). Power-Law Distributions in Empirical Data SIAM Review, 51 (4) DOI: 10.1137/070710111

Free version of the article available on the ArXiv, here: http://arxiv.org/abs/0706.1062

Thanks for the plug. Two small comments/corrections:

1. The conventional wisdom for a long time was that income distributions (at least in developed capitalist democracies where we actually have data, etc.) tend to have a body which looks pretty log-normal, but with a Pareto-style right or upper tail. (Meaning that the truly rich are vastly wealthier than one would expect from a mere log-normal fit to the bulk of the population.) This may have changed recently, for all I know.

2. The free version of our paper is arxiv:0706.1062.

Cool. I’ve added updates to the text on both points.

Thanks, Cosma!

Nice article – I was irritated by Alok’s implicit assumption that Benford’s law was a Law not just an empirical regularity.

BTW, talk of long tails always reminds me of George Box’s figure at the bottom of this page.

I have the impression that focusing on inequality is the wrong approach. This Paul Graham’s essay capture the idea:

http://www.paulgraham.com/inequality.html

This is an excerpt:

“I realize startups are not the main target of those who want to eliminate economic inequality. What they really dislike is the sort of wealth that becomes self-perpetuating through an alliance with power. For example, construction firms that fund politicians’ campaigns in return for government contracts, or rich parents who get their children into good colleges by sending them to expensive schools designed for that purpose. But if you try to attack this type of wealth through economic policy, it’s hard to hit without destroying startups as collateral damage.

The problem here is not wealth, but corruption. So why not go after corruption?”

Alejandro,

I’m with you 100% on that. I guess one of the points that I was trying to make is that when you see extreme wealth inequality, it is suggestive of a self-perpetuating, corrupt positive-feedback mechanism.

Eliminating the corruption is the preferable route, but an alternative (or, rather, a supplement) is to acknowledge that there is a degree of positive feedback involved in our system, and to use tools like a progressive tax structure to help re-level the playing field.

The way that I view the problem(s), either way of framing and approaching things should actually help, rather than hurt, startups and other small companies.

The way that I view the problem(s), either way of framing and approaching things should actually help, rather than part a site it hurt, startups and other small companies.