Self-organization and wealth distribution

The question of wealth inequality has been making headlines, in everything from the Occupy Wall Street movement and their decrying the wealth of the 1%, to discussion in the Republican Presidential-Candidate Popularity Contest currently ongoing in the US.

There have always been voices clamouring for equal wealth for everyone, but the real world doesn't work like that. Wealth inequality doesn't seem particularly unfair given the inequalities in natural abilities and access to capital or resources. Intuitively, it seems that the distribution of wealth in society will follow a power-law distribution. A power-law distribution is one in which the observations show a 1/f distribution, as described in this article.

Recent modeling studies suggest a 1/f distribution over most of the population, but wealth distribution becomes exponential near the tails. The model distribution is described as Pareto-like, with a relatively few super-wealthy floating over an ever-changing middle class.

So wealth inequality should be expected in any society, no matter how even the playing field. The skills necessary to navigate through the economy are not evenly distributed. Some individuals play better than others. Therefore, some individuals will be wealthier than others. Let's take a look through some public data and see if we can recognize a power-law distribution.

According to Wolff (2010), the breakdown of wealth among different quintiles (and finer groups) is:

Fraction of                        Fraction of 

population                         wealth

Lowest 40%                      0.2%

40 - 60%                            4.0%

60 - 80%                          10.9%

80 - 90%                          12.0%

90 - 95%                          11.2%

95 - 99%                          27.1%

99 - 100%                        34.6%

Given that the wealth of Americans in 2007 was reported by the Fed to be $79.482 trillion, and the population of the US at that time was 299,398,400 (roughly), we can plot a logarithmic graph of individual wealth vs population to check for self-organization in wealth distribution.

In order to do this, I have estimated that the wealth of the individual in the middle of each group to have the average wealth of the group. Based on past experience, this estimate will tend to be biased--however given the number of orders of magnitudes on the resulting graph, the errors are so small as to be unnoticed.

To interpret this graph, consider the first two points--they suggest that roughly 80 million people have less than about $2,500, and about 130 million people have less than about $75,000. Most of the data appear to lie along a line of fit, but there are a few exceptionally rich individuals, including some on the Forbes 400 list, who plot far above the line. 

Also note that "the 99%" includes people that have about $8 million in assets.

The observed distribution agrees somewhat with the models described above--a few super-wealthy lording it over the rest. However, there is a significant difference between our observed slope and the slope of the models--the models suggest a slope for the straight-line of about 2. On our graph, the slope of the straight line is over 4 (meaning four orders of magnitude in wealth over one order of magnitude of population).

On our graph, roughly 290,000,000 people have less than $1 million, and 29,000,000 have less than $100. Seems a tad steep. With a slope of 2, the 29,000,000 would have less than $10,000.

If the wealth of the entire population were described by a 1/f distribution, then the richest American would have a wealth of only about $1.5 million. We here at the World Complex think it would be difficult to manage that summer home in the Hamptons with such a paltry sum.

The Ebert and Paul (2009) paper linked to above attempts to explain the semi-permanent nature of the super-rich. The super-rich have benefited from leverage in the system, and remain at the top due to the ongoing access to greater leverage than is possible for the average citizen. 

A poor geologist like me can only wonder--what happens when leverage becomes wealth-destroying rather than wealth-enhancing? Unfortunately, the answer we are seeing is that the super-rich get bailed out of their losing positions by everyone else.

And here we come to the question of fairness in the system. A fair system with an even playing-field will always result in inequalities--but even extreme inequalities will be tolerated to the extent that the system is perceived as being fair. In the past, during times when the system was fair(er), people tended to respect that someone had earned money and was able to enjoy the fruits of success. Under the present system, there is a widespread and growing skepticism that unusually wealth individuals have obtained their wealth not through production of wealth but through gaming the system and even stealing wealth from those lower down the socio-economic ladder.

Lastly we see the same plot as above, but with the estimated and "ideal" wealth distributions as determined from a series of nationwide interviews with over 5500 respondents reported in Norton and Ariely (2010).

Clearly most Americans thought the system was more equitable that was actually the case, and interestingly, they seemed to wish the system were more equitable still. I would like to point out that the "ideal" distribution is actually mathematically impossible (the third and fourth quintiles had equal wealth), which seems fitting. 

In an ideal world, according to the survey, only 10 million Americans would have less than $100,000 in assets, and no one would have as much as a million.

Unfortunately the survey neglected to ask respondents what they felt the wealth of Mssrs Gates and Snyder (no. 1 and 400 on the Forbes 400 list) should be in an ideal world, which might have been very interesting.