Fractals, fractiles and inequality

I finally got around to reading Mandelbrot and Hudson’s The Misbehavior of Markets, and can’t recommend it highly enough. I hope to have more to say about Mandelbrot’s ideas of investment risk, particularly as they relate to Luhmann (whose systems theory is, I think, incredibly well-suited to thinking about modern financial markets).

In the meantime, I was struck by Mandelbrot’s statement that fractals appear not only in the physical world but have also been found in the social world. And since the book’s focus is financial markets, I wanted to see whether it was possible to find other social examples.

First a word of definition – fractals are forms that have several properties, key among which is that they look similar at both the micro and macro levels. One example is the Sierpinski gasket below. The generating process begins with a single equilateral triangle and then, in each successive stage, inserts smaller equilateral triangles whose corners bisect the sides of the larger triangle containing them.

Source: Wikipedia

Mandelbrot’s primary social example in the book is Pareto’s discovery of power laws in income distributions, which were the basis for his 80/20 rule (as in, 80% of the wealth/income of a society is typically held by only 20% of the population). These power laws are implicit in discussions of growing income inequality, though those conversations tend to be framed in terms of fractiles (top 1%, top 5%, etc.) rather than fractals.

At the risk of over-reach, I thought I would try to combine the two ideas by treating fractiles as fractals, in essence looking for power-law relationships within the power law distributions of income. I make no claims to econometric rigor here – this is simply a thought experiment – but I thought it would be interesting to see whether rising inequality can be related to the fractal concept. More specifically, I wanted to test for the possibility of decoupling – defined here as increasing concentration of wealth – not only in terms of the top X% of the total population but also within successively smaller income tiers.

I started with the World Top Income Database and rather than income, used share of total national income for the top 10%, 1%, 0.1% and 0.01% of the US population. Taking the income share of the top 10% as a proxy for macro inequality, the total series shows a pronounced drop in inequality from the post-War boom through the mid/late 1970s, followed by a steep increase from the late 1970s to today.

That is hardly news, but it does allow for a closer look at the relative changes within income tiers. If those relationships remained stable, then the decoupling only occurred at the macro level, as the top 10 in total just took more of the national income. If they changed, however, then there is potentially more information in the data.

Given the data available, I decided to use successively smaller tenths as the fractiles to be treated as fractals. For lack of a better term, “share of share” (SoS) in the following charts refers to the proportion of a given share relative to the next larger 10th. The share of share for the top 10% vs. the entire population remains the share of the top 10% by definition, while the share of share for the top 1% as a percentage of the top 10% becomes the SoS of the top 1%, and so on (this is presumably kosher since they all have the same denominator).  It’s not the most elegant solution, but it seemed a reasonable way to measure decoupling at the level of successively (and consistently) smaller groups.

The results for the US are interesting. Flat lines in each case would indicate that the successively smaller groups remained constant in proportion, but that’s not the case. If I’m correct, then not only did the top 10% claim a much larger share of the total income (39.3% more, to be exact) but each successively smaller 10th claimed significantly more of its “fractal” over the time period.

Since the US and the UK tend to be lumped together in these stratification comparisons, I show them side by side here as the Neoliberal States. The levels themselves are not directly comparable (the time periods are slightly different, the UK does not include a top 0.01%, and the UK income data don’t include cap gains), so this is less of a between analysis and more of a within. The important point is the changes within the UK over time, which if anything indicate an even more intense decoupling of relative wealth.

I also wanted to find a counterexample that was not Scandinavian, so Germany and Japan seemed good bets. Japan’s data did not include a top 10% measure, however, so I left that aside. The data for Germany are frustratingly incomplete, and end at 1998. Still, over the period from 1978 to 1998 it’s clear that stratification within German society, at least by income, remained stable.

So the upshot is that the decoupling I referred to earlier in the first definition of the word appears to be happening fractally across the top of the income scale in the US and UK. This is in a sense just a different way of stating that the tiers in the US and UK remain in a power law relationship that is simply getting steeper, albeit with more specific information about changes in the shape of that curve.

As for What It All Means, I have no idea, but it does bring to mind the Berkeley/UCSF study released last week, which found that:

Upper-class people are different, Keltner says. “What wealth and education and prestige and a higher station in life gives you is the freedom to focus on the self.” In psychology experiments, wealthier people don’t read other people’s emotions as well. They hoard resources and are less generous than they could be.


The ability to rise in class is the great promise of the American Dream. But studies have found that, as people rise in the classes, they become less empathetic. Studies have also found that as people rise in wealth, they become happier—but not as much as you’d expect. “I think one of the reasons why is the human psyche stops feeling the need to connect and be closer to others, and we know that’s one of the greatest sources of happiness science can study,” Keltner says.

It’s not a huge stretch to think that the loss of empathy, and the reverence for social Darwinism enshrined in neoliberalism, have to do with the intense competition even (or perhaps especially) among those at the top. I’d imagine that it’s a lot less palatable to think of paying taxes for “those people” if you perceive yourself as fighting tooth and nail just to maintain your place in the pecking order, especially if you’re psychologically blinded to the fact that your place is unattainable to the vast majority of people on the planet.

I will stop here before I wade any further into topics that are beyond my virtual pay grade. If anyone reads this and finds any mathematical or other gaffes, please let me know in comments or at

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