# Why Are Billionaires So Rich?

## A story of Power Law, Broccoli, and Billionaires

*“For to every one who has will more be given, and he will have abundance” — Mathew 25:29*

## Of Billionaires and Broccoli

When you break off a piece of broccoli, what you get is an even tinier broccoli. The shape of the constituent parts is the same as the shape of the whole. This is true of all fractals, of which broccoli is perhaps the most delicious. We say that fractals are ‘self-similar’, or ‘scale-invariant’¹.

Scale invariance is also a defining feature of one of the most important mathematical functions in modern society — power law.

Power Law patterns are everywhere in our socially-linked lives². These patterns are the reason that people live in bigger cities, that tech companies are so powerful, and that billionaires are making more money than ever. Like broccoli, power law distributions look the same no matter how you scale them. The mathematical proof of this is so short, I’ll risk putting it in the introduction:

To take one example, we know that the distribution of wealth in a society follows a power law pattern³. Is wealth distribution scale invariant? What if we multiplied it by some constant, say a billion? Would we get the same unequal pattern?

## Data on Billionaires

Luckily for us, we have pretty good data on the net worth of the world’s billionaires. The Forbes list from 2017 shows us the familiar media moguls and tech titans, each elite members of the 3 comma club. Below are the top 20.

If we plot all of the billionaires on the list, we get the familiar power law curve.

This proves that even among the insanely wealthy, the same inequality persists. After all, Jeff Bezos has 100x as much money as that long tail of losers who have only a single billion dollars to their name.

We can see the same curve when looking at wealth distribution among all people. Using data from the British Office of National Statistics, we get a picture of net household wealth by percentile. The below graph shows clearly that the top 1% have as much wealth as the bottom 46% combined.

## Data on Companies

It’s not just people that are unequal in terms of how much money they have. Companies ranging from large cap giants to small businesses vary equally in terms of how much they make. Regardless of the scale, there are winners and losers in each category.

The plot below shows all companies in the S&P500 organized by market cap⁴.

This data is a few months old, before Apple cracked the trillion-dollar mark. Still, it shows that a few companies tower over the rest in a classic power law pattern.

If we limit ourselves to smaller and younger companies like those that IPO’d in the last year, we see the same pattern. A few big fish raised most of the capital, while a long tail raised only a little.

The dollar values involved are off by a factor of 500 or so, but the shape is similar.

## Data on Cities and Countries

The fact that the population of cities follows a power law relationship has been known for a long time⁵. Plotting the top 20 cities in the US by GDP instead of population shows the same pattern.

In the US, we see that the big urban centers of New York and Los Angeles contain most of the economic activity in the country.

On a global scale, the same pattern emerges, with the United States and China dominating global GPD.

Once again, we can plot all U.S. cities and all nations by GDP. In each case we see an almost identical shape, even though the scale of the charts differ.

## Conclusions

Whenever we talk about the distribution of money, chances are that we will be talking about Power Law. Since power law is scale invariant, it operates the same way at the level of normal people and at the level of billionaires. It governs the size of companies big and small, and can tell us about how wealth will be distributed among cities and among nations.

In the below plot, you can see the fitted power law curve for people, companies, and cities/nations. Like broccoli, no matter the scale involved, it always looks the same.

The fact that capital accumulation always seems to follow a power law pattern may seem slightly disturbing. It suggests that inequality is built into the process of wealth accumulation itself. In cities, people draw more people. In corporations, capital brings in more capital. Profits generates greater profit.

Ultimately, wealth begets wealth. As the Mathew principle suggests, winners will gain more over time while losers are left with little. If this natural process remains unchecked, the gap between billionaires and the rest of us will only grow larger.

## Notes

All code and data associated with this post can be found on github at https://github.com/taubergm/PowerLawScaling

1- Technically self-similarity is a general case of scale invariance — https://en.wikipedia.org/wiki/Self-similarity. For some cool examples of self-similar fractals, see https://mathigon.org/world/Fractals

2 — For more information on why power laws form, see my previous post — https://medium.com/@michaeltauberg/power-law-in-popular-media-7d7efef3fb7c

3 — https://en.wikipedia.org/wiki/Pareto_distribution

4 — data from https://www.barchart.com/stocks/indices/sp/sp500 taken on 06/29/2018

5 — A good article on power law and city size - https://io9.gizmodo.com/the-mysterious-law-that-governs-the-size-of-your-city-1479244159