Scalability and inequality

May 15 JDN 2459715

Why are some molecules (e.g. DNA) billions of times larger than others (e.g. H2O), but all atoms are within a much narrower range of sizes (only a few hundred)?

Why are some animals (e.g. elephants) millions of times as heavy as other (e.g. mice), but their cells are basically the same size?

Why does capital income vary so much more (factors of thousands or millions) than wages (factors of tens or hundreds)?

These three questions turn out to have much the same answer: Scalability.

Atoms are not very scalable: Adding another proton to a nucleus causes interactions with all the other protons, which makes the whole atom unstable after a hundred protons or so. But molecules, particularly organic polymers such as DNA, are tremendously scalable: You can add another piece to one end without affecting anything else in the molecule, and keep on doing that more or less forever.

Cells are not very scalable: Even with the aid of active transport mechanisms and complex cellular machinery, a cell’s functionality is still very much limited by its surface area. But animals are tremendously scalable: The same exponential growth that got you from a zygote to a mouse only needs to continue a couple years longer and it’ll get you all the way to an elephant. (A baby elephant, anyway; an adult will require a dozen or so years—remarkably comparable to humans, in fact.)

Labor income is not very scalable: There are only so many hours in a day, and the more hours you work the less productive you’ll be in each additional hour. But capital income is perfectly scalable: We can add another digit to that brokerage account with nothing more than a few milliseconds of electronic pulses, and keep doing that basically forever (due to the way integer storage works, above 2^63 it would require special coding, but it can be done; and seeing as that’s over 9 quintillion, it’s not likely to be a problem any time soon—though I am vaguely tempted to write a short story about an interplanetary corporation that gets thrown into turmoil by an integer overflow error).

This isn’t just an effect of our accounting either. Capital is scalable in a way that labor is not. When your contribution to production is owning a factory, there’s really nothing to stop you from owning another factory, and then another, and another. But when your contribution is working at a factory, you can only work so hard for so many hours.

When a phenomenon is highly scalable, it can take on a wide range of outcomes—as we see in molecules, animals, and capital income. When it’s not, it will only take on a narrow range of outcomes—as we see in atoms, cells, and labor income.

Exponential growth is also part of the story here: Animals certainly grow exponentially, and so can capital when invested; even some polymers function that way (e.g. under polymerase chain reaction). But I think the scalability is actually more important: Growing rapidly isn’t so useful if you’re going to immediately be blocked by a scalability constraint. (This actually relates to the difference between r- and K- evolutionary strategies, and offers further insight into the differences between mice and elephants.) Conversely, even if you grow slowly, given enough time, you’ll reach whatever constraint you’re up against.

Indeed, we can even say something about the probability distribution we are likely to get from random processes that are scalable or non-scalable.

A non-scalable random process will generally converge toward the familiar normal distribution, a “bell curve”:

[Image from Wikipedia: By Inductiveload – self-made, Mathematica, Inkscape, Public Domain,]

The normal distribution has most of its weight near the middle; most of the population ends up near there. This is clearly the case for labor income: Most people are middle class, while some are poor and a few are rich.

But a scalable random process will typically converge toward quite a different distribution, a Pareto distribution:

[Image from Wikipedia: By Danvildanvil – Own work, CC BY-SA 3.0,]

A Pareto distribution has most of its weight near zero, but covers an extremely wide range. Indeed it is what we call fat tailed, meaning that really extreme events occur often enough to have a meaningful effect on the average. A Pareto distribution has most of the people at the bottom, but the ones at the top are really on top.

And indeed, that’s exactly how capital income works: Most people have little or no capital income (indeed only about half of Americans and only a third(!) of Brits own any stocks at all), while a handful of hectobillionaires make utterly ludicrous amounts of money literally in their sleep.

Indeed, it turns out that income in general is pretty close to distributed normally (or maybe lognormally) for most of the income range, and then becomes very much Pareto at the top—where nearly all the income is capital income.

This fundamental difference in scalability between capital and labor underlies much of what makes income inequality so difficult to fight. Capital is scalable, and begets more capital. Labor is non-scalable, and we only have to much to give.

It would require a radically different system of capital ownership to really eliminate this gap—and, well, that’s been tried, and so far, it hasn’t worked out so well. Our best option is probably to let people continue to own whatever amounts of capital, and then tax the proceeds in order to redistribute the resulting income. That certainly has its own downsides, but they seem to be a lot more manageable than either unfettered anarcho-capitalism or totalitarian communism.

Billionaires bear the burden of proof

Sep 15 JDN 2458743

A king sits atop a golden throne, surrounded by a thousand stacks of gold coins six feet high. A hundred starving peasants beseech him for just one gold coin each, so that they might buy enough food to eat and clothes for the winter. The king responds: “How dare you take my hard-earned money!”

This is essentially the world we live in today. I really cannot emphasize enough how astonishingly, horrifically, mind-bogglingly rich billionares are. I am writing this sentence at 13:00 PDT on September 8, 2019. A thousand seconds ago was 12:43, about when I started this post. A million seconds ago was Wednesday, August 28. A billion seconds ago was 1987. I will be a billion seconds old this October.

Jeff Bezos has $170 billion. 170 billion seconds ago was a thousand years before the construction of the Great Pyramid. To get as much money as he has gaining one dollar per second (that’s $3600 an hour!), Jeff Bezos would have had to work for as long as human civilization has existed.

At a more sensible wage like $30 per hour (still better than most people get), how long would it take to amass $170 billion? Oh, just about 600,000 years—or about twice the length of time that Homo sapiens has existed on Earth.

How does this compare to my fictional king with a thousand stacks of gold? A typical gold coin is worth about $500, depending on its age and condition. Coins are about 2 millimeters thick. So a thousand stacks, each 2 meters high, would be about $500*1000*1000 = $500 million. This king isn’t even a billionaire! Jeff Bezos has three hundred times as much as him.

Coins are about 30 millimeters in diameter, so assuming they are packed in neat rows, these thousand stacks of gold coins would fill a square about 0.9 meters to a side—in our silly Imperial units, that’s 3 feet wide, 3 feet deep, 6 feet tall. If Jeff Bezo’s stock portfolio were liquidated into gold coins (which would require about 2% of the world’s entire gold supply and surely tank the market), the neat rows of coins stacked a thousand high would fill a square over 16 meters to a side—that’s a 50-foot-wide block of gold coins. Smaug’s hoard in The Hobbit was probably about the same amount of money as what Jeff Bezos has.

And yet, somehow there are still people who believe that he deserves this money, that he earned it, that to take even a fraction of it away would be a crime tantamount to theft or even slavery.

Their arguments can be quite seductive: How would you feel about the government taking your hard-earned money? Entrepreneurs are brilliant, dedicated, hard-working people; why shouldn’t they be rewarded? What crime do CEOs commit by selling products at low prices?

The way to cut through these arguments is to never lose sight of the numbers. In defense of a man who had $5 million or even $20 million, such an argument might make sense. I can imagine how someone could contribute enough to humanity to legitimately deserve $20 million. I can understand how a talented person might work hard enough to earn $5 million. But it’s simply not possible for any human being to be so brilliant, so dedicated, so hard-working, or make such a contribution to the world, that they deserve to have more dollars than there have been seconds since the Great Pyramid.

It’s not necessary to find specific unethical behaviors that brought a billionaire to where he (and yes, it’s nearly always he) is. They are generally there to be found: At best, one becomes a billionaire by sheer luck. Typically, one becomes a billionaire by exerting monopoly power. At worst, one can become a billionaire by ruthless exploitation or even mass murder. But it’s not our responsibility to point out a specific crime for every specific billionaire.

The burden of proof is on billionaires: Explain how you can possibly deserve that much money.

It’s not enough to point to some good things you did, or emphasize what a bold innovator you are: You need to explain what you did that was so good that it deserves to be rewarded with Smaug-level hoards of wealth. Did you save the world from a catastrophic plague? Did you end world hunger? Did you personally prevent a global nuclear war? I could almost see the case for Norman Borlaug or Jonas Salk earning a billion dollars (neither did, by the way). But Jeff Bezos? You didn’t save the world. You made a company that sells things cheaply and ships them quickly. Get over yourself.

Where exactly do we draw that line? That’s a fair question. $20 million? $100 million? $500 million? Maybe there shouldn’t even be a hard cap. There are many other approaches we could take to reducing this staggering inequality. Previously I have proposed a tax system that gets continuously more progressive forever, as well as a CEO compensation cap based on the pay of the lowliest employees. We could impose a wealth tax, as Elizabeth Warren has proposed. Or we could simply raise the top marginal rate on income tax to something more like what it was in the 1960s. Or as Republicans today would call it, radical socialism.