Pareto distribution

Scalability and inequality

pnrj core principles, inequality, mathematical models, statistics , , , , , , , , , , , , , ,

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, https://commons.wikimedia.org/w/index.php?curid=3817954]

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, https://commons.wikimedia.org/w/index.php?curid=31096324]

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.

Russia has invaded Ukraine.

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Mar 6 JDN 2459645

Russia has invaded Ukraine. No doubt you have heard it by now, as it’s all over the news now in dozens of outlets, from CNN to NBC to The Guardian to Al-Jazeera. And as well it should be, as this is the first time in history that a nuclear power has annexed another country. Yes, nuclear powers have fought wars before—the US just got out of one in Afghanistan as you may recall. They have even started wars and led invasions—the US did that in Iraq. And certainly, countries have been annexing and conquering other countries for millennia. But never before—never before, in human history—has a nuclear-armed state invaded another country simply to claim it as part of itself. (Trump said he thought the US should have done something like that, and the world was rightly horrified.)

Ukraine is not a nuclear power—not anymore. The Soviet Union built up a great deal of its nuclear production in Ukraine, and in 1991 when Ukraine became independent it still had a sizable nuclear arsenal. But starting in 1994 Ukraine began disarming that arsenal, and now it is gone. Now that Russia has invaded them, the government of Ukraine has begun publicly reconsidering their agreements to disarm their nuclear arsenal.

Russia’s invasion of Ukraine has just disproved the most optimistic models of international relations, which basically said that major power wars for territory were over at the end of WW2. Some thought it was nuclear weapons, others the United Nations, still others a general improvement in trade integration and living standards around the world. But they’ve all turned out to be wrong; maybe such wars are rarer, but they can clearly still happen, because one just did.

I would say that only two major theories of the Long Peace are still left standing in light of this invasion, and that is nuclear deterrence and the democratic peace. Ukraine gave up its nuclear arsenal and later got attacked—that’s consistent with nuclear deterrence. Russia under Putin is nearly as authoritarian as the Soviet Union, and Ukraine is a “hybrid regime” (let’s call it a solid D), so there’s no reason the democratic peace would stop this invasion. But any model which posits that trade or the UN prevent war is pretty much off the table now, as Ukraine had very extensive trade with both Russia and the EU and the UN has been utterly toothless so far. (Maybe we could say the UN prevents wars except those led by permanent Security Council members.)

Well, then, what if the nuclear deterrence theory is right? What would have happened if Ukraine had kept its nuclear weapons? Would that have made this situation better, or worse? It could have made it better, if it acted as a deterrent against Russian aggression. But it could also have made it much, much worse, if it resulted in a nuclear exchange between Russia and Ukraine.

This is the problem with nukes. They are not a guarantee of safety. They are a guarantee of fat tails. To explain what I mean by that, let’s take a brief detour into statistics.

A fat-tailed distribution is one for which very extreme events have non-negligible probability. For some distributions, like a uniform distribution, events are clearly contained within a certain interval and nothing outside is even possible. For others, like a normal distribution or lognormal distribution, extreme events are theoretically possible, but so vanishingly improbable they aren’t worth worrying about. But for fat-tailed distributions like a Cauchy distribution or a Pareto distribution, extreme events are not so improbable. They may be unlikely, but they are not so unlikely they can simply be ignored. Indeed, they can actually dominate the average—most of what happens, happens in a handful of extreme events.

Deaths in war seem to be fat-tailed, even in conventional warfare. They seem to follow a Pareto distribution. There are lots of tiny skirmishes, relatively frequent regional conflicts, occasional major wars, and a handful of super-deadly global wars. This kind of pattern tends to emerge when a phenomenon is self-reinforcing by positive feedback—hence why we also see it in distributions of income and wildfire intensity.

Fat-tailed distributions typically (though not always—it’s easy to construct counterexamples, like the Cauchy distribution with low values truncated off) have another property as well, which is that minor events are common. More common, in fact, than they would be under a normal distribution. What seems to happen is that the probability mass moves away from the moderate outcomes and shifts to both the extreme outcomes and the minor ones.

Nuclear weapons fit this pattern perfectly. They may in fact reduce the probability of moderate, regional conflicts, in favor of increasing the probability of tiny skirmishes or peaceful negotiations. But they also increase the probability of utterly catastrophic outcomes—a full-scale nuclear war could kill billions of people. It probably wouldn’t wipe out all of humanity, and more recent analyses suggest that a catastrophic “nuclear winter” is unlikely. But even 2 billion people dead would be literally the worst thing that has ever happened, and nukes could make it happen in hours when such a death toll by conventional weapons would take years.

If we could somehow guarantee that such an outcome would never occur, then the lower rate of moderate conflicts nuclear weapons provide would justify their existence. But we can’t. It hasn’t happened yet, but it doesn’t have to happen often to be terrible. Really, just once would be bad enough.

Let us hope, then, that the democratic peace turns out to be the theory that’s right. Because a more democratic world would clearly be better—while a more nuclearized world could be better, but could also be much, much worse.

What we lose by aggregating

pnrj core principles, critique of neoclassical economics, inequality, macroeconomics , , , , , , , , , , , , , , ,

Jun 25, JDN 2457930

One of the central premises of current neoclassical macroeconomics is the representative agent: Rather than trying to keep track of all the thousands of firms, millions of people, and billions of goods and in a national economy, we aggregate everything up into a single worker/consumer and a single firm producing and consuming a single commodity.

This sometimes goes under the baffling misnomer of microfoundations, which would seem to suggest that it carries detailed information about the microeconomic behavior underlying it; in fact what this means is that the large-scale behavior is determined by some sort of (perfectly) rational optimization process as if there were just one person running the entire economy optimally.

First of all, let me say that some degree of aggregation is obviously necessary. Literally keeping track of every single transaction by every single person in an entire economy would require absurd amounts of data and calculation. We might have enough computing power to theoretically try this nowadays, but then again we might not—and in any case such a model would very rapidly lose sight of the forest for the trees.

But it is also clearly possible to aggregate too much, and most economists don’t seem to appreciate this. They cite a couple of famous theorems (like the Gorman Aggregation Theorem) involving perfectly-competitive firms and perfectly-rational identical consumers that offer a thin veneer of justification for aggregating everything into one, and then go on with their work as if this meant everything were fine.

What’s wrong with such an approach?

Well, first of all, a representative agent model can’t talk about inequality at all. It’s not even that a representative agent model says inequality is good, or not a problem; it lacks the capacity to even formulate the concept. Trying to talk about income or wealth inequality in a representative agent model would be like trying to decide whether your left hand is richer than your right hand.

It’s also nearly impossible to talk about poverty in a representative agent model; the best you can do is talk about a country’s overall level of development, and assume (not without reason) that a country with a per-capita GDP of $1,000 probably has a lot more poverty than a country with a per-capita GDP of $50,000. But two countries with the same per-capita GDP can have very different poverty rates—and indeed, the cynic in me wonders if the reason we’re reluctant to use inequality-adjusted measures of development is precisely that many American economists fear where this might put the US in the rankings. The Human Development Index was a step in the right direction because it includes things other than money (and as a result Saudi Arabia looks much worse and Cuba much better), but it still aggregates and averages everything, so as long as your rich people are doing well enough they can compensate for how badly your poor people are doing.

Nor can you talk about oligopoly in a representative agent model, as there is always only one firm, which for some reason chooses to act as if it were facing competition instead of rationally behaving as a monopoly. (This is not quite as nonsensical as it sounds, as the aggregation actually does kind of work if there truly are so many firms that they are all forced down to zero profit by fierce competition—but then again, what market is actually like that?) There is no market share, no market power; all are at the mercy of the One True Price.

You can still talk about externalities, sort of; but in order to do so you have to set up this weird doublethink phenomenon where the representative consumer keeps polluting their backyard and then can’t figure out why their backyard is so darn polluted. (I suppose humans do seem to behave like that sometimes; but wait, I thought you believed people were rational?) I think this probably confuses many an undergrad, in fact; the models we teach them about externalities generally use this baffling assumption that people consider one set of costs when making their decisions and then bear a different set of costs from the outcome. If you can conceptualize the idea that we’re aggregating across people and thinking “as if” there were a representative agent, you can ultimately make sense of this; but I think a lot of students get really confused by it.

Indeed, what can you talk about with a representative agent model? Economic growth and business cycles. That’s… about it. These are not minor issues, of course; indeed, as Robert Lucas famously said:

The consequences for human welfare involved in questions like these [on economic growth] are simply staggering: once one starts to think about them, it is hard to think about anything else.

I certainly do think that studying economic growth and business cycles should be among the top priorities of macroeconomics. But then, I also think that poverty and inequality should be among the top priorities, and they haven’t been—perhaps because the obsession with representative agent models make that basically impossible.

I want to be constructive here; I appreciate that aggregating makes things much easier. So what could we do to include some heterogeneity without too much cost in complexity?

Here’s one: How about we have p firms, making q types of goods, sold to n consumers? If you want you can start by setting all these numbers equal to 2; simply going from 1 to 2 has an enormous effect, as it allows you to at least say something about inequality. Getting them as high as 100 or even 1000 still shouldn’t be a problem for computing the model on an ordinary laptop. (There are “econophysicists” who like to use these sorts of agent-based models, but so far very few economists take them seriously. Partly that is justified by their lack of foundational knowledge in economics—the arrogance of physicists taking on a new field is legendary—but partly it is also interdepartmental turf war, as economists don’t like the idea of physicists treading on their sacred ground.) One thing that really baffles me about this is that economists routinely use computers to solve models that can’t be calculated by hand, but it never seems to occur to them that they could have started at the beginning planning to make the model solvable only by computer, and that would spare them from making the sort of heroic assumptions they are accustomed to making—assumptions that only made sense when they were used to make a model solvable that otherwise wouldn’t be.

You could also assign a probability distribution over incomes; that can get messy quickly, but we actually are fortunate that the constant relative risk aversion utility function and the Pareto distribution over incomes seem to fit the data quite well—as the product of those two things is integrable by hand. As long as you can model how your policy affects this distribution without making that integral impossible (which is surprisingly tricky), you can aggregate over utility instead of over income, which is a lot more reasonable as a measure of welfare.

And really I’m only scratching the surface here. There are a vast array of possible new approaches that would allow us to extend macroeconomic models to cover heterogeneity; the real problem is an apparent lack of will in the community to make such an attempt. Most economists still seem very happy with representative agent models, and reluctant to consider anything else—often arguing, in fact, that anything else would make the model less microfounded when plainly the opposite is the case.