“The cake is a lie”: The fundamental distortions of inequality

July 13, JDN 2457583

Inequality of wealth and income, especially when it is very large, fundamentally and radically distorts outcomes in a capitalist market. I’ve already alluded to this matter in previous posts on externalities and marginal utility of wealth, but it is so important I think it deserves to have its own post. In many ways this marks a paradigm shift: You can’t think about economics the same way once you realize it is true.

To motivate what I’m getting at, I’ll expand upon an example from a previous post.

Suppose there are only two goods in the world; let’s call them “cake” (K) and “money” (M). Then suppose there are three people, Baker, who makes cakes, Richie, who is very rich, and Hungry, who is very poor. Furthermore, suppose that Baker, Richie and Hungry all have exactly the same utility function, which exhibits diminishing marginal utility in cake and money. To make it more concrete, let’s suppose that this utility function is logarithmic, specifically: U = 10*ln(K+1) + ln(M+1)

The only difference between them is in their initial endowments: Baker starts with 10 cakes, Richie starts with $100,000, and Hungry starts with $10.

Therefore their starting utilities are:

U(B) = 10*ln(10+1)= 23.98

U(R) = ln(100,000+1) = 11.51

U(H) = ln(10+1) = 2.40

Thus, the total happiness is the sum of these: U = 37.89

Now let’s ask two very simple questions:

1. What redistribution would maximize overall happiness?
2. What redistribution will actually occur if the three agents trade rationally?

If multiple agents have the same diminishing marginal utility function, it’s actually a simple and deep theorem that the total will be maximized if they split the wealth exactly evenly. In the following blockquote I’ll prove the simplest case, which is two agents and one good; it’s an incredibly elegant proof:

Given: for all x, f(x) > 0, f'(x) > 0, f”(x) < 0.

Maximize: f(x) + f(A-x) for fixed A

f'(x) – f'(A – x) = 0

f'(x) = f'(A – x)

Since f”(x) < 0, this is a maximum.

Since f'(x) > 0, f is monotonic; therefore f is injective.

x = A – x

QED

This can be generalized to any number of agents, and for multiple goods. Thus, in this case overall happiness is maximized if the cakes and money are both evenly distributed, so that each person gets 3 1/3 cakes and $33,336.66.

The total utility in that case is:

3 * (10 ln(10/3+1) + ln(33,336.66+1)) = 3 * (14.66 + 10.414) = 3 (25.074) =75.22

That’s considerably better than our initial distribution (almost twice as good). Now, how close do we get by rational trade?

Each person is willing to trade up until the point where their marginal utility of cake is equal to their marginal utility of money. The price of cake will be set by the respective marginal utilities.

In particular, let’s look at the trade that will occur between Baker and Richie. They will trade until their marginal rate of substitution is the same.

The actual algebra involved is obnoxious (if you’re really curious, here are some solved exercises of similar trade problems), so let’s just skip to the end. (I rushed through, so I’m not actually totally sure I got it right, but to make my point the precise numbers aren’t important.)
Basically what happens is that Richie pays an exorbitant price of $10,000 per cake, buying half the cakes with half of his money.

Baker’s new utility and Richie’s new utility are thus the same:
U(R) = U(B) = 10*ln(5+1) + ln(50,000+1) = 17.92 + 10.82 = 28.74
What about Hungry? Yeah, well, he doesn’t have $10,000. If cakes are infinitely divisible, he can buy up to 1/1000 of a cake. But it turns out that even that isn’t worth doing (it would cost too much for what he gains from it), so he may as well buy nothing, and his utility remains 2.40.

Hungry wanted cake just as much as Richie, and because Richie has so much more Hungry would have gotten more happiness from each new bite. Neoclassical economists promised him that markets were efficient and optimal, and so he thought he’d get the cake he needs—but the cake is a lie.

The total utility is therefore:

U = U(B) + U(R) + U(H)

U = 28.74 + 28.74 + 2.40

U = 59.88

Note three things about this result: First, it is more than where we started at 37.89—trade increases utility. Second, both Richie and Baker are better off than they were—trade is Pareto-improving. Third, the total is less than the optimal value of 75.22—trade is not utility-maximizing in the presence of inequality. This is a general theorem that I could prove formally, if I wanted to bore and confuse all my readers. (Perhaps someday I will try to publish a paper doing that.)

This result is incredibly radical—it basically goes against the core of neoclassical welfare theory, or at least of all its applications to real-world policy—so let me be absolutely clear about what I’m saying, and what assumptions I had to make to get there.

I am saying that if people start with different amounts of wealth, the trades they would willfully engage in, acting purely under their own self interest, would not maximize the total happiness of the population. Redistribution of wealth toward equality would increase total happiness.

First, I had to assume that we could simply redistribute goods however we like without affecting the total amount of goods. This is wildly unrealistic, which is why I’m not actually saying we should reduce inequality to zero (as would follow if you took this result completely literally). Ironically, this is an assumption that most neoclassical welfare theory agrees with—the Second Welfare Theorem only makes any sense in a world where wealth can be magically redistributed between people without any harmful economic effects. If you weaken this assumption, what you find is basically that we should redistribute wealth toward equality, but beware of the tradeoff between too much redistribution and too little.

Second, I had to assume that there’s such a thing as “utility”—specifically, interpersonally comparable cardinal utility. In other words, I had to assume that there’s some way of measuring how much happiness each person has, and meaningfully comparing them so that I can say whether taking something from one person and giving it to someone else is good or bad in any given circumstance.

This is the assumption neoclassical welfare theory generally does not accept; instead they use ordinal utility, on which we can only say whether things are better or worse, but never by how much. Thus, their only way of determining whether a situation is better or worse is Pareto efficiency, which I discussed in a post a couple years ago. The change from the situation where Baker and Richie trade and Hungry is left in the lurch to the situation where all share cake and money equally in socialist utopia is not a Pareto-improvement. Richie and Baker are slightly worse off with 25.07 utilons in the latter scenario, while they had 28.74 utilons in the former.

Third, I had to assume selfishness—which is again fairly unrealistic, but again not something neoclassical theory disagrees with. If you weaken this assumption and say that people are at least partially altruistic, you can get the result where instead of buying things for themselves, people donate money to help others out, and eventually the whole system achieves optimal utility by willful actions. (It depends just how altruistic people are, as well as how unequal the initial endowments are.) This actually is basically what I’m trying to make happen in the real world—I want to show people that markets won’t do it on their own, but we have the chance to do it ourselves. But even then, it would go a lot faster if we used the power of government instead of waiting on private donations.

Also, I’m ignoring externalities, which are a different type of market failure which in no way conflicts with this type of failure. Indeed, there are three basic functions of government in my view: One is to maintain security. The second is to cancel externalities. The third is to redistribute wealth. The DOD, the EPA, and the SSA, basically. One could also add macroeconomic stability as a fourth core function—the Fed.

One way to escape my theorem would be to deny interpersonally comparable utility, but this makes measuring welfare in any way (including the usual methods of consumer surplus and GDP) meaningless, and furthermore results in the ridiculous claim that we have no way of being sure whether Bill Gates is happier than a child starving and dying of malaria in Burkina Faso, because they are two different people and we can’t compare different people. Far more reasonable is not to believe in cardinal utility, meaning that we can say an extra dollar makes you better off, but we can’t put a number on how much.

And indeed, the difficulty of even finding a unit of measure for utility would seem to support this view: Should I use QALY? DALY? A Likert scale from 0 to 10? There is no known measure of utility that is without serious flaws and limitations.

But it’s important to understand just how strong your denial of cardinal utility needs to be in order for this theorem to fail. It’s not enough that we can’t measure precisely; it’s not even enough that we can’t measure with current knowledge and technology. It must be fundamentally impossible to measure. It must be literally meaningless to say that taking a dollar from Bill Gates and giving it to the starving Burkinabe would do more good than harm, as if you were asserting that triangles are greener than schadenfreude.

Indeed, the whole project of welfare theory doesn’t make a whole lot of sense if all you have to work with is ordinal utility. Yes, in principle there are policy changes that could make absolutely everyone better off, or make some better off while harming absolutely no one; and the Pareto criterion can indeed tell you that those would be good things to do.

But in reality, such policies almost never exist. In the real world, almost anything you do is going to harm someone. The Nuremburg trials harmed Nazi war criminals. The invention of the automobile harmed horse trainers. The discovery of scientific medicine took jobs away from witch doctors. Inversely, almost any policy is going to benefit someone. The Great Leap Forward was a pretty good deal for Mao. The purges advanced the self-interest of Stalin. Slavery was profitable for plantation owners. So if you can only evaluate policy outcomes based on the Pareto criterion, you are literally committed to saying that there is no difference in welfare between the Great Leap Forward and the invention of the polio vaccine.

One way around it (that might actually be a good kludge for now, until we get better at measuring utility) is to broaden the Pareto criterion: We could use a majoritarian criterion, where you care about the number of people benefited versus harmed, without worrying about magnitudes—but this can lead to Tyranny of the Majority. Or you could use the Difference Principle developed by Rawls: find an ordering where we can say that some people are better or worse off than others, and then make the system so that the worst-off people are benefited as much as possible. I can think of a few cases where I wouldn’t want to apply this criterion (essentially they are circumstances where autonomy and consent are vital), but in general it’s a very good approach.

Neither of these depends upon cardinal utility, so have you escaped my theorem? Well, no, actually. You’ve weakened it, to be sure—it is no longer a statement about the fundamental impossibility of welfare-maximizing markets. But applied to the real world, people in Third World poverty are obviously the worst off, and therefore worthy of our help by the Difference Principle; and there are an awful lot of them and very few billionaires, so majority rule says take from the billionaires. The basic conclusion that it is a moral imperative to dramatically reduce global inequality remains—as does the realization that the “efficiency” and “optimality” of unregulated capitalism is a chimera.

How do we measure happiness?

JDN 2457028 EST 20:33.

No, really, I’m asking. I strongly encourage my readers to offer in the comments any ideas they have about the measurement of happiness in the real world; this has been a stumbling block in one of my ongoing research projects.

In one sense the measurement of happiness—or more formally utility—is absolutely fundamental to economics; in another it’s something most economists are astonishingly afraid of even trying to do.

The basic question of economics has nothing to do with money, and is really only incidentally related to “scarce resources” or “the production of goods” (though many textbooks will define economics in this way—apparently implying that a post-scarcity economy is not an economy). The basic question of economics is really this: How do we make people happy?

This must always be the goal in any economic decision, and if we lose sight of that fact we can make some truly awful decisions. Other goals may work sometimes, but they inevitably fail: If you conceive of the goal as “maximize GDP”, then you’ll try to do any policy that will increase the amount of production, even if that production comes at the expense of stress, injury, disease, or pollution. (And doesn’t that sound awfully familiar, particularly here in the US? 40% of Americans report their jobs as “very stressful” or “extremely stressful”.) If you were to conceive of the goal as “maximize the amount of money”, you’d print money as fast as possible and end up with hyperinflation and total economic collapse ala Zimbabwe. If you were to conceive of the goal as “maximize human life”, you’d support methods of increasing population to the point where we had a hundred billion people whose lives were barely worth living. Even if you were to conceive of the goal as “save as many lives as possible”, you’d find yourself investing in whatever would extend lifespan even if it meant enormous pain and suffering—which is a major problem in end-of-life care around the world. No, there is one goal and one goal only: Maximize happiness.

I suppose technically it should be “maximize utility”, but those are in fact basically the same thing as long as “happiness” is broadly conceived as eudaimoniathe joy of a life well-lived—and not a narrow concept of just adding up pleasure and subtracting out pain. The goal is not to maximize the quantity of dopamine and endorphins in your brain; the goal is to achieve a world where people are safe from danger, free to express themselves, with friends and family who love them, who participate in a world that is just and peaceful. We do not want merely the illusion of these things—we want to actually have them. So let me be clear that this is what I mean when I say “maximize happiness”.

The challenge, therefore, is how we figure out if we are doing that. Things like money and GDP are easy to measure; but how do you measure happiness?
Early economists like Adam Smith and John Stuart Mill tried to deal with this question, and while they were not very successful I think they deserve credit for recognizing its importance and trying to resolve it. But sometime around the rise of modern neoclassical economics, economists gave up on the project and instead sought a narrower task, to measure preferences.

This is often called technically ordinal utility, as opposed to cardinal utility; but this terminology obscures the fundamental distinction. Cardinal utility is actual utility; ordinal utility is just preferences.

(The notion that cardinal utility is defined “up to a linear transformation” is really an eminently trivial observation, and it shows just how little physics the physics-envious economists really understand. All we’re talking about here is units of measurement—the same distance is 10.0 inches or 25.4 centimeters, so is distance only defined “up to a linear transformation”? It’s sometimes argued that there is no clear zero—like Fahrenheit and Celsius—but actually it’s pretty clear to me that there is: Zero utility is not existing. So there you go, now you have Kelvin.)

Preferences are a bit easier to measure than happiness, but not by as much as most economists seem to think. If you imagine a small number of options, you can just put them in order from most to least preferred and there you go; and we could imagine asking someone to do that, or—the technique of revealed preferenceuse the choices they make to infer their preferences by assuming that when given the choice of X and Y, choosing X means you prefer X to Y.

Like much of neoclassical theory, this sounds good in principle and utterly collapses when applied to the real world. Above all: How many options do you have? It’s not easy to say, but the number is definitely huge—and both of those facts pose serious problems for a theory of preferences.

The fact that it’s not easy to say means that we don’t have a well-defined set of choices; even if Y is theoretically on the table, people might not realize it, or they might not see that it’s better even though it actually is. Much of our cognitive effort in any decision is actually spent narrowing the decision space—when deciding who to date or where to go to college or even what groceries to buy, simply generating a list of viable options involves a great deal of effort and extremely complex computation. If you have a true utility function, you can satisficechoosing the first option that is above a certain threshold—or engage in constrained optimizationchoosing whether to continue searching or accept your current choice based on how good it is. Under preference theory, there is no such “how good it is” and no such thresholds. You either search forever or choose a cutoff arbitrarily.

Even if we could decide how many options there are in any given choice, in order for this to form a complete guide for human behavior we would need an enormous amount of information. Suppose there are 10 different items I could have or not have; then there are 10! = 3.6 million possible preference orderings. If there were 100 items, there would be 100! = 9e157 possible orderings. It won’t do simply to decide on each item whether I’d like to have it or not. Some things are complements: I prefer to have shoes, but I probably prefer to have $100 and no shoes at all rather than $50 and just a left shoe. Other things are substitutes: I generally prefer eating either a bowl of spaghetti or a pizza, rather than both at the same time. No, the combinations matter, and that means that we have an exponentially increasing decision space every time we add a new option. If there really is no more structure to preferences than this, we have an absurd computational task to make even the most basic decisions.

This is in fact most likely why we have happiness in the first place. Happiness did not emerge from a vacuum; it evolved by natural selection. Why make an organism have feelings? Why make it care about things? Wouldn’t it be easier to just hard-code a list of decisions it should make? No, on the contrary, it would be exponentially more complex. Utility exists precisely because it is more efficient for an organism to like or dislike things by certain amounts rather than trying to define arbitrary preference orderings. Adding a new item means assigning it an emotional value and then slotting it in, instead of comparing it to every single other possibility.

To illustrate this: I like Coke more than I like Pepsi. (Let the flame wars begin?) I also like getting massages more than I like being stabbed. (I imagine less controversy on this point.) But the difference in my mind between massages and stabbings is an awful lot larger than the difference between Coke and Pepsi. Yet according to preference theory (“ordinal utility”), that difference is not meaningful; instead I have to say that I prefer the pair “drink Pepsi and get a massage” to the pair “drink Coke and get stabbed”. There’s no such thing as “a little better” or “a lot worse”; there is only what I prefer over what I do not prefer, and since these can be assigned arbitrarily there is an impossible computational task before me to make even the most basic decisions.

Real utility also allows you to make decisions under risk, to decide when it’s worth taking a chance. Is a 50% chance of $100 worth giving up a guaranteed $50? Probably. Is a 50% chance of $10 million worth giving up a guaranteed $5 million? Not for me. Maybe for Bill Gates. How do I make that decision? It’s not about what I prefer—I do in fact prefer $10 million to $5 million. It’s about how much difference there is in terms of my real happiness—$5 million is almost as good as $10 million, but $100 is a lot better than $50. My marginal utility of wealth—as I discussed in my post on progressive taxation—is a lot steeper at $50 than it is at $5 million. There’s actually a way to use revealed preferences under risk to estimate true (“cardinal”) utility, developed by Von Neumann and Morgenstern. In fact they proved a remarkably strong theorem: If you don’t have a cardinal utility function that you’re maximizing, you can’t make rational decisions under risk. (In fact many of our risk decisions clearly aren’t rational, because we aren’t actually maximizing an expected utility; what we’re actually doing is something more like cumulative prospect theory, the leading cognitive economic theory of risk decisions. We overrespond to extreme but improbable events—like lightning strikes and terrorist attacks—and underrespond to moderate but probable events—like heart attacks and car crashes. We play the lottery but still buy health insurance. We fear Ebola—which has never killed a single American—but not influenza—which kills 10,000 Americans every year.)

A lot of economists would argue that it’s “unscientific”—Kenneth Arrow said “impossible”—to assign this sort of cardinal distance between our choices. But assigning distances between preferences is something we do all the time. Amazon.com lets us vote on a 5-star scale, and very few people send in error reports saying that cardinal utility is meaningless and only preference orderings exist. In 2000 I would have said “I like Gore best, Nader is almost as good, and Bush is pretty awful; but of course they’re all a lot better than the Fascist Party.” If we had simply been able to express those feelings on the 2000 ballot according to a range vote, either Nader would have won and the United States would now have a three-party system (and possibly a nationalized banking system!), or Gore would have won and we would be a decade ahead of where we currently are in preventing and mitigating global warming. Either one of these things would benefit millions of people.

This is extremely important because of another thing that Arrow said was “impossible”—namely, “Arrow’s Impossibility Theorem”. It should be called Arrow’s Range Voting Theorem, because simply by restricting preferences to a well-defined utility and allowing people to make range votes according to that utility, we can fulfill all the requirements that are supposedly “impossible”. The theorem doesn’t say—as it is commonly paraphrased—that there is no fair voting system; it says that range voting is the only fair voting system. A better claim is that there is no perfect voting system, which is true if you mean that there is no way to vote strategically that doesn’t accurately reflect your true beliefs. The Myerson-Satterthwaithe Theorem is then the proper theorem to use; if you could design a voting system that would force you to reveal your beliefs, you could design a market auction that would force you to reveal your optimal price. But the least expressive way to vote in a range vote is to pick your favorite and give them 100% while giving everyone else 0%—which is identical to our current plurality vote system. The worst-case scenario in range voting is our current system.

But the fact that utility exists and matters, unfortunately doesn’t tell us how to measure it. The current state-of-the-art in economics is what’s called “willingness-to-pay”, where we arrange (or observe) decisions people make involving money and try to assign dollar values to each of their choices. This is how you get disturbing calculations like “the lives lost due to air pollution are worth $10.2 billion.”

Why are these calculations disturbing? Because they have the whole thing backwards—people aren’t valuable because they are worth money; money is valuable because it helps people. It’s also really bizarre because it has to be adjusted for inflation. Finally—and this is the point that far too few people appreciate—the value of a dollar is not constant across people. Because different people have different marginal utilities of wealth, something that I would only be willing to pay $1000 for, Bill Gates might be willing to pay $1 million for—and a child in Africa might only be willing to pay $10, because that is all he has to spend. This makes the “willingness-to-pay” a basically meaningless concept independent of whose wealth we are spending.

Utility, on the other hand, might differ between people—but, at least in principle, it can still be added up between them on the same scale. The problem is that “in principle” part: How do we actually measure it?

So far, the best I’ve come up with is to borrow from public health policy and use the QALY, or quality-adjusted life year. By asking people macabre questions like “What is the maximum number of years of your life you would give up to not have a severe migraine every day?” (I’d say about 20—that’s where I feel ambivalent. At 10 I definitely would; at 30 I definitely wouldn’t.) or “What chance of total paralysis would you take in order to avoid being paralyzed from the waist down?” (I’d say about 20%.) we assign utility values: 80 years of migraines is worth giving up 20 years to avoid, so chronic migraine is a quality of life factor of 0.75. Total paralysis is 5 times as bad as paralysis from the waist down, so if waist-down paralysis is a quality of life factor of 0.90 then total paralysis is 0.50.

You can probably already see that there are lots of problems: What if people don’t agree? What if due to framing effects the same person gives different answers to slightly different phrasing? Some conditions will directly bias our judgments—depression being the obvious example. How many years of your life would you give up to not be depressed? Suicide means some people say all of them. How well do we really know our preferences on these sorts of decisions, given that most of them are decisions we will never have to make? It’s difficult enough to make the actual decisions in our lives, let alone hypothetical decisions we’ve never encountered.

Another problem is often suggested as well: How do we apply this methodology outside questions of health? Does it really make sense to ask you how many years of your life drinking Coke or driving your car is worth?
Well, actually… it better, because you make that sort of decision all the time. You drive instead of staying home, because you value where you’re going more than the risk of dying in a car accident. You drive instead of walking because getting there on time is worth that additional risk as well. You eat foods you know aren’t good for you because you think the taste is worth the cost. Indeed, most of us aren’t making most of these decisions very well—maybe you shouldn’t actually drive or drink that Coke. But in order to know that, we need to know how many years of your life a Coke is worth.

As a very rough estimate, I figure you can convert from willingness-to-pay to QALY by dividing by your annual consumption spending Say you spend annually about $20,000—pretty typical for a First World individual. Then $1 is worth about 50 microQALY, or about 26 quality-adjusted life-minutes. Now suppose you are in Third World poverty; your consumption might be only $200 a year, so $1 becomes worth 5 milliQALY, or 1.8 quality-adjusted life-days. The very richest individuals might spend as much as $10 million on consumption, so $1 to them is only worth 100 nanoQALY, or 3 quality-adjusted life-seconds.

That’s an extremely rough estimate, of course; it assumes you are in perfect health, all your time is equally valuable and all your purchasing decisions are optimized by purchasing at marginal utility. Don’t take it too literally; based on the above estimate, an hour to you is worth about $2.30, so it would be worth your while to work for even $3 an hour. Here’s a simple correction we should probably make: if only a third of your time is really usable for work, you should expect at least $6.90 an hour—and hey, that’s a little less than the US minimum wage. So I think we’re in the right order of magnitude, but the details have a long way to go.

So let’s hear it, readers: How do you think we can best measure happiness?