# Inequality-adjusted GDP and median income

Dec 11 JDN 2459925

There are many problems with GDP as a measure of a nation’s prosperity. For one, GDP ignores natural resources and ecological degradation; so a tree is only counted in GDP once it is cut down. For another, it doesn’t value unpaid work, so caring for a child only increases GDP if you are a paid nanny rather than the child’s parents.

But one of the most obvious problems is the use of an average to evaluate overall prosperity, without considering the level of inequality.

Consider two countries. In Alphania, everyone has an income of about \$50,000. In Betavia, 99% of people have an income of \$1,000 and 1% have an income of \$10 million. What is the per-capita GDP of each country? Alphania’s is \$50,000 of course; but Betavia’s is \$100,990. Does it really make sense to say that Betavia is a more prosperous country? Maybe it has more wealth overall, but its huge inequality means that it is really not at a high level of development. It honestly sounds like an awful place to live.

A much more sensible measure would be something like median income: How much does a typical person have? In Alphania this is still \$50,000; but in Betavia it is only \$1,000.

Yet even this leaves out most of the actual distribution; by definition a median is only determined by what is the 50th percentile. We could vary all other incomes a great deal without changing the median.

A better measure would be some sort of inequality-adjusted per-capita GDP, which rescales GDP based on the level of inequality in a country. But we would need a good way of making that adjustment.

I contend that the most sensible way would be to adopt some kind of model of marginal utility of income, and then figure out what income would correspond to the overall average level of utility.

In other words, average over the level of happiness that people in a country get from their income, and then figure out what level of income would correspond to that level of happiness. If we magically gave everyone the same amount of money, how much would they need to get in order for the average happiness in the country to remain the same?

This is clearly going to be less than the average level of income, because marginal utility of income is decreasing; a dollar is not worth as much in real terms to a rich person as it is to a poor person. So if we could somehow redistribute all income evenly while keeping the average the same, that would actually increase overall happiness (though, for many reasons, we can’t simply do that).

For example, suppose that utility of income is logarithmic: U = ln(I).

This means that the marginal utility of an additional dollar is inversely proportional to how many dollars you already have: U'(I) = 1/I.

It also means that a 1% gain or loss in your income feels about the same regardless of how much income you have: ln((1+r)Y) = ln(Y) + ln(1+r). This seems like a quite reasonable, maybe even a bit conservative, assumption; I suspect that losing 1% of your income actually hurts more when you are poor than when you are rich.

Then the inequality adjusted GDP Y is a value such that ln(Y) is equal to the overall average level of utility: E[U] = ln(Y), so Y = exp(E[U]).

This sounds like a very difficult thing to calculate. But fortunately, the distribution of actual income seems to quite closely follow a log-normal distribution. This means that when we take the logarithm of income to get utility, we just get back a very nice, convenient normal distribution!

In fact, it turns out that for a log-normal distribution, the following holds: exp(E[ln(Y)]) = median(Y)

The income which corresponds to the average utility turns out to simply be the median income! We went looking for a better measure than median income, and ended up finding out that median income was the right measure all along.

This wouldn’t hold for most other distributions; and since real-world economies don’t perfectly follow a log-normal distribution, a more precise estimate would need to be adjusted accordingly. But the approximation is quite good for most countries we have good data on, so even for the ones we don’t, median income is likely a very good estimate.

The ranking of countries by median income isn’t radically different from the ranking by per-capita GDP; rich countries are still rich and poor countries are still poor. But it is different enough to matter.

Luxembourg is in 1st place on both lists. Scandinavian countries and the US are in the top 10 in both cases. So it’s fair to say that #ScandinaviaIsBetter for real, and the US really is so rich that our higher inequality doesn’t make our median income lower than the rest of the First World.

But some countries are quite different. Ireland looks quite good in per-capita GDP, but quite bad in median income. This is because a lot of the GDP in Ireland is actually profits by corporations that are only nominally headquartered in Ireland and don’t actually employ very many people there.

The comparison between the US, the UK, and Canada seems particularly instructive. If you look at per-capita GDP PPP, the US looks much richer at \$75,000 compared to Canada’s \$57,800 (a difference of 29% or 26 log points). But if you look at median personal income, they are nearly equal: \$19,300 in the US and \$18,600 in Canada (3.7% or 3.7 log points).

On the other hand, in per-capita GDP PPP, the UK looks close to Canada at \$55,800 (3.6% or 3.6 lp); but in median income it is dramatically worse, at only \$14,800 (26% or 23 lp). So Canada and the UK have similar overall levels of wealth, but life for a typical Canadian is much better than life for a typical Briton because of the higher inequality in Britain. And the US has more wealth than Canada, but it doesn’t meaningfully improve the lifestyle of a typical American relative to a typical Canadian.

# Why risking nuclear war should be a war crime

Nov 19, JDN 2458078

“What is the value of a human life?” is a notoriously difficult question, probably because people keep trying to answer it in terms of dollars, and it rightfully offends our moral sensibilities to do so. We shouldn’t be valuing people in terms of dollars—we should be valuing dollars in terms of their benefits to people.

So let me ask a simpler question: Does the value of an individual human life increase, decrease, or stay the same, as we increase the number of people in the world?

A case can be made that it should stay the same: Why should my value as a person depend upon how many other people there are? Everything that I am, I still am, whether there are a billion other people or a thousand.

But in fact I think the correct answer is that it decreases. This is for two reasons: First, anything that I can do is less valuable if there are other people who can do it better. This is true whether we’re talking about writing blog posts or ending world hunger. Second, and most importantly, if the number of humans in the world gets small enough, we begin to face danger of total permanent extinction.

If the value of a human life is constant, then 1,000 deaths is equally bad whether it happens in a population of 10,000 or a population of 10 billion. That doesn’t seem right, does it? It seems more reasonable to say that losing ten percent should have a roughly constant effect; in that case losing 1,000 people in a population of 10,000 is equally bad as losing 1 billion in a population of 10 billion. If that seems too strong, we could choose some value in between, and say perhaps that losing 1,000 out of 10,000 is equally bad as losing 1 million out of 1 billion. This would mean that the value of 1 person’s life today is about 1/1,000 of what it was immediately after the Toba Event.

Of course, with such uncertainty, perhaps it’s safest to assume constant value. This seems the fairest, and it is certainly a reasonable approximation.

In any case, I think it should be obvious that the inherent value of a human life does not increase as you add more human lives. Losing 1,000 people out of a population of 7 billion is not worse than losing 1,000 people out of a population of 10,000. That way lies nonsense.

Yet if we agree that the value of a human life is not increasing, this has a very important counter-intuitive consequence: It means that increasing the risk of a global catastrophe is at least as bad as causing a proportional number of deaths. Specifically, it implies that a 1% risk of global nuclear war is worse than killing 10 million people outright.

The calculation is simple: If the value of a human life is a constant V, then the expected utility (admittedly, expected utility theory has its flaws) from killing 10 million people is -10 million V. But the expected utility from a 1% risk of global nuclear war is 1% times -V times the expected number of deaths from such a nuclear war—and I think even 2 billion is a conservative estimate. (0.01)(-2 billion) V = -20 million V.

This probably sounds too abstract, or even cold, so let me put it another way. Suppose we had the choice between two worlds, and these were the only worlds we could choose from. In world A, there are 100 leaders who each make choices that result in 10 million deaths. In world B, there are 100 leaders who each make choices that result in a 1% chance of nuclear war. Which world should we choose?

The choice is a terrible one, to be sure.

In world A, 1 billion people die.

Yet what happens in world B?

If the risks are independent, we can’t just multiply by 100 to get a guarantee of nuclear war. The actual probability is 1-(1-0.01)^100 = 63%. Yet even so, (0.63)(2 billion) = 1.26 billion. The expected number of deaths is higher in world B. Indeed, the most likely scenario is that 2 billion people die.

Yet this is probably too conservative. The risks are most likely positively correlated; two world leaders who each take a 1% chance of nuclear war probably do so in response to one another. Therefore maybe adding up the chances isn’t actually so unreasonable—for all practical intents and purposes, we may be best off considering nuclear war in world B as guaranteed to happen. In that case, world B is even worse.

And that is all assuming that the nuclear war is relatively contained. Major cities are hit, then a peace treaty is signed, and we manage to rebuild human civilization more or less as it was. This is what most experts on the issue believe would happen; but I for one am not so sure. The nuclear winter and total collapse of institutions and infrastructure could result in a global apocalypse that would result in human extinctionnot 2 billion deaths but 7 billion, and an end to all of humanity’s projects once and forever This is the kind of outcome we should be prepared to do almost anything to prevent.

What does this imply for global policy? It means that we should be far more aggressive in punishing any action that seems to bring the world closer to nuclear war. Even tiny increases in risk, of the sort that would ordinarily be considered negligible, are as bad as murder. A measurably large increase is as bad as genocide.

Of course, in practice, we have to be able to measure something in order to punish it. We can’t have politicians imprisoned over 0.000001% chances of nuclear war, because such a chance is so tiny that there would be no way to attain even reasonable certainty that such a change had even occurred, much less who was responsible.

Even for very large chances—and in this context, 1% is very large—it would be highly problematic to directly penalize increasing the probability, as we have no consistent, fair, objective measure of that probability.

Therefore in practice what I think we must do is severely and mercilessly penalize certain types of actions that would be reasonably expected to increase the probability of catastrophic nuclear war.

If we had the chance to start over from the Manhattan Project, maybe simply building a nuclear weapon should be considered a war crime. But at this point, nuclear proliferation has already proceeded far enough that this is no longer a viable option. At least the US and Russia for the time being seem poised to maintain their nuclear arsenals, and in fact it’s probably better for them to keep maintaining and updating them rather than leaving decades-old ICBMs to rot.

First, we probably need to penalize speech that would tend to incite war between nuclear powers. Normally I am fiercely opposed to restrictions on speech, but this is nuclear war we’re talking about. We can’t take any chances on this one. If there is even a slight chance that a leader’s rhetoric might trigger a nuclear conflict, they should be censored, punished, and probably even imprisoned. Making even a veiled threat of nuclear war is like pointing a gun at someone’s head and threatening to shoot them—only the gun is pointed at everyone’s head simultaneously. This isn’t just yelling “fire” in a crowded theater; it’s literally threatening to burn down every theater in the world at once.

Such a regulation must be designed to allow speech that is necessary for diplomatic negotiations, as conflicts will invariably arise between any two countries. We need to find a way to draw the line so that it’s possible for a US President to criticize Russia’s intervention in the Ukraine or for a Chinese President to challenge US trade policy, without being accused of inciting war between nuclear powers. But one thing is quite clear: Wherever we draw that line, President Trump’s statement about “fire and fury” definitely crosses it. This is a direct threat of nuclear war, and it should be considered a war crime. That reason by itself—let alone his web of Russian entanglements and violations of the Emoluments Clause—should be sufficient to not only have Trump removed from office, but to have him tried at the Hague. Impulsiveness and incompetence are no excuse when weapons of mass destruction are involved.

Second, any nuclear policy that would tend to increase first-strike capability rather than second-strike capability should be considered a violation of international law. In case you are unfamiliar with such terms: First-strike capability consists of weapons such as ICBMs that are only viable to use as the opening salvo of an attack, because their launch sites can be easily located and targeted. Second-strike capability consists of weapons such as submarines that are more concealable, so it’s much more likely that they could wait for an attack to happen, confirm who was responsible and how much damage was done, and then retaliate afterward.
Even that retaliation would be difficult to justify: It’s effectively answering genocide with genocide, the ultimate expression of “an eye for an eye” writ large upon humanity’s future. I’ve previously written about my Credible Targeted Conventional Response strategy that makes it both more ethical and more credible to respond to a nuclear attack with a non-nuclear retaliation. But at least second-strike weapons are not inherently only functional at starting a nuclear war. A first-strike weapon can theoretically be fired in response to a surprise attack, but only before the attack hits you—which gives you literally minutes to decide the fate of the world, most likely with only the sketchiest of information upon which to base your decision. Second-strike weapons allow deliberation. They give us a chance to think carefully for a moment before we unleash irrevocable devastation.

All the launch codes should of course be randomized onetime pads for utmost security. But in addition to the launch codes themselves, I believe that anyone who wants to launch a nuclear weapon should be required to type, letter by letter (no copy-pasting), and then have the machine read aloud, Oppenheimer’s line about Shiva, “Now I am become Death, the destroyer of worlds.” Perhaps the passphrase should conclude with something like “I hereby sentence millions of innocent children to death by fire, and millions more to death by cancer.” I want it to be as salient as possible in the heads of every single soldier and technician just exactly how many innocent people they are killing. And if that means they won’t turn the key—so be it. (Indeed, I wouldn’t mind if every Hellfire missile required a passphrase of “By the authority vested in me by the United States of America, I hereby sentence you to death or dismemberment.” Somehow I think our drone strike numbers might go down. And don’t tell me they couldn’t; this isn’t like shooting a rifle in a firefight. These strikes are planned days in advance and specifically designed to be unpredictable by their targets.)

If everyone is going to have guns pointed at each other, at least in a second-strike world they’re wearing body armor and the first one to pull the trigger won’t automatically be the last one left standing.

Third, nuclear non-proliferation treaties need to be strengthened into disarmament treaties, with rapid but achievable timelines for disarmament of all nuclear weapons, starting with the nations that have the largest arsenals. Random inspections of the disarmament should be performed without warning on a frequent schedule. Any nation that is so much as a day late on their disarmament deadlines needs to have its leaders likewise hauled off to the Hague. If there is any doubt at all in your mind whether your government will meet its deadlines, you need to double your disarmament budget. And if your government is too corrupt or too bureaucratic to meet its deadlines even if they try, well, you’d better shape up fast. We’ll keep removing and imprisoning your leaders until you do. Once again, nothing can be left to chance.

We might want to maintain some small nuclear arsenal for the sole purpose of deflecting asteroids from colliding with the Earth. If so, that arsenal should be jointly owned and frequently inspected by both the United States and Russia—not just the nuclear superpowers, but also the only two nations with sufficient rocket launch capability in any case. The launch of the deflection missiles should require joint authorization from the presidents of both nations. But in fact nuclear weapons are probably not necessary for such a deflection; nuclear rockets would probably be a better option. Vaporizing the asteroid wouldn’t accomplish much, even if you could do it; what you actually want to do is impart as much sideways momentum as possible.

What I’m saying probably sounds extreme. It may even seem unjust or irrational. But look at those numbers again. Think carefully about the value of a human life. When we are talking about a risk of total human extinction, this is what rationality looks like. Zero tolerance for drug abuse or even terrorism is a ridiculous policy that does more harm than good. Zero tolerance for risk of nuclear war may be the only hope for humanity’s ongoing survival.

Throughout the vastness of the universe, there are probably billions of civilizations—I need only assume one civilization for every hundred galaxies. Of the civilizations that were unwilling to adopt zero tolerance policies on weapons of mass destruction and bear any cost, however unthinkable, to prevent their own extinction, there is almost boundless diversity, but they all have one thing in common: None of them will exist much longer. The only civilizations that last are the ones that refuse to tolerate weapons of mass destruction.

# Markets value rich people more

Feb 26, JDN 2457811

Competitive markets are optimal at maximizing utility, as long as you value rich people more.

That is literally a theorem in neoclassical economics. I had previously thought that this was something most economists didn’t realize; I had delusions of grandeur that maybe I could finally convince them that this is the case. But no, it turns out this is actually a well-known finding; it’s just that somehow nobody seems to care. Or if they do care, they never talk about it. For all the thousands of papers and articles about the distortions created by minimum wage and capital gains tax, you’d think someone could spare the time to talk about the vastly larger fundamental distortions created by the structure of the market itself.

It’s not as if this is something completely hopeless we could never deal with. A basic income would go a long way toward correcting this distortion, especially if coupled with highly progressive taxes. By creating a hard floor and a soft ceiling on income, you can reduce the inequality that makes these distortions so large.

The basics of the theorem are quite straightforward, so I think it’s worth explaining them here. It’s extremely general; it applies anywhere that goods are allocated by market prices and different individuals have wildly different amounts of wealth.

Suppose that each person has a certain amount of wealth W to spend. Person 1 has W1, person 2 has W2, and so on. They all have some amount of happiness, defined by a utility function, which I’ll assume is only dependent on wealth; this is a massive oversimplification of course, but it wouldn’t substantially change my conclusions to include other factors—it would just make everything more complicated. (In fact, including altruistic motives would make the whole argument stronger, not weaker.) Thus I can write each person’s utility as a function U(W). The rate of change of this utility as wealth increases, the marginal utility of wealth, is denoted U'(W).

By the law of diminishing marginal utility, the marginal utility of wealth U'(W) is decreasing. That is, the more wealth you have, the less each new dollar is worth to you.

Now suppose people are buying goods. Each good C provides some amount of marginal utility U'(C) to the person who buys it. This can vary across individuals; some people like Pepsi, others Coke. This marginal utility is also decreasing; a house is worth a lot more to you if you are living in the street than if you already have a mansion. Ideally we would want the goods to go to the people who want them the most—but as you’ll see in a moment, markets systematically fail to do this.

If people are making their purchases rationally, each person’s willingness-to-pay P for a given good C will be equal to their marginal utility of that good, divided by their marginal utility of wealth:

P = U'(C)/U'(W)

Now consider this from the perspective of society as a whole. If you wanted to maximize utility, you’d equalize marginal utility across individuals (by the Extreme Value Theorem). The idea is that if marginal utility is higher for one person, you should give that person more, because the benefit of what you give them will be larger that way; and if marginal utility is lower for another person, you should give that person less, because the benefit of what you give them will be smaller. When everyone is equal, you are at the maximum.

But market prices don’t actually do this. Instead they equalize over willingness-to-pay. So if you’ve got two individuals 1 and 2, instead of having this:

U'(C1) = U'(C2)

you have this:

P1 = P2

which translates to:

U'(C1)/U'(W1) = U'(C2)/U'(W2)

If the marginal utilities were the same, U'(W1) = U'(W2), we’d be fine; these would give the same results. But that would only happen if W1 = W2, that is, if the two individuals had the same amount of wealth.

Now suppose we were instead maximizing weighted utility, where each person gets a weighting factor A based on how “important” they are or something. If your A is higher, your utility matters more. If we maximized this new weighted utility, we would end up like this:

A1*U'(C1) = A2*U'(C2)

Because person 1’s utility counts for more, their marginal utility also counts for more. This seems very strange; why are we valuing some people more than others? On what grounds?

Yet this is effectively what we’ve already done by using market prices.
Just set:
A = 1/U'(W)

Since marginal utility of wealth is decreasing, 1/U'(W) is higher precisely when W is higher.

How much higher? Well, that depends on the utility function. The two utility functions I find most plausible are logarithmic and harmonic. (Actually I think both apply, one to other-directed spending and the other to self-directed spending.)

If utility is logarithmic:

U = ln(W)

Then marginal utility is inversely proportional:

U'(W) = 1/W

In that case, your value as a human being, as spoken by the One True Market, is precisely equal to your wealth:

A = 1/U'(W) = W

If utility is harmonic, matters are even more severe.

U(W) = 1-1/W

Marginal utility goes as the inverse square of wealth:

U'(W) = 1/W^2

And thus your value, according to the market, is equal to the square of your wealth:

A = 1/U'(W) = W^2

What are we really saying here? Hopefully no one actually believes that Bill Gates is really morally worth 400 trillion times as much as a starving child in Malawi, as the calculation from harmonic utility would imply. (Bill Gates himself certainly doesn’t!) Even the logarithmic utility estimate saying that he’s worth 20 million times as much is pretty hard to believe.

But implicitly, the market “believes” that, because when it decides how to allocate resources, something that is worth 1 microQALY to Bill Gates (about the value a nickel dropped on the floor to you or I) but worth 20 QALY (twenty years of life!) to the Malawian child, will in either case be priced at \$8,000, and since the child doesn’t have \$8,000, it will probably go to Mr. Gates. Perhaps a middle-class American could purchase it, provided it was worth some 0.3 QALY to them.

Now consider that this is happening in every transaction, for every good, in every market. Goods are not being sold to the people who get the most value out of them; they are being sold to the people who have the most money.

And suddenly, the entire edifice of “market efficiency” comes crashing down like a house of cards. A global market that quite efficiently maximizes willingness-to-pay is so thoroughly out of whack when it comes to actually maximizing utility that massive redistribution of wealth could enormously increase human welfare, even if it turned out to cut our total output in half—if utility is harmonic, even if it cut our total output to one-tenth its current value.

The only way to escape this is to argue that marginal utility of wealth is not decreasing, or at least decreasing very, very slowly. Suppose for instance that utility goes as the 0.9 power of wealth:

U(W) = W^0.9

Then marginal utility goes as the -0.1 power of wealth:

U'(W) = 0.9 W^(-0.1)

On this scale, Bill Gates is only worth about 5 times as much as the Malawian child, which in his particular case might actually be too small—if a trolley is about to kill either Bill Gates or 5 Malawian children, I think I save Bill Gates, because he’ll go on to save many more than 5 Malawian children. (Of course, substitute Donald Trump or Charles Koch and I’d let the trolley run over him without a second thought if even a single child is at stake, so it’s not actually a function of wealth.) In any case, a 5 to 1 range across the whole range of human wealth is really not that big a deal. It would introduce some distortions, but not enough to justify any redistribution that would meaningfully reduce overall output.

Of course, that commits you to saying that \$1 to a Malawian child is only worth about \$1.50 to you or I and \$5 to Bill Gates. If you can truly believe this, then perhaps you can sleep at night accepting the outcomes of neoclassical economics. But can you, really, believe that? If you had the choice between an intervention that would give \$100 to each of 10,000 children in Malawi, and another that would give \$50,000 to each of 100 billionaires, would you really choose the billionaires? Do you really think that the world would be better off if you did?

We don’t have precise measurements of marginal utility of wealth, unfortunately. At the moment, I think logarithmic utility is the safest assumption; it’s about the slowest decrease that is consistent with the data we have and it is very intuitive and mathematically tractable. Perhaps I’m wrong and the decrease is even slower than that, say W^(-0.5) (then the market only values billionaires as worth thousands of times as much as starving children). But there’s no way you can go as far as it would take to justify our current distribution of wealth. W^(-0.1) is simply not a plausible value.

And this means that free markets, left to their own devices, will systematically fail to maximize human welfare. We need redistribution—a lot of redistribution. Don’t take my word for it; the math says so.

# “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.

# Two terms in marginal utility of wealth

JDN 2457569

This post is going to be a little wonkier than most; I’m actually trying to sort out my thoughts and draw some public comment on a theory that has been dancing around my head for awhile. The original idea of separating terms in marginal utility of wealth was actually suggested by my boyfriend, and from there I’ve been trying to give it some more mathematical precision to see if I can come up with a way to test it experimentally. My thinking is also influenced by a paper Miles Kimball wrote about the distinction between happiness and utility.

There are lots of ways one could conceivably spend money—everything from watching football games to buying refrigerators to building museums to inventing vaccines. But insofar as we are rational (and we are after all about 90% rational), we’re going to try to spend our money in such a way that its marginal utility is approximately equal across various activities. You’ll buy one refrigerator, maybe two, but not seven, because the marginal utility of refrigerators drops off pretty fast; instead you’ll spend that money elsewhere. You probably won’t buy a house that’s twice as large if it means you can’t afford groceries anymore. I don’t think our spending is truly optimal at maximizing utility, but I think it’s fairly good.

Therefore, it doesn’t make much sense to break down marginal utility of wealth into all these different categories—cars, refrigerators, football games, shoes, and so on—because we already do a fairly good job of equalizing marginal utility across all those different categories. I could see breaking it down into a few specific categories, such as food, housing, transportation, medicine, and entertainment (and this definitely seems useful for making your own household budget); but even then, I don’t get the impression that most people routinely spend too much on one of these categories and not enough on the others.

However, I can think of two quite different fundamental motives behind spending money, which I think are distinct enough to be worth separating.

One way to spend money is on yourself, raising your own standard of living, making yourself more comfortable. This would include both football games and refrigerators, really anything that makes your life better. We could call this the consumption motive, or maybe simply the self-directed motive.

The other way is to spend it on other people, which, depending on your personality can take either the form of philanthropy to help others, or as a means of self-aggrandizement to raise your own relative status. It’s also possible to do both at the same time in various combinations; while the Gates Foundation is almost entirely philanthropic and Trump Tower is almost entirely self-aggrandizing, Carnegie Hall falls somewhere in between, being at once a significant contribution to our society and an obvious attempt to bring praise and adulation to himself. I would also include spending on Veblen goods that are mainly to show off your own wealth and status in this category. We can call this spending the philanthropic/status motive, or simply the other-directed motive.

There is some spending which combines both motives: A car is surely useful, but a Ferrari is mainly for show—but then, a Lexus or a BMW could be either to show off or really because you like the car better. Some form of housing is a basic human need, and bigger, fancier houses are often better, but the main reason one builds mansions in Beverly Hills is to demonstrate to the world that one is fabulously rich. This complicates the theory somewhat, but basically I think the best approach is to try to separate a sort of “spending proportion” on such goods, so that say \$20,000 of the Lexus is for usefulness and \$15,000 is for show. Empirically this might be hard to do, but theoretically it makes sense.

One of the central mysteries in cognitive economics right now is the fact that while self-reported happiness rises very little, if at all, as income increases, a finding which was recently replicated even in poor countries where we might not expect it to be true, nonetheless self-reported satisfaction continues to rise indefinitely. A number of theories have been proposed to explain this apparent paradox.

This model might just be able to account for that, if by “happiness” we’re really talking about the self-directed motive, and by “satisfaction” we’re talking about the other-directed motive. Self-reported happiness seems to obey a rule that \$100 is worth as much to someone with \$10,000 as \$25 is to someone with \$5,000, or \$400 to someone with \$20,000.

Self-reported satisfaction seems to obey a different rule, such that each unit of additional satisfaction requires a roughly equal proportional increase in income.

By having a utility function with two terms, we can account for both of these effects. Total utility will be u(x), happiness h(x), and satisfaction s(x).

u(x) = h(x) + s(x)

To obey the above rule, happiness must obey harmonic utility, like this, for some constants h0 and r:

h(x) = h0 – r/x

Proof of this is straightforward, though to keep it simple I’ve hand-waved why it’s a power law:

Given

h'(2x) = 1/4 h'(x)

Let

h'(x) = r x^n

h'(2x) = r (2x)^n

r (2x)^n = 1/4 r x^n

n = -2

h'(x) = r/x^2

h(x) = – r x^(-1) + C

h(x) = h0 – r/x

Miles Kimball also has some more discussion on his blog about how a utility function of this form works. (His statement about redistribution at the end is kind of baffling though; sure, dollar for dollar, redistributing wealth from the middle class to the poor would produce a higher gain in utility than redistributing wealth from the rich to the middle class. But neither is as good as redistributing from the rich to the poor, and the rich have a lot more dollars to redistribute.)

Satisfaction, however, must obey logarithmic utility, like this, for some constants s0 and k.

The x+1 means that it takes slightly less proportionally to have the same effect as your wealth increases, but it allows the function to be equal to s0 at x=0 instead of going to negative infinity:

s(x) = s0 + k ln(x)

Proof of this is very simple, almost trivial:

Given

s'(x) = k/x

s(x) = k ln(x) + s0

Both of these functions actually have a serious problem that as x approaches zero, they go to negative infinity. For self-directed utility this almost makes sense (if your real consumption goes to zero, you die), but it makes no sense at all for other-directed utility, and since there are causes most of us would willingly die for, the disutility of dying should be large, but not infinite.

Therefore I think it’s probably better to use x +1 in place of x:

h(x) = h0 – r/(x+1)

s(x) = s0 + k ln(x+1)

This makes s0 the baseline satisfaction of having no other-directed spending, though the baseline happiness of zero self-directed spending is actually h0 – r rather than just h0. If we want it to be h0, we could use this form instead:

h(x) = h0 + r x/(x+1)

This looks quite different, but actually only differs by a constant.

Therefore, my final answer for the utility of wealth (or possibly income, or spending? I’m not sure which interpretation is best just yet) is actually this:

u(x) = h(x) + s(x)

h(x) = h0 + r x/(x+1)

s(x) = s0 + k ln(x+1)

Marginal utility is then the derivatives of these:

h'(x) = r/(x+1)^2

s'(x) = k/(x+1)

Let’s assign some values to the constants so that we can actually graph these.

Let h0 = s0 = 0, so our baseline is just zero.

Furthermore, let r = k = 1, which would mean that the value of \$1 is the same whether spent either on yourself or on others, if \$1 is all you have. (This is probably wrong, actually, but it’s the simplest to start with. Shortly I’ll discuss what happens as you vary the ratio k/r.)

Here is the result graphed on a linear scale:

And now, graphed with wealth on a logarithmic scale:

As you can see, self-directed marginal utility drops off much faster than other-directed marginal utility, so the amount you spend on others relative to yourself rapidly increases as your wealth increases. If that doesn’t sound right, remember that I’m including Veblen goods as “other-directed”; when you buy a Ferrari, it’s not really for yourself. While proportional rates of charitable donation do not increase as wealth increases (it’s actually a U-shaped pattern, largely driven by poor people giving to religious institutions), they probably should (people should really stop giving to religious institutions! Even the good ones aren’t cost-effective, and some are very, very bad.). Furthermore, if you include spending on relative power and status as the other-directed motive, that kind of spending clearly does proportionally increase as wealth increases—gotta keep up with those Joneses.

If r/k = 1, that basically means you value others exactly as much as yourself, which I think is implausible (maybe some extreme altruists do that, and Peter Singer seems to think this would be morally optimal). r/k < 1 would mean you should never spend anything on yourself, which not even Peter Singer believes. I think r/k = 10 is a more reasonable estimate.

For any given value of r/k, there is an optimal ratio of self-directed versus other-directed spending, which can vary based on your total wealth.

Actually deriving what the optimal proportion would be requires a whole lot of algebra in a post that probably already has too much algebra, but the point is, there is one, and it will depend strongly on the ratio r/k, that is, the overall relative importance of self-directed versus other-directed motivation.

Take a look at this graph, which uses r/k = 10.

If you only have 2 to spend, you should spend it entirely on yourself, because up to that point the marginal utility of self-directed spending is always higher. If you have 3 to spend, you should spend most of it on yourself, but a little bit on other people, because after you’ve spent about 2.2 on yourself there is more marginal utility for spending on others than on yourself.

If your available wealth is W, you would spend some amount x on yourself, and then W-x on others:

u(x) = h(x) + s(W-x)

u(x) = r x/(x+1) + k ln(W – x + 1)

Then you take the derivative and set it equal to zero to find the local maximum. I’ll spare you the algebra, but this is the result of that optimization:

x = – 1 – r/(2k) + sqrt(r/k) sqrt(2 + W + r/(4k))

As long as k <= r (which more or less means that you care at least as much about yourself as about others—I think this is true of basically everyone) then as long as W > 0 (as long as you have some money to spend) we also have x > 0 (you will spend at least something on yourself).

Below a certain threshold (depending on r/k), the optimal value of x is greater than W, which means that, if possible, you should be receiving donations from other people and spending them on yourself. (Otherwise, just spend everything on yourself). After that, x < W, which means that you should be donating to others. The proportion that you should be donating smoothly increases as W increases, as you can see on this graph (which uses r/k = 10, a figure I find fairly plausible):

While I’m sure no one literally does this calculation, most people do seem to have an intuitive sense that you should donate an increasing proportion of your income to others as your income increases, and similarly that you should pay a higher proportion in taxes. This utility function would justify that—which is something that most proposed utility functions cannot do. In most models there is a hard cutoff where you should donate nothing up to the point where your marginal utility is equal to the marginal utility of donating, and then from that point forward you should donate absolutely everything. Maybe a case can be made for that ethically, but psychologically I think it’s a non-starter.

I’m still not sure exactly how to test this empirically. It’s already quite difficult to get people to answer questions about marginal utility in a way that is meaningful and coherent (people just don’t think about questions like “Which is worth more? \$4 to me now or \$10 if I had twice as much wealth?” on a regular basis). I’m thinking maybe they could play some sort of game where they have the opportunity to make money at the game, but must perform tasks or bear risks to do so, and can then keep the money or donate it to charity. The biggest problem I see with that is that the amounts would probably be too small to really cover a significant part of anyone’s total wealth, and therefore couldn’t cover much of their marginal utility of wealth function either. (This is actually a big problem with a lot of experiments that use risk aversion to try to tease out marginal utility of wealth.) But maybe with a variety of experimental participants, all of whom we get income figures on?

# The difference between price, cost, and value

JDN 2457559

This topic has been on the voting list for my Patreons for several months, but it never quite seems to win the vote. Well, this time it did. I’m glad, because I was tempted to do it anyway.

“Price”, “cost”, and “value”; the words are often used more or less interchangeably, not only by regular people but even by economists. I’ve read papers that talked about “rising labor costs” when what they clearly meant was rising wages—rising labor prices. I’ve read papers that tried to assess the projected “cost” of climate change by using the prices of different commodity futures. And hardly a day goes buy that I don’t see a TV commercial listing one (purely theoretical) price, cutting it in half (to the actual price), and saying they’re now giving you “more value”.

As I’ll get to, there are reasons to think they would be approximately the same for some purposes. Indeed, they would be equal, at the margin, in a perfectly efficient market—that may be why so many economists use them this way, because they implicitly or explicitly assume efficient markets. But they are fundamentally different concepts, and it’s dangerous to equate them casually.

Price

Price is exactly what you think it is: The number of dollars you must pay to purchase something. Most of the time when we talk about “cost” or “value” and then give a dollar figure, we’re actually talking about some notion of price.

Generally we speak in terms of nominal prices, which are the usual concept of prices in actual dollars paid, but sometimes we do also speak in terms of real prices, which are relative prices of different things once you’ve adjusted for overall inflation. “Inflation-adjusted price” can be a somewhat counter-intuitive concept; if a good’s (nominal) price rises, but by less than most other prices have risen, its real price has actually fallen.

You also need to be careful about just what price you’re looking at. When we look at labor prices, for example, we need to consider not only cash wages, but also fringe benefits and other compensation such as stock options. But other than that, prices are fairly straightforward.

Cost

Cost is probably not at all what you think it is. The real cost of something has nothing to do with money; saying that a candy bar “costs \$2” or a computer “costs \$2,000” is at best a somewhat sloppy shorthand and at worst a fundamental distortion of what cost is and why it matters. No, those are prices. The cost of a candy bar is the toil of children in cocoa farms in Cote d’Ivoire. The cost of a computer is the ecological damage and displaced indigenous people caused by coltan mining in Congo.

The cost of something is the harm that it does to human well-being (or for that matter to the well-being of any sentient being). It is not measured in money but in “the sweat of our laborers, the genius of our scientists, the hopes of our children” (to quote Eisenhower, who understood real cost better than most economists). There is also opportunity cost, the real cost we pay not by what we did, but by what we didn’t do—what we could have done instead.

This is important precisely because while costs should always be reduced when possible, prices can in fact be too low—and indeed, artificially low prices of goods due to externalities are probably the leading reason why humanity bears so many excess real costs. If the price of that chocolate bar accurately reflected the suffering of those African children (perhaps by—Gasp! Paying them a fair wage?), and the price of that computer accurately reflected the ecological damage of those coltan mines (a carbon tax, at least?), you might not want to buy them anymore; in which case, you should not have bought them. In fact, as I’ll get to once I discuss value, there is reason to think that even if you would buy them at a price that accurately reflected the dollar value of the real cost to their producers, we would still buy more than we should.

There is a point at which we should still buy things even though people get hurt making them; if you deny this, stop buying literally anything ever again. We don’t like to think about it, but any product we buy did cause some person, in some place, some degree of discomfort or unpleasantness in production. And many quite useful products will in fact cause death to a nonzero number of human beings.

For some products this is only barely true—it’s hard to feel bad for bestselling authors and artists who sell their work for millions, for whatever toil they may put into their work, whatever their elevated suicide rate (which is clearly endogenous; people aren’t randomly assigned to be writers), they also surely enjoy it a good deal of the time, and even if they didn’t, their work sells for millions. But for many products it is quite obviously true: A certain proportion of roofers, steelworkers, and truck drivers will die doing their jobs. We can either accept that, recognizing that it’s worth it to have roofs, steel, and trucking—and by extension, industrial capitalism, and its whole babies not dying thing—or we can give up on the entire project of human civilization, and go back to hunting and gathering; even if we somehow managed to avoid the direct homicide most hunter-gatherers engage in, far more people would simply die of disease or get eaten by predators.

Of course, we should have safety standards; but the benefits of higher safety must be carefully weighed against the potential costs of inefficiency, unemployment, and poverty. Safety regulations can reduce some real costs and increase others, even if they almost always increase prices. A good balance is struck when real cost is minimized, where any additional regulation would increase inefficiency more than it improves safety.

Actually OSHA are unsung heroes for their excellent performance at striking this balance, just as EPA are unsung heroes for their balance in environmental regulations (and that whole cutting crime in half business). If activists are mad at you for not banning everything bad and business owners are mad at you for not letting them do whatever they want, you’re probably doing it right. Would you rather people saved from fires, or fires prevented by good safety procedures? Would you rather murderers imprisoned, or boys who grow up healthy and never become murderers? If an ounce of prevention is worth a pound of cure, why does everyone love firefighters and hate safety regulators?So let me take this opportunity to say thank you, OSHA and EPA, for doing the jobs of firefighters and police way better than they do, and unlike them, never expecting to be lauded for it.

And now back to our regularly scheduled programming. Markets are supposed to reflect costs in prices, which is why it’s not totally nonsensical to say “cost” when you mean “price”; but in fact they aren’t very good at that, for reasons I’ll get to in a moment.

Value

Value is how much something is worth—not to sell it (that’s the price again), but to use it. One of the core principles of economics is that trade is nonzero-sum, because people can exchange goods that they value differently and thereby make everyone better off. They can’t price them differently—the buyer and the seller must agree upon a price to make the trade. But they can value them differently.

To see how this works, let’s look at a very simple toy model, the simplest essence of trade: Alice likes chocolate ice cream, but all she has is a gallon of vanilla ice cream. Bob likes vanilla ice cream, but all he has is a gallon of chocolate ice cream. So Alice and Bob agree to trade their ice cream, and both of them are happier.

We can measure value in “willingness-to-pay” (WTP), the highest price you’d willingly pay for something. That makes value look more like a price; but there are several reasons we must be careful when we do that. The obvious reason is that WTP is obviously going to vary based on overall inflation; since \$5 isn’t worth as much in 2016 as it was in 1956, something with a WTP of \$5 in 1956 would have a much higher WTP in 2016. The not-so-obvious reason is that money is worth less to you the more you have, so we also need to take into account the effect of wealth, and the marginal utility of wealth. The more money you have, the more money you’ll be willing to pay in order to get the same amount of real benefit. (This actually creates some very serious market distortions in the presence of high income inequality, which I may make the subject of a post or even a paper at some point.) Similarly there is “willingness-to-accept” (WTA), the lowest price you’d willingly accept for it. In theory these should be equal; in practice, WTA is usually slightly higher than WTP in what’s called endowment effect.

So to make our model a bit more quantitative, we could suppose that Alice values vanilla at \$5 per gallon and chocolate at \$10 per gallon, while Bob also values vanilla at \$5 per gallon but only values chocolate at \$4 per gallon. (I’m using these numbers to point out that not all the valuations have to be different for trade to be beneficial, as long as some are.) Therefore, if Alice sells her vanilla ice cream to Bob for \$5, both will (just barely) accept that deal; and then Alice can buy chocolate ice cream from Bob for anywhere between \$4 and \$10 and still make both people better off. Let’s say they agree to also sell for \$5, so that no net money is exchanged and it is effectively the same as just trading ice cream for ice cream. In that case, Alice has gained \$5 in consumer surplus (her WTP of \$10 minus the \$5 she paid) while Bob has gained \$1 in producer surplus (the \$5 he received minus his \$4 WTP). The total surplus will be \$6 no matter what price they choose, which we can compute directly from Alice’s WTP of \$10 minus Bob’s WTA of \$4. The price ultimately decides how that total surplus is distributed between the two parties, and in the real world it would very likely be the result of which one is the better negotiator.

The enormous cost of our distorted understanding

(See what I did there?) If markets were perfectly efficient, prices would automatically adjust so that, at the margin, value is equal to price is equal to cost. What I mean by “at the margin” might be clearer with an example: Suppose we’re selling apples. How many apples do you decide to buy? Well, the value of each successive apple to you is lower, the more apples you have (the law of diminishing marginal utility, which unlike most “laws” in economics is actually almost always true). At some point, the value of the next apple will be just barely above what you have to pay for it, so you’ll stop there. By a similar argument, the cost of producing apples increases the more apples you produce (the law of diminishing returns, which is a lot less reliable, more like the Pirate Code), and the producers of apples will keep selling them until the price they can get is only just barely larger than the cost of production. Thus, in the theoretical limit of infinitely-divisible apples and perfect rationality, marginal value = price = marginal cost. In such a world, markets are perfectly efficient and they maximize surplus, which is the difference between value and cost.

But in the real world of course, none of those assumptions are true. No product is infinitely divisible (though the gasoline in a car is obviously a lot more divisible than the car itself). No one is perfectly rational. And worst of all, we’re not measuring value in the same units. As a result, there is basically no reason to think that markets are optimizing anything; their optimization mechanism is setting two things equal that aren’t measured the same way, like trying to achieve thermal equilibrium by matching the temperature of one thing in Celsius to the temperature of other things in Fahrenheit.

An implicit assumption of the above argument that didn’t even seem worth mentioning was that when I set value equal to price and set price equal to cost, I’m setting value equal to cost; transitive property of equality, right? Wrong. The value is equal to the price, as measured by the buyer. The cost is equal to the price, as measured by the seller.

If the buyer and seller have the same marginal utility of wealth, no problem; they are measuring in the same units. But if not, we convert from utility to money and then back to utility, using a different function to convert each time. In the real world, wealth inequality is massive, so it’s wildly implausible that we all have anything close to the same marginal utility of wealth. Maybe that’s close enough if you restrict yourself to middle-class people in the First World; so when a tutoring client pays me, we might really be getting close to setting marginal value equal to marginal cost. But once you include corporations that are owned by billionaires and people who live on \$2 per day, there’s simply no way that those price-to-utility conversions are the same at each end. For Bill Gates, a million dollars is a rounding error. For me, it would buy a house, give me more flexible work options, and keep me out of debt, but not radically change the course of my life. For a child on a cocoa farm in Cote d’Ivoire, it could change her life in ways she can probably not even comprehend.

The market distortions created by this are huge; indeed, most of the fundamental flaws in capitalism as we know it are ultimately traceable to this. Why do Americans throw away enough food to feed all the starving children in Africa? Marginal utility of wealth. Why are Silicon Valley programmers driving the prices for homes in San Francisco higher than most Americans will make in their lifetimes? Marginal utility of wealth. Why are the Koch brothers spending more on this year’s elections than the nominal GDP of the Gambia? Marginal utility of wealth. It’s the sort of pattern that once you see it suddenly seems obvious and undeniable, a paradigm shift a bit like the heliocentric model of the solar system. Forget trade barriers, immigration laws, and taxes; the most important market distortions around the world are all created by wealth inequality. Indeed, the wonder is that markets work as well as they do.

The real challenge is what to do about it, how to reduce this huge inequality of wealth and therefore marginal utility of wealth, without giving up entirely on the undeniable successes of free market capitalism. My hope is that once more people fully appreciate the difference between price, cost, and value, this paradigm shift will be much easier to make; and then perhaps we can all work together to find a solution.

# Is Equal Unfair?

JDN 2457492

Much as you are officially a professional when people start paying you for what you do, I think you are officially a book reviewer when people start sending you books for free asking you to review them for publicity. This has now happened to me, with the book Equal Is Unfair by Don Watkins and Yaron Brook. This post is longer than usual, but in order to be fair to the book’s virtues as well as its flaws, I felt a need to explain quite thoroughly.

It’s a very frustrating book, because at times I find myself agreeing quite strongly with the first part of a paragraph, and then reaching the end of that same paragraph and wanting to press my forehead firmly into the desk in front of me. It makes some really good points, and for the most part uses economic statistics reasonably accurately—but then it rides gleefully down a slippery slope fallacy like a waterslide. But I guess that’s what I should have expected; it’s by leaders of the Ayn Rand Institute, and my experience with reading Ayn Rand is similar to that of Randall Monroe (I’m mainly referring to the alt-text, which uses slightly foul language).

As I kept being jostled between “That’s a very good point.”, “Hmm, that’s an interesting perspective.”, and “How can anyone as educated as you believe anything that stupid!?” I realized that there are actually three books here, interleaved:

1. A decent economics text on the downsides of taxation and regulation and the great success of technology and capitalism at raising the standard of living in the United States, which could have been written by just about any mainstream centrist neoclassical economist—I’d say it reads most like John Taylor or Ken Galbraith. My reactions to this book were things like “That’s a very good point.”, and “Sure, but any economist would agree with that.”

2. An interesting philosophical treatise on the meanings of “equality” and “opportunity” and their application to normative economic policy, as well as about the limitations of statistical data in making political and ethical judgments. It could have been written by Robert Nozick (actually I think much of it was based on Robert Nozick). Some of the arguments are convincing, others are not, and many of the conclusions are taken too far; but it’s well within the space of reasonable philosophical arguments. My reactions to this book were things like “Hmm, that’s an interesting perspective.” and “Your argument is valid, but I think I reject the second premise.”

3. A delusional rant of the sort that could only be penned by a True Believer in the One True Gospel of Ayn Rand, about how poor people are lazy moochers, billionaires are world-changing geniuses whose superior talent and great generosity we should all bow down before, and anyone who would dare suggest that perhaps Steve Jobs got lucky or owes something to the rest of society is an authoritarian Communist who hates all achievement and wants to destroy the American Dream. It was this book that gave me reactions like “How can anyone as educated as you believe anything that stupid!?” and “You clearly have no idea what poverty is like, do you?” and “[expletive] you, you narcissistic ingrate!”

Given that the two co-authors are Executive Director and a fellow of the Ayn Rand Institute, I suppose I should really be pleasantly surprised that books 1 and 2 exist, rather than disappointed by book 3.

As evidence of each of the three books interleaved, I offer the following quotations:

Book 1:

“All else being equal, taxes discourage production and prosperity.” (p. 30)

No reasonable economist would disagree. The key is all else being equal—it rarely is.

“For most of human history, our most pressing problem was getting enough food. Now food is abundant and affordable.” (p.84)

Correct! And worth pointing out, especially to anyone who thinks that economic progress is an illusion or we should go back to pre-industrial farming practices—and such people do exist.

“Wealth creation is first and foremost knowledge creation. And this is why you can add to the list of people who have created the modern world, great thinkers: people such as Euclid, Aristotle, Galileo, Newton, Darwin, Einstein, and a relative handful of others.” (p.90, emph. in orig.)

Absolutely right, though as I’ll get to below there’s something rather notable about that list.

“To be sure, there is competition in an economy, but it’s not a zero-sum game in which some have to lose so that others can win—not in the big picture.” (p. 97)

Yes! Precisely! I wish I could explain to more people—on both the Left and the Right, by the way—that economics is nonzero-sum, and that in the long run competitive markets improve the standard of living of society as a whole, not just the people who win that competition.

Book 2:

“Even opportunities that may come to us without effort on our part—affluent parents, valuable personal connections, a good education—require enormous effort to capitalize on.” (p. 66)

This is sometimes true, but clearly doesn’t apply to things like the Waltons’ inherited billions, for which all they had to do was be born in the right family and not waste their money too extravagantly.

“But life is not a game, and achieving equality of initial chances means forcing people to play by different rules.” (p. 79)

This is an interesting point, and one that I think we should acknowledge; we must treat those born rich differently from those born poor, because their unequal starting positions mean that treating them equally from this point forward would lead to a wildly unfair outcome. If my grandfather stole your grandfather’s wealth and passed it on to me, the fair thing to do is not to treat you and I equally from this point forward—it’s to force me to return what was stolen, insofar as that is possible. And even if we suppose that my grandfather earned far vaster wealth than yours, I think a more limited redistribution remains justified simply to put you and I on a level playing field and ensure fair competition and economic efficiency.

“The key error in this argument is that it totally mischaracterizes what it means to earn something. For the egalitarians, the results of our actions don’t merely have to be under our control, but entirely of our own making. […] But there is nothing like that in reality, and so what the egalitarians are ultimately doing is wiping out the very possibility of earning something.” (p. 193)

The way they use “egalitarian” as an insult is a bit grating, but there clearly are some actual egalitarian philosophers whose views are this extreme, such as G.A. Cohen, James Kwak and Peter Singer. I strongly agree that we need to make a principled distinction between gains that are earned and gains that are unearned, such that both sets are nonempty. Yet while Cohen would seem to make “earned” an empty set, Watkins and Brook very nearly make “unearned” empty—you get what you get, and you deserve it. The only exceptions they seem willing to make are outright theft and, what they consider equivalent, taxation. They have no concept of exploitation, excessive market power, or arbitrage—and while they claim they oppose fraud, they seem to think that only government is capable of it.

Book 3:

“What about government handouts (usually referred to as ‘transfer payments’)?” (p. 23)

Because Social Security is totally just a handout—it’s not like you pay into it your whole life or anything.

“No one cares whether the person who fixes his car or performs his brain surgery or applies for a job at his company is male or female, Indian or Pakistani—he wants to know whether they are competent.” (p.61)

Yes they do. We have direct experimental evidence of this.

“The notion that ‘spending drives the economy’ and that rich people spend less than others isn’t a view seriously entertained by economists,[…]” (p. 110)

The New Synthesis is Keynesian! This is what Milton Friedman was talking about when he said, “We’re all Keynesians now.”

“Because mobility statistics don’t distinguish between those who don’t rise and those who can’t, they are useless when it comes to assessing how healthy mobility is.” (p. 119)

So, if Black people have much lower odds of achieving high incomes even controlling for education, we can’t assume that they are disadvantaged or discriminated against; maybe Black people are just lazy or stupid? Is that what you’re saying here? (I think it might be.)

“Payroll taxes alone amount to 15.3 percent of your income; money that is taken from you and handed out to the elderly. This means that you have to spend more than a month and a half each year working without pay in order to fund other people’s retirement and medical care.” (p. 127)

That is not even close to how taxes work. Taxes are not “taken” from money you’d otherwise get—taxation changes prices and the monetary system depends upon taxation.

“People are poor, in the end, because they have not created enough wealth to make themselves prosperous.” (p. 144)

This sentence was so awful that when I showed it to my boyfriend, he assumed it must be out of context. When I showed him the context, he started swearing the most I’ve heard him swear in a long time, because the context was even worse than it sounds. Yes, this book is literally arguing that the reason people are poor is that they’re just too lazy and stupid to work their way out of poverty.

“No society has fully implemented the egalitarian doctrine, but one came as close as any society can come: Cambodia’s Khmer Rouge.” (p. 207)

Because obviously the problem with the Khmer Rouge was their capital gains taxes. They were just too darn fair, and if they’d been more selfish they would never have committed genocide. (The authors literally appear to believe this.)

So there are my extensive quotations, to show that this really is what the book is saying. Now, a little more summary of the good, the bad, and the ugly.

One good thing is that the authors really do seem to understand fairly well the arguments of their opponents. They quote their opponents extensively, and only a few times did it feel meaningfully out of context. Their use of economic statistics is also fairly good, though occasionally they present misleading numbers or compare two obviously incomparable measures.

One of the core points in Equal is Unfair is quite weak: They argue against the “shared-pie assumption”, which is that we create wealth as a society, and thus the rest of society is owed some portion of the fruits of our efforts. They maintain that this is fundamentally authoritarian and immoral; essentially they believe a totalizing false dichotomy between either absolute laissez-faire or Stalinist Communism.

But the “shared-pie assumption” is not false; we do create wealth as a society. Human cognition is fundamentally social cognition; they said themselves that we depend upon the discoveries of people like Newton and Einstein for our way of life. But it should be obvious we can never pay Einstein back; so instead we must pay forward, to help some child born in the ghetto to rise to become the next Einstein. I agree that we must build a society where opportunity is maximized—and that means, necessarily, redistributing wealth from its current state of absurd and immoral inequality.

I do however agree with another core point, which is that most discussions of inequality rely upon a tacit assumption which is false: They call it the “fixed-pie assumption”.

When you talk about the share of income going to different groups in a population, you have to be careful about the fact that there is not a fixed amount of wealth in a society to be distributed—not a “fixed pie” that we are cutting up and giving around. If it were really true that the rising income share of the top 1% were necessary to maximize the absolute benefits of the bottom 99%, we probably should tolerate that, because the alternative means harming everyone. (In arguing this they quote John Rawls several times with disapprobation, which is baffling because that is exactly what Rawls says.)

Even if that’s true, there is still a case to be made against inequality, because too much wealth in the hands of a few people will give them more power—and unequal power can be dangerous even if wealth is earned, exchanges are uncoerced, and the distribution is optimally efficient. (Watkins and Brook dismiss this contention out of hand, essentially defining beneficent exploitation out of existence.)

Of course, in the real world, there’s no reason to think that the ballooning income share of the top 0.01% in the US is actually associated with improved standard of living for everyone else.

I’ve shown these graphs before, but they bear repeating:

Income shares for the top 1% and especially the top 0.1% and 0.01% have risen dramatically in the last 30 years.

But real median income has only slightly increased during the same period.

Thus, mean income has risen much faster than median income.

While theoretically it could be that the nature of our productivity technology has shifted in such a way that it suddenly became necessary to heap more and more wealth on the top 1% in order to continue increasing national output, there is actually very little evidence of this. On the contrary, as Joseph Stiglitz (Nobel Laureate, you may recall) has documented, the leading cause of our rising inequality appears to be a dramatic increase in rent-seeking, which is to say corruption, exploitation, and monopoly power. (This probably has something to do with why I found in my master’s thesis that rising top income shares correlate quite strongly with rising levels of corruption.)

Now to be fair, the authors of Equal is Unfair do say that they are opposed to rent-seeking, and would like to see it removed. But they have a very odd concept of what rent-seeking entails, and it basically seems to amount to saying that whatever the government does is rent-seeking, whatever corporations do is fair free-market competition. On page 38 they warn us not to assume that government is good and corporations are bad—but actually it’s much more that they assume that government is bad and corporations are good. (The mainstream opinion appears to be actually that both are bad, and we should replace them both with… er… something.)

They do make some other good points I wish more leftists would appreciate, such as the point that while colonialism and imperialism can damage countries that suffer them and make them poorer, they generally do not benefit the countries that commit them and make them richer. The notion that Europe is rich because of imperialism is simply wrong; Europe is rich because of education, technology, and good governance. Indeed, the greatest surge in Europe’s economic growth occurred as the period of imperialism was winding down—when Europeans realized that they would be better off trying to actually invent and produce things rather than stealing them from others.

Likewise, they rightfully demolish notions of primitivism and anti-globalization that I often see bouncing around from folks like Naomi Klein. But these are book 1 messages; any economist would agree that primitivism is a terrible idea, and very few are opposed to globalization per se.

The end of Equal is Unfair gives a five-part plan for unleashing opportunity in America:

1. Abolish all forms of corporate welfare so that no business can gain unfair advantage.

2. Abolish government barriers to work so that every individual can enjoy the dignity of earned success.

3. Phase out the welfare state so that America can once again become the land of self-reliance.

4. Unleash the power of innovation in education by ending the government monopoly on schooling.

5. Liberate innovators from the regulatory shackles that are strangling them.

Number 1 is hard to disagree with, except that they include literally everything the government does that benefits a corporation as corporate welfare, including things like subsidies for solar power that the world desperately needs (or millions of people will die).

Number 2 sounds really great until you realize that they are including all labor standards, environmental standards and safety regulations as “barriers to work”; because it’s such a barrier for children to not be able to work in a factory where your arm can get cut off, and such a barrier that we’ve eliminated lead from gasoline emissions and thereby cut crime in half.

Number 3 could mean a lot of things; if it means replacing the existing system with a basic income I’m all for it. But in fact it seems to mean removing all social insurance whatsoever. Indeed, Watkins and Brook do not appear to believe in social insurance at all. The whole concept of “less fortunate”, “there but for the grace of God go I” seems to elude them. They have no sense that being fortunate in their own lives gives them some duty to help others who were not; they feel no pang of moral obligation whatsoever to help anyone else who needs help. Indeed, they literally mock the idea that human beings are “all in this together”.

They also don’t even seem to believe in public goods, or somehow imagine that rational self-interest could lead people to pay for public goods without any enforcement whatsoever despite the overwhelming incentives to free-ride. (What if you allow people to freely enter a contract that provides such enforcement mechanisms? Oh, you mean like social democracy?)

Regarding number 4, I’d first like to point out that private schools exist. Moreover, so do charter schools in most states, and in states without charter schools there are usually vouchers parents can use to offset the cost of private schools. So while the government has a monopoly in the market share sense—the vast majority of education in the US is public—it does not actually appear to be enforcing a monopoly in the anti-competitive sense—you can go to private school, it’s just too expensive or not as good. Why, it’s almost as if education is a public good or a natural monopoly.

Number 5 also sounds all right, until you see that they actually seem most opposed to antitrust laws of all things. Why would antitrust laws be the ones that bother you? They are designed to increase competition and lower barriers, and largely succeed in doing so (when they are actually enforced, which is rare of late). If you really want to end barriers to innovation and government-granted monopolies, why is it not patents that draw your ire?

They also seem to have trouble with the difference between handicapping and redistribution—they seem to think that the only way to make outcomes more equal is to bring the top down and leave the bottom where it is, and they often use ridiculous examples like “Should we ban reading to your children, because some people don’t?” But of course no serious egalitarian would suggest such a thing. Education isn’t fungible, so it can’t be redistributed. You can take it away (and sometimes you can add it, e.g. public education, which Watkins and Brooks adamantly oppose); but you can’t simply transfer it from one person to another. Money on the other hand, is by definition fungible—that’s kind of what makes it money, really. So when we take a dollar from a rich person and give it to a poor person, the poor person now has an extra dollar. We’ve not simply lowered; we’ve also raised. (In practice it’s a bit more complicated than that, as redistribution can introduce inefficiencies. So realistically maybe we take \$1.00 and give \$0.90; that’s still worth doing in a lot of cases.)

If attributes like intelligence were fungible, I think we’d have a very serious moral question on our hands! It is not obvious to me that the world is better off with its current range of intelligence, compared to a world where geniuses had their excess IQ somehow sucked out and transferred to mentally disabled people. Or if you think that the marginal utility of intelligence is increasing, then maybe we should redistribute IQ upward—take it from some mentally disabled children who aren’t really using it for much and add it onto some geniuses to make them super-geniuses. Of course, the whole notion is ridiculous; you can’t do that. But whereas Watkins and Brook seem to think it’s obvious that we shouldn’t even if we could, I don’t find that obvious at all. You didn’t earn your IQ (for the most part); you don’t seem to deserve it in any deep sense; so why should you get to keep it, if the world would be much better off if you didn’t? Why should other people barely be able to feed themselves so I can be good at calculus? At best, maybe I’m free to keep it—but given the stakes, I’m not even sure that would be justifiable. Peter Singer is right about one thing: You’re not free to let a child drown in a lake just to keep your suit from getting wet.

Ultimately, if you really want to understand what’s going on with Equal is Unfair, consider the following sentence, which I find deeply revealing as to the true objectives of these Objectivists:

“Today, meanwhile, although we have far more liberty than our feudal ancestors, there are countless ways in which the government restricts our freedom to produce and trade including minimum wage laws, rent control, occupational licensing laws, tariffs, union shop laws, antitrust laws, government monopolies such as those granted to the post office and education system, subsidies for industries such as agriculture or wind and solar power, eminent domain laws, wealth redistribution via the welfare state, and the progressive income tax.” (p. 114)

Some of these are things no serious economist would disagree with: We should stop subsidizing agriculture and tariffs should be reduced or removed. Many occupational licenses are clearly unnecessary (though this has a very small impact on inequality in real terms—licensing may stop you from becoming a barber, but it’s not what stops you from becoming a CEO). Others are legitimately controversial: Economists are currently quite divided over whether minimum wage is beneficial or harmful (I lean toward beneficial, but I’d prefer a better solution), as well as how to properly regulate unions so that they give workers much-needed bargaining power without giving unions too much power. But a couple of these are totally backward, exactly contrary to what any mainstream economist would say: Antitrust laws need to be enforced more, not eliminated (don’t take it from me; take it from that well-known Marxist rag The Economist). Subsidies for wind and solar power make the economy more efficient, not less—and suspiciously Watkins and Brook omitted the competing subsidies that actually are harmful, namely those to coal and oil.

Moreover, I think it’s very revealing that they included the word progressive when talking about taxation. In what sense does making a tax progressive undermine our freedom? None, so far as I can tell. The presence of a tax undermines freedom—your freedom to spend that money some other way. Making the tax higher undermines freedom—it’s more money you lose control over. But making the tax progressive increases freedom for some and decreases it for others—and since rich people have lower marginal utility of wealth and are generally more free in substantive terms in general, it really makes the most sense that, holding revenue constant, making a tax progressive generally makes your people more free.

But there’s one thing that making taxes progressive does do: It benefits poor people and hurts rich people. And thus the true agenda of Equal is Unfair becomes clear: They aren’t actually interested in maximizing freedom—if they were, they wouldn’t be complaining about occupational licensing and progressive taxation, they’d be outraged by forced labor, mass incarceration, indefinite detention, and the very real loss of substantive freedom that comes from being born into poverty. They wouldn’t want less redistribution, they’d want more efficient and transparent redistribution—a shift from the current hodgepodge welfare state to a basic income system. They would be less concerned about the “freedom” to pollute the air and water with impunity, and more concerned about the freedom to breathe clean air and drink clean water.

No, what they really believe is rich people are better. They believe that billionaires attained their status not by luck or circumstance, not by corruption or ruthlessness, but by the sheer force of their genius. (This is essentially the entire subject of chapter 6, “The Money-Makers and the Money-Appropriators”, and it’s nauseating.) They describe our financial industry as “fundamentally moral and productive” (p.156)—the industry that you may recall stole millions of homes and laundered money for terrorists. They assert that no sane person could believe that Steve Wozniack got lucky—I maintain no sane person could think otherwise. Yes, he was brilliant; yes, he invented good things. But he had to be at the right place at the right time, in a society that supported and educated him and provided him with customers and employees. You didn’t build that.

Indeed, perhaps most baffling is that they themselves seem to admit that the really great innovators, such as Newton, Einstein, and Darwin, were scientists—but scientists are almost never billionaires. Even the common counterexample, Thomas Edison, is largely false; he mainly plagiarized from Nikola Tesla and appropriated the ideas of his employees. Newton, Einstein and Darwin were all at least upper-middle class (as was Tesla, by the way—he did not die poor as is sometimes portrayed), but they weren’t spectacularly mind-bogglingly rich the way that Steve Jobs and Andrew Carnegie were and Bill Gates and Jeff Bezos are.

Some people clearly have more talent than others, and some people clearly work harder than others, and some people clearly produce more than others. But I just can’t wrap my head around the idea that a single man can work so hard, be so talented, produce so much that he can deserve to have as much wealth as a nation of millions of people produces in a year. Yet, Mark Zuckerberg has that much wealth. Remind me again what he did? Did he cure a disease that was killing millions? Did he colonize another planet? Did he discover a fundamental law of nature? Oh yes, he made a piece of software that’s particularly convenient for talking to your friends. Clearly that is worth the GDP of Latvia. Not that silly Darwin fellow, who only uncovered the fundamental laws of life itself.

In the grand tradition of reducing complex systems to simple numerical values, I give book 1 a 7/10, book 2 a 5/10, and book 3 a 2/10. Equal is Unfair is about 25% book 1, 25% book 2, and 50% book 3, so altogether their final score is, drumroll please: 4/10. Maybe read the first half, I guess? That’s where most of the good stuff is.

# Do we always want to internalize externalities?

JDN 2457437

I often talk about the importance of externalitiesa full discussion in this earlier post, and one of their important implications, the tragedy of the commons, in another. Briefly, externalities are consequences of actions incurred upon people who did not perform those actions. Anything I do affecting you that you had no say in, is an externality.

Usually I’m talking about how we want to internalize externalities, meaning that we set up a system of incentives to make it so that the consequences fall upon the people who chose the actions instead of anyone else. If you pollute a river, you should have to pay to clean it up. If you assault someone, you should serve jail time as punishment. If you invent a new technology, you should be rewarded for it. These are all attempts to internalize externalities.

But today I’m going to push back a little, and ask whether we really always want to internalize externalities. If you think carefully, it’s not hard to come up with scenarios where it actually seems fairer to leave the externality in place, or perhaps reduce it somewhat without eliminating it.

For example, suppose indeed that someone invents a great new technology. To be specific, let’s think about Jonas Salk, inventing the polio vaccine. This vaccine saved the lives of thousands of people and saved millions more from pain and suffering. Its value to society is enormous, and of course Salk deserved to be rewarded for it.

But we did not actually fully internalize the externality. If we had, every family whose child was saved from polio would have had to pay Jonas Salk an amount equal to what they saved on medical treatments as a result, or even an amount somehow equal to the value of their child’s life (imagine how offended people would get if you asked that on a survey!). Those millions of people spared from suffering would need to each pay, at minimum, thousands of dollars to Jonas Salk, making him of course a billionaire.

And indeed this is more or less what would have happened, if he had been willing and able to enforce a patent on the vaccine. The inability of some to pay for the vaccine at its monopoly prices would add some deadweight loss, but even that could be removed if Salk Industries had found a way to offer targeted price vouchers that let them precisely price-discriminate so that every single customer paid exactly what they could afford to pay. If that had happened, we would have fully internalized the externality and therefore maximized economic efficiency.

But doesn’t that sound awful? Doesn’t it sound much worse than what we actually did, where Jonas Salk received a great deal of funding and support from governments and universities, and lived out his life comfortably upper-middle class as a tenured university professor?

Now, perhaps he should have been awarded a Nobel Prize—I take that back, there’s no “perhaps” about it, he definitely should have been awarded a Nobel Prize in Medicine, it’s absurd that he did not—which means that I at least do feel the externality should have been internalized a bit more than it was. But a Nobel Prize is only 10 million SEK, about \$1.1 million. That’s about enough to be independently wealthy and live comfortably for the rest of your life; but it’s a small fraction of the roughly \$7 billion he could have gotten if he had patented the vaccine. Yet while the possible world in which he wins a Nobel is better than this one, I’m fairly well convinced that the possible world in which he patents the vaccine and becomes a billionaire is considerably worse.

Internalizing externalities makes sense if your goal is to maximize total surplus (a concept I explain further in the linked post), but total surplus is actually a terrible measure of human welfare.

Total surplus counts every dollar of willingness-to-pay exactly the same across different people, regardless of whether they live on \$400 per year or \$4 billion.

It also takes no account whatsoever of how wealth is distributed. Suppose a new technology adds \$10 billion in wealth to the world. As far as total surplus, it makes no difference whether that \$10 billion is spread evenly across the entire planet, distributed among a city of a million people, concentrated in a small town of 2,000, or even held entirely in the bank account of a single man.

Particularly a propos of the Salk example, total surplus makes no distinction between these two scenarios: a perfectly-competitive market where everything is sold at a fair price, and a perfectly price-discriminating monopoly, where everything is sold at the very highest possible price each person would be willing to pay.

This is a perfectly-competitive market, where the benefits are more or less equally (in this case exactly equally, but that need not be true in real life) between sellers and buyers:

This is a perfectly price-discriminating monopoly, where the benefits accrue entirely to the corporation selling the good:

In the former case, the company profits, consumers are better off, everyone is happy. In the latter case, the company reaps all the benefits and everyone else is left exactly as they were. In real terms those are obviously very different outcomes—the former being what we want, the latter being the cyberpunk dystopia we seem to be hurtling mercilessly toward. But in terms of total surplus, and therefore the kind of “efficiency” that is maximize by internalizing all externalities, they are indistinguishable.

In fact (as I hope to publish a paper about at some point), the way willingness-to-pay works, it weights rich people more. Redistributing goods from the poor to the rich will typically increase total surplus.

Here’s an example. Suppose there is a cake, which is sufficiently delicious that it offers 2 milliQALY in utility to whoever consumes it (this is a truly fabulous cake). Suppose there are two people to whom we might give this cake: Richie, who has \$10 million in annual income, and Hungry, who has only \$1,000 in annual income. How much will each of them be willing to pay?

Well, assuming logarithmic marginal utility of wealth (which is itself probably biasing slightly in favor of the rich), 1 milliQALY is about \$1 to Hungry, so Hungry will be willing to pay \$2 for the cake. To Richie, however, 1 milliQALY is about \$10,000; so he will be willing to pay a whopping \$20,000 for this cake.

What this means is that the cake will almost certainly be sold to Richie; and if we proposed a policy to redistribute the cake from Richie to Hungry, economists would emerge to tell us that we have just reduced total surplus by \$19,998 and thereby committed a great sin against economic efficiency. They will cajole us into returning the cake to Richie and thus raising total surplus by \$19,998 once more.

This despite the fact that I stipulated that the cake is worth just as much in real terms to Hungry as it is to Richie; the difference is due to their wildly differing marginal utility of wealth.

Indeed, it gets worse, because even if we suppose that the cake is worth much more in real utility to Hungry—because he is in fact hungry—it can still easily turn out that Richie’s willingness-to-pay is substantially higher. Suppose that Hungry actually gets 20 milliQALY out of eating the cake, while Richie still only gets 2 milliQALY. Hungry’s willingness-to-pay is now \$20, but Richie is still going to end up with the cake.

Now, if your thought is, “Why would Richie pay \$20,000, when he can go to another store and get another cake that’s just as good for \$20?” Well, he wouldn’t—but in the sense we mean for total surplus, willingness-to-pay isn’t just what you’d actually be willing to pay given the actual prices of the goods, but the absolute maximum price you’d be willing to pay to get that good under any circumstances. It is instead the marginal utility of the good divided by your marginal utility of wealth. In this sense the cake is “worth” \$20,000 to Richie, and “worth” substantially less to Hungry—but not because it’s actually worth less in real terms, but simply because Richie has so much more money.

Even economists often equate these two, implicitly assuming that we are spending our money up to the point where our marginal willingness-to-pay is the actual price we choose to pay; but in general our willingness-to-pay is higher than the price if we are willing to buy the good at all. The consumer surplus we get from goods is in fact equal to the difference between willingness-to-pay and actual price paid, summed up over all the goods we have purchased.

Internalizing all externalities would definitely maximize total surplus—but would it actually maximize happiness? Probably not.

If you asked most people what their marginal utility of wealth is, they’d have no idea what you’re talking about. But most people do actually have an intuitive sense that a dollar is worth more to a homeless person than it is to a millionaire, and that’s really all we mean by diminishing marginal utility of wealth.

I think the reason we’re uncomfortable with the idea of Jonas Salk getting \$7 billion from selling the polio vaccine, rather than the same number of people getting the polio vaccine and Jonas Salk only getting the \$1.1 million from a Nobel Prize, is that we intuitively grasp that after that \$1.1 million makes him independently wealthy, the rest of the money is just going to sit in some stock account and continue making even more money, while if we’d let the families keep it they would have put it to much better use raising their children who are now protected from polio. We do want to reward Salk for his great accomplishment, but we don’t see why we should keep throwing cash at him when it could obviously be spent in better ways.

And indeed I think this intuition is correct; great accomplishments—which is to say, large positive externalities—should be rewarded, but not in direct proportion. Maybe there should be some threshold above which we say, “You know what? You’re rich enough now; we can stop giving you money.” Or maybe it should simply damp down very quickly, so that a contribution which is worth \$10 billion to the world pays only slightly more than one that is worth \$100 million, but a contribution that is worth \$100,000 pays considerably more than one which is only worth \$10,000.

What it ultimately comes down to is that if we make all the benefits incur to the person who did it, there aren’t any benefits anymore. The whole point of Jonas Salk inventing the polio vaccine (or Einstein discovering relativity, or Darwin figuring out natural selection, or any great achievement) is that it will benefit the rest of humanity, preferably on to future generations. If you managed to fully internalize that externality, this would no longer be true; Salk and Einstein and Darwin would have become fabulously wealthy, and then somehow we’d all have to continue paying into their estates or something an amount equal to the benefits we received from their discoveries. (Every time you use your GPS, pay a royalty to the Einsteins. Every time you take a pill, pay a royalty to the Darwins.) At some point we’d probably get fed up and decide we’re no better off with them than without them—which is exactly by construction how we should feel if the externality were fully internalized.

Internalizing negative externalities is much less problematic—it’s your mess, clean it up. We don’t want other people to be harmed by your actions, and if we can pull that off that’s fantastic. (In reality, we usually can’t fully internalize negative externalities, but we can at least try.)

But maybe internalizing positive externalities really isn’t so great after all.

# Bet five dollars for maximum performance

JDN 2457433

One of the more surprising findings from the study of human behavior under stress is the Yerkes-Dodson curve:

This curve shows how well humans perform at a given task, as a function of how high the stakes are on whether or not they do it properly.

For simple tasks, it says what most people intuitively expect—and what neoclassical economists appear to believe: As the stakes rise, the more highly incentivized you are to do it, and the better you do it.

But for complex tasks, it says something quite different: While increased stakes do raise performance to a point—with nothing at stake at all, people hardly work at all—it is possible to become too incentivized. Formally we say the curve is not monotonic; it has a local maximum.

This is one of many reasons why it’s ridiculous to say that top CEOs should make tens of millions of dollars a year on the rise and fall of their company’s stock price (as a great many economists do in fact say). Even if I believed that stock prices accurately reflect the company’s viability (they do not), and believed that the CEO has a great deal to do with the company’s success, it would still be a case of overincentivizing. When a million dollars rides on a decision, that decision is going to be worse than if the stakes had only been \$100. With this in mind, it’s really not surprising that higher CEO pay is correlated with worse company performance. Stock options are terrible motivators, but do offer a subtle way of making wages adjust to the business cycle.

The reason for this is that as the stakes get higher, we become stressed, and that stress response inhibits our ability to use higher cognitive functions. The sympathetic nervous system evolved to make us very good at fighting or running away in the face of danger, which works well should you ever be attacked by a tiger. It did not evolve to make us good at complex tasks under high stakes, the sort of skill we’d need when calculating the trajectory of an errant spacecraft or disarming a nuclear warhead.

To be fair, most of us never have to worry about piloting errant spacecraft or disarming nuclear warheads—indeed, you’re about as likely to get attacked by a tiger even in today’s world. (The rate of tiger attacks in the US is just under 2 per year, and the rate of manned space launches in the US was about 5 per year until the Space Shuttle was terminated.)

There are certain professions, such as pilots and surgeons, where performing complex tasks under life-or-death pressure is commonplace, but only a small fraction of people take such professions for precisely that reason. And if you’ve ever wondered why we use checklists for pilots and there is discussion of also using checklists for surgeons, this is why—checklists convert a single complex task into many simple tasks, allowing high performance even at extreme stakes.

But we do have to do a fair number of quite complex tasks with stakes that are, if not urgent life-or-death scenarios, then at least actions that affect our long-term life prospects substantially. In my tutoring business I encounter one in particular quite frequently: Standardized tests.

Tests like the SAT, ACT, GRE, LSAT, GMAT, and other assorted acronyms are not literally life-or-death, but they often feel that way to students because they really do have a powerful impact on where you’ll end up in life. Will you get into a good college? Will you get into grad school? Will you get the job you want? Even subtle deviations from the path of optimal academic success can make it much harder to achieve career success in the future.

Of course, these are hardly the only examples. Many jobs require us to complete tasks properly on tight deadlines, or else risk being fired. Working in academia infamously requires publishing in journals in time to rise up the tenure track, or else falling off the track entirely. (This incentivizes the production of huge numbers of papers, whether they’re worth writing or not; yes, the number of papers published goes down after tenure, but is that a bad thing? What we need to know is whether the number of good papers goes down. My suspicion is that most if not all of the reduction in publications is due to not publishing things that weren’t worth publishing.)

So if you are faced with this sort of task, what can you do? If you realize that you are faced with a high-stakes complex task, you know your performance will be bad—which only makes your stress worse!

My advice is to pretend you’re betting five dollars on the outcome.

Ignore all other stakes, and pretend you’re betting five dollars. \$5.00 USD. Do it right and you get a Lincoln; do it wrong and you lose one.
What this does is ensures that you care enough—you don’t want to lose \$5 for no reason—but not too much—if you do lose \$5, you don’t feel like your life is ending. We want to put you near that peak of the Yerkes-Dodson curve.

The great irony here is that you most want to do this when it is most untrue. If you actually do have a task for which you’ve bet \$5 and nothing else rides on it, you don’t need this technique, and any technique to improve your performance is not particularly worthwhile. It’s when you have a standardized test to pass that you really want to use this—and part of me even hopes that people know to do this whenever they have nuclear warheads to disarm. It is precisely when the stakes are highest that you must put those stakes out of your mind.

Why five dollars? Well, the exact amount is arbitrary, but this is at least about the right order of magnitude for most First World individuals. If you really want to get precise, I think the optimal stakes level for maximum performance is something like 100 microQALY per task, and assuming logarithmic utility of wealth, \$5 at the US median household income of \$53,600 is approximately 100 microQALY. If you have a particularly low or high income, feel free to adjust accordingly. Literally you should be prepared to bet about an hour of your life; but we are not accustomed to thinking that way, so use \$5. (I think most people, if asked outright, would radically overestimate what an hour of life is worth to them. “I wouldn’t give up an hour of my life for \$1,000!” Then why do you work at \$20 an hour?)

It’s a simple heuristic, easy to remember, and sometimes effective. Give it a try.

# The moral—and economic—case for progressive taxation

JDN 2456935 PDT 09:44.

Broadly speaking, there are three ways a tax system can be arranged: It can be flat, in which every person pays the same tax rate; it can be regressive, in which people with higher incomes pay lower rates; or it can be progressive, in which case people with higher incomes pay higher rates.

There are certain benefits to a flat tax: Above all, it’s extremely easy to calculate. It’s easy to determine how much revenue a given tax rate will raise; multiply the rate times your GDP. It’s also easy to determine how much a given person should owe; multiply the rate times their income. This also makes the tax withholding process much easier; a fixed proportion can be withheld from all income everyone makes without worrying about how much they made before or are expected to make later. If your goal is minimal bureaucracy, a flat tax does have something to be said for it.

A regressive tax, on the other hand, is just as complicated as a progressive tax but has none of the benefits. It’s unfair because you’re actually taking more from people who can afford the least. (Note that this is true even if the rich actually pay a higher total; the key point, which I will explain in detail shortly, is that a dollar is worth more to you if you don’t have very many.) There is basically no reason you would ever want to have a regressive tax system—and yet, all US states have regressive tax systems. This is mainly because they rely upon sales taxes, which are regressive because rich people spend a smaller portion of what they have. If you make \$10,000 per year, you probably spend \$9,500 (you may even spend \$15,000 and rack up the difference in debt!). If you make \$50,000, you probably spend \$40,000. But if you make \$10 million, you probably only spend \$4 million. Since sales taxes only tax on what you spend, the rich effectively pay a lower rate. This could be corrected to some extent by raising the sales tax on luxury goods—say a 20% rate on wine and a 50% rate on yachts—but this is awkward and very few states even try. Not even my beloved California; they fear drawing the ire of wineries and Silicon Valley.

The best option is to make the tax system progressive. Thomas Piketty has been called a “Communist” for favoring strongly progressive taxation, but in fact most Americans—including Republicans—agree that our tax system should be progressive. (Most Americans also favor cutting the Department of Defense rather than Medicare. This then raises the question: Why isn’t Congress doing that? Why aren’t people voting in representatives to Congress who will do that?) Most people judge whether taxes are fair based on what they themselves pay—which is why, in surveys, the marginal rate on the top 1% is basically unrelated to whether people think taxes are too high, even though that one bracket is the critical decision in deciding any tax system—you can raise about 20% of your revenue by hurting about 1% of your people. In a typical sample of 1,000 respondents, only about 10 are in the top 1%. If you want to run for Congress, the implication is clear: Cut taxes on all but the top 1%, raise them enormously on the top 0.1%, 0.01%, and 0.001%, and leave the 1% the same. People will feel that you’ve made the taxes more fair, and you’ve also raised more revenue. In other words, make the tax system more progressive.

The good news on this front is that the US federal tax system is progressive—barely. Actually the US tax system is especially progressive over the whole distribution—by some measures the most progressive in the world—but the problem is that it’s not nearly progressive enough at the very top, where the real money is. The usual measure based on our Gini coefficient ignores the fact that Warren Buffett pays a lower rate than his secretary. The Gini is based on population, and billionaires are a tiny portion of the population—but they are not a tiny portion of the money. Net wealth of the 400 richest people (the top 0.0001%) adds up to about \$2 trillion (13% of our \$15 trillion GDP, or about 4% of our \$54 trillion net wealth). It also matters of course how you spend your tax revenue; even though Sweden’s tax system is no more progressive than ours and their pre-tax inequality is about the same, their spending is much more targeted at reducing inequality.

Progressive taxation is inherently more fair, because the value of a dollar decreases the more you have. We call this diminishing marginal utility of wealth. There is a debate within the cognitive economics literature about just how quickly the marginal utility of wealth decreases. On the low end, Easterlin argues that it drops off extremely fast, becoming almost negligible as low as \$75,000 per year. This paper is on the high end, arguing that marginal utility decreases “only” as the logarithm of how much you have. That’s what I’ll use in this post, because it’s the most conservative reasonable estimate. I actually think the truth is somewhere in between, with marginal utility decreasing about exponentially.

Logarithms are also really easy to work with, once you get used to them. So let’s say that the amount of happiness (utility) U you get from an amount of income I is like this: U = ln(I)

Now let’s suppose the IRS comes along and taxes your money at a rate r. We must have r < 1, or otherwise they’re trying to take money you don’t have. We don’t need to have r > 0; r < 0 would just mean that you receive more in transfers than you lose in taxes. For the poor we should have r < 0.

Now your happiness is U = ln((1-r)I).

By the magic of logarithms, this is U = ln(I) + ln(1-r).

If r is between 0 and 1, ln(1-r) is negative and you’re losing happiness. (If r < 0, you’re gaining happiness.) The amount of happiness you lose, ln(1-r), is independent of your income. So if your goal is to take a fixed amount of happiness, you should tax at a fixed rate of income—a flat tax.

But that really isn’t fair, is it? If I’m getting 100 utilons of happiness from my money and you’re only getting 2 utilons from your money, then taking that 1 utilon, while it hurts the same—that’s the whole point of utility—leaves you an awful lot worse off than I. It actually makes the ratio between us worse, going from 50 to 1, all the way up to 99 to 1.

Notice how if we had a regressive tax, it would be obviously unfair—we’d actually take more utility from poor people than rich people. I have 100 utilons, you have 2 utilons; the taxes take 1.5 of yours but only 0.5 of mine. That seems frankly outrageous; but it’s what all US states have.

Most of the money you have is ultimately dependent on your society. Let’s say you own a business and made your wealth selling products; it seems like you deserve to have that wealth, doesn’t it? (Don’t get me started on people who inherited their wealth!) Well, in order to do that, you need to have strong institutions of civil government; you need security against invasion; you need protection of property rights and control of crime; you need a customer base who can afford your products (that’s our problem in the Second Depression); you need workers who are healthy and skilled; you need a financial system that provides reliable credit (also a problem). I’m having trouble finding any good research on exactly what proportion of individual wealth is dependent upon the surrounding society, but let’s just say Bill Gates wouldn’t be spending billions fighting malaria in villages in Ghana if he had been born in a village in Ghana. It doesn’t matter how brilliant or determined or hard-working you are, if you live in a society that can’t support economic activity.

In other words, society is giving you a lot of happiness you wouldn’t otherwise have. Because of this, it makes sense that in order to pay for all that stuff society is doing for you (and maintain a stable monetary system), they would tax you according to how much happiness they’re giving you. Hence we shouldn’t tax your money at a constant rate; we should tax your utility at a constant rate and then convert back to money. This defines a new sort of “tax rate” which I’ll call p. Like our tax rate r, p needs to be less than 1, but it doesn’t need to be greater than 0.

Of the U = ln(I) utility you get from your money, you will get to keep U = (1-p) ln(I). Say it’s 10%; then if I have 100 utilons, they take 10 utilons and leave me with 90. If you have 2 utilons, they take 0.2 and leave you with 1.8. The ratio between us remains the same: 50 to 1.

What does this mean for the actual tax rate? It has to be progressive. Very progressive, as a matter of fact. And in particular, progressive all the way up—there is no maximum tax bracket.

The amount of money you had before is just I.

The amount of money you have now can be found as the amount of money I’ that gives you the right amount of utility. U = ln(I’) = (1-p) ln(I). Take the exponential of both sides: I’ = I^(1-p).

The units on this are a bit weird, “dollars to the 0.8 power”? Oddly, this rarely seems to bother economists when they use Cobb-Douglas functions which are like K^(1/3) L^(2/3). It bothers me though; to really make this tax system in practice you’d need to fix the units of measurement, probably using some subsistence level. Say that’s set at \$10,000; instead of saying you make \$2 million, we’d say you make 200 subsistence levels.

The tax rate you pay is then r = 1 – I’/I, which is r = 1 – I^-p. As I increases, I^-p decreases, so r gets closer and closer to 1. It never actually hits 1 (that would be a 100% tax rate, which hardly anyone thinks is fair), but for very large income is does get quite close.

Here, let’s use some actual numbers. Suppose as I said we make the subsistence level \$10,000. Let’s also set p = 0.1, meaning we tax 10% of your utility. Then, if you make the US median individual income, that’s about \$30,000 which would be I = 3. US per-capita GDP of \$55,000 would be I = 5.5, and so on. I’ll ignore incomes below the subsistence level for now—basically what you want to do there is establish a basic income so that nobody is below the subsistence level.

I made a table of tax rates and after-tax incomes that would result:

 Pre-tax income Tax rate After-tax income \$10,000 0.0% \$10,000 \$20,000 6.7% \$18,661 \$30,000 10.4% \$26,879 \$40,000 12.9% \$34,822 \$50,000 14.9% \$42,567 \$60,000 16.4% \$50,158 \$70,000 17.7% \$57,622 \$80,000 18.8% \$64,980 \$90,000 19.7% \$72,247 \$100,000 20.6% \$79,433 \$1,000,000 36.9% \$630,957 \$10,000,000 49.9% \$5,011,872 \$100,000,000 60.2% \$39,810,717 \$1,000,000,000 68.4% \$316,227,766

What if that’s not enough revenue? We could raise to p = 0.2:

 Pre-tax income Tax rate After-tax income \$10,000 0.0% \$10,000 \$20,000 12.9% \$17,411 \$30,000 19.7% \$24,082 \$40,000 24.2% \$30,314 \$50,000 27.5% \$36,239 \$60,000 30.1% \$41,930 \$70,000 32.2% \$47,433 \$80,000 34.0% \$52,780 \$90,000 35.6% \$57,995 \$100,000 36.9% \$63,096 \$1,000,000 60.2% \$398,107 \$10,000,000 74.9% \$2,511,886 \$100,000,000 84.2% \$15,848,932 \$1,000,000,000 90.0% \$100,000,000

The richest 400 people in the US have a combined net wealth of about \$2.2 trillion. If we assume that billionaires make about a 10% return on their net wealth, this 90% rate would raise over \$200 billion just from those 400 billionaires alone, enough to pay all interest on the national debt. Let me say that again: This tax system would raise enough money from a group of people who could fit in a large lecture hall to provide for servicing the national debt. And it could do so indefinitely, because we are only taxing the interest, not the principal.

And what if that’s still not enough? We could raise it even further, to p = 0.3. Now the tax rates look a bit high for most people, but not absurdly so—and notice how the person at the poverty line is still paying nothing, as it should be. The millionaire is unhappy with 75%, but the billionaire is really unhappy with his 97% rate. But the government now has plenty of money.

 Pre-tax income Tax rate After-tax income \$10,000 0.0% \$10,000 \$20,000 18.8% \$16,245 \$30,000 28.1% \$21,577 \$40,000 34.0% \$26,390 \$50,000 38.3% \$30,852 \$60,000 41.6% \$35,051 \$70,000 44.2% \$39,045 \$80,000 46.4% \$42,871 \$90,000 48.3% \$46,555 \$100,000 49.9% \$50,119 \$1,000,000 74.9% \$251,189 \$10,000,000 87.4% \$1,258,925 \$100,000,000 93.7% \$6,309,573 \$1,000,000,000 96.8% \$31,622,777

Is it fair to tax the super-rich at such extreme rates? Well, why wouldn’t it be? They are living fabulously well, and most of their opportunity to do so is dependent upon living in our society. It’s actually not at all unreasonable to think that over 97% of the wealth a billionaire has is dependent upon society in this way—indeed, I think it’s unreasonable to imagine that it’s any less than 99.9%. If you say that the portion a billionaire receives from society is less than 99.9%, you are claiming that it is possible to become a millionaire while living on a desert island. (Remember, 0.1% of \$1 billion is \$1 million.) Forget the money system; do you really think that anything remotely like a millionaire standard of living is possible from catching your own fish and cutting down your own trees?Another fun fact is that this tax system will not change the ordering of income at all. If you were the 37,824th richest person yesterday, you will be the 37,824th richest person today; you’ll just have a lot less money while you do so. And if you were the 300,120,916th richest person, you’ll still be the 300,120,916th person, and probably still have the same amount of money you did before (or even more, if the basic income is doled out on tax day).

And these figures, remember, are based on a conservative estimate of how quickly the marginal utility of wealth decreases. I’m actually pretty well convinced that it’s much faster than that, in which case even these tax rates may not be progressive enough.

Many economists worry that taxes reduce the incentive to work. If you are taxed at 30%, that’s like having a wage that’s 30% lower. It’s not hard to imagine why someone might not work as much if they were being paid 30% less.

But there are actually two effects here. One is the substitution effect: a higher wage gives you more reason to work. The other is the income effect: having more money means that you can meet your needs without working as much.

For low incomes, the substitution effect dominates; if your pay rises from \$12,000 a year to \$15,000, you’re probably going to work more, because you get paid more to work and you’re still hardly wealthy enough to rest on your laurels.

For moderate incomes, the effects actually balance quite well; people who make \$40,000 work about the same number of hours as people who make \$50,000.

For high incomes, the income effect dominates; if your pay rises from \$300,000 to \$400,000, you’re probably going to work less, because you can pay all your bills while putting in less work.

So if you want to maximize work incentives, what should you do? You want to raise the wages of poor people and lower the wages of rich people. In other words, you want very low—or negative—taxes on the lower brackets, and very high taxes on the upper brackets. If you’re genuinely worried about taxes distorting incentives to work, you should be absolutely in favor of progressive taxation.

In conclusion: Because money is worth less to you the more of it you have, in order to take a fixed proportion of the happiness, we should be taking an increasing proportion of the money. In order to be fair in terms of real utility, taxes should be progressive. And this would actually increase work incentives.