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 37,824^th richest person yesterday, you will be the 37,824^th richest person today; you’ll just have a lot less money while you do so. And if you were the 300,120,916^th richest person, you’ll still be the 300,120,916^th 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.