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Just in case anyone is actually wondering if it really is the case that some poor households do in fact face marginal income tax rates above 100%, check out this post.

Great link Stephen! Plus, that 2.5% of the (Quebec) population that faces 100% MTRs is in some sense an underestimate. Since there is a *very* strong incentive for people who would face a 100% MTR to reduce their income until they get to a point in the distribution where it's less than 100%.

Does this argument work?:

There is a density function of some unobserved individual attribute A (wage rate, or earning capacity) with a lower bound Amin and upper bound Amax. By What'shisname's revelation principle, we can think of tax being contingent on A subject to people revealing their true A.

Proof by contradiction.

Suppose it were optimal to have the same level of (edit) after-tax income for all individuals from the lower bound Amin to some medium level Amed (this corresponds to a 100% MTR on a range of low levels of income). Then all individuals from Amin to Amed would reveal the same attribute. By some sort of envelope theorem fisrt order of small changes in the tax system around the optimal should only have second order effects on the value of the Social Welfare Function.

Now consider a very small change in the level of taxes on all those at or above above Amed. Cut them all by 1 penny. But the person who has A=Amed (call him Ahmed) now jumps from earning $0 to earning (say) $10,000, and this saves the government $10,000 in benefits. Which is a first order improvement, and should outweigh all the second order costs to the SWF. Which is a contradiction.

I'm out of my depth, of course.

Hmmm. Seem to me that 100% marginal tax rates imposed on the "poor" simply incentivizes crime by cutting off the escape route from poverty. This entails increased government expenditure on security in addition to losses incurred by victims. Unintended consequences of a failed policy, or just bad behavior? The left points to the former and the right to the latter.

The left argues that it is a misperception to say that people have a choice about work, and many shirk their responsibilities voluntarily owing to the availability of public assistance. The position of the left is that in modern societies, employment is not something that most people can create for themselves for a variety of reasons. They are therefore dependent on the economic system for employment above subsistence. or else assistance and a chance to escape from poverty. The notion of individual responsibility in such a system is misplaced, and economic policy needs to address this as a societal responsibility. If people are to be expected to get off assistance, then there has to be a realistic path to it. If the private sector does not provide this, which it manifestly doesn't, then government policy must if the problem is to be solved, and it doesn't yet do so.

One cost-effective and socially responsible solution is for the government as employer of last resort to provide a job guarantee above subsistence level (including income and benefits). The minimum wage is most places is below subsistence level, and marginal taxes on the income of the poor simply incentive cheating, especially when a system is manifestly unfair. After all, the government as lender of last resort had no trouble providing trillions of funding, loans, and guarantees to rescue the financial system from collapse due to "economic and social necessity." What about the millions in the US (and billions around the globe) stuck in poverty?

If you believe in a minimum living standard, the only way to avoid a 100% MTR is to subsidize incomes over greater than than that amount. So if you wanted to limit MTR to 50%, you would provide the minimum at zero income and reduce the benefit by half for anything they earned over it. Not until they earned twice the minimum would benefits cease and taxes start. Lower that to a 33% MTR and you would have to subsidize those up to three times the minimum. This would approach median in most cases assuming a minimum around the poverty level (which isn't really the case since the poverty level doesn't include the value of those benefits and one that did would be a fair amount higher). It wouldn't take long for this subsidy to exceed the total benefit to those below the minimum. Assuming a linear income distribution, a 50% MTR would require double the subsidy, a quarter of which would go to those above the minimum, while a 33% MTR would require triple the subsidy with 44% going to those above the minimum, and these would be underestimates using a more realistic distribution. Still, I wouldn't think 100% optimum unless any work would be more than the minimum but it seems doubtful anything less than 50% would be either due to the size of the subsidy and how much is above minimum. On the other end, I don't think anyone making over $1B does it for the money.

Here's the question: can it ever be part of an optimal tax system to have 100% marginal tax rates on some part of the income distribution?
Sure but they aren't marginal tax-rates as the term is usually used. Rather, an optimal tax system taxes 100% of the first dollars created (including benefits). Because this raises the minimum subsistence income, it stimulates rather than discourages work.

The optimal tax system also has rates that go to zero at the margin. Thus counters the natural marginal disutility to work.

Marginal rates of 100% only make sense if, for some reason, you do not want anyone to earn a wage greater than the cutoff amount. For top incomes, that's a bit harsh, even during the 90% marginal rate era of our past. For bottom incomes, it is perverse, and simply the result of miscoordination. You see a similar effect in the U.S. with the mortgage interest deduction -- there is a dip in taxes paid by the middle class as a result of interest payment tax shelters, resulting in poorer people paying more in taxes than wealthier people that own housing. All these "wrinkles" should be smoothed out, so that the net taxes paid (net of benefit payments) per marginal dollar is a positive concave function of income.

Well, a marginal rate of 100% may not be so bad if it applies to only a small slice of income. In the case of poor households, your chart shows a 100%+ rate applying to those with incomes between $12,500 and $15,000. After that, rates go down.

While the 100% rate would give me a disincentive to grow my income between $12,500 and $15,000, if I expected to earn $30,000 total, it wouldn't do much to keep me at $12,500.

Imagine if, instead of a marginal tax rate of 50%, there was a 100% rate on the first dollar, a 0% rate on the second dollar, a 100% rate on the third dollar, and so on. The effect would be the same as the 50% rate, obviously, so the "temporary" 100% rate wouldn't hurt.

This is probably not what you're looking for, though ... I'm not an economist, and I'm not really sure what you're getting at.

I can't understand why the "left" ... and the "right" ... don't join forces to change this stupid policy.

The electorate (left and right) doesn't understand. Given the time of year, I've been at several social gatherings recently and more than once have had to explain to people who really ought to know better the difference between marginal and average tax rates - it usually comes-up in an anti-government rant when someone claims that if they earned more they'd be poorer because they'd be in a higher tax bracket. Ugh.

I'm both out of my depth ( as always ) and tired, but how can this problem still exist? Wasn't it addressed by Milton Friedman in "The Case of for the Negative Income Tax", published in 1968? See "The Essence of Friedman", p. 60.

On Cigarette Island and Asbestos Town. In Germany in the 1940s a 10000% corporate rate for their chemical company concentration camp suppliers isn't high enough. I with the above who said is too vague. Are just asking if some taxable activities are really bad.

RSJ said: "Marginal rates of 100% only make sense if, for some reason, you do not want anyone to earn a wage greater than the cutoff amount. For top incomes, that's a bit harsh, even during the 90% marginal rate era of our past."

I don't think it is harsh. IMO, it is what is needed to eliminate some excess savers who, whether knowingly or not, are trying to "steal" everyone else's retirement.

Don, the 100%+ marginal tax rate exists because politicians have an incentive to make sure it does. Not sure, if they consciously do this, but it results in the type of candidates who gain their votes from the poor being the ones who most support policies that result in 100%+ marginal tax rates on the poor.

I'd like to back up a bit.

My understanding of optimal tax theory is that it attempts to design taxes that have no influence on behaviour. That would seem to imply no constraints on the (generally) positive feedback loop involved in the gathering of wealth and power.

Why do you want to design a tax system like that?

Keeping Physicists off Wall Street

Income tax can play policy role beyond gov't revenue collection (and redistribution).

A simple stylized model highlights the role that mtr's can have in sorting workers of different abilities. Consider 2 firms and 2 workers, a high ability (h) physicist and a low ability (l) economist.

Let firm 1's profit equal: pft1=f_high-w_high=f_low-w_low, where f is output and w is wage cost. Firm 2's profit is prf2=f_high-w_high>f_low-w_low.

Assume that the physicist's has one unit of labour and solves the following integer programming problem: U=w_1(1+a)(1-t1)+w_2(1-t2) subject to w_1+w_2=<1 and w_1,w_2>=0, where t1 and t2 are the mtr's and w's are wages from working for firm 1 and 2 resp. The "a" is the additional value that the physicist receives from working at firm 1. Let the low ability economist's utility function be identical except for the (1+a) term.

Next assume that society (a third party) values the output-profit combinations where the high ability physicist works for firm 1 while the economist works for firm 2: W(pft1(high),f1_high,pft2(low),f2_low)>W(pft1(low),f1_low,pft2(high),f2_high).

As long as firm 2 can offer the high ability worker a wage equal to or greater than w_2>=w_1(1+a)(1-t1)/(1-t2) she will choose to work for firm 2 even though i) it is not her preferred choice if wages were equal, and ii) it generates less social welfare. This result may mimic physicists working on Wall Street.

In this scenario, it is theoretically possible to use the income tax system to provide incentive for workers to choose their socially optimal employment contract - i.e., if the government can adjust t1 and t2, then they can ensure that the high ability worker is employed by firm 1 and the low ability economist is employed at firm 2.

All this to say: if the income tax policy is used for purposes beyond revenue generation, it is possible to justify mtrs which are greater than 100%.

Let me explain the basic idea behind optimal tax theory and the income tax system.

The "government" is trying to choose an income tax function, T=T(Y), where T is tax paid as a function of income Y. It wants to maximise a Social Welfare Function, where social welfare depends on individuals' utilities. (Simple Utilitarianism, where the govt tries to maximise the sum of individuals' utilities would be one example.) It maximises the SWF subject to its budget constraint, and subject to individuals' response to the Tax system.

Each individual i has a characteristic, Wi, (maybe the wage rate, or earning capacity), that is not observed by the government, which only observes actual income. The individual maximises utility, which depends positively on after tax income, Y-T, and negatively on effort to earn income.

The government faces a trade-off. A high marginal tax rate redistributes from rich (who have low marginal utility of income) to the poor (who have high marginal utility of income), and so tends to increase the SWF. But at the same time, a high marginal tax rate leads individuals to choose less effort to earn income, so reduces the SWF. The optimal tax system is a function T(Y) that picks the best point on that trade-off.

Could it ever be optimal to choose a 100% marginal tax rate?

Brandon's example, and Phillip Huggan's example, are cases where there's a negative externality to the taxed activity. So we want to discourage that activity, even if we tax it so high we lose tax revenue (we are on the wrong side of the Laffer curve). Fair enough, but I don't think it's plausible that we want the poor to stay on welfare. If anything, there's a positive externality to giving them an incentive to work, since they won't turn to crime.

Doc's explanation of why 100% MTRs exist makes some sense, but does assume the poor are too ignorant to figure it out. Some must get pissed off because they are stuck in a poverty trap. Though Patrick's anecdote suggests I may be too optimistic about people's knowledge.

My preferred explanation, like RSJ, is some sort of political coordination problem. It's only when you add together all the different tax-benefits programs that you see the problem.

But these are all positive explanations. What about the normative question?

Yes, for brothel janitor and playboy photographer. I suspect you are looking for a more general application.
Generally low wage jobs are unsatisfying and self-selected careers are. Positions that report high levels of self-satisfaction, say an entrepreneur, could be taxed higher without leakage than jobs like nurses that have high dropout rates. But I'd doubt you'd want to penalize job happiness and good career planning.

- at a 100% marginal tax rate, there's no incentive to work. But there's not much of an incentive to take a minimum wage job at a 99% mtr either. At what mtr would 99% of people decide it's not worth taking a minimum wage job? Let's assume that this MTR is, say, 60%. What that tells you is that, unless you can keep the MTR below (say) 60%, you might as well jack it all the way up to 100%, because you have no work incentives in your system anyways, and you want the lack of work incentives to apply to as small a part of the income distribution as possible. You typically wouldn't get this out of a formal model, because formal models assume away these type of corner solutions.

- now take that maximum possible incentive-preserving m.t.r. and ask: is it possible to, at this m.t.r. (a) provide people with no other means of support with a basic subsistence (b) achieve budget balance without creating unacceptably high m.t.rs elsewhere? If not, then from a practical point of view, a 100% m.t.r. on a small portion of the income distribution may be the best compromise. Sure, a small portion of the income distribution has no work incentives, but it's not like there's huge numbers of jobs available for unskilled workers in many parts of rural Canada anyways.

- it's easy to dwell on income assistance programs' high marginal tax rates - if for no other reasons than (a) they make really cool exam problems (great exercises for drawing budget constraints) and (b) we can model them. What we typically can't model is the effect of asset tests, because we don't have very good wealth data, and asset tests aren't marginal - either you have assets above a certain level and you aren't eligible for income assistance, or you don't and you are. (see, for example, http://www.peelregion.ca/ow/applying/calculator.asp to figure out how various levels of income and assets affect income assistance eligibility).

But I think these savings effects are if anything a greater policy concern.

Frances: At 100% MTR we *know* we are on the wrong side of the Laffer Curve. I interpret you as saying that at (say) 60% MTR we might be on the wrong side of the Laffer Curve too, in some parts of the income distribution. OK. Sounds entirely plausible. But then I would re-phrase my question as: "Is it ever optimal to have a tax system that puts some slice of the income distribution on the wrong side of the Laffer Curve?"

And I think you are saying the answer is "maybe, yes"? And (given you are more familiar with that literature than me, even if it's not exactly your cup of tea), I take that to mean that optimal tax theorists haven't explicitly addressed this problem?

Suppose there were a gap in the attribute distribution. Then I could understand that you might have 100% or higher MTRs in that gap. Because nobody would be affected by the disincentives. By extension, if one part of the distribution were rather thin (if it were bi-modal for example), you might get the same result?

Damn! I have to get my head around this. It just doesn't seem right.

The asset-testing may indeed be a bigger deal, policy-wise. But I've got my teeth on the income angle, and can't let go!


The curve Mr. Laffer famously drew on the napkin shows how government revenue changes as a single tax rate changes. But the taxation function specifies a tax rate at every income level. If incomes range from $1 to $1 million, you could draw a million laffer curves if you just had enough napkins. For example, you could draw a Laffer curve showing how government revenue changes as the tax rate on, say, the 40,000th dollar earned changes.

Did you ever play those arcade games where a bunch of stuff pops out of a a bunch of different holes and you have to hammer them down with a mallet? That's the way to think of marginal tax rates. If you bang one down another one pops up.

Here's another way to think of the argument I was making earlier. Suppose the tax rate is 100% at one point in the income distribution and 50% at another point. Is it obviously a good idea to replace this with a marginal tax rate of 75% or even 65% at both points in the distribution? Not if incentive effects go to zero at a mtr of, say, 60%.

(and you can't replace the 100% with 60% and keep the 50% unless you reduce either the benefits paid to the poor or the taxes raised. It's like ripping off a bandaid quickly rather than slowly - though I've heard about some recent research that actually suggests ripping bandaids off slowly might be a better strategy).

Frances: You are right.

Let me have another go at it.

For any given income tax function, T(Y), you can (in principle) solve for the mapping from the distribution of individual's attributes Ai into a distribution of individual's incomes Yi. So that an individual with attribute Ai will choose, given that tax function, to earn income Yi.

What I am saying (my conjecture) is that an optimal tax function can never create "gaps" in that mapping. In other words, If the distribution F(A) is continuous, the distribution of income F(Y) must also be continuous. If two individuals i and j are arbitrarily close to one another in attributes Ai and Aj, they ought also be arbitrarily close to one another in incomes Yi and Yj. When you have a 100% MTR over some range of incomes then, theoretically, there should be a "gap" in the distribution of income over that range. (But you would also probably get a "gap" at less than 100% MTR, for some parameter values.)

If there were a "gap" in the mapping from attributes into income, then it would not be possible to create a reverse mapping from income into attributes. My conjecture is that an optimal tax function would always have a reversible mapping.

And the "proof" of my conjecture is that if there were a "gap", an infinitesimally small change in the tax system would cause some individuals to make a large "jump" to a higher level of income. And that "jump" would violate the FOC for an optimum. The costs of changing the tax function would be infinitesimal, but the benefits would be finite.

Normally, when you start with the optimal tax function, and make an infinitesimally small reduction in the MTR at some point, the costs are an infinitesimally small increase in inequality, and the benefits are an infinitesimally small increase in efficiency (as people work a tiny bit harder). But if there's a "gap", some people jump to the top of the gap, so the benefits are not infinitesimal.

I wish I understood optimal control theory!

Where are all the keen young maths types when you need them??

But my intuition is starting to firm up.

No doubt a high MTR can be a disincentive to work, but so can a low income. A 100% MTR is a complete disincentive to work for less than the amount it is effective over; they could as well donate their time if they are working for other purposes. This would diminish greatly if their income would be substantially greater. A 50% MTR is also a disincentive but would still represent a subsidy to those in it, so would it really act as more of a disincentive than the lower incomes they would have received otherwise? I am doubtful. On the other hand, if these subsidies fall on more workers, how much of it falls to them and how much to their employers? Reducing subsidies may result in higher nominal incomes for them.

Nick, if you look at some of the papers in the optimal tax literature, e.g. this classic by Diamond, http://www.jourdan.ens.fr/piketty/fichiers/enseig/ecoineg/articl/Diamond1998.pdf, you'll see that with not unreasonable parameter values the optimal marginal tax rate can easily reach 80 or 90%. Basically this is because many estimates find that the elasticity of male labour supply is close to zero, so with any concern at all for equity it's easy to get a very high marginal tax rate.

The argument that you're making above seems to rely upon having continuous distributions etc. If you assume that things are nice and smooth then, no, you're not going to get a corner solution, which is what a 100% mtr is. But with a perfectly inelastic labour supply it's easy to generate a 100% m.t.r. - that's the classic utilitarian argument for equalized post-tax income distribution.

If you're really seriously curious about this, Stan Winer's work on probabilistic voting models (which is related to Steve's recent post) is analytically very similar to optimal tax theory.

For me, the work/reward ratio from optimal tax theory is just way too high.


I would say it's possible. Suppose there's a class of workers who have significant bargaining power (like, say, CEOs or professional athletes). Wages for this group aren't driven by the marginal product of their work for the company, rather they are set to keep these workers from going to another company. No additional work takes place if they are paid more (we assume these people are already working as well/hard as they can).

If one company raises the salary they are willing to offer, others must follow suit. No company benefits from increasing wages to this class of workers. (The situation is in fact comparable to an arms race, where the other side's willingness to increase expenditures forces you to keep up or lose something.)

In this situation a 100% MTR or salary cap has a pure re-distribution effect, where that money that might have been used to raise a small number of peoples' salary even further is used either to retain workers lower down the scale, invest, or paid out to shareholders. If we set the cap really high, and marginal utility is strictly diminishing, all of those potential outcomes is a net benefit to society. (The cap works like mutually binding arms control imposed by a third party, which is beneficial to all participants in an arms race even though it restricts their theoretical freedom of action.)

My understanding of optimal tax theory is the same as that Facebook guy's. The objective, according to the theory, should simply be to create a tax system that distorts behaviour as little as possible while still collecting necessary revenues.

Would 100% marginal tax rates be more distortionary than other possible taxation options? My guess is yes — if you're not any better off earning extra money, why would you earn it? The exception would be if your effort cost in earning the money is 0 or negative (e.g. the "utility from working" option you said to exclude).

Still, you might be able to make a case for 0 effort cost at really high income levels; if you're really rich, your income probably comes more from investments than labour, and in that case it may be pretty costless to earn additional profits since your investments basically do the work themselves. If all my income just comes from bonds, for example, it may be costlier for me to sell my bonds than to hang on to them. So you might be able to argue that a 100% marginal tax rate for the really rich is less distortionary than other taxes, but it's a real stretch. Especially if we have an open economy, where such a tax would likely distort behaviour by incentivizing someone to move to a different tax jurisdiction.

Nick, 100%+ marginal tax rates, usually masquerade as handouts, so its very difficult for an individual to realize that they are being harmed by them.

Once someone develops the thought processes and mentality required to figure that out on their own, they are probably about to leave the bottom quintile anyway. So, I don't think it makes that much of a difference.

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