The Real Bracketology

“Hooray, it’s tax season!” said nobody ever, except for the clinically ill and the clinically sarcastic. But I’m here, in this season of paperwork and low spirits, to offer a hymn of praise to the poor, misunderstood public servants that make income taxes work. No, not IRS agents, although goodness knows those sorry devils could use a defender or two.

Tax brackets enact a simple idea: that not all income should be taxed equally. The first \$8000 that I earn is likely to be precious—rent and grocery money.

For me, income above \$30,000 is a little less precious—it’s vacation, savings, and new computer money. And if my teaching/blogging career somehow pulled in over \$250,000 (I can dream, right?), those last few thousand bucks would be nice, but not terribly precious.

That’s why the tax code grabs a larger share from higher income levels. The more money you have, the less precious an extra dollar becomes. But this is where tax brackets have been maligned and misinterpreted. They don’t apply to people. They apply to money.

To see the difference, consider a simple two-bracket world, where all income below \$8000 is taxed at 10% and all income above \$8000 is taxed at 20%.

If I earn \$1, I pay that low 10% tax rate, which comes out to \$0.10.

If I earn \$2, I’m still paying just 10%, so my total tax is \$0.20.

If I earn \$3, I’m still paying 10%, so I pay \$0.30 total.

You get the idea. This continues until I earn \$8000. All that income is taxed at 10%, so my total income tax bill comes out to \$800.

But then my boss comes up and says, “Hey, great work teaching/blogging, Ben! Here’s an extra dollar!” Now I’ve made \$8001.

I assume this means I pay 20% tax now… but 20% of \$8001 comes out to roughly \$1600. That’s double the total tax I was paying before the measly bonus. Did an extra \$1 in earnings really just cost me \$800 in extra taxes?!

No. That’d be crazy.

The 20% tax applies only to my last dollar earned. That’s \$0.20. So my total tax didn’t skyrocket from \$800 to \$1600. It just nudged upward slightly, from \$800 to \$800.20.

As I said, brackets don’t make distinctions among people. They only distinguish between different types of money. The first \$8000 I earn will always face a 10% tax rate, no matter how much more I go on to earn. Even if I earned a billion dollars, the first \$8000 would get taxed at 10%. You, I, and Bill Gates all pay that same 10% on our first \$8000.

This system avoids perverse scenarios where earning extra income actually winds up costing me money. In the good ol’ US of A, earning an extra dollar will always increase your taxes, but it’ll always increase your after-tax income, too. When people fret, “Oh no, that’ll bump me up into the next tax bracket!” they’re either joking, crazy, or confused about how taxes work.

When Congress debates, say, a 40% tax rate for people earning more than \$250,000, this doesn’t mean a family making \$250,001 has to pay 40% of their income in taxes. They’ll only pay 40% on that very last dollar earned.

Does this mean a progressive income tax is better than other systems, like a value-added tax? Not necessarily. But while you’re grumbling about taxes this month, remember to tip your cap to the humble tax bracket, the unsung hero of the IRS.

P.S. Check out HAL 10000’s comment below for a more nuanced take on the rare situations when earning \$1 extra really can increase your taxes by more than \$1.

29 thoughts on “The Real Bracketology”

1. robly18 says:

“Hate to bracket to you”
Remind me to plagiarize that one someday.

In all seriousness, where do you get these puns?

1. You just wait quietly in the bushes and wait for the puns to come to you, really.

1. Trey says:

I prefer waiting in less conspicuous places, like a classroom.

2. If you have a dry sense of humor, you typically are a very punny person. That’s only on a good day, though. On a not-so-good day, you are either sarcastic or a smart-aleck. I guess the comedians are right: timing is everything!

2. I agree with your basic point, but this *sometimes* isn’t true because of the complications of the tax code. Before 1986, our tax code was so complicated and had so many phase-outs and phase-ins that it was possible, at certain inflection points, to pay a marginal rate over 100%. My dad gave some of his secretaries raises only to see their take-home pay drop because of those wrinkles in the code.

That may be marginally true now, as some PPACA credits phase out or cut off at certain income levels, so there are some narrow ranges, around \$30k or so, where the self-insured will face 100%+ marginal rates. The AMT can also do this because you have to figure your taxes two ways. Those windows of income are somewhat narrow, but they do exist.

1. Thanks for the extra details and more nuanced take. I’ll add a postscript directing people towards your comment.

1. Anthony says:

The other complication comes from withholdings, especially if it’s done poorly.

The typical withholding schedule assumes that whatever you made for this 2-week period will be your income for every two-week period all year, and directs you to withhold taxes at the appropriate rate. If you’re on straight salary, or get exactly 40 hours every week, that’s fine. But if you get overtime, or are unemployed for part of the year (lots of teachers get 10 monthly checks, for example), getting a short-term pay bump can really mess with this. For example, if you normally make \$25/hour (\$52k/year), but in one pay period, you worked 120 hours, getting 40 hours OT, and getting \$3500 for that check instead of \$2000, the withholdings are going to eat a *lot* of the additional pay. It’s worse if the payroll department uses some combination of “what I earned this period” and “how much has been withheld so far this year”; that could actually leave you a lower paycheck. The difference would come back when you got your refund, but that’s a long way off.

It’s also bad if you have highly variable income – say you have those 120-hour fortnights interspersed with ones where you get only 30 hours for both weeks, and so your pay in the slow fortnights is \$750. Over four weeks, you’re getting \$4250, but the withholdings on \$750 for two weeks are really low (because so much of it is in the zero bracket), and that may overbalance the additional withholding from the busy periods, leaving you under-withheld.

3. An interesting and comical look at taxes. However in my libertarian eyes I still say increasing taxes based on how much is earned is not right. I say an equal tax rate for everyone and the Government can compensate with what’s “lost” by not spending so much on pointless things (such as the ten of thousands of unused government buildings that they pay to have maintained and guarded or funding to military bases throughout the world that give America that “World Police” image that we all hate).

1. I prefer a progressive tax, but I can see the appeal of a flat tax. My understanding is that most “fair tax” systems involve a flat income or sales tax, plus a standard per-person rebate (which will lead to an effectively lower tax rate among those with low income, but would be closer to “flat” than our current system).

There are some aspects of public policy where I find it important to stand on principle, but the tax code isn’t really one of them. I think there are lots of plausible ways we could design the tax system, and as long as we’re (1) Not running big deficits, (2) Adequately funding valuable government programs, and (3) Not imposing a crippling tax burden on any citizens, I’m fine with a variety of approaches.

1. I humbly disagree. This country was formed in a revolution due to the lack of freedoms and increased taxes. I think how taxes are collected are one of the most important things in this country. I’m paying an arm and a leg and I’m in the lower tax bracket. What’s my money being used for? An American empire, 11 different spy agencies (more than a few are specifically targeting U.S. citizens), and hundreds of programs that should never have existed or at the very least they should have been privatized. I don’t make a lot of money therefore I am very picky about what I spend it on; I feel the same way about my taxes.

1. Well, we’re probably not going to find a middle ground, but thanks for sharing your thoughts. I think of the American Revolution as being about lack of representation in Parliament more than the specifics of taxation. I suspect that a similar set of taxes might have been accepted by the colonies if they’d felt like they had a voice in the process. I also like to separate the question of how money is collected from how it’s used, since it’s quite possible to reallocate spending without altering the relative tax burden among citizens (or vice versa).

2. Well I appreciate your civil attitude about it. I try not to talk much about politics on WordPress because I don’t want my blog to be thought of as a soapbox for politics and because many people would rather start insulting and getting mad about it rather than talk.

3. The population of United States before the war if I remember correctly was about 1/3 of England’s. Which means they would take at most 25% of the seats and will still get taxes slapped onto them. Also they probably won’t be able to get any seats in the House of Lords, thus making their representation symbolic and meaningless.
Who ever wants to pay taxes? So why does this government have a bias against high income earners? This discourages growth and people striving for a better income. The government already collects more money from rich people even if they were taxed at the same rate, but upping the rates as income grows seems to me like taxing people’s success. The government relocating wealth is often inefficient with lots of leaks, and seems to me that it encourages people to stay where they are at to avoid further taxing of success or lost of government money.

4. You sir are thinking like a libertarian. I wholly believe that both the GOP and the democrats allow this to encourage class warfare and maintain a lower class.

4. The reason that progressive taxes are fair is simply that \$1 is worth more to those who have less. As a simple example, a “flat amount” tax is obviously unfair. If a global tax of \$1000 per year were imposed on everyone, that would be very little to me and a pinprick to Bill Gates, but for any of the 1.1 billion people in the world in absolute poverty, it would amount to a confiscation of 100% of their income, leaving them to starve. Furthermore, the discounted value of a marginal dollar drops faster than linearly, which is why the “flat rate” tax is not fair either.

That said, the U.S. income tax has been seriously perverted from its original purposes. Back in 1913 when the modern income tax was established, there were two brackets: 0% for the first \$500,000 and 7% for the rest. But \$500,000 in 1913 would be more than \$10 million today! As a result, essentially all wage income was exempt from tax, and the tax fell mostly on income from inheritances and on economic rent from land. Now, of course, it falls most heavily on the middle class, most of whose income is wages.

As a Georgist libertarian, I hold that taxes should fall on the value of natural resources. Land (especially land in cities) is the most important variety, but things like oil, first-growth forest, and so on also count, as do the value of goverment-granted monopolies. Such a taxation system is effective, because evading it is very difficult. It is also just, because it allows the efficient exploitation of shared natural resources while returning their value to all of us. Finally, it is unshiftable: a tax on resource value cannot be transferred to the consumer of the resource (the “Henry George theorem” of economics).

1. Trey says:

I believe you’ve mistaken a lump sum tax for a flat rate tax. A flat rate tax would tax say 10% to everyone for every dollar earned. A lump sum tax is when you’re taxed say \$1000 a year.

1. jt512 says:

He did confuse them, but his doing so illustrates the absurdity of the libertarian position that a uniform income tax rate is fair. After all, what could be “fairer” than to have every citizen pay exact same absolute amount of tax.

2. I didn’t confuse them, actually; see the last sentence of my first paragraph. It is not all one whether you make \$10,000 a year (far less than minimum wage) and have to pay \$1000, or you make \$1,000,000 a year and have to pay \$100,000. The first person suffers a far greater impact from the flat-rate tax. Sales taxes are flat-rate in many states (in New York, where I live, food is exempt) and have far more impact on the poor than on the rich.

1. ericastor says:

Sure, but combine a flat tax rate with a basic income system, also known as a “negative income tax”, and you get something… well, at least interesting. Most proposals start somewhere just above \$10,000 for a single person with no dependents, and use a tax somewhere near 25% – either income or sales tax; the latter is much easier to collect, but less fair by some standards. Let’s use a simple standard – \$10,000 basic income to every person, regardless of circumstances, and a flat 25% tax rate on income (regardless of source – effectively reducing basic income to \$7,500, but we’ll cope for this example).

Applying this to your hypothetical examples: one person makes \$10,000 per year by working, receives a “prebate” of \$10,000 per year, and pays an overall 25% in taxes – netting \$15,000 per year, with an effective income tax of -50%. Another person makes \$1,000,000 per year, receives her \$10,000 prebate, and pays an overall 25% in taxes – netting \$757,500 per year, with an effective income tax of 24.25%.

You get a natural progressive tax, plus a system of guaranteed income that some believe could replace welfare, while being much, much easier to administer. No loopholes, no deductions, and a tax return is essentially trivial to fill out… and several actual experiments have shown little reduction in the workforce.

I’m not convinced it’s the best idea – but there’s some appeal, isn’t there?

3. No, he’s not *mistaken* a lump sum tax for a flat rate tax; he carefully described a “flat amount” tax (and carefully made clear he was doing so) in order to highlight a defect (more acute in it, but) that it shares with a flat *rate* tax.

5. I love tax brackets, umm, as a math concept that most people do not get. It makes a super-neat calculus problem. Depending on the strength of my students, I’ll have them merely find instantaneous and average rate and net taxes from current U.S. tax brackets (pretty easy to do), or if they’re really strong, I’ll have them try to model the current brackets with a function (hard) or design an “ideal tax” that doesn’t violate a set of given conditions (wicked hard). Here’s a description of these kinds of questions:

http://art-of-logic.blogspot.tw/2013/05/taxes-as-calculus-problem.html

Anyway, I remember reading this news story about a small business owner who didn’t open a new store because of the fear it would bump him to a higher tax bracket (not quite getting the concept). The reporter pointed out it was, conservatively, a \$10k-a-year mistake.

1. As a comment on your linked article points out (with links to their own paper on the subject), a logarithmic tax rate makes quite a bit of sense; and (in a world where everyone uses a calculator, and any half-way decent calculator knows how to compute logarithms) is easier to compute than the present system.

When considering the problem generally, I prefer to think about the function, F, from “raw income” (before government intervention) to “net income” (after). One desideratum is, naturally, that this function always be increasing, F’ > 0 – the more you earn, the more you get. One might reasonably also require F” = m for some chosen “enough to survive on” amount m; F’ > 0 then ensures F(x) > m for all x > m (so anyone who earns at least enough to live on will have at least enough to live on).

We can then combine this function (as x-F(x), the tax paid or (when negative) subsidy given) with the income-distribution of the population to determine the net income of the government from this intervention in incomes. This has to cover all the government expenditures (except for the subsidies it already takes into account). The income-distribution, as a function of raw income, is some p for which the integral from x=a to x=b of p(x).dx is the number of folk with raw incomes in the range from a to b. The state’s gross income then integrates p(x).(x -F(x)).dx across the whole raw-income range (from zero to astronomical); the subtract the budget from that to get the increase in the national debt.

The article I mention above uses F(x) = x.(1 -R.log(x/P)) in which P is some low-income threshold (the author chose the poverty line) and R is the national budget divided by the integral of x.log(x/P).p(x).dx. (A rescaling u = x/P turns that into P.P.u.log(u).p(P.u).du; in so far as the metaphor “a rising tide lifts all boats” applies to real-world economics (albeit I’m sceptical), we can expect p(P.u).P, as a function of u, to be somewhat immune to fluctuations, as long as P is adjusted in line with income growth (a.k.a. inflation). In so far as that holds, the integral of u.log(u).p(P.u).P.du will remain unchanged and our over-all integral, P times this, will thus grow in proportion to P. The budget likely also grows in the same proportion, so R should remain stable as long as P remains pegged to the same point in p’s shape, as p changes from year to year; if P is defined as the top of the bottom income pentile, for example, changes to R would mean real changes in government spending relative to inflation, or real shifts in the distribution of income among the population.)

Now, with F(x) = x.(1 -R.log(x/P)), the income intervention has two parameters, P and R. This is the essence of its simplicity, and makes it easy to understand the impact of changing the values of these two parameters. The choice of P says, roughly, how much the government believes it’s possible to live on; in effect, it says it’s reasonable to expect anyone with an income above P to be living well enough to be able to contribute at least something to the joint cause of keeping the nation running. Given a value of P, R is just a scaling to make the total income match the budget. So you can characterise the income-intervention within the tax system by the per-capita budget (how much the government spends, per tax-payer) and by how poor you have to be for the government to let you keep everything you earn; these are both quantities folk can intuitively understand, in particular on scales they can sensibly compare to their own income and those of their peers, so this would improve transparency in the tax system.

Let’s see if this F satisfies my constraints; it has F(P) = P, meeting the “F(m) >= m for some m it’s possible to survive on” condition, provided P *is* enough to survive on. It has F'(x) = 1 -R.(1 +log(x/P)), which is headed for F'(x) P.exp(1/R -1), which is apt to be a problem, unless exp(1/R -1) is so huge that no-one is affected by this. (The author noted above used base-ten logs, where I’m using natural logs, and got 8.83% as R, so would get R = 8.83%/ln(10) = 3.85% in my form of it; this gives 1/R -1 = 25.08, so the problem only hits if x/P > 77,756 million; with P = 11161, this would affect someone whose annual income dwarfs the US national debt.) We also get F”(x) = -R/x < 0, satisfying this fairness constraint. All told, not bad, albeit with a theoretical concern if anyone's income ever dwarfs (the rest of) GDP.

You might be a bit nervous about x x for all 0 < x < P, so how big are those subsidies to folk earning less than P ? We're multiplying by x; and x.log(x) does tend to zero as x does (from above), so the huge negative log won't actually be a problem. Indeed, brute numerical computation shows F(x) tends to 0 as x tends to 0 from above, at least with R = 3.85% and P = 11161. With F(P) = P and F'(P) = 1 -R = .9615 or so, those earning P get what they earned and enjoy almost all of any increase. The maximum subsidy goes to those with incomes around \$4k; some get almost \$160 (per year). So the payout to those earning too little to live on is hopefully small enough to satisfy even the most ungenerous tax-payer that they're not coddling the indigent. Those who earn nothing get an exaggerated improvement in their lot as they earn more; their subsidies are around one fifth of their earnings on the first \$100 or so; that falls off, as a proportion, even as it grows towards the maximum just described.

So, that's how F(x) = x.(1 -R.log(x/P)) plays out. It has one (highly) theoretical defect but otherwise provides a reasonable proposal (albeit some improvement for those below the poverty line might be prudent), whose income-scale parameter P is intuitively intelligible; and whose other parameter is easily associated with the per-capita budget, albeit via a scaling that depends on which income-percentile P falls in; that scaling shall also change as the shape of the population's income distribution shifts, which might make folk more aware of income disparities, among other things.

1. d’oh – typo: “One might reasonably also require F” = m for some” I meant “also require F(m) >= m” there …

2. ericastor says:

So… if we change the condition F(P) >= P to F(0) >= P, thus incorporating welfare into the analysis and forbidding benefit cliffs automatically, what results?

Presumably we’d use a lower value of P, to be the bare minimum standard of living. I’ll have to try seeing whether the condition can be met with a small tweak, and what effects it has on the rest of the analysis.

6. Nevin says:

Here’s a thought: right now in the US, if you want to pay someone enough that they keep \$50k after federal taxes, you have to spend \$75k to do so (plus insurance and a host of other things, not to mention state taxes, but let’s just look at federal tax for the moment.) We can think of this abstractly as one person selling \$50k of labor, and the government charging \$25k tax on that transaction. This is equivalent to a 50% sales tax on labor.

Conversely, suppose someone buys shares of a company, holds onto them for a year, and sells them at a 7% gain. They pay at most a 15% tax (the top marginal capital gains rate) on that 7% gain, which amounts to, essentially, a 1% sales tax on stock.

1. This contrast illustrates the rather crucial point that “capital gains” are mostly made by the most wealthy, who have most influence over tax policy; so they’re taxed less.

The excuse given for this is the importance of encouraging investment to enable business to prosper; but a significant part of capital gains goes to short-term speculators, who cause volatility, whose harms to businesses (and distortions of how they prioritise and plan) outweigh their (indirect) contribution to the system rewarding those who actually enable a business to get started or to grow towards prosperity. Some countries have interesting rules (which, unfortunately, get complicated fast) about the tax rate on capital gains depending on how long you held the asset, so as to tax speculators more than those who make a long-term commitment to helping a business grow and prosper.

In the end, tax is tricky. It’s how we pay for there to be a not-too-unfair playing field in which we at least know the rules well enough that we can make somewhat rational decisions about how to conduct our business. Finding a way to do that fairly is complicated by everyone having biases that are more apt to think something is fair in so far as it benefits self (or self perceives it as benefiting self; clever politicians have been known to mislead voters about their own self-interest).

Those who are prosperous and satisfied to be so, while unlikely to complain at what gives them more, tend to be willing to pay what it takes to sustain the system that enables them to remain prosperous; they are also apt to be willing to let that system take less from those who are struggling to attain prosperity, at the expense of those who have ample beyond what it takes to prosper. Those who are not satisfied with mere prosperity, when they have it, tend to complain about anything that interferes with their ability to get richer; but it is quite easy for those who prosper on less than they have to disdain their pleas for less tax. Those who struggle to get by, much less attain prosperity, feel every penny they are taxed, for it makes their struggle harder; and it is easy to sympathise with them in this. A progressive taxation system thus tends to sit well with the sympathies of, at least, those who are satisfied with their lot.

7. Jay Patel says: