With respect to the countries about which I am familiar, income tax is partitioned into tax "brackets", meaning that the rate of taxation varies discretely. Different tax rates are therefore applied to different "blocks" of an individual's annual income. The UK, for example, also has a personal allowance - a pot of money an individual can earn tax-free.

This system seems counter-intuitive to me - it complicates annual tax calculation, as income needs to first be partitioned. Why not apply a continuous and smooth (but obviously non-linear) scale to taxation. The rate would approach a limit and start slow, forming a kind of sigmoid-shaped distribution. The tax rate could then be applied in one motion and could be arguably more "fair."

I'm editing this because it seems to have received many responses - which I didn't anticipate, so I'll offer some of my reasoning behind the question.

In my eyes, and from going through some of the comments & answers which skewed my opinion slightly, I see the following pros and cons:


  • The commonly held perspective of "There's no point in taking that job for $N/annum because I'd be taxed through the roof." at some salary thresholds would be partially diminished.

  • There's no two ways about it - it would give finer-grained control over income taxation at the expense of ease-of-understanding. It could offer a fairer tax system, in spite of public impression of complexity.

  • Tax is digital in many countries - we shouldn't have to be narrow-minded about the possibilities for tax any more. Tax tables are (or at least should be) a thing of the past.


  • Personal finance has hitherto been blessed with ease of calculation, even with pen and paper. This would throw a spanner in the works.

  • Would there be a limit to how complex the function could become? Could governments exploit the system by making the function so convoluted or employ devious mathematical techniques (e.g. around the rounding of pennies) as to extort certain parts of the populous?

  • The system cannot be changed to accommodate one group without affecting all, however marginally. (Actually, this isn't completely true if a piecewise function is adopted, but that's going even further down the rabbit hole.) This would seriously impact political campaigns for example, as intention can not be so easily conveyed.

  • The fragile tax legal framework would be seriously threatened by such a major change.

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    But the resulting overall tax rate is still a continuous function! Only the marginal tax rate is non-continuous, i.e. the tax rate function is not differentiable. Also, calculating tax in a bracket system is very simple using a pocket calculator and a table: find the highest tax bracket you're in, calculate the tax from that bracket, then add the maximum tax from all lower brackets. – amon May 23 '20 at 16:42
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    Take a look at a progressive bracket system like the USA's and create a smooth function that generally follows that line. (as close as reasonably possible to it, since I gather you are not aiming to change the intent of tax law). a) can it be made simple and doable with pencil and paper by most people? – Harper - Reinstate Monica May 24 '20 at 17:51
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    I wonder now more fundamentally why income is taxed at all. Now that for the last 13 years or so we live in an economy completely dependent on stimulus from the govt and central banks, and now that households need direct stimulus too, and that the govt borrows but the fed/central bank writes off the debt, what value is tax? – Frank May 25 '20 at 5:09
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    Cross-site duplicate: Why isn’t the tax system continuous rather than bracketed? – gerrit May 25 '20 at 8:38

Before we discuss why the system is the way it is, it needs to be pointed out that the effective tax rate is indeed continuous (although it is not mathematically smooth). The tax you pay does not jump when you cross a bracket threshold.

The main advantage of the marginal system over a continuously changing tax rate is that it is easier to understand.

For example, let's say that my annual income puts me right in the middle of the 22% tax bracket. I know that if I contribute an extra $1000 to my HSA, I will save exactly $220 on my taxes. Conversely, if I take a part time job and earn an extra $1000, it will cost me $220 in taxes. If the rate was continuously changing, it would be much less clear what the tax implications would be for these actions. Any extra money I earn would not only affect the rate I pay on those dollars, but all the other dollars I had already earned that year.

To address some of the "pros" of the proposed system that have been added to your question:

The commonly held perspective of "There's no point in taking that job for $N/annum because I'd be taxed through the roof." at some salary thresholds would be partially diminished.

This would be true of any progressive tax system, whether it is marginal or a continuously variable rate. Remember, again, that marginal brackets do not result in a sudden jump in tax paid.

There's no two ways about it - it [the proposed continuous-spectrum tax rate] would give finer-grained control over income taxation at the expense of ease-of-understanding. It could offer a fairer tax system, in spite of public impression of complexity.

"Fairness" is a loaded term. There are people who believe that the only fair tax system would be a flat rate for everybody. There are others who believe that those who earn more should pay a higher rate than those that earn less (progressive taxing, which is what we have now in the U.S.). But what the marginal rate brackets do is attempt to appease both of those groups. Those who earn more do indeed pay a higher effective rate on their income than those who pay less. And yet, everyone, rich or poor, has the first ~$10k of their annual taxable income taxed at 10%, the next ~$30k taxed at 12%, etc. In that sense, it is quite fair.

Tax is digital in many countries - we shouldn't have to be narrow-minded about the possibilities for tax any more. Tax tables are (or at least should be) a thing of the past.

As I explained in my answer to "What's the point of tax tables?", the fact that most people use a computer to calculate their taxes goes both ways. If you are using a computer, you don't care whether that program is using a formula to calculate your tax or looking it up in a table. This isn't really an argument to get rid of tax tables.

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    It’s not technically continuous, because no function on money can be continuous. Money is quantised. You cannot pay tax equal to exactly 1/3 of $10,000. – Mike Scott May 24 '20 at 6:19
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    @MikeScott It’s worse than that: Tax returns are done in whole dollars (rounding off the cents), and then the tax you pay is determined by tax tables, where there are many little tax brackets of income with rounded tax amounts. – Ben Miller - Remember Monica May 24 '20 at 11:25
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    But my point is that the tax table is based on marginal tax rates that produce a continuous function for tax paid and effective tax rate before any rounding due to quantized currency, dollar rounding, or the tax tables. – Ben Miller - Remember Monica May 24 '20 at 11:26
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    Algebraic continuity is like Ɐ x, ε > 0 ∃ δ > 0 : |y - x| < δ → |f(y) - f(x)| < ε . To make sense for taxes we need to modify the definition of "continuity" to accept a certain level of granularity like Ɐ x, ε > $25 ∃ δ > $1. – krubo May 24 '20 at 17:26
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    Really, continuity depends on the topology of the domain and range of the function we're considering. Why bother worrying about the technicality of dollars being discrete? Under any useful definition of continuity of functions of dollars, the effective tax rate is continuous. – James Otto May 25 '20 at 1:10

Historically, taxes had to be calculated (or tax tables compiled) by hand. Taxes in a system of discrete rates and brackets are probably easier to compute by hand.

Nowadays, we could easily compute other mathematical functions using pocket calculators or computers. But to do that we'd have to change the tax code substantially.

First, the legal language to describe brackets is well established, and has been tested in court and refined over decades to avoid challenges by unwilling taxpayers. Changing the language dramatically would risk ambiguous language slipping in to the code, leading to legal challenges and lost revenue under the new code.

Second, other parts of the tax code depend on the bracket structure we have now. So any change to an entirely new system of computing taxes would require scrubbing the entire rest of the tax code (6550 pages of legalese for the US, according to this) to make other aspects of the code sensible under the new system. Even laws outside the tax code (for example, who is eligible for welfare programs or higher education grants) might depend on the brackets in the tax code.

Given the sensitivity of the tax code to politics (meaning, many voters have opinions on how they'd like the tax code adjusted to favor them, so all the legislators have an opinion on it), even small changes in the code are difficult to make. It's not surprising nobody wants to make such a fundamental change to the way taxes are calculated.


I suspect it's because a lot of people (including many politicians & bureaucrats) are math-challenged, and find brackets easier to understand than a continuous function. And at least in the US, most people just look up the amount in the tax tables, anyway :-)

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    And even allowing that the quantitative specifics are out of reach, what about simply qualitatively describing such a tax? How many members of Congress, if asked to sketch a “sigmoid-shaped distribution”, would be able to do so? I expect not many. Brackets are easier to explain in legislative language. – C8H10N4O2 May 24 '20 at 1:19
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    Totally agree. That's the problem of allowing math-oblivious people to write laws involving numbers. They should just write generic laws instead, and having technically competent people refine them. – o0'. May 24 '20 at 13:42
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    @o0'.: I disagree. As other answers note, brackets & tax tables is much simpler than having to compute a complex function. It's essentially the same reason that I sometimes use lookup tables rather than expensive computations in performance-critical software. – jamesqf May 25 '20 at 3:41
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    @C8H10N4O2 "Easier to explain" and easier to change independently. – Michael May 25 '20 at 14:08
  • I don't think "because people are bad at maths" is a good enough excuse to dismiss the question. I also disagree that using brackets and tax tables is any "simpler" than plugging a figure into an equation at the end of the year. Most employees' income tax (+ NI in UK) is handled at the employer end. We have highly integrated digital payroll and tax systems now which would make short work of even "complex functions." If the individuals are paranoid about it, they can do the maths themselves. – Benjamin Crawford Ctrl-Alt-Tut May 25 '20 at 20:39

Having tax brackets allows you to vary the tax rate for one portion of incomes independently of the other incomes. Say you want to introduce a tax-free part, or increase taxes for the ultra-rich. A continuous and smooth function would either have fewer free parameters, in which case you would change also the other tax rates. Or you have at least one parameter for each income region but then you would have to do a matching between the regions, so that the function is still continuous and smooth, and you would have brackets again.


Because in practice, it's actually quite simple. It works like this (hypothetical example):

If your taxable income is between $125,001 and $170,000:
Your tax is $16,208 plus 25% of the amount over $125,000.

What's happening is that the tax at exactly $125,000 is known to be $16,208. So for those inside the 25% bracket running from $125,000 to $170,000, they are simply saying to add 25% of the amount in that bracket,

  • I understand how the bracket system works and I'm not suggesting it isn't simple - I'm suggesting it doesn't make sense. How is this any simpler from substituting annual gross income into a formula at the end of the year, a task which would likely be automated by your employer's computer system anyway? – Benjamin Crawford Ctrl-Alt-Tut May 25 '20 at 21:05

If the tax rate is computed in some way, i.e. it is expressed as some sigmoid function of income, the resulting system is neither intuitive nor simple. In reality, a switch to a system like that is unlikely to be understood as a simplification or perceived as fair.

Another solution is to create a table of effective tax rates, depending on income. That's how income tax is (or was) computed in Geneva, Switzerland. Two drawbacks: that table becomes very long and, unlike brackets, you can still have threshold effects (lower after-tax income following an increase in before-tax income). As a limiting case, you could have an entry in the table for each integer dollar/pound amount but the resulting table would be too unwieldy to handle and would probably have to be used electronically. This too would seem to make the system more complex, not simpler.

At the end of the day, even if many people do struggle to fully understand it, a simple bracket system seems like a decent compromise while providing some element of progressive taxation that many people find desirable and associate with fairness. The only approach that is unambiguously simpler is a flat tax, which has its proponents.

  • Flat tax is "constant slope with a zero intercept". There's also "constant slope with a non-zero intercept" which is just about as easy to comprehend and is somewhat progressive and I will abbreviate as "non-zero intercept" or NZI. Another way to look at it is that flat tax reduces today's system to a single tax bracket removing all others, and NZI tax removes all but one non-zero bracket in today's system but keeps the zero-tax bracket as well. If you take this NZI tax and implement it by collecting sales tax instead of income tax, you basically have the so-called "FairTax". – Ben Voigt May 25 '20 at 17:14
  • I disagree with practically all of your first paragraph. How it's perceived now, in the long run, isn't important. I think it is intuitive to have a continuously varying tax - the more you earn, the higher proportion of tax you pay (to a limit). No part of that statement suggests abrupt jumps in tax rate. – Benjamin Crawford Ctrl-Alt-Tut May 25 '20 at 21:09
  • @BenVoigt Effectively that means two brackets. You add the same complexity while losing much of the progressive taxation aspect so I fail to see what the point would be or why it deserves to be discussed separately. I am also unclear on how you can implement a zero bracket through a sales tax. – Relaxed May 25 '20 at 22:22
  • @BenjaminCrawfordCtrl-Alt-Tut It doesn't really matter, my point is that the very fact that “the more you earn, the higher proportion of tax you pay” wouldn't be intuitively clear in the setup you propose. Brackets have exactly the same property and are much more intuitive. – Relaxed May 25 '20 at 22:23
  • I also think that how it is perceived now is hugely important, possibly more than in a long run that would never come about if you cannot articulate the benefits of the change. However, I don't think the system would become more intuitive in the long run either. That's why it wouldn't be perceived as fair: not because people have a problem with the tax rate increasing with income but because the system would be completely opaque. – Relaxed May 25 '20 at 22:27

In all countries where I have looked at income tax, it is indeed a smooth function. Germany has one of the more complicate once, using a third degree polynomial (which is probably beyond what most politicians can understand except the odd one with a degree in quantum chemistry), but that was never a problem: They just printed a table with taxes over a reasonable range, and you looked it up in the table. Countries with simpler rules are still too complex for most people to understand.

Most countries (but not all) avoid discontinuities. That is where say one dollar more income would cost you $1000 more in taxes. The obvious reason is that people would then have to worry about how much money you make. Boss offers a 4.5% raise, and you have to go back to him and say "please can you make that 4.4%" because otherwise it costs you more money. And of course it's basically unfair. The UK is very good at unfair taxes. You can save a lot of tax here if you have kids and not a high income. One person working making £45,000 a year is "high income". Two people working making £44,000 each for a total of £88,000 is not "high income".

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    You might save a lot of taxes with kids, but you will have less money, because kids cost money. This is also not an unintentional effect, but instead it is an tax instrument used in many countries to ease the burden of families. – dirkk May 25 '20 at 10:16
  • It's not a smooth function in the UK, in the mathematical sense of "smooth" that all derivatives exist and are continuous. – Steve Jessop May 25 '20 at 21:20
  • Dirkk The point was £45,000 = “high income”, 2x£44,000 = “low income” according to U.K. tax law. – gnasher729 May 26 '20 at 19:18

it complicates annual tax calculation, as income needs to first be partitioned

Well, the crux of the issue then is, complicates it compared to what?

In practice there are at least two things people want to know about their tax: how much to pay and how changes in gross income convert to changes in net income.

A piecewise-linear function is easy to differentiate without even knowing what calculus is, and a likewise a piece-wise constant function to integrate. I won't attempt to estimate what proportion of the population are comfortable with calculus, but it's not high. With a bit of strain many people can even do calculations involving marginal rates of tax in their heads: "if I get a payrise of 100 a month, how much extra is that in my pocket?". We can sometimes even get close to the right answer in the case where we remember to deduct NI. People don't need to know that they're integrating their (locally constant) tax rate when they do this.

Now, there are people who can't do this at all, and just trust their tax software or their accountant. There are people who could cope with much more calculus, and would happily evaluate the definite integrate of your specified function on the range [salary, salary + 100]. But, broadly speaking, a piece-wise linear function has the property that most people, most of the time, kind of understand what tax payable and marginal tax rate mean. Many know (in imprecise terms, perhaps) how they relate.

What other functions have the property that people can do that? I don't have a mathematical proof, but I strongly suspect the answer is that only locally-constant and locally-linear functions are any good here. So, to comprehend both tax payable and marginal rate, I think most people will find a piecewise-linear function less complicated to deal with, not more complicated, than any smooth non-linear function you care to specify. So, if you introduced one, someone would just approximate it with a piece-wise linear function (a "tax table") for ease of use. Use of that tax table would simplify, not complicate the everyday person's interaction with tax, compared with doing calculus.

Of course, compared with a simple linear function (or "flat tax rate" as it's known in this context), sure, multiple tax brackets are more complex. The reasons against flat income tax rates are political, and basically tax brackets amount to the "simplest" possible income tax system that is progressive (i.e. marginal rate is not everywhere constant and is everywhere non-decreasing).

Two tax brackets would be enough to achieve that (for example you could have a 0% bracket and then one flat rate beyond that). As for why that's not used in the UK: again political, but it amounts to because politicians like to tinker with income tax rates in a bit more detail than that, but not a lot more detail than that.

I honestly don't think any of your three listed pros would appeal to very many politicians:

  • The tax bracket isn't the reason some people might be disincentivised to seek high salaries: it's the high average tax rate. Making the function smoother doesn't hide that, only flattening the tax rate would. It might remove some irregularities where, say, someone might make a pension payment that not-coincidentally takes their taxable income exactly to a bracket boundary, but I don't think anyone considers those really problematic.

  • The practical political use for nightmarish complexity is in the rules for what is taxable, not the conversion from taxable income to tax payable. So, politicians don't especially want finer-grained control of the latter at the expense of understanding. They're way more interested in fiddling around with tax allowances for particular uses of money (annual changes to pensions, ISAs, allowable business expenses, sketchy investment vehicles) than they are in adding more tax brackets (two in my lifetime in the UK IIRC), let alone making it smooth. Actual fairness debates are around how progressive the system should be and/or what types of person should pay what tax, not around the smoothness of the tax function itself.

  • While it's true that tax is digital, and many people will use software for convenience, it doesn't follow that we want it to be impracticable for taxpayers to understand and even check those calculations (especially not after they've dealt with what is currently the tricky part, deciding what is taxable).

Short answer: tidiness is not considered a virtue in politics. Your proposal primarily adds tidiness, and therefore will not recommend itself to politicians.

  • All good practical and political points. The precise mathematical nature of any proposed function for marginal tax rate is perhaps over-specified in my question and therefore in the resultant answers - SE pedantries. None of my points were necessarily aiming to win over politicians, this is purely a question of fairness and precision and could be in many ways seen as a bit of a thought experiment. – Benjamin Crawford Ctrl-Alt-Tut May 25 '20 at 21:22
  • Well, I don't think you can realistically define "fairness" without reference to politics, except in the sense used in a dictatorship, "fairness means whatever one person thinks is fair" :-) My basic point on the mathematical definition is just that anything not piecewise linear is going to be intractable to the average person armed with a calculator: whether what you come up with is really smooth or just has 1 or 2 continuous deriatives doesn't make any difference. – Steve Jessop May 25 '20 at 21:25
  • Unless you're a flat tax proponent (or hey, even a 0 tax proponent), fair insofar as: "You earn more, you pay more tax (with bounds at either end - or not!)." I don't remember the last Great Income Tax Referendum we had, but I could imagine the outcome :-) – Benjamin Crawford Ctrl-Alt-Tut May 25 '20 at 21:36
  • Yeah, you could say that the reason we rarely have referenda in the UK is that it's very rare for politicians to be willing to prioritise one issue over all others, to the point where they will implement the result of the referendum "come what may". A referendum on income tax in effect would remove the ability of the government to mess about with income tax on a semi-regular basis, which is precisely the opposite of what they want (and also probably not what most voters want, given that promises to alter taxes are frequently the reason they vote for one party over another). – Steve Jessop May 25 '20 at 23:10

What continuous function would you suggest and how would you describe it to an ordinary tax payer?

If you want simplicity and equality, I would suggest a living wage and a flat rate income tax. If all income after the living wage is taxed at a flat rate of 10%, then it wouldn't matter if you lived in the Bowery or Beverly Hills, you would pay an equal proportion of taxes. This should apply equally to ALL commercially obtained income. No discounts for losses in past years, for political or charitable donations or for pension investments. If the tax authorities think a charity or pension fund is legitimate and deserves tax refunds, then the tax authorities can pay those refunds. In other words, no loop holes for the wealthy to exploit, after all, the poor can't afford the tax lawyers. You do not need to charge the super-rich a higher rate of tax then the working class, as this has been proven to be counter productive, but the current approach of charging the rich and super rich lower taxes then poorer people is both unjust and unfair and is just increasing the "us and them" divide.

This idea works just as well for commercial ventures as it does for individuals. The difference between the purchase of raw materials and the income from sales is the gross commercial income (gross profit). That income is taxed at the same rate as private income. All of a sudden you have no more mega-corps (Google, Apple, Starbucks, Amazon etc.) getting away with paying no taxes. The company can do what it likes with the remaining profit, re-invest it, pay dividends, but any person or entity receiving that income pays income tax on what they receive.

  • downvotes without comments are not very helpful. – Paul Smith Jun 2 '20 at 22:01

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