In some countries like the US the tax system is bracketed rather than continuous.

There would be some obvious trade offs, but would it be worth considering eliminating the marginal, bracketed tax system in favor of a continuous, non-marginal tax rate system without brackets?

Picture a chart with income on the X axis and tax rate on the Y axis. In the current bracketed tax system the tax rate line increases as steps in a ladder. In a continuous tax system the tax rate would instead increase along some line or curve as income increased. This would shift the tax debate from,

“Where should we place the brackets and how many brackets should there be?”


“Where should the line or curve start and how fast should it rise?”


Thank you to all who have taken time to propose answers and comment so far. Your tremendous feedback unburdened some of my misunderstandings and helped me recognize some confusing points and distractions in how the original question was posed. To that end I offer this attempt to clarify the question.

The focus of the question was intended to be a generic comparison of bracketed vs continuous systems that ignores whether the bracketed system is marginal. However, I don’t know enough about tax systems to know if any relevant bracketed tax systems are non-marginal. As such, despite the discussion of incentives in the original question, non-marginal bracketed tax systems were not intended to be the focus of the question.

In an attempt to avoid this confusion I’ve updated the text in the question above to reduce the scope of the question such that it includes only the comparison between bracketed, marginal tax systems versus continuous, non-marginal, non-bracketed tax systems.

Also, the wording of my original question included possible incentives for earners to not pass an income bracket. This wording understandably caused some answers to focus on an implied misunderstanding of marginal brackets. Even in the cases where bracketed tax systems are marginal I think some incentives (real or perceived) for an earner not passing a bracket could exist. However, given that any possible incentives for avoiding crossing a tax bracket are a minor point and not the focus of the question, I’ve moved this discussion from the question’s introduction to the “Assumptions” section below.

With these clarifications in mind please also note that I know next to nothing about tax systems. Where you notice an error in the following assumptions, please point them out and I will try to update to help others.


Rightly or wrongly, the following aspects of a bracketed, marginal tax rate system may seem confusing, be perceived as unfair, or disincentivize earning for some people in some cases:

  1. There seems to be a common public misunderstanding of how marginal tax brackets work. For example, the belief that earning into the next bracket may subject a person’s entire income to that bracket’s tax rate rather than just the amount earned within the bracket. For people who have such a misunderstanding, they may be reluctant to earn into the next tax bracket in order to avoid what they think will result in significantly higher taxes across their entire income.

  2. For those who do understand how the marginal system works, earning just under each bracket may still have a perceived advantage over going over a bracket in some cases. For example, while likely a minor concern, if the tax rate for the next bracket is believed to be significantly higher then the final tax percentage for all earnings may be seen to increase disproportionately for the amount earned when going over the next bracket.

  3. There may be cases where a discrete, all or nothing tax incentive is offered only to those whose earnings do not reach the next bracket. While tax incentives won’t necessarily have a 1 to 1 mapping to a tax bracket, this may happen for simplicity in some marginal tax systems.

  4. Calculating the base taxes owed (before applying tax incentives, loopholes, etc.) requires summing the taxable area under each bracket by first knowing the tax rate for each bracket where earnings were made, determining the amount earned within each bracket, multiplication of the rate and the total earnings for each bracket, summing each bracket’s taxable income with taxes owed for all the other relevant brackets, etc. While these calculations aren’t difficult, per se, they may require additional tax resources and make the system feel more complicated than necessary for some people – especially when brackets boundaries or their tax rates change.

A continuous, non-marginal tax rate system without brackets would surely have it’s own set of problems but it would seem to avoid all points listed above. For example, as opposed to 1-4 above, in a continuous, non-marginal tax rate system without brackets:

  1. The public would be more likely to intuitively understand their tax rate because it is in line with what they expect when they hear “tax rate for X income is Y percent”. For example, they could find their income on the tax chart (a continuous curve) to find their tax rate. Multiplying their income by the percentage on the chart would yield their base taxes owed before any tax incentives are applied. There would be no need to use complex maths like calculus to derive the area under the curve because the continuous tax system in question is not marginal.

  2. Since the tax rate change is continuous there would be no specific point in the curve that could be seen to disproportionately increase the final tax percentage.

  3. While point 3 above could happen in a continuous system as well depending on how the tax laws are written, a continuous system would be more likely to change the way politicians communicate and how legislation creates tax incentives such that they would begin to apply in a continuously increasing or decreasing income range rather than applying in an all or nothing fashion at income X. This may cause the public to view the tax system as more fair.

  4. Determining the taxes owed (at a base level prior to any incentives, loopholes, etc.) would only require knowing the single tax percentage for the earned income in question. This would ease the burden on communication resources required to help the public determine their base tax rate prior to specific tax incentives.

While a continuous, non-marginal, non-bracketed tax rate system seems like it may be perceived to be more fair and easier for the public to understand it would certainly have serious negative trade offs worth discussing. This was the impetus and background for the original question and should be seen as context for the question as posed above.

  • 3
    Is there a particular country you're asking about? For those countries where I have lived, the tax bracket system is continuous.
    – Joe C
    Jun 21, 2019 at 23:02
  • 4
    Firstly this question belongs in personal finance. If you are intending to ask about tax brackets in the United States, the tax liability versus income is actually continuous curve (after an initial zero tax liability for very low income levels)
    – BobE
    Jun 21, 2019 at 23:45
  • 3
    cross-site discussion of this issue: money.stackexchange.com/questions/101575/… Jun 22, 2019 at 8:49
  • 2
    Discrete brackets may also disincentivize earnings in some situations if an individual is just below a bracket That's a misunderstanding of how marginal tax rates work. Having a higher income will never result in less money (at least at far as the tax system is concerned). Jun 25, 2019 at 7:08
  • 3
    You're using the wrong graph - the relevant graph is gross income vs net income; the tax rate is the derivative. Brackets cause a change in slope, not a step.
    – MSalters
    Jun 25, 2019 at 10:39

8 Answers 8


In most countries where this system is used, there is a level of continuity between brackets.

For example, in the United Kingdom, where the tax brackets are defined as follows:

  • 0% up to £12,500
  • 20% from £12,501 to £50,000
  • 40% from £50,001 to £150,000
  • 50% over £150,000

In this scenario, if I were on a £200,000 salary, I would not pay 50% on the whole £200,000. Rather, my tax bill would be as follows:

0% up to £12,500             = 0.00 * £ 12,500 = £     0
20% from £12,501 to £50,000  = 0.20 * £ 37,500 = £ 7,500
40% from £50,001 to £150,000 = 0.40 * £100,000 = £40,000
50% over £150,000            = 0.50 * £ 50,000 = £25,000
TOTAL                                 £200,000   £72,500

Talking specifically about the scenario on the edge of a bracket, if my salary were to go from £49,999 to £50,001, this £2 increase in salary would result in a 60p increase in my tax bill (20p for the first pound, 40p for the second pound).

  • 6
    This is the key factor as it gives the government a lot more flexibility to charge less for lower income amounts where people can't afford to pay as much and more for people who can afford to pay it.
    – Joe W
    Jun 22, 2019 at 14:59
  • 6
    Technically the UK tax system is considerably more complicated than the above explanation, as the personal allowance (the 0% band) is withdrawn progressively once your earnings go over £100,000. Additionally, you start to lose tax credits from pension contributions at some point, and for those with children, child benefit is withdrawn between £50K and £60K.
    – al45tair
    Jun 23, 2019 at 13:46
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    @pipe If you have a better answer, then I'm sure the OP would be very interested to read it.
    – Joe C
    Jun 23, 2019 at 16:34
  • 25
    @pipe The OP says "It seems tax brackets are beneficial to those earning just under each the tax bracket while being detrimental to those earning just over a bracket. Discrete brackets may also disincentivize earnings in some situations if an individual is just below a bracket." This answer says "that's not how tax brackets work, so your argument is invalid". You can also write an answer on arguments against continuous tax systems, but that doesn't make this answer "not answer the question".
    – Luaan
    Jun 24, 2019 at 7:03
  • 5
    @Luaan So you're the other guy who didn't get it. Discrete steps can disincentivize even if you don't earn less. It may make the difference between staying another hour or going home earlier, because you know you won't keep as much as the previous hour. But the question isn't about that. That's not OPs problem. This isn't a frame challenge, it's just off the mark.
    – pipe
    Jun 24, 2019 at 17:15

You want people to be able to understand their tax returns. Addition and multiplication are hard enough; finding your place on a curve could involve square roots or (gasp) logarithms. Even some college graduates might have trouble applying that in the real world, without the formalism of calculus homeworks.

  • 5
    Every country that I tried has some website where you enter your taxable income and it tells you the amount you have to pay.
    – gnasher729
    Jun 22, 2019 at 14:09
  • 30
    @gnasher729, justice must be done, and it must be seen to be done. An algorithm people don't understand is problematic for that reason, even if there are tools.
    – o.m.
    Jun 22, 2019 at 16:34
  • 3
    @gnasher729 If you are happy with that, you do not need any continuity requirement (and in fact would even ignore it if the function is not increasing) Jun 22, 2019 at 21:42
  • 12
    @o.m. Enough people don't understand how tax brackets work already that I'm not sure most people would find a continuous curve more confusing at all.
    – Turksarama
    Jun 24, 2019 at 1:40
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    @gnasher729: Meet the United States. The IRS has contracted a rather outrageous agreement with various tax-preparation companies to not develop such a thing, and in exchange the private companies offer similar services on their websites for free (but they aggressively try to upsell you to the paid product regardless of whether you need it, while simultaneously advertising the "free" product's existence to anyone who will listen).
    – Kevin
    Jun 24, 2019 at 6:19

(The original phrasing of the question seemed to indicate a common misconception of the way that the tax brackets work in the US and in many other countries, in thinking that one will lose money by crossing a bracket boundary. The edit cleared up that misunderstanding, but I'm leaving my response up since many people seemed to find it useful.)

The way tax brackets work is that the tax rate of each bracket is marginal - that is, it is only applied to the income that is "inside" the limits of that bracket. "Entering" a bracket does not affect your tax rate for the income below the bracket. Thus, the brackets actually produce a continuous curve. Consider this hypothetical tax bracket setup:

| Gross taxable income | Tax for bracket |
|         $0 - $10,000 |              0% |
|    $10,000 - $20,000 |             10% |
|    $20,000 - $30,000 |             25% |
|    $30,000 - $40,000 |             50% |
|    $40,000 and above |             75% |

The 10% only applies to income above $10,000, so once you start earning more than $10,000, your first $10,000 are "safe" - if you earn $12,000, you'll only pay $200 in taxes (10% of the $2,000 that are the excess of $10,000). Once you break into the $20,000 range, your taxes will be $2,000 plus 25% of what exceeds $20,000. Here's a graph illustrating the brackets above:

Tax based on gross taxable income, with hypothetical brackets from above

A real tax bracket setup is typically quite a bit smoother (except for the very lowermost portion) - here is the lower range of the real 2019 US federal tax brackets:

Tax based on gross taxable income, with 2019 US federal brackets

You can see the "breaks" in the curve at $9,700, $39,475, and $84,200, which are the delimiters between the first four brackets. So there is nothing to gain from trying to avoid entering a bracket. Below are a hypothetical and a real graph showing now your post-tax income develops - as you see, there is never any "drop"; your post-tax income just starts growing a bit more slowly as you enter new brackets.

Hypothetical brakcets:

Net income based on gross taxable income, with hypothetical brackets from above

In particular, note how even as you're $10,000 into the 75% bracket, the taxes are only 32% of your gross income, so you'll still get to take home 68%. As you keep earning money, your overall tax percentage will keep increasing smoothly. This example is very relevant to the current political climate: the politicians suggesting maximum taxes in the seventies are not at all trying to burden everyone with taking three-fourths of their income - they're only targeting the richest people, and even as a millionaire starts entering the top bracket, they're not going to pay nearly that much tax until they're several millions in.

Real 2019 US federal brackets:

Net income based on gross taxable income, with 2019 US federal brackets

Finally, per request, here is the tax, the post-tax income, and the ratio between the tax and the gross income (the "tax percentage") for the real 2019 US federal brackets for gross income up to half a million dollars. Note that this income is eight times the median US household income, so the vast majority of Americans are miles away from it - according to the same article, only 3% of households have incomes exceeding a quarter million (and that's for the combined household; this graph is for single filers). As we see, the tax percentage graph has already almost flattened out at 30%.

Tax, net income, and tax percentage based on gross taxable income, with 2019 US federal brackets

If we had continued the graph from there, the tax percentage would grow very slowly from there - only at about $2,600,000 does it reach 36%, and it reaches 36.5% around $7,750,000, and it will never actually reach 37%.

The real tax calculations are more complicated, taking a number of deductions and special situations into account, but the above are the "core" curves.

As for why governments don't tend to use smoother curves, based on formulas rather than brackets, other posters have explained why it's beneficial to have a tax system that non-math people can understand.

Source: my own graphs, based on my own example and on tax bracket data from the IRS.

  • Nice answer. Could you add one graph using actual 2019 rates on taxable income up to $10M? Another nice addition would be to have a second y axis showing realized tax rate (as you noted at 32% in the paragraph after the second to last graph.)
    – CramerTV
    Jun 25, 2019 at 1:00
  • I thought in the US you actually have discontinuities. Jun 25, 2019 at 19:42
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    @Trilarion: The discontinuities are in the marginal tax rate. Up to $9,700, every dollar you earn will be taxed at 10%. If you have earned $9,700 so far and then earn one dollar more, that dollar will be taxed at 12%, but the first $9,700 will still be taxed at 10%. So the only thing that jumped was the tax rate on the money you earn on top of what you earned in the lower bracket(s)_. Jun 25, 2019 at 22:05
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    @Trilarion: In practice, of course, the tax that is withheld from every paycheck is based not on how much you've earned so far this year (in which case you'd get less and less on every paycheck), but on an estimation of how much you're going to earn. But the end result is the same: If you earn $9,700 one year, your tax will be $970 (10%), and if you earn $9,701 the next year, your tax will be $970.12 (10% of $9,700 plus 12% of $1). Jun 25, 2019 at 22:10
  • @CramerTV: Thanks! Good idea, although the graph would look a bit boring if we extend it all the way to $10M - the tax percentage is pretty flat already at half a million, so I stopped there. Jun 28, 2019 at 2:33

Most tax system are continuous (or try to be continuous). Systems like the US system with tax brackets apply different tax rates to income within each bracket, but there is no penalty going from a lower bracket to a higher one.

Where things are discontinuous are often additional advantages: As an example, in Germany it used to be that you could deduct a certain percentage of the purchase price of a new home from tax - but only if your income was below some (rather high) threshold - so someone on low income saved a little bit, someone on a high income saved more, but someone on a very high income saved nothing at all - which I always found very strange, because no matter where you stand politically, that's not an outcome that you want.

I think in the UK there are similar effects today, where you can get tax benefits for your children, unless you make too much - so a salary increase of £1 a year could mean you lose these benefits and lose significant amounts of money. (What's weirder is that two parents each earning just below that threshold get the tax benefits, while a family with one income, just above the threshold, gets nothing even when they are financially much less well off).

Why do these things happen? I suspect it happens because politicians either are exceedingly bad at maths, or because they don't care, or both.

  • 9
    I was about to upvote this, then read the last sentence, which feels unnecessary and biased.
    – IMSoP
    Jun 22, 2019 at 18:14
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    I though the last sentence was spot-on, so I upvoted for you.
    – EvilSnack
    Jun 22, 2019 at 18:29
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    I'm neutral toward the last sentence, but I upvoted for the rest of it. ;-)
    – ruakh
    Jun 22, 2019 at 21:35
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    Given that a Federal Secretary of Finance(!) once could not answer the question how many zeroes there are in a billion, the last sentence has its merits. (Note that it was clear from the national context which of the two internationally possible values of billion were meant) Jun 22, 2019 at 21:47
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    @Delioth The "round half to odd" rule is to prevent the rounding from introducing bias that could occur if everything were rounded up or down.
    – JAB
    Jun 24, 2019 at 20:27

Lawmakers like to be able to set the marginal tax rates for different income groups independently of each other.

A politician who is negotiating an increase or decrease in taxes wants to be able to communicate very specifically to his electorate about who is hit by, or benefits from, the changes he's voting for.

A single unified curve that defines marginal tax rates for everybody based on just a few parameters would make political fine-tuning impossible. You can't, for example, give low-income groups a tax break without shifting the entire curve a bit, so your tax breaks would have knock-on effects for the the middle class, and thus be more expensive than if you could lower just one of the tax rates.

Having a smooth curve looks appealing from a strictly technocratic point of view, but I can't see any political advantage of it for anyone. Especially when the smoothness comes at a net cost in how much of the citizenry would understand the effect of a quadratic curve intuitively. (Being able to predict your own taxes by following a cookbook recipe is one thing; forming an opinion of the entire tax system and whether you consider it fair requires a deeper understanding).

Copied from an answer to a similar question on Money.SE

  • 1
    That problem exists for marginal rates, though. Generally, if you give a tax break in a lower marginal rate band, it "gives money" to everyone, including "the rich". That's why some left-wing progressives oppose tax breaks on principle, and would rather use benefits to target particular segments of the population.
    – al45tair
    Jun 23, 2019 at 13:52
  • 2
    @alastair: If you want, you can move the thresholds simultaneously such that everyone above a certain income pay the same tax as they did before. Jun 23, 2019 at 14:07
  • @alastair The problem is that this way you create wage gaps which tend to keep (relatively) poor people poor - at some point, increasing wages yield decreasing net income (or at least, very little increase in net income). Mind, I'm not in favour of tax breaks - but I'm also not in favour of benefits, especially as a way to replace tax breaks (or, given the context, more accurately - tax brackets) :D
    – Luaan
    Jun 24, 2019 at 7:11
  • 2
    @Luann Totally agree. I didn't say I'm a left-wing progressive. I'm not. I'd rather have a simple and relatively flat tax system (personal allowance plus single band), and try to abolish most tax breaks and also most benefits.
    – al45tair
    Jun 24, 2019 at 10:00
  • 1
    Political communication is definitely a factor. Thank you for bringing this up. I’ve updated the question with some clarification. I’m imagining a continuous curve that is non-marginal and likely always either stays the same or moves up. I would think politicians could raise any part of the curve arbitrarily since they could raise a future part less to avoid knock-on effects – but I may be failing to recognize them. My assumption may be wrong but I was thinking that in this case we could just find the earnings amount, multiply it by the tax rate depicted for that earnings amount and be done.
    – Eric Buess
    Jun 30, 2019 at 17:26

Look at the history of income taxes: until about twenty years ago, virtually everybody calculated their taxes using pencil and paper, possibly with the assistance of a calculator. Prior to the 1960s, even banks and other large corporations that could afford a computer did their taxes that way.

Bracketed taxes are far easier to calculate by hand than continuous taxes are. Once you've got a bracketed system in place, simple inertia keeps it in place.

  • 2
    @sleske That's a 1600-page document. Did every taxpayer really get a copy of it so they could calculate their taxes?
    – Barmar
    Jun 25, 2019 at 0:10
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    @Barmar: I'm not sure. I think normally only the tax office used these lists - you just declared your stuff, and the tax office ran the numbers. I believe a shorter table was published for interested taxpayers.
    – sleske
    Jun 25, 2019 at 9:53
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    @sleske Of course considerations are different if the government calculates your takes for you. The US system is based on taxpayers doing the calculations themselves. The instructions here are something simple like "If your taxable income is higher than X, subtract X, multiply by Y%, add that to Z, that's your tax"
    – Barmar
    Jun 25, 2019 at 16:34
  • 1
    @Barmar: Thanks, didn't know that. Seems weird to let individuals calculate their taxes - the tax office can do it more easily, as they have the software already (they need it anyway). Anyway, different countries, different approaches.
    – sleske
    Jun 26, 2019 at 8:41
  • 1
    @sleske It's been suggested, but the tax preparation industry has successfully lobbied against it.
    – Barmar
    Jun 26, 2019 at 15:29

The reason why stepped tax curves are used, rather than continuous, is because of mathematics, and a desire to keep things simple.

The amount of tax you will pay on your earnings is equal to the area under the curve of tax rate on the y-axis vs salary on the x-axis. Having a stepped curve with discrete levels is the easiest type of curve to calculate the area of, because the area under the curve just consists of several simple rectangles.

If you were to use a smooth 'linear' curve (i.e. constant gradient), then that is not much more complicated to calculate, as you have to use the area of a triangle. However, it then makes it essentially impossible to target tax policy on specific income levels, because you've fixed the curve as linear (the only parameter available to tweak is the gradient).

So, the next step would be to use a smooth, arbitrary non-linear curve. This would allow tax policy to target specific income levels, by adjusting the shape of the curve. However, calculating the area (i.e. amount of tax paid) will now require integration, which is significantly more complicated mathematically. This may be beyond the abilities/training of many people to understand and the calculations would also be significantly less clear on tax returns and other documents.

So, a discrete, stepped, bracketed curve is a good choice to keep things simple, but still allow a level of policy adjustment.

  • 1
    This comment is helpful. Thank you! It makes sense to me that finding the amount you would pay on your earnings would require calculating the area under the curve in a bracketed tax system that is marginal. However, regarding your fourth paragraph, is there are reason we’d need to calculate the area under the curve if the curve is non-marginal and always either stays the same or moves up? My assumption may be wrong but I was thinking that in this case we could just find the earnings amount, multiply it by the tax rate depicted for that earnings amount and be done. Perhaps this would not work?
    – Eric Buess
    Jun 30, 2019 at 16:44
  • @EricBuess sorry, I'm not sure what you mean by 'marginal'?
    – Time4Tea
    Jul 3, 2019 at 13:40
  • I’m referring to the marginal tax rates that often apply to tax system that uses brackets. I clarify this in Assumption 1 of the Update portion of the question above.
    – Eric Buess
    Jul 3, 2019 at 13:48
  • It doesn't require the individual filing taxes to do integration. Once a smooth tax rate function is set, the antiderivative can be worked out once. Thus turning a function for marginal tax rate for a given taxable income into a function of tax owed for a given taxable income. The result would arguably arguably be simpler as it would remove steps but harder to explain to people who forgot middle school level mathematics as it would involve some form of non-linear function.
    – smithkm
    Dec 24, 2019 at 0:58

Actually some countries has had step-bracketed tax systems where the 'curve' isn't continuous. It was changed a long time ago especially to avoid the penalties when getting a raise and now it's exactly like the US bracketed system.

Now, here in Denmark we also do our tax returns but for the last decade (or more) it has been more or less automated, which means that for most people the basically just have to look it over and sign it. The IRS-equivalent (SKAT) already has all the numbers if you're a typical employee (income tax is automatically deducted before you get your pay, they know your property taxes etc.) and then it's just a matter of filling in the data in the tax return.

The problem now is the question: Where does those numbers come from? Most common people have no idea and if you try to edit something you'll have to prove the new number and even trained accountants have a hard time validating the numbers. But it's trade-off in return for the ease of just signing it.

  • That makes sense and is good to know. I’ve updated the question to address some confusion in the original wording. I’ve also added some notes about the lack of public understanding and how a continuous system might address some of it. Or maybe not.
    – Eric Buess
    Jun 30, 2019 at 17:33

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