Why is income tax generally not in a continuous "spectrum"?

Question

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:

Pros

• 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.

Cons

• 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.

Answer 1

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.

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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.

Answer 2

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.

Answer 3

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 :-)

Answer 4

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.

Answer 5

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,

Answer 6

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.

Answer 7

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".

Answer 8

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.

Answer 9

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.