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In Canada (and many other countries), high-earners pay a greater percentage of their income in taxes than low-earners. Here is our Schedule 1 form:

Canadian tax brackets

The marginal utility of money decreases smoothly with increased income: If I earn $91,830, an additional dollar would be slightly less useful to me than the last one. The same is true if I earn $91,831 and earn an extra dollar, but the difference between the two cases is minuscule.

However, my marginal tax rate jumps up by 5.5%, which is not minuscule. Why is that? It seems there is nothing special about the value 91,831. Couldn't the marginal tax rate increase smoothly with income? Here is a made up example where the marginal tax rate increases linearly from 13% at $0 to 33% at $200,000 and then stays constant. The actual function used could be different, but the point is that it would reflect the "smooth" nature of the decrease in the utility of money. The blue line represents the actual function used in Canada.

Graph of marginal income taxes

Are brackets are easier to use than a smooth curve? Is it just a "good enough" approximation to make the calculation simpler? The Schedule 1 form would probably look something like this, which is not that bad: you could do this with a 4-function calculator.

Schedule 1 form mockup

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    This is not simply a theoretical question about economics. It is a question about how the income tax works. It is directly related to personal finance and should be reopened. – Ben Miller - Reinstate Monica Oct 31 '18 at 9:44
  • I think this question should remain closed - it's a question about why a certain policy has been chosen that is very remote from people's personal financial planning. – GS - Apologise to Monica Mar 26 at 8:26
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    I edited this question to remove the "why" and focus on the "how" I think that it can be reopened now that it is not asking a political question, but instead asks for help understanding. – NL - Apologize to Monica May 12 at 2:51
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However, my marginal tax rate jumps up by 5.5%, which is not minuscule.

Although it is true that your marginal tax rate jumps as you go up in income to the next bracket, your effective tax rate, which is found by dividing your total tax by your total income, does not jump. It is a gradual transition as you go up in income.

For example, in the Canadian tax brackets, you start out at 15% marginal rate, and you stay at that rate through your first $45,916. If your income is $45,000, your effective rate is also 15%. However, if you make $46,000, you are now in the 20.5% bracket, but your effective rate has not jumped. Instead, your effective rate is 15.01%.

The nature of the marginal rate system provides a continuous curve of effective rates across all incomes. The curve begins at 15%, starts to increase once your taxable income crosses the first bracket threshold, and eventually approaches 33% as the taxable income gets very large.

Canadian effective tax rate vs income graph.

The chart above shows the basic rate calculation (grey) which shows a reasonably smooth effective rate increase above the first bracket. After including the basic personal deduction of (this year) $11,635 which reduces the effective rate to zero below that threshold, we recover the standard curve most people experience (blue).

One of the advantages to the marginal system is that it is easy to figure out how much tax you will save when applying deductions. I’m more familiar with the U.S. deductions, so I’ll use a U.S. example. Let’s say that I’m in the 25% tax bracket. My effective rate is less than 25%, but because of my tax bracket, I know that every $1,000 I contribute to my tax-deductible retirement account will result in $250 off my tax bill.

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    Just a quick nitpick - using the strict mathematical definitions of the terms, the marginal rate system produces a continuous curve, but not a smooth one, of effective rates. – bobajob Oct 30 '18 at 9:57
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    I don't know what you mean by "almost", but the curve will definitely kick up every time you change brackets. – bobajob Oct 30 '18 at 10:33
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    @bobajob: It is piecewise smooth (with a small number of pieces). It is also piecewise linear. – tomasz Oct 30 '18 at 11:28
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    @bobajob If we want to be really pedantic, I don't think I would consider this continuous either. Money would be discrete, because there is some minimum denomination that you can transfer, along with not being able to have irrational amounts of money. – JMac Oct 30 '18 at 11:43
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    @BenMiller: I think what bobajob is trying to say is that the derivative is not contiguous (there's an "angle" which will produce a gap in the derivative), and therefore the curve is continuous (no gap) but not smooth (the derivative has gaps). – Matthieu M. Oct 30 '18 at 13:07
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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).

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    But the current system does actually will give everyone a small tax break if you decrease the lowest tax bracket's rate, it's just not proportional. – Chieron Oct 30 '18 at 14:03
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    @Chieron: But you can, if you want, correct for that by moving the thresholds simultaneously, such that everyone over a certain income pay exactly the same in tax as they did before. – Henning Makholm Oct 30 '18 at 16:46
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TL;DR The effective tax rate curve is continuous (no jumps) anyway, so why complicate things?

Continuous effective tax rate

The system does yield a "smooth" (in fact you mean "continuous", no jumps) effective tax rate curve. You can have all the jumps/brackets/discontinuities you want in the curve for the marginal tax rate, and it will always produce a "smooth" curve for the effective tax rate.

Let's take an example:

  • Marginal tax rate below $500: 5%
  • Marginal tax rate between $500 and $2000: 10%
  • Marginal tax rate above $2000: 20%

This marginal tax rate is shown below as the dotted blue curve:

Marginal tax rate (blue) vs. effective tax rate (red)

And it yields the red curve as effective tax rate.

So how is that? Well, up to $500 dollar this is trivial. Everything you have is taxed 5%, so that's the effective tax rate as well. So what happens when you have $501? The first $500 are still taxed at the lower rate of 5% ($25), and only the additional $1 is taxed at the higher rate of 10% ($0.10). In total, you have to pay $25.10 in taxes, which is only very slightly higher than the $25.05 you would have to pay at a rate of 5%. So there was no jump!

Of course, that's only because there was only a tiny fraction of your money that fell in the second bracket. Once you earn more, that proportion is going to increase, bringing the total effective tax rate closer and closer to the tax rate of the second bracket. You can see that in the curve: Between $500 and $2000 the red curve gets closer and closer to the blue one.

That changes again when we have another jump in the marginal tax rate (at $2000). The red curve then tries to get closer to that new tax rate (20%). Once there are no more jumps (because you are in the highest bracket), the curves will get infinitely close.

Steps vs. linear function in marginal tax rate

However, my marginal tax rate jumps up by 5.5%, which is not minuscule.

What's wrong with that? There is nothing to fix here because what matters in the end is the continuity of the curve for the effective tax rate. If the effective tax rate had jumps, that would be a problem because it would mean extremely high marginal rates at those points, leading to situation where earning more money before taxes could mean having less money after taxes. But that's not the case here.

Couldn't the marginal tax rate increase smoothly with income?

So I don't see what a linear progression of the marginal tax rate achieves. It will only make things much more complicated to calculate: You will have a quadratic function as a result for the total tax owed, which is a result of the integration of your linear function. That's not too nice to calculate for non-mathematicians.

Reasons for defining marginal tax rate (vs. continuous curve for effective tax rate)

The reasons for defining marginal tax rates with brackets (steps, blue curve) instead of defining a continuous (and possibly smooth) effective tax rate curve (red curve) directly are mainly simplicity:

  • Simpler math to figure out the total tax burden. Polynomials are not flexible enough for the type of curved that is typically desired as effective tax rate (horizontal asymptote), so you have to introduce more complex functions. People will mess this up, for sure!
  • Trivial way of determining the marginal tax rate (just look at the right bracket): If I try to get that raise, how much of that additional money will go to taxes? Well, if you are in the tax bracket for 30% and your employer pays you $1000 more, you will pay 30% * $1000 = $300 in taxes and therefore have $700 more. Try to figure that out with an algebraically defined smooth effective tax rate curve! You either have to make the complicated calculus twice, or figure out the marginal tax rate by calculating a derivative of the complicated function.
  • Trivial way to make sure there is no marginal tax rate of more than 100% (and not too close to it, either). You want to avoid that situation, otherwise a raise in salary can result in a lower salary after taxes, and that kind of incentive to work less is generally not appreciated in our economic system. With a complicated effective tax rate curve, you'd have to analyze the first derivative to make sure it has a low enough bound.
  • Simpler political discussion about changes in the parameters: It's more understandable to say you want to raise the tax rate in the highest bracket from 40% to 42%, than to say you want to change the second and third factor of the polynomial function from 0.4 to 0.41 and 0.02 to 0.018, respectively. Who would immediately see the implications? Also, this way you can define tax rates for different income groups (even though that's an oversimplification) and the "smooth" transition happens automatically.
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This would be an excellent idea, except people are dumb. Taxpayers have to calculate their taxes themselves, so you have to explain the calculation in a way that a high-school drop-out can figure it out. The manual will say something like

  • If the amount on Line 25 is between $41,000 and $48,999 then subtract $41,000 from the amount on Line 25, and write the result on Line 26; multiply the amount on Line 26 by 0.38, and write the result on Line 27; add $7300 to the amount on Line 27 and write the result on Line 28.

(You could even perform the calculation in two steps, but in my experience, they specify three.)

A dozen lines like that and you have the whole (step-)graduated income tax.

But a smooth graduation? Hard to see how to do that with just a subtraction, a multiplication, and an addition.

And it's not just the filing. The law has to be written and (more or less) understood by lawmakers.

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    OP explicitly showed that the smooth tax result can be computed with just a few worksheet lines and a few operations of addition, subtraction, and multiplication, nothing else. All it does is compute a quadratic function. The problem is that taxpayers, accountants, and politicians also want to be able to intuitively see "what's my (or my clients'/constituents') marginal tax rate", and this would be nonintuitive for everyone who doesn't know calculus. It might be a great way to force an increase in calculus understanding in the general population, though! – nanoman Oct 30 '18 at 2:34
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    People are not dumb, but taxes are incredibly complicated. Especially in the US. – DJClayworth Oct 30 '18 at 3:23
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    Surely two multiplications are easier than a subtraction, multiplication and addition in sequence? – Fred Stark Oct 30 '18 at 6:25
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    "Taxpayers have to calculate their taxes themselves". It's the 21st century. Developed countries have web interfaces to file taxes. There's no reason for that anymore. In fact, I've filed taxes for about 20 years now, and I have never needed to do the sort of steps you describe. It's always been automated. – MSalters Oct 30 '18 at 10:54
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    I smell selection bias. Reality check: a quick Google suggests about 20% of US returns are filed on paper, approximately the same proportion don't use the internet. Take a guess at how many people won't pay for tax preparation because they need that money for gas, or food, or rent. – Phil Frost Oct 30 '18 at 12:25
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The effective tax rate referred to by Henning Makholm and Weirdo is in fact the average tax rate (or the effective average tax rate). Maservant's original question is framed in terms of the marginal utility of income for which the marginal tax rate is the relevant one. It seems to be a good point in principle to argue that if the marginal utility is always (and presumably continuously) declining, then the marginal tax rate should also be always (and continuously?) increasing.

There are a number of points here. First, the obvious conclusion may not be so obvious in economic terms. Modern optimal tax theory, as first developed in the 1970s by Peter Diamond and James Mirrlees, suggests that in the optimal tax rate schedule the marginal rate might peak at some point (it is actually more complicated than this).

More practically, other responders have noted that a smooth marginal tax rate requires more complicated calculations. With electronic filing, or even with a calculator, this need not present a great problem in completing the return, but it does create problems of transparency. First, for the taxpayer. Under the current system, if your taxable income is $100,000, you know that your marginal tax rate is 26%. With a rate structure such as that proposed by the OP, you have to know your exact income to know your exact marginal tax rate. Does this matter? Do taxpayers respond to how they see their after tax income change in fact, or is it important for their behaviour that they actually know what their marginal rate is?

In addition, the proposed structure would make it more difficult for legislators to understand what tax rates they were adopting, and to understand the effects of a proposed rate change. In the OP's proposal there are three parameters: 0.00005%, 13% and 33%. You would need a graph to see easily the effect of changing one of them, and how it worked might not be easy to describe to the non-mathematically minded.

In short, maservant['s suggestion is, at least initially, logical and attractive, but the current system, while arguably crude, has advantages of practicality and transparency. It may not do the right thing, but at least it is reasonably easy to understand what it is doing.

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The tax bands mean that your marginal tax jumps up. The marginal tax determines your willingness to do extra work for extra money.

In the UK, this can be extreme: If your income is £44,000, and your boss asks you to work £1,000 worth of overtime, you pay £200 in tax. If you make £45,000, you pay £400 in tax. (The numbers may have changed a bit). So the second employee will be less willing to do the overtime.

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