<p>The way tax brackets currently work is a fixed marginal rate that increments at set intervals. Solving the amount of tax to be paid can be done with a pocket calculator and basic algebra. Using a curve implies a constant change to the marginal rate implying every dollar earned will have a slighly different tax rate and you need to use calculus to figure out how much to pay. </p>

<p>At the end of the day, the people running the tax office are not going to care about having a beautiful curve for tax rates as they are dealing with all of us incompetent tax payers paying the wrong amount of taxes. As well, complicated tax calculations hurt small business owners and the self-employed, who can't be expected to be using calculus just to pay their taxes.</p>

<p>Politicians are smart enough to realize just because it's pretty that doesn't mean it's practical. You've got to consider the audience of people actually implementing the rules.</p>

CC BY-SA 3.0
2017-11-05T23:21:44.363Z
Hey, yes, all valid points and it's indeed something I considered. I keep thinking that we're technologically progressing at such pace that in the future it won't matter how complicated the graph is or whether you can calculate your amount due in your head. As such, my original question assumes software would be used to perform these calculations and there would be no-to-low human intervention. E.g. software like FreshBooks would automatically do all of this for you, so as a business owner you don't care about the complexity behind the tax formulae.
lmenus
CC BY-SA 3.0
2017-11-03T17:36:20.310Z
2017-11-03T17:36:20.310Z
serakfalcon
<p><strong>Why not use a smoothly changing incremental rate? "A steady curve"</strong></p>

<p>First, as it has been clarified that the bracket changes are on incremental income then there are no major discontinuities in the <em>average</em> rate even though there may be in the <em>marginal</em> rate. So one might put the question in reverse: why <em>would</em> you have smoothly changing marginal rates? </p>

<p>There is more interest in the simplification and flattening of income tax. </p>

<p><strong>Example: UK margin and average rates of Income Tax &amp; NI</strong></p>

<p>UK taxes on employee income have two components which together comprise the margin rate at which tax is taken. National Insurance was a 'welfare stamp' with an upper income limit, though it has since been used to increase higher rate income tax by 2% without affecting the headline rates that gets most attention</p>

<p>There is also a 'claw back' of the tax free allowance between £100,000 &amp; £125,000 making the peak UK income tax rate be 62%!</p>

<p>There are some difference in tax in Scotland &amp; Wales, so strictly these numbers apply to England and Northern Ireland.</p>

<p>It is pretty bonkers!</p>

<p><a href="https://i.sstatic.net/LDkSm.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/LDkSm.png" alt="Margin % rates of Income Tax &amp; Employee National Insurance"></a></p>

<p>But even with all this, when you look at overall amount taken as a % of income ("the average % rate") this is relatively smooth and curved even given the bonkers changes in margin rates.</p>

<p><a href="https://i.sstatic.net/xPve8.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/xPve8.png" alt="enter image description here"></a></p>

<p><strong>How would you implement algebraic rates</strong></p>

<p>If you are after smoothly changing average % rates as you can see this is entirely possible with a piecemeal set of marginal rates.</p>

<p>So it is not so clear to me why you would want smoothly changing marginal rates - you can have smoothly changing average rates without them. Further, two particular problems occur to me:</p>

<ul>
<li><p>Any simple formula has all sorts of political and ethical
implications embedded in it that may not correspond to any political
consensus and it would be hard to adjust to reflect any evolution of
public concerns (this might be an advantage from an economists point
of view!) and it would be hard to manage any kind of public debate.
Creating a piecemeal continuous margin rate curve (piecewise splines
perhaps!) would add massively to the complexity of the system with
little clear benefit.</p></li>
<li><p>The is also more "plumbing" behind the scenes in the tax system that
may not be obvious at once - rules rounding off, different kinds of
income having different marginal rates, etc that would mean a great
deal of established regulation and system would have to be rebuilt
more the ground up. Again: for what benefit.</p></li>
</ul>

<p><strong>What reforms to income tax make sense?</strong></p>

<p>Two current themes in tax reform are:</p>

<ul>
<li><strong>Simplification</strong> - The US tax code is c. 9,000 pages; the UK's is c. 17,000 and allegedly the longest in the world. This compared to say Hong Kong at c. 275 pages. Complexity begets complexity as each change creates new problems, new transitional measures and new anti-avoidance measures. </li>
</ul>

<p><a href="https://www.theguardian.com/commentisfree/2015/feb/13/britain-tax-code-17000-pages-long-dog-whistle-very-rich" rel="nofollow noreferrer">Guardian - UK has longest tax code in the world</a></p>

<p><a href="https://taxfoundation.org/how-many-words-are-tax-code/" rel="nofollow noreferrer">Tax Foundation - length of the US's tax code</a></p>

<ul>
<li><strong>Flattening</strong> - the greater the variation in rates the more benefit there is to finding ways of changing the tax status of some income. These schemes will both distort economic decision making, lead to pointless or damaging but tax efficient investments, lead to far to much money and effort going into developing such schemes, and lead to lots of effort and bureaucracy in an effort to stop such schemes. A simple tax code with single low rate but few exemptions could lead to high compliance, lower collection costs and less damage to the economy.</li>
</ul>

CC BY-SA 4.0
2019-11-16T13:55:36.313Z
Like it, thank you for your answer!
lmenus
CC BY-SA 4.0
2019-11-27T02:46:43.220Z
Good analysis. Another way to look at it is that tax brackets are a "technology" for writing a function for a kind of smoothly increasing average rate. Smoother curves are attainable but somebody would have to find a practical benefit to improving something that is already smooth.
H2ONaCl
CC BY-SA 4.0
2019-11-15T12:03:13.500Z
2020-05-14T07:37:39.580Z
2020-05-14T07:37:39.580Z
Duke Bouvier
<p>I have been asking myself this question for a long time. </p>

<p>Indeed using a steady curve (instead of brackets) for linking income to tax rates would be simpler and clearer.  </p>

<p>Everybody could easily calculate their tax rates at a given income. Or if he or she is traumatised by maths in school: government could publish look-up tables anyway. </p>

<p>So why don't states do this?</p>

<p>The only explanation I can imagine is that </p>

<p>a) politicians are often maths- and formula-averse, or </p>

<p>b) they are not but they feel that a majority of population is formula-averse and that they would not meet the necesssary level of understanding in the electorate, so it is unattractive for them to propagate such an improved system, or </p>

<p>c) bureaucrats in financial authorities are change-averse and do not suggest that solution to politicians. </p>

<p>All of these are weak reasons. But still I do not see better ones. </p>

CC BY-SA 3.0
2017-11-03T13:46:19.747Z
"you fall into a different bracket, and your taxes eat up your entire pay raise and more" That is not [how tax brackets work](https://en.wikipedia.org/wiki/Tax_bracket#Example).
Giskard
CC BY-SA 3.0
2017-11-03T15:21:06.363Z
Okay. Thanks. I removed the respective section as it was indeed inadequate. - Still the main question remains: why not use a steady curve?
Christian Geiselmann
CC BY-SA 3.0
2017-11-03T12:36:09.697Z
2017-11-03T15:20:13.220Z
2017-11-03T15:20:13.220Z
Christian Geiselmann
<p>I think you've misunderstood how the (UK) income tax brackets work. They work differently to how stamp duty used to work.</p>

<p>Stamp duty brackets used to be absolute rates: so when the purchase price crossed over the bracket threshold by £1, the marginal rate would be huge, as the whole purchase price was taxed in the higher rate. Hence the clustering.</p>

<p>That's not what happens with income tax. (and it's no longer what happens with stamp duty, either)</p>

<p>With income tax, only the amount of income that's above the threshold, gets taxed at the higher rate. So, the marginal rate is always equal to the band rate. i.e. if the rates are as in your example, and your income rises from 10,000 to 10,001 then you would pay 10% on the first 10,000 and 20% on the extra 1. You would not pay 20% on all 10,001.</p>

CC BY-SA 3.0
2017-11-05T23:15:23.570Z
Hey, you're right, I actually wasn't aware this was how the rate is calculated. I didn't get to see Christian's original reply below, but I assume he had the same misunderstanding. Thanks for this!
lmenus
CC BY-SA 3.0
2017-11-03T16:20:28.157Z
2017-11-03T16:20:28.157Z
410 gone
<p>Majority of countries (if not all) use tax brackets to define how much tax should the earning workforce pay. So we could have something like this.</p>

<pre><code>Pay | Tax
0 - 10,000 | 10%
10,001 - 20,000 | 20%
20,001 - 30,000 | 30%
etc.
</code></pre>

<p>The breakpoints are what I am going to ask about. It has been proven in other schemes, such as the UK's <a href="https://www.gov.uk/stamp-duty-land-tax/overview" rel="noreferrer">Stamp Duty Land Tax</a> that separating tax into categories like this leads to clusters on the lower bound of the breakpoints. My understanding is that this is not a desired effect and the distribution on the scale should be more fluid.</p>

<p>I come more from the software engineering side as opposed to the economic side, so I assume that in the past it was easier to calculate tax using this system and that's why it was chosen. Humans would not be able or willing to do complex tax calculations had we used rules that can be charted for example as S curves.</p>

<p>However, I wonder if we could have a better system today? (Assuming the current system is not sufficient.) Are there any proposals or significant movements that are not satisfied with the current tax system?</p>

<p>Note: I am not making assumptions about how the collected money is being spent, or how much is being collected. This is purely about the curves behind the tax brackets and clustering.</p>

CC BY-SA 3.0
2017-07-25T19:46:57.543Z
Part of the reason for using brackets is to make planning easier. If I know which bracket I fall into I can figure out about how much tax I own with a pencil and paper. If the marginal rate changed for each dollar I earn, it turns into a complex problem. It is much easier to plan when i can make a rough but good enough guess at my tax liability. And ability to plan is a huge deal in economic endeavors.
zeta-band
CC BY-SA 3.0
2017-11-03T12:19:50.587Z
@zeta-band   I understand your idea about easier planning for those who pay taxes. But on the other hand: I do not see what the problem would be if there was a formula-based (steady) curve to link income with tax rates (instead of stepped brackets). The tax payer could anyway make a "rough but good enough guess" (as you said) on his tax liability. Or it would even allow for a *better* guess, because a steady curve by definition does not have steps (like if you earn 99,999 vs 100,000 and you fall into a completely different bracket just because of the difference of 1 [currency unit]).
Christian Geiselmann
CC BY-SA 3.0
2017-07-10T08:24:03.723Z
The income tax bracket is base on assumption that subsequent bracket reaping/exploits more resources from the society. However, there is tax brakes/incentive to reduce reaching the bracket or reduce amount to be paid.
mootmoot
CC BY-SA 3.0
2017-07-10T15:01:30.357Z
The way the question is currently formulated it is asking for opinion on a hypothetical tax system. Unfortunately we do not deal in opinion-based answers.
Giskard
<taxation>
Why do we use tax brackets?
lmenus
# Why do we use tax brackets?

(Asked by: username_0 on 2017-07-10)

Majority of countries (if not all) use tax brackets to define how much tax should the earning workforce pay. So we could have something like this.

```
Pay | Tax
0 - 10,000 | 10%
10,001 - 20,000 | 20%
20,001 - 30,000 | 30%
etc.

```

The breakpoints are what I am going to ask about. It has been proven in other schemes, such as the UK's [Stamp Duty Land Tax](https://www.gov.uk/stamp-duty-land-tax/overview) that separating tax into categories like this leads to clusters on the lower bound of the breakpoints. My understanding is that this is not a desired effect and the distribution on the scale should be more fluid.

I come more from the software engineering side as opposed to the economic side, so I assume that in the past it was easier to calculate tax using this system and that's why it was chosen. Humans would not be able or willing to do complex tax calculations had we used rules that can be charted for example as S curves.

However, I wonder if we could have a better system today? (Assuming the current system is not sufficient.) Are there any proposals or significant movements that are not satisfied with the current tax system?

Note: I am not making assumptions about how the collected money is being spent, or how much is being collected. This is purely about the curves behind the tax brackets and clustering.

**Tags:** taxation

> **Comments:**
> **username_3**: The income tax bracket is base on assumption that subsequent bracket reaping/exploits more resources from the society. However, there is tax brakes/incentive to reduce reaching the bracket or reduce amount to be paid.  
> **username_4**: The way the question is currently formulated it is asking for opinion on a hypothetical tax system. Unfortunately we do not deal in opinion-based answers.  
> **username_1**: Part of the reason for using brackets is to make planning easier. If I know which bracket I fall into I can figure out about how much tax I own with a pencil and paper. If the marginal rate changed for each dollar I earn, it turns into a complex problem. It is much easier to plan when i can make a rough but good enough guess at my tax liability. And ability to plan is a huge deal in economic endeavors.  
> **username_2**: @username_1   I understand your idea about easier planning for those who pay taxes. But on the other hand: I do not see what the problem would be if there was a formula-based (steady) curve to link income with tax rates (instead of stepped brackets). The tax payer could anyway make a "rough but good enough guess" (as you said) on his tax liability. Or it would even allow for a *better* guess, because a steady curve by definition does not have steps (like if you earn 99,999 vs 100,000 and you fall into a completely different bracket just because of the difference of 1 [currency unit]).  

---

## 4 answers

### Answer by username_8 on 2017-11-03. Score: 7 | Accepted answer

I think you've misunderstood how the (UK) income tax brackets work. They work differently to how stamp duty used to work.

Stamp duty brackets used to be absolute rates: so when the purchase price crossed over the bracket threshold by £1, the marginal rate would be huge, as the whole purchase price was taxed in the higher rate. Hence the clustering.

That's not what happens with income tax. (and it's no longer what happens with stamp duty, either)

With income tax, only the amount of income that's above the threshold, gets taxed at the higher rate. So, the marginal rate is always equal to the band rate. i.e. if the rates are as in your example, and your income rises from 10,000 to 10,001 then you would pay 10% on the first 10,000 and 20% on the extra 1. You would not pay 20% on all 10,001.

> **Comments:**
> **username_0**: Hey, you're right, I actually wasn't aware this was how the rate is calculated. I didn't get to see username_2's original reply below, but I assume he had the same misunderstanding. Thanks for this!  

---

### Answer by username_5 on 2017-11-03. Score: 3

The way tax brackets currently work is a fixed marginal rate that increments at set intervals. Solving the amount of tax to be paid can be done with a pocket calculator and basic algebra. Using a curve implies a constant change to the marginal rate implying every dollar earned will have a slighly different tax rate and you need to use calculus to figure out how much to pay.

At the end of the day, the people running the tax office are not going to care about having a beautiful curve for tax rates as they are dealing with all of us incompetent tax payers paying the wrong amount of taxes. As well, complicated tax calculations hurt small business owners and the self-employed, who can't be expected to be using calculus just to pay their taxes.

Politicians are smart enough to realize just because it's pretty that doesn't mean it's practical. You've got to consider the audience of people actually implementing the rules.

> **Comments:**
> **username_0**: Hey, yes, all valid points and it's indeed something I considered. I keep thinking that we're technologically progressing at such pace that in the future it won't matter how complicated the graph is or whether you can calculate your amount due in your head. As such, my original question assumes software would be used to perform these calculations and there would be no-to-low human intervention. E.g. software like FreshBooks would automatically do all of this for you, so as a business owner you don't care about the complexity behind the tax formulae.  

---

### Answer by username_6 on 2019-11-15. Score: 2

**Why not use a smoothly changing incremental rate? "A steady curve"**

First, as it has been clarified that the bracket changes are on incremental income then there are no major discontinuities in the *average* rate even though there may be in the *marginal* rate. So one might put the question in reverse: why *would* you have smoothly changing marginal rates?

There is more interest in the simplification and flattening of income tax.

**Example: UK margin and average rates of Income Tax & NI**

UK taxes on employee income have two components which together comprise the margin rate at which tax is taken. National Insurance was a 'welfare stamp' with an upper income limit, though it has since been used to increase higher rate income tax by 2% without affecting the headline rates that gets most attention

There is also a 'claw back' of the tax free allowance between £100,000 & £125,000 making the peak UK income tax rate be 62%!

There are some difference in tax in Scotland & Wales, so strictly these numbers apply to England and Northern Ireland.

It is pretty bonkers!

[![Margin % rates of Income Tax & Employee National Insurance](https://i.sstatic.net/LDkSm.png)](https://i.sstatic.net/LDkSm.png)

But even with all this, when you look at overall amount taken as a % of income ("the average % rate") this is relatively smooth and curved even given the bonkers changes in margin rates.

[![enter image description here](https://i.sstatic.net/xPve8.png)](https://i.sstatic.net/xPve8.png)

**How would you implement algebraic rates**

If you are after smoothly changing average % rates as you can see this is entirely possible with a piecemeal set of marginal rates.

So it is not so clear to me why you would want smoothly changing marginal rates - you can have smoothly changing average rates without them. Further, two particular problems occur to me:

* Any simple formula has all sorts of political and ethical
  implications embedded in it that may not correspond to any political
  consensus and it would be hard to adjust to reflect any evolution of
  public concerns (this might be an advantage from an economists point
  of view!) and it would be hard to manage any kind of public debate.
  Creating a piecemeal continuous margin rate curve (piecewise splines
  perhaps!) would add massively to the complexity of the system with
  little clear benefit.
* The is also more "plumbing" behind the scenes in the tax system that
  may not be obvious at once - rules rounding off, different kinds of
  income having different marginal rates, etc that would mean a great
  deal of established regulation and system would have to be rebuilt
  more the ground up. Again: for what benefit.

**What reforms to income tax make sense?**

Two current themes in tax reform are:

* **Simplification** - The US tax code is c. 9,000 pages; the UK's is c. 17,000 and allegedly the longest in the world. This compared to say Hong Kong at c. 275 pages. Complexity begets complexity as each change creates new problems, new transitional measures and new anti-avoidance measures.

[Guardian - UK has longest tax code in the world](https://www.theguardian.com/commentisfree/2015/feb/13/britain-tax-code-17000-pages-long-dog-whistle-very-rich)

[Tax Foundation - length of the US's tax code](https://taxfoundation.org/how-many-words-are-tax-code/)

* **Flattening** - the greater the variation in rates the more benefit there is to finding ways of changing the tax status of some income. These schemes will both distort economic decision making, lead to pointless or damaging but tax efficient investments, lead to far to much money and effort going into developing such schemes, and lead to lots of effort and bureaucracy in an effort to stop such schemes. A simple tax code with single low rate but few exemptions could lead to high compliance, lower collection costs and less damage to the economy.

> **Comments:**
> **username_0**: Like it, thank you for your answer!  
> **username_7**: Good analysis. Another way to look at it is that tax brackets are a "technology" for writing a function for a kind of smoothly increasing average rate. Smoother curves are attainable but somebody would have to find a practical benefit to improving something that is already smooth.  

---

### Answer by username_2 on 2017-11-03. Score: 0

I have been asking myself this question for a long time.

Indeed using a steady curve (instead of brackets) for linking income to tax rates would be simpler and clearer.

Everybody could easily calculate their tax rates at a given income. Or if he or she is traumatised by maths in school: government could publish look-up tables anyway.

So why don't states do this?

The only explanation I can imagine is that

a) politicians are often maths- and formula-averse, or

b) they are not but they feel that a majority of population is formula-averse and that they would not meet the necesssary level of understanding in the electorate, so it is unattractive for them to propagate such an improved system, or

c) bureaucrats in financial authorities are change-averse and do not suggest that solution to politicians.

All of these are weak reasons. But still I do not see better ones.

> **Comments:**
> **username_4**: "you fall into a different bracket, and your taxes eat up your entire pay raise and more" That is not [how tax brackets work](https://en.wikipedia.org/wiki/Tax_bracket#Example).  
> **username_2**: Okay. Thanks. I removed the respective section as it was indeed inadequate. - Still the main question remains: why not use a steady curve?  

---

