<h2>tl;dr</h2>
<p>No they do not for several reasons:</p>
<ol>
<li>Your post is based on misinformation, I know of no work on optimal income taxation that would aim to 'equalize sacrifice' (heck you would not find such philosophy even among most relevant moral/political philosophies). Consequently, the whole premise is false, and you would not even expect to see arguments in favor of implementation of and implementation itself of such tax system.</li>
<li>Even if your question would be rephrased to whether governments generally set marginal tax rates as dictated by optimal taxation literature the answer would be again negative. Generally speaking, politicians for better or worse are simply not interested in optimizing income tax rates (from societal perspective - of course, they likely do optimize it in a way to curry political favors).</li>
</ol>
<h2>Correction of Misinformation in The Question</h2>
<p>Let me first address some misconceptions/misinformation in your question before answering the question:</p>
<blockquote>
<p>From my understanding, the application of a progressive income tax (increasing rate based on income level) follows from the assumptions of: &quot;Equal Sacrifice&quot; - everyone's &quot;sacrifice&quot; should be the same, and the &quot;Decreasing marginal value of wealth&quot; - the actual value inherent in receiving an additional unit of money is less for someone with more money.</p>
</blockquote>
<p>This is incorrect understanding of optimal taxation literature.</p>
<ol>
<li><p>We are talking about optimal income taxes so marginal value of <strong>wealth</strong> does not even enter the calculation here. Optimal income taxation is as the name says about optimizing <strong>income</strong> taxation, and thus what enters optimal income taxation problems is marginal value of <strong>income</strong>.</p>
</li>
<li>
<blockquote>
<p>From my understanding, the application of a progressive income tax (increasing rate based on income level) follows from the assumptions of: &quot;Equal Sacrifice&quot;</p>
</blockquote>
</li>
</ol>
<p>I am not sure where you (mis)heard the above, but it is extremely inaccurate statement.</p>
<p>They are based on the following optimal non-linear tax function which comes from the seminal works of <a href="https://www.jstor.org/stable/2296779?seq=1#metadata_info_tab_contents" rel="nofollow noreferrer">Mirrlees (1971)</a> (who in fact got Nobel Prize in Economics for this contribution), <a href="https://www.jstor.org/stable/116819?seq=1#metadata_info_tab_contents" rel="nofollow noreferrer">Diamond (1998)</a> and <a href="http://www.piketty.pse.ens.fr/files/Saez2001a.pdf" rel="nofollow noreferrer">Saez (2001)</a>. The optimal marginal income tax will be given by:</p>
<p><span class="math-container">$$ \frac{T'(z_n)}{1-T'(z_n)} = \left( 1 + \frac{1}{\epsilon_{lT}} \right)\frac{\int (1-b_m)f(z_m)dz_m}{1-F(z_n)} \frac{1-F(z_n)}{z_nf(z_n0)}$$</span>,</p>
<p>with <span class="math-container">$b_n \equiv \frac{\Psi'(u_n)u_c}{\eta}+ nT'(z_n) \frac{\partial l_n}{\partial \rho} $</span>.</p>
<p>I wont go over every single term in the formula as this would turn this answer into a book but broadly speaking the first part <span class="math-container">$\left( 1 + \frac{1}{\epsilon_{lT}^*} \right)$</span> is given by elasticity of labor supply to income taxes and you can think of it as an 'efficiency' parameter, the second part <span class="math-container">$\frac{\int (1-b_m)f(z_m)dzm}{1-F(z_n)}$</span> tells us what the marginal benefit of redistribution is and this marginal benefit factors in underlying actual welfare which is captured by <span class="math-container">$b_n$</span> which depends on both utility of consumers and the societal utility function,  and finally <span class="math-container">$\frac{1-F(z_n)}{z_nf(z_n0)}$</span> is the part that captures the relative magnitude of distortions created by this taxation. Again since we are talking about improvements in actual underlying utility what the aggregate price level is does not matter.</p>
<p>Now utility does enter the problem via <span class="math-container">$b$</span>, but there is no principle of equal sacrifice or anything like that.</p>
<ul>
<li>It is correct to say that people with concave utilities will have diminishing marginal utility of <strong>income</strong> (they will also have diminishing marginal utility of wealth but this is optimization of <em>income</em> taxes not <em>wealth</em> taxes). However, this has almost no effect on the progressiveness of optimal tax rates and actually does not affect the shape of marginal tax rate schedule that much.</li>
<li>What is far more important is what is the societal welfare function which assigns welfare weights to each individual. Here most commonly used societal welfare functions are (e.g. see Saez 2011):</li>
</ul>
<p>A) <em>Utilitarian social welfare function</em> -  This social welfare function is based on the idea that poorer people value on a margin 1 dollar more than richer people - but by no means it implies 'equal sacrifice'. As a matter of fact if there would be no efficiency cost to taxation a utilitarian would make all incomes completely equal since if there is any person anywhere with higher income than someone else that means aggregate utility could be increased by moving that dollar from higher income person to poor person. Utilitarians  are not trying to make sure everyone sacrifices equal utility - they care about maximizing total utility.</p>
<p>B) <em>Rawlsian social welfare function</em> - This is based on Rawlsian MaxMin idea which states that it is moral to maximize the welfare of the poorest members of society (Rawls (1971), Theory of Justice p. 152). This welfare function gives welfare weight of 1 to the poorest member of society and everyone above zero (of course, in process of redistribution someone else becomes the poorest one - don't interpret this in a way that all redistribution goes to single person). Tax systems based on this function are as far from idea of equal sacrifice as possible. This social welfare function literally does not give a damn about sacrifice of anyone else than a poorest member of society and taxation and redistribution will occur to the point that it generates so much distortions that government is simply not able to make a positive redistribution (in terms of welfare) to the poor anymore.</p>
<p>C) <em>Charitable conservative/Libertarian social welfare function</em> - This one is typically also based on MaxMin criterion with a twist that although poorest still get welfare weight of 1 people who are not poorest also get some non-zero welfare weights (Atkinson 1995 and Diamond 1998). Again nothing to do with idea of equal sacrifice when it comes to utility.</p>
<ul>
<li>What has the major impact on whether tax system is progressive or not actually does not depend that much on either of the above mentioned but primarily on the shape of income distribution.</li>
</ul>
<p>For example, early <a href="https://www.jstor.org/stable/2296779?seq=1" rel="nofollow noreferrer">Mirrlees (1971)</a> and <a href="https://www.sciencedirect.com/science/article/abs/pii/0047272784900355" rel="nofollow noreferrer">Tuomala (1984)</a> simulations based on assumption of log-normal distribution yielded regressive (but approximately linear) marginal tax rates even under maxmin or utilitarian scenarios (e.g. see example below).</p>
<p><a href="https://i.sstatic.net/EWzWO.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/EWzWO.png" alt="enter image description here" /></a></p>
<p>The true reason why modern optimal income taxation shows that tax system is progressive is that we later empirically discovered that log-normal distribution does not fit data nicely because income distributions seem to have pareto-tail (e.g. see Saez 2001). If we switch from log-normal income distribution to log-normal distribution with pareto tail suddenly we get progressive tax rates as you can see below under both utilitarian and maxmin scenario (the marginal tax rates are high for poor due to redistribution but average tax rates are progressive here).</p>
<p><a href="https://i.sstatic.net/LRaQc.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/LRaQc.png" alt="enter image description here" /></a></p>
<p>So the premises of your question are prima facie wrong. Even though it is probably at least in principle possible to derive some new social welfare function that would be about  equal sacrifice, I know of no published research or even famous economist advocating such principle. The shape of an optimal marginal income taxes is primarily determined by shape of income distribution, different moral philosophies typically stretch or squeeze it or move it up and down (of course, one can create arbitrary moral philosophy and some would probably change a shape itself significantly but that is not what you will see with the usual ones).</p>
<h2>Answer</h2>
<blockquote>
<p>My question is, then, to what extent do modern progressive income tax systems match money's &quot;actual&quot; quantity/value curve?</p>
</blockquote>
<p>If I reinterpret this question about asking whether modern progressive income tax systems match the ones prescribed by optimal income tax calculations, the answer is that they simply generally don't. If they do it is often by accident rather than an intent.</p>
<p>For example, below you can see simulation of utilitarian and Rawlsian optimal marginal tax schedules in the Netherlands compared to the real observed ones from Zoutman et al (2012):</p>
<p><a href="https://i.sstatic.net/KPliS.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/KPliS.png" alt="enter image description here" /></a></p>
<p><a href="https://i.sstatic.net/lADb5.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/lADb5.png" alt="enter image description here" /></a></p>
<p>First as you can clearly see the utilitarian optimal marginal tax rate is completely different from the real observed one (although they intersect at 1 point).</p>
<p>Second, the optimal income taxation under Rawlsian scenario also intersects the real marginal tax rates only at a single point, although here above about 80000 euro income the optimal marginal income taxes are more close to the real observed ones, they still not match, and below 80000 euro they completely diverge.</p>
<p>The result that you see above for the Netherlands is not unique (it is not possible to go over all literature and all simulations in scope of SE answer but I never seen work that would show real life progressive marginal tax rates match the optimal ones), even though many countries have different tax codes in most countries you will get match between optimal marginal income tax schedule and real marginal income tax schedule at best at a single point (in case they cross) but often not even that. This is typically explained by the fact that taxes are determined via democratic processes so we should not even expect them to match anything what is predicted by optimal taxation literature. In fact, as shown and argued by <a href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2769798" rel="nofollow noreferrer">Zoutman, Jacobs &amp; Jongen (2016)</a> real world tax systems are often set up to give some extra weights to middle/upper middle class and this is argued to be due to their political importance.</p>

CC BY-SA 4.0
2021-05-11T21:34:52.297Z
Not taxation or political philosophy, but in cooperative game theory there are certain cost sharing rules which fulfill the "equal loss property".
Giskard
CC BY-SA 4.0
2021-05-11T21:38:57.997Z
@Giskard thanks for that addition, I did not know about them, although they are not applied to optimal taxation
1muflon1
CC BY-SA 4.0
2021-05-11T21:06:02.647Z
2021-05-11T21:29:19.780Z
2021-05-11T21:29:19.780Z
1muflon1
<p>From my understanding, the application of a progressive income tax (increasing rate based on income level) follows from the assumptions of: &quot;Equal Sacrifice&quot; - everyone's &quot;sacrifice&quot; should be the same, and the &quot;Decreasing marginal value of wealth&quot; - the actual <em>value</em> inherent in receiving an additional unit of money is less for someone with more money.</p>
<p>My question is, then, to what extent do modern progressive income tax systems match money's &quot;actual&quot; quantity/value curve? Presumably this curve is something that could be estimated/calculated/measured, correct? But as far as I'm aware, no modern taxation system is set with respect to any theoretical or actual quantity/value curve for money.</p>

CC BY-SA 4.0
2021-05-11T17:36:23.850Z
> *progressive income tax [...] follows from the assumptions of: "Equal Sacrifice" - everyone's "sacrifice" should be the same* Can you back this up with a source please?
Giskard
CC BY-SA 4.0
2021-05-11T19:19:04.593Z
Oh, I suppose I don't actually have a source for that assertion. I was just under the impression that a typical "goal" of a taxation system is "Fairness". A priori this would seem to imply that a flat tax would be ideal, but the fact that wealth has a decreasing marginal value actually means that a progressive tax rate is required to achieve the desired "fairness".
Sean Ericson
CC BY-SA 4.0
2021-05-11T19:34:28.157Z
https://www.nber.org/system/files/working_papers/w10005/w10005.pdf
EB3112
CC BY-SA 4.0
2021-05-11T20:11:59.363Z
@EB3112 What part of my question is that paper supposed to address? It seems to be more concerned with "The marginal social value of income redistribution", which isn't really what I'm asking about.
Sean Ericson
CC BY-SA 4.0
2021-05-11T20:24:59.130Z
Well that's fair mate. But no need for the hostile tone. I thought it had appeal, or else I wouldn't have spent my leisure time attempting to help a stranger.
EB3112
CC BY-SA 4.0
2021-05-11T20:36:15.353Z
No hostility intended, friend. I'm just trying to better understand a particular aspect of the field.
Sean Ericson
<taxation><nonexpert><optimal-taxation>
Does Progressive Taxation Actually Follow The Decreasing Marginal Value of Wealth?
Sean Ericson
# Does Progressive Taxation Actually Follow The Decreasing Marginal Value of Wealth?

(Asked by: username_0 on 2021-05-11)

From my understanding, the application of a progressive income tax (increasing rate based on income level) follows from the assumptions of: "Equal Sacrifice" - everyone's "sacrifice" should be the same, and the "Decreasing marginal value of wealth" - the actual *value* inherent in receiving an additional unit of money is less for someone with more money.

My question is, then, to what extent do modern progressive income tax systems match money's "actual" quantity/value curve? Presumably this curve is something that could be estimated/calculated/measured, correct? But as far as I'm aware, no modern taxation system is set with respect to any theoretical or actual quantity/value curve for money.

**Tags:** taxation, nonexpert, optimal-taxation
> **Comments:**
> **username_1**: > *progressive income tax [...] follows from the assumptions of: "Equal Sacrifice" - everyone's "sacrifice" should be the same* Can you back this up with a source please?  
> **username_0**: Oh, I suppose I don't actually have a source for that assertion. I was just under the impression that a typical "goal" of a taxation system is "Fairness". A priori this would seem to imply that a flat tax would be ideal, but the fact that wealth has a decreasing marginal value actually means that a progressive tax rate is required to achieve the desired "fairness".  
> **username_2**: https://www.nber.org/system/files/working_papers/w10005/w10005.pdf  
> **username_0**: @EB3112 What part of my question is that paper supposed to address? It seems to be more concerned with "The marginal social value of income redistribution", which isn't really what I'm asking about.  
> **username_2**: Well that's fair mate. But no need for the hostile tone. I thought it had appeal, or else I wouldn't have spent my leisure time attempting to help a stranger.  
> **username_0**: No hostility intended, friend. I'm just trying to better understand a particular aspect of the field.  

---

## 1 answers

### Answer by username_3 on 2021-05-11. Score: 1

## tl;dr

No they do not for several reasons:

1. Your post is based on misinformation, I know of no work on optimal income taxation that would aim to 'equalize sacrifice' (heck you would not find such philosophy even among most relevant moral/political philosophies). Consequently, the whole premise is false, and you would not even expect to see arguments in favor of implementation of and implementation itself of such tax system.
2. Even if your question would be rephrased to whether governments generally set marginal tax rates as dictated by optimal taxation literature the answer would be again negative. Generally speaking, politicians for better or worse are simply not interested in optimizing income tax rates (from societal perspective - of course, they likely do optimize it in a way to curry political favors).

## Correction of Misinformation in The Question

Let me first address some misconceptions/misinformation in your question before answering the question:

> From my understanding, the application of a progressive income tax (increasing rate based on income level) follows from the assumptions of: "Equal Sacrifice" - everyone's "sacrifice" should be the same, and the "Decreasing marginal value of wealth" - the actual value inherent in receiving an additional unit of money is less for someone with more money.

This is incorrect understanding of optimal taxation literature.

1. We are talking about optimal income taxes so marginal value of **wealth** does not even enter the calculation here. Optimal income taxation is as the name says about optimizing **income** taxation, and thus what enters optimal income taxation problems is marginal value of **income**.
2. > From my understanding, the application of a progressive income tax (increasing rate based on income level) follows from the assumptions of: "Equal Sacrifice"

I am not sure where you (mis)heard the above, but it is extremely inaccurate statement.

They are based on the following optimal non-linear tax function which comes from the seminal works of [Mirrlees (1971)](https://www.jstor.org/stable/2296779?seq=1#metadata_info_tab_contents) (who in fact got Nobel Prize in Economics for this contribution), [Diamond (1998)](https://www.jstor.org/stable/116819?seq=1#metadata_info_tab_contents) and [Saez (2001)](http://www.piketty.pse.ens.fr/files/Saez2001a.pdf). The optimal marginal income tax will be given by:

$$ \frac{T'(z\_n)}{1-T'(z\_n)} = \left( 1 + \frac{1}{\epsilon\_{lT}} \right)\frac{\int (1-b\_m)f(z\_m)dz\_m}{1-F(z\_n)} \frac{1-F(z\_n)}{z\_nf(z\_n0)}$$,

with $b\_n \equiv \frac{\Psi'(u\_n)u\_c}{\eta}+ nT'(z\_n) \frac{\partial l\_n}{\partial \rho} $.

I wont go over every single term in the formula as this would turn this answer into a book but broadly speaking the first part $\left( 1 + \frac{1}{\epsilon\_{lT}^\*} \right)$ is given by elasticity of labor supply to income taxes and you can think of it as an 'efficiency' parameter, the second part $\frac{\int (1-b\_m)f(z\_m)dzm}{1-F(z\_n)}$ tells us what the marginal benefit of redistribution is and this marginal benefit factors in underlying actual welfare which is captured by $b\_n$ which depends on both utility of consumers and the societal utility function, and finally $\frac{1-F(z\_n)}{z\_nf(z\_n0)}$ is the part that captures the relative magnitude of distortions created by this taxation. Again since we are talking about improvements in actual underlying utility what the aggregate price level is does not matter.

Now utility does enter the problem via $b$, but there is no principle of equal sacrifice or anything like that.

* It is correct to say that people with concave utilities will have diminishing marginal utility of **income** (they will also have diminishing marginal utility of wealth but this is optimization of *income* taxes not *wealth* taxes). However, this has almost no effect on the progressiveness of optimal tax rates and actually does not affect the shape of marginal tax rate schedule that much.
* What is far more important is what is the societal welfare function which assigns welfare weights to each individual. Here most commonly used societal welfare functions are (e.g. see Saez 2011):

A) *Utilitarian social welfare function* - This social welfare function is based on the idea that poorer people value on a margin 1 dollar more than richer people - but by no means it implies 'equal sacrifice'. As a matter of fact if there would be no efficiency cost to taxation a utilitarian would make all incomes completely equal since if there is any person anywhere with higher income than someone else that means aggregate utility could be increased by moving that dollar from higher income person to poor person. Utilitarians are not trying to make sure everyone sacrifices equal utility - they care about maximizing total utility.

B) *Rawlsian social welfare function* - This is based on Rawlsian MaxMin idea which states that it is moral to maximize the welfare of the poorest members of society (Rawls (1971), Theory of Justice p. 152). This welfare function gives welfare weight of 1 to the poorest member of society and everyone above zero (of course, in process of redistribution someone else becomes the poorest one - don't interpret this in a way that all redistribution goes to single person). Tax systems based on this function are as far from idea of equal sacrifice as possible. This social welfare function literally does not give a damn about sacrifice of anyone else than a poorest member of society and taxation and redistribution will occur to the point that it generates so much distortions that government is simply not able to make a positive redistribution (in terms of welfare) to the poor anymore.

C) *Charitable conservative/Libertarian social welfare function* - This one is typically also based on MaxMin criterion with a twist that although poorest still get welfare weight of 1 people who are not poorest also get some non-zero welfare weights (Atkinson 1995 and Diamond 1998). Again nothing to do with idea of equal sacrifice when it comes to utility.

* What has the major impact on whether tax system is progressive or not actually does not depend that much on either of the above mentioned but primarily on the shape of income distribution.

For example, early [Mirrlees (1971)](https://www.jstor.org/stable/2296779?seq=1) and [Tuomala (1984)](https://www.sciencedirect.com/science/article/abs/pii/0047272784900355) simulations based on assumption of log-normal distribution yielded regressive (but approximately linear) marginal tax rates even under maxmin or utilitarian scenarios (e.g. see example below).

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

The true reason why modern optimal income taxation shows that tax system is progressive is that we later empirically discovered that log-normal distribution does not fit data nicely because income distributions seem to have pareto-tail (e.g. see Saez 2001). If we switch from log-normal income distribution to log-normal distribution with pareto tail suddenly we get progressive tax rates as you can see below under both utilitarian and maxmin scenario (the marginal tax rates are high for poor due to redistribution but average tax rates are progressive here).

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

So the premises of your question are prima facie wrong. Even though it is probably at least in principle possible to derive some new social welfare function that would be about equal sacrifice, I know of no published research or even famous economist advocating such principle. The shape of an optimal marginal income taxes is primarily determined by shape of income distribution, different moral philosophies typically stretch or squeeze it or move it up and down (of course, one can create arbitrary moral philosophy and some would probably change a shape itself significantly but that is not what you will see with the usual ones).

## Answer

> My question is, then, to what extent do modern progressive income tax systems match money's "actual" quantity/value curve?

If I reinterpret this question about asking whether modern progressive income tax systems match the ones prescribed by optimal income tax calculations, the answer is that they simply generally don't. If they do it is often by accident rather than an intent.

For example, below you can see simulation of utilitarian and Rawlsian optimal marginal tax schedules in the Netherlands compared to the real observed ones from Zoutman et al (2012):

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

[![enter image description here](https://i.sstatic.net/lADb5.png)](https://i.sstatic.net/lADb5.png)

First as you can clearly see the utilitarian optimal marginal tax rate is completely different from the real observed one (although they intersect at 1 point).

Second, the optimal income taxation under Rawlsian scenario also intersects the real marginal tax rates only at a single point, although here above about 80000 euro income the optimal marginal income taxes are more close to the real observed ones, they still not match, and below 80000 euro they completely diverge.

The result that you see above for the Netherlands is not unique (it is not possible to go over all literature and all simulations in scope of SE answer but I never seen work that would show real life progressive marginal tax rates match the optimal ones), even though many countries have different tax codes in most countries you will get match between optimal marginal income tax schedule and real marginal income tax schedule at best at a single point (in case they cross) but often not even that. This is typically explained by the fact that taxes are determined via democratic processes so we should not even expect them to match anything what is predicted by optimal taxation literature. In fact, as shown and argued by [Zoutman, Jacobs & Jongen (2016)](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2769798) real world tax systems are often set up to give some extra weights to middle/upper middle class and this is argued to be due to their political importance.

> **Comments:**
> **username_1**: Not taxation or political philosophy, but in cooperative game theory there are certain cost sharing rules which fulfill the "equal loss property".  
> **username_3**: @Giskard thanks for that addition, I did not know about them, although they are not applied to optimal taxation  

---

