<p>It does determine the optimum tax rate when the weighted average elasticity is constant.</p>
<p>If <span class="math-container">$e (1-\tau) = \frac{1-\tau}{z} \frac{dZ}{d(1-\tau)} = e$</span> there is no issue, because then regardless of a tax rate the elasticity is constant number. This is also what <a href="https://eml.berkeley.edu/%7Esaez/derive.pdf" rel="nofollow noreferrer">Saez 2001</a> does in his simulations (see last chapter).</p>
<p>This is what makes the formula useful,  in fact the intent of Saez paper was to derive the optimal tax rate <strong>as functions of elasticities</strong>. The paper title is literally 'using elasticities to derive optimal income tax rates'.</p>
<p>On a first sight, this might seem as foolish, but it is actually very useful to have these formulas, since in empirical estimations typically all you get is just point estimate for elasticity. Using this formula you can just plug the point estimate, assume elasticity is constant (which is not 100% realistic but given we talk about the average elasticity of labor supply wrt net income also not completely foolish).</p>
<p>You can also use the same formulas with non-constant elasticities, because even though the top rate is defined only implicitly, it can be shown to converge to some rate (see Saez 2001), but then you cannot just plug in the numbers, but you could still find numerically the optimal rate using numerical optimization.</p>

CC BY-SA 4.0
2021-12-09T14:16:24.300Z
Great, that's what I thought. So I guess the assumption in the background is that $Z(\tau) = k\tau^{-e}$ for some $k > 0$ (iso-elastic function)?
afreelunch
CC BY-SA 4.0
2021-12-09T14:29:09.053Z
@afreelunch the assumption is that elasticity is constant but you can get constant elasticity with more than just that one single function (if I am not mistaken there are other functions that also give constant elasticity as special case etc). The paper does not make any assumptions on its functional form, just assumes its constant. Also it’s income elasticity  wrt $(1-\tau)$ not $\tau$ - but I had the  same mistake in answer (now corrected)
1muflon1
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2021-12-09T15:38:41.177Z
If I may ask a separate question, does $g$ also need to evaluated at $\tau^*$? If it reflects marginal utilities evaluated at the resulting consumption profile, then these would also seem to depend on the tax. (Happy to open this as a separate question.)
afreelunch
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2021-12-09T18:31:07.973Z
@afreelunch no, $g$ does not depend on tau
1muflon1
CC BY-SA 4.0
2021-12-10T03:41:16.427Z
But doesn't $g$ depends on the marginal utility of income, which depends on the income level, which depends on $\tau$?
afreelunch
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2021-12-10T07:43:19.770Z
@afreelunch but it depends on marginal utility of gross income (pre tax)
1muflon1
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2021-12-10T11:18:43.970Z
If you go through the derivation, you'll see that $g$ comes from a first order condition, so needs to be evaluated at the optimal consumption profile (not gross income).
afreelunch
CC BY-SA 4.0
2021-12-10T11:20:13.267Z
From the Saez/Piketty textbook: "Under a standard utilitarian social welfare criterion with concave utility of consumption, $\bar{g}$ increases with $\tau$ as the
need for redistribution (i.e., the variation of the $g_i$ with $z_i$) decreases with the level of taxation $\tau$."
afreelunch
CC BY-SA 4.0
2021-12-10T11:54:18.440Z
@afreelunch ok that’s a good point, I thought it’s just gross income, then open it as a new question, I would have to go over the derivation maybe I will answer it if I will figure it later or someone else will
1muflon1
CC BY-SA 4.0
2021-12-12T18:24:18.627Z
You may wish to alter the first sentence of this. (As established, even if $e$ is a constant, this is still not an explicit formula.)
afreelunch
CC BY-SA 4.0
2021-12-12T19:27:27.057Z
@afreelunch no, I investigated this further and the solution is still correct, I thought that $\bar{g}$ means average $g$ but recently I discovered that some people in econ use $\bar{}$ to indicate fixed variables (see Sydsaeter et al EMEA pp 38). This is actually why they also use $\bar{e}$ in the notes. So I believe its just fixed
1muflon1
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2021-12-09T12:27:08.900Z
2021-12-09T14:36:07.317Z
2021-12-09T14:36:07.317Z
1muflon1
<p>According to a famous formula (due I believe to Saez), the optimal maximum tax rate <span class="math-container">$\tau^*$</span> is given by
<span class="math-container">$$ \tau^* = \frac{1 - \bar{g}}{1 - \bar{g} + ae}$$</span>
where <span class="math-container">$g$</span> is the 'average social marginal welfare weight', <span class="math-container">$a$</span> measures the thickness of the tail of the income distribution, and <span class="math-container">$e$</span> is the elasticity of aggregate income with respect to the net of tax rate. (See <a href="https://eml.berkeley.edu/%7Esaez/course/optimallabortax/tax-redistribution_slides.pdf" rel="nofollow noreferrer">these very helpful slides</a>.)</p>
<p><strong>Question:</strong> Given the logic of the derivation, should the formula not be
<span class="math-container">$$ \tau^* = \frac{1 - \bar{g}}{1 - \bar{g} + ae(\tau^*)}$$</span></p>
<p>i.e. do we not need to evaluate the elasticity of aggregate income at the optimal tax rate?</p>
<p>If the answer is 'yes', this leads to some natural follow up questions:</p>
<ul>
<li>If <span class="math-container">$e$</span> is a function of <span class="math-container">$\tau$</span>, in what sense does this formula <em>determine</em> the optimal tax rate? After all, <span class="math-container">$\tau^*$</span> shows up in both the LHS and RHS (so is only defined implicitly)</li>
<li>As I understand matters, empirical applications try to estimate <span class="math-container">$e$</span> by asking how much labour supply would change if we slightly change tax rates. But couldn't such exercises yield misleading answers if the elasticity evaluated at the <em>optimal</em> tax rate differs from the elasticity evaluated at the <em>actual</em> tax rate?</li>
</ul>

<public-economics>
The optimal tax rate
afreelunch
# The optimal tax rate

(Asked by: username_0 on 2021-12-09)

According to a famous formula (due I believe to Saez), the optimal maximum tax rate $\tau^\*$ is given by
$$ \tau^\* = \frac{1 - \bar{g}}{1 - \bar{g} + ae}$$
where $g$ is the 'average social marginal welfare weight', $a$ measures the thickness of the tail of the income distribution, and $e$ is the elasticity of aggregate income with respect to the net of tax rate. (See [these very helpful slides](https://eml.berkeley.edu/%7Esaez/course/optimallabortax/tax-redistribution_slides.pdf).)

**Question:** Given the logic of the derivation, should the formula not be
$$ \tau^\* = \frac{1 - \bar{g}}{1 - \bar{g} + ae(\tau^\*)}$$

i.e. do we not need to evaluate the elasticity of aggregate income at the optimal tax rate?

If the answer is 'yes', this leads to some natural follow up questions:

* If $e$ is a function of $\tau$, in what sense does this formula *determine* the optimal tax rate? After all, $\tau^\*$ shows up in both the LHS and RHS (so is only defined implicitly)
* As I understand matters, empirical applications try to estimate $e$ by asking how much labour supply would change if we slightly change tax rates. But couldn't such exercises yield misleading answers if the elasticity evaluated at the *optimal* tax rate differs from the elasticity evaluated at the *actual* tax rate?

**Tags:** public-economics


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## 1 answers

### Answer by username_1 on 2021-12-09. Score: 3 | Accepted answer

It does determine the optimum tax rate when the weighted average elasticity is constant.

If $e (1-\tau) = \frac{1-\tau}{z} \frac{dZ}{d(1-\tau)} = e$ there is no issue, because then regardless of a tax rate the elasticity is constant number. This is also what [Saez 2001](https://eml.berkeley.edu/%7Esaez/derive.pdf) does in his simulations (see last chapter).

This is what makes the formula useful, in fact the intent of Saez paper was to derive the optimal tax rate **as functions of elasticities**. The paper title is literally 'using elasticities to derive optimal income tax rates'.

On a first sight, this might seem as foolish, but it is actually very useful to have these formulas, since in empirical estimations typically all you get is just point estimate for elasticity. Using this formula you can just plug the point estimate, assume elasticity is constant (which is not 100% realistic but given we talk about the average elasticity of labor supply wrt net income also not completely foolish).

You can also use the same formulas with non-constant elasticities, because even though the top rate is defined only implicitly, it can be shown to converge to some rate (see Saez 2001), but then you cannot just plug in the numbers, but you could still find numerically the optimal rate using numerical optimization.

> **Comments:**
> **username_0**: Great, that's what I thought. So I guess the assumption in the background is that $Z(\tau) = k\tau^{-e}$ for some $k > 0$ (iso-elastic function)?  
> **username_1**: @username_0 the assumption is that elasticity is constant but you can get constant elasticity with more than just that one single function (if I am not mistaken there are other functions that also give constant elasticity as special case etc). The paper does not make any assumptions on its functional form, just assumes its constant. Also it’s income elasticity  wrt $(1-\tau)$ not $\tau$ - but I had the  same mistake in answer (now corrected)  
> **username_0**: If I may ask a separate question, does $g$ also need to evaluated at $\tau^*$? If it reflects marginal utilities evaluated at the resulting consumption profile, then these would also seem to depend on the tax. (Happy to open this as a separate question.)  
> **username_1**: @username_0 no, $g$ does not depend on tau  
> **username_0**: But doesn't $g$ depends on the marginal utility of income, which depends on the income level, which depends on $\tau$?  
> **username_1**: @username_0 but it depends on marginal utility of gross income (pre tax)  
> **username_0**: If you go through the derivation, you'll see that $g$ comes from a first order condition, so needs to be evaluated at the optimal consumption profile (not gross income).  
> **username_0**: From the Saez/Piketty textbook: "Under a standard utilitarian social welfare criterion with concave utility of consumption, $\bar{g}$ increases with $\tau$ as the need for redistribution (i.e., the variation of the $g_i$ with $z_i$) decreases with the level of taxation $\tau$."  
> **username_1**: @username_0 ok that’s a good point, I thought it’s just gross income, then open it as a new question, I would have to go over the derivation maybe I will answer it if I will figure it later or someone else will  
> **username_0**: You may wish to alter the first sentence of this. (As established, even if $e$ is a constant, this is still not an explicit formula.)  
> **username_1**: @username_0 no, I investigated this further and the solution is still correct, I thought that $\bar{g}$ means average $g$ but recently I discovered that some people in econ use $\bar{}$ to indicate fixed variables (see Sydsaeter et al EMEA pp 38). This is actually why they also use $\bar{e}$ in the notes. So I believe its just fixed  

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