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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.)

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?
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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 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.

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  • $\begingroup$ 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)? $\endgroup$
    – afreelunch
    Commented Dec 9, 2021 at 14:16
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    $\begingroup$ @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) $\endgroup$
    – 1muflon1 ♦
    Commented Dec 9, 2021 at 14:29
  • $\begingroup$ 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.) $\endgroup$
    – afreelunch
    Commented Dec 9, 2021 at 15:38
  • $\begingroup$ @afreelunch no, $g$ does not depend on tau $\endgroup$
    – 1muflon1 ♦
    Commented Dec 9, 2021 at 18:31
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    $\begingroup$ 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 $\tau$." $\endgroup$
    – afreelunch
    Commented Dec 10, 2021 at 11:20

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