# Which model produces the correct revenue-maximizing rate?

Model 1:

The tax rate $\tau$ which maximizes tax revenue equals $\frac{1}{1 + a\epsilon}$ where $\epsilon$ is the elasticity of how taxable income responds to changes in the tax rate across and $a$ is some parameter. See this paper: https://eml.berkeley.edu/~saez/saez-slemrod-giertzNBER09.pdf (see page 6)

We can estimate $a$ to be about 1.5 and $\epsilon$ to be about 0.4 or less, hence the maximizing rate is about 60 % or more.

Model 2:

Tax revenue is $\tau Y(\tau)$ where $Y(\tau)$ is total production which depends on the tax rate. If we say that a 1 % increase in the tax rate leads to a 2.5 % decrease in total output, we get that $Y'(\tau) = -2.5Y(\tau)$. So, $Y(\tau) = Ce^{-2.5\tau}$.

So revenue is $\tau C e^{-2.5\tau}$. Maximizing this simple equation in $\tau$ gives a tax maximizing rate of 40 %.

These two models give highly different results. Which one makes more "sense"?

Model 2 is simple to understand, but I have not read the paper which uses Model 1, it seems a bit advanced for me.

• What context(s) are you considering? Model 1 seems to be based on micro-level household behavior, while Model 2 appears to be related to macro-level national income and tax variables. – Herr K. Mar 1 '18 at 17:51
• If the difference is due to the parameters, then question depends on those magic numbers too. It seems $60\%=1.5\times 0.4$ and $40\% = \frac{1}{2.5}$ – Henry Mar 1 '18 at 18:43