How to use the bunching estimator to estimate the elasticity of taxable income Today my professor told us about a recent estimator that non-parametrically estimates the elasticity of taxable income. I understood why doing this gets around problems of previous studies that use the difference in the differences or similar.
He said that the bunching estimator uses the mass of people after the tax threshold in order to estimate the elasticity epsilon. I don't understand the mathematical/statistical idea behind this estimator. My professor only mentioned it briefly as a more advanced topic but he did not go into detail. I would like to know more about it because I might use it in my master thesis.
Therefore I am looking for an explanation of how the bunching estimator works, especially the statistical part. He said the reference is Saez 2010, Do Taxpayers bunch at kink points.
 A: The best thing to do would be to look at Emanuel Saez' original paper.
He came up with this bunching estimator but you need to be aware of the assumptions which are also stated in the paper. He assumes that the marginal tax rate $t$ is the same for everyone, individuals have perfect knowledge of the tax system (i.e. there are no adjustment frictions, inattention, etc.), and that individuals maximize utility at the point where their indifference curve is tangent to the budget constraint. Then the tax rate changes for people who earn above a certain threshold $z^*$ from $t$ to $t+dt$ where $dt$ is a small quantity (this will be important later). So the tax rate then is
$$
\tau = \begin{cases} t &\mbox{if } z \leq z^* \\ z + dt &\mbox{if } z > z^* \end{cases}
$$
The change in utility for both cases is shown by Saez in the figure below where $L$ is the individual who is unaffected by the tax change and $H$ is subject to the new marginal tax rate after the reform, i.e. she reduces her taxable income in order to move from $z^* + dz^*$ to $z^*$. This is what Saez calls "bunching" because the interval $(z^*,z^* + dz^*)$ is strictly dominated in the sense that anybody in this range is better off by decreasing their taxable income to $z^*$ (which is the bunching point). Again, the graph comes directly from the paper which is very well written so you should definitely read it if you want to use something like this in your thesis.

Now Saez makes another assumption which is that income is distributed according to some smooth function $h(z)$. The number of individuals that bunch at $z^*$ is then given by the area under the density in the dominated region which is approximately the height of the density at the bunch point,
$$
\begin{align}
B &= \int^{z^* + dz^*}_{z^*} h(z) dz \newline
&\approx h(z^*)dz^*
\end{align}
$$
This is where the earlier assumption on the size of the tax change comes in because this approximation only holds for small changes. In order to estimate the elasticity of taxable income recall the definition:
$$
\begin{align}
\epsilon &= \frac{dz}{d(1-t)}\frac{(1-t)}{z} \newline
&= \frac{\frac{dz}{z}}{\frac{d(1-t)}{1-t}}
\end{align}
$$
where the denominator is the change in the net-of-tax rate. Evaluate this elasticity at the bunch point $z^*$ and use the above approximation (re-arranged) $dz^* = \frac{B}{h(z^*)}$, the bunching estimator is
$$
\epsilon^* = \frac{\frac{B}{z^* \cdot h(z^*)}}{\frac{d(1-t)}{1-t}}
$$
Note that you need high quality data and many data points in order to estimate the density properly. If I remember correctly, Saez uses administrative data from the Internal Revenue Service of the United States. The Stata code for Saez' paper should be available on the website of the AEJ: Policy section here. 
Once you understood Saez' estimator you might find the paper by Chetty et al. (2011) interesting as they relax an important assumption of Saez' bunching estimator. Namely they allow for optimization frictions given that people cannot immediately adjust their income in order to reach the kink point or there is too much inattention/reluctance to move to the kink.
