# Bunching Estimator with Heterogeneous Elasticities - Saez (2010)

The bunching estimator is an estimator developed by Saez (2010) for kinks and Kleven and Waseem (2013) for notches in order to estimate the elasticity of taxable income.

I understand the estimator for the case of homogeneous elasticities, which is also discussed here and here.

I do not understand the case for heterogeneous elasticities. Kleven and Waseem (2013) argue that for heterogeneous elasticities excess bunching is equal to:

$B=\int_e \int_{z^*}^{z^* + \Delta z_{e}^{*}} \tilde{h}_0(z,e)\ dz\ de \ \approx\ h_0(z^*)E[ z_{e}^{*}]$

Here $h_0(z)$ is the counterfactual density and $h_0(z) \equiv \int_e \tilde{h_0}(z,e)\ de$. Taxable income is $z$, the elasticity is $e$ and $\Delta z_{e}^{*}$ is increasing in $e$. They also assume that $\tilde{h_0}(z,e)$ is locally constant in $z$.

How do they mathematically arrive at this approximation? Also, what is the intuition behind this result?

References

• Kleven, H. and Waseem, M. (2013). Using notches to uncover optimization frictions and structural elasticities: theory and evidence from Pakistan. Quarterly Journal of Economics. 128:669–723

• Saez, E. (2010). Do taxpayers bunch at kink points? American Economic Journal: Economic Policy. 2:180–212