reform changed the policy from **p** = _{b, τ }_ = _{_ 70 _,_ 70 _}_ to **p**<sup>_′_</sup> = _{b_<sup>_′_</sup> _, τ_<sup>_′_</sup> _}_ = _{_ 77 _,_ 77 _}_ . Thus, we seek to calculate: where _t_<sup> ̄</sup> denotes the end of our four-year-window, i.e., _t_<sup> ̄</sup> =June 2016. We will then use the bounds on the MVPF to bound the welfare impact of the reform for June 2016.<sup>31</sup> Thus, we need to estimate the number of jumping and bunching households as of June 2016, which, in turn, requires us to estimate the ̄ ̄ number of households locating below the old and new notch under both **p** and **p**<sup>_′_</sup> at _t_<sup> ̄</sup> : _Gt_ (70; **p** ), _Gt_ (70; **p**<sup>_′_</sup> ), ̄ ̄ ̄ ̄ _Gt_ (77; **p** ), and _Gt_ (77; **p**<sup>_′_</sup> ). However, because this was a national reform, only _Gt_ (77; **p**<sup>_′_</sup> ) and _Gt_ (70; **p**<sup>_′_</sup> ) are observed directly because only policy **p**<sup>_′_</sup> was offered in period _t_<sup> ̄</sup> . Thus, our goal is to estimate the number of households reporting less than or equal to R$70 and R$77 in period _t_<sup> ̄</sup> had the reform not happened: _Gt_ ̄(77; **p** ) ̄ and _Gt_ (70; **p** ).

Ideally, we would estimate _Gt_ ̄(77; **p** ) and _Gt_ ̄(70; **p** ) using control groups from random experimental variation (e.g., from staggering the implementation of the reform randomly across geographies); however, as mentioned above, the BF reform was a national reform implemented everywhere starting June 2014. Consequently, our identification strategy will rely on using regions of the reported income distribution that _were not_ impacted by the reform to control for underlying time trends in the portions of the reported income distribution that _were_ impacted by the reform. This brings us to our two identification assumptions.

# _4.1 Identification Assumption 1_

Our identification strategy relies on first finding a region of the reported income distribution which was not impacted by the reform. In the context of the baseline model in Section 2.1, the number of single individual households reporting an income strictly below R$70 should be unchanged by the reform. The intuition for this is that anyone who has a true income below R$70 always reports truthfully both pre- and post-reform and anyone who has a true income above R$70 prefers to misreport at the threshold rather than misreport to an income level below the threshold (because misreporting costs are increasing in the distance between true and reported incomes). While in this baseline model bunching should occur precisely at R$70, in reality bunching is typically more diffuse due to small optimization errors and/or frictions (Kleven, 2016). Thus, our first identification assumption is that the number of people reporting incomes at or below R$63 is unaffected by the reform (i.e., we assume that R$63 is sufficiently far below R$70 such that there are no “bunchers” at or below R$63):

**Identification Assumption 1.** _The distribution of reported incomes below R$63 is unaffected by the reform: Gt_ ( _x_ ; **p** ) = _Gt_ ( _x_ ; **p**<sup>_′_</sup> ) _∀ x ≤_ 63 _, t._

To provide suggestive evidence that Assumption 1 is reasonable, Figure 3 plots the number of single individual households reporting in income bins of size 7 from R$0 to R$63 (with the numbers in each bin

> 31. Implicitly, we assume that agents are not making dynamic decisions (e.g., agents are myopic) so that we can apply Proposition 3 and calculate the welfare impact of the reform in a given time period. Note, there is another reform to the schedule in June 2016 so we cannot estimate the welfare impact of the reform beyond June 2016.

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