cash; if the transfer is paid in-kind, one would require an estimate of the average WTP for $1 worth of the in-kind good to apply an analogue to Proposition 3.

Lastly, we discuss the advantages of bounding the welfare impacts (as opposed to exactly characterizing the welfare impacts) of a notch reform. First, because we cannot apply envelope conditions in our setting, exactly characterizing welfare impacts requires one to take a stance on which margins households are responding, the functional form of their utility function, the sorts of frictions they face, etc., whereas bounding welfare impacts requires very little structure on the household optimization problem. In fact, we prove in Appendix A.4 that the bounds in Proposition 3 are as tight as possible without making additional assumptions on primitives. Second, our bounds on the welfare impact of a notch change are expressed in terms of estimable reducedform objects, whereas exactly characterizing welfare impacts would require one to estimate a potentially large number of structural parameters.

We now turn to our empirical application: estimating the number of bunching and jumping households for the June 2014 Bolsa Fam ́ılia reform, and, in turn, calculating bounds for the welfare impact of this reform. We begin by describing the structure of the Bolsa Fam ́ılia program, our data, and the June 2014 reform.

# 3 The Bolsa Fam ́ılia Program

The Brazilian anti-poverty program Bolsa Fam ́ılia (BF) provides cash transfers to poor households based on their reported, monthly per-capita income. BF was implemented in October 2003 and is administered by the social development ministry (Minist ́erio do Desenvolvimento Social, or MDS). BF is one of the world’s largest cash transfer programs with around 14 million households receiving benefits in 2014 (Gazola Hellmann, 2015). Applicants report their information, including information on household income, expenditures, assets, socioeconomic characteristics, and demographic characteristics, to interviewers at Cadastro Unico<sup> ́</sup> agencies, which are program offices spread across Brazil’s 5,570 municipalities. Information is entered by interviewers into the Cadastro Unico<sup> ́</sup> registry. Beneficiaries of the BF program are required to update their information once every 2 years to maintain their benefits.

Eligibility for the BF program is based on reported household per-capita income. Household per-capita income is calculated in five steps. First, the applicant is asked to report labor income for the last month as well as average monthly labor income over the past year for each member in the household (the applicant must present a government-issued ID for herself and for each family member thus making it difficult to register fictitious family members). Second, the applicant is asked to report average monthly income received from five additional sources for each household member (see Appendix B.1, which shows the questionnaire used to calculate monthly income). Third, for each individual, the computer calculates the minimum between the average monthly labor income over the past year and last month’s labor income. Fourth, the computer sums this minimum monthly labor income along with income from the five additional sources to get a measure of government spending under policy **p** (exclusive of spending on BF), and ∆ _R_ = _R_ ( **p**<sup>_′_</sup> ) _− R_ ( **p** ) captures the fiscal externalities of the reform. In this case, the lower and upper bounds for the MVPF are given by:

> <sup><u>[</u></sup><sup>_G_</sup><sup><u>(</u></sup><sup>_τ ′_</sup><sup><u>;</u></sup><sup>**<u>p</u>**</sup><sup>_′_</sup><sup><u>)</u></sup><sup>_−G_</sup><sup><u>(</u></sup><sup>_τ ′_</sup><sup><u>;</u></sup><sup>**<u>p</u>**</sup><sup><u>)]</u></sup><sup>_−_∆</sup><sup>_R_</sup> _J_ ∆ _R MV PFL ≡_ 1 _−_<sup>_b′_</sup> _b_<sup>_′_</sup> _G_ ( _τ_<sup>_′_</sup> ; **p**<sup>_′_</sup> ) _− bG_ ( _τ_ ; **p** ) + ∆ _R_<sup>= 1</sup><sup>_−b′_</sup> Total Cost<sup>_−_</sup> Total Cost

> <sup>**<u>p</u>**</sup><sup><u>)</u></sup><sup>_−G_</sup><sup><u>(</u></sup><sup>_τ_</sup><sup><u>;</u></sup><sup>**<u>p</u>**</sup><sup>_′_</sup><sup><u>)] + ∆</u></sup><sup>_R_</sup> _B_ ∆ _R MV PFU ≡_ 1 +<sup>_b_</sup><sup><u>[</u></sup><sup>_G_</sup><sup><u>(</u></sup><sup>_τ_</sup><sup><u>;</u></sup> _b_<sup>_′_</sup> _G_ ( _τ_<sup>_′_</sup> ; **p**<sup>_′_</sup> ) _− bG_ ( _τ_ ; **p** ) _−_ ∆ _R_<sup>= 1 +</sup><sup>_b_</sup> Total Cost<sup>_−_</sup> Total Cost Thus, in addition to _B_ and _J_ , we need to observe ∆ _R_ to assess the welfare impact of the reform.

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