Next, we introduce a simplified approximation of how the government specifies the SNAP-benefits formula. The approximation is meant to isolate the intuition; more details follow in Appendix Section B, where we formalize the SNAP algorithm in its full complexity.

Let _Xit_ = _{Nit, Sit, Iit, Rit}_ represent the set of potential SNAP recipient _i_ ’s household characteristics observed by the government at time _t_ . Characteristics important to the government include _Nit_ , the person’s household size; _Sit_ , the person’s state of residence; _Iit_ , the person’s gross income minus earned income deduction; _Rit_ , the person’s rent.<sup>15</sup> Let _pt_ ( _Xit_ ) = _{bt, ot, ut}_ represent SNAP’s formulaic parameters, which are functions of _Xit_ . _bt_ represents the maximum-benefits formula; _ut_ , the utility-cost deduction formula; and _ot_ , the deduction formulae for various basic needs other than housing. The SNAP benefits per recipient, denoted by _B_<sup> ̃</sup> _it_ , can be approximated by As described before, _bt_ decreases in _Nit_ . _ot_ also generally decreases in _Nit_ . _ust_ is determined by the state-determined SUA in which each state sets a uniform utility-deduction standard intended to reflect the average state utility cost per household; these deductions apply uniformly across the state to households with a positive reported utility cost.

The problem with using state-level SNAP benefits per population directly is that variables such as _Iit_ and _Rit_ are endogenously correlated with trends in unobservable economic fundamentals that vary over time. In order to address this problem, we propose a simulated instrument: the benefits’ process that relies only on (1) household characteristics of the program participants that are fixed at pre-recession (2006) levels, and (2) the way that these fixed characteristics interact with changes in formulaic parameters determined by the state and federal governments.<sup>16</sup> Instrumenting for the state-level benefits per population, the IV _Z_<sup> ̃</sup> _st_ is constructed by aggregating the simulated benefits per recipient, _B_<sup> ̃</sup> _it_ ( _Xi_ 0; _pst_ ( _Xi_ 0)), up to the state-level using sampling weights in the QC data and then dividing by _ns,_ 0, the population of state _s_

> 15The government uses other observed characteristics as well, but in this subsection, we do not focus on these characteristics, which include legal child-support expenses, dependent-care expenses, and medical expenses for households with elderly or disabled members; in 2008, deductions based on these expenses comprised only 2.2% of total expense deductions granted by the program.

> 16As mentioned in Ganong and Liebman (2018), the impact of state-level reforms such as electronic applications on SNAP enrollment mostly occurred before the recession in the early 2000s. During the recession, there was indeed increased state-level adoption of Broad-Based Categorical Eligibility (BBCE), which they calculate to only account for about 8% (1.57/19.1 in Table 2) of SNAP enrollment increases. They discuss that the benefit calculation rule sharply limited the scope of this eligibility expansion. Given these facts and our sample period, we focus on intensive margin changes in this paper.

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