state and period fixed effects, then the total change in the residualized log synthetic benefits per population during the Farm Bill and the ARRA for each state is used to separate the states into quantiles. The first quantile denotes the 24 states with larger changes in residualized log synthetic benefits per population and the second quantile denotes the 24 states with smaller changes. The figure shows the pre-trends are parallel and the price indices, which are normalized to 1 in January 2006, diverge just around the Farm Bill and become roughly parallel again after the ARRA, implying price growth in states with larger changes in synthetic benefits per population is around 2%-3% larger than states with smaller changes. Likewise, we plot these figures for population-weighted averages of housing prices and unemployment rates in Appendix Figure O4 to check how observable economic covariates vary across quantiles. Both housing-price and unemployment-rate trends look nearly identical across the two quantiles, implying the variation generated by the IV is largely uncorrelated with observable economic variables.<sup>65</sup>

# **H Within-Chain Price Rigidity**

To understand why the pass-through elasticity estimates are heterogeneous across store types as shown in Appendix Table N10, we first show the proportion of revenue generated by each of five product departments across store types in Appendix Table N9. Drug stores earn most of their revenue from health and beauty care products, whereas both grocery and merchandise stores earn most of their revenue from food, although grocery stores earn a lot more from food at around 77%. If consumers respond to SNAP-benefit increases mostly by changing demand for products such as food but not other types of products as shown in Section 4.2, we would see smaller price responses in drug and merchandise stores, both of which do not derive the majority of their revenue from selling food.

Next, we follow DVG to measure the extent of price rigidity for each of the retail chains in the data. First, we pick the top UPC from each of 12 product modules with high revenue: canned soup, cat food, chocolate, coffee, cookies, carbonated soft drinks, yogurt, orange juice, bleach, toilet tissue, paper towel, and tooth cleaners. Next, we calculate the weekly correlation in prices for each store pair as a similarity measure, first demeaning the price at the store-quarter-product level before calculating the correlations over all weeks and products that are not missing both store pairs. For each chain, we define the flexibility measure as the percentage difference between the average correlation for store pairs within the same state and the average correlation for store pairs across different states. The closer the flexibility measure is to zero, the more rigid the chain pricing. Because multiple product modules exist, we take either the mean or median flexibility measure across product modules for each chain. DVG perform the same exercise using an alternative similarity measure.

The pass-through elasticity estimated from local variation in SNAP benefits per population should be affected by the extent to which chains are pricing rigidly. Chains that price flexibly across states as well as chains that price rigidly but are located primarily in one state should exhibit local pricing and react to local shocks, whereas chains that price rigidly and locate across many states should exhibit national pricing and will not react to

> 65In fact, the slight divergence in trends implies the states with smaller changes in the IV have slightly better economic outcomes, creating, if any, a slightly negative bias in our results.

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