level.<sup>9</sup> The majority of coefficients in columns 1–3 are negative and statistically significant. There is only one positive and significant slope coefficients, on cooking bananas (column 2), but it is statistically indistinguishable from zero when we look within vendor (column 3).

For comparison purposes we estimate similar regressions using the diary data. Columns 4 and 5 of Table 3 show the slope coefficients. These estimates are not directly comparable with those in columns 1–3, because for some items the quantity support is different across the two data sets. Our interest is primarily in the signs of the estimated coefficients.

In columns 4 of Table 3, all but two estimated slope coefficient are negative, and none are positive and statistically significant. Looking within household (column 5), all coefficients are negative and 16 of 19 are statistically significant.<sup>10</sup> Taken together, the clear pattern in the market survey and diary data sets is that unit price is decreasing in quantity for many items. This finding holds within village-day, within-vendor, and within-household.<sup>11</sup> Having established that bulk discounts exist, we turn to the non-parametric approach of Section 2 to estimate expenditure schedules. The focal-point approach is less susceptible to measurement error than one based on parametric approximations of price schedules, and it reflects the reality of shopping in these markets. We designate a quantity as focal if it accounts for at least 5% of all observations at the item-district level.<sup>12</sup> By this definition there are 1-8 focal quantities per item-district, with 3.4 on average. Overall, 69% of purchases are at focal quantities. We use the median unit price at the focal quantity to estimate the focal price.

To provide a visual example of the estimated schedules, Figure 2 depicts expenditures and unit prices by quantity for 702 purchases of kerosene in one of the study districts. The

> 9For these descriptive regressions there would be no benefit to pooling and using a full set of interactions. The units and scale vary across items, so we allow the levels of the fixed effects to vary as well. 10To verify that the negative relationship between quantity and unit price is not due to division bias (from constructing unit price as the quotient of two variables measured with error), we also estimate expenditure schedules by regressing transaction-level expenditure on quantity and its square, suppressing the constant and fixed effects to enforce regression through the origin. The coefficient on _q_<sup>2</sup> is negative and significant for 13 of 19 items, and never positive and significant. This indicates that expenditure schedules are generally concave, which is consistent with bulk discounts. Results in Appendix S2.

> 11In Appendix S3 we show that the extensive margin probability that households purchase an item is not related to the degree of bulk discounts, as measured by the slope coefficients from column 5 of Table 3.

> 12We drop the roughly 1 in 5 candidate focal quantities that either require greater total expenditure than a larger-quantity focal point, or that have a higher unit price than a smaller-quantity focal point, because these points can never be part of an optimal counterfactual purchase. This is a conservative step. Because this also impacts adjusted expenditure, dropping these points slightly attenuates losses.

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