money-metric utility function associated with the reference utility level defined in equation 1. Let h denote the preimage of _u_ , that is, the set of households zz _h_ that attain uu the same utility level _u_ . Hence, in a multiple poverty line setting, the referencing problem amounts ff<sup>−1</sup> ({uu}) = {h: uu = uu} to identifying correctly the preimage of for every stratum _s_ , as follows: −1 zz uu <mark>(13)</mark> ss zz ss′ Ravallion (2016) emphasizes that the identification of ff ({uu }) = RR ∀ ss is a complicated empirical issue, and −1 increasing the number of strata increases the risk of violating condition (13), with ss<sup>′</sup> }) ≠ RR .<sup>7</sup> This means that multiple poverty lines ) may be inconsistent because they are ss zz SS<sup>′</sup> ff ({uu based on standards of living ss<sup>′</sup> that are different from ss ss<sup>′</sup> .<sup>8</sup> The risk then turns into certainty if the RR zz (SS<sup>′</sup> , JJ<sup>′</sup> , RR maximum level of disaggregation is reached by choosing ss zz . In such a case, the reference uu uu group is reduced, by construction, to one household, and condition (13) may be satisfied only SS<sup>′</sup> = HH if all households perceived the same standard of living ss<sup>′</sup> , which is clearly unlikely. Thus, as a RR rule, . In practice, multiple poverty lines and implicit spatial deflators are computed zz uu at the regional level or even at a higher level of aggregation. SS<sup>′</sup> < SS≤HH If the focus is on the coverage of the spatial price index, implicit deflators S ) have undeniable advantages. The multiple poverty lines ) are, by construction, consistent ss ss<sup>′</sup> �(SSSS<sup>′</sup> , JJ<sup>′</sup> , RR with the consumption aggregate and include all the commodity and services consumed by ss ss<sup>′</sup> zz (SS<sup>′</sup> , JJ<sup>′</sup> , RR households, that is, . Hence, even though the implicit deflator S ) does not specify the prices and quantities of all goods, it can serve as a full-coverage index. The same is ss ss<sup>′</sup> JJ<sup>′</sup> = GG �(SSSS<sup>′</sup> , JJ<sup>′</sup> , RR not true of S , the spatial price indexes that are often calculated by proxying market prices through unit values and relying on household-level budget shares. While surveys often ss SS (SS, JJ, RR) provide accurate information on food items, that task is challenging for nonfood items: the <mark>quantities consumed are rarely reported, and the high heterogeneity in the quality of goods makes it impossible to construct a nonfood price index . Furthermore, for certain components of the consumption aggregate, such as the consumption flow from durable goods, defining a notion of unitary prices is conceptually challenging. This explains why, typically,</mark> , <mark>and, in practice, the coverage of index S ) is limited to food consumption goods, i.e. J</mark> . JJ< JJ<sup>′</sup> = GG ss In summary, compared with the implicit index SS (SS, JJ, RR S ), the index S allows for J= FF a higher level of disaggregation, properly accounts for the specificity of household consumption ss ss<sup>′</sup> ss �(SSSS<sup>′</sup> , JJ<sup>′</sup> , RR SS (SS, JJ, RR)

> 7 The most widely accepted approach to addressing the referencing issue is based on the caloric intake required to meet a minimum daily energy threshold and on per capita or per adult equivalent food expenditure (Ravallion 2016).

> 8 See also Arndt and Simler (2010); Ravallion and Lokshin (2006).

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