metadata: authors: 0: author: Amendola,Nicola
1: author: Belotti,Federico
2: author: Edo,María
3: author: Ranzani,Marco
4: author: Vecchi,Giovanni
count: World
docna: 0: docna: Poverty Lines and Spatial Differences in the
            Cost of Living
docty: Policy Research Working Paper
abstracts: cdata!: This paper proposes a new method for
            estimating a full-coverage spatial price index using data
            typically available in household budget surveys. The food
            component of the index is estimated at the household level
            using reported expenditures and quantities, while the
            nonfood component is derived indirectly as a ratio among
            subnational poverty lines. The paper extends the analytical
            framework described in Deaton and Zaidi (2002) and discusses
            the advantages of this new methodology.
display_title: Poverty Lines and Spatial
            Differences in the Cost of Living
listing_relative_url: /research/2024/11/34424631/poverty-lines-spatial-differences-cost-living
topicv3: Poverty m1326687 695,Spatial Inequality m1327181 1166,Inequality and Shared Prosperity m1327655 700
publishtoextweb_dt: 2024-11-19T00:00:00Z
dois: 10.1596/1813-9450-10978
origu: Off of Sr VP Dev Econ/Chief Econ (DECVP)
no_of_pages: 27
author_keys: 0: author_key: 998611
1: author_key: 1831729
2: author_key: NA
3: author_key: 927986
4: author_key: 458686
model_extractions: Policy Research Working Paper 10978 Poverty Lines and Spatial Differences in the Cost of Living _Nicola Amendola Federico Belotti María Edo Marco Ranzani Giovanni Vecchi_ Poverty and Equity Global Practice November 2024
Policy Research Working Paper 10978

# **Abstract**

This paper proposes a new method for estimating a full-coverage spatial price index using data typically available in household budget surveys. The food component of the index is estimated at the household level using reported expenditures and quantities, while the nonfood component is derived indirectly as a ratio among subnational poverty lines. The paper extends the analytical framework described in Deaton and Zaidi (2002) and discusses the advantages of this new methodology.

This paper is a product of the Poverty and Equity Global Practice. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at mranzani@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Poverty Lines and Spatial Differences in the Cost of Living

Nicola Amendola (Tor Vergata University of Rome) Federico Belotti (Tor Vergata University of Rome) María Edo (Universidad de San Andrés) Marco Ranzani (The World Bank) Giovanni Vecchi (Tor Vergata University of Rome)

**Keywords** : cost-of-living differences, poverty lines, welfare analysis. **JEL codes** : D63, I32, R20.

**Acknowledgements** : Corresponding Author: Marco Ranzani, mranzani@worldbank.org. The authors would like to thank Benoit Decerf and Nobuo Yoshida for their thorough review of the manuscript and for providing valuable and constructive suggestions. The authors thank Dean Jolliffe for his insightful comments. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
# **1 Introduction**

In April 2024, the average price of one gallon of fresh fortified milk in US cities was $3.868.<sup>1</sup> Consumers in the northeast paid, on average, $4.395, and consumers in the west purchased milk at an average price per gallon of about $0.60 less ($3.757). In the same month, the urban consumer index of rent on a primary residence was about 5.4 percent higher in the west than in the northeast.<sup>2</sup> The existence of differences in the prices of consumption goods and services across space is well established. These variations are sometimes considerable between urban and rural areas, across regions, provinces or states, cities, and so on. The simple statistical evidence has, however, important effects on the purchasing power of money. The higher the prices, the lower the purchasing power of households.

Welfare economists, practitioners, statisticians, and analysts interested in living standards are aware that any meaningful comparison of material well-being among individuals living in different geographical areas requires some adjustment to transform a nominal welfare aggregate, be it consumption or income, into a metric that does not conceal spatial differences in the cost of living.<sup>3</sup> First, adjusting for spatial price differentials has important implications for monetary poverty and inequality. Examples of relevant statistics that are affected by spatial differences in prices include poverty headcount ratios, the depth of poverty, the number of the poor, poverty profiles, and indicators of inequality. Second, such statistics on living standards are key inputs to poverty reduction strategies. Adjustments for spatial price differences are therefore relevant to the design of policies aimed at helping people move out of poverty and to the allocation across ministries or across subnational state or provincial governments of resources derived from national budgets, international donors, or multilateral development banks.

Somewhat paradoxically, it is precisely low-income countries, in which the statistical information needed to measure price variability adequately is rarely available, that are characterized by a

> 1 See BLS Data Viewer (graphic), time series: Milk, fresh, whole, fortified, per gal. (3.8 lit), time period: 2024, series ID: APU0000709112.

> 2 The base period of the index is 1982–84, and the index is not seasonally adjusted. For an overview, see Measuring Price Change in the CPI: Rent and Rental Equivalence (web page), US Bureau of Labor Statistics, Washington, DC, https://www.bls.gov/cpi/factsheets/owners-equivalent-rent-and-rent.htm.

3 This is in addition to any required correction for differences in the prices of consumption goods and services over time (inflation) and differences in household size and composition (adjusted through the use of equivalence scales).

2
<!-- Start of picture text --> = |w (spi) +(4-w — )(SPI ) |<br><!-- End of picture text -->
prices, it relies on the highly problematic assumption of the stability of the Engel curve across regions. By contrast, the method here does not impose any restrictions on household behavior or preferences. Instead, it relies on the accuracy and consistency of the referencing used in constructing multiple poverty lines. It is therefore essential that regional poverty lines be utility consistent, that is, based on a common reference standard of living (Ravallion and Lokshin 2006).

The rest of this paper is structured as follows. Section 2 introduces a theoretical framework for spatial deflation based on ratios among poverty lines. Section 3 discusses the challenges involved in estimating multiple poverty lines versus estimating a single national poverty line and a real consumption aggregate. Section 4 introduces a full-coverage household-level spatial price index, together with two slightly different estimation methods. Section 5 concludes.

# **2 Spatial deflation, poverty lines, and poverty measurement: A theoretical framework**

From a welfarist perspective (Ravallion 1998, 2016), the measurement of poverty and inequality relies on the assessment of individual well-being, typically derived from household-level consumption expenditure.<sup>4</sup> To ensure meaningful welfare comparisons across individuals, a form of price adjustment is needed to maintain consistency with the analytical microeconomic framework that underpins the definition of poverty concepts and measures of well-being (Deaton and Muellbauer 1980; Deaton and Zaidi 2002).

This section aims to provide a theoretical framework that demonstrates the equivalence of two strategies for poverty measurement: one based on the spatial deflation of expenditures relative to a national poverty line and another relying on nominal expenditures using multiple spatial poverty lines under the condition of homothetic preferences. The price adjustment mechanism thus acts as a bridge between an empirical strategy that estimates poverty using a single poverty line and one that uses multiple lines.

Household expenditure can be interpreted as a money-metric utility function representing the minimum cost sustained by households to achieve a specific standard of living or, equivalently, a certain level of utility. Let h denote the market price vector faced by household _h_ that achieves utility level h. The household’s well-being is then measured by an expenditure function pp h h), that is, by the minimum cost to achieve the utility level h which, keeping constant the uu ee(pp , uu uu

> 4 <mark>In this section, household and individual are used interchangeably, that is, it is assumed that the preferences of household members may be specified by the preferences of a representative household member.</mark>

4
price vector, is a utility function representing the household’s preferences.<sup>5</sup> Consider now two households, say _i_ and _j_ , enjoying the same utility level _u_ . In general, one may observe that , indicating that households facing different market prices will spend different amounts of money to attain the identical utility level ii jj _u_ . This is where price adjustments come into ee(pp , uu) ≠ee�pp , uu� play. A strategy to compare household well-being through household expenditure consists in choosing a reference price vector 0, whereby, if , then 0 0 �= 0 and vice versa. ii jj ii jj pp uu = uu = uu ee(pp , uu ) = ee�pp , uu ee(ppA similar challenge arises in poverty measurement, where the standard of living of each household , uu) needs to be compared with a common minimum standard of living. Let represent the utility level that defines such a minimum standard of living. A household _h_ is deemed to be poor if and zz uu only if h . To express this identification criterion in monetary terms, expenditure functions are used wherein the minimum expenditure to attain the standard of living zz is referred to as the uu ≤uu poverty line. To maintain the poverty line constant across households, a reference price vector zz uu must be chosen. Household _h_ is poor if and only if the following is true: 0 h 0 ) zz ee(p , uu ) ≤ee(p , uu Using a reference price vector is analytically equivalent to applying a price adjustment factor or a price index to convert nominal expenditure into real terms. <mark>Alternatively, a theoretically equivalent strategy involves adopting multiple household-specific poverty lines. These two strategies are inherently linked, establishing an analytical connection between multiple poverty lines and price deflators. The following sections make this relationship explicit.</mark> <mark>Equation 1 makes clear that identifying poverty status through a single national level poverty line requires the establishment of a reference price system p</mark> 0<sup><mark>. The challenge lies in the fact that</mark></sup> <mark>observed expenditures depend on the actual prices p</mark> h<sup><mark>faced by households rather than the</mark></sup> <mark>reference price system used to define the poverty line z = e(p</mark> 0 <mark>, u simple solution to this issue. Define a “true” cost-of-living index for household</mark> zz<sup><mark>). Conceptually, there is a</mark></sup> _<mark>h</mark>_ <mark>as follows: e(ph, u)</mark> T h <mark>(2) e(p0, u)</mark> TTSS (uu) = <mark>Equation 2 is the ratio between the minimum expenditure faced by household</mark> _<mark>h</mark>_ <mark>in achieving the utility level u and the same minimum expenditure at the reference price vector p</mark> 0<sup><mark>. This allows</mark></sup> equation 1 <mark>to be rewritten</mark> as follows:

> 5 The expenditure function depends on household preferences and should be indexed by _h_ . To simplify the notation, this index is omitted here and reintroduced in section 4.1, where it becomes essential.

5
One problem with equation 3 is that the estimation of a true cost-of-living index represents a complex econometric challenge mainly because of the constraints posed by the available data (Ray 2018). Deaton and Zaidi (2002) suggest approximating T h by employing a price index, such as a Paasche or Laspeyres index, provided that spatial price variability and TTSS (uu) substitution effects remain relatively low. Hence, if h is the nominal expenditure of household _h_ , the empirical counterpart of equation 3 may be expressed as follows: xx h <mark>(4)</mark> S h xx ≤zz where S h denotes a household-level spatial price index. Equation 4 describes the standard SS practice for estimating poverty. SS An alternative strategy consists in using multiple poverty lines and price-unadjusted household expenditures. The idea, in this case, is to choose h as a reference price vector, which implies that poverty lines are now household specific and given by h h. Household _h_ is considered pp poor if and only if the following is true: zz ee(pp , uu ) = zz h h h <mark>(5)</mark> And the equivalent of equation 4 is as follows: ee(p , uu ) ≤zz h h <mark>(6)</mark> Unlike in equation 4, in which poverty status is determined by a single scalar poverty line xx ≤zz applied to the distribution of a household-level price-adjusted expenditure aggregate h⁄S h), zz poverty status in equation 6 is identified by applying household-level poverty lines h (as many (xx SS lines as the number of households) to price-unadjusted household expenditure ( h). zz The link between strategy 1 (real expenditure applied to a single poverty line) and strategy 2 xx (nominal expenditure applied to multiple poverty lines) may be established by rewriting equation 5 as follows: T h ) where T h ) is the true cost-of-living index for household _h_ calculated at utility level . zz zz Equation 7 may be compared with equation 3. If the preferences are homothetic, the true cost-of-TTSS (uu uu living index does not depend on _u_ and T h ) = T h . Equation (7) is therefore zz TTSS (uu TTSS (uu) 6
equivalent to equation 3. Thus, if preferences are homothetic, strategy 1 and strategy 2 lead to identical results (Blackorby and Donaldson, 1988).

<mark>The central finding of this section—the equivalence of two strategies for setting poverty lines— holds within the outlined theoretical framework. However, the empirical application of the two strategies entails several deviations from the theoretical tenets, resulting in the nonequivalence of equations 4 and 6. The next section explores these deviations in detail.</mark>

# **3 From theory to practice: Challenges in estimating one or multiple poverty lines**

Consider first the empirical strategy based on a single national poverty line. In this case, as shown in equation 4, it is necessary to estimate a spatial price index that allows nominal expenditures to be transformed into real expenditures comparable with the single national poverty line. <mark>To enhance precision in defining the deflator, it is beneficial to introduce additional notation. Denote the spatial price index</mark> as S , where the three arguments refer to (a) the aggregation level of the index (whereby ss represents the number of strata), (b) the coverage of SS (SS, JJ, RR) the index ( indicates the number of commodities and services considered), and (c) the choice of ss= 1, . . . . , SS the underlying reference group (where shows the reference group of households used in JJ computing the index). To illustrate, consider a Paasche price index whereby is the total RR number of households in the sample; is the total number of goods included in the SS= HH consumption aggregate; and , that is, the reference group is a subset of all households, for JJ= GG instance, households in the bottom two deciles of the expenditure distribution ( 1,2). It RR⊃HH follows then that RR= DD −1 ∙<sup>EE</sup> <u>h�</u> S h 1,2 GG h jj jj HH<sup><u>�</u></sup> h<sup>pp</sup> jj ~~�~~ SS �HH, GG, DD �= ��ww jj=1 pp where h denotes the ’s good budget share for household h, h is the price of good faced by jj jj household ww h and hjj� is the average price of good _j_ at the national level. Such an index has the pp jj jj lowest possible level of aggregation (the household level), provides full coverage in terms of HH EE �pp consumption items, and is based on a reference group that includes the poorest 20 percent of households. Different countries implement different choices. Recent poverty assessments produced by the World Bank make use of a Paasche index for food items (for example, Indonesia, Kenya, and Mozambique), regional or provincial (and urban or rural) consumer price indexes (CPIs) (for instance, Ethiopia, Gabon, Ghana, and Viet Nam), a mix of Paasche indexes for food 7
items and CPIs for nonfood items (such as Cambodia and the Lao People’s Democratic Republic), and, in some cases, the ratio between regional and national poverty lines (for example, Central African Republic, Chad, Niger, and Uganda). A common choice would be S 2,3�, where denotes the number of Primary Sampling Units, which would be read as follows: a Paasche pp SS �SS, FF, DD index calculated at the PSU-level, covering food items only ( _J=F_ ), and based on consumption SS patterns of households in the bottom second and third deciles of the per capita expenditure distribution.

distribution. Consistently with the notation introduced for the SPI, the national level poverty line becomes to emphasize that it depends on both coverage and reference group _R_ . Accordingly, zz equation (4) can be rewritten as: zz(JJ, RR) JJ h <mark>(8)</mark> S xx ss ≤zz(JJ, RR) In equation 8, households are identified as poor based on a single poverty line. The same equation SS (SS, JJ, RR) indirectly identifies a set of implicit multiple poverty lines. Indeed, it is convenient to distinguish implicit or indirectly estimated variables from directly estimated variables by denoting the former with a tilde: Equations 8 and 9 are interchangeable. One may use the real consumption aggregate compared with a single poverty line or the nominal consumption aggregate compared with a set of indirect poverty lines. The number of indirect poverty lines depends on the aggregation level S, while the consistency of with respect to the definition of the consumption aggregate ss zz̃ (SS, JJ, RR) depends on the coverage of the index. ss zz̃ (SS, JJ, RR) There is also a second empirical strategy based on the _direct_ estimation of a set of multiple poverty lines. Let ) denote the poverty line directly estimated for stratum , where is the number of strata considered; ss ss<sup>′</sup> is the implicit coverage of the line; and is the reference group zz (SS<sup>′</sup> , JJ<sup>′</sup> , RR ss SS<sup>′</sup> utilized in stratum _s_ . A household h belonging to stratum _s_ is poor if an only if the following is ss<sup>′</sup> JJ<sup>′</sup> RR true: h ) <mark>(10)</mark> ss ss<sup>′</sup> A key point is that a set of multiple lines xx ≤zz (SS<sup>′</sup> , JJ<sup>′</sup> , RR ) implicitly defines a set of spatial price deflators, as follows: ss ss<sup>′</sup> zz (SS<sup>′</sup> , JJ<sup>′</sup> , RR ) S ) = =<sup>zzss(SS′, JJ′, RRss′)</sup> <mark>(11)</mark> )] ) ss ss<sup>′</sup> zz (SS<sup>′</sup> , JJ<sup>′</sup> , RR ss ss<sup>′</sup> �(SSSS<sup>′</sup> , JJ<sup>′</sup> , RR ss ss<sup>′</sup> EE[zz (SS<sup>′</sup> , JJ<sup>′</sup> , RR zz̃(SS<sup>′</sup> , JJ<sup>′</sup> , RR<sup>′</sup> 8
C (S (Ss (Ss (Ss (s
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).

10
patterns, and minimizes the risk of inconsistently setting the reference standard of living: in terms of our notation and . By contrast, S has partial coverage and is likely to be inconsistent with respect to the composition of the consumption aggregate (i.e., zz ss SS> SS<sup>′</sup> RR<sup>−1</sup> (RR) = uu SS (SS, JJ, RR) ). JJ< JJSimilar conclusions hold for the other terms in table 1. The number of directly estimated multiple<sup>′</sup> = GG poverty lines ) depends on the aggregation level and is positively correlated with the ability to capture the specificity of household consumption patterns, but also with the risk of ss ss<sup>′</sup> zz (SS<sup>′</sup> , JJ<sup>′</sup> , RR SS<sup>′</sup> inconsistency regarding the standards of living . There is a trade-off between maximizing and minimizing the risk of inconsistency. The implicit poverty lines ss may be computed at uu SS the household level without risk of inconsistency, but there is a price to be paid in terms of ss zz̃ (SS, JJ, RR) coverage. All these issues also affect the single poverty lines, ), that cannot be evaluated separately from the spatial deflators, S and S ), respectively. zz(JJ, RR) and zz̃(SS<sup>′</sup> , JJ<sup>′</sup> , RR<sup>′</sup> ss ss ss<sup>′</sup> The analysis in this section indicates a preference for the empirical strategy centered on a single SS (SS, JJ, RR) �(SSSS<sup>′</sup> , JJ<sup>′</sup> , RR poverty line over the one grounded on multiple lines. Within the single-line strategy, specificity and consistency in the standard of living, , may be independently pursued by increasing _S_ and selecting the appropriate unique reference group, zz , respectively. In the strategy involving uu multiple lines, raising the value of entails a trade-off, potentially jeopardizing consistency with RR respect to the standard of living, . This trade-off is inevitable and cannot be circumvented. SS<sup>′</sup> ss However, there are significant empirical issues that restrict the coverage of a directly estimated uu spatial index and undermine its consistency with the composition of the consumption aggregate. One might therefore argue that, to make an empirical strategy based on a single poverty line appealing, it would still be necessary to establish an acceptable degree of consistency between the composition of household expenditure and the coverage of the price index. This objective can be achieved in two ways: (a) directly, by increasing the coverage level _J_ of the spatial price index S , or (b) indirectly, by adjusting the composition of the consumption aggregate to make it consistent with the given coverage level ss _J_ of the spatial price index S . SS (SS, JJ, RR) ss Consider the second option. The idea is that, in poverty measurement, what matters is not the SS (SS, JJ, RR) composition or comprehensiveness of the consumption aggregate, but the consistency of the aggregate with the poverty line. Lanjouw and Lanjouw (2001) show that poverty rates across countries or over time may be compared even if the underlying consumption aggregates are different, provided that the poverty lines are constructed consistently with the aggregates. However, the conditions necessary for the validity of the outcome are rather restrictive. Food consumption aggregates must still be consistent across countries; there must be a monotonic relationship between the share of household food expenditure and total household expenditure; 11
and this relationship must be stable across groups or over time. Moreover, the invariance of the poverty measures with respect to the composition of the consumption aggregate holds true for the poverty headcount, but does not extend to all the other Foster, Greer, and Thorbecke (1984) poverty measures. Hence, even if option (b) is simple and viable, it does not guarantee empirical robustness in poverty profiles and poverty dynamics.

In summary, the assumption is highly plausible that narrowing the consumption aggregate to only food expenditures would result in a nonnegligible bias in poverty estimates and poverty profiles. For this reason, option (a), which involves extending the coverage of S to enhance the alignment with the consumption aggregate, albeit more demanding in empirical terms, seems to ss SS (SS, JJ, RR) be the preferable choice. The next section is therefore dedicated to possible ways of implementing option (a) by expanding the coverage of the spatial price index, which is typically limited to food items.

# **4 A full-coverage household-level spatial price index**

Two possible ways may be conceived to extend the coverage of spatial price indexes. The first way is direct and involves the estimation of the price variability of additional components of the consumption aggregate beyond food items. The empirical implementation of this approach is often challenging because of data scarcity (Chen et. al. 2020). The Appendix presents two price indexes that include two additional components of the consumption aggregate, housing and durable goods. In both cases, the index may be biased because of the heterogeneity of goods, difficulties in detecting market prices, and the possible absence of prices (and necessary imputation). This approach does not allow a price index to be obtained that is fully consistent with the consumption aggregate.

Alternatively, indirect estimation methods may be employed. Such approaches have been suggested in the literature. Perhaps the most well-known is the method based on Engel curves, originally proposed for temporal price indexes (Hamilton 2001) and later extended to spatial indexes by Almås (2012). The Engel curve method indirectly reconstructs the entire price index based on expenditure ratios and hinges on the highly restrictive assumption of a single homogeneous Engel curve among strata or regions. However, Gibson, Le, and Kim (2017) have strongly criticized this approach, highlighting not only the theoretical limitations, but also the empirical shortcomings.

12
# **4.1 The indirect method as a ratio of poverty lines**

In this subsection, the approach described involves using (a) the implicit spatial deflator S ) defined as the ratio between the poverty line ) for stratum _s_ and the average poverty line ss ss<sup>′</sup> )], and (b) the food Paasche deflator ss ss<sup>′</sup> S h to obtain �(SSSS<sup>′</sup> , JJ<sup>′</sup> , RR zz (SS<sup>′</sup> , JJ<sup>′</sup> , RR a household level spatial price index for all nonfood goods and services included in the ss ss<sup>′</sup> EE[zz (SS<sup>′</sup> , JJ<sup>′</sup> , RR SS (HH, FF, RR) consumption aggregate. This nonfood- _SPI_ serves as a component that can be combined with a household-level food Paasche spatial price index, S h , to produce a full-coverage S that captures both food and nonfood items. The underlying idea is that poverty lines contain SS (HH, FF, RR) SS information on the variability of the cost of non-food items that cannot be directly reconstructed due to the lack of data on prices and quantities. This information is contingent upon the methodology used to estimate the poverty lines. For example, the food share method (Ravallion, 2016), the most commonly used approach for estimating the non-food component of poverty lines, leverages data on the consumption patterns of poor households and in the direct estimated cost of a basic need food basket.

The advantage of this method is that it allows a full-coverage price index to be estimated, potentially at the household level, that is perfectly consistent, by construction, with the consumption aggregate used for poverty and inequality analysis. The d <mark>rawbacks are that the strategy overlooks the price variability in nonfood items within stratum</mark> _<mark>s</mark>_ <mark>and that the price variability pertains to the subset of poor households.</mark><sup>9</sup> A true cost-of-living index T for households that belong in stratum _s_ and attain a utility level may be written as follows: ss TTSS zz f f f f f f uu ) ) T ) =<sup>eess(pss, uuzz)</sup> =<sup>ppss</sup> +<sup>ppss</sup> nnf nnf (p0 ) 0 ss 0 ss )zz 0 ss 0 ss ) zz ∙qq (pp , uu ∙qq (pp , uu ss zz ttfftt ttfftt ttfftt ttfftt TTSS (uu ss zz ss zz ss zz where is the average price vector in stratum ee , uu pp ∙qq _s_ and (pp , uu0 is the average price vector at the national pp ∙qq (pp , uu level, while ss and 0 are the corresponding price vectors for _k = food, nfood, tot_ ; is the pp pp kk average quantity vector in stratum ss<sup>kk</sup> _s_ , while is the quantity vectors for _k = food, nfood, tot._ ss Now, pp pp qq define the average food budget share calculated at the national level for a household around the ss<sup>kk</sup> qq utility level , that is, around the poverty line, as follows: zz uu

> 9 <mark>The latter aspect may not be necessarily deemed a weakness, and, in any case, its significance is negligible if the preferences are homothetic</mark> .

13
”<sup>simpli</sup> dn o forz )), (S ~~:~~ “12 [p | | ew | ~~ed~~
foe ~~D~~ aler) y=[w (spr) +QQ-w )(ser_—)
Hence, recalling equation (11), we can write the implicit spatial price index for stratum _s_ as follows: Assume now that the food budget share of poor households may be approximated by the ratio between the average food poverty line and the average total poverty line: Substituting (19) in equation (16) produces the following: Equation (20) differs from equation (16), first, because equation (19) might not hold, i.e. the budget share is different from the ratio between the food component of the poverty line and � f f the total poverty line and, second, because the food price index zz S , derived implicitly as the ww ratio of the food poverty line for stratum _s_ and their average across strata, is different from the ss SS Paasche food price index defined in equation (15).<sup>12</sup> Subsection 4.3 compares the two methods to determine if a systematic bias exists in using the direct method instead of the indirect method.

# **4.3 A comparison of the two methods**

Assume that equation (19) holds with equality. Subtracting (20) from (15) gives the following: The difference between the two methods depends on the difference between the two possible f f estimates of a food spatial price index whereby S is a Paasche price index, typically ss SS

> 12 The ratio between the average food poverty line and the average total poverty line is a good approximation of the average food budget share of poor households if and only if the relationship between total expenditure and the food budget share is a monotonic increasing function and not a correspondence or a nonmonotonic function (Lanjouw and Lanjouw 2001).

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� f f computed over all households, while S is an implicit price deflator based on the consumption pattern of a reference group of households with total expenditure approximately ss SS around the poverty line. Insofar as the typical consumption bundle of poor households includes a small number of items of more homogeneous quality compared with the consumption bundle of f f the entire population, the estimated food price index S may exhibit more variability than � f f � f f f f � f f S . Hence, when S greater than 1, we would expect SSss S −S > 0, and vice versa. That means that the estimation method described in equation (16) tends to capture ss ss ss ss SS SS iiss SS SS more spatial price variability than the estimation method described in equation (20).

In conclusion, it is crucial to discuss the advantages and limitations of the approach proposed in this paper. The primary advantage is the ability to extend the coverage of the index, thus ensuring full consistency with the composition of the consumption aggregate. Theoretically, this is equivalent to constructing an index with the highest level of disaggregation ( ), encompassing all goods and services included in the welfare indicator ( , thereby SS= HH minimizing the risk of identifying inconsistent standards of living. JJ= GG) <mark>On this last point, however, it is necessary to be more precise. The index</mark> S� f <mark>, which captures</mark> nnf <mark>the variability of prices of nonfood items, is an implicit index that does not rely on the direct</mark> ss SS <mark>observation of prices and quantities. Thus, there is no direct control over the potential heterogeneity of goods. The underlying assumption is that the quality of commodities and services is closely related to the standards of living of households. Thus, households that perceive the same level of utility purchase goods of similar quality. At this juncture, the reliability of the referencing, that is, the consistency of the underlying utility levels used in estimating multiple poverty lines, becomes crucial.</mark> <mark>becomes crucial.</mark> To clarify the point, refer to equation (13) in section 2. Let be the underlying utility level for multiple poverty lines, and let , the average utility for all households ss<sup>′</sup> ss zz ff(RR ) = αα uu belonging to the reference groups . If utility levels, that is, the standards of living, were entirely zz ss zz uu� = EE[αα ] ∙uu consistent, = 1 for every _s_ and ss<sup>′</sup> . For the sake of simplicity, suppose also that RR preferences are homothetic. In this case, the Hicksian demand functions are linear in utility, and ss zz zz αα uu� = uu the following is therefore true: Hence, the variability of the nonfood price index used to complement the food index may not hinge solely on the variability in the cost of living, but also on potential fluctuations in standards of living among strata. These fluctuations might arise from errors or inconsistencies in the 17
referencing underlying multiple poverty lines. In other words, if higher standards of living were presumed in wealthier regions (or strata), [ ] would occur, thus leading the nonfood component of the full-coverage index to overstate the cost of living in those regions. ss ss αα > EE αα For this reason, it is advisable to run the utility-consistency tests that have been proposed in the literature, starting with Gibson and Rozelle (1999), and based on the theory of revealed preferences. Ravallion and Lokshin (2006) use quantity indexes in comparing alternative price and quantity combinations. Another option is to follow the approach proposed by Arndt and Simler (2010), based on information theory. In brief, Arndt and Simler impose on households’ behavior a rationality constraint, represented by the revealed preference conditions on the poverty bundles underlying multiple poverty lines. The rationality constraint guarantees that the consumption bundles of poor households are consistent with at least one preferences’ system, which can be then interpreted normatively as the “representative” preference system of rational poor households. At the same time, they take into account the actual consumption of poor households by minimizing the distance of the utility consistent budget shares from the original budget shares.<sup>13</sup> The advantage of this last option is that utility consistency is satisfied by construction; the drawback is that the utility consistent budget shares could differ significantly from the actual consumption shares even if poor households behave rationally.<sup>14</sup>

# **5 Conclusion**

This paper proposes a methodology to adjust welfare aggregates for differences in prices across geographical areas, treating poverty lines as spatial deflators. Poverty lines may be compared against the nominal expenditure of households to measure poverty status. When used indirectly, poverty lines facilitate the conversion of household nominal expenditure into real expenditure, thus adjusting the welfare aggregate for differences in prices across geographical areas. This real expenditure aggregate can then be compared against a single national level poverty line. The paper 13 Additional examples of applications of the method proposed by Arndt and Simler (2010) are Campenhout et al. (2016), Stifel and Woldehanna (2016), and Stifel et al. (2016).

14 Availability of suitable data is key for the implementation of the Arndt and Simler (2010) procedure. The maximum entropy approach is reasonable and (hopefully) feasible for the food component; in contrast, one would expect systematic violation of the utility consistency constraint once the nonfood component is added, due to a number of reasons, including differences in needs or heterogeneous preferences. See Arndt and Tarp (2017) for additional material.

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leverages this analytical property to develop a method for estimating a household-level fullcoverage spatial deflator.

The method relies on a mixed empirical strategy. The estimation of the food component of the index is based on observed prices and quantities, typically derived from household surveys. By contrast, the nonfood component of the index is derived indirectly from multiple poverty lines, usually defined at the subnational level. For nonfood items, prices and quantities are rarely available, thus the proposed approach relies on expenditure, particularly the minimum required to achieve a fixed standard of living.

The proposed method offers two main advantages. First, it retains detailed information on the prices (unit values) and quantities involved in food expenditure. Second, it integrates this data with nonfood information derived from subnational poverty lines. This results in a deflator that, by construction, is fully consistent in terms of coverage with the consumption aggregate used for poverty and inequality estimates. This deflator may also be estimated at the household level.

A key limitation of the method is that the ability to capture accurately the spatial variability of the prices of nonfood items depends on the quality of the identification process in the estimation of multiple poverty lines (Ravallion 2016). The underlying assumption is that multiple poverty lines are utility consistent (Ravallion and Lokshin 2006). This assumption may be tested or analytically imposed (Arndt and Simler 2010).

Overall, the proposed method represents a significant advancement in measuring living standards by accounting for spatial differences in the prices of nonfood items, which comprise an increasingly large share of household consumption. This enhancement leads to more accurate measures of poverty and inequality.

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