The World Bank Economic Review, 36(2), 2022, 413–432 https://doi.org10.1093/wber/lhab021 Article Consumption Subaggregates Should Not Be Used to Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 Measure Poverty Luc Christiaensen, Ethan Ligon , and Thomas Pave Sohnesen Abstract Frequent measurement of poverty is challenging because measurement often relies on complex and expensive expenditure surveys that try to measure expenditures on a comprehensive consumption aggregate. This paper investigates the use of consumption “subaggregates” instead. The use of consumption subaggregates is theo- retically justified if and only if all Engel curves are linear for any realization of prices. This is very stringent. However, it may be possible to empirically identify certain goods that happen to have linear Engel curves given prevailing prices, and when the effect of price changes is small, such a subaggregate might work in practice. The paper constructs such linear subaggregates using data from Rwanda, Tanzania, and Uganda. The findings show that using subaggregates is ill advised in practice as well as in theory. This also raises questions about the consistency of the poverty tracking efforts currently applied across countries, since obtaining exhaustive consumption measures remains an unmet challenge. JEL classification: D11, D12, I32 Keywords: consumption subaggregates, poverty, Engel curves 1. Introduction Directly and comprehensively recording nondurable consumption expenditures via large-scale (face-to- face) household surveys is generally regarded as the best way to assess the prevalence of poverty in low- income countries. Comprehensive checklists prompt respondents to remember their consumption more completely and accurately. But administering lengthy in-person questionnaires is costly, and long ques- tionnaires are often not compatible with other, cheaper survey formats, such as phone surveys, or simply not feasible, as in fragile and conflict-ridden contexts (Pape and Mistiaen 2018). As a result, comprehen- sive consumption expenditure surveys (or survey modules) are conducted in fewer places, with smaller samples, and with lower frequency than is desirable if our aim is to accurately track changes in poverty over time (Beegle et al. 2016). This study considers one possible approach to reducing the cost of measuring poverty, using what we call consumption “subaggregates.” Rather than collecting data on a comprehensive set of consump- tion expenditures and summing these, perhaps one could collect data on an inexpensively measured sub- set and consider only the sum of these. The theory behind this approach builds on an idea found in Luc Christiaensen is a Lead Agriculture Economist in the Jobs Group of the World Bank; his email address is lchristiaensen@worldbank.org. Ethan Ligon (corresponding author) is Professor of Agricultural & Resource Economics at the University of California, Berkeley; his email address is ligon@berkeley.edu. Thomas Pave Sohnesen is an Independent Consultant, Copenhagen, Denmark; his email address is sohnesen@gmail.com. © The Author(s) 2021. Published by Oxford University Press on behalf of the International Bank for Reconstruction and Development / THE WORLD BANK. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com 414 Christiaensen, Ligon, and Sohnesen Lanjouw and Lanjouw (2001), who imagine constructing poverty measures by observing only food ex- penditures (a particular possible subaggregate), and establish conditions under which the Foster–Greer– Thorbecke (FGT) family of poverty measures will yield weakly smaller measured poverty when one uses only food expenditures rather than all expenditures. So, should one use consumption subaggregates to measure poverty? In our view, the answer should only be yes if the idea passes three necessary tests. First, is the idea reasonable, in the sense that it is broadly Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 consistent with economic and statistical principles? Second, is it stable in the sense that it not only works in delivering estimates of poverty that are similar to what one would obtain from the comprehensive approach in cases where one could compare the two, but also works “out of sample” when the population or time period is at least somewhat different? Third and finally, is the approach actually cheaper than collecting comprehensive data on consumption expenditures? After some consideration of the existing literature, this paper tackles two of these three questions. To see whether the use of subaggregates is reasonable, it draws on demand theory to establish the conditions under which it is possible to correctly measure poverty using subaggregates. This study demonstrates that there are conditions on consumer preferences under which the approach is reasonable, but that these conditions are quite stringent, and seem not to hold in data from the three countries (Rwanda, Tanzania, and Uganda) that this study explores. This study subsequently investigates whether it is stable empirically by trying to construct subaggregates using rules devised at one date that can replicate the poverty measures obtained using comprehensive datasets at a later date. This study finds that they cannot reliably do so. Having answered the first two necessary tests in the negative, we do not feel obliged to answer the third (which in any case already has a small literature devoted to it, discussed below). This study concludes that one should not attempt to measure poverty by constructing subaggregates of consumption. 2. Related Approaches With the related aims of more frequent and less costly measurement of poverty, researchers have con- sidered a variety of alternative approaches. This study neither endorses nor condemns these alternative approaches, except to note that each should be required to pass the same three necessary tests (reasonable, stable, cheaper) described above. One approach involves statistical methods that use existing comprehensive consumption surveys and observed characteristics that are easier to collect (or that have already been collected for other purposes) to predict total consumption. Elbers, Lanjouw, and Lanjouw (2003) establish conditions under which this approach is reasonable, and by avoiding the need for new consumption surveys, the approach may be much cheaper. Whether the method is stable is harder to answer. First, the method is only possible in cases where there are existing comprehensive surveys to begin with. It is stable if the joint probability dis- tribution between measures of total household consumption and other observed characteristics obtained from those existing comprehensive surveys (collected on some particular population at some particular time) is stationary, in the sense that the the same joint probability distribution holds for a different popu- lation at a different time. A number of recent papers are concerned with exploring whether this property of stability is satisfied when one attempts to predict over longer or shorter intervals of time and space, or when there are shocks to the economy (Christiaensen et al. 2012; Mathiassen 2013; Douidich et al. 2016), or indeed nothing more than changes to questionnaire design (Kilic and Sohnesen 2019). Prediction methods of this sort can be made more stable by using not just static observed charac- teristics for prediction, but by also controlling for information on dynamic changes in certain features. Features of interest include those that might affect welfare, and those that, if neglected, might upset the relationship between consumption and observed household characteristics. Examples of such forces in- clude changes in rainfall patterns, especially in agricultural societies, or changes in infrastructure, or local economic activity more broadly. With more high-resolution earth observation data now readily The World Bank Economic Review 415 available at high frequency, tracking the evolution of these forces has become increasingly feasible without the need for additional surveys. A new class of models thus predicts consumption and poverty by apply- ing machine learning techniques to a combination of satellite imagery (especially daytime imagery) and existing household data. Prediction accuracy often exceeds benchmarks from more standard regression- based approaches, especially in estimating poverty levels at very disaggregated levels—so-called poverty mapping (Jean et al. 2016; Head et al. 2017; Sohnesen, Fisker, and Malmgren-Hansen 2021; Watmough Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 et al. 2019). However, stability—whether these models trained on data for a certain year can make decent out-of-sample predictions in another year—remains largely untested.1 Despite a number of encouraging results, concerns about model stability in the poverty prediction approach remain (Dang 2020). Another class of approaches therefore proposes to continue to collect some consumption information, either on a small subset of the respondents and on nonconsumption proxies for all (Ahmed et al. 2014), or to collect a different subset of consumption for different households, including a core set common for all households (Pape and Mistiaen 2018). The results of Ahmed et al. (2014) suggest improved accuracy, at modest additional costs, but ad hoc and untestable assumptions about stability remain; at the same time, Fujii and van der Weide (2020) argue that possible gains from “double sampling” are probably illusory. A distinct set of papers bears on our third question: How much cheaper is it to collect a subaggregate rather than a comprehensive measure of total household consumption? In a consumption measurement survey experiment in Tanzania, Beegle et al. (2012) recorded a reduction in interview time of 15 percent (from 50 to 41 minutes on average) when leaving out nonregularly consumed food items (reducing the number of consumption items on the checklist from 79 to 38). Since there are also large fixed costs associated with travel and locating each household, the magnitude of savings is smaller (perhaps much smaller) than this 15 percent. On the other hand, as the authors indicate, time savings might be larger if regularly consumed food items are excluded, and potential item reduction could be substantially larger in Household Budget Surveys focused on consumption data collection. There, checklists sometimes extend to hundreds of items. Furthermore, even small reductions in interview times in long multipurpose surveys may at the margin have oversized effects on the likelihood that a consumption module is included to begin with. An additional benefit of shorter consumption modules is that they can leverage important survey cost savings by facilitating consumption data collection through phone surveys, which are increasingly used in low-income countries. Sometimes these are used because physical visits are not possible, as during the COVID-19 pandemic (World Bank 2020). But phone surveys are also generally less expensive than in-person surveys, and must be relatively short. Collecting information on consumption subaggregates might be feasible in a phone survey where completing a comprehensive consumption modules might not be. Requiring less interview time, the collection of information on consumption subaggregates also stands to suffer less from respondent fatigue (Roberts et al. 2010), improving data quality. 3. Questions and Approach Next, this study now tries to answer the first two questions raised above: Is it reasonable to try to track poverty,2 ordinarily defined as a function of total expenditures, by constructing a subaggregate? And are rules constructed at one date (or place) for mapping the subaggregate into poverty measures stable, in the sense that the same rules work at a different date (or place)? 1 An interesting exception is Bansal (2020) who demonstrates temporal transferability of a simple machine learning model applied to satellite data to estimate the evolution of a district-level development index (akin to the Human Development Index) using temporal satellite data as input. 2 In this paper, the tracking of monetary poverty is based on the commonly used FGT family of poverty measures described by Foster, Greer, and Thorbecke (1984). 416 Christiaensen, Ligon, and Sohnesen Extending the Lanjouw and Lanjouw (2001) result mentioned above, this study shows that so long as household welfare can be summarized by total consumption expenditures, then one can instead construct the welfare measure using only a subaggregate if and only if the Engel curve (subaggregate expenditures versus total expenditures) for the subaggregate is linear. These are the conditions under which the sub- aggregate expenditure approach is reasonable. Yet the conditions seem very stringent, and it is not clear whether they hold for real-world households. Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 Accordingly, this study describes a search for subaggregate expenditures that are linear in total ex- penditures, and finds some that are close in at least some periods. However, if relative prices within the subaggregate change, then the resulting subaggregate expenditures are only guaranteed to remain linear if all of the constituent goods in the aggregate also have linear Engel curves.3 This is the theory reminding us of an important way in which our approach may not be stable. Thus, when prices are changing, the key to constructing a robust proxy for total expenditures that can be used to track changes in poverty over time (or place) is to identify particular goods that have linear Engel curves that are fairly insensitive to changes in prices.4 Using Living Standards Measurement Surveys (LSMS) data from Rwanda, Tanzania, and Uganda, this study documents a simple way to identify goods with (nearly) linear estimated Engel curves us- ing data in an initial period. Then one can form a linear subaggregate of such goods and ask whether this subaggregate can be used to accurately measure poverty in a subsequent period. Theory pre- dicts that this will not work if there are significant changes in relative prices: our method will not be stable. The evidence bears out this prediction—the constructed linear subaggregates do not perform well. 3.1. Theory of Demand and the Aggregation of Commodities This study exploits some basic consumer theory to establish conditions under which it seems reasonable to hope that some subaggregate could be used to measure poverty. The goal is to identify a particular set of goods over which expenditures can be added to form a subaggregate, so that this subaggregate can subsequently be used as a proxy for total expenditures to measure poverty and changes in poverty over time. Theory delivers two simple results that provide a set of conditions that are both necessary and sufficient for a particular subaggregate to serve as a proxy for total expenditures. To fix ideas, this section first introduces some notation, and recalls some of the standard results from demand theory. The link between consumption aggregation and subaggregates with different FGT poverty measures is subsequently explored. 3.1.1. Consumption Aggregates Suppose that a consumer (or household) values n distinct nondurable commodities indexed by i = 1, …, n; call the bundle of these commodities consumed by the consumer c ∈ X ⊆ Rn . Note our implicit as- sumption that consumption is continuous in all these goods. Assume also that there is a vector of prices p ∈ Rn + , so that the consumer’s total expenditures on the consumption bundle c is x = p c. Because it comprises expenditures on all nondurable commodities, call x the consumer’s total expen- ditures, or aggregate expenditures. However, there are many possible subaggregates. Consider some par- tition or aggregation of the n different commodities A = (X1 , …, Xm ), with Xa ∩ Xb = ∅ for indices a = b and X = ∪m a=1 Xa ; thus, each good i belongs to one and only one subaggregate. 3 Changes in relative prices over time are the focus here, but of course relative prices might also vary across space, in which case a subaggregate that is linear in one place might not be in another. 4 If relative prices can be relied upon not to change, then one can instead rely on Hicks’ composite commodity theorem. The question of whether relative prices vary is considered by Gibson and Kim (2015) (over space within Vietnam) and below (over time for the case of Uganda). The World Bank Economic Review 417 Just as x is the consumer’s total expenditures on all goods in X, given the aggregation A let xa denote the sum of expenditures on all goods in aggregate Xa . Similarly, let ca denote the vector of consumption goods in Xa . 3.1.2. Separable Aggregates Let a consumer’s utility function u : X → R map consumption of these n goods into utility. The consumer’s demand for these different goods is said to be separable in the aggregation A if for any aggregate Xa ∈ A Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 and any two consumption goods ci and cj that are both in Xa , the consumer’s demand for ci and cj can be written as a function just of prices for goods in Xa and of total expenditures xa on the aggregate; demands for goods in aggregate Xa written in this form then will not depend on the prices of commodities in other aggregates, except to the extent that these prices affect xa . A sufficient condition (Gorman 1961) for the aggregation A to be separable is that the consumer’s utility function can be written as a set of m subutility functions {ua }m a=1 and an aggregating utility function u0 : Rm → R such that u(c1 , c2 , . . . , cn ) ≡ u0 u1 (c1 ), u2 (c2 ), . . . , um (cm ) , where ca = (ci )i∈Xa . 3.1.3. Consumption Aggregation and FGT Poverty Measures Lanjouw and Lanjouw (2001) consider the case in which total nondurable expenditures x can be divided into an aggregation (X1 , X2 ) with corresponding expenditures (x1 , x2 ); each of these aggregates is then assumed to depend only on total nondurable expenditures x, so that x ≡ x1 (x) + x2 (x).5 Call the function relating expenditures on a particular aggregate to total expenditures the Engel curve for the aggregate; then the Lanjouws assume that the Engel curve x1 (x) is a continuous and strictly increasing function of x so that, observing some level of expenditures x ˆ 1 , one can invert x1 (x) to infer what the overall level of ˆ = g1 ( x expenditures is, obtaining, say, x 1 ˆ ). This study extends the ideas of Lanjouw and Lanjouw (2001) in two important ways. First, the Lanjouws’ study gives conditions under which headcount poverty statistics for a population will not depend on whether one uses total expenditures or expenditures on a smaller aggregate as a proxy for total expenditures. The present study provides conditions under which all FGT poverty measures will be similarly invariant to the use of an aggregate. Second, by assuming that expenditures on different aggregates depend only on total expenditures, the Lanjouws implicitly assume that relative prices are un- changing (which was sensible given their application, which focused on changes in survey design within a single time period). A simple application of demand theory shows how to relax this assumption as well, allowing one to apply these methods to data on expenditures over time, where one may expect changes in economic conditions to lead to changes in relative prices. We are interested in identifying a particular aggregate that can be used to measure poverty and changes in poverty over time or space. The two extensions described above leave us with a set of conditions that are both necessary and sufficient for such a particular subaggregate to serve as a proxy for total expenditures. We then turn our attention to a search over different possible subaggregates to identify subaggregates that may be valid proxies for total expenditures. 3.1.4. Subaggregates and Headcount Poverty Consider an aggregation A with two or more subaggregates (i.e., with m ≥ 2). Let Pα be an FGT poverty measure, with parameter α ≥ 0; P0 is called “headcount poverty.” Let z1 be a “poverty line” in expenditures 5 In what follows, it is useful to adopt some common notational conventions from functional analysis that blur the lines between variables and functions. For example, x1 will always take a value, but may implicitly or explicitly be regarded as a function of other quantities, as in x1 (x). 418 Christiaensen, Ligon, and Sohnesen Figure 1. Illustration of the Construction of the “Upper Bound” Poverty Line Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 Source: Authors’ construction. Note: Here z is the poverty line corresponding to total expenditures, while z1 is the corresponding poverty line for a subaggregate x1 . on the first subaggregate, x1 (in the Lanjouws’ application x1 is food and z1 is a “food poverty line”). Then the z that satisfies x1 (z) = z1 is what Ravallion (1994) calls the “upper bound” poverty line: interpret it as the total expenditures of a household that choose food expenditures equal to the food poverty line. Let g1 (x1 ) be the inverse of the Engel curve x1 (x), so that z ≡ g1 (x1 (z)); this inverse is guaranteed to exist for aggregates in A provided that x1 is strictly increasing. A main result from Lanjouw and Lanjouw (2001) is that the headcount poverty statistic does not depend on whether one uses expenditures xa on a subaggregate or total expenditures x; the only trick is moving from the subaggregate poverty line z1 to the poverty line z = g1 (z1 ) relevant to total expenditures. Thus, provided one knows the mapping g1 from the subaggregate poverty line to the “full” poverty line, one ought to be able to use any monotonic subaggregate to measure headcount poverty. The idea of the method is illustrated in fig. 1. Here there are two subaggregates with expenditures x1 and x2 ; the way in which these vary with total expenditures x is illustrated by the two lower curves in fig. 1. Then define a subaggregate poverty line z1 = x1 (z) (marked as z1 on the vertical axis), and by construction we have z = g1 (z1 ), which gives the mapping between the poverty line corresponding to total expenditures z and the poverty line corresponding to expenditures on subaggregate 1, z1 . Consider first an extension of the Lanjouws’ invariance result, which allows one to move not only from a subaggregate to total expenditures, but from any monotonic subaggregate to any other. Proposition 1 Let A be an aggregation, and let M be the subset of aggregates Xa ∈ A such that expen- ditures xa are continuous, strictly increasing functions of total expenditures x. Pick any two aggregates Xa , Xb in M. Suppose that a consumer is regarded as poor if and only if expenditures xa on aggregate Xa are less than a poverty line za . Then there exists another poverty line zb such that headcount poverty using expenditures xa with poverty line za will be equal to headcount poverty using expenditures xb with poverty line zb = xb (ga (za )). The World Bank Economic Review 419 Proof. The result follows directly from the existence of the inverse functions ga and gb , and the existence of these is guaranteed by our assumption that Xa and Xb are in M. The poverty measure P0 (xa , za ) is equal to the proportion of consumers with xa ≤ za , and by the definition of the inverse functions the consumer has xa ≤ za if and only if xb ≤ xb (ga (za )). 3.1.5. Subaggregates and General FGT Measures (Pα ) Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 Note that Proposition 1 does not require that expenditures on all aggregates are increasing; if some ex- penditure aggregates are decreasing one can simply set them aside. More generally, moving from one aggregate to another will matter. Lanjouw and Lanjouw (2001) consider the class of FGT poverty measures, and show that (the headcount measure aside) using two different aggregates will not generally give the same poverty statistics, even if one adjusts the poverty line as indicated above. Define the poverty measure over a population of N households by N 1 Pα (x, z ) = pα (x j , z ), N j=1 where x is an N-vector of expenditures for all households and where xj denotes total expenditures of the jth household. The FGT class of measures can then be obtained by defining pα (x, z ) = 1(x < z )(1 − x/z )α , where 1 is an indicator function that takes the value of 1 if x < z, and 0 otherwise. How measured poverty changes using the FGT measure depends on how relative expenditure shares change in the different ag- gregates one considers. Some possibilities are classified in the following proposition. Proposition 2 Let x1 and x2 be the expenditures corresponding to two different nondurable consumption aggregates X1 and X2 in an aggregation A, let x be total expenditures on all goods, and assume that x1 and x2 are both strictly increasing functions of x. Then for any z > 0 there exists a z∗ ∗ 1 ∗ 2 ∗ 1 and a z2 such that P0 (x, z ) = P0 (x , z1 ) = P0 (x , z2 ), and the following statements hold: (1) If the expenditure share of X1 is nonincreasing in total expenditures, then Pα (x1 , z∗ 1 ) ≤ Pα (x, z ). (This is the Lanjouws’ result.) (2) If the expenditure share of X1 is nondecreasing in total expenditures, then Pα (x1 , z∗ 1 ) ≥ Pα (x, z ). (This is the obvious converse.) (3) If the ratio of expenditure shares of X1 to X2 is nonincreasing in total expenditures, then Pα (x1 , z∗ 1) ≤ 2 ∗ Pα (x , z2 ). (This is an extension of the Lanjouws’ second result to any two aggregates.) (4) If the ratio of expenditure shares of X1 to X2 is nondecreasing in total expenditures, then Pα (x1 , z∗ 1) ≥ Pα (x2 , z∗ 2 ). (This is the converse of the extension.) (5) If the ratio of expenditure shares of X1 to X2 does not vary with total expenditures, then Pα (x1 , z∗ 1) = Pα (x2 , z∗ 2 ). (This is the case of affine Engel curves.) Proof. We prove case (3); the remaining cases are all either immediate consequences or follow mutatis mutandis. The case of α = 0 has already been addressed in Proposition 1. So fix α > 0 and consider the difference (1−x1 (x )/z∗ α 2 ∗ 1 (z )) − (1−x (x )/z2 (z )) α for x < z, pα (x1 , z∗ 2 ∗ 1 ) − pα (x , z2 ) = 0 for x ≥ z. 420 Christiaensen, Ligon, and Sohnesen If x < z then the sign of this difference is negative if and only if (1 − x1 (x )/z∗ α 2 ∗ 1 (z )) < (1 − x (x )/z2 (z )) , α ∗ ∗ which in turn is satisfied if and only if x (x )/z1 (z ) > x (x )/z2 (z ), which is again satisfied if and only if 1 2 x1 (x )/x2 (x ) > z∗ ∗ 1 (z )/z2 (z ). At x = z the two sides of this expression are equal by construction; by assump- tion the ratio on the left is nonincreasing in x, so for any x < z one has pα (x1 , z∗ 2 ∗ 1 ) ≤ pα (x , z2 ), and the result follows. Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 This result is, like the Lanjouws’ original result, rather general. The only critical assumption is that expenditures on the aggregates considered need to be strictly increasing in total expenditures. This in turn is promised by demand theory: provided only that consumers are not satiated, this monotonicity property must hold for some aggregate, must hold for any separable aggregate, and more generally will hold for any aggregate of normal goods (that need not be separable). Beyond this, nothing further needs to be known or assumed regarding consumer preferences. The result also appears to be quite useful, and perhaps an important step toward the Lanjouws’ goal of devising methods for constructing possibly small consumption aggregates that would nevertheless al- low one to construct the same poverty statistics one would obtain from measuring total expenditures. In particular, one could either use Proposition 1 to motivate using any small aggregate and construct- ing comparable headcount poverty statistics or, using part (5) of Proposition 2, find a single small linear aggregate to obtain comparable statistics for any of the FGT poverty measures. This last idea is summa- rized in the following corollary. Corollary 1 If expenditures on a separable aggregate X1 are proportional to total expenditures x, then Pα (x1 (x ), z∗ 1 (z )) = Pα (x, z ) for all x, for all z > 0, and for all α ≥ 0. Separability here implies an increasing relationship between x1 and x, so that case (5) of Proposition 2 obtains, taking as x2 total expenditures x, and one needs to find just some aggregate that varies in a linear way with total expenditures. Note that while the property of proportionality here is presumed to always hold, the factor of proportion could be permitted to change over time, so it seems that by simply identifying an aggregate with the necessary linearity we would have a very useful result. However, while the result is general, this hoped-for usefulness may at the same time be limited be- cause the result is also both incomplete and somewhat unstable. First the incompleteness: though modest restrictions on consumer preferences can be invoked to guarantee monotonicity of x1 and x2 , the ratio x1 /x2 need not be monotone, and any nonmonotone behavior of this ratio for x < z gives us a case not covered by Proposition 2. Further, there is simply no theoretical guarantee that a subaggregate that varies in a linear way with x (the requirement of our corollary) exists. Second, stability: demand theory forcefully tells us that expenditure aggregates will be a function not only of total expenditures x, but also of prices. And even if the ratio x1 /x2 is monotone for the prices that prevail at one time and place, it need not be monotone for other prices. This makes using this result to try to identify a linear aggregate a worrisome exercise: using data to identify an aggregate that is linear now is no guarantee that it will be linear later, when relative prices change. This point is revisited below in Proposition 3, at which point we are in a position to specify conditions under which one can guarantee the desired linearity, regardless of relative prices. 3.2. Linearity of Engel Curves for Aggregates If a particular aggregate of goods has the property that a consumer’s demand for the aggregate depends only on the prices of the goods that comprise it, then say that the aggregate is separable. In this section consider the circumstances under which expenditures on any such aggregate will be linear in total ex- penditures. Note that the relevant thought experiment involves manipulating a given consumer’s total expenditures. The World Bank Economic Review 421 Figure 2. Example of Linear Engel Curves (Homothetic Preferences) Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 Source: Authors’ calculations using data from the 2005 Living Standards Measurement Survey for Uganda. Note: Shares are constant. Lines for different goods are ordered from top to bottom as in the legend. Assume that demand is separable in the aggregation A, and consider expenditures on an aggregate Xa . These expenditures will generally depend on (all) prices, and on total expenditures x. We are interested in understanding when xa is linear in x—that is, in the case in which expenditures on a consumption aggregate will be a fixed share of total expenditures. Adding up implies that there are only two cases in which some aggregate can be linear in total expen- ditures. The first is when expenditures for every aggregate Xa ∈ A are linear in total expenditures; the second allows some aggregates to have nonlinear Engel curves, but for nonlinearities in the Engel curves for one aggregate to be offset by nonlinearities in some other, so that the Engel curve for an aggregation of these two is in fact linear. These two cases are considered in turn. 3.2.1. Linear Engel Curves The consumer will have linear Engel curves for the aggregation A if and only if the aggregating utility function u0 is quasi-homothetic (Gorman 1959). This case is illustrated with homothetic demands in fig. 2, using expenditure shares for different aggregates in the UN “Classification of Individual Consumption According to Purpose” (COICOP) aggregation of goods for Uganda in 2005. More generally, Engel curves from quasi-homothetic (rather than homothetic) preferences need not pass through the origin, a case that is illustrated in fig. 3. Here prices of different goods are all equal to 1, and the consumer has Stone-Geary type utility given by 1 U (c1 , c2 , c3 ) = log(c1 − 1) + log(c2 + 1) + 2 log(c3 ). 2 A consequence of having linear Engel curves that do not pass through the origin is that budget balance requires that for any good for which consumption is positive when total expenditures are 0, others must be negative. This is illustrated in fig. 3, where a subsistence demand of 1 for good 1 is financed by consuming a quantity of good 2 equal to −1 when total expenditures are equal to 0. Requiring a consumer to have positive levels of consumption of, for example, food seems perfectly sensible, but allowing a consumer with no resources to finance their food consumption through nega- tive consumption of some other good may be problematical. In such cases the consumer’s Engel curves will simply not be defined for levels of total expenditures below that necessary to finance subsistence 422 Christiaensen, Ligon, and Sohnesen Figure 3. Example of Linear Engel Curves (Quasi-Homothetic Preferences) Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 Source: Authors’ construction. Note: Shares are not constant. Note that quantities can be negative. Figure 4. Example of Linear Engel Curves (Quasi-Homothetic Preferences) with Corner Solutions Source: Authors’ construction. Note: Shares are not constant. Note that quantities cannot be negative. consumption. This circumstance is illustrated in fig. 4. Here a minimum expenditure of 1 is required for good 1, so that Engel curves are not defined for levels of expenditures below this. Related, consumption of good 2 is 0 at the lowest levels of income because of the nonnegativity constraint on consumption. Thus, when the corner constraint is no longer binding this induces a “kink” in the Engel curves for other goods. These kinks in the Engel curves created by corner solutions for demand for some goods violate the claim made above that a quasi-homothetic aggregating utility u0 is sufficient for linear Engel curves. To extend the claim to cover the case in which consumption is required to be nonnegative, restrict the domain of total expenditures so that over this domain the consumer never finds themselves at a corner. The requirement that the consumer must not find themselves at a corner is less restrictive than one might suppose, because the separability of the aggregation A implies that demand for every Xa ∈ A is The World Bank Economic Review 423 Figure 5. Example of Nonlinear Engel Curves, with Approximately Linear Aggregates Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 Source: Authors’ construction. Note: Shares are not constant. Note that quantities cannot be negative. normal. Accordingly, there exists a lower bound on total expenditures x such that Engel curves for all Xa ∈ A are linear for all x ≥ x. 3.2.2. Nonlinear Engel Curves The second case allows for some Engel curves to be nonlinear, but for the nonlinearity of an aggregate Xa to be offset by nonlinearities in the Engel curves of the remaining aggregates, as illustrated in fig. 5. However, in cases where two or more nonlinear curves sum (or average) to a straight line it is a bit of a balancing act. The weights of the curves in the sum depend on prices, and if relative prices change at all then the curves will no longer deliver a linear sum. This point is made formally in the following proposition. Proposition 3 If the sum of the Engel curves of some set of goods is a linear function of total expenditures x for all prices p, then either (a) the Engel curves of each good in the sum are linear, or (b) all goods are included in the sum. Proof. If the sum is over a single good, then case (a) obtains trivially. Consider the case of sum- ming the Engel curves of two goods 1 and 2. Assume that the sum of Engel curves is linear, i.e., that p1 c1 + p2 c2 = a(p) + b(p)x for all p for some functions a and b that may depend on prices. Taking deriva- tives with respect to x delivers b(p) = p1 ∂ c1 /∂ x + p2 ∂ c2 /∂ x. Taking derivatives with respect to prices delivers c1 c11 c21 p1 a1 ( p) b1 ( p) 1 + = , c2 c12 c22 p2 a2 ( p) b2 ( p) x where ai (p) and bi (p) indicate the partial derivatives of these functions with respect to the price of the ith good, and where cij indicates the partial derivative of demand for good i with respect to price pj . 424 Christiaensen, Ligon, and Sohnesen Now, using the Slutsky decomposition to substitute for these last partial derivatives, one obtains c1 a1 ( p) b1 ( p) 1 p1 c1 p1 = − + ∂ c1 /∂ x ∂ c2 /∂ x , c2 a2 ( p) b2 ( p) x p2 c2 p2 where is the Slutsky substitution matrix for compensated demands, which is independent of x by con- struction. We’ve already demonstrated that the last inner product in this expression is equal to b(p). Using Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 this fact and rearranging, c1 a1 ( p) b1 ( p) 1 p1 (1 − b( p)) = − . c2 a2 ( p) b2 ( p) x p2 There are then two cases to consider. First, if b(p) = 1 then both c1 and c2 are linear functions of x, by inspection, corresponding to part (a) of the proposition. Second, if b(p) = 1 then the budget constraint implies that there are only the two goods c1 and c2 , corresponding to the second case. This proposition establishes that a linear expenditure aggregate must be exclusively comprised of linear goods. The following corollary establishes that if any good is linear in total expenditures, then all goods must also be. Corollary 2 If the Engel curve of a good is a linear function of total expenditures x for all prices p, then the Engel curves of all goods are also linear functions of total expenditures. Proof. Suppose that good 1 has linear Engel curves, with p1 c1 (p, x) = a1 (p) + b1 (p)x. Then from the budget constraint it follows that an aggregate of the remaining goods also has a linear Engel curve, since n i=2 pi ci = x − p1 c1 = −a ( p) + (1 − b ( p))x. But then by Proposition 3 all goods i = 2, …, n also have 1 1 linear Engel curves. 3.3. Subaggregates and Poverty in Practice Corollary 1 tells us that we can use any separable subaggregate provided expenditures on the subag- gregate vary in proportion to total expenditures. But Proposition 3 and its corollary tell us that such a subaggregate will exist only under very special (and highly implausible) conditions. In particular, even if one can identify goods with linear Engel curves given prevailing prices, under different relative prices this linearity may fail. One possible justification for using a subaggregate of goods with linear Engel curves is that perhaps relative prices do not change much. Engel curves are functions of total expenditures given prices, but if relative prices are more or less fixed over time then the issues raised in the previous section will remain theoretical. So how much should one expect relative prices to change? Consider changes in unit values6 just for different sorts of food for Uganda, using the sequence of LSMS’s conducted in that country in 2005, 2009, 2010, 2011, 2013, and 2015. For each good calculate the median unit value among all the households who purchased a positive quantity of the good in a given year. Next compute the change in the logarithm of these median prices (approximating the growth rate of each price), and decompose it into common time components and a residual that reflects changes in relative prices, i.e., log pit = μt + rit , where μt is chosen so that the expectation Erit = 0 for all t. Then μt can be interpreted as (the log of) Jevons’ index for food prices, while the rit can be interpreted as the real changes in log relative prices. The question of how important relative price changes are relative to changes in the level of food prices can 6 Another real concern left unexplored has to do with the possibility of unmeasured variation in quality, in which case one would need relative prices for different qualities to also remain fixed. But these issues have already been explored in the literature (Deaton 1988; McKelvey 2011; Gibson and Kim 2019). The World Bank Economic Review 425 then be answered by decomposing the variance; since μt is orthogonal to rit by construction, it follows that Var( log pit ) = Var(μt ) + Var(rit ). In the event, the total variance in changes in log prices is 0.25, while the variance of the common compo- nent is 0.05, implying that variation in relative prices is responsible for 80 percent of all variation. Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 Thus, one cannot reasonably justify use of a linear subaggregate by arguing that there is little variation in relative prices.7 However, from a practical point of view perhaps it may be possible to identify goods with linear Engel curves in one period, and then get lucky: in a subsequent period perhaps relative prices will not have changed in such a way that linearity will be compromised, and our subaggregate will work for measuring poverty. 3.3.1. Data The remainder of this paper tests this luck using data from expenditure surveys in Rwanda, Uganda, and Tanzania.8 All surveys are large multipurpose household consumption surveys, representative at both na- tional and urban/rural levels with large sample sizes. In Uganda about 3,000 households were interviewed each year, in Tanzania between 3,000 and 4,000 each year, and in Rwanda the number of households in- creased substantially from 6,900 to 14,300 household observations per year. The goods included in the consumption aggregates follow the guidelines for consumption aggregates found in Deaton and Zaidi (2002).9 In all three countries, FGT measures rely on consumption data col- lected by recall questions with probing for each consumption good. The number of consumption goods probed for in the questionnaire varies across countries, with 112 in Tanzania, 126 in Uganda, and 284 in Rwanda.10 Beegle et al. (2012) review recall-based consumption modules in multipurpose LSMS’s com- monly used for national poverty analysis and report ranges from 37 to 305 goods, with a mean of 137 goods in total. Thus, the sample of countries seems to represent typical recall consumption surveys for poverty analysis, even if the total number of goods does not actually encompass all nondurable con- sumption. In each country, consumption goods are valued in year 1’s prices corrected by the same price adjustment as used to evaluate the trend in poverty in the surveys. In each year, values are spatially and temporally deflated. The three countries are chosen to display variation in level and trend in poverty headcount. Headcount poverty stood at 15 percent in Tanzania in year 1, compared to 57 percent in Rwanda, and poverty fell substantially in Uganda and Rwanda, while it increased in Tanzania (Development Research Group of the World Bank 2017). The five-year time span between surveys in Rwanda can also be seen as an upper bound for the approach, as best practice calls for countries to implement full consumption surveys every five years or less. 7 The different seasonality patterns in prices across food groups, both in terms of timing and intensity, documented by Bai, Naumova, and Masters (2020) in three countries (Ethiopia, Malawi, and Tanzania), further suggest that relative prices are likely to vary even over shorter time periods (i.e., within the year). 8 Rwanda: Enquete Intégrale sur les Conditions de Vie des ménages de Rwanda (EICV1) 2001 and (EICV2) 2006. Uganda: Uganda National Household Survey (UNHS) 2005/06 and 2009/10. Tanzania: Tanzania National Panel Survey (NPS) 2008/09 and 2010/11. All of our code and relevant extracts of these datasets are available on-line from the Center for Open Science at https://dx.doi.org/10.17605/OSF.IO/UYGS2. 9 The included goods in the aggregates are very similar to those defined by the national statistical agencies, but are not necessarily identical. 10 In Tanzania and Uganda, the questionnaire is designed so that each row captures consumption of own production, gifts and purchased, summed into total consumption of items. In Rwanda, own consumption, and gifts and purchased items are split into two separate questionnaires. Hence, in the data it appears as if Rwanda has twice as many items (2 × 284). The analysis here combines the values of each item from both questionnaires into one value, following the same principle as in Tanzania and Uganda. 426 Christiaensen, Ligon, and Sohnesen 3.3.2. Looking for Linearity Moving from theory to practice, one might wish to gauge the consumption patterns of households, as some goods might be more suitable than others for inclusion in a subaggregate. Figure 2 illustrates how a sum of linear goods will always lead to a linear aggregate, but fig. 5 also shows that a sum of expenditures on nonlinear goods can be linear for a particular set of prices. To gauge properties of goods in relation to the total aggregate, consider an OLS regression of the form Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 log(xi ) = γi + βi log(x ) + i , (1) where (xi , x, i ) are all understood to vary across households. Note that this equation can be rewritten to give an expression for expenditure shares: xi = eγi + i xβi −1 . x In this regression one would expect that the linear goods we are looking for will have β i = 1, and a constant share of consumption expenditures.11 Any goods with expenditure share decreasing in total expenditures will have β i < 1. Goods with β i > 1 will have an increasing share of total consumption expenditures, while any goods with β i < 0 will actually decrease in levels with total expenditures. Only for goods with income elasticity of β i = 1 will the consumption share remain constant. Note that these regressions describe a temporary relationship, since theory tells us to expect the estimated coefficients to be stable across periods only if prices do not change (much) or if all Engel curves are linear. The empirical question is whether in real-world data actual price changes and preferences are such that equation (1) provides a reasonable approximation or not. If so, then one may use goods with estimated β i close to 1 to form a subaggregate. If not, then the construction of subaggregates for the purpose of tracking poverty fails our test of stability. Thus, estimate the regression equation (1) for the many different consumption goods across three countries described above. Then use the estimated values of β i to classify goods: β i < 1: shares decreasing; β i > 1: shares increasing; β i = 1: shares constant. The idea is then to use goods for which expenditure shares are constant to construct a subaggregate. If one can construct such a linear subaggregate and if it remains linear when relative prices change then one could confidently use the subaggregate and result of Proposition 2(5) to construct any of the FGT poverty measures. To implement this classification, one must first choose two different rounds for each of the three coun- tries considered. Exclude any observations of zero expenditures, then exclude any consumption goods that have fewer than 10 observations. Some estimated point values of β i should be regarded as insignif- icant; take these to be those having a t-statistic (associated with the test that the point estimate is 0) less than 2.326 (the critical value associated with having the probability of a type I error of 1 percent). Stan- dard errors are computed clustering at the level of the enumeration area. Classify the remaining point estimates according to two simple criteria: First, are they greater than or less than 1? Second, are they “approximately linear,” in the sense that the point estimates lie within the interval (0.9,1.1)? Table 1 reports this classification of goods, initially using only data from the first round of data for each country, to estimate and classify the β i . For Tanzania, 98 of all 112 goods can be classified (i.e., have enough observations and are significant); of these 72 percent (71/98) are less than 1, so that their 11 This argument rests implicitly on the assumption that preferences and prices are common across households within a country, or that any heterogeneities take rather special forms (Brown and Walker 1989). If different preference structures corresponded to different observable characteristics one could take account of this in the estimation. The World Bank Economic Review 427 Table 1. Consumption Goods and Linearity: Counts of Goods for Each Country by Classification Classification Tanzania Rwanda Uganda Fewer than 10 observations 2 (0)a 13 (0) 0 (0) Insignificant relationship 12 (6) 91 (60) 24 (17) Inferior goods (β < 0) 2 (0) 0 (0) 0 (0) Decreasing shares (0 < β < 0.9) 69 (59) 160 (126) 74 (65) Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 Increasing (β > 1.1) 15 (9) 4 (1) 8 (4) Approximately linear (0.9 < β < 1.1) 12 (11) 16 (9) 20 (16) Total number of goods 112 284 126 Source: Authors’ analysis using Living Standards Measurement Survey data for indicated countries. Note: a Numbers in parentheses are counts of goods in the second year conditional on having been in the same category the first year. expenditure shares fall on average with increasing total expenditures; 15 percent (15/98) have β i greater than 1, and so increasing shares, while 12 percent (12/98) indicate an approximately linear good (i.e., β i is within 0.1 of unity). Similarly, for Rwanda, 180 of 284 goods can be classified, of which 88 percent (160/180) have decreasing shares, while 9 percent (16/180) are approximately linear. And for Uganda, 92 of 126 goods can be classified, of which 80 percent (74/92) have decreasing shares, while 22 percent (20/92) are approximately linear. This exercise allows for the construction of linear subaggregates for each country, consisting respec- tively of 12, 16, and 20 goods, and an appeal to Proposition 2(5) provides assurance that these subaggre- gates can be used to compute the same FGT poverty measures one could have computed using all observed expenditures. Further, if one assumes that the estimated parameters (γ i , β i ) obtained at a different time (for the same population) remained constant, then one could use the quite small amount of data required on expenditures on the subaggregate to track FGT poverty measures. However, as discussed in previous sections, there is no good theoretical justification to suppose that these parameters will be stable over time; and indeed, if relative prices change we should expect the estimated β i to also change. So what do the data say about this? To explore this, reestimate equation (1) using data from a second round for each country. Then table 1 reports (in parentheses) the number of goods where estimates of β i are unchanged. Of chief interest is the number of goods that remain approximately linear. In Tanzania this is 11 of 12, in Rwanda 9 of 16, and in Uganda 16 of 20. Overall only three-quarters of the goods initially classified as “approximately linear” received the same classification in a second round of data, and while the number of reclassifications was modest in Tanzania and Uganda, it was quite large in Rwanda. It is impossible to comfortably conclude that the linear classification is stable over time. Why are these parameters not stable? Corollary 2 tells us that any given good can have a linear Engel curve (for all prices) if and only if all goods have linear Engel curves. In this situation (and only in this sit- uation) will changes in relative prices not affect relative expenditures. The classification exercise reported in table 1 provides evidence that some goods are not linear given prevailing prices; it follows that no good is linear in expenditures for all prices. Or, put differently, the parameters β i must be functions of prices. 3.3.3. Performance of Subaggregates in Measuring Poverty To further explore stability, perhaps deviations from linearity are quantitatively unimportant for con- structing FGT poverty measures? Guided by the theory above, consider identifying expenditure items that are linear in a base period and using these to construct a subaggregate for measuring poverty in a subsequent period. In our exercise there is also data on other observed expenditures, so one can evaluate the performance of the subaggregate in measuring poverty. To make this work, one needs to identify the ga functions of Proposition 1. Here this amounts to a simple rescaling. For example, if a poverty headcount 428 Christiaensen, Ligon, and Sohnesen Figure 6. Performance of Subaggregates for Measuring Poverty Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 Source: Authors’ analysis using Living Standards Measurement Survey data for indicated countries. Note: Poverty measures are “headcount” (P0 ). Each subaggregate is constructed by progressively adding an additional good, starting with the most linear. Dashed lines indicate 95 percent confidence intervals about P0 using the full consumption aggregate. Filled dots correspond to subaggregates for which one cannot reject the hypothesis that poverty measured using the full consumption aggregate is equal to that measured using the subaggregate. of 20 percent was observed for the full aggregate in year 1, the poverty line for the reduced aggregate is set to the value at the 20th percentile of the reduced aggregate in the same year. Then valuing year 2 quantities using year 1’s prices, one can use exactly this same poverty line in the second year. Construct a subaggregate by beginning with the “best” good, i.e., the good i with β i closest to 1 in absolute value. Then the second-best good, and so on, until all goods have been added to the aggregate. Then for each such subaggregate evaluate its ability to replicate measured poverty levels against poverty levels measured using the complete set of expenditures. This procedure does not provide good approximations of poverty headcounts in the second year (fig. 6). In all three countries the estimates are systematically off in one direction, until almost all goods are added to the aggregate. The approximated poverty headcounts are clearly outside the 95 percent confidence interval of the poverty estimates for the full consumption aggregate, illustrated by the dotted lines. Filled dots correspond to subaggregates for which one cannot reject the hypothesis that poverty measured using the full consumption aggregate is equal to that measured using the subaggregate, taking into account the sampling variation of both measures. One reason for this pattern is the exclusion of goods with large expenditure shares, usually staple foods, that have values of β i far from 1. Examples are cassava and sweet potatoes in Uganda and maize in Tanzania. The latter changes from 12 percent of total consumption to 9 percent. These are examples of goods that have large changes in their relative share of consumption over time—these changes could be due to changes in relative prices, or changes in the distribution of wealth with nonlinear Engel curves. Either way, these large changes in expenditure shares illustrate the fragility of the approach. 4. Conclusion A standard approach to measuring consumer or household welfare involves first constructing a measure of total aggregate consumption expenditures. These totals are usually elicited by asking about many detailed individual expenditures, and then summing over these to obtain the desired consumption aggregate. The data collection involved is both expensive and time consuming. The World Bank Economic Review 429 This paper explores the idea of choosing, ex ante, a subset of all consumption goods, and adding up to form a “subaggregate” consisting only of expenditures on this subset. If such a subaggregate could be used to reliably measure welfare instead of total expenditures this could lead to less expensive data collection (perhaps particularly if used in combination with phone surveys). A result due to Lanjouw and Lanjouw (2001) is encouraging in this regard, but requires the Engel curve for the subaggregate to be stable over time or across populations. This study extends their result and shows that the critical feature Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 is that the Engel curve for the subaggregate be linear regardless of prevailing prices. Basic consumer theory allows one to establish necessary and sufficient conditions for the linearity of the subaggregate Engel curve. Unfortunately these conditions are quite stringent if relative prices vary over time—effectively all Engel curves must be linear. It remains possible that Engel curves in fact possess the desired linearity, or that changes in Engel curves due to changes in relative prices are negligible in real-world data. Accordingly, this study uses data from Rwanda, Uganda, and Tanzania, and searches across different goods to identify those that feature (nearly) linear Engel curves in a base period. It then uses these goods to construct a subaggregate in a subsequent period. Consistent with theory, this method turns out to perform quite poorly. One should not construct subaggregates for the purpose of measuring poverty by relying on the fact that Engel curves may be linear at some time and place, because they are unlikely to be linear in a different time or place since relative prices are likely to be different; the method is not stable. Perhaps there are better ways to construct a subaggregate? Our conditions are both necessary and suf- ficient, so theory tells us that the answer is no. Another idea often used in practice involves constructing a subaggregate by choosing goods with large expenditure shares. If one can do this in such a way that a very large percentage of all households expenditures were in the subaggregate this would hold promise. But if Engel curves are not linear then there will be variation in expenditure shares across households, and identifying goods that have large shares on average will tend to select goods that have a low income elas- ticity (such as staple foods). This is more or less the opposite of what one would wish, since expenditures on these goods will convey little information about underlying household resources. There are other practical reasons for not pursuing the course of trying to construct a linear subaggre- gate. First, the possible savings from having a shorter survey are limited, at least within the framework of a conventional comprehensive in-person household survey, since the total cost of the survey will generally not be proportional to the number of expenditure items. Further, travel between interviews, even in a clustered sample design, means that there is a fixed travel cost associated with obtaining each interview, and these travel costs may dominate in surveys using face-to-face interviews (see Cochran 1963 and Fujii and van der Weide 2020 for an analysis of sample design for the case of a clustered sample). Second, suppose that one observed that relative prices seemed not to change much in some country over several rounds of surveys, so that one was encouraged to construct a linear subaggregate. If a major macroeconomic shock were to occur (say a famine, a currency crisis, or an epidemic), then that is exactly when policy makers would be most interested in quickly having new measures of welfare, but it would be exceedingly likely that the large shock would change relative prices and upset the previously observed stability of the method. A final over-arching worry is that given current practice any measured “total” consumption aggregate is likely to actually be a subaggregate, simply because some goods and services are extremely difficult to measure. For instance, the number of consumption items (or categories) for which data are collected from households in LSMS’s ranges from 37 to 305, with the mean being 137 and the median 130 (Beegle et al. 2012). Services from consumer durables ought to be included, but are difficult to price, and often are not. The same holds for foods eaten away from home, which are increasingly important in urban settings. And arguably the value of public goods and time and leisure should also be included in a com- prehensive or “full” consumption expenditure aggregate for measurement of poverty, but in most cases this consumption is excluded. The consumption of publicly provided health and education, as well as the 430 Christiaensen, Ligon, and Sohnesen use value of durable goods, is currently included in some countries, but excluded in others. These mat- ters have been largely ignored in poverty tracking, but our results suggest that they may be of first-order importance. This paper has been principally concerned with whether subaggregates are useful in constructing poverty measures such as the FGT family, but it is worth noting that the same issues will affect any other sort of welfare measure that depends on measuring some sort of “total consumption.” Attempts to Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 get at household welfare by measuring consumption expenditures are very common in many randomized- control trials (RCTs) in low-income countries, for example, but approaches to collecting these are highly variable, and often focus on food. Measuring subaggregates in these circumstances would not compro- mise the ability to estimate, for example, average treatment effects on measured consumption, but would complicate the interpretation of the magnitudes of these effects (and of course would be fatal to efforts to measure treatment effect on poverty status). The main message of this study is negative: consumption subaggregates should not be used to track poverty. But we wish to end on a constructive note, and suggest ways in which, as Ravallion (1996) puts it, one might let “method substitute for data.” The methods discussed here suggest all involve the use of demand theory to interpret variation in expenditures as being due to variation in prices, households’ budgets, or households’ characteristics. In principle, these methods could allow one to measure changes in poverty even if one collects data on a subset of expenditures. Note, however, that these approaches involve measuring demand for different goods within a subset of expenditures, not constructing a subaggregate from this subset. The principal difficulty in using subaggregates pertains to the fact that expenditures on the subag- gregate in general depend on the prices of all goods—theory tells us that because relative prices might change that a measure will be inherently unstable. We suggest three ideas or methods for dealing with this dependence in a way that could reduce the demands of data collection. The first idea is essentially the idea of Lanjouw and Lanjouw (2001) described earlier: if relative prices are fixed and people have common preferences, then one could exploit this, say by collecting comprehen- sive expenditures from one sample, and collecting only data for a subset for a second sample from the same population at the same time. Then one could estimate Engel curves for the goods in the subset as a function of total expenditures using the first sample, and invert these curves to estimate total expenditures for the second sample. Second, one could extend the idea of inverting Engel curves to cover the case of variable prices by estimating a complete demand system using data on all expenditures and prices for some sample, so that total real expenditures could be obtained from the inverse Engel curve. There are a number of applications that take this approach to produce correct cost of living indices (Almås 2012; Almås, Beatty, and Crossley 2018; Dabalen, Gaddis, and Nguyen 2020), or to construct welfare measures (Atkin et al. 2020; Tagliati 2021). The same general approach could be easily adapted to measuring rates of poverty. Third, one could estimate an incomplete demand system (e.g., LaFrance, Beatty, and Pope 2006) to avoid the initial difficulty of collecting data on all goods. For example, Ligon (2019) develops an approach that uses LSMS data from Uganda to estimate an incomplete Frisch demand system, and shows how this can be exploited to track changes in headcount poverty over time. All three of these ideas can be thought of as substituting modeling assumptions of various sorts for actual data on complete expenditures, and of course their success depends on how well the assumed models actually describe demand. On the other hand, this paper has pointed out that using subaggregates to measure poverty is only stable if prices are unchanging or if all Engel curves are linear, and both of these assumptions are considerably stronger than what is typically assumed in applied demand estimation. It is our hope that the present paper will encourage people interested in the measurement of poverty to experiment with the estimation of demand systems. The World Bank Economic Review 431 References Ahmed, F., C. Dorji, S. Takamatsu, and N. Yoshida. 2014. “Hybrid Survey to Improve the Reliability of Poverty Statistics in a Cost-Effective Manner.” Policy Research Working Paper 6909, World Bank. Almås, I. 2012. “International Income Inequality: Measuring PPP Bias by Estimating Engel Curves for Food.” Amer- ican Economic Review 102 (2): 1093–117. Almås, I., T. K. Beatty, and T. Crossley. 2018. “Lost in Translation: What Do Engel Curves Tell Us about the Cost of Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 Living?” CESifo Working Paper Series 6886, CESifo Group Munich. Atkin, D., B. Faber, T. Fally, and M. Gonzalez-Navarro. 2020. “Measuring Welfare and Inequality with Incomplete Price Information.” Manuscript. Bai, Y., E. N. Naumova, and W. A. Masters. 2020. “Seasonality of Diet Costs Reveals Food System Performance in East Africa.” Science Advances 6 (49): eabc2162. Bansal, C. 2020. “Describing Patterns of Socio-Economic Development at Fine Spatial and Temporal Resolutions.” In Proceedings of the 7th ACM IKDD CoDS and 25th COMAD, 338–39. Beegle, K., L. Christiaensen, A. Dabalen, and I. Gaddis. 2016. Poverty in a Rising Africa. World Bank. Beegle, K., J. De Weerdt, J. Friedman, and J. Gibson. 2012. “Methods of Household Consumption Measurement through Surveys: Experimental Results from Tanzania.” Journal of Development Economics 98 (1): 3–18. Brown, B. W., and M. B. Walker. 1989. “The Random Utility Hypothesis and Inference in Demand Systems.” Econo- metrica 57 (4): 815–29. Christiaensen, L., P. Lanjouw, J. Luoto, and D. Stifel. 2012. “Small Area Estimation-Based Prediction Methods to Track Poverty: Validation and Applications.” Journal of Economic Inequality 10 (2): 267–97. Cochran, W. G. 1963. Sampling Techniques (2nd ed.). John Wiley & Sons. Dabalen, A., I. Gaddis, and N. T. V. Nguyen. 2020. “CPI Bias and Its Implications for Poverty Reduction in Africa.” Journal of Economic Inequality 18 (1): 13–44. Dang, H.-A. H. 2020. “To Impute or Not to Impute, and How? A Review of Poverty-Estimation Methods in the Absence of Consumption Data.” Development Policy Review (Forthcoming). Deaton, A. 1988. “Quality, Quantity, and Spatial Variation of Price.” American Economic Review 78 (3): 418–30. Deaton, A., and S. Zaidi. 2002. Guidelines for Constructing Consumption Aggregates for Welfare Analysis. World Bank Publications. Development Research Group of the World Bank. Downloaded 2017. “PovcalNet: The On-Line Tool for Poverty Measurement.”. Douidich, M., A. Ezzrari, R. Van der Weide, and P. Verme. 2016. “Estimating Quarterly Poverty Rates Using Labor Force Surveys: A Primer.” World Bank Economic Review 30 (3): 475–500. Elbers, C., J. Lanjouw, and P. Lanjouw. 2003. “Micro-Level Estimation of Poverty and Inequality.” Econometrica 71 (1): 355–64. Foster, J., J. Greer, and E. Thorbecke. 1984. “A Class of Decomposable Poverty Measures.” Econometrica 52 (3): 761–66. Fujii, T., and R. van der Weide. 2020. “Is Predicted Data a Viable Alternative to Real Data?” World Bank Economic Review 34 (2): 485–508. Gibson, J., and B. Kim. 2015. “Hicksian Separability Does Not Hold over Space: Implications for the Design of Household Surveys and Price Questionnaires.” Journal of Development Economics 114: 34–40. ———. 2019. “Quality, Quantity, and Spatial Variation of Price: Back to the Bog.” Journal of Development Economics 137: 66–77. Gorman, W. M. 1959. “Separable Utility and Aggregation.” Econometrica 27 (3): 469–81. ———. 1961. “On a Class of Preference Fields.” Metroeconomica 13 (2): 53–56. Head, A., M. Manguin, N. Tran, and J. E. Blumenstock. 2017. “Can Human Development Be Measured with Satellite Imagery?” In ICTD. Jean, N., M. Burke, M. Xie, W. M. Davis, D. B. Lobell, and S. Ermon. 2016. “Combining Satellite Imagery and Machine Learning to Predict Poverty.” Science 353 (6301): 790–94. Kilic, T., and T. P. Sohnesen. 2019. “Same Question but Different Answer: Experimental Evidence on Questionnaire Design’s Impact on Poverty Measured by Proxies.” Review of Income and Wealth 65 (1): 144–65. LaFrance, J. T., T. K. Beatty, and R. D. Pope. 2006. “Gorman Engel Curves for Incomplete Demand Systems.” In Essays in Honor of Stanley R. Johnson, edited by M. T. Holt and J. P. Chavas. BE Press. 432 Christiaensen, Ligon, and Sohnesen Lanjouw, J. O., and P. Lanjouw. 2001. “How to Compare Apples and Oranges: Poverty Measurement Based on Different Definitions of Consumption.” Review of Income and Wealth 47 (1): 25–42. Ligon, E. 2019. “Estimating Household Welfare from Disaggregate Expenditures.” Unpublished manuscript. Mathiassen, A. 2013. “Testing Prediction Performance of Poverty Models: Empirical Evidence from Uganda.” Review of Income and Wealth 59 (1): 91–112. McKelvey, C. 2011. “Price, Unit Value, and Quality Demanded.” Journal of Development Economics 95 (2): 157–69. Pape, U., and J. Mistiaen. 2018. “Household Expenditure and Poverty Measures in 60 Minutes: A New Approach Downloaded from https://academic.oup.com/wber/article/36/2/413/6364497 by LEGVP Law Library user on 08 December 2023 with Results from Mogadishu.” Working Paper Series 8430, World Bank. Ravallion, M. 1994. Poverty Comparisons, volume 56. Taylor & Francis US. ———. 1996. “How Well Can Method Substitute for Data? Five Experiments in Poverty Analysis.” World Bank Research Observer 11 (2): 199–221. Roberts, C., G. Eva, N. Allum, and P. Lynn. 2010. “Data Quality in Telephone Surveys and the Effect of Questionnaire Length: A Cross-National Experiment.” Working Paper Series 2010-36, Institute for Social and Economic Research. Sohnesen, T. P., P. Fisker, and D. Malmgren-Hansen. 2021. “Using Satellite Data to Guide Urban Poverty Reduction.” Review of Income and Wealth (Forthcoming). Tagliati, F. 2021. “Welfare Effects of an In-Kind Transfer Program: Evidence from Mexico.” Manuscript. Watmough, G. R., C. L. Marcinko, C. Sullivan, K. Tschirhart, P. K. Mutuo, C. A. Palm, and J.-C. Svenning. 2019. “Socioecologically Informed Use of Remote Sensing Data to Predict Rural Household Poverty.” Proceedings of the National Academy of Sciences 116 (4): 1213–18. World Bank. 2020. “High-Frequency Monitoring Systems to Track the Impacts of the COVID-19 Pandemic.” Accessed August 27, 2021. https://www.worldbank.org/en/topic/poverty/brief/high-frequency-monitoring-surveys.