metadata: authors: 0: author: Christiaensen,Luc
1: author: Ligon,Ethan
2: author: Pave Sohnesen,Thomas
count: Tanzania,Rwanda,Uganda
docna: 0: docna: Should Consumption Sub-Aggregates Be Used to
            Measure Poverty ?
docty: Policy Research Working Paper
subtopic: Inequality,Poverty Lines,Poverty Assessment,Poverty Impact Evaluation,Small Area Estimation Poverty Mapping,Poverty Monitoring & Analysis,Poverty Diagnostics,Labor Markets,Social Protections & Assistance,Educational Sciences
teratopic: Poverty Reduction,Social Protections and Labor,Education
abstracts: cdata!: Frequent measurement of poverty is
            challenging, as 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
            "sub-aggregates" instead. The use of consumption
            sub-aggregates is theoretically justified if and only if all
            the 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 sub-aggregate might
            work in practice. The paper constructs such linear
            sub-aggregates using data from Rwanda, Tanzania, and Uganda.
            The findings show that using sub-aggregates is ill-advised
            in practice as well as in theory. This raises questions
            about the consistency of the poverty-tracking efforts
            currently applied across countries, since obtaining
            exhaustive consumption measures remains an unmet challenge.
geo_reg_key: 517196,517191
repnme: repnme: Should Consumption Sub-Aggregates Be Used
            to Measure Poverty ?
display_title: Should Consumption Sub-Aggregates Be
            Used to Measure Poverty
listing_relative_url: /research/2020/07/32207503/consumption-sub-aggregates-used-measure-poverty
topicv3: Poverty m1326687 695,Poverty Measurement and Analysis m1326941 707,Education m1327048 470,Social Protection and Labor m1327194 697,Jobs m1503931 2670
publishtoextweb_dt: 2020-07-01T00:00:00Z
historic_topic: Poverty Reduction,Social Protections and Labor,Education
origu: JOBS- CCSA - IBRD (HSPJB)
no_of_pages: 25
author_keys: 0: author_key: 480482
1: author_key: 688061
2: author_key: 664321
model_extractions: Policy Research Working Paper 9312 Should Consumption Sub-Aggregates Be Used to Measure Poverty?

_Luc Christiaensen Ethan Ligon Thomas Pave Sohnesen_ Social Protection and Jobs Global Practice July 2020
Policy Research Working Paper 9312

# **Abstract**

Frequent measurement of poverty is challenging, as 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 “sub-aggregates” instead. The use of consumption sub-aggregates is theoretically justified if and only if all the 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 sub-aggregate might work in practice. The paper constructs such linear sub-aggregates using data from Rwanda, Tanzania, and Uganda. The findings show that using sub-aggregates is ill-advised in practice as well as in theory. This raises questions about the consistency of the poverty-tracking efforts currently applied across countries, since obtaining exhaustive consumption measures remains an unmet challenge.

This paper is a product of the Social Protection and Jobs 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 lchristiaensen@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
# **SHOULD CONSUMPTION SUB-AGGREGATES BE USED TO MEASURE POVERTY?**

# LUC CHRISTIAENSEN

_World Bank_ ETHAN LIGON _University of California, Berkeley_ THOMAS PAVE SOHNESEN _World Bank_

> . _Date_ : June 30, 2020.

> . _Keywords._ Consumption sub-aggregates, poverty, Engel curves. . JEL Classification: D11, D12, I32.
# 1. Introduction

Directly and comprehensively measuring non-durable consumption expenditures via large-scale household surveys such as the Living Standards Measurement Surveys (LSMS) is generally regarded as the best way to assess the prevalence of poverty in low income countries. But such surveys are complex and costly, and so are conducted in fewer places, with smaller samples, and with lower frequency than is desirable if our aim is to measure changes in poverty over time (Beegle et al. 2016).

In response, researchers have been looking for alternative approaches to track poverty at reduced cost and effort. One approach involves the construction of models that use observed characteristics that are easier to collect (or that have already been collected for other purposes) to predict consumption. The success of this approach hinges critically on the stability of the model parameters over time and across different survey designs. One way to mitigate this concern is to identify empirically the conditions under which this assumption is more likely to hold (Christiaensen et al. 2012; Mathiassen 2013; Kilic and Sohnesen 2019). Another is to combine the survey collection, with more complete consumption aggregates for some households, while only collecting proxies for consumption for others (Pape and Mistiaen 2018; Yoshida et al. 2015). Nonetheless, model parameter instability cannot be excluded. With more high-resolution earth observation data and more powerful machine learning expertise now readily available, another class of models predicts consumption and poverty by applying artificial intelligence techniques to a combination of satellite imagery (especially daytime imagery) and existing household data. Prediction accuracy often exceeds benchmarks from more standard regressionbased approaches, especially in estimating poverty levels at very disaggregated levels—so-called poverty mapping (Head et al. 2017; Jean et al. 2016; Watmough et al. 2019). However, whether these models calibrated on data for a certain year are better able to make accurate predictions on data for another year remains largely untested.<sup>1</sup> The issue of model stability remains.

This paper explores the potential of a different approach, without the need for prediction and model parameters. Instead of falling back

> 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.
CONSUMPTION SUB-AGGREGATES 2 on other, more readily available data sources and consumption prediction methods, we explore whether less comprehensive consumption measures, or consumption sub-aggregates, that are also easier and less expensive to collect, might not suffice as an information base to track poverty over time. We draw on demand theory to establish the conditions under which this might be the case, and subsequently explore empirically whether these conditions hold in practice given the experience from a couple of different country settings.

The idea is inspired by Lanjouw and Lanjouw (2001) who imagine predicting poverty with food consumption only. Provided that Engel’s law holds (so that the food share is larger for the poor than for the nonpoor) and given a particular method of adjusting the poverty line when moving between expenditure aggregates,<sup>2</sup> the Lanjouws demonstrate that 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.

This raises the question: Under what circumstances can one construct consistent poverty measures using different expenditure aggregates?<sup>3</sup> We extend the Lanjouws’ result, and show that so long as household welfare can be summarized by total non-durable expenditures, then one can construct consistent welfare measures using smaller aggregates provided that the Engel curve (quantities vs. total expenditures) for the sub-aggregate is linear.

Accordingly, we search for sub-aggregate expenditures that are linear in total expenditure. However, if relative prices within the subaggregate change, then the resulting sub-aggregate expenditures are only guaranteed to _remain_ linear if all of the constituent goods in the aggregate also have linear Engel curves. Thus, the key to constructing a robust proxy for total expenditures that can be used to track changes in poverty over time when prices are changing is to identify particular goods that have linear Engel curves that are fairly insensitive to changes in prices.

We use LSMS data from Rwanda, Tanzania, and Uganda and devise a simple way to identify goods with (nearly) linear Engel curves using data in an initial period. We form a linear sub-aggregate of such goods, and then ask whether this sub-aggregate can be used to accurately measure poverty in a subsequent period. Theory suggests that this will not work if there are significant changes in relative prices. Unfortunately, 2. The “upper bound” method described by Ravallion (1994).

3. In this paper, the tracking of monetary poverty is based on the commonly used “Foster-Greer-Thorbecke” (FGT) family of poverty measures described by Foster, Greer, and Thorbecke (1984).
CONSUMPTION SUB-AGGREGATES 3 the evidence bears out this suggestion—our constructed linear subaggregate does not perform well, and we conclude that one should not attempt to track poverty by trying to construct a single linear subaggregate. To the extent that comprehensive expenditures measures are still incomplete in practice, this also counsels caution about the validity of expenditure-based poverty tracking more broadly.

# 2. Theory of Demand and The Aggregation of Commodities

We are interested in identifying a particular set of expenditure goods that can be added to form a sub-aggregate, that can then be used as a proxy for total expenditures to measure poverty and changes in poverty over time. In this section we present two 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, we first introduce some notation (2.1) and recall some of the standard results from demand theory (2.2). The link between consumption aggregation and sub-aggregates with different FGT poverty measures is then explored in subsections 2.3–2.5.

2.1. **Consumption Aggregates.** Suppose that a consumer (or household) values _n_ distinct non-durable commodities indexed by _i_ = 1 _, . . . , n_ ; call the bundle of these commodities consumed by the consumer _c ∈ X ⊆_ R<sup>_n_</sup> . Note our implicit assumption that consumption is continuous in all these goods. Assume also that there is a vector of prices _p ∈_ R<sup>_n_</sup> +<sup>,sothattheconsumer’stotalexpendituresonthe</sup> consumption bundle _c_ is _x_ = _p_<sup>_′_</sup> _c_ .

Because it comprises expenditures on all non-durable commodities, we call _x_ the consumer’s _total expenditures_ , or _aggregate expenditures_ . However, there are many possible sub-aggregates. Consider some partition or _aggregation_ of the _n_ different commodities _A_ = ( _X_ 1 _, . . . , Xm_ ), with _Xj ∩ Xk_ = _∅_ for _j_ = _k_ and _X_ = _∪_<sup>_m_</sup> _j_ =1<sup>_Xj_,sothatgood</sup><sup>_i_isan</sup> element of the _j_ th aggregate if _i ∈ Xj_ .

Just as _x_ is the consumer’s total expenditures on all goods in _X_ , given the aggregation _A_ let _x_<sup>_j_</sup> denote sum of expenditures on all goods in aggregate _Xj_ . Similarly, let _c_<sup>_j_</sup> denote the consumption bundle of goods in the _j_ th aggregate.

2.2. **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 _Xi ∈ A_ and any two consumption
CONSUMPTION SUB-AGGREGATES 4 goods _cj_ and _ck_ that are both in _Xi_ the consumer’s demand for _cj_ and _ck_ can be written as a function just of prices for goods in _Xi_ and of expenditures on the aggregate _x_<sup>_i_</sup> ; demands for goods in aggregate _Xi_ written in this form then will not depend on the prices of commodities in _other_ aggregates, except to the extent that these prices affect _xi_ .

A sufficient condition that the aggregation _A_ be separable is that the consumers’ utility function can be written as a set of _m_ sub-utility functions _{uj}_<sup>_m_</sup> _j_ =1<sup>andanaggregatingutilityfunction</sup><sup>_u_0:R</sup><sup>_m→_R</sup> such that where _c_<sup>_j_</sup> = ( _ci_ ) _i∈Xj_ is a set of consumption aggregates (Gorman 1961).

2.3. **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 ( _X_ 1 _, X_ 2) with corresponding expenditures ( _x_<sup>1</sup> _, x_<sup>2</sup> ); each of these aggregates is then assumed to depend only on total non-durable expenditures _x_ , so that _x ≡ x_<sup>1</sup> ( _x_ ) + _x_<sup>2</sup> ( _x_ ).<sup>4</sup> 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 _x_<sup>1</sup> ( _x_ ) is a continuous and strictly increasing function of _x_ , so that, observing some level of expenditures _x_ ˆ<sup>1</sup> one can invert _x_<sup>1</sup> ( _x_ ) to infer what the ˆ overall level of expenditures is, obtaining, say, _x_ = _g_ 1(ˆ _x_<sup>1</sup> ).

We extend the ideas of Lanjouw and Lanjouw (2001) in two important ways. First, they give conditions under which head-count poverty statistics for a population will not depend on whether one uses total expenditures or a expenditures on a smaller aggregate as a proxy for total expenditures. We provide 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 unchanging. In a simple application of demand theory we show how to relax this assumption as well, allowing one to apply these methods to data on expenditures over time, where we 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. Our two

> 4. In what follows it’s useful to follow some common notational conventions in functional analysis, that blur the lines between variables and functions. For example, _x_<sup>1</sup> will always take a value, but may implicitly or explicitly be regarded as a function of other quantities, as in _x_<sup>1</sup> ( _x_ ).
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CONSUMPTION SUB-AGGREGATES 6 is moving from the sub-aggregate poverty line _z_ 1 to the poverty line _z_ = _g_ 1( _z_ 1) relevant to total expenditures. Thus, provided one knows the mapping _g_ 1 from the sub-aggregate poverty line to the ‘full’ poverty line, one ought to be able to use any monotonic sub-aggregate to measure head-count poverty.

The idea of the method is illustrated in Figure 1. Here there are two sub-aggregates with expenditures _x_<sup>1</sup> and _x_<sup>2</sup> ; the way in which these vary with total expenditures _x_ is illustrated by the two lower curves in Figure 1. Then we can define a sub-aggregate poverty line _z_ 1 = _x_ 1( _z_ ) (marked as _z_ 1 on the vertical axis), and by construction we have _z_ = _g_ 1( _z_ 1), that gives the mapping between the poverty line corresponding to total expenditures _z_ and the poverty line corresponding to expenditures on sub-aggregate one, _z_ 1.

We first extend the Lanjouws’ invariance result, allowing us to move not only from a sub-aggregate 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 Xj ∈ A such that expenditures x_<sup>_j_</sup> _are continuous, strictly increasing functions of total expenditures x. Pick any two aggregates Xj, Xk in M . Suppose that a consumer is regarded as poor if and only if expenditures x_<sup>_j_</sup> _on aggregate Xj are less than a poverty line zj. Then there exists another poverty line zk such that headcount poverty using expenditures x_<sup>_j_</sup> _with poverty line zj will be equal to head count poverty using expenditures x_<sup>_k_</sup> _with poverty line zk_ = _x_<sup>_k_</sup> ( _gj_ ( _zj_ )) _._

_Proof._ The result follows directly from the existence of the inverse functions _g_<sup>_j_</sup> and _g_<sup>_k_</sup> , and the existence of these is guaranteed by our assumption that _Xj_ and _Xk_ are in _M_ . The poverty measure _P_ 0( _x_<sup>_j_</sup> _, zj_ ) is equal to the proportion of consumers with _x_<sup>_j_</sup> _≤ zj_ , and by the definition of the inverse functions the consumer has _x_<sup>_j_</sup> _≤ zj_ if and only if _x_<sup>_k_</sup> _≤ x_<sup>_k_</sup> ( _gj_ ( _zj_ )). □ 2.5. **Sub-aggregates and general FGT measures** ( _Pα_ ) **.** Note that Proposition 1 does _not_ require that expenditures on all aggregates are increasing; if some expenditure aggregates are decreasing we can simply set them aside.

More generally, moving from one aggregate to another _will_ matter. Lanjouw and Lanjouw (2001) consider the class of Foster, Greer, and Thorbecke (1984) (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.
CONSUMPTION SUB-AGGREGATES 7 Define the poverty measure over a population of _N_ households by where **x** is an _N_ -vector of expenditures for all households and where _xj_ denotes total expenditures of the _j_ th household.

The FGT class of measures can then be obtained by defining where 1 is an indicator function that takes the value of one if _x < z_ , and zero otherwise. How measured poverty changes using the FGT measure depends on how relative expenditure shares change in the different aggregates one considers. We classify some possibilities in the following proposition.

**Proposition 2.** _Let x_<sup>1</sup> _and x_<sup>2</sup> _be the expenditures corresponding to two different non-durable consumption aggregates X_ 1 _and X_ 2 _in an aggregation A, let x be total expenditures on all goods, and assume that x_<sup>1</sup> _and x_<sup>2</sup> _are both strictly increasing functions of x._

_Then for any z >_ 0 _there exists a z_ 1<sup>_∗andaz_</sup> 2<sup>_∗suchthatP_0(</sup><sup>_x, z_)=</sup> _P_ 0( _x_<sup>1</sup> _, z_ 1<sup>_∗_) =</sup><sup>_P_0(</sup><sup>_x_2</sup><sup>_, z_</sup> 2<sup>_∗_)</sup><sup>_,and_</sup>

- (1) _If the expenditure share of X_ 1 _is non-increasing in total expenditures, then Pα_ ( _x_<sup>1</sup> _, z_ 1<sup>_∗_)</sup><sup>_≤Pα_(</sup><sup>_x, z_)</sup><sup>_.[ThisistheLanjouws’_</sup> _result.]_

- (2) _If the expenditure share of X_ 1 _is non-decreasing in total expenditures, then Pα_ ( _x_<sup>1</sup> _, z_ 1<sup>_∗_)</sup><sup>_≥Pα_(</sup><sup>_x, z_)</sup><sup>_._</sup> _[This is the obvious converse.]_

- (3) _If the ratio of expenditure shares of X_ 1 _to X_ 2 _is non-increasing in total expenditures, then Pα_ ( _x_<sup>1</sup> _, z_ 1<sup>_∗_)</sup><sup>_≤Pα_(</sup><sup>_x_2</sup><sup>_, z_</sup> 2<sup>_∗_)</sup><sup>_.[Thisisan_</sup> _extension of the Lanjouws’ second result to any two aggregates.]_

- (4) _If the ratio of expenditure shares of X_ 1 _to X_ 2 _is non-decreasing in total expenditures, then Pα_ ( _x_<sup>1</sup> _, z_ 1<sup>_∗_))</sup><sup>_≥Pα_(</sup><sup>_x_2</sup><sup>_, z_</sup> 2<sup>_∗_)</sup><sup>_.[This is the_</sup> _converse of the extension.]_

- (5) _If the ratio of expenditure shares of X_ 1 _to X_ 2 _does not vary with total expenditures, then Pα_ ( _x_<sup>1</sup> _, z_ 1<sup>_∗_)=</sup><sup>_Pα_(</sup><sup>_x_2</sup><sup>_, z_</sup> 2<sup>_∗_)</sup><sup>_.[Thisis_</sup> _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
CONSUMPTION SUB-AGGREGATES 8 If _x < z_ then the sign of this difference is negative if and only if (1 _− x_<sup>1</sup> ( _x_ ) _/z_ 1<sup>_∗_(</sup><sup>_z_))</sup><sup>_α<_(1</sup><sup>_−x_2(</sup><sup>_x_)</sup><sup>_/z_</sup> 2<sup>_∗_(</sup><sup>_z_))</sup><sup>_α_which in turn is satisfied if and</sup> only if _x_<sup>1</sup> ( _x_ ) _/z_ 1<sup>_∗_(</sup><sup>_z_)</sup><sup>_>x_2(</sup><sup>_x_)</sup><sup>_/z_</sup> 2<sup>_∗_(</sup><sup>_z_),whichisagainsatisfiedifandonly</sup> if _x_<sup>1</sup> ( _x_ ) _/x_<sup>2</sup> ( _x_ ) _> z_ 1<sup>_∗_(</sup><sup>_z_)</sup><sup>_/z_</sup> 2<sup>_∗_(</sup><sup>_z_).At</sup><sup>_x_=</sup><sup>_z_thetwosidesofthisexpression</sup> are equal by construction; by assumption the ratio on the left is nonincreasing in _x_ , so for any _x < z_ we have _pα_ ( _x_<sup>1</sup> _, z_ 1<sup>_∗_)</sup><sup>_≤pα_(</sup><sup>_x_2</sup><sup>_, z_</sup> 2<sup>_∗_),and</sup> the result follows. □ This result is, like the Lanjouws’ original result, rather general. The only critical assumption is that expenditures on the aggregates we consider 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 allow 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 constructing 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 summarized in the following Corollary:

**Corollary 1.** _If expenditures on a separable aggregate X_ 1 _are proportional to total expenditures x, then Pα_ ( _x_<sup>1</sup> ( _x_ ) _, z_ 1<sup>_∗_(</sup><sup>_z_))=</sup><sup>_Pα_(</sup><sup>_x, z_)</sup><sup>_forall_</sup> _x, for all z >_ 0 _and for all α ≥_ 0 _._

Separability here implies an increasing relationship between _x_<sup>1</sup> and _x_ , so that case (5) of Proposition 2 obtains, taking as _x_<sup>2</sup> total expenditures _x_ , and we need 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 if we could simply
CONSUMPTION SUB-AGGREGATES 9 identify 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 because the result is also both incomplete and somewhat fragile. First the incompleteness: though modest restrictions on consumer preferences can be invoked to guarantee monotonicity of _x_<sup>1</sup> and _x_<sup>2</sup> , the ratio _x_<sup>1</sup> _/x_<sup>2</sup> 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’s simply no theoretical guarantee that a sub-aggregate that varies in a linear way with _x_ (the requirement of our Corollary) exists.

Second, the fragility: 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 _x_<sup>1</sup> _/x_<sup>2</sup> is monotone for the prices that obtain in one place and one time, it needn’t 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. We revisit this point below in Proposition 3, at which point we’re in a position to specify conditions under which we _can_ guarantee the desired linearity.

# 3. Linearity of Engel Curves for Aggregates

If a particular aggregate of goods has the property that demand for the aggregate depends only on the prices of the goods that comprise it, then we say that the aggregate is _separable_ . In this section we consider the circumstances under which expenditures on any such aggregate will be linear in total expenditures.

Assume that demand is separable in the aggregation _A_ , and consider expenditures on an aggregate _Xj_ . These expenditures will generally depend on (all) prices, and on total expenditures _x_ . We’re interested in understanding when _x_<sup>_j_</sup> is linear in _x_ —that is, in the case in which expenditures on the _j_ th 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 expenditures. The first is when expenditures for _every_ aggregate _Xj ∈ A_ are linear in total expenditures; the second allows _some_ aggregates to have non-linear Engel curves, but for non-linearities 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. We consider these two cases in turn.
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CONSUMPTION SUB-AGGREGATES 14 case; here suppose that there are only the two goods _c_ 1 and _c_ 2. Then from the consumer’s budget constraint it follows that _b_ ( _p_ ) = 1. □ 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 expenditure._

_Proof._ Suppose that good 1 has linear Engel curves, with _p_ 1 _c_ 1( _p, x_ ) = _a_<sup>1</sup> ( _p_ ) + _b_<sup>1</sup> ( _p_ ) _x_ . Then from the budget constraint it follows that an aggregate of the remaining goods also have linear Engel curves, since � _ni_ =2<sup>_pici_=</sup><sup>_x −p_1</sup><sup>_c_1=</sup><sup>_−a_1(</sup><sup>_p_) + (1</sup><sup>_−b_1(</sup><sup>_p_))</sup><sup>_x_.ButthenbyProposition</sup> 3 all goods _i_ = 2 _, . . . , n_ also have linear Engel curves. □

# 4. Sub-aggregates and Poverty in Practice

Corollary 1 tells us that we can use any separable sub-aggregate provided expenditures on the sub-aggregate vary in proportion to total expenditures. But Proposition 3 and its corollary tell us that such a sub-aggregate 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 prices this linearity may fail.

However, from a practical point of view it may be possible to identify goods with linear Engel curves in one period, and then we may get lucky: in a subsequent period perhaps relative prices will not have changed in such a way that linearity will be compromised, and our sub-aggregate will work for measuring poverty.

4.1. **Data.** In the remainder of this paper we test our luck using data from expenditure surveys in Rwanda, Uganda, and Tanzania.<sup>5</sup> All surveys are large multipurpose household consumption surveys, representative at both national and urban/rural levels with large sample sizes. In Uganda about 3000 households were interviewed each year, in Tanzania between 3000 and 4000 each year, and in Rwanda the number of households increased substantially from 6900 to 14300 household observations per year.

> 5. Rwanda: Enquete Int ́egrale sur les Conditions de Vie des m ́enages 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.
CONSUMPTION SUB-AGGREGATES 15 The goods included in the consumption aggregates follow the guidelines for consumption aggregates found in Deaton and Zaidi (2002).<sup>6</sup> In all three countries, FGT measures rely on consumption data collected 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.<sup>7</sup> Beegle et al. (2012) review recall-based consumption surveys 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, even if the total number of goods doesn’t actually encompass all non-durable consumption. In each country, consumption goods are valued in year one’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 one, 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, Downloaded 2017). The five year time span between surveys in Rwanda can also be seen as an upper bound for the approach, as most countries implement full consumption surveys every five years or less.

4.2. **Looking for Linearity.** Moving from theory to practice, we would like to gauge the consumption patterns of households, as some goods might be more suitable than others. Figure 2 illustrates how a sum of linear goods will always lead to a linear aggregate, but Figure 5 also shows that a sum of expenditures on non-linear goods can be linear for a particular set of prices. To gauge properties of goods in relation to the total aggregate, we run an OLS regression of householdlevel log consumption expenditures for each good on a constant and the log total aggregate: 6. The included goods in the aggregates are very similar to those defined by the national statstical agencies, but are not necessarily identical.

7. 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). For the analysis we have combined the values of each item from both questionnaires into one value, following the same principle as in Tanzania and Uganda.
CONSUMPTION SUB-AGGREGATES 16 Note that this equation can be re-written to give an expression for expenditure shares: In this regression we would expect that the linear goods we are looking for will have _βi_ = 1, and a constant share of consumption expenditures. Any goods with expenditures decreasing in total expenditures will have a negative _βi_ . Goods with _βi >_ 1 will have an increasing share of total consumption expenditures, while other goods will have 0 _< βi <_ 1 (and a declining share of total consumption). Only for goods with income elasticity of _βi_ = 1 will the consumption share remain constant. We emphasize that these regressions should be regarded as _temporary_ , 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 (1) provides a reasonable approximation or not. If so, then we may use goods with estimated _βi_ close to one to form a sub-aggregate. If not, then the construction of sub-aggregates for the purpose of tracking poverty may simply not be wise.

Thus, we estimate the regression equation (1) for the many different consumption goods across three countries described above. We then use the estimated values of _βi_ to classify goods: _βi <_ 1 **:** Shares decreasing; _βi >_ 1 **:** Shares increasing; _βi_ = 1 **:** Shares constant.

Our idea is then to use goods for which expenditures shares are constant to construct a sub-aggregate. If we can construct such a linear subaggregate and if it _remains_ linear when relative prices change then we can confidently use the sub-aggregate and result of Proposition 2.5 to construct any of the FGT poverty measures.

To implement this classification, we first choose two different rounds for each of the three countries we consider. We first exclude any observations of zero expenditures, then exclude any consumption goods that have fewer than five observations. We regard some estimated point values of _βi_ as insignificant; these are those having a _t_ -statistic (associated with the test that the point estimate is zero) less than 2.326 (the critical value associated with having the probability of a type I error of one percent). We classify the remaining point estimates according to two simple criteria: first, are they greater or less than one; and second, are they “approximately linear”, in the sense that the point estimates lie within the interval (0 _._ 9 _,_ 1 _._ 1)?
CONSUMPTION SUB-AGGREGATES 17 Table 1 gives results of this exercise. We first use just data from the first round of data for each country to estimate and classify the _βi_ . For Tanzania 83% of all 112 goods can be classified (i.e., have enough observations and are significant); of these 80% are less than one, so that their expenditure shares fall on average with increasing total expenditures, while the remainder have _βi_ greater than one, and so increasing shares. Of these, 18% indicate an approximately linear good (i.e., _βi_ is within 0.1 of unity). Similarly, for Rwanda 88% of 284 goods can be classified, of which 92% have decreasing shares, while 11% are approximately linear. And for Uganda 95% of 126 goods can be classified, of which 81% have decreasing shares, while 15% are approximately linear.

This exercise allows us to construct linear sub-aggregates for each country consisting respectively of 17, 28, and 18 goods, and an appeal to Proposition 2.5 assures us that these sub-aggregates can be used to compute the same FGT poverty measures we could have computed using all observed expenditures. Further, if we were to assume that the estimated parameters ( _αi, βi_ ) obtained at a _different_ time (for the same population) remained constant, we could use the quite small amount of data required on expenditures on the sub-aggregate to compute FGT poverty measures.

Table 1. Consumption goods and linearity. Counts of goods for each country by classification; numbers in parentheses are counts of goods with the same classification in the second year as in the first.

|Classifcation|Tanzania|Rwanda|Uganda|
|---|---|---|---|
|Fewer than 5 observations|1 (1)|3 (1)|0 (0)|
|Insignifcant relationship|18 (7)|31 (12)|6 (5)|
|Decreasing shares (0_< β <_1)|74 (69)|229 (209)|97 (89)|
|Increasing goods (_β >_1)|18 (14)|21 (9)|23 (13)|
|Approximately linear (0_._9_< β <_1_._1)|17 (10)|28 (12)|18 (7)|
|Total number of expenditure goods|112|284|126|

However, as discussed in previous sections we have 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 _βi_ to also change. So what do the data tell us?

We re-estimate (1) using data from a _second_ round for each country, and Table 1 reports (in parentheses) the number of goods where our
CONSUMPTION SUB-AGGREGATES 18 estimate of _βi_ is unchanged. Of chief interest is the number of goods that remain approximately linear. In Tanzania this is 10 of 17; in Rwanda 12 of 28; and in Uganda 7 of 18. Overall less than half of the goods initially classified as “approximately linear” received the same classification in a second round of data. It is impossible to comfortably conclude that the linear 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 situation) changes in relative prices will not affect relative expenditures. The classification exercise we report 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 _βi_ must be functions of prices.

4.3. **Performance of sub-aggregates in measuring poverty.** Perhaps deviations from linearity are quantitatively unimportant for constructing FGT poverty measures? Guided by the theory above, we consider identifying expenditure items that are linear in a base period and using these to construct a sub-aggregate used for measuring poverty in a subsequent period. In our exercise we _also_ have data on all other expenditures, so we can evaluate the performance of the sub-aggregate in measuring poverty. To make this work, we need to identify the _gi_ functions of Proposition 1. Here this amounts to a simple rescaling. For example, if a poverty headcount of 20 percent was observed for the full aggregate in year one, 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 two quantities using year one’s prices, we can use exactly this same poverty line in the second year.

We construct our sub-aggregate by beginning with the “best” good; i.e., the good _i_ with _βi_ closest to one in absolute value. Then the second best good, and so on, until all goods have been added to the aggregate. For each such sub-aggregate we then 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 (Figure 6). In all three countries the estimates are systematically off in one direction, until almost all goods are added to the aggregate. We see this by the approximated poverty headcounts being clearly outside the 95 percent confidence interval of the poverty estimates for the full consumption aggregate, illustrated by the dotted lines. The reason for this pattern is the exclusion of
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CONSUMPTION SUB-AGGREGATES 20 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, we use data from Rwanda, Uganda, and Tanzania and search across different goods to identify those that feature (nearly) linear Engel curves in a base period. We then use these goods to construct a sub-aggregate in a subsequent period. Consistent with theory, the performance of this method turns out to be extremely bad at measuring head-count poverty rates. One should _not_ construct sub-aggregates for this purpose 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 prices are likely to be different.

Perhaps there are better ways to construct a sub-aggregate? Our conditions are both necessary and sufficient, so theory tells us that the answer is “no”. Another idea often used in practice involves constructing a sub-aggregate 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 sub-aggregate this would hold promise. But if Engel curves aren’t 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 elasticity. 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.

Finally, as a practical matter, any measured consumption aggregate is likely to be an incomplete or a reduced aggregate approximation to the full aggregate. For instance, the number of consumption items (or categories) for which data are collected from households in LSMS surveys ranges from 37 to 305, with the mean being 137 and the median 130 (Beegle et al. 2010). Arguably the value of public goods and time and leisure should also be included in a comprehensive 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 use value of durable goods are currently included in some countries, but excluded in others. These matters have been largely ignored in poverty tracking, but our results strongly suggest that they may be of first order importance.
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