metadata: authors: 0: author: Decerf,Benoit Marie A
1: author: Ferrando,Mery
2: author: Quinn,Natalie N.
count: World
docna: 0: docna: Global Income Poverty Measurement with
            Preference Heterogeneity : Theory and Application
docty: Policy Research Working Paper
subtopic: Inequality,International Trade and Trade Rules,Poverty Impact Evaluation,Small Area Estimation Poverty Mapping,Poverty Monitoring & Analysis,Poverty Diagnostics,Poverty Lines,Poverty Assessment,Services & Transfers to Poor,Disability,Access of Poor to Social Services,Economic Assistance,Social Development & Poverty
teratopic: International Economics and Trade,Macroeconomics and Economic Growth,Social Protections and Labor,Poverty Reduction
abstracts: cdata!: There is growing support for monitoring
            global poverty using a measure that accounts for both own
            and relative income. This paper shows how&#x2014;in the
            context of heterogeneous preferences over these
            factors&#x2014;the well-known conflict between fairness
            and welfare-consistency can be resolved, establishing the
            first preference-based foundation for both the established
            societal global poverty line and recently proposed
            hierarchical poverty indices. The paper reformulates one
            hierarchical index as a modified headcount ratio. Unlike all
            classic poverty indices, this index is necessarily reduced
            when an individual escapes poverty. The application
            highlights that the proposed index substantially changes the
            assessment of global poverty reduction.
repnme: repnme: Global Income Poverty Measurement with
            Preference Heterogeneity : Theory and Application
display_title: Global Income Poverty Measurement
            with Preference Heterogeneity : Theory and Application
listing_relative_url: /research/2021/11/33593423/global-income-poverty-measurement-preference-heterogeneity-theory-application
topicv3: Social Inclusion m1326031 421,Poverty m1326687 695,Trade m1326924 698,Poverty Measurement and Analysis m1326941 707,Social Assistance m1327201 1466
publishtoextweb_dt: 2021-11-08T00:00:00Z
dois: 10.1596/1813-9450-9844
historic_topic: International Economics and Trade,Macroeconomics and Economic Growth,Social Protections and Labor,Poverty Reduction
origu: DECRG: Poverty & Inequality (DECPI)
no_of_pages: 85
author_keys: 0: author_key: 1947055
1: author_key: NA
2: author_key: NA
model_extractions: Policy Research Working Paper 9844 Global Income Poverty Measurement with Preference Heterogeneity Theory and Application _Benoit Decerf Mery Ferrando Natalie N. Quinn_ Development Economics Development Research Group November 2021
Policy Research Working Paper 9844

# **Abstract**

There is growing support for monitoring global poverty using a measure that accounts for both own and relative income. This paper shows how—in the context of heterogeneous preferences over these factors—the well-known conflict between fairness and welfare-consistency can be resolved, establishing the first preference-based foundation for both the established societal global poverty line and recently proposed hierarchical poverty indices. The paper reformulates one hierarchical index as a modified headcount ratio. Unlike all classic poverty indices, this index is necessarily reduced when an individual escapes poverty. The application highlights that the proposed index substantially changes the assessment of global poverty reduction.

This paper is a product of the Development Research Group, Development Economics. 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 bdecerf@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
# Global Income Poverty Measurement with Preference Heterogeneity: Theory and Application<sup>∗</sup>

Benoit Decerf,<sup>†</sup> Mery Ferrando,<sup>‡</sup> and Natalie Naïri Quinn<sup>§</sup> Originally published in the Policy Research Working Paper Series on _November 2021._ This version is updated on _January 2024._ To obtain the originally published version, please email prwp@worldbank.org.

**JEL** : I32, C43, N30. **Keywords** : Global Income Poverty, Preference Heterogeneity, Welfare-Consistency, Relative Poverty, Absolute Poverty.

> ∗Acknowledgments: We thank Peter Lanjouw, Daniel G. Malher, Berk Ozler, and Roy van der Weide, who provided useful comments on earlier versions of this paper. We are grateful to Martin Ravallion for a discussion in Namur that was at the origin of this research project, as well as for his encouraging comments on an earlier version of this paper. We thank all the participants at the Welfare Economics and Economic Policy webinar, the Opportunity, Mobility and Well-being Conference in Munich, the Ninth ECINEQ Meeting at LSE, SSCW 2022 in Mexico City, internal seminars at the World Bank and Tilburg University, and, in particular, Francisco Ferreira and Tim Goedeme. This work was supported by: FNRS Excellence of Science (EOS) Research project #O020918F. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors and should not be attributed in any manner to the World Bank, its affiliated organizations, or members of its Board of Executive Directors or the countries they represent. The World Bank does not guarantee the accuracy of the data included in this paper and accepts no responsibility for any consequence of their use. All remaining mistakes are, of course, our own.

> †World Bank, Development Research Group. Email: bdecerf@worldbank.org.

> ‡Tilburg University, Department of Economics. Email: m.ferrando@tilburguniversity.edu.

> §University of Oxford, Department of Economics. Email: natalie.quinn@economics.ox.ac.uk.
# **1 Introduction**

Poverty reduction is the first Sustainable Development Goal, adopted by the United Nations in 2015. Both the design of effective poverty-reduction policies and the monitoring of progress require that poverty be measured meaningfully. There is growing support for the idea that global income poverty should be assessed with a measure that accounts for _both_ own income and relative income.<sup>1</sup> Two related justifications for this have been proposed. First, Atkinson and Bourguignon (2001) argue that taking a global perspective requires accounting for both subsistence and social inclusion, the two functionings underpinning poverty measurement practices in developing countries and developed countries, respectively. While the real cost of subsistence is typically assumed fixed, that of social inclusion increases with standards of living and therefore depends on relative income (Smith, 1776; Townsend, 1985). Second, Ravallion and Chen (2011) argue that relative income is, like own income, an important determinant of the concept of economic welfare that is relevant for global poverty measurement; there is now extensive evidence that relative income is an important determinant of subjective well-being (Clark and Oswald, 1996; Luttmer, 2005; Perez-Truglia, 2020). These two justifications are related if individuals care about both own income and relative income because they care about subsistence and social inclusion (Ravallion, 2020).

When both own income and relative income matter, the trade-offs that a poverty measure makes between them becomes a key question. Altering these trade-offs may reverse cross-country comparisons or poverty trends, so they should not be made arbitrarily. Rather, welfare-consistency requires that these trade-offs be related to individual preferences over own income and relative income.<sup>2</sup> Current research efforts on global income poverty measurement, recently reviewed in Ravallion (2020), have two important limitations that are directly related to these trade-offs.<sup>3</sup> The first limitation is the absence of robust

> 1As recommended by Atkinson (2016), the World Bank now also reports global poverty estimates based on a measure that takes both own income and relative income into account. 2 Following Ravallion (2020), we assume that an individual’s preference over own income and relative income serves as a ‘reduced form’ for her deeper preference over her levels of nutrition and social participation (see online Appendix S1 for details). Hence, malicious aspects of the other-regarding preference are assumed to be laundered away.

> 3There are also other important limitations not directly related to these trade-offs, for example, the comparability of own income across households and across countries. Producing a comparable own income variable is challenging when dealing with heterogeneous relative prices between goods and heterogeneous preferences over these goods (Van Veelen and van der Weide,

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theoretical justification for the trade-offs embedded in its poverty measures. This literature always assumes a common welfare function, which makes the same trade-offs between own income and relative income for all individuals. However, existing evidence shows that individuals hold heterogeneous preferences. Previous studies have shown that even poor individuals hold heterogeneous preferences over necessities (Atkin, 2013, 2016), while a growing body of field and experimental evidence documents that social preferences are heterogeneous (Eckel and Grossman, 1998; Andreoni and Vesterlund, 2001; Blanco et al., 2011). The heterogeneity of preferences does not necessarily rule out the use of a common welfare function. However, it begs the question: which common welfare function should be used given individual preferences? We provide the first theoretical answer to this open question.

The second limitation is that the recent literature focuses on the design of welfare-consistent global poverty _lines_ .<sup>4</sup> However, a poverty measure is always the combination of a poverty line and a poverty index (Sen, 1976). Thus far, the literature has not investigated the implications of welfare-consistency for the poverty index. These implications could matter because the choice of poverty index may impact the evaluation of the global poverty trend at least as much as the design of the global poverty line (Decerf and Ferrando, 2022).

In this paper, we develop a theory of global income poverty measurement that makes progress on these two limitations. We show that a fair and welfareconsistent aggregation of heterogeneous preferences not only justifies the use of a common welfare function but also pins down the trade-offs that this welfare function should make between own income and relative income. Our results not only characterize the shape of the global poverty line but also characterize the type of poverty indexes to be used with the global line.

More precisely, we study the constraints that two requirements, one about welfare-consistency and one about fairness, impose on the poverty measure. Welfare-consistency requires that the poverty score attributed to a poor individual should decrease when she becomes better-off. A fairness requirement is called for in the presence of heterogeneous preferences to rule out counter-intuitive poverty comparisons. In particular, when comparing two individuals living in the same society, the measure should not attribute a _greater_ 2008; Dimri and Maniquet, 2020). In line with Atkinson and Bourguignon (2001) and Ravallion and Chen (2011), we abstract from this issue and assume that a comparable own income variable is available.

> 4 Following Atkinson and Bourguignon (2001), a _global_ poverty line should be consistent with a common framework applied to all countries of the world.

3
poverty score to the individual with _greater_ income on the basis that she is more sensitive to relative income. Fairness rules out this possibility. Unsurprisingly, these two requirements are incompatible under heterogeneous preferences. We side with fairness and thus impose a weaker welfare-consistency requirement, which still requires that an individual’s poverty score is reduced when she escapes poverty. Our main theoretical results, which rely heavily on preference heterogeneity, fully characterize the joint implications that these two requirements have on the global poverty line and on the poverty index (Theorems 1 and 2 in Section 5). We summarize these key implications below.

First, we show that the global poverty line must be _societal_ , that is, it should be absolute in low-income countries and relative in more developed countries. In low-income countries, the global line is determined by the preference least sensitive to relative income. In middle- and high-income countries, the global line is determined by the preference most sensitive to relative income. This result provides theoretical guidance for the design of the global line and a complete preference-based foundation for the societal lines proposed in the literature.<sup>5</sup> Second, we show that classical indices such as the societal head-count ratio<sup>6</sup> violate our very basic welfare-consistency property when preferences are heterogeneous. While the headcount ratio is well-known for violating many properties (Sen, 1976), such limitations have often not been deemed sufficient for discarding this index. One key justification is that, at least, the headcount ratio is reduced when an individual escapes poverty. We show this is no longer the case for the societal headcount ratio when preferences are heterogeneous. In contrast, our two requirements characterize the class of _hierarchical_ poverty indices recently proposed by Decerf (2017). The particularity of hierarchical indices is that they always attribute a larger poverty score to an absolutely poor individual in a low-income country than to an only-relatively poor individual in a middle- or high-income country. We reformulate one particular hierarchical index, showing that this index can be expressed as a modified headcount ratio. This modified index sums up the fraction of absolutely poor individuals with the fraction of only-relatively poor individuals, multiplied by a weight smaller than one. This index is straightforward and ready to apply, and we show that it may receive an interesting interpretation under preference heterogeneity. Importantly, the trade-offs that this index makes between own income and relative income

> 5In practice, the exact design of proposed global lines has been informed by regressing national poverty lines on standards of living (Atkinson and Bourguignon, 2001; Ravallion and Chen, 2011; Jolliffe and Prydz, 2021). Virtually all global lines proposed by these authors are societal.

> 6The societal headcount ratio is the fraction of individuals who are below the societal line.

4
correspond to those characterized in our main theoretical results (Theorem 1 and 2).

Our theoretical results allow us to elucidate the implicit normative judgements embedded in the the societal poverty line of the World Bank (2018). Conversely, the World Bank’s line gives us an anchor for the normative factors that are exogenous to our theory. We compare the empirical distribution and evolution of global income poverty using three different poverty measures: the absolute headcount ratio, the societal headcount ratio, and our proposed index. These three measures provide markedly different evaluations of global income poverty. For the period 1999-2015, our proposed index assesses global poverty reduction to have been 50% higher than current estimates based on the (societal) headcount ratio, while 25% lower than estimates based on the absolute headcount ratio.

We make several theoretical contributions to the literatures on social choice and on income poverty measurement. First, our results establish that the assumption of preference homogeneity in the context of income poverty measurement is an unnecessary modelling restriction. Indeed, we show that, despite the tension between welfare-consistency and fairness, we can derive meaningful and ready-to-use poverty measures by aggregating heterogeneous references. In so doing, we contribute to the branch of the social choice literature that aims to characterise normative indices that aggregate heterogeneous preferences while satisfying welfare-consistency and fairness requirements (Fleurbaey and Maniquet, 2011). To the best of our knowledge, we are the first to study the implications for income poverty measurement of aggregating (other-regarding) preferences over own income and relative income.<sup>7</sup> Second, our results provide the first complete preference-based justification for global poverty lines of the societal type. Ravallion and Chen (2011) investigate the implications of a weak relativity axiom (WRA), which requires that a poverty measure is reduced when all incomes grow by the same proportion. The WRA is weaker than our welfare-consistency requirement. Our results extend and improve on the results of Ravallion and Chen (2011) in at least two ways: (i) we do not assume, but rather show, that the relative segment of the global line must be 7 We are also the first to resolve the tension by weakening welfare-consistency rather than fairness, which we argue is more appropriate in our context. Decancq et al. (2019) also consider poverty measurement. As in their approach, the exogenous normative choice of a reference bundle plays an important role in our theory, but their framework features self-centered preferences over multidimensional goods. Treibich (2019) also considers a setting with other-regarding preferences over own income and relative income, but he studies social welfare measurement. Both studies weaken fairness.

5
linear and (ii) our theoretical results explain not only the relative segment of the global line, but also its absolute segment.

Third, we provide the first preference-based justification for _hierarchical_ poverty indices. In contrast, Decerf (2017) merely argues that hierarchical indices yield poverty comparisons that are more in line with intuition than the comparisons associated with standard indices.<sup>8</sup> Our results thus provide a strong conceptual motivation for discarding the societal headcount ratio and replacing it with a hierarchical index. To facilitate their adoption, we propose a simple reformulation of one of these indices as a hierarchical headcount ratio, provide some justification for this particular index and assess empirically the value of the weight it attributes to the only-relatively income poor.

The remainder of the paper is organized as follows. In Section 2, we provide intuitive explanations for our main theoretical results in a simplified setting and introduce the hierarchical headcount ratio, explaining how it captures the main aspects of our theory. In Section 3, we introduce our full framework and formal definitions of the fairness and welfare-consistency properties considered. In Section 4, we show that these two properties clash under heterogeneous preferences and weaken the latter. In Section 5, we characterize their joint implications for the global poverty line and index, on nested sets of preferences. In Section 6, we present our application to global poverty measurement. In Section 7, we provide concluding comments.

# **2 Main Theoretical Findings**

## **2.1 The Basic Framework**

Let **y** := ( _y_ 1 _, . . . , yn_ ( **y** )) denote an income distribution, and let _<u>y</u>_ denote the median income in distribution **y** .<sup>9</sup> In line with Ravallion (2020), an individual has preferences over bundles comprising both her own income _y_ and her relative income _y/y_ ~~.~~ Her preference relation can be represented by a utility function _u_ ( _y, y/y_ <u>)</u> that is strictly increasing in its first argument and weakly increasing in the second. For instance, an individual could have her preferences represented by

> 8The axiomatic result of Decerf (2021) does not account for individual preferences, which are absent from his framework, and thus does not provide a non-paternalistic justification for the trade-offs these indices make between own income and relative income.

> 9We use _income_ as shorthand for a comparable individual monetary welfare indicator. In principle, such an indicator would be adjusted to account for household composition and individuals’ needs, for example, due to disability. In practice, data limitations mean that needsadjustments are rarely implemented.

6
a utility function in the following family: where the parameter _σ ≥_ 0 tunes the sensitivity to relative income. Self-centered preferences correspond to the case _σ_ = 0, that is, to _u_<sup>0</sup> .<sup>10</sup> This utility function is strictly increasing in relative income when _σ >_ 0. Without loss of generality, we often write the two arguments of utility to be own income and median income, that is, we write _u_ ( _y,_ _<u>y</u>_ <u>)</u> instead of _u_ ( _y, y/y_ <u>),</u> and refer to ( _y,_ _<u>y</u>_ <u>)</u> as a _bundle_ . Our main results are robust to more general sets of preferences (see Section 5.3).

An individual is **welfare poor** if she is worse off than at a reference bundle (Ravallion, 1998). The selection of this reference bundle is exogenous to our theory, and we assume it is selected by some social planner. We denote the reference bundle by ( _za,_ _<u>y</u>_<sup>_~~z~~_</sup> ), where _za >_ 0 and _<u>y</u>_<sup>_~~z~~_</sup> _>_ 0. Parameter _za_ is the subsistence income, which we take in our empirical application to be the World Bank’s extreme poverty line (Ferreira et al., 2016). Parameter _<u>y</u>_<sup>_~~z~~_</sup> is the reference median income, which we interpret as the largest value of median income at which the subsistence income is sufficient for social inclusion. We emphasize that our definition of the welfare poor is _not_ based on an exogenously given global poverty line.

Figure 1 illustrates the definition of welfare poverty. An individual is welfare poor when her bundle lies below her indifference curve passing through the reference bundle.<sup>11</sup> Indifference curves have non-negative slopes because relative income is positively valued. Individual 1 is not welfare poor because she is self-centered ( _u_ 1 = _u_<sup>0</sup> ) and her income is larger than the subsistence income (Figure 1.a). Individual 2 prefers the reference bundle over her bundle and is thus welfare poor (Figure 1.b).

The literature traditionally assumes that preferences are homogeneous. When the common preference is self-centered ( _u_<sup>0</sup> ), only own income matters. The selfcentered case provides the foundation for absolute poverty lines. To see this, assume that the absolute line is set at the subsistence income. With a common self-centered preference, all individuals below the subsistence income are welfare poor and all those above it are not welfare poor. In this sense, the absolute line perfectly identifies the welfare poor. In this case, the absolute headcount ratio,

> 10In our terminology, _self-centered_ has no negative connotation. It means that the individual only cares about own income.

> 11 When the preference is _uσ_ , its indifference curves are straight lines because _uσ_ is ordinally equivalent to _−_ ( _u_<sup>_σ_</sup> )<sup>_−_1</sup> = _y_ <u>1</u><sup>+</sup> <u>1</u> _yσ/y_<sup>=</sup> 1+ _<u>yσy</u>_<sup>.</sup>

7
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heterogeneous, welfare-consistency and fairness may clash. In this paper, we side with fairness to rule out poverty measures that make counter-intuitive comparisons.

For this reason, we consider additive poverty indices that satisfy a classical fairness requirement. This fairness requirement has strong implications, because indices that satisfy it must treat equally any two individuals who live in the same society and earn the same income. Therefore, the definition of these _fair_ poverty indices cannot depend on the specific profile **u** of society ( **y** _,_ **u** ), and hence, these indices can be implemented without eliciting individuals’ preferences. But this does not mean that preferences play no role. The set _U_ of admissible utility functions will play a central role in our analysis. Formally, a fair poverty index is where _p_ ( _yi,_ _<u>y</u>_ <u>)</u> is individual _i_ ’s poverty score, that is, the non-negative amount she contributes to the poverty measure. The poverty score _p_ ( _yi,_ _<u>y</u>_ <u>) may depend on the</u> set _U_ , but equal treatment is guaranteed because _p_ ( _yi,_ _<u>y</u>_ <u>)</u> does _not_ depend on _i_ ’s preference _ui_ .

Importantly, this equal treatment extends to the poverty line, which we will also describe as _fair_ . The poverty line is by definition the minimal income level for which an individual’s poverty score is zero. The value of the poverty line, denoted by _z_ <u>(</u> _<u>y</u>_ <u>),</u> may depend on median income _<u>y</u>_ ~~,~~ but not on _i_ ’s preference _ui_ . An individual _i_ whose income _yi_ is smaller than the poverty line _z_ <u>(</u> _<u>y</u>_ <u>)</u> is **income poor** and her poverty score _p_ ( _yi,_ _<u>y</u>_ <u>)</u> is strictly positive. An individual’s income poverty status is ‘objective’ in the sense that it only depends on the poverty line. An individual’s welfare poverty status is (partly) ‘subjective’ in the sense that it depends on her preference as well as the reference bundle.

When preferences are heterogeneous, an individual’s income poverty status may differ from her welfare poverty status. This is the reason why the societal headcount ratio violates our basic welfare-consistency requirement: any poverty measure should be reduced when a welfare poor individual escapes welfare poverty, at least when others are not negatively affected. We call this minimal welfare-consistency requirement the **escaping-poverty property** . It is arguably the most fundamental property in the poverty measurement literature. Before illustrating that the societal headcount ratio violates the escaping-poverty property, we show how this property endogenously determines the poverty line 9
_z_ <u>(</u> _<u>y</u>_ <u>).</u>

# **2.2 Implications for the poverty line**

We show that, in the context of heterogeneous preferences, the escaping-poverty property entails that the (fair) global poverty line must be _societal_ , that is, absolute in low-income societies and then relative in higher-income societies.<sup>13</sup> Such a societal global poverty line is illustrated by the solid red line in Figure 2. Here is the intuition. Any welfare poor individual must be attributed a strictly positive poverty score in order for the measure to be reduced when she escapes welfare poverty. Indeed, if a welfare poor individual’s poverty score is zero, her poverty score cannot further decrease when she escapes welfare poverty. Hence, any bundle ( _y,_ _<u>y</u>_ <u>) at which some individual could be welfare poor must be attributed a strictly</u> positive poverty score, which by definition of the global line implies _y < z_ <u>(</u> _<u>y</u>_ <u>).</u> Thus, the escaping-poverty property implies that for any value of median income _<u>y</u>_ ~~,~~ the global line _z_ <u>(</u> _<u>y</u>_ <u>)</u> must be equal to (or greater than) the smallest income for which no individual can be welfare poor, that is, _u_ ( _z_ <u>(</u> _<u>y</u>_ <u>)</u> _,_ _<u>y</u>_ <u>)</u> _≥ u_ ( _za,_ _<u>y</u>_<sup>_~~z~~_</sup> ) for all _u ∈ U_ . If the global line _z_ <u>(</u> _<u>y</u>_ <u>)</u> is equal to the smallest income for which no individual can be welfare poor then _u_<sup>_′_</sup> ( _z_ <u>(</u> _<u>y</u>_ <u>)</u> _,_ _<u>y</u>_ <u>)</u> = _u_<sup>_′_</sup> ( _za,_ _<u>y</u>_<sup>_~~z~~_</sup> ) for some _u_<sup>_′_</sup> _∈ U_ and we call _z_ <u>(</u> _<u>y</u>_ <u>)</u> _maximal_ . The definition of a maximal line depends on the set of admissible preferences _U_ .

For the three sets of heterogeneous preferences that we consider in Section 5, the maximal line is a societal line. We illustrate this by considering an even simpler set of preferences, which only contains the two utility functions _u_<sup>0</sup> and _u_<sup>_σ_ ̄</sup> . As explained above, for given _<u>y</u>_ ~~,~~ the global line _z_ <u>(</u> _<u>y</u>_ <u>)</u> corresponds to the smallest income for which no individual can be welfare poor. Graphically, this means that the global line is the upper contour of the two indifference curves through the reference bundle respectively associated with _u_<sup>0</sup> and _u_<sup>_σ_ ̄</sup> , which are drawn in Figure 1. For low-income societies, whose median income is _smaller_ than the reference median income _<u>y</u>_<sup>_~~z~~_</sup> , the smallest income at which no one is welfare poor is determined by the preference _least_ sensitive to relative income, that is, the selfcentered preference _u_<sup>0</sup> . Hence, this smallest income corresponds to the subsistence income _za_ . Indeed, an individual with preference _u_<sup>0</sup> and own income just below _za_ who lives in a low-income society is welfare poor. For higher-income societies, whose median income is _larger_ than the reference median income _<u>y</u>_<sup>_~~z~~_</sup> , this smallest

> 13The literature defines a poverty line as _absolute_ when its value _z_ <u>(</u> _<u>y</u>_ <u>)</u> is independent of _<u>y</u>_ and as _relative_ when its value _z_ <u>(</u> _<u>y</u>_ <u>)</u> depends on _<u>y</u>_ <u>.</u>

10
<!-- Start of picture text --> Y<br>Reference | |<br>bundle I, | /<br>. \ evi ett<br>Za See eee<br>evi 2,<br>:||<br>——7]<br><!-- End of picture text -->
# **2.3 Implications for the Poverty Index**

We now turn to the implications that the escaping-poverty property has on the poverty index. These implications are non-trivial in the context of heterogeneous preferences because some individuals are income poor but not welfare poor. The main implication of this property is that the poverty index is _hierarchical_ : it must attribute a larger poverty score to absolutely income poor individuals than to only-relatively income poor individuals.

We begin by examining the implications of the escaping-poverty property for a value of median income _<u>y</u>_ that is fixed and larger than the reference median income _<u>y</u>_<sup>_~~z~~_</sup> . For such _<u>y</u>_ ~~,~~ the poverty line is higher than the subsistence income ( _z_ <u>(</u> _<u>y</u>_ <u>)</u> _> za_ ) and some income poor individuals are only-relatively income poor. The key observation is that some income poor individuals escape welfare poverty when their incomes increase, even if they remain only-relatively income poor. In particular, this may happen when their income surpasses the subsistence income. For instance, in Figure 2, the self-centered individual 1 is absolutely income poor in distribution **y** and thus welfare poor, but she is only-relatively income poor in distribution **y**<sup>_′_</sup> , which has the same median income as **y** , and thus not welfare poor. Hence, the escaping-poverty property is satisfied only if any only-relatively income poor individual is attributed a strictly smaller poverty score than any absolutely income poor individual living in the same society.

The above reasoning shows that the societal headcount ratio, which captures the fraction of individuals who are income poor, violates the escaping-poverty property when preferences are heterogeneous. The societal headcount ratio can be decomposed as: where _HA_ ( **y** ) and _HR_ ( **y** ) respectively denote the fraction of individuals who are absolutely income poor (which we call the absolute headcount ratio) and the fraction of individuals who are only-relatively income poor. (Our notation for specific poverty indices omits their dependence on the set _U_ .) This decomposition reveals that the societal headcount ratio is not reduced when an absolutely income poor individual becomes only-relatively income poor, even when she escapes welfare poverty. Fundamentally, the issue is that the societal headcount ratio _HS_ measures not the fraction of individuals who are welfare poor but rather the fraction of individuals who are income poor.

When considering a fixed _<u>y</u>_ ~~,~~ the escaping-poverty property is satisfied by the 12
_expected_ fraction of individuals who are welfare poor. Assume that income distributions are observable but preferences are not observable. For some belief that an observer may hold with respect to the distribution of preferences, we denote by EH( **y** ) the fraction of individuals that the observer expects to be welfare poor in distribution **y** . For instance, if she believes there is a 50% probability that any individual has preference _u_<sup>0</sup> and a 50% probability that the individual has preference _u_<sup>_σ_ ̄</sup> , then for any distribution **y** with _<u>y</u> >_ _<u>y</u>_<sup>_~~z~~_</sup> , EH corresponds to the index _HS_<sup>1</sup><sup>_/_2</sup> defined as where the poverty score of any absolutely income poor individual is one and that of any only-relatively income poor is one-half. Index _HS_<sup>1</sup><sup>_/_2</sup> is reduced when a selfcentered individual escapes welfare poverty. The reason is that her income must surpass the subsistence income to escape welfare poverty, implying her poverty score is reduced by one half.

Index _HS_<sup>1</sup><sup>_/_2</sup> is arguably too simplistic since it will also violate the escapingpoverty property as soon as some individuals have a sensitivity to relative income that is intermediate between those of _u_<sup>0</sup> and _u_<sup>_σ_ ̄</sup> . Indeed, an individual with preference _u_<sup>_σ/_ ̄2</sup> can have two different welfare poverty statuses for two different bundles that both yield only-relatively income poverty status. For instance, this individual could be welfare poor when consuming the bundle of individual 1 but not welfare poor when consuming the bundle of individual 2 (see Figure 1). She would thus escape welfare poverty when she moves from the former bundle to the latter, even if the index _HS_<sup>1</sup><sup>_/_2</sup> attributes the same poverty score to both.

We therefore favor another index that better reflects more realistic distributions of preferences. We call this second index the _hierarchical headcount ratio_ ( _HHS_ ). Index _HHS_ is again defined as the sum of the fraction of absolutely income poor and the fraction of only-relatively income poor individuals multiplied by some weight smaller than one. The particularity of _HHS_ is that its weight is endogenous to the income distribution. For the societal poverty line _z_ <u>(</u> _<u>y</u>_ <u>),</u> the _hierarchical headcount ratio_ is defined as: 13
where the endogenous weight _ω_ ( **y** ) _∈_ [0 _,_ 1] has the following linear expression: for _<u>y</u> >_ _<u>y</u>_<sup>_~~z~~_</sup> , where _y_ ˆ<sup>_R_</sup> is the average income among only-relatively income poor individuals;<sup>14</sup> _ω_ ( **y** ) := 0 for _<u>y</u> ≤_ _<u>y</u>_<sup>_~~z~~_</sup> . The closer the average income among the only-relatively income poor is to _za_ (respectively _z_ <u>(</u> _<u>y</u>_ <u>)),</u> the closer their weight is to one (respectively zero). Contrasting Equations (3) and (5) reveals that poverty as measured by _HHS_ always lies between the absolute headcount ratio and the societal headcount ratio. More precisely, we have Under some assumption on the distribution of preferences, index _HHS_ corresponds to the expected fraction of individuals who are welfare poor when _<u>y</u> >_ _<u>y</u>_<sup>_~~z~~_</sup> . The probability that an individual who is not income poor is welfare poor is zero, because the poverty line corresponds to the smallest income for which no individual can be welfare poor. The probability that an absolutely income poor individual is welfare poor is one because her own income is smaller than _za_ and her relative income is smaller than _za/y_<sup>_~~z~~_</sup> . The assumption under which index _HHS_ corresponds to EH is that the probability that an only-relatively income poor individual is welfare poor increases _linearly_ between zero and one as her income decreases from the poverty line to the subsistence income. Under this assumption, the probability that any individual _i_ is welfare poor corresponds to the poverty score that _HHS_ attributes to _i_ , which is: As an illustration, we show in online Appendix S2 that, for a certain probability distribution on the class of utility functions _u_<sup>_σ_</sup> , index _HHS_ is equal to EH for all distributions **y** such that _<u>y</u> >_ _<u>y</u>_<sup>_~~z~~_</sup> .<sup>15</sup>

> 14 ˆ _R_ <u>1</u> _y_ := _n_ ( **y** ) _HR_ ( **y** ) <u>�</u> _i∈NR_ ( **y** )<sup>_yi_,where</sup><sup>_NR_(</sup><sup>**y**)isthesetofonly-relativelyincomepoor</sup> individuals.

> 15We also discuss in online Appendix S2 the relationship between our theory and the literature on the fuzzy measurement of poverty.

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We consider now the implication of the escaping-poverty property when the value of median income _<u>y</u>_ varies. The key observation is that self-centered individuals escape welfare poverty when their income surpasses the subsistence income, even if median income is simultaneously increased. For instance, in Figure 2, the self-centered individual 1 is absolutely income poor in distribution **y** , and thus welfare poor. In contrast, individual 1 is only-relatively income poor in distribution **y**<sup>_′′_</sup> , and thus not welfare poor. Hence, the escaping-poverty property is satisfied only if any only-relatively income poor individual is attributed a strictly smaller poverty score than any absolutely income poor individual, even if they live in societies with different values of median income. The view that only-relatively income poor individuals should always be attributed a smaller poverty score than absolutely income poor individuals seems to be largely shared, as survey evidence suggests (Decerf and Ferrando, 2022). We have thus shown that the escaping-poverty property provides a preference-based foundation for this view.

This second implication of the escaping-poverty property is violated by the societal headcount ratio but satisfied by the hierarchical headcount ratio. Index _HS_ violates it because _HS_ attributes a poverty score equal to one to all income poor individuals. Index _HHS_ satisfies it because _HHS_ attributes a poverty score equal to one to absolutely income poor individuals and a smaller poverty score to only-relatively income poor individuals.

We conclude this section with three remarks. First, index _HHS_ need not always satisfy the escaping-poverty property. Indeed, _HHS_ violates that property if some absolutely income poor individuals are not welfare poor.<sup>16</sup> Thus, even if index _HHS_ comes closer than index _HS_ to satisfying the escaping-poverty property, it does not fully do so.<sup>17</sup> Unfortunately, this is the price to pay for considering additive indices that do not treat individuals differently based on their sensitivity to relative income. No index defined by Equation (2) fully satisfies the escapingpoverty property.<sup>18</sup> Importantly, the interpersonal comparisons across different

> 16This could, for instance, be the case for an individual with preference _uσ_  ̄ when the value of median income is smaller than the reference median income _<u>y</u>_<sup>_~~z~~_</sup> . Such individual could escape welfare poverty without changing her absolute income poverty status, while index _HHS_ attributes a poverty score equal to one to all absolutely income poor individuals.

> 17As we show in online Appendix S3, _HHS_ satisfies a weak version of the escaping-poverty property that _HS_ violates.

> 18Even the expected fraction of poor EH( **y** ) violates the second implication of the escapingpoverty property when some absolutely income poor individuals are not welfare poor under some preference _u_<sup>_σ_</sup> . Indeed, one can then find two bundles, ( _y,_ _<u>y</u>_ <u>)</u> and ( _y_<sup>_′_</sup> _,_ _<u>y</u>_<sup>_~~′~~_</sup> ), with _y < za < y_<sup>_′_</sup> and _<u>y</u> <_ _<u>y</u>_<sup>_~~z~~_</sup> _<_ _<u>y</u>_<sup>_~~′~~_</sup> such that a self-centered individual is welfare poor in ( _y,_ _<u>y</u>_ <u>)</u> but not in ( _y_<sup>_′_</sup> _,_ _<u>y</u>_<sup>_~~′~~_</sup> ), whereas an individual with preference _u_<sup>_σ_</sup> is welfare poor in ( _y_<sup>_′_</sup> _,_ _<u>y</u>_<sup>_~~′~~_</sup> ) but not in ( _y,_ _<u>y</u>_ <u>).</u>

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societies made by index _HHS_ are in line with our theory. Our main theoretical results (see Theorems 1 and 2 in Section 5) show that the trade-offs that index _HHS_ makes between own income and relative income correspond to those implied by a fair and welfare-consistent aggregation of heterogeneous preferences.

Second, our results do not require that some individuals be completely selfcentered. It is, in fact, sufficient that some individuals are not affected by relative income when their income is below the subsistence income _za_ . Alternatively, it is also sufficient that the sensitivity to relative income is arbitrarily small.

Third, some characteristics of the hierarchical headcount ratio, in particular its bounds as stated in Equation (6), have an interesting connection with the proposal in Ravallion and Chen (2019) to use lower and upper bounds on the fraction of individuals who are welfare poor. Ravallion and Chen assume the existence of a _common_ welfare function that is unknown. They observe that the exact shape of the global poverty line depends on the specification of this common welfare function, whose exact sensitivity to relative income _σ_ is unknown but lies in an interval [0 _,_ ̄ _σ_ ]. For median income larger than the reference median income _<u>y</u>_<sup>_~~z~~_</sup> , the poverty line must be in an income range bounded below by _za_ and bounded above by the income level associated with the maximal sensitivity to relative income ( ̄ _σ_ ). This implies two bounds on the fraction of individuals who are welfare poor, namely _HA_ and _HS_ . The main conceptual difference with our approach is that we assume preference heterogeneity, whereas they assume a common welfare function that is unknown. As a consequence, the weight _ω_ that should be given to the only-relatively income poor is endogenous in our theory whereas it is unknown in Ravallion and Chen (2019). Another difference is that our framework allows characterization of the trade-offs that the measure should make _below_ the global line (Theorems 1 and 2), which is absent in Ravallion and Chen (2019).

# **3 The Complete Framework**

In Sections 3, 4 and 5, we present the complete theoretical results. The application of the measure developed in Section 2 to global poverty measurement follows in Section 6.

## **3.1 Income Distributions and Preference Profiles**

An income distribution, **y** := ( _y_ 1 _, . . . , yn_ ( **y** )), is a list of non-negative incomes. Let _N_ ( **y** ) := _{_ 1 _, . . . , n_ ( **y** ) _}_ denote the set of individuals in distribution **y** , where 16
_n_ ( **y** ) _∈_ N. Let _<u>y</u>_ denote the median income in distribution **y** . (As explained in online Appendix S5, all of our results still hold when _<u>y</u>_ denotes mean income instead of median income.) When we wish to emphasize the median income in a particular distribution **y** , we write **<u>y</u>** = _<u>y</u>_ ~~.~~ We assume that every individual _i_ has a complete, transitive, and continuous preference that can be represented by the utility function _ui_ ( _yi, yi/y_ <u>).</u> We often drop subscript _i_ when discussing points that are not specific to a given individual, for example, writing _u_ ( _y, y/y_ <u>)</u> instead of _ui_ ( _yi, yi/y_ <u>).</u> We impose two (ordinal) monotonicity assumptions on preferences. First, utility functions are strictly increasing in own income when holding relative income constant; that is, _∂_ 1 _u >_ 0 whenever _u_ is differentiable. It follows that, when own income and the median income are multiplied by a common factor _δ >_ 1, we have _u_ ( _δy, y/y_ <u>)</u> _≥ u_ ( _y, y/y_ <u>),</u> and the inequality is strict when _y >_ 0. Second, utility functions are weakly increasing in relative income when holding own income constant; that is, _∂_ 2 _u ≥_ 0 whenever _u_ is differentiable. (Our results also hold if _u_ is strictly increasing in its second argument.) These two monotonicity assumptions together imply that utility functions are strictly increasing in own income when holding median income constant.

We find it convenient to represent preferences with utility functions, but our theory only relies on ordinal preferences. Hence, our terminology uses utility functions and preferences interchangeably, even if they are different formal objects.

Let _U_<sup>_B_</sup> denote the set of individual utility functions representing preferences that satisfy these basic restrictions. Some of our results are based on narrower sets of preferences. Let _U ⊆ U_<sup>_B_</sup> denote a generic subset of utility functions.

Let **u** := ( _u_ 1 _, . . . , un_ ( **u** )) denote a profile of utility functions (or preferences profile). For a given set _U_ , the domain of distribution-profile pairs ( **y** _,_ **u** ) is _XU_ := � _n∈_ N<sup>_Yn×U n_where</sup><sup>_Yn_:=</sup> � **y** _∈_ R<sup>_n_</sup> +<sup>_~~|~~_</sup> _<u>y</u> ≥ za_ �. The mild restriction _<u>y</u> ≥ za_ is necessary for our results.<sup>19</sup> We respectively denote the set of bundles that an individual can consume and the subset of bundles with income smaller than _za_ by

where the subscript _A_ reflects that the frontier of this subset is the ‘absolute’

> 19Observe that although our theoretical results require that _<u>y</u> ≥ za_ , the poverty indices characterized can readily be applied for countries with _<u>y</u> < za_ . In our sample, there are 11 countries for which _<u>y</u> < za_ in 2015.

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subsistence income _za_ .

# **3.2 Definition of the Poor**

As is standard in poverty measurement (Ravallion, 1998; Alkire and Foster, 2011), the identification of the poor is based on a reference bundle, specified exogenously by a social planner. The reference bundle ( _za,_ _<u>y</u>_<sup>_~~z~~_</sup> ) _∈_ R++ _×_ R++ comprises a subsistence income and the maximal value of median income at which the social planner considers the subsistence income sufficient for social inclusion. It is subject to the restriction that _za/y_<sup>_~~z~~_</sup> _≤_ 1, which is needed for our results. We consider this a weak restriction, as if it were violated then the social planner would consider all individuals socially excluded in the equal distribution ( _za, . . . , za_ ).<sup>20</sup> Furthermore, as is standard in frameworks with heterogeneous preferences (Decancq et al., 2019; Dimri and Maniquet, 2020), an individual is deemed welfare poor if she is worse off than at the reference bundle, that is, if she prefers the reference bundle over the bundle she consumes. We have two remarks with regard to this definition. First, this is the standard ‘welfarist’ definition of poverty, according to which an individual is welfare poor if her well-being is below that associated with some reference bundle (Ravallion, 1998). Second, the reference bundle is the cornerstone of interpersonal comparisons. Indeed, any two individuals consuming the reference bundle are not welfare poor, even when they have different preferences.

The set of individuals who are welfare poor in the distribution-profile pair ( **y** _,_ **u** ) is denoted by _Q_ ( **y** _,_ **u** ) := _{i ∈ N_ ( **y** ) _|ui_ ( _yi,_ _<u>y</u>_ <u>)</u> _< ui_ ( _za,_ _<u>y</u>_<sup>_~~z~~_</sup> ) _}_ . For an individual with utility function _u_ , we denote by

the set of bundles whose consumption leaves her in welfare poverty. The union of these sets of bundles for a given set of preferences _U_ is denoted by _XQ_ ( _U_ ) := � _u∈U_<sup>_XQ_(</sup><sup>_u_).</sup> For some bundles, whether or not an individual is welfare poor does not depend on her preference. For instance, all individuals whose income is smaller than _za_ when median income is larger than _<u>y</u>_<sup>_~~z~~_</sup> are welfare poor. In turn, all individuals whose bundle is not in _XQ_ ( _U_<sup>_B_</sup> ) are not welfare poor, regardless of their preferences.

> 20Observe that the reference bundle ( _za,_ _<u>y</u>_<sup>_~~z~~_</sup> ) = (1 _._ 9 _,_ 1 _._ 8) associated with the societal line of the World Bank slightly violates this restriction. We do not view this as a major issue, though, given that our main objective in the empirical section is to illustrate the impact of replacing the societal headcount ratio with the hierarchical headcount ratio.

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# **3.3 Poverty Indices**

A poverty index is a function, _PU_ : _XU →_ R, that represents a poverty ranking on _XU_ . That is, _PU_ ( **y** _,_ **u** ) _≥ PU_ ( **y**<sup>_′_</sup> _,_ **u**<sup>_′_</sup> ) indicates that there is (weakly) more poverty in ( **y** _,_ **u** ) than in ( **y**<sup>_′_</sup> _,_ **u**<sup>_′_</sup> ). This definition of the poverty index allows cross-country poverty comparisons to be made, since the index is able to compare across different preference profiles.

We restrict our attention to a family of additive indices that sum up individual poverty scores. For our purposes, an individual’s poverty score depends solely on her utility function and her bundle. We allow an individual’s poverty score to depend on her utility function at this point for the sake of generality. However, our results will preclude such dependence.

**Definition 1** (Additive index) **.** _Given any U ⊆ U_<sup>_B_</sup> _, we say that PU_ : _XU →_ R _is an additive index if_

_where for every u ∈ U , the poverty score function pu_ : _X →_ R _is well-defined on X and continuous on XQ_ ( _u_ ) _._ This definition introduces a poverty score function _p_ : _X × U →_ R such that _p_ : ( _y,_ _<u>y, u</u>_ ) _�→ pu_ ( _y,_ _<u>y</u>_ <u>),</u> where, in line with the definition of utility functions, the poverty score of an individual depends on the incomes of other individuals only through median income _<u>y</u>_ ~~.~~ While this is a strong requirement, virtually all applications resort to such additive indices. More fundamentally, Decerf (2021) shows in a similar framework that axioms à la Foster and Shorrocks (1991) similarly imply such an additive expression.

Nevertheless, besides imposing a mild form of continuity, the definition of an additive index places no constraint on the way in which the poverty score function compares different bundles.<sup>21</sup> This is important, because the definition of the poverty score function governs the trade-offs that the measure makes between own income and relative income, and thus also the interpersonal comparisons across societies with different median incomes.

We constrain the poverty score function by imposing two properties that encapsulate fairness and welfare-consistency requirements. First, our

> 21The index “fraction of individuals who are welfare poor” is not ruled out by such mild continuity.

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welfare-consistency requirement adapts the Pareto principle to our poverty measurement setting. In a social welfare measurement setting, the Pareto principle requires that social welfare improves when the utility of all individuals increases. In our setting, _Pareto_ requires that poverty cannot increase when the utility of all individuals increases.<sup>22</sup> Moreover, poverty is strictly reduced when the utility of some welfare poor individual is strictly improved.

**Axiom 1** ( _Pareto_ ) **.** _For all_ ( **y** _,_ **u** ) _,_ ( **y**<sup>_′_</sup> _,_ **u** ) _∈XU such that n_ ( **y** ) = _n_ ( **y**<sup>_′_</sup> ) _, if ui_ ( _yi_<sup>_′,_</sup> _<u>y</u>_<sup>_~~′~~_</sup> ) _≥ ui_ ( _yi,_ _<u>y</u>_ <u>)</u> _for all i ∈ N_ ( **y** ) _, then PU_ ( **y**<sup>_′_</sup> _,_ **u** ) _≤ PU_ ( **y** _,_ **u** ) _. If, in addition, ul_ ( _yl_<sup>_′,_</sup> _<u>y</u>_<sup>_~~′~~_</sup> ) _> ul_ ( _yl,_ _<u>y</u>_ <u>)</u> _for some l ∈ Q_ ( **y** _,_ **u** ) _, then PU_ ( **y**<sup>_′_</sup> _,_ **u** ) _< PU_ ( **y** _,_ **u** ) _._

_Pareto_ is stronger than but encapsulates a version of the escaping-poverty property, which we discussed in Section 2. Indeed, when a welfare poor individual escapes welfare poverty, her utility is strictly increased. In that case, _Pareto_ requires that the index is strictly reduced, at least when no other individual is made worse off.

Second, our fairness requirement adapts the Domination principle to our setting. When measuring social welfare, the Domination principle requires that social welfare is improved when the bundle of each individual is improved according to all relevant preferences.<sup>23</sup> When measuring poverty, we are interested in the trade-offs that welfare poor individuals make. In our setting, _Domination_ requires that if the bundle of each individual is improved according to all utility functions under which they are welfare poor in the final distribution (if any), then poverty cannot increase, regardless of the exact preferences profiles.

**Axiom 2** ( _Domination_ ) **.** _For all_ ( **y** _,_ **u** ) _,_ ( **y**<sup>_′_</sup> _,_ **u**<sup>_′_</sup> ) _∈XU such that n_ ( **y** ) = _n_ ( **y**<sup>_′_</sup> ) _, if u_ ( _yi_<sup>_′,_</sup> _<u>y</u>_<sup>_~~′~~_</sup> ) _≥ u_ ( _yi,_ _<u>y</u>_ <u>)</u> _for all i ∈ N_ ( **y** ) _and all u ∈ U such that_ ( _yi_<sup>_′,_</sup> _<u>y</u>_<sup>_~~′~~_</sup> ) _∈ XQ_ ( _u_ ) _, then PU_ ( **y**<sup>_′_</sup> _,_ **u**<sup>_′_</sup> ) _≤ PU_ ( **y** _,_ **u** ) _._

In contrast to _Pareto_ , _Domination_ does not hold the preference profile fixed. _Domination_ is a strong axiom that, we will show, implies that two different individuals with the same own income in the same society must be attributed the same poverty score, even if they have different sensitivities to relative income (Proposition 1). _Domination_ thus prevents an individual who is particularly sensitive to relative income being attributed a _larger_ poverty score than another individual who lives in the same society and has a _smaller_ income.<sup>24</sup> 22It would also be natural to require that poverty cannot increase when the utility of all _welfare poor_ individuals increases. This stronger version of Pareto would only exacerbate the clash with the fairness property.

> 23The Domination principle was originally called the Intersection principle in Sen (1985).

> 24We emphasize that the implications of _Domination_ are satisfied by all the standard poverty

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# **4.1 Fair Additive Poverty Indices**

In Proposition 1 we show that any additive index satisfying _Domination_ must be a _fair additive index_ .

The main characteristic of a fair additive index is that the poverty score of an individual does not depend on her preference. As a result, fair additive indices do not make the counter-intuitive interpersonal comparisons discussed in the Introduction. Also, a fair additive index is based on a global poverty line that cannot admit an individual’s preference among its arguments. Mathematically, the value of the global line is given by a function _z_ : [ _za, ∞_ ) _→_ R+ whose sole argument is the median income _<u>y</u>_ ~~.~~ We sometimes call function _z_ the global line, even if the global line _z_ <u>(</u> _<u>y</u>_ <u>)</u> is formally a different object than function _z_ . Function _z_ partitions the set of bundles between those below the global line, and those on and above the global line, _X\Xz_ . The poverty score of bundles below the global line is strictly positive and the poverty score of bundles above the global line is zero.<sup>26</sup>

**Definition 2** (Fair additive index) **.** _Given any U ⊆ U_<sup>_B_</sup> _, we say that PU_ : _XU →_ R _is a fair additive index if_

_for some k_<sup>ˆ</sup> _∈_ R _where the (degenerate)_<sup>27</sup> _poverty score function p_ : _X →_ R _is such that, for some continuous function z_ : [ _za, ∞_ ) _→_ R+ _such that Xz ⊆ XQ_ ( _U_ ) _, we have:_ ( _i_ ) _p_ ( _y,_ _<u>y</u>_ <u>)</u> = 0 _on X\Xz,_ ( _ii_ ) _p_ ( _y,_ _<u>y</u>_ <u>)</u> _>_ 0 _on Xz,_ ( _iii_ ) _p is continuous on Xz,_ ( _iv_ ) _p is weakly decreasing in its first argument on Xz, and_ ( _v_ ) _p is weakly increasing in its second argument on Xz._

**Proposition 1.** _Given any U ⊆ U_<sup>_B_</sup> _, an additive index PU satisfies Domination only if PU is a fair additive index._

_Proof._ See online Appendix S6.

> 26The parameter _k_ ˆ in Eq. (8) permits the anchoring to zero without any loss of generality.

> 27We call _p_ a degenerate poverty score function because this object is formally distinct from a poverty score function _pu_ (see Definition 1).

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Proposition 1 may not be surprising, but it has fundamental implications. _Domination_ implies that the measure is based on some _preference-independent_ global line, which is the frontier of an _objectively_ defined income poverty status. This is in contrast to the welfare poverty status, which has an element of subjectivity because it depends on the individual’s preference.

**Definition 3** (Preference-independent global poverty line and income poverty status) **.** _Given any U ⊆ U_<sup>_B_</sup> _, we say that the additive index PU is based on a preference-independent global poverty line if there is a function z_ : [ _za, ∞_ ) _→_ R+ _such that for all u ∈ U , we have pu_ ( _y,_ _<u>y</u>_ <u>)</u> _>_ 0 _for all_ ( _y,_ _<u>y</u>_ <u>)</u> _∈ Xz and pu_ ( _y,_ _<u>y</u>_ <u>)</u> = 0 _for all_ ( _y,_ _<u>y</u>_ <u>)</u> _∈ X\Xz. If this is the case, we say that any individual with bundle_ ( _y,_ _<u>y</u>_ <u>)</u> _is income poor if_ ( _y,_ _<u>y</u>_ <u>)</u> _∈ Xz, absolutely income poor if_ ( _y,_ _<u>y</u>_ <u>)</u> _∈ Xz ∩ XA, and only-relatively income poor if_ ( _y,_ _<u>y</u>_ <u>)</u> _∈ Xz\XA._

We emphasize that we do not assume that the global line is preference-independent, but this is rather a characteristic of indices satisfying _Domination_ . For instance, the additive index “fraction of individuals who are welfare poor” is not based on a preference-independent global line. Indeed, as soon as preferences are heterogeneous, this index requires a non-degenerate poverty score function; that is, the poverty score of an individual must depend on her preference.

The definition of a fair additive index does not constrain the trade-offs that the (degenerate) poverty score function makes between own income and relative income. In the remainder of our theory, we investigate how welfare-consistency requirements tie these trade-offs to those embedded in individual preferences.

# **4.2 Benchmark: Homogeneous Preferences**

As a benchmark, we consider the case for which all individuals have the same preference. It is well-known that, when measuring social welfare, the Pareto principle is compatible with the Domination principle when preferences are homogeneous (Fleurbaey and Trannoy, 2003). This is also the case in our setting where we measure poverty. Proposition 2 shows that any additive index satisfying the two properties of fairness and welfare-consistency is such that its poverty score function is a (negative) representation of the common preference.

**Proposition 2.** _Given any {u} ⊂ U_<sup>_B_</sup> _, the additive index P{u} satisfies Domination and Pareto if and only if P{u} is a fair additive index with a global_

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_line z such that Xz_ = _XQ_ ( _{u}_ ) _and a poverty score function p such that for all_ ( _y,_ _<u>y</u>_ <u>)</u> _,_ ( _y_<sup>_′_</sup> _,_ _<u>y</u>_<sup>_~~′~~_</sup> ) _∈ Xz,_ _Proof._ See online Appendix S7.

■ We emphasize two implications of Proposition 2 for the properties of the global poverty line. First, the global line corresponds to the indifference curve passing through the reference bundle (see Figure 4). That is, the global line yields, for each individual (separately), the same utility in all countries.<sup>28</sup> Second, the global line yields a _perfect identification_ of the welfare poor; that is, all individuals below the global line are welfare poor and all individuals above the global line are not welfare poor. This is because all individuals hold the common utility function.

The trade-offs that the measure makes between own income and relative income can be graphically illustrated by means of its iso-poverty-score map (see Figure 4). An iso-poverty-score map is a collection of iso-poverty-score curves, which are sets of bundles that yield a constant poverty score. The global line is one iso-poverty-score curve, or, more precisely, it is the frontier of a ‘thick’ iso-poverty-score curve. Below the global line, iso-poverty-score curves exactly correspond to indifference curves. The two properties (fairness and welfare-consistency) completely characterize the trade-offs made by the measure; that is, the comparison of any two bundles is determined.

# **4.3 General Impossibility and Weak Pareto**

In many settings with heterogeneous preferences, the Pareto principle conflicts with the Domination principle (Fleurbaey and Trannoy, 2003; Brun and Tungodden, 2004). This conflict also arises in our setting, as we show in Proposition 3. Intuitively, the reason is that the Pareto principle requires that the poverty score function (negatively) represents individual preferences, while the Domination principle requires that the poverty score function does not depend on preferences.

We say that the set _U_ is _heterogeneous_ if there exist two _u, u_<sup>_′_</sup> _∈ U_ and some ( _y,_ _<u>y</u>_ <u>)</u> _∈ X_ with _<u>y</u> ≥_ _<u>y</u>_<sup>_~~z~~_</sup> such that ( _y,_ _<u>y</u>_ <u>)</u> _∈ XQ_ ( _u_ ) and ( _y,_ _<u>y</u>_ <u>)</u> _∈/ XQ_ ( _u_<sup>_′_</sup> ).<sup>29</sup>

> 28Our assumption of homogeneous ordinal preferences does not imply interpersonal levelcomparability of utility.

> 29Our definition of a heterogeneous set _U_ is not the complement of that of an homogeneous set _{u}_ . For instance, set _{u, u_<sup>_′_</sup> _}_ with _u_ = _u_<sup>_′_</sup> does not meet our definition of an heterogeneous set

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income is reduced, at least one individual is made worse off, because her own income is decreased, while her relative income does not increase.

**Lemma 1.** _Given any U ⊆ U_<sup>_B_</sup> _, for all_ ( **y** _,_ **u** ) _,_ ( **y**<sup>_′_</sup> _,_ **u** ) _∈XU with N_ ( **y** ) = _N_ ( **y**<sup>_′_</sup> ) _and_ _<u>y</u> <_ _<u>y</u>_<sup>_~~′~~_</sup> _, there exists j ∈ N_ ( **y** ) _for whom uj_ ( _yj_<sup>_′,_</sup> _<u>y</u>_<sup>_~~′~~_</sup> ) _> uj_ ( _yj,_ _<u>y</u>_ <u>)</u> _. Proof._ See Appendix A3. ■

The incompatibility presented in Proposition 3 arises because all individuals may prefer a distribution that has more individuals below the global line. To see this, assume that some distribution **y** has no individual below the global line _z_ <u>(</u> _<u>y</u>_ <u>).</u> There may exist another unanimously preferred distribution **y**<sup>_′_</sup> in which one individual is below the global line _z_ <u>(</u> _<u>y</u>_<sup>_~~′~~_</sup> ). For instance, this may happen when the global line _z_ <u>(</u> _<u>y</u>_<sup>_~~′~~_</sup> ) is higher than _z_ <u>(</u> _<u>y</u>_ <u>),</u> because median income is higher in **y**<sup>_′_</sup> than in **y** . In that case, a self-centered individual _j_ who is not welfare poor may prefer her bundle ( _yj_<sup>_′,_</sup> _<u>y</u>_<sup>_~~′~~_</sup> ) below the global line _z_ <u>(</u> _<u>y</u>_<sup>_~~′~~_</sup> ) over her bundle ( _yj,_ _<u>y</u>_ <u>)</u> above the global line _z_ <u>(</u> _<u>y</u>_ <u>).</u> _Pareto_ requires that the poverty index cannot be larger for the unanimously preferred distribution **y**<sup>_′_</sup> , even if _j_ is income poor in **y**<sup>_′_</sup> but not in **y** . In contrast, _Domination_ requires that the measure be based on a fair additive index, which attributes a poverty score equal to zero to individuals who are not income poor and a positive poverty score to individuals who are income poor. Therefore, the poverty index must be strictly larger for distribution **y**<sup>_′_</sup> . Thus, the incompatibility arises because _Pareto_ requires that the poverty index is reduced _even when_ the unanimously preferred situation features more individuals below the global line.

In order to escape the impossibility, we consider a weak version of _Pareto_ that remains silent when the number of income poor individuals is increased. To do this, we need to add a precondition requiring that individuals who are not welfare poor in the initial distribution do not fall below the global line in the final distribution. Of course, the global line is not a primitive of our framework and we can thus not express this precondition by referring to the global line. Fortunately, we can rely on the fact that individuals who are not income poor cannot fall below the global line when their income grows in the same proportion as the median income. Formally, _Weak Pareto_ is a weakening of _Pareto_ that adds a precondition for individuals who are not welfare poor in the initial distribution. Their rate of income growth is required to be non smaller than the rate of growth of the median income.<sup>31</sup>

> 31If this weakening is deemed too favourable to individuals who are not welfare poor, one

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**Axiom 3** ( _Weak Pareto_ ) **.** _For all_ ( **y** _,_ **u** ) _,_ ( **y**<sup>_′_</sup> _,_ **u** ) _∈XU such that n_ ( **y** ) = _n_ ( **y**<sup>_′_</sup> ) _,_ _<u>y</u>_<sup>_~~′~~_</sup> _if ui_ ( _yi_<sup>_′,_</sup> _<u>y</u>_<sup>_~~′~~_</sup> ) _≥ ui_ ( _yi,_ _<u>y</u>_ <u>)</u> _for all i ∈ N_ ( **y** ) _and yj_<sup>_′≥_</sup> _<u>y</u>_<sup>_yjforallj∈/Q_(</sup><sup>**y**</sup><sup>_,_</sup><sup>**u**)</sup><sup>_,then_</sup> _PU_ ( **y**<sup>_′_</sup> _,_ **u** ) _≤ PU_ ( **y** _,_ **u** ) _. If, in addition, ul_ ( _yl_<sup>_′,_</sup> _<u>y</u>_<sup>_~~′~~_</sup> ) _> ul_ ( _yl,_ _<u>y</u>_ <u>)</u> _for some l ∈ Q_ ( **y** _,_ **u** ) _, then PU_ ( **y**<sup>_′_</sup> _,_ **u** ) _< PU_ ( **y** _,_ **u** ) _._

Admittedly, _Weak Pareto_ is not a fully satisfactory weakening because this axiom remains silent on _some_ pairs for which one distribution is unanimously preferred and does not have more individuals who are income poor than the other distribution.<sup>32</sup> In spite of this shortcoming, we deem _Weak Pareto_ fit for our purpose. First, _Weak Pareto_ is logically stronger than the welfare-consistency requirement used by Ravallion and Chen (2011). These authors note that the poverty index must be reduced when all incomes grow by the same proportion. Indeed, equi-proportionate growth makes every individual strictly better off because own income increases, while relative income is unchanged. They encapsulate this requirement in a weak relativity axiom (WRA). As we show in online Appendix S8, the WRA is itself a weakening of _Weak Pareto_ . As a result, the WRA also remains silent on the pairs for which _Weak Pareto_ remains silent. Second, _Weak Pareto_ is compatible with _Domination_ and it retains enough ‘bite’ in order to fully characterize the trade-offs made by the measure.

# **5 Results under Heterogeneous Preferences**

## **5.1 The Case for Hierarchical Indices**

In this section, we show that when _some_ utility function does not depend on relative income _below the subsistence income_ , the poverty index must be hierarchical. That is, the index must attribute a larger poverty score to individuals who are absolutely income poor than to individuals who are only-relatively income poor. Formally, we denote by _U_<sup>_∗_</sup> _⊂ U_<sup>_B_</sup> the subset of utility functions that are independent of relative income below the subsistence

> could restrict the additional precondition to the subset of individuals among them whose income is no greater than the median income in the final distribution, that is, for all _j ∈/ Q_ ( **y** _,_ **u** ) with _yj ≤_ _<u>y</u>_<sup>_~~′~~_</sup> . Our results would be unchanged.

> 32Indeed, _Weak Pareto_ remains silent as soon as the unanimously preferred distribution has a individual who is not welfare poor, not income poor and whose income does not grow as fast as the median income. Although _Weak Pareto_ remains silent for some pairs on which the poverty comparison should be unambiguous, the poverty measures that _Weak Pareto_ and _Domination_ characterize do make the affirmative comparisons that we would like to impose for such pairs.

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income; that is, for all _u ∈ U_<sup>_∗_</sup> , we have _u_ ( _y,_ _<u>y</u>_ <u>)</u> = _u_ ( _y,_ _<u>y</u>_<sup>_~~′~~_</sup> ) for all _y ∈_ [0 _, za_ ) and _<u>y, y</u>_<sup>_~~′~~_</sup> _≥_ 0. For instance, self-centered preferences belong to _U_<sup>_∗_</sup> .

We consider the subsets of utility functions _U_ that contain at least one member of _U_<sup>_∗_</sup> . Those subsets _U_ are such that _XA ⊆ XQ_ ( _U_ ), because all individuals with a utility function in _U_<sup>_∗_</sup> prefer the reference bundle over any bundle with income below the subsistence income. This implies that the global line is never smaller than the subsistence income.

On these subsets, Proposition 4 shows that indices satisfying _Domination_ and _Weak Pareto_ must belong to the family of ‘hierarchical’ indices (Decerf, 2021).

**Definition 4** (Hierarchical poverty index) **.** _Given any U ⊆ U_<sup>_B_</sup> _, we say that the additive index PU_ : _XU →_ R _is a hierarchical poverty index if PU is a fair additive index with global line z such that Xz_ = _XQ_ ( _U_ ) _and for which_ ( _i_ ) _p is strictly decreasing in its first argument on Xz and_ ( _ii_ ) _p is constant in its second argument on XA ∩ Xz._

Hierarchical indices are based on a ‘maximal’ global line, defined by _Xz_ = _XQ_ ( _U_ ). This means that a bundle provides income poverty status when there is _some_ preference in _U_ for which the reference bundle is preferred to that bundle.

The trade-offs between own income and relative income associated with hierarchical indices are illustrated in Figure 5.a. Crucially, all iso-poverty-score curves below the subsistence income are flat lines. As a result, hierarchical indices systematically attribute a larger poverty score to an individual who is absolutely income poor than to an individual who is only-relatively income poor, regardless of the median income in their respective societies. Moreover, when comparing two absolutely income poor individuals, hierarchical indices attribute a larger poverty score to the one who earns the smaller income. The indices standardly used in the global poverty literature are not hierarchical (Decerf, 2017).

**Proposition 4.** _Given any U ⊆ U_<sup>_B_</sup> _with U ∩ U_<sup>_∗_</sup> = _∅, the additive index PU satisfies Domination and Weak Pareto only if PU is a hierarchical index. Proof._ See Appendix A4. ■

■ The intuitive explanation for Proposition 4 is as follows. _Weak Pareto_ requires that some unanimous improvements (weakly) reduce poverty. If, in addition, the utility of a welfare poor individual increases, poverty must strictly decrease. For simplicity, assume that there is only one welfare poor individual and that 28
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that are flat above _za_ , these individuals need not be welfare poor. For instance, self-centered individuals are not welfare poor above _za_ .

# **5.2 Characterization on Set** _U_<sup>_σ_ ̄</sup>

We characterize the trade-offs made by poverty indices satisfying _Domination_ and _Weak Pareto_ under a particular subset of _U_<sup>_B_</sup> . Let _U_<sup>_σ_ ̄</sup> be the subset of utility functions _u_<sup>_σ_</sup> defined by Equation (1) for which 0 _≤ σ < σ_ ̄ for some _σ_ ̄ _>_ 0. _U_<sup>_σ_ ̄</sup> is much smaller than _U_<sup>_B_</sup> , but _U_<sup>_σ_ ̄</sup> features an upper bound on the sensitivity to relative income _σ_ .

Theorem 1 shows that these measures have two key properties. First, the global line is a societal line defined as the upper contour of the subsistence income _za_ and a _weakly_ relative line (Ravallion and Chen, 2011) passing through the reference bundle. This global line is illustrated in Figure 5.b. On the heterogeneous set _U_<sup>_σ_ ̄</sup> , the global line yields an _imperfect identification_ of the welfare poor. Although all individuals who are not income poor are not welfare poor, some income poor individuals are _not_ welfare poor.

Second, the trade-offs that the indices must make between own income and relative income are completely characterized. These trade-offs are graphically illustrated in Figure 5.b, which shows their common iso-poverty-score map.<sup>35</sup> Below the subsistence income, the poverty score function does not depend on relative income, and thus iso-poverty-score curves are flat lines. Above the subsistence income, the poverty score function does depend on relative income. The iso-poverty-score curves are straight lines pointing to the reference bundle.

**Theorem 1.** _The additive index PU_  ̄ _σ satisfies Domination and Weak Pareto if and only if PU_  ̄ _σ is a hierarchical index with global line z_<sup>_∗_</sup> _defined as_

_for all_ _<u>y</u> ≥ za, where R_ := 1+ ̄ _<u>zσay</u>_<sup>_~~<u>z</u>~~,andwhosepovertyscorefunctionpissuchthat_</sup> _for all_ ( _y,_ _<u>y</u>_ <u>)</u> _,_ ( _y_<sup>_′_</sup> _,_ _<u>y</u>_<sup>_~~′~~_</sup> ) _∈ Xz∗\XA_<sup>36</sup>

> 35The indices characterized in Theorem 1 are non-continuous at the reference bundle ( _za,_ _<u>y</u>_<sup>_~~z~~_</sup> ). Recall that the poverty score function of additive indices is required to be continuous only on _XQ_ ( _u_ ), which never includes ( _za,_ _<u>y</u>_<sup>_~~z~~_</sup> ). If the poverty score function is required to be continuous on the whole domain _X_ , then the two axioms are incompatible.

> 36A hierarchical index has for all ( _y,_ _<u>y</u>_ <u>)</u> _,_ ( _y_<sup>_′_</sup> _,_ _<u>y</u>_<sup>_~~′~~_</sup> ) _∈ XA_ that _p_ ( _y,_ _<u>y</u>_ <u>) =</u> _p_ ( _y_<sup>_′_</sup> _,_ _<u>y</u>_<sup>_~~′~~_</sup> ) when _y_ = _y_<sup>_′_</sup> .

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_Proof._ See Appendix A5.

■ We emphasize two interesting features of Theorem 1 that relate to the global line. First, our axioms imply the use of a _strongly_ relative global line only when the sensitivity to relative income has no upper bound. Indeed, when _σ_ ̄ _→∞_ , we have _R →_ 0 and _Rσ_ ̄ _→ za/y_<sup>_~~z~~_</sup> , and thus the relative part of _z_<sup>_∗_</sup> tends to a strongly relative line, which points to the origin. Hence, on _U_<sup>_σ_ ̄</sup> , the global line is weakly relative when this sensitivity is bounded above ( _σ_ ̄ _∈_ R++), but strongly relative when this sensitivity has no upper bound ( _σ_ ̄ _→∞_ ). Second, under heterogeneous preferences, the global line need not correspond to an indifference curve. Indeed, there is no preference in _U_<sup>_σ_ ̄</sup> that has an indifference curve corresponding to the global line _z_<sup>_∗_</sup> . This means that there is no individual for whom the global line provides the same utility in (all) different societies. Instead, the global line defines an ‘objective’ income poverty status, which is attached to any individual whose income is smaller than the global line.

Observe that the trade-offs between own income and relative income made by index _HHS_ under the global line _z_<sup>_∗_</sup> correspond to those characterized in Theorem 1. Above the subsistence income, the poverty score function of index _HHS_ has the same iso-poverty-score curves.<sup>37</sup> Below the subsistence income, the poverty score function of index _HHS_ does not depend on relative income.<sup>38</sup> Interestingly, Decerf (2017) proposes a family of hierarchical indices whose isopoverty-score curves satisfy the constraints derived in Theorem 1. For a given global line _z_ , index _P_<sup>_H_</sup> is defined by the following poverty score function: where _α >_ 0 and _λ ∈_ (0 _,_ 1). The hierarchical headcount ratio corresponds to the index that would be obtained when setting _α_ = 1 and _λ_ = 0, the latter value being on the frontier of acceptable values.

> 37On _Xz∗ \XA_ , we have that _<u>y</u>_<sup>_<u>y</u>_</sup> _<u>−</u>_<sup>_−_</sup> _<u>y</u>_<sup>_z_</sup> _<u>a</u>_<sup>_~~<u>z</u>~~_=</sup> _<u>y</u>_<sup>_<u>y</u>_</sup><sup>_~~<u>′</u>~~′_</sup> _<u>−</u>_<sup>_−_</sup> _<u>y</u>_<sup>_za_</sup><sup>_~~<u>z</u>~~⇔_</sup> _zz_<sup>_∗∗_</sup> (( _<u>yy</u>_ <u>))</u> _−−zya_<sup>=</sup> _z_<sup>_z∗∗_</sup> (<sup><u>(</u></sup> _<u>yy</u>_<sup>_~~<u>′′</u>~~_</sup> )) _−−zya_<sup>_′_.</sup>

> 38Below _za_ , _HHS_ attributes a poverty score equal to one regardless of own income, which is thus not strictly decreasing in own income as required by Theorem 1.

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# **5.3 Robustness**

The trade-offs characterized in Theorem 1 are valid under subset _U_<sup>_σ_ ̄</sup> . In general, the trade-offs characterized by _Domination_ and _Weak Pareto_ may depend on the set _U_ considered. We show that some of their key characteristics are preserved on larger sets of preferences. For robustness, we consider the whole set _U_<sup>_B_</sup> .

**Theorem 2.** _The additive index PU B satisfies Domination and Weak Pareto if and only if PU B is a hierarchical index with global line z_<sup>_∗∗_</sup> _defined as_

_for all_ _<u>y</u> ≥ za, whose poverty score function p is such that for all_ ( _y,_ _<u>y</u>_ <u>)</u> _,_ ( _y_<sup>_′_</sup> _,_ _<u>y</u>_<sup>_~~′~~_</sup> ) _∈ Xz∗∗\XA,_ _Proof._ See online Appendix S9.

Interestingly, Theorem 2 shows that the iso-poverty-score curves and the relative part of the global line must be straight lines even when the set _U_<sup>_B_</sup> contains preferences whose indifference curves are _not_ straight lines.<sup>39</sup> There are two key differences between Theorems 1 and 2. First, the global line of the latter is _z_<sup>_∗∗_</sup> instead of _z_<sup>_∗_</sup> . This reflects the fact that there is no upper bound on the sensitivity to relative income in _U_<sup>_B_</sup> . Second, below the global line, the poverty score function in Theorem 2 only depends on own income. This does not mean that relative income plays no role, but its role is limited to the definition of the global line. This second difference reflects the fact that _U_<sup>_B_</sup> contains utility functions that exhibit extreme forms of concavities. This difference disappears on the subset _U_<sup>_C_</sup> of utility functions in _U_<sup>_B_</sup> whose indifference curves are weakly convex in the space of bundles. On _U_<sup>_C_</sup> , the iso-poverty-score map characterized by _Domination_ and _Weak Pareto_ corresponds to that illustrated in Figure 5.b, except that the global line is then _z_<sup>_∗∗_</sup> because no upper bound is placed on the sensitivity to relative income in _U_<sup>_C_</sup> .<sup>40</sup>

> 39For the relative part of the global line, the explanation follows from the fact that the global line is maximal. When _<u>y</u> ≥_ _<u>y</u>_<sup>_~~z~~_</sup> , the shape of the global line is defined by the utility function most sensitive to relative income. On _U_<sup>_B_</sup> , the strict monotonicity to own income when holding relative income constant constrains the sensitivity to relative income. Graphically, the slope of the indifference curve passing through a bundle cannot exceed the slope of the ray passing through this bundle and the origin.

> 40Details can be found in online Appendix S10.

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We emphasize that our results also hold when assuming that utility is _strictly_ increasing in relative income, instead of _weakly_ increasing. Indeed, Theorem 1 would be unchanged if the subset _U_<sup>_σ_ ̄</sup> excludes _u_<sup>0</sup> .<sup>41</sup>

# **6 Empirical Application**

In this section we apply our proposed index, the hierarchical headcount ratio ( _HHS_ ), to measure global poverty. The goal of this exercise is to illustrate how the switch of index can affect the assessment of poverty. We compare the evaluation of poverty according to the hierarchical headcount ratio both with the societal headcount ratio ( _HS_ ) using the same global poverty line, and with the absolute headcount ratio ( _HA_ ), arguably the most well-known poverty measure. We compare poverty at the country level as well as the evolution and distribution of global poverty. Besides the theoretical arguments for moving away from the societal headcount ratio, ultimately, such a change of index would be justified only if it significantly affects the measurement of global poverty.

## **6.1 Data and Parameters**

Our source of data is PovcalNet,<sup>42</sup> the on-line tool for poverty measurement developed by the World Bank, which offers income or consumption data from more than 1500 household surveys across more than 160 countries. In order to allow for cross-country comparisons, the World Bank converts income or consumption using the 2011 PPP exchange rates for household consumption from the International Comparison Program. Moreover, for the purpose of performing multi-country aggregations the World Bank defines reference years aligning survey estimates from different years. To analyze poverty at the global or regional level, we only use reference years.<sup>43</sup> We use data from 1999 up to 2015, the most recent reference year available. We take 1999 as our base year because before that, we have _HHS_ = _HS_ in several populous countries that still had median income below _<u>y</u>_<sup>_~~z~~_</sup> . Our dataset includes 168 countries.<sup>44</sup>

> 41Details can be found in online Appendix S11.

> 42 PovcalNet can be found at: http://iresearch.worldbank.org/PovcalNet/povOnDemand.aspx. The data was retrieved in April 2021. All data is in per-capita terms.

> 43The reference years available from 1999 on are: 1999, 2002, 2005, 2008, 2010-2013 and 2015.

> 44China, India, and Indonesia compile data for rural and urban areas separately. Thus, we have 171 units in total. A few countries do not have data for all reference years. These countries are: Kosovo (missing data in 1999), Nauru (missing data before 2005), Somalia (missing data before 2011), South Sudan (missing data before 2008), Timor-Leste (missing data in 1999), and

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To estimate the hierarchical headcount ratio, we need to choose a societal poverty line. In our theoretical framework, two exogenous elements determine the societal poverty line: the reference bundle ( _za,_ _<u>y</u>_<sup>_~~z~~_</sup> ) and the set of admissible (ordinal) utility functions _U_ representing preferences over own and relative income. The specification of the reference bundle is clearly a normative choice, while the identification of _U_ is more subtle. It could be taken as the set of all preferences actually held by some individual, which is, in principle, empirically observable. Under this approach, identification of _U_ is a positive exercise. It could, alternatively, be taken as another normative choice to be made by the social planner. The social planner might make some normative evaluation of ‘reasonableness’ of preferences and only include in _U_ preferences that she considers to be ‘reasonable’. The choice of approach is of course, itself, a normative choice.

For the purposes of our empirical analysis, we infer both the reference bundle ( _za,_ _<u>y</u>_<sup>_~~z~~_</sup> ) and the set of admissible utility functions _U_ from the societal poverty line currently used by the World Bank to assess income poverty, combining its absolute and relative aspects. In doing so, we elucidate the normative choices implicit in the World Bank approach from the perspective of our theoretical framework. This has the additional advantage that the societal headcount ratio ( _HS_ ) corresponds exactly to the World Bank’s official societal poverty measure, while the absolute headcount ratio ( _HA_ ) corresponds to its official extreme poverty measure, thus facilitating comparison of existing global poverty assessments with those of our proposed measure.

The World Bank’s societal poverty line, _max_ ($1 _._ 90 _,_ $1 _._ 00 + 0 _._ <u>5</u> _<u>y</u>_ <u>)</u> where _<u>y</u>_ is country median income, is the upper contour of an absolute poverty line and a weakly relative line (World Bank, 2018). The absolute line of $1 _._ 90 per person per day, in 2011 PPP, has been the official World Bank extreme poverty line since 2015 (Ferreira et al., 2016). The weakly relative line $1 _._ 00 + 0 _._ <u>5</u> _<u>y</u>_ was estimated from regressions of 699 (national) poverty lines against median income (Jolliffe and Prydz, 2021).

The components of the reference bundle are thus straightforwardly inferred from the intersection of the absolute and weakly relative lines as subsistence income _za_ = $1 _._ 90 and reference median income _<u>y</u>_<sup>_~~z~~_</sup> = $1 _._ 80, the latter being the maximal median income at which the subsistence income is considered sufficient for social inclusion. We observe with interest that the reference median income is close to, but slightly less than, the subsistence income. It follows that an individual whose Venezuela (missing data in 2015). Our results are robust to excluding this set of countries.

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income is _za_ , who lives in a country where the median income is also equal to _za_ , is not considered to have sufficient income for social inclusion. Meanwhile, the set of admissible preferences _U_ corresponds to members of the parametric family specified by Equation (1) with parameter values _σ_ in the interval [0 _,_ 0 _._ 5]. The extent to which this set aligns with the actual heterogeneity of preferences in the global population remains an open empirical question, beyond the scope of the present study.

# **6.2 Empirical Values for the Weight of** _HHS_

We first provide some empirical insights on the hierarchical headcount ratio ( _HHS_ ). As seen in Equation (5), the hierarchical headcount ratio can be decomposed as the sum of the fraction of individuals who are absolutely income poor ( _HA_ ) and the fraction of individuals who are only-relatively income poor ( _HR_ ) multiplied by a weight _ω_ ( **y** ) _∈_ [0 _,_ 1].

Table 1 displays the median weight by country income group as defined by the World Bank. The results highlight two main points. First, we observe that for low- and middle-income countries, the median weight is close to 0.5 and there is little variation within each group, especially in the low-income and lower-middleincome groups. This means that the median poverty score of the only-relatively income poor individuals in low- and middle-income countries is approximately 0.5. More specifically, we observe that more than 60% of countries in our sample have a median endogenous weight between 0.4 and 0.55. Except in two cases, all these are low- or middle-income countries (see also Figure S.1 in online Appendix S4). This implies that for many countries in our sample, _HHS_ and the simpler index _HS_<sup>1</sup><sup>_/_2</sup> would yield similar poverty evaluations. Recall that _HS_<sup>1</sup><sup>_/_2</sup> is defined similarly to _HHS_ but with a fixed weight of 0.5 (see Section 2).

The fact that _ω_ ( **y** ) is close to 0.5 for many low- and middle-income countries can be explained intuitively. In those countries, the relative line is close to the absolute line. As a result, the density function of the income distribution is close to being constant in the small interval between lines. In other words, the distribution of individual incomes between _za_ and _z_ <u>(</u> _<u>y</u>_ <u>)</u> is close to being uniform. In such a ˆ _<u>y</u>_ <u>)</u> case, the average income for the only-relatively income poor is _y_<sup>_R_</sup> _≈_<sup>_za_</sup><sup><u>+</u></sup> 2<sup>_z_</sup><sup><u>(</u></sup> , which yields _ω_ ( **y** ) _≈_ 0 _._ 5 when _za ≈ z_ <u>(</u> _<u>y</u>_ <u>).</u> Second, we observe that the weight decreases as income grows along country groups. Indeed, the median weight among high-income countries is 0.31, considerably smaller than in the other groups. This implies that the incomes of 35
the only-relatively income poor in richer countries are, on average, proportionally closer to the relative line than to the absolute line than the incomes of the only-relatively income poor in low-income countries.

The results summarized in Table 1 help us to understand why, when we compute global poverty (see Section 6.4), _HHS_ is about halfway between _HS_ and _HA_ . The reason is that income poverty is concentrated in highly populated developing countries such as India and Indonesia, where the endogenous weight is close to 0.5.

**Table 1:** Endogenous weight _ω_ by country income group

|Country income group|Median|SD|
|---|---|---|
|Low income|0.49|0.06|
|Lower-middle income|0.48|0.05|
|Upper-middle income|0.42|0.08|
|High income|0.31|0.06|

_Note:_ Country income groups as defined by the World Bank.

# **6.3 Country-Level Poverty**

We begin by comparing poverty at the country level between the societal headcount ratio ( _HS_ ), the absolute headcount ratio ( _HA_ ), and our proposed index ( _HHS_ ).

Our first example illustrates how the change of index that we propose can affect the poverty diagnoses of individual countries. Table 2 displays the level of poverty as well as the rankings by those three poverty measures for Botswana and Egypt in 2008. According to _HS_ , Egypt has more poverty than Botswana (the fraction of income poor individuals is 33% and 30%, respectively). However, this comparison masks the very different composition of their income poor populations. The vast majority of income poor individuals in Egypt are only-relatively income poor, whereas about half of the income poor individuals in Botswana are absolutely income poor. Specifically, the fractions of absolutely and only-relatively income poor individuals are, respectively, 5% and 28% in Egypt, and 16% and 14% in Botswana. Similarly to _HA_ , Botswana has more poverty than Egypt by _HHS_ . The reason for these opposite diagnoses between _HS_ and _HHS_ is that while _HS_ gives the same poverty score to all income poor individuals, _HHS_ down-weights individuals who are only-relatively income poor to around 0.5. As explained in Section 2, a weight smaller than one reflects the fact that not all only-relatively income poor individuals are welfare poor.

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These differences imply that the country-rankings of Egypt and Botswana differ considerably between _HS_ and _HHS_ . While Egypt is 11 places below Botswana by _HS_ , it is 18 places higher by _HHS_ . More generally, we could also estimate how many re-rankings might occur overall, on average, for all countries. When considering all countries and all years in the sample, we observe an average absolute difference in rankings between _HS_ and _HHS_ of 3.

**Table 2:** Egypt and Botswana: Poverty values and rankings in 2008

|||Values|||Rankings||_ω_(**y**)|Median inc.|
|---|---|---|---|---|---|---|---|---|
||_HS_|_HHS_|_HA_|_HS_|_HHS_|_HA_||(PPP$)|
|Botswana|29.7|22.9|15.8|89|104|112|0.51|117|
|Egypt|32.9|17.1|4.7|100|86|83|0.44|132|

Our next example illustrates how trends can diverge across the three poverty measures. Figure 6 displays the evolution of poverty in Samoa over 2002-2008, as captured by _HA_ , _HS_ , and _HHS_ . Samoa experienced unequal income growth over 2002-2008; that is, both the standards of living (measured by either mean or median income) and inequality (measured by the Gini index) increased. The greater inequality resulted in the fraction of individuals who were income poor, as measured by _HS_ , increasing by 7% over this period. However, the economic growth meant that over time, a considerably smaller fraction of those income poor remained absolutely income poor. In fact, the absolute headcount ratio decreased by 70% over this period. Thus, many absolutely income poor individuals became only-relatively income poor, an evolution not captured by _HS_ . _HHS_ , on the other hand, does capture the shift from absolute to only-relative income poverty, decreasing by 9% over 2002-2008. Moving beyond this illustrative example, we can estimate how often these opposite conclusions between _HS_ and _HHS_ occur in our sample. Considering only countries with _za < z_ <u>(</u> _<u>y</u>_ <u>), for which</u> _HS_ and _HHS_ do not coincide, we compare the evolution of _HS_ and _HHS_ over all _t/t −_ 4 periods for all countries in the sample.<sup>45</sup> The results show that the share of opposite trends between _HS_ and _HHS_ is 9.7%.

# **6.4 Global Poverty: Trends and Distribution**

We turn now to studying how global poverty varies between the absolute headcount ratio ( _HA_ ), the societal headcount ratio ( _HS_ ), and the hierarchical headcount ratio ( _HHS_ ). First, we analyze the trends in global poverty according to these three measures. Figure 7a shows that _HHS_ decreases by 48% between 1999 and 2015,

> 45We use the following reference years: 1999, 2003, 2007, 2011, and 2015.

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the standard indices such as the societal headcount ratio violate the most basic welfare-consistency property when preferences are heterogeneous. We show that a simple modification of the societal headcount is better aligned with the theory, and that this proposed switch yields a different picture of the evolution and distribution of income poverty both at the country and world level.

Our paper leaves several questions unanswered and additional research is called for. First, the reference bundle is taken to be exogenously chosen by some social planner. Although such exogeneity is sufficient to provide the foundations for global lines of the ‘societal’ type, it implies that our theory cannot pin down the exact design of the global line. Thus, the construction of a fair and welfareconsistent global poverty measure requires bringing together the insights of our results with those of the literature on global lines (Ravallion, 2020). Furthermore, we demonstrated that the design of the global line depends on the set of admissible preferences as well as the specification of the reference bundle. While this set of preferences could also be taken as an exogenous choice of the social planner, it could alternatively represent the empirical diversity of preferences over own and relative income in the global population. We leave elicitation of such preferences for future research. Second, our results rule out the use of standard poverty indices in our framework, but do not pin down a unique index to replace them. We argue that the hierarchical headcount ratio would be a natural candidate to replace the societal headcount ratio because it strikes a good compromise between simplicity and having desirable properties. However, this is a mere proposal and the selection of the ideal index for our framework remains an open question.

Finally, we motivate our paper based on _global_ income poverty measurement. More generally, our theory would be relevant for any government or regional entity concerned with both subsistence and social inclusion, such as, for example, the European Union (European Commission, 2015).

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