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 recently proposed hierarchical poverty indices. The paper using a measure that accounts for both own and relative reformulates one hierarchical index as a modified headcount income. This paper shows how—in the context of hetero- ratio. Unlike all classic poverty indices, this index is nec- geneous preferences over these factors—the well-known essarily reduced when an individual escapes poverty. The conflict between fairness and welfare-consistency can be application highlights that the proposed index substantially resolved, establishing the first preference-based foundation changes the assessment of global poverty reduction. for both the established societal global poverty line and 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∗ Benoit Decerf,† Mery Ferrando,‡ and Natalie Naïri Quinn§ 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.1 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.2 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.3 The first limitation is the absence of robust 1 As 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. 3 There 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, 2 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.4 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 welfare- consistent 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.5 Second, we show that classical indices such as the societal head-count ratio6 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 5 In 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. 6 The 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.7 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.8 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 := (y1 , . . . , yn(y) ) denote an income distribution, and let y denote the median income in distribution y.9 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 ) that is strictly increasing in its first argument and weakly increasing in the second. For instance, an individual could have her preferences represented by 8 The 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. 9 We 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 needs- adjustments are rarely implemented. 6 a utility function in the following family: 1 σ uσ (y, y/y ) := − + , (1) y y/y where the parameter σ ≥ 0 tunes the sensitivity to relative income. Self-centered preferences correspond to the case σ = 0, that is, to u0 .10 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, y ) instead of u(y, y/y ), and refer to (y, y ) 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 , y z ), where za > 0 and y z > 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 y z 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.11 Indifference curves have non-negative slopes because relative income is positively valued. Individual 1 is not welfare poor because she is self-centered (u1 = u0 ) 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 (u0 ), only own income matters. The self- centered 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, 10 In 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 y equivalent to −(uσ )−1 = 1 +1 σ = 1+ σy . y y/y 7 Figure 1: Definition of welfare poverty. Individual 1 is not welfare poor (a). Individual 2 is welfare poor (b). Notes: The blue lines are indifference curves passing through the reference bundle. Individual 1’s ¯ is self-centered because u1 = u0 . Individual 2 is not self-centered because u2 = uσ ¯ > 0. with σ that is, the fraction of individuals below the subsistence income, is equal to the fraction of individuals who are welfare poor. When the common preference is not self-centered (e.g., uσ ¯ for some σ ¯ > 0), it is still possible that the poverty line perfectly identifies the welfare poor. For any given value of median income y , the poverty line corresponds to the income level z (y ) that provides the same utility as the reference bundle, i.e., uσ ¯ (z (y ), y ) = uσ ¯ (za , y z ).12 This case provides the foundation for relative lines. The (relative) headcount ratio below the poverty line z (y ) is again equal to the fraction of individuals who are welfare poor. We depart from the literature by considering heterogeneous preferences. A society is thus characterized by a distribution-profile pair (y, u), where u := (u1 , . . . , un(y) ) is the profile of utility functions. In this setting, welfare-consistent measures may provide highly counter-intuitive comparisons. Take, for instance, the measure “fraction of individuals who are welfare poor” and consider again Figure 1. Individuals 1 and 2 live in the same society and the former has a smaller income than the latter. However, individual 1 is not welfare poor because her preference is u1 = u0 , whereas individual 2 is welfare poor because her preference is u2 = uσ ¯ . In that case, the measure “fraction of individuals who are welfare poor” attributes a smaller poverty score to individual 1 (whose score is equal to zero) than to individual 2 (whose score is equal to one), because the former is less sensitive to relative income than the latter. However, this seems unfair because individual 1 has a worse objective situation than individual 2. This example illustrates that when preferences are 12 We can interpret z (y ) − za > 0 as the extra income needed by a person who lives in a society with y > y z and earns the subsistence income za to also be socially included. 8 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 n(y) 1 PU (y, u) := p(yi , y ), (2) n(y) i=1 where p(yi , y ) is individual i’s poverty score, that is, the non-negative amount she contributes to the poverty measure. The poverty score p(yi , y ) may depend on the set U , but equal treatment is guaranteed because p(yi , y ) 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 (y ), may depend on median income y , but not on i’s preference ui . An individual i whose income yi is smaller than the poverty line z (y ) is income poor and her poverty score p(yi , y ) 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 (y ). 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.13 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, y ) at which some individual could be welfare poor must be attributed a strictly positive poverty score, which by definition of the global line implies y < z (y ). Thus, the escaping-poverty property implies that for any value of median income y , the global line z (y ) must be equal to (or greater than) the smallest income for which no individual can be welfare poor, that is, u(z (y ), y ) ≥ u(za , y z ) for all u ∈ U . If the global line z (y ) is equal to the smallest income for which no individual can be welfare poor then u′ (z (y ), y ) = u′ (za , y z ) for some u′ ∈ U and we call z (y ) 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 u0 and uσ ¯ . As explained above, for given y , the global line z (y ) 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 u0 and uσ ¯ , which are drawn in Figure 1. For low-income societies, whose median income is smaller than the reference median income y z , the smallest income at which no one is welfare poor is determined by the preference least sensitive to relative income, that is, the self- centered preference u0 . Hence, this smallest income corresponds to the subsistence income za . Indeed, an individual with preference u0 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 y z , this smallest 13 The literature defines a poverty line as absolute when its value z (y ) is independent of y and as relative when its value z (y ) depends on y . 10 income is determined by the preference most sensitive to relative income, that is, uσ¯ . Indeed, an individual with preference uσ ¯ and income just below z (y ) who lives in a higher-income society is welfare poor. Figure 2: Societal poverty line and the distinction between income poverty and welfare poverty Notes: The red line labelled z is the global poverty line z (y ) and the dashed red line is the subsistence income za . Individual 1 is self-centered (u1 = u0 ). In distribution y, individual 1 is absolutely income poor and welfare poor. In distributions y′ and y′′ , individual 1 is only- relatively income poor, but not welfare poor. We emphasize that this reasoning hinges on preference heterogeneity. Indeed, the global line is absolute in low-income countries because some individuals may not be sensitive to relative income. In turn, the global line is relative in higher- income countries because some individuals are sensitive to relative income. The above reasoning also implies that some income poor individuals are not welfare poor. When median income is larger than the reference median income y z , the global line is higher than the subsistence income. We describe as absolutely income poor the individuals whose incomes are below the subsistence income and only-relatively income poor the individuals whose incomes are above the subsistence income but below the global line z (y ). By definition, these are two mutually exclusive forms of income poverty. As illustrated in Figure 2, some individuals who are only-relatively income poor are not welfare poor. For instance, individual 1 is only-relatively income poor in distribution y′ because za < y1 ′ < z (y ′ ), but she is not welfare poor because her preference is self-centered (u1 = u0 ) and thus u0 (y1 ′ , y ′ ) > u0 (za , y z ). 11 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 y that is fixed and larger than the reference median income y z . For such y , the poverty line is higher than the subsistence income (z (y ) > 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′ , 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: HS (y) := HA (y) + HR (y), (3) 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 y , 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 u0 and a 50% probability that the individual has preference uσ ¯ , then for any distribution y with y > y z , EH 1/2 corresponds to the index HS defined as 1/2 1 HS (y) := HA (y) + HR (y), (4) 2 where the poverty score of any absolutely income poor individual is one and that 1/2 of any only-relatively income poor is one-half. Index HS is reduced when a self- centered 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. 1/2 Index HS is arguably too simplistic since it will also violate the escaping- poverty property as soon as some individuals have a sensitivity to relative income that is intermediate between those of u0 and uσ ¯ . Indeed, an individual with preference u σ¯ /2 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 1/2 latter, even if the index HS 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 (y ), the hierarchical headcount ratio is defined as: HHS (y) := HA (y) + ω (y)HR (y), (5) 13 where the endogenous weight ω (y) ∈ [0, 1] has the following linear expression: ˆR z (y ) − y ω (y) := z (y ) − za ˆR is the average income among only-relatively income poor for y > y z , where y individuals;14 ω (y) := 0 for y ≤ y z . The closer the average income among the only-relatively income poor is to za (respectively z (y )), 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 HA (y) = HHS (y) = HS (y) if y ≤ yz , (6) HA (y) ≤ HHS (y) ≤ HS (y) if y > yz . Under some assumption on the distribution of preferences, index HHS corresponds to the expected fraction of individuals who are welfare poor when y > y z . 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 z . 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: if yi < za ,    1 HHS p (yi , y ) := z (y )−yi if za ≤ yi < z (y ).   z (y )−za As an illustration, we show in online Appendix S2 that, for a certain probability distribution on the class of utility functions uσ , index HHS is equal to EH for all distributions y such that y > y z .15 14 R 1 ˆ := n(y)H y R (y) i∈NR (y) yi , where NR (y) is the set of only-relatively income poor individuals. 15 We also discuss in online Appendix S2 the relationship between our theory and the literature on the fuzzy measurement of poverty. 14 We consider now the implication of the escaping-poverty property when the value of median income y 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′′ , 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.16 Thus, even if index HHS comes closer than index HS to satisfying the escaping-poverty property, it does not fully do so.17 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 escaping- poverty property.18 Importantly, the interpersonal comparisons across different 16 This could, for instance, be the case for an individual with preference uσ ¯ when the value z of median income is smaller than the reference median income y . 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. 17 As we show in online Appendix S3, HHS satisfies a weak version of the escaping-poverty property that HS violates. 18 Even the expected fraction of poor EH(y) violates the second implication of the escaping- poverty property when some absolutely income poor individuals are not welfare poor under some preference uσ . Indeed, one can then find two bundles, (y, y ) and (y ′ , y ′ ), with y < za < y ′ and y < y z < y ′ such that a self-centered individual is welfare poor in (y, y ) but not in (y ′ , y ′ ), whereas an individual with preference uσ is welfare poor in (y ′ , y ′ ) but not in (y, y ). 15 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 self- centered. 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 ¯ ]. For median income larger than the reference median income y z , the interval [0, σ 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 := (y1 , . . . , 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 y denote the median income in distribution y. (As explained in online Appendix S5, all of our results still hold when y denotes mean income instead of median income.) When we wish to emphasize the median income in a particular distribution y, we write y = y . 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 ). We often drop subscript i when discussing points that are not specific to a given individual, for example, writing u(y, y/y ) instead of ui (yi , yi /y ). 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(y, y/y ), 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 B 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 B denote a generic subset of utility functions. Let u := (u1 , . . . , 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∈N Y × U n where Y n := y ∈ Rn + |y ≥ za . The mild restriction y ≥ za is necessary for our results.19 We respectively denote the set of bundles that an individual can consume and the subset of bundles with income smaller than za by X := {(y, y ) ∈ R+ × [za , ∞)} XA := {(y, y ) ∈ [0, za ) × [za , ∞)}, where the subscript A reflects that the frontier of this subset is the ‘absolute’ 19 Observe that although our theoretical results require that y ≥ za , the poverty indices characterized can readily be applied for countries with y < za . In our sample, there are 11 countries for which y < za in 2015. 17 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 , y z ) ∈ 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 z ≤ 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 ).20 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 , y ) < ui (za , y z )}. For an individual with utility function u, we denote by XQ (u) := {(y, y ) ∈ X |u(y, y ) < u(za , y z )}. 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 XQ (u). 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 y z are welfare poor. In turn, all individuals whose bundle is not in XQ (U B ) are not welfare poor, regardless of their preferences. 20 Observe that the reference bundle (za , y z ) = (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. 18 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′ , u′ ) indicates that there is (weakly) more poverty in (y, u) than in (y′ , u′ ). 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 B , we say that PU : XU → R is an additive index if n(y) 1 PU (y, u) := pui (yi , y ), (7) n(y) i=1 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, y, u) → pu (y, y ), 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 y . 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.21 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 21 The index “fraction of individuals who are welfare poor” is not ruled out by such mild continuity. 19 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.22 Moreover, poverty is strictly reduced when the utility of some welfare poor individual is strictly improved. Axiom 1 (Pareto ). For all (y, u), (y′ , u) ∈ XU such that n(y) = n(y′ ), if ′ ui (yi , y ′ ) ≥ ui (yi , y ) for all i ∈ N (y), then PU (y′ , u) ≤ PU (y, u). If, in addition, ′ uℓ (yℓ , y ′ ) > uℓ (yℓ , y ) for some ℓ ∈ Q(y, u), then PU (y′ , 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.23 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′ , u′ ) ∈ XU such that n(y) = n(y′ ), if u(yi′ , y ′ ) ≥ u(yi , y ) for all i ∈ N (y) and all u ∈ U such that (yi ′ , y ′ ) ∈ XQ (u), then PU (y′ , u′ ) ≤ 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.24 22 It 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. 23 The Domination principle was originally called the Intersection principle in Sen (1985). 24 We emphasize that the implications of Domination are satisfied by all the standard poverty 20 We illustrate in Figure 3 how the Domination axiom works. Assume that there are only two different utility functions in set U ; that is, U = {u, u0 }. We show that Domination implies that poverty in distribution y′ cannot be larger than in distribution y. First, the bundles of individuals 2 and 3 are irrelevant for the comparison because these individuals cannot be worse off than at the reference bundle, regardless of whether their utility functions are u or u0 .25 Second, individual 1 is worse off than at the reference bundle in both y and y′ , regardless of whether her utility function is u or u0 . So, the poverty comparison of y′ and y depends on the comparison of individual 1’s bundles under each of these two utility functions. Domination implies that poverty is no greater in y′ because bundle (y1 ′ , y ′ ) yields a higher utility than bundle (y1 , y ) for both utility functions (u or u0 ). Observe that the axiom would have remained silent if the two utility functions had implied opposite comparisons of the bundles of individual 1 (which cannot be the case in Figure 3 since individual 1’s own income and relative income are both greater in y′ ). Figure 3: Comparing poverty in distributions y and y′ based on Domination . Notes: The blue curves are indifference curves. Under Domination, poverty cannot be larger in y′ than in y when U = {u, u0 }. 4 Baseline Results and Impossibility This section prepares the stage for our main results (presented in Section 5). measures. But Domination also implies that the poverty measure must focus on the preferences of individuals who are welfare poor. As a result, all bundles that cannot be consumed by an individual who is welfare poor can, without loss of generality, be attributed a poverty score equal to zero. 25 ′ ′ Formally, we have that (yi , y ), (yi / XQ (U ) for all i ∈ {2, 3}. ,y ) ∈ 21 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 y . We sometimes call function z the global line, even if the global line z (y ) is formally a different object than function z . Function z partitions the set of bundles between those below the global line, Xz := {(y, y ) ∈ X |y < z (y )}, 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.26 Definition 2 (Fair additive index). Given any U ⊆ U B , we say that PU : XU → R is a fair additive index if n(y) ˆ+ 1 PU (y, u) := k p(yi , y ) (8) n(y) i=1 for some k ˆ ∈ R where the (degenerate)27 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, y ) = 0 on X \Xz , (ii) p(y, y ) > 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 B , an additive index PU satisfies Domination only if PU is a fair additive index. Proof. See online Appendix S6. ■ 26 The parameter k ˆ in Eq. (8) permits the anchoring to zero without any loss of generality. 27 We call p a degenerate poverty score function because this object is formally distinct from a poverty score function pu (see Definition 1). 22 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 B , 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, y ) > 0 for all (y, y ) ∈ Xz and pu (y, y ) = 0 for all (y, y ) ∈ X \Xz . If this is the case, we say that any individual with bundle (y, y ) is income poor if (y, y ) ∈ Xz , absolutely income poor if (y, y ) ∈ Xz ∩ XA , and only-relatively income poor if (y, y ) ∈ 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 B , the additive index P{u} satisfies Domination and Pareto if and only if P{u} is a fair additive index with a global 23 line z such that Xz = XQ ({u}) and a poverty score function p such that for all (y, y ), (y ′ , y ′ ) ∈ Xz , p(y ′ , y ′ ) ≤ p(y, y ) ⇔ u(y ′ , y ′ ) ≥ u(y, y ). 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.28 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′ ∈ U and some / XQ (u′ ).29 (y, y ) ∈ X with y ≥ y z such that (y, y ) ∈ XQ (u) and (y, y ) ∈ 28 Our assumption of homogeneous ordinal preferences does not imply interpersonal level- comparability of utility. 29 Our definition of a heterogeneous set U is not the complement of that of an homogeneous set {u}. For instance, set {u, u′ } with u ̸= u′ does not meet our definition of an heterogeneous set 24 Figure 4: Iso-poverty-score curves under homogeneous preferences Notes: Plain black curves are iso-poverty-score curves. Dashed blue curves are indifference curves. The thick red curve is the global line. Under homogeneous preferences, iso-poverty-score curves correspond to indifference curves below the global line. Proposition 3. Given any heterogeneous U ⊆ U B , no additive index PU satisfies Domination and Pareto. Proof. See Appendix A2. ■ When confronted with similar incompatibilities in other settings, authors have taken one of either two routes: weaken the Pareto principle or weaken the Domination principle.30 We follow the former route. We believe that poverty indices violating Domination would not garner much support, because they make interpersonal comparisons that many would consider counter-intuitive. This route also has a non-negligible pragmatic advantage. Poverty indices satisfying the Domination principle are easy to implement in practice because they do not require elicitation of individual preferences (Proposition 1). Therefore, our objective for the remainder of this section is to identify the ‘lightest’ weakening of Pareto that allows us to escape the impossibility (Proposition 3). The incompatibility is less deep in our setting with other-regarding preferences, because Pareto has limited ‘bite’. The reason is that there is never a unanimous improvement when the median income is reduced. When median when u and u′ share the same indifference curve through the reference bundle. Our definition is convenient for our results. 30 For instance, in social welfare measurement settings, Fleurbaey and Maniquet (2006, 2011) and Decancq et al. (2015) weaken the Domination principle, while Sprumont (2012) weakens the Pareto principle. Sprumont (2012) defines a Consensus axiom, whose precondition for recording a social improvement is that everybody finds that everyone’s bundle is better in the new allocation. Consensus is a rather ‘heavy’ weakening because Pareto’s precondition only requires that everybody finds her own bundle better in the new allocation. 25 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 B , for all (y, u), (y′ , u) ∈ XU with N (y) = N (y′ ) and y < y ′ , there exists j ∈ N (y) for whom uj (yj ′ , y ′ ) > uj (yj , y ). 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 (y ). There may exist another unanimously preferred distribution y′ in which one individual is below the global line z (y ′ ). For instance, this may happen when the global line z (y ′ ) is higher than z (y ), because median income is higher in y′ than in y. In that case, a self-centered individual j who is not welfare poor may prefer her bundle (yj ′ , y ′ ) below the global line z (y ′ ) over her bundle (yj , y ) above the global line z (y ). Pareto requires that the poverty index cannot be larger for the unanimously preferred distribution y′ , even if j is income poor in y′ 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′ . 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.31 31 If this weakening is deemed too favourable to individuals who are not welfare poor, one 26 Axiom 3 (Weak Pareto ). For all (y, u), (y′ , u) ∈ XU such that n(y) = n(y′ ), ′ ′ if ui (yi , y ′ ) ≥ ui (yi , y ) for all i ∈ N (y) and yj ′ ≥ yy j y for all j ∈ / Q(y, u), then PU (y′ , u) ≤ PU (y, u). If, in addition, uℓ (yℓ ′ , y ′ ) > uℓ (yℓ , y ) for some ℓ ∈ Q(y, u), then PU (y′ , 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.32 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 ∗ ⊂ U B 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 ≤ y ′ . Our results would be unchanged. 32 Indeed, 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. 27 income; that is, for all u ∈ U ∗ , we have u(y, y ) = u(y, y ′ ) for all y ∈ [0, za ) and y, y ′ ≥ 0. For instance, self-centered preferences belong to U ∗ . We consider the subsets of utility functions U that contain at least one member of U ∗ . Those subsets U are such that XA ⊆ XQ (U ), because all individuals with a utility function in U ∗ 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 B , 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 B with U ∩ U ∗ ̸= ∅, 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 Figure 5: Iso-poverty-score curves under heterogeneous preferences. Notes: Plain black curves are iso-poverty-score curves. The red curve is the global line. all other individuals have a poverty score equal to zero. Under this assumption, the poverty index must be reduced when the welfare poor individual moves to a bundle she prefers, at least if no-one is worse off in the new distribution.33 Such unanimous improvement requires that the median income in the new distribution is not smaller than the median income in the initial distribution (Lemma 1). Thus, when the welfare poor individual prefers a bundle with a larger median income, the poverty score attributed to this bundle must be strictly smaller (Lemma 3 in Appendix A1). Importantly, this reasoning holds true for all the preferences in U that the welfare poor individual may hold. Graphically, at any bundle, the slope of the iso-poverty-score curve cannot be steeper than the slope of the indifference curve of an individual who is welfare poor at this bundle. This implies that the iso-poverty-score curves are flat below the subsistence income. Indeed, the indifference curves of a self-centered individual are flat, and moreover, said self- centered individual is welfare poor when she is absolutely income poor.34 Thus, the iso-poverty-score curves must also be flat below za , because Domination prevents iso-poverty-score curves from having negative slopes. Observe that the above reasoning does not necessarily imply that iso-poverty- score curves above za are flat. Even if some individuals have indifference curves 33 The other precondition for Weak Pareto is that none of the other individuals becomes income poor in the new distribution. It is always possible to find an appropriate distribution-profile pair such that this other precondition is met, as we show in Lemma 3 in Appendix A1. 34 There need not be self-centered preferences in the set U , but the same reasoning holds for preferences in the set U ∗ . 29 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 σ We characterize the trade-offs made by poverty indices satisfying Domination and Weak Pareto under a particular subset of U B . Let U σ ¯ be the subset of utility functions uσ defined by Equation (1) for which 0 ≤ σ < σ ¯ for some σ¯ > 0. U σ¯ is much smaller than U B , but U σ ¯ 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 σ ¯ , 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.35 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 ∗ ¯ is a hierarchical index with global line z and only if PU σ defined as z ∗ (y ) := max(za , R + Rσ ¯ y) za for all y ≥ za , where R := 1+¯ σyz , and whose poverty score function p is such that for all (y, y ), (y ′ , y ′ ) ∈ Xz∗ \XA 36 ′ ′ y − za y ′ − za p(y, y ) = p(y , y ) when = ′ . y − yz y − yz 35 The indices characterized in Theorem 1 are non-continuous at the reference bundle (za , y z ). Recall that the poverty score function of additive indices is required to be continuous only on XQ (u), which never includes (za , y z ). If the poverty score function is required to be continuous on the whole domain X , then the two axioms are incompatible. 36 A hierarchical index has for all (y, y ), (y ′ , y ′ ) ∈ XA that p(y, y ) = p(y ′ , y ′ ) when y = y ′ . 30 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 , and thus the relative part of z tends to a strongly z ∗ relative line, which points to the origin. Hence, on U σ ¯ , 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 σ ¯ that has an indifference curve corresponding to the global line z . 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 ∗ 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.37 Below the subsistence income, the poverty score function of index HHS does not depend on relative income.38 Interestingly, Decerf (2017) proposes a family of hierarchical indices whose iso- poverty-score curves satisfy the constraints derived in Theorem 1. For a given global line z , index P H is defined by the following poverty score function:  α    yi 1 − λz a if yi < za , pH αλ (yi , y ) := (9)  α λ) z y i − za if za ≤ yi < z (y ),   (1 − λ) − (1 − (y )−za 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. ′ ∗ ∗ ′ ′ 37 y −za y −za z (y )−y z (y )−y On Xz∗ \XA , we have that y −y z = y ′ −y z ⇔ z ∗ (y )−za = z ∗ (y ′ )−za . 38 Below 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. 31 5.3 Robustness The trade-offs characterized in Theorem 1 are valid under subset U σ ¯ . 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 B . 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 ∗∗ defined as za z ∗∗ (y ) := max za , y yz for all y ≥ za , whose poverty score function p is such that for all (y, y ), (y ′ , y ′ ) ∈ Xz∗∗ \XA , p(y, y ) = p(y ′ , y ′ ) when y = y′. 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 B contains preferences whose indifference curves are not straight lines.39 There are two key differences between Theorems 1 and 2. First, the global line of the latter is z ∗∗ instead of z ∗ . This reflects the fact that there is no upper bound on the sensitivity to relative income in U B . 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 B contains utility functions that exhibit extreme forms of concavities. This difference disappears on the subset U C of utility functions in U B whose indifference curves are weakly convex in the space of bundles. On U C , 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 ∗∗ because no upper bound is placed on the sensitivity to relative income in U C .40 39 For the relative part of the global line, the explanation follows from the fact that the global line is maximal. When y ≥ y z , the shape of the global line is defined by the utility function most sensitive to relative income. On U B , 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. 40 Details can be found in online Appendix S10. 32 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 σ¯ excludes u0 .41 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,42 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.43 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 y z . Our dataset includes 168 countries.44 41 Details 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. 43 The reference years available from 1999 on are: 1999, 2002, 2005, 2008, 2010-2013 and 2015. 44 China, 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 33 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 , y z ) 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 , y z ) 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 + 0.5y ) where y 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 + 0.5y 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 y z = $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. 34 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-middle- income 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 1/2 1/2 HS would yield similar poverty evaluations. Recall that HS 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 (y ) is close to being uniform. In such a case, the average income for the only-relatively income poor is y ˆR ≈ za +2z(y) , which yields ω (y) ≈ 0.5 when za ≈ z (y ). 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. 36 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 (y ), for which 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.45 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, 45 We use the following reference years: 1999, 2003, 2007, 2011, and 2015. 37 Figure 6: Evolution of poverty in Samoa, 2002-2008. Note: The graph plots the evolution of poverty relative to 2002 for all years through 2008. while HS decreases by ‘only’ 32% over the same period (see also Table S.1 in online Appendix S4). This result, albeit meaningful, is not surprising given the definitions of HS and HHS , and the values of ω that we observe in the data. The main reason for this difference is that the sharp decrease in HA over 1999-2015, which amounts to 64%, has a stronger effect on HHS than on HS . Indeed, HHS systematically decreases when an individual leaves absolute income poverty, while HS is unchanged if the individual becomes only-relatively income poor. Turning the focus to regional poverty, we show that and how the proposed adoption of HHS affects the distribution of global poverty across regions. Figure 7b displays the regional distribution of global poverty in 2015 for the three measures. We highlight two major differences in the current distribution of poverty across regions. First, the share of global poverty in East Asia and the Pacific is considerably smaller for HHS than for HS . In 2015, this region accounted for 18% of global poverty according to HHS while accounting for 25% according to HS . This difference in the weight of East Asia and the Pacific is to a large extent explained by the distribution of absolute income poverty. Indeed, in 2015, only 6% of the world’s population suffering from absolute income poverty lived in East Asia and the Pacific. Second, Sub-Saharan Africa has a considerably larger share of global poverty according to HHS than according to HS . In 2015, this region accounted for the largest share of global poverty according to HHS , amounting to 36%, while being the third region according to HS , at 24%. Again, this increase in the share of global poverty in Sub-Saharan African when HS is replaced by HHS is to a large extent explained by the fact that this region was host to almost 60% of absolutely 38 income poor individuals. (a) Evolution of global poverty, 1999-2015 (b) Distribution by region, 2015 Figure 7: Evolution and distribution of global poverty Notes: Panel a plots the evolution of poverty relative to 1999 for all reference years through 2015. It includes all countries with available information in each reference year. See Footnote 44 for a list of countries with missing information in given reference years. Panel b plots the contribution to global poverty for each of the following regions: East Asia & Pacific (EAS), Europe & Central Asia (ECS), Latin America & Caribbean (LCN), Middle East & North Africa (MEA), North America (NAC), South Asia (SAS), and Sub-Saharan Africa (SSF). As discussed in Section 6.2 and easily seen in Figure 7a, at the worldwide level, HHS lies very much in between HS and HA because many low- and middle- income countries have a ω ≈ 0.5. Hence, one could wonder whether we could 1/2 obtain similar results to HHS if we instead use the even simpler index HS . As expected, given our previous analysis, Figures S.3a and S.3b in online Appendix 1/2 S4 show that using HS yields very similar results as HHS when measuring both the evolution of global poverty and its distribution across regions, respectively. We can then safely conclude that if HHS were to be deemed too complex to 1/2 implement and HS used instead, it would make little difference in terms of the revised poverty diagnoses for both many low- and middle-income countries (for which ω ≈ 0.5) and the world as a whole (at least in the near future). 7 Concluding Remarks We have developed a theory of global income poverty measurement with preference heterogeneity over own income and relative income. This theory fills three gaps in the literature. First, it shows how the poverty measure can aggregate heterogeneous individual preferences. Second, it provides a welfarist foundation for the societal lines proposed in the literature. Third, it shows that 39 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 welfare- consistent 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. 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We have u(y, y ) ≥ u(za , za ) for all u ∈ U because utility functions are increasing in own income when holding relative income constant and because we assume y ≥ za . By transitivity, we have u(y, y ) ≥ u(za , y z ). This, in turn, implies that u(y, y ) ≥ u(za , y z ) for all u ∈ U because y ≥ y . By definition this yields (y, y ) ∈/ XQ (U ), the desired result. ■ Lemma 3. Consider any U ⊆ U B , any fair additive index PU satisfying Weak Pareto and any two bundles (y, y ), (y ′ , y ′ ) ∈ X with y ≤ y ′ . If there exists some u ∈ U such that (y, y ) ∈ XQ (u) and u(y ′ , y ′ ) ≥ u(y, y ), then p(y ′ , y ′ ) ≤ p(y, y ). If, in addition, u(y ′ , y ′ ) > u(y, y ), then p(y ′ , y ′ ) < p(y, y ). Proof. Consider the two distributions y := (y, y, y ) and y′ := (y ′ , y ′ , y ′ ), for which y = y and y′ = y ′ . For some u′ ∈ U , consider the preferences profile u := (u, u′ , u′ ) ∈ U 3 . By construction, we have (y1 , y ) = (y, y ), (y1 ′ , y ′ ) = (y ′ , y ′ ) and u1 = u. Also, we have (yi , y ) = (y, y ), (yi ′ , y ′ ) = (y ′ , y ′ ) and ui = u′ for all i ∈ {2, 3}. First, we show that p(yi , y ) = p(yi ′ , y ′ ) = 0 for all i ∈ {2, 3}. We have ′ (yi , y ), (yi , y′) ∈/ XQ (U ) for all i ∈ {2, 3} since (y, y ), (y ′ , y ′ ) ∈ / XQ (U ) (Lemma 2). This implies that (yi , y ), (yi ′ / Xz because a fair additive index has , y′) ∈ Xz ⊆ XQ (U ). In turn, the fact that (yi , y ), (yi ′ / Xz for all i ∈ {2, 3} implies , y′) ∈ p(yi , y ) = p(yi ′ , y ′ ) = 0 for all i ∈ {2, 3} by definition of a fair additive index. Second, we show that P (y, u) ≥ P (y′ , u). We have 1 ∈ Q(y, u) because (y1 , y ) ∈ XQ (u1 ) since (y, y ) ∈ XQ (u). We have u1 (y1 , y ) ≤ u1 (y1 ′ , y ′ ) because u(y, y ) ≤ u(y ′ , y ′ ). Consider λ := y ′ /y , which is such that λ ≥ 1. We have ′ (yi , y ′ ) = (λyi , λy ) for all i ∈ {2, 3}. This implies ui (yi , y ) ≤ ui (λyi , λy ) for all i ∈ {2, 3} because utility is increasing in own income when relative income is kept constant. We directly get ui (yi , y ) ≤ ui (yi ′ , y ′ ) for all i ∈ {2, 3}. Hence, Weak Pareto implies that P (y, u) ≥ P (y′ , u). 44 We are now equipped to prove the statement. The fact that P (y, u) ≥ P (y′ , u) and p(yi , y ) = p(yi ′ , y ′ ) = 0 for all i ∈ {2, 3} implies that p(y1 , y ) ≥ p(y1 ′ , y ′ ). This directly yields p(y, y ) ≥ p(y ′ , y ′ ). If, in addition, u(y, y ) < u(y ′ , y ′ ), then the same argument shows that Weak Pareto implies P (y, u) > P (y′ , u) and thus p(y, y ) > p(y ′ , y ′ ), the desired result. ■ Lemma 4. Given any U ⊆ U B , any fair additive index PU satisfying Weak Pareto has Xz = XQ (U ). Proof. Assume to the contrary that Xz ̸= XQ (U ). This implies that Xz ⊂ XQ (U ) because a fair additive index must have Xz ⊆ XQ (U ). Hence, there must exist some (y, y ) ∈ XQ (U ) with y > 0 for which (y, y ) ∈ / Xz . As (y, y ) ∈/ Xz , we have p(y, y ) = 0 because PU is a fair additive index. As (y, y ) ∈ XQ (U ), there exists some u′ ∈ U such that (y, y ) ∈ XQ (u′ ). For any λ > 1, we have that bundle (λy, λy ) ∈ X is such that u′ (λy, λy ) > u′ (y, y ) because utility is strictly increasing in own income when relative income is constant. As PU satisfies Weak Pareto , Lemma 3 implies that p(λy, λy ) < p(y, y ). This is a contradiction to p(y, y ) = 0 because 0 is the minimum value that function p can take. ■ Lemma 5. Consider any σ ¯ > 0, any y > y z , and any y ∈ [za , z ∗ ] where z ∗ := za ¯ y and R := 1+¯ R + Rσ σyz . The value of sensitivity to relative income y − za σ ∗ := (A.1) za y − yy z ∗ ∗ is such that (i) σ ∗ ∈ [0, σ ¯ ], (ii) uσ (y, y ) = uσ (za , y z ), and (iii) (y, y ) ∈ XQ (uσ ) ⇔ y ′ −za y −za σ > σ ∗ . Moreover, for any y ′ ≥ za and y ′ > y z with y ′ −y z = y −y z , we have (iv) ∗ ∗ uσ (y ′ , y ′ ) = uσ (y, y ), and (v) if y ′ > y , then uσ (y ′ , y ′ ) < uσ (y, y ) for all σ > σ ∗ . Proof. The proof uses the observation that uσ is ordinally equivalent to ˆσ := −(uσ )−1 . From Equation (1), we get u u ˆσ (y, y ) = 1+y σy . Part (i). In order to show that σ ∗ ≥ 0, it is sufficient to prove that its denominator is positive because by assumption y ≥ za . Thus, we must show that za y > yy z . By assumption, we have y ≤ z ∗ , so it is sufficient to show that za y > z ∗ y z . Replacing z ∗ by its expression in last inequality yields yyz > 1+¯ σy 1+¯ σyz . This inequality holds because y > y z and σ ¯ > 0, which proves that σ ∗ ≥ 0. In order to show that ∗ σ∗ ≤ σ ¯ , it is sufficient to show that it holds for y = z ∗ , because ∂σ ∂y ≥ 0 and y ≤ z ∗ . Replacing y by z ∗ in the expression for σ ∗ yields after some manipulations σ∗ = σ¯ , the desired result. 45 Part (ii). Let y z ≥ 0 be the income level implicitly defined by ∗ ∗ uσ (y z , y z ) = uσ (y, y ). ∗ In words, an individual with preference uσ is indifferent between bundle (y, y ) and ∗ ∗ earning y z when median income is y z . In order to show that uσ (za , y z ) = uσ (y, y ), ∗ ∗ we show that y z = za . As uσ is ordinally equivalent to u ˆσ , the definition of y z ∗ ∗ σ∗ yz yields u ˆσ (y, y ). By the definition of u ˆσ (y z , y z ) = u ˆσ , this yields y z = y 1+1+σ ∗ y . By replacing σ by its expression in that of y , we get after some manipulations ∗ z y z = za , as desired. Part (iii). By definition, we have (y, y ) ∈ XQ (uσ ) if and only if uσ (y, y ) < uσ (za , y z ). As uσ is ordinally equivalent to u ˆσ , last inequality is equivalent to 1+ y σy < 1+za σy z . A few manipulations yield zayy−za −yy z < σ , as desired. Part (iv). For all σ ≥ 0, let y σ ≥ 0 be the income level implicitly defined by uσ (y σ , y ′ ) = uσ (y, y ). ′ As uσ is ordinally equivalent to u ˆσ , we get y σ = y 1+ σy 1+σy . For the case σ = σ ∗ , we ∗ ∗ ∗ ∗ σ∗ y′ have uσ (y σ , y ′ ) = uσ (y, y ) when y σ = y 1+ 1+σ ∗ y . In order to show that σ∗ ′ ′ σ∗ σ∗ u (y , y ) = u (y, y ), we show that y = y . By replacing σ ∗ by its expression ′ ∗ in that of y σ , we get, with a few manipulations, ∗ za (y − y ′ ) + y (y ′ − y z ) yσ = . (A.2) y − yz ′ ′ We also have y = za + y ′ −y z (y − y ) because y −y z = y ′ −y z . By replacing y in y −za z y −za y − za ∗ Equation (A.2) by its expression, we get with a few manipulations, y σ = y ′ , as desired. Part (v). By the definition of y σ (see proof of Part (iv)), we have uσ (y ′ , y ′ ) < uσ (y, y ) if and only if y ′ < y σ , because utility is strictly increasing in own income. ∗ The proof of Part (iv) shows that y σ = y ′ . Thus, we have uσ (y ′ , y ′ ) < uσ (y, y ) for σ σ y ′ −y all σ > σ ∗ if we have ∂y ∂σ > 0. From the expression of y σ , we get ∂y ∂σ = y (1+ σy )2 . ∂y σ We therefore have ∂σ > 0 because y ≥ za and y > y > 0, the desired result. ■ ′ A2 Proof of Proposition 3 As set U is heterogeneous, there exist two u, u′ ∈ U and some (y, y ) with y ≥ y z / XQ (u′ ). PU is a fair additive index because such that (y, y ) ∈ XQ (u) and (y, y ) ∈ 46 PU satisfies Domination (Proposition 1). As PU also satisfies Pareto , Lemma 4 implies that Xz = XQ (U ).46 Therefore, (y, y ) ∈ Xz because (y, y ) ∈ XQ (u). Consider the two distributions y := (y, y, y ) and y′ := (za , y z , y z ), for which y = y and y′ = y z . By construction, we have (y1 , y ) = (y, y ), (y1 ′ , y ′ ) = (za , y z ). Also, we have (yi , y ) = (y, y ) and (yi ′ , y ′ ) = (y z , y z ) for all i ∈ {2, 3}. We have (y, y ), (y z , y z ) ∈ / XQ (U ) (Lemma 2). And we have by construction of y′ that (yi ′ , y′) ∈ / Xz for all i ∈ {1, 2, 3} because (za , y z ) ∈ / XQ (U ) and Xz = XQ (U ). Therefore, we have PU (y , u) = k ′ ˆ for all u ∈ U because PU is a fair additive index 3 (since it satisfies Domination ). In contrast, (y1 , y ) ∈ Xz because (y, y ) ∈ Xz . This implies that p(y1 , y ) > 0, and so, P (y, u) > k ˆ for all u ∈ U 3 . Together, this implies that P (y′ , u) < P (y, u) for all u ∈ U 3 . In order to prove the incompatibility, we show that Pareto implies that P (y′ , u′ ) ≥ P (y, u′ ) for the profile u′ := (u′ , u′ , u′ ) ∈ U 3 . This is indeed an implication of Pareto if we have u′ (yi , y ) ≥ u′ (yi ′ , y ′ ) for all i ∈ {1, 2, 3}. We have u′ (y, y ) ≥ u′ (za , y z ) because (y, y ) ∈ / XQ (u′ ). This implies that u (y1 , y ) ≥ u (y1 , y ), because (y1 , y ) = (y, y ) and (y1 , y ) = (za , y z ). We have ′ ′ ′ ′ ′ ′ u′ (yi , y ) ≥ u′ (yi ′ , y ′ ) for all i ∈ {2, 3} because both i’s own income and i’s relative income is weakly larger under y than under y′ since (yi , y ) = (y, y ), (yi ′ , y ′ ) = (y z , y z ) and y ≥ y z . The desired result. A3 Proof of Lemma 1 y y′ We show that there exists some j ∈ N (y) for which yj < yj ′ and yj ≤ yj′ . If that is the case, then the monotonicity properties of utility functions imply that ′ uj (yj , yj /y ) < uj (yj ′ , yj /y ′ ) because uj ∈ U B . Assume for simplicity that the number of individuals n(y) is odd.47 As y < y ′ , more than half of the individuals earn an income no larger than y in distribution y, while more than half of the individuals earn an income no smaller than y ′ in distribution y′ . Therefore, there must be an individual j with yj ≤ y < y ′ ≤ yj′ . ′ yj yj This implies that yj < yj ′ and y ≤1≤ y′ , as desired. 46 We define Weak Pareto in Section 4.3. PU satisfies Weak Pareto because PU satisfies Pareto . Again, we allow ourselves to simply reference Lemma 4 in order to avoid duplicating the argument. 47 The argument is the same if n(y) is even, but is less easily exposed. 47 A4 Proof of Proposition 4 PU is a fair additive index because PU satisfies Domination (Proposition 1). Furthermore, we have Xz = XQ (U ) because PU is a fair additive index that satisfies Weak Pareto (Lemma 4). First, assume to the contrary that Condition (i) in the definition of a hierarchical index does not hold; that is, function p is not strictly decreasing in its first argument on Xz . As PU is a fair additive index, p is weakly decreasing in its first argument on Xz . Thus, the contradiction assumption implies that there exist two bundles (y, y ), (y ′ , y ) ∈ Xz with 0 < y < y ′ < z (y ) such that p(y, y ) = p(y ′ , y ). As (y, y ) ∈ XQ (U ), there exists some u′ ∈ U such that (y, y ) ∈ XQ (u′ ). Let λ′ := y ′ /y , which is such that λ′ > 1, and let y ′ := λ′ y . Bundle (y ′ , y ′ ) := (λ′ y, λ′ y ) is such that u′ (y ′ , y ′ ) > u′ (y, y ) because utility is strictly increasing in own income when relative income is constant. As PU satisfies Weak Pareto , Lemma 3 implies that p(y ′ , y ′ ) < p(y, y ). We also have p(y ′ , y ) ≤ p(y ′ , y ′ ) because function p is weakly increasing in its second argument and y ′ ≥ y . Transitivity then implies that p(y ′ , y ) < p(y, y ), which yields the desired contradiction. There remains to show that Condition (ii) in the definition of a hierarchical index holds; that is, for all (y, y ), (y, y ′ ) ∈ XA ∩ Xz , we have p(y, y ) = p(y, y ′ ). If y = y ′ , then that is trivially the case, as PU is a fair additive index. Thus, consider without loss of generality that y < y ′ . By assumption, there exists some u′ ∈ U ∩ U ∗ . Since u′ ∈ U ∗ , we have XA = XQ (u′ ). Therefore, we have XA ⊆ Xz because Xz = XQ (U ). Consider the contradiction assumption that for some y ∈ [0, za ), we have p(y, y ) ̸= p(y, y ′ ). We must then have p(y, y ) < p(y, y ′ ), because function p is weakly increasing in its second argument on Xz , given that PU is a fair additive index. Now, as u′ ∈ U ∗ , we have (y, y ) ∈ XQ (u′ ) and u′ (y, y ) = u′ (y, y ′ ). We can thus apply Lemma 3, because we also have that PU satisfies Weak Pareto and y < y ′ . As u′ (y, y ) = u′ (y, y ′ ), Lemma 3 implies that p(y, y ) ≥ p(y, y ′ ), the desired contradiction. A5 Proof of Theorem 1 ¯ satisfying these two axioms has the required ⇒. We show that any PU σ properties. 48 As the self-centered preference u0 ∈ U σ ¯ , we have U ∗ ∩ U σ ¯ ̸= ∅. By ¯ satisfies Domination and Weak Pareto , then PU σ Proposition 4, if PU σ ¯ is a hierarchical index. By definition of a hierarchical index, we have Xz = XQ (U σ ¯ ). We show that Xz = Xz∗ ; that is, z (y ) = z ∗ (y ) for all y ≥ za . The proof exploits the fact that Xz = XQ (U σ ¯ ). We first show that z (y ) = za for all y ≤ y z . Take any y ≤ y z . We start by showing that z (y ) ≥ za . For all y ′ < za we have (y ′ , y ) ∈ XQ (u0 ), i.e., a self-centered individual is welfare poor when she earns less than the subsistence income. This implies that (y ′ , y ) ∈ XQ (U σ ¯ ) for all y ′ < za because u0 ∈ U σ ¯ . Therefore, we have z (y ) ≥ za because Xz = XQ (U σ ¯ ). There remains to show that z (y ) ≤ za . We must have z (y ) ≤ y for any y ≥ 0 such that (y ′ , y ) ∈ ′ ′ / X Q (U σ¯ ) because Xz = XQ (U σ ¯ ). It is thus sufficient to show that (za , y ) ∈ / XQ ( U σ¯ ). We have u(za , y ) ≥ u(za , y ) for all u ∈ U because y ≤ y and individual utility is z σ ¯ z weakly decreasing in the median income. By definition of XQ (U σ ¯ ), we thus have (za , y ) ∈/ XQ (U ), as desired. ¯ σ We then show that z (y ) = R + Rσ ¯ y for all y > y z . Take any y > y z . First, we show that z (y ) ≤ R + Rσ ¯ y . Letting y := R + Rσ ¯ y , we have by Lemma 5, Part (iii), that (y, y ) ∈ XQ (uσ ) ⇔ σ > σ ∗ for σ ∗ := zayy− za −yy z . Replacing y and R by their expressions in the definition of σ yields σ = σ ∗ ∗ ¯ . This implies that (y, y ) ∈ XQ (uσ ) ⇔ σ > σ ¯ . As σ ≤ σ ¯ for all uσ ∈ U σ ¯ , we have thus shown that (y, y ) ∈ / XQ (U ). This implies that z (y ) ≤ y because Xz = XQ (U σ ¯ σ ¯ ). By the definition of y , we get z (y ) ≤ R + Rσ ¯ y . There remains to show that z (y ) ≥ R + Rσ ¯ y . Letting y := R + Rσ ¯ y , we have by Lemma 5, Part (ii), that σ∗ σ∗ u (y, y ) = u (za , y ) for σ z ∗ := y −za za y −yy z . This implies that ¯ σ u (R + Rσ ¯ y, y ) = u (za , y ) because σ = σ ¯ σ z ∗ ¯ (as shown above). As utility is strictly increasing in own income, we thus have for any y ′ < R + Rσ ¯ y that u (y , y ) < u (y, y ). By transitivity, this yields u (y , y ) < u (za , y ); that is, ¯ ′ σ σ ¯ ¯ ′ σ σ ¯ z (y ′ , y ) ∈ XQ (uσ ¯ ). Therefore, we have (y ′ , y ) ∈ XQ (U σ ¯ ) for any y ′ < R + Rσ ¯y because uσ ¯ ∈ Uσ ¯ . This shows that z (y ) ≥ R + Rσ ¯ y because Xz = XQ (U σ ¯ ). Hence, we have shown that z (y ) = z (y ) for all y ≥ za . ∗ ′ There remains to show that for all (y, y ), (y ′ , y ′ ) ∈ Xz∗ \XA with y y − za −y z y −za = y ′ −y z we have p(y, y ) = p(y , y ). Assume to the contrary that there are two bundles ′ ′ y ′ − za (y, y ), (y ′ , y ′ ) ∈ Xz∗ \XA with y y − za −y z =y ′ −y z but p(y, y ) ̸= p(y , y ). ′ ′ If y = y ′ , then the two bundles must be identical, a contradiction to p(y, y ) ̸= p(y ′ , y ′ ), because PU σ ¯ is a fair additive index. Therefore, we have y ̸= y . Without ′ 49 loss of generality, assume that y < y ′ . We must have y ≥ za and y ′ ≥ za because / XA . We must also have y > y z because (y, y ) ∈ Xz∗ and y ≥ za . (y, y ), (y ′ , y ′ ) ∈ There are two cases: • Case 1: p(y, y ) < p(y ′ , y ′ ). This case is such that there exists some y ′′ < y for which p(y ′′ , y ) < p(y ′ , y ′ ). Indeed, function p is continuous on XQ (U σ ¯ ) and (y, y ) ∈ XQ (U σ ¯ ) because XQ (U σ ¯ ) = Xz ∗ . We show that there exists a preference uσ ∈ U σ ¯ for which (y ′′ , y ) ∈ XQ (uσ ) and uσ (y ′′ , y ) < uσ (y ′ , y ′ ). If such a preference exists, we can resort to Lemma 3 because PU σ ¯ is a fair additive index that satisfies Weak Pareto and y < y . ′ Lemma 3 then implies that p(y ′′ , y ) > p(y ′ , y ′ ), the desired contradiction. There remains to show the existence of such uσ ∈ U σ ¯ . The conditions for Lemma 5 are met by bundle (y ′ , y ′ ). Indeed, we have y ′ > y z because y ′ > y > y z . We also have y ′ ∈ [za , R + Rσ ¯ y ′ ] because (y ′ , y ′ ) ∈ Xz∗ \XA . By Lemma 5, Part ∗ ∗ ′ −z y ′ −za (ii), we have uσ (y ′ , y ′ ) = uσ (za , y z ) for σ ∗ := zay a y ′ −y ′ y z . As y y −za −y z = y ′ −y z , ∗ ∗ Lemma 5, Part (iv), further implies that uσ (y, y ) = uσ (y ′ , y ′ ) for the same ∗ ∗ σ ∗ .48 Therefore, we have uσ (y ′′ , y ) < uσ (za , y z ) because y ′′ < y . This shows ∗ ∗ ∗ that (y ′′ , y ) ∈ XQ (uσ ). By transitivity, we also have uσ (y ′′ , y ) < uσ (y ′ , y ′ ). ∗ Finally, we have uσ ∈ U σ ¯ because Lemma 5, Part (i), implies that σ ∗ ∈ [0, σ ¯ ], ∗ which proves that u has the required properties. σ • Case 2: p(y, y ) > p(y ′ , y ′ ). This case is such that there exists some y ′′ > y for which p(y ′′ , y ) > p(y ′ , y ′ ). Indeed, function p is continuous on (the open) set XQ (U σ¯ ), and (y, y ) ∈ XQ (U σ ¯ ) because XQ (U ) = Xz∗ . We must have p(y , y ) > 0 because p(y , y ) ≥ 0 as σ ¯ ′′ ′ ′ ¯ is a fair additive index. This, in turn, implies that y < z ∗ (y ) and thus ′′ PU σ y ′′ ∈ [za , R + Rσ¯ y ). Consider the two distributions y := (y ′′ , y, y ) and y′ := (y ′ , y ′ , y ′ ), which are, respectively, such that y = y and y′ = y ′ . By construction, we have (y1 , y ) = (y ′′ , y ), (y1 ′ , y ′ ) = (y ′ , y ′ ). Also, we have (yi , y ) = (y, y ) and (yi ′ , y ′ ) = (y ′ , y ′ ) for all i ∈ {2, 3}. We have that bundles (y, y ), (y ′ , y ′ ) ∈ / XQ ( U σ ¯ ) (Lemma 2). Therefore, p(y, y ) = p(y ′ , y ′ ) = 0 because PU σ ¯ is a fair additive index. Consider any u := (u, u, u). If we have PU σ ¯ (y , u) ≤ PU σ ¯ (y , u), then we get ′ a contradiction to p(y ′′ , y ) > p(y ′ , y ′ ). Indeed, PU σ ¯ (y , u) ≤ PU σ ¯ (y , u) implies ′ that p(y ′′ , y ) ≤ p(y ′ , y ′ ) because p(yi , y ) = p(yi ′ , y ′ ) = 0 for all i ∈ {2, 3}. 48 The conditions for Lemma 5, Part (iv), are met because y ≥ za and y > y z . 50 We show that Domination implies that PU σ ¯ (y , u) ≤ PU σ¯ (y , u). Domination ′ implies PU σ ¯ (y , u) ≤ PU σ ¯ (y , u) if u (yi , y ) ≥ u (y , y ) for all i ∈ {1, 2, 3} and all ′ σ σ ′ i ′ uσ ∈ U σ ¯ such that (yi , y ) ∈ XQ (uσ ). As (y, y ), (y ′ , y ′ ) ∈ / XQ (U σ¯ ), there remains to show that u (y , y ) ≥ u (y , y ) for all u ∈ U for which (y ′′ , y ) ∈ XQ (uσ ). σ ′′ σ ′ ′ σ ¯ σ Lemma 5, Part (iii), implies that (y ′′ , y ) ∈ XQ (uσ ) if and only if σ > σ ∗ , where ′′ −z σ ∗ := zayy− a y ′′ y z .49 Hence, we must show that uσ (y ′′ , y ) ≥ uσ (y ′ , y ′ ) for all uσ ∈ U σ ¯ for which σ > σ ∗ . e ′′ Let the income level y e ≥ 0 be defined as y − za y ′ −y z = yy− −za yz . This definition is such y ′ − za y e −za ′′ −z that y > y because y > y and y′ −yz = y−yz . As y′ −yz = yy− e ′ ′′ y − za yz a , Lemma 5, Part σ∗ e ′ σ ∗ ′′ (iv), implies that u (y , y ) = u (y , y ) for the same σ . As y < y ′ , Lemma ∗ 5, Part (v), further implies that uσ (y e , y ′ ) < uσ (y ′′ , y ) for all σ > σ ∗ . We also have uσ (y ′ , y ′ ) < uσ (y e , y ′ ) for all σ ≥ 0 because y e > y ′ . By transitivity, we get uσ (y ′ , y ′ ) < uσ (y ′′ , y ) for all σ > σ ∗ , the desired result. ¯ satisfies the two axioms. ⇐. We show that PU σ Domination : Take any (y, u), (y′ , u′ ) ∈ XU σ ¯ that satisfy the preconditions under which Domination implies that PU σ ′ ′ ¯ (y , u ) ≤ PU σ ¯ (y , u); that is, we have n(y) = n(y′ ) and u(yi ′ , y ′ ) ≥ u(yi , y ) for all i ∈ N (y′ ) and all u ∈ U σ ¯ such that (yi , y ) ∈ XQ (u). In order to prove that PU σ ′ ′ ′ ′ ¯ (y , u ) ≤ PU σ ¯ (y , u), we show that p(yi ′ , y ′ ) ≤ p(yi , y ) for all i ∈ N (y′ ). Take any i ∈ N (y′ ) for whom (yi ′ , y′) ∈ / Xz∗ . Since PU σ ¯ is a fair additive index, we have p(yi ′ , y ′ ) = 0 and thus p(yi ′ , y ′ ) ≤ p(yi , y ). Take any i ∈ N (y′ ) for whom (yi ′ , y ′ ) ∈ XA . If za > yi ′ ≥ yi , then we directly have p(yi ′ , y ′ ) ≤ p(yi , y ) because PU σ ¯ is a hierarchical index. We have za > y , ′ i as (yi ′ , y ′ ) ∈ XA . There remains to show that yi ′ ≥ yi . As za > yi ′ , we have (yi ′ , y ′ ) ∈ XQ (u0 ) for the self-centered preference u0 . The precondition then implies that u0 (yi ′ , y ′ ) ≥ u0 (yi , y ) because u0 ∈ U σ ¯ . This inequality implies that yi ′ ≥ yi , as desired. Finally, take any i ∈ N (y′ ) for whom (yi ′ , y ′ ) ∈ Xz∗ \XA . If yi < za , then we directly have that p(yi ′ , y ′ ) ≤ p(yi , y ) because yi ′ ≥ za , as (yi ′ / XA and PU σ , y′) ∈ ¯ is a hierarchical index. There remains the alternative case yi ≥ za . If we have yi′ −z ′ y −y z a ≥y i − za y −y z , then the properties of function p on Xz∗ \XA imply that p(yi ′ , y′) ≤ y ′ −z p(yi , y ), as desired. Assume to the contrary that yi ′ −y z < y −y z . We show that a yi −za there exists some uσ ∈ U σ ¯ for which (yi ′ , y ′ ) ∈ XQ (uσ ) and uσ (yi , y ) > uσ (yi ′ , y ′ ), a contradiction to the precondition of Domination . There are two cases. 49 The conditions for Lemma 5 are met because y ′′ ∈ [za , R + Rσ ¯ y ) and y > y z . 51 • Case 1: za ≤ yi < z ∗ (y ). This case is such that y > y z , because za < z ∗ (y ), and we also have y ′ > ∗ y z because (yi ′ , y ′ ) ∈ Xz∗ \XA . By Lemma 5, Part (ii), we have uσ (yi , y ) = ∗ e −z uσ (za , y z ) for σ ∗ := zayyi− − za yi y z . Let the income level y e ≥ 0 be defined as y a y ′ −y z = y ′ −z e yi −za y −y z . This definition is such that y e > yi ′ because yi ′ −y z < y −y z . As y ′ −y z = a yi − z a y −za ∗ ∗ yi −za y −y z Lemma 5, Part (iv), further implies that uσ (y e , y ′ ) = uσ (yi , y ) for the , ∗ ∗ same σ ∗ . As y e > yi ′ , this yields uσ (yi ′ , y ′ ) < uσ (yi , y ). By transitivity, we also ∗ ∗ ∗ ∗ get uσ (yi′ , y ′ ) < uσ (za , y z ) and thus (yi ′ , y ′ ) ∈ XQ (uσ ). Preference uσ thus has ∗ the required properties because uσ ∈ U σ ¯ , as σ ∗ ∈ [0, σ¯ ] (Lemma 5, Part (i)). • Case 2: za ≤ z ∗ (y ) ≤ yi . As (yi ′ , y ′ ) ∈ Xz∗ , there exists some preference uσ ∈ U σ ¯ for which (yi ′ , y′) ∈ XQ (uσ ) because Xz∗ = XQ (U σ ¯ ). By definition of XQ (uσ ), we have uσ (za , y z ) > uσ (yi ′ , y ′ ). We must have uσ (za , y z ) ≤ uσ (yi , y ) because this case is such that (yi , y ) ∈ / Xz∗ and Xz∗ = XQ (U σ ¯ ). By transitivity, we have uσ (yi , y ) > uσ (yi ′ , y ′ ). Preference uσ ∈ U σ ¯ thus has all the desired properties. Weak Pareto : Take any (y, u), (y′ , u′ ) ∈ XU σ ¯ that satisfy the preconditions under which Weak Pareto implies PU σ ′ ′ ¯ (y , u ) ≤ PU σ ¯ (y , u). That is, we have n(y) = ′ n(y ), ui (yi , y ) ≥ ui (yi , y ) for all i ∈ N (y ), and yj ≥ y ′ ′ ′ ′ ′ y j y for all j ∈ / Q(y, u). The unanimous preference for distribution y′ implies that y ≤ y ′ (Lemma 1). In order to prove PU σ ′ ′ ¯ (y , u ) ≤ PU σ ¯ (y , u), we show that p(y , y ) ≤ p(yi , y ) for all i ′ ′ i ∈ N (y′ ). Take any i ∈ N (y′ ) for whom (yi ′ , y′) ∈ / Xz∗ . Since PU σ ¯ is a fair additive index, we have p(yi ′ , y ′ ) = 0 and thus p(yi ′ , y ′ ) ≤ p(yi , y ). Take any i ∈ N (y′ ) for whom (yi ′ , y ′ ) ∈ XA . We first show that yi ≤ yi ′ . This follows from ui (yi ′ , y ′ ) ≥ ui (yi , y ), because y ≤ y ′ and utility functions are non-increasing in the median income. This implies that yi ≤ yi ′ < za because ′ (yi , y ′ ) ∈ XA . We therefore have p(yi ′ , y ′ ) ≤ p(yi , y ), because PU σ ¯ is hierarchical. Finally, take any i ∈ N (y ) for whom (yi , y ) ∈ Xz∗ \XA . We have yi ′ ′ ′ ′ ≥ za because (yi ′ , y′) ∈ / XA . We also have y ′ > y z , because (yi ′ , y ′ ) ∈ Xz∗ \XA . If yi < za , then we directly have p(yi ′ , y ′ ) ≤ p(yi , y ), because PU σ ¯ is a hierarchical index. There remains the case yi ≥ za . If we have both (yi , y ) ∈ Xz∗ \XA and ′ yi − za yi − za ′ z ≥ , (E.3) y −y y − yz 52 then the properties of function p imply that p(yi ′ , y ′ ) ≤ p(yi , y ), the desired result. There remains to show that both hold. We first show that there exists some uσ ˆ ∈ Uσ ¯ for which (yi , y ) ∈ XQ (uσ ˆ ) and u (yi , y ) ≥ u (yi , y ). First, assume that i ∈ ˆ ′ σ ′ σ ˆ / Q(y, u). There exists some uσˆ ∈ Uσ ¯ for which (yi ′ , y ′ ) ∈ XQ (uσ ˆ ) because (yi ′ , y ′ ) ∈ Xz∗ = XQ (U σ ¯ ). The ′ precondition of Weak Pareto requires that yi ≥ y yi because i ∈ ′ y / Q(y, u). We thus ′ ′ have uσ (yi , y ′ ) ≥ uσ ˆ ′ ˆ (yi , y ), because yi ′ ≥ y y , y ≥ 1 and utility is increasing in y i y own income when holding relative income constant. We also have (yi , y ) ∈ XQ (uσ ˆ ) because (yi ′ , y ′ ) ∈ XQ (uσ ˆ ) and uσ (yi , y ′ ) ≥ uσ ˆ ′ ˆ (yi , y ), showing that uσ ˆ has all the desired properties. Second, assume that i ∈ Q(y, u). Consider u σ ˆ := ui . We have (yi , y ) ∈ XQ (uσ ˆ ) because i ∈ Q(y, u). We further have uσ (yi , y ′ ) ≥ uσ ˆ ′ ˆ (yi , y ), because the preconditions of Weak Pareto imply that ui (yi , y ) ≥ ui (yi , y ), i.e., uσ ′ ′ ˆ has all the desired properties. We have (yi , y ) ∈ Xz∗ because Xz∗ = XQ (U σ ¯ ), and there exists some uσ ˆ ∈ Uσ ¯ for which (yi , y ) ∈ XQ (uσ ˆ ). This yields (yi , y ) ∈ Xz∗ \XA because yi ≥ za . There remains to show that Inequality (E.3) holds. If y = y ′ , then Inequality (E.3) directly follows because uσ (yi , y ′ ) ≥ uσ ˆ ′ ˆ (yi , y ). If, instead, y ̸= y ′ , then we have y < y ′ because y ≤ y ′ . We have shown above that (yi , y ) ∈ XQ (uσ ˆ ) and uσ (yi , y ′ ) ≥ uσ ˆ ′ ˆ (yi , y ). By Lemma 5, Part (iii), we have (yi , y ) ∈ XQ (uσ ˆ )⇔σ ˆ > σ∗ for σ ∗ := zayyi− −za 50 yi y z . Thus, we have σ ˆ > σ ∗ . Let the income level y e ≥ 0 be defined e −z ∗ ∗ as y a y ′ −y z =y i − za y −y z . Lemma 5, Part (iv), implies uσ (y e , y ′ ) = uσ (yi , y ) for the same σ ∗ . As y < y ′ , Lemma 5, Part (v), further implies that uσ ˆ e ′ (y , y ) < uσ ˆ (yi , y ), because σ ˆ > σ ∗ . We must thus have yi ′ ≥ y e , because uσ ˆ ′ ′ (yi , y ) ≥ uσ ˆ (yi , y ). e Inequality (E.3) then directly follows from the fact that yi ≥ y and y′ −yz = y ′ e y −za i − za y −y z , the desired result. Take any (y, u), (y′ , u′ ) ∈ XU σ ¯ that satisfy the preconditions under which Weak Pareto implies PU σ ′ ′ ¯ (y , u ) < PU σ ¯ (y , u). The proof is a straightforward adaptation of arguments used above and is thus omitted. 50 Lemma 5 applies because yi ∈ [za , z ∗ (y )] and y > y z because (yi , y ) ∈ Xz∗ \XA . 53 Online Appendix Global Income Poverty Measurement with Preference Heterogeneity: Theory and Application Benoit Decerf Mery Ferrando Natalie Naïri Quinn July 4, 2023 S1 Relationship with the Capability Approach In this section, we explain in more details that the individual’s preference over own income and relative income, which are the basis of our framework, can be understood as a “reduced form” for the individual’s preference over underlying functionings (unrelated to misanthropic feelings). We also show that our definition of the welfare poor, based on a reference bundle, is equivalent to a definition of the welfare poor based on a reference capability (Sen, 1985). Our framework builds on the theoretical foundations provided by Ravallion (2020) for global income poverty measurement.1 As we build on Ravallion (2020), our premises are slightly different from those of the approach of Atkinson and Bourguignon (2001).2 Sen argues that the space in which individual welfare should be measured is the space of functionings and capabilities. Functionings are, loosely speaking, what a person can do and be. In turn, capabilities are sets of functioning vectors, i.e. opportunity sets in the space of functionings. After Atkinson and Bourguignon (2001), two major functionings are deemed key for the measurement of global poverty. First is subsistence, which is typically implemented through nutritional status. In a first approximation, the real cost of consuming a given amount of calories does not evolve with a society’s median 1 Unlike Ravallion (2020), who surveys some of the major issues in global poverty measure- ment, we ignore here important problems associated to prices and individual characteristics other than preferences. 2 As explained by Ravallion (2020), Atkinson and Bourguignon (2001) take a non-welfarist approach that does not consider the trade-offs that individuals make between different function- ings. Instead, their approach respect the functioning that is deemed relevant in a given society, without demanding welfare-consistency. 1 income. Hence, the larger an individual’s income, the higher the number of calories she can purchase, independently of her society’s median income. Second is social inclusion, which can be achieved through the consumption of clothing and housing, as well as food diets.3 The level of social participation that a given bundle of goods provides depends on the society’s median income. Typically, the larger the median income, the smaller the level of social participation that an individual can reach with fixed income. This idea borrows from Smith (1776) and Townsend (1979): individuals whose income is too far below median income in their society are at risk of social exclusion because their income might not be sufficient for them to participate in the everyday activities of their society. Let f = (f1 , f2 ) denote a vector of functionings, where the first component f1 captures the level of nutrition and the second component f2 captures the level of social participation. To be sure, each of these components reflects a continuous score on its associated functioning, and not a binary status. Individual i’s budget set in the space of functionings B (yi , y ), i.e. her capability, depends on her income yi and the median income y in her society. This capability is the set of functioning vectors she can potentially achieve. Given this capability B (yi , y ), the vector of functionings she achieves fi∗ (yi , y ) depends on her expenditure choices. Individual i could achieve a relatively high level of nutrition if she spends large amounts on cheap calories. Alternatively, she could reach a relatively high level of social participation is she spends large amounts on clothes and housing, or expensive calories. The expenditure choices of i depends on her primal utility function wi (fi ), which captures the trade-off she makes between different functionings. We extend the theoretical framework laid out in Ravallion (2020) by allowing this primal utility function to be individual specific, whereas he assumes this function to be stable and interpersonally comparable. We argue that this extension is called for when individuals with the same capability select different expenditure patterns, thereby revealing the different trade-offs they make between nutrition and social participation. The primal utility function implies that fi∗ (yi , y ) = arg max wi (fi ). fi ∈B (yi ,y ) Our analysis based on utility functions ui is bridged to the primal utility function 3 As discussed by Ravallion (2020), food consumption can also play a role in social inclusion. Poor individuals do not systematically consume the cheapest calories. 2 wi when ui (yi , y ) = wi (fi∗ (yi , y )). In our extended framework, we do not assume comparability of the primal util- ity function across individuals. Rather, interpersonal comparisons emerge from the reference bundle. Our approach based on a reference bundle (za , y z ) is equivalent to the selection of a reference capability B (za , y z ). Individual i is welfare poor if her capability B (yi , y ) does not allow reaching the level of primal welfare that she could reach under the reference capability B (za , y z ). Hence, individual i is welfare poor if her capability B (yi , y ) does not contain a vector of functionings that provides her a level of welfare as large as fi∗ (za , y z ), i.e. if wi (fi∗ (yi , y )) < wi (fi∗ (za , y z )). Observe that two individuals i and j who have different primal utility functions wi and wj may have different reference vectors of functionings, i.e. fi∗ (za , y z ) ̸= fj∗ (za , y z ). However, when wi = wj , as assumed by Ravallion (2020), we have fi∗ (za , y z ) = fj∗ (za , y z ). In that case, assuming a reference bundle (za , y z ), i.e. assuming a reference capability, is the same as assuming a reference vector of functioning f ∗ , as he proceeds. S2 Relation Between Index HHS and EH Assume that individual utility functions are distributed i.i.d in the set U σ ¯ , which is the subset of utility functions uσ for which 0 ≤ σ < σ ¯ .4 Each individual i draws ui in U σ ¯ according to some probability measure P : F → [0, 1], where F is a sigma-algebra on U σ . ¯ 5 Proposition S.1. ¯ ¯ ∈ [0, ∞). There exist a sigma-algebra F on U σ Consider any σ and a probability ∗ measure P : F → [0, 1] such that #Q(y, u) HHS (y) = E∗ for all y ∈ Y n with y ≥ y z , n(y) 4 That is, individual utility functions are drawn independently from the bundles consumed and independently from all other individuals’ utility functions. 5 F is thus a set of subsets of U that contains U and is closed under complements and countable unions; the probability measure satisfies P(U ) = 1, P(∅) = 0 and is additive over countable collections of pairwise disjoint elements of F . 3 where E∗ is the expectation operator under P∗ and # is the set cardinality operator. Proof. The proof is by construction. First, we define the probability space (U σ ¯ , F , P∗ ). Let U x := {uσ ∈ U σ ¯ |0 ≤ ¯ ]}, and let F be the closure of F under complements σ ≤ x}, let F := {U x |x ∈ [0, σ and countable unions. Define P∗ : F → [0, 1] such that x 1+σ¯ yz P∗ (U x ) := , 1 + xy z ¯ σ interpreted as the probability that σ ≤ x, and extend P∗ to all events in F through its additivity property.6 Second, we show that this probability space has the required property. Recall we can write uσ (y, y ¯) = − 1+y ¯ σy . Utility function uσ is strictly increasing in y ≥ 0 and decreasing in y ¯ ≥ 0 and σ ≥ 0. Let the income level ζ (¯ y ) := za1+¯ (1+¯ σy ¯z σy ¯) , noting that uσ ¯ (ζ (¯ ¯) = uσ y ), y ¯ ¯z ) for all y (za , y ¯ ≥ 0 and, furthermore, that ζ (¯ y ) is the highest income level for which an indifference curve of some u ∈ U passes σ ¯ σ through both bundles (ζ (¯ ¯) and (za , y y ), y ¯z ). Given any y ∈ Y n with y ≥ y z , note that ζ (¯ y ), where z (¯ y ) = z (¯ y ) is the function such that Xz = XQ (U σ ¯ ). Letting I be the indicator function, taking the value 1 if its argument is true and 0 if false, we have   n(y) #Q(y, u) 1 ∗ E∗ = E I (ui (yi , y ¯z )) ¯) < ui (za , y n(y) n(y) i=1 n(y) 1 = P∗ (ui (yi , y ¯z )) ¯) < ui (za , y n(y) i=1 because n(y) is non-random and by linearity of expectation. There are three cases to consider. Case 1. If yi < za then uσ (yi , y ¯) < uσ (za , y ¯z ) for all σ ≥ 0 because ¯) ≤ uσ (za , y y ≥ y z , so {uσ ∈ U σ ¯ σ |u (yi , y ¯) < uσ (za , y ¯z )} = U σ ¯ and P∗ (ui (yi , y ¯z )) = ¯) < ui (za , y P∗ (U σ ¯ ) = 1 = pHHS (yi , y¯). Case 2. If yi ≥ z (¯ y ) then uσ (yi , y ¯) ≥ uσ (z (¯ ¯) ≥ uσ y ), y ¯ (z (¯ ¯) = uσ y ), y ¯ ¯z ) for (za , y ¯ ], so {uσ ∈ U σ all σ ∈ [0, σ ¯ σ ¯z )} = ∅ and P∗ (ui (yi , y ¯) < uσ (za , y |u (yi , y ¯) < ui (za , y¯z )) = P∗ (∅) = 0 = pHHS (yi , y¯). 6 ¯ Note that S : U σ ¯ ] such that S : uσ → σ is a random variable on this probability → [0, σ space, whose probability density function is smoothly decreasing in σ . 4 Case 3. If za ≤ yi < z (¯ y ) then uσ (yi , y ¯) < uσ (za , y¯z ) if and only if σ (za y ¯− ¯ ) > yi −za if and only if σ > za y yi y z yi − z a ¯−yi y ¯z , as yyi ¯ < y z (¯ ¯ y) za ≤y ¯z . So {u ∈ U |u (yi , y σ ¯ σ σ ¯) < uσ (za , y¯z )} = U σ ¯ \ U xi where xi = zayy i −za ¯−yi y¯z , and P∗ (ui (yi , y ¯z )) = ¯) < ui (za , y 1 − P∗ (U xi ) which, after straightforward manipulations, can be shown to equal y )−yi z (¯ y )−za z (¯ ¯). = pHHS (yi , y In all three cases we have established that P∗ (ui (yi , y ¯z )) = pHHS (yi , y ¯) < ui (za , y ¯), so n(y) ∗ #Q(y, u) 1 E = pHHS (yi , y ¯) = HHS (y). n(y) n(y) i=1 ■ Proposition S.1 relates to a strain of research on the uncertain identification of the poor. The literature on fuzzy poverty measures starts from the assumption that the poverty line lies in some income range, but its exact value is not precisely known (Cerioli and Zani, 1990). One possible reason is that people have different perceptions about what constitutes poverty (Zheng, 2015). Proposition S.1 has strong similarities with that approach if we assume that the individual utility functions are not precisely known. Indeed, all individuals below the global line are attributed positive poverty scores, even if they might not be welfare poor. Our results are conceptually different because we investigate the trade-offs between own income and relative income. Also, the trade-offs that we characterize do not depend on a probability distribution on individual preferences. 5 S3 Index HHS satisfies a minimal version of Pareto In this appendix, we refer to material presented in sections 3, 4 and 5. No fair additive index is fully welfare-consistent when preferences are hetero- geneous (Proposition 3). We show that, in contrast to HS , HHS is minimally welfare-consistent and thus grants a minimal role to preferences. Formally, the (societal) headcount ratio violates Minimal Pareto . This minimal welfare-consistency property is a weakening of Weak Pareto .7 For poverty to be reduced, Minimal Pareto adds the precondition that a welfare poor individual ℓ escapes welfare poverty and earns an income above the subsistence income za . Axiom S.1 (Minimal Pareto ). For all (y, u), (y′ , u) ∈ XU such that n(y) = n(y′ ), if ui (yi ′ , y ′ ) ≥ ui (yi , y ) for all ′ i ∈ N (y), yj′ ≥ yy j / Q(y, u) and there is some ℓ ∈ Q(y, u) for whom y for all j ∈ ℓ∈ ′ ′ / Q(y , u) and yℓ > za , then PU (y′ , u) < PU (y, u). Since Minimal Pareto is a weakening of Weak Pareto , all indices characterized in Theorems 1, 2 and S.1 satisfy this property on their respective sets of utility functions. However, Proposition S.2 shows that the (societal) headcount ratio violates Minimal Pareto on heterogeneous sets of utility functions. Hence, not only is the (societal) headcount ratio not monotonic, but it is not even minimally welfare-consistent. Proposition S.2. On any heterogeneous U ⊆ U B , the headcount ratio below the global line z with Xz = XQ (U ) violates Minimal Pareto. Proof. We construct two distributions y, y′ ∈ Y 3 and show that for some u′ ∈ U 3 Minimal Pareto implies PU (y, u′ ) < PU (y′ , u′ ) but HS (y, u′ ) = HS (y′ , u′ ). Since U is heterogeneous, there exist two u, u′ ∈ U and some (y, y ) ∈ X with y ≥ y z such that (y, y ) ∈ XQ (u) and (y, y ) ∈ / XQ (u′ ) and / XQ (u′ ). As (y, y ) ∈ y ≥ y z , we must have y ≥ za . If y = za , then by the continuity of u there exists / XQ (u′ ). Therefore, we can assume y ′ > y such that (y ′ , y ) ∈ XQ (u) and (y ′ , y ) ∈ without loss of generality that y > za . Consider the income distribution y := (y, y, y ) ∈ Y 3 , which is such that z / XQ (U ). For δ = yy , we define a second distribution y′ ∈ Y 3 by (y2 , y) = (y3 , y) ∈ letting yi′ = δyi for all i ∈ {1, 2, 3}. By construction, we have y′ = y z . 7 Indices HS and HHS both violate Weak Pareto . Index HHS does not satisfy Weak Pareto because its poverty score function is constant for all levels of own income smaller than za . 6 We show that the preconditions for Minimal Pareto are met for u′ = (u′ , u′ , u′ ). First, we have u′i (yi , y ) > u′i (yi ′ , y ′ ) for all i ∈ {1, 2, 3} if we have δ < 1. As (y, y ) ∈ XQ (u) and y > za , we must have y > y z , which yields δ < 1. Second, we have by construction that yi = y ′ yi for all i ∈ {2, 3}. Finally, we show that y ′ 1 ∈ Q(y′ , u′ ), 1 ∈ / Q(y, u′ ) and y1 > za . We have y1 > za because y1 = y > za . We have 1 ∈ / Q(y, u′ ) because u′1 = u′ and (y, y ) ∈ / XQ (u′ ). There remains to show that 1 ∈ Q(y′ , u′ ). It is sufficient to show that y1 ′ < za because y′ = y z . As z ′ y1 = y yy , we have y1 ′ < za if we can show y y < y za z . As (y, y ) ∈ XQ (u), we have u(y, y ) < u(za , y z ), which implies together with y > y z that y < za yyz because utility functions are strictly increasing in own income when relative income is held constant. We have shown that all the preconditions for Minimal Pareto are met and thus PU (y, u′ ) < PU (y′ , u′ ). To conclude, we show that HS (y, u′ ) = HS (y′ , u′ ) = 1/3 when Xz = XQ (U ). To do this, it is sufficient to prove that (y1 , y), (y1 ′ , y′ ) ∈ Xz and that (y2 , y), (y3 , y), (y2′ , y′ ), (y3 ′ / Xz . We have (y1 , y) ∈ Xz because (y, y ) ∈ XQ (u). We have , y′ ) ∈ (y1 , y ) ∈ Xz because y1 ′ ′ ′ < za and y′ = y z . We have (y2 , y), (y3 , y) ∈ / Xz by Lemma 2 because y2 = y3 ≥ y . In order to show (y2 ′ , y′ ), (y3 ′ , y′ ) ∈/ Xz , we show z that y2 = y3 ≥ za (this is sufficient because y = y ). As yi = yy yi for all ′ ′ ′ z ′ i ∈ {1, 2, 3} and y2 = y3 ≥ y , we have y2 ′ = y3 ′ ≥ y z . We have y2 ′ = y3 ′ ≥ za because we assume y ≥ za , the desired result. z ■ The HHS satisfies Minimal Pareto on any subset of U C , where U C is the subset of U B on which indifference curves are convex. Proposition S.3. On any U ⊆ U C , the hierarchical headcount ratio below the global line z with Xz = XQ (U ) satisfies Minimal Pareto. Proof. Take any (y, u), (y′ , u) ∈ XU C that satisfy the preconditions under which Minimal Pareto implies PU (y′ , u) < PU (y, u). These preconditions require that ′ ui (yi , y ′ ) ≥ ui (yi , y ) for all i ∈ N (y). By Lemma 1, this implies in turn that y ≤ y ′ . First, we show that pHHS (yi ′ , y ′ ) ≤ pHHS (yi , y ) for all i ∈ N (y) = N (y′ ). For all i ∈ Q(y, u) for whom (yi , y ) ∈ XA , the two inequalities y ≤ y ′ and ui (yi ′ , y ′ ) ≥ ui (yi , y ) imply that yi ′ ≥ yi . This implies that pHHS (yi ′ , y′) ≤ pHHS (yi , y ). For all i ∈ Q(y, u) for whom (yi , y ) ∈ Xz \XA , the convexity of the indiffer- ence curves of ui together with the inequalities y ≤ y ′ , ui (yi ′ , y ′ ) ≥ ui (yi , y ) and 7 ui (za , y z ) > ui (yi , y ) imply that8 ′ yi − za ′ z yi ≥ za + z (y − y ). y−y y ′ −z Last inequality can be rewritten yi ′ −y z ≥ y −y z , which by the definition of p implies a yi − z a that pHHS (yi ′ , y ′ ) ≤ pHHS (yi , y ). ′ ′ For all i ∈ / Q(y, u), the preconditions require that yi ′ ≥ y y i y , where y y ≥ 1. This requirement directly implies that pHHS (yi ′ , y ′ ) ≤ pHHS (yi , y ) if yi < za . For the case yi ≥ za , this requirement also implies pHHS (yi ′ , y ′ ) ≤ pHHS (yi , y ) under ′ the assumption that z (y ′ ) ≤ y y z (y ). There remains to derive a contradiction when y′ assuming that z (y ) > y z (y ). As y ′ ≥ y , this contradiction assumption implies ′ ′ that y ′ > y and thus y y > 1. As Xz = XQ (U ), this contradiction assumption ′ implies that there exists a utility function u ∈ U such that ( y y z (y ), y ′ ) ∈ XQ (u). As by definition of z (y ) there exists no u′ ∈ U for which (z (y ), y ) ∈ XQ (u′ ), we ′ must have (z (y ), y ) ∈ / XQ (u). Therefore we have u(z (y ), y ) ≥ u( y y z (y ), y ′ ). This is ′ ′ equivalent to u(z (y ), y ) ≥ u( y y z (y ), yy y ), a contradiction to utility function being strictly increasing in own income when relative income is kept constant. Second, the preconditions also require that yℓ ′ > za and uℓ (yℓ ′ , y ′ ) ≥ uℓ (za , y z ) for some ℓ ∈ Q(y, u). As ℓ ∈ Q(y, u), we must have (yℓ , y ) ∈ Xz because Xz = XQ (U ). For the case (yℓ , y ) ∈ XA , we have pHHS (yℓ , y ) = 1. Since yℓ ′ > za , we directly get pHHS (yℓ ′ , y ′ ) < 1, and so pHHS (yℓ ′ , y ′ ) < pHHS (yℓ , y ). For the alternative case (yℓ , y ) ∈ Xz \XA , the argument used above (for the case i ∈ Q(y, u) for whom y ′ −za (yi , y ) ∈ Xz \XA ) yields yℓ ′ −y z > y −y z , and therefore p yℓ −za HHS ′ (yℓ , y ′ ) < pHHS (yℓ , y ). In both cases, we obtain HHS (y′ ) < HHS (y), the desired result. ■ ′ yi −za ′ ′ ′ 8 Assume to the contrary that yi < za + y −y z (y − y z ). The inequality ui (yi , y ) ≥ ui (yi , y ) and the convexity of ui imply that ′ ′ yi − yi ′ ui (yi , y ) ≥ ui yi − (y − y z ), y z . y′ − y ′ yi −za ′ Then, the inequality yi < za + y −y z ( y − y z ) implies that ui (yi , y ) ≥ ui (za , y z ) , showing that i ∈ / Q(y, u), a contradiction. 8 S4 Additional Tables and Figures Figure S.1: Endogenous weight as a function of the ratio of global lines Notes: The graph plots the median endogenous weight ω as a function of the median ratio between the weakly relative line and the absolute line by country. We only consider country- year pairs with z (y ) ≥ za . The following countries with a median ratio of global lines below 1 are excluded: Burundi, Central African Republic, Democratic Republic of Congo, Madagascar, Malawi, Mozambique, and Rwanda. Full markers display the median values within each country income group as defined by the World Bank. Table S.1: Poverty by region, 1999 & 2015 1999 2015 HS HHS HA Mean inc. Pop. HS HHS HA Mean inc. Pop. (PPP$) (million) (PPP$) (million) East Asia & Pacific 43.4 38.3 33.8 210 1977 22.7 10.3 1.9 421 2223 Europe & Central Asia 20.5 10.7 4.5 692 858 15.8 6.1 1 935 906 Latin America & Caribbean 34.4 24.1 13.9 324 493 26.7 14.1 3.7 496 572 Middle East & North Africa 28.3 14.8 3.6 274 279 22.6 11.2 4.1 340 376 North America 18.8 6.8 .7 1838 309 18.4 6.8 1.1 1995 356 South Asia 51.2 46.1 40.7 83 1345 33 22.8 13.2 128 1715 Sub-Saharan Africa 62.3 60.5 58.7 81 631 49.7 46.1 42 108 991 World 41.1 34.5 29 335 5892 28.1 17.9 10.3 453 7139 Notes: Mean income per capita is expressed in PPP$ per month. Regions as defined by the World Bank. 9 (a) East Asia and Pacific (b) Europe and Central Asia (c) Latin America and the (d) Middle East and North Africa Caribbean (e) North America (f ) South Asia (g) Sub-Saharan Africa Figure S.2: Evolution of poverty by region, 1990-2015 Notes: The graph plots the evolution of poverty relative to 1999 for all reference years through 2015. Each region includes all countries with available information in each reference year. 10 (a) Evolution of global poverty, 1999- (b) Distribution by region, 2015 2015 Figure S.3: Evolution and distribution of global poverty Notes: Panel a plots the evolution of poverty relative to 1999 for all reference years through 2015. It includes all countries with available information in each reference year. Panel b plots the contribution to global poverty for each of the following regions: East Asia & Pacific (EAS), Europe & Central Asia (ECS), Latin America & Caribbean (LCN), Middle East & North Africa (MEA), North America (NAC), South Asia (SAS), and Sub-Saharan Africa (SSF). S5 Defining Relative Income Using Mean Income Although we assume throughout that relative income is defined with respect to median income, all our results also hold when relative income is defined with respect to mean income. That is, our results also hold when y denotes mean income in the distribution. The formal proofs for these claims can be found in our discussion paper, where we prove our results for both definitions of y . In this section, we do not repeat all proofs when assuming that y denotes mean income. Rather, we do two things. First, we explain how to adapt the typical argument that deals with the other- regarding aspect of individual preferences. Second, we provide the argument that is the most challenging to adapt when y denotes mean income. This argument is used in Step 2 of the proof that only fair additive indices satisfy Domination (Proposition 1). S5.1 Dealing with ORP with Respect to Mean Income Our results show how our axioms constrain the trade-off that the measure makes between own income and relative income. That is, they show how the poverty score function compares the bundles of individuals living in societies with different values for y . The difficulty is that our axioms do not constrain the comparison of 11 bundles, but rather the comparison of income distributions.9 Hence, our axioms do not constrain how the poverty measure compares two bundles (yi , y ) and (yi ′ , y ′ ), but rather two distributions y and y′ . Yet, the problem vanishes when the bundles (yj , y ) and (yj ′ , y ′ ) for all j ̸= i are attributed a zero poverty score by function p. When y denotes median income, the difficulty is easily circumvented when considering distributions y and y′ for which all other individuals earn an income equal to the median income, i.e., when yj = y and yj ′ = y ′ for all j ̸= i. Indeed, such definition will be such that for all (yi , y ), (yi ′ , y ′ ) ∈ XQ (U ) for which y ′ ≥ y we have that these distributions are such that • y = y and y′ = y ′ , • (yj , y ), (yj ′ / XQ (U ) and thus p(yj , y ) = p(yj , y′) ∈ ′ , y ′ ) = 0, y′ • yj ′ ≥ y j y, • uj (yj ′ , y ′ ) > uj (yj , y ) for all uj ∈ U B because utility is strictly increasing in own income when relative income is kept constant, which shows that the precondition for our axioms are met. As Lemma S.1 shows, one can also find distributions with the required prop- erties when y denotes mean income. We consider here the case of two bundles ′ (y, y ), (y ′ , y ′ ) ∈ XQ (U ) with y ′ ≥ y such that y ′ ≤ y y y , because this is the rele- vant case for which there exist two preferences u, u ∈ U B that disagree on the ′ comparison of these bundles, i.e., u(y, y ) > u(y ′ , y ′ ) and u′ (y, y ) ≤ u′ (y ′ , y ′ ).10 Lemma S.1. Consider any U ⊆ U B and any (y, y ), (y ′ , y ′ ) ∈ XQ (U ) such that y ′ ≥ y and ′ y′ ≤ y y y . There exist distributions y, y′ ∈ Y n such that (i) y = y and y′ = y ′ , (ii) (y1 , y ) = (y, y ), (y1′ , y ′ ) = (y ′ , y ′ ), (iii) (yj , y ), (yj ′ , y′) ∈ / XQ (U ) for all j ̸= i, (iv) y ′ ′ ′ ′ yj ≥ y yj and (v) uj (yj , y ) > uj (yj , y ) for all j ̸= i and all uj ∈ U B . Proof. We show that when (y1 , y ) := (y, y ), (y1 ′ , y ′ ) := (y ′ , y ′ ) and yj := y + (y − y1 )/(n − 1), ′ yj ′ := y ′ + (y ′ − y1 )/(n − 1), for all j ̸= 1, the distributions y, y′ ∈ Y n have the desired properties. 9 More precisely, they constrain the comparison of distribution-profile pairs. 10 Additional arguments are required in order to extend Lemma S.1 for the comparison of bundles (y, y ) and (y ′ , y ′ ) for which u(y, y ) < u(y ′ , y ′ ) for all u ∈ U B . 12 Part (i) is immediate given that ny = i yi when y denotes mean income. Part (ii) is by construction. / XQ (U ). By Lemma 2, it is sufficient to show Part (iii) requires that (yj , y ) ∈ that yj ≥ y . As (y1 , y ) ∈ XQ (U ), we have that y1 < y (Lemma 2). As all j ̸= 1 earn the same income and y1 < y , we have that ny = i yi only when yj ≥ y , as desired. ′ ′ Part (iv) requires that yj ′ ≥ yy j y . By construction, we have that y1 ′ ≤ yy 1 y y′ because y ≤ y y . As all j ̸= 1 earn the same income yj in y and yj in y , we have ′ ′ ′ ′ y′ y′ that i yi ′ =y y i yi only if yj ≥ y yj because y1 ≤ y y1 , as desired. ′ ′ ′ Part (v) is a direct consequence of the fact that yj ′ ≥ y y j y because utility is strictly increasing in own income when relative income is kept constant. ■ S5.2 Adapting the Proof for Proposition 1 We show how to adapt the key argument provided in Step 2 of Proposition 1. Step 2. PU satisfies Domination only if for all (y, u) ∈ XU 1 PU (y, u) = p(za , za ) + p′ (yi , y ) (S.1) n(y) i∈Q∗ (y) where Q∗ (y) = {i ∈ N (y)|(yi , y ) ∈ XQ (U )} and p′ : X → R is well-defined on X and continuous on XQ (U ). Recall the definition of the supremum function z ˆ : [za , ∞) → R+ ˆ(y ) := sup{y ≥ 0|u(za , y z ) = u(y, y ) for some u ∈ U }, z which is such that Xz ˆ(y ). ˆ = XQ (U ) because (y, y ) ∈ XQ (U ) if and only if y < z We do not repeat the argument showing that p(y, y ) = p(za , za ) for all y ≥ za . Rather, we only establish Eq. (S.1) for the case for which y denotes mean income. We first show, for all (y, y ) ∈ X \XQ (U ), that p(y, y ) = α(y )(y − y ) + p(za , za ) (S.2) for some continuous function α : [za , ∞) → R. We start by showing Eq. (S.2) for an arbitrary but fixed level of mean income ˆ = α(y ) ∈ R such that y ∈ [za , ∞). We derive a contradiction if there exists no α Eq. (S.2) holds. Under this contradiction assumption, there must exist δ > 0 and 13 z (y ), ∞) for which11 two income levels y, y ′ ∈ [ˆ p(y + δ, y ) − p(y, y ) ̸= p(y ′ + δ, y ) − p(y ′ , y ). (S.3) Consider any two distributions y, y′ ∈ Y n such that y = y′ = y , y1 = y + δ , y2 = y ′ , y1 ′ = y , y2 ′ = y ′ + δ , and yj = yj ′ for all j ∈ {3, . . . , n}.12 Since y ≥ z ˆ(y ), ′ y ≥z ˆ(y ) and δ > 0, bundles (y, y ), (y + δ, y ), (y , y ), (y + δ, y ) are all in X \XQ (U ). ′ ′ Therefore, Domination implies for all u ∈ U n that PU (y, u) = PU (y′ , u). Using Eq. (S.5), we get p(y + δ, y ) + p(y ′ , y ) = p(y ′ + δ, y ) + p(y, y ), a contradiction to Eq. (S.3). There remains to show that function α is continuous. Assume to the contrary that α is discontinuous at some level of mean income y ∗ ∈ [za , ∞). For the two income levels y p := z ˆ(y ∗ ) and ˆ(y ∗ )/2 and y r := 2y ∗ − y p , which are such that y p < z y r > y ∗ , consider the income distribution y := (y p , y r ). By construction, we have y = y ∗ . First, consider the case for which p(y r , y ∗ ) < limϵ→0 p(y r , y ∗ + ϵ) for ϵ > 0. For some γ > 0, consider the income distribution yγ := (y p + γ, y r ), which is such that yγ = y ∗ + γ/2. For γ small enough, we have bundle (y p + γ, y ∗ + γ/2) in XQ (U ) and bundle (y r , y ∗ + γ/2) in X \XQ (U ). Also, the monotonicity properties of utility functions imply that u(y p + γ, y ∗ + γ/2) > u(y p , y ∗ ) for all u ∈ U B because the relative income of the former bundle is larger. Therefore, Domination implies for all u ∈ U 2 that PU (y, u) ≥ PU (yγ , u) for all γ > 0 sufficiently small, which by Eq. (S.5) means p(y p , y ∗ ) + p(y r , y ∗ ) ≥ p(y p + γ, y ∗ + γ/2) + p(y r , y ∗ + γ/2). (S.4) However, since (y p , y ∗ ), (y p + γ, y ∗ + γ/2) ∈ XQ (U ), the continuity of function p on XQ (U ) implies that limγ →0 p(y p + γ, y ∗ + γ/2) = p(y p , y ∗ ). This leads to a contra- diction to Eq. (S.4) because this case is such that p(y r , y ∗ ) < limγ →0 p(y r , y ∗ +γ/2). The proof for the alternative case p(y r , y ∗ ) > limϵ→0 p(y r , y ∗ + ϵ) for ϵ > 0 is based ′ ′ on another distribution yγ constructed as yγ := (y p − γ, y r +2γ ). As the reasoning is very similar (the main difference being the direction of inequalities) we do not develop this case. Finally, the cases for which the discontinuity of function α at 11 ˆ(y ) is such that (y ′′ , y ) ∈ Any income level y ′′ < z / X \XQ (U ). 12 Observe that, for all values of y , y ′ and δ , there exists a value for n sufficiently large to ′ ensure that yj = yj ≥ 0. 14 y ∗ comes “from the left” use the same reasoning. This proves that α is continuous, and Eq. (S.2) holds. We use Eq. (S.2) in order to prove Eq. (S.1). Let p′ : X → R be defined for all (y, y ) ∈ X as p′ (y, y ) := p(y, y ) − α(y )(y − y ) + p(za , za ) , where function p′ is continuous on XQ (U ) because function α is continuous and function p is continuous on XQ (U ). Letting PU ′ (y, u) := i =1 p (y, y ), we get from n(y) ′ the definition of p′ that PU ′ (y, u) = PU (y, u) − p(za , za ) because i (yi − y ) = 0 when y denotes mean income. The definition of p′ together with Eq. (S.2) implies that p′ (y, y ) = 0 for all (y, y ) ∈ X \XQ (U ), which shows that p′ (yi , y ) = 0 for all i ∈ N (y)\Q∗ (y). Together, we obtain Eq. (S.1). S6 Proof of Proposition 1 In this section, we show that an additive index satisfies Domination only if it is a fair additive index. Take any additive index PU : XU → R. The proof of Proposition 1 has three steps. In Step 1, we show that index PU satisfies Domination only if it is independent of the particular preference profile u. In Step 2, we show that index PU satisfies Domination only if its poverty score function is constant on X \XQ (U ) and weakly decreasing in own income on XQ (U ). In Step 3, we construct a particular function z and show that PU satisfies Domination only if it satisfies the definition of a fair additive index. Step 1. PU satisfies Domination only if for all (y, u) ∈ XU n(y) 1 PU (y, u) = p(yi , y ), (S.5) n(y) i=1 where function p : X → R is well-defined on X and continuous on XQ (U ). Being an additive index, PU is defined as n(y) 1 PU (y, u) = pui (yi , y ), (S.6) n(y) i=1 15 where for every u ∈ U , pu : X → R is well-defined on X and continuous on XQ (u). First, we show that the poverty score function p : X × U → R such that p : (y, y, u) → pu (y, y ) is independent of u. Suppose, to the contrary, that there exist u, u′ ∈ U such that, for some (y, y ) ∈ X , pu (y, y ) ̸= pu′ (y, y ). It is possible to construct two pairs (y, u), (y, u′ ) ∈ XU such that y has y1 = y and y = y and such that u1 = u, u′1 = u′ and uj = u′j for all j ∈ N (y) \ {1}. By Eq. (S.6), we have PU (y, u) ̸= PU (y, u′ ) because by construction pu1 (y1 , y ) ̸= pu′1 (y1 , y ) and puj (yj , y ) = pu′j (yj , y ) for all j ∈ N (y) \ {1}. As the two distribution-profile pairs feature the same distribution y, Domination implies that both PU (y, u′ ) ≤ PU (y, u) and PU (y, u′ ) ≥ PU (y, u). Therefore we have PU (y, u) = PU (y, u′ ), yielding the desired contradiction. We have thus shown that PU is based on a degenerate poverty score function p : X → R, which is well-defined on X . Second, we show that the function p : X → R is continuous on XQ (U ). For any bundle (y, y ) ∈ XQ (U ) there exists some u ∈ U such that (y, y ) ∈ XQ (u) (by the definition of XQ (U )). By the definition of an additive index, pu is continuous on XQ (u). Thus p is continuous at (y, y ) because (y, y ) ∈ XQ (u) and p = pu . This concludes the proof for Step 1. Step 2. PU satisfies Domination only if for all (y, u) ∈ XU 1 PU (y, u) = p(za , za ) + p′ (yi , y ) (S.7) n(y) i∈Q∗ (y) where Q∗ (y) := {i ∈ N (y)|(yi , y ) ∈ XQ (U )} and p′ : X → R is well-defined on X and continuous on XQ (U ). Moreover, p′ is weakly decreasing in its first argument on XQ (U ) and p′ (yi , y ) ≥ 0 for all i ∈ Q∗ (y). First, we establish Eq. (S.7). For any (y, u) ∈ XU , Eq. (S.5) implies n(y) 1 PU (y, u) = p(yi , y ) n(y) i=1 n(y) 1 = p(za , za ) + (p(yi , y ) − p(za , za )) . n(y) i=1 Let function p′ be defined as p′ (y, y ) := p(y, y ) − p(za , za ) for all (y, y ) ∈ X . Function p′ inherits continuity on XQ (U ) from function p. If p′ (y, y ) = 0 for all 16 (y, y ) ∈ X \XQ (U ), then we would obtain 1 PU (y, u) = p(za , za ) + p′ (yi , y ) n(y) i∈Q∗ (y) because Q∗ (y) = {i ∈ N (y)|(yi , y ) ∈ XQ (U )}. In order to prove that p′ (y, y ) = 0 for all (y, y ) ∈ X \XQ (U ), we show that p(y, y ) = p(za , za ) for any (y, y ) ∈ X \XQ (U ). Consider the two distributions y, y′ ∈ Y 3 defined as y := (y, y, y ) and y′ := (y, y, y ). Note that y = y′ = y . We show that (yi , y ), (yi ′ / XQ (u) for all i ∈ {1, 2, 3} and all u ∈ U . This is , y′) ∈ immediate for (y1 , y ) = (y, y ) because we assumed (y, y ) ∈ X \XQ (U ). All other bundles correspond to (y, y ), which by Lemma 2 is such that (y, y ) ∈ / XQ (U ) because y ≥ za . As (yi , y ), (yi ′ / XQ (u) for all i ∈ {1, 2, 3} and all u ∈ U , , y′) ∈ Domination implies PU (y, u) ≤ PU (y′ , u) and PU (y, u) ≥ PU (y′ , u) for all u ∈ U 3 . Therefore, we have PU (y, u) = PU (y′ , u). By Eq. (S.5), we also have that PU (y, u) = 3 1 p(y, y ) + 2 3 p(y, y ) and PU (y′ , u) = p(y, y ) for all u ∈ U 3 . Therefore, we have p(y, y ) = p(y, y ) for any (y, y ) ∈ X \XQ (U ) because PU (y, u) = PU (y′ , u). Thus, Eq. (S.7) holds if we can show that p(y, y ) = p(za , za ) for all y ≥ za . For any y ≥ za and for any n ∈ N, consider the distributions za := (za , . . . , za ) ∈ Y n and y := (y, . . . , y ) ∈ Y n . By construction, we have za = za and y = y . By Lemma / XQ (U ), which by definition implies that (za , za ), (y, y ) ∈ 2, (za , za ), (y, y ) ∈ / XQ (u) for all u ∈ U . It follows from Domination that PU (za , u) ≤ PU (y, u) and PU (za , u) ≥ PU (y, u), so PU (za , u) = PU (y, u) for all u ∈ U n . But by Eq. (S.5), PU (za , u) = p(za , za ) and PU (y, u) = p(y, y ), so p(y, y ) = p(za , za ) as required. Second, we show that function p′ is weakly decreasing in its first argument on XQ (U ) and that p′ (yi , y ) ≥ 0 for all i ∈ Q∗ (y). Let the “supremum” function z ˆ : [za , ∞) → R+ be defined as ˆ(y ) := sup{y ≥ 0|u(za , y z ) = u(y, y ) for some u ∈ U }. z The definition of function z ˆ is such that for any (y, y ) ∈ X we have y < z ˆ(y ) if and only if (y, y ) ∈ XQ (U ). In our notation, this means that Xz ˆ = XQ (U ). Therefore, we have i ∈ Q (y) if and only if yi < z ∗ ˆ(y ). We start by showing that function p′ is weakly decreasing in its first argument on XQ (U ). We must thus show that p′ (y, y ) ≥ p′ (y ′ , y ) for all (y ′ , y ) ∈ XQ (U ) for which y ′ ∈ [y, z ˆ(y )). Consider the two distributions y, y′ ∈ Y 3 defined as y := (y, y, y ) and y′ := (y ′ , y, y ). By construction, we have y = y′ = y . By 17 Lemma 2, we have that (y2 , y ), (y2 ′ , y ′ ), (y3 , y ), (y3′ , y′) ∈/ XQ (U ). We also have u(y1 , y ) < u(y1 , y ) for all u ∈ U because y1 < y1 and y = y ′ . Thus, Domination ′ ′ ′ implies that PU (y, u) ≥ PU (y′ , u) for all u. Therefore, Eq. (S.7) implies that p′ (y, y ) ≥ p′ (y ′ , y ), as desired. We now show that p′ (yi , y ) ≥ 0 for all i ∈ Q∗ (y). We must thus show that p′ (yi , y ) ≥ 0 for all i ∈ N (y) for whom yi < z ˆ(y ). We have shown above that p′ (y, y ) = 0 when (y, y ) ∈ / XQ (U ), i.e., when y ≥ z ˆ(y ). There remains to show that p (yi , y ) ≥ p (y, y ) when yi < z ′ ′ ˆ(y ) ≤ y . This is implied by the reasoning used in the previous paragraph when modifying the definitions of distributions y and y′ in such a way that y1 = yi and y1 ′ = y , the desired result. This concludes the proof for Step 2. Step 3. PU satisfies Domination only if it satisfies the definition of a fair additive index. We construct a continuous function z : [za , ∞) → R+ for which function p′ defined in Step 2 satisfies the definition of a fair additive index. This function z is defined using function zˆ and function p′ as follows ˆ(y )]|p′ (y, y ) = 0}. z (y ) := min{y ∈ [0, z We show that function z is well-defined. Recall that p′ (y, y ) = 0 for all (y, y ) ∈ / XQ (U ). We therefore have p (ˆ ′ z (y ), y ) = 0 for all y ≥ za because the definition of function z ˆ is such that (ˆ / XQ (U ). We then get that z is well-defined z (y ), y ) ∈ because function p is continuous in its first argument for all y ∈ [0, z ′ ˆ(y )). The definition of z is such that Xz ⊆ Xz ˆ, which implies Xz ⊆ XQ (U ). Function z is continuous because function z ˆ is continuous13 and function p′ is continuous on XQ (U ). Consider first the special case for which Xz = ∅. This case is such that p′ (y, y ) = 0 for all (y, y ) ∈ X . Hence, we have z (y ) = 0 for all y ≥ za . All the necessary properties are trivially satisfied when Xz = ∅. There remains the case for which Xz ̸= ∅. We show properties (i) to (v) in turn. Property (iii): function p′ is continuous on Xz because Step 2 shows that p′ is continuous on XQ (U ) and Xz ⊆ XQ (U ). Property (iv): function p′ is weakly decreasing in its first argument on Xz because Step 2 shows that p′ has this property on XQ (U ) and Xz ⊆ XQ (U ). Properties (i) and (ii): the definition 13 Function z ˆ(y ) := sup{k ≥ 0|u(za , y z ) = u(k, y ) for some u ∈ U } ˆ is continuous in y because z and all functions u ∈ U are continuous. 18 of z implies that p′ (y, y ) = 0 when y ≥ z (y ) and p′ (y, y ) > 0 when y < z (y ). This implies that p′ (y, y ) = 0 for all (y, y ) ∈ X \Xz and p′ (y, y ) > 0 for all (y, y ) ∈ Xz . There remains to show property (v), i.e., that p′ is weakly increasing in its second argument on Xz . Assume to the contrary that there are two bundles (y, y ), (y, y ′ ) ∈ Xz with y < y ′ such that p′ (y, y ) > p′ (y, y ′ ). Consider the two dis- tributions y, y′ ∈ Y 3 defined as y := (y, y, y ) and y′ := (y, y ′ , y ′ ). By construction, we have y = y and y′ = y ′ . By Lemma 2, we have that (y2 , y ), (y2 ′ , y ′ ), (y3 , y ), (y3 ′ , y′) ∈ / XQ (U ). We also have u(y1 , y ) ≥ u(y1 , y ) for all u ∈ U because utility func- ′ ′ tions are weakly decreasing in the median income. Thus, Domination implies that PU (y, u) ≤ PU (y′ , u) for all u. We also have p′ (yi , y ) = p′ (yi ′ , y′) = 0 for all i ∈ {2, 3} because Xz ⊆ XQ (U ). Therefore, Eq. (S.7) implies that p′ (y1 , y ) ≤ p′ (y1′ , y ′ ) because PU (y, u) ≤ PU (y′ , u). This yields a contradiction to p′ (y, y ) > p′ (y, y ′ ). We have thus shown that properties (i) to (v) are all satis- fied. S7 Proof of Proposition 2 ⇒. We show that any P{u} satisfying the two axioms has the required properties. By Proposition 1, P{u} satisfies Domination only if P{u} is a fair additive index. As P{u} satisfies Pareto , P{u} also satisfies Weak Pareto .14 By Lemma 4, we have Xz = XQ ({u}) = XQ (u). There remains to show that u(y ′ , y ′ ) ≥ u(y, y ) ⇔ p(y ′ , y ′ ) ≤ p(y, y ) for all (y, y ), (y ′ , y ′ ) ∈ Xz . As u represents a complete ordering on Xz , it is sufficient to show that for all (y, y ), (y ′ , y ′ ) ∈ Xz we have u(y ′ , y ′ ) = u(y, y ) ⇒ p(y ′ , y ′ ) = p(y, y ) and u(y ′ , y ′ ) > u(y, y ) ⇒ p(y ′ , y ′ ) < p(y, y ). We start by showing that for any two (y, y ), (y ′ , y ′ ) ∈ Xz we have u(y ′ , y ′ ) = u(y, y ) ⇒ p(y ′ , y ′ ) = p(y, y ). Consider the two distributions y := (y, y, y ) and y′ := / XQ ({u}) (Lemma (y ′ , y ′ , y ′ ), for which y = y and y′ = y ′ . We have (y, y ), (y ′ , y ′ ) ∈ 2). Consider profile u ∈ {u}3 . As (y, y ), (y ′ , y ′ ) ∈/ XQ ({u}) and u(y ′ , y ′ ) = u(y, y ), Domination implies P{u} (y, u) ≥ P{u} (y′ , u) and P{u} (y, u) ≤ P{u} (y′ , u), showing that P{u} (y, u) = P{u} (y′ , u). We have (y, y ), (y ′ , y ′ ) ∈/ Xz because (y, y ), (y ′ , y ′ ) ∈ / XQ ({u}) and Xz = XQ (u). Therefore, we have p(y, y ) = p(y , y ) = 0 because P{u} ′ ′ 14 Weak Pareto is a weakening of Pareto that we define in Section 4.3. Any fair additive index PU that satisfies Pareto also satisfies Weak Pareto . We prove in Lemma 4 that a fair additive index PU that satisfies Weak Pareto has Xz = XQ (U ). We allow ourselves to already reference Lemma 4 in order to avoid duplicating the argument. 19 is a fair additive index and (y, y ), (y ′ , y ′ ) ∈ / Xz . Therefore, P{u} (y, u) = P{u} (y′ , u) implies p(y, y ) = p(y ′ , y ′ ) because P{u} is a fair additive index, the desired result. We then show that for any two bundles (y, y ), (y ′ , y ′ ) ∈ Xz we have u(y ′ , y ′ ) > u(y, y ) ⇒ p(y ′ , y ′ ) < p(y, y ). Take any bundle (y ′′ , y ′′ ) ∈ Xz such that y ′′ > y and u(y ′′ , y ′′ ) = u(y ′ , y ′ ). By transitivity, we also have u(y ′′ , y ′′ ) > u(y, y ). As P{u} satisfies Weak Pareto , Lemma 3 implies that p(y ′′ , y ′′ ) < p(y, y ). Since u(y ′′ , y ′′ ) = u(y ′ , y ′ ), we must have p(y ′′ , y ′′ ) = p(y ′ , y ′ ), which shows that p(y ′ , y ′ ) < p(y, y ), the desired result. ⇐. We show that P{u} satisfies the two axioms. Domination : Take any (y, u), (y′ , u′ ) ∈ X{u} that satisfy the preconditions under which Domination implies PU (y′ , u′ ) ≤ PU (y, u). That is, we have n(y) = n(y′ ) and u(yi ′ , y ′ ) ≥ u(yi , y ) for all i ∈ N (y′ ) for whom (yi ′ , y ′ ) ∈ XQ (u).15 In order to prove P{u} (y′ , u′ ) ≤ P{u} (y, u), we show that p(yi ′ , y ′ ) ≤ p(yi , y ) for all i ∈ N (y′ ). First, consider any i ∈ N (y′ ) for whom (yi ′ / XQ (u). Since Xz = XQ (u) we , y′) ∈ have that (yi , y ) ∈ ′ ′ / Xz . Then, the definition of a fair additive index implies that p(yi ′ , y ′ ) = 0. This implies p(yi ′ , y ′ ) ≤ p(yi , y ) since p(yi , y ) ≥ 0 by the definition of a fair additive index. Second, consider any i ∈ N (y′ ) for whom (yi ′ , y ′ ) ∈ XQ (u). The precondi- tions of Domination imply that u(yi ′ , y ′ ) ≥ u(yi , y ). In turn, this shows that (yi , y ) ∈ XQ (u). Therefore, we have that (yi , y ), (yi ′ , y ′ ) ∈ Xz because Xz = XQ (u). This yields the result as u(yi ′ , y ′ ) ≥ u(yi , y ) ⇒ p(yi ′ , y ′ ) ≤ p(yi , y ) when (yi , y ), (yi ′ , y ′ ) ∈ Xz . Pareto : Take any (y, u), (y′ , u) ∈ X{u} that satisfy the preconditions under which Pareto implies PU (y′ , u) ≤ PU (y, u). That is, we have n(y) = n(y′ ), ui (yi′ , y ′ ) ≥ ui (yi , y ) for all i ∈ N (y′ ). The unanimous preference for distribution y′ implies that y ≤ y ′ (Lemma 1). In order to prove P{u} (y′ , u′ ) ≤ P{u} (y, u), we show p(yi ′ , y ′ ) ≤ p(yi , y ) for all i ∈ N (y′ ). Consider any i ∈ N (y′ ). The preconditions of Pareto imply that u(yi ′ , y′) ≥ u(yi , y ). The argument is the same as that given in the proof of Domination . Take any (y, u), (y′ , u) ∈ X{u} that satisfy the preconditions under which Pareto implies PU (y′ , u) < PU (y, u). Following the same argument, we have that p(yi′ , y ′ ) ≤ p(yi , y ) for all i ∈ N (y) = N (y′ ). In addition, we have uℓ (yℓ ′ , y′) > 15 On {u}, we have u = u′ = (u, . . . , u). 20 uℓ (yℓ , y ) for some ℓ ∈ Q(y, u). We must show that p(yℓ ′ , y ′ ) < p(yℓ , y ). As ℓ ∈ Q(y, u), we have (yℓ , y ) ∈ XQ (u). Therefore we have p(yℓ , y ) > 0 because (yℓ , y ) ∈ Xz since Xz = XQ (u). If (yℓ ′ / Xz , then p(yℓ , y′) ∈ ′ , y ′ ) = 0, which yields the result. Otherwise (yℓ ′ , y ′ ) ∈ Xz , and we then have u(yℓ ′ , y ′ ) > u(yℓ , y ) ⇒ p(yℓ ′ , y ′ ) < p(yi , y ) because (yℓ , y ), (yℓ ′ , y ′ ) ∈ Xz . S8 Relation between Weak Pareto and the Weak Relativity Axiom Ravallion and Chen (2011) use another welfare-consistency requirement. These authors note that the poverty index must be reduced when all incomes grow in the same proportion. This requirement can be expressed in our framework as follows. Axiom S.2 (Weak Relativity ). For all (y, u) ∈ XU and λ > 1, if y′ = λy and yℓ > 0 for some ℓ ∈ Q(y, u), then PU (y′ , u) < PU (y, u). Proposition S.4 shows that Weak Relativity is a weakening of Weak Pareto . Proposition S.4. Given any U ⊆ U B , the additive index PU satisfies Weak Pareto only if PU satisfies Weak Relativity. Proof. We show that the preconditions of Weak Pareto for a strict comparison are met when the preconditions of Weak Relativity are met. Take any (y, u), (y′ , u) ∈ XU B that satisfy the preconditions under which Weak Relativity implies PU (y′ , u) < PU (y, u). That is, y′ = λy for some λ > 1 and yℓ > 0 for some ℓ ∈ Q(y, u). ′ ′ First, we show that yj ′ ≥y y j y for all j ∈ / Q(y, u). As y′ = λy, we have yi ′ =y y i y for all i ∈ N (y). Second, we show that ui (yi ′ , y ′ ) ≥ ui (yi , y ) for all i ∈ N (y). For all i ∈ N (y) for ′ whom yi > 0 we have ui (yi ′ , y ′ ) > ui (yi , y ) because yi ′ = yy i y and utility functions are strictly increasing in own income when relative income is held constant. For all i ∈ N (y) for whom yi = 0 we have ui (yi ′ , y ′ ) = ui (yi , y ) because we have yi′ = yi = 0, which implies that ui (yi ′ ′ , yi /y ′ ) = ui (yi , yi /y ) = ui (0, 0). Finally, we show that uℓ (yℓ , y ′ ) > uℓ (yℓ , y ) for some ℓ ∈ Q(y, u). This follows ′ from the fact that there is yℓ > 0 for some ℓ ∈ Q(y, u), for whom yℓ ′ = y y ℓ y > yℓ . ■ 21 S9 Proof of Theorem 2 The proof is based on Lemmas S.2 and S.3. Lemma S.2. za Given any U ⊆ U B , for all y ≥ y z we have ( y z y, y ) ∈ / XQ (U ). Proof. Let λ := y/y z , which is such that λ ≥ 1. Since utility is increasing in own income when holding relative income constant, we have u(λza , λy z ) ≥ u(za , y z ) for all u ∈ U . This shows that u( yyz za , yyz y z ) ≥ u(za , y z ) for all u ∈ U and thus za (y / XQ (U ). z y, y ) ∈ ■ Lemma S.3. For all (y, y ) ∈ XQ (U B )\XA , there exists a utility function u ∈ U B such that (y, y ) ∈ XQ (u) and u(y, y ′ ) = u(y, y ) for all y ′ ≥ y . Proof. The proof is by construction. In Step 1, we construct a particular indiffer- ence curve that passes through bundle (y, y ), below the reference bundle (za , y z ) and that is flat for all y ′ ≥ y . In Step 2, we construct a utility function u ∈ U B that has one of its indifference curve that corresponds to the indifference curve constructed in the first step. If it is the case, we have indeed that (y, y ) ∈ XQ (u) and u(y, y ′ ) = u(y, y ) for all y ′ ≥ y . Step 1. The construction of the indifference curve is illustrated in Figure S.4. Take any s ∈ y y − za y , −y z y and let R := y − sy .16 Observe that we have s > 0 because y ≥ za as (y, y ) ∈ / XA . We also have R > 0 because s < y/y . The indifference curve is defined by the function w′ : R+ → R++ , if y ′ < y,    R + sy ′ w′ (y ′ ) := (S.8) if y ′ ≥ y.  R + sy  This indifference curve passes through (y, y ) since w′ (y ) = y . This indifference curve passes below the reference bundle because we have w′ (y z ) < za since s > y −za y −y z . This indifference curve is flat beyond (y, y ) because w′ is constant for all y′ ≥ y. Step 2. We construct a utility function u ∈ U B such that u(w′ (y ′ ), y ′ ) = u(w′ (y ′′ ), y ′′ ) for all y ′ , y ′′ ∈ [za , ∞). The utility function u is defined for all 16 y −z a y za We have that y −y z < y because we have y < y z y , which itself follows from the fact that za (y, y ) ∈ XQ (U B )\XA and Xz∗∗ = XQ (U B ) and z ∗∗ (y ) = y z y. 22 Figure S.4: Construction of the indifference curve w′ (in blue) passing through bundle (y, y ). (y ′ , y ′ ) ∈ X as y′ u(y ′ , y ′ ) := . (S.9) w′ (y ′ ) The construction is such that u(w′ (y ′ ), y ′ ) = u(w′ (y ′′ ), y ′′ ) = 1 for all y ′ , y ′′ ∈ [za , ∞). There remains to show that u ∈ U B . First, u is continuous because w′ is continuous. Then, u is strictly increasing in own income. Also, u is weakly de- creasing in the median income because w′ is a weakly increasing function. Finally, we must show that u is strictly increasing in own income when holding relative in- come constant. For all y ′ ≥ y , this follows from the fact that u is strictly increasing in own income and independent on the median income, i.e., independent on rela- tive income. For all y ′ < y , we can rewrite u for any y ′ ≥ 0 as u(y ′ , y ′ ) = R +1 s , y′ y ′ /y ′ which shows that u has the required property. ■ ⇒. We show that any PU B satisfying these two axioms has the required prop- erties. As the self-centered preference u0 ∈ U B , we have U ∗ ∩ U B ̸= ∅. By Proposition 4, if PU B satisfies Domination and Weak Pareto , then PU B is a hierarchical index. By definition of a hierarchical index, we have Xz = XQ (U B ). We show that Xz = Xz∗∗ , i.e. z (y ) = z ∗∗ (y ) for all y ≥ za . The proof exploits the fact that Xz = XQ (U B ). The argument showing that z (y ) = za for all y ≤ y z is the same as that given in the proof of Theorem 1. We then show that z (y ) = y z y for all y > y . Take any y > y . We start za z z by showing that z (y ) ≤ y z y. za As Xz = XQ (U B ), we must have z (y ) ≤ y ′ for / XQ (U B ). By Lemma S.2, we have ( y all y ′ such that (y ′ , y ) ∈ za z y, y ) ∈/ XQ (U B ), which implies z (y ) ≤ y z y . There remains to show that z (y ) ≥ y z y . Take any za za 23 z y ), it is sufficient to show that there exists some u ∈ U B such that za σ y ∈ [za , y (y, y ) ∈ XQ (uσ ). Indeed, we would then have (y, y ) ∈ XQ (U B ), which directly im- plies z (y ) > y as Xz = XQ (U B ). This in turn yields z (y ) ≥ y z y because z (y ) > y za for all y ∈ [za , yz y ). There remains to prove that such u ∈ U exists. There exists za σ B some y ′ ∈ (y, y z y ) because y ∈ [za , y z y ). By Lemma 5, Parts (i) and (ii), we have za za ∗ ∗ ∗ ∗ uσ (y ′ , y ) = uσ (za , y z ) for some σ ∗ ≥ 0.17 This implies that uσ (y, y ) < uσ (za , y z ) ∗ because y ′ > y . The preference uσ has the required properties, as desired. We have thus shown that z (y ) = z ∗∗ (y ) for all y ≥ za . There remains to show that the poverty score function p has the required prop- erties. Assume to the contrary that there are two bundles (y, y ), (y, y ′ ) ∈ Xz \XA with y < y ′ such that p(y, y ) ̸= p(y, y ′ ). As Xz = XQ (U B ), we thus have (y, y ), (y, y ′ ) ∈ XQ (U B )\XA . As function p is weakly increasing in the median income, this implies that p(y, y ) < p(y, y ′ ). By Lemma S.3, there exists a pref- erence u′ ∈ U B such that (y, y ) ∈ XQ (u′ ) and u′ (y, y ′ ) = u′ (y, y ). As y < y ′ and PU B is a fair additive index satisfying Weak Pareto , Lemma 3 implies that p(y, y ) ≥ p(y, y ′ ), the desired contradiction. ⇐. We show that PU B satisfies the two axioms. Domination : Take any (y, u), (y′ , u′ ) ∈ XU B that satisfy the preconditions under which Domination implies PU B (y′ , u′ ) ≤ PU B (y, u). That is, we have n(y) = n(y′ ) and u(yi ′ , y ′ ) ≥ u(yi , y ) for all i ∈ N (y′ ) and all u ∈ U B such that (yi ′ , y ′ ) ∈ XQ (u). In order to prove PU B (y′ , u′ ) ≤ PU B (y, u), we show that p(yi ′ , y ′ ) ≤ p(yi , y ) for all i ∈ N (y′ ). Take any i ∈ N (y′ ) for whom (yi ′ , y′) ∈ / Xz . Since PU B is a fair additive index, we have p(yi , y ) = 0 and thus p(yi , y ) ≤ p(yi , y ). ′ ′ ′ ′ Take any i ∈ N (y′ ) for whom (yi ′ , y ′ ) ∈ Xz . We show that yi ′ ≥ yi and (yi , y ), (yi , y ) ∈ Xz , which implies p(yi , y ) ≤ p(yi , y ) by the properties of func- ′ ′ ′ ′ tion p. First, we show that (yi , y ) ∈ Xz . As Xz = XQ (U B ), there exists some u′ ∈ U B such that (yi ′ , y ′ ) ∈ XQ (u′ ). Therefore, the precondition of Domina- tion implies that u′ (yi ′ , y ′ ) ≥ u′ (yi , y ). In turn, we have (yi , y ) ∈ XQ (u′ ) because (yi′ , y ′ ) ∈ XQ (u′ ) and u′ (yi ′ , y ′ ) ≥ u′ (yi , y ). This shows that (yi , y ) ∈ Xz because Xz = XQ (U B ). There remains to show that yi ′ ≥ yi . Assume to the contrary that za 17 We can invoke Lemma 5 if y ′ < z ∗ where z ∗ := R + Rσ ¯ y and R := 1+¯σ y z . By definition, za z∗ → y z y as σ¯ → ∞. Hence, there must exist some large enough σ ¯ > 0 for which y ′ < z ∗ za because y ′ < y z y. 24 yi > yi′ . We have (yi ′ , y ) ∈ Xz because (yi , y ) ∈ Xz and yi > yi ′ . By Lemma S.3, there exists some u ∈ U such that u(yi , y ) = u(yi , y ) and (yi , y ′ ) ∈ XQ (u). Since B ′ ′ ′ ′ ′ yi > yi , we must have u(yi ′ , y ′ ) < u(yi , y ). This is a contradiction to the precondi- tion of Domination that requires u(yi ′ , y ′ ) ≥ u(yi , y ) because (yi ′ , y ′ ) ∈ XQ (u), the desired result. Weak Pareto : Take any (y, u), (y′ , u) ∈ XU B that satisfy the preconditions under which Weak Pareto implies PU B (y′ , u) ≤ PU B (y, u). That is, we have ′ n(y) = n(y′ ), ui (yi ′ , y ′ ) ≥ ui (yi , y ) for all i ∈ N (y′ ) and yj ′ ≥ y y j y for all j ∈ / Q(y, u). The unanimous preference for distribution y implies that y ≤ y (Lemma ′ ′ 1). In order to prove PU B (y′ , u) ≤ PU B (y, u), we show that p(yi ′ , y ′ ) ≤ p(yi , y ) for all i ∈ N (y′ ). Take any i ∈ N (y′ ) for whom (yi ′ , y′) ∈ / Xz . Since PU B is a fair additive index, we have p(yi , y ) = 0 and thus p(yi , y ) ≤ p(yi , y ). ′ ′ ′ ′ Finally, take any i ∈ N (y′ ) for whom (yi ′ , y ′ ) ∈ Xz . First, we show this case is such that (yi , y ) ∈ Xz . Indeed, if (yi , y ) ∈ / Xz , then / XQ (U B ). Therefore, i ∈ (yi , y ) ∈ / Q(y, u) and the precondition of Weak Pareto y′ requires yi ≥ y yi . This implies in turn that u(yi ′ ′ , y ′ ) ≥ u(yi , y ) for all u ∈ U B because y ′ ≥ y and utility is increasing in own income when relative income is kept constant. As (yi , y ) ∈ / XQ (U B ) and u(yi ′ , y ′ ) ≥ u(yi , y ) for all u ∈ U B , we must have that (yi ′ / XQ (U B ). We therefore get a contradiction to (yi , y′) ∈ ′ , y ′ ) ∈ Xz because Xz = XQ (U B ) Now, the other precondition of Weak Pareto requires that ui (yi ′ , y ′ ) ≥ ui (yi , y ) for all i ∈ N (y′ ). The two inequalities y ≤ y ′ and ui (yi ′ , y ′ ) ≥ ui (yi , y ) together imply that yi ′ ≥ yi because individual utility functions are weakly decreasing in the median income. As (yi , y ), (yi ′ , y ′ ) ∈ Xz and yi ′ ≥ yi , the properties of function p yield p(yi , y ) ≤ p(yi , y ), the desired result. ′ ′ Take any (y, u), (y′ , u) ∈ XU B that satisfy the preconditions under which Weak Pareto implies PU B (y′ , u) < PU B (y, u). The proof is a straightforward adaptation of arguments used above, and is thus omitted. S10 Robustness of main theorem on U C Theorem S.1. The additive index PU C satisfies Domination and Weak Pareto if and only if PU C 25 is a hierarchical index with global line z ∗∗ defined for all y ≥ za as za z ∗∗ (y ) := max za , y , yz where R := za 1+¯σyz , and whose poverty score function p is such that for all (y, y ), (y ′ , y ′ ) ∈ Xz∗∗ \XA 18 y − za y ′ − za p(y, y ) = p(y ′ , y ′ ) when = . y − yz y′ − yz Proof. ⇒. We show that any PU C satisfying these two axioms has the required properties. As the self-centered preference u0 ∈ U C , we have U ∗ ∩ U C ̸= ∅. By Proposition 4, if PU C satisfies Domination and Weak Pareto , then PU C is a hierarchical index. By definition of a hierarchical index, we have Xz = XQ (U C ). The proof that Xz = Xz∗∗ , i.e. z (y ) = z ∗∗ (y ) for all y ≥ za , follows the same ar- gument as that presented for Theorem 1 (for the case σ ¯ → ∞) and is thus omitted. ′ There remains to show that for all (y, y ), (y ′ , y ′ ) ∈ Xz∗∗ \XA with y y −za −y z y −za =y ′ −y z we have p(y, y ) = p(y , y ). Assume to the contrary that there are two bundles ′ ′ y ′ −za (y, y ), (y ′ , y ′ ) ∈ Xz∗∗ \XA with y y −za −y z = y ′ −y z but p(y, y ) ̸= p(y , y ). Without loss ′ ′ of generality, assume that y < y ′ . Since (y, y ) ∈ Xz∗∗ \XA , we have y > y z and y ≥ za . Since y < y ′ , we have y ≤ y ′ . • Case 1: p(y, y ) < p(y ′ , y ′ ). Since U σ ¯ ⊂ U C , the argument yielding a contradiction is the same as that given in the proof of Theorem 1. • Case 2: p(y, y ) > p(y ′ , y ′ ). Consider the two distributions y := (y, y, y ) and y′ := (y ′ , y ′ , y ′ ), which are, respectively, such that y = y and y′ = y ′ . We have that bundles / XQ (U C ) (Lemma 2). (y, y ), (y ′ , y ′ ) ∈ There exists some u ∈ U C for which (y, y ) ∈ XQ (u) because (y, y ) ∈ Xz∗∗ and Xz∗∗ = XQ (U C ) (Lemma 4). Let u = (u, u, u) ∈ (U C )3 . 18 Recall that, by definition, a hierarchical index also has for all (y, y ), (y ′ , y ′ ) ∈ XA that p(y, y ) = p(y ′ , y ′ ) when y = y ′ . 26 As Xz∗∗ = XQ (U C ), we have p(yi , y ) = p(yi ′ , y ′ ) = 0 for all i ∈ {2, 3}. Since ′ p(y1 , y ) > p(y1 , y ′ ), we must have PU C (y, u) > PU C (y′ , u). We derive a contradiction by showing that Domination implies PU C (y, u) ≤ PU C (y′ , u). By construction, we have Q(y′ , u′ ) ⊆ {1} for all u′ ∈ (U C )3 . Therefore, Domination implies PU C (y, u) ≤ PU C (y′ , u) if for all u′ ∈ (U C )3 for which Q(y′ , u′ ) = {1} (i.e. such that (y ′ , y ′ ) ∈ XQ (u′1 )) we have u′1 (y1 , y ) > ′ u′1 (y1 , y ′ ). Letting u′′ be any profile in (U C )3 for which Q(y′ , u′′ ) = {1}, we show that u′′ 1 (y1 , y ) > u1 (y1 , y ). The constructions are illustrated in Figure ′′ ′ ′ S.5. Let the income level y z be implicitly defined by u′′ 1 (y , y ) = u1 (y1 , y ). z z ′′ ′ ′ We must have y z < za because u′′ 1 (y1 , y ) < u1 (za , y ) as Q(y , u ) = {1}. ′ ′ ′′ z ′ ′′ The indifference curves associated to u′′ 1 are convex because u1 ∈ U . As ′′ C shown in Figure S.5, the convexity of the indifference curve of u′′ 1 passing through (y1 ′ , y ′ ) and (y z , y z ) implies that y′ − yz u′′ z 1 y + (y − y z ), y ≥ u′′ ′ ′ 1 (y1 , y ) y′ − yz because y z < y < y ′ . This implies that u′′ 1 (y1 , y ) > u1 (y1 , y ) because ′′ ′ ′ y′ − yz y > yz + (y − y z ), y′ − yz y ′ −y z where the last inequality follows from y = za + y ′ −y z (y − y z ) and y z < za . Figure S.5: Implications of convexity of the indifference curve (in blue) passing through (y z , y z ) and (y ′ , y ′ ). ⇐. We show that PU C satisfies the two axioms. Domination : Since U σ ¯ ⊂ U C , the argument is the same as that given in the proof of Theorem 1. 27 Weak Pareto : Since U σ ¯ ⊂ U C , the argument is almost the same as that given in the proof of Theorem 1. The only difference lies in proving that for any i ∈ N (y′ ) for whom (yi′ , y′) ∈ Xz∗∗ \XA , (yi , y ) ∈ Xz∗∗ \XA and i ∈ Q(y, u), the preconditions of Weak Pareto imply that19 ′ yi − za yi − za ′ z ≥ . y −y y − yz The preconditions of Weak Pareto imply that y ≤ y ′ (Lemma 1). As i ∈ Q(y, u), these preconditions also imply that ui (yi ′ , y ′ ) ≥ ui (yi , y ). The reasoning is illustrated in Figure S.6. As i ∈ Q(y, u), the indifference curve of ui pass- ing through (yi , y ) must pass below the reference bundle (za , y z ). As ui (yi ′ , y′) ≥ ui (yi , y ), the indifference curve of ui passing through (yi , y ) cannot pass above bun- dle (yi ′ , y ′ ). The indifference curves associated to ui are convex because ui ∈ U C . Together, the indifference curve of ui passing through (yi , y ) satisfies these con- straints and is convex only if ′ yi − za ′ yi ≥ za + (y − y z ), y − yz ′ −z yi because y ≤ y ′ . A few manipulations yield ′ y −y z a ≥ yi −za y −y z . Figure S.6: Implications of the convexity of the indifference curve (in blue) passing through (yi , y ). ■ 19 More precisely, in the proof of Theorem 1, we can no longer consider uσ ˆ := ui in order ˆ σ to prove the existence the utility function u with the claimed properties because there is no ¯ guarantee that ui ∈ U σ . 28 S11 Utility strictly increasing in relative income We explain why Theorem 1 is robust to a framework for which utility is strictly increasing in relative income. Let U σ ¯0 be the subset of utility functions uσ defined by Eq. (1) for which 0 < σ < σ ¯ for some σ ¯ > 0. The set U σ ¯0 does not contain self-centered preferences (because σ ̸= 0). The proof of an equivalent of Theorem 1 for set U σ ¯0 is almost identical, except when we show that PU σ ¯ 0 is a hierarchical index. We cannot invoke Proposition 4 because U ∩ U = ∅ since σ > 0. By Proposition 1, PU σ ∗ σ ¯0 ¯ 0 satisfies Domination only if PU σ ¯ 0 is a fair additive index. By Lemma 4, any fair additive index that satisfies Weak Pareto is such that Xz = XQ (U σ ¯0 ). The proof that p is strictly decreasing in its first argument on Xz is the same as in the proof of Proposition 4 and is thus omitted. There remains to show that for all (y, y ), (y, y ′ ) ∈ XA ∩ Xz , we have p(y, y ) = p(y, y ′ ). Consider the contradiction assumption that for y ∈ [0, za ) and y < y ′ we have p(y, y ) ̸= p(y, y ′ ). Since PU σ ¯ 0 is a fair additive index, p is weakly increasing in its second argument, and thus we must have p(y, y ) < p(y, y ′ ). As p is continuous on Xz , we have that p is continuous on XA because XA ⊆ Xz .20 As p is continuous on XA , there exists some ϵ ∈ (0, za − y ) such that p(y, y ) < p(y + ϵ, y ′ ). As ϵ > 0 and y < y ′ , there exists some sufficiently small σ ′ > 0 such ′ ′ that uσ (y, y ) < uσ (y + ϵ, y ′ ). 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