Conciliating absolute and relative poverty: Income poverty measurement with two poverty lines.

Transcription

1 Conciliating absolute and relative poverty: Income poverty measurement with two poverty lines. Benoit Decerf May 16, 2018 Abstract I study income poverty indices in a framework considering two poverty lines: one absolute line capturing subsistence and one relative line capturing social exclusion. I show that poverty indices accounting for these two lines should be hierarchical additive. Hierarchical additive indices grant a form of priority to subsistence: they always implicitly consider that absolutely poor individuals are worse-off than relatively poor individuals. Importantly, classical additive indices are not hierarchical. As a result, they yield debatable poverty comparisons of societies with different standards of living. I derive a new hierarchical index that generalizes the ubiquitous Head-Count Ratio. This index sums the fraction of absolutely poor individuals with the fraction of relatively poor individuals multiplied by an endogenous weight. An empirical application illustrates how to apply the new index and contrasts its poverty comparisons with those obtained using the extreme poverty measure of the World Bank. JEL: D63, I32. Keywords: Income Poverty Measurement, Poverty Line, Relative Poverty, Absolute poverty, Additive Indices. Acknowledgments : I express all my gratitude to Francois Maniquet for his many comments and suggestions. I am grateful to Mery Ferrando, Martin Van der Linden and John Weymark who extensively commented on earlier versions of this document. I thank all the participants to the 5th ECINEQ conference (2013), the 12th meeting of the Society for Social Choice and Welfare (2014) and later seminars at UCLouvain, KULeuven, U. Laval and U. Bielefeld, in particular C. d Aspremont, K. Decancq, J. Foster, I. Gilboa, M. Ravallion, F. Riedel, E. Schokkaert and P. van Parijs. I am grateful to CORE, my host institution for most of the time during which I worked on this research. All remaining mistakes are of course mine. Fundings from the European Research Council under the EU s Seventh Framework Program (FP/ ) / ERC Grant Agreement n and from the Fond National de la Recherche Scientifique (Belgium, mandat d aspirant FC 95720) are gratefully acknowledged. University of Namur. benoit.decerf@unamur.be 1

2 1 Introduction There are two different approaches to income poverty measurement: the absolute and the relative approach. An individual is deemed absolutely poor if her income is insufficient to cover her subsistence needs. Covering these needs requires consuming goods such as food, clothes or a dwelling. As the real cost for these goods should not evolve substantially with standards of living, the threshold of an absolute poverty line is independent on standards of living. This is for instance the approach underlying the extreme poverty line of the World Bank, whose threshold is $1.9 per person per day (Ferreira et al., 2016). In turn, an individual is deemed relatively poor if her income is so much smaller than the income standard in her society that she is at risk of social exclusion. 1 The real cost of social participation evolves with standards of living. Therefore, the threshold of a relative poverty line depends on the income standard. A well-known example are strongly relative lines, whose income threshold corresponds to a given fraction of mean or median income. Strongly relative lines have now been adopted in most OECD countries. Many policymakers care for both subsistence and social exclusion. This is for instance the case of the World Bank and the European Commission who aim at reducing both absolute and relative poverty (World Bank, 2015; European Commission, 2015). The difficulty they face is that selecting poverty alleviation policies often implies making a trade-off between absolute and relative poverty. Typically, policies promoting growth reduce absolute poverty but may at the same time increase income inequality and relative poverty. In contrast, redistributive policies reduce relative poverty but may reduce growth. Therefore, evaluating the poverty impact of policies requires using a measure that combines the absolute and relative aspects of poverty. The research efforts aimed at combining absolute and relative poverty have mostly focused on the design of new poverty lines (Foster, 1998; Ravallion and Chen, 2011; Jolliffe and Prydz, 2017). This line of research has developed procedures to construct a hybrid line that combines an absolute and a relative line. Yet, a poverty measure is defined with two components: a poverty line and a poverty index (Sen, 1976). Poverty indices, like the Head-Count Ratio or the Poverty-Gap Ratio, aggregate the contributions to poverty of all poor individuals in a distribution. The properties of poverty indices have been extensively studied under the assumption that the poverty line is absolute (Zheng, 1997). Unfortunately, when combined with a hybrid line, these indices provide highly counterintuitive poverty comparisons because they deny a minimal priority to absolute poverty. Consider Table 1 that contrasts the situations of two poor individuals living in societies with different standards of living. The poor individual living in the low-income society earns $1.5 a day and is extremely poor. The poor individual living in the middle-income society earns $3 a day and is not extremely poor. The definition of classical poverty indices implies that the contribution to poverty of an individual decreases with her normalized income, i.e. her income divided by the poverty threshold. In Table 1, the normalized income of the extremely poor individual is larger because the hybrid threshold is smaller in her society. As a result, her contribution to poverty is smaller than the contribution of the individual earning $3 a day. Table 1: Classical indices may deny a minimal priority to absolute poverty Income Hybrid threshold Normalized income Low-income distribution Middle-income distribution This counterintuitive implicit ranking of individual situations has drastic implications for poverty comparisons. An unequal growth process that transforms the low-income distribution into the middle-income distribution is deemed poverty increasing even if this process eradicates extreme poverty. Decerf (2017) illustrates this issue empirically by contrasting the poverty figures for Brazil and Ivory Coast in Importantly, the same problem arises when evaluating such unequal growth process using a poverty 1 See Ravallion (2008) for a review of the normative foundations of the relative approach to poverty. 2

3 measure that weighs an absolute index with a relative index. Thus, producing two separate indices each capturing one aspect of income poverty and then aggregating these indices does not necessarily yield a sound yardstick for the evaluation of unequal growth. In this paper, I study poverty indices based on two poverty lines: one absolute line and one relative line. So far, only poverty indices based on an absolute line have been rigorously studied. My first main contribution is an axiomatic result showing that indices based on two lines should be hierarchical additive. Additive indices sum the contributions to poverty of all poor individuals in a distribution. Hierarchical indices are such that absolutely poor individuals always contribute more to poverty than relatively poor individuals. As a result, hierarchical additive indices avoid the limitation illustrated in Table 1. Importantly, classical indices are not hierarchical except in trivial cases. My second main contribution is the derivation of a new hierarchical additive index that generalizes the Head-Count Ratio. This new index adds the fraction of absolutely poor individuals to the fraction of relatively poor individual multiplied by an endogenous weight. An empirical illustration shows how this index affects cross-country poverty comparisons and demonstrates that the index yields intuitive judgments on unequal growth. First, I show that indices based on two lines should be hierarchical additive. The two poverty lines define two different poverty status. An individual is deemed absolutely poor if her income is below the absolute threshold; relatively poor if her income is above the absolute threshold but below the relative threshold and non-poor otherwise. The specificity of poverty indices is to satisfy a focus property forcing them to disregard the income of non-poor individuals. In the presence of two poverty lines, two focus properties should be satisfied. Each focus property is specific to the need captured by its associated poverty line. On one hand, in the absence of relatively poor individuals, the income of non-poor individuals is irrelevant. On the other hand, if some individuals are relatively poor, the income of non-poor individuals is irrelevant only as long as the income standard does not change. Together with weak versions of classical properties, these two focus properties characterize the family of hierarchical additive indices. This result shows that the contribution of a poor individual may depend on both her income and the income standard. However, the contribution of an absolutely poor individual only depends on her income. The poverty contribution function implicitly ranks individual situations across societies with different standards of living. The particularity of hierarchical indices is to always rank absolutely poor individuals below relatively poor individuals, regardless of income standards. The desirability of such a hierarchical ranking of individual situations is expressed in Atkinson and Bourguignon (2001). Moreover, this hierarchical ranking is largely reflected in the answers to a questionnaire experiment conducted all over the world by Corazzini et al. (2011). The characterization of hierarchical additive indices also holds when considering a single poverty line. In my framework, the two poverty lines reduce to a single relative line when the absolute threshold is equal to zero. Then, my main result provides a foundation to relative poverty measures based on additive indices. In turn, the two lines reduce to a single absolute line when the relative threshold does not depend on the income standard. Thus, the characterization of additive indices obtained by Foster and Shorrocks (1991) is nested in my main result. In both cases, hierarchical additive indices are trivially equivalent to additive indices. Therefore, my main result does not preclude the use of classical additive indices when a single poverty line is considered. Second, I show that any hierarchical additive index violates the Pigou-Dalton transfer principle and another basic property. This impossibility result reveals that, when both the absolute and relative aspects of poverty matter, selecting a poverty index requires making a trade-off between basic properties and hierarchical comparisons of individual situations. Either the index violates basic properties or the index implicitly considers that some absolutely poor individuals are less poor than some relatively poor individuals. As the limitation illustrated in Table 1 reveals, one may want to use a hierarchical index when evaluating policies impacting growth. In that case, the violation of basic properties is unavoidable. Third, I study a particular family of hierarchical additive indices whose mathematical 3

4 expression features two normative parameters. This family is a hierarchical version of the popular Foster-Greer-Thorbecke (FGT) family of indices (Foster et al., 1984). My last result shows that only one member of this family satisfies classical robustness and monotonicity properties. This index stands out by its simple interpretation as it is a particular generalization of the Head-Count Ratio for a framework considering two poverty lines. The value returned by this index is equal to the fraction of absolutely poor individuals plus the fraction of relatively poor individuals multiplied by an endogenous weight. This weight evolves linearly between zero and one as a function of the average income among relatively poor individuals. The closer this average income is to the relative (absolute) threshold, the closer the weight is to zero (one). All results are derived assuming that the income standard affecting the relative poverty threshold is mean income. Yet, most results are robust to the use of other income standards such as censored means, quantiles or even harmonic means. I detail in the text which results are not robust and explain how to adapt the proofs of results that are robust. Finally, I conduct an empirical application of the new index using World Bank data. This application illustrates how a hierarchical measure can be implemented and how its use leads to different poverty comparisons than those obtained with a classical measure based on a single poverty line. As expected, the hierarchical measure emphasizes poverty in countries with high-inequality more than a measure based on an absolute line, but less than a measure based on a relative line. When using the hierarchical measure, developed countries are the least poor, sub-saharan countries are the most poor and latin-american countries are poorer than conventionally reported. The poverty evaluation of unequal growth periods with the hierarchical measure can go in either direction. Typically, unequal growth is deemed poverty increasing if there is little growth and a strong increase in inequality and if there is little absolute poverty in the initial distribution. In contrast, the unequal growth process taking place in urban China over the period is a particularly telling example of poverty reduction. This process almost entirely eradicates the initial absolute poverty, but if the fraction of absolutely poor decreases from 20.9% to 1.3%, the fraction of relatively poor increases from 4.8% to 25%, implying that the total fraction of poor slightly increases from 25.7% to 26.4%. However, the new measure attributes to the relatively poor in urban China a weight of about 0.5. As a result, the value taken by this measure is halved over the period, from 23.3% to 11.5%. The paper is organized as follows. A succinct literature review is provided in Section 2. I present the framework in Section 3. I characterize the family of hierarchical additive indices and expose an impossibility result for these indices in Section 4. I characterize the new index in a hierarchical version of the FGT family in Section 5. The empirical illustration is presented in Section 6. I make some concluding remarks in Section 7. 2 Literature review The literature on income poverty measurement studies indicators called poverty measures that rank income distributions as a function of poverty. Any poverty measure is composed of two elements: a poverty line and an index. In his groundbreaking paper, Sen (1976) proposes a framework allowing to study the properties inherent to these indices. Following Sen, many authors have proposed particular families of indices and characterized their properties. Among other proposals are the indices studied by Foster et al. (1984), Foster and Shorrocks (1991), Kakwani (1980), Chakravarty (1983) or Duclos and Gregoire (2002). The major results derived in this literature are reviewed in Zheng (1997). This paper extends this literature by departing from the assumption that poverty indices are combined with an absolute line. A small literature launched by Atkinson and Bourguignon (2001) investigates indices that combine the absolute and relative aspects of income poverty. Atkinson and Bourguignon (2001) suggest to use two poverty lines, an absolute line capturing subsistence and a relative line capturing social exclusion. They propose a family of additive indices which are not hierarchical but do not study their properties. The same holds for Anderson and Esposito (2013). Finally, Decerf (2017) considers a different framework with one absolute threshold and one hybrid line whose threshold is everywhere above 4

5 the absolute threshold. He starts from the assumption that being absolutely poor is worse than being relatively poor and proposes a particular hierarchical index. There are three key differences between Decerf (2017) and this paper. First, the priority given to absolutely poor individuals is not assumed but derived from fundamental properties. Second, the current framework with two poverty lines that cross each other is more in line with the literature and its implementation requires making less normative assumptions. Decerf (2017) shows that, when the hybrid line crosses the absolute line, his index entirely disregards the relative aspect of income poverty. Third, the absolute component of the new index corresponds to the fraction of absolutely poor individuals. As a result, this index can easily complement any absolute head-count based measure, such as the extreme poverty measure of the World Bank or the US official poverty measure. The design of appropriate absolute, relative and hybrid poverty lines is still an active area of research (Foster, 1998; Ravallion and Chen, 2011, 2017; Allen, 2017). This paper does not contribute to such design. Rather, my starting point is to consider two poverty lines that cross each other, a premise in agreement with the literature designing hybrid lines. 3 The framework Let an income distribution y = (y 1,...,y n ) be a list of non-negative incomes sorted in non-decreasing order y 1 y n. The number of individuals in distribution y is denoted by n(y). Two poverty lines are considered. First, there is an absolute line whose threshold defines the minimal income necessary to cover an individual s subsistence needs. Its absolute threshold is denoted by z a 0. Second, there is a relative line whose threshold defines the minimal income necessary to be able to participate in social life. The relative line is defined by a threshold function z r : R + R + whose unique argument is the income standard in the society considered. There is currently no consensus about which statistic should be used as income standard when defining a relative line. In practice, most relative lines depend either on median income or on mean income. Yet, both statistics have been criticized. 2 Therefore, other statistics such as censored means or harmonic means could be considered in order to avoid the limitations of the median and the mean. 3 The selection of an appropriate income standard is the subject of ongoing research efforts (Ravallion and Chen, 2017). In order to simplify the exposition, I henceforth use mean income as the income standard. Yet, as discussed below, the main results are robust to other statistics such as quantiles, censored means and harmonic means. Letting y = distribution y, the threshold function considered is z r (y) = b+sy, yi n denote the mean income in where s [0,1) defines the slope of the relative line and b [0,z a (1 s)] defines the lower bound to social participation costs. As I impose that b z a (1 s), no individual is relatively poor when all individuals earn z a. This restriction also implies that the two poverty lines cross at a given mean income y c 0. This crossing property is necessary for some results to hold. Let Z denote the set of acceptable threshold functions. Following Atkinson and Bourguignon (2001) and Ravallion and Chen (2011), the upper contour of the absolute line and the relative line defines a hybrid line whose 2 Using the median as income standard may lead to paradoxical conclusions when inequality varies over time (de Mesnard, 2007). For instance, policies inducing regressive transfers from the middle class to the rich may actually decrease poverty. Easton (2002) reports that the official relative measure of New-Zealand exhibited this behavior over the period 1981 and Another concern is that the median only reflects one income in the distribution. Using the mean may also lead to strange conclusions because this statistic is influenced by the income of the super-rich. Moreover, the mean is less robust than the median to measurement errors. 3 For instance, a censored mean such as mean income among the 90% least rich is not influenced by the super-rich. Another possibility is to use a harmonic mean as this statistic depends on all incomes and weighs these incomes as a function of their position in the distribution. 5

6 y i y i (a) (b) (c) y i z r z r z r z a z a ȳ c ȳ b ȳ c ȳ z a ȳ Figure 1: Several pairs of absolute and relative poverty lines. (a) Positive absolute threshold and strongly relative line. (b) Positive absolute threshold and weakly relative line. (c) Null absolute threshold and strongly relative line. threshold function z : R + R + is z(y) = max{z a,z r (y)}. Particular cases of this hybrid line include strongly relative lines (z a = 0 and b = 0) and absolute lines (s = 0). Figure 1 illustrates several pairs of poverty lines. Individual i qualifies as poor in distribution y if her income is smaller than the hybrid threshold, i.e. if y i < z(y). The number of poor individuals in y is denoted by q(y). As income distributions are sorted, if i q(y) then individual i is poor in distribution y. Individual i qualifies as absolutely poor if her income is smaller than the absolute threshold or if her income is zero, i.e. if y i < z a or y i = 0. 4 The number of absolutely poor individuals in y is denoted by q a (y). The poor individual i q(y) is deemed relatively poor if her income is larger than the absolute threshold, i.e. if y i z a. An individual whose income is smaller than both thresholds is considered as absolutely poor. Therefore, the number of individuals considered as relatively poor in y is equal to q(y) q a (y). This classification of poor individuals implies that the two lines defining poverty status are the absolute line and the hybrid line. Letting N = {n N n 4}, the set of income distributions considered is Y = { y R N + y i y i+1, i = 1,...,n 1 and y n z(y) where n = n(y) }. This set excludes distributions for which all individuals are poor. This restriction on the set of distributions is necessary for the derivation of the main result (Theorem 1). A poverty index is a real-valued function that ranks income distributions using the two poverty lines as parameters. Formally, a poverty index is a function P : P R + where P = {(y,z a,z r ) R N + R + Z y Y and b z a (1 s)}. The poverty in distribution y is simply denoted by P(y) when this notation does not create confusion. For any two distributions y and y, there is strictly more poverty in y than in y if P(y) > P(y ), and weakly more if P(y) P(y ). 4 Hierarchical additive poverty indices I study which indices should be used to compare the poverty in different income distributions. So far, the only difference with the classical literature is the presence of a second poverty line, whose threshold depends on mean income. The indices I consider acknowledge this presence in two ways. First, the relevance of each of the two poverty lines is established in a separate focus axiom. Each of these two focus axioms is specific to the particular need captured by its associated line. Second, several properties are weakened to the comparison of income distributions that have the same value of mean 4 For the particular case z a = 0, this definition implies that individual i is absolutely poor if y i = 0. This convention allows Theorem 1 to cover the particular case z a = 0. 6

7 income. Theorem 1 shows that these adapted properties together define a particular family of additive poverty indices. I present these properties in turn. Focus is the central property in poverty measurement. Focus requires that only the situations of poor individuals matter to poverty comparisons. Two kinds of poverty are considered here: absolute poverty and relative poverty. The absolute line defines who is absolutely poor and the hybrid line defines who is relatively poor. Absolute Focus and Hybrid Focus specify how the incomes of non-poor individuals affect absolutely and relatively poor individuals. An individual is absolutely poor if her income is insufficient to meet her subsistence needs. By assumption, the minimal income necessary to cover subsistence needs does not depend on the income of non-poor individuals. Absolute Focus requires that, when a distribution does not feature any relatively poor individual, the exact income earned by its non-poor individuals is irrelevant. The formal statement of this property is based on Y A = {y Y q a (y) = q(y)}, where Y A is the set of distributions that do not feature relatively poor individuals. Poverty axiom 1 (Absolute Focus). For all z a R +, z r Z and y,y Y A with n(y) = n(y ) and q(y) = q(y ), if y i = y i for all i q(y), then P(y,z a,z r ) = P(y,z a,z r ). The income of absolutely poor individuals is typically insufficient to cover social participation costs. The income necessary for social participation depends on the income of non-poor individuals. Therefore, when none of the poor individuals can cover her subsistence needs, Absolute Focus requires that some priority is given to subsistence over social participation. Indices that satisfy Absolute Focus and a basic monotonicity property never conclude that having all poor individuals in absolute poverty is better than having them in relative poverty. Hence, these indices avoid the problem illustrated in Table 1. In contrast, the focus axiom associated to the hybrid line does not completely disregard the income of non-poor individuals. The income necessary for an individual to meet her social participation needs depends on standards of living. Thus, relatively poor individuals are affected by the mean income, which, in turn, depends on the income of non-poor individuals. Hybrid Focus requires that the exact income earned by the non-poor individuals is irrelevant only as long as mean income is not changed. Poverty axiom 2 (Hybrid Focus). For all z a R +, z r Z and y,y Y with n(y) = n(y ) and q(y) = q(y ), if y i = y i for all i q(y) and y = y, then P(y,z a,z r ) = P(y,z a,z r ). Weak Monotonicity is a weakening of the classical monotonicity property for which poverty is reduced when a poor individual earns an additional amount of income. Weak Monotonicity adds the precondition that the other poor individuals are not affected. This precondition is guaranteed by restricting monotonicity comparisons to distributions that have the same value of mean income. Poverty axiom 3 (Weak Monotonicity). For all z a R +, z r Z and y,y Y with n(y) = n(y ), if y j < y j for some j q(y), y i = y i for all i q(y) with i j and y = y, then P(y,z a,z r ) > P(y,z a,z r ). Subgroup Consistency is a standard axiom requiring that, if poverty decreases in a subgroup while it remains constant in the rest of the distribution, overall poverty must decline. Sen (1992) questioned the desirability of this axiom by arguing that the incomes in one subgroup may affect poverty in another subgroup. Foster and Sen (1997) recommend not to use this axiom when the index aims at capturing relative aspects of income poverty. I subscribe to this point of view. The issue becomes transparent once the channel through which one subgroup affects the other is modeled. In this framework, the incomes in a subgroup impact the mean income, which in turn affects poor individuals in another subgroup. If the relative income matters, then it is not always meaningful to 7

8 extrapolate the judgments made on subgroups to the whole population. Weak Subgroup Consistency restricts such extrapolations to cases for which the two subgroups of a population have the same mean income. Then, the mean income in a subgroup is equal to the mean income in the total population. In such cases, poverty judgments made on subgroups are relevant for the total population. Poverty axiom 4 (Weak Subgroup Consistency). For all z a R +, z r Z and y 1,y 2,y 3,y 4 Y, if n(y 1 ) = n(y 3 ), n(y 2 ) = n(y 4 ), y 1 = y 2 = y 3 = y 4 and P(y 1,z a,z r ) > P(y 3,z a,z r ) and P(y 2,z a,z r ) = P(y 4,z a,z r ), then P((y 1,y 2 ),z a,z r ) > P((y 3,y 4 ),z a,z r ). The remaining three auxiliary axioms are straightforward adaptations of their classic counterparts. Symmetry requires that individuals identities do not matter. Working with sorted distributions is therefore without loss of generality. Symmetry implies that the preferences over bundles of the concerned individuals are irrelevant to the poverty index. This property generates little debate when only the level of income appears in preferences. If preferences are monotonic, then the property does not override individual preferences. When both the level of income and the relative situation matter, the monotonicity of preferences does not entirely define individual preferences and Symmetry explicitly requires to completely disregard these preferences. This form of paternalism can be defended on the ground that it prevents poverty indices from giving priority to individuals that are more other-regarding. 5 Weak Continuity requires indices to be continuous in all incomes on a subset of distributions. Such continuity requirement is important in empirical applications in order to avoid that measurement errors have an excessive impact on poverty judgments. Let y c denote the value of mean income at which the relative line crosses the absolute line. If s > 0, then its value is y c = za b s. If s = 0, the two poverty lines may not cross, and I define y c = z a. Weak Continuity requires indices to be continuous on Y c = { y Y y > y c }, where Y c is the set of distributions whose associated hybrid threshold is larger than the absolute threshold. 6 Finally, Replication Invariance specifies how to compare poverty in distributions of different population sizes. If a distribution is obtained by replicating another distribution several times, then the two distributions have equal poverty. Theorem 1 shows that indices satisfying these axioms sum the contributions to poverty of all poor individuals. Importantly, the poverty contribution of any individual potentially depends on both her income and the mean income. As the poverty contribution function is the same for all individuals, these indices implicitly rank individual situations across societies with different values of mean income. Crucially, the implicit ranking of individual situations must grant a particular form of priority to the absolute over the relative aspect of poverty: below the absolute threshold, the relative aspect of income poverty is irrelevant. The formal statement of Theorem 1 requires introducing a new object that summarizes the implicit comparisons of individual situations. The situation of individual i in distribution y is defined by the bundle (y i,y) that she enjoys. Her two-dimensional 5 For an illustration of the issue, consider two poor individuals living in the same society. Assume that individual 1 has a smaller income than individual 2 but individual 2 has preferences that are more affected by relative income than the preferences of individual 1. If individual preferences matter for the poverty index, it could be that individual 2 contributes more to poverty than individual 1. Potentially, transferring an amount of income from individual 1 to individual 2 decreases poverty. Such conclusion is debatable given that both individuals would agree that individual 2 is better-off than individual 1. 6 The continuity requirement is restricted to distributions in Y c in order to get around an impossibility result: indices that are continuous on the whole domain Y cannot simultaneously satisfy Absolute Focus and Weak Monotonicity when z a > 0 and s > 0. The proof of this impossibility result is omitted. 8

9 bundle is composed of her income y i and mean income y in her society. Hence, this bundle summarizes the absolute and relative aspects of her income. The set of bundles implicitly ranked is X = {(x 1,x 2 ) R + R ++ x 1 z(x 2 )}. An ethical ordering (EO) is an ordering on the space of bundles. 7 Therefore, the EO implicitly defined by an index summarizes how this index aggregates the absolute and relative aspects of income poverty at the individual level. Formally, an EO is a form of other-regarding preferences, as defined by the Behavioral literature (Fehr and Schmidt, 1999; Bolton, G., Ockenfels, 2000). Yet, other-regarding preferences and an EO should be interpreted differently. The former are the preferences held by an individual over her own situation, whereas the latter summarizes the normative judgments formed by an external observer when comparing individual situations. Theorem 1 shows that poverty indices satisfying the axioms previously stated can only rank individual situations based on a hierarchical EO. The particularity of a hierarchical EO is that it completely disregards the value of mean income as long as the individual income is below the absolute threshold. Definition 1 (Hierarchical ethical ordering). A hierarchical ethical ordering, denoted by, is an ordering on X that is continuous on X\{(z a,y c )} and for which 1. x x if x 1 > x 1 and x 2 = x 2 (monotonicity in own income), 2. x x if x 1 = z(x 2 ) and x 1 = z(x 2 ) (hybrid line), 3. x x if x 1 = x 1 < z a (priority to absolute poverty). Three different hierarchical EOs are illustrated in Figure 2. Graphically, an EO is depicted by its associated iso-poverty map, which is a collection of iso-poverty curves. An iso-poverty curve connects all the bundles that the EO deems equivalent. Observe that the hybrid line is a particular iso-poverty curve. Crucially, the iso-poverty curves below the absolute threshold are all flat. Therefore, a hierarchical EO always deems that an absolutely poor individual is worse-off than a relatively poor individual. y i y i (a) (b) (c) y i z r z r z r z a z a ȳ c ȳ b ȳ c ȳ z a ȳ Figure 2: Iso-poverty maps of hierarchical ethical orderings. Note: (a) Positive absolute threshold and strongly relative line. (b) Positive absolute threshold and weakly relative line. (c) Null absolute threshold and strongly relative line. The hierarchical EO associated to a given poverty index is implicitly defined by its poverty contribution function. The poverty contribution function is a (normalized) numerical representation of an ethical ordering. Definition 2 (Poverty contribution function). The poverty contribution function p : X R + is a continuous function on X\{(z a,y c )} such that for some ethical ordering we have and p(x) = 0 when x 1 = z(x 2 ). x x p(x ) p(x) for all x,x X. 7 An ordering is a reflexive, transitive and complete binary relation. 9

10 Observe that a poverty contribution function differs from a utility function, even when they both represent the same EO. The value returned by the poverty contribution function is smaller for better bundles. The value returned by this function corresponds to the individual contribution to poverty, i.e. the opposite of utility. The definition of a hierarchical additive poverty index requires both a poverty contribution function and a hierarchical EO. Definition 3 (Hierarchical additive poverty index). Index P is a hierarchical additive poverty index if P is ordinally equivalent to an index P : P R + defined by P (y) = 1 q(y) p(y i,y), (1) n(y) where the poverty contribution function p represents a hierarchical ethical ordering. The seven axioms presented above jointly characterize the family of hierarchical additive poverty indices. Theorem 1 (Characterization of hierarchical additive poverty indices). The following two statements are equivalent. 1. P is a hierarchical additive poverty index. 2. P satisfies Absolute Focus, Hybrid Focus, Weak Monotonicity, Weak Subgroup Consistency, Symmetry, Weak Continuity and Replication Invariance. Proof. It is easy to check that hierarchical additive poverty indices satisfy these seven axioms, so the proof that statement 1 implies statement 2 is omitted. The proof of the converse implication is in Appendix 8.2. The proof of Theorem 1 assumes that mean income is the income standard affecting the level of the relative threshold. Similar proofs can be constructed for other statistics that meet a subgroup consistency property. This is for instance the case of censored means, quantiles or even most harmonic means. 8 Depending on the relevant statistic, the domain of distributionsy on which the index is defined needs to be adapted. In order to construct a version of the proof of Theorem 1, domain Y must exclude distributions for which the statistic is not influenced by the income of non-poor individuals. 9 I emphasize that classical FGT indices, which are pervasive in empirical applications, are hierarchical additive indices only in two trivial cases. The first case is when they are combined with a relative line (z a = 0). The second case is when they are combined with an absolute line (s = 0). In all other cases, they regularly attribute smaller poverty contributions to absolutely poor individuals in low-income societies than to relatively poor individuals in middle-income societies. As illustrated in Table 1, the reason is that their contribution function depends only on the normalized income. Therefore, beyond these two trivial cases, Theorem 1 shows that classical indices should not be used when measuring poverty. Observe that Theorem 1 holds even in the trivial cases for which the hybrid line is an absolute line or a strongly relative line. When the hybrid line is an absolute line (s = 0), poverty contributions do not depend on mean income. Therefore, the additive separability result of Foster and Shorrocks (1991) is nested in Theorem 1. When the hybrid line is a strongly relative line (z a = 0 and b = 0), the family of hierarchical additive indices coincides with the family of additive indices. Therefore, Theorem 1 provides a foundation for the widespread practice consisting in using additive indices in 8 The geometric mean is an harmonic mean for which the proof of Theorem 1 does not hold. The particularity of this statistic is that any distribution with at least one individual earning zero income has a geometric mean equal to zero. 9 For instance, if the income standard is mean income among the 90% least rich individuals, then domain Y excludes distributions that have more than 90% of individuals who are poor. In turn, if the income standard is median income, then domain Y excludes distributions that have more than 50% of individuals who are poor. Finally, notice that the definition of domain Y needs to be adapted in order for statement 2 to imply statement 1. The reversed implication requires no such adaptation and holds for an even wider set of statistics capturing income standard. 10

11 combination with strongly relative lines. Indeed, this result reveals that additive indices can be justified by a weak subgroup consistency property that accounts for the critic of Sen (1992) and Foster and Sen (1997). From here onwards, I consider that the hybrid line is neither an absolute line nor a strongly relative line. Henceforth, the absolute threshold and the slope of the relative line are strictly positive (z a > 0 and s > 0). Finally, I assume that the absolute and relative lines cross at a strictly positive value of mean income (z a > b). These additional restrictions imply a slightly narrower domain for poverty indices. An impossibility result Theorem 1 places no restriction on the poverty contribution function other than representing a hierarchical EO. Such restrictions emerge from axioms constraining how the index must trade-off the incomes of different individuals. I consider two such axioms and show that, unfortunately, hierarchical indices violate both of them. First, Weak Transfer is a classical axiom requiring that a Pigou-Dalton transfer taking place between two poor individuals never unambiguously increases poverty. Poverty axiom 5 (Weak Transfer). For all z a R +, z r Z and y,y Y with n(y) = n(y ) and all δ > 0, if y j < z(y), y j δ = y j > y k = y k + δ, y i = y i for all i j,k and y = y, then P(y,z a,z r ) P(y,z a,z r ). The weak version of Transfer requires that the transfer does not alter the income standard. However, when mean income is the income standard, balanced transfers do not alter the mean. As well-known, poverty indices satisfying Weak Transfer are based on convex poverty contribution functions. Second, Strong Monotonicity considers an increase in the income of some poor individual. Poverty indices based on a relative line capture the relative aspects of income. For such indices, an increase in the income of a poor individual has opposing effects. On one hand, her contribution to poverty decreases as both her absolute and relative situations improve. This direct effect reduces poverty. On the other hand, mean income increases. 10 If the hybrid threshold increases, then the poverty contributions of the other relatively poor individuals increase. 11 Moreover, some individuals who were non-poor might become poor. Strong Monotonicity requires that these indirect adverse effects are dominated by the direct effect. Hence, this property imposes that decreasing the income of some poor individual never leads to an unambiguous poverty reduction. For this reason, a policy whose unique impact is to decrease the incomes of some poor individuals cannot be deemed poverty reducing. Poverty axiom 6 (Strong Monotonicity). For all z a R +, z r Z and y,y Y with n(y) = n(y ), if y i < y i < z(y ) and y j = y j for all j i, then P(y,z a,z r ) > P(y,z a,z r ). Observe that, Weak Monotonicity and Strong Monotonicity are logically related when the hybrid line is an absolute line (s = 0). In that case, all iso-poverty curves are flat and poor individuals are never affected by an increase in mean income. As a result, an increase in the income of a poor individual has no indirect effect and indices satisfying Weak Monotonicity also satisfy Strong Monotonicity. Theorem 2 shows that all hierarchical additive indices violate both Strong Monotonicity and Weak Transfer. 12 Again, the proofs for Theorem 2 can be adapted to relative lines sensitive to other statistics than mean income Observe that the larger the number of individuals, the lower is the impact of such an income increase on mean income and, hence, on the poverty contributions of others. 11 The contribution to poverty of an absolutely poor individual only depends on her level of income and is therefore not affected by the income of other poor individuals. 12 This impossibility result is strong as it is robust to a version of Strong Monotonicity where the inequality sign of the implication is weak. 13 Regarding part 2 in Theorem 2, these other statistics include censored means, quantiles or harmonic means. For part 1, these other statistics include censored means or harmonic means. 11

12 Theorem 2 (Hierarchical additive indices violate two basic properties). Let P be a hierarchical additive index. 1. P violates Strong Monotonicity. 2. P violates Weak Transfer. Proof. See Appendix 8.3. Theorem 2 shows that, when both the absolute and relative aspects of poverty matter, selecting a poverty index requires to make a trade-off between basic properties and the shape of the implicit iso-poverty curves. There are three main ways out of this impossibility. One way out is to discard hierarchical indices. It can be shown that some nonhierarchical additive indices based on the hybrid line such as the Poverty Gap Ratio satisfy both Strong Monotonicity and Weak Transfer on the domain considered. However, non-hierarchical indices deny a minimal form of priority to absolute poverty. As a result, they yield highly counterintuitive poverty comparisons as shown in the example of Table 1 and as illustrated empirically in Decerf (2017). Another way out is to change the definitions of the two poverty lines. The proof of Theorem 2 reveals that this impossibility follows from the fact that the relative line crosses the absolute line. Decerf (2017) considers a different framework with two lines, one absolute line and one hybrid line, which do not cross each other. In this alternative framework, the index proposed by Decerf (2017) satisfies both Strong Monotonicity and Weak Transfer. Yet, this alternative definition of the two lines is at odds with the literature on the design of hybrid poverty lines. Also, when the relative line crosses the absolute line, his index completely disregards the relative poverty line. Finally, it is possible to get around Theorem 2 by weakening the two axioms or restricting the domain of distributions on which they are imposed. For the reasons explained above, this pragmatic route is favored below. In the next section, I consider an FGT mathematical expression for the poverty contribution function that depends on two normative parameters. This expression defines a particular family of hierarchical additive indices. I show that one of its members stands out both for its good properties and for its simple interpretation. 5 A generalization of the Head-Count Ratio In this section, I define a parametric family of hierarchical additive indices and show that its parameters values can be determined from basic properties. The results derived are much less general than Theorem 1 and 2, but still provide a decent foundation for the simple hierarchical index identified. The poverty contribution function considered is inspired by the FGT subfamily of additive indices (Foster et al., 1984). FGT indices have an exponential expression whose properties have been studied by Foster and Shorrocks (1991) as well as Ebert and Moyes (2002). The poverty contribution ˆp(y i,y) of individual i in distribution y is given by ˆp(y i,y) = (1 u λ (y i,y)) α, (2) whereα 0 is the poverty aversion parameter andu λ is a utility function that represents an EO on the space of individual bundles. Classical FGT indices define this utility function with the normalized income, i.e. u λ (y i,y) = yi z(y). The implicit EO associated to these indices is not hierarchical in the narrower domain considered. The limitation illustrated in Table 1 reveals that the iso-poverty curves associated to bundles below the absolute threshold eventually cross the absolute threshold when mean income is sufficiently large. In contrast, the implicit EO is hierarchical if this utility function is defined in two parts as u λ (y i,y) = λ yi z a if y i < z a, λ+(1 λ)g(y i,y) if z a y i < z(y). (3) 12

13 where λ [0,1] is a parameter tuning which fraction of the [0,1] utility interval is attributed to absolute poverty and g : X [0,1] is a utility function on the subset of bundles whose income is larger than the absolute threshold. I assume that function g is linear in its first argument. By definition, an index is hierarchical additive only if its contribution function ˆp is continuous in its first argument and ˆp(z(y),y) = 0. Therefore, if function g is linear in its first argument, then we have g(y i,y) = y i z a z(y) z a. (4) For any relatively poor individual, function g measures the relative gap between the two poverty thresholds. The expression for g defines the shape of iso-poverty curves above the absolute threshold. The linear expression for g implies that iso-poverty curves between the absolute threshold and the hybrid line are homothetic, as illustrated in Figure 2. This assumption is strong but is also a natural default option. Together, equations (2) to (4) define a hierarchical version of the FGT family of indices. Let ˆP denote a generic index in this version of the FGT family. A particular index ˆP is defined by a pair of values for the two parameters α and λ, except in the special case α = 0 for which the value of λ is irrelevant. Observe that index ˆP violates Weak Monotonicity for some values of the parameters. This if for instance the case of the Head-Count Ratio (α = 0), the index most used in practice. However, this should not be viewed as a problem as all ˆP indices satisfy a weaker version of Weak Monotonicity where the inequality sign of the implication is weak. Yet, some ˆP indices should be dismissed on more fundamental ground. There is no point in considering a relative line if the index used does not acknowledge the relevance of social participation needs. Therefore, an index should attribute a positive contribution to relatively poor individuals. This is not the case of ˆP when λ = 1. Second, a hierarchical index should at least grant a minimal precedence to absolute poverty over relative poverty. That is, an index should attribute a strictly larger contribution to absolutely poor individuals than to relatively poor individuals. This is not the case of ˆP when α = 0. In my terminology, an index conciliates absolute and relative poverty if it satisfies these two minimal requirements. Definition 4 (Absolute and relative poverty conciliation). An additive index P conciliates absolute and relative poverty if its contribution function p is such that for any two bundles (x 1,x 2 ),(x 1,x 2) X with x 1 < z a and z a < x 1 < z(x 2 ) we have p(x 1,x 2 ) > p(x 1,x 2 ) > 0. By definition, all hierarchical additive indices conciliate absolute and relative poverty. In contrast, classical FGT indices do not conciliate absolute and relative poverty as they do not grant a minimal priority to absolute poverty. Lemma 1 follows directly from the above discussion. Lemma 1. ˆP conciliates absolute and relative poverty if and only if α 0 and λ 1. Further requirements allow discriminating among the remaining values for the parameters of index ˆP. Let ˆP denote a generic index ˆP that conciliates absolute and relative poverty. Theorem 2 reveals that any index ˆP violates both Strong Monotonicity and Weak Transfer. Yet, Lemma 2 shows that some of these indices satisfy Strong Monotonicity on the domain Y R = {y Y y 1 z a }, where Y R is the domain of distributions that feature no absolutely poor individuals. Lemma 2. ˆP satisfies Strong Monotonicity on Y R if and only if α = 1. Proof. See Appendix