Policy Research Working Paper 10519 A Meta-Theory for Absolute Poverty Lines Benoit Decerf Development Economics Development Research Group July 2023 Policy Research Working Paper 10519 Abstract Absolute poverty lines aim to track a fixed poverty standard arguments about their respective theoretical validity. This consistently. There are two main approaches for the construc- paper proposes a meta-theory for the consistency properties tion of absolute poverty lines. The “welfaristic” approach of absolute poverty lines under heterogeneous prices and tracks a fixed level of utility, and the “objective” approach heterogeneous preferences. The results identify the sets of tracks a fixed list of achievements. As they yield different consistency properties that fully characterize the poverty poverty comparisons, longstanding debates between their lines underpinned by these approaches. Which approach respective proponents take place both at global and national has better consistency properties depends on two aspects of levels. However, these debates only provide informal the application for which poverty is monitored. 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 author 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 A Meta-Theory for Absolute Poverty Lines.∗ Benoit Decerf† JEL: I32, D63, E31. Keywords: Absolute Poverty, Poverty line, Global poverty, Heterogeneous preferences. ∗ Acknowledgments : I am grateful to Aart Kraay, Daniel Malher, William Masters, Berk Ozler and Roy Van der Weide for helpful discussions. I thank all the participants at two internal seminars at the World Bank, and in particular Dean Jolliffe, Christoph Lakner, Vijayendra Rao and Hee Kwon Seo. The findings, interpretations, and conclusions expressed in this paper are entirely those of the author and should not be attributed in any manner to the World Bank, to its affiliated organizations, or to members of its Board of Executive Directors or the countries they represent. All errors remain mine. † World Bank, Development Research Group. Email: bdecerf@worldbank.org. 1 Introduction Absolute poverty lines aim for consistency. Consistency means that, although its nominal monetary amount may depend on prevailing prices, an absolute poverty line captures a fixed poverty standard. There are two main approaches to the construction of absolute poverty lines. Each has its own definition of a fixed poverty standard. Under the “ welfaristic ” approach, an absolute poverty line aims at capturing a fixed value in the utility space (Ravallion, 1998; Arndt and Simler, 2010). Its nominal monetary amount corresponds to the minimal cost of reaching the reference utility level given pre- vailing prices. Under the “ objective ” approach, an absolute poverty line aims at capturing a fixed list of achievements (Stigler, 1945; Allen, 2017; Herforth et al., 2022). Its nominal monetary amount corresponds to the minimal cost of satisfying a fixed list of basic needs given prevailing prices. The fixed list most often cap- tures nutritional needs, but may also include non-food needs like shelter or social inclusion. The two approaches yield different poverty comparisons because they rely on different price-adjustment methods. We illustrate this using a stylized example with two “goods”, food and housing.1 Assume that under prices p∗ both approaches yield the same nominal monetary amount p∗ z , which corresponds to the cost of the poverty bundle z that consists of 2,000 calories and 3 square meters of housing (Figure 1). When relative prices change from p∗ to p, the two approaches use different price-adjustment methods. Under the objective approach, the nominal monetary amount becomes pz so as to capture the same list of achievements z . Under the welfaristic approach, the nominal monetary amount becomes px so as to capture the same utility level u(z ). Poverty comparisons differ because the welfaristic approach accounts for substitution effects while the objective approach does not. Longstanding debates oppose the proponents of these two approaches. At global level, the welfaristic methodology underpinning the World Bank’s Interna- tional Poverty Line (IPL)2 have been criticized by Reddy and Pogge (2009) and 1 For simplicity, we abstract from the distinction between goods consumption and basic needs satisfaction. We discuss this simplification in Section 4. 2 See Ravallion et al. (2009a); Ferreira et al. (2016); Jolliffe et al. (2022) for details. 2 Housing u (# m2) Objective approach x Welfaristic approach z 3 p∗ p p 2,000 Food (# calories) Figure 1: Two alternative adjustments of the poverty line to price p. Allen (2017). The objective methodology underpinning the alternative proposals from the latter authors has in turn been criticized by Ferreira (2017) and Ravallion (2020). At national level, a similar debate opposes proponents of a fixed poverty basket to proponents of context-specific poverty baskets.3 However, these debates only provide informal arguments on the theoretical validity of each approach.4 Un- fortunately, informal arguments need not offer rigorous answers to well-specified questions and may generate misunderstandings, as evidenced by the exchange be- tween Ravallion (2002) and Reddy and Pogge (2002). To this day, there exists no formal comparison of the consistency properties of these two approaches. In this paper, we propose a meta-theory for absolute poverty lines with which we compare the consistency properties of the welfaristic and objective approaches. We propose three consistency properties and derive the respective combinations of these properties that fully characterize each type of absolute poverty line. Our framework considers heterogeneous prices and/or heterogeneous preferences.5 Our 3 For instance, India modified the definition of its official poverty lines following the recom- mendations from its expert group (Tendulkar et al., 2009) that the substitution effects behind observed consumption behavior be accounted for. 4 These debates also cover the practical limitations specific to their empirical applications. The IPL is for instance criticized for the construction and revision of its PPPs indices (Heston and Summers, 1996; Deaton, 2010). In turn, the global objective line proposed by Allen (2017) is criticized for the robustness of the procedures used to estimate the cost of specific needs, e.g., Ferreira (2017) questions this robustness for housing and energy needs. See Section 4. 5 There exists some empirical evidence that individuals hold heterogeneous preferences over necessities, even across groups of disadvantaged individuals who live in the same country (Atkin, 3 results shed light on the circumstances under which each approach may have better consistency properties than the other. Our three consistency properties encapsulate views that are pervasive in the poverty measurement literature. The first property – Dominated Command – re- quires that the poverty line consistently reflects individuals’ command over com- modities. The second property – Respect for Preference – requires that the poverty line consistently reflects individual preferences. The third property – Objective Anchorage – requires that the poverty line consistently reflects a universal list of achievements. When abstracting from the distinction between goods consumption and basic needs satisfaction (see Section 4 for discussion), the poverty line should consistently reflect some poverty bundle. Under heterogeneous prices and homogeneous preferences, our results show that the welfaristic approach enjoys better consistency properties. The reason is that the welfaristic approach satisfies all three consistency properties while the objective approach violates Respect for Preference . Under heterogeneous prices and heterogeneous preferences, our results identify the conditions under which the objective approach may enjoy better consistency properties. No poverty line satisfies all three consistency properties under this dou- ble heterogeneity (Theorem 1). The most common types of poverty lines under- pinned by the welfaristic and objective approaches – respectively, “representative- preference” lines and “achievements-list” lines – both violate Respect for Prefer- ence .6 However, these two types of poverty lines both satisfy a weak version of this property – Weak Respect for Preference . Together with Dominated Com- mand , Weak Respect for Preference exactly characterizes the non-paternalistic 2016; Dimri and Maniquet, 2019). One mechanism explaining such heterogeneity is habit for- mation (Atkin, 2013). 6 Examples of representative-preference lines are the IPL of the World Bank or the official poverty lines in India since 2009 or yet the poverty lines constructed with CBN-method pre- sented in Ravallion (1998); Ravallion and Bidani (1994) and refined by Arndt and Simler (2010). Examples of achievements-list lines are the poverty lines constructed with the least-cost linear- programming methods developed by Stigler (1945); Allen (2017); Herforth et al. (2022). The FAO recently commited to regularly update on the affordability of healthy diets (FAO, 2020), whose measurement is based on achievements-list lines. Similarly, the US Supplemental Nu- trition Assistance Program (SNAP) determines its monetary benefits from the Thrifty Food Plan (USDA, 2021), which computes the cost of achieving a healthy diet and is thus akin to an achievements-list line. 4 view that being poor depends on an individual’s budget set, but not on the exact bundle she selects in that set (Proposition 4). The consistency difference between representative-preference lines and achievements-list lines is that, unlike the latter, the former cannot consistently reflect the same poverty bundle for all preferences. A poverty line satisfies Dominated Command , Weak Respect for Preference and Objective Anchorage if and only if it is an achievements-list line (Theorem 3). A poverty line satisfies Dominated Command , Weak Respect for Preference and Sub- jective Anchorage “if and only if” it is a representative-preference line (Theorem 4). Subjective Anchorage is a weak version of Objective Anchorage that allows the poverty bundle to be preference-specific. Our results clarify some informal arguments made by proponents of each ap- proach. For instance, Reddy and Pogge (2009) and Allen (2017) criticize the meaning of the IPL, e.g., asking what it means to live on $1 a day in terms of basic needs satisfaction. Atkinson (2016) suggests that the World Bank develops “basic needs-based estimates of extreme poverty ... to provide an interpretation of what the IPL would buy”. As the IPL is essentially a representative-preference line, our results imply that the answer to this question is country-specific,7 at least if preferences over different basic needs (at $1 a day) vary substantially across coun- tries. To take another example, Ferreira (2017) and Ravallion (2020) criticize the arbitrariness of some elements of the achievements list underpinning the objective global line proposed by Allen (2017). Our results reveal that, under heterogeneous preferences, achievements-list lines may have better consistency properties than representative-preference lines if Objective Anchorage seems more appealing than Subjective Anchorage . However, this condition will not be met if the achievements list cannot make a strong claim for its “universality”. More importantly, our results reveal that the most consistent approach depends on two aspects of the application for which poverty is monitored. The first aspect is the extent of preferences heterogeneity. The second aspect is the claim for “universality” associated to the poverty bundle / achievement list underpinning an objective poverty line. We discuss the factors influencing these two aspects – like the austerity of the poverty standard or the diversity of contexts over which poverty 7 More precisely, the list of basic needs achievements reflected in the purchasing power of the international poverty line is preference-specific (Theorem 4) and thus country-specific. 5 is monitored – and how they could play out for global poverty measurement in Section 3.3. Our main contribution is to characterize the consistency properties of the wel- faristic and objective approaches. The paper closest related to ours is Dimri and Maniquet (2019). These authors also study monetary poverty measurement un- der heterogeneous preferences, but they do not study the consistency properties associated to representative-preference lines and achievements-list lines, which are ubiquitous in practice.8 Our results also speak to the literature on real incomes and price indices (Diew- ert, 1979; Van Veelen, 2002; Van Veelen and van der Weide, 2008). We emphasize that our absolute poverty line problem is not exactly dual to the classical problem of constructing a measure of real income. Our problem is in fact less ambitious. Indeed, an absolute poverty line must construct comparability only to a unique reference economic situation, which is (z, p∗ , u) in the example illustrated in Fig- ure 1. In contrast, a measure of real income must construct comparability to all possible economic situations.9 Clearly, the latter problem is more complex, if only because it involves potential conflicts of transitivity. A priori, our problem is afflicted by fewer impossibilities. We discuss the normative differences with the price index literature in Section 5.1. The remainder of this paper is organized as follows. The formal framework is presented in Section 2, starting with basic definitions, following with the three types of absolute poverty lines and ending with the axioms encapsulating the three basic principles. The characterization results for the three types of lines are presented and discussed in Section 3. Concluding comments are presented in Section 4. 8 Relatedly, Decancq et al. (2019) characterize a family of multidimensional poverty measures under heterogeneous preferences. Their framework focuses on non-market goods and thus ignores prices, which play a central role when considering market goods. Decerf et al. (2022) study monetary poverty measurement in a framework where individuals hold heterogeneous preferences over own income and relative income. Their framework also ignores prices as they focus on the trade-off between own and relative income. 9 In our notation, an absolute poverty line compares any economic situation (x′ , p′ , u′ ) to a unique reference economic situation (z, p∗ , u). In contrast, a measure of real income must compare any economic situation (x′ , p′ , u′ ) to any other economic situation (x, p, u). 6 2 Framework 2.1 Basic definitions There are m goods, indexed by j . Let x := (x1 , . . . , xm ) denote a bundle whose set is denoted by X := Rm + . Let 0 := (0, . . . , 0) ∈ X denote the zero bundle and let X=0 := X \{0}. We denote by x ≪ x′ the fact that x′ strictly dominates x, i.e., xj < x′j for all j , by x ≤ x′ the fact that x′ weakly dominates x, i.e., xj ≤ x′j for all j , and by x x′ the fact that x′ weakly dominates x and x = x′ . Let u denote a utility function that represents a preference relation over X , whose generic set is denoted by U . Although we use utility functions, our theory is ordinal in the sense that it only relies on the underlying preferences. Let p denote a normalized price vector, whose generic set is denoted by P . An economic situation is a triple s := (x, p, u), whose generic set is denoted by S := X × P × U . An individual’s economic situation (x, p, u) captures all variables that are potentially relevant to evaluate whether she is poor or not. In line with the literature, we restrict attention to the subset of equilibrium situations, whose bundle is optimal given the price vector and preference. Formally, any situation (x, p, u) is in equilibrium if bundle x is the demand given p and u, i.e., if we have x ∈ arg max u(x′ ). px ≤px ′ Let S ∗ denote the set of equilibrium situations. An absolute poverty line partitions S ∗ into the situations identified as poor and those that are not. Formally, we define a poverty line to be an identification function ½ : S ∗ → {0, 1}, where s ∈ S ∗ is identified as poor when ½(s) = 1 and identified as non-poor when ½(s) = 0. This definition calls for two remarks. First, defining the poverty line as an identification function is equivalent to defining the poverty line as a price adjustment function Z : P × U → R+ that computes the nominal amount of the poverty line. More precisely, when the identification 7 function ½ is weakly decreasing in x,10 we have for all (x, p, u) ∈ S ∗ that ½(x, p, u) = 1 ⇔ px < Z (p, u), where Z is the price adjustment function associated to ½, which computes the poverty line in local prices.11 In the remainder, we use the terminology “poverty line” to refer to the identification function. Second, the value returned by the identification function ½ can be interpretated as the (binary) poverty score attributed to an individual whose economic situa- tion is s.12 Under this interpretation, the average poverty score in a population corresponds to the fraction of individuals indentified as poor. We define a poverty line as an identification function ½ (rather than as a price adjustment function Z ) to make our normative analysis more transparent. We derive results under different heterogeneity assumptions for prices and pref- erences. Formally, our results consider different sets of equilibrium situations. Let U denote the set of all preferences that are continuous, strictly monotonic (in all goods) and weakly convex. Let the simplex of all normalized price vectors be de- noted by ∆ := {(p1 , . . . , pm ) ∈ Rm +| j pj = 1}. Clearly, the extent of heterogeneity in sets U and ∆ is large. The set of equilibrium situations with heterogeneous pref- ∗ erences is denoted by SU := X × {p} × U and the set with heterogeneous price ∗ := vectors is denoted by S∆ X × ∆ × {u}, where p ∈ ∆ and u ∈ U . The set with ∗ both types of heterogeneity is denoted by SU ∆ := X × ∆ × U . By definition, we ∗ ∗ ∗ ∗ have SU ⊂ SU ∆ and S∆ ⊂ SU ∆ . ∗ Observe that, for any u ∈ U , there is no equilibrium situation (x, p, u) ∈ SU ∆ whose price vector p is on the frontier of ∆. Indeed, any p on the frontier of ∆ has pj = 0 for some good j , which implies an infinite demand for good j . Let ∆+ 10 The axioms we consider – Dominated Command or Respect for Preference – imply that function ½ is weakly decreasing in all goods. 11 As we restrict attention to equilibrium situations, for given p and u, any bundle x ∈ X for which (x, p, u) ∈ S ∗ and ½(x, p, u) = 1 is on an increasing path of bundles that contains the zero bundle 0. Moreover, for any monetary amount y > 0, exactly one bundle x∗ on this increasing path is such that px∗ = y . 12 Considering a binary poverty score allows sidestepping the issue of how the poverty score should evolve with the monetary gap to the poverty line. This assumption thus allows focusing on the identification of the poor. However, our theory readily extends to non-binary poverty scores, as considered by Dimri and Maniquet (2019). 8 denote the interior of ∆, i.e., the set of p ∈ ∆ for which pj > 0 for all j . 2.2 Types of absolute poverty lines We define three types of absolute poverty lines that respectively correspond to the welfaristic and objective approaches. Each of these types are a stylized theoretical blueprint for some methods used in practice to construct absolute poverty lines. We define each of these three types of poverty lines based on an exogenously given poverty bundle, denoted by z . To avoid trivialities, we assume that z is optimal for some preference u ∈ U and price vector p ∈ P ; i.e., (z, p, u) ∈ S ∗ . Taking z to be exogeneous allows comparing the two approaches by contrasting the consistency properties of their respective price-adjustment methods. Yet, the two approaches may also differ in the austerity of their poverty standards. That is, they may select a different poverty bundle z even when prices are homogeneous (P = {p}) and there are no price-adjustments. We abstract from such “austerity” differences to focus on the consistency of price-adjustments methods. As we discuss in Section 4, there is an irreducible arbitrary/political element in the austerity selected for the poverty standard.13 We start with the welfaristic approach. First, an individual-preference poverty line considers an individual as poor when the individual prefers the poverty bundle z over the bundle she consumes. This type of poverty line thus accounts for individual-specific substitution effects. Definition 1 (Individual-Preference poverty line). The poverty line ½IP is individual- preference if and only if there exists z ∈ X=0 such that for all (x, p, u) ∈ S ∗ ½IP (x, p, u) = 1 ⇔ u(x) < u(z ). By definition, any individual-preference line ½IP does not account for price vector p, i.e., ½IP (x, p, u) = ½IP (x, u). In a context with heterogeneous preferences, Dimri and Maniquet (2019) provide a conceptual foundation for poverty measures based on individual-preference poverty lines. 13 Any approach can justify any austerity for its poverty standard. For instance, one can modify this austerity by changing the number of calories required or by requiring a healthy diet instead of a mere dietary energy standard, as recently suggested by Mahrt et al. (2022). 9 The second type of poverty line also belongs to the welfaristic approach. This type is distinct from the individual-preference type when preferences are hetero- geneous. A representative-preference poverty line considers an individual as poor when her income is smaller than the minimal expenditure necessary to reach the ˜ is a “welfare” associated to the poverty bundle z , where the “welfare” function u representative preference that may differ from the individual’s preference.14 Let eu (p, u(z )) denote the minimal expenditure that is necessary under price vector p in order to reach the utility level u(z ) associated to bundle z , where eu : ∆ × R → R+ denotes the expenditure function associated to preference u, i.e., eu (p, u(z )) := min px′ . {x′ ∈X |u(x′ )≥u(z )} Definition 2 (Representative-Preference poverty line). The poverty line ½RP is ˜ ∈ U and z ∈ X=0 representative-preference if and only if there exist a preference u ∗ such that for all (x, p, u) ∈ S ½RP (x, p, u) = 1 ⇔ ˜ (p, u px < eu ˜(z )). By definition, any representative-preference line ½RP does not account for the specific preference u of the individual, i.e., ½RP (x, p, u) = ½RP (x, p). Under hetero- geneous preferences, the World Bank’s extreme poverty line or the official poverty lines in India are rooted in representative-preference poverty lines. As we argue in Appendix 5.3, the CBN-method proposed by Ravallion and Bidani (1994), whose context-specific poverty bundles are tailored to local consumption behavior, is also rooted in representative-preference poverty lines. This is clearly the case when its context-specific bundles are submitted to the revealed-preference adjustments de- veloped by Arndt and Simler (2010) and Mahrt et al. (2022),15 because then the adjusted bundles can be rationalized.16 14 To avoid trivialities, we require for representative-preference lines that there exists p ∈ P ˜) ∈ S ∗ . such that (z, p, u 15 Relatively poor individuals living in more affluent contexts could on average consume more expensive calories. As a result, the CBN food bundle in those regions may yield a higher welfare than the CBN food bundle in less affluent contexts. 16 When examining official poverty lines (derived through a wide range of methodologies), violations of revealed-preference conditions have been found (Ravallion and Sen, 1996; Wodon, 10 The third type captures the objective approach. An achievements-list line considers an individual as poor when the poverty bundle z is not in her budget set. Definition 3 (Achievements-list poverty line). The poverty line ½AL is achievements- list if and only if there exists z ∈ X=0 such that for all (x, p, u) ∈ S ∗ ½AL (x, p, u) = 1 ⇔ px < pz. By definition, any achievements-list line ½AL does not account for the indi- vidual’s preference u, i.e., ½AL (x, p, u) = ½AL (x, p). When abstracting from the distinction between goods consumption and basic needs satisfaction, the least-cost poverty measurement practices developed by Stigler (1945) and Allen (2017) are achievements-list poverty lines. In practice, although least-cost poverty lines ig- nore preference-based substitution effects between basic needs requirements, they do account for price-based substitution effects between goods satisfying the same basic need, such as rice and wheat in the case of nutrition (see Section 4). Figure 2 illustrates that these three types of poverty lines yield different clas- sification of equilibrium economic situations. Situation (ˆ x, p, u) is poor according x) < u(z ) – and to the achievements- to both the individual-preference line – as u(ˆ list line – as px ˆ < pz , i.e., bundle z does not belong to the budget set asso- ciated to (ˆ x, p, u) is non-poor according to the x, p, u). In contrast, situation (ˆ representative-preference line because the indifference curve of the representative preference u˜ through z passes through the budget set associated to (ˆ x, p, u). In ′ ′ x, p , u ) is poor according to both the achievements-list line – as turn, situation (ˆ ˆ < p′ z – and the representative-preference line, since the indifference curve of p′ x the representative preference u ˜ through z has no intersection with the budget set ′ ′ associated to (ˆ x, p, u) is non-poor according to the x, p , u ). In contrast, situation (ˆ x) > u′ (z ). These classifications are summarized individual-preference line – as u′(ˆ in Table 1. A more exhaustive graphical illustration of the three types of lines lies in Appendix 5.2. 1997; Tarp et al., 2002; Ravallion and Lokshin, 2006). 11 u ˜ x2 u x ˆ u′ z u ˜ p′ p x1 Figure 2: Different identifications by the three types of poverty lines. Note: Budget sets associated to bundle x ˆ and price vectors p and p′ are in red. Indifference curves are in blue. The indifference curve of the representative preference u ˜ through z is dashed. The classifications of (ˆ x, p′ , u′ ) ∈ S ∗ by the three lines are recorded in Table 1. x, p, u), (ˆ Table 1: Poverty classification in Fig. 2 by types of poverty line. Type of poverty line Poor Non-poor Individual-Preference (ˆ x, p, u) x, p′ , u′ ) (ˆ Representative-Preference u ˜ x, p′ , u′ ) (ˆ x, p, u) (ˆ ′ ′ Achievements-List x, p, u), (ˆ (ˆ x, p , u ) Conceptually, these three types of poverty lines have different meanings. That is, they differ in the space in which the poverty line is kept constant across different price vectors. For an individual-preference line, different nominal amounts of the poverty line all refer to the same utility level u(z ). For an achievements-list line, different nominal amounts all permit to just afford the reference bundle z . For a representative-preference line, different nominal amounts all permit to just afford ˜(z ). Table 2 summarizes our discussion of these three types. the “welfare” level u 12 Table 2: Three stylized types of absolute poverty lines in theory and in practice. Individual-Preference Representative-Preference u ˜ Achievements-List Poverty space utility ˜(z )-affordability u z -affordability Threshold u (z ) ˜ (p, u eu ˜(z )) pz Standard method Group-specific Price indices (eg PPPs), Least-cost method demand systems CBN method for fixed basic needs (Ravallion-Bidani) requirements Applications Dimri-Maniquet $1 a day (global), Allen (global), (India) Official line (India) FAO (global) Note: The references used as examples of different types of poverty lines are respectively Dimri and Maniquet (2019) for an individual-preference line, Allen (2017) and FAO (2020) for an achievements-list global line and Ravallion et al. (2009b); Ferreira et al. (2016); Jolliffe et al. (2022) for the World Bank’s representative-preference $1 a day global extreme line, Tendulkar et al. (2009); Rangarajan et al. (2014) for the representative-preference official poverty measure- ment methodology in India and Ravallion and Bidani (1994) for the CBN method tailored to local consumption. 2.3 Three consistency axioms for poverty lines In this section, we propose three axioms that encapsulate the basic consistency principles that absolute poverty lines should satisfy. These axioms constrain the price-adjustment method of acceptable absolute poverty lines. First, Dominated Command requires the poverty line to consistently reflect the command that individuals exert over commodities. This axiom limits the role that preferences can play by requiring that an individual cannot be considered as poor when she commands more commodities than an individual who is not considered as poor. Dominated Command compares two individuals who face the same price vector. If the first individual is non-poor and the second individual consumes a bundle that has more of every goods than the bundle of the first individual, then the second individual must be considered non-poor, even if they hold different preferences. Indeed, the budget set of the first individual is a subset of the budget set of the second individual. Axiom 1 (Dominated Command ). For all (x, p, u), (x′, p′ , u′ ) ∈ S ∗ with p = p′ 13 and x ≤ x′ , ½(x, p, u) = 0 ⇒ ½(x′ , p′ , u′ ) = 0. Second, Respect for Preference encapsulates non-paternalism, i.e., the poverty line should consistently reflect individual preferences. This axiom requires that an individual who is not poor under an initial bundle must be considered non-poor when she consumes another bundle that she prefers to her initial bundle. Axiom 2 (Respect for Preference ). For all (x, p, u), (x′ , p′ , u′ ) ∈ S ∗ with u = u′ and u(x) ≤ u(x′ ) ½(x, p, u) = 0 ⇒ ½(x′ , p′ , u′ ) = 0. Third, Objective Anchorage requires that the poverty line consistently reflects some universal anchor in the space of bundles, i.e., in the space of basic needs achievements as we do not distinguish between the two (see Section 4). More precisely, Objective Anchorage requires the existence of one “anchor” bundle z ∗ such that any individual whose bundle has less of every good than z ∗ is poor and any individual whose bundle has more of every good than z ∗ is non-poor. Axiom 3 (Objective Anchorage ). There exists z ∗ ∈ X=0 such that for all (x, p, u) ∈ S∗ x z∗ ⇒ ½(x, p, u) = 1. z∗ ≤ x ⇒ ½(x, p, u) = 0. Objective Anchorage requires the anchor bundle to be the same for all individ- uals who have the same needs, even when they hold heterogeneous preferences. The anchor bundle thus provides a universal objective meaning to being poor. Importantly, Objective Anchorage does not constrain the substitution effects allowed at the anchor bundle z ∗ . Indeed, Objective Anchorage does not constrain the poverty status of any bundle x that lies in the North-West quadrant or in the ∗ South-East quadrant with respect to z ∗ , i.e., x1 > z1 ∗ and x2 < z2 (or vice versa). Rather, Objective Anchorage merely requires that individuals whose bundle lies in 14 the South-West (resp., North-East) quadrant are poor (resp., non-poor), regardless of their preferences. 3 Results In this section, we derive the implications that the axioms have on poverty lines. When prices are homogeneous, the welfaristic and objective approaches both share the same consistency properties. Indeed, representative-preference lines are equivalent to achievement-list lines and they satisfy all three consistency axioms. We relegate this case to Appendix 5.5 and assume henceforth that prices are heterogeneous. 3.1 Homogeneous preferences We show that the welfaristic approach has better consistency properties than the objective approach under heterogeneous prices and homogeneous preferences. When preferences are homogeneous, all individuals share a common preference u, i.e., U = {u} for some arbitrary but fixed preference u ∈ U . This case simplifies the analysis on two grounds. First, individual-preference lines and representative- preference lines become equivalent. Indeed, the representative preference u ˜ must ˜ ∈ U. be the common individual preference u because by assumption u Observation 1. Under homogeneous preferences (S∆ ∗ ), the poverty line ½ is individual- preference if and only if ½ is representative-preference. Second, the homogeneous preference case is such that Dominated Command is implied by Respect for Preference .17 Observation 2. Under homogeneous preferences (S∆ ∗ ), the poverty line ½ satisfies Respect for Preference only if ½ satisfies Dominated Command. 17 Under homogeneous preferences, the two preconditions for Respect for Preference always hold when the precondition for Dominated Command holds. The first precondition for Respect for Preference , i.e., that the two situations considered share the same preference, is met because all individuals hold the same preference. The second precondition for Respect for Preference , i.e., that one bundle is preferred to the other, is automatically met when the former dominates the latter. Hence, by Respect for Preference , the implication of Dominated Command must hold. 15 Proposition 1 formalizes the theoretical foundations for individual-preference poverty lines under homogeneous preferences (Ravallion, 2015). This proposition shows that the three axioms considered are compatible under homogeneous pref- erences and they jointly characterize the welfaristic approach. ∗ Proposition 1. Under homogeneous preferences (S∆ ), the poverty line ½ satisfies Objective Anchorage and Respect for Preference if and only if ½ is individual- preference. Proof. See Appendix 5.6. No achievements-list poverty line is individual-preference because preferences are assumed to be strictly monotonic.18 Therefore, the characterization in Propo- sition 1 tells us that any achievements-list poverty line must violate at least one of these two axioms. Unsurprisingly, achievements-list poverty lines do not respect individual preferences (Ravallion, 2015; Dimri and Maniquet, 2019). Proposition 2 confirms again this fact by showing that achievements-list poverty lines violate Respect for Preference . The intuition is easily grasped from Figure 3, which shows that a bundle x may be preferred to the reference poverty bundle z even when bundle z is not affordable in the budget set sustaining x. x2 x z u p x1 Figure 3: Achievements-list poverty lines violate Respect for Preference . Note: The frontier of the budget set sustaining (x, p, u) ∈ S ∗ is in red and the indifference curve through x of preference u is in blue. 18 An achievements-list poverty line could be individual-preference if Leontiev preferences belong to domain U ⊆ U . However, Leontiev preferences are not strictly monotonic. 16 ∗ Proposition 2. Under homogeneous preferences (S∆ ), any achievements-list poverty line ½ satisfies Objective Anchorage and Dominated Command but violates Respect for Preference. Proof. See Appendix 5.7. When preferences are homogeneous, Proposition 1 and Proposition 2 together show that the welfaristic approach enjoys better consistency properties than the objective approach. 3.2 Heterogeneous preferences We show that the welfaristic approach does not have better consistency properties than the objective approach when both prices and preferences are heterogeneous. To start with, no poverty line jointly satisfies the three consistency axioms when both prices and preferences are heterogeneous, as stated in Theorem 1. ∗ Theorem 1. Under heterogeneous prices and heterogeneous preferences (SU ∆ ), no poverty line jointly satisfies Objective Anchorage, Respect for Preference and Dominated Command. Proof. See Appendix 5.8. This impossibility result echoes similar impossibility results. When preferences are heterogeneous, no social welfare function respects preferences and is egalitar- ian (Fleurbaey and Trannoy, 2003; Brun and Tungodden, 2004; Fleurbaey and Maniquet, 2011). No multilateral price index satisfies a minimal set of desirable properties (Van Veelen, 2002). Theorem 1 implies that any poverty line must violate at least one of the three axioms. Value judgments are thus a theoretical necessity when designing an ab- solute poverty line in the presence of both heterogeneities. We next characterize the combination of axioms that define each of the three types of poverty lines. First, individual-preference poverty line are again fully characterized by the same two axioms. This result is similar to a conceptualization provided by Dimri and Maniquet (2019). 17 ∗ Theorem 2. Under heterogeneous prices and heterogeneous preferences (SU ∆ ), the poverty line ½ satisfies Objective Anchorage and Respect for Preference if and only if ½ is individual-preference. Proof. See Appendix 5.9. However, Theorem 2 does not imply that individual-preference poverty lines satisfy Dominated Command when preferences are heterogeneous. In that case, Respect for Preference does not imply Dominated Command . Figure 4 illustrates that individual-preference lines violate Dominated Command when preferences are heterogeneous. In the figure, an individual with preference u demands x under price vector p, but she is considered poor by an individual-preference line because she prefers the poverty bundle z . An individual with preference u′ demands x′ under the same price vector p, but she is not considered poor by an individual- preference line because she finds bundle z worse than x′ . However, bundle x′ has less of every good than bundle x. Hence, under price vector p, the command over resources associated to x′ is smaller than that associated to bundle x. The fact that an individual-preference line considers the individual consuming x poor and the individual consuming x′ non-poor thus violates Dominated Command . x2 x′ x u′ z u p p x1 Figure 4: Individual-Preference poverty lines violate Dominated Command when preferences are heterogeneous. Note: The frontiers of the budget sets sustaining (x, p, u), (x′ , p, u′ ) ∈ S ∗ are in red and indiffer- ence curves are in blue. ∗ Proposition 3. Under heterogeneous preferences (SU ), individual-preference poverty lines violate Dominated Command. 18 Proof. The proof is illustrated in Figure 4. Individual-preference lines are barely used in practice. Applying individual- preference lines requires eliciting preferences, a complex and data-demanding en- deavor that is typically performed by solving demand-systems a la Deaton and Muellbauer (1980). Proposition 3 provides another potential reason, namely that these lines do not consistently capture individuals’ command over economic re- sources. We now turn to representative-preference lines and achievements-list lines. These two stylized types underpin most of the absolute poverty lines used in practice. These two types of lines both identify the poor without accounting for the exact bundle that the individual selects in her budget set. In that impor- tant aspect, they differ from individual-preference lines. Because they disregard the exact bundle consumed, representative-preference lines and achievements-list lines violate Respect for Preference .19 However, although they violate Respect for Preference , they still obey a restricted version of non-paternalism. Indeed, they hold individuals responsible for the bundle they chose in their budget set. As we next show, this form of non-paternalism can be encapsulated in a weak version of Respect for Preference . Weak Respect for Preference is a weakening of Respect for Preference that adds the precondition that the two economic situations considered must share the same price vector.20 Axiom 4 (Weak Respect for Preference ). For all (x, p, u), (x′, p′ , u′ ) ∈ S ∗ with u = u′ , u(x) ≤ u(x′ ) and p = p′ , ½(x, p, u) = 0 ⇒ ½(x′ , p′ , u′ ) = 0. 19 Some trivial poverty lines satisfy both Respect for Preference and Dominated Command . This is for instance the case of a poverty line according to which no situation is poor. However, this poverty line satisfies both Respect for Preference and Dominated Command in a trivial way, because these axioms never require that one situation is poor and the other is not poor. If these axioms did involve strict comparisons, they would become mutually incompatible, even without imposing Objective Anchorage . 20 Observe that, even if Weak Respect for Preference and Dominated Command seem similar, the three axioms Objective Anchorage , Weak Respect for Preference and Dominated Command are mutually independent as we show in Appendix 5.11. 19 Proposition 4 shows that Weak Respect for Preference and Dominated Com- mand fully characterize the non-paternalistic view that whether an individual is poor or not depends on her budget set rather than on her actual consumption.21 ∗ Proposition 4. Under heterogeneous prices and heterogeneous preferences (SU ∆ ), the poverty line ½ satisfies Weak Respect for Preference and Dominated Command if and only if for any two (x, p, u), (x′, p′ , u′ ) ∈ S ∗ with p = p′ and px ≤ px′ we have ½(x, p, u) ≥ ½(x′ , p′ , u′ ). Proof. See Appendix 5.10. Observe that the union approach, which identifies an individual as poor when xj < zj for some dimension j , does not satisfy the non-paternalism view charac- terized in Proposition 4. Although the union approach satisfies Dominated Com- mand , it violates Weak Respect for Preference . Theorem 3 identifies the exact combination of consistency properties that char- acterize achievements-list poverty lines. These lines satisfy both Weak Respect for Preference and Dominated Command , which implies they satisfy the version of non-paternalism presented in Proposition 4. They also satisfy Objective Anchor- age , which implies their reference poverty bundle z is an anchor bundle z ∗ . ∗ Theorem 3. Under heterogeneous prices and heterogeneous preferences (SU ∆ ), the poverty line ½ satisfies Objective Anchorage, Weak Respect for Preference and Dominated Command if and only if ½ is an achievements-list poverty line. Proof. See Appendix 5.12. 21 Ravallion and Bidani (1994) write “The criterion for defining poverty is rarely that one attains too little of each basic need. Rather, it is that one cannot afford the cost of a given vector of basic needs. [. . . ] A person who consumes less food (say) than the stipulated basic needs is not considered poor if the person’s budget allocation could be rearranged to cover the basic needs.” Atkinson and Bourguignon (2001) write “It is not taken to mean that we should require that people actually possess the goods in question. Poverty is not being defined by the absence of a mobile phone; rather it is that the level of resources should permit the purchase of a phone, if that is deemed necessary to compete in the labour market.” 20 We finally turn to representative-preference poverty lines. Observe that representative- preference lines and achievements-list line are disjoint types. An achievements-list line corresponds to the limit case of a representative-preference line whose repre- ˜ is a Leontiev preference. However, Leontiev preferences are sentative preference u not strictly monotonic. Therefore, any representative-preference line ½ must vio- late at least one of the three axioms in Theorem 3 because ½ is not an achievement line. When preferences and prices are heterogeneous, representative-preference poverty lines violate Objective Anchorage . Figure 5 illustrates for a representative-preference line that no poverty bundle z meets the requirements of Objective Anchorage and thus no bundle z can be an anchor bundle z ∗ . To understand why, consider in Figure 5 the nearby bundle x that lies on the indifference curve of the representative preference u ˜ passing through z . Let price vector p correspond to the marginal rate of substitution of u ˜ at x. By definition we have px = eu ˜(z )). Consider the budget set defined by ˜ (p, u price vector p and bundle x. There exists a third bundle x′ with x′ z that lies on the frontier of the budget set defined by p and x. There also exists a preference u for which x′ is optimal given p, which implies that the economic situation (x′ , p, u) ∗ ′ belongs to SU ∆ . By construction, we have px = eu ˜(z )), which implies that ˜ (p, u (x′ , p, u) is not poor according to the representative-preference line. This shows that bundle z cannot meet the requirements of Objective Anchorage as x′ z . x2 p x z ′ x u ˜ u x1 Figure 5: Representative-Preference poverty lines violate Objective Anchorage when preferences are heterogeneous. ˜), (x′ , p, u) ∈ S ∗ is in red and indifference Note: The frontier of the budget set sustaining (x, p, u curves are in blue. 21 ∗ Proposition 5. Under heterogeneous prices and heterogeneous preferences (SU ∆ ), any representative-preference poverty line ½ violates Objective Anchorage. Proof. See Appendix 5.13. Our main result uncovers the theoretical foundations for representative-preference poverty lines when preferences are heterogeneous. Like achievements-list lines, representative-preference lines satisfy the version of non-paternalism presented in Proposition 4. In contrast to achievements-list lines, representative-preference lines are not based on a “universal” anchor bundle z ∗ . Rather, representative- preference lines must allow the anchor bundle to be preference-specific. Formally, they satisfy a weak version of Objective Anchorage , which we call Subjective An- chorage . More precisely, Subjective Anchorage requires that for each preference uˆ ∗ ˆ such that any individual with prefer- there exists one associated poverty bundle zu ∗ ˆ and whose bundle has less of every good than zu ence u ˆ is poor and any individual ∗ ˆ and whose bundle has more of every good than zu with preference u ˆ is non-poor. ∗ ˆ ∈ U there exists zu Axiom 5 (Subjective Anchorage ). For all u ˆ ∈ X=0 such that for all (x, p, u) ∈ S ∗ with u = u ˆ, x ∗ zu ˆ ⇒ ½(x, p, u) = 1. ∗ zu ˆ ≤ x ⇒ ½(x, p, u) = 0. Theorem 4 shows that the three axioms Subjective Anchorage , Weak Respect for Preference and Dominated Command are arbitrarily close to fully characterizing the class of representative-preference lines. The characterization in Theorem 4 is not exact because some poverty lines that are not representative-preference – like achievements-list lines – also satisfy these three axioms. Also, some representative-preference lines may violate Subjective Anchorage . However, Theorem 4 shows that both of these cases are always irrele- vant in the sense that they never affect the classification of equilibrium situations as poor or non-poor. For instance, the fact that some representative-preference line violates Subjective Anchorage does not prevent it from classifying situations as poor or non-poor in the same way as some representative-preference line that satisfies this axiom. 22 Formally, let Sˆ∗ ⊂ S ∗ denote a subset containing a finite number of equi- U∆ U∆ librium situations. We say that the line ½′ is equivalent to the line ½ on S ˆ∗ when U∆ ½ (x, p, u) = ½(x, p, u) for all (x, p, u) ∈ SU ∆ . ′ ˆ ∗ ∗ Theorem 4. Assume that preferences and prices are both heterogeneous (SU ∆) and consider any finite-sized S ˆ ⊂ S . (i) For all poverty line ½ that satisfies ∗ ∗ U∆ U∆ Subjective Anchorage, Weak Respect for Preference and Dominated Command on S ∗ , there exists a poverty line ½′ that is equivalent to ½ on S U∆ ˆ∗ and U∆ (a) ½′ satisfies Subjective Anchorage, Weak Respect for Preference and Domi- ∗ nated Command on SU ∆, (b) ½′ is representative preference. (ii) For all representative-preference poverty line ½ there exists a poverty line ½′ that is equivalent to ½ on Sˆ∗ and for which conditions (a) and (b) hold. U∆ Proof. See Appendix 5.14. Theorems 3 and 4 reveal that a key consistency difference between representative- preference lines and achievements-list lines is captured by Objective Anchorage vs Subjective Anchorage . Interestingly, these two axioms formalize and clarify infor- mal arguments used by the proponents of either approaches. First, Reddy and Pogge (2009) criticize the welfaristic approach underpinning the World Bank’s In- ternational poverty line by calling it “meaningless” and only “defined in abstract money units and to local currency amounts that it deems to be equivalent”. Simi- larly, Allen (2017) asserts “there is no persuasive answer to the question: how can you live on $ 1 a day?”. Echoing these criticisms, recommendation 15 in Atkinson (2016) suggests that the World Bank develops “basic needs-based estimates of ex- treme poverty ... to provide an interpretation of what the International Poverty Line would buy”. Proposition 5 reveals that a representative-preference line ad- mits no universal interpretation in terms of basic needs. As a result, any such interpretation of the International poverty line is bound to depend on preferences, and thus, will be country-specific at best. Second, Ravallion (2020) and Ferreira (2017) point to the arbitrariness and insufficiently universal basis for the defini- tion of the anchor of the achievements-list global poverty line proposed by Allen 23 (2017). Their attack on the universality of its anchor aims at casting doubt on the normative advantage of satisfying Objective Anchorage . 3.3 Discussion The welfaristic and objective approaches only differ in their consistency properties when prices are heterogeneous. In that case, the circumstances under which one approach has better consistency properties than the other are revealed by our results, which are summarized in Table 3. Table 3: Consistency properties of the two main types of absolute poverty lines. Homogeneous Pref. Heterogeneous Pref. Rep-Pref line Ach-List line Rep-Pref line Ach-List line Dominated Command + + + + Respect for Preference + - - - Objective Anchorage + + - + Two practical aspects must be successively considered when evaluating which approach is more consistent. The first aspect is the extent to which preferences are heterogeneous. As shown in Table 3, the welfaristic approach has better consis- tency properties under homogeneous preferences while the reverse may hold under heterogeneous preferences.22 The extent of preferences heterogeneity depends on the application for which the absolute poverty measure is designed. In particular, it depends on size of the geographic area considered, its cultural diversity or yet the length of the period over which poverty is monitored. Clearly, preferences are more heterogeneous when evaluating poverty at the global rather than at the coun- try level. The extent of preferences heterogeneity also depends on the austerity of the poverty standard used. Indeed, what matters for our results is the heterogene- ity that preferences exhibit around the poverty standard.23 Arguably, one would expect less preferences heterogeneity when considering a standard close to subsis- 22 We set aside individual-preference lines because they are barely used in practice. 23 In the particular case that all preferences share the same indifference curve through the poverty bundle z , then the welfaristic approach would satisfy all three consistency axioms, even if these preferences differ elsewhere. 24 tence needs, as used in low-income countries, than when considering a standard largely driven by social inclusion needs, as used in higher-income countries. One rationale for this expectation could be that “necessity displaces desire”. The second aspect is the “universality” commanded by the anchor defining the proposed achievements-list line. When preferences are heterogeneous, Table 3 re- minds that achievements-list lines satisfy Objective Anchorage while representative- preference lines only satisfy Subjective Anchorage . Hence, the former are under- pinned by a universal anchor while the latter by tastes-specific anchors. There is unambiguous added value to a universal anchor only if this anchor derives its legitimacy from sources unrelated to preferences. Minimal nutritional achieve- ments, e.g., amounts of calories or micro-nutrients, arguably command a higher “universality” than social inclusion achievements, e.g., possession of a linen shirt (Smith, 1776). Again, the austerity of the poverty standard used may matter as more austere standards seem more likely to derive legitimacy from sources un- related to preferences. Indeed, an individual failing to meet minimal nutritional needs may suffer health consequences, regardless of food-tastes in her society. Meeting minimal nutritional needs is thus not a mere matter of preference satis- faction, but is rather a condition to avoid diet-related diseases (Willett et al., 2019) and thus bad-health-related externalities on society. This suggests that austere poverty standards based on healthy nutrition, like the one considered by Herforth et al. (2022), may “improve” the consistency case for the objective approach by increasing its potential “universality”. As discussed above, more austere poverty standards may “deteriorate” the consistency case for the objective approach by decreasing the extent of preferences heterogeneity. The above discussion suggests that, for the purpose of global extreme poverty monitoring, no approach seems unambiguously more consistent than the other. An achievements-list poverty line would not necessarily have unambiguously bet- ter consistency properties than the International poverty line. Indeed, it is unclear whether preferences are substantially heterogeneous over basic needs satisfaction at its austere $ 1 a day standard. Even if it was the case, a sufficiently universal anchor defining the alternative achievements-list global line – say only based on nutritional achievements – may provide an insufficient coverage of human needs. This subtle point is made by Ravallion (2020) who writes that “credible measures 25 require that we allow for the functioning of social inclusion” on top of nutritional requirements. Social inclusion goods can hardly derive legitimacy from sources completely unrelated to preferences. In turn, the affordability of healthy diet esti- mates monitored by the FAO (2020) offer a useful complement to the International poverty line. Each of these two approaches has valid consistency objections against the other (Proposition 2 and 5). However, these valid objections do not seem suffi- cient to disqualify any of the two approaches, especially when no absolute poverty line can be fully consistent (Theorem 1). The same two aspects also shed light on the question whether region-specific poverty baskets should be used instead of a unique national poverty basket. We do not repeat here our discussion on the aspects influencing which type of poverty line is the most consistent. Rather, we discuss how each type of poverty line relates to national vs region-specific baskets. Both achievements-list and representative- preference lines can be constructed using a unique national poverty basket. The type of poverty line obtained with a unique basket depends on the spatial-deflation method used. If the spatial deflation is based on a price index that accounts for substitution effects, like Paasche or Fisher, then it implements a representative- preference line. If instead the spatial deflation is based on a Laspeyres index, which does not account for substitution effects, then it implements an achievements-list line. Representative-preference lines can also be constructed using region-specific poverty baskets.24 In that case, one should check that the region-specific baskets satisfy revealed preference conditions (Ravallion and Sen, 1996; Arndt and Simler, 2010). 4 Concluding comments We have contrasted the consistency properties of two main approaches used to construct absolute poverty lines. Our results identify the conditions under which one approach can be considered more consistent than the other. We now discuss two strong assumptions underlying our results. 24 Achievements-list lines can also be constructed using region-specific poverty baskets if needs differ across regions. 26 First, our theory abstracts from the distinction between goods consumption and basic needs satisfaction. Under this simplification, the “least-cost” poverty lines developed by Stigler (1945), Allen (2017) or yet Herforth et al. (2022) directly correspond to achievements-list lines. In practice, “least-cost” poverty lines do account for one type of substitution effect, which is related to the mapping from goods consumption to basic needs satisfaction. For instance, when rice provides cheaper calories than wheat, the former may replace the latter in the poverty bundle implicit in a “least-cost” poverty line. To some extent, our results are robust to this type of substitution effect, which is not tastes-based. Our theory remains robust if (1) preferences over commodity bundles derive from primal preferences over basic needs achievements25 and (2) we assume that the consumption of each good proportionally increases the satisfaction of a unique basic need. Assumption (2) implies for instance that rice and wheat may both provide calories, but their consumption cannot provide housing achievements. Second, when contrasting the welfaristic and objective approaches, our meta- theory only considers the differences that relate to price adjustments. Beyond price adjustments, these approaches may also differ in the austerity of their poverty standard and thus differ in their poverty bundle z . We illustrate this by contrasting the (objective) “least-cost” method to the (welfaristic) CBN method of Ravallion and Bidani (1994). Assume that both methods derive their poverty standard with the same basic needs requirement, say 2,000 calories. The commodity bundle considered by the “least-cost” method to reach 2,000 calories will be cheaper than that of the CBN method of Ravallion and Bidani (1994) because the latter adds the constraint that the commodity bundle must reflect local consumption behavior. As a result, the objective approach yields a more austere poverty bundle z O than that of the welfaristic approach z W . One reason we abstract from these differences is the irreducible arbitrary/political element in the austerity of the poverty standard selected.26 Indeed, both approaches can justify any level of austerity for its poverty 25 Under assumption (1), the Objective Anchorage axiom must be adapted to reflect that an absolute poverty line is anchored by a fixed list of basic needs achievements, rather than by a fixed commodity bundle. 26 To be sure, we do not mean that this selection is entirely arbitrary or politically-driven. Indeed, researchers can help guide this selection by showing how to map a given list of basic needs requirements to a poverty bundle z . 27 standard. More precisely, for any given price vector p, any monetary amount y > 0 can be justified by each approach. For instance, we can have y = pz O = pz W if the respective poverty bundles z O and z W are not based on the same number of calories.27 We discuss in greater details the methods used in practice to construct absolute poverty lines in Appendix 5.3. Finally, besides their respective conceptual merits, other considerations also matter when selecting among alternative approaches to absolute poverty lines. In particular, each type of poverty line generates its own set of issues related to empirical applications. For instance, the International poverty line must deal with practical issues related to PPP indices while the objective approach must deal with the mapping from commodities to basic needs satisfaction. We further discuss these practical issues in Appendix 5.4. 27 Relatedly, comparing the respective total number of poor yielded by two different approaches is not highly informative if the alternative approaches select the austerity of their poverty stan- dard in uncoordinated ways. Indeed, the difference in the numbers of poor can always be attributed to different arbitrary/political standards. What is arguably more interesting is to compare the respective poverty trends or the respective distributions of poverty across space. Such investigation typically requires selecting, for all approaches, the austerity of their poverty standard in a coordinated way, e.g., in a way that yields the same total number of poor. 28 5 Appendix 5.1 Relation to the price index literature We contrast our axioms with the main properties imposed on measures of real income by Van Veelen (2002) and Van Veelen and van der Weide (2008). The similarity between our “absolute poverty line” problem and the “real in- come” problem implies that some normative requirements should be the same. For instance, our Dominated Command axiom is a weakening of both the Weak Ranking Restriction (Van Veelen, 2002) and Budget Set Ranking (Van Vee- len and van der Weide, 2008).28 However, the difference between the two problems imply that some of our axioms differ from the properties used in that literature. As mentioned in the In- troduction, our “absolute poverty line” problem is not dual with the “real income” problem. Indeed, defining a measure of real income requires constructing compa- rability between any two situations (x, p, u) and (x′ , p′ , u′ ), whereas our problem only requires constructing comparability between any situation (x, p, u) and one reference situation (z, p∗ , u∗ ). This difference has normative implications. For in- stance, transitivity is an issue that must be considered when building a measure of real income,29 whereas it is irrelevant in our problem because comparability is only constructed with respect to a fixed reference. Other normative differences emerge from the distinct formal objects respec- tively considered in the two problems. The conceptual question we tackle re- quires considering an individual’s situation (x, p, u), whose preference u is assumed known. As a result, we can impose the Respect for Preference axiom. In contrast, the application-oriented literature on price indices starts from the acknowledge- ment that preferences are not observed. The basic object is not the individual situation (x, p, u) but rather the dataset (xi , pi )i∈N , which is a list of observed 28 Weak Ranking Restriction requires that the real income of an individual is larger than that of another individual if the bundle of the former has more of every good than the bundle of the latter. Budget Set Ranking requires that if the budget set of one individual includes the budget set of another individual, then the real income of the former is at least as high as that of the latter. 29 The “real income” problem must impose transitivity in order to make sure that if real income is larger in (x, p, u) than in (x′ , p′ , u′ ), and also larger in (x′ , p′ , u′ ) than in (x′′ , p′′ , u′′ ), we get that real income is larger in (x, p, u) than in (x′′ , p′′ , u′′ ). 29 pairs of bundles and price vectors.30 Typically, a true price index requires that its associated measure of real income is a money-metric utility representation of a preference that rationalizes the dataset, at least when the dataset can be ra- tionalized by a common preference. Such requirement has a similar flavor to the Respect for Preference axiom, but they are not the same. First, the price index must “recover” the common preference from the dataset. The recovered preference then depends on all pairs in the dataset, which may imply that the price index between two countries characterized by different price vectors depends on the pres- ence of a third countries whose pairs are also included in the dataset. Hence, the price index literature considers an Independence of Irrelevant Country axiom (Van Veelen, 2002), which is absent from our framework where individual pref- erences are assumed known. Second, the Respect for Preference axiom remains compelling when preferences are heterogeneous. In contrast, the true price index requirement seems tailored to homogeneous preferences settings. Our “absolute poverty line” problem also comes with specific properties, like the Objective Anchorage axiom. This axiom seems less relevant for the real income problem, which admits no a priori reference. 5.2 Graphical illustration of the three types of lines The three types of absolute poverty lines are graphically illustrated in Figure 6. The description of this figure requires introducing the notion of poverty frontier on the set of bundles. For any given pair (p, u) ∈ ∆ × U , the poverty line ½ partitions the set of bundles x ∈ X for which situation (x, p, u) is in equilibrium between the bundles identified as “poor” and the bundles identified as “non-poor”. As a result, poverty line ½ can be graphically illustrated by the frontier between these subsets of “poor” and “non-poor” bundles.31 30 The “Dependence on Price” property (Van Veelen, 2002), which requires that prices have some influence on the measure of real income, should be adapted in our problem as an axiom requiring that either prices or preferences have some influence on the poverty line. Indeed, individual-preference poverty lines do not formally depend on the price vector, but still make perfect sense. Such adapted version of the “Dependence on Price or Preference” axiom is implied by the Respect for Preference axiom. 31 Formally, for given p and u, the poverty frontier F ½ (p, u) is defined as F ½ (p, u) := {x ∈ X |(x, p, u) ∈ S ∗ , ½(x, p, u) = 0 and ½(x′ , p, u) = 1 ∀ x′ ≪ x s. t. (x′ , p, u) ∈ S ∗ }. 30 (a) (b) (c) Individual-Preference Achievements-list Representative-Preference x2 x2 x2 p p F IP (u′) F AL (p ) F IP (u ) p′ F AL(p′) p′ z z z u u ˜ RP ′ RP u ′ F (p ) F (p ) x1 x1 x1 Figure 6: Poverty frontiers for three types of absolute poverty lines. Note: The frontiers of bundles at which individuals are considered poor are in purple. (a) The frontiers F IP (u) of a individual-preference line only depends on the preference u. (b) The frontiers F AL (p) of an achievements-list line only depends on the price vector p. (c) The frontiers F RP (p) of a representative-preference line only depends on the price vector p. For any individual-preference poverty line ½IP , we can draw a “summary” fron- tier F IP (u) that does not depend on price vector p.32 As illustrated in Figure 6.a, frontier F IP (u) corresponds to the indifference curve of preference u that passes through the poverty bundle z . For any achievements-list poverty line ½AL , we can draw a “summary” frontier F AL (p) that does not depend on the preference u.33 As illustrated in Figure 6.b, frontier F AL (p) corresponds to the frontier of the budget set for which the poverty bundle z is just affordable under price vector p. For any representative-preference poverty line ½RP , we can draw a “summary” frontier F RP (p) that does not depend on the preference u.34 As illustrated in Figure 6.c, frontier F RP (p) corresponds to the frontier of the budget set for which ˜(z ) is just affordable under price vector p. Hence, F RP (p) is the the welfare level u frontier of the budget set defined by price vector p and income eu ˜(z )). ˜ (p, u 32 Frontier F IP (u) summarizes all frontiers F ½ (p, u) in the sense that F IP (u) is consistent with F ½ (p, u) for all p ∈ ∆. 33 Frontier F AL (p) summarizes all frontiers F ½ (p, u) in the sense that F AL (p) is consistent with F ½ (p, u) for all u ∈ U . 34 See footnote 33. 31 5.3 Methods for constructing absolute poverty lines We present here in greater details some of the major methods used in practice to construct absolute poverty lines. We consider first the selection of their nominal amount in a reference context, then price adjustments, specificities of global lines and finally come back to the relation between these methods and our three stylized types of poverty lines. Selection of the nominal amount under p∗ There exist alternative methods to the selection of a nominal monetary amount in a given reference context (p∗ ) when starting from a list of basic needs requirements. Assume for simplicity that an individual is considered poor if she cannot purchase a food bundle that yields 2,000 calories per day.35 The poverty line is then typically defined as the cost of some food bundle that yields 2,000 calories.36 As mentioned by Ravallion (1998), there exists an infinite set of bundles that achieve a given list of basic needs requirements. For instance, one could reach 2,000 calories solely by consuming low quality rice or by consuming more expensive animal-sourced calories. These alternative (food) bundles need not be equivalent in terms of utility. Hence, the method for selecting the (food) bundle affects the reference nominal monetary amount. The least-cost method (based on linear programming) identifies the cheapest bundle that provides the desired number of calories. This bundle needs not correspond to the consumption patterns of the poor (Stigler, 1945; Ravallion, 2020). The least-cost method has been used and refined in the literature on the global affordability of nutrition (Masters et al., 2018; Hirvonen et al., 2020; Herforth et al., 2022).37 The cost-of-basic-needs (CBN) method of Ravallion and Bidani (1994) identifies a bundle that provides the desired number of calories and reflects the consumption behavior typical of individuals who are poor. 35 In practice, less simplistic lists of basic needs requirements are used (Ravallion, 2020). Sev- eral papers contain arguments in defense of longer list of basic needs requirements (Allen, 2017; Mahrt et al., 2022). 36 An alternative is to use the Food Energy Intake method (Dandekar and Rath, 1971; Greer and Thorbecke, 1986a,b), which uses a regression in order to compute the income level at which individuals living in the reference context consume on average 2,000 calories. Ravallion and Bidani (1994) make the point that the Food Energy Intake method may perform worse in terms of utility consistency than alternative methods. 37 The FAO is commited to regularly update on the affordability of healthy diets (FAO, 2020). 32 To do so, the CBN method first identifies the typical food bundle from relatively poor individuals, i.e., individuals in the bottom of the income distribution. Second, the CBN method scales the quantities in that bundle until the desired number of calories is reached. As mentioned in the Conclusion, for fixed basic needs requirements, the least-cost method yields a more austere monetary amount than the CBN method. Adjusting the nominal amount to other prices There exist different methods to adjust the reference nominal monetary amount of the poverty line across contexts with different prices. First, one could start from the poverty bundle and compute, in each context, the income that would leave individuals indifferent to the reference bundle. Ideally, doing this would require eliciting preferences using demand systems a la Deaton and Muellbauer (1980), as for instance done in Dimri and Maniquet (2019). In practice, price indices are typically used as a simpler price-adjustment method that provides a reasonable approximation (Deaton and Zaidi, 2002; Mancini and Vecchi, 2022).38 Second, one could disregard substitution effects and directly compute the poverty line in each context as the cost of purchasing the poverty bundle. This approach was for instance used in India before 2009 (Tendulkar et al., 2009). Third, one could use the implicit price-adjustment method that consists in computing the poverty line’s nominal monetary amount from the basic needs requirements, in each con- text, through the least-cost method (or through the CBN-method). In our earlier example, in any context, the nominal monetary amount would correspond to the cost of the context-specific food bundle that yields 2,000 calories. As discussed in Section 2.2, the exact method used to identify the context-specific bundles (least-cost or CBN) matters for the implicit price-adjustment method obtained. As argued by Ravallion and Bidani (1994), the CBN-method in each context per- forms well in terms of specificity, because the bundles corresponds to the behavior observed in each context. In order to mitigate the risk that context-specific bun- 38 A review of the conditions under which different price indices are correct approximations can be found in Diewert (1979). 33 dles violate utility-consistency,39 Arndt and Simler (2010) and Mahrt et al. (2022) suggest adding revealed-preference adjustments to the CBN-method. Particularities of global poverty lines Constructing global poverty lines is typically more challenging than national poverty lines because the former face a larger variety of contexts. Obviously, prices and preferences are more heterogeneous across countries than within countries. The global context also exacerbates the issue of goods unavailability. Indeed, the goods consumed by the poor in one country may not be available or consumed by the poor in another country. Importantly, the absence of a global government complicates the selection of the arbitrary elements inherent to an absolute line. The solution implemented by the World Bank consists in defining its international poverty line as the median value in US$ of the national poverty lines in low-income countries (Ravallion et al., 2009b; Ferreira et al., 2016; Jolliffe et al., 2022).40 This method is thus not based on a global poverty bundle. Interestingly, this solution of taking some “average” of national standards is also used by the least-cost approach in the case of healthy diets (Hirvonen et al., 2020; Herforth et al., 2022). Practical methods and our three types of lines Our three types of poverty lines constitute stylized formalizations for different methods to construct absolute poverty lines. Of course, there is a large variety of methods used in practice and some may not exactly correspond to any of these three stylized types. We now discuss in greater detail how some major methods might be rooted to some of these three types. First, a common practice consists in defining the poverty line in a reference context, and then adjusting for price vectors using price indices that account for tastes-based substitution effects, like Paasche, Fisher or yet PPP indices. The most well-known example is the international poverty line of the World Bank 39 Relatively poor individuals living in more affluent contexts could on average consume more expensive calories. As a result, the CBN food bundle in those regions may yield a higher welfare than the CBN food bundle in less affluent contexts. 40 More precisely, national poverty lines of low-income countries are translated into $US through PPP indices. Then, the median value (in $US) among these national lines defines the international poverty line. 34 (Ravallion et al., 2009b; Ferreira et al., 2016; Jolliffe et al., 2022). The post-2009 official poverty lines of India constitute another prominent example (Tendulkar et al., 2009). The price index is typically defined at the “context level” – a unique price index is relevant for all individuals living in the same region and time – which means that such practices are rooted in representative-preference poverty lines. The common preference embodied by the price index can be thought of as the representative preference that rationalizes the average consumption entering into the construction of the price index. Another practice consists in defining the poverty line simultaneously in all con- texts, which implicitly defines a method for adjusting its nominal amount across contexts.41 As mentioned above, there are several methods for computing a nomi- nal monetary amount from a list of basic needs. We contrast here the CBN method of Ravallion and Bidani (1994) with the least-cost linear-programming method. For our purpose, the relevant difference between these two methods is that the former accounts for observed consumption behavior while the latter does not. As a result, the former accounts for tastes-based substitution effects in a way that the latter does not. We argue that this implies that the CBN method is best rooted in representative-preference lines while the least-cost method is best rooted in achievements-list lines. There is little controversy that the CBN method is aimed at producing representative-preference poverty lines.42 It is less immediate that the least-cost method is rooted in achievements-list lines. Indeed, the least-cost method may yield different implicit poverty bundles when relative prices change. Yet, even if implicit poverty bundles may differ across contexts, they do not allow for preference-based substitution effects. To see this, consider food and non-food needs and assume that the relative price between food and non-food varies across contexts. When comparing expenditure patterns across contexts, one may find that a relatively poor person living in a context where the relative price for food is low consumes more animal-sourced calories and consumes less of some non-food goods (than their counterparts living in a context where the relative price for food 41 Such practice cannot be rooted in individual-preference poverty lines when preferences are heterogeneous. The reason is that this practice yields a unique nominal amount in each context. 42 An indirect evidence for this claim is that some authors have developed revealed preference adjustments for the nominal amounts obtained in different contexts through the CBN method (Arndt and Simler, 2010). 35 is high). These substitution effects between food and non-food are captured by the CBN method, whose nominal amounts account for food habits and share of ex- penditure on non-food. However, they are not captured by the least-cost method, which considers in each context the cheapest way of achieving, say, 2,000 calories and 3 m2 of housing. In other words, the least-cost method does not allow for any substitution between the number (or quality) of calories and the surface of housing consumed. 5.4 Practical issues of alternative types of absolute poverty lines We consider three types of poverty lines – individual-preference, achievements-list and representative-preference – that we formally define in Section 2.2. Beyond having different consistency properties, the three types of absolute poverty lines generate different difficulties in their practical implementations, which have been pointed out in the debates between advocates of alternative approaches. First, individual-preference poverty lines are arguably the most challenging to apply under preference heterogeneity. The main difficulty is that they require eliciting preferences. In principle, demand systems a la Deaton and Muellbauer (1980) can be used for this purpose. However, eliciting preferences through de- mand systems is highly data-intensive and requires judgment (it is far from being a routine task). Other techniques to elicit preferences exist (Decancq et al., 2019), but they are also challenging to apply. As a result, individual-preference poverty lines are typically confined in research papers. Second, achievements-list poverty lines typically rely on a narrow set of prices, which makes them more vulnerable to price-collection errors. Consider here the case of global poverty measurement. As argued in Section 2.2, poverty lines a la Allen (2017), which are based on the least-cost of satisfying a small list of basic needs requirements, fall into the achievements-list type. The goods non- availability issue implies that a common global poverty basket cannot be defined in the space of goods, but must instead be defined in the space of basic needs requirements. For instance, this global list of achievements could include 2,000 calories per person per day and 3 m2 of housing. The least-cost of this list of 36 achievements may rely on wheat and caravan in one country and on rice and yurt in another country. Clearly this least-cost method relies heavily on the prices observed for a limited number of goods and is thus vulnerable to outliers. A decrease in the price of one cereal may strongly affect the result, even if this cereal is scarcely consumed in the country. Finally, representative-preference poverty lines typically rely on imperfect price indices. Consider again the case of global poverty measurement. The World Bank’s international poverty line is based on the PPPs price indices, whose limitations are well-known (Heston and Summers, 1996; Deaton, 2010; Deaton and Heston, 2010; Ravallion, 2018), see also Deaton and Dupriez (2011). One serious practical difficulty is that every new round of ICP data introduces new PPPs, which po- tentially modifies the distribution and trend of international global poverty. Most often, the changes are relatively marginal in practice, but these changes may gen- erate confusion (Ravallion, 2020). Unlike the least-cost method, PPPs incorporate a large array of different commodities, which provides some robustness to the in- ternational poverty line. Now, representative-preference poverty lines need not depend on price indices. As discussed in Section 2.2, their nominal amounts can be constructed as the cost of context-specific poverty bundles obtained through the CBN method. In this case, a practical issue is that it is always unclear whether the nominal amounts obtained in different contexts correspond to the same level of utility. This issue can be mitigated by checking revealed preference conditions (Ravallion and Sen, 1996; Wodon, 1997) or adjusting the region-specific bundles in a way that guarantees they meet revealed preference conditions (Arndt and Simler, 2010). However, these solutions may at best mitigate this issue, which never completely disappears. Furthermore, one could think of conditions under which it is unclear whether such revealed preference adjustments are necessarily an improvement.43 43 Here are two hypothetical cases where revealed preference adjustments may go against the ideal of utility-consistency. The first case is when social inclusion is among the needs to be captured. In that case, relativist considerations may enter the picture. For instance, it may be that a linen shirt should be added to the poverty bundle when considering a context with higher standard of living. Obviously, the poverty bundle z in this context will be revealed preferred to the poverty bundle z ′ of another context with lower standard of living. However, this does not mean that bundles z and z ′ need to be adjusted, because they may both yield the same utility (as relative income may matter for utility when social inclusion is to be captured). 37 Other practical issues have been raised in the debate between alternative types of poverty line. Some of them in fact apply to all three types of poverty lines, as we illustrate in the case of global poverty measurement. One example is the element of arbitrariness of any global poverty line. Ferreira (2017) argues that some elements in the list of basic needs requirements considered by Allen (2017) may be arbitrary, while the World Bank’s international line is anchored in national poverty lines (because the WB’s line is defined by “averaging” the national poverty lines of low-income countries). In turn, Allen (2017) argues that the World Bank also relies on a somewhat arbitrary selection of the set of countries whose national poverty lines matter for the definition of its international line (Allen, 2017). More importantly, global achievements-list lines can similarly be defined by “averaging” national baskets. This has for instance been done by Herforth et al. (2022) for the case of nutrition, where their income threshold is the least cost of achieving a healthy diet that “averages” recommended healthy diets in 10 countries. However, the number of “arbitrary” choices underpinning the extreme poverty line seems smaller than the number of “arbitrary” choices underpinning a least-cost global line a la Allen (2017). Ultimately, any type of poverty line must make some arbitrary choice(s) when defining the poverty standard in a reference context, which requires making a choice of a political nature. Both types of global lines can potentially make this choice as an average of countries’ choices. Another example is that global poverty lines do not score well in terms of specificity, in the sense that they are not well rooted in the consumption behav- ior of the poor. This is well known in the case of a least-cost global line a la Allen (2017). Ravallion (2020) reminds that such line relies in practice on implicit poverty bundles that may significantly deviate from observed consumption behav- ior. But some version of this criticism similarly applies to the extreme line of the World Bank. As observed by Allen (2017) and Atkinson (2016), in most countries, The second case is when only a subset of the poverty bundle is explicitly identified. This typically happens when using the CBN method because only the food bundle is explicitly iden- tified while the non-food bundle is not explicitly identified. In practice, a non-food component is usually computed as a fraction of the food poverty line. Consider a fixed preference and two contexts that differ in the relative prices between food and non-food. If the relative price of food is smaller, an optimal behavior may be to consume more expensive calories and consume less non-food goods. In that case, applying revealed preference adjustments based on the food bundle only needs not be wise, as such adjustments may go against utility-consistency. 38 it is unclear what it means to be living at the international poverty line.44 More fundamentally, as revealed by Theorem 4, the extreme line cannot be anchored in a unique global poverty bundle because it is a representative-preference poverty line. 5.5 Results for homogeneous price vectors When both prices and preferences are homogeneous, all three types of poverty lines are equivalent. Observation 3. Under homogeneous prices and homogeneous preferences (X × {p} × {u}), the three types of poverty line are equivalent. Also, the three types of poverty lines satisfy all three axioms. Proposition 6. Under homogeneous prices and homogeneous preferences (X × {p} × {u}), the three types of poverty line satisfy Dominated Command, Respect for Preference and Objective Anchorage. Proof. Representative-preference and achievements-list poverty lines satisfy all three axioms by Proposition 7. Then, individual-preference lines satisfy all three axioms because they are equivalent to representative-preference lines (Observation 3), which satisfy all three axioms. When prices are homogeneous but preferences are heterogeneous, representative- preference lines and achievements-list lines are equivalent. This follows from the fact that their respective price-adjustment functions Z do not depend on prefer- ences and the price vector is fixed. ∗ Observation 4. Under homogeneous prices and heterogeneous preferences (SU ), representative-preference and achievements-list poverty lines are equivalent. 44 It is only in the subset of low-income countries whose national line is close enough to the extreme line (in PPPs terms) that the international line may score well in terms of specificity, i.e., that its meaning can be confidently related to the satisfaction of a few basic needs by relying on a bundle reflecting local consumption behavior. A related issue is that different countries in this subset may rely on different basic needs requirements when defining their national poverty line. 39 Proposition 7 shows that both representative-preference lines and achievements- list lines satisfy all three consistency axioms. ∗ Proposition 7. Under homogeneous prices and heterogeneous preferences (SU ), representative-preference and achievements-list poverty lines satisfy Dominated Command, Respect for Preference and Objective Anchorage. Proof. By Observation 4, we only need to prove this for achievements-list lines. By Theorem 3, any achievements-list line satisfies both Dominated Command ∗ ∗ and Objective Anchorage on SU ∆ and thus the same also holds on SU because ∗ ∗ SU ⊂ SU ∆. There remains to show that any achievements-list line ½ satisfies Respect for ∗ Preference on SU ∗ . Consider any two (x, p, u), (x′, p′ , u′ ) ∈ SU with u = u′ , u(x) ≤ u(x′ ) and ½(x, p, u) = 0. We must show that ½(x′ , p′ , u′) = 0. We have px ≥ pz because ½(x, p, u) = 0 and ½ is an achievements-list line. We also have that p = p′ ∗ because (x, p, u), (x′ , p′ , u′) ∈ SU . We immediately get that p′ x ≥ p′ z . We also have that p′ x′ ≥ p′ x because the two equilibrium situations are such that u = u′ , p = p′ and u(x′ ) ≥ u(x). By transitivity we get that p′ x′ ≥ p′ z and thus ½(x′ , p′ , u′ ) = 0 because ½ is an achievements-list line, as desired. In turn, individual-preference lines satisfy Respect for Preference and Objective Anchorage (Theorem 3), but they violate Dominated Command (Proposition 3). 5.6 Proof of Proposition 1 ⇐. We show that any individual-preference ½ satisfies Objective Anchorage and ∗ Respect for Preference on S∆ . As ½ is individual-preference, there exists z ∈ X=0 such that for all (x, p, u) ∈ ∗ S∆ ½(x, p, u) = 1 ⇔ u(x) < u(z ). (1) First we show that ½ satisfies Objective Anchorage for bundle z ∗ := z . Consider ∗ any (x, p, u), (x′ , p′ , u′) ∈ S∆ that are such that x z ≤ x′ . We have u(x) < u(z ) and u′ (z ) ≤ u′ (x′ ) because preferences are strictly monotonic and x z ≤ x′ . By 40 Eq. (1), we thus have ½(x, p, u) = 1 and ½(x′ , p′ , u′ ) = 0, which proves Objective Anchorage . Second we show that ½ satisfies Respect for Preference . Consider any two situations (x, p, u), (x′ , p′, u) ∈ S∆ ∗ with u(x) ≤ u(x′ ) and ½(x, p, u) = 0. As ½(x, p, u) = 0, Eq. (1) implies that u(x) ≥ u(z ). This implies in turn that u(x′ ) ≥ u(z ) because u(x) ≤ u(x′ ). Eq. (1) thus implies that ½(x′ , p′ , u) = 0 because u(x′ ) ≥ u(z ), as desired. ⇒. We show that any poverty line ½ that satisfies Objective Anchorage and ∗ Respect for Preference on S∆ is individual-preference. As ½ satisfies Objective Anchorage there exists a bundle z ∗ ∈ X=0 with the properties defined in that axiom. To prove that ½ is individual-preference, we ∗ show that for all (x, p, u) ∈ S∆ we have u(x) < u(z ∗ ) ⇒ ½(x, p, u) = 1, u(x) ≥ u(z ∗ ) ⇒ ½(x, p, u) = 0. Consider first the case u(x) < u(z ∗ ). This case is such that there exists a bundle x˜ such that x˜ z ∗ and u(˜ ˜ ∈ ∆ such x) = u(x) and a price vector p ∗ 45 that the situation (˜ ˜, u) belongs to S∆ . x, p By Objective Anchorage we have that ½(˜ ˜, u) = 1 because x x, p ˜ z ∗ . By Respect for Preference we have that ½(x, p, u) = ½(˜ ˜, u) and thus ½(x, p, u) = 1, as desired. x, p Consider then the case u(x) ≥ u(z ∗ ). This case is such that there exists a bundle x˜ such that x ˜ ≥ z ∗ and u(˜ ˜ ∈ ∆ such that x) = u(x) and a price p ∗ 46 the situation (˜ ˜, u) belongs to S∆ x, p . By Objective Anchorage we have that ½(˜ ˜ ≥ z ∗ . By Respect for Preference we have that ½(x, p, u) = ˜, u) = 0 because x x, p ½(˜ ˜, u) and thus ½(x, p, u) = 0, as desired. x, p 45 Indeed, for all u ∈ U and all x′ ∈ X there exists a p′ ∈ ∆ for which the situation (x′ , p′ , u) ∗ belongs to S∆ because u is strictly monotonic in all goods (Lemma 3). 46 See Footnote 45. 41 5.7 Proof of Proposition 2 The proofs that the achievements-list line ½ satisfies Objective Anchorage and Dominated Command are straightforward and are thus omitted. We show that ½ violates Respect for Preference on S∆∗ . Let z denote the poverty bundle for the achievements-list line ½. As u is strictly monotonic, there exist a price vector p ∈ ∆+ and a sufficiently small ǫ > 0 such that the budget set defined by price vector p and income y := pz − ǫ contains a bundle x such that u(x) ≥ u(z ). This implies that the (optimal) demand x∗ in this budget set is also such that u(x∗ ) ≥ u(z ), and moreover (x∗ , p, u) ∈ S∆ ∗ . As ½ is an achievements-list line, we must have ½(x , p, u) = 1 because px = y < pz as ǫ > 0. Also, we ∗ ∗ must have ½(z, p′ , u) = 0 for any p′ ∈ ∆ such that (z, p′ , u) ∈ S∆ ∗ because z is the poverty bundle of this achievements-list line. This yields a contradiction to Respect for Preference , which implies that ½(x∗ , p, u) = 0 because u(x∗ ) ≥ u(z ) and ½(z, p′ , u) = 0. 5.8 Proof of Theorem 1 We derive a contradiction when assuming that ½ satisfies the three axioms. The construction is illustrated in Figure 7. As ½ satisfies Objective Anchorage , there exists a bundle z ∗ ∈ X=0 that satisfies the requirements of this axiom. Consider two distinct linear preferences u ˜′ ∈ U whose constant marginal rates of substi- ˜, u tution are respectively proportional to some price vectors p ˜′ ∈ ∆+ where p ˜, p ˜′ . ˜= p As p˜= p˜′ , there exists two situations (x, p ˜), (x′ , p ˜, u ˜′ , u ˜ ′ ) ∈ SU∗ ∆ that are such that ∗ ′ ′ ′ ∗ ′ 47 ′ ′ ˜(x) = u u ˜ (z ), u ˜ (x ) < u ˜ (z ) and x ≪ x . As u ˜ (x ) < u ˜′ (z ∗ ), there exists a situation (˜ ˜′ , u x′ , p ∗ ˜ ′ ) ∈ SU ∆ that is such that u˜′ (˜ x′ ) = u ˜′ (x′ ) and x ˜′ z ∗ . By Ob- jective Anchorage , we have ½(z ∗ , p ˜) = 0 and ½(˜ ˜, u ˜′ , u x′ , p ˜′) = 1. By Respect for Preference , we have48 • ½(x, p ˜) = ½(z ∗ , p ˜, u ˜) and ½(x′ , p ˜, u ˜′) = ½(˜ ˜′ , u x′ , p ˜ ′ ), ˜′ , u 47 The situations (x, p ˜) and (x′ , p ˜, u ˜′ ) belong to SU ˜′ , u ∗ ∆ because prices p ˜ and p˜′ respectively ′ correspond to the constant marginal rates of substitution of u ˜ and u˜. 48 This follows from the fact that u ˜(x) = u ˜(z ∗ ) and u x′ ) = u ˜′ (˜ ˜′ (x′ ). For the sake of com- pleteness, we show ½(x, p ˜) = ½(z ∗ , p ˜, u ˜, u˜). If ½(x, p ˜) = 0, then Respect for Preference directly ˜, u yields ½(z , p ∗ ˜) = 0 because u ˜, u ˜(x) ≤ u ˜(z ∗ ). If instead ½(x, p ˜) = 1, then Respect for Preference ˜, u prevents that ½(z ∗ , p ˜) = 0 because u ˜, u ˜(x) ≥ u ˜(z ∗ ), and thus ½(z ∗ , p ˜ ) = 1. ˜, u 42 which yields ½(x, p ˜) = 0 and ½(x′ , p ˜, u ˜ ′ ) = 1. ˜′ , u x2 uL x′ x′′ uL p ˜ ∗ x z p ˜ ˜′ x ˜′ p u ˜ ′ u ˜ x1 Figure 7: Construction for the proof of Theorem 1. Note: The frontiers of budget sets are in red and indifference curves are in blue. The frontiers of budget sets sustaining (x, p ˜, u ˜′ , u ˜), (x′ , p ∗ ˜ ′ ) ∈ SU ∆ are in blue because they are also indifference ′ ˜ and u curves for preferences u ˜. ′′ ′′ ′′ Consider a bundle x ∈ X such that p ˜x > p ˜′ x ≥ p ˜x and p ˜′ x′ . (Observe that ′′ such bundle exists because x := x′ satisfies these two requirements as x ≪ x′ .) ′′ ′′ As p˜x > p˜x and p ˜′ x′ , there exists a preference uL ∈ U , which we illustrate ˜′ x ≥ p in Figure 7,49 such that • uL (x′ ) = uL (x′′ ), and ∗ ′ ′ ∗ ′′ ∗ ˜, uL ) ∈ SU • (x, p ˜ , u L ) ∈ SU ∆ , (x , p ˜, uL ) ∈ SU ∆ and (x , p ∆. By Dominated Command , we have50 ˜, uL ) = ½(x, p • ½(x, p ˜′ , uL ) = ½(x′ , p ˜) and ½(x′ , p ˜, u ˜ ′ ), ˜′ , u ˜, uL ) = 0 and ½(x′ , p which yields ½(x, p ˜′ , uL ) = 1. As illustrated in Figure L L L ′′ ′′ 7, preference u must be such that u (x) < u (x ) because p ˜x > p ˜x and ˜, uL ), (x′′ , p (x, p ˜, uL ) ∈ SU ∗ L ∆ . By Respect for Preference , the facts that u (x) < uL (x′′ ) and ½(x, p ˜, uL ) = 0 imply that ′′ 49 We can easily prove that such preference uL exists for the case x := x′ . Take uL to be such that uL ’s indifference curves through x corresponds to the upper contour of the two indifference curves of u ˜′ through x, and uL ’s indifference curves through x′ corresponds to the upper ˜ and u contour of the two indifference curves of u˜ and u˜′ through x′ . It is straightforward to check that L this preference u meets all requirements. 50 For the sake of completeness, we show ½(x, p ˜, uL ) = ½(x, p ˜). If ½(x, p ˜, u ˜, uL ) = 0, then Dominated Command directly yields ½(x, p ˜) = 0 because x ≥ x. If instead ½(x, p ˜, u ˜, uL ) = 1, then Dominated Command prevents that ½(x, p ˜) = 0 because x ≤ x, and thus ½(x, p ˜, u ˜) = 1. ˜, u 43 • ½(x′′ , p ˜, uL ) = 0. However, as uL (x′ ) = uL (x′′ ), Respect for Preference also implies that ˜, uL ) = ½(x′ , p • ½(x′′ , p ˜′ , uL ), ˜, uL ) = 1, a contradiction to ½(x′′ , p which yields ½(x′′ , p ˜, uL ) = 0. 5.9 Proof of Theorem 2 This follows from the proof of Proposition 1, whose argument does not depend on ∗ ∗ the assumption of homogeneous preferences (simply replace S∆ by SU ∆ ). 5.10 Proof of Proposition 4 ∗ ⇒. Consider any two (x, p, u), (x′, p′ , u′ ) ∈ SU ′ ′ ∆ with p = p and px ≤ px . We must show that ½(x, p, u) ≥ ½(x′ , p′ , u′ ). Consider the linear preference u ∈ U whose constant marginal rate of substitution is proportional to p. Preference u is strictly monotonic because p ∈ ∆+ as (x, p, u), (x′ , p′, u′ ) ∈ SU ∗ ∆ . The economic ∗ situations (x, p, u) and (x′ , p′ , u) belong to SU ′ ∆ because p = p correspond to the constant marginal rate of substitution of preference u. For the same reason, we must have u(x′ ) ≥ u(x) because px ≤ px′ . By Weak Respect for Preference , we have ½(x, p, u) ≥ ½(x′ , p′ , u). By Dominated Command , we have ½(x, p, u) = ½(x, p, u) and ½(x′ , p′ , u) = ½(x′ , p′ , u′ ). This yields by transivity that ½(x, p, u) ≥ ½(x′ , p′ , u′), as desired. ⇐. First, we show that the poverty line ½ satisfies Dominated Command . Take any two (x, p, u), (x′ , p′ , u′) ∈ SU ∆ with p = p , x ≤ x and ½(x, p, u) = 0. We ∗ ′ ′ must show that ½(x′ , p′ , u′ ) = 0. We have px ≤ px′ because p = p′ and x ≤ x′ . Therefore, the precondition directly yields ½(x, p, u) ≥ ½(x′ , p′ , u′) because p = p′ and px ≤ px′ . We therefore get ½(x′ , p′ , u′ ) = 0 because ½(x, p, u) = 0, as desired. Second, we show that the poverty line ½ satisfies Weak Respect for Preference . ∗ Take any two (x, p, u), (x′, p′ , u′ ) ∈ SU ′ ′ ′ ∆ with u = u , u(x) ≤ u(x ), p = p and ½(x, p, u) = 0. We must show that ½(x′ , p′ , u′) = 0. We have px ≤ px′ because the two equilibrium situations are such that p = p′ , u = u′ and u(x) ≤ u(x′ ). Therefore, the precondition directly yields ½(x, p, u) ≥ ½(x′ , p′ , u′) because p = p′ and px ≤ px′ . We therefore get ½(x′ , p′ , u′ ) = 0 because ½(x, p, u) = 0, as desired. 44 5.11 Independence of axioms in Theorem 3 ∗ Lemma 1. Under heterogeneous prices and heterogeneous preferences (SU ∆ ), Ob- jective Anchorage, Weak Respect for Preference and Dominated Command are mutually independent. Proof. We provide a poverty line ½∗ that satisfies Objective Anchorage and Dom- ∆ . Line ½ is ∗ ∗ inated Command but violates Weak Respect for Preference on SU ∗ defined by a poverty bundle z ∈ X with z ≫ 0 such that for all (x, p, u) ∈ SU ∆ ½(x, p, u) = 0 ⇔ x ≥ z. Line ½∗ violates Weak Respect for Preference because it is possible to find (x′ , p, u), (x′′, p, u) ∈ ∗ ′ ′ ′′ ′′ SU ∆ such that x ≥ z , u(x ) < u(x ) and xj < zj for some dimension j . Indeed, we would have ½∗ (x′ , p, u) = 0 and ½∗ (x′′ , p, u) = 1 even though Weak Respect for Preference implies ½(x′ , p, u) ≥ ½(x′′ , p, u). We provide a line ½∗∗ that satisfies Weak Respect for Preference and Dominated ∆ . Line ½ ∗ ∗∗ Command but violates Objective Anchorage on SU is defined such that ½(x, p, u) = 0 for all (x, p, u) ∈ S . ∗ Individual-Preference poverty lines satisfy Weak Respect for Preference and ∗ Objective Anchorage but violate Dominated Command on SU ∆ (Proposition 3). 5.12 Proof of Theorem 3 ⇐. We must show that any achievements-list line ½ satisfies Objective Anchorage , ∗ Weak Respect for Preference and Dominated Command on SU ∆. The proofs that an achievements-list line satisfies Objective Anchorage and Dominated Command are straightforward and thus omitted. We only show that ½ satisfies Weak Respect for Preference . Consider any (x, p, u), (x′ , p′ , u′) ∈ SU ∆ such that u = u , p = p , u(x) ≤ u(x ) and ½(x, p, u) = 0. ∗ ′ ′ ′ We must show that ½(x′ , p′ , u′) = 0. As ½ is an achievements-list line, this case is such that pz ≤ px for the poverty bundle z . We have px ≤ px′ because u(x) ≤ u(x′ ) and bundles x and x′ are both optimal given price vector p and 45 preference u. We therefore get pz ≤ px′ . As p′ = p, we obtain p′ z ≤ p′ x′ . This yields ½(x′ , p′ , u′ ) = 0 because ½ is an achievements-list line, the desired result. ⇒. We must show that any line ½ that satisfies Objective Anchorage , Weak Respect ∗ for Preference and Dominated Command on SU ∆ is an achievements-list line. As ½ satisfies Objective Anchorage , there exists a bundle z ∗ ∈ X=0 that satisfies the requirements of this axiom. To prove that ½ is an achievements-list line, we ∗ show that for all (x, p, u) ∈ SU ∆ we have px < pz ∗ ⇒ ½(x, p, u) = 1, px ≥ pz ∗ ⇒ ½(x, p, u) = 0. ∗ Observe that p ∈ ∆+ because (x, p, u) ∈ SU ∆. ∗ Consider first the case px < pz . This case is such that there exists a bundle x′ ∈ X with px′ = px and x′ z ∗ . As p ∈ ∆+ , there exists some preference u′ ∈ U for which (x′ , p, u′) ∈ SU∗ ∆. 51 By Objective Anchorage , the situation (x′ , p, u′) is such that ½(x′ , p, u′) = 1 because x′ z ∗ . By Lemma 2, we then get that ½(x, p, u) = ½(x′ , p, u′) because px′ = px, which yields ½(x, p, u) = 1 as desired. Consider then the case px ≥ pz ∗ . This case is such that there exists a bundle x′′ ∈ X with px′′ = px and x′′ ≥ z ∗ . Again, there exists some preference u′′ ∈ U ∗ for which (x′′ , p, u′′ ) ∈ SU ′′ ′′ ∆ . By Objective Anchorage , the situation (x , p, u ) is such that ½(x′′ , p, u′′) = 0 because x′′ ≥ z ∗ . By Lemma 2, we then get that ½(x, p, u) = ½(x′′ , p, u′′) because px′′ = px, which yields ½(x, p, u) = 0 as desired. Lemma 2. On U , if the poverty line ½ satisfies Weak Respect for Preference and ∗ Dominated Command, then for any two (x, p, u), (x′, p′ , u′ ) ∈ SU ′ ∆ with p = p and px ≤ px′ we have ½(x, p, u) ≥ ½(x′ , p′ , u′ ). Proof. Consider the linear preference u ∈ U whose constant marginal rate of sub- stitution is proportional to p. Preference u is strictly monotonic because p ∈ ∆+ as (x, p, u), (x′ , p′ , u′) ∈ SU∗ ′ ′ ∆ . The economic situations (x, p, u) and (x , p , u) belong ∗ ′ to SU ∆ because p = p correspond to the constant marginal rate of substitution of 51 ∗ The situation (x′ , p, u′ ) belongs to SU ′ ∆ if, for instance, u is a linear preference with constant marginal rate of substitution equal to p. Such linear preference u′ would be strictly monotonic because p must lie in the interior of ∆. 46 preference u. For the same reason, we must have u(x′ ) ≥ u(x) because px ≤ px′ . By Weak Respect for Preference , we have ½(x, p, u) ≥ ½(x′ , p′ , u). By Dominated Command , we have ½(x, p, u) = ½(x, p, u) and ½(x′ , p′ , u) = ½(x′ , p′ , u′ ). This yields ½(x, p, u) ≥ ½(x′ , p′ , u′ ) as desired. 5.13 Proof of Proposition 5 By Theorem 3, any poverty line that satisfies Objective Anchorage , Dominated ∗ Command and Weak Respect for Preference on SU ∆ is an achievements-list line. Any representative-preference poverty line must violate at least one of these three axioms because, as all preferences in U are strictly monotonic, no representative- preference line is an achievements-list line. We directly get that a representative- ∗ preference poverty line violates Objective Anchorage on SU ∆ because such line satisfies both Dominated Command and Weak Respect for Preference (Step 1 of the proof of Theorem 4). 5.14 Proof of Theorem 4 Let Pˆ ∗ denote the finite-sized subset of price vectors that belong to some situation ˆ ∗ := {p ∈ ∆|p = p′ for some (x, p′ , u) ∈ S ˆ∗ , i.e., P in S ˆ∗ }. Observe that P ˆ∗ ⊂ U∆ U∆ ∆+ because any price vector p ∈ P ˆ∗ , ˆ ∗ is such that p = p′ for some (x, p′ , u) ∈ S U∆ which implies that p′ ∈ ∆+ (Lemma 3). Lemma 3. For all u ∈ U and x ∈ X , there exists some p ∈ ∆+ such that ∗ (x, p, u) ∈ S∆ and if p′ ∈ ∆\∆+ then (x, p′ , u) ∈ ∗ / S∆ . Proof. The existence of such p ∈ ∆+ directly follows from the fact that u must admit a finite marginal rate of substitution at x because u is strictly monotonic in all goods. The impossibility for (x, p′ , u) to be an equilibrium situation directly follows from the fact that p′ ∈ ∆\∆+ implies an infinite demand in the good j for which p′j = 0 while x has finite quantities in all goods because x ∈ X . ˜ For any representative-preference line, we denote by I u (z ) the indifference curve ˜ ˜ through the poverty bundle z , i.e., I u of its representative preference u (z ) := {x′ ∈ ˜(x′ ) = u X |u ˜ (z )} . 47 Part (ii). The proof of Part (ii) is based on Steps 1, 2, 3 and 4. Observe that Steps 1 and 2 are sufficient to prove Part (ii) for any representative-preference line ½ whose ˜ indifference curve I u (z ) is compact. Step 1 : Any representative-preference line ½ satisfies Dominated Command and ∗ Weak Respect for Preference on SU ∆. First, we show that a representative-preference line ½ satisfies Dominated Com- ∗ mand . Consider any two situations (x, p, u), (x′ , p′ , u′) ∈ SU ′ ∆ such that p = p , x ≤ x′ and ½(x, p, u) = 0. We must show that ½(x′ , p′, u′ ) = 0. As ½ is representative-preference, we have px ≥ eu ˜(z )) because ½(x, p, u) = 0. This ˜ (p, u implies that px′ ≥ eu ˜(z )) because px′ ≥ px as x ≤ x′ . As ½ is representative- ˜ (p, u preference, we obtain ½(x′ , p′ , u′ ) = 0 as desired. Second, we show that a representative-preference line ½ satisfies Weak Respect ∗ for Preference . Consider any two situations (x, p, u), (x′ , p′ , u′) ∈ SU ∆ such that p = p′ , u = u′, u(x) ≤ u(x′ ) and ½(x, p, u) = 0. We must show that ½(x′ , p′ , u′ ) = 0. As ½ is representative-preference, we have px ≥ eu ˜(z )) because ½(x, p, u) = 0. ˜ (p, u We have px′ ≥ px because u(x) ≤ u(x′ ) and bundle x′ is optimal for preference u′ = u and price vector p′ = p. Therefore, we get px′ ≥ eu ˜(z )). As ½ is ˜ (p, u representative-preference, we obtain ½(x′ , p′ , u′ ) = 0 as desired. This concludes the proof of Step 1. Step 2 : Any representative-preference line ½ whose indifference curve I u ˜ (z ) is ∗ compact satisfies Subjective Anchorage on SU ∆. ∗ We must show that for all u ∈ U there exists zu ∈ X=0 such that for all 48 ∗ (x, p, u) ∈ SU ∆ x ∗ zu ⇒ ½(x, p, u) = 1, (2) ∗ x ≥ zu ⇒ ½(x, p, u) = 0. (3) As ½ is a representative-preference line, we have for all (x, p, u) ∈ SU ∗ ∆ that ½(x, p, u) = 1 ⇔ ˜ (p, u px < eu ˜(z )). ˜ By assumption, indifference curve I u (z ) is compact, i.e., closed and bounded. ˜ u Loosely speaking, indifference curve I (z ) “touches” all axes. This implies that any preference u ∈ U admits an indifference curve that shares a common bundle xu ˜ ˜ ˜ with I u (z ) but does not “cross” I u (z ). Formally, the compactness of I u (z ) implies ˜ that for all (strictly monotonic) u ∈ U there exists some bundle xu ∈ I u (z ) such that ˜(xu ) for all x′ ∈ X for which u(x′ ) = u(xu ). ˜(x′ ) ≤ u (i) u Property (i) is illustrated in Figure 8. An immediate corollary of Property (i) is ˜(xu ) for all x′′ ∈ X for which u(x′′ ) < u(xu ). ˜(x′′ ) < u that u x2 ˜ Compact indifference curve I u (z ) xu (“touches both axes”) z u u ˜ x1 ˜ Figure 8: Illustration of the bundle xu ∈ I u (z ) associated to preference u. ∗ := ∗ ˜ Take zu xu . Such definition is such that zu ∈ X=0 because xu ∈ I u (z ) and ˜ ∗ / Iu 0∈ (z ).52 We need to prove that Eq. (2) and Eq. (3) hold for all (x, p, u) ∈ SU ∆ ∗ when zu = xu . 52 ˜ / Iu We have 0 ∈ ˜ is strictly monotonic. (z ) because z ∈ X=0 and preference u 49 Consider first the case x ≥ xu . This case is such that px ≥ pxu . We have pxu ≥ eu ˜(xu )) because an individual with preference u ˜ (p, u ˜ and income pxu can consume xu and thus can (at least) reach utility u˜(xu ). This implies that pxu ≥ eu ˜(z )) ˜ (p, u because u˜(xu ) = u˜(z ). This shows that px ≥ eu ˜(z )) because x ≥ xu , which ˜ (p, u yields ½(x, p, u) = 0 because ½ is representative-preference. This proves Eq. (3). Consider then the case x xu . To derive a contradiction to Eq. (2), assume that ½(x, p, u) = 0. This implies that px ≥ eu ˜(z )) because ½ is representative- ˜ (p, u preference. The fact that px ≥ eu ˜(z )) implies in turn that there exists x ˜ (p, u ˆ∈X with pxˆ ≤ px for which u x) ≥ u ˜(ˆ ˜(z ). Last inequality implies u ˜(ˆ x) ≥ u˜(xu ) be- ∗ cause u ˜(z ). Now, as (x, p, u) ∈ SU ∆ , bundle x is optimal given price p and ˜(xu ) = u preference u. Therefore, bundle x x) ≤ u(x). As x xu , we have ˆ is such that u(ˆ u(x) < u(xu ) and thus u(ˆx) < u(xu ). Hence, bundle x ˆ is such that u(ˆ x) < u(xu ) and u˜(ˆ x) ≥ u˜(xu ), a contradiction to the corollary of Property (i). Hence, Eq. (2) must hold. We have thus shown that ½ satisfies Subjective Anchorage . This concludes the proof of Step 2. Step 3 : If the poverty line ½ admits a function x ˆ → X=0 such that ˆ:P ∆ with p ∈ P : ½(x, p, u) = 1 ⇔ px < px ∗ (A) for all (x, p, u) ∈ SU ˆ ˆ(p) , and ( B ) p′ x ˆ ( p) ≥ p′ x ˆ, ˆ(p′ ) for all p, p′ ∈ P then there exists a poverty line ½′ that is equivalent to ½ on S ˆ∗ and U∆ • ½′ is representative-preference, and (z ′ ) of ½′ is compact. ˜ ′ • the indifference curve I u To prove Step 3, we first construct poverty line ½′ from function x ˆ. For any ˆ price vector p ∈ P , let H (p) denote the hyperplane in space X defined by price vector p and bundle x ˆ(p), i.e., ˆ (p) := {x ∈ X |px = px Hx ˆ ( p) } . 50 Let C x ˆ ˆ (P ) denote the upper contour of the hyperplanes associated to all price vectors in Pˆ , i.e., Cx ˆ ˆ ˆ (p) for some p ∈ P and ∄x ˆ (P ) := {x ∈ X |x ∈ Hx ′ x s.t. x′ ∈ Hx ′ ′ ˆ ˆ (p ) for some p ∈ P }. By construction, upper contour C x ˆ ˆ (P ) has the following properties. First, it is such ˆ ˆ (P ) ⊂ X=0 because px that C x ˆ(p) > 0 for all p ∈ P ˆ .53 Second, it is weakly convex because it is defined as the upper contour of hyperplanes. Third, it is compact because all hyperplanes Hx ˆ ˆ (p) are such that p ∈ ∆+ because p ∈ P . The first two properties imply that upper contour C x ˆ ˆ (P ) corresponds to one indifference curve of some weakly convex preference in U , which we denote by u ˜′ . Taking some p∈P ˆ , consider the bundle z ′ := x ˆ(p). The indifference curve I u˜′ ′ (z ) of preference ′ ′ ˜ through z corresponds to upper contour C x u ˆ ˆ (P ). 54 The third property implies (z ′ ) is compact. We define poverty line ½′ to be the ˜ ′ that indifference curve I u ˜′ ∈ U and the representative-preference poverty line defined by the preference u poverty bundle z ′ ∈ X=0 . We show that poverty line ½′ has the desired properties. By construction, line ½′ is representative-preference and its indifference curve I u ˜ ′ (z ′ ) is compact. There remains to show that line ½′ is equivalent to line ½ on S ˆ∗ . Take any U∆ (x, p, u) ∈ S . We must show that ½ (x, p, u) = ½(x, p, u). By condition (A) for ˆ ∗ U∆ ′ ˆ, we have ½(x, p, u) = 1 ⇔ px < px function x ˆ(p). As line ½′ is representative- preference, we have ½′ (x, p, u) = 1 ⇔ px < eu ˜′(z ′ )). We must thus show ˜′ (p, u that eu ˜′(z ′ )) = px ˜′ (p, u ˆ(p). By definition, eu ˜′ (z ′ )) corresponds to the mini- ˜′ (p, u ˜′ ′ mal expenditure necessary to afford a bundle on indifference curve I u (z ) under price vector p. This minimal expenditure is equal to px ˆ(p) if bundle x ˆ(p) be- ˜′ ′ u longs to indifference curve I (z ) and the hyperplane Hx ˆ (p) is tangent to indif- ˜′ ′ u ference curve I (z ) at bundle x ˆ(p). By condition (B) for function x ˆ, we have that p x′ ′ ˆ ( p) ≥ p x ′ ˆ(p ) for all p ∈ P′ ˆ . This implies that bundle x ˆ(p) belongs to Cx ˆ ˆ (P ) because C x ˆ ˆ (P ) is the upper contour of all hyperplanes Hx ′ ′ ˆ ˆ (p ) with p ∈ P . Moreover, hyperplane Hx ˆ (p) is tangent to C x ˆ ˆ (P ) at bundle xˆ(p) because C x ˆ ˆ (P ) is defined as the upper contour of these hyperplanes. By construction of preference 53 We have pxˆ(p) > 0 because xˆ(p) ∈ X=0 and p ∈ ∆+ . 54 Condition (B) implies that bundle xˆ(p) must be on the upper contour C x ˆ ˆ ˆ (P ) when p ∈ P . 51 ˜′ and poverty bundle z ′ , upper contour C x u ˆ ˆ (P ) corresponds to indifference curve ˜′ ′ ˜′ ′ Iu (z ). This shows that bundle x ˆ(p) belongs to indifference curve I u (z ) and the ˜ u ′ ′ hyperplane Hx ˆ (p) is tangent to indifference curve I (z ) at bundle xˆ(p). We have thus shown that eu ˆ(p) and thus line ½′ is equivalent to line ½ on ˜′ (z ′ )) = px ˜′ (p, u ˆ∗ . This concludes the proof of Step 3. S U∆ Step 4 : Any representative-preference poverty line ½ admits a function x ˆ→ ˆ:P X=0 such that ∆ with p ∈ P : ½(x, p, u) = 1 ⇔ px < px ∗ (A) for all (x, p, u) ∈ SU ˆ ˆ(p), and ( B ) p′ x ˆ ( p) ≥ p′ x ˆ. ˆ(p′ ) for all p, p′ ∈ P To prove Step 4, we first construct function x ˆ:P ˆ → X=0 . We define function ˆ by constructing bundle x x ˆ(p) for any p ∈ P ˆ . Consider any (x, p, u) ∈ S ∗ such U∆ that p ∈ P . As line ½ is representative-poverty, we have ½(x, p, u) = 1 ⇔ px < ˆ eu ˜ denotes its representative preference and z denotes its poverty ˜(z )) where u ˜ (p, u bundle. By definition, eu ˜(z )) corresponds to the minimal expenditure neces- ˜ (p, u sary to afford a bundle xp such that u ˜(xp ) = u˜(z ) under price vector p. Bundle 55 xp belongs to X=0 because eu ˜(z )) > 0. The marginal rate of substitution ˜ (p, u ˜ at bundle xp is proportional to the price vector p because any of preference u smaller expenditure does not allow reaching utility level u ˜(z ) under price vector p. Formally, this implies that eu ˆ(p) := xp , which completes ˜(z )) = pxp . We let x ˜ (p, u ˆ. the construction of function x We show that function x ˆ has the required properties. We first prove condi- tion (A) for function xˆ. Consider any (x, p, u) ∈ S ∗ with p ∈ P ˆ . We must U∆ show that ½(x, p, u) = 1 ⇔ ˆ(p). As line ½ is representative-poverty, px < px we must thus show that eu ˜ (p, u ˆ(p). This follows from the construc- ˜(z )) = px ˆ because x tion of function x ˆ(p) = xp where eu ˜(z )) = pxp . We then prove ˜ (p, u ˆ. Consider any two p, p′ ∈ P condition (B) for function x ˆ . We must show that ˆ ( p) ≥ p′ x p′ x ˆ(p′ ). By construction of function x ˆ, last inequality is equivalent to ′ ′ ′ p xp ≥ p xp , where these two bundles are such that u ˜(xp ) = u ˜ (z ), ˜(xp′ ) = u ′ eu ˜(z )) = pxp and eu ˜ (p, u ˜(z )) = p′ xp′ . ˜ (p , u By definition of the expenditure 55 The minimal expenditure eu ˜ (p, u ˆ and ˜(z )) is a real number because p ∈ ∆+ as p ∈ P ˜ (p, u eu ˜(z )) is strictly larger than zero because z ∈ X=0 . 52 ′ function, we have eu ˜(xp )) ≤ p′ xp . This yields eu ˜ (p , u ′ ˜(z )) ≤ p′ xp because ˜ (p , u u ˜(xp ) = u ˜(z ). The contradiction assumption p′ xp < p′ xp′ then implies that ′ eu ˜(z )) < p′ xp′ , a contradiction to eu ˜ (p , u ′ ˜(z )) = p′ xp′ . This concludes the ˜ (p , u proof of Step 4. There remains to prove Part (ii) using Steps 1 to 4. By assumption, poverty line ½ is representative-preference. By Step 4, poverty line ½ admits a function x ˆ:P ˆ → X satisfying conditions (A) and (B). Therefore, by Step 3, there exists a ˆ∗ and ½′ is representative-preference poverty line ½′ that is equivalent to ½ on S U∆ (z ′ ) of ½′ is compact. By Step 2, the representative- ˜ ′ and the indifference curve I u preference ½′ satisfies Subjective Anchorage on S ˆ∗ . By Step 1, the representative- U∆ preference ½′ also satisfies Dominated Command and Weak Respect for Preference on Sˆ∗ , as desired. This concludes the proof of Part (ii). U∆ Part (i). We show that any poverty line satisfying the three axioms admits a function x ˆ:P ˆ → X that satisfies the conditions (A) and (B ) stated in Step 3 of part (ii). If that is indeed the case, then Step 3 of part (ii) applies and there exists a poverty line ½′ that is equivalent to ½ on S ˆ∗ and ½′ is representative-preference and its U∆ ˜ ′ indifference curve I u (z ′ ) is compact. By Step 1 of part (ii), the representative- preference poverty line ½′ satisfies Dominated Command and Weak Respect for ∗ Preference on SU ∆ . By Step 2 of part (ii), the representative-preference poverty line ½ also satisfies Subjective Anchorage on SU ′ ∗ ∆ . Hence, line ½ has all properties ′ required for part (i). There remains to show that any poverty line ½ that satisfies Subjective An- ∗ chorage , Weak Respect for Preference and Dominated Command on SU ∆ admits a function x ˆ ˆ : P → X such that ∆ with p ∈ P : ½(x, p, u) = 1 ⇔ px < px ∗ (A) for all (x, p, u) ∈ SU ˆ ˆ(p), and ( B ) p′ x ˆ ( p) ≥ p′ x ˆ. ˆ(p′ ) for all p, p′ ∈ P 53 First, we construct function x ˆ → X=0 . This construction requires defining ˆ:P particular hyperplanes and an upper-contour on these hyperplanes. ˆ , let up denote the We start by defining particular hyperplanes. For all p ∈ P linear preference whose constant marginal rate of substitution is proportional to p. Preference up is such that up ∈ U because p ∈ ∆+ . Subjective Anchorage implies p ∈ X=0 such that ½(x, p, u ) = 1 if x ∗ p ∗ that there exists a bundle zu zu p and ½(x, p, up ) = 0 if x ≥ zu ∗ p . Let H½ (p) denote the hyperplane in space X defined by price vector p ∈ P ˆ and bundle z ∗p , i.e., u ∗ H½ (p) := {x ∈ X |px = pzu p }. ˆ , Claim 1 shows that hyperplane H½ (p) characterizes line ½. For p ∈ P ∗ Claim 1 : for any equilibrium situation (x, p, u) ∈ SU ˆ ∆ with p ∈ P we have ½(x, p, u) = 1 ⇔ px < pzu ∗ p. p ⇒ ½(x, p, u) = 1. The fact that px < pzup ∗ ∗ We start by showing that px < pzu ∗ implies that there exists another bundle xˆ such that x ˆ zu ˆ = px. p and px p p ∗ x, p, u ) ∈ SU ∆ . By Subjec- By definition of the linear preference u , we have (ˆ tive Anchorage , we have ½(ˆ x, p, up ) = 1 because x ˆ ∗ zu p . As ½ satisfies Weak Respect for Preference and Dominated Command , Proposition 2 implies that ½(x, p, u) = ½(ˆ x, p, up ) because pxˆ = px. This yields ½(x, p, u) = 1, as desired. p ⇒ ½(x, p, u) = 0 follows the same argument and is thus ∗ The proof that px ≥ pzu omitted. This concludes the proof of Claim 1. Let C ½ (Pˆ ) denote the upper contour of the hyperplanes associated to all price vectors in Pˆ , i.e., ˆ and ∄x′ ˆ ) := {x ∈ X |x ∈ H½ (p) for some p ∈ P C ½ (P ˆ }. x s.t. x′ ∈ H½ (p′ ) for some p′ ∈ P ˆ ) has the following properties. First, it is By construction, upper contour C ½ (P ˆ ) ⊂ X=0 because pzup > 0 for all p ∈ P such that C ½ (P ˆ .56 Second, it is weakly 56 We have pzup > 0 because zup ∈ X=0 and p ∈ ∆+ . 54 convex because it is defined as the upper contour of hyperplanes. These two prop- ˆ ) corresponds to one indifference curve of erties imply that upper contour C ½ (P some weakly convex preference in U . Claim 2 is used for the construction of func- tion x ˆ:P ˆ → X= 0 . ˆ there exists a bundle xp ∈ X such that xp ∈ H½ (p) ∩ C ½ (P Claim 2 : for all p ∈ P ˆ ). We prove Claim 2 by contradiction. Assume that for some p ˆ∈ Pˆ , there exists no xp p) such that xp ˆ ∈ H½ (ˆ ˆ ˆ ∈ C ½ (P ). ˆ ∈ U . We (partially) construct The proof is based on a particular preference u ˆ by providing two of its (weakly convex) indifference curves. The preference u construction of uˆ’s indifference curves, which is illustrated in Figure 9 for the case m = 2, is as follows: ˆ u • The first indifference curve, denoted by I1 , corresponds to the hyperplane p) : H½ (ˆ ˆ u I1 := H½ (ˆ p) ˆ u • The second indifference curve, denoted by I2 , is a translation of the upper- ˆ ˆ u contour set C ½ (P ) such that I2 is strictly dominated by C ½ (P ˆ ) but I u ˆ 2 strictly ˆ 57 u dominates I1 , namely ˆ u I2 ˆ )} := {x ∈ X |(x1 + ǫ, x2 , . . . , xm ) ∈ C ½ (P where ǫ > 0 is smaller than the smallest distance along axis-1 between H½ (ˆ p) ˆ ), namely and C ½ (P 58 0<ǫ< min x ˜′1 . ˜1 − x ˆ ), ˜∈C ½ (P x ˜′ ∈H x ½ (ˆ p) 57 By definition of upper contour C ½ (P ˆ ), the contradiction assumption implies that hyperplane H½ (ˆ ˆ p) is dominated by C ½ (P ). 58 There exists ǫ > 0 satisfying this definition because hyperplane H½ (ˆ p) is not arbitrarily close ˆ ) since there is finite number of price vectors in P to C ½ (P ˆ. 55 ε x x′ p′ x2 H 1 (p ′ ) ˆ u I1 H 1 (p ) = ˆ) H1(ˆ p) p C 1 (P p ˆ ˆ u I2 x1 ˆ u ˆ u Figure 9: Construction of the two indifference curves I1 and I2 ˆ. of preference u Note: Hyperplanes H½ (p), H½ (p′ ) and H½ (ˆ ˆ u p) are in red. Indifference curves I1 and I2uˆ are in p) is part red and part blue because H½ (ˆ blue. The hyperplane H½ (ˆ p) corresponds to indifference u ˆ curve I1 ˆ ) of all hyperplanes is in green. . The upper contour C ½ (P Observe that there exists a preference uˆ ∈ U that admits two indifference ˆ u ˆ u ˆ u ˆ u curves that respectively correspond to I1 and I2 . Indeed, both I1 and I2 are 59 ˆ u ˆ u weakly convex. Furthermore, I1 and I2 do not intersect as guaranteed by the bounds on the parameter ǫ that defines the translation. Therefore, there exists a ˆ u ˆ u ˆ ∈ U that admits I1 preference u and I2 as two of its indifference curves. Finally, preference u ˆ are such that p ∈ ∆+ . ˆ is strictly monotonic because all p ∈ P We are now equipped to derive a contradiction. We show that line ½ must ˆ that there exists violate Subjective Anchorage . To do so, we show for preference u ∗ ∗ ˆ ∈ X such that for all (x, p, u no bundle zu ˆ ) ∈ SU ∆ we have x ∗ zu ˆ ⇒ ½(x, p, u ˆ) = 1, (4) ∗ x ≥ zu ˆ ⇒ ½(x, p, u ˆ) = 0. (5) ∗ ˆ ∈ X for which Eq. (4) holds must be such To do this, we show that any bundle zu 59 u ˆ I2 ˆ u is weakly convex because I2 ˆ ). is a translation of the weakly convex C ½ (P 56 that60 ∗ ˆ u u ˆ(zu ˆ) ≤ u ˆ ( I1 ), (6) ∗ ˆ ∈ X for which Eq. (5) holds must be such that and any bundle zu ∗ ˆ u u ˆ(zu ˆ) ≥ u ˆ ( I2 ). (7) If both conditions (6) and (7) hold, then we get a contradiction. Indeed, the con- ˆ u ˆ u ˆ u ˆ u struction of indifference curves I1 and I2 is such that u ˆ ( I1 ) u ˆ ( I1 ). This case is such that ˆ u ∗ ∗ there exists a bundle x ∈ I1 such that x ˆ = 0). The situation ˆ (because zu zu ∗ (x, p ˆ) belongs to SU ˆ, u ∆ because the x belongs to the hyperplane H½ (ˆ p) – recall ˆ u that I1 = H½ (ˆp) – which corresponds to an indifference curve of u p) , ˆ. As x ∈ H½ (ˆ ∗ ∗ ˆ p we have p ˆx = pˆzu ˆ , where zup p ˆ is the anchor bundle for the linear preference u . By Claim 1, we thus have that ½(x, p ˆ) = 0. This is in contradiction to Eq. (4), ˆ, u which implies ½(x, p ˆ) = 1 because x zu ˆ, u ∗ ˆ . Thus, Eq. (6) must hold. ∗ ˆ u We then prove Eq. (7). Assume to the contrary that u ˆ(zu ˆ) < u ˆ ( I2 ). This ˆ u ∗ ˆ . By construction, case is such that there exists a bundle x ∈ I2 such that x ≥ zu ˆ u ˆ ). Consider indifference I is an ǫ-translation along axis-1 of upper contour C ½ (P 2 the bundle xǫ defined from x as xǫ := (x1 + ǫ, x2 , . . . , xm ). We have xǫ ∈ C ½ (P ˆ) because x ∈ I2 ˆ u . The fact that xǫ belongs to upper contour C ½ (P ˆ ) implies that there exists some p′ ∈ P ˆ such that xǫ ∈ H½ (p′ ). We thus have xǫ ∈ H½ (p′ ) ∩ C ½ (Pˆ ). Hence, bundle xǫ is optimal for price vector p′ for any preference that admits the upper contour C ½ (P ˆ ) as one of its indifference curve. This implies that bundle x is optimal for price vector p′ for preference u ˆ because its indifference curve I2 ˆ u is an ǫ-translation along axis-1 of upper contour C ½ (P ˆ ). Formally, we have shown that (x, p′ , u ˆ) ∈ S ∗ for some p′ ∈ P ˆ . We also have p′ x < p′ xǫ by the construction U∆ 60 ˆ u We abuse slightly notation by writing u ˆ(I1 ˆ(x) for any bundle ), which we take to mean u ˆ u x ∈ I1 . 57 of xǫ from x. This implies that p′ x < p′ zu ∗ ∗ p′ , where zup′ is the anchor bundle for ′ the linear preference up , because p′ xǫ = p′ zu ∗ ′ p′ as xǫ ∈ H½ (p ). By Claim 1, we get ½(x, p′ , u ∗ ˆ) = 1 because p′ x < p′ zu p′ . This is in contradiction to Eq. (5), which implies ½(x, p′ , u ∗ ˆ . Thus, Eq. (7) must hold. This concludes ˆ) = 0 because x ≥ zu the proof of Claim 2. We are now equipped for the construction of function x ˆ:Pˆ → X=0 . We define ˆ by constructing bundle x function x ˆ(p) for any p ∈ P ˆ . By ˆ . Consider any p ∈ P ˆ ). Bundle xp belongs to X=0 Claim 2, there exists a bundle xp ∈ H½ (p) ∩ C ½ (P because H½ (p) ⊂ X=0 . We let x ˆ(p) := xp , which completes the construction of ˆ. function x There remains to show that function x ˆ has the required properties. We first prove condition (A) for function x ∗ ˆ. Consider any (x, p, u) ∈ SU ˆ ∆ with p ∈ P . We must show that ½(x, p, u) = 1 ⇔ px < px ˆ(p). By definition, xˆ(p) ∈ H½ (p). By ∗ definition of hyperplane H½ (p), we have px ˆ(p) = pzu p . By Claim 1, we directly get that ½(x, p, u) = 1 ⇔ px < px ˆ(p), as desired. We then prove condition (B) for function x ˆ. Consider any two p, p′ ∈ Pˆ. ˆ ( p) ≥ p′ x We must show that p′ x ˆ(p′ ). We derive a contradiction when assuming ˆ ( p) < p′ x p′ x ˆ(p′ ). By definition, x ˆ ( p) < p′ x ˆ(p′ ) ∈ H½ (p′ ). Inequality p′ x ˆ(p′ ) implies that bundle x ˆ(p) is dominated by hyperplane H½ (p′ ) and thus x ˆ(p) cannot belong to the upper contour of hyperplanes C ½ (P ˆ ). 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