## Abstract

When measuring multidimensional poverty it is reasonable to expect that the trade-offs between variable pairs can differ depending on whether the concerned pairs are complements or substitutes. Yet, currently existing approaches based on deprivation count distributions unrealistically assume that all pairs of variables are related in the same way—an unfortunate circumstance that undermines the possibilities of identifying the poor, aggregating their poverty levels and modeling non-trivial interactions between variables in highly flexible ways. This paper, which aims at modeling non-trivial relational structures across variables both in the identification and aggregation steps, is a first contribution towards addressing these inadequacies. The approach has been axiomatically characterized to flesh out the normative foundations upon which it is based and has a vast potential for application.

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## Notes

Yalonetzky (2014) makes room for defining identification functions that can potentially take into account the relationship between variables when defining the poverty status of individuals. Yet, in the context of three or more variables (which is the one we are interested in this paper), the author only explores the extreme cases of the ‘union’ and ‘intersection’ approaches—see definitions in Sect. 2.

The terms ‘variable’, ‘indicator’ or ‘attribute’ will be used interchangeably in this paper.

The alternative approach advocated by Ravallion (2011) of working in the space of attainments is not followed in this paper because (1) it might be possible for a poor person to be lifted out of poverty as a result of an increment in a nondeprived dimension, (2) it does not allow keeping track of the dimension-specific deprivations simultaneously.

There exist other definitions of deprivation gaps (see Table 1 in Permanyer (2014: 4) for other examples). Since alternative definitions do not alter the findings of the paper, we have chosen the one that is more commonly used in the literature.

The naming ‘consistent identification functions’ is reminiscent of the so-called ‘poverty consistency’ property introduced by Lasso de la Vega (2010) in the context of the counting approach (see Sect. 2.1). In that paper an identification function is ‘poverty consistent’ if, when identifying a person with a deprivation score equal to

*s*as poor, it also considers as poor anybody whose deprivation score count is at least as high as*s*. Clearly, the consistency condition proposed here is more general than the ‘poverty consistency’ property in that the former applies to*any*identification function, not just those relying on scores.This excludes trivial partitions in which each dimension is composed of one variable only and forces \(d\ge 4\).

The class of weighted generalized means is well-known and has been widely used in welfare analysis. Higher values of \(\theta \) give more importance to the upper tails of the distribution and vice versa. In the limit, as \(\theta \rightarrow \infty \) (resp. \(\theta \rightarrow -\infty \)) the generalized mean converges towards the maximum (resp. minimum) of the distribution.

Yet, these two measures and the one shown in (13) differ in the identification function they use to select the members of \(Q_{\rho }\). While Datt (2017) uses the union approach to identify the poor, Pattanaik and Xu (2018) propose a novel identification function based on individuals’ weighted sum of deprivation gaps \(\sum _{j}a_{j}\gamma _{ij}\) (rather than the traditional weighted proportion of deprivations \(\sum _{j}a_{j}\omega (\gamma _{ij})\) used in the counting approach).

Yet, the lack of axiomatic characterization should not necessarily preclude the use of \(\Pi _{\theta }\) or \(\Pi _{\varvec{\theta }}^{G}\) in empirical applications where the variables one is dealing with are ordinal. Indeed, the \(M_{0}\) index suggested by Alkire and Foster (2011) for the ordinal setting is not axiomatically characterized but is massively employed in practice.

Consider a scenario with two individuals

*a*,*b*and two variables*u*,*v*, where*a*has more of*u*but less of*v*.than*b*. A ‘correlation increasing switch’ occurs when we interchange the amounts of attribute*v*between the two persons. After such change, the marginal distributions are unchanged but*a*has more of*u*and*v*than*b*.In this paper we use the standard ALEP definition of complementarity/substitutability (see Kannai Kannai 1980). That is: when the cross partial derivative of the individual poverty function is positive (resp. negative), the attributes are considered complements (resp. substitutes).

Yet, it is currently difficult to ascertain whether identification and aggregation functions treat the complementarity/substitutability among variables in a truly coherent way. While it is clear that the ALEP criterion can be used for aggregation functions, we are not aware of any analogous method that can be implemented in the discrete setting of identification functions—an important topic that should be explored in future research.

If one had that \(\rho ({\varvec{0}})=1\), then MON would imply that \(\rho ( {\varvec{x}})=1\) for all \({\varvec{x}}\in X^{d}\), and if \(\rho ({\varvec{1}})=0\), then MON would imply \(\rho ({\varvec{x}})=0\) for all \({\varvec{x}}\in X^{d}\)—in both cases contradicting NTR.

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## Acknowledgements

Different versions of this paper have been presented at several conferences and seminars (SCW 2014, ECINEQ 2015, SCW 2016): I am grateful to its participants for their valuable comments and suggestions. In particular, I am indebted to Eugenio Peluso, Casilda Lasso de la Vega, Buhong Zheng, Salvador Barberà, Jordi Massó, Sabina Alkire, Suman Seth, Gastón Yalonetzky, Xavi Ramos, Coral del Rio, Olga Alonso-Villar, Carlos Gradín, Rafael Salas, Juan Gabriel Rodriguez and, more generally, to the members of EQUALITAS. The research leading to these results has received funding from the European Research Council (ERC-2014-StG-637768, EQUALIZE project); from the Spanish Ministry of Science, Innovation and Universities ‘Ramón y Cajal’ Research Grant Program (RYC-2013-14196); and its National R&D&I Plan GLOBFAM (RTI2018-096730-B-I00).

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## Appendix

### Appendix

### Proof of Proposition 1

We start with the ‘if’ part of the proof. Assume that \(\rho \in {\mathcal {I}}_{d}\). We have to prove that \(({\mathcal {L}} \left( P_{\rho }\right) )^{\uparrow }=P_{\rho }\).

(1) We start proving \(({\mathcal {L}}\left( P_{\rho }\right) )^{\uparrow }\subset P_{\rho }\). Take \({\varvec{x}}\in ({\mathcal {L}}\left( P_{\rho }\right) )^{\uparrow }\). Then, there exists some \({\varvec{z}}\in {\mathcal {L}}\left( P_{\rho }\right) \) such that \({\varvec{z}}\le {\varvec{x}}\) (if \({\varvec{x}}\in {\mathcal {L}}\left( P_{\rho }\right) \), then \({\varvec{z}}={\varvec{x}}\)). Since \( {\mathcal {L}}\left( P_{\rho }\right) \subset P_{\rho }\), \({\varvec{z}}\in P_{\rho }\). In addition, since \({\varvec{x}}\in {\varvec{z}}^{\uparrow }\) and \( \rho \in {\mathcal {I}}_{d}\), one can conclude that \({\varvec{x}}\in P_{\rho }\).

(2) We now prove \(\left( {\mathcal {L}}\left( P_{\rho }\right) \right) ^{\uparrow }\supset P_{\rho }\). Take \({\varvec{x}}\in P_{\rho }\). If it turns out that \({\varvec{x}}\in {\mathcal {L}}\left( P_{\rho }\right) \) then we are done. If \({\varvec{x}}\notin {\mathcal {L}}\left( P_{\rho }\right) \) then there must exist some \({\varvec{y}}\in P_{\rho }\backslash \{{\varvec{x}}\}\) such that \({\varvec{y}}\le {\varvec{x}}\). Now, if \({\varvec{y}}\in {\mathcal {L}}\left( P_{\rho }\right) \subset P_{\rho }\) then \({\varvec{x}}\in {\varvec{y}}^{\uparrow }\). Since \(\rho \in {\mathcal {I}}_{d}\), one can conclude that \({\varvec{x}}\in \left( {\mathcal {L}}\left( P_{\rho }\right) \right) ^{\uparrow }\). Otherwise, if \({\varvec{y}}\notin {\mathcal {L}}\left( P_{\rho }\right) \) then we can proceed iteratively until reaching an element belonging to \({\mathcal {L}} \left( P_{\rho }\right) \). That is: since \(X^{d}\) is finite (\(\left| X^{d}\right| =2^{d}\)) there must exist a finite sequence of vector dominations \({\varvec{z}}_{i}\le {\varvec{z}}_{i+1}\) from some element \(\varvec{ z}_{1}\in {\mathcal {L}}\left( P_{\rho }\right) \) up to \({\varvec{x}}\) (i.e.: \( {\varvec{z}}_{1}\le {\varvec{z}}_{2}\cdots \le {\varvec{z}}_{n}\le {\varvec{x}}\) ), so that \({\varvec{x}}\in {\varvec{z}}_{1}{}^{\uparrow }\). Since \(\rho \in {\mathcal {I}}_{d}\), one can conclude that \({\varvec{x}}\in \left( {\mathcal {L}} \left( P_{\rho }\right) \right) ^{\uparrow }\).

This proves the ‘if’ part of the proposition. The ‘only if’ part of the proof goes as follows. Assume \(P_{\rho }\) is a subset of \(X^{d}\) such that \(\left( {\mathcal {L}}\left( P_{\rho }\right) \right) ^{\uparrow }=P_{\rho }\). We have to prove that \(\rho \in {\mathcal {I}}_{d}\). Take any \({\varvec{x}}\in P_{\rho }\). Since \(\left( {\mathcal {L}}\left( P_{\rho }\right) \right) ^{\uparrow }=P_{\rho }\) we can say that \({\varvec{x}}\in {\varvec{z}}^{\uparrow }\) for some \({\varvec{z}}\in {\mathcal {L}}\left( P_{\rho }\right) \). Consider now any \({\varvec{y}}\in {\varvec{x}}^{\uparrow }\). By the transitivity of \(\le \) one has that \({\varvec{y}}\in {\varvec{z}}^{\uparrow }\). Since \(\left( {\mathcal {L}}\left( P_{\rho }\right) \right) ^{\uparrow }=P_{\rho }\), we can conclude that \({\varvec{y}}\in P_{\rho }\). \(\square \)

### Proof of Theorem 1

It is easy to verify that when \(S_{d}=\mathcal {C }_{d}\), then any \(\rho \in S\) satisfies MON, IND, VAN and NTR. Therefore, we will prove that when a group of identification functions \(S_{d}\subseteq \Omega _{d}\) satisfies these four axioms then it must be equal to \(\mathcal {C }_{d}\).

We will first prove the following auxiliary lemma:

### Auxiliary Lemma 1

Whenever IND and VAN hold for some \(\rho \in S_{d}\subseteq \Omega _{d}\), then, one has that \(\left| {\varvec{x}} \right| =\left| {\varvec{y}}\right| \Rightarrow \rho ({\varvec{x}} )=\rho ({\varvec{y}})\) for all \({\varvec{x}},{\varvec{y}}\in X^{d}\) and all \(\rho \in S_{d}\).

Let \(\rho \in S_{d}\) and let \(m>1\) be an integer for which \(\left| {\varvec{x}}\right| =\left| {\varvec{y}}\right| \) entails \(\rho ( {\varvec{x}})=\rho ({\varvec{y}})\) for all \({\varvec{x}},{\varvec{y}}\in X^{d}\) such that \(\left| {\varvec{x}}\right| =\left| {\varvec{y}} \right| <m\). We will now prove the result also holds true whenever \( \left| {\varvec{x}}\right| =\left| {\varvec{y}}\right| =m\). Let \( {\varvec{x}},{\varvec{y}}\in X^{d}\) be two vectors with \(\left| {\varvec{x}} \right| =\left| {\varvec{y}}\right| =m\). There are now two mutually exclusive cases.

**Case 1** Assume there exists a variable \(j\in \{1,\ldots ,d\}\) such that \(x_{j}=y_{j}=1\). Consider the vectors \({\varvec{x}}-{\varvec{e}}_{j}, {\varvec{y}}-{\varvec{e}}_{j}\). Since \(\left| {\varvec{x}}-{\varvec{e}} _{j}\right| =\left| {\varvec{y}}-{\varvec{e}}_{j}\right| =m-1<m\), the induction hypothesis holds, so \(\rho ({\varvec{x}}-{\varvec{e}}_{j})=\rho ( {\varvec{y}}-{\varvec{e}}_{j})\). Now, by IND one has that \(\rho ({\varvec{x}} )=\rho ({\varvec{y}})\), as desired.

**Case 2** Assume there does not exist any variable \(j\in \{1,\ldots ,d\} \) such that \(x_{j}=y_{j}=1\). Consider two different variables \(j,l\in \{1,\ldots ,d\}\) such that \(x_{j}=0,y_{j}=1\) and \(x_{l}=1,y_{l}=0\). Since \( \left| {\varvec{x}}-{\varvec{e}}_{l}\right| =\left| {\varvec{y}}- {\varvec{e}}_{j}\right| =m-1<m\), the induction hypothesis holds, so \(\rho ( {\varvec{x}}-{\varvec{e}}_{l})=\rho ({\varvec{y}}-{\varvec{e}}_{j})\). Now, by IND one has that

Observe that

where \(\delta ({\varvec{y}}):=\{i\in \{1,\ldots ,d\}|y_{i}=1\}\ \)is the subset of variables in profile \({\varvec{y}}\) where \(y_{i}=1\). By VAN, one has that \( \rho ({\varvec{e}}_{j})=\rho ({\varvec{e}}_{l})\). Applying now IND several times (once per variable \(i\in \delta ({\varvec{y}})\backslash \{j\}\)), one can conclude that

By (A2) and (A3), (A4) can be rewritten as

Comparing (A1) with (A5), we can conclude that \(\rho ({\varvec{x}} )=\rho ({\varvec{y}})\), as desired. This proves the auxiliary lemma 1. \(\square \)

Take now any two vectors \({\varvec{x}},{\varvec{y}}\in X^{d}\) such that \( \left| {\varvec{x}}\right| \ge \left| {\varvec{y}}\right| \). Define now \({\varvec{w}}\in X^{d}\) in such a way that \(\left| {\varvec{w}} \right| =\left| {\varvec{y}}\right| \) and \(\delta ({\varvec{w}} )\subseteq \delta ({\varvec{x}})\). By auxiliary lemma 1, one has \(\rho ( {\varvec{w}})=\rho ({\varvec{y}})\). On the other hand, by MON \(\rho ({\varvec{x}} )\ge \rho ({\varvec{w}})\), so one can conclude that \(\rho ({\varvec{x}})\ge \rho ({\varvec{y}})\). This ensures that \(\rho \) is a counting measure for all \(\rho \in S_{d}\).

Observe that NTR and MON imply that \(\rho ({\varvec{0}})=0\) and \(\rho ( {\varvec{1}})=1\) (if \(\rho ({\varvec{0}})=1\), then \(\rho ({\varvec{x}})=1\) for all \({\varvec{x}}\in X^{d}\) and if \(\rho ^{b}({\varvec{1}})=0\), then \(\rho ^{b}( {\varvec{x}})=0\) for all \({\varvec{x}}\in X^{d}\)—in both cases contradicting NTR). Lastly, by MON and NTR there must exist a \(k\in \{1,\ldots ,d\}\) such that \(\rho ({\varvec{x}})=0\) whenever \(\left| {\varvec{x}}\right| <k\) and \(\rho ({\varvec{x}})=1\) whenever \(\left| {\varvec{x}}\right| \ge k\) . Therefore, \(\rho \in {\mathcal {C}}_{d}\), as desired. This proves Theorem 1. \(\square \)

**Statement**: If COM holds for a given \(S_{d}\subseteq \Omega _{d}\), then IND holds as well for all \(\rho \in S_{d}\); however, the opposite is not necessarily true.

### Proof of the statement

To verify this claim let’s start assuming that COM applies for a certain set of identification functions \( S_{d}\subseteq \Omega _{d}\). Consider now the deprivation vectors \({\varvec{x}} ,{\varvec{y}},{\varvec{x}}^{\prime },{\varvec{y}}^{\prime }\in X^{d}\) as in the statement of IND. Then, it is trivial to verify that \(({\varvec{x}},{\varvec{y}} ^{\prime })\) and \(({\varvec{y}},{\varvec{x}}^{\prime })\) are equivalent societies (with \(m=2\)). Imposing COM, one has that \(\rho ({\varvec{x}})\ge \rho ({\varvec{y}})\) implies \(\rho ({\varvec{x}}^{\prime })\ge \rho ({\varvec{y}} ^{\prime })\) for all \(\rho \in S_{d}\): this is precisely what IND states. On the other hand, one can find infinitely many examples of sets of identification functions satisfying IND but failing to satisfy COM. A very simple example for the case \(d=3\) can be \(S_{d}=\{\rho _{0}\}\), where

It is trivial to check that \(S_{d}=\{\rho _{0}\}\) satisfies IND. However, it does not satisfy COM. To verify this, consider the following pair of three-person societies \(({\varvec{x}}_{1},{\varvec{x}}_{2},{\varvec{x}}_{3})\) and \( ({\varvec{y}}_{1},{\varvec{y}}_{2},{\varvec{y}}_{3})\) with \({\varvec{x}}_{1}=(111), {\varvec{x}}_{2}=(101),{\varvec{x}}_{3}=(001),{\varvec{y}}_{1}=(101),{\varvec{y}} _{2}=(011),{\varvec{y}}_{3}=(101)\). Clearly, both societies are equivalent (the number of individuals experiencing deprivation in each variable coincides). However, we have that \(\rho _{0}({\varvec{x}}_{1})\le \rho _{0}( {\varvec{y}}_{1})\) and \(\rho _{0}({\varvec{x}}_{2})\le \rho _{0}({\varvec{y}} _{2})\) for all \(\rho \in S_{d}\) and yet \(\rho _{0}({\varvec{x}}_{3})<\rho _{0}({\varvec{y}}_{3})\), thus contradicting COM. \(\square \)

### Proof of Theorem 2

It is easy to verify that when \(S_{d}=\mathcal {W }_{d}\), then any \(\rho \in S_{d}\) satisfies MON, COM and NTR. Therefore, we will prove that when a group of identification functions \(S_{d}\subseteq \Omega _{d}\) satisfies these three axioms then it must be equal to \(\mathcal { W}_{d}(k)\) for some \(k\in (0,1]\). Since (i) \(X^{d}\) is finite, (ii) each \( \rho \in S_{d}\) induces a complete ordering in \(X^{d}\times X^{d}\), and (iii) COM holds for all \(\rho \in S_{d}\), the hypotheses of Theorem 4.1.B in Fishburn (1970) are satisfied. Therefore, for all \({\varvec{x}},{\varvec{y}}\in X^{d}\) and for all \(\rho \in S_{d}\) one has that

for some real-valued functions \(u_{1},\ldots ,u_{d}\) on \(\{0,1\}\). One can rewrite the last expression as follows

In turn, this expression can be rewritten as

where \({\widetilde{u}}_{j}(x_{j}):=u_{j}(x_{j})-u_{j}(0)\). Clearly, \( {\widetilde{u}}_{j}(0)=0\). If we define \(w_{j}:={\widetilde{u}}_{j}(1)\), (A9) can be written as

By MON, one must have that \(w_{j}\ge 0\forall j\). This ensures that \(\rho \) is a counting measure for all \(\rho \in S\). By MON and NTR, \(\rho (\varvec{ 0})=0\) and \(\rho ({\varvec{1}})=1\).^{Footnote 16} Lastly, by MON and NTR there must exist a real number \(q\in (0,\sum _{i}w_{i}]\) such that \(\rho ({\varvec{x}})=0\) whenever \(\sum _{j=1}^{d}w_{j}x_{j}<q\) and \(\rho ({\varvec{x}})=1\) whenever \( \sum _{j=1}^{d}w_{j}x_{j}\ge q\). Defining \(a_{i}:=w_{i}/\sum _{i}w_{i}\) and \( k:=q/\sum _{i}w_{i}\) we have found a vector of weights \({\varvec{a}}\in \Delta _{d}\) and a deprivation threshold \(k\in (0,1]\) such that \(\rho \in \mathcal { W}_{d}\), as desired. This proves Theorem 2. \(\square \)

### Proof of Proposition 2

It is straightforward to prove that if \( S_{d}\in \{{\mathcal {I}}_{d},{\mathcal {W}}_{d},{\mathcal {C}}_{d}\}\) then \( S_{d}^{\psi }\) satisfies NTR and MON. We will only show that \(S_{d}^{\psi }\) does not satisfy COM. For that purpose we need to prove an auxiliary lemma (see below). According to (11) \({\mathcal {C}}_{d}^{\psi }\) is the set of identification functions for *d* variables where the unweighted counting approach is used within- and between-dimensions (given \(\psi \in \Psi _{G}\) ). To allow proper labelling, this set will be rewritten as \({\mathcal {C}} _{d}^{\psi }(k_{1},\ldots ,k_{G};k^{b}),\) where \(k_{1},\ldots ,k_{G}\) denote the domain specific deprivation thresholds and \(k^{b}\) the between domain deprivation threshold. Within this set, define

The set \(\widehat{{\mathcal {C}}}_{d}^{\psi }\) contains identification functions where poor individuals do not have to experience deprivation in all dimensions simultaneously and in some of them they must be deprived in at least two variables. The set \(\widetilde{{\mathcal {C}}}_{d}^{\psi }\) contains identification functions where poor individuals have to experience deprivation in all dimensions simultaneously but where deprivation needs not to be universal within at least two of these dimensions. The sets \(\widehat{ {\mathcal {C}}}_{d}^{\psi }\) and \(\widetilde{{\mathcal {C}}}_{d}^{\psi }\ \)are generalizations of Examples 1 and 2 to the multiple dimension context.

### Auxiliary Lemma 2

\(\widehat{{\mathcal {C}}}_{d}^{\psi }\cap \mathcal {W }_{d}=\emptyset \) and \(\widetilde{{\mathcal {C}}}_{d}^{\psi }\cap {\mathcal {W}} _{d}=\emptyset .\)

### Proof of Auxiliary Lemma 2

In both cases we follow the same strategy: if \(\rho \in \widehat{{\mathcal {C}}}_{d}^{\psi }\) or \(\rho \in \widetilde{{\mathcal {C}}}_{d}^{\psi }\) we start assuming that there *is* a weighting scheme \({\varvec{a}}\in \Delta _{d}\) and a deprivation threshold *k* such that \(\rho \in {\mathcal {W}}_{d}\) to arrive at a contradiction. Given the partition of *D* in *G* dimensions \((D_{1},\ldots ,D_{G})\in \Psi _{G}\), we will denote the elements of the weighting vector \({\varvec{a}}\) as \(a_{gv}\), where \(g\in \{1,\ldots ,G\}\) indexes the member of the partition \( D_{g}\) to which the weight belongs and \(v\in \{1,\ldots ,d_{g}\}\) indexes the members within domain \(D_{g}\). We can assume without loss of generality that *within* each domain \(D_{g}\) the weights are sorted in a non-ascending order, i.e.: \(a_{gv}\ge a_{gv+1}\) for all \(g\in \{1,\ldots ,G\}\) and all \(v\in \{1,\ldots ,d_{g}-1\}\).

We start with \(\widehat{{\mathcal {C}}}_{d}^{\psi }\). Without loss of generality, we can assume that the two dimensions \(g_{1},g_{2}\in \{1,\ldots ,G\}\) with \(k_{g_{1}}\ge 2/d_{g_{1}}\) and \(k_{g_{2}}\ge 2/d_{g_{2}}\ \)are \(g_{1}=1\) and \(g_{2}=2\). Since \(\rho \in \widehat{{\mathcal {C}}}_{d}^{\psi }\), there exist \(\rho ^{b}\in {\mathcal {C}}_{G}(k^{b})\) and \(\rho _{g}^{w}\in {\mathcal {C}}_{d_{g}}(k_{g})\) such that \(\rho ({\varvec{x}})=\rho ^{b}(\rho _{1}^{w}({\varvec{x}}_{1}),\ldots ,\rho _{G}^{w}({\varvec{x}}_{G}))\). Let \( {\mathcal {L}}^{b}\subset X^{G}\) be the set of *least* deprived profiles in \(\left( \rho ^{b}\right) ^{-1}(1)\) (i.e. if \({\varvec{x}}\in {\mathcal {L}} ^{b} \) and \({\varvec{y}}\in X^{G}\) is such that \({\varvec{y}}<{\varvec{x}}\), then \({\varvec{y}}\in \left( \rho ^{b}\right) ^{-1}(0)\)). Consider \({\varvec{u}}, {\varvec{u}}^{\prime }\in {\mathcal {L}}^{b}\). Without loss of generality we will write them as \({\varvec{u}}=(1\ 0\)\(u_{3}\ldots u_{G})\), \({\varvec{u}}^{\prime }=(0\ 1\)\(u_{3}\ldots u_{G}).\) By definition, the following inequalities must hold:

where \(m_{g}:=\lfloor k_{g}d_{g}\rfloor \) and \(\delta (u_{3}\ldots u_{G}):=\{g|u_{g}=1\}\ \)is the subset of elements in vector \((u_{3}\ldots u_{G})\) where \(u_{g}=1\). Consider now a third vector \({\varvec{u}}^{\prime \prime }=(0\ 0\)\(u_{3}\ldots u_{G}).\) Since \({\varvec{u}},{\varvec{u}}^{\prime }\in {\mathcal {L}}^{b}\), one has that \({\varvec{u}}^{\prime \prime }\in \left( \rho ^{b}\right) ^{-1}(0)\). This implies that the following inequalities must hold.

If one defines

the inequalities (A13)–(A18) can be rewritten as

It is trivial to show that the inequalities system shown in (A20) does not have feasible solutions. In the first inequality of the system, either \( a_{11}\) or \(a_{12}\) must be greater or equal than \(k^{\prime }/2\). The same goes for \(a_{21},a_{22}\) in the second inequality of the system: at least one of them must be greater or equal than \(k^{\prime }/2\). Picking the largest elements between \(a_{11},a_{12}\) and \(a_{21},a_{22}\) and adding them up results in a number that is greater or equal than \(k^{\prime }\), therefore contradicting at least one of the four last inequalities of the system. We have reached the contradiction we were looking for.

Let us now consider case \(\widetilde{{\mathcal {C}}}_{d}^{\psi }\). Without loss of generality, we can assume that the two dimensions \(g_{1},g_{2}\in \{1,\ldots ,G\}\) with \(k_{g_{1}}<1\) and \(k_{g_{2}}<1\ \)are \(g_{1}=1\) and \( g_{2}=2\). Since \(\rho \in \widetilde{{\mathcal {C}}}_{d}^{\psi }\), there exist \( \rho ^{b}\in {\mathcal {C}}_{G}(1)\) and \(\rho _{g}^{w}\in {\mathcal {C}} _{d_{g}}(k_{g})\) such that \(\rho ({\varvec{x}})=\rho ^{b}(\rho _{1}^{w}( {\varvec{x}}_{1}),\ldots ,\rho _{G}^{w}({\varvec{x}}_{G}))\). By definition, the following inequalities must hold:

where \(m_{g}:=\lfloor k_{g}d_{g}\rfloor \). Consider now the following *G*-dimensional binary vectors: \({\varvec{v}}=(1\ 0\)\(1\ldots 1)\), \({\varvec{v}} ^{\prime }=(0\ 1\)\(1\ldots 1).\) Since \({\varvec{v}},{\varvec{v}}^{\prime }\in \left( \rho ^{b}\right) ^{-1}(0)\), the following inequalities must hold:

Defining

the inequalities (A21)–(A26) can be rewritten as

Again, it is trivial to prove that the inequalities system shown in (A28) does not have feasible solutions. In the second to last inequality of the system, either \(a_{11}\) or \(a_{12}\) must be smaller than \(k^{\prime }/2\) . The same goes for \(a_{21},a_{22}\) in the last inequality of the system: at least one of them must be smaller than \(k^{\prime }/2\). Picking the smallest elements between \(a_{11},a_{12}\) and \(a_{21},a_{22}\) and adding them up results in a number that is smaller than \(k^{\prime }\), therefore contradicting at least one of the four first inequalities of the system. We have reached the contradiction we were looking for. This proves auxiliary Lemma 2. \(\square \)

The proof of Proposition 1 is now almost immediate. According to Auxiliary Lemma 2, \({\mathcal {C}}_{d}^{\psi }\backslash {\mathcal {W}}_{d}\ne \emptyset \) (essentially, it is only when the union or intersection approaches are used both within and across dimensions –i.e. either \(k_{g}=1/d_{g}\forall g,k^{b}=1/G\) or \(k_{g}=1\forall g,k^{b}=1\)—that the counting approach \( {\mathcal {W}}_{d}\) is able to generate some identification functions included in \({\mathcal {C}}_{d}^{\psi }\)). Since COM uniquely characterizes \({\mathcal {W}} _{d}\) (see Theorem 2), \({\mathcal {C}}_{d}^{\psi }\) cannot satisfy that axiom as well. Finally, since \({\mathcal {I}}_{d}^{\psi }\supset {\mathcal {W}} _{d}^{\psi }\supset {\mathcal {C}}_{d}^{\psi }\) and \({\mathcal {C}}_{d}^{\psi }\) does not satisfy COM, neither \({\mathcal {W}}_{d}^{\psi }\) nor \({\mathcal {I}} _{d}^{\psi }\) can satisfy COM. This proves Proposition 2. \(\square \)

### Proof of Theorem 3

It is clear that the multivariate poverty index shown in Eq. (12) satisfies SDC, MOA, NRM, FOC, SEP, HMG and CON. We are going to prove the opposite implication. Since \({\varvec{f}} =\{f_{d}\}_{d\in {\mathbb {N}} }\) satisfies SDC, it can be written as

for some function \(p:[0,1]^{d}\rightarrow {\mathbb {R}} \). Clearly, for any \(\varvec{\gamma }\in [0,1]^{d},p(\varvec{\gamma } )=f_{d}([\varvec{\gamma }])\) for some deprivation matrix \([\varvec{\gamma } ]\in {\mathcal {G}}_{n\times d}^{S}\) where all rows are equal to \(\varvec{ \gamma }\). Therefore, since \(f_{d}\) satisfies SDC, CON, HMG, SEP and MOA, *p* will satisfy them too. It can be shown that MOA implies minimal increasingness and strict essentiality (see Blackorby and Donaldson 1982: 251). Moreover, the domain of *p* is \([0,1]^{d}\), which is connected and topologically separable. In an analogous way to Blackorby and Donaldson (1982: 252), based on Gorman (1968: 369) and Blackorby, Primont and Russel (1978: 127) it can be shown that *p* is additively separable and can be written as

where \(p^{*}\) and \(p_{j},j\in \{1,\ldots ,d\}\) are continuous real-valued functions and \(p^{*}\) is increasing. By HMG, one has that

for any \(\lambda \in (0,1]\). Using Eq. (A30), for each \(l\in \{1,\ldots ,d\}\) we can define the functions

Since *p* is linearly homogeneous on \(\varvec{\gamma }\), so are the functions \(h_{l}(\gamma _{l}).\) Therefore

for any \(\lambda \in (0,1]\) and for all \(l\in \{1,\ldots ,d\}\). As a consequence, there exist constants \(c_{l}\) such that

Plugging Eqs. (A32) and (A34) we have that

Hence

Substituting Eq. (A36) in Eq. (A30), one has that

for some constants \(c_{j},\varsigma \) and a continuous increasing function \( p^{*}\). Equation (A37) is essentially the same as Eq. (34) in Blackorby and Donaldson (1982: 260). Therefore, following those authors–who in turn draw from Eichhorn (1978: 32–34)—it can be proven that \(p^{*-1}=:f\) must satisfy the following functional equation

Without the domain restrictions on \(\lambda \) and *u*, the solutions to Eq. (A38) are well-known (Aczél et al. 1986). It is straightforward to show that the solution for Eq. (A38) on the present restricted domain is

for some parameters \(a,b,c,d,\theta \) (with \(\theta \ne 0\)). Since continuity of *f* at 0 precludes the logarithmic solution, the general solution of Eq. (A37) can be written as

Since *p* is linearly homogeneous on \(\varvec{\gamma }\), one must have that \(\left( b(d-1)+\varsigma \right) /a=0\). Therefore, Eq. (A30) can be rewritten as

for some continuous increasing function \(\psi \). Since *p* satisfies HMG one has

Equation (A42) implies that

for all \(x\ge 0,\lambda \in (0,1]\). As a consequence, there exists a constant *q* such that

Imposing NRM and MOA and rewriting accordingly, one obtains the desired functional form. \(\square \)

### Proof of Theorem 4

Given its similarity with Theorem 3, we will simply present a brief sketch of the proof (the complete proof is available upon request). The sufficiency part of the theorem is clear, so we focus on the reverse implication. Axioms SDC, FOC, MOA and NRM imply that \( f_{d}(\Gamma )\) can be written as

for some increasing function \(\Psi :[0,1]^{d}\rightarrow [0,1]\) with \( \Psi ({\varvec{1}})=1\) and \(\Psi ({\varvec{0}})=0\). Imposing CON, HMG, MOA and WDS, it turns out that \(\Psi (\gamma _{i1},\ldots ,\gamma _{id})=\Phi (P_{1}( \varvec{\gamma }_{1}),\ldots ,P_{G}(\varvec{\gamma }_{G}))\), where

for some \(\theta _{g}>0,w_{gv}>0\). Lastly, CON, HMG, MOA and BDS imply that

for some \(\theta>0,a_{g\cdot }>0\), so we obtain the desired functional form. \(\square \)

### Proof of Proposition 3

Let

be the individual level poverty function corresponding to (16). Therefore, one has that

After several algebraic manipulations it is easy to show that

The last expression can be rearranged and written as follows:

Therefore, one can basically say that

for some real constants \(A,B>0\). Hence, whenever \(\theta _{g}<\min \{1,\theta \}\), \(\left( \partial ^{2}\pi _{\varvec{\theta }}\right) /\left( \partial \gamma _{gv}\partial \gamma _{gu}\right) >0\), so the attributes *u*, *v* belonging to the same dimension are complements. On the other hand, whenever \(\theta _{g}>\max \{1,\theta \}\), \(\left( \partial ^{2}\pi _{ \varvec{\theta }}\right) /\left( \partial \gamma _{gv}\partial \gamma _{gu}\right) <0\), so the attributes *u*, *v* belonging to the same dimension are substitutes. This proves part (i). For part (ii), we need to compute \( \left( \partial ^{2}\pi _{\varvec{\theta }}\right) /\left( \partial \gamma _{gv}\partial \gamma _{hu}\right) \). After algebraic manipulations it can be shown that

From the previous equation we can say that

for some real constant \(C>0\). Therefore, whenever \(\theta <1\), \(\left( \partial ^{2}\pi _{\varvec{\theta }}\right) /\left( \partial \gamma _{gv}\partial \gamma _{hu}\right) >0\), so the attributes *u*, *v* belonging to different dimensions are complements. Analogously, when \(\theta >1\), \(\left( \partial ^{2}\pi _{\varvec{\theta }}\right) /\left( \partial \gamma _{gv}\partial \gamma _{hu}\right) <0\), so the attributes *u*, *v* belonging to different dimensions are substitutes. This proves part (ii). \(\square \)

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Permanyer, I. Measuring poverty in multidimensional contexts.
*Soc Choice Welf* **53**, 677–708 (2019). https://doi.org/10.1007/s00355-019-01207-4

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DOI: https://doi.org/10.1007/s00355-019-01207-4