The decompositions of rank-dependent poverty measures using ordered weighted averaging operators

International Journal of Approximate Reasoning 76 (2016) 47–62

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International Journal of Approximate Reasoning www.elsevier.com/locate/ijar

The decompositions of rank-dependent poverty measures using ordered weighted averaging operators Oihana Aristondo a,∗ , Mariateresa Ciommi b a

BRIDGE Research Group, Departamento de Matemática Aplicada, Universidad del País Vasco, Avenida de Otaola 29, 20600 Eibar, Gipuzkoa, Spain b Department of Economics and Social Science, Università Politecnica delle Marche, Piazzale Martelli 8, 60121 Ancona, Italy

a r t i c l e

i n f o

Article history: Received 26 June 2015 Received in revised form 22 March 2016 Accepted 17 April 2016 Available online 3 May 2016 Keywords: Aggregation functions OWA operators Dual decomposition Rank-dependent poverty measure Intensity, incidence and inequality among the poor

a b s t r a c t This paper is concerned with rank-dependent poverty measures and shows that an ordered weighted averaging, hereafter OWA, operator underlies in the definition of these indices. The dual decomposition of an OWA operator into the self-dual core and the anti-selfdual remainder allows us to propose a decomposition for all the rank-dependent poverty measures in terms of incidence, intensity and inequality. In fact, in the poverty field, it is well known that every poverty index should be sensitive to the incidence of poverty, the intensity of poverty and the inequality among the poor individuals. However, the inequality among the poor can be analyzed in terms of either incomes or gaps of the distribution of the poor. And, depending on the side we focus on, contradictory results can be obtained. Nevertheless, the properties inherited by the proposed decompositions from the OWA operators oblige the inequality components to measure equally the inequality of income and inequality of gap overcoming one of the main drawbacks in poverty and inequality measurement. Finally, we provide an empirical illustration showing the appeal of our decompositions for some European Countries in 2005 and 2011. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Recently, there has been an increasing interest by scholars of applying the Ordered Weighted Averaging, hereafter OWA, operators to several contexts, and in particular, into the economic field. Specifically, the decomposition of the OWA operators as the sum of two parts, proposed by García-Lapresta and Marques Pereira [17], namely the self-dual core and the anti-selfdual remainder, has been applied in a social framework (see García-Lapresta et al. [18] and Aristondo et al. [2], among others). In this paper we show that there exists a connection between a class of poverty measures and OWA operators. In the evaluation of poverty, two different steps must be taken into account: the identification of the poor and the aggregation of poverty information in a numerical value. The identification problem has been solved by considering an income threshold, the so-called poverty line, which divides a society into poor, whose incomes are below the poverty line, and non-poor, whose incomes are above the poverty line. On the other hand, the aggregation step should essentially be the choice of an appropriate poverty measure. Since the seminal work of Sen [26], a great number of poverty measures have been introduced in the literature, see Clark et al. [11], Chakravarty [7], Foster, Greer and Thorbecke [16], Shorrocks [27], Kakwani [22] among others.

*

Corresponding author. E-mail addresses: [email protected] (O. Aristondo), [email protected] (M. Ciommi).

http://dx.doi.org/10.1016/j.ijar.2016.04.008 0888-613X/© 2016 Elsevier Inc. All rights reserved.

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This paper focuses on the class of rank-dependent poverty measures that are those indices defined on poverty gaps for which the weight associated to each individual depends on its position in the distribution.1 We prove that the normalized version of all the rank-dependent poverty measures, such as the poverty gap ratio, the two popular indices introduced by Sen [26], the index and the consequent class of indices proposed by Thon [28] and [29], the Kakwani family of indices [21], the Shorrocks index analyzed by Shorrocks [27] and Chakravarty [8] and the S-Gini class introduced by Weymark [30], can be interpreted as OWA operators. We adopt the methodology proposed by García-Lapresta and Marques Pereira [17] decomposing the OWA operators, underlying in the poverty measures, as the sum of the self-dual core and the anti-self-dual remainder of an OWA operator. In particular, we show that the self-dual core and the anti-self-dual remainder can be reinterpreted as a measure of intensity of poverty and the inequality among the poor, respectively. In fact, Sen [26] mentions in his seminal work that every poverty index should combine three components: the incidence of poverty, the intensity of poverty and the inequality among the poor. In other words, any poverty measure should be a function of the number of poor people in the society, the incidence, the extent of the shortfall of the poor, the intensity, and finally, it should take into account the inequality among the poor. Therefore, we will be able to decompose all the rank dependent poverty measures into their three underlying components. Firstly, we normalize all the rank-dependent poverty measures in order to transform them into OWA operators, and then, we decompose the OWA operators into the self-dual core and the anti-self-dual remainder. The incidence component is obtained from the normalization factor, whereas the OWA decomposition gives the remaining two components, namely the intensity and the inequality part. The archetypical measures for incidence and intensity are the headcount ratio and the income gap ratio, respectively.2 However, the inequality among the poor component can also be measured using different inequality measures. In addition, inequality of the poor could refer to the inequality of the income of the poor or to the inequality of the gap of the poor. A crucial requirement in the measurement of inequality is the Pigou–Dalton principle. The axiom requires that a transfer of income from a poor individual to a richer one entails an increase in the inequality of the society. This axiom could be interpreted as the counterpart of Sen’s [26] transfer axiom which demands that a regressive transfer of income has to increase the level of poverty.3 However, a regressive transfer of income could be also interpreted as a regressive transfer of gap. That is, a transfer of income from a richer to a poorer individuals entails a transfer of gap from the richer on gaps to the poorer on gaps. However, if we focus on shortfalls, the richer on incomes is now the poorer on gaps, and the poorer on incomes is now richer on gaps. Consequently, as we have mentioned, the inequality component obtained decomposing a poverty measure could be defined in terms of incomes or gaps. In this respect, in the literature, different poverty decompositions have been proposed, also for the same poverty index, in terms of incomes or gaps of the poor (see Osberg and Xu [25] and Aristondo et al. [3] among others). In addition, the choice between income and gap inequality is not innocuous and different choices may lead to contradictory results. That is, the inequality of incomes and the inequality of gaps may have opposite results for the same distribution. However, from the properties of OWA operators inherited by their components, we can conclude that the antiself-dual remainder, namely the inequality component in our decompositions, is a perfect complementary indicator, i.e., an inequality index that measures equally the inequality of income and the inequality of gap. A poverty measure that can be decomposed in this way becomes quite important from policy perspective. In fact, policymakers will be able to analyze the sources of changes in poverty focusing on their three components. In addition, these decompositions allow us to obtain consistent results for incomes and gaps when analyzing the inequality among the poor individuals. Therefore, we provide an empirical illustration of how our decompositions could be a good instrument for policy-makers so as to allow them to better understand the sources that cause poverty. Using individual level data from the European Union Survey on Income and Living Conditions (EU-SILC), we compute some rank-dependent poverty indices and their decompositions for 25 European countries in two different periods, 2005 and 2011. The analysis will bring out different situations since the increment or reduction on the poverty level for a country could result from different causes. That is, for example, an increase on poverty can be due to an increment on the number of poor, an increase on the intensity of poverty, a growth on the inequality among the poor or some combination of the three. For this reason, we believe that our proposal allows us to identify the source that causes a variation in the poverty index and in consequence to address more efficiency policy aiming at the reduction of poverty. Our paper is naturally related to two different strands of the literature. Firstly, it is concerned with the literature on aggregation functions and on OWA operators and in particular on their decomposition into the self-dual core and the anti-self-dual remainder. García-Lapresta et al. [18] prove that the normalized version of a class of poverty measures based on the one-parameter family of exponential means can be decomposed using the dual decomposition of an aggregation function into a self-dual core and anti-self-dual remainder. This decomposition lets them to decompose this poverty family into the three poverty components that can be interpreted as incidence, intensity and the inequality among the poor,

1 The poverty gap of an individual below the poverty line is the difference between the poverty line and its income level. For the ones above the poverty line it is 0. 2 The headcount ratio is the percentage of poor people and the income gap ratio is the mean of the relative gap of the poor. 3 A regressive transfer is a transfer from a poorer to a richer individual when both are poor.

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respectively. Aristondo et al. [2] prove that the decomposition of the Sen poverty index in terms of the headcount ratio, the mean of the normalized poverty gap and the Gini index, can also be achieved using the dual decomposition of OWA operators. On the other hand, Aristondo et al. [1] analyze the welfare and illfare functions associated with three classical inequality measures, namely the Gini index, the Bonferroni index and the De Vergottini index, proving that these functions can be interpreted as OWA operators and decomposing them in the corresponding two factors. Secondly, it is associated with the literature on poverty measurement and, in particular, with the decomposition of poverty measures in terms of what Jenkins and Lambert [20] called the three ‘I ’s of poverty: incidence, intensity and inequality among the poor. And also with the correlated research on the consistent decomposition of poverty indices in terms of incomes or gaps. In this respect, in the literature, different poverty decompositions have been proposed in terms of incomes or gaps of the poor, see Osberg and Xu [25] and Aristondo et al. [3]. This difficulty also arises in other economic fields in which bounded variables are involved. For instance, Erreygers [14] characterizes two indicators which measure achievements and shortfall inequality equally. On the other hand, Lambert and Zheng [23] introduce a weakener version ensuring that achievement and shortfall inequality ranking should not be reversed. Lasso de la Vega and Aristondo [24] and Aristondo and Lasso de la Vega [4] propose a unified framework in which income and gap distributions can be jointly analyzed. Chakravarty et al. [10] show that under a decomposability postulate, which is weaker than population subgroup decomposability, the Lambert–Zheng condition follows from some well-known properties of inequality indices. The remainder of the paper is organized as follows. Section 2 presents basic notation and definitions both for aggregation functions and OWA operators, as well as the dual decompositions. In addition, the definitions of poverty measurement and rank-dependent poverty measures complete this section. Our proposal for the decomposition of rank-dependent poverty measures based on incidence, intensity and inequality are provided in Section 3. Section 4 is devoted to the empirical illustration of our proposal and Section 5 concludes with a summary of our results and concluding remarks. 2. Poverty measures In what follows, we introduce basic notations and definitions used in poverty measurement. 2.1. Notation and definitions Points in [0, ∞]n are denoted x = (x1 , . . . , xn ), with 1 = (1, . . . , 1), 0 = (0, . . . , 0). Accordingly, for every x ∈ [0, ∞], we have x · 1 = (x, . . . , x). Given x, y ∈ [0, ∞]n , by x ≥ y we mean xi ≥ y i for every i ∈ {1, . . . , n}, and by x > y we mean x ≥ y of n ≥ 2 individuals such that and x = y. With this notation, x represents the income distribution vector of a population  xi stands for the income of i-th individual. The set of income distributions is D = n∈N [0, ∞)n . For a given x ∈ D, let us denote by x(1) ≤ · · · ≤ x(n) and x[1] ≥ · · · ≥ x[n] the non-decreasing and non-increasing rearrangement of the coordinates of x, respectively. In particular, x(1) = min{x1 , . . . , xn } = x[n] and x(n) = max{x1 , . . . , xn } = x[1] . A permutation δ on {1, . . . , n} is denoted as xδ = (xδ(1) , . . . , xδ(n) ) and the arithmetic mean as μ(x) = (x1 + · · · + xn )/n. 2.2. Aggregation functions and OWA operators We summarize basic notation on aggregation functions, OWA operators and the decomposition of OWA operators into the self-dual core and the anti-self-dual remainder. We begin by defining standard properties of real functions defined on [0, 1]n . For more details see Fodor and Roubens [15], Calvo et al. [6], Beliakov et al. [5], Grabish et al. [19] and García-Lapresta and Marques Pereira [17]. Definition 1. Let A : D n −→ R be a function with D = [0, 1] or D = [0, ∞] 1. A is symmetric if A (xδ ) = A (x), for any permutation δ on {1, . . . , n} and all x ∈ D n . 2. A is monotonic if for any x, y ∈ D n whenever x ≥ y holds, then A (x) ≥ A ( y ) holds. Moreover, A is strictly monotonic whenever x < y holds, then A (x) < A ( y ) holds. 3. A is invariant for translations if A (x + t · 1) = A (x), for all t ∈ R and x ∈ D n such that x + t · 1 ∈ D n . On the other hand, A is stable for translations if A (x + t · 1) = A (x) + t, for all t ∈ R and x ∈ D n such that x + t · 1 ∈ D n . 4. A is invariant for dilations if A (λ · x) = A (x), for all λ > 0 and x ∈ D n such that λ · x ∈ D n . On the other hand, A is stable for dilations if A (λ · x) = λ A (x), for all λ > 0 and x ∈ D n such that λ · x ∈ D n . 5. A is idempotent if A (x · 1) = x, for all x ∈ D. 6. A is compensative if x(1) ≤ A (x) ≤ x(n) , for all x ∈ D n . 7. A is self-dual if A (1 − x) = 1 − A (x), for all x ∈ D n , where D = [0, 1]. 8. A is anti-self-dual if A (1 − x) = A (x), for all x ∈ D n , where D = [0, 1]. 9. A is S-convex if A ( y ) ≤ A (x), for all x, y ∈ D n where y is obtained from x by rank preserving progressive transfer. Moreover a function A is strictly S-convex if strict inequality holds.

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Definition 2. Let { A (k) }k∈N be a sequence of functions, with A (k) : D k −→ R and A (1) (x) = x for every x ∈ D, where D = [0, 1] or D = [0, ∞]. We say that { A (k) }k∈N is invariant for replications if it holds that m

A

(mn)

   (x, . . . , x) = A (n) (x) ,

for all x ∈ D n and any number of replications m ≥ 2 of x. The following definition presents the aggregation functions that can only be defined in the [0, 1] interval. In what follows, in this section, we will restrict the domain to [0, 1]n . Definition 3. A function A : [0, 1]n −→ [0, 1] is called an n-ary aggregation function if it is monotonic and A (0) = 0, A (1) = 1. An aggregation function is said to be strict if it is strictly monotonic.4 Aggregation functions in general do not satisfy neither self-duality nor anti-self-duality properties. Nevertheless, selfduality, anti-self-duality and stability for translations are important properties of aggregation functions and will play an important role in this paper. 2.3. Dual decomposition and OWA operators It is known that any aggregation function A, can be decomposed as the sum of a self-dual function and an anti-self-dual function as follows: Definition 4. Let A : [0, 1]n −→ [0, 1] be an aggregation function. The functions  A,  A : [0, 1]n −→ [0, 1] defined as

A (x) − A (1 − x) + 1  A ( x) =

(1)

 A ( x) =

(2)

2 A (x) + A (1 − x) − 1 2

are called the core and the remainder of aggregation function A, respectively.

 A is self-dual and it is called the self-dual core of the aggregation function A. Note that  A is also an aggregation function since  A (0) = 0,  A (1) = 1 and A satisfies monotonicity. On the other hand,  A is called the anti-self-dual remainder of the aggregation function A since it satisfies the anti-self-duality property. However, it is not an aggregation function since  A (0) =  A (1) = 0. Moreover, −0.5 ≤  A (x) ≤ 0.5 for every x ∈ [0, 1]n . For more information see García-Lapresta and Marques Pereira [17]. Remark 1. Following García-Lapresta and Marques Pereira [17] any aggregation function can be decomposed as the sum of the self-dual core,  A, and the anti-self-dual remainder,  A, as follows A (x) =  A (x) +  A (x). As mentioned, the self-dual core  A is also an aggregation function and inherits from the aggregation function A the properties of continuity, idempotency, compensativeness, symmetry, strict monotonicity, stability for translations, and invariance for replications, whenever A has these properties. On the other hand, the anti-self-dual remainder  A is not an aggregation function. In this case, the anti-self-dual remainder  A inherits from the aggregation function A the properties of continuity, symmetry, invariance for replications, and also (strict) S-convexity, whenever A has these properties. Following with the definitions, Yager [31] introduces OWA operators that are special aggregation functions similar to weighted means. Definition 5. Given a weighting vector w = ( w 1 , . . . , w n ) ∈ [0, 1]n satisfying w is the aggregation function A w : [0, 1]n −→ [0, 1] defined as follows,

A w ( x) =

n

n

i =1

w i = 1, the OWA operator associated with

w i x[i ]

i =1

In addition, following Chakravarty [9], if the weights are ordered in a (strictly) non-increasing way, (w 1 > · · · > w n ) w 1 ≥ · · · ≥ w n , then the OWA operator A w will be (strictly) S-convex. 4

For simplicity, the n-arity is omitted whenever it is clear from the context.

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Remark 2. García-Lapresta and Marques Pereira [17] prove that the self-dual core and the anti-self-dual remainder of an OWA operator can be defined in the same way as follows:

 A w ( x) =

n

w i + w n − i +1

2

i =1

 A w ( x) =

n

w i − w n − i +1

2

i =1

x[i ]

(3)

x[i ]

(4)

n

In particular, the self-dual core  A w is an OWA operator since +1 )/2 = 1. However, the anti-self-dual i =1 ( w i + w n−i n remainder  A w is not an OWA operator since  A w is not an aggregation function and i =1 ( w i − w n−i +1 )/2 = 0. 2.4. Poverty measures As argued by Sen [26], the analysis of poverty involves two steps: the identification of the poor and the aggregation of their individual poverty levels into a composite poverty measure. Given an income distribution x ∈ [0, ∞]n , the identification requires the choice of a poverty line z ∈ (0, ∞) that establishes a cut-off point for poor and non-poor. An individual i is identified as poor if xi < z and as non-poor if xi ≥ z.5 We denote by q = q(x, z) the number of the poor people in the society. For a distribution x, we define the poor distribution and its mean as, xq = (x(1) , . . . , x(q) ) and μ(xq ) = (x(1) + · · · + x(q) )/q, respectively. The second step, the aggregation, assigns a numerical value to each distribution that determines the overall level of poverty. That is, a poverty measure is a non-constant function P : D × [0, ∞) → R whose value P (x, z) denotes the degree of intensity of the poverty associated with an income distribution x and the poverty line z. The most common measure of poverty is the so-called headcount ratio, defined as the ratio between the number of poor people in society q and the total number of individuals n.

H = H (x, z) =

q

(5)

n

It ranges from zero, nobody is poor, to one, everybody is poor. This poverty measure satisfies the focus, replication invariance and symmetry axioms.6 However, the most important limitation is that it neither captures how far the poor are from a given poverty line, intensity of the poverty, nor the inequality among the poor since it violates both the monotonicity and transfer axioms.7 With the intention of analyzing the individual shortfall, normalized gaps are introduced. For incomes that are below the poverty line, the normalized gap is the relative distance between the income value and the poverty line and for incomes above, it is zero. Formally:

 g i = max

z − xi z

,0

We denote the censored normalized income gap vector as g = ( g 1 , . . . , gn ) and the normalized gap vector of the poor as g q = ( g 1 , . . . , gq ). If we compute the mean of the normalized poverty gaps, we obtain an intensity measure of poverty, defined as:

1

q

M = M (x, z) = μ( g q ) = 1

q

=

q

i =1

z − x(i ) z



q

g [i ]

i =1

1 x(i ) q

=1−

q

i =1

z

=1−

μ(xq ) z

(6)

where g [1] ≥ · · · ≥ g [q] and x(1) ≤ · · · ≤ x(q) < z. However, this index does not reflect the inequality among the poor. In addition, even if the poverty gap measure satisfies the monotonicity axiom, it violates the transfer axiom.

5

Donaldson and Weymark [13] define two different ways to identify the poor: the weak and the strong definition. In particular, we use the weak form. The focus axiom demands independence of the index from the non-poor people. The replication invariance axiom claims that replications of the distribution do not change the index value and the symmetry axiom entails that the name and the position do not matter. 7 The Monotonicity axiom requires an increase in poverty with a reduction in the income level of a poor individual. The Transfer axiom states that a poverty measure decreases with a (progressive) transfer of income from a poor to another poorer individual. 6

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2.4.1. Rank-dependent poverty measures In what follows, we review the distribution sensitive or rank-dependent poverty measures. In economic literature, rankdependent poverty measures are those whose individual’s weight depends only on its place in the distribution with respect to the others. A poverty measure P w (x, z) is rank-dependent if for each income distribution x in D n = [0, ∞]n and any fixed poverty line z:

P w ( x, z ) =

q

w i (q, n)

z − x(i )

=

z

i =1

q

w i (q, n) g [i ]

(7)

i =1

where w 1 (q, n) ≥ w 2 (q, n) ≥ · · · ≥ w q (q, n). In addition, if the weights decrease strictly then the transfer axiom is satisfied. The most simple index is the so-called poverty gap ratio, hereafter P G R which represents the mean of the poverty gap ratios among all individuals in the population. q

1

P R G ( x, z ) =

i =1

n

(8)

g [i ]

It is the average of the normalized gap vector of the poor. This index satisfies the focus, replication invariance, symmetry and monotonicity axioms. However, it violates the transfer axiom. Among the indices that depend on the individual ranking, Sen [26] proposes a new poverty measure that places a greater weight on the income of a poor individual. We refer to it as the Sen poverty index, S : [0, ∞)n × (0, ∞) → [0, 1]. Formally, it is defined as follows:

S ( x, z ) =

q

2(q + 1 − i )

n(q + 1)

i =1

g [i ]

(9)

S (x, z) is a weighted sum of the normalized gaps, where the poorest of the group has a weight of q and the richest has a weight equal to 1. This measure satisfies the focus, symmetry and scale invariance axioms. However, it violates the continuity, replication invariance and transfer axioms.8 Sen proposes a second index, a replication invariance version, S , approximating the S index by its asymptotic value when the population size is large. In the literature the S index is the official version of the Sen index.9

S ( x, z ) =

q

2(q + 0.5 − i )

nq

i =1

g [i ]

(10)

However, the violation of the transfer axiom and the lack of continuity do not disappear. The fact that the Sen index does not satisfy the transfer axiom motivated Thon [28] to propose a variant of Sen’s poverty measure, replacing the rank of the poor individual among the poor, with the rank of each poor individual among the total population distribution.

T ( x, z ) =

q

2(n + 1 − i ) i =1

n(n + 1)

g [i ]

(11)

This simple change allows the Thon index, T , to satisfy the continuity and transfer axioms. However, this measure is a violator of the replication invariance axiom. Afterwards, Shorrocks [27] proposes a new poverty measure, S S T , that overcomes the lack of the replication invariance property suffered by the Thon index.

S S T (x, z) =

q

2(n + 0.5 − i )

n2

i =1

g [i ]

(12)

The Shorrocks index is similar to the Thon poverty measure since for greater values of n and q the Thon measure becomes the Shorrocks measure. Furthermore, this measure also overcomes the lack of the replication invariance property suffered by the Thon poverty index. Retaining that a poverty measure should also be sensitive to the absolute rank of the individuals involved in any income transfer, Kakwani [21] proposes a weaker version of the Sen index by raising the weighting function to the power of k.

K k ( x, z ) =

q

q(q + 1 − i )k g [i ] k ≥ 0 q k i =1

8 9

n

j =1

j

Continuity axiom states that the poverty index is continuous in the income distribution. Sen’s indices satisfy a weaker version of the transfer axiom, when no one crosses the poverty line as the consequence of the transfer.

(13)

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53

In particular, for k = 0 and k = 1, K k (x, z) reduces to the PGR and the first S index, respectively. The value of the parameter k may be chosen according to our preference about the sensitivity of the measure to income transfers at different income positions. That is, for higher values of k the measure will be more sensitive for transfers at the bottom of the distribution. Nevertheless, if we want to study extreme poverty we should take high values of k. Later, Thon [29] proposes a family of poverty indices depending on a parameter τ , T τ index, that is based on the idea that transfers should have the same impact irrespective of their location. For τ = 2 we have the S S T index. Formally:

T τ ( x, z ) =

q

τ n + 1 − 2i i =1

(τ − 1)n2

τ ≥2

g [i ]

(14)

Unlike Kakwani’s family’s parameter, when the parameter of this measure τ increase the poverty measure is more sensitive for transfer at the top of the distribution, that is, for transfers between the richest of the poor. Finally, the S-Gini class of poverty measures, S σ , is obtained by combining the S-Gini social welfare measure proposed by Donaldson and Weymark [12] and Chakravarty’s [7] welfare-based poverty measure.

S σ ( x, z ) =

σ q 

n+1−i i =1

n

−

n−i n

σ  g [i ]

σ ≥1

(15)

In particular, for σ = 1 and σ = 2 we have the P G R and S S T indices, respectively. Analogously to Kakwani index, the measure will be more sensitive for transfers at the bottom of the distribution for higher values of σ . In fact, if we would want to measure extreme poverty we should use Kakwani or S-Gini indices for high values of their parameter. However, if we want to know what happens with the poor near the poverty line we should choose a member of the Thon family for a high value of τ . 3. Alternative decompositions using OWA operators Since Sen [26] argued that every poverty measure should be sensitive to incidence, intensity and inequality, in the last years, there has been an increasing interest in decomposing indices in these three components of poverty. In addition, Kakwani [22] establish that every poverty measure should be expressed as an increasing function of the headcount ratio, H, the mean of the normalized gap vector, M, and an inequality measure of the poor, I. That is:

P = φ( H , M , I ) where φ is an increasing function in the three components. However, the inequality among the poor component could refer both to the inequality of the income of the poor and to the inequality of the gap of the poor. And the choice between the inequality index of the poor incomes, and the inequality index of the gaps of the poor is not innocuous, since it may lead to contradictory conclusions. In this work we propose alternative decompositions for every rank-dependent poverty indices on incidence, intensity and inequality based on ordered weighted averaging, OWA, operators. In addition, from the basic properties of OWA operators, the inequality term of these decompositions will be a consistent inequality measure, that is, it will measure equally the inequality of incomes and the inequality of gaps. 3.1. Decomposing the rank-dependent poverty measures in the three components of poverty The rank-dependent poverty indices discussed in the previous section do not automatically fulfil the requirements described in Definition 5. In fact, they are not defined in the interval [0, 1]n and the sum of the weights differs from 1. To overcome the second one, let us introduce a normalization factor C P w for each poverty measure P w and we denote by A P w (gq ) the corresponding normalized poverty index. Table 1 shows for each poverty index, the corresponding normalization factor and the normalized poverty measure. However, it is easy to prove that A P w (gq ) is an OWA operator applied to the normalized gap vector. Note that the normalized gap vectors are restricted to the [0, 1]n domain. In consequence, each rank-dependent poverty measure P w can be written as the product of the normalization factor and the OWA operator A P w (gq ), as follows:

P w = C P w · A P w (gq ) Proposition 1 shows all the properties satisfied by the OWA operator A P w (gq ). Proposition 1. The OWA operator A P w (gq ) satisfies continuity, idempotency (hence, compensativeness), symmetry, monotonicity, stability for translations and S-convexity, for each rank-dependent poverty index P . Proof. Following OWA literature we know that in general, OWA operators are continuous, idempotent (hence, compensative), symmetric and stable for translations. The positivity of weights implies monotonicity. Finally, since PGR the weights are ordered

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Table 1 Normalization factor and normalized poverty indices. Pw

CPw

PGR

H

S

H

S

H

T

H [n(2− H )+1] n+1

SST

H (2 − H )

Kk

H

Tτ

H (τ − H ) τ −1

A P w (gq )

q

1 i =1 q g [i ] 2(q+1−i ) g [i ] i =1 q(q+1) q 2(q+0.5−i ) g [i ] 2 i =1 q q 2(n+1−i ) g i =1 q(2n−q+1) [i ]

q

n

i =1

q

g [i ]

(q+1−i )k q k g [i ] i =1 i q τ n+i=11−2i i =1 q(τ n−q) g [i ]

q

1 − (1 − H )σ

Sσ

2(n+0.5−i ) q(2n−q)

i =1

(n+1−i )σ −(n−i )σ nσ −(n−q)

σ

g [i ]

Table 2 Decompositions. Index

 A P w ( gq )

 A P w ( gq )

PGR

M (x, z)

0

S

M (x, z)

S

q G ( gq ) q+1 A

M (x, z)

G A ( gq )

T

M (x, z)

SST

M (x, z) q (q+1−i )k +ik

nH (x, z) G ( gq ) n(2− H (x, z))+1 A H (x, z) G ( gq ) A 2− H (x, z))

Kk

i =1

Tτ

2

q

i =1

ik

M (x, z) q (n+1−i )σ −(n−i )σ +(n−q+i )σ −(n−q−1+i )σ

Sσ

i =1

q

g [i ]

2(nσ −(n−q)σ )

g [i ]

(q+1−i )k −ik i =1 2 q ik g [i ] i =1 H ( x, z ) G τ − H ( x, z ) A ( g q ) q (n+1−i )σ −(n−i )σ −(n−q+i )σ +(n−q−1+i )σ i =1 2(nσ −(n−q)σ )

g [i ]

in a non-increasing way and they are strictly decreasing for the others, then S-convexity is satisfied for PGR and strict S-convexity for the others. 2 Using the definition of the self-dual core and the anti-self-dual reminder of an OWA operator, we propose an alternative decomposition for all the rank-dependent poverty measures in terms of the three components of poverty. The following proposition formalizes our intuition. Proposition 2. All rank-dependent poverty measures can be rewritten as follows:

P w (x, z) = C P w · A P w (gq ) = C P w · ( A P w (gq ) +  A P w (gq ))

(16)

where, for each index,  A P w (gq ) and  A P w (gq ) are the self-dual core and the anti-self-dual remainder of A P w (gq ) defined in (3) and (4) and reported in Table 2. Proof. The proof is straightforward.

2

The previous proposition shows that the poverty measures of PGR, S, S , T, SST and T τ can be written as increasing function of H, M and G A (gq ), headcount ratio, income gap ratio and the absolute Gini index of the gap of the poor, respectively. They can be written as follows:

PGR = HM S = HM + H

(17) q q+1

G A (gq )

(18)

S = H M + H G A (gq ) T=

nH (2 − H ) + H n+1

(19)

M+

nH

2

n+1

G A (gq )

S S T = H (2 − H ) M + H 2 G A (gq ) Tτ =

H (τ − H )

τ −1

M+

H

(20) (21)

2

τ −1

G A (gq )

(22)

O. Aristondo, M. Ciommi / International Journal of Approximate Reasoning 76 (2016) 47–62

55

Nevertheless, the decompositions for the poverty families of K k and G σ are written in terms of an increasing function of the headcount ratio and two new indicators  A P w and  A P w that we will see in the following remark that they can be considered as intensity and inequality indicators, respectively. Remark 3. By Proposition 1 and Remark 1, A P w is idempotent, symmetric, strict monotonic and stable for translations. As mentioned before, the self-dual core  A P w inherits the properties of idempotency, symmetry, strict monotonicity and stability for translations from the OWA operator A P . The strictly monotonicity implies that  A P w is increasing in the gap of a poor person. The stability for translations means that equal absolute changes in all poor gaps lead to the same absolute change in  A P w . However,  A P w is not S-convex, and then it goes against the Pigou–Dalton transfer principle.10 Consequently, these properties can be regarded as basic properties of a poverty intensity index. A P w is symmetric, fulfils  A P w ( g 1 , . . . , gq ) = 0 if and only if g 1 = · · · = gq . From On the other hand, the remainder  Proposition 1,  A P w is also strictly S-convex, and consequently the anti-self-dual reminder  A P w satisfies the Pigou–Dalton transfer principle. Therefore,  A P w can be interpreted as a measure of inequality among the poor individuals. As we have mentioned before, when we want to measure the inequality of the poor people, we could focus on inequality of income or on inequality of gap. And as we have mentioned before this election can lead to contradictory result. Nevertheless, the following proposition show that the inequality component  A P w is consistent with respect to incomes and gaps, since it measures equally the both concepts. Proposition 3. For inequality measure  A P w , the following equivalence holds:

 A P w ( gq) =  A P w (xq / z)

(23)

Proof. Since  A P w is an anti-self-dual function it satisfies

 A P w ( x) =  A P w (1 − x), then

 A P w ( gq) =  A P w (1 − g q ) =  A P w (xq / z) that concludes the proof.

2

As mentioned above, the main result of Proposition 3 is that the inequality components of all the rank-dependent poverty indices measure income and gap inequality equally. This result overcomes one of the common problems in the literature of inequality of income and inequality of gap. A P w defined before. The last proposition shows the invariance properties that are satisfied by the inequality measure  A P w is invariant for translations and stable for dilations. Proposition 4. The inequality measure  Proof. The proof is straightforward.

2

That is, the  A P w component remains invariant if the gaps of all the poor individuals are increased by the same amount. Hence,  A P w can be considered an absolute inequality measure. 4. Empirical findings In this section, we illustrate the methodology developed in the paper with data from European Union countries for two different years: 2005 and 2011. We analyze the changes of poverty and the sources of these changes between 2005 and 2011 for 25 European countries. For this purpose, we use the European Union Survey on Income and Living Conditions (EU-SILC). The variable of interest we use in this application is the household disposable income, however, the individuals are the unit of analysis. Therefore, the equivalence scale, defined as the square root of the number of individuals in each household, is here used to convert household income into individual income. In order to take into account the differences in the purchasing power of different national currencies, monetary values are converted to Purchasing Power Standards (PPS), including for those countries that share a common currency, for example the Euro. In addition, a time series exercise is done for the period 2005–2011. Taking into account the inflation, the monetary values are also deflated by the Harmonized Index of Consumer Prices (HICP), based 2005. We stress that, these changes only affect the mean income of the poor and the poverty line, and not the poverty and inequality measures, as they are relative measures. 10

Pigou–Dalton is a crucial axiom in inequality measurement. It requires that a transfer from a rich person to a poorer one decreases inequality.

56

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The last considerations that should be made are with regard to the poverty line and poverty index selection. Accordingly, individuals are considered poor if they live in a household whose income is below a threshold. In this work, we choose relative poverty measures. In particular, the 60 percent of the median national equivalized income for each country and each year. Finally, we need to choose the poverty measure. For example, as we have mentioned before, Kakwani. K k , and S-Gini, S σ , families are more sensitive for transfers at the bottom of the distribution for higher values of their parameters k and σ . Therefore, if we want to measure extreme poverty we should take the poverty measures of Kakwani or S-Gini for high parameter values. On the other hand, Thon family, T τ , is more sensitive for transfers at the top of the distribution when the value of parameter τ grows. So, if we want to know what happens at the top of the distribution, with the richest of the poor, we should choose Thon index for high parameter values. In this work, first of all we focus on poverty in general and finally we analyze extreme poverty. 4.1. Poverty decomposition in incidence, intensity and inequality For the first purpose, we choose the poverty measures of S, S , T and S S T . The results obtained for S and S and T and S S T are almost equal. Consequently, we only show the results for the poverty measures of P G R, S and S S T . In order to know the sources of poverty changes we also compute the three poverty components. Note that as the three measures are increasing functions of the Headcount ratio, H, the Income gap ratio, M, and the absolute Gini index, GA, we only need to focus in these three components. Finally, standard errors for both the poverty measures and the corresponding poverty components, incidence, intensity and inequality, have been computed via a bootstrap procedure. More in details, to compute the bootstrap standard errors for all the estimators, we randomly re-sample with replacement the income variable, obtaining a new sample of size the same size denoted as x∗ . Then, the new sample is used to compute each new index. Of course, having a new distribution implies the computation of a new poverty line. The procedure is thus repeated 1,000 times. Finally, for each index, the bootstrap standard deviation has been computed as the sample standard deviation of the estimated values from the new samples. The results for the poverty decompositions in the two periods are reported in Table 3. In column 1 and column 2, we can see the year of study and the poverty lines used in this application, respectively. Between columns three and five we can observe the poverty values for the poverty measures of P G R, S and S S T and the last three columns show the corresponding three components, incidence, intensity and inequality. 4.1.1. Sources of poverty increase First of all, we focus on the countries for with poverty increases between 2005 and 2011. Focusing on Table 3, we can conclude that, for the poverty measures chosen, poverty increase for Austria, Belgium, Czech Republic, Germany, Denmark, Spain, Finland, Greece, Italy, Luxembourg, Sweden, Slovenia and Slovakia. However, these increments on poverty could have different sources. Poverty could increase because all the components increase or because the increment on some components compensates the decrease of the others. Fig. 1 shows, in three graphs, the three poverty components of these 13 countries. Focusing on Fig. 1 we can conclude that for Austria, Belgium, Czech Republic, Denmark, Spain, and Finland the increase on poverty is due to increases in all the components. However, for the other countries at least one of the component decreases. For Italy, Germany and Slovakia the Headcount ratio decreases, nevertheless, the increase on the other two components compensate the decrease on incidence. Luxembourg also suffers a drop in one of its components, the inequality, that is compensated with the other two components. On the other hand, we want to add that the unique origin of the growth on poverty for Greece, Slovenia and Sweden is an increment on the number of poor people, that compensate the fall in the other two components. On the other hand, we want to add that there exist countries for witch the unique origin of the poverty growth is a component. For example, the unique origin of the growth on poverty for Greece, Slovenia and Sweden is an increment on the number of poor people, that compensate the fall in the other two components. 4.1.2. Sources of poverty decrease With respect to Cyprus, Estonia, France, Hungary, Iceland, Lithuania, Latvia, Netherlands, Norway, Poland, Portugal and United Kingdom, Table 3 shows that poverty decreases between 2005 and 2011 for all the poverty measures. Nevertheless, if we want to know the sources of the fall in poverty we should focus on the three poverty components. Fig. 2 shows the decomposition components for these countries. The graphs show us that poverty and all the components decrease for five countries: Hungary, Netherlands, Poland, Portugal and United Kingdom. For the other seven countries at least one of the component grows. Cyprus suffers an increment on the number of the poor, however the other two components decrease concluding with a decrease on poverty. With respect to France, Estonia and Lithuania, the inequality components increase in this period, but, the fall on the other two components, incidence and intensity, compensate the increment on the inequality among the poor. Finally, the countries of Norway, Latvia and Iceland suffer an increment on the last two components, intensity and inequality, however, the drop on incidence concludes with a final decrease on poverty. Finally, the countries of Norway Latvia and Iceland suffer an increment on the last two components, intensity and inequality, however, the big drop on incidence conclude with a final decrease on poverty.

O. Aristondo, M. Ciommi / International Journal of Approximate Reasoning 76 (2016) 47–62

57

It is very important to note that all the results related with inequality of the poor are consistent with respect to incomes and gaps. That is, the inequality values of the poor for incomes and gaps are equal. 4.2. Extreme poverty decomposition in incidence, intensity and inequality On the other hand, if we want to analyze what happens at the bottom of the distribution, we should compute other poverty measures appropriate to measure extreme poverty. For this purpose, we choose poverty measures that focus on the poorest of the poor. We compute two members of the poverty family of Kakwani, for k = 4, 5, K 4 and K 5, and other two members of the family of indices S-Gini, for τ = 4, 5, S 4 and S 5 . In this study we want to analyze the sources of extreme poverty changes, so we also compute the decompositions of these poverty measures. The results are shown in Table 4. From the third to the sixth column of this table, we can see the results for the four poverty measures: K 4, K 5, S 4 and S 5 . Column 7 shows the Headcount ratio values and columns between 8 and 11 and columns between 12 and 15 show the corresponding intensity and inequality components for K 4, K 5, S 4 and S 5 , respectively. 4.2.1. Sources of extreme poverty increase Table 4 shows that extreme poverty, as poverty, also increases for Austria, Belgium, Czech Republic, Germany, Denmark, Spain, Finland, Greece, Italy, Luxembourg, Sweden and Slovakia. The unique country for which poverty increases and extreme poverty does not increase is Slovenia. As we want to focus on the sources of these growths, we decompose each poverty measure into their three poverty components. However, unlike for PGR, S and SST, the poverty components obtained for the four extreme poverty measures are different. That is, the incidence indicator is the Headcount ratio for every index, nevertheless, the intensity components and the inequality components are different for each measure. Therefore, if we want to know the sources of the increase on extreme poverty we should focus of the different intensity and inequality indicators. For example, if we focus on the intensity components of Germany, contradictory results are obtained for the Kakwani indices and S-Gini indices. In fact, the intensity components for the Kakwani indices increase in this period and for the S-Gini indices decrease. Therefore, in this application we only focus on the countries for which the intensity and the inequality components have the same ordinations for the different measures. These countries include Austria, Belgium, Czech Republic, Denmark, Spain, Finland, Italy and Slovakia and their decomposition for the poverty measure of K 5 in the three poverty components is shown in Fig. 3.11 On the other hand, nothing can be concluded for the remaining countries: Germany, Greece, Luxembourg and Sweden. In fact, contradictory results are obtained for different poverty decompositions. Focusing in the poverty decompositions of the mentioned countries, Fig. 3 exhibits that poverty, intensity and inequality components increase between 2005 and 2011 for all these countries. However, the Headcount ratio decreases for two countries: Slovakia and Italy. 4.2.2. Sources of extreme poverty increase Table 4 also shows that extreme poverty, as poverty, suffers a drop between 2005 and 2011 for the countries of Cyprus, Estonia, France, Hungary, Iceland, Lithuania, Latvia, Netherlands, Norway, Poland, Portugal and United Kingdom. In addition, as we have mentioned before, Slovenia’s poverty level also decreases in this period. The decomposition for the poverty measure of K 5 in the three components of poverty, for some of the countries, is shown in Fig. 4. In this figure we can observe the results for Cyprus, Hungary, Latvia, Netherlands, Norway, Poland, Portugal, Slovenia and United Kingdom. We do not need to show the decompositions for the other poverty measures because the results are consistent with the ones shown in Fig. 4. We do not graph the decomposition results for Estonia, France, Iceland and Lithuania because contradictory results are obtained for different poverty decompositions. Now, focusing on the sources of poverty in Fig. 4, it can be concluded that poverty decreases due to falls in all the components for Hungary, Netherlands, Poland, Portugal and United Kingdom. Slovenia and Cyprus suffer an increment on the number of the poor people, nevertheless, this growth is compensated by the other two components. On the other hand, Latvia and Norway countries’ poverty decrement is due to the fall of the incidence components that is higher than the growths on the intensity and inequality components. 5. Concluding remarks The recent literature on poverty measurement stresses the importance of an index to take into account intensity, incidence and inequality. In this paper, we have proposed alternative decompositions for the rank-dependent poverty indices in terms of the above mentioned components using OWA literature. We have shown that an OWA operator is underlying in the definition of the rank-dependent poverty indices and the decomposition of the OWA operators in the self-dual core and the anti-self-dual remainder components have been applied. This decomposition have provided us a decomposition of all the rank-dependent poverty measures in terms of incidence, intensity and inequality among the poor. The properties inherited by each component from the OWA operators have allowed us to obtain consistent inequality components. That is, inequality indices that measure equally the inequality of the income of the poor, and the inequality of the gap of the poor. Finally, the illustration using EU-SILC data for some European Countries in 2005 and 2011 has shown the changes on poverty and the sources of these changes in the period of study.

11

The results for the other extreme poverty measures are identical.

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O. Aristondo, M. Ciommi / International Journal of Approximate Reasoning 76 (2016) 47–62

Acknowledgements O. Aristondo and M. Ciommi gratefully acknowledge the funding support of Departamento de Educación, Política Lingüistica y Cultura del Gobierno Vasco under the project IT568-13. O. Aristondo gratefully acknowledges also the funding support of the Spanish Ministerio de Educación y Ciencia under the project ECO2015-67519, cofunded by FEDER and UPV/EHU, UFI 11/46 BETS. The authors would like to express their sincere thanks to the Editor-in-Chief Professor Thierry Denoeux and anonymous reviewers for their valuable comments. Authors are grateful to Casilda Lasso de la Vega for her helpful suggestions for extending and improving the paper and to the participants to the EUSFLAT (2015) and ECINEQ (2015). Appendix A Table 3 EU-SILC 2005–2011. Monetary variable in PPS. Poverty indices

Austria Belgium Cyprus Czech Republic Germany Denmark Estonia Spain Finland France Greece Hungary Iceland Italy Lithuania Luxembourg Latvia Netherlands Norway Poland Portugal Sweden Slovenia Slovakia United Kingdom

2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011

z

PGR

13806 16522 12442 15264 10821 9907 1730 3493 15421 13004 22046 25195 1382 2224 8415 8302 15583 18106 14746 15637 6991 6469 1562 1497 25452 10797 11097 11285 956 391 22046 30023 914 1496 13538 16318 25636 41966 1211 1862 4899 5038 15526 20806 5084 6905 1507 2875 14761 12639

0.0378∗ 0.0406∗ 0.0365∗ 0.0410∗ 0.0584∗ 0.0503∗ 0.0242∗ 0.0258∗ 0.0444∗ 0.0466∗∗ 0.0390∗ 0.0488∗ 0.0628 0.0570∗ 0.0662∗ 0.0693∗ 0.0381∗ 0.0418∗ 0.0397∗ ∗ ∗ 0.0342∗ 0.0621∗ 0.0627∗∗ 0.0303∗ 0.0269∗ 0.0382∗ 0.0341∗ 0.0571∗ 0.0584∗ 0.0715∗ 0.0616 0.0280∗ 0.0291∗ 0.0669∗∗ 0.0645∗ 0.0325∗ 0.0284∗ 0.0462∗ 0.0443∗ ∗ ∗ 0.0659∗ ∗ ∗ 0.0498∗∗ 0.0665∗ 0.0521∗ 0.0371∗ 0.0462 0.0404∗ 0.0415∗ 0.0344∗ 0.0352∗ 0.0597∗ ∗ ∗ 0.0507∗

S 0.0542∗ 0.0589∗∗ 0.0519∗ 0.0586∗ 0.0799∗ 0.0691∗ ∗ ∗ 0.0353∗ 0.0378∗ 0.0631∗ 0.0652∗ 0.0572∗ ∗ ∗ 0.0722∗ 0.0889 0.0831∗ 0.0927∗ 0.0981∗ 0.0539∗ 0.0593∗ 0.0562∗∗ 0.0493∗ 0.0867∗∗ 0.0883∗∗ 0.0435∗ 0.0390∗ 0.0546∗ 0.0504∗ 0.0808∗ 0.0831∗ 0.0997∗ 0.0882 0.0408∗ 0.0420∗ 0.0943∗ 0.0925∗∗ 0.0481∗ 0.0421∗∗ 0.0656∗ 0.0642∗ ∗ ∗ 0.0932 0.0712∗ 0.0926∗ 0.0742∗∗ 0.0549∗ 0.0662 0.0574∗∗ 0.0582∗ 0.0496∗ 0.0517∗ 0.0846∗∗ 0.0723∗

∗ , ∗∗ , and ∗ ∗ ∗ significant at the 0.05, 0.01 and 0.005 probability level, respectively.

Note: Own elaboration from EU-SILC data and HICP from Eurostatweb page.

Incidence

Intensity

Inequality

SST

H

M

GA

0.0726∗ 0.0779∗ 0.0694∗ 0.0777∗ 0.1098∗ 0.0945 0.0469∗ 0.0499∗ 0.0845∗ 0.0886∗∗ 0.0744∗∗ 0.0931∗ 0.1178 0.1076∗ 0.1241∗ 0.1299∗ 0.0724∗ 0.0793∗ 0.0757∗ ∗ ∗ 0.0657∗ 0.1164∗ 0.1175∗∗ 0.0584∗ 0.052∗ 0.0729∗ 0.0658∗ 0.1077∗ 0.1106∗ 0.1333∗ 0.1161 0.0542∗ 0.0560∗ 0.1250∗∗ 0.1214∗ 0.0627∗ 0.0548∗ 0.088∗ 0.0850∗ ∗ ∗ 0.1238∗ ∗ ∗ 0.0945∗ 0.1242∗ 0.0982∗ 0.0715∗ 0.0876 0.0773∗∗ 0.0791∗ 0.066∗ 0.0680∗ 0.1122∗ ∗ ∗ 0.0961∗

14.08%∗ 14.85% 16.91%∗ 18.27%∗ ∗ ∗ 18.82%∗ 19.23%∗∗ 11.64%∗ 11.93%∗ 16.76%∗ 16.56%∗ 16.94% 17.53%∗ 21.37% 20.47%∗∗ 20.98%∗ 21.36%∗ 16.85%∗ 17.92%∗ 16.18%∗ 14.23%∗ 20.62%∗ 21.39%∗ 13.14%∗ 12.70%∗ 15.75%∗ 13.72%∗ 19.38%∗∗ 18.45%∗∗ 22.31%∗ 20.20%∗ ∗ ∗ 12.39%∗ 12.83%∗ 22.25%∗∗ 20.54%∗ 13.3%∗ 13.04% 16.31%∗ 14.56%∗ 20.61%∗ ∗ ∗ 18.09%∗ 21.72%∗ 20.30% 14.51%∗ ∗ ∗ 18.23%∗ ∗ ∗ 14.84%∗∗ 15.99%∗ 14.30% 13.08% 20.57% 18.45%∗

0.2684∗ 0.2733 0.2157∗ 0.2243∗ 0.3100∗ 0.2614 0.2079∗ ∗ ∗ 0.2160∗ 0.2651∗∗ 0.2816 0.2300∗ 0.2784∗∗ 0.2939 0.2783∗∗ 0.3158∗ 0.3244∗∗ 0.2259∗ 0.2335∗ ∗ ∗ 0.2454 0.2403∗∗ 0.3010∗ 0.2932∗∗ 0.2307∗ 0.2121∗ 0.2425∗ 0.2488∗∗ 0.2944∗ 0.3167∗ 0.3205∗ 0.3048∗ ∗ ∗ 0.2262∗ 0.2265∗ 0.3006∗ 0.3139∗ 0.2441∗ 0.2175∗∗ 0.2832∗ 0.3039 0.3196∗ 0.2753∗ ∗ ∗ 0.3060∗ 0.2567 0.2559∗ 0.2532∗∗ 0.2720∗ 0.2596∗ 0.2402 0.2692∗ ∗ ∗ 0.2902∗ 0.2749∗

0.1168∗ ∗ ∗ 0.1234∗ 0.0910∗∗ 0.0968∗ ∗ ∗ 0.1143 0.0977∗ ∗ ∗ 0.0953 0.1005∗ ∗ ∗ 0.1113 0.1122∗ 0.1077 0.1333∗ ∗ ∗ 0.1222 0.1279∗∗ 0.1260 0.1350∗ ∗ ∗ 0.0943∗ ∗ ∗ 0.0975∗ ∗ ∗ 0.1021∗∗ 0.1064 0.1197∗ ∗ ∗ 0.1195∗ 0.1001 0.0950 0.1041∗ 0.1184 0.1223∗ 0.1336∗ ∗ ∗ 0.1262∗∗ 0.1319∗ ∗ ∗ 0.1026∗∗ 0.1012∗∗ 0.1231∗ ∗ ∗ 0.1364 0.1178 0.1055∗ 0.1193∗ ∗ ∗ 0.1372∗ ∗ ∗ 0.1327∗ ∗ ∗ 0.1180∗ ∗ ∗ 0.1205 0.1088 0.1229∗∗ 0.1101∗ ∗ ∗ 0.1149∗ ∗ ∗ 0.1042∗ ∗ ∗ 0.1067 0.1257∗ 0.1213∗ ∗ ∗ 0.1169∗∗

Table 4 EU-SILC 2005–2011. Monetary variable in PPS. Poverty indices

AT BE CY CZ

DK EE ES FI FR GR HU IS IT LT LU LV NL NO PL PT SE SI SK UK

Intensity

K5

S4

S5

H

MK4

MK5

MS4

MS5

IK4

Inequality IK5

IS4

IS5

0.0776∗ 0.0855 0.0732∗∗ 0.0844∗ 0.1045∗ 0.0926∗ 0.0512∗∗ 0.0548∗ 0.0879∗∗ 0.0900∗ 0.0842∗ ∗ ∗ 0.1060∗ 0.1250∗ 0.1199 0.1263∗ ∗ ∗ 0.1364∗ ∗ ∗ 0.0760 0.0834 0.0789∗ ∗ ∗ 0.0704 0.1183∗ 0.1223 0.0613 0.0562∗ ∗ ∗ 0.0785∗ 0.0744∗ 0.1132∗∗ 0.1168∗ 0.1359∗ 0.1236∗ 0.059∗ 0.0614∗ 0.1310∗ 0.1290∗∗ 0.0720∗ 0.0628 0.0925∗ 0.0921∗∗ 0.1292 0.0998∗∗ 0.1275∗ 0.1041 0.0822∗ 0.0963∗∗ 0.0801∗ 0.0794∗ 0.0698 0.0751 0.1183∗ ∗ ∗ 0.1009∗

0.0822∗ 0.0909 0.0776∗∗ 0.0897∗ 0.1087∗∗ 0.0971∗ 0.0543∗∗ 0.0581∗ 0.0926∗∗ 0.0948∗ 0.0896∗ ∗ ∗ 0.1125∗∗ 0.1319∗ 0.1269 0.1325∗ ∗ ∗ 0.1435∗ ∗ ∗ 0.0803 0.0882 0.0834 0.0745 0.1241∗ 0.1287 0.0647 0.0596 0.0835∗ 0.0791∗ 0.1195∗∗ 0.1232∗ 0.1426∗ 0.1301∗ 0.0627∗ 0.0655∗ 0.1381∗ 0.1357∗ ∗ ∗ 0.0768∗ 0.0670 0.0977∗ 0.0973∗∗ 0.1358 0.1052∗∗ 0.1342∗ 0.1098 0.0877∗ 0.1026∗∗ 0.0844∗ 0.0834∗ 0.0735 0.0796 0.1248∗ ∗ ∗ 0.1063∗

0.1343∗ 0.1438 0.1262∗ 0.1405∗ 0.1955∗ 0.1681∗ 0.0881∗ 0.0937∗ 0.1538∗ 0.1608∗ 0.1364∗ 0.1705∗ 0.2089∗ 0.1934∗ 0.2197∗ 0.2303∗ 0.1316∗∗ 0.1434∗ 0.1380∗ ∗ ∗ 0.1215∗ 0.2064∗ 0.2079 0.1086∗ ∗ ∗ 0.0971∗ 0.1337∗ 0.1227∗ 0.1929∗ 0.1995∗ 0.2340∗ 0.2078∗ 0.1014∗ 0.1046∗ 0.2202∗ 0.2170∗ 0.1173∗ 0.1027∗ 0.1605∗ 0.1572∗∗ 0.2203∗ 0.1709∗ 0.2187∗ 0.1753 0.1328∗ 0.1585∗ 0.1421∗ 0.144∗ 0.1221∗ 0.1270∗ 0.1999 0.1733∗

0.1617∗ 0.1731 0.1507∗ 0.1675∗ 0.2313∗ 0.1988∗ 0.1069∗ 0.1137∗ 0.1838∗ 0.1919∗ 0.1636∗ 0.2043∗ 0.2468∗ 0.2299∗ 0.2593∗ 0.2721∗ 0.1572∗∗ 0.1708∗ 0.1652∗ ∗ ∗ 0.1464∗ 0.2438∗ 0.2454 0.1310 0.1174∗ 0.1603∗ 0.1483∗ 0.2290∗ 0.2375∗ 0.2752∗ 0.2465∗ 0.1227∗ 0.1264∗ 0.2595∗ 0.2572∗ 0.1419∗ 0.1244∗ 0.1921∗ 0.1893∗∗ 0.2607∗ 0.2037∗ 0.2577∗ 0.2078 0.1603∗ 0.1891∗ 0.1707∗ 0.1722∗ 0.1470∗ 0.1536∗∗ 0.2367 0.2063∗

14.08%∗ 14.85% 16.91%∗ 18.27%∗ ∗ ∗ 18.82%∗ 19.23%∗∗ 11.64%∗ 11.93%∗ 16.76%∗ 16.56%∗ 16.94% 17.53%∗ 21.37% 20.47%∗∗ 20.98%∗ 21.36%∗ 16.85%∗ 17.92%∗ 16.18%∗ 14.23%∗ 20.62%∗ 21.39%∗ 13.14%∗ 12.70%∗ 15.75%∗ 13.72%∗ 19.38%∗∗ 18.45%∗∗ 22.31%∗ 20.20%∗ ∗ ∗ 12.39%∗ 12.83%∗ 22.25%∗∗ 20.54%∗ 13.3%∗ 13.04% 16.31%∗ 14.56%∗ 20.61%∗ ∗ ∗ 18.09%∗ 21.72%∗ 0.203 14.51%∗ ∗ ∗ 18.23%∗ ∗ ∗ 14.84%∗∗ 15.99%∗ 14.30% 13.08% 20.57% 18.45%∗

0.3103∗∗ 0.3205∗∗ 0.2455∗ 0.2608∗ 0.3239∗ 0.2816∗ 0.2440∗ 0.2539∗ 0.2966∗ 0.3126 0.2740∗ 0.3321∗ 0.3332∗∗ 0.3249∗ 0.3453∗ 0.3627∗ 0.2562∗ ∗ ∗ 0.2643∗ 0.2772 0.2756∗ 0.3297∗∗ 0.3272 0.2619 0.2470∗∗ 0.2825∗ 0.2989∗ 0.3324∗ 0.3590∗ 0.3514∗∗ 0.3428∗∗ 0.2645∗ 0.2681∗ 0.3362∗ 0.3510∗ 0.2977∗ 0.2644∗ 0.3222∗ 0.3522 0.3557∗ 0.3104∗ 0.3394∗ 0.2899∗ 0.3121 0.2987∗ 0.3046∗ 0.2842∗ ∗ ∗ 0.2712∗ 0.3164∗ 0.3263∗ 0.3081∗

0.3216∗ ∗ ∗ 0.3334∗ ∗ ∗ 0.2540∗∗ 0.2710∗ 0.3279∗ 0.2874∗ 0.2535∗ 0.2640∗ 0.3053∗ 0.3214 0.2859∗ 0.3458∗ 0.3437∗ ∗ ∗ 0.3369∗ 0.3534∗ 0.3729∗ 0.2646∗ ∗ ∗ 0.2728∗ 0.2860 0.2854∗ 0.3375∗ ∗ ∗ 0.3364 0.2703 0.2565∗ ∗ ∗ 0.2935∗ 0.3119∗ 0.3428∗ 0.3701∗∗ 0.3597∗ ∗ ∗ 0.3503∗ 0.2751∗ 0.2796∗ 0.3459∗ 0.3609∗ 0.3117∗ 0.2767∗∗ 0.3327∗∗ 0.3646 0.3654∗ 0.3199∗ 0.3483∗ 0.2988∗ 0.3269 0.3111∗ 0.3137∗ 0.2911 0.2794∗ 0.3288∗∗ 0.3362∗ 0.3172∗

0.0264∗ 0.0296 0.0250∗ 0.0290∗ 0.0353∗ ∗ ∗ 0.0309∗ 0.0176∗ ∗ ∗ 0.0191∗ 0.0303∗ ∗ ∗ 0.0302∗ 0.0297 0.0381∗∗ 0.0434∗ 0.0433 0.0438 0.0479 0.0259∗ ∗ ∗ 0.0286 0.0268∗ 0.0243 0.0408∗ 0.0424∗ ∗ ∗ 0.0210 0.0193 0.0265∗ 0.0261∗ 0.0390∗ ∗ ∗ 0.0404∗ 0.0470∗ 0.0440∗∗ 0.0203∗ 0.0207∗ 0.0457∗ 0.0463 0.0251∗ 0.022 0.0316∗ 0.0322∗ 0.0453 0.0349∗ ∗ ∗ 0.0436∗ 0.0365 0.0287∗ 0.0329∗ 0.0275∗ 0.0270∗ 0.0245 0.0263 0.0413∗ ∗ ∗ 0.0353∗

0.0351∗ 0.0393 0.0332∗ 0.0384∗ 0.0468∗ ∗ ∗ 0.0410∗ 0.0234∗ ∗ ∗ 0.0254∗ 0.0403∗ ∗ ∗ 0.0401∗ 0.0394 0.0506∗∗ 0.0574∗ 0.0573 0.0580 0.0633 0.0343∗ ∗ ∗ 0.0379 0.0356∗ 0.0323 0.0540∗ 0.0561∗ ∗ ∗ 0.0280 0.0256 0.0352∗ 0.0346∗ 0.0517∗ ∗ ∗ 0.0536∗ 0.0621∗ 0.0583∗∗ 0.0270∗ 0.0276∗ 0.0604∗ 0.0613 0.0333∗ 0.0292 0.0419∗ 0.0427 0.0599 0.0463∗ ∗ ∗ 0.0576∗ 0.0483 0.0381∗ 0.0436∗ 0.0365∗ 0.0358∗ 0.0326 0.0350 0.0546∗ ∗ ∗ 0.0469∗

0.2408∗ ∗ ∗ 0.2553 0.1878 0.2011 0.2314 0.2002 0.1961 0.2059∗ ∗ ∗ 0.2276 0.2308∗∗ 0.2226 0.2728 0.2514∗ ∗ ∗ 0.2611∗ ∗ ∗ 0.2567 0.2757 0.1947 0.2012 0.2103 0.2188 0.2440 0.2445 0.2049 0.1957 0.2161∗∗ 0.2433∗∗ 0.2517 0.2740 0.2579 0.2688 0.2115∗∗ 0.2106∗ 0.2527 0.2772 0.2436∗ ∗ ∗ 0.2174 0.2451 0.2801 0.2709 0.2411 0.2476 0.2229 0.2543 0.2294∗∗ 0.2351∗∗ 0.2125 0.2169 0.2575 0.2489 0.2389

0.2624 0.2786 0.2048 0.2202 0.2496 0.2174 0.2134 0.2235∗ ∗ ∗ 0.2470 0.2513∗∗ 0.2429 0.296 0.2736 0.2831∗ ∗ ∗ 0.2780 0.2989 0.2123 0.2193 0.2291 0.2382 0.2642 0.2654 0.2223 0.2132 0.2364∗∗ 0.2647∗∗ 0.2740 0.2976 0.2795 0.2910 0.2305∗∗ 0.2307∗∗ 0.2747 0.2997 0.2659 0.2367 0.2665 0.3036 0.2935 0.2615 0.2693 0.2420 0.2776 0.2514∗∗ 0.2552∗∗ 0.2302 0.2346 0.2796 0.2706 0.2592

0.2687∗ 0.2737 0.2160∗ 0.2248∗ 0.3102∗ 0.2617∗ 0.2081 0.2162∗ 0.2654∗∗ 0.2819 0.2305∗ ∗ ∗ 0.2791∗∗ 0.2947∗ 0.2792 0.3163∗∗ 0.3251∗∗ 0.2263∗ 0.2339∗ ∗ ∗ 0.2458 0.2406∗∗ 0.3015∗∗ 0.2938 0.2309∗ 0.2123∗ 0.2429∗ 0.2492∗ 0.2950∗ 0.3173∗ 0.3211∗ 0.3055 0.2264∗ 0.2268∗ ∗ ∗ 0.3013∗ 0.3146∗∗ 0.2445∗∗ 0.2178∗ 0.2836∗ 0.3043 0.3202∗ 0.2758∗∗ 0.3066∗ 0.2573∗∗ 0.2564∗∗ 0.2538∗ 0.2723∗ 0.2599∗∗ 0.2404 0.2695∗ 0.2909∗ 0.2754∗

0.2690∗ 0.2741 0.2164∗ 0.2253∗ 0.3104∗ 0.2620∗ 0.2083 0.2165∗ 0.2658∗∗ 0.2822 0.2310∗ ∗ ∗ 0.2798∗∗ 0.2954∗ 0.2800 0.3169∗∗ 0.3258∗∗ 0.2266∗ 0.2343∗ ∗ ∗ 0.2461 0.2409∗∗ 0.3020∗∗ 0.2945 0.2311∗ 0.2125∗ 0.2433∗ 0.2496∗ 0.2956∗ 0.3179∗ 0.3218∗ 0.3061 0.2266∗ 0.2270∗ ∗ ∗ 0.3021∗ 0.3152∗∗ 0.2448∗∗ 0.2182∗ 0.284∗ 0.3047 0.3208∗ 0.2762∗∗ 0.3073∗ 0.2579∗∗ 0.2569∗∗ 0.2545∗∗ 0.2726∗ 0.2601∗∗ 0.2407 0.2699∗ 0.2915∗ 0.2759∗

59

AT = Austria, BE = Belgium, CY = Cyprus, CZ = Czech Republic, DE = Germany, DK = Denmark, EE = Estonia, ES = Spain, FI = Finland, FR = France, GR = Greece, HU = Hungary, IS = Iceland, IT = Italy, LT = Lithuania, LU = Luxembourg, LV = Latvia, NL = Netherlands, NO = Norway, PL = Poland, PT = Portugal, SE = Sweden, SI = Slovenia, SK = Slovakia, UK = United Kingdom. ∗ , ∗∗ , and ∗ ∗ ∗ significant at the 0.05, 0.01 and 0.005 probability level, respectively. Note: Own elaboration from EU-SILC data and HICP from Eurostatweb page.

O. Aristondo, M. Ciommi / International Journal of Approximate Reasoning 76 (2016) 47–62

DE

2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011 2005 2011

Incidence

K4

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O. Aristondo, M. Ciommi / International Journal of Approximate Reasoning 76 (2016) 47–62

Fig. 1. Poverty components for Austria (AT), Belgium (BE), Czech Republic (CZ), Germany (DE), Denmark (DK), Spain (ES), Finland (FI), Greece (GR), Italy (IT), Luxembourg (LU), Sweden (SE), Slovenia (SI), Slovakia (SK).

Fig. 2. Poverty components for Cyprus (CY), Estonia (EE), France (FR), Hungary (HU), Iceland (IS), Lithuania (LT), Latvia (LV), Netherlands (NL), Norway (NO), Poland (PL), Portugal (PT), United Kingdom (UK).

O. Aristondo, M. Ciommi / International Journal of Approximate Reasoning 76 (2016) 47–62

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Fig. 3. Poverty components for Austria (AT), Belgium (BE), Czech Republic (CZ), Denmark (DK), Spain (ES), Finland (FI), Italy (IT) and Slovakia (SK).

Fig. 4. Poverty components for Cyprus (CY), Hungary (HU), Latvia (LV), Netherlands (NL), Norway (NO), Poland (PL), Portugal (PT), Slovenia (SI) and United Kingdom (UK).

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