Unidimensional and multidimensional fuzzy poverty measures: New approach

Unidimensional and multidimensional fuzzy poverty measures: New approach

Economic Modelling 29 (2012) 995–1002 Contents lists available at SciVerse ScienceDirect Economic Modelling journal homepage: www.elsevier.com/locat...

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Economic Modelling 29 (2012) 995–1002

Contents lists available at SciVerse ScienceDirect

Economic Modelling journal homepage: www.elsevier.com/locate/ecmod

Unidimensional and multidimensional fuzzy poverty measures: New approach Besma Belhadj a,⁎, Mohamed Limam b a b

ISGT, University of Tunis, Tunis, Tunisia LARODEC, ISGT, University of Tunis, Tunis, Tunisia

a r t i c l e

i n f o

Article history: Accepted 7 March 2012 JEL classification: P46 I32 D81 C00 Keywords: Unidimensional poverty Multidimensional poverty Poverty line Membership function of poor Membership function of non-poor Fuzzy set operators

a b s t r a c t The analysis of deprivation is usually seen as a unidimensional condition. However, recently it is considered to be a multidimensional one. A useful tool for such analysis is to view deprivation as a degree providing a quantitative expression to its intensity for individuals. Such fuzzy conceptualisation has been widely adopted in poverty and deprivation research. This paper aims to further develop and refine this strand of research. First, we re-examine two aspects introduced by the use of fuzzy measures, as opposed to conventional poor/non-poor dichotomous measures, namely the choice of membership functions and the rules to manipulate, resulting fuzzy sets. Secondly, we propose fuzzy monetary and non-monetary measures with the membership functions of poor and non-poor. An application based on individual well-being data from Tunisian households in 1990 is presented to illustrate use of one of the proposed concept. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Poverty measurement is very sensitive on how the poor are identified. Traditional approaches are based on a rigid poor/non-poor dichotomy, and most of the literature on poverty measurement uses poverty thresholds (Rodgers and Rodgers, 2000), or poverty line, as the minimum level of income deemed necessary to achieve an adequate standard of living in a given country. Yet it is quite obvious that such a clear-cut division causes a loss of information and removes the nuances that do exist between the two extremes of substantial welfare on one hand and the distinct material hardship on the other. Nowadays many authors recognise the fact that poverty should be considered as a degree rather than as an attribute, or, i.e., that is simply present or absent among individuals in the population (Betti et al., 2006). An early attempt to incorporate this concept at a methodological level and in a multidimensional framework was made by Cerioli and Zani (1990) based on Fuzzy Sets theory (FS). This method was later developed by Cheli and Lemmi (1995) yielding the so called Totally Fuzzy and Relative (TFR) approach. Both methods have been applied by many authors subsequently, with a preference to the original TFR version1. Moreover, the TFR method was refined by Cheli ⁎ Corresponding author. E-mail addresses: [email protected] (B. Belhadj), [email protected] (M. Limam). 0264-9993/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2012.03.009

(1995) and used it to analyze poverty in fuzzy terms in a dynamic context represented by two consecutive panel waves. Dagum et al. (1992), Chiappero Martinetti (1994), Chiappero Martinetti (2009) Betti and Verma (2008) and Silber (2011), among others, made further contributions and applications. When poverty is viewed as a degree, i.e., as a fuzzy measure, two additional aspects are considered in the analysis, as compared to conventional poor/non-poor dichotomous approach. First, the choice of membership functions, i.e., quantitative specification of households' degree of poverty and deprivation. Second, the choice of rules for manipulating resulting fuzzy sets such as defining complements, intersections, union and averaging. In this paper, some conceptual and theoretical aspects concerning fuzzy set logic are discussed and used to propose fuzzy measurements for monetary and nonmonetary poverty (capabilities approach). Moreover, we focus on the relationship between membership functions of poor and nonpoor. It is useful to clarify the concept of poverty and various forms of deprivation as a degree. Hence, we need to replace the conventional classification of a population by a simple dichotomy. Basically, all individuals in a population are subject to poverty with varying degrees. We say that each individual has a certain propensity to be poor, the population covering the whole range [0,1]. The conventional approach is a special case where the population is dichotomised as {0,1}. Those with an income below a certain threshold are deemed to be poor and are all assigned a constant propensity equal to 1.

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Others with income equal or above that threshold are deemed to be non-poor and are all assigned a constant propensity equal to zero. The basic idea of fuzzy sets is as follows. Given a set Y of elements y ∈ Y, any fuzzy subset A of Y, defined as A = {y, μA(y)}, where μA(y): Y → [0,1], is called the membership function (m.f.) in the fuzzy subset A. The value μA(y) indicates the degree of membership of y inA. Thus, μA(y) = 0 means that y does not belong toA, whereas μ(y) = 1 means that y belongs to A completely. On the other hand when 0 b μA(y) b 1 then y partially belongs to Aand its degree of membership of A increases proportionally to the proximity of μA(y) to 1 (Kaufmann and Gupta, 1991). This paper is structured as follows. Section 2 presents a brief description of the theory of FS and its application to poverty. Section 3 uses FS approach to derive alternative families of unidimensional and multidimensional poverty indices. Section 4 illustrates the use of unidimensional measures using Tunisian household data of 1990, and Section 5 concludes. 2. FS approach Fuzzy logic is a form of multivalued logic derived from fuzzy set theory to deal with approximate reasoning rather than the precise one. In binary logic named also crisp logic, in contrast to fuzzy logic, the variables may have a membership value of 0 or 1. In fuzzy set theory with fuzzy logic the set membership values can range (inclusively) between 0 and 1. Also, the degree of truth of a statement can range between 0 and 1 and is not constrained to the two true values (true (1), false (0)) as in classic predicate logic (Novák et al., 1990). And when linguistic variables are used, those degrees may be managed by specific functions as discussed below. The term “fuzzy logic” emerged as a consequence of the development of fuzzy sets theory proposed by Zadeh (1965). A paper introducing the concept without using the term was published by Wilkinson (1963) and thus preceded fuzzy set theory. Wilkinson was the first to redefine and generalize the earlier multivalued logics in terms of set theory. The main purpose of his paper is to show how a mathematical function is simulated using hardwire analog of electronic circuits. He first created various linear voltage ramps which were then selected in a “logic block” using diodes and resistor circuits which implemented the maximum and minimum Fuzzy Logic rules of the INCLUSE, OR and AND operations, respectively. He called this logic “analog logic”. 2.1. Fuzzy poverty Apart from various methodological choices involved in the construction of conventional poverty measures, the introduction of fuzzy measures brings in additional factors on which choices have to be made. These factors concern at least two aspects. First, the choice of membership functions meaning a quantitative specification of the propensity to poverty, or deprivation of each person, given the level and distribution of income and other aspects of living conditions of the population. Second, the choice of rules for manipulating resulting fuzzy sets, specially the rules defining complement, intersection, union and aggregation of sets. To be meaningful both of these choices should meet some basic logical and substantive requirements mainly by being useful and elucidating aspects of the situation not captured, or not captured adequately, by the conventional approach. We begin with the issue of the choice of the poverty membership function. In the conventional head count ratio H, the m.f. could be given by  μ ðyi Þ ¼

1 if yi bz 0 if yi ≥z;

where yi is the equivalised income of household i, and z is the poverty line. In order to move away from the poor/non-poor dichotomy, Cerioli and Zani (1990) proposed the introduction of a transition zone z1, z2 between the two states, a zone over which the m.f. declines from 1 to 0 linearly. This m.f. is given by 8 1 > < z −y 2 i μ ðyi Þ ¼ > : z2 −z1 0

if yi bz1 if z1 ≤yi bz2

ð1Þ

if yi ≥z2 :

3. Measuring poverty 3.1. Traditional approach The traditional approach involves establishing an income threshold z and calculating how many individuals, families or households fall below it. The measurement of poverty involves three steps: selecting an appropriate indicator to represent individuals' well-being; choosing z which identifies the lower part of the distribution; and finally, selecting some function of the level of well-being of ‘poor’ individuals relative to the poverty line (Sen, 1976). The traditional approach to poverty uses income or consumption expenditure yi as an indicator of well-being. It identifies the poor as those with insufficient income to attain minimum basic needs z, and aggregates their shortfall from the poverty line into a poverty index (Deaton, 1997). Poverty headcount, poverty gap, and severity of poverty are the most common indices used in the literature. All these indices belong to the family of Foster–Greer–Thorbecke (FGT) poverty measures (Foster et al., 1984). In the traditional approach to poverty, the individual poverty function is ( Si ¼

0 z−yi z

yi > z yi bz;

and the aggregator function is the FGT index given by: FGT α ¼ P ðX; zÞ ¼

n 1X α ðS Þ IðSi > 0Þ; n i¼1 i

ð2Þ

where I is the indicator function, X is n × q strictly positive real-valued matrix whose element xijrepresents the attainments of individual i = 1, …, n in dimension j = 1, …, m and α is a parameter indicating the sensitivity of the index to the distribution among the poor. The higher α the more sensitive the index is to the poorest persons in the economy. For α = 0, FGT is the headcount, for α = 1 it is the poverty gap, and for α = 2 it represents the severity of poverty. 3.2. Fuzzy approach This section presents an alternative approach to the derivation of multidimensional and unidimensional poverty indices using instruments from FS. Two sequential decisions need to be made. First, we need to choose between a poverty line that represents the ‘shortfall of well-being’ and the one based on the ‘well-being of the shortfalls.’ Second, we present alternative measures of unidimensional and multidimensional poverty while checking some required properties. 3.2.1. Confidence interval for the poverty line In this section we calculate a bootstrap interval ½z^−l; ^z þ l ¼ ½z1 ; z2 , where ^z represents an estimate of the poverty line. Computing z^ is a delicate step since it depends on the socioeconomic context in

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which the individual is established and should take into account the particular characteristics of the choice of deprivation indicator. So, to calculate z^, we consider a general approach adopted by Ravallion and Bidani (1994). This approach consists in determining the minimum income to satisfy basic food needs and then estimate the minimum income to satisfy non food needs. These minimum incomes constitute respectively the food and non food poverty lines. Basic food needs are computed on a regional basis depending on the local food consumer behaviour so that the typical consumption basket ensures a minimal calorific intake as determined by nutritionists. Then, this basket is evaluated using local prices so that the food poverty line can be calculated. The valuation of the non food component is based on a method suggested by Ravallion (1994). This approach uses the intuitive argument that the definition of `basic non food needs' requires the valuation of the willingness to give up a necessary food product in order to purchase the required item. Ravallion (1994) estimates the value of the food component based on AIDS (Almost Ideal System of Deaton and Muellbauer (1980)) class of functions. This class is given by

0

ωij ¼ α j þ βj log

yij zj f nθij

! k k

þ ∑ δj dij þ εij ;

ð3Þ

k

where ωij is the food share of household i in dimension j, yij is its total per capita expenditure, zjf is the already established food poverty line in dimension j, nijθ is the equivalent size of household i in dimension j, θ is the elasticity size that corresponds to the elasticity of equivalence scale compared to the size of household, dijk are socioeconomic variables such as the age of household head, the number of children, the number of working women, etc.…, and εij is a disturbance term. k

The value of λj ¼ α 0j þ ∑k δkj d j estimates the expected non food shares of households with per capita expenditure that reaches the k

food poverty line, i.e., yij = zjf. The evaluation of d j is made by means of the sub-sample with per capita expenditure around the poverty   line. Then, the poverty line is given by z^j ¼ 2−λj zfj and includes minimum expenditure to satisfy basic food and non food needs. To calculate poverty line interval we use the bootstrap method. The bootstrap (for a general discussion of the bootstrap, see Hall, 1992; Shao and Tu, 1995) provides an estimate of the sampling distribution of a given statistic by resampling from the original sample, thus simulating the original sampling procedure. Besides having advantages in small samples under certain circumstances (see e.g. Efron and Tibshirani, 1993), the bootstrap can take into account stochastic dependencies in multivariate data without explicitly dealing with its covariance structure. In the context of inequality measurement, the bootstrap was first applied by Mills and Zandvakili (1997). They calculated bootstrap confidence intervals for some inequality indices as well as for the components of a decomposition of Theil's coefficient by subgroup and compared them with intervals obtained by the normal approximation method. We take M random samples of sizenwith replacement from our original sample. The larger the value of M the better is the approximation. Values of M between 100 and 200 are commonly used. Each of the M samples is called a bootstrap sample. We calculate z^ for every bootstrap sample and let z^m be the value of the poverty line for the mth bootstrap sample. Then, the bootstrap standard error of ^ z is the standard deviation of the bootstrap poverty lines, given by

seðz^Þ ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ∑M ðz^ −z^Þ2 : M m¼1 m

997

The standard bootstrap confidence interval for the poverty line is defined as follows: z^  Φð1−α=2Þ seðz^Þ; where Φ(1 − α/2) is the (1 − α/2) percentile of the standard Gauss distribution. 3.2.2. The membership function of poor Let P be the set of poor households andP its complement. The standard practice is to define this set as follows P = {i : yi ≤ z} and P ¼ fi : yi > zg. With these definitions, either i ∈ P and i∉P , or i ∉ P and i∈P . However, if there is an uncertainty in the definition of the poverty line, and thereby of the poor, we may specify a membership function which indicates the degree to which a household is considered as poor. In this case, we consider the possibility of two thresholds as described above: a minimum threshold for extreme poverty z1 under which everyone agrees that the membership function μ(yi) = 1, and a maximum threshold for moderate poverty z2 over which everyone agrees that the membership function μ(yi) = 0. The membership function being continuous differentiable and non-increasing (Fig. 1), to be used in the empirical illustration, is proposed as follows:

μ ðyi Þ ¼

8 1 > > <

z2 −yi > z > : 2 −z1 0



if yi bz1 ð4Þ

if z1 ≤yi bz2 if yi ≥z2 ;

where δ is the concavity of the underlying individual poverty function and is related to the extreme poverty aversion parameter involved in Foster et al. (1984). As noted in Atkinson (2003), it seems reasonable in practice to use the traditional values 1 and 2 for δ. A value of 3 for δ is sometimes used in the unidimensional context to get a measure which is more sensitive to transfers involving the poorest members of the population. In the simulation study δ = 2 is used. For a lower value of δ some of axioms are violated, while for high values of δ the shortfalls of the poorer segments are weighted more heavily. Therefore, the intensity of deprivation by the poorer segments, in particular the poorest, is magnified for values of δ greater than 2. For these values of δ both the monotonicity and transfer axioms of Sen (1976) are satisfied. We may recall that both of these axioms have to do with the sensitivity of the index to the incomes of the poor as opposed to simply the number of poor. Thus, the monotonicity axiom states that, ceteris paribus, a decrease in the income of a poor person should increase the poverty index. The transfer axiom states that a transfer of income from a lower income poor person to a higher income poor person increases the poverty index.

( yi ) ( yi )

z* Fig. 1. Membership functions of poor and non-poor.

yi

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If δ = 0, we get the standard definition of a crisp set with the membership functions taking a value of 1 for all those with an income below z2. 3.2.3. The membership function of non-poor Fuzzy set operations are a generalization of the corresponding crisp set operations in the sense that the former reduce to the latter when the fuzzy membership functions, being in the whole range [0,1], are reduced to a {0,1} dichotomy. However, there is more than one way in which fuzzy set operations are formulated, each representing an equally valid generalization of the corresponding crisp set operations. The choice among alternative formulations has to be made primarily on substantive grounds. Some options are more appropriate than others, depending on the context and objectives of the application. While the rules of fuzzy set operations are not discussed in this paper in detail, it is essential to clarify their application specifically for the study of poverty and deprivation. Since fuzzy sets are completely specified by their membership functions, any operation on them, such as union, intersection, complement or aggregation, is defined in terms of the membership functions of the original fuzzy sets involved. In this work, we define the membership function μ ðyi Þ to represent the degree up to which a household i with incomeyiis considered as non-poor (Fig. 1). Since μ ðyi Þ is the complement of μ(yi), it is defined as μ ðyi Þ ¼ 1−μ ðyi Þ. The membership function μ ðyi Þ associated with Eq. (4) is defined as: 8 0 > >   < z −yi δ μ ðyi Þ ¼ 1− 2 > z2 −z1 > : 1

m n h i  X 1X  w μ ðyi Þ∧μ ðyi Þ I z ≥yi n j¼1 j i¼1

P 2 ðY; zÞ ¼

m n h i  X 1X  wj μ ðyi Þ∨μ ðyi Þ I z ≤yi : n j¼1 i¼1

0 P iM ðY; zÞ

ð5Þ

if z1 ≤yi bz2

ð9Þ

ð10Þ

3.2.4.2. Multidimensional poverty measure. Let IDt, v … be the deprivation space related to the attributes {t, v …} and corresponds m to the set of points yi ∈ IR+ such that yij b zj, ∀ j ∈ {t, v …}. A k m k-deprivation space ID is the subset of IR+ corresponding to the set of vectors yi such that yij b zj for at least k attributes, i.e., informal terms ID 2 = ∪ t ≠ uIDt, u ∀ {t, u} {1, …, m}. By definition, we observe ID m ⊂ ID m − 1 … ⊂ ID 1. The poverty m domain, that is the set of all admissible points in IR+ for which individuals deemed poor, is denoted by IP. The individual multi-attribute poverty function is ¼ ε@

1δ=β

    β  m  X  wj μ yij ∨μ yij I zj ≥yij A j¼1

0

þ ð1−εÞ@

1δ=β

    β  m  X  wj μ yij ∧μ yij I zj ≤yij A ;

ð11Þ

j¼1

if yi ≥z2 :

3.2.4. The aggregator function Two alternative measures of unidimensional and multidimensional poverty are proposed. 3.2.4.1. Unidimensional poverty measure. The aggregator function across individuals and attributes is P U ðY; zÞ ¼ ε P 1 ðY; zÞ þ ð1−εÞ P 2 ðY; zÞ;

ð6Þ

where

ε¼

P 1 ðY; zÞ ¼

if yi bz1

Even though it is difficult to define poverty, it is easier to define the “richness.” It is the whole flow of incomes, monetary and nonmonetary, current and future, brought up to date at the interest rate of the market. It could be easier to approach the problem of poverty by its negation: “richness” (Hayek, 1960). Admittedly, one can measure fuzzy richness starting from the membership function given by Eq. (5).

8 <

This measure complies with extended strong focus, monotonicity, restricted strong monotonicity, restricted continuity, non-decreasingness in poverty domain, Subgroup consistency, anonymity and population invariance. It can be shown that it satisfies the transfer and transfer sensitivity axioms, while being continuous and even decomposable. To measure the richness, Eqs. (7) and (8) become:

where Yis an n × m strictly positive real-valued matrix whose element yijrepresents the jth attribute of the ith individual,  1wj > 0 ∀ j ∈ {1,...., m},  − are attributes' weights and zj ¼ z2j − z2j −z1j 2 δ represents the  in tersection point of the two membership functions μ(yij) and μ yij in dimension j = 1, …, m. The latter functions are defined as follows:   μ yij ¼

8 1 > > > <

z2j −yij > z2j −z1j > > : 0



if yij bz1j if z1j ≤yij bz2j

ð12Þ

if yij ≥z2j

and 8 0 > > !δ <   > z −yij μ yij ¼ 1− 2j > z2j −z1j > > : 1

if yij bz1j if z1j ≤yij bz2j

ð13Þ

if yij ≥z2j :

And the aggregator function (across individuals) is given by 

1 1 if yi ≤z  P ðY; zÞ ¼ : 0 if yi ≥z 1 n

m X

n h X

j¼1

i¼1

wj

i   μ ðyi Þ∨μ ðyi Þ I z ≥yi

m n h i  X 1X  wj μ ðyi Þ∧μ ðyi Þ I z ≤yi ; P 2 ðY; zÞ ¼ n j¼1 i¼1

ð7Þ

ð8Þ

with Y being an n × m strictly positive real-valued matrix whose element yi denote the m-vectors that correspond to the vector of individual i vector of attributes, wj > 0, ∀ j ∈ {1,...., m} the weights, ∧ and ∨ are respectively the minimal and the maximal value of μ(yi) and 1 μ ðyi Þ and z ¼ z2 −ðz2 −z1 Þ2−δ represents the intersection point (see Fig. 1) of the two membership functions μ(yi) and μ ðyi Þ.

0 1δ=β

    β  n m  X εX  @ w μ yij ∨μ yij P M ðY; zÞ ¼ I zj ≥yij A n i¼1 j¼1 j

0 1δ=β

    β  n m  X ð1−εÞ X  @ wj μ yij ∧μ yij þ I zj ≤yij A n i¼1 j¼1

ð14Þ

The parameterβ defines the degree of substitutability between different attributes. The higher β is the lower is the degree of substitutability between them. Here, we have two interesting special cases: when β tends to infinity, relative deprivations are non-substitutes; and when β = 1 attributes are perfect substitutes. In both situations, poverty is defined unidimensionally, in the first case by the attribute

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deprivation with the highest level of deprivation, and in the second, as a simple weighted sum of attributes. The second option resembles the standard monetary approach to poverty if the weights are equal to market prices. Convexity of attributes, that is concavity in the space of deprivations, requires β ≥ 1. 3.2.5. Transfer axioms and correlation of distributions The specificity of multidimensional poverty is that it necessitates the use of axioms that do not echo those employed in the unidimensional context. In particular, it may be ethically relevant to take into account the correlation between the observed distributions of each attribute. Using Atkinson and Bourguignon's (1982) study on multidimensional inequality measurement, Tsui (2002) and Bourguignon and Chakravarty (2003) propose the following axioms to deal with situations in which transfers do not alter the marginal distribution of each attribute (see Bresson, 2009): Axiom 1. Non-decreasingness under correlation increasing switch ∀ {i, i '} IP with yi ≠ yi ', PM(X, z) ≥ PM(Y, z) if X is obtained from Yof transfers between two poor persons i and i ' such that xi = yi ∨ yi ' and xi' = yi ∧ yi '. Axiom 1s. Strong non-decreasingness under correlation increasing switch ∀ {i, i '} IP with yi ≠ yi ', PM(X, z) ≥ PM(Y, z) if X is obtained from Y by a sequence of transfers between two poor persons i and i ' such that xi = yi ∨ yi ' and xi' = yi ∧ yi ' ∀ j, j ' {1, …, m} such that yij > yi ' j and yij' b yi ' j'. Axiom 2. Non-increasingness under correlation increasing switch ∀ {i, i '} IP with yi ≠ yi ', PM(X, z) ≤ PM(Y, z) if X is obtained from Yof transfers between two poor persons i and i ' such that xi = yi ∨ yi ' and xi' = yi ∨ yi '. Axiom 2s. Strong non-increasingness under correlation increasing switch ∀ {i, i '} IP with yi ≠ yi ', PM(X, z) b PM(Y, z) if X is obtained from Y by a sequence of transfers between two poor persons i and i ' such that x' i = yi ∨ yi ' and xi' = yi ∧ yi ' ∀ j, j {1, …, m} such that yij > yi ' j and yij' b yi ' j'. Axiom 1 (2) means that increasing the correlation between all series of attributes do not cause a fall of the poverty level. With axioms 1 and 2, X is obtained by switching the values of the different attributes of two poor individuals i and i ' to make i ' unambiguously poorer than i. Tsui (2002, proposition 4) shows that Axiom 1 is verified if and only if the individual poverty function is supermodular on IP in the general case, and if ∂ 2/(∂ yij ∂ yij') ≥ 0 , ∀ j ≠ j ', in the case of twice differentiable poverty measures. This result can directly be extended to any view of poverty identification. The same line of reasoning can be used to show that Axiom 2 requires individual poverty function to be L-subadditive or to exhibit non-positive cross-derivatives on P. Axioms 1 and 2 can be expressed in stronger forms, respectively 1s and 2s, if PM(Y, z) and PM(X, z) can be compared using strict inequalities. We may also require the following property: Axiom 3. Attribute decomposability   m P M ðY; zÞ ¼ m1 ∑m j¼1 wj θj y:j ; zj with ∑j¼1 wj ¼ m. Axiom 3 means that the global level of poverty is a weighted sum of poverty levels measured for each attribute distribution. The estimation of the contribution to overall poverty of observed deprivations in a given dimension for a given group is straightforward. As stressed by Sen (1976), the definition of any poverty measure should be governed by concerns about inequalities among the poor.

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The main reason for this emphasis on inequalities is that it generally favours the use of weighing patterns that foster attention to the situation of the worst off. So we turn to the more general case of transfers that induce changes in the marginal distributions of some attributes. Many equivalences (see for instance Dasgupta et al., 1973 in the context of inequality measurement) make the treatment of that issue easy in the unidimensional context. These equivalences generally do not hold when multiple dimensions are considered. At least four alternative extensions of the transfer principle do exist in the multidimensional poverty literature. We mention them in the following: Axiom 4. Simple transfer PM(X, z) ≤ PM(Y, z) if X can be obtained from Y by opering a progressive transfer εk ∈ IR+ + for the kth attribute between two individuals i and i ', {i, i '} IP, with |xi'k − xik| b |yi'k − yik| and yij = yi'j, ∀ j ≠ k. Axiom 4s. Strong simple transfer PM(X, z) b PM(Y, z) if X can be obtained from Y by opering a progressive transfer εk ∈ IR+ + for the kth attribute between two individuals i and i ', {i, i '} IP, with |xi'k − xik| b |yi'k − yik| and yij = yi'j, ∀ j ≠ k. Axiom 4R. Restricted strong simple transfer PM(X, z) b PM(Y, z) if X can be obtained from X by opering a progressive transfer εk ∈ IR+ + for the kth attribute between two individuals i and i ', {i, i '} S m, with |yi'k − yik| b |xi'k − xik| and yij = yi'j, ∀ j ≠ k. Axiom 5. Non-ambiguous transfer PM(X, z) ≤ PM(Y, z) if X can be obtained from Y by opering a prom gressive transfer ε ∈ IR+ /{0} between two individuals i and i ', {i, i '} IP, with ε such that |yi'k − yik| ≤ |xi'k − xik| ∀ j ∈ {1, …, m}. Axiom 5s. Strong non-ambiguous transfer PM(X, z) b PM(Y, z) if X can be obtained from Y by opering a progresm sive transfer ε ∈ IR+ /{0} between two individuals i and i ', {i, i '} IP, ' with ε such that |xi k − xik| ≤ |yi'k − yik| ∀ j ∈ {1, …, m}. Axiom 5R. Restricted strong non-ambiguous transfer PM(X, z) b PM(Y, z) if X can be obtained from Y by transfering m ε ∈ IR+ /{0} between two individuals i and i ', {i, i '} S m, with ε such that |yi'j − yij| ≤ |xi'j − xij| ∀ j ∈ {1, …, m}. Axiom 6. Transfer in the sense of Schur PM(X, z) ≤ PM(Y, z), if X = DY with B being any n × n bistochastic matrix whose elements dii' differ from 0 and 1 only for {i, i '} IP. Axiom 6s. Strong transfer in the sense of Schur PM(X, z) b PM(Y, z), if X = DY with B being any n × n bistochastic matrix whose elements dii' differ from 0 and 1 only for {i, i '} IP and ∃ j ∈ {1, …, m} such that yij ≠ yij'. Axiom 6R. Restricted strong transfer in the sense of Schur PM(X, z) b PM(Y, z), if X = DYwith D being any n × n bistochastic matrix which elements dii' differ from 0 and 1 only for {i, i '} S m and ∃ j ∈ {1, …, m} such that yij ≠ yij'. Axiom 7. Independent transfer PM(X, z) ≤ PM(Y, z), if X can be obtained from Y by premultiplying y. j, ∀ j ∈ {1, …, m}, by any n × n bistochastic matrix Dwhose elements dii' differ from 0 and 1 only for {i, i '} IP. Axiom 7s. Strong independent transfer PM(X, z) b PM(Y, z), if X can be obtained from Y by premultiplying y. j, ∀ j ∈ {1, …, m}, by any n × n bistochastic matrix Dwhose elements dii' differ from 0 and 1 only for {i, i '} IP with yij ≠ yij'. Axiom 7R. Restricted strong independent transfer PM(X, z) b PM(Y, z), if X can be obtained from Y by premultiplying y. j, ∀ j ∈ {1, …, m}, by any n × n bistochastic matrix D which elements dii' differ from 0 and 1 only for {i, i '} S m with yij ≠ yij'.

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Axiom 4 means that a progressive transfer between two poor individuals, that differ only with respect to the value of the jth attribute, does not increase the level of poverty. The axiom 5 is stronger than Axiom 4 since the non-increasing effect concerns all pairs of poor individuals i and i ' that can be ranked unambiguously in terms of poverty since one of the two individuals exhibits attribute levels that are no greater than those of the other one. The approach adopted for the axiom 6 differs from the one used for axiom 5 with respect to two fundamental aspects. On one hand, it is stronger since it deals with transfers between any pair of poor individuals, and on the other hand, it is weaker as it deals with transfers that are simultaneously performed in each dimension and in the same proportions. As a consequence, no causal relationship can be found between Axiom 5 and Axiom 6. However, it can be noted that Axiom 6 also implies Axiom 4. Finally, Axiom 7 states that any progressive transfer in any dimension between two poor individuals cannot increase poverty. Proving that Axiom 7 entails both Axioms 5 and 6 is straightforward. While Axioms 4, 5, 6 and 7 do not conflict with the approach used for the identification of the poor nor with the version adopted for the focus axiom (Zheng, 1997), it is not the case for stronger versions. For instance, Tsui (2002, proposition 2) has shown that it is not possible to get a non-trivial poverty measure that simultaneously complies with strong focus (Zheng, 1997), subgroup decomposability and Axiom 6S within a “union” approach. This incompatibility is caused by the non-sensitivity of the measure PM to increases of xij above the related poverty line but corresponding to vectors xi of poor persons. In fact, Tsui's observation also holds with Axioms 4S, 5S and 7S, and is still observed if strong focus is replaced by extended strong focus, except for S m = IP, where Sk, k ∈ {2, …, m}is the set in which the poverty effect of any decrease in the level of an attribute can be partially compensated by some increase of the quantity of another attribute. In order to get poverty measures that keep the main intuition of the strong version of different poverty axioms and complies with either strong focus or extended strong focus, a slight modification in the definition of these strong transfer axioms is introduced. More precisely, assuming that progressive transfers do not increase the overall poverty level, we just need to restrict our attention to transfers between individuals whose vectors of attributes belong to the substitution space S m instead of the poverty domain P. We then obtain Axioms 4R, 5R, 6R and 7R. Any poverty measure complies with Ax ioms 4R, 5R, 6R and 7R if and only if: θm ðX; zÞ ¼ ξ 1n ∑i∈Ρ θ ðyi ; zÞ , with ξ : IR → IR being some increasing continuous function and θ : m m IR+ + × IR+ → IR a continuous non-increasing function such that ∂ θ/ ∂ yi b 0,yi ∈ P, and ∂ θ/∂ yi = 0∀ yi ∉ P. Finally, some authors (Kakwani, 1980) have stressed the potential relevance of assuming that the non-increasing effect of progressive transfers is a decreasing function of the beneficiary initial situation. As this type of axioms is based on transfer principles, their application in the multidimensional context entails the existence of rival expressions that are given in the following: Axiom 8. Simple transfer sensitivity PM(X, z) ≤ PM(X ', z) if Xand X 'are obtained from Y by transfering εj ∈ IR+ + respectively from poor individuals u tosand v to t such ' that yuj b yvj, ys, − j = yt, − j = yu, − j = yv, − j and xuj − xsj = xvj − xtj' b εj, where yi, − jdenotes the (m − 1)vector obtained from yiafter dropping its jthelement. Axiom 9. Sensitivity to independent transfers ∀ {ys, yu, yt, yv} IP, PM(X, z) ≤ PM(X ', z) if X and X ' are obtained from Y by transfering εj ∈ IR+ + respectively from poor individuals u to s and v to t such that yuj b yvj. Axiom 10. Sensitivity to Schur transfers PM(X, z) ≤ PM(X ', z) ∀ X, X ' obtained from Y by premultiplying respectively Yu, s Yv, t, {u, s, v, t} IP, by the 2 × 2 bistochastic matrix D2,

with yu − ys = yv − yt ≠ 0, if yu b yv; Yu, s is the 2 × m matrix obtained with the individual vectors yu and yv. Axiom 11. Non-ambiguous transfer sensitivity m PM(X, z) ≤ PM(X ', z) ∀ X, X ' obtained from Y by transferring ε ∈ IR+ \{0} respectively from poor individuals u to s and v to t with yu − ys = yv − yt > ε, |yu − ys| > |xu − xs| and xs b xt. The simple transfer sensitivity of Axiom 10 means that the effects of a given transfer between two poor persons that differ only with respect to the quantity of the jth attribute should not increase if initial values of this attribute are set to a higher level, other things being equal. As with Axiom 5, Axiom 11 extends Axiom 8 to pairs of individuals whose poorest (or richest) members could be unambiguously ranked in terms of well-being. In order to get a measure that complies with Axiom 1 (1R), we need δ (Eq. (14)) to be strictly greater than β, and Axiom 2 (resp. 2R) is observed in the converse situation. As the restrictions imposed on individual poverty function for Axiom 5 are the same as the one for NDS, the principle of nonambiguous transfers is logically respected for δ ≥ β. To get a measure that satisfies axiom 5R, we also have to impose β > 1. Interestingly, with δ >1 and β >1 the individual poverty measure is strictly convex in the substitution space ID2. With respect to meagre attributes, and so Axiom 6R is fulfilled. On the other hand, Axiom 4R requires only to be strictly greater than 1. Perfect substitutability is obtained for δ =β, so that Axiom 3 is satisfied in this particular case. Once this equality is observed, the measure also complies with 7, and with 7R for δ ≥1. Considering transfer sensitivity, Axiom 8 requires δ to be (strictly) greater than 2 and β ∉ (1, 2). Axiom 9 is observed for δ = β ≥ 2. Finally, it can be shown that Axiom 10 is fulfilled if and only if δ = β = 1. 4. Empirical illustration In this section, due to lack of data, we limit our illustration to a unidimensional measure using data from Tunisia. The information used is supplied from the household survey data conducted by the Tunisian Statistical Institute (INS) in 1990 involving 7734 households. The sampling scheme and the results of the survey are detailed in INS (1990). The survey also provides demographic characteristics of households. In order to take into account different geographical and socioeconomic characteristics of regions in Tunisia, we split the country into 5 different homogenous regions. To make use of the characteristics of different regions in Tunisia, we separated the households according to their location with respect to 5 different regions. Tunisia is traditionally subdivided into three natural regions: North, Center and South. This decomposition is motivated by the geographical characteristics of the country. However, from an economic point of view, it is more appropriate to divide Tunisia into three parts: The Greater Tunis and two homogenous namely the Littoral and the Interior. The Greater Tunis area, which covers almost 25% of the total population, is characterised by very special administrative, social and economic properties. The Tunisian Littoral (Bizerte, Cap-Bon, Sahel, Sfax and Gabes) have known since the independence an economic and social prosperity. This coastal fringe extending from North to South contains, along with the Greater Tunis area, the essential touristic, industrial and urban activity of the economy. Despite a certain economic progress, the Interior region witnesses some social and economic problems which distinguish it from the other two regions. This subdivision is justified if we compare the per capita expenditure during 1990. In addition to this regional decomposition, it is necessary to take into account the rural-urban distinction. We also aggregated the rural part of the Greater Tunis and the rural part of the littoral. Two reasons support this

B. Belhadj, M. Limam / Economic Modelling 29 (2012) 995–1002 Table 1 Fuzzy poverty by area (1990).

Greater Tunis Urban littoral Urban interior Rural littoral Rural interior Fuzzy poverty

z^

[z1, z2]

ωR

PU(X, z)

270 243 202 162 159

[263–277] [235–251] [193–211] [157–167] [151–167]

0.25 0.15 0.20 0.22 0.18 1

0.057 0.048 0.099 0.092 0.139 0.074

ωR represents the part of each area

aggregation. First, the size of the rural Greater Tunis is very small, only 167 households, second the rural of Greater Tunis and those of the rest of the littoral are very similar and can be lumped together to form a homogenous spatial set. This leads us to five homogenous regions, namely the urban Greater Tunis, the urban littoral, the urban interior, the rural littoral and rural Interior. The estimated intervals, using α = 0, 05, for the 5 regions of Tunisia are presented in the second column of Table 1. We note that, in 1990, for the “Greater Tunis,” for example, any household whose annual expenditure is lower than 263DT is considered poor and its degree of membership to the fuzzy subset “Strong Privation” is very high. On the other hand, any household whose annual total expenditure exceeds 277DT is considered as non-poor and its degree of membership to the fuzzy subset “Weak Privation” is high. To compute the consumer's basket and since the respective samples were relatively small, we grouped Greater Tunis and Littoral urban together. However, we considered different unit values and therefore poverty lines for the two regions. The estimated poverty lines are lower in the poorest regions (urban interior, rural interior and rural littoral. We notice the difference between urban and rural lines. The urban/rural ratio in the littoral region is equal to 1.5. On the other hand, the ratio urban/rural in the Interior is equal to 1.27. Indeed, the urban littoral witnessed a rapid economic development compared to the interior which led to an increase in living costs and provide an explanation why the urban/rural difference in the littoral region is greater than that of the interior region. By examining the fourth column of Table 1, we notice that poverty in Tunisia during the year 1990, is mainly a phenomenon that affects severely rural areas than urban ones. In each region the rural poverty index exceeds that of the urban one. We can observe conspicuously that the ratio of the rural over the urban in the interior region amounts to 140%. Moreover, it reaches the peak of 192% for the littoral region. Results show that total fuzzy poverty is about 0,074 and that areas “urban littoral” and “Greater Tunis” present on average living conditions that are different from the others and better than the national average. 5. Concluding remarks In the poverty literature, a poverty threshold is often established below which households/individuals are considered to be poor. Unfortunately, the choice of a poverty threshold is an arbitrary one (Filippone et al., 2001) establishing an artificial dichotomy between poor and non-poor. As pointed out by Cerioli and Zani (1990) and Cheli et al. (1994), the problem is that a sharp division of households between poor and non-poor is unrealistic. This has led Qizilbash (2001) to characterise poverty as a vague concept since there seams to be no clear cut-line between the poor and the non-poor. This calls for a mathematical framework capable of modelling vague concepts such as poverty, were FS theory seems particularly appropriate. As opposed to the conventional poor/non-poor dichotomy, two additional aspects are introduced in the analysis: the choice of the membership functions and the choice of rules for the manipulation

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of the resulting fuzzy sets, rules defining their complements, intersections, union and aggregation. This paper proposes measures of unidimensional and multidimensional poverty using the FS approach. A general rule for the construction of fuzzy set intersections, i.e., for the construction of a total poverty measure from a sequence of “poor” and “non-poor” sets, is used. The proposed fuzzy unidimensional and multidimensional poverty measures constructed satisfies many desirable properties. An application using individual well-being data from Tunisian households in 1990 is presented to illustrate use of unidimensional poverty. Results show that in 1990 poverty in Tunisia was clearly a rural phenomenon. In order to take full advantage of our fuzzy methodology, it is of interest to measure multivariate poverty introducing population poverty ratios by attributes and determine their contributions to the total poverty ratio. These ratios provide basic information about the causes of poverty. They are of paramount importance to the design and implementation of a structural socioeconomic policy to abate the causes of poverty. Hence, they purport to break the mechanism responsible for its intergenerational transmission. This policy, being structural, not cyclical, should aim at a process of structural change with the scope of generating stable, efficient and equitable socioeconomic processes. References Atkinson, A., 2003. Multidimensional deprivation: contrasting social welfare and counting approaches. Journal of Economic Inequality 1, 51–65. Atkinson, A., Bourguignon, F., 1982. The comparison of multi-dimensioned distributions of economic status. Review of Economic Studies 49, 183–201. Betti, G., Verma, V., 2008. Fuzzy measures of the incidence of relative poverty and deprivation: a multi-dimensional perspective. Statistical Methods & Applications 17 (2), 225–250. Betti, G., Cheli, B., Lemmi, A., Verma, V., 2006. Multidimensional and longitudinal poverty: an integrated fuzzy approach, fuzzy set approach to multidimensional poverty measurement: Economic Studies in Inequality, Social Exclusion and Well-Being, vol. 3, pp. 115–137. Bourguignon, F., Chakravarty, S., 2003. The measurement of multidimensional poverty. Journal of Economic Inequality 1, 25–49. Bresson, F., 2009. Multidimensional Poverty Measurement with the Weak Focus Axiom, Document de Travail. Cerioli, A., Zani, S., 1990. A Fuzzy Approach to the Measurement of Poverty. In: Dagum, C., Zenga, M. (Eds.), Income and Wealth Distribution, Inequality and Poverty, Studies in Contemporary Economics. Spinger Verlag, Berlin, pp. 272–284. Cheli, B., 1995. Totally fuzzy and relative measures of poverty in dynamics context. Metron 53, 183–205. Cheli, B., Lemmi, A., 1995. Totally fuzzy and relative approach to the multidimensional analysis of poverty. Economic Notes 24, 115–134. Cheli, B., Ghellini, A., Lemmi, A., Pannuzi, N., 1994. Measuring poverty in the countries in transition, via TFR method: the case of Poland in 1990–1991. Statistics in Transition 1 (5), 585–636. Chiappero Martinetti, E., 1994. A new approach to evaluation of well-being and poverty by fuzzy set theory. Giornale degli Economisti e Annali di Economia 53, 367–388. Chiappero Martinetti, E., 2009. Operationalization of the capability approach, from theory to practice: a review of techniques and empirical applications. Contribution to “Debating Global Society: reach and limits of the CA”, Fondazione Feltrinelli Part IICh1. Dagum, C., Gambassi, R., Lemmi, A., 1992. New approaches to the measurement of poverty. Poverty Measurement of Economics in Transition. Polish statistical Association & Central Statistical office, Warsaw. Dasgupta, P., Sen, A., Starett, D., 1973. Notes on the measurement of inequality. Journal of Economic Theory 6, 180–187. Deaton, A., 1997. The analysis of household surveys: a microeconometric approach to development policy. Johns Hopkins University Press, Baltimore. Deaton, A., Muellbauer, J., 1980. An almost ideal demand system. American Economic Review 70, 312–326. Efron, B., Tibshirani, R.J., 1993. An introduction to the bootstrap. Chapman and Hall, New York. Filippone, A., Cheli, B., D'Agostino, A., 2001. Addressing the Interpretation and the Aggregation Problems in Totally Fuzzy and Relative Poverty Measures. Working Papers of the Institute for Social and Economic Research, paper 2001-22. University of Essex. Foster, J., Greer, J., Thorbecke, F., 1984. A class of decomposable poverty measures. Econometrica 52, 761–765. Hall, P., 1992. The Bootstrap and Edgeworth Expansion. Springer, New York. Hayek, F.A., 1960. The Constitution of Liberty. University of Chicago, Chicago. Enquête sur le budget et la consommation des ménages en Tunisie. Ministère du plan, Tunis.

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Besma Belhadj, Assistant professor, she teaches at ISG of the University of Tunis. Her research interests are applied statistics, Economic Systems; Microeconomics; Mathematical and Quantitative Methods. She is the author of many research studies published in Empirical Economics, Journal of Quantitative Economics, Swiss Journal of Economics and Statistics, Research in Economics. She is the author of many books in Descriptive Statistics, Mathematical Statistics, Inference Statistics, Probability and Mathematics.

Mohamed Limam received a PhD in Statistics from Oregon State University, he teaches at ISG of the University of Tunis, and his research interests are applied statistics, Data Mining and Quality Control. He is the author of many research studies published in JASA, Machine Learning, Expert systems with applications, Communications in Statistics, Quantitative Finance, International Journal of Production Research, Quality and Reliability Engineering International. He is a co-founder of the Tunisian Management Science Society, and founder of the Tunisian Association of Statistics and its Applications.