Fuzzy poverty measurement: Multidimensional and unidimensional indicators

Costa, Michele Working Paper Fuzzy poverty measurement: Multidimensional and unidimensional indicators Quaderni - Working Paper DSE, No. 1156 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Costa, Michele (2020) : Fuzzy poverty measurement: Multidimensional and unidimensional indicators, Quaderni - Working Paper DSE, No. 1156, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/6562 This Version is available at: https://hdl.handle.net/10419/245897 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ ISSN 2282-6483! ! ! ! ! ! Fuzzy poverty measurement: multidimensional and unidimensional indicators Michele Costa Quaderni - Working Paper DSE N°1156 ! ! Fuzzy poverty measurement: multidimensional and unidimensional indicators Michele Costa Abstract This paper proposes a comparison between multidimensional and unidimensional poverty indicators. Sets of poor units identified by traditional head count ratio, fuzzy unidimensional and fuzzy multidimensional indices are compared by means of a rank correlation analysis. The robustness of the comparison is ensured by a simulation study, which allows to address several issues related not only to fuzzy sets based methods, such as the subjective choice of membership to the poor set, but also to the multidimensional measurement, such as the effect of the weighting system. Our results stress that the unidimensional indicators provide partial information on poverty condition. Key words: Multidimensional poverty index, Fuzzy indicators, Simulation study JEL classification: D63; I32; C1 Michele Costa Department of Economics, University of Bologna, e-mail: [email protected] 1 2 Michele Costa Non-technical summary Poverty is a multidimensional condition, not exclusively related to an insufficient provision of income or wealth, but the result of the contemporaneous occurrence of many factors. Multidimensional poverty is coherent with the concept of social exclusion, which is the target of modern welfare policies and which also represents a key element within the debate on alternative welfare measures to GDP. Notwithstanding the general consensus about the multidimensional view of poverty, its quantitative measurement is frequently operated on the basis of a single indicator, usually the income, thus implying that the empirical assessments of unidimensional and multidimensional approach generally coincide. In this case, the multidimensional approach would be an elegant and theoretically useful development, without the need to implement it empirically. Since the unidimensional indicator is extremely simpler and faster to calculate, if it is also fully informative, it represents a suitable solution for the measurement of poverty. When, on the other hand, the two approaches indicate two different sets of poor units, it is important to evaluate the different information contained in the two sets. A relevant implication of the multidimensional approach is the opportunity to divide total population not simply into just two subgroups, poor and non poor, as it is done with poverty lines, but to detect a multiplicity of populations subgroups, each related to different poverty levels, gradually moving from the completely poor to the absolutely non poor condition. A simple but effective method to detect different degrees of poverty, overcoming the dichotomy poor and non poor, is given by the fuzzy sets. Within fuzzy sets based methods it is possible to derive and to compose both multiple unidimensional and also multidimensional indicators, thus allowing a more general and flexible approach to poverty measurement. In this work we aim to address some issues related to the choice of the degrees of membership to the poor of the different population subgroups: this choice belongs to the researchers and, therefore, is frequently view as arbitrary and questionable. We propose a simulation study in order to assess the effects on the set of poor units of different degrees of membership and to evaluate the robustness of the fuzzy poverty indicators. We implement a rank correlation analysis able to provide an exhaustive comparison between the sets of poor units defined by the different methods. The result of rank correlation analysis allow to demonstrate that the two approaches define two different sets of poor households, and we also show how this difference increases as poorer population subgroups are targeted. Overall our results suggest not only two different theoretical frameworks, but also mismatched empirical findings, with the unidimensional indicator providing only partial information on poverty condition. Relying on our conclusions, any socio-economic policy to reduce poverty developed on the basis of income information is likely to no achieve its proposed goals, being addressed to socioeconomic units which are, in effect, non-poor. Only in the framework of the multidimensional approach it is possible to correctly individuate the set of the poor and to formulate actions able to reduce poverty. Fuzzy poverty measurement: multidimensional and unidimensional indicators 3 1 Introduction Poverty is a multidimensional condition, not exclusively related to an insufficient provision of income or wealth, but the result of the contemporaneous occurrence of many factors [12], [15]. Multidimensional poverty is coherent with the concept of social exclusion [1] [14], which is the target of modern welfare policies and which also represents a key element within the debate on alternative welfare measures to GDP. Notwithstanding the general consensus about the multidimensional view of poverty, its quantitative measurement is frequently operated on the basis of a single indicator, usually the income, thus implying that the empirical assessments of unidimensional and multidimensional approach generally coincide. In this case, the multidimensional approach would be an elegant and theoretically useful development, without the need to implement it empirically. Since the unidimensional indicator is extremely simpler and faster to calculate, if it is also fully informative, it represents a suitable solution for the measurement of poverty. When, on the other hand, the two approaches indicate two different sets of poor units, it is important to evaluate the different information contained in the two sets [5]. Our main contribution is to compare the sets of poor units defined by traditional unidimensional and multidimensional approach. For this purpose we develop a rank correlation analysis, able to investigate in depth the similarities between the different indicators. A relevant implication of the multidimensional approach is the opportunity to divide total population not simply into just two subgroups, poor and non poor, as it is done with poverty lines, but to detect a multiplicity of populations subgroups, each related to different poverty levels, gradually moving from the completely poor to the absolutely non poor condition. Only 2 subgroups can represent a severe limit with just one inequality factor, but they are likely to be inadequate by using many inequality factors. A simple but effective method to detect different degrees of poverty, overcoming the dichotomy poor and non poor, is given by the fuzzy sets. Within fuzzy sets based methods it is possible to derive and to compose both multiple unidimensional and also multidimensional indicators, thus allowing a more general and flexible approach to poverty measurement. In this work we also aim to address some issues related to the choice of the degrees of membership to the poor of the different population subgroups: this choice belongs to the researchers and, therefore, is frequently view as arbitrary and questionable. The weakness implied by subjective choices refers to their influence on the stability of the results, that is, on the robustness of the method. Despite the variety of studies on multidimensional poverty, analyses on their robustness are still at a preliminary stage. Literature offers some interesting papers, among the others Alkire and Santos [2] analyze the effects on robustness related to a wide variety of elements, while Duclos et al. [10] develop a meaningful proposal in the framework of stochastic dominance. In the following we propose a simulation study in order to 4 Michele Costa assess the effects on the set of poor units of different degrees of membership and to evaluate the robustness of the fuzzy poverty indicators. Our purpose is to contribute to the theory of multidimensional poverty measurement by providing evidence in favor of fuzzy sets methods which we suggest as an efficient and rigorous solution. A second aim of the simulation study is to evaluate the effects of different sets of weights, first on the final indicator, and, most importantly, on the definition of the set of poor units. We implement the rank correlation analysis by referring also to the simulated indicators, thus providing an exhaustive comparison between the sets of poor units defined by the different methods. Besides the methodological aspects, our results are also interesting for applied poverty analysis and for poverty reduction policies, where the focus is the correct definition of the set of poor units. The paper is organized as follows. Section 2 critically introduces the basic of fuzzy poverty indicators, Section 3 illustrates the Monte Carlo study developed to assess the properties of fuzzy indicators, Section 4 compares poor sets defined by traditional, fuzzy unidimensional and fuzzy multidimensional indicators, Section 5 presents a case study on Italian data and Section 6 concludes. 2 Fuzzy poverty indicators Many Authors proposed, generalized or applied fuzzy sets based methods to the poverty analysis, among the others [7], [4], [8], [13]. In the variety of the meaningful contributions, we make a specific reference to the totally fuzzy approach by Cerioli and Zani [6] and to the relative fuzzy approach by [7]. Let be Athe set of the nunits {a1,a2,...,an},Bthe subset of the poor units, and µ(X)an indicator function, which determines the degree of membership of the i-th unit to the set Bwith respect to a certain attribute X. The indicator function transforms the achievement of a unit with respect to Xto the deprivation of that unit. A low or null achievement implies a maximum deprivation, that is µ(X) = 1, while a high achievement leads to an absence of deprivation, that is µ(X) = 0. Traditionally µ(X)can assume only two values, 1 for the poor units and 0 for the non poor units, and Xis the (equivalent) income; the well known head count ratio His the most famous example of this case, obtained when the population is divided in two non overlapping subgroups, poor and non poor, on the basis of a poverty line zwhich uniquely defines the subset of poor units B: µ(X) = µ1with µ1=1 if X<z µ2with µ2=0 if X≥z(1) and B=[(ai|µ(X)i=1). We obtain the head count ratio Has the ratio between the number of poor and the size of the population Fuzzy poverty measurement: multidimensional and unidimensional indicators 5 H= n ∑ i=1 µ(X)ini/ n ∑ i=1 ni(2) where µ(X)iand niare the value of the indicator function and the sample weight of the i-th unit, respectively. The presence of only two groups and the use of a poverty line zintroduce a high rigidity in the analysis, since, given an εas small as desired, any unit with income xi=z−εis considered as a poor unit, while a unit with income xi=z+εis classified as a non poor unit. In order to overcome this limit, which also strongly affects policies and actions aimed to poverty reduction, it is necessary a more flexible approach, such as the fuzzy method, able to allow more than two groups. 2.1 Fuzzy unidimensional poverty indicators Within the fuzzy sets, µ(X)is no longer only equal to 0 or 1, but ranges between 0 and 1, thus allowing intermediate or partial membership to B: if µ(X) = 1 and µ(X) = 0 still indicate a poor and a non poor unit, respectively, values as µ(X) = 0.8 and µ(X) = 0.2 refer to an almost poor and to an almost non poor unit, and µ(X) = 0.5 to a neither poor or non poor unit. The first problem encountered in the framework of fuzzy indicators concerns the choice of the number of groups to consider: while, in the classic case, two groups, poor and non-poor, represent a forced choice, moving to k>2 groups A1,A2,...,Ak with A=∪Ajand Aj∩Ah=0∀i,j, the choice of kdepends on the preferences of the researchers. In order to obtain the groups, the possible outcomes of X, ranging between Xmin and Xmax, are divided in kintervals, one for each of the kgroups; the intervals are defined by a set of (k−1)values {X1,X2,...,Xk−1}, chosen by the researchers with Xmin <X1<X2< ... < Xk−1<Xmax. The poverty attribute Xcan be a polytomous or a continuous variable (such as educational level or income) or also a composite indicator specifically derived by the researchers combining different variables (such as ownership of the dwelling, number of rooms and presence of toilet). The second issue related to fuzzy indicators refers to the necessity to associate to each group a value of the indicator function, which therefore assumes kvalues {µ1,µ2,...,µk}. In this way the indicator function takes on a more complex form than in (1) and can be represented as µ(X) =            µ1with µ1=1 if X∈[Xmin,X1] µ2if X∈[X1,X2] ... µk−1if X∈[Xk−2,Xk−1] µkwith µk=0 if X∈[Xk−1,Xmax]. (3) 6 Michele Costa A fuzzy unidimensional index evaluates the poverty condition with respect to a certain attribute Xand is obtained as in (4): I(X) = n ∑ i=1 µ(X)ini/ n ∑ i=1 ni.(4) where the main innovation is that µ(X)can also assume intermediate values between 0 and 1. The head count ratio His a special case of I(X), obtained when k=2 and the indicator function can be only 0 or 1. The fuzzy unidimensional index I(X), as the head count ratio, ranges between 0 and 1, and indicates the fraction of poor units on the total. The fuzzy approach does not possess a poverty line partitioning total population into poor and non-poor; in order to identify poor units, it is sufficient to have the list of the µ(X)iin decreasing order and to obtain its cumulative distribution F(µ(X)): if F(µ(X)i)<I(X)we have a poor household, while F(µ(X)i)≥I(X)indicates a non-poor household. The subset of poor units Bis given by the union of the subgroups Ajassociated to the highest values of µ(X)from A1up to the subgroup Alto which belongs the unit which fulfills the condition F(µ(X)i) = I(X): B= l [ j=1 Aj. Since this unit is usually not the last of its group, also in the unidimensional fuzzy framework we face a awkward situation: to the same subgroup Albelong poor units up to the i-th and non poor units from the i-th. A finer classification with respect to only two cases, 0 and 1, is a relevant improvement, but the choice of the intermediate values depends on the individual researchers, thus introducing into the analysis a certain degree of subjectivity and, potentially, of uncertainty and unstability. Overall, in the analysis of fuzzy unidimensional indicators, we face two sources of subjectivity, the choice of kand of the values {µ2,...,µk−1}and it is therefore relevant to assess the impact of both these elements on the final indicator. 2.2 Fuzzy multidimensional poverty indicators Moving from the unidimensional to the multidimensional framework requires to identify the different inequality factors and to define their number m, thus obtaining the structural economic, demographic and social factors {X1,X2,...,Xm}related to some form of social exclusion and able to describe and interpret the poverty condition. Furthermore, within a multidimensional framework, it is essential to derive a weighting system which allows to rank the minequality factors. Fuzzy poverty measurement: multidimensional and unidimensional indicators 7 The presence of minequality factors leads to introduce a second subscript j, where µ(Xj)idenotes the degree of membership to Bof the i-th unit with respect to the j-th inequality factor. Furthermore, minequality factors imply mindicator functions µ(Xj) = (µj1,µj2,...,µjkj)j=1,...,m where kjis the number of groups identified for the j-th inequality factor. The result of the matching between the nunits and the mindicator functions is a matrix M(X)with nrows and mcolumns, where the generic element represents the degree of membership to the subset of poor units for the i-th unit with respect to the j-th factor. By columns of the matrix M(X), we derive the unidimensional poverty indices I(X1),I(X2),...,I(Xm)where I(Xj) = n ∑ i=1 µ(Xj)ini/ n ∑ i=1 ni(5) measures the poverty level, that is the fraction of poor units, with respect to the j-th inequality factor. By rows of M(X), we obtain a multidimensional index for the i-th unit, I(ai), which measures the multidimensional degree of membership to the subset of poor units for the i-th unit. I(ai)is obtained as a weighted sum of the µ(Xj)i I(ai) = m ∑ j=1 µ(Xj)iwj/ m ∑ j=1 wj. and ranges between 0, when the i-th unit is considered non-poor with respect to all minequality factors, and 1, when the i-th unit is poor with respect to all mfactors. 2.2.1 The weighting structure The weights wjrepresent the distinctive feature of the multidimensional indicators and measure the intensity of deprivation and social exclusion related to the j-th inequality factor: the underlying guideline is that a factor not possessed by any unit has no effect on the social exclusion, while, on the contrary, if all the units but a few possess the factor, it represents a relevant source of inequality and social exclusion. A weighting system consistent with that is [6] wCZ j =log(n/ n ∑ i=1 µ(Xj)ini)(6) where wjis equal to 0 for µ(Xj)i=1, i=1,...,n, while wjincreases for decreasing ∑n i=1µ(Xj)i. 14 Michele Costa Table 1 Mean µ, standard deviation σ, skewness γ1and kurtosis γ2indices for 10000 simulated unidimensional fuzzy indicators I(X)by k, number of subgroups, and h, minimum distance between two consecutive values of µ(X). h k =3k=4k=5 µ σ γ1γ2µ σ γ1γ2µ σ γ1γ2 0.05 0.50 0.08 0.01 -1.16 0.48 0,07 -0.38 -0.50 0.47 0.05 -0.59 0.02 0.10 0.50 0.08 0.00 -1.19 0.48 0,05 -0.39 -0.46 0.48 0.04 -0.61 0.05 0.20 0.50 0.06 0.01 -1.14 0.49 0,03 -0.37 -0.42 0.49 0.01 -0.45 0.01 As h, the minimum distance between two consecutive values of µ(X), increases, a decreasing standard deviation is observed, as expected, that is a lower variability of the simulated unidimensional fuzzy indices I(X). The variability of the simulated I(X)is, as intuitively, also negatively affected by the number of subgroups k, which also influence the skewness γ1of the I(X): for k=3 we get a symmetric distribution, while for k>3 the scheme (10) leads to skewed distributions, with a fat tail for smaller values, i.e. for richer units. For k=3 there is an almost uniform distribution, while, as kincreases, there is both a decrease in the importance of the tails, and the presence of negative skewness. In the following, we exploit the potential of scheme (10) with the aim of carrying out an in-depth analysis of the fuzzy indicators. 5.3 Unidimensional poverty indicator The second step of our results concerns to the unidimensional poverty indicators, where we compare the traditional head count ratio to both a fuzzy income-based indicator and simulated fuzzy indicators. We use the OECD equivalence scale and a poverty line zbeing the 60% of the median ¯xme of the equivalent income: z=0.6∗¯xme. An income based fuzzy poverty indicator, instead to divide total population in only two subgroups, is based on k>2 subgroups: for example µ(X1) =        1.0 if x1i<0.4 ¯x1me 0.9 if 0.4 ¯x1me ≤x1i<0.6 ¯x1me 0.5 if 0.6 ¯x1me ≤x1i<0.8 ¯x1me 0 if 0.8 ¯x1me ≤x1i (15) In (15) we allow a greater flexibility with respect to (1): under the poverty line z we assume two classes, introducing a difference among the poor, and also above the poverty line we differentiate among the non poor, assigning a positive membership (µ13 =0.5) to units with income below 0.8 ¯x1me. In order to vary the scores of the indicator function µ(X1), we run a Monte Carlo experiment, by using the scheme (10) with k=4 and h=0.1: Fuzzy poverty measurement: multidimensional and unidimensional indicators 15 µ(X1) =        µ11 with µ11 =1 µ12 with µ12 ∈[µ13 +0.1,0.9] µ13 with µ13 ∈[0.1,0.45] µ14 with µ14 =0 (16) Given the µ(X1), we derive the fuzzy unidimensional indicator I(X1)as in (4), and we compare the results related to (15) and to (16), thus investigating the effects of µ(X)on the poverty indicator. Table 1 reports in the first column the head count ratio, equal to 0.202, in the second the fuzzy indicator based on (15), equal to 0.264, while in the following columns are summarized the results of 100000 random extractions of µ(X1)as in (16). We report, in the first row some values of I(X1), ranging from 0.14 to 0.25, and from the third to the fifth row the mean of the respective simulated µ(X). It is possible to observe how extreme values of I(X1)occur only in correspondence with unlikely values of µ(X):I(X1) = 0.14, for example, requires µ12 =0.30, which is clearly an inadmissible value for incomes below the poverty line. When, however, µ(X)remains within acceptable ranges, the index I(X1)does not show significant changes. Table 2 Head count ratio with µ(X)=(2), fuzzy poverty indicator with µ(X)=(5) and fuzzy poverty indicator with simulated µ(X)=(6), Italian households 2016 I(X1)0.202 0.264 0.142 0.161 0.181 0.201 0.219 0.238 0.252 µ(X1)(1) (15) (16) (16) (16) (16) (16) (16) (16) µ11 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 µ12 1.00 0.90 0.300 0.420 0.535 0.650 0.725 0.813 0.879 µ13 0 0.50 0.133 0.170 0.212 0.257 0.331 0.389 0.434 µ14 0 0 0 0 0 0 0 0 0 We repeat the same procedure also for the other indicators, still obtaining similar results: the scores of the indicator functions µ(Xj)and the values of the unidimensional fuzzy indicators I(Xj)are validated through a Monte Carlo study that ensures their robustness. The adopted values of the indicator functions µ(Xj)are shown in the Appendix for the entire set of indicators used in the paper, without adding the simulations results, which would only repeat the scheme presented in this paragraph for the first indicator. Overall, we derive a set of 10 unidimensional fuzzy indicators which provide a powerful basis for the development of the multidimensional index. 16 Michele Costa 5.4 Multidimensional poverty indicator The multidimensional fuzzy poverty index I, obtained as (7), is a function of unidimensional fuzzy indicators and weights. Our first goal is to evaluate the effects of the weighting system on the multidimensional indicator, for this purpose we analyze three different types of weights: the case of equal weights we, the structure adopted by the United Nations wUN , a system based on social exclusion and proposed by Cerioli and Zani wCZ. In Table 3 are reported, together with the 10 unidimensional indices I(X), the results for the three (normalized) weighting systems and the related multidimensional fuzzy poverty index Ias in (7). Table 3 Unidimensional poverty indicators, weights and multidimensional poverty index - Italian households 2016. unidimensional indices multidim. I II III IV V VI VII VIII IX X index I(X)0.264 0.037 0.163 0.341 0.184 0.033 0.128 0.094 0.329 0.306 we0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.188 wUN 0.111 0.111 0.111 0.167 0.167 0.067 0.067 0.067 0.067 0.067 0.199 wCZ 0.069 0.170 0.094 0.056 0.088 0.176 0.106 0.122 0.058 0.061 0.144 The unidimensional poverty indicators I(X)in the first row of the Table indicate the fraction of poor units on the total according to the different inequality factors, ranging from 34.1% of the fourth indicator (educational achievement of the household head and his/her father) to 3.3% 3per cent of the sixth (gender, age, occupational status of the household head). Consistent with the intuition underlying the weights related to social exclusion, wCZ, the higher weights correspond to the smaller indices. The last column of the table shows the multidimensional index I, which is equal to Ie=0.188 by using equal weights, to IUN =0.199 by referring to the United Nations weighting system, and to ICZ =0.144 on the basis of the weights (6) related to social exclusion. Results provided by Table 3 allow interesting insights on the poverty structure. Multiplying the unidimensional indices I(Xj)in the first row by the weights and dividing by the multidimensional index Iin the last column, we obtain the quantity I(Xj)wj/I, that is the influence of the single inequality factors on I. The ratios I(Xj)wj/I, together with the related indicator, are shown in descending order in Table 4. The fourth indicator, educational achievement of the household head and his/her father, represents the main source of inequality, followed by a group of four indicators: dimension of the residence (IX), job status of the household head and the other components of the household (X), education of the spouse (V), equivalent income (I). The last 5 indicators in order of relevance are the value and the occupancy title of the household residence (III), the geographical area of residence Fuzzy poverty measurement: multidimensional and unidimensional indicators 17 (VII), the household size and number of bathrooms of the residence (VIII), the ratio between the number of the household members perceiving an income and the household size (II), gender, age and occupational status of the household head (VI). Equal weights weand social-exclusion related weights wCZ suggest the same ranking of the inequality factors, while on the basis of United Nations weights wUN we observe some differences from the second to the fifth position. Overall, in the second panel of Table 4 we observe how the living conditions domain accounts for over 45 per cent of the inequality for weand wCZ , while for wUN the main source of inequality is the educational domain. Table 4 Ratios I(Xj)wj/Iand poverty indicator - Italian households 2016. I(Xj)we j/IeI(Xj)wUN j/IUN I(Xj)wCZ j/ICZ 0.181 IV 0.286 IV 0.133 IV 0.175 IX 0.155 V 0.133 IX 0.163 X 0.148 I 0.130 X 0.141 I 0.111 IX 0.127 I 0.098 V 0.103 X 0.113 V 0.087 III 0.091 III 0.107 III 0.068 VII 0.043 VII 0.094 VII 0.050 VIII 0.032 VIII 0.080 VIII 0.020 II 0.021 II 0.044 II 0.018 VI 0.011 VI 0.040 VI 0.474 living 0.441 education 0.477 living 0.279 education 0.299 living 0.277 wealth 0.247 wealth 0.260 wealth 0.246 education In order to address the robustness issue for the multidimensional index, we further develop the simulation study, randomly extracting the indicator function µ(Xj) as in (10) for the ten fuzzy indicators and obtaining the multidimensional index on the basis of the three weighting system used in the paper. The results related to 30000 simulated indices are illustrated in Figure 1 and indicate how weights have relevant effects on the multidimensional index: the use of wCZ leads to the smallest values of the index I, while the highest values are linked to wUN , thus confirming the pattern already observed in Table 3. Besides the unidimensional fuzzy indices I(Xj)and the multidimensional I, we also derive the list obtained from the unidimensional approach and the list resulted from the multidimensional approach, thus allowing a comparison between the two approaches. 18 Michele Costa Fig. 1 Frequency histogram of 30000 simulated multidimensional fuzzy poverty indices by weights wCZ ,weandwUN . 5.5 Unidimensional and multidimensional comparison In order to compare the unidimensional and the multidimensional approaches, we start with Tables 5, which reports a subset of observations for the four variables of interest: the household equivalent income X1i, its rank ui, the multidimensional poverty index I(ai), its rank vi. Table 5 Household equivalent income X1i, its rank ui, multidimensional poverty index I(ai), its rank viItalian households 2016. uiX1iviIiviIiuiX1i 1 -6763 3785 0.109 1 0.779 6 0 10 0 7 0.706 10 0.667 336 4960 25 0 128 0.463 25 0.605 87 1429 50 0 1556 0.215 50 0.550 424 5840 75 802 2890 0.145 75 0.515 21 0 100 1667 301 0.383 100 0.493 960 8820 7321 57052 5156 0.061 7321 0.00 6940 37333 7346 62854 7160 0.012 7346 0.00 7182 45859 7371 69769 7402 0.00 7371 0.00 7279 52070 7396 85375 7407 0.00 7396 0.00 7343 62000 7411 131831 7414 0.00 7411 0.00 7406 113684 7421 218805 7421 0.00 7421 0.00 7421 218805 First, we rank the observations on the basis of ui, with the left part of Table 5 showing X1i,ui,I(ai), and vifor a subset of selected observations. Observations presenting the same value of X1iare sorted by decreasing size of the multidimensional indicator I(ai). By comparing column 2 to column 4 in Table 5, we can observe a strong similarity between the highest ranks, while, on the contrary, the lowest ranks are quite different. Fuzzy poverty measurement: multidimensional and unidimensional indicators 19 Our second step is to rank the observations on the basis of vi, with the right part of Table 5 showing X1i,ui,I(ai), and vifor the same subset of selected observations. Analogously to the previous case, observations with the same value of I(a) are sorted by increasing size of X1i. As before, the highest ranks are almost identical, while the lowest ranks are quite different. It is also interesting to note that, for the lowest ranks, the difference between the values uiand viis clearly higher in the left part of Table 5 than in the right part. Further information on the comparison between the two approaches can be provided by the similarity indices Suand Smintroduced in Section 4 and reported in Table 6. By referring to wCZ, for the whole population, Su=0.559 indicates that only 55.9% of the units with income below the poverty line are also poor according to the multidimensional approach, while Sm=0.889 indicates that 88.9% of the poor units on the basis of Iare also poor on the basis of income. Even analyzing particular subsets, a difference between the two methods is confirmed: for the poorest 5% of the population, Su=0.775 implies that over 20% of the poorest on the basis of income are not poor in the multidimensional approach. The values of the similarity indices Suand Smobtained by using weor wUN confirm the previous difference between the indices, with Susteadily lower than Sm. Table 6 Similarity indices Suand SmItalian households 2016. SuSmSuSmSuSm αwewUN wCZ 0.05 0.841 0.978 0.771 0.948 0.775 0.975 0.10 0.767 0.941 0.686 0.887 0.667 0.941 0.15 0.717 0.908 0.656 0.811 0.621 0.889 1.00 0.671 0.867 0.616 0.754 0.559 0.889 From Tables 5 and 6 we get a first indication that unidimensional and multidimensional approaches seem to define two different sets of poor households: a more complete analysis, carried out on the basis of rank correlation, is illustrated in the following. 5.6 Rank correlation analysis The highlight of the comparison between the unidimensional and the multidimensional approach is the rank correlation analysis, both overall, and in reference to subsets of the poorest. The first results refer to the correlation between the multidimensional poverty index Iand the equivalent income X1. Table 7 shows the indices of Bravais-Pearson r, Kendall τ, Spearman Sand Gini GPearson; the first column reports selected values of the cumulative distribution of I, where Iis in decreasing order, and the 20 Michele Costa following columns show the values of the four correlation indices with respect to the three weighting systems used. The last rows of the Table contains the indices for the whole population (F(I) = 100), while the first row refers to the poorest 5% of total population (F(I) = 5). Moving from the bottom to the top of the Table it is clearly observable a strong decrease in rank correlation, thus suggesting that multidimensional and unidimensional approaches define two different subsets of poor units. Kendall’s and Gini’s measures give quite similar results and report a lower correlation between uiand vithan Bravais-Pearson and Spearman indices. The United Nations weighting system indicates the lowest values, especially for the poorest subgroups, while weand wCZ provide substantially similar results. All indices, however, clearly signal a decreasing rank correlation for decreasing values of F(I). Table 7 Rank correlation between multidimensional poverty index Iand equivalent income X1Italian households 2016. wewUN wCZ F(I)rτS G r τS G r τS G 5 0.227 0.178 0.334 0.168 0.157 0.117 0.285 0.119 0.229 0.164 0.328 0.159 10 0.383 0.232 0.504 0.259 0.299 0.153 0.439 0.199 0.375 0.236 0.492 0.259 25 0.484 0.350 0.568 0.353 0.387 0.258 0.481 0.283 0.469 0.347 0.555 0.339 50 0.691 0.514 0.705 0.552 0.603 0.433 0.618 0.471 0.677 0.504 0.689 0.543 100 0.775 0.589 0.768 0.654 0.725 0.534 0.717 0.601 0.765 0.577 0.755 0.649 Also in the case of correlation analysis, the robustness of the results is assessed by analyzing the correlation between simulated multidimensional indices Iand equivalent income X1. Table 8 summarizes the results of 30,000 simulated indices (10,000 for each of the 3 weighting systems), showing the average of the correlations obtained. The resulting picture closely resembles that outlined in Table 7, with the confirmation of all the main results and, above all, of the decreasing correlation for decreasing values of F(I). Table 8 Rank correlation between simulated multidimensional poverty index and equivalent income X1, average of 10000 simulations - Italian households 2016. wewUN wCZ F(I)rτS G r τS G r τS G 5 0.202 0.151 0.314 0.147 0.140 0.097 0.274 0.098 0.224 0.158 0.329 0.161 10 0.384 0.230 0.505 0.262 0.318 0.165 0.455 0.214 0.401 0.255 0.513 0.282 25 0.499 0.360 0.582 0.364 0.418 0.283 0.508 0.305 0.520 0.387 0.600 0.377 50 0.643 0.469 0.659 0.511 0.563 0.400 0.580 0.435 0.671 0.496 0.686 0.536 100 0.748 0.560 0.739 0.630 0.701 0.512 0.692 0.577 0.757 0.569 0.747 0.642 Fuzzy poverty measurement: multidimensional and unidimensional indicators 21 In the last step of the rank correlation analysis we compare the rankings derived by the simulated multidimensional indices and by I. As before, Table 9 summarizes the results of 30,000 simulated indices (10,000 for each of the 3 weighting systems), showing the average of the correlations obtained. From Table 9 we can observe extremely high values, which suggest that the subsets of poor units identified by the simulated indices are almost coincident, lending further support to the robustness of the multidimensional fuzzy indicator. Table 9 Rank correlation between simulated multidimensional poverty index and multidimensional poverty index I, average of 10000 simulationsItalian households 2016. wewUN wCZ F(I)rτS G r τS G r τS G 5 0.974 0.847 0.702 0.830 0.979 0.871 0.654 0.833 0.979 0.869 0.730 0.849 10 0.973 0.846 0.621 0.811 0.980 0.871 0.612 0.828 0.979 0.867 0.660 0.838 25 0.970 0.842 0.697 0.824 0.979 0.866 0.739 0.861 0.976 0.862 0.701 0.834 50 0.973 0.852 0.913 0.884 0.981 0.874 0.929 0.905 0.982 0.881 0.932 0.909 100 0.981 0.867 0.967 0.932 0.988 0.893 0.947 0.943 0.988 0.895 0.997 0.959 Overall rank correlation analysis results point out to two different subsets of poor units derived from the unidimensional and the multidimensional indicators, suggesting that the two approaches differ not only theoretically, but also as regards data analysis and empirical findings. 6 Conclusions The comparison between unidimensional and multidimensional approach to poverty measurement is carried out by means of a rank correlation analysis, aimed at evaluating the intersection between the set of poor units indicated by the two methods. The robustness of the comparison is analyzed through a Monte Carlo study, with particular emphasis on the issues related to the choice of µ(X)and their effects on the set of poor units. The simulated values suggest an extremely satisfactory robustness of the fuzzy poverty indicators, thus overcoming the objection related to the subjectivity of the choice of µ(X). Furthermore, the simulation study proves to be a particularly powerful tool to investigate further characteristics of the multidimensional fuzzy poverty indicators, such as the weighting system. The main drivers of poverty, identified and ranked on 2016 Italian household data, are educational achievement, dimension of the residence, job status and equivalent income, and they remain remarkably stable over the simulations. A key point in poverty analyses is not to establish how many are the poor households, but who are they. We contribute to poverty measurement by comparing the 22 Michele Costa sets of poor units identified by traditional unidimensional, fuzzy unidimensional and fuzzy multidimensional indicators. The result of rank correlation analysis allow to demonstrate that the two approaches define two different sets of poor households, and we also show how this difference increases as poorer population subgroups are targeted. Overall our results suggest not only two different theoretical frameworks, but also mismatched empirical findings, with the unidimensional indicator providing only partial information on poverty condition. Relying on our conclusions, any socio-economic policy to reduce poverty developed on the basis of income information is likely to no achieve its proposed goals, being addressed to socioeconomic units which are, in effect, non-poor. Only in the framework of the multidimensional approach it is possible to correctly individuate the set of the poor and to formulate actions able to reduce poverty. Appendix Table 10 Indicator function µ(X1)for the I indicator: Household equivalent disposable income µ(X1) x1i<0.4¯x1me 1 0.4¯x1me ≤x1i<0.6¯x1me 0.9 0.6¯x1me ≤x1i<0.8¯x1me 0.5 0.8¯x1me ≤x1i0 References 1. Aaberge, R., Brandolini, A.: Multidimensional poverty and inequality. Bank of Italy Working Papers, 976, 1–133 (2014) 2. Alkire, S., Santos, M.E.: Measuring acute poverty in the developing world: robustness and scope of the multidimensional poverty index. World Development, 59, 251–274 (2014) 3. Andreoli, F., Zoli, C.: From unidimensional to multidimensional inequality: a review. Metron, 78, 5–42 (2020) 4. Betti, G., Cheli, B., Gambini, R.: A statistical model for the dynamics between two fuzzy states: theory and an application to poverty analysis. Metron, 62, 391–411 (2004) 5. Bossert, W., Chakravarty, S. R., DAmbrosio, C.: Multidimensional poverty and material deprivation. Review of Income and Wealth, 59, 29–43 (2013) 6. Cerioli, A., Zani, S.: A fuzzy approach to the measurement of poverty. In Dagum, C., Xenga, M.: Income and wealth distribution, inequality and poverty. Springer, Berlin, 272–284 (1990) 7. Cheli, B., Lemmi, A.: A totally fuzzy and relative approach to the multidimensional analysis of poverty. Economic Notes, 24, 115–134 (1995) Fuzzy poverty measurement: multidimensional and unidimensional indicators 23 Table 11 Indicator function µ(X2)for the II indicator: Ratio between the number of the household members perceiving an income and the household size Household size Value of the ratio µ(X2) 101 110 201 2≥0.50 0 301 3≥0.33 0 401 4 0.25 0.4 4≥0.50 0 501 5 0.20 0.5 5≥0.40 0 601 6 1/6 0.75 6 2/6 0.25 6≥0.50 0 ≥7 0 1 ≥7 0.14 - 0.29 0.75 ≥7 0.300.58 0.25 ≥7>0.58 0 Table 12 Indicator function µ(X3)for the III indicator: Occupancy title and value of the household residence Value Occupancy title >150000 75000 - 150000 ≤75000 Owned 0 0 0.5 Rented 0 0.3 1 Occupied under redemtion agreement 0 0.2 0.5 Occupied in usufruct 0 0.2 0.5 Occupied free of charge 0 0.3 1 Table 13 Indicator function µ(X4)for the IV indicator: Educational achievement of the household head and his/her father Head of the household Father None Elementary school J. high school S. high school University None 1 0.6 0.2 0 0 Elementary school 1 1 0.5 0 0 Junior high school 1 1 1 0 0 Senior high school 1 1 1 0.4 0 University degree 1 1 1 0.5 0