A new multiplicative decomposition for the Foster-Greer-Thorbecke poverty indices.

A new multiplicative decomposition for the Foster-Greer-Thorbecke poverty indices. Mª Casilda Lasso de la Vega University of the Basque Country Ana Marta Urrutia University of the Basque Country and Oihana Aristondo University of the Basque Country Draft: June/2006 Correspondence: Mª Casilda Lasso de la Vega, University of the Basque Country, Department of Applied Economics IV, Av. Lehendakari Aguirre, 83, 48015-Bilbao (Spain). E-mail: casilda.lassodelavega@ehu.es. Ana Marta Urrutia, University of the Basque Country, Department of Applied Economics IV, Av. Lehendakari Aguirre, 83, 48015-Bilbao (Spain). E-mail: anamarta.urrutia@ehhu.es. Oihana Aristondo, University of the Basque Country, Department of Applied Economics IV, Av. Lehendakari Aguirre, 83, 48015-Bilbao (Spain). E-mail: oihana.aristondo@ehu.es. The research was carried out in conjunction with the University of the Basque Country Research Project UPV05/117. 1

ABSTRACT This paper explores a unified framework for the study of changes in poverty over time as measured by the Foster-Greer-Thorbecke poverty indices. First we identify a multiplicative decomposition for the indices of this family as a product of the three components which should be involved in every poverty index: the incidence, the intensity and the inequality in the distribution of the income of the poor. Then we establish a connection between this procedure and the methodology developed by Nanak Kakwani (2000) to decompose the change in poverty into the growth and inequality effects. Finally, taking the Spanish Household Budget Surveys (SHB) for 1973/74, 1980/81 and 1990/91 as a basis we show the advantages and possibilities of this framework in regard to completing and detailing information in studies of poverty over time. Key words: Poverty Measurement, Foster-Greer-Thorbecke poverty indices, Multiplicative Decomposition. JEL Classification: D63, I30, I32 2

1. INTRODUCTION. In a seminal article Sen (1976) argued that a poverty index should be sensitive to the number of people below the poverty line, to the extent of the short-fall of the income of the poor from the poverty line and to the exact pattern of distribution of the incomes of the poor. Hence, every poverty measure should be expressed as a function of these three poverty indicators, indicating the incidence of poverty, the intensity of the poverty and the inequality between the poor respectively. When analysing the sources of changes in poverty it is therefore of interest to ask how much of a change in poverty is due to changes in these components and to know if increasing poverty is due to more people becoming poor, or increasing deprivation of the poor, or because income shortfall below the poverty line have become more unequal, or some combination of the above. Thus, poverty changes are more meaningful and easily understandable if poverty indices can be decomposed into their underlying contributing factors. There are in the literature indices for which it is possible to know this decomposition explicitly. For instance, Kakwani (1999) provides examples for several poverty measures, such as Sen s measure (Sen, 1976), Kakwani s measures (Kakwani 1980, 1984, 1999), Watts measure (Watts 1968), Chakravarty index (Chakravarty 1983) and others. Alternative decompositions for the Sen index and the modified Sen index (Shorrocks 1995) are also proposed by Xu and Osberg (2002a, 2002b) and Osberg and Xu (1999). It may be interesting to note that, up to now, only for one index of the Foster, Greer and Thorbecke family (1984), henceforth FGT family, a decomposition in these three components is identified. Moreover, in some applications of poverty it may be appropriate to consider populations classified by characteristics such as age, gender or area of residence. In these cases studies rely heavily on measures which must increase when poverty increases in any subgroup without decrease in poverty elsewhere. In other words, they must meet the subgroup consistent requirement as defined by Foster and Shorrocks (1991). Subgroup inconsistent measures may misguide policy analysis. Although the Sen index, the modified Sen index and their generalizations, such as Blackorby and Donaldson index (Blackorby and Donaldson (1980)) are grounded on a set of well founded and commonly agreed axioms they do not have this property. In turn, Foster 3

and Shorrocks (1991) establish that the FGT class contains the canonical forms of almost all subgroup consistent poverty indices. In this respect it would be helpful and worth-while to examine these measures in terms of their contributing components: incidence of poverty, intensity of the poverty and the inequality between the poor. In this paper we show that the FGT indices may be multiplicatively decomposed. Specifically this paper proves that each index in the FGT family can be viewed as a product of the three commonly used poverty and inequality measures: the headcount ratio, the aggregate income gap ratio and one plus an increasing transformation of the Generalized Entropy index of the income gap of the poor. The main advantage of this new decomposition is that it allows the FGT indices to be decomposed in a different way from those existing decompositions, completing information provided by them and disentangling the contributions of the factors affecting the poverty trends. This straightforward decomposition into three meaningful and familiar poverty indicators can be readily interpreted by policy makers, clarifies the understanding of this family and turns out to be a useful result for empirical comparisons and policy analysis. Moreover, this decomposition may be used jointly with other existing theoretical decomposition formulae in order to quantify how much of change in poverty is due to variations in distributions, and how much is due to growth in average living standard. Economic growth is typically expected to lead to a reduction in the poverty. If economic growth does not operate proportionally, but is instead accompanied by a reduction in inequality, it is likely to be a further reduction in the poverty. The total poverty changes may therefore be decomposed into the growth and redistribution components. A number of recent papers have examined the relative importance of growth versus redistribution in poverty change (Datt and Ravallion (1991), Kakwani and Subbarao (1992), Kakwani (1993), Kakwani (2000), Shorrocks (1999), Kolenikov and Shorrocks (2005). Following the Kakwani axiomatic approach (Kakwani (2000)), which is equivalent to the Shapley approach (Shorrocks (1999) and Kolenikov and Shorrocks (2005)) for this decomposition, we can measure the growth and the redistribution effects in poverty changes as measured by this family. Then the multiplicative decomposition for the FGT indices identified in this paper can be used. We propose going further in it and decomposing the calculated growth and the redistribution effects into their respective 4

effects on the basic poverty indicators composing these poverty indices. In this way, we propose an exact decomposition in which both growth and inequality effects in changes in the FGT poverty index are decomposed in order to evaluate consistently their effects in the basic poverty indicators that compose the FGT index. This paper is structured as follows. The next section presents the notation and the basic definitions used. In Section 3 we identify the new multiplicative decomposition for the FGT poverty indices and provide a unified framework where changes in the incidence, intensity and inequality between the poor on the one hand and growth, and redistribution on the other hand may be jointly analysed. In Section 4 we use the foregoing approach to analysing the poverty in Spain for the period 1973/74 to 1980/81 and 1980/81 to 1990/91 using the Spanish Household Budget Surveys (HBS). Finally, Section 5 offers some concluding remarks. 2. NOTATION AND DEFINITIONS. We consider a population consisting of n 2 individuals. Individual i s income is denoted by y = (0, ), i = 1,2,,n. An income distribution is represented by a i vector y =(y 1, y 2,..., y n) n arranged in increasing order. We let D = represent the set of all finite dimensional income distribution. For any given poverty line z and distribution y D we define as poor all incomes y z. We denote by n= n( y ) and q q( z) = y; the population size and the number of the poor respectively, and by µ(y) the mean income of y. Let be ( ) th poor and = ( g 1,...,gq) i i i n= 1 n g = z y / z the income gap ratio of the i- g the poverty gap ratio vector. We use the vector y to signify the equalised version of y, defined by n( ) = n( ) i=1,..., n n( ) = y. y y and ( ) y =µ y for all i 5

We say that ( y' ) is obtained from ( ) ;z' D change if ( ;z') =λ( ;z for some ( ) ( ) ( y;z D by a relative y' y ) λ > 0; and by an absolute change if y' ;z' = y;z + λ1 ; λ) for some λ > 0, where 1 is an appropriate sized vector of 1 s. Suppose that the population of n individuals is split into J 2 mutually exclusive subgroups with income distribution y = (y, y,..., y ), population sizes j j j j 1 2 n j n n( y, j j = ) numbers of poor ( j j qj = q y; z), and mean incomes µ j =µ ( y ) for all j = 1,2,,J. Let p j be the population share of subgroup j. A poverty index P is a real valued function P:D. Examples include the headcount ratio H ( ;z) = q( z ) /n( ) ( ) ( ) y y; y, the aggregate income gap ratio µ g y;z = z y /qz, the poverty gap ratio which is equal to the product of the 1 i q i headcount ratio and the aggregate income gap ratio, i. e. proposed by Foster et al. (1984), henceforth FGT family, given by Hµ g, and the class of indices q 1 z yi α FGT α ( y;z) = α > 1 (1) n i= 1 z Particularly interesting is to note that when α = 0 the measure FGT 0 is simply the headcount ratio H and when α= 1 FGT1 is the poverty gap ratio, Hµ g. Moreover, Foster and al. (1984) propose a decomposition for FGT2 using the headcount ratio, the aggregate income gap ratio and the squared coefficient of variation. The following properties for a poverty index will be mentioned in the paper. A poverty index P is (i) continuous if, for every z P ( ;z) a function of y on D;. (ii) subgroup consistent if, for every z and for which n y is continuous as ( x) = n( x' ) and n( y) = n( y' ), we have P (, ;z) > P (, ;z) x, x', y, y' D xy x'y' whenever P ( x;z) > P ( x' ;z) and P ( y;z) = P ( y' ;z) ; (iii) additively decomposable if, for every J 2 and any J j 1 J j y D, j=1,,j, P ( y,..., y ;z) = p j P ( y ;z) j1 = 6

A poverty index P corresponds to a concept of relative poverty if it is scale invariant, that is ( y ) ;z D translation invariant, that is from ( ) P ( x' ;z') = P ( x ;z) whenever ( y' ;z') D is obtained from by a relative change; and to a concept of absolute poverty if it is ( x' ) = ( x ) whenever ( ) P ;z' P ;z y' ;z' D is obtained y;z D by an absolute change. Finally the poverty indices P and P are compatible if for all P' ( x;z) P' ( y ;z). z and x, y D, P ( x;z) P ( y ;z) if and only if Each index of the FGT class given in (1) is additively decomposable. In addition, Foster and Shorrocks (1991) establish that the FGT class contains the canonical forms of all continuous, subgroup consistent, relative poverty indices which are compatible with some continuous, subgroup consistent, absolute poverty index. Finally, in the following the inequality indices of Generalised Entropy class, henceforth GE class, (Bourguignon (1979), Shorrocks (1980, 1984), Cowell, (1980), Cowell and Kuga (1981a, 1981b)) will play a role. The GE family is given by: ( y) α 2 (( yi ) 1) n( ) ( i ) 1 i n ( ) ( ) µ α α α 0,1 1 i n = µ α= y 1 i n i µ log yi µ n α= 1 GE Iα log y n 0 (2) 3. DECOMPOSING CHANGES IN POVERTY AS MEASURED BY THE FGT FAMILY. 3.1. Incidence, Intensity and Inequality in the FGT family. The following proposition shows that every FGT index allows a multiplicative decomposition in terms of incidence, intensity and inequality. Following Seild (1988), we consider the headcount ratio as the archetypical measure of the incidence of poverty and the aggregate income gap ratio as the archetypical measure of the intensity of 7

poverty. With respect to the inequality of the poor, an index of the GE family measuring inequality of the income gap ratios of the poor is involved in the decomposition. PROPOSITION 1: For each 1 multiplicative decomposition: ( ) ( ) ( ) α >, FGT ( ; z) ( ) α 2 GE ( ) α ( ) α y satisfies the following ( ) FGT y; z = H y; z µ g y; z 1+ α α I g (3) α Proof. The proof is straightforward after a few lines of standard computations and rearrangements. Q.E.D. This proposition shows that each FGT α poverty index can be decomposed as the product of the headcount ratio, the aggregate income gap ratio to the power of α and one plus the corresponding GE inequality index of poverty gap ratios of the poor multiplied by 2 α α. It may be worth comparing the multiplicative decomposition for the Sen family of poverty indices proposed by Xu and Osberg (2002a, 2002b) and Osberg and Xu (1999) with the decomposition presented in (3). Indeed, the major difference between their decomposition and ours has to do with the inequality term. Thus the role played by the Gini coefficient in their decomposition is replaced here by an index of the GE family. As well known, for parameter values large but positive the GE index is sensitive to what happens to large values of the distribution, in this case to large values of income gap ratios, therefore to the poorest of the poor. From now on we simplify notation unless misinterpretation arises, and we denote * I α * 2 GE Iα = 1+ ( α α) Iα ( g ) the inequality component in the decomposition in (3), that is The multiplicative decomposition for these poverty indices given in (3) is, as usual, a starting point for the derivation of the impact of marginal changes in a given component on overall poverty. Indeed the multiplicative decomposition of these indices 8

can be transformed, through the logarithmic transformation, so that it is additive in a simple form. The marginal effects derived from multiplicative decomposition appear in the following equation α * α = H + (µ g ) Iα (4) FGT + where in general x = lnxt lnx t 1 (xt x t 1) xt 1approximates the percentage change in x. This equation shows that the overall percentage rate of change in poverty can be expressed as the sum of the percentage changes in the headcount ratio, the aggregate income gap ratio and inequality of the income gap ratios of the poor. 3.2. Measuring growth and inequality components of changes in poverty. Recently researchers have been concerned with the decomposition of change in poverty into growth and inequality components. In particular Kakwani (2000) derives a new decomposition method using an axiomatic approach and explores a decomposition of the total change in poverty as the sum of average growth and inequality effects. Actually this procedure gives the same result as the one suggested by Shorrocks (1999) and Kolenikov and Shorrocks (2005) based on the Shapley value. The growth effect measures changes in poverty when mean income varies but inequality remains the same. Similarly the inequality effect measures changes in poverty when inequality changes but the mean income remains unchanged. Bearing in mind the decomposition given in (4) we intend to propose a unified framework where the effects of growth and redistribution on poverty changes are analysed. Following Kakwani (2000) let consider a general poverty measure characterized ( ) as P= P z, µ,l( p) where z is the poverty line, µ is the mean income and L(p) is the Lorenz curve of the distribution. The total proportional change in poverty between period i and j can be written as [ P( z,µ, L (p))] Ln[ P( z,µ, L (p)) P = Ln ] (5) j j i i 9

where and µ and (p) and (p) are, respectively, the mean income and the µ i j L i L j Lorenz curves in years i and j. In addition, total proportional change in poverty is equal to the sum of growth and inequality proportional changes ( P) G ( P) I P = + (6) where the growth effect, is the proportional change in poverty if mean income changes from period i to period j but the Lorenz curve remains unchanged, and the inequality effect, is the proportional change in poverty if the Lorenz curve changes but mean income remains the same. Thus, Kakwani and Pernia (2000) arrive at the following expressions for the growth and inequality effec ( P) G = ( P) I ( P) G 1 Ln P ( z,µ j,l i( p )) Ln P ( z,µ i,l i( p )) + Ln P ( z,µ j,l j( p )) Ln P ( z,µ i,l j( p )) 2 (7) ( P) I = 1 Ln P ( z,µ i, L j( p )) Ln P ( z,µ i, L i( p )) + Ln P ( z,µ j, L j( p )) Ln P ( z,µ j, L i( p )) 2 (8) The above decomposition given in (6) allows us to derive the growth and inequality effect on proportional changes in every index of the FGT class and in each of its three components. Combining equations (4) and (6) we obtain the following equation: ( ) ( ) FGT = FGT + FGT = α α G α I α α * * G I g G g I α G α) I ( ) + ( ) + ( ) + ( ) + ( ) + ( H H (µ ) (µ ) I I (9) In short, in empirical applications the poverty change decomposition may be performed in two steps. The first step consists in decomposing poverty changes into the total change in the headcount ratio, aggregate income gap ratio and income gap inequality, using the new multiplicative decomposition given by equation (4). The 10

second one is to decompose changes in FGT poverty index into growth and redistribution contributions, applying the Kakwani approach and equation (8). Finally, the Kakwani procedure may be applied, of course, to decompose each of these poverty indicators that yield contributions for the growth and redistribution effects given respectively by the first step. 4. AN EMPIRICAL ILUSTRATION. DECOMPOSING CHANGES IN POVERTY IN SPAIN 1973/74-1980/81 AND 1980/81-1990/91. The methodology developed in this paper will now be applied to analyse changes in poverty in Spain during the 1970s and 1980s 1. We make use of Spanish Household Budget Surveys (SHB) for 1973/74, 1980/81 and 1990/91. The variable we use is the equivalent net expenditure, defined as the total expenditure, monetary expenditure plus self-consumption, self-supply, free meals, in-kind salary and imputed rents for house ownership, minus the investment expenditures and transfers to other households and Institutions. The equivalence scale we use for each household is n 0.5 where n is the number of individual of the household. The unit of study is the individual so the equivalence net expenditure is assigned to every person in the household and then for each person we apply the elevation factor of the household they belong. The poverty line we use in this illustration is 50% of the mean income of the base year 1980/81 and we transform all bases into 1981 s prices. Table1 presents the FGT0, FGT1, FGT2 and FGT3 indices in Spain for the years 1973, 1980 and 1990. The aggregate income gap ratio, I 2 GE GE income gaps of the poor, and are also presented in this table. I 3 µ g, and the inequality of the 1 Del Río and Ruiz-Castillo (2001) apply the methodology developed by Jenkins and Lambert (1997, 1998a, 1998b) to the study of the evolution of poverty in Spain in the same period. 11

Table1. FGT 0, FGT 1, FGT 2 and FGT 3 decomposition in incidence, intensity and inequality. Years FGT 0 =H FGT 1 =H µ g FGT 2 FGT 3 µ g I GE 2 I GE 3 1973/74 21,9% 0,014 0,027 0,014 0,064 14,78 40,87 1980/81 16,2% 0,007 0,018 0,009 0,044 28,59 111,64 1990/91 6,1% 0,001 0,005 0,003 0,014 215,78 2395,65 It can be observed from Table1 that FGT 0, FGT 1, FGT 2 and FGT 3 show a monotonic reduction on poverty between 1973/74 and 1991/91. Indeed, the percentage of poor has decreased from 21,9 in 1973/74 to 6,1 in 1990/91 and the poverty gap ratio, FGT 1, has also decreased from 0,014 to 0.001. A similar pattern emerges when we measure poverty by FGT 2 and FGT 3. The main results of our proposed decomposition in equation (3) are also presented in Table1. As already mentioned, the incidence term, as measured by the headcount ratio fell. The second decomposition term, intensity, as measured by the income gap ratio also decreased. Unfortunately, the inequality components do not show a reduction. As a matter of fact, these terms increase between 1973/74 and 1991/91. Using equation (9) the growth and inequality effects have been computed and the main results are presented in Table2 and Table3 from 1973/74 to 1980/81 and from 1980/81 to 1991/91 respectively. In these tables the contributions of each effect, growth and inequality, to the total percentage change are also showed. 12

Table2. FGT 0, FGT 1, FGT 2 and FGT 3 decomposition in growth and inequality effects from 1973/74 to 1980/81. Indicators Total Change Growth Component Inequality Component FGT 0 =H -0,30-0,12 (41%) -0,18 (59%) FGT 1-0,68-0,29 (43%) -0,38 (57%) FGT 2-0,41-0,19 (45%) -0,22 (55%) FGT 3-0,43-0,20 (46%) -0,23 (54%) µ g -0,38-0,17 (45%) -0,21 (55%) µ g 2-0,76-0,34 (45%) -0,42 (55%) µ g 3-1,14-0,51 (45%) -0,63 (55%) I* 2 0,64 0,28 (43%) 0,37 (57%) I* 3 1,00 0,43 (43%) 0,57 (57%) Table3. FGT 0, FGT 1, FGT 2 and FGT 3 decomposition in growth and inequality effects from 1980/81 to 1990/91. Indicators Total Change Growth Component Inequality Component FGT 0 =H -0,99-0,18 (18%) -0,81 (82%) FGT1-2,11-0,44 (21%) -1,67 (79%) FGT 2-1,22-0,32 (26%) -0,90 (74%) FGT 3-1,28-0,37 (29%) -0,92 (71%) µ g -1,12-0,26 (23%) -0,86 (77%) µ g 2-2,24-0,52 (23%) -1,73 (77%) µ g 3-3,36-0,77 (23%) -2,59 (77%) I* 2 2,01 0,37 (19%) 1,63 (81%) I* 3 3,06 0,58 (19%) 2,48 (81%) It can be noted that the growth and inequality effects are negative for all the poverty indicators in both periods except for the inequality terms. These results imply that the redistribution of income which has occurred with the economic growth has 13

benefited the poor more than the rich. For instance, if inequality did not change, the percentage of poor would have reduced by 12 percent points from 1973 to 1980. But the actual reduction is 30 percent points. Thus the change in redistribution has contributed to a decrease in the percentage of poor by 18 percent points in the same period. 5. CONCLUDING REMARKS. This paper highlights the properties and the underlying possibilities of the FGT poverty family. Although this family represents a considerable advance in both the theory and measurement of poverty, neither theory nor the measures have had much impact in part because such indices do not have a readily intuitive interpretation. We have identified a multiplicative decomposition for these indices being the product of the headcount ratio, the aggregate gap ratio to the power of α and one plus the corresponding GE inequality index of the income gap ratios of the poor multiplied by 2 α α. This decomposition allows us to shed light on this family in order to better understand its underlying structure and makes these indices easier to compute. With regard to changes in poverty over time two quite different questions may be asked. On the one hand, how changes in the incidence, intensity or inequality between the poor do affect overall poverty. On the other hand, how changes in growth and inequality affect overall poverty. We have answered these two questions by providing a unified framework where changes in all these components may be jointly analysed. Finally we have illustrated the methodology proposed with an application for Spain from the HBS for 1980/81 and 1990/91. 14

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