A New Multiplicative Decomposition For The Foster-Greer-Thorbecke Poverty Indices

Transcription

1 Workng Paper Seres A New Multplcatve Decomposton For The Foster-Greer-Thorbecke Poverty Indces Ohana Arstondo Caslda Lasso de la Vega Ana Urruta ECINEQ WP

2 ECINEC August A New Multplcatve Decomposton For The Foster-Greer-Thorbecke Poverty Indces * Ohana Arstondo Caslda Lasso de la Vega and Ana Urruta Unversty of the Basque Country. Span Abstract Ths paper dentfes a multplcatve decomposton for the Foster-Greer-Thorbecke poverty ndces as a product of the three components whch should be nvolved n every poverty ndex: the ncdence of poverty, measured by the headcount rato, the ntensty of poverty, measured by the aggregate ncome gap rato and the nequalty among the poor measured by an ncreasng transformaton of the correspondng nequalty ndex of the Generalzed Entropy famly. Then, takng data from the Spansh Household Budget Surveys (SHB) as a bass we show the advantages and possbltes of ths framework n regard to completng and detalng nformaton n studes of poverty over tme. Key words: Poverty Measurement, Foster-Greer-Thorbecke poverty ndces, Multplcatve Decomposton. JEL Classfcaton: I30, I3, D63 * We are grateful to P. Lambert for hs detaled comments and suggestons. A prelmnary verson of ths paper was presented n the 5th Internatonal Conference on Logc, Game Theory and Socal Choce (Blbao, 007). We also wsh to thank the partcpants n ths conference for ther suggestons and comments. Ths research has been partally supported by the Spansh Mnstero de Educacón y Cenca under project SEJ , cofunded by FEDER and by the Unversty of the Basque Country under the projects GIU06/44 and UPV05/117. Address for correspondence: Ohana Arstondo, Unversty of the Basque Country, Department of Appled Economcs IV, Av. Lehendakar Agurre, 83, Blbao (Span). E-mal: ohana.arstondo@ehu.es Mª Caslda Lasso de la Vega, Unversty of the Basque Country, Department of Appled Economcs IV, Av. Lehendakar Agurre, 83, Blbao (Span). E-mal: caslda.lassodelavega@ehu.es Ana Marta Urruta, Unversty of the Basque Country, Department of Appled Economcs IV, Av. Lehendakar Agurre, 83, Blbao (Span). E-mal: anamarta.urruta@ehhu.es

3 I. INTRODUCTION In a semnal artcle Sen (1976) argued that a poverty ndex should be senstve to the number of people below the poverty lne, to the extent of the short-fall of the ncome of the poor from the poverty lne and to the exact pattern of dstrbuton of the ncomes of the poor. In other words, every poverty measure should be expressed as a functon of these three poverty ndcators, showng the ncdence, the ntensty and the nequalty of the poverty respectvely, the three I s of poverty accordng to Jenkns and Lambert s desgnaton (1998b). When analyzng the sources of changes n poverty t s therefore of nterest to ask how much of a change n poverty s due to changes n these components and to know f ncreasng poverty s due to more people becomng poor, or ncreasng deprvaton of the poor, or because ncome short-fall below the poverty lne have become more unequal, or some combnaton of the above. Thus, poverty changes are more meanngful and easly understandable f poverty ndces can be decomposed nto ther underlyng contrbutng factors. There are ndces n the lterature for whch t s possble to know ths functon explctly. Besdes Clark et al. (1981) and Osberg and Xu (000), some of them may be found n Kakwan (1999). However, t may be nterestng to note that, up to now, a decomposton n these three components s dentfed for only one ndex of the Foster, Greer and Thorbecke famly (1984), henceforth FGT famly. One of the characterstcs of the ndces of the FGT class s that they are addtvely decomposable. In addton, Foster and Shorrocks (1991) establsh that the FGT class contans the canoncal forms of one nterestng famly of subgroup consstent relatve poverty ndces. Moreover Foster and Shorrocks (1988a, 1988b and 1988c) prove a useful lnk between the FGT ndces and stochastc domnance whch makes the analyss of the poverty domnance easer. In ths connecton t would be helpful and worthwhle to examne these measures n terms of ther contrbutng components. Specfcally n Secton III ths paper proves that each ndex n the FGT famly can be vewed as a product of the headcount rato, the aggregate ncome gap rato, and an ncreasng transformaton of the Generalzed Entropy nequalty ndex of the ncome gaps of the poor. A bref llustraton takng data from the Spansh Household Budget Surveys (HBS) for 1973/74, 1980/81 and 1990/91 and the more recent contnuous HBS for 000 shows the possblty of ths new decomposton n regard to completng and detalng nformaton n studes of poverty over tme. The notaton s presented n Secton II whle Secton V offers some concludng remarks. II. NOTATION The subgroup consstent prncple as defned by Foster and Shorrocks (1991) requres that poverty must ncrease when poverty ncreases n any subgroup wthout decrease n poverty elsewhere.

4 We consder a populaton consstng of n by ndvduals. Indvdual s ncome s denoted y (0, ), 1,,,n. An ncome dstrbuton s represented by a vector n y ( y 1, y,..., y n ) arranged n ncreasng order. We let D n represent the set n 1 of all fnte dmensonal ncome dstrbutons. For any gven poverty lne z and dstrbuton y D we defne as poor all ncomes y z. We denote by n n y and q q y; z the populaton sze and the number of the poor respectvely, and by y) the mean ncome of y. Let g z y / z be the ncome gap rato of the -th poor and g g,..., g the ncome gap rato vector. 1 q Poverty ndces whch wll be mentoned n ths paper are the headcount rato H H y; z q y; z / n y ; the aggregate ncome gap rato g g y ; z z y / qz 1 q ; the poverty gap rato whch s equal to the product of the headcount rato and the aggregate ncome gap rato,. e. H g, and the FGT famly proposed by Foster et al. (1984), gven by q 1 z y FG T ; z 1 y (1) n 1 z Partcularly nterestng s to note that when 0 the measure FGT 0 s smply the headcount rato H and when 1 FGT 1 s the poverty gap rato, H g. Moreover, Foster et al. (1984) propose a decomposton for FGT usng the headcount rato, the aggregate ncome gap rato and the squared coeffcent of varaton, accordng to the followng expresson: FG T y ; z H g 1 g C p where C s the squared coeffcent of varaton of the poor ncome. p

5 Fnally, n the followng the nequalty ndces of Generalsed Entropy class, henceforth GE class, (among others Bourgugnon (1979), Shorrocks (1984), Cowell, (1980)) wll play a role. The GE famly s gven by: GE 1 n 1 n y 1 n 0,1 I lo g y n 0 1 n y lo g y n 1 y () III. THE THREE I S OF POVERTY IN THE FGT FAMILY Snce Sen (1976) any poverty ndex can be expressed as P H, g, I (3) where H s the headcount rato, consdered as the archetypcal measure of the ncdence of poverty, g s the aggregate ncome gap rato, measurng the ntensty of poverty and I captures nequalty among the poor and s a non-decreasng functon n ts arguments. The proposton below shows that every FGT ndex allows a multplcatve decomposton n these three terms Wth respect to the nequalty of the poor, whch can be equvalently measured n terms of ether ncome or gaps of the poor, n the decomposton we propose, an ndex of the GE class, measurng nequalty of the ncome gap ratos of the poor, s nvolved. PROPOSITION 1: For each 1, FGT y ; z satsfes the followng multplcatve decomposton: GE y ; y ; y ; 1 g (4) F G T z H z g z I Proof. The proof s straghtforward after a few lnes of standard computatons and rearrangements. Q.E.D. The poverty gap rato s also used to measure ntensty of poverty (Jenkns and Lambert (1998a, 1998b) for nstance).

6 Ths proposton shows that each FGT poverty ndex can be decomposed n a very smple way just as the product of the headcount rato, the aggregate ncome gap rato to the power of and one plus the correspondng GE nequalty ndex of vector of the ncome gap ratos of the poor multpled by. As s well known, for large postve parameter values the GE ndex s senstve to what happens to hgh values of the dstrbuton, n ths case to large values of ncome gap ratos, therefore to the poorest of the poor. The smplcty of ths decomposton s the major dfference as regards prevous decomposton formulae for varous poverty measures dsplayed by Kakwan (1999). Ths smplcty s shared by the decomposton of the Sen poverty ndex proposed by Clark et al. (1981) and that of the Sen-Shorrocks-Thon poverty ndex proposed by Osberg and Xu (000). As usual, the multplcatve decomposton n (4) s a startng pont for the dervaton of the mpact of margnal changes n a gven component on overall poverty. Indeed the multplcatve decomposton of these ndces can be transformed, through the logarthmc transformaton, so that t s addtve n a smple form. The margnal effects derved from multplcatve decomposton appear n the followng equaton α * ΔFGT α ΔH Δ(μ g ) ΔI (5) α where * G E I 1 I g, and n general Δ x lnx lnx (x x ) x approxmates the percentage rate of change n x. t t 1 t t 1 t 1 Ths equaton shows that the overall percentage rate of change n poverty can be expressed as the sum of the percentage rate of changes n the headcount rato, the aggregate ncome gap rato and nequalty of the ncome gap ratos of the poor, and allows us to determne f ncreasng poverty s due to ether more people becomng poor, or ncreasng deprvaton of the poor, or because ncome short-falls below the poverty lne have become more unequal, or some combnaton of the above. IV. AN EMPIRICAL ILUSTRATION The methodology developed n ths paper wll now be llustrated wth data from Spansh Household Budget Surveys (HBS) for 1973/74, 1980/81 and 1990/91 and the contnuous

7 HBS for 000 **. The varable we use s the net expendture per capta and the poverty lne s 50% of the mean ncome of each year. Although statstcal analyss wth explct confdence ntervals s desrable to gauge to what extent measured changes correspond to real changes, an analyss of that knd s beyond the scope of ths bref note. Takng nto account, as already mentoned, that FGT s the headcount rato,.e. H, 0 and FGT, s the poverty gap rato,.e. H 1 g, we decompose FGT and FGT nto 3 ncdence, H, ntensty, g, and nequalty as measured by GE I and GE I respectvely. 3 TABLE 1 The three I s n FGT 0, FGT 1, FGT and FGT 3. FGT 0 =H FGT 1 =H * g FGT FGT 3 g I GE I GE % % % % As can be seen n Table 1 FGT 0, FGT 1, FGT and FGT 3 show monotonc reductons n ths perod. Indeed, the percentage of poor has decreased from 0.% n 1973 to 13.4% n 000 and the poverty gap rato, FGT 1, has also decreased from to A smlar pattern emerges when we measure poverty by FGT and FGT 3. The nequalty components correspondng to both FGT and FGT 3 have ncreased, but not by enough to reverse the downward overall trends. TABLE Decomposton of poverty trend. Contrbuton to the change n poverty as measured by FGT Contrbuton to the change n poverty as measured by FGT 3 FGT 0 =H ( g ) I* total FGT 0 =H ( g ) 3 I* 3 total 1973 to to to to ** An n-depth study of the evoluton of poverty n Span n the same perod has been carred out by del Río and Ruz-Castllo (001) applyng the methodology developed by Jenkns and Lambert (1998a, 1998b). The results are bascally the same f another equvalence of scale s used.

8 Usng equaton (5), Table offers a decomposton of the poverty trend nto the contrbutons assocated wth these three components, for the ndex FGT (analogous results can be obtaned by analysng FGT 3 ). Frst of all t s noteworthy that the declne n the headcount has decreased the poverty rate by 40.5 percentage ponts over the perod Decreasng the poverty gap rato has contrbuted a further 3 ponts, whle a hgher nequalty has caused the poverty change to ncrease by 5.6 ponts. The concluson that follows from these results s that the declne n poverty by 67.4 percentage ponts over the whole perod s due manly to the declnes n the headcount and n the poverty gap ratos. If we look at the results decade by decade, we can see that the man sources of the poverty changes are not the same. In fact, n the perods and the poverty decrease dsplays smlar values although the orgns of these reductons are qute dfferent. The prncpal cause of lower poverty n was the decrease n the poverty gap rato, whereas n the t was the declne n the headcount rato. It s the new decomposton (4) and the breakdown nto mpacts (5), whch enable these nsghts. V. CONCLUDING REMARKS Ths paper proposes a new decomposton for the FGT poverty ndces allowng us to better understand the underlyng structure of ths famly and to analyse sources of poverty change. It would be of nterest to explore a unfed framework for the study of changes n poverty over tme establshng a connecton between ths procedure and the recent pro-poor growth lterature n order to decompose the change n poverty nto growth and nequalty effects. Ths s an mportant ssue for further research. REFERENCES Bourgugnon, F., 1979, Decomposable ncome nequalty measures, Econometrca, 47, Clark, S., R. Hemmng and D. Ulph, 1981, On ndces for the measurement of poverty, Economc Journal, 91,

9 Cowell, F. A., 1980, On the structure of addtve nequalty measures, Revew of Economc Studes, 47, del Río C. and J. Ruíz-Castllo, 001, TIPs for poverty analyss. The case of Span to , Investgacones Económcas, vol. XXV (1), Foster, J., J. Greer and E. Thorbecke, 1984, A class of decomposable poverty measures, Econometrca, 5, Foster, J. and A. Shorrocks, 1988a, Poverty Orderngs, Econometrca, 56, Foster, J. and A. Shorrocks, 1988b, Poverty Orderngs and Welfare Domnance, Socal Choce and Welfare, 5, Foster, J. and A. Shorrocks, 1988c, Inequalty and Poverty Orderngs, European Economc Revew, 3, Foster, J. and A. Shorrocks, 1991, Subgroup consstent poverty ndces, Econometrca, 59, Jenkns, S. and P. Lambert, 1998a, Rankng poverty gap dstrbutons: further TIPs for poverty analyss. Research on Economc Inequalty 8, (JAI Press, Greenwch),, Jenkns, S. and P. Lambert, 1998b, Three I s of poverty curves and poverty domnance: TIPs for poverty analyss, Research on Economc Inequalty 8, (JAI Press, Greenwch), Kakwan, N., 1999, Inequalty, Welfare and Poverty, n J. Slber, eds., Handbook of ncome nequalty measurement, (Kluwer Academc Publshers, Massachusetts), Sedl, C., 1988, Poverty Measurement: A Survey, n D. Bös, M. Rose and C. Seld, eds., Welfare and Effcency n Publc Economcs, (Sprnger-Verlag, Berln). Sen, A. K., 1976, Poverty: An Ordnal Approach to Measurement, Econometrca, 44, Shorrocks, A. F., 1984, Inequalty decomposton by populaton subgroups, Econometrca, 5, Osberg, L. and K. Xu, 000, Internatonal comparsons of poverty ntensty: Index decomposton and bootstrap nference, The Journal of Human Resources, XXXV, 1-81