STRATIFICATION AND BETWEEN-GROUP INEQUALITY: A NEW INTERPRETATION. and

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1 roiw_ Review of Icome d Welth Series 57, Number 3, Setember 20 DOI: 0./j x STRATIFICATION AND BETWEEN-GROUP INEQUALITY: A NEW INTERPRETATION by Mri Moti* DSGE, Uiversity of Mil-Bicocc d DEAS, Uiversity of Mil d Alessdro Storo DSGE, Uiversity of Mil-Bicocc d Ecoubblic Trditiolly, the literture hs see strtifictio s liked closely to withi-grou iequlity. More recetly, some ers hve focused o mesurig the imct of strtifictio o betwee-grou iequlity. I this er, we show tht whe two grous re ivolved, such imct c be mesured by simle comriso of the two cumultive distributio fuctios. This roch llows iterrettio of strtifictio i terms of robbilities d ves the wy for et d simle grhicl illustrtio. We ly it to the lysis of betwee-cotiet iequlity.. Itroductio Strtifictio mes grou s isoltio from members of other grous (Yitzhki d Lerm, 99,. 39). A grou is sid to be strtified whe it teds to form erfect strtum i the overll distributio. Strtifictio hs bee used trditiolly i sociologicl studies, but its rigorous defiitio d mesuremet re owed to Yitzhki d Lerm (99) d Yitzhki (994). To this ed, they roose decomositio of the Gii idex ito two rts: comoet tht is weighted sum of grous Giis, d betwee-grou iequlity mesure. I the first comoet, the weight deeds ositively o the vlue of the overlig idex for ech grou. I tur, the overlig idex for every grou mesures the extet to which the icome rges of the members of tht grou overl with those of members of other grous. The less grou is strtified, the more it overls with other grous, d therefore Yitzhki d Lerm (99,. 323) coclude tht iequlity d strtifictio re iversely relted. Yitzhki d Lerm rgue tht, though couterituitive, this result is cosistet with the reltive derivtio theory. The ide is tht grouigs re legues, so tht ech erso cofies his sirtios to his ssiged legue (Yitzhki d Lerm, 99,. 323). To ut it other wy, idividul (i.e. grou member) Note: We thk the Editor, two oymous referees, Cochit d Ambrosio d Bruo Bosco for their commets o revious versio of this er. We lso thk rticits from semirs held t the Stte Uiversity of Mil d the Uiversity of Mil-Bicocc. The usul disclimers ly. Ficil suort from the Itli Miistry of Eductio d Reserch (PRIN 2007HEWTBE) is grtefully ckowledged. *Corresodece to: Mri Moti, DSGE, Uiversità degli studi di Milo-Bicocc, Pizz dell Ateeo Nuovo, 2026 Mil, Itly (mri.moti@guest.uimi.it). Review of Icome d Welth 20 Itertiol Associtio for Reserch i Icome d Welth Published by Blckwell Publishig, 9600 Grsigto Rod, Oxford OX4 2DQ, UK d 350 Mi St, Mlde, MA, 0248, USA. 42

2 Review of Icome d Welth, Series 57, Number 3, Setember 20 cres more bout the distributio i his/her ow grou th i other grous. Thus, strtified societies c tolerte higher iequlity th ustrtified societies sice, s eole become more (less) egged with ech other, they hve less (more) tolerce for give level of iequlity (Yitzhki d Lerm, 99,. 323). Yitzhki d Lerm (99,. 35) reiforce this iterrettio, observig tht geerlly rise i subgrou s iequlity will reduce the subgrou s strtifictio, so tht, i geerl, if withi-grou iequlity is low, overlig is lso low or, equivletly, strtifictio is high. Note tht i this lie of resoig, the betwee-grou iequlity, which is the secod comoet of the Gii decomositio i Yitzhki (994), is left comletely side: both Yitzhki d Lerm (99) d Yitzhki (994) tret strtifictio d betwee-grou iequlity s two comletely serte objects. Nevertheless, Milovic d Yitzhki (2002), wre of the imct of strtifictio o betwee-grou iequlity, suggest evlutio by the rtio of the Yitzhki d Lerm (99) betwee-grou Gii d the covetiol betwee-grou Gii coefficiet, s obtied by Pytt (976). I this er, we tke this rtio s our strtig oit. Whe two grous re cosidered, we show tht such rtio, i.e. the mesure of the imct of strtifictio o betwee-grou iequlity, is fuctio of the robbility tht rdom member of the oorer (o verge) grou is richer th rdom member drw from the (o verge) richer grou. Thus, this mesure deeds exclusively o the cumultive distributio fuctios of the two grous. The vlue of the rtio roosed by Milovic d Yitzhki (2002) c ctully be exressed s oe mius twice the re uder the cumultive distributio fuctio for the richer (o verge) grou, evluted withi rge of vlues of the cumultive distributio fuctio of the oorer (o verge) grou. This exressio turlly suggests grhicl iterrettio. We rovide the ltter by lyig our roch to the lysis of betwee-cotiet iequlity usig the dt reorted by Milovic d Yitzhki (2002). This is the first d mi cotributio of the er. The secod cotributio of the er is the lik we estblish betwee the literture o strtifictio d the Gii cocet of trsvritio. Buildig o Moti d Storo (2009), it c be show tht the rtio roosed by Milovic d Yitzhki (2002) deeds o the totl umber of trsvritios occurrig betwee the members of the two grous. Thus, such rtio is ultimtely fuctio of the robbility of trsvritio, d this offers dditiol iterrettio of the results obtied here. The er is orgized s follows. Sectio 2 summrizes the cotributio of Yitzhki d his collegues to the mesuremet of strtifictio. Sectio 3 cotis the theoreticl results of the er. We derive the mesures of the imct of strtifictio o betwee-grou iequlity i terms of the cumultive distributio fuctios of the comred grous d rovide lysis of the rge of vritio of these mesures. Sectio 4 lies these results to the lysis of betwee-cotiet iequlity usig mily grhicl roch. Sectio 5 cocludes. Give two grous with differet verge icomes, trsvritio occurs wheever member of the oorer (o verge) grou hs icome higher th member of the richer (o verge) grou. Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth 20 43

3 Review of Icome d Welth, Series 57, Number 3, Setember Overview of the Literture The cocet of strtifictio is frequetly used i socil sciece literture. Its defiitio c be trced bck t lest to Lsswell (965,. 0): A strtum is horizotl lyer. Strtifictio is the rocess of formig observble lyers... where the mss of society is costructed of lyer uo lyer of cogeled oultio qulities. Util Yitzhki d Lerm (99), however, rigorous roch to defiitio d mesuremet ws lckig. Yitzhki d Lerm s (99) cotributio is threefold. First, they obti idex of reltive strtifictio, idex Q, which ctures the extet of strtifictio of every grou with resect to the etire oultio, tkig ito ccout the size of the grou. Secod, they derive idex of bsolute overlig, O, which is iversely relted to Q. Third, they decomose the Gii idex ito three rts: withi-iequlity comoet, comoet tht reflects the imct of strtifictio, d mesure of betwee-grou iequlity. Whe commetig uo the dymics of these three rts, Yitzhki d Lerm (99,. 323; our emhsis) ote tht some chges i Q i s my leve Giis uchged, d ifluece oly comoet two. This imlies tht, i geerl, there is o reltioshi betwee strtifictio d betwee-grou iequlity d tht these c be treted s serte cocets. Yitzhki (994) further develos the idex of overlig O, focusig o overlig betwee suboultios. He obtis decomositio of the Gii idex ito two comoets: the betwee-grou iequlity mesure, defied by Yitzhki d Lerm (99), d term tht is the sum of the roducts of icome shres, Giis d overls for ll grous. This decomositio is used by Milovic d Yitzhki (2002) to mesure the world s icome iequlity. To summrize the mi results obtied by Yitzhki (994) let us itroduce some ottio. We cosider oultio of idividuls d we cofie our ttetio to the cse of two oultio grous, 2 which std for give socioecoomic rtitio of the oultio bsed o the idividuls chrcteristics. We cll ffluet, deoted by, the grou with the higher verge icome d we cll oor, deoted by, the other grou. The oultio size is + =, with,,, where is the umber of idividuls belogig to grou, d is the umber of idividuls belogig to grou. Byy ih we deote the icome of idividul h belogig to grou i (i =, ), d by m, m, d m, the overll, the grou, d the grou verge icome, resectively. 3 Cosistetly with the ottio bove, we hve m > m. Filly, {y} is the set of ll icome uits. Followig Yitzhki (994), the Gii idex decomoses s () Gy ( ) = ( ω GO+ ω GO)+ G, b where (2) Oi = si + sjoji, j i 2 As suggested by oymous referee, it is worth emhsizig tht oly whe two grous re cosidered do strtifictio d betwee-grou iequlity relte so directly. 3 We ssume tht these mes re ll ositive. Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth 20 44

4 Review of Icome d Welth, Series 57, Number 3, Setember 20 (3) ( ) ( ) O = cov y, F ( y) cov ( y, F y ), ji i j i i d (4) G = 2cov ( μ, F ) μ. b i Oi I ( 3), F i(y), m i, G i, d s i rereset the cumultive distributio, the verge icome, the Gii idex, d the shre of grou i i the overll distributio, resectively. Let w i = s im i/m deote the shre of totl icome owed by grou i, d O i deote the overlig idex of the sme grou i with the oultio s distributio. The idex O i is fuctio of the overlig of grou j by grou i, O ji, which i tur, is equl to the rtio betwee the covrice betwee icomes of grou i d their rk, hd they bee cosidered s belogig to the grou j (Yitzhki, 994,. 49) d the covrice betwee icomes d ow rkig i grou i, the ltter beig ormlizig fctor. 4 Filly, i exressio (4), G b is twice the covrice betwee ech grou s verge icome d grou s verge rk i the overll oultio (F Oi ), divided by the overll me icome. The overlig idex O i reflects the overlig of grou i with itself d with the other grous, d c be iterreted s mesure of strtifictio. I (), i the roud brckets, the subgrou Gii idices, G i, d the overl idices, O i, hve symmetricl imcts o overll iequlity, sice iequlity rises i both. Nevertheless, high strtifictio imlies low overlig so tht if G b is igored, oe cocludes tht iequlity d strtifictio re iversely relted (Yitzhki d Lerm, 99,. 323). Accordig to Milovic d Yitzhki (2002,. 6), however, more overlig (i.e. less strtifictio) leds to lower correltio betwee verge icome d verge rk d this decreses the betweegrou comoet. To mesure the imct of strtifictio o betwee-grou iequlity, Milovic d Yitzhki (2002) refer to the covetiol decomositio of the Gii idex s roosed by Pytt (976): (5) G( y)= G + G + R, W B where, (6) G 2 2 = ( Gμ + Gμ ) G = ( ) > 2 2 μ d μ μ μ ; μ μ. W B I (6), the terms G d G deote the grou d grou Gii idex, resectively, so tht G W mesures withi-grou iequlity, G B ctures betweegrou iequlity, d R is the residul, which deeds o the overlig betwee the two grou icome distributios. I the covetiol Gii decomositio, G B is differet from G b. I both G B d G b, ech grou is joitly rereseted by its me icome d rk. I G B, however, the rk of grou is the rk of its me wheres i G b oe tkes ccout of ech observtio s rkig i the overll 4 To iterret these defiitios recll tht Yitzhki d Lerm (99,. 32) estimte the cumultive distributio, F(y), by the rk of y. Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth 20 45

5 Review of Icome d Welth, Series 57, Number 3, Setember 20 distributio by vergig these rkigs withi ech grou (Yitzhki d Lerm, 99,. 322). The two betwee-grou iequlity comoets re equl if there is o overl betwee grous d it c be show tht G b < G B whe grous overl (we retur to this i Sectio 3). The rtio G b/g B is suggested by Milovic d Yitzhki (2002,. 6) s idex reresetig the loss i betwee-grou iequlity owig to icrese (decrese) i overlig (strtifictio). Immeditely G b rewrites s (7) G b ( μ μ ) = I, 2 μ where 5 (8) Gb I =. G B I the Aedix, we show tht (9) = ( ) Rh r h h I = 2. I exressio (9), usig the sme ottio of Yitzhki d Lerm (99), we deote by R h the rk of y h i the overll distributio d by r k the rk of y h i the distributio of grou. The differece R h - r h evlutes the umber of members of the ffluet grou with icome lower th y h so tht the sum Rh r h= ( h ) is the umber of istces where icome of grou is greter th icome of grou. Note tht this umber is equl to the umber of istces where icome of grou is lower th icome of grou. 3. Idex I Here, we focus o I d o its exressio (9). Sice is the totl umber of comrisos betwee members of the two grous, the rtio ( Rh rh ) h= c be iterreted s the robbility tht rdom member of the grou which is o verge oorer is richer th rdom member drw from the (o verge) richer grou. The, recllig from the revious sectio tht m > m, we rereset the icome sets of the two grous s two discrete rdom vribles deoted by Y d Y, resectively, so tht 6 5 This rtio is used by Frick et l. (2006), who lso suggest sttisticl tests tht re ot covered i the reset er. 6 We observe tht i the defiitio of the Gii idex d i its decomositio, there is imlicit ssumtio of ideedece betwee Y d Y. For ll j d l, the robbility of the differece (y j - y l) is the roduct of the robbility to observe y j i the distributio Y d the robbility to observe y l i distributio Y. Tht is, give Pr(Y = y j) = / d Pr(Y = y l) = /, oe hs Pr ( Y = yj, Y = yl)= Pr( Y = yj) Pr ( Y = yl)=. Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth 20 46

6 Review of Icome d Welth, Series 57, Number 3, Setember 20 (0) Prob[ Y > Y ] h= ( R r ) h h d we c write () [ ] I = 2Prob Y > Y. We c ow derive umber of results cocerig the rge of I. First, we ote tht idex I ssumes its mximum, I = (G b = G B), whe strtifictio is erfect: (2) I = Prob [ Y > Y ]= 0. (3) Secod, we ote tht I = 0 Prob[ Y > Y]= 2 ( R r )= h = h h 2. The idex I is equl to zero if, d oly if, the robbility tht rdom member of the oorer grou is richer th rdom member drw from the (o verge) richer grou is exctly equl to 50 ercet. Third, we ote tht the idex I = G b/g B is t its miimum whe the ltter robbility reches its mximum, i.e. (4) Gb mi I = mx Y Y, G = 2 { Prob[ > ]} B where: (5) mx { Prob [ Y > Y ] mx }= Rh rh. h ( ) = c By deotig q s the umber of members of grou whose icome is higher th m, d l s the umber of members of grou whose icome is (wekly) lower th m, oe obtis 7 (6) mx ( Rh rh )= ql, h= d the miimum vlue of I = G b/g B is (7) G q b mi I =. G = + 2 l B Exressio (7) suggests immeditely writig both the idex I d its miimum i cotiuous form. I wht follows, we ssume tht (see footote 6) 7 The roof of this result c be obtied from the uthors uo request. Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth 20 47

7 Review of Icome d Welth, Series 57, Number 3, Setember 20 (8) Prob( Y < y, Y < y )= Prob( Y < y ) Prob( Y < y )= F( y ) G( y ), where F(y ) d G(y ) re the cumultive distributios of Y d Y, resectively. The, sice (9) Prob( Y > μ, Y < μ )=[ F( μ )] G( μ ), exressio (5) rewrites s (20) [ ] mx { Prob Y > Y }= [ F( μ )] G( μ ), d exressio (7) becomes (2) Gb mi I = F( ) G. G = 2 [ μ ] ( μ ) B Usig (2), we c see tht mi I c be egtive d tht its vlue deeds o the skewess of the two distributios. If the two distributios re both symmetric with resect to their me, the miimum vlue of idex I is -/2. O the other hd, oe hs mi I > -/2 if either the distributio of Y is symmetric with resect to m d the distributio of Y hs ositive symmetry (m > medi, right obliquity), or if the distributio of Y hs egtive symmetry (m < medi, left obliquity) d the distributio of Y is either symmetric or symmetric with ositive symmetry. Moreover, oe hs mi I < -/2 if either the distributio of Y is symmetric with resect to m d the distributio of Y hs egtive symmetry, or if the distributio of Y hs ositive symmetry (m > medi, right obliquity) d the distributio of Y is either symmetric or symmetric with egtive symmetry. Nothig c be sid bout mi I if the two distributios re symmetric with the sme symmetry. Let us ow cosider the cotiuous exressio of idex I (exressios (0) d ()). Observe tht (22) Prob( Y > Y )= Prob( Y Y < 0. ) The, if the differece vrible (Y - Y ) is deoted by Z, d its cumultive distributio is deoted by H(z), oe hs (23) z+ y Prob(Z < z ) = H( z)= df( y ) dg( y )= F z+ y dg y, ( ) ( ) d exressio (0) becomes (24) Prob(Z 0) H 0 F y dg y. < = ( )= ( ) ( ) Usig (24) bck i (), we c write Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth 20 48

8 Review of Icome d Welth, Series 57, Number 3, Setember 20 (25) I 2 F y dg y. = ( ) ( ) Exressio (25) sys tht the mesure of imct of overlig o betweeiequlity c be exressed s fuctio of the cumultive distributio fuctios of the two grous. More recisely, exressio (25) sys tht I = G b/g B is equl to oe mius twice the re uder the cumultive distributio fuctio of the richer grou, evluted s fuctio of the cumultive distributio fuctio of the oorer grou. For reders fmilir with Gii s cocet of trsvritio (Gii, 959; Giorgi, 2005; Moti d Storo, 2009), dditiol isights c be rovided. I geerl, trsvritio occurs wheever member of the oorer (o verge) grou is richer th member of the richer (o verge) grou (Gii, 959). It imlies tht the sig of differece betwee the two icomes is oosite with resect to the sig of the differece betwee the mes of the two grous. I our cse, sice m > m, trsvritio occurs wheever member of grou is richer th member of grou. Now, the term R h - r h reresets the umber of trsvritios i which y h is ivolved so tht the term Rh r h= ( h ) is simly equl to the totl umber of trsvritios. Therefore idex I rewrites s (26) TR N I = 2. where N TR is the totl umber of trsvritios. Sice is the totl umber of comrisos betwee members of the two grous, the rtio N TR / c be iterreted s the robbility tht the sig of differece betwee two icomes belogig to differet grous is oosite with resect to the differece betwee the mes of the two grous. I other words, this rtio corresods to the robbility of trsvritio. 8 We ow discuss the imlictios of exressio (25) usig grhicl roch where we lyze betwee-cotiet iequlity i wy tht immeditely reltes to the reserch of Milovic d Yitzhki (2002). 4. A Grhicl Iterrettio Usig the tiol icome/exediture distributio dt from coutries, Milovic d Yitzhki (2002) decomosed totl iequlity betwee idividuls i the world by cotiets d regios. I rticulr, they rtitioed the world ito five cotiets: Afric; Asi; Wester Euroe, North Americ, d Ocei (WENAO); Ester Euroe d the former Soviet Uio (EUFSU); d Lti Americ d the Cribbe (LAC). Commetig o the results reltig to betwee-cotiet iequlity, they ote tht betwee-cotiet iequlity Gii is 0.309; hd we used Pytt s betwee-grou comoet, we would hve gotte 8 See Gii (959,. 8) o this oit. Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth 20 49

9 Review of Icome d Welth, Series 57, Number 3, Setember 20 Me Icome TABLE Vlues of I for Cotiet-by-Cotiet Comrisos Afric (m = 30) Asi (m = 594.6) EUFSU (m = ) LAC (m = ) Afric Asi 32.6% EUFSU 32.0% -4.5% LAC 77.4% 30.0% 45.0% WENAO 99.0% 72.2% 96.4% 92.7% WENAO (m = 002.4) Notes: m = me icome i $PPP (993), weighted for the size of the oultio. EUFSU, Ester Euroe d former Soviet Uio; LAC, Lti Americ d Cribbe; WENAO, Wester Euroe, North Americ, d Ocei. Source: Authors clcultio from Milovic d Yitzhki (2002). betwee-cotiet Gii of which mes tht overlig hs decresed the betwee-cotiet comoet by bout 9 Gii oits (Milovic d Yitzhki, 2002,. 63). It is iterestig to verify how cotiet-by-cotiet comrisos hve cotributed to this result. We tret coutries s uits of observtio d cotiets s grous. 9 Orderig the cotiets by their er cit verge icome i itertiol dollrs, i Tble we reort the vlues of I, i.e. the vlues of the rtio G b/g B, for ech ir of cotiets. Recllig the discussio i Sectio 3, we ote tht Tble covers the whole rge of ossible vlues of I. Whe WENAO is comred with other cotiets, I reches very high vlues. I rticulr, I is very close to uity whe WENAO is cotrsted with Afric (99 ercet), EUFSU (96.4 ercet), d LAC (92.7 ercet). Accordig to exressio (2) bove, i ll these cses, Prob (Y > Y )is close to zero, so strtifictio domites d there is virtully o overlig betwee WENAO coutries d coutries belogig to other cotiets. Thus, i these cses, usig Pytt s (976) decomositio d Yitzhki s (994) decomositio, we would obti lmost equivlet mesures of betwee-cotiet iequlity. The excetios ivolvig WENAO rise i the comriso with Asi coutries. I this cse, I equls 72.2 ercet, so tht we kow from () tht the robbility of Asi coutry hvig me icome higher th WENAO coutry is equl to 3.9 ercet. At the other extreme, I reches (smll) egtive vlue (i.e ercet) whe EUFSU d Asi re comred. Accordig to exressio (3), this mes tht, lthough the me icome of EUFSU is 75 ercet higher th the me icome of Asi, it is more likely tht Asi coutry hs me icome higher th EUFSU coutry th the reverse. More recisely, usig (), this robbility mouts to 52.2 ercet. This result is ssocited clerly with high olriztio withi both these cotiets, which geertes egtive vlue of I. This sigls low strtifictio d high overlig. 9 Note, however, tht usig this level of ggregtio we cot fully exli fidigs by Milovic d Yitzhki (2002), sice, i this er, coutries re reseted by deciles d eve smller ggregted observtios, d strtifictio is ffected by the level of ggregtio. Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth

10 Review of Icome d Welth, Series 57, Number 3, Setember 20 Filly, remiig comrisos re somewht i betwee these two extremes. For exmle, whe LAC d EUFSU re comred, the vlue of I is close to 50 ercet, which idictes, gi from (), tht the robbility of EUFSU coutry hvig me icome higher th LAC coutry is roud 25 ercet. Accordig to Tble d exressio (), the robbility of Afric coutry hvig me icome higher th either Asi or EUFSU coutry, d the robbility of Asi coutry hvig me icome higher th LAC coutry, rges betwee 25 d 50 ercet. These, d more results, c be illustrted reresetig, for ech ir of cotiets, the cumultive distributio of the richer cotiet s fuctio of the cumultive distributio of the oorer oe, usig uweighted me icome for ech coutry ivolved i the comriso. I Figure, we reset four comrisos tht rereset the grhicl couterrt of exressio (25). I ech of the four digrms, the kiked lie reresets the cumultive distributio of the richer cotiet (fuctio F(y) usig the ottio of exressio (25)) lotted gist the cumultive distributio of the oorer cotiet (fuctio G(y)). It follows tht the coordites of ech oit o the kiked lie re give by: (i) the roortio of coutries, which, i the oorer cotiet, hve verge icome smller th y o the horizotl xis; d (ii) the ercetge of coutries, which, i the richer cotiet, hve verge icome smller th y o the verticl xis. I ech digrm, res uder the F(y) curves re equl to Prob (Y > Y ) betwee coutries belogig to the corresodig cotiets, s c be verified by simle umericl comuttio. The 45 lie (whe reset) idictes the vlues tht the cumultive distributio of the richer cotiet should ossess to be exctly equl to the cumultive distributio of the oorer cotiet [F(y) = G(y)] t the sme verge icome level. Where the sloe of F(y) is higher th oe (the sloe of the 45 lie) i give rge of G(y), it mes tht, for ech oit belogig to tht regio, the ercetge of coutries hvig verge icome smller th y i the richer cotiet is higher th the ercetge of coutries hvig verge icome smller th the sme vlue y i the oorer cotiet. We choose to reset two comrisos ivolvig EUFSU coutries d two comrisos ivolvig WENAO coutries; these cotiets re both comred with Asi d Afric coutries. The olriztio mog EUFSU coutries is visible i the she of its cumultive distributio whe lotted gist Asi d, to some extet, gist Afric. I both these cses, t low icome levels, F(y) is bove the 45 lie sice my of the bsolute oorest coutries belog to EUFSU (Georgi, Uzbekist, Armei; see Milovic d Yitzhki, 2002,. 76). This mes tht the miimum verge icome level, i.e. the lower boudry of the regio R, where the itegrl i (25) is evluted, belogs to the richer (o verge) cotiet, EUFSU i both cses. As higher icome levels re cosidered, the cumultive distributio for EUFSU flls behid the 45 degree lie with resect to both Afric d Asi. This sigls tht the robbility of EUFSU coutry hvig verge icome below give level is lower th the robbility of fidig Afric or Asi coutry with verge icome below the sme level. I the comriso with Asi, however, there is other regio, t middlehigh icome levels, where F(y) lies bove the 45 lie t the to. The ltter reflects Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth 20 42

11 Review of Icome d Welth, Series 57, Number 3, Setember 20 20,00% 00,00% 80,00% WENAO vs ASIA WENAO 60,00% 40,00% 20,00% P=3.8% 0,00% 5% 0% 5% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65,0% 70,0% 75,0% 80,0% 85,0% 90,0% 95,0% 00,0 Asi 5% 5% 4% 4% 3% 3% 2% 2% % % 0% WENAO vs AFRICA WENAO P=0.5% 4% % 9% 26% 33% 4% 48% 56% 63% 70% 78% 85% 89% 96% Afric 20,0% EUFSU vs ASIA 00,0% 80,0% EUFSU 60,0% 40,0% 20,0% P=52.3% 0,0% 0% 5% 0% 5% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 90% 95% 00% Asi 20,0% EUFSU vs AFRICA 00,0% 80,0% 60,0% EUFSU 40,0% 20,0% 0,0% P=34.0% 0% 4% 7% % 5% 9% 22% 26% 30% 33% 37% 4% 44% 48% 52% 56% 59% 63% 67% 70% 74% 78% 8% 85% 89% 93% 96% 00 Afric Figure. Cotiet-by-Cotiet Comrisos Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth

12 Review of Icome d Welth, Series 57, Number 3, Setember 20 olriztio of icome cross Asi coutries, mely the resece of high-icome coutries, such s Sigore, Tiw, Kore, J, d Hog Kog. Digrms ivolvig WENAO re much more covetiol. Sice the oorest WENAO coutry (Turkey) is lwys fr richer th the oorest Afric or Asi coutry, F(y) (i these cses) lies o the horizotl xes for lrge itervl of vlues of G(y). More recisely, there is o WENAO coutry with me icome lower th Asi coutry util the seveth decile of the Asi distributio. At higher me icome levels, some Asi coutries re richer th WENAO coutries, but the kiked lie ever crosses the 45 lie. Overll the robbility tht Asi coutry is richer th WENAO coutry mouts to 3.8 ercet, gi, vlue tht c be roximted by clcultig the re uder the kiked curve. The comriso betwee WENAO d Afric coutries is much more drmtic, sice the strtifictio, s idicted by the vlue of I t 99 ercet i Tble, is lmost bsolute. The robbility of Afric coutry beig richer th WENAO coutry is oly mrgilly differet from zero; thus the 45 degree lie cot be rereseted d F(y) lies lmost everywhere o the horizotl xis. The etire discussio bove c be reformulted i terms of trsvritios. Ideed, whe WENAO coutry is cosidered, the robbility of trsvritio is geerlly egligible or very low, the excetio beig the ossibility tht Asi coutry is richer. O the other hd, whe EUFSU d Asi re comred, the robbility of trsvritio is high, gi becuse of the existece of very rich Asi coutries. Filly, i the itermedite cses such s LAC vs EUFSU, the vlue of I is close to 50 ercet sice the robbility of trsvritio is bout 4. Also, i the figures bove, the sloe of F(y) becomig higher th oe sigls tht i tht regio trsvritios re beig origited, becuse, t the sme levels of verge icome, there re coutries belogig to the richer cotiet whose icomes re lower th t lest oe of the coutries of the oorer cotiet. 5. Cocludig Remrks The trditiol literture o strtifictio mesuremet (Yitzhki d Lerm, 99; Yitzhki, 994) teds to see strtifictio s iversely relted to iequlity. This view derives from the fct tht higher strtifictio, i.e. lower overlig, is usully ssocited with lower withi-grou iequlity. The most recet literture, however, focuses o the imct of strtifictio o betweegrou iequlity d rooses mesure to evlute it (Milovic d Yitzhki, 2002; Moti d Storo, 2009). This mesure is such tht, ceteris ribus, higher strtifictio is ssocited with higher betwee-grou iequlity. I this er, we iterret this mesure s fuctio of the robbility tht rdom member of the oorer (o verge) grou is richer th rdom member drw from the (o verge) richer grou. We show tht whe two grous re cosidered, this roch leds to rewritig such mesure s oe mius twice the re uder the cumultive distributio fuctio of the richer grou, exressed s fuctio of the cumultive distributio fuctio of the oorer grou. This formul is, to some extet, similr to the exressio of the Gii idex d turlly suggests the grhicl illustrtio tht we rovide to lyze betwee-cotiet iequlity. The mjor dvtge of our roch is tht lot of iformtio bout the imct Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth

13 Review of Icome d Welth, Series 57, Number 3, Setember 20 of strtifictio o betwee-grou iequlity c be obtied by simle grhicl isectio of the lot of the cumultive distributio fuctio of the grou with higher me icome gist the cumultive distributio fuctio of the grou with lower me icome. Moreover, such isectio immeditely suggests where the robbility of trsvritio s defied by Gii i 96 (Gii, 959) is icresig. Wht rtiole c be rovided for this iterrettio of the imct of strtifictio o betwee-grou iequlity? We thik swer to this questio c be rovided by the cocet of grou derivtio; it lies whe grou members shre strog body of commo morl, socil, d culturl vlues. By grou derivtio we me the feelig of derivtio tht grou hs wheever the icome of y of its members is lower th the icome of y member of the other grou. I such cse, every member of grou feels emthy for y other member of their ow grou, the grou s whole beig ffected by the robbility tht y of its members is richer th y of the member of the other grou. Grou derivtio, thus icreses i this robbility d this drives the imct of strtifictio o betwee-grou iequlity. To rovide exmle, we refer bck to the lysis of betwee-cotiet iequlity d cosider the viewoit of reresettive idividul of Afric coutry. By reresettive idividul, we me idividul whose icome is exctly equl to the me icome of their coutry. Suose this idividul feels he/she belogs to the Afric cotiet, ot oly to his/her ow coutry. Whe comrig Afric with y other cotiet, this idividul would therefore cre bout the ossibility tht y reresettive Afric is richer th the reresettive idividul of Wester or Asi coutry. This ossibility corresods to the robbility tht y reresettive Afric is richer th reresettive idividul of other (richer) cotiet. The higher this robbility, i.e. the robbility of trsvritio, the lower the feelig of grou derivtio d betwee-grou iequlity. Aedix I this Aedix we show how exressio (9) is obtied. We strt with ottio. We defie Y-orderig s the orderig of icome uits by their icome level, where: R ih is the rk of the icome y ih i the overll oultio, R ih =...; F O(y) is the overll cumultive distributio of Y, emiriclly estimted by R ih/; R i is the verge rk of the grou i i the overll oultio, or R = i i i R k = ik; F Oi is the me of the cumultive fuctio vlues for the grou i, estimted by Ri. Also, we defie g-orderig (grou-orderig) s the order i which the grous re lied u followig the o-decresig orderig of their mes d, withi the grous, icomes re ordered by their o-decresig order. I this orderig: Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth

14 Review of Icome d Welth, Series 57, Number 3, Setember 20 r ih is the rk of the icome y ih i its ow grou; r i is the verge rk of grou i, or r i r = i i h = ih; F g ( O y ) is the overll cumultive distributio, emiriclly estimted by r ih/ [i =, ; h =...,...,( + ),... ( + j)...]; g F Oi is the me of the cumultive distributio vlues for the grou i, estimted by ri. The roof of (9) is equivlet to the roof of the followig equivlece: (A) ( ) = ( ) B b h h h G = G + R 2 2 μ μ r μ where m, m, m, re the me icome of the grou, of the grou d of the g overll oultio resectively. GB = 2cov ( μi, FOi) μ is the betwee comoet of the covetiol Gii decomositio, Gb = 2cov ( μi, FOi) μ is the betwee comoet of the Yitzhki (994) Gii idex decomositio, d Rh r k= ( h ) is the umber of istces where member of grou hs icome higher th member of grou. From the defiitio of covrice oe hs ( ) ( + + ) g g g cov ( μi, FOi)= μ FO + μ FO. 2 2 Usig rks to estimte the cumultive distributio fuctio we write g r h cov ( μi, FOi)= μ + μ h = + k 2 ( + ) + = μ 2 2 ( + ) ( + ) + μ = ( μ μ ). 2 Thus we c write r + k = + 2 (A2) G B = ( ) = ( ) μ μ g 2cov μi, FOi μ. 2 μ Tht is, the betwee comoet i the covetiol Gii decomositio, G B,is twice the covrice betwee the grou s verge icome d the verge rk of the grou divided by the overll me. From the defiitio of covrice oe hs (A3) ( ) ( + + ) g g g cov ( μi, FOi)= μ FO + μ FO. 2 2 The usig the rk to estimte the cumultive distributio, we obti Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth

15 Review of Icome d Welth, Series 57, Number 3, Setember 20 (A4) g r h rk cov ( μi, FOi)= μ + μ h + + = k 2 = +. 2 Addig d subtrctig R ik i the sums of (A4), exressio (A3) rewrites: (A5) (A6) g rh Rh R h cov ( μi, FOi)= μ + + h = 2 + μ rk Rk + Rk + k c = + 2, ( ) ( + + ) g cov ( μi, FOi)= μ R + μ R 2 2 μ μ ( R h h rh )+ ( r k k Rk ) =. = + We observe tht the differece betwee the rk of y k i Y-orderig (R k) d the rk of y k i g-orderig (r k) reresets the umber of icomes belogig to grou less th y k. Alogously, lookig t the comrisos betwee icomes (belogig to differet grous) from the ersective of grou, the differece r k - R k reresets the umber of elemets of grou greter th y k. The sum Rh r h= ( h ) is the umber of times tht icome of grou is greter th icome of grou, d the sum ( rk R ) k= k c + is the umber of times tht icome of grou is lesser th icome of grou, so tht k= + k k h= ( ) ( r R )= R r Moreover, from Yitzhki d Lerm (99,. 32) we hve The we c write: ( ) ( + + ) R 2 R μ + μ μi FOi 2 = cov (, ). ( i Oi) = ( i Oi) + ( ) ( h h h) = g 2 2cov μ, F μ 2cov μ, F μ 2 μ μ R r μ or equivletly ( i Oi) = ( i Oi) + ( ) = + g G 2 2cov μ, F μ 2cov μ, F μ 2 μ μ ( r R ) μ which roves (A) d thus (9). h h k k k. Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth

16 Review of Icome d Welth, Series 57, Number 3, Setember 20 Refereces Frick, J. R., J. Goebel, E. Schechtm, G. G. Wger, d S. Yitzhki, Usig Alysis of Gii (ANOGI) for Detectig Whether Two Subsmles Rereset the Sme Uiverse: The Germ Socio-Ecoomic Pel Study (SOEP) Exeriece, Sociologicl Methods d Reserch, 34, , Gii, C., Il Cocetto di Trsvrizioe e le Sue Prime Aliczioi, 96, reroduced i Corrdo Gii Memorie di Metodologi Sttistic, Volume II: Trsvrizioe, Uiversità degli Studi di Rom, 55, 959. Giorgi, G. M., Gii s Scietific Work: A Evergree, Metro Itertiol Jourl of Sttistics, 63, , Lsswell, T. E., Clss d Strtum, Houghto Miffli, Bosto, MA, 965. Milovic, B. d S. Yitzhki, Decomosig World Icome Distributio: Does the World hve Middle Clss? Review of Icome d Welth, 48, 55 78, Moti, M. d A. Storo, A Note o Betwee-Grou Iequlity with Alictio to Households, Jourl of Icome Distributio, 8(3 4), 34 48, Pytt, G., O the Iterrettio d Disggregtio of Gii Coefficiet, Ecoomic Jourl, 86, , 976. Yitzhki, S., Ecoomic Distce d Overlig of Distributios, Jourl of Ecoometrics, 6, 47 59, 994. Yitzhki, S. d R. Lerm, Icome Strtifictio d Icome Iequlity, Review of Icome d Welth, 37, 33 29, 99. Review of Icome d Welth Itertiol Associtio for Reserch i Icome d Welth