How Has the Literature on Gini s Index Evolved in the Past 80 Years?

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1 How Has the Literature o Gii s Idex Evolved i the Past 80 Years? Kua Xu Departmet of Ecoomics Dalhousie Uiversity Halifax, Nova Scotia Caada B3H 3J5 Jauary 2004 The author started this survey paper whe visitig the Graduate School, the People s Bak of Chia ad the School of Fiace, Remi Uiversity i the summer of The author would like to thak the two istitutios for playig the hosts. The author also wishes to thak Professor Yag Yao at the Chia Ceter for Ecoomic Research of Pekig Uiversity for his ecouragemet ad suggestios, ad Zhiyog Huag, a studet at Na Kai Uiversity for correctig some errors i the earlier draft of this paper. I additio to what have bee writte here, the origial pla also icludes two sectios o the geeralizatio ad statistical ifereces of the Gii idex. This would make this paper too legthy. The author chooses to leave those topics i aother survey paper. A brief survey of this kid always rus the risk of ievitable omissios. The author wishes to thak all authors, cited ad u-cited, for their cotributios to this large ad sigificat literature. 1

2 Abstract The Gii coefficiet or idex is perhaps oe of the most used idicators of social ad ecoomic coditios. From its first proposal i Eglish i 1921 to the preset, a large umber of papers o the Gii idex has bee writte ad published. Goig through these papers represets a demadig task. The aim of this survey paper is to help the reader to avigate through the major developmets of the literature ad to icorporate recet theoretical research results with a particular focus o differet formulatios ad iterpretatios of the Gii idex, its social welfare implicatio, ad source ad subgroup decompositio. 2

3 1 Itroductio Sice the Gii coefficiet or idex as a summary statistics bore Gii (1912, 1914, 1921) s ame as we ow kow it, the theoretical literature has evolved for more tha 80 years. Durig the past 80 years, the Gii idex gradually became oe of the pricipal iequality measures i the disciplie of ecoomics. This measure is uderstood by may ecoomists ad has bee applied i umerous empirical studies ad policy research. 1 As may are aware, research o iequality ad poverty measuremet cotiues to evolve. May ecoomists, experieced ad ewly mited, always wish to have the collectio of the theoretical results o the Gii idex hady ad accessible. Aad (1983) ad Chakravarty (1990) provided comprehesive surveys o the measures of iequality icludig Gii idex. But the literature is i a costat state of flux eve i the research area which is cosidered to be well established. Other authors such as Lambert (1989), Silber (1999), ad Atkiso ad Bourguigo (2000) also provided comprehesive refereces for icome iequality ad poverty with the Gii idex as oe of may iequality measures. This survey paper differs from those refereces i that it collects the theoretical results oly o the Gii idex, old ad more recet, i oe place. 2 1 May ecoomists i Chia such as Mr. Li Shi also worked o the Gii measures usig the Chiese data. Accordig to the ews coferece give by Premier Zhu Rogji of the People s Republic of Chia i March 15, 2001, the Gii idex calculated i 1999 i Chia reached 0.39 which was cosidered at a alarmig level by the iteratioal stadard. The Chiese Govermet vowed to solve the problem durig the developmet process. 2 Thak Buhog Zheg for poitig out a excellet techical survey paper o the Gii idex by Yizhaki (1998), which presets the results oly based o the cotiuous distributios. Hece the curret paper may still be justifiable give it focuses o both the history ad techical results ad cosiders both discrete ad cotiuous distributios. 3

4 This survey paper is a atural cotiuatio of Aad (1983) ad Chakravarty (1990) o this specific literature ad attempts to icorporate additioal research results o the Gii idex. It is hoped that this paper will ot oly provide readers a summary of mai theoretical literature but also cover some related issues such as differet formulatios ad iterpretatios of the Gii idex, its social welfare implicatio, ad source ad subgroup decompositio. The Gii idex ca be used to measure the dispersio of a distributio of icome, or cosumptio, or wealth, or a distributio of ay other kids. But the kid of distributios where the Gii idex is used most is the distributio of icome. For this reaso, ad for simplicity, this paper will focus o the Gii idex i the cotext of icome distributios although its applicatios should ot be limited to icome distributios. A icome distributio may be for differet icomes: household icomes or idividual icomes. The choice of the icome uit is ofte determied by the purpose of research. For simplicity, the discussio i this paper is based o icome distributios of the idividuals withi the populatio. Eve the cocept of icome distributio ca vary from the icomes that are pre tax ad other fiscal trasfers to those that are after tax ad other fiscal trasfers. As a covetio, this is ot the focus of the paper. Istead, the paper focuses o theoretical results ad iterpretatios. Similarly, the trasformatio from a family icome to idividual icomes via equivalet scale will ot be discussed here. This paper will show that the Gii idex has may differet formulatios ad iterestig iterpretatios. It ca be expressed as a ratio of two regios defied by a 45 degree lie ad a Lorez curve i a uit box, or a fuctio of Gii s mea differece, or a covariace betwee icomes ad their raks, 4

5 or a matrix form of a special kid. Each formulatio has its ow appeal i a specific cotext. The Gii idex was proposed as a summary statistics of dispersio of a distributio. It was viewed, for a quite log time, ot too differet from other dispersio measures such as variace ad stadard deviatio. But whe comig to a decisio as to which iequality measure should be adopted i a study, ecoomists foud that it was rather difficult to select oe statistics over others without ay justificatio i terms of social welfare implicatio. Thus, they started to search the lik betwee the existig iequality measures ad their uderlyig social welfare fuctios. It is ow kow that may well-kow iequality measures ideed have direct, although implicit, relatios with social welfare fuctios ad that the measured iequality ca be iterpreted as social welfare loss due to iequality. With this itellectual premise, the social welfare implicatio of a Gii idex value ca ow be iterpreted with a greater clarity. Ecoomists also examied how the Gii idex as a aggregate iequality measure could be decomposed accordig to either icome sources or subpopulatio groups. A great effort has bee made to specify the coditios uder which such decompositios are feasible. Eve whe decompositios are feasible, it is ot always clear what meaigful iterpretatio each ad every decomposed parts of the Gii idex has. More specifically, whe subgroup decompositio is made of the Gii idex, oe term called the crossover term appears difficult to iterpret. Over time, this view has chaged. Now may ecoomists foud that this term ca be viewed as a measure of icome stratificatio or the degree to which the icomes of differet social groups 5

6 cluster. The remaiig paper is orgaized as follows. Sectio 2 itroduces ecessary mathematical symbols ad basic defiitios. Sectio 3 reviews the evolutio of computatio methods of the Gii idex. Sectio 4 revisits the literature o Pigou-Dalto s priciple of trasfers ad social welfare implicatio of the Gii idex. Icome source ad subgroup decompositio are discussed i Sectio 5. Fially, Sectio 6 cocludes. 2 Mathematical Symbols ad Defiitios There are geerally two differet approaches for aalyzig theoretical results of the Gii idex: oe is based o discrete distributios; the other o cotiuous distributios. The latter demads certai coditios o cotiuity while the former does ot require such coditios. The discrete icome distributio is easy to uderstad i some cases while the cotiuous icome distributio ca simplify some derivatios i some situatios. The two ca be uified as show i Dorfma (1979). Whe the distributio fuctio is discrete, y takes values that ca be deoted by a 1 colum vector y = [y 1, y 2,..., y ] such that the elemets i the vector are arraged i o-decreasig order: y 1 y 2 y. The values of y are bouded below by a = 0 ad above by b < +, or y i [a, b] for i = 1, 2,...,. The otatio is used to sort y i the opposite order; i.e., the elemets of ỹ = [ỹ 1, ỹ 2,..., ỹ ] are arraged i o-icreasig order: ỹ 1 ỹ 2 ỹ. The discrete cumulative distributio fuctio is F i = i, while the probability of y takig o the value y i is f i = 1 for i = 1, 2,...,. 6

7 F (y k ) = k i i=1 is the cumulative probability up to y k ad ca be iterpreted as the populatio share of those whose icomes are less tha or equal to y k. 3 The mea icome, µ y, is give by µ y = 1 i=1 y i. The cumulative icome shares of the populatio up to the idividual whose icome is raked ith from the lowest to the highest are give by L i = 1 µ y i y j, (1) for i = 1, 2,...,. L 0 is defied as zero while L = 1. Note that L i s are arraged i o-decreasig order. Sometimes, ecoomists wish to use L i to idicate L i s that are arraged i o-icreasig order [see equatio (29)]. Whe the icome distributio is cotiuous, y ca be viewed as a value of the cumulative distributio fuctio of icome F (y) or a distributio fuctio of icome f(y). j=1 I geeral, y is bouded below by a = 0 ad above by b < +. F (a) = 0 while F (b) = 1. F (y ) = y f(y)dy is the cumulative probability up to y. The mea icome, µ y, is give by µ y = b ydf (y) = b yf(y)dy. The cumulative icome share of the popula- a a tio up to the idividual whose icome is y is give by L(p ) = L(F (y )) = 1 y yf(y)dy. (2) µ y The Lorez curve was hited by Sir Leo Chiozza Moey (1905) ad origially proposed by Mr. M. O. Lorez i It is deoted as L(p) = 3 Whe the complex samplig survey desigs are used to collect icome data, the researcher must deal with the issues of statistical iferece ad samplig weights. 4 See Publicatios of the America Statistical Associatio, Vol. ix, pp. 209ff. a a 7

8 L(F (y)), the proportio of the total icome of the ecoomy that is received by the lowest 100p of the populatio for all possible values of p. I other words, the graph of F ad L is the Lorez curve with 0 F 1 ad 0 L 1. For a discrete distributio, k = F (y k), F 1 ( k ) = y k, ad the Lorez curve is L k = L( k ) = L(F (y k)) for 0 k 1. For a cotiuous distributio, p = F (y), F 1 (p) = y, ad the Lorez curve is L(p) = L(F (y)) for 0 p 1. Figure 1 shows a Lorez curve is below the 45 degree lie. This reflects that the icome share grows at a much slower rate as the populatio share icreases ad that there exists a higher degree of icome cocetratio withi the populatio. 1 Figure 1. Lorez Curve % of the icome Lie of Perfect Equality Area A Area B Lorez Curve % of the populatio 3 Evolutio of Computatioal Methods The Gii idex as is called today was, accordig to Dalto (1920, p. 354), amed after the fact that a remarkable relatio has bee established be- 8

9 twee this measure of iequality ad the relative mea differece, the former measure beig always equal to half the latter. This remarkable relatio was first give by Gii i Dalto (1920, p. 353) therefore called this mea differece as Professor Gii s mea differece. The computatioal methods for the Gii idex iclude the geometric approach, Gii s mea differece approach (or the relative mea differece approach), covariace approach, ad matrix form approach. Each approach has its ow appeal ad is desirable i some way but all ca be uified ad are cosistet with oe aother. These methods ad their techical justificatios are examied i the followig. 3.1 Geometric Approach The attractiveess of the Gii idex to may ecoomists is that it has a ituitive geometric iterpretatio. That is, the Gii idex ca be, as i Figure 1, defied geometrically as the ratio of two geometrical areas i the uit box: (a) the area betwee the lie of perfect equality (45 degree lie i the uit box) ad the Lorez curve, which is called Area A ad (b) the area uder the 45 degree lie, or Areas A + B. Because Areas A + B represets the half of the uit box, that is, A+B = 1, the Gii idex, G, ca be writte 2 as G = A A+B = 2A = 1 2B. If oe works with a discrete icome distributio, he or she ca compute (3) 9

10 F i s ad L i s ad the the area below the Lorez curve B = 1 1 (F i+1 F i ) (L i+1 + L i ). (4) 2 i=0 Substitutig equatio (4) ito equatio (3) yields the Gii idex G: 5 1 G = 1 (F i+1 F i ) (L i+1 + L i ). (5) i=0 To illustrate how to use the above defiitio, let the hypothetical icome distributio be y 1 = 0, y 2 = 1, y 3 = 2. For this distributio, the Lorez curve ca be described by the poits (L 1 = 0, F 1 = 1 3 ), (L 2 = 1 3, F 2 = 2 3 ), ad (L 3 = 1, F 3 = 1) i the uit box. As idicated i Figure 2, Area B is the sum of the area of a small triagle ( 1 ), the area of a square ( 1 ), ad the 18 9 area of a large triagle ( 1 9 ): B = = The Gii idex, as idicated i equatio (5), G = 1 [( ) 1 (0) + 3 ( 1 3 ) ( ) ( 1 3 ) ( )] 4 = There are various expressios of this defiitio. For example, Yao [1999, p. 1251, equatio (1)] adopted a spread sheet approach usig this method. Osberg ad Xu [2000, p. 59, equatio (14)] modified the defiitio to accommodate the complex samplig survey data. 10

11 Figure 2. Lorez Curve ad Gii Idex % of the icome Lie of Perfect Equality Area A Area B Lorez Curve % of the populatio Several alterative formulatios i fact follow the same traditio, Rao (1969) showed that the Gii idex ca be defied as 1 G = (F i L i+1 F i+1 L i ). (6) i=1 This formulatio ca be show to be cosistet to equatio (5): give F = L = 1 ad F 0 = L 0 = 0, the Gii idex defied i equatio (5) ca be rewritte as 1 1 G = 1 + (F i L i+1 F i+1 L i ) (F i+1 L i+1 F i L i ). i=0 i=0 Sice the last term o the right-had side is oe, we have equatio (6). 11

12 Se (1973) defied the Gii idex as G = µ y ( + 1 i) y i. (7) This formulatio illustrates that the icome-rak-based weights are iversely associated with the sizes of icomes. That is, i the idex the richer s icomes get lower weights while the poor s icome get higher weights. Se s defiitio 6 i=1 ca be derived from equatio (6) by otig G = 1 i=1 (F il i+1 F i+1 L i ) = i=1 (F i 1L i F i L i 1 ) = i=1 (F i (L i L i 1 ) (F i F i 1 ) L i ) ( 1 = 2 µ y i=1 iy i ) i j=1 y j (8) give F i = i, L i = 1 µ y i j=1 y j, F i F i 1 = 1, ad L i L i 1 = y i µ y. The expressio for G ca be further maipulated as [ 1 G = 2 µ y i=1 iy i ] i i=1 j=1 y j = 1 2 µ y [ i=1 iy i i=1 ( + 1 i)y i] = 1 2 µ y [ i=1 ( + 1)y i 2 i=1 ( + 1 i)y i] = µ y i=1 ( + 1 i)y i. (9) The last equality is cosistet with equatio (7). Fei ad Rais (1974) ad Fei, Rais, Kuo (1978) defied the Gii idex 6 Se (1997) gives a slightly differet defiitio usig the icomes sorted i oicreasig order. 12

13 as a liear fuctio of a u y -idex: where the u y -idex is give by G = 2 u y + 1. (10) u y = i=i iy i i=1 y. i Substitutig u y = i=1 iy i i=1 y i ito equatio (10) yields 2 G = 2 µ y i=1 iy i +1 = +1 2(+1) µ y i=1 iy i = µ y i=1 ( + 1 i)y i, (11) which is cosistet with equatio (7). If oe deals with a cotiuous icome distributio, he or she ca express the area uder the Lorez curve as B = 1 0 L(p)dp. (12) Substitutig equatio (12) ito equatio (3) yields the Gii idex for the cotiuous icome distributio as G = L(p)dp. (13) Clearly, it is much simpler to uderstad the Gii idex geometrically. However, its computatio may be tedious. 13

14 3.2 Gii s Mea Differece Approach Differig from the geometric approach, Gii (1912) showed that the geometric approach is i fact related to the statistical approach via a cocept called the (absolute ad relative) mea differece. That is, the Gii idex as a ratio of two areas defied above is always equal to the half of the relative mea differece that will be explaied later. Accordig to David (1968), the relative mea differece discussed by ad amed after Gii (1912) was i fact discussed much earlier by F. R. Helmert ad other Germa writers i the 1870 s. I 1912, Gii s book was published i Italia ad hece was ot accessible to Eglish-speakig ecoomists at the time. I 1921 whe commetig o Dalto s (1920) work, Gii (1921) explaied his work (1912) ad related literature i Eglish i a short Ecoomic Joural article. Sice the, the Gii idex ad Gii s relative mea differece were made kow i the literature of icome iequality measuremet i the Eglish-speakig world. Followig Gii (1912), Kedall ad Stuart (1958) i their well-kow book Advaced Theory of Statistics stated the Gii idex as the half of Gii s relative mea differece because it was ideed a importat statistical result at that time. There is o doubt that may geeratios of statisticias leared this result through the classical work of Kedall ad Stuart. Gii s absolute mea differece for a discrete distributio is defied as = 1 2 y i y j. (14) i=1 j=1 where y i ad y j are the variates from the same distributio. The absolute 14

15 mea differece for a cotiuous distributio is defied similarly as the mea differece betwee ay two variates of the same distributios: = E y i y j (15) where E is the mathematical expectatio operator. The relative mea differece is defied as µ y = E y i y j µ y. (16) That is, the relative mea differece equals the absolute mea differece divided by the mea of the icome distributio. I additio to the above result, Shalit ad Yitzhaki (1984) also provided several alterative ways to express Gii relative mea differece. The Gii idex is the oe-half of Gii s relative mea differece The above expressio is also writte as G = G = 2µ y. (17) 1 2 i=1 i=1 max(0, y i y j ) (18) µ y because i=1 j=1 y i y j = 2 i=1 i=1 max(0, y i y j ) [see Pyatt (1976)]. Aad (1983) showed that equatio (17) is cosistet with the geometric defiitio give i equatio (5). For a discrete distributio, the absolute mea differece ca be rewritte as 2 2 i=1 j i (y i y i ) ad the Gii 15

16 idex ca be expressed as G = 1 2 µ y i=1 = 1 2 µ y i=1 j i (y i y i ) ( iy i i j=1 y j ). (19) This result is cosistet with equatio (8). I other words, Gii s mea differece approach is cosistet with the geometric approach. Followig the traditio of Kedall ad Stuart, Dorfma (1979) proposed a simple formula for the Gii idex for the cotiuous icome distributio, that is G = 2µ y = 1 1 µ y b a (1 F (y)) 2 dy (20) where Gii s absolute mea differece for a cotiuous icome distributio is give by equatio (15). They also oted that Gastwirth (1972) proposed a similar formula without a proof which was attributed to Kedall ad Stuart (1977) who also omitted the proof. This formula ca be derived as follows. Because y i y j = y i + y j 2 mi(y i, y j ), Gii s absolute mea differece is writte as = E y i y j = 2µ y 2E mi(y i, y j ). To lear more about the term E mi(y i, y j ), it is ecessary to kow the probability for mi(y i, y j ), that is Pr [mi(y i, y j ) y] = 1 Pr (y i > y) Pr (y j > y) = 1 (1 F (y)) 2. 16

17 Icorporatig this probability ito Gii s absolute mea differece yields = E y i y j = 2µ y 2 b Substitutig ito G yields yd ( 1 (1 F (y)) 2) = 2µ y +2 b a a yd (1 F (y)) 2. G = = 2µ b y + 2 yd (1 F (y))2 a. 2µ y 2µ y Sice a = 0 ad b is fiite, a(1 F (a)) 2 = b(1 F (b)) 2 = 0, G = 1+ 1 µ y (y(1 F (y)) 2 b a b a ) b (1 F (y)) 2 dy = 1 1µy (1 F (y)) 2 dy. a The defiitio of the Gii idex based o the relative mea differece has its root i the statistics. However, its computatio could be complex. It is the covariace approach, which will be discussed below, that ca facilitate the computatio of the Gii idex usig the commoly used covariace procedure i most statistical software packages. 3.3 Covariace Approach It was kow that the Gii s absolute mea differece ca be expressed as a fuctio of the covariace betwee variate-values ad raks as oted i Stuart (1954, 1955): = 4 b a ( y F (y) 1 ) f(y)dy. (21) 2 17

18 But, fidig of the lik betwee this fact ad the computatio of the Gii idex occurred much later. I the cotext of discrete icome distributios, Aad (1983) showed the Gii idex ca be computed by 7 G = 2cov(y i,i) µ y ; (22) that is, the Gii idex ca be expressed as a fuctio of the covariace betwee icomes ad their raks. I the cotext of the discrete icome distributio, Aad demostrated how the defiitio of equatio (22) is justified. He oted that the mea of the raks is give by i = 1 i=1 i = ad the covariace is expressed as cov(y i, i) = 1 i=1 Thus, the Gii idex ca be writte as ( yi µ y ) ( i i ) = 1 i=1 iy i +1 2 µ y. G = 2cov(y i,i) µ y = 2 2 µ y which is cosistet with equatio (11). i=1 iy i + 1 Idepedetly, Lerma ad Yitzhaki (1984) also reported the same result 7 Based o the author s commuicatio with Sudhir Aad, the author leared that Sudhir Aad s thesis, which is the basis of Aad (1983), was completed i

19 for cotiuous icome distributios. Yitzhaki (1982) oted that accordig to Lomicki (1952) the Gii s absolute mea differece of the distributio F ca be writte as = b b a Lerma ad Yitzhaki (1984) 8 a x y f(x)f(y)dxdy = 2 b u = F (y) [1 F (y)] ad v = y, ca be writte as 9 = 4 b a a F (x) [1 F (x)] dx. showed that, by itegratio by parts, with ( y F (y) 1 ) f(y)dy. 2 By trasformatio of variables, write f(y)dy = df ad chage from the bouds [a, b] for y to the bouds [0, 1] for F, = ( y(f ) F 1 ) df. 2 Note that F is uiformly distributed betwee 0 ad 1 so that its mea is 1 2. Thus, Gii s absolute mea differece is give by = 4cov [y, F (y)] ad the Gii idex is defied as G = 2µ y G = 2cov [y, F (y)] µ y. (23) 8 This result i Charavarty (1990, p. 88) is called the Stuart(1954)-Lerma-Yitzhaki (1984) propositio. 9 This is cosistet to Stuart[1954, pp , equatios (13) (15)]. 19

20 I the cotext of cotiuous icome distributios, Lambert (1989) used a slightly differet approach to the same problem. He first oted that the Lorez curve L(p) has the followig property: L (p) = dl(p) dp = dl(p)/dy dp/dy = yf(y)/µ y f(y) = y µ y. (24) The usig itegratio by parts he rewrote the Gii idex from equatio (13) as G = L(p)dp = pl (p)dp 1 = 2 b a yf (y) µ y f(y)dy 1. (25) Sice cov[y, F (y)] = E[yF (y)] E(y)E[F (y)] ad E[F (y)] = 1, the Gii 2 idex ca be derived from equatio (25) as G = [ ] b 2 yf (y)f(y)dy µ y a 2 = µ y 2Cov[y, F (y)] µ y. (26) Although the derivatios of Aad (1983), Lerma ad Yizhaki (1984) ad Lambert (1989) differ, each result is a variat of the other. To compute the Gii idex usig Aad s approach, first obtai rak for each observatio y i, i ; the, compute the covariace betwee y i ad i, cov(y i, i). The resultig covariace must be divided by the umber of the observatios, cov(y i, i ) = 1 cov(y i, i), sice i/ terms are empirical cumulative distributio of F (y). Fially, the Gii idex is computed by G = 2 µ y cov(y i, i). This is cosistet with the result of Lerma ad Yitzhaki (1984) ad Lambert (1989). This approach is also exteded by Shalit (1985) so that a regressio ca be used 20

21 to compute the Gii idex. The advatage of the covariace approach is that the computatio of the Gii idex ca be facilitated by usig the covariace procedure i existig statistical software packages. 3.4 Matrix Form Approach I the literature, the matrix form approach was proposed by Pyatt (1976) ad Silber (1989) for decompositio purposes. 10 Pyatt (1976) focused o equatio (18) i which the Gii idex is iterpreted as a ratio of two terms: (a) the umerator 1 2 i=1 i=1 max(0, y i y j ) is the average expected gai to be expected by the populatio if each ad every idividual i the populatio is allowed to compare his or her icome y i with ay other chose idividual s icome y j ad to take y j whe y j is greater tha y i ad (b) the deomiator is the mea icome µ y. Cosider that the populatio ca be divided ito k subpopulatio groups with group i has p i proportio of the populatio. It is possible to express the average expected gai as k i=1 k E(gai i j) Pr(i j) (27) i=1 where Pr(i j) = p i p j for all i, j = 1, 2,..., k ad k i=1 p i = 1. Let E be a k k matrix with elemets beig E ij = E(gai i j). Stack p i s ito a k 1 colum vector p. Let the average icome for group i be m i. Stack m i s ito a k 1 colum vector m. Hece m p = k i=1 m ip i = µ y. Thus, the Gii 10 Also see Yao (1999) for a spread sheet applicatio. 21

22 idex, correspodig to equatio (18), ca be defied as G = (m p) 1 p Ep. (28) Silber (1989) proposed aother elegat approach for computig the Gii idex. The derivatio starts from the Gii idex s defiitio: G = j=1 L j [ ( j) ] (j 1), (29) where L i is the proportio of total icome eared by the idividual whose icome has the ith rak i the icome distributio with L 1 L 2 L j L. (This meas that the richest idividual is raked 1st while the poorest idividual is raked th.) First of all, examie how this defiitio is liked to the previous oes. Note that L j = y j j=1 y j implies that y j s are arraged i o-icreasig order while y i s i our previous discussio are arraged i o-decreasig order. Therefore, it is useful to fid out the system lik betwee two idex systems. It turs out that i = j + 1 ad that equatio (29) ca be rewritte as G = i=1 [ y i i 1 i=1 y i i ], 22

23 which i tur equals [ 1 G = i=1 y i i=1 y i i=1 = µ y i=1 (2 2i + 1) y i ( 2 2i+1 ) ] yi = µ y i=1 ( + 1 i) y i. The last equality i the above is cosistet with equatio (7). Secod, Equatio (29) ca be readily show to be G = [ L i j i i=1 1 j i ] 1. (30) Equatio (30) is equivalet to equatio (29) because the sum i the former is idetical to the sum i the latter as show below: j Lj [ ( j) ] (j 1) i Li [ j i 1 j i ] 1 j = 1 j = 2. j = L1 [ 1 L2 [ 2. L [ ] ( = 1 ) L1 ] = L2 ( 3 ) ] ( = 1 ) L [ i = 1 L1 ( 1 ] = 1 ) L1 [ i = 2 1 L2 ] ( 2 = 3 ) L2.. i = L [ 1 ] = L ( 1 ) Third, equatio (30) ca be readily writte compactly i matrix form as G = e G L (31) 23

24 or [ 1 G =, 1,..., 1 ] L 1 L 2. L. where e is a colum vector of elemets of 1/, L is a colum vector of elemets beig respectively equal to L 1, L 2,..., L, ad G is the G-matrix whose elemets G ij are equal to -1 whe i < j, to 1 whe i > j, to 0 whe i = j. 4 Social Welfare Implicatio From the statistics poit of view, the Gii idex is a fuctio of Gii s mea differece ad hece it was iitially, ad still is, iterpreted a measure of dispersio. Pyatt (1976), however, wet a bit further ad gave the Gii idex a iterpretatio as the average gai to be expected, if each ad every idividual is allowed to compare his or her icome with the icome of aother idividual ad to keep the icome that is higher. But this iterpretatio is statistical i ature ad more coveiet for subpopulatio group decompositio rather tha for measurig social welfare (loss) due to iequality. I fact, Dalto i his 1920 paper, followig Pigou (1912, p. 24), had attempted to raise a miimum criterio for a iequality measure. It is ow called Pigou-Dalto s priciple of trasfers. To establish this priciple, he said: 24

25 We have, first, what may be called the priciple of trasfers.... we may safely say that, if there are oly two icome-receivers, ad a trasfer of icome takes place from the richer to the poorer, iequality is dimiished. There is, ideed, a obvious limitig coditio. For the trasfer must ot be so large, as more tha to reverse the relative positios of the two icome-receivers, ad it will produce its maximum result, that is to say, create equality. Ad we may safely go further ad say that, however great the umber of icome-receivers ad whatever the amout of their icomes, ay trasfer betwee ay two of them, or, i geeral, ay series of such trasfers, subject to the above coditio, will dimiish iequality. (Dalto, 1920, p. 351) Dalto (1920) also oted that the Gii idex ca be viewed as half of the Gii s relative mea differece. Accordig to Dalto, as the relative mea differece satisfies the priciple of trasfers, the Gii idex must satisfies the same priciple ad be judged as a desirable iequality measure. Jekis (1991), amog others, used the total differetial approach to evaluate whether the Gii idex ideed satisfies the priciple of trasfers whe the trasfers are very small. To do so he assumed that the trasfer is meapreservig (i.e., µ y is fixed) ad that there is a trasfer from to the richer idividual i to the poorer idividual j but this trasfer will ot chage the fact the relative positios of the rich ad the poor i the icome distributio. Takig the total differetial of equatio (7) with respective to y i ad y j yields G = ( G/ y j )dy j ( G/ y i )dy i = 2(j i) 2 µ y dy < 0 (32) 25

26 give that dy i = dy j, j < i, ad dy i = dy j = dy. Thus, the Gii idex ideed satisfies the priciple of trasfer. That is, whe the trasfer occurs, the value of the Gii idex will decrease. Although the Gii idex ideed satisfies the priciple of trasfers, there was little discussio about the social welfare implicatio of iequality measures icludig the Gii idex after Dalto (1920). For example, Gii (1921) himself, i respose to Dalto s work (1920), suggested that the measure of iequality (such as the oe he proposed) was of icome ad wealth ot of ecoomic welfare. The ormative approach, which relates a iequality measure directly to a uderlyig social welfare fuctio, appeared much later. Kolm (1969) advocated the use of social welfare fuctio i measurig icome iequality. Atkiso (1970) oted that the social welfare implicatio was particularly importat whe oe came to select a summary statistics of icome iequality. He wrote: Firstly, the use of these measures ofte serves to obscure that fact that a complete rakig of distributios caot be reached without fully specifyig the form of the social welfare fuctio. Secodly, examiatio of the social welfare fuctios implicit i these measures shows that i a umber of cases they have properties which are ulikely to be acceptable, ad i geeral these are o grouds for believig that they would accord with social values. For these reasos, I hope that these covetioal measures will be rejected i favour of direct cosideratio of the properties that we should like the social welfare fuctio to display. 26

27 (Atkiso, 1970, p. 262) Se (1973) also discussed this approach as a geeralizatio of Atkiso s measure. It is Blackorby ad Doaldso (1978) who examied the issue further i a systematical fashio, established the geeral results ad applied them to iequality measures icludig the Gii idex. The way to lik the Gii idex to its uderlyig social welfare fuctio is to defie the Gii idex i terms of the equally-distributed-equivaleticome (EDEI), or the represetative icome proposed by Atkiso (1970), Kolm (1969), ad Se (1973). Usig this approach, a iequality measure, I, ca be writte as a fuctio of the EDEI icome, ξ, ad the mea icome, µ y. I = 1 ξ. (33) µ y If I is defied based o the Gii social welfare fuctio, the I is deoted by I G or, simply, G. Give this setup, if ξ is idetical to µ y, the I is zero. That is, there is o iequality i the icome distributio from which ξ ad µ y are computed. If, o the other had, ξ is less tha µ y (say, the former is oly 70% of the latter), the I will be greater tha zero but bouded by 1 (I will take a value of 0.3). That is, there is some degree of iequality. Of course, it is crucial to kow how ξ is derived. Geerally speakig, for a particular social welfare fuctio or social evaluatio fuctio, a EDEI give to every idividual could be viewed as idetical i terms of social welfare to a actual icome distributio. To explai the idea further, let W (y) φ(w (y)) be a homothetic (ordial) social welfare fuctio of icome with φ beig a icreasig fuctio ad 27

28 W beig a liearly homogeeous fuctio. Let 1 be a colum vector of oes with a appropriate dimesio. The, W (ξ 1) = W (y) or W (ξ 1) = W (y). Give that W is positively liearly homogeeous, a EDEI is computed by ξ = W (y) W (1) = Ξ(y). The social welfare fuctio, W, ad the EDEI, Ξ, have a oe-to-oe correspodig relatioship. The homotheticity of the social welfare fuctio makes the idifferece curves blowig-out or -i proportioally. Uder this coditio, the EDEI fuctio, Ξ(y), is liearly homogeeous i y; that is, doublig y will also double ξ. The above idea ca be further explaied by the case of a icome distributio of two idividuals. Figure 3 shows a actual icome distributio deoted by the poit y (that is, the icome of the first idividual, y 1, is 2 while that of the secod idividual, y 2, is 5). I Figure 3, icome distributios o the 45 degree lie from the origi represet perfect equality (that is, y 1 = y 2 ) of the two-perso world. The social welfare fuctio W has idifferece curves like I 1 (the dotted covex curve) ad I 2 (the solid covex curve). Sice the actual icome distributio y is o idifferece curve I 2, EDEI, the poit at which I 2 ad the 45 degree lie cross, will give the same level of social welfare. Thus, with a particular social welfare fuctio W, a actual icome distributio y ad its correspodig EDEI are idifferet. 28

29 Figure 3. Social Welfare Fuctio ad EDEI 10 8 y y I 2 EDEI y 1 = y 2 2 I y 1 The Gii idex ca be defied with the Gii EDEI 11 Ξ G (y) 1 2 (2 2i + 1)y i (35) i=1 that correspods to the Gii social welfare fuctio, W G (y) = 1 2 i=1 (2 2i + 1)y i. 12 The Gii social welfare fuctio attaches a higher weight to a lower level of icome ad vice versa. The weight is determied by the rak of a icome rather tha the size of the icome. The Gii idex ca be defied 11 If we sort the elemets i y i o-icreasig order ad deote the ew vector as ỹ, the Ξ G(ỹ) = 1 2 (2i 1)ỹ i (34) with Ξ G (y) = Ξ G(ỹ). This is because y i = ỹ i+1 ad ỹ i = y i This is because W (1) = 1 2 i=1 i=1 (2 2i + 1) = 1. 29

30 i terms of the Gii EDEI ad the mea icome as 13 I G G 1 Ξ G(y) µ y = µ y (2 2i + 1)y i. (37) i=1 Note that while G 1 Ξ G(y) µ y takig values betwee 0 (perfect equality) ad 1 (perfect iequality) measures iequality; 1 G Ξ G(y) µ y also takig values betwee 0 (perfect iequality) ad 1 (perfect equality) measures equality. The term 1 G ca be viewed as a measure of equality. The social welfare implicatio of the Gii idex ca be better appreciated by the above formulatio. For example, whe the Gii idex is.3, this meas that accordig to the Gii social welfare fuctio the iequality reduces the social welfare to the level that oly 70% of the curret total icome if distributed equally amog the populatio ca achieve. The iequality ca be reduced to zero if the curret total icome ca be distributed equally amog the populatio. Whe this uderstadig is established, a society which is iequality averse would prefer the icome distributio with a lower Gii idex value to the oe with a higher Gii idex value. Of course, differet iequality measures may have very differet social welfare implicatios. Comparisos betwee the Gii idex ad other measures such as the coefficiet of variatio ad Theil idices 14 were made based 13 Alteratively, G(ỹ) 1 Ξ G(ỹ) µ y = µ y (2i 1)ỹ i, (36) where ỹ has elemets i o-icreasig order.the two equatios are idetical because y=ỹ, y i = ỹ i+1, ad ỹ i = y i+1. Note that i equatios (37) ad (36) G(y) = G(ỹ) but G( ) ad G( ) have differet fuctioal forms ad the elemets i y ad ỹ are sorted differetly. 14 See Theil (1967). i=1 30

31 o the uit-simplex by Se (1973, pp , Figures 3.3 ad 3.4), Blackorby ad Doaldso (1978, p. 73, Figure 4), ad Se (1997, pp , Figures A4.1 ad A4.2). 5 Subgroup ad Source Decompositio The Gii idex like ay other idices is a aggregate summary statistics of icome iequality ad it ca apply to a atio or regios/subgroups withi the atio. However, researchers frequetly wish to explore how iequality statistics i regios/subgroups cotribute to the atioal iequality. Similarly, the Gii idex is also applicable to each ad every icome compoet. How the iequalities i all icome compoets cotribute the overall icome iequality is also of iterest to social scietists. Chakravarty (1990) detailed subgroup ad source decompositio i Sectio 2.6 of his book. I geeral, the decompositio of a iequality measure ca be coducted either o some kid partitio of the populatio or o some divisio of the icome. The former is referred to as subgroup decompositio ad the latter is called as source decompositio. For subgroup decompositio, oe wishes to see how subgroup iequality measures ca be effectively related to the populatio iequality. By the same toke, for source decompositio, oe wishes to examie relatioship betwee the aggregated iequality measure ad the iequality measures from the compoets or sources. Whe a researcher does source decompositio, he or she must apply the Gii idex equatio to each ad every icome compoets. Presetig these Gii idices themselves will ot pose ay problem. It is how to relate these 31

32 Gii idices for differet icome compoets to the Gii idex of the aggregate icomes that becomes very challegig [see Silber (1993) ad refereces therei]. Because the Gii idex does ot always permit a clear ad explicit form of source decompositio, ormally pseudo-gii idices are used [see Chakravarty (1990)]. Subgroup decompositio is differet. Whe the populatio is divided ito K subgroups, the icomes for subgroup i also costitute a distributio ad hece the Gii idex for that subgroup, say G i, ca be computed. Similarly, the Gii idex of the distributio of the mea icomes of these subgroups ca also be computed ad is called the betwee-group Gii idex, or G B. Now let the weight for each group i, b i, be the product of the proportio of the populatio i subgroup i ad the proportio of the aggregate icome of the populatio i subgroup i. The, the followig relatioship ca be established: K G = G B + b i G i + R (38) i=1 where R is ormally called the crossover term. Bhatacharya ad Mahalaois (1967) are perhaps the first ecoomists workig o the subgroup decompositio of the Gii idex. Pyatt (1976) ad Das ad Parikh (1982) foud the same result usig the matrix form approach. Mookherjee ad Shorrocks (1982) oted that the crossover term is a awkward iteractio effect... impossible to iterpret with ay precisio. Shorrocks (1984) foud that the Gii idex is kow to be decomposable (without the crossover term) whe rakig icomes based o icome sizes leads to the subgroup icomes cluster ito subgroup icome rages without overlappig across subgroups. Silber (1989), however, suggested that the crossover effect of subgroup 32

33 decompositio of the Gii idex is ot troublesome ad, i fact, has a clear ad ituitive iterpretatio. It measures the itesity of the permutatio which occur whe istead of rakig all the idividual shares by decreasig (or icreasig) icome shares, oe raks them, firstly by decreasig (or icreasig) value of the average icome of the populatio subgroup to which they belog, ad secodly, withi each subgroup, by decreasig (or icreasig) idividual icome share. Lambert ad Aroso (1993) used a geometric approach to explai the crossover term ad gave it a similar ad good geometric iterpretatio. 15 Yitzhaki ad Lerma (1991) ad Yitzhaki (1994) wet a bit further suggestig that the crossover term would be a good idicator of icome stratificatio. They oted that sociologists ofte fid that the lack of the stratificatio dimesio, whe usig the Theil idex for subgroup decompositio, warrats some recosideratio whe the issue of stratificatio is of iterest to researchers. To examie how stratificatio amog populatio subgroups to overall icome iequality, Yitzhaki (1988, 1994) ad Yizhaki ad Lerma (1991) have also developed a pseudo-gii coefficiet, which mimics the Gii idex i some respects. Their idex decomposes iequality across populatio subgroups i a way which is differet from the covetioal way but is superficially similar: a crossover term, over ad above betwee- ad withi-group terms, measures stratificatio i a precisely defied sese ad has ituitively appealig properties [also see Lambert ad Aroso (1993), ]. Of course, some scholars have some reservatios about usig the Gii 15 Sastry ad Kelkar (1994) proposed a decompositio that is slight differet from oe proposed by Silber (1989). 33

34 idex for subgroup decompositio. Cowell (1988) used specific examples to show that the Gii idex could be a bad iequality measure because it allows the three thigs happe at the same time after some trasfers have occurred: (a) mea icome i every subgroup is costat; (b) iequality i every subgroup goes up; ad (c) overall iequality for the populatio of all subgroups goes dow. After a careful cosideratio, oe ca fid that these ca happe because withi-subgroup, ot across-subgroup, trasfers ca be made as such so that the coditios (a), (b), ad (c) ca be satisfied at the same time. Thus, this is ulikely to be a problem of the Gii idex. A decreasig overall iequality reflects a social welfare gai i the populatio as a whole, while icreasig iequality i every subgroup should be iterpreted as a social welfare loss for each subgroup whe evaluated idepedet of other subgroups of the populatio. Hece, whe the social welfare implicatio is cosidered, the Gii idex ad its subgroup decompositio appear to have a clear iterpretatio. 6 Coclusio Geerally, it ca be see that over the past 80 years sice Gii (1921) made his idex kow beyod Italy, our uderstadig of this idex has improved ad deepeed substatially. Now ecoomists have leared that there are may ways to formulate ad iterpret the Gii idex. That is, the Gii idex ca be computed based o the geometric approach, Gii s mea differece approach, covariace approach ad matrix form approach. The Gii idex is closely related to the 34

35 uderlyig Gii social welfare fuctio. I additio, source ad subgroup decompositio ca assist the aalysis of the iequality i icome compoets ad i subgroups. Now ecoomists has a much better uderstadig of the crossover term whe we deal with the subgroup decompositio of the Gii idex. The Gii idex is also a importat compoet of the Se idex of poverty itesity ad the modified Se idex of poverty itesity [see, for example, Xu ad Osberg (2001)]. The Gii idex has bee geeralized to the S-Gii ad E-Gii idex to reflect various level of the iequality aversio [see, for example, Xu (2000) ad the refereces therei]. Because the computatio of the Gii idex ad other iequality measures is ofte made based o sample data. The statistical ifereces based o the Gii idex become more ad more importat. [see, for example, Osberg ad Xu (2000), Xu (2000) ad Biewe(2002) ad the refereces therei]. Give the space of this survey, it is ot possible to iclude these topics which will be left as aother project i the future. Because the mai focus of this survey is o the methodology, may good empirical studies usig the Gii idex are ot cited here. 35

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