Measurement of Income Inequality: A Survey

Transcription

1 Forma Joural of Ecoomc Studes Vol. 3, 07 (Jauary December) pp. -3 Measuremet of Icome Iequalty: A Survey Muhammad Idrees ad Eatzaz Ahmad Abstract Ths paper revews varous equalty measures ad fds that oly few measures possess desrable propertes of a equalty measure. G coeffcet ad the coeffcet of varato are decomposable both addtve ad o-addtve forms ad possess all the propertes, except the strget Pgou-Dalto codto gve by Dmshg Trasfers Axom. Kawa dex ad four popular geeralzed G dces posses all the propertes but they are ot decomposable addtvely or o-addtvely. Sce sestvty of a equalty measure to the locato of come trasfers also vares across varous measures, hardly ay measure ca serve all the purposes ad t s desrable to employ more tha oe measure a emprcal aalyss of come equalty. Keywords: Icome Iequalty, Measuremet, Survey JEL Classfcato: E0, E5, C80. Itroducto Measuremet of equalty has bee a area of great terest for statstcas ad ecoomsts. Tll the ed of eghteeth cetury pure statstcal measures le rage ad mea devato were used to measure come equalty. However the early eteeth cetury a few specfc measures of equalty were proposed. I 905, Max Otto Lorez proposed a revolutoary graphcal measure of equalty ow as Lorez curve, from whch G 9 derved a parametrc measure of equalty, ow as G coeffcet. Sce the a szable lterature o the measuremet of come equalty has emerged. I aother remarable cotrbuto 90 Dalto led equalty to ecoomc welfare ad thereby orgates the dea of ormatve equalty measures, whch was further polshed by Atso, (970). Thel, (967) derved equalty measure from the oto of etropy formato theory. Later o t was realzed that G coeffcet ad some other measures are characterzed by certa rgdtes. For example, G coeffcet attaches more weght to come trasfers affectg mddle-come classes ad ot much weght The authors are respectvely Assocate Professor, School of Ecoomcs, Quad--Azam Uversty, Islamabad ad Professor o SBP Memoral Char, Uversty of Peshawar, Peshawar. Correspodg author s emal: mdrees@qau.edu.p

2 Idrees ad Ahmad to come trasfers wth extreme come classes. I case of Lorez curves the cocluso regardg the degree of equalty becomes ambguous whe the curves represetg two dfferet come dstrbutos tersect each other. Moreover for comparg socal welfare of two or more come dstrbutos Lorez curves are useful oly whe dstrbutos have the same mea comes. Such rgdtes were serously addressed ad a umber of geeralzed measures were proposed to overcome these flexbltes durg the 980s. Amog such geeralzatos basc focus of aalyss was o the geeralzato of Lorez curve ad G coeffcet. Shorrocs, (983) troduced geeralzato of Lorez curve, whle a large umber of geeralzatos of G coeffcet were proposed, amogst whch the oes proposed by Kawa, (980b), Doaldso ad Weymar, (980, 983) ad tzha, (983) became popular. Shorrocs (980) also proposed geeralzato of etropy dces. Apart from measuremet of equalty, aother vtal ssue has bee the splttg up of overall equalty to sub-compoets or sub-groups. There ca be at least two ways to coduct decomposto of equalty,.e. addtve ad oaddtve. A measure s sad to be addtve decomposable whe total equalty the populato uder cosderato ca be broe to a weghted average of the equaltes exstg betwee ad wth sub-groups of the populato. I oaddtve decomposto the focus of aalyss s o the cotrbuto of sub categores of the varable uder cosderato to total equalty. The lterature shows that ot all measures of equalty are decomposable. For example, geeralzed etropy dces ad Ebert, (999) dces are addtve decomposable oly, whle G coeffcet ca be decomposed both ways. Ths study presets a comprehesve revew of equalty measures. It cossts of fve sectos. Secto proposes a classfcato of equalty measures to two groups that ca be labeled as statstcal measures ad regular measures. The statstcal measures of equalty clude all the measures of dsperso that ca also be used to measure equalty. The regular measures, o the other had, clude the measures that are meat for measurg equalty. These ca further be sub-classfed to four groups, amely ordary measures, Lorez curve ad related measures; etropy measures ad pure welfare based measures. Secto 3 explas decomposg of equalty measure to sub groups ad sources. Secto 4 explas the desrable propertes that a good equalty measure s supposed to possess ad t evaluates all the measures cosdered o the bass of these propertes. Fally, secto 5 summarzes the etre dscusso.

3 Measuremet of Icome Iequalty: A Survey. Classfcato of Iequalty Measures Measures of equalty ca be classfed a umber of ways. Se, (973) classfed them to two categores amely postve ad ormatve measures. Postve measures are those, whch quatfy the extet of equalty a obectve sese usually by employg statstcal measures of dsperso. Normatve measures are based o explct formulato of socal welfare fucto that dcates the welfare loss arsg from a uequal dstrbuto of come. Thus postve measures see to descrbe the exstg patter of come dstrbuto as a sgle statstc wthout volvg ay value udgmet, whle ormatve measures base equalty o value udgmets. Postve measures clude rage, relatve mea devato, varace, coeffcet of varato, G dex, Lorez curve, etc. Some well-ow ormatve measures are Dalto measure ad Atso dex. As Se, (973) has poted out, o frm le ca be draw betwee postve ad ormatve measures ad may of the postve measures are specal cases of ormatve measures. Ths paper proposes the classfcato of equalty measures to the groups of statstcal measures of dsperso ad regular measures of equalty. Statstcal measures are desged to measure dsperso ay data ad are also useable for measurg come equalty. These are postve measures of equalty ad clude, rage, mea devato, relatve mea devato, varace, coeffcet of varato ad varace of logarthms. The measures of equalty, purely meat for measurg equalty are hereby referred to as regular measures. These measures ca be further classfed as ordary measures, Lorez curve ad related measures, etropy measures ad pure welfare based measures... Statstcal Measures... Rage Some of the well-ow statstcal measures of dsperso are as follows. Rage s defed as the dfferece betwee extreme values of a varable. Whle frst tme usg rage as a measure of equalty, Se, (973) dvded t by arthmetc mea order to mae t a ut free measure. Deotg the come of Strctly speag all the measures of equalty ca be regarded as ormatve measures because frst, choosg ay partcular so-called postve measure oe maes the value udgmet that the chose measure s a true dcator of equalty ad secod, may of the postve measures are specal cases of certa ormatve measures. 3

4 Idrees ad Ahmad come ut by ad arthmetc mea of come by, the rage as a measure of equalty s wrtte as: ( Max M ) R = () The rage taes values of zero ad for the extreme cases of perfect equalty ad perfect equalty respectvely. Rage s a smple measure of equalty but t completely gores the dstrbuto betwee the extreme come levels.... Mea Devato ad Relatve Mea Devato I order to cosder the etre dstrbuto, Bortewcz (889) proposed mea devato, defed below, whch depeds o the scale of measuremet. M = = A related measure proposed by Turro (90), whch s scale depedet, s the relatve mea devato: RM = = M = = If come s dstrbuted equally, the value of RM wll be equal to zero, whle case of perfect equalty where oe come ut holds the etre come, ts value wll be equal to ( ). Ths measure (alog wth mea devato) s ot sestve to come trasfers betwee come uts lyg o ay oe sde of mea come ad t assgs equal weght to small as well as large devatos from the mea. Both these problems are overcome by varace ad related measures...3. Varace ad Related Measures Istead of gorg sgs, varace taes squares of mea devatos before averagg. It s gve by V = = ( ) A related measure s stadard devato, whch s the square root of varace. Both these measures are scale depedet. Ths defcecy s overcome by coeffcet of varato: () (3) (4) 4

5 Measuremet of Icome Iequalty: A Survey CV = V (5) I case of perfect equalty the value of CV wll be equal to zero ad case of perfect equalty ts value wll be equal to. A problem assocated wth ths measure s that t s more sestve to dffereces amog rch or amog poor come-uts as compared to the dffereces amog mddle come-uts. Varace of the logarthm of come s aother measure of equalty. Ule varace, t s come scale depedet ad t maes equalty as sestve to dffereces amog rch or amog poor come-uts as to the dffereces amog mddle come-uts. Deotg geometrc mea of come by ~, the varace of log-come ca be wrtte as V L = = ~ ( l ) l (6) May studes (see Se 973) have used arthmetc mea place of geometrc mea ths formula. I ay case a problem wth ths measures s that t become udefed whe come of ay ut equals zero... Regular Measures The regular measures of equalty meat purely for the measuremet of equalty ca be classfed to four categores, whch are dscussed below.... Ordary measures Ordary equalty measures ca also be labeled as adhoc equalty measures, as they are featured wth umerous lmtatos. 3 We clude Elteto ad Frgyes dces ths category.. Elteto ad Frgyes Idces Elteto ad Frgyes (968) dvded the etre populato to two groups; those whose come s less tha mea come ad those whose come s equal to or greater tha mea come ad proposed the followg three measures of equalty. u =, L v =, w = (7) G L G 3 These measures do ot fulfll may of the desrable propertes of the equalty measures. For more detals see Secto 4 ad Table. 5

6 Idrees ad Ahmad where s the mea come the etre populato, the mea come of L those whose come s less tha ad the mea come of those whose come s greater tha or equal to. The dex v s a overall measure of equalty; whle u ad w dcate dsparty of poor ad rch come groups from the overall mea come. The three measures have the lower lmt of oe, whle oly w has a upper lmt, whch s. Elteto ad Frgyes further proposed the followg trasformed measures. L u = =, u G L v = =, v G G G w = = (8) w The lower lmt of each of these measures s zero. The upper lmt of u ad v s oe, whle that of w s equal to. Clearly v = uw, ad v = u + w u w, that s oly two of the three measures u, v ad w or u, v ad w are mutually depedet. Kodor (97) has show that these three dces ca be combed as follows to yeld a value equal to oe-half of the relatve mea devato, that s ( u )( w ) p s p s L L L L = = p s = ( v ) s p s p L L RM L L L where p ad s are respectvely the populato ad come shares of the poor L L group of populato. 4. Lorez Curve ad Related Measures Lorez curve s oe of the most wdely used tools to descrbe state of equalty. A large umbers of equalty measures are drectly based o Lorez curve. The most commo amog them are G coeffcet, Schultz dex ad Kawa dex. 5 L G (9) 4 Schultz s (95) measured, dscussed the sub-secto.., s also equal to half of relatve mea devato. 5 Schultz Coeffcet s smply equal to oe half of the relatve mea devato so the later ca also be based o Lorez curve. However, sce relatve mea devato has bee derved depedetly, t wll ot be dscussed here. Smlarly the measures proposed by Elteto & Frgyes ca also be related to relatve mea devato. But aga these measures wll ot be dscussed here, as they have ot bee derved from Lorez curve. 6

7 Measuremet of Icome Iequalty: A Survey Lorez curve, amed after a US statstca Max Otto Lorez ad troduced 905 s the relatoshp betwee the cumulatve percetages of comes (placed ascedg order) measured alog the vertcal axs ad the correspodg cumulatve percetages of come uts measured alog the horzotal axs. 6 Let comes be deoted by,..., such that 0 <,... ad the correspodg come shares by are s,...,s ad q = s = tag combatos of. The cumulatve shares of come uts ad comes respectvely. The Lorez curve ca ow be costructed by ad q. ( 0, 0 ),, q,, q,...,, q,, q, (, ) The Lorez curve, as plotted Fgure, shows that the curve closer to the le of perfect equalty (the dagoal le) represets a more equal dstrbuto of come as compared to the oe that s relatvely away from the le of perfect equalty. Fgure : Lorez Curve (0) Cumulatve Icome Share Cumulatve Populato Share Although Lorez curve s smple ad a wdely accepted measure of equalty, yet t s ot free from lmtatos. For example, cocluso regardg the degree of equalty becomes ambguous whe the curves represetg two dfferet come 6 Kua Xu (003) has poted out that the cocept of Lorez curve was tally hted by Sr Leo Chozza Moey (905) but t was formally proposed by Lorez. 7

8 Idrees ad Ahmad dstrbutos tersect each other. Furthermore the Lorez curve does ot provde a umerc measure of equalty. Atso (970) explaed the ethcal stregth of Lorez curve by relatg t to socal welfare. I ts modfed form, the Lorez curve s related to the basc characterstcs of socal welfare fucto. Ths s best explaed Atso s Theorem, called Lorez Domace Crtero, stated as follows. Theorem I: Ths theorem states that for two come dstrbutos A ad B, havg detcal meas, socal welfare dstrbuto A s greater tha socal welfare dstrbuto B f Lorez curve of dstrbuto A les everywhere above the Lorez curve of dstrbuto B, provded that the uderlyg socal welfare fucto s dvdualstc, o-decreasg, symmetrc, addtve ad strctly cocave. Although Lorez Domace s a useful ad mportat theorem, but as a crtero of welfare comparso t has two lmtatos; t permts comparso oly whe dstrbutos have same mea comes ad t does ot provde comparso betwee tersectg Lorez curves.. Geeralzed Lorez Curve Ths curve was proposed by Shorrocs (983) to overcome the lmtatos of Lorez Domace crtera to some extet. It s obtaed by scalg up Lorez curve by mea come. It s obtaed by plottg the combatos of cumulatve populato shares ad cumulatve come shares multpled by mea come q, that s, ( ) q q, 0,,,,,...,, q,, q, ( ) 0, () Heght of the pot where the geeralzed Lorez curve termates shows the mea come ad covexty measures the extet of equalty. As wth the ordary Lorez curve, the hgher the degree of covexty, the hgher wll be the extet of equalty ad vce-versa. As a example three geeralzed Lorez curves are show Fgure. The lowest curve represets equal dstrbuto, whle the upper-most curve dcates the maxmum degree of equalty. v. Socal Welfare ad Geeralzed Lorez Curve Shorrocs (983) has proposed Geeralzed Lorez Domace crtero by whch welfare comparso ca be made eve betwee the dstrbutos havg dfferet mea come ad/or tersectg ordary Lorez curves. 8

9 Measuremet of Icome Iequalty: A Survey Theorem : For two come dstrbutos A ad B, socal welfare the dstrbuto A s greater tha socal welfare the dstrbuto B f the geeralzed Lorez curve of dstrbuto A les everywhere above the geeralzed Lorez curve of dstrbuto B, provded that uderlyg socal welfare fucto s dvdualstc, o decreasg, symmetrc, addtve ad strctly cocave. Fgure : Geeralzed Lorez Curve Cumulatve Icome Share Tmes Mea Icome Note, however, that eve Geeralzed Lorez Domace crtero fals f the geeralzed Lorez curves tersect each other. Thus both Lorez Domace ad Geeralzed Lorez Domace crtera provde complete rag of welfare states. We ow cosder the parametrc measures of equalty that ca be derved from Lorez curve. v. Kawa Idex Kawa, (980a) troduced the followg measure of equalty based o Lorez curve. Cumulatve Populato Share ( ) ( ) K = l () where 'l' s the legth of Lorez curve. If each come ut receves the same come, the legth wll be equal to ad f oe come ut holds the etre 9

10 Idrees ad Ahmad come, the legth wll be equal to. Thus the value of Kawa dex les betwee zero ad oe. 7 v. Schultz Idex Schultz, (95) proposed aother measure of equalty based o Lorez curve. It s defed as the value of the maxmum dscrepacy (measured by horzotal dstace) betwee the le of perfect equalty ad Lorez curve. It s gve by: S = Schultz coeffcet measures the proporto of total come that would have to be trasferred from those whose come s above mea come to those whose come s below mea order to atta perfect equalty. That s why t s also ow as maxmum equalzato percetage or Rob Hood dex. Schultz coeffcet s equal to oe half of relatve mea devato, so t shares all the merts ad demerts of relatve mea devato. v. G Coeffcet G coeffcet, attrbuted to G, (9), s by far the most popular measures of come equalty. 8 There are at least three approaches to defe G coeffcet. The frst oe, called geometrc approach, expresses G coeffcet as the rato of area betwee the le of absolute equalty ad the Lorez curve to the total area below the le of absolute equalty. Rao, (969) has gve followg formula to calculate G coeffcet through geometrc approach: where = = ( P q P q ) + + (3) G (4) P s the cumulatve populato share ad q s the cumulatve come share of the come ut, whe all come uts are arraged ascedg order of come. 7 G coeffcet attaches more weght to trasfers of come ear the mode of the dstrbuto tha ay oe of the tals, whle Kawa dex attaches more weght to trasfers at the lower ed tha at the mddle ad upper eds of dstrbuto (See Kawa 980a). Dscusso o G coeffcet s gve later ths secto. 8 Davd (968) has poted out that G Coeffcet, as gve by relatve mea dfferece, was developed much earler by F. R. Helmert 870s. However, ts l wth Lorez curve was establshed by G hmself. 0

11 Measuremet of Icome Iequalty: A Survey The secod approach s attrbuted to G hmself who referred to the G coeffcet as cocetrato rato. I the words of G, (9) the cocetrato rato s the quotet of the mea dfferece by the twce the arthmetc mea. Deotg comes of the come uts ad by ad, the mea come by ad the umber of come uts by, G coeffcet accordg to ths approach ca be wrtte as (see Kedall ad Stuart 963): G = (5) The thrd approach expresses G coeffcet as a fucto of covarace betwee comes ad ther ras ad t s gve by (see Aad 983): 9 (, ) G = Cov (6) G coeffcet les betwee zero ad oe; zero represetg perfect equalty ad oe perfect equalty. It provdes a meagful terpretato of Lorez curve. Moreover t s based o more drect approach ad does ot tae arbtrary squares, as case of varace ad related measures. However, a problem wth G coeffcet s that t attaches more weght to come trasfers affectg mddle-come classes ad ot much weght to come trasfers wth extreme come classes. Ths problem s somewhat solved by Geeralzed G dces. v. Geeralzed G Idces Geeralzed G dces are ormatve ature, as these ca be made more or less sestve to come trasfers at ay part of come dstrbuto. Geeralzed G dces are based o weghted gaps betwee the le of perfect equalty ad Lorez curve alog dfferet locatos of a gve come dstrbuto ad varous geeralzatos dffers ther weghtg scheme. Ths property s ofte referred to as ethcal flexblty. Kawa, (980b) troduced the followg geeralzato, whch the parameter α represets sestvty of the equalty dex to come trasfers betwee dfferet come uts. 9 G coeffcet ca also be calculated by may other ways. A good descrpto s avalable Aad (983).

12 Idrees ad Ahmad G α = ( ) ( )( ) ( ) + α = = For α > ( α < ) more weght s gve to trasfers lower (upper) tal of come dstrbuto. It cocdes wth ordary G coeffcet wheα =. Doaldso ad Weymar, (980) ad tzha, (983) have also proposed geeralzato of G coeffcet, but these are cardally equvalet to Kawa s geeralzed G dex (see Charawarty, 988 ad tzha, 983). More recetly Chotapach ad Grffths (00) proposed the followg geeralzato, where s, p ad P are respectvely come share, populato α (7) share ad cumulatve populato share of the th come ut ad υ s a equalty averso parameter: G s v v = + [( ) ( ) ] ( ) P P v, v > (8) = p For v < ( v > ) more weght s gve to trasfers upper (lower) tal of the come dstrbuto. The dex cocdes wth ordary G coeffcet whe v =.... Etropy Measures Aother popular class of equalty measures s ow as etropy measure, whch s derved from the oto of etropy formato theory. It cludes two types of measures amely Thel etropy measures ad geeralzed etropy dces. The basc dea behd etropy s that evets that dffer from what was expected, should receve more weght tha the evets that cofrm wth pror expectatos.. Thel s Etropy Measures If s s the probablty that a certa evet wll occur, the formato cotet h ( s) of otcg that the evet has occurred must be a decreasg fucto of s. Oe possble way to express such a fucto s the form of logarthm of recprocals, that s h( s) = l( s). Wth possble evets wth probabltes s,..., s, the etropy ca be defed as sum of the formato cotets of all the evets weghted by ther respectve probabltes: H ( s) = s h( s ) = s l( s ) = =

13 Measuremet of Icome Iequalty: A Survey The formato cotet s zero whe oe of the evets has probablty equal to oe; that s oe draws o formato from the occurrece of a evet that was atcpated wth certaty. The formato cotet s at ts maxmum whe s = ad, hece H = l ( ). If s s terpreted as the come share of the come ut, ( s) loo le a measure of equalty. Thus subtractg etropy ( s) maxmum value l ( ) H wll H from ts, the latter represetg perfect equalty, yelds a dex of equalty, ow as Thel s frst etropy dex of equalty: T [ ] ( ) H ( s) = s l s ( ) = l. Sce s =, we ca further wrte Thel s dex as: T = = ( ) l( ) = Thel, (967) has terpreted T as the expected formato of a message that trasforms populato shares to come shares. I case of perfect equalty the come share of each come ut s equal to the correspodg populato share ad hece the dex taes the value equal to zero. O the other had, f come share of oe come ut s equal to oe ad that of everyoe else s equal to zero the l. Furthermore, hgher the T assumes the value equal to ( ) dfferece betwee come shares ad populato shares, the hgher wll be the value of Thel dex. Cosderg ths prcple, Aad (983) restated Thel dex as a geeral dstace fucto that measures dvergece betwee come ad populato shares. Although Thel dex s frequetly used for measurg come equalty, Se (973) oped that t lacs tutve sese ad s ust a arbtrary formula. (9) Thel s secod measure s obtaed by terchagg the roles of [ ] populato ad come shares the formula T = s s ( ) ( ) l ( ) [ s ] = l[ ( ) s ] = l to yeld T =. Sce s =, Thel s secod = = measure ca be wrtte as 3

14 Idrees ad Ahmad T = [ l( )] = It s apparet from the above that Thel s secod measure of equalty s equal to mea log devato or log of the rato of arthmetc mea to geometrc mea of come. The lower lmt of Thel s secod measure s zero but t has o upper lmt. I actual practce both the measures of Thel do ot cosder zero comes, as log of zero s udetermed.. Geeralzed Etropy Idces (0) Shorrocs, (980) preseted the followg class of geeralzed etropy dces. c [( ), ] c 0, c = I c (a) c( c ) = = = = = ( ) l( ) = T, c = [ l( )] = T, c = 0 (b) (c) Note that for c = the dex becomes oe half of the squared coeffcet of varato ad cardally equvalet to Herfdal dex, whch s a measure of dustral cocetrato. As the value of c creases, the measure becomes more sestve to chages the upper tal of the come dstrbuto. The lower boud of I s zero, whle the upper boud vares wth c. Wth c = 0 the measure has o c upper lmt, whle wth = c ts upper lmt s ( ) comes are postve the the upper bod of..3. Pure Welfare Based Measures c l. If c > 0 ad c ad all I wll be ( ) c( c ) c. The measures of equalty dscussed so far are postve measures ther specfc orgal forms. The geeralzed forms of these measures base equalty o value udgmet about the sestvty parameter. I ths sese the geeralzed measures become ormatve ature. Now we descrbe the class of equalty measures, whch are ot geeralzed forms of postve measures but are pure ormatve measures. 4

15 Measuremet of Icome Iequalty: A Survey. Dalto s Measure Dalto, (90) was the frst to troduce the dea that equalty measuremet should relate to ecoomc welfare. Hs measure s based o utltara framewor ad t uses the tuto that come equalty results loss of socal welfare. As show Kawa (980a), Dalto s measure ca be wrtte as ( ) U ( ) D = U () Wth perfect equalty U ( ) = U ( ) =, hece D = 0. Assumg dmshg margal utlty f comes are uequally dstrbuted, we ll have U ( ) > U ( ) = > betwee U ( ) = be the degree of equalty. 0 ad, hece, 0 < D <. It follows that greater the dfferece ad U ( ), the greater wll be the value of D ad hgher wll Dalto s measure provdes a geeral rule for defg equalty terms of welfare. For actual measuremet of equalty the utlty fucto eeds to be parameterzed. Usg a equalty averso parameter ε, Cowell (000) redefed Dalto s dex as follows. D = = ε ε [ ], ε > 0 The codto ε > 0 mples socal preferece for equalty. The larger the value of ε, the greater wll be the weght attached to trasfers at the lower ed of the dstrbuto. As ε, the welfare fucto becomes Rawalsa,.e., welfare depeds o come of the poorest member of socety. O the other had as ε 0, the welfare fucto becomes lear come ad, hece, varat to redstrbuto of come. Cowell has poted out the lmtato of Dalto s dex that ts value does ot ecessarly crease wth ε, the presumed equalty averso parameter.. Atso s Measure Atso, (970) crtczed Dalto s dex o the grouds that t s varat wth respect to postve lear trasformatos of utlty fucto. Atso (3) 5

16 Idrees ad Ahmad suggested a alteratve measure based o the cocept of equally dstrbuted equvalet come, whch f equally dstrbuted wll mae the welfare level e exactly equal to the level geerated by actual dstrbuto of the gve aggregate come. That s e = U ( ) = U ( ). If the fucto U ( ) s cocave, e caot be larger tha the mea come. The dfferece betwee these two ca be terpreted as the welfare loss due to equalty. Thus greater s the dvergece betwee ad, the greater wll be the level of equalty ad vce versa. e Atso s measure ca be obtaed by dvdg the dfferece betwee ad by, that s: 0 e A ( ) = e e = (4) The specfc form of Atso s dex s gve by: A ε ε ( ε ) = ( ), ε > 0, ε (5a) = A ( ε ) = ( ), ε =, (5b) = If comes are equally dstrbuted ad hece e e = the the value of Atso s dex wll be equal to zero. If all come s gve to ust oe come ut, e wll approach to zero ad Atso s dex wll tae the value equal to oe. I geeral whe comes are uequally dstrbuted we shall have 0 < A <. Ebert (999) has terpreted A as the fracto of mea come that s lost per come ut due to equalty. Atso s dex also has certa lmtatos. For example, ts values are ot comparable across socetes eve for a gve value of the equalty averso parameter ε because oe caot clam that all socetes have the same atttude towards equalty. However, ths argumet ca equally be appled to other equalty measures geeral ad the oes volvg equalty averso parameters partcular. 0 Se (97) poted out that Atso s measure requres that the fucto U() be cocave but ot strctly cocave,..e. U > 0 ad U 0. 6

17 Measuremet of Icome Iequalty: A Survey. Ebert s Measure Ebert (999) troduced the followg measure based o the cocept of equally dstrbuted equvalet come, whch Ebert terpreted as represetatve come or average stadard of lvg evaluated by the uderlyg socal welfare orderg. ( ) = E = (6) e e e I specfc terms Ebert (999) proposed followg formula for the measuremet of equalty. E ( ε ) = E ( ε ) = ε ε ( ) = ( ) =,, ε = ε > 0, ε (7a) (7b) Ebert s dex has the lower lmt equal to zero ad has o upper lmt. Note that whle Atso s dex measures welfare loss due to equalty as a proporto to mea come, Ebert s dex expresses the welfare loss as a proporto to equally dstrbuted equvalet come. Obvously Ebert s dex ad Atso s dex are ordally equvalet because E = A [ A ]. Ebert ( ε ) ( ε ) ( ε ) (999) has further poted out that every Ebert s dex s also ordally equvalet to the correspodg geeralzed etropy measure for ε = c <. Ths cocludes the descrpto of the measuremet of equalty. The ext secto s focused o decomposto aalyss of the equalty measures. 3. Decomposto of Iequalty Measures A mportat tas aalyzg equalty s to wor out ts structure ad sources. For example, t s worthwhle to ow how the come equalty a coutry s accouted for by equalty wth ts dfferet regos ad the equalty betwee the regos. Smlarly t s worthwhle to express equalty to dfferet sources of come, such as wages, rets, etc. Iequalty decomposto s a stadard techque for examg the cotrbuto of subgroups of populato, come sources/types ad characterstcs of come uts to the overall equalty. Decomposto aalyss s helpful potg out the sources ad cdece of equalty. 7

18 Idrees ad Ahmad There ca be at least two ways to coduct decomposto of equalty,.e. ether by splttg up of the populato, called sub-group decomposto or by dvso of come (or other such varables), ow as source-decomposto. It s the obectve of decomposto, whch determes what type of decomposto s to be carred out. The sub-group ad source decomposto ca approprately be dstgushed as addtve ad o-addtve decomposto. A measure s sad to be addtve decomposable whe total equalty of populato ca be broe dow to a weghted average of the equaltes exstg betwee ad wth subgroups of populatos. I o-addtve decomposto the focus of aalyss s o the cotrbuto of sub-categores of come (or other such varables) to total equalty, rather tha o how total equalty s sub-dvded betwee ad wth sub-groups. Further detal o the two types of decomposto techques are dscussed as follows. 3.. Addtve Decomposto A addtve decomposable measure s the oe that ca splt total equalty I to a weghted average of the equalty exstg wth sub-groups of the ( ) T populatos ( ) I ad the equalty exstg betwee the sub-groups ( I ) W,, g are the K sub-groups of populato the a addtve decomposable K measure ca be wrtte as follows. I ( g,..., g ) [ I ( g ) I ( g )] + [ I ( g,..., g )] T K W W K B K where I ( g,..., g ), I ( g ) ad I ( g,..., g ) K W B K = (8) B. If g deote equalty the etre populato, wth sub-group ad betwee sub-groups to K respectvely. The betwee-group compoet ca be defed as the level of equalty whe comes wth each sub-group have bee equalzed,.e., each come ut wth a subgroup s gve the mea come of the sub-group. Lewse the wth subgroup compoet ca be defed as the value of the equalty dex whe all the betwee group come dffereces are suppressed. I order to elmate the betwee group come dffereces mea comes across the sub-groups are equalzed to the overall mea come through equ-proportoate chages comes of the dvdual come uts. There are two vews regardg the equalzato of all wth group comes. The tradtoal vew followed by Shorrocs (980, 984) ad Cowell (980) s to assume that each come ut receves the mea come of ts sub-group. Ebert (999) has, however, used equally dstrbuted equvalet come for ths equalzato. 8

19 Measuremet of Icome Iequalty: A Survey Aad (983) has poted out that Thel s secod measure ad varace of logcomes are the oly two measures whch are addtve decomposable uder ths strct defto. Cowell has further show that practce the varace of logcomes s ot properly decomposable because of the complexty dsetaglg the wth-group ad betwee-group equalty compoets. A more relaxed defto of the wth group compoet, whch has broader applcato, s that t s the weghted sum of equalty dces of all the sub-groups. I the followg aalyss we shall cosder ths relaxed defto order to add more equalty measures the class of addtve decomposable measures, whch alog wth decomposablty also eed to satsfy other desrable propertes of a equalty measure. Shorrocs (980) has show that ay equalty measure that satsfes dmshg-trasfer axom, prcple of populato, come-scale depedece ad decomposablty must belog to geeralzed etropy class or ts ordal trasformatos. Shorrocs has, however, poted out that wth the excepto of Thel s two measures, o etropy dex satsfes addg-up codto,.e., wthgroup compoet weghts do ot sum to oe ad the decomposto coeffcets are depedet o the betwee groups cotrbuto. A related problem les terpretg cotrbutos of the two compoets to total equalty: I words of Shorrocs Iterpretato () suggests a comparso of total equalty wth the value whch would arse f equalty was zero wth each age group, but the dfferece mea come betwee age groups remaed the same. For the addtve decomposable dces ths would elmate the total wth group-term ad leave oly the betwee group cotrbuto. Iterpretato () suggests a comparso of total equalty wth the value, whch would result f the mea comes of the age groups were made detcal, but equalty wth each group remaed uchaged. Ths elmates the betwee group term the decomposto equato; but the reducto equalty s ot smply betwee groups equalty, because geeral, chagg the age-group meas wll also affect the decomposto coeffcets ad hece the total wth group cotrbuto. Shorrocs further poted out that the two terpretatos are recoclable f ad oly f the weghts assged to come uts are depedet of come shares. Oly Thel s secod measure (also called mea log devato), whch weghts are populato share, satsfes ths requremet. Hece Thel s secod measure s the most satsfactory addtve decomposable measure amog the class of geeralzed etropy dces. These are the desrable propertes of equalty measures, whch wll be explaed secto 4. 9

20 Idrees ad Ahmad Foster ad Sheyerov (000) have explored path depedet decomposto property for eat decomposto whch wth group ad betwee groups compoets of total equalty are mutually depedet. They coclude that mea log devato or Thel s secod measure has a path depedet decomposto whe devatos are tae from arthmetc mea, ad varace of logarthms s path depedet whe devatos are tae from geometrc mea. G coeffcet s aother addtve decomposable equalty measure. However, t s eat addtve decomposto remaed a usettled ssue for a umber of years. Bhatacharya ad Mahalaos (967) were frst to wor o the sub-group decomposto of G coeffcet ad they eded up wth a addtoal cross term. Paytt (976) ad Das ad Parh (98) also foud the same results. Mooheree ad Shorrocs (98) arrved at the cocluso that there s o meagful terpretato of ths cross term. Shorrocs (984) showed that eat addtve decomposto of G coeffcet, wthout cross term s possble f comes of all come uts oe sub-group are less tha those the other subgroup. Slber (989), tzha ad Lerma (99) ad tzha (994), however, suggested dfferet terpretatos of cross term (see Kua 003). Dagum (997) has show that G coeffcet s eatly decomposable wthout cross term f subgroups of populatos do ot overlap ad f they do overlap the G coeffcet ca be addtvely decomposed to three compoets; equalty wth subgroups of populato, the et cotrbuto of exteded G equalty betwee the sub-groups ad the cotrbuto of the testy of tras-varato (overlappg effect) betwee sub groups of populato. The word tras-varato stads to the fact that the dffereces comes across sub-groups cosdered are of opposte sg tha the dfferece mea comes of the correspodg sub-groups. More recetly Ebert (999) has preseted a ew famly of addtvely decomposable measures. Ebert s measures, geeralzed etropy measures ad Atso s dces are ordally equvalet, so they provde the same rag of equalty across a set of populatos. Atso s dces are ot addtvely decomposable; whle Ebert s ad geeralzed etropy dces are decomposable. 3 The selecto betwee Ebert s measures ad geeralzed etropy dces for decomposto depeds upo obectve of decomposto. I words of Ebert (999) f the focus s o come (descrbg opportutes) geeralzed etropy measures seem to be more sutable; f the dstrbuto of lvg stadard s 3 de la Vega ad Urruta (003) have, however, preseted factoral decomposto of Atso s dces whch total equalty ca be wrtte as product of wth group ad betwee groups equalty. 0

21 Measuremet of Icome Iequalty: A Survey relevat Ebert s dces should be preferred. The addtve decomposto of each addtve decomposable measure s gve Table: A (Appedx A). 3.. No-Addtve Decomposto No-addtve decomposto focuses o the cotrbuto of compoets of the varable aalyzed, such as come or cosumpto, to total equalty. Shorrocs (98) has show that G coeffcet, varace ad coeffcet of varato are the oly well-ow measures of equalty that ca be decomposed by ths crtero. The decomposto of G coeffcet s straghtforward. Deotg the cumulatve shares of come compoet by q, the cocetrato coeffcet for the come compoet, whch s le G coeffcet gve by (5) but after placg the come compoet the ascedg order of come, ca be calculated as = = ( P q P q ) C (9) + + It s straghtforward to show that G coeffcet of come s equal to the weghted sum of the cocetrato coeffcets of come compoets, where weghts are the shares of aggregate come compoets the aggregate come, that s, G = K = [ ( )] (30) s C The cotrbuto of come compoet to total equalty (deoted by O ) ca be obtaed as: G G C O = s G Ths effect wll be postve, zero ad egatve whe C < G respectvely. (3) C > G, C = G ad Next, the varace V ( ) ad the squared coeffcet of varato [ ( )] decomposed, followg Shorrocs (98), as V ( ) = Cov( ) = V ( ) + V ( ) [ ] / [ V ( )] /, ρ (3) CV ca be

22 Idrees ad Ahmad [ CV ( )] = = Cov (, ) V ( K ) = Cov (, ) = + ρ [ ] [ V ( )] / V ( ) [ ][ CV ( )] [ CV ( )] + ρ CV ( ) where ρ s the correlato coeffcet betwee come compoets ad. It s easy to verfy that the cotrbuto of each come compoet to the overall equalty for varace ad the square of coeffcet of varato s the same ad s gve by:. O V (, ) ( ) ( ) ( ) ( ) ( ) = C Cov V V O = = ρ = + ρ V V V V V ( ) ( ) where ρ s the correlato coeffcet betwee ad. Ths cocludes our dscusso o the decomposto of equalty measure. 4. Desrable Propertes of a Iequalty Measure Each equalty measure has certa qualtes of ts ow. Oe way of selectg a desrable equalty measure s to adopt axomatc approach, accordg to whch a deal measure should possess certa characterstcs. A summary of these propertes, mostly based o Ltchfeld (999), s gve as follows. 4.. The Pgou-Dalto Trasfer Prcple The equalty measure should dcate crease (decrease) equalty as a result of regressve (progressve) trasfers of come. That s a come trasfer from a poor (rch) to a rcher (poorer) come ut should crease (decrease) the value of the equalty measure or at least leave t uchaged, provded ther ras do ot chage. A stroger verso of ths axom s Dmshg Trasfers Axom (DTA), accordg to whch f equal amouts of comes are tae from two come uts wth comes ad, where < ad gve to come uts wth comes c ad c, where c > 0, such that come trasfers are ra preservg, the come trasfer from / (33) (34) wll reduce equalty by a greater extet as compared to the reducto caused by come trasfer from

23 Measuremet of Icome Iequalty: A Survey 4.. Prcple of Populato A equalty measure should be varat to replcato of populato. Thus mergg two or more detcal dstrbutos should ot alter the degree of equalty. Ths axom dcates that the extet of measured equalty should ot deped o sze of the populato Symmetry Iequalty measure should be depedet of ay characterstcs of come uts other tha ther comes or other welfare dcators beg measured Icome Scale Idepedece Iequalty measure should be varat to uform proportoal chages comes. Felds ad Fe (978) have show that a dex that satsfes the above four propertes (excludg the stroger Dmshg Trasfer axom) wll also fulfll Lorez Domace crtero. Such a measure s also referred to as Lorez Cosstet measure Prcple of Addto If a postve (egatve) costat s added to the comes of the all come uts, the value of equalty measure should dcate decrease (crease) the degree of equalty. The basc dea s that f uequally dstrbuted comes are supplemeted wth equally dstrbuted trasfers, the comes of the poor wll rse relatve to the comes of rch, thereby reducg the degree of equalty. It s easy to verfy that Icome Scale Idepedece ad Pgou-Dalto Trasfer Prcple are suffcet, though ot ecessary, for the fulfllmet of the Prcple of Addto axom Decomposablty A equalty measure should be decomposable, both addtvely ad oaddtvely Defed Lmts A equalty measure should have defed ad terpretable lmts depedet of the sze of populato. I most cases the lower lmt of a equalty measure s zero, showg perfect equalty ad upper lmt s oe, showg perfect equalty. Ths property allows terpretable assessmet of the degree of equalty ad ts comparso across populatos. 3

24 Idrees ad Ahmad Not all measures of equalty satsfy the above propertes. Table shows that the prcple of populato ad symmetry s the oly crtera that satsfy all measures. Furthermore wth the excepto of mea devato, varace ad Dalto measure, all the measures are also come scale depedet. As far as Pgou-Dalto Trasfer codto s cocered rage satsfes ths codto f ad oly f the recpet s the poorest come ut ad/or the door s the rchest come ut, as t taes to accout oly two extreme comes. Mea devato, relatve mea devato ad the measures proposed by Elteto ad Frgyes ad Shultz dex satsfy ths codto f come of the recpet s below mea come ad come of the door s above mea come, that s, these measures are ot sestve to trasfers betwee come uts lyg o oe sde of the mea come. Aad (983) has show that varace of log-comes also does ot satsfy Pgou-Dalto codto for trasfers wth comes above e ~, where e s the base of atural logarthm ad ~ s geometrc mea of comes. All other measures satsfy ths property. Varace, coeffcet of varato, G coeffcet ad geeralzed G coeffcets that satsfy the basc Pgou-Dalto trasfer codto, fal to meet codtos for the stroger verso, that s, Dmshg Trasfer axom. Oly oe of the geeralzatos of G coeffcet preseted by Chotapach ad Grffths satsfes the dmshg trasfer axom ad that too for the sestvty parameter v >. It follows from the dscusso so far that most of the well wo measures, specfcally coeffcet of varato, Kawa dex, the G ad Etropy classes, Atso dex ad Ebert dces, are Lorez Cosstet. As far as Prcple of Addto s cocered oly mea devato, varace ad the dces of Elteto ad Frgyes do ot satsfy ths codto. The lower lmt of all the measures s zero, whle there are oly few dces that have meagful upper lmt. Elteto ad Frgyes dces, Kawa dex, G coeffcet, geeralzed G dces, Delto s dex ad Atso s dces are the few measures that have upper lmt of oe. It may, however, be oted that ay measure wth fte lower ad upper lmts ca be coverted to a [0, ] rage through a approprate lear trasformato. Comg ow to the decomposto of equalty measure, varace, coeffcet of varato ad G coeffcet are the oly three measures that are decomposable both addtve ad o-addtve forms, whle Thel s two measures, geeralzed etropy dces ad Ebert s dces are oly addtvely decomposable. 4

25 Measuremet of Icome Iequalty: A Survey Table : Comparso of Iequalty Measures Terms of Propertes ad Lmts Iequalty Measure Pgou-Dalto Codto N o r ma l Stroger Prcple of Populato & Symmetry Icome Scale Idepedece Prcple Lorez of Cosstet Addto Decomposablty Addtve No Addtve Lower Defed Lmts Upper Rage No No Satsfy Satsfy No Satsfy No No Zero Mea Devato Relatve Mea Devato No No Satsfy No No No No No Zero ( ) No No Satsfy Satsfy No Satsfy No No Zero ( ) Varace Satsfy No Satsfy No No No Satsfy Satsfy Zero ( ) Coeffcet of Varato Varace of Satsfy No Satsfy Satsfy Satsfy Satsfy Satsfy Satsfy Zero Log-comes No No Satsfy Satsfy No Satsfy No No Zero No lmt Elteto & Frgyes Idces No No Satsfy Satsfy No No No No Zero Oe Kawa Idex Satsfy Satsfy Satsfy Satsfy Satsfy Satsfy No No Zero Oe Schultz Idex No No Satsfy Satsfy No Satsfy No No Zero ( ) G Coeffcet Geeralzed G Idces (Kawa) Geeralzed G Idces (Doaldso- Weymar) Geeralzed G Idces (tzha) Geeralzed G Idces (Chotapac h-grffth) Satsfy No Satsfy Satsfy Satsfy Satsfy Satsfy Satsfy Zero Oe Satsfy No Satsfy Satsfy Satsfy Satsfy No No Zero Oe Satsfy No Satsfy Satsfy Satsfy Satsfy No No Zero Oe Satsfy No Satsfy Satsfy Satsfy Satsfy No No Zero Oe Satsfy No Satsfy Satsfy Satsfy Satsfy No No Zero Oe 5

26 Idrees ad Ahmad Thel s Frst Measure Satsfy Satsfy Satsfy Satsfy Satsfy Satsfy Satsfy No Zero l ( ) Thel s Secod Measure Satsfy Satsfy Satsfy Satsfy Satsfy Satsfy Satsfy No Zero No lmt Geeralzed Etropy Idces Satsfy Satsfy Satsfy Satsfy Satsfy Satsfy Satsfy No Zero ( ) ( ) c c c Dalto s Idces Satsfy Satsfy Satsfy No No Satsfy No No Zero Oe Atso s Idces Satsfy Satsfy Satsfy Satsfy Satsfy Satsfy No No Zero Oe Ebert s Idces Satsfy Satsfy Satsfy Satsfy Satsfy Satsfy Satsfy No Zero No lmt Dfferet measures have dfferet resposes to the trasfers of come from oe come ut or group to aother. Some measures are more sestve to come trasfers at upper come tal, whle others are more sestve to trasfers at the lower ed of the dstrbuto. The measures that are more sestve to trasfers of come betwee rch classes are called alpha type measures. The coeffcet of varato falls ths category. The measures that are more sestve to trasfers of come betwee poor classes are called gema type measures. Varace of logarthms ad Thel etropy measure fall ths group. The measures, whch are more sestve to trasfers of come the mddle-come rage, are called beta type measures. Ths category cludes G coeffcet. Fally, ote that the sestvty of all pure ormatve measures ad geeralzed dces depeds o the values of the relevat sestvty parameters Summary It appears that although there s a large lterature avalable o the measures of come equalty, but there are oly few measures that meet the crtera of desrable propertes that a deal equalty measure should fulfll. As far as the measuremet of come equalty s cocered, coeffcet of varato, Kawa dex, G coeffcet, geeralzed G dces, Thel s two measures, geeralzed etropy dces, Atso s dces ad Ebert s dces ca be cosdered as the best measures. However, amog these measures Kawa dex, geeralzed G dces ad Atso s dces are ot decomposable addtvely or o-addtvely, whle Thel s two measures ad geeralzed etropy dces 4 For the emprcal estmates of these equalty measures see Idrees M ad E. Ahmad (00), Idrees M. ad Ahmad E. (0) ad Idrees M. (0). 6

27 Measuremet of Icome Iequalty: A Survey ad Ebert s dces are decomposable addtvely oly. Thus for a thorough aalyss of come equalty coeffcet of varato ad G coeffcet are the oly avalable measures that posses all the desrable propertes. Fally, sce sestvty of a equalty measure to the locato of come trasfers also vares across varous measures, hardly ay measure ca serve all purposes ad t s desrable to employ more tha oe measure a emprcal aalyss of come equalty. I utshell t s cocluded that each equalty measure loos at come equalty from dfferet dmeso. The selecto of a approprate equalty measure depeds upo the obectve of researcher. If obectve s merely to measure come equalty the G coeffcet s the most approprate measure, f the obectve s decomposto the alog wth G Coeffcet, geeralzed etropy measures are the best choces ad f the obectve s to corporate value udgmet the Atso s dces are the best choce. 7

28 Idrees ad Ahmad Refereces Aad, S. (983). Iequalty ad Poverty Malaysa, Measuremet ad Decomposto. Oxford Uversty Press, Iteratoal Ba for Recostructo ad Developmet, Washgto, D.C. U.S.A. Atso, A. B. (970). O the measuremet of equalty. Joural of Ecoomc Theory,, Bhatacharya & Mahalaos (967). Regoal dspartes household cosumpto Ida. Joural of the Amerca Statstcal Assocato, 6, Charavarty, S. R. (988). Exteded G dces of equalty. Iteratoal Ecoomc Revew, 9, Chotapach, D., & Grffths, W. (00). O the calculato of the exteded G Coeffcet. Revew of Icome ad Wealth, 47, Cowell, F. A. (980). O the structure of addtve equalty measures. Revew of Ecoomc Studes, 47, Cowell, F. A. (000). Measurg Iequalty, Thrd Edto, Oxford Uversty Press. Dagum, C. (997). A ew approach to the decomposto of G come equalty rato. Emprcal Ecoomcs,, Dalto, H. (90). The measuremet of the equalty of comes. Ecoomc Joural, 30, Das, T. & Parh, (98). Decomposto of equalty measures ad a comparatve aalyss. Emprcal Ecoomcs, 7, Doaldso, D., & Weymar, J. A. (980). A sgle-parametrc geeralzato of the G dces of equalty. Joural of Ecoomc Theory,, Doaldso, D., & Weymar, J. A. (983). Ethcally flexble G dces for come dstrbutos the cotuum. Joural of Ecoomc Theory, 9, Davd, H. A. (968). G's mea dfferece redscovered. Bometra, 55, Ebert, U. (999). Dual decomposable equalty measures. Caada Joural of Ecoomcs, 3,

29 Measuremet of Icome Iequalty: A Survey Elteto, U., & Frgyes, E. (968). New come equalty measures as effcet tools for causal aalyss ad plag. Ecoometrca, 36, Feld, G., & Fe, (978). O equalty comparso. Ecoometrca, 46, Foster, J. E., & Sheyerov, A. A. (000). Path depedet equalty measures. Joural of Ecoomc Theory, 9, 99-. G, C. (9). Measuremet of equalty of comes. Ecoomc Joural, 3, 4-6. G, C. (9). Varabltae Mutablta Bologa: Tpografa d Paola Cupp. Idrees, M. (00). A aalyss of dstrct level earg equaltes Pasta. Joural of Busess & Ecoomcs, 4(), 4-4. Idrees, M & Ahmad, E. (00). Measuremet ad decomposto of cosumpto equalty Pasta. The Lahore Joural of Ecoomcs, 5(), 97-. Idrees, M & Ahmad, E. (0). A aalyss of teratoal come equalty. Forma Joural of Ecoomc Studes, 8, -. Kawa, N. C. (980a). Icome equalty ad poverty: Methods of Estmato ad Polcy Applcatos, Oxford Uversty Iteratoal Ba for Recostructo ad Developmet, Washgto, D.C. U.S.A. Kawa, N. C. (980b). O a class of poverty measures. Ecoometrca, 48, Kedall, M. G., & Stuart, A. (963). The Advaced Theory of Statstcs, Dstrbuto Theory, Secod Edto, Lodo: Grff. Kodor,. (97). A old-ew measures of come equalty. Ecoometrca, 6, Kua, Xu, (003). How has the lterature o G's dex evolved the past 80 years?," Worg Papers Archve Departmet of Ecoomc, Dalhouse Uversty. Ltchfeld, J. A. (999). Iequalty: Methods ad Tools. World Ba's Web ste o Iequalty, Poverty ad Soco-ecoomc Performace, Lorez, M. C. (905). Methods of measurg the cocetrato wealth. Joural of The Amerca Statstcal Assocato, 9, Moey, L. G. (905). Rches ad Poverty, Lodo: Methew & Compay Ltd. 9

30 Idrees ad Ahmad Mooheree, D., & Shorrocs, A. F. (98). A decomposto aalyss of the tred UK come equalty. The Ecoomc Joural, 9, Pyatt, G. (976). O the terpretato ad dsaggregato of G coeffcets. Ecoomc Joural, 86, Rao, V. M. (969). Two decompostos of cocetrato rato. Joural of the Royal Statstcal Socety, 3. Schultz, R. R. (95). O the measuremet of come equalty. Amerca Ecoomc Revew, 4, 07-. Se, A. (973). O Ecoomc Iequalty, Oxford: Claredo Press, New or. Shorrocs, A. F. (980). The class of addtvely decomposable equalty measures. Ecoometrca, 48, Shorrocs, A. F. (98). Iequalty decomposto by factor compoets. Ecoometrca, 50, 93-. Shorrocs, A. F. (983). Rag Icome Dstrbutos. Ecoometrca, 50, 3-7. Shorrocs, A. F. (984). Iequalty decomposto by populato sub-groups. Ecoometrca, 5, Slber, J. (989). Factor compoets, populato subgroups ad the G dex of equalty. Revew of Ecoomcs ad Statstcs, 7, Thel, H. (967). Ecoomcs ad Iformato Theory (North-Hollad, Amsterdam). Turro, B. C. (90). D u dce msuratore della dsuguaglaza della rcchezza. I Stud Oore d oere B. Brug, Palermo. Vega. & Urruta, A. M. (003). A ew factoral decomposto for the Atso measure. Ecoomc Bullet, 4, -. tzha, S. (983). O a exteso of the G equalty dex. Iteratoal Ecoomc Revew, 4, tzha, S. (99). Calculatg acfe varace estmators for parameters of the G method. Joural of Busess & Ecoomc Statstcs, 9, tzha, S. (994). Ecoomc dstace ad overlappg of dstrbutos. Joural of Ecoometrcs, 6,