World Bank Reprint Sorls: Number Thirty-eight

Grahawi Pyatt

*n the Interpretationand a isaggregationof : sini Coefficients

Reprinted from The Economic. Journal 86 (June 1W97e

Pub

lic D

iscl

osur

e A

utho

rized

Pub

lic D

iscl

osur

e A

utho

rized

Pub

lic D

iscl

osur

e A

utho

rized

Pub

lic D

iscl

osur

e A

utho

rized

Pub

lic D

iscl

osur

e A

utho

rized

Pub

lic D

iscl

osur

e A

utho

rized

Pub

lic D

iscl

osur

e A

utho

rized

Pub

lic D

iscl

osur

e A

utho

rized

The most recent editions of Catalog of Publicatlons, describing the fullrange of World Bank publications, and World Bank Research Program,describing each of the continuing research programs of the Bank, are avaii-able without charge from: The World Bank, Publications Unit, 1818 H Street,N.W., Washington, D.C. 20433 U.S.A.

WORLD BANK BOOKS ABOUT DEVELOPMENT

Research Publications

Unskilled Labor for Detelopment: Its Economic Cost by Orville McDiarmid,published by The Johns Hopkins University Press, 1977

Housing for the Urban Poor: Economics and Policy in the Developing World byOrville F. Grimes, Jr., published by The Johns Hopkins University Press, 1976

Electricity Economics: Essays and Case Studies by Ralph Turvey and DennisAnderson, published by The Johns Hopkins University Press, 1977

Village Water Supply: Economics and Policy in the Developing World by RobertSaunders and Jeremy Warford, published by The Johns Hopkins UniversityPress, 1976

Economic Analysis of Projects by Lyn Squire and Herman G. van der Tak,published by The Johns Hopkins University Press, 1975

The Design of Rural Development: Lessons from Africa by Uma Lele, publishedby The Johns Hopkins University Press, 1975

Economy-Wide Models and Development Planning edited by Charles R. Blitzer,Peter B. Clark, and Lance Taylor, published by Oxford University Press, 1975

Pattems of Development, 1950-1970 by Hollis Chenery and Moises Syrquin withHazel Elkington, published by Oxford University Press, 1975

A System of International Comparisons of Gross Product and Purchasing Powerby Irving B. Kravis, Zoltan Kenessey, Alan Heston, and Robert Summers,published by The Johns Hopkins University Press, 1975

Country Economic Reports

Chad: Development Potential and Contraints by Richard Westebbe and others,distributed by The Johns Hopkins University Press, 1974

Economic Growth of Colombia: Problems and Prospects by DragoslavAvramovic and others, published by The Johns Hopkins University Press, 1972

The Current Economic Position and Prospects of Ecuador by Roberto Echeverriaand others, distributed by The Johns Hopkins University Press, 1973

Kenya: Into the Second Decade by John Burrows and others, published by TheJohns Hopkins University Press, 1975

Korea: Problems and Issues in a Rapidly Growing Economy by Parvez Hasan,published by The Johns Hopkins University Press, 1976

(continued on inside back cover)

The Economic Joutrnal, 86 (June iyii .-i

Printed in Great B3ritaint

ON THIE RNT 1R ERETATr1)N ANDDISAGGREGATIC)N (OF GrINI COEFFICIENTS1

I. IN'rRoD VCTtoN

The contimt'niporu;yay inltelrest in qtiestio0iis of inectialit), lhas resulted in a greatmany papers couiceriwedl with problems of niiicas ,reenwn t and ofin terl-pe taltion.Meaiwhile, nd IIot lh.l't as a'1 r, stilt of niajor cm itribtitions by Atkinson (1970)and Seni (1973), the tiaclditiwiaull f'.aniv %rk of LolrClZ (nli ves and Gini coef-ficients has retained its adflicieciits, althouhi Trheil (1 9 (67 ) has shown that ameasure of inequality hascd oni eitropy has tdvaIintags. Specifically, Theil'smeasuire canl be iicattlv clectimp)o:dl so tlihat, in a gouiped population, total in-equality depenids oni ilIL'(ieqnabtv wN'itlhiln groups and inequality between them.No such simple (ltomllpsitiOli is lavalilu)le for the Gini coeflfiient, yet its directrelat ioiIlip to the Lo ieIOI (iil( lias rcsilldtti in j)c.rSistil attemnipts to derixe adisaguireg;atioii wlhic can be -hc(tl in cmp)ii t,l i .(k. A ilnii w conicern in thepresent paper is to) f1t thcr such iidc'avi,ii'.

Il on litl no, tlhe pruw wi ,t r is in three main st (aittinsl uwing thliS jut1en(dLuC-tion. The first se( tiOil ilv-e, anl ii r ilt i'ejiu 't.I of the Ginii cotffCi ic t in terms ofthe expectici d l ul ( iii the c.ttbi it il 5I of.(` a gamnie in whicich each iniidiCviLdal

is able to conmpare himself'with some other drawn at ria(lt iii from the totalpopulationi. This provridt- a basis for the Gniii covfIT1liClt, wvhich has somenovelty and is of potctntial use in nal. a i. Ihi cSi't)ui(i section illustrates thisby decomnpositioui of the Gini t ow'liei itt ill n l 111.in of cI ulitiilial expeetat ionsof the valute of the game. A (h'fflikpiucurit 4' 1k result shows tlhliat thiet decom-positioul can be expre'sse(d as lvig tllree terlnIIS. The first (lepemlds on theGini mlLS.ltLu1T Of iTIC(1eq1ulit% Witin Stu Ilb iplls ofl tlhe- tota.l puptulation. The lastdepeIIds oII differenices in the average v'alue of' incomei betweeni groups. Inbetween, there is a term tluii depen(lds oi tile extent to wlichli the income dis-tributions for differeit guotips ovci'l,ip each other. This same disaggregationhas been (lerive(l previotls!v by BLat ti.liai'.% aard Ma;h,-ihlmohis (I967) usinga mnore ( apnpr'itijiia l ape1. h in ter1111s of aiLhu.Jl1te (lifle e eTes. The justifica-tion for tw pu.Seuitt paper niuu.:t rest, therecfore, oni the siniplicity of the proofsand tlhe novelty of tlle ap)J)1 tt It. 0nle inujpli'.:ttiOli of thle d luc(ompositill is toob)tainl a ufeastuli' of the cxIteul to xlliuii the total can be ac(onnitc(I for simplyby comlSi(l(vrati(.ii of (difleuv u.'es in ut.cis IS li t% Ct e II gr1ou0PS.

Fiiially, in se(-tioni Ibur, s01 V)Ukultidi .11)1plii tLi0iHS of the (decomposition1algoriotillm to tllhe aInlysis of 1i1igi ati iii andII di i.- li i tiiti1 ii;atr suiggested. Thus,

1 I amt grateful to a nunmber of people who hiave sent counnents and related papier, in rtsponse to anearlier draft, and would particularly like to idiiik ( li%', Bell and John F'ei in this context, as well asNanak Kakwani, Mark leisersoni Fartiac MNelhran, Gus It,iiii , \ e ,iii) rl. )tii. tn lWouter van (inneken.Of course I arn fully responsible for this final version.

2 It is j,j'niventijj to discuss thie subjelt-rnatter itn termn'; T1' inene dhi.itrihutiilh if only becatuseincome irwepi;lil lihas been, and rensai,s. thie Imain ntiotivationl for1 work in this area. I-lowever, theresubi (an of ourse be applied to a widi i11-1 ii iisstes ad are niot ple'( ill to incoIne.

L '243 3 9.2

244 TEIE ECONOMIC JOURNAI. [JUNE

while reservations about the tise of Gilli COffiCit'll(S to IMeSUITc inetqaility mayremain, it is hoped that the presenit paper may encouirage emnipiricafl aniailysisdirected towards imdlerstaii(li nc causes of iineqotali ty, allbeit withlinl thelimitaitions of the Gini firanieworkl.

1I, AN INTERPRETATION O1 TIE GINI COEFIFICIENT

The Ginii coefficient of iniequialit) among a set of nmnlbers" yl, y2, 2 . .y, anbe ex\presse(d in -ariouis ways. One such way whiclh brini,gs out the rlchtiolslhipof the coefficient to initerpersonial compaiisons is: 2

(I/2n2) y' VG - - lj -1_ (I)

(i/n) yi

i.e. the ratio of the inean ahl)swlte differce between all pairs %ti, y,) to twiceethe mean level of tlhe variable y. 'Io (1\e,(-loJ) tlhis, note that

IN I yi -uj=2 1ax (aI,oY -. j (2)i lj 1 n 11-l

where max lo, y; -yj) (denioes the higler of thl two things wit lil the )l)wack(tIt follows that

ar 7V,2 n V max (o, y. -y,)

- i ,-7 -I n (3)(I/n) Ž y

i -1

This last expr essiotn is the basis for the initelr-prvtation of the Gini coefficient onwhich our suibsequiet analysis rests.

Conisider the following statistical game. For each itidividtial we cond(uIct anexperiment. First, some in1co)1met', y, is selected at random from thle populhitionof incomes Yl, . , q,. If the income seleclted is greater than thle actual incomeof the in(idi(vidul then lie can retain th-e value selected : oltlerwz%ist lhe retainishis actiual inicomie. Clearly, no indiVi(dIual coutld lose From p)articip)ating inthis experimenit; and all indclixvicdiuals apart from the riciest wouldl have aniatheniatical expectation of gaininig from it. For in(lixvi(dual i the (Xpcctelgain is given by

I t

in ima.x (c, yj-yi) > o for all i, (4i)

and if wve now average these *\>;p)e Led gains over all individtials, i, we obtainan expressioni which is the numeratio of equationi1 (3).

If follows from these arguments that equiatioii (3) can b)e initerpreted asstating that the Gini coefficienlt is the average gain to be expected, if eaci

choice of formulation in ternis of a discrete set of numbers is, a matter of style. All the resultsin this 1aper can be derived via the tlheoretical disi ibtutioni of a onitiiuti(lsly disLtriblited variate y.

2 This result is proved by Kendal and Stuart (1963), pp. 48i .

I976] DISAGGREGATION OF GINI COEFFICIENTS 245

individual has the choice of being himself or some other member of thepopulation drawn at random, expressed as a proportion of the average level ofincome. And in these terms the link between the Gini coefficient and inter-personal comparisons is imm ."-.-te and obvious.'

III. DISAGGREGATION BY GROUPS

The average expected gain defined in the previous section can be disaggregatedin a variety of ways especially if the population can be divided into a numberof interesting groups (e.g. by geographical area). In particulars we can write

k kaverage expected gain = , E(gain/i -j) Pr (i j (5)

where the event "i -*-j" refers to an individual being in population group iand drawing a member of group j to compare himself with in the hypotheticalgame previoLusly specified. In this notation E(gain/i ->j) is the average, takenover all individuals in group i, of their expected gain, given that they draw amember of group j to compare with in the game. Since the game speci fies thatsampling is randomn, we lave

Pr (i ÷j) = ptpj for all i,j = i.. .k, (6)

kwhere E pi-I,

i -1

so that pi is the proportion of the population that is in group i, and the totalpopulation is divided into k mutually exclusive and exhaustive groups.

These results can be combined in the matrix eqUat[ion

G = (m'p>'1p'Ep, (7)

where p and m are both k-element columin vectors. The tth element of p is thepopulation proportion pi: the ith clermietnt of m is the average income ofindividuals in populationi group i. Hence

I Xl=S 71 (8)

n I.-I

i.e. m'p is the average incormc of the total population. Finally, E is a k x kmatrix with (i,j)th clementi giveni by E(gain1i -÷j) as defined in cquaitio n (5).

An empirical illustrationi of the clisaggregationi (7) is given in Table i. Thisis taken from a 1973 survey of inicomIc distiibutioni in Sri Lanka2 in which thetotal poppulationi of ificoime receivcr.s is classified geographically according totheir location in urban, rural or estate areas. The table shows the populationproportions, pi, in each area; the average income, mi, of each subgroup; and

1 A symmetric treatment in terms of the minimum of a pair of incomes is, of course, possible, Such

an approach would be a developrrent of Sen's ( 973) observation, p. 33. "In any pair-wise co-iparison

the man with the lower income can be thought to be sflfrering from some depression on finding his

income to be lower. Let this depression be proportional to the cldiff rMient l in incone. The sum total of all

such depressions in all possible pair-wise comparisons takes tis to tlir' (.ini eoefficient."

2 The data are from Central Bank or Ceylon (1974) as zeported by Karunatilake (I974).

246 THE ECONOMIC JOURNAL [JUNE

Table I

Distribution of Income (in Rupees per Month) of Income Receivers inSri Lanka, 1973, by Geographical Areas

Conditional cxpectationsLocation Mean Population E(gain/i-,j) = E,J Conditional

of income iitcome, proportion, ,-- . expectations,receiver m p Estate Rural UTrban Ep (m'p)-' Ep

Estates 227.4 0.177 84-1 277-4 410-2 266-5 o 6oRural 457 5 o0648 47 2 169.3 279 3 1670 0°38Urban 6oo'6 Ox75 36-9 136-1 240'3 136 9 0°31

m'p = 441'9 Total i ooo pEp = 179'3 G = 0-41

the matrix, E, of conditional expected gains. From this the vector Ep is easilycalculated: its elements are the expected gains for groups of individuals in ourhypothetical game conditional only on the group in which they are located.As might be expected, these gains are inversely correlated with the group meanincomes, mi: a member of the poorest group stands to gain most from thechance of being someone else.

Table 2 illustrates the fact that the inverse relationship between the elementsof m and those of Ep does not have to be monotonic. The table is derived fromthe work of Soltow (i960) on the relationship between income inequality andeducation usi-g U.S. data for 1956,1 and shows a break in monotonicity forthose who have I3-15 years of schooling. The conditions under which such abreak in monotonicity may occur are discussed later in the argument.

Table 2

Distribution of Family Income ($ per annum) in the United States, 1956, by years ofSchooling Completed by Head of HIousehold

Mea-n Population ConditionalYears at Income, proportion, expectatiorns, Ginischool m p Ep (m'p) - Ep coefficient

< 8 3,170 0-232 4,388 o 89 0 458 4,240 0-185 2,564 0'52 0.41

9-l I 4,830 o0x80 1,873 0o38 0o3412 5,530 0-233 I,134 0'23 0'33

13-15 6,340 0°077 1,430 0'29 0B3716+ 8,260 0o093 740 0O15 0°39

m'p 4,930 Total i ooo pEp = 1,972 G = 0-40

A variant of equaticn (7) is obtained by defining

7C = (m'p) -1 xmp, (9)

so that the ith element of the column vector 7- is the proportion of aggregate

I Soltow's analysis is based on U.S. Government (i956).

I976] DISAGGREGATION OF GINI COEFFICIENTS 247

Tabl,: 3Distribution of Income (in Rupees per 3 fonth) of Xncome ':., eivers in Sri Lanka,

by Geographical Areas

Location Income Share of m- E-U*of income received, income, I -.

receiver p'm 7 Estates Rural Urban E*p

EstatCs 40'2 o091 0*37 1'22 i 8o 1'17R(ural 29 6

-4 0o671 010 0°37 oG6i 0'37Urban 105-4 0-238 owo6 0-23 °'40 0 23

m'p = 441 9 Total l ooo 'A'E*p = 0 41

income which accrues to members of population group i. With this notation (7)becomes G = 7r'E*p, (7a)

where E* = mA -E. (IO)Table 3 shows the dlecommposition of G in terms of equtiatio n (7 a) for the Sri Lankadata used in comnpiling Table I.

The matrix E* defined in equation (io) is a -ormalisatilol of the imatir^ix Eobtainied by dividing the eleentis of each row of E by the nmcan inconle for thecorlespondlilng population group. Thus the tal)le shovs, for exaniple, that theexpected gain for a rural liotuselhold in being given the option of having theincome of an urban lhouselhold within our game framevork is equal to o-6Itimes the average income of a rural household.

A urtliher implication of the definilion of E* is that its diagonial elementsare themselves Gini coefficierits: the ithi such element is the expected gain for aniiidividuial in group i who draws some (other) member of group i within thegame fi-amieork, expressed as a proportion of average income in group i.This tlieii is the Gini coefficient of inequality for the subpopulation which isin group i. For example, from Table 3, the Gini coefficient of inequality amongestate households is 0-37.

An interl)Petatiorn of the off-cliagonial elements of E* can be approached byC(ollsi(delling iniequality as measuired simply by clifferences in the means, mi,b)etWeeI su1bgrL1ou)s. Thus, given the data in the vcciors p andil m, and hence7,an a.SsumInin1g that all individuals in each group have the same income, mi,an cstimiiate of G can I)e obtainied from first principles, i.e. by plottin,g the Lorenzcurve and calculatinig the proportion of the area below the 450 line which isabove the curve. If the elements of m are ordered from smallest to largest, andthe same orderling, adopted for p and 7t, this comes out matlhematically as:'

; = 7c'(m 1 A' m- A')p (II)

w%here G is an approximation to G; A is a k x k matrix with all letmenits on andb(elo\w thle main diagonal equial to one, and all other elements zero. Hence all

1 lt is1 ;ilatik'(1ly easy to sliow from first principles that G - n'(2A-I) p- i. This can be reduced tor'('A-A')p fromt the fact tlhat t'(A+A'-I)p -I. T'le result (i ) follows from the equivalence of

r'Ap and t'mA 'A'mp.

248 THE ECONOMIC JOURNAL [JUNE

Table 4

Inequality in Sri Lanka Due to DJfl'renccs in AIean Incomesbetween Geogreaphici Areas

Location Sliare of Populatioin E* i m A'm -A'of income income, prorlioll,prreceiver p Estate Rural Urban E*p

Estates ot1 oq1 T 01 1 64 (194Rural o0671 o)64b - 0,31 (005Urban 0-238 175 ---

Total i *oo0 Total Isooo 7,']E*p 2

elements of (Ma-1 A'Mx -A') lying below the main diagonal are zero: theexpected gain from the opportunity of beconmiing a memiber of a poorer groupwhen there is no intra-group variation must be zero. Sinilarly, the diagonalelemiients of (tM"- Am'- A') must be zero. And this result can also be dcri%,e-c 1 asa special case of conisiderinig the (i, j) tlh elemnit of ( M'- A' m -A') wlherei < j. Such eleimeiints have the formii

(i nz !i) - = (?j -j 1 in1) , in, (I 2)

so they (dep1cjd simply on mean inoe (n i fferceiiCes, explrcsseCd as a rattio of,groiipmean incomres. And tL.ey must be positive, since l)y (kefinitioni grouLp j has ahigher avcrage income than group i ifj > i.

It is important to note at this point that while the result iII) can be dlerivedfrom first principles as suiggested, it canl also be obtained directly fromequation (7a). If tlhere is no ine(qluality within groups, tlhein the diagonal elemnentsof E* (which are the Gini co(efficients for s itliin-group ineqi.ialiLy) must be

zero. Further, elements of E (and hence of E*) below the cliagonal rnust bezero since there is no advantage from the clhatnice to have an incomle from apoorer group. And an element of E albo%e the diagonal rmutst be of form

M -Mi > o, (13)

since the gain in the chance to join a riclher grouip must be the (liffeieice ingroup micaii incomies in this case. Hience

E = A'm-MA' (I 4)

by direct application of the pi niciples for deriving E in the special case whlecre(there is no intra-group variatioii.

At this point we have an exact clisaggregation of G given by (7a); and ancxpression (i I) which depends only on cliflerenices in mean incomiles, mi, andwhich would be exact if there were no initr;a-grotl) variation. As Table 4shows, this latter yields a valuie of C of oI2 for our Sri Lanka data, Mforcover,intra-group variation as nmcasured by the Gini coefficienits within groups con-tributes only oxI8 to the aggr-egatc Gini coefficient of Oi.$1 Clearl), then, tlhesctwo components do not accouint for all inequality as measurned by G in thiscase.

1976] DISAGGREGATION OF GINI COEFFICIENTS 249

The third component which is missing depends on whether the distributionsfor different subgroups of the population overlap. In other words on whether,given mj > mi, the minimum income within group j is less than the maximumincome within group i.

To explore this further we need to consider the exact relationship betweenthe conditional expectations E(gain/i -÷j) as defined in equation (5) and thedifference in means mi -ini. From first principles

E(gain/i -*j) E(y1 -yi/yj > yi) Pr (y1 > yi) (I5)

and similarly

E(gain/j - i) = E(y -y1 /j y > yj) Pr (yi > yj). (i6)

It is now easily shownl that

Mi-Mj = E(gain/j -> i) - E(gain/i -*j) ( 7)

or, in other words, that the difference between the (j, i)th and (i,j)th elementsof the matrix E is the difference in means mj - mi. We can therefore write.

E E+E2 (I8)

=E1+A A'm-mA' (I l )

where El is a symmetric matrix with (i, j)th element given by the minimum ofthe (i, j)th and (j, i)th elements of E.

Thus E* = m El m-1EAA' m-A' (20)

and G = 7'[E + (m- A'mA-A')] p, (2 I)

where E = m (22)

and the ediagonal elements of E* are the Gini coefficients of inequality withingroups. Moreover, as is apparent from (I5) and (i6), the off-diagonal elementsof E* will be zero if the distributions within groups do not overlap: wilhoutoverlaps one or other of the probabilities Pr (yi > yj) and Pr (y1 > yi) must bezero, and hence the minimum of (I5) and (i6) must be zero. In the moregeneral case, when overlaps exist, then by definition some members of thericlher group (group j) are poorer than some members of the poorer group(group i). Accordingly, the expected gain for such mnbl)crs of group j fromthe chance of becoming a member of group i will be positive. Since for all othermembers of groupj the expected gain is zero, it follows that the wveighted averagefor group j as a whole is positive, i.e. that the expressioni (i6) is positive eventhough m1 > mi. Thus the (j, i)th clement of E is positive and, from (I7), mustbe smaller than the (i, j)th element of E. Accordingly, this positive (j, i)thelement of E is the corrcsponding (i,j)th (and (j, i)th) clement ofthe symmetricmatrix El.

Given (i5) and (I 6), the result (I7) follows directly from the fact that

mj-mj = E(y -yjlyi > yi) Pr (yi > y1)+ E(y-yjlyi < yf) Pr (yi < yi)

and noting that the second term is the negative of the expression (x6). It is interesting to note that thesum of the two expressions (X5) and (i 6) is E(Iy -yJ,).

250 THE ECONOMIC JOURNAL [JUNE

The result (2I) is our fina1 disaggregation of the Gini coefficienL by popula-tion subgroups. It should be noted that all its components are non-negl-itiveand that it is essentially in two parts: the first arises from variations withingroups while the second depenids enitirely on differences b)etweein groullmeans.'

At this point we recall that the data in Table 2 show that the c cments of Epneed not be monotonically related to group mean incomes. The ith element ofEp is the average, taken over all individuals in group i, of the mathermaticalexpectation of the gain for each such individual from having the chance ofreceiving the incomc of some other population member. It is naturalto anticipate that the poorer group i is, as measured by the averageincome of the group, mi, then the more its members might expect to gainfrom our hypothetical statistical experiment. Thus an inverse relationislhipbetween elements of m and the corresponding elements of Ep is intuitivelyplausible. Table 2 illustrates the fact that this relationship is sUbject to exccp-tions.

To clarify the nature of such exceptions, it can be noted that if there is onlyone individual in each group, then the (i, j)th clement of E is simplymax (o,yj-yi Since all elements of p are equial to (iln) in this case, it rollmos

that the ith element of Ep is (i/n) E (yj-yi). Ience the ith elemnent. of Ep

exceeds the (i+ i)th element by (i/n) (y+1-yj); which must be positive, sinceindividuals are ordered in terms of increasing income. Thus there can be nocounter-intuitive ordering of elements of Ep in the absence of grouping: ele-ments of Ep decrease as individual incomes increase. By extenision, if individualincomes are grouped in such a way that there is no overlap betveen groupsthen the moniotonicityr is maintained. Given such grouping, elements of Epmust decline as group mean incomes inicrease. It follows that exceptiorns sucLCas that observed in Table 2 must derive from overlaps between groups asopposed to variations within them.

The way in wvhich counter-intuitive results may arise, given overlappinggroups, can be approached by considering a population divided into twogroups. SLuppose that the first group has the lower mean and is highly con-ecnt rated arouLnld it. The second group has the higlher mean and wide dis-persioii. Thlus, the first element on the diagonal of E is small, while the secoCndIelement is large. If almost all the population are concentrated in the firstgroup, then only the first column of E matters in calculatinig Ep. And the firstclemenit of this column can approach zero arbitrarily closely as the concntn'lt-tion in the first group increases. By contrast, the second eleniclt of the first

I An obvious corollary of the result (21) arises when a Gini co-efficient is calculated from populationgroups defined by income ranges. Clearly, in this case there are no overlaps between groups, so that theoff-diagonal elements of E* are all zero. If curnulative proportions of income are plotted on a Lorenzdiagram against cumulative population proportions, and the points obtained are joined by straightlines, then the elementary graphical methods lead to an estimate oFC as given by Gin (i x). The straightlines implicitly assume that there is no inequality within groups, i.e. that the diagonal elements of Elare also zero. By contrast, joining the points on the Lorenz diagram by a snmootlh cuirve is ')ne (albeitprimitive) method of attempting to capture variations wvithin classes, and hence allows for the con-iribUtion of thiese diagonal elements to the total Gini measure.

1976] DISAGGREGATION OF GINI COEFFICIENTS 25I

column of E can be quite large. This element is the expected gain for a group 2individual who draws a group I individual in the statistical game. Most suchindividuals may expect to gain very little, or nothing, since they will have anincome greater than the mean value around which group one is concentrated.However, the wide dispersion of group two will imply a significant number ofindividuals in that group with income below the central value for group one,especially if the difference in average incomes between the two groups is notlarge. All individuals in this second category will accordingly have a significantexpected gain if given the choice of having a group I income. Hence the secondelement in the first column of E may not be small, leading to the exceptionaliesult that the first element of Ep is less than the second.1

This informal discussion suggests that three considerations may lead to ex-ceptional ordering of the elements of Ep. If group i is to show a break in mono-tonicity, then this is most likely when m -mi- 1 is not large; when pi is small;and when the Gini coefficient for group i is large relative to the Gini coefficientsfor other groups. Data in Table 2 illustrate the fact that some element of eachof these conditions is present for the group defined as having I3-15 years ofschooling. This group has pi = 0o077 and is the smallest grouping. The meandifference mi-mi-I of 8Io is the second smallest, while the Gini coefficient forthe group (0o37) is larger than that for the two groups which rank below it interms of means.

A result such as (21) is contained in the work of Bhattacharya andMahalanobis (I967) as noted in the Introduction. However, their analysis doesnot use the "game" concept developed here or matrix notation, They work en-tirely in terms of mean absolute differences with the emphasis on the scalarcontribution of each component to the aggregate Gini coefficient. The result(2 I) is perhaps, therefore, an extension of theirs if only at the level of psychologi-cal novelty.2

One obvious implication of (2i) is that G cannot be decomposed into a Ginimeasure of inequality within groups and a Gini measure of inequality betweengroup means; in addition, there are the off-diagonal terms of E* due to over-laps. This is in effect the point made by Theil (I967) in arguing the superiorityof his entropy measure. But we can now suggest that the argument is not

1 It is, of course, possible to produce an exact mathematical condition for a break in monotonicity tooccur. However, none of the various possible arrangements of this condition adds significantly to theintcrpretation given in the text.

2 Various other attempts at Gini disaggregation have been made. In the published literature Soltow(1960) working with absolute differences rather than Gini coefficients, separates out what is, in cfcct,the contributiion of the diagonal elements of the matrix E*, i.e. the Gini coefficients of inequality withingroups. An unipublished paper by Mchran (i974) goes further, giving an alternative derivation in termsof the three components and extending the decompositon to a two-way population disaggregation. Yetanother proof is provided in a private communication from John Fei. Finally, Mangahas (1975) andRao (1969) provide an alternative formulation which results in separating out the Gini inequalitywithin groups and compounds other effects in an expression of form (m'p)-1 p'Dp, where the matrix Din the Mangahas version has zeros on and below the main diagonial. This can be rcconciled with thepresent formulation by noting that, since E1 is a symmetric matrix:

p'El p = p'6j p+2p'E+p,

where 61 is the matrix formed by setting off-diagonal elements of E, equal to zero; and Et is formed bysetting to zero all elements of El on and below the main diagonal.

252 THE ECONOMIC JOURNAI [LJUNE

necessarily persuasive. In expected-gain terms there is a simple (lichotolmybetween the part due to diffeienccs within groups ancl the part du11e to dif-ferences between their means. The fact that the former, given by p'E1 p, is niota simple average of its diagonal elements does not seem to l)e so iml)rtant.Indeed, there are positive advantages in anal)ytic ternis sehich are dis.cusscdl inthe next section. Meanwhile, the cYruCial (qUestion is wVIItller or nlot thie (derivcd

statistics indicated by our disaggi,egatioii are interesting. Anld in these terms itis suggested that a decomposition such as in Tables i-q does pass thte test, notleast because of the interpretation in terms of expectedl gains via iiiterpcrsonalcomparisons which undler-lies it.

IV. SOME ]X.\T; 1NSIONS

The main result in this paper is conitained in eq(uation (2I), wvhere it is shownthat the Gini index can be expr-essed as

G = 7(c2E*p) ( 3)

where 7t is the -.,ector of inicomiie proportions over l)opulatioln sLbgrojl)s, p isthe vector of population proportions in subgroups, andl E* is a inltrix wlhichcan usefully be interpreted and dconrposc(d. Tlle intcrprtdti)ioll suggested isthat the (i,j)th elemenit of E* is the gain to be expected by tfle typical in(lividulalin group i ifhe (or she) could have the free choice oflweini a nieniber of popiula-tion group j. In terms of E*, this gain is expressed as a proportion of axelrageincomes in group i.

One application we have not developecl is that the elenents of E* are aningredient in the decision to nligrate. Here " Iigration" sliou l(d be inwterpretcdin terms of changing population groupl, and not necessaril)y as .illlple geo-graphical migration. Thus, in a classificationi based on years of schooling, therelevant decision is whether to stay at schlool longer. This brings up a numberof further points we need to develop.

For an individual considering migration, the relevant population groupingis their peer group; the population group to be joiniedl as per(eivd l)y the in-dividual; and the rest of the population. If the group to bejoined all lhave hiighlerincomes then no problems of inlterglroup Clistribution overlaps arise. If this isnot so, then the formulatioin adoptecd here assumnes thlat the inidividiial can rejectthe new situation and revert to the initial one. A\not her way of puitting it is tllhatthe migration is perceived as being reversible in the se(nse that the inldivi(vidullwill not be worse off as a iresult. This may well be closer to I annuau psycolilogythan the assumption that the migrant may reckoni a priori that he ma)y be stuickwith an iniferior situation. Thus the modlel does not assumiie that the potentialmigrant thinks it is certain he will be better off, but only that he can revcrt, andhence be at no risk pf being worse off.

Of course, to complete a model of miigratioin wvould require specification ofcosts as well as benefits, and some dynamic considerations might also lue(d tobe taken 1-ao account. Meanwhile, it seems potenitiall) fruiitifl to poilnt outthe link between migration and the Gini disaggregationi.

1976] DISAGGREGATION OF GINI COEFFICIENTS 253Going further, the disaggregation applied to migration implies freedom of

entry into some other population group. This is qualified if costs, such asacquiring more education, are introduced. By implication, such costs and otherrestrictions on mobility are a cause of observed inequalities in a "qociety: in theabsence of the costs mobility would take place on a greater scale and differencesin opportunity would be reduced thereby. In practice, there are other barriersto mobility derived from sex or religion, for example. Such discrimination isa further source of inequality. The framework provided here suggests a way inwhich the importance of such barriers might be quantified. This would be toreplace the matrix E* in equation (23) by some other, Ej* derived from E* bysetting to zero all elements of E* which correspond to expected gains fromchanges to a different group which are in fact precluded by discriminatory bar-riers. Such a procedure can be applied to our Sri Lanka data on the assumptionthat the Estate population is distinct from that in Rural and Urban areas. Theresult is a reduction in the total Gini coefficient from 04I to 0o29. This impliesthat owI2 or 29 % of the aggregate Gini can be associated with immobility be-tween Estate workers and others in Sri Lanka. Clearly some comparativestudies are needed to know whether this is a large fiLgure or not, and in anyevent its significance depends on the extent to which the barriers in a particularcase are ones which individuals would like to cross. Meanwvhile, the illustrationserves as a starting point for quantificationi and incdicates the relevance of theframework to such aspects of inequality.

V. SUMMARY AND CONCLUSIONS

This paper shows, first, that the Gini-coefficient mealsuri-e of income inequalitycan be interpreted in terms of a simple statisticil ganim. In this game eachindividual draws an income at random from the dlistr iblutior of incomes whichobtains in the society. If the income so selected is higher than the actual incomeof the individual, then this higher inicome is recorded. Otherwise the individual'sactual income is recorded. Thus for all individuals except the richest there isa mathematical expectation of a positive gain of recorded income over actualincome: for tlhe richest person the expected gaiin is zero. In section II of thepaper it is shown that if these expected gains are averaged over all inidixidtials,then the result, ex:pressed as a proportion of average income, is equal to theGinii coefficient. Hence the latter can be interpreted in Ltermzs of the averageexpected gain from having the option of receiving the inicome of someonie elseselected at ranMdom.

On the basis of this rIesultl an expressiuni is obtained, in section TII, for the Ginicoefficient within a population which is sub-dli\idecd into groups by somecriterion, such as location (but which could be inicomiie itself). This expressionis the sum of three parts, none of which can be negati% e. The simplest is due todlifercnces in mean incomes between groups, and wouild be the only com-ponent if there was no variation in income vithii n groups. The second and thirdternms both depend on variation withini groups. The distinctiorn between themis this. If there are no overlaps between. the incomie ranges in diffrecjt groups

254 THE ECONOMIC JOITRNAL [JUNE

then the third term is zero: otherwise the third term is positive, reflecting thefact that there is now a positive expectation of gain in the statistical game forpoorer members of a rich group who draw richer members of a poorer groupas a result of the random sampling. Meanwhile, irrespective of whether grouipsoverlap, variation within them contributes to overall inequal ity and leads tothe second term in the deconiposition whichi depend(s on the Gini coefficientsof inequality within each group.

This decomposition is partially illustrated by reference to the work of Soltow(1960) on earnings according to years of formal education. A full illuistrationis provided by analysing some income distribution data for Sri Lanka in whichhouseholds are disaggregated by geographical location.

Finally, in section IV, it is suggested that the disaggregation may haveparticular relevance to studies of migrationi and discrimination. Here migrationis to be understood as movement from one group to another and tlicieforeincludes non-geographical mobility, such as a change in edUcational status.If such mobility is based on a difference between the costs and benefits of ashift from one group to another, then it may be reasonable to eluate thebenefits with the expected gain for individuals in the former group who drawan individual from the latter group in the statistical game. This leaves out-standing the specification of costs, but nieanwhile SUggests that the costs repre-sent barriers to mobility which would otherwvise take place. If these barrielstake the extreme form of total discrimination, wlhereby mobility from somegroups to others is precluded, then a measure of their importance can be ob-tained by setting to zero all expected gains which arise within the game frame-work from comparisons between individuals who are in fact scparated by suchtotal discrimination.

GRAHAM PYATTDevelopment Research Center, Wforld Bank:

on leavefrom University of TZVarwick

Date of receipt offinal typescript: October 1975

REFLRENCES

Atkinson, A. B. (1970). "On the Measurement of Inequality." Joutrnal of Econ2omic rhleory, vol. 2.Bhattacharya, N. and Mahalanobis, B. (1967). "Rcgional l)isparitirs in hIouschold Consumption in

India." Journal of the American Statistical .Issocieztionz, vol. 62, no. 317 (March).Central Bank ofCeylon (1974). Department of Economnit P. esear I. Surtt'rl of ,Sri Lanka's (onsut tmer l'inances

1973. Colombo.Karunatilake, H. N. S. (1974.). "(1lhanges in Income Distribution in Sri Lanka." TheJapan E oiinnnti

Research Centre, Seminar on Income Distribution, Employmenrt and Economic Development inSoutheast and East Asia, December (firncograph).

Kendal, M. G. and Stuart, A. (i963). The Adtanced Theory of .Staliitics. Vol. I. Distribution Theory, 2nded. London: Griffin.

Mangahas, M. (1975). "Income Inequality in the Philippines: A Decomposition Analysis." WorldEmployment Programme Research Jl'orking Papers. Population and Employment Working Paper, no. i2,I.L.O., Geneva.

Mchran, F. (1974). "Decomposition of the Gini Index: A Statistical Analysis of Income Inequality."I.L,O., Geneva, September (Mimeograph).

1976] DISAGGREGATION OF GINI COEFFICIENTS 255

Rao, V. M. (I969). "Two Decompositions of Concentration Ratio." Journal of the Royal StatisticalSociety, series A, vol. 132, part 3.

Sen, A. K. (1973)- On Economic Inequality. Oxford: Clarendon Press.Soltow, Lee ( 196o). "The Distribution of Income Related to Changes in the Distributions of Education,

Age, and Occupation." Review of Economics and Statistics, vol. XLII.Theil, H. (1967). Economics and Information Theory. Amsterdam: North-Holland.U.S. Government (1956). "Income of Families and Persons in tL United States: 1956." Current Popu-

lation Reports, series P-6o, no. 27.

Lesotho: A Development Challenge by Willem Maane, distributed by The JohnsHopkins University Press, 1975

Nigeria: Options for Long-Term Development by Wouter Tims and others,published by The Johns Hopkins University Press, 1974

The Current Economic Position and Prospects of Peru by Jose Guerra andothers, distributed by The Johns Hopkins University Press, 1973

The Philippines: Priorities and Prospects for Development by Russell Cheetham,Edward Hawkins, and others, distributed by The Johns Hopkisn UniversityPress, 1976

Senegal: Tradition, Diversification, and Economic Development by HeinzBachrimann and others, distributed by The Johns Hopkins University Press,1974

Turkey: Prospects and Problems of an Expanding Economy by Edmond Asfourand others, distributed by The Johns Hopkins University Press, 1975

Yugoslavia: Development with Decentralization by Vinod Dubey and others,published by The Johns Hopkins University Press, 1975

World Bank Staff Occasional Papers

Coffee, Tea, and Cocoa: Marks? Prospects and Development Lending byShamsher Singh and others, published by The Johns Hopkins UniversityPress, 1977

Malnutrition and Poverty: Magnitude and Policy Options by Shlomo Reutlingerand Marcelo Selowsky, publishecd by The Johns Hopkins University Press,

1976Economic Evaluation of Vocational Training Programs by Manuel Zymeiman,

published by The Johns Hopkins University Press, 1976A Development Model for the Agricultural Sector of Portugal by Alvin C. Egbert

and Hyung M. Kim, published by The Johns Hopkins University Press, 1975The Future for Hard Fibers and Competition from Synthetics by Enzo R. Grilli, dis-

tributed by The Johns Hopkins University Press, 1975Public Expenditures on Education and Income Distribution in Colombia by Jean-

Pierre Jallade, distributed by The Johns Hopkins University Press, 1974Tropical Hardwood Trade in the Asia-Pacific Region by Kenji Takeuchi, dis-

tributed by The JIohns Hopkins University Press, 1974Methods of Project Analysis: A Review by Deepak Lal, distributed by The Johns

Hopkins University Press, 1974

Other Publications

World Tables 1976, published by The Johns Hopkins University Press, 1976The Tropics and Economic Development: A Provocative Inquiry Into the Poverty

of Nations by Andrew Kamarck, published by The Johns Hopkins UniversityPress, 1976

Size Distribution of Income: A Compilation of Data by Shail Jain, distributed byThe Johns Hopkins University Press, 1975

Redistribution with Growth by Hollis Chenery. Montek S. Ahluwalia, C. L. G. Bell,John H. Duloy, and Richard Jolly, published by Oxford University Press, 1974

THE WORLD BANKHeadquarters U1818 H Street, N.W.Washington, D.C. 20433 U.S.A.European Office66, avenue d'lena75116 Paris, FranceTokyo OfficeKokusai Building1-1 Marunouchi 3-chomeChiyoda-ku, Tokyo 1 00, Japan

World Bank reprintsNo. 21. V. V. Bhatt, "Some Aspects of Financial Policies and Central Banking in De-

veloping Countries," World Development

No. 22. Bela Balassa, "Reforming the System of Incentives in Developing Coun-tries," World Development

No. 23. John H. Duloy and Roger D. Norton, "Prices and Incomes in Linear Program-ming Models," American Journal of Agricultural Economics

No. 24. P.B.R. Hazell and P. L. Scandizzo, "Market Intervention Policies When Pro-duction Is Risky," American ,Journal of Agricultural Economics

No. 25. Martin Karcher, "Unemployment and Underemployment in the People's Re-public of China," China Report

No. 26. Luz Maria Bassoco and Roger D. Norton, "A Quantitative Approach to Agri-cultural Policy Planning," Annals of Economic and Social Measurement

No. 27. Efrain Friedmann, "Financing Energy in Developing Countries," EnergyPolicy

No. 28. Norman L. Hicks, "A Model of Trade and Growth for the Developing World,"European Economic Review

No. 29. V. V. Bhatt, "On Technology Policy and its Institutional Frame," World Devel-opment

No. 30, Bela Balassa and Ardy Stoutjesdijk, "Economic Integration among Develop-ing Countries," Journal of Common Market Studies

No. 31. Constantino Lluch and Ross Williams, "Cross Country Demand and SavingsPatterns: An Application of the Extended Linear Expenditure System,"The Review of Economics and Statistics

No. 32. Marcelo Selowsky, "A Note on Preschool-Age Investment in Human Capitalin Developing Countries," Economic Development and Cultural Change

No. 33. Shankar Acharya, "Fiscal Financial Intervention, Factor Prices and FactorPropositions: A Review of Issues," Bangladesh Development Studies

No.34. Shlomo Reutlinger, "A Simulation Model for Evalutating Wurldwide BufferStocks of Wheat," American Journal of Agricultural Economics

No. 35. John Simmons, "Retention of Cognitive Skills Acquired in Primary School,"Comparative Education Review

No. 36. Montel; S. Ahluwalia, "Inequality, Poverty, and Development," Journal of De-velopment Economics