ON CALCULATION OF THE EXTENDED GINI COEFFICIENT

Transcription

1 ON CALCULATION OF THE EXTENDED GINI COEFFICIENT Duangkamon Chotkapanch Curtn Unersty of Technology and Wllam Grffths Unersty of elbourne Abstract The conentonal formula for estmatng the extended Gn coeffcent s a coarance formula proded by Lerman and Ytzhak (989). We suggest an alternate estmator obtaned by approxmatng the Lorenz cure by a seres of lnear segments. In a onte Carlo experment desgned to assess the relate bas and effcency of the two estmators, we fnd that, when usng grouped data wth 0 or less groups, our new estmator has less bas and lower mean squared error than the coarance estmator. When nddual obseratons are used, or the number of groups s 30 or more, there s lttle or no dfference n the performance of the two estmators.

2 . INTRODUCTION The Gn coeffcent s a popular measure of ncome nequalty. A generalsaton of t, known as the extended Gn coeffcent, was ntroduced by Ytzhak (983) to accommodate dfferng aersons to nequalty. Whle a number of algebracallyequalent formulas hae been descrbed n the lterature for estmatng the orgnal Gn coeffcent (for example, Nygård and Sandström 98, Table 8.; Creedy 996, p.0, 0), estmaton of the extended Gn coeffcent seems to hae been confned to a coarance formula suggested by Lerman and Ytzhak (989). We suggest an alternate estmator obtaned by approxmatng the Lorenz cure by a seres of lnear segments. The coarance formula and our lnear-segment estmator are dentcal for the orgnal Gn coeffcent, but are not equal n general for the extended Gn coeffcent. Thus, for the orgnal Gn coeffcent, any choce between the two estmators s made on the bass of computatonal conenence only. For the extended Gn coeffcent, howeer, both computatonal conenence and estmator samplng propertes are mportant consderatons. In a onte Carlo experment that we conduct, the two estmators hae smlar propertes when calculated from nddual obseratons; when calculated from grouped data, our new estmator outperforms the coarance estmator n terms of both bas and mean-squared error. Our results hae releance not just for estmaton of the extended Gn coeffcent, but also for estmaton of socal welfare measures that are dependent on the extended Gn coeffcent. See, for example, Lambert (993, p.3-30). In Secton we ntroduce requred notaton and descrbe two ersons of the orgnal Gn coeffcent. In Secton 3 we present the extended Gn coeffcent and ts correspondng coarance estmator, and go on to dere our alternate estmator,

3 3 leang some of the detals to an appendx. The setups and results of the onte Carlo experment are descrbed n Secton 4 and some summary remarks are made n Secton 5.. THE GINI COEFFICIENT Let π = F(x) represent the dstrbuton functon for ncome x and let η = F ( x) be the correspondng frst moment dstrbuton functon. The relatonshp between η and π, defned for 0 x < s the Lorenz cure. We denote t by η = L (π). The much-used Gn coeffcent s equal to twce the area between a 45-degree lne and the Lorenz cure. That s, () G = L( π) dπ 0 It can also be wrtten as (see, for example, Lambert 993, p. 43) G = + xf( x) f( x) dx µ 0 x = co, ( ) µ () { x F x } x where µ E(x) s mean ncome and f ( x) = df( x) dx s the densty functon for ncome. x = Algebracally-equalent dscrete ersons of equatons () and () are often used to estmate G. To ntroduce the notaton necessary to descrbe these two estmators, suppose that ncome data hae been sampled and classfed nto ncome groups. The estmators that we descrbe can be used wth grouped data or wth nddual obseratons. In the case of nddual obseratons, s the number of obseratons, and, n what follows, there s one obseraton n each group, wth the

4 4 proporton of obseratons n each group beng p = /. Gen ths leel of generalty, we assume the followng nformaton s aalable for the -th group:. Aerage ncome x.. The proporton of obseratons p. 3. The cumulate proporton of obseratons π = p + p + + p. 4. The proporton of ncome φ= p x p x. j j j= 5. The cumulate proporton of ncome η = φ + φ + + φ. Also, let x = px denote the sample mean ncome. = As noted by Lerman and Ytzhak (989), the dscrete erson of () that prodes an estmator for G, s (3) G = p ( x x)( π π) x = where π = ( π + π ) and π= pπ. = To obtan a dscrete erson of equaton () to use as an estmator for G, the Lorenz cure L (π) s approxmated by a number of lnear segments, wth the -th lnear segment beng a straght lne jonng ( π, η ) to ( π, η ). Then, the area defned by equaton () can be estmated by aggregatng the areas between the lnear segments and the 45-degree lne. Ths process leads to another famlar expresson for the Gn coeffcent (4) G = η π ηπ + + = =

5 5 It can be shown that G = G. Howeer, when the estmaton prncples used to obtan Ĝ and Ĝ are appled to the extended Gn coeffcent ntroduced by Ytzhak (983), they yeld estmators that are, n general, not dentcal. Preous lterature has focused on a coarance formula smlar to Ĝ (Lerman and Ytzhak 989). The purpose of our note s to dere an expresson for the extended-gn counterpart of Ĝ and to compare the bas and effcency of the two alternate estmators a a onte Carlo experment. 3. A NEW ESTIATOR FOR THE EXTENDED GINI COEFFICIENT The extended Gn coeffcent can be wrtten as (5) () ( ) ( ) () 0 G = π Lπ dπ = x F x f x dx µ [ ( )] ( ) 0 x (6) co {, [ ( )] = x F x } µ x where s an nequalty aerson parameter. The coeffcent G ( ) s defned for > and s equal to the orgnal Gn coeffcent when =. The coarance-formula estmator, gen by the emprcal dscrete erson of equaton (6) s (Lerman and Ytzhak 989) (7) G p x x m ( ) = ( )[( π ) ] x = where p( ). = m = π To dere an alternate estmator obtaned by approxmatng the Lorenz cure n equaton (5) wth a seres of lnear segments, we wrte the equaton of a lnear

6 6 segment from π, η ) to ( π, η ) as η = c π + d where c =φ / p and ( d = ( πη π η )/ p. Then, a lnear-segment approxmaton to G () s gen by (8) π () ( ) ( ) π ( ) = G = π c π+ d dπ In the appendx we show that ths expresson reduces to (9) φ () [( ) ( )] G = + π π = p Ths expresson s a relately smple one whch s easy to calculate, despte the tedous algebra necessary to dere t. Its samplng propertes are assessed n Secton 4. It can be shown that G ( ) = G ( ) f =. Howeer, n general, the two estmators are not dentcal. 4. THE RELATIVE PERFORANCE OF THE TWO ESTIATORS Gen the exstence of two reasonable alternate estmators for the extended Gn coeffcent, ther relate samplng performance s of nterest. To ealuate ths performance, we report the results of a onte Carlo experment wth two hypothetcal ncome dstrbutons. One dstrbuton s a lognormal dstrbuton where log( x ) s normally dstrbuted wth mean µ = 5 and standard deaton σ=.5. The second dstrbuton s one suggested by Sngh and addala (976), wth dstrbuton functon π = F( x) = a = 0.84, b= 400, q =.4 a q x + b Both these dstrbutons exhbt a smlar and relately hgh leel of nequalty wth, approxmately, G (.33) = 0.43, G () = 0.7 and G (5) = 0.9. onte Carlo results

7 7 were also obtaned for other parametersatons, wth lower leels of nequalty. These results are aalable from the authors upon request. They lead to the same conclusons as the results reported here. The other dmensons oer whch senstty was assessed were the alue of and the number of ncome groups. For, we used = (.33,.67,, 3, 5) Samplng performance was ealuated by drawng 5000 samples, each of sze 000, from each dstrbuton. In addton to usng the nddual obseratons ( = 000), results were obtaned for three ncome groupngs = (0, 0, 30). The results from the onte Carlo experment appear n Tables and. The bas of the two estmators appears n Table. Ther relate arance, and ther relate mean-squared error appear n Table. Values of relate arance and mean-squared error greater than one mply the coarance estmator G ( ) s outperformng our lnear-segment estmator G ( ). [Insert Tables and near here.] From Table we can make the followng obseratons about bas:. The bas of both estmators s always negate, reflectng the fact they mplctly assume no nequalty wthn each group.. When = 000, both estmators hae neglgble and almost dentcal bas; the bas s also relately small for = The absolute bas of the coarance estmator s neer less, and often substantally more, than the absolute bas of the lnear-segment estmator. 4. The relate performance of the lnear-segment estmator mproes the further s the departure of from, and the smaller the number of groups.

8 8 From the results n Table, we see that the lower bas for the lnear-segment estmator comes at a cost of hgher arance. Snce a comparson of bases faors the lnear-segment estmator, and a comparson of arances faors the coarance estmator, a mean-squared error comparson s useful. The results usng ths crteron appear n parentheses n Table. These results show that:. For = 30 and = 000 the performance of the two estmators s ery smlar except when = 5 and = 30, where the lnear-segment estmator s notceably better.. For = 0 and = 0 the lnear segment estmator s always better, and sometmes ery much better than the coarance estmator. 5. SUARY An estmator for the extended Gn coeffcent has been dered by approxmatng the Lorenz cure by a seres of lnear segments. Ths estmator s smple to compute and has less bas than a coarance-based estmator that has been used n the lterature. For grouped data where the number of groups s 0 or less, t also has lower mean-squared error than the coarance estmator. The expermental edence s suffcently strong to recommend that, for grouped data where the number of groups s 0 or less, practtoners should use our new estmator n preference to the coarance estmator. If the number of groups s 30 or more, or nddual obseratons are aalable, both estmators perform equally well. Fnally, t should be emphaszed that both estmators requre knowledge of arthmetc mean ncome n each group; these alues are not always aalable.

9 9 APPENDIX In ths appendx we show that equaton (8) can be smplfed to equaton (9). The summaton n equaton (8) can be wrtten as where π ( π) ( c π+ d ) dπ [ I() I()] π = + = = π I () = c ( ) d π π π π c = [ π( π) π ( π ) ] c [( π) ( π ) ] ( ) π I () = d ( π) d π d π = [( π ) + ( π ) ] Substtutng for c and d, and addng these two equatons, yelds, after some algebra, I() + I() = [ η( π) η ( π ) ] φ [( π) ( π ) ] ( ) p Summng ths expresson oer all groups, we obtan φ [ I( ) + I( )] = [( π) ( π ) ] = ( ) = p Substtutng ths expresson nto equaton (8) ges the desred result.

10 0 REFERENCES Creedy, J., Fscal Polcy and Socal Welfare, Edward Elgar, Cheltenham 996. Lambert, P.J., The Dstrbuton and Redstrbuton of Income: A athematcal Analyss, nd edton, anchester Unersty Press, anchester, 993. Lerman, R.I. and S. Ytzhak, Improng the Accuracy of Estmates of Gn Coeffcents, Journal of Econometrcs, 4(), 43-47, September, 989. Nygård, F. and A. Sandström, easurng Income Inequalty, Almqst & Wksell, Stockholm, 98. Sngh, S.K. and G.S. addala, A Functon for Sze Dstrbuton of Incomes, Econometrca, 44(5), , September, 976. Ytzhak, S., On an Extenson of the Gn Inequalty Index, Internatonal Economc Reew, 4(3), 67-68, October, 983.

11 TABLE BIAS OF THE ESTIATORS Groups Estmator Lognormal = 0 G ( ) G ( ) = 0 G ( ) G ( ) = 30 G ( ) G ( ) = 000 G ( ) G ( ) Sngh-addala = 0 G ( ) G ( ) = 0 G ( ) G ( ) = 30 G ( ) G ( ) = 000 G ( ) G ( )

12 TABLE ar[ RELATIVE VARIANCE G ( )] / ar[ G ( )] SE[ G ( )] SE[ G ( )] AND RELATIVE EAN SQUARED ERROR Groups Lognormal = (0.85) (0.964) (.000) (0.75) (0.78) = (0.955) (0.996) (.000) (0.940) (0.494) = (0.993) (0.999) (.000) (0.983) (0.79) = (.006) (.000) (.000) (.000) (.000) Sngh-addala = (0.874) (0.975) (.000) (0.77) (0.04) = (0.978) (0.998) (.000) (0.954) (0.537) = (.003) (.000) (.000) (0.986) (0.84) = (.00) (.00) (.000) (.000) (.000) Note: The relate SEs appear n parentheses below the relate arances.