ON CALCULATION OF THE EXTENDED GINI COEFFICIENT. Duangkamon Chotikapanich and William Griffiths. No May 2000

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1 Unersty of New England School of Economc Studes ON CALCULATION OF THE EXTENDED GINI COEFFICIENT by Duangkamon Chotkapanch and Wllam Grffths No ay 000 Workng Paper Seres n Econometrcs and Appled Statstcs ISSN Copyrght 000 by Duangkamon Chotkapanch and Wllam Grffths. All rghts resered. Readers may make erbatm copes of ths document for non-commercal purposes by any means, proded ths copyrght notce appears on all such copes. ISBN

2 ON CALCULATION OF THE EXTENDED GINI COEFFICIENT Duangkamon Chotkapanch and Wllam Grffths 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. The two estmators are dentcal for the orgnal Gn coeffcent, where the nequalty aerson parameter s =. Howeer, for, they are dfferent. In a onte Carlo experment desgned to assess the relate bas and effcency of the two estmators, we fnd that, when usng grouped data, our new estmator has less bas and lower mean squared error than the coarance estmator. When sngle obseratons are used, there s lttle or no dfference n the performance of the two estmators. Key Words: Lorenz cure, ncome nequalty Duangkamon Chotkapanch s a Senor Lecturer n the School of Economcs and Fnance at Curtn Unersty of Technology. Wllam Grffths s a Professor n the School of Economc Studes at the Unersty of New England. Contact nformaton: School of Economc Studes, Unersty of New England, Armdale, NSW 35, Australa. Emal: wgrfft@metz.une.edu.au.

3 ON CALCULATION OF THE EXTENDED GINI COEFFICIENT. otaton 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). A muchused measure of ncome nequalty s the Gn coeffcent whch s equal to twce the area between a 45-degree lne and the Lorenz cure. That s, 0 G = L( π) dπ () 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 { xfx} where µ E(x) s mean ncome and f ( x) = df( x) dx s the densty functon for x = ncome. Gen a sample of obseratons on ncome, whch may be aalable as sngle obseratons, or grouped nto ncome classes, seeral algebracally equalent formulas for estmatng G hae been descrbed n the lterature. See, for example, Table 8. n Nygård and Sandström (98) or Creedy (996, p.0, 0). The focus n ths paper s on dscrete ersons of the expressons n equatons () and (), and on generalsatons of them that can be used to estmate the extended Gn coeffcent ntroduced by Ytzhak (983) to accommodate dfferng aersons to nequalty. Because the dscrete ersons of () and () are algebracally equalent, any choce between them s made smply on the bass of computatonal conenence. Howeer, as we wll see, for estmatng the extended Gn coeffcent, generalsatons of the dscrete formulas lead to two dfferent estmators whch are not algebracally equalent. In these crcumstances, when choosng a formula for estmaton, computatonal conenence and estmator samplng propertes are mportant consderatons. A generalsaton of the dscrete erson of the coarance formula n 3

4 () was suggested by Lerman and Ytzhak (989). In ths paper we dere an alternate estmator whch s based on a generalsaton of a dscrete erson of equaton (). Ths alternate estmator s smple to calculate and has good samplng propertes. In a onte Carlo experment that we conduct, the two estmators hae smlar propertes when calculated from sngle 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 the remander of ths secton we ntroduce requred notaton and ge the dscrete ersons of () and () that hae been used for estmatng G. In Secton we present the extended Gn coeffcent and ts correspondng coarance estmator, and go on to dere our alternate estmator. The setups and results of the onte Carlo experment are descrbed n Secton 3 and some summary remarks are made n Secton 4. To ntroduce the notaton necessary to descrbe the arous 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 sngle obseratons. In the case of sngle ungrouped obseratons, s the number of obseratons, and, n what follows, there s one obseraton n each group, wth the 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 + K + p. 4. The proporton of ncome φ= px px. j j j= 5. The cumulate proporton of ncome η = φ + φ + K + φ. Also, let x = px denote the sample mean ncome. = 4

5 As noted by Lerman and Ytzhak (989), the dscrete erson of (), that prodes an estmator for G, s where G = p( x x)( π π ) (3) x = π = ( π + π ) and π= p π = Another way to approach the estmaton problem s a equaton (). If 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 G (4) = η π ηπ + + = = 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.. A New Estmator for the Extended Gn Coeffcent The extended Gn coeffcent, proposed by Ytzhak (983) to accommodate the fact that dfferent ndduals can hae dfferent aersons to nequalty, can be wrtten as 5

6 0 G( ) = ( ) ( π) L( π) dπ (5) = x F x f xdx µ [ ( )] ( ) 0 x = co,[ ( )] µ x { x F x } (6) 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) where G p x x m (7) ( ) = ( )[( π) ] x = m = p ( π ) = In the remander of ths secton, we dere an alternate estmator obtaned by approxmatng the Lorenz cure n equaton (5) wth a seres of lnear segments. To begn, note that the equaton of a straght lne jonng ( π, η ) to ( π, η ) can be wrtten as where η = c π + (8) d c = η π η π = (9) φ p π η π η d = (0) π π Now, denote the ntegral n equaton (5) by H (). That s, 0 H ( ) = ( π) L( π) dπ () The lnear-segment approxmaton to H () s 6

7 ( ) π ( ) ( ) π ( ) = H = π c π+ d dπ [ I() I()] () = = + where I π ( ) = c π( π) π dπ c = [ π( π) π ( π ) ] c [( π) ( π )] ( ) (3) I π ( ) = d ( π) π dπ d = [( π ) + ( π ) ] (4) Substtutng for c and d n equatons (3) and (4), and addng these two equatons, yelds, after some algebra, I() + I() = [ η( π) η ( π ) ] φ π π ( ) p [( ) ( )] (5) Thus, [ ] H () = I () + I () = φ = π π ( ) [( ) ( ) ] (6) = p and the alternate estmator for the extended Gn coeffcent s G ( ) = ( ) H ( ) 7

8 = + φ [( π ) ( π )] (7) = 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. 3. The Relate Performance of the Two Estmators Gen the exstence of two ery reasonable alternate estmators for the extended Gn coeffcent, ther relate samplng performance s of nterest. To ealuate ths performance, we set up a onte Carlo experment wth 4 hypothetcal ncome dstrbutons. The detals of those dstrbutons appear n Table. Two are lognormal dstrbutons where log(x) s normally dstrbuted wth mean µ and standard deaton σ. The other two are parametersatons of the dstrbuton suggested by Sngh and addala (976); ts dstrbuton functon s π = F ( x) = x + b The parameter alues were chosen to ge one lognormal and one Sngh-addala dstrbuton wth relately hgh nequalty, and another par of dstrbutons wth relately low nequalty. Setups and hae relately hgh nequalty wth approxmate Gn coeffcents of G ( ) = 0.7. Setups 3 and 4 hae relately low nequalty wth approxmate Gn coeffcents of G ( ) = Relate to the lognormal dstrbuton wth a smlar alue of the Gn coeffcent, the Sngh-addala dstrbuton has a thcker tal, wth extreme alues of ncome more lkely. Desgnng the experment n ths way ges nformaton on the senstty of performance to the type of ncome dstrbuton and the leel of nequalty. a The other dmensons oer whch senstty was assessed were the alue of and the number of ncome groups. The chosen alues of are = (.33,.67,, 3, 5, 0 ). Values of the extended Gn coeffcent correspondng to these alues were q 8

9 computed a numercal ntegraton of equaton (6), and are reported n Table. Samplng performance was ealuated by drawng 5000 samples, each of sze 000, from each of the four dstrbutons. In addton to usng the sngle obseratons ( = 000), results were obtaned for two ncome groupngs = (0, 0). The results from the onte Carlo experment appear n Tables, 3 and 4 whch contan, respectely, the bas of the two estmators, ther relate arance, and ther relate mean-squared error. Values of relate arance and mean-squared error greater than one mply the coarance estmator G ( ) s outperformng our lnearsegment estmator G ( ). 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. 3. 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. From the results n Table 3, we see that the lower bas for the lnear-segment estmator usually comes at a cost of hgher arance. Exceptons occur wth the frst three setups when = 0 and = 0. Snce these exceptons look atypcal relate to the remander of the table, we carred out further experments for alues of between 5 and 0 and greater than 0. These experments reealed that, as ncreases, there s a alue for beyond whch the arance of the lnear-segment estmator s less than the arance of the coarance estmator. The alue depends on the setup; n setup t s lower than for setups, 3 and 4. Also, t s larger for = 0 than for = 0. Howeer, for = 0 or 0, and for large enough, there s always a reersal n the relate magntudes of the arances. When = 000, there s no notceable dfference. 9

10 Snce a comparson of bases faors the lnear-segment estmator, and a comparson of arances faors the coarance estmator (except for large ), a meansquared error comparson s useful. Usng ths crteron, the results n Table 4 show the performance of the lnear-segment estmator s seldom worse, and sometmes ery much better, than the coarance estmator. In the nstances where the lnear-segment estmator s worse (for example, =.33, = 000, setup ), t s only margnally worse, wth the least faorable comparson beng a mean-squared error whch s only 0.8% larger. 4. Summary 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 and lower mean-squared error than a coarancebased estmator that has been used n the lterature. The expermental edence s suffcently strong to recommend that practtoners use our new estmator n preference to the coarance estmator n future emprcal studes. 0

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

12 Table. Setups for onte Carlo Experment Setup 3 4 Dstrbuton Lognormal Sngh addala Lognormal Sngh addala Parameters µ = 5 σ =.5 b = 400 a = 0.84 q =.4 µ = 6.4 σ = 0.7 b = 550 a =.9 q = 0.85 µ x G(.33) G(.67) G() G(3) G(5) G(0)

13 Table. Bas of the Estmators Groups Estmator Setup = 0 G ( ) G ( ) = 0 G ( ) G ( ) = 000 G ( ) G ( ) Setup = 0 G ( ) G ( ) = 0 G ( ) G ( ) = 000 G ( ) G ( ) Setup 3 = 0 G ( ) G ( ) = 0 G ( ) G ( ) = 000 G ( ) G ( ) Setup 4 = 0 G ( ) G ( ) = 0 G ( ) G ( ) = 000 G ( ) G ( ) 3

14 Table 3. Relate Varance: ar[ G ( )] / ar[ G( )] Groups Setup = = = Setup = = = Setup 3 = = = Setup 4 = = =

15 Table 4. Relate SE: SE[ G ()] SE[ G( )] Groups Setup = = = Setup = = = Setup 3 = = = Setup 4 = = =