Decomposition of Gini and the generalized entropy inequality measures. Abstract

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1 Decompositio of Gii ad the geeralized etropy iequality measures Mussard Stéphae LAMETA Uiversity of Motpellier I Terraza Michel LAMETA Uiversity of Motpellier I Seyte Fraçoise LAMETA Uiversity of Motpellier I Abstract I this article we provide a overview of the Gii decompositio ad the geeralized etropy iequality measures a free access to their computatio a applicatio o Frech wages ad a differet way tha Dagum to demostrate that the Gii idex is a more coveiet measure tha those issued from etropy: Theil Hirschma Herfidahl ad Bourguigo Citatio: Stéphae Mussard Terraza Michel ad Seyte Fraçoise (2003) "Decompositio of Gii ad the geeralized etropy iequality measures" Ecoomics Bulleti Vol 4 No 3 pp 5 Submitted: December Accepted: Jauary URL: 02D60003Apdf

2 Itroductio This paper aalyses Dagum s (997a 997b 998) articles about the Gii idex decompositio ad three particular cases of the etropy coefficiet (Theil Hirschma- Herfidahl ad Bourguigo decompositios) The aim of this article is multiple First it facilitates the access to the computatio of these decomposed measures ad gives a theoretical overview (Sectio 2) Secodly it presets a applicatio o the Frech wages (Sectio 3) Fially it gives a differet approach to Dagum s (998) assessmet of the Gii ratio lie a more complete measure i studyig the ormative cocepts of iequality measuremets 2 The Decompositios of Gii Theil Hirschma-Herfidahl ad Bourguigo O the web site: we propose a free program to estimate the four decompositios ad all the eeded directives to use it 2 The Gii Decompositio Let us cosider a populatio P with icome uits yi (i ) where F(y) µ ad G are respectively the cumulative distributio the mea ad the Gii idex calculated o P which is partitioed i subpopulatios P ( ) The size ad the icome average of P are give by ad µ The Gii idex measured o P is: yi - yr i r G () 2 ²µ The Gii withi the subpopulatio P (withi-group Gii) is give by: G yi - yr (2) i r 2 ²µ ad the betwee-group Gii (that calculates the iequalities betwee P ad Ph) is: h i r h yi - yhr G µ + µ h (3) All these ratios are icluded i the iterval [0] If they ted towards the icome repartitio is uequal ad if they ted towards 0 the repartitio is equal Now let us itroduce two fudametal cocepts O oe had the gross ecoomic affluece (see Dagum 997b) is expressed i the form: y h h h h i hr i hr h df (y) (y - x)df (x) d y - y y > y ad µ µ 0 0 i r d > It is the expected icome differece betwee the groups ad h such as: yi > yhr ad µ > µh O the other had the first order momet of trasvariatio is the expected icome differece betwee P ad Ph give that yi < yhr ad µ > µh: y h h h h i hr i hr h df (y) (y - x) df (x) p y - y y < y ad µ µ 0 0 i r p > Accordig to (4) ad (5) we ca itroduce the relative ecoomic affluece (Dagum 980) It is a ormalized idex that idicates the distace betwee P ad Ph: Dh ( dh - ph) / h ( dh - ph) / ( dh + ph) (6) (4) (5)

3 Calculatig Gh Dh we proceed to the et measure of the betwee-group Gii It symbolizes the iequalities derived from the o-overlap of the distributios ad h The expressio Gh (-Dh) is the trasvariatio betwee P ad Ph which is the part of the iequality issued from the overlap of the distributios ad h If p ad s are respectively the percetage of the idividuals belogig to P ad the icome share of the subpopulatio we have: µ p s (7) µ Accordig to (3) (6) ad (7) we ca defie the first compoet of the Gii decompositio It is the et cotributio of the betwee-group iequalities to the overall Gii measured o P: - G b Gh Dh(p sh + ph s) (8) 2 h The secod compoet is the cotributio of the trasvariatio betwee the subpopulatios to G: - G t Gh ( - Dh)(p sh + ph s) (9) 2 h The third elemet is the cotributio of the withi-group iequalities to G: G w G p s (0) Fially give (8) (9) ad (0) the fudametal equatio of the Gii decompositio i three compoets is: G Gw + Gb + Gt () 22 The Theil Hirschma-Herfidahl ad Bourguigo Decompositios The Theil Hirschma-Herfidahl (H-H) ad Bourguigo idexes are three particular cases of the geeralized etropy ratio give by: yi yi I + - ( ) i µ µ real (2) The geeralized etropy idex I ca be decomposed i a withi-group cotributio Iw ad a betwee-group cotributio Ib: µ µ Iw Iw µ µ (3) µ µ I + - (4) b ( ) µ µ such as I is separable i two compoets I Iw + Ib (5) 22 The Theil Decompositio The Theil idex T is the geeralized etropy ratio whe teds towards 0: yi yi T lim I log 0 (6) i µ µ The betwee-group cotributio Tb ad the withi-group cotributio Tw are: µ µ Tb lim I log (7) 0 µ µ 2

4 µ yi yi Tw lim I log 0 µ i µ µ (8) such as T Tw + Tb (9) 222 The Hirschma-Herfidahl Decompositio The H-H idex I is the particular case of the geeralized etropy whe teds towards : 2 yi yi I lim I - i µ µ (20) The withi-group cotributio Iw ad the betwee-group cotributio Ib are: I w µ ² Var y µ ² lim Iw V²(y ) 2 µ² µ ² 2 µ² (2) µ µ Var µ I b lim Ib - 2 µ µ (22) 2µ² where Var ad V² are respectively the variace ad the coefficiet of variatio Therefore the breadow of the Hirschma-Herfidahl idex is: I Iw + Ib (23) 223 The Bourguigo Decompositio Bourguigo (979) presets a ew coefficiet B Dagum (997b) demostrates that it is the limit of the etropy idex whe teds towards -: B lim I logµ - log Mg (24) - I the same way tha the two precedet ratios the Bourguigo coefficiet is separated i a withi-group cotributio Bw ad a betwee-group cotributio Bb: B w lim Iw ( log µ - log Mg) (25) - µ B b lim Ib log log µ - log Mgµ (26) - µ The expressios M g M g ad Mgµ are the geometric mea respectively measured o P P ad o the vector µ ( ) So the breadow of the Bourguigo idex is: B Bw + Bb (27) 3 Applicatio Let us tae a 996-year wage sample of the south area of Frace It represets idividuals raed by sex (5394 me ad 2266 wome) The methods itroduced above allow oe to measure the compoets of the four decompositios Furthermore it is possible to ow if the iequalities are geerated by the wage gaps withi the two groups or if the iequalities are egedered by the wage gaps betwee me ad wome Table shows these results i givig the percetage of each elemet i the global iequality The three etropic idexes give the same cotributio Ideed the We should attribute the paterity of the measure (24) to Hart (970) p80 3

5 differeces betwee the me ad the wome represet 2% of the global iequality ad the cotributio withi the subpopulatios represets 98% of the overall iequality whereas the Gii idex grats as much importace to the withi-group elemet (509%) as to the betwee-group elemet (the et betwee-group compoet ad the trasvariatio represet 49%) Oly the Gii decompositio ca provide the itesity of trasvariatio (356%) which is the part of the betwee-group disparities issued from the overlap betwee the two distributios Table 2 idicates that all the measures grat two times more wage gaps betwee the me tha betwee the wome Nevertheless the differeces of results betwee the Gii ad the etropic idexes are importat So it is ecessary to direct the choice of the users of iequality coefficiets i examiig the property they chec 4 Commet The mai cocer of Dagum s article (998) we wat to commet is the properties of the social choice theory that the four idexes itegrate Dagum chooses to discuss about the followig priciples: - (A) the iterpersoal utility comparisos; - (B) the iequality aversio (the utility fuctio U is cocave: U <0); - (C) ad a icreasig utility fuctio (U >0) I his paper Dagum demostrates that the Gii ratio satisfies with (A) (B) ad (C) requiremets ad cocludes we should retai the Gii coefficiet as the pricipal measure because the idexes issued from the etropy do ot itegrate the criteria of the iterpersoal utility comparisos Nevertheless uder may coditios the iterpersoal utility comparisos ad the cocavity of the utility fuctio are icompatible If the iterpersoal utility comparisos are permitted idividuals have cosciece of the crucial last dollar eared by the other idividuals (for istace the poor persos) Ideed persos have more satisfactio because they ow that whe their icomes rise they actively participate to the future redistributio i order to decrease iequalities ad poverty So the growth of the utility fuctio ca ot decrease whe icomes icrease Fially we ca doubt about the complemetarity betwee the (A) ad (B) priciples So it is more coveiet for icome iequality idexes to satisfy oly oe of these two properties Therefore i a world where oly the (A) (B) ad (C) priciples exist we should accept the coefficiets derived from the geeralized etropy idex However because this world does ot exist ad because the betwee-group cotributios (4) (7) (22) are obtaied lie a residual (Ib I - Iw) it is preferable to use the Gii decompositio because the betwee-group idex (Gh) is specified ad also because Gb ad Gt ca ot be cosidered as residuals 5 Coclusio We have provided the way for the computatio of the Gii decompositio ad for the etropic idexes with their specificatios The we firstly see that i the south area of Frace the Gii idex attributes as much importace to the cotributio betwee groups as to the withi-group compoet whereas the Theil H-H ad Bourguigo coefficiets show that the iequalities are geerated withi the groups (98%) The Gii ad the three particular cases of the etropy coefficiet idicate that the me are two 4

6 times more cocered with the disparities tha the wome are Nevertheless these two types of measures are too distat ad i order to motivate the choice of users of these measures we show i a differet way tha Dagum (998) that the Gii decompositio is a better idex Eve if Theil H-H ad Bourguigo idexes chec the (B) ad (C) properties which are more importat tha the combiatio of (A) (B) (C) we icite to privilege the Gii decompositio i particular because it is built o a better betweegroup specificatio Refereces Bourguigo F (979) Decomposable Iequality Measures Ecoometrica Dagum C (980) Iequality Measures betwee Icome Distributios with Applicatios Ecoometrica Dagum C (997a) A New Approach to the Decompositio of the Gii Icome Iequality Ratio Empirical Ecoomics 22(4) Dagum C (997b) Decompositio ad Iterpretatio of Gii ad the Geeralized Etropy Iequality Measures Proceedigs of the America Statistical Associatio Busiess ad Ecoomic Statistics Sectio 57 th Meetig Dagum C (998) Fodemets de bie-être social et décompositio des mesures d iégalité das la répartitio du reveu Ecoomie Appliquée Hart P E (970) Etropy ad Other Measures of Cocetratio Joural of the Royal Statistical Society A Mussard S Seyte F ad M Terraza (2002) Program for Dagum s Gii Decompositio Theil H (967) Ecoomics ad Iformatio Theory North-Hollad Publishig Compay Amsterdam Table : Cotributios of each elemet of the four decompositios to the overall iequality Idexes % of the withi-group compoet % of the betwee-group compoet % of trasvariatio G T NA* I NA* B NA* *NA: No available for this type of idex Table 2: Cotributios of the me ad the wome to the global iequality Idexes Wage iequalities withi the me group (%) Wage iequalities withi the wome group (%) G T I B