Third handout: On the Gini Index
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1 Thrd hadout: O the dex Corrado, a tala statstca, proposed (, 9, 96) to measure absolute equalt va the mea dfferece whch s defed as ( / ) where refers to the total umber of dvduals socet. Assume that. The expresso elemets defed below s the equal to the sum of all the Assumg that 3, t s the eas to derve that the geeral case (wth, rather tha four dvduals) oe ca prove, assumg aga that......, that ( ) t s the eas to derve that the mea dfferece ca be wrtte as (/ ) ( ) As a relatve measure of equalt (9,96) proposed to defe what he called a cocetrato rato, kow also as the dex, wth ( / )( / ) where s the (arthmetc) mea of the comes.
2 Note frst the parallelsm betwee ths defto of the dex ad that of the coeffcet of varato CV defed as the rato of the stadard devato over the mea come. Both the dex ad the coeffcet of varato are measures of relatve dsperso (relatve to the mea come, whch appears the deomator of the formulato of both dces). But whereas the coeffcet of varato compares each come wth the mea come (remember that the stadard devato s defed as comes. [(/ ) ( ) ]), the dex compares each come wth all other Moreover ote that the dex ma be also expressed as ( )(/ )( / ) [(( ) / ) s [(( ) / )( / )] where s refers to the share of dvdual total come. The prevous expresso ma be also wrtte as [(( ) / ) (( ) / )] s Sce (( ) / ) refers to the proporto the total populato of the dvduals who ear less tha dvdual (or at least who do ot ear more tha hm/her) ad (( ) / ) to the proporto the total populato of the dvduals who ear more tha dvdual (or at least who do ot ear less tha hm/her), the expresso [(( ) / ) (( ) / )] ma be cosdered as a measure of the "et satsfacto" of dvdual, ths "et satsfacto" beg the dfferece betwee a dvdual "gross satsfacto" (( ) / ) ad a dvdual "deprvato" (( ) / ). The dex ma the be defed as beg equal to the weghted average of these degrees of dvdual "et satsfacto", the weghts beg equal to the share of each dvdual total come. There s therefore a lk betwee the dex ad the cocept of "deprvato" or "satsfacto". Formulato for the dex whe there s more tha observato for each come value Whe there s more tha observato for each come value, t ca be show that the
3 expresso [(( ) / ) (( ) / )] s ma be geeralzed. Assume, for example, that we wat to compute the equalt of come the whole world. We wll the express the dex as { [(( ) ( ) ) / ]( / )} where s the DP per capta coutr s the average DP per capta the world, that s, ( ) / s the sze of the populato coutr s the total world populato represets the total umber of coutres Note that whe we appl ths formulato to the case where there s ol oe dvdual per come we wll get N { [(( ()) { { [(( [( ( ( ()) ) ( ) ) / ))/ ]( ) / ]( ]( / )} / )} / )} { (( ) / )( (( ) / ) where s ( / ) s / )} The related welfare fucto The defto gve prevousl to the dex allows us also to express the dex as [(( ) / )( / )] 3
4 ( / )( / ) [(( ) / )( / )] ( / ) where, the "equall dstrbuted equvalet level of come" correspodg to the dex, s defed as (( ) / ) Let us ow tr to represet graphcall the socal welfare fucto correspodg to the dex. We wll evdetl draw a graph for the case where there are ol two dvduals whose comes are such that. From the expresso ( / ) (3/ ) whch mples that (( ) / ) we derve that, whe, 3 Let us represet ths equato o a graph whose vertcal axs refers to the come of dvdual whle the horzotal axs refers to the come of dvdual. frst dvdual's come + C - 3 M C' B Secod dvdual's come Assumg, for example, that 0, we coclude that 80 3 The socal dfferece curves correspodg to the dex are therefore straght les wth a slope equal to-3. Note that the pots C, ad C' are smmetrc.
5 The pot refers to the "equall dstrbuted equvalet level of come clearl there s the same level of welfare pots, C ad C'. ad Note fall that, as stressed b Doaldso ad Wemark (980), the expresso (( ) / ma be also wrtte as [(( ) ( ) ) / ] ) Ths last expresso ma be geeralzed to derve the "equall dstrbuted level of come" correspodg to a geeralzed socal welfare fucto where ' ' (( / ) ( ) ) Oe ma ote that whe oe obtas the tradtoal case of the dex. Ths geeralzato allows oe however to defe a geeralzed dex as. {[ (( / ) (( ) / ) ) ]/ ]} t s ot dffcult to observe that whe, the geeralzed dex gves us the equalt dex correspodg to Rawls' approach, that s, ' M{,..., }. A useful algorthm: Note that the defto of the dex gve prevousl, wth [(( ) / ) (( ) / )] s ma be also wrtte ( the case of a eve umber of dvduals) as ( / ) (( ) / )( s s ) A smlar expresso ma be derved the case of a odd umber of dvduals. Ths s a ver useful algorthm. Take, for example, the case of sx dvduals wth comes, 3, 0, 6, 60, 00. The followg graph shows how to quckl compute the dex such a case. S + S
6 We have therefore to compare the frst dvdual wth ts smmetrc the dstrbuto, that s, the sxth dvdual. Smlarl we compare the secod dvdual wth the ffth ad the thrd wth the fourth. We the derve the dex as = What wll happe f there s a odd umber of dvduals, sa, fve? We wll proceed the same wa ad wrte the dex as = The dex ad the Lorez curve: Let us ow dscuss aother algorthm to compute the dex, oe whch s lked to the cocept of Lorez curve. The followg table gves the comes of three dvduals, ther share total come ad the cumulatve populato ad come shares, assumg we rak ths tme the dvduals b creasg come. dvduals come Shares Cumulatve Shares Cumulatve raked b total shares total come shares creasg populato total total come come populato (/3) (/3) (/0) (/0) (/3) (/3) (/0) (3/0) 3 7 (/3) (3/3) (7/0) (0/0) Let us ow put the cumulatve populato shares o the horzotal axs ad the cumulatve come shares o the vertcal axs. The graph obtaed s called Lorez curve ad was frst proposed 905 b Lorez. 6
7 Cumulatve come Shares Lorez Curve Cumulatve Populato Shares t ca easl be prove that the (9) dex s fact equal to twce the area betwee the Lorez (905) curve ad the dagoal. So the further awa from the dagoal the Lorez curve s, the more equalt there s (the greater the dex). Whe there s perfect equalt (everoe has the same come) the Lorez curve wll be detcal to the dagoal. O aother terestg formulato of the dex Startg from the followg defto of the dex (( ) / )( / ) where ad s the (arthmetc) mea come, we ca easl whe : derve the followg expresso for {[(( ( ) ) / ]( / )} (( ) / )( / ) ( / )(/ ) [( / ) (( ) / )] (/ )(/ ) [( / ) ((( ) / ) / )] (/ ){(/ ) sce [( / ) ((( ) / ) / )] 0 [( / ) ((( ) / ) / )][ ]} Sce ( ) ad ( / ) (( ) / ) / where refers to the expectac of a varable, we ed up wth 7
8 (/ ) {[( / ) ( / )][ ( )]} (/ ) Cov( F, ) where F ( / ) refers to the relatve rak of dvdual. Rememberg that ( / )( / ) where s s mea dfferece, we also coclude that Cov( F, ) 8
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