Groupe de Recherche en Économie et Développement International. Cahier de Recherche / Working Paper 10-18

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1 Groupe de Recherche e Écoomie et Développemet Iteratioal Cahier de Recherche / Workig Paper 0-8 Quadratic Pe's Parade ad the Computatio of the Gii idex Stéphae Mussard, Jules Sadefo Kamdem Fraçoise Seyte Michel Terraza

2 Quadratic Pe s Parade ad the Computatio of the Gii idex Stéphae Mussard Uiversité Motpellier I Fraçoise Seyte Uiversité Motpellier I J.Sadefo Kamdem Uiversité Motpellier I Michel Terraza Uiversité Motpellier I Abstract Followig Milaovic s (997) paper [Ecoomics Letters, vol., p. 4-49], we propose a simple way to compute the Gii idex whe icome y is a quadratic fuctio of its rak amog idividuals. Key-words ad phrases: Gii, Icome iequality, Quadratic parade, Raks. JEL Classificatio: D3, D3, C. Correspodig author: Uiversité de Motpellier I, UFR Scieces Ecoomiques, Aveue Raymod Dugrad - Site Richter - CS 790, F-3490 Motpellier Cedex, Frace; sadefo@lameta.uiv-motp.fr

3 Itroductio Suppose that positive icomes, expressed as a vector y, deped o idividuals raks r y i ay give icome distributio of size. Suppose that icomes are raked by ascedig order ad let r y for the poorest idividual ad r y for the richest oe. Hece, followig Lerma ad Yitzhaki (984), the Gii idex may be rewritte as follows: G cov(y, r y) ȳ. () Here, cov(y, r y ) represets the covariace betwee icomes ad raks ad ȳ the mea icome. It is straightforward to rewrite () as: G σ ry ρ(y, r y ) ȳ, () where ρ(y, r y ) is Pearso s correlatio coefficiet betwee icomes y ad idividuals raks r y, where is the stadard deviatio of y ad where σ ry is the stadard deviatio of r y. Followig () ad uder the assumptio of a liear Pe s parade (i.e. y a + b r y ), Milaovic (997) demostrates that for a sufficietly large, the Gii idex ca be further expressed as: G 3y ρ(y, r y ). (3) Milaovic s result is very iterestig sice it yields a simple way to compute the Gii idex. However, as metioed by Milaovic (993) himself, "i almost all real world cases, Pe s parade is covex: icomes at first rise very slowly, ad the their absolute icrease, ad fially eve the rate of icrease, accelerates". Thus, ρ(y, r y ) which measures liear correlatio will be less tha. Agai, from Milaovic (993), a covex Pe s parade may be derived from a liear Pe s parade throughout regressive trasfers (poorto-rich icome trasfers). Ispired from Milaovic s fidig, we demostrate i the sequel, without takig recourse to regressive trasfers, that the Gii idex ca be computed with a quite geeral quadratic Pe s parade. Simple Gii Idex with Quadratic Pe s Parade Cosider a quadratic relatio betwee icomes ad raks: y a + b r y + c r y. (4)

4 The covariace betwee y ad r y is give by: cov(y, r y ) b cov(r y, r y ) + c cov(r y, r y ) b σ r y + c cov(r y, r y ). () The mea icome ȳ is the: y a+br y +c r y a+ i [b i+c i ] a+b + ( + )( + ) +c. (). The coefficiet of variatio Sice icomes y are positive, we use () by assumig that c > 0 ad b 4ac < 0. We are ow able to compute the coefficiet of variatio of icomes as follows: y The stadard deviatio of r y is: b σr y + c σr + bc cov(r y y, ry) a + b (+) + c (+)(+). (7) ( ) σ r y i 4 i i i ( + )( + )(3 + 3 ) ( + ) ( + ) 30 3 ( + )( + )(8 + 3 ). (8) The stadard deviatio of r y is: σ ry ( ) i i i i ( + )( + ) ( + ) 4. (9) 3

5 The covariace betwee r y ad r y is: cov(r y, r y) i i 3 ( ) ( ) i i ( + ) + ( + )( + ) 4 ( + ). 4 3 (0) Thereby, the coefficiet of variatio is expressed as: y c c b 3 3( ) + c 3 i i (+)(+)(8 +3 ) + bc (+) 3 a + b (+) + c (+)(+) b 3( ) c (+)(+) (+)(+) + b c a c (+)(+) + b c ( ) + + (+). () Assumig that max(a, b) c, sice c > 0, we deduce the followig limit: lim y. () Therefore, whe is sufficietly large, we have the followig approximatio for the coefficiet of variatio: y. (3) Remark. Uder the quadratic Pe s parade assumptio, we obtai a approximatio for the coefficiet of variatio valued to be /, while Milaovic (997) obtaied / 3 uder the liear Pe s parade assumptio. O the other had, followig Milaovic (997): lim σ r y lim 3. (4) 3 The product of (4), (3) ad ρ(y, r y ) etails the followig result: Theorem. Uder the assumptio of a quadratic Pe s parade, i.e., y a + b r y + c r y, the Gii idex approximatio is: G ρ(y, r y ), () if is sufficietly large, max(a, b) c, c > 0 ad b 4ac < 0. 4

6 . Applicatio Followig Milaovic s data (997), we obtai the followig results: Coutry (year) ρ(y, r y ) G ρ(y, r y ) G 3 ρ(y, r y) Hugary (993; aual) Polad (993; aual) Romaia (994; mothly) Bulgaria (994; aual) Estoia(99; quarterly) UK (98; aual) Germay (889; aual) US (99; aual) Russia (993-4; quarterly)) Kyrgyzsta (993; quarterly) Remark. As ca be see i the previous Table, Milaovic s Gii idex based o the liear Pe s parade uderestimates the Gii idex obtaied uder the quadratic Pe s parade. 3 Cocludig Remarks Followig Milaovic (997), we have proposed aother simple way to calculate the Gii coefficiet uder the assumptio of a quadratic Pe s parade. Two immediate ad practical implicatios result from this ew Gii expressio. First, the possibility to address a simplified sigificatio test sice our Gii idex (as well as Milaovic s oe) is based o Pearso s correlatio coefficiet. Thereby, testig for the Gii idex sigificatio reduces to testig for the sigificatio of Pearso s correlatio coefficiet (up to the costat / ). This test relies o the well-kow Studet statistics based o the hyperbolic taget trasformatio. Secod, estimatig the coefficiets â, ˆb ad ĉ, e.g. with Yitzhaki s Gii regressio aalysis, eables a parametric Gii idex to be obtaied that depeds o parameters reflectig the curvature of Pe s parade, which may be of iterest whe oe compares the shape of two icome distributios.

7 Refereces [] Milaovic B A simple way to calculate the Gii coefficiet, ad some implicatios. Ecoomics Letters, [] Lerma R.I., Yitzhaki S., 984. A ote o the calculatio ad iterpretatio of the Gii idex. Ecoomics Letters,