Gii Idex ad Polyomial Pe s Parade Jules Sadefo Kamdem To cite this versio: Jules Sadefo Kamdem. Gii Idex ad Polyomial Pe s Parade. 2011. <hal-00582625> HAL Id: hal-00582625 https://hal.archives-ouvertes.fr/hal-00582625 Submitted o 2 Apr 2011 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.
Gii Idex ad Polyomial Pe's Parade Jules SADEFO KAMDEM Uiversité Motpellier I Lameta CNRS UMR 5474, FRANCE Abstract I this paper, we propose a simple way to compute the Gii idex whe icome y is a ite order k N polyomial fuctio of its rak amog idividuals. Key-words ad phrases: Gii, Icome iequality, Polyomial pe's parade, Raks. JEL Classicatio: D63, D31, C15. Correspodig author: Uiversité de Motpellier I, UFR Scieces Ecoomiques, Aveue Raymod Dugrad - Site Richter - CS 79606, F-34960 Motpellier Cedex 2, Frace; E-mail: sadefo@lameta.uiv-motp1.fr 1
1 Itroductio Iterest i the lik betwee icome ad its rak is kow i the icome distributio literature as Pe's parade followig Pe (1971, 1973). The precede has motivated research o the relatioship betwee Pe's parade ad the Gii idex that is a very importat iequality measure. However, this research has so far focused o the liear Pe's parade for which icome icreases by a costat amout as its rak icreases by oe uit (see Milaovic (1997). However, a liear parade does ot closely t may real world icome distributios, Pe's parade, which is covex i the absece of egative icomes (i.e., icomes that icrease by a greater amout as its rak icreases by each additioal uit). Mussard et al. (2010) has recetly itroduced the computatio of Gii idex with a covex quadratic Pe's parade (or secod degree polyomial) parade for which icome is a quadratic fuctio of its rak. I this paper, we exted the computatio of gii idex by usig a more geeral ad empirically more realistic case i which Pe's parade is a covex polyomial of ite order k N. Hece, the Gii idexes for a liear Pe's parade ad for quadratic pe's parade becomes a special case of that for a higher degree polyomial Pe's parade uder some costraits o parameters to keep covexity. The rest of the paper is orgaized as follows: I Sectio 2, the specicatio for a higher degree polyomial Pe's parade is provided. I Sectio 3, the problem of ttig a higher degree polyomial Pe's parade to real world data sets of Milaovic (1997) is discussed. Cocludig is i Sectio 4. 2 High degree Polyomial Pe's Parade 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 i ascedig order ad let r y 1 for the poorest idividual ad r y for the richest oe. Hece, followig Lerma ad Yitzhaki (1984), the Gii idex may be rewritte as follows: G 2 cov(y, r y) ȳ. (1) 2
Here, cov(y, r y ) represets the covariace betwee icomes ad raks ad ȳ the mea icome. It is straightforward to rewrite (13) as: G 2 σ y σ ry ρ(y, r y ) ȳ, (2) where ρ(y, r y ) is Pearso's correlatio coeciet betwee icomes y ad idividuals' raks r y, where σ y is the stadard deviatio of y ad where σ ry is the stadard deviatio of r y. Followig (2) ad uder the assumptio of a liear Pe's parade (i.e. y a + b r y ), Milaovic (1997) demostrates that for a sucietly large, the Gii idex ca be further expressed as: G σ y ρ(y, r y ). (3) 3y Milaovic's result is very iterestig sice it yields a simple way to compute the Gii idex. However, as metioed by Milaovic (1997, page 48) himself, "i almost all real world cases, Pe's parade is covex: icomes rise very slowly at the begiig, the go up by their absolute icrease, ad ally icrease eve at the rate of acceleratio". Thus, ρ(y, r y ) which measures liear correlatio will be less tha 1. Agai, from Milaovic (1997), a covex Pe's parade may be derived from a liear Pe's parade throughout regressive trasfers (poor-to-rich icome trasfers). Ispired by Milaovic's dig, we demostrate i the sequel, without takig recourse to regressive trasfers, that the Gii idex ca be computed with a geeral oliear polyomial fuctio Pe's parade. The computatio of the Gii Idex usig a polyomial fuctio of order 2 (i.e. quadratic fuctio) Pe's Parade has bee itroduced i Mussard et al.(2010). I this paper, we geeralize the precedure to compute the Gii Idex usig a polyomial fuctio of order k N pe's parade. Note that a liear pe's parade correspods to k 1 ad quadratic pe's parade correspods to k 2. Our result here will be available for each ite k N. 2.1 Simple Gii Idex with oliear power Pe's Parade Cosider a power fuctio relatio betwee icomes ad raks: k 1 y b i ry i + b k ry k. (4) 3
with k N. The covariace betwee y ad r j y for j N is give by: cov(y, r j y) b i cov ( ) ry, i ry j b i b i ri+j+1 y r y1 b i 2 r i+j+1 y ry j ry i r y1 r y1 b i ry i ry j r y1 r y1 (5) The mea icome ȳ is the: [ y b i ry i 1 ] b i r i (6) 2.2 The coeciet of variatio Sice the icomes y are positive, we use (13) by assumig that b k > 0 ad b j for j 1,..., k 1 are chose such that y > 0. For istace, if k 2 the we ca use b j for j 1,..., k such that b 2 1 4b 2 b 0 < 0. We are ow able to compute the coeciet of variatio of icomes as follows: σ y y k j0 b ib j cov(r i y, r j y) b i r i y (7) k b i j0 b j cov(r i y, r j y) b i r i y (8) k j0 b j cov(y, r j y) b i r i y (9) k j0 b j [ b i r y1 ri+j+1 y 1 [ b i ] 2 r y1 ri y r y1 rj y ( k b ir i )]. 4
where the variace of r k y is cov(r k y, r k y) σ 2 r k y 1 r 2k ( 1 the covariace betwee r j y ad r i y for 0 i, j k, is: cov ( ) ry, i ry j 1 After a double summatio ( ) cov b i ry, i b j ry j j0 r i+j 1 2 b i b j [ 1 ) 2 r k, (10) r i r j. (11) r i+j 1 2 r i ] r j. (12) Lemma 2.1 Whe, for q N ad r N, we have that r q q+1 q + 1. (13) Based o the precedig lemma, the variace of y whe is equivalet to: ( ) cov ry, i ry j b2 k k+k+1 k + k + 1 b2 k k+1 k+1 2 k + 1 k + 1 ( k b k k) 2 (2k + 1)(k + 1). 2 j0 (14) Therefore whe the stadard deviatio of y is equivalet to ( σ y k b k k) 2 (2k + 1)(k + 1) k b k 2 (k + 1) 2k + 1 k. (15) Whe, the mea of y is equivalet to y 1 b i r i b k k+1 k k + 1 b k k + 1 Thereby, as the coeciet of variatio is is equivalet expressed as: (16) σ y y k b k (k+1) 2k+1 k b k k k+1 (17) 5
the we have the followig limit which depeds o k ad the sig of b k : σ y lim y b k We have the proved the followig theorem: b k k k sig(b k ). (18) 2k + 1 2k + 1 Theorem 2.1 Uder the assumptio of a oliear polyomial Pe's parade, i.e., y b i ry, i with b k 0, whe, the coeciet of variatio of the reveue y has the followig limit: σ y lim y b k b k k (19) 2k + 1 O the other had, followig Milaovic (1997): lim 2 σ r y lim 2 1 1. (20) 3 2 3 The product of (20), (18) ad ρ(y, r y ) etails the followig result: Theorem 2.2 Uder the assumptio of a oliear polyomial Pe's parade, i.e., y b i ry, i with b k 0, whe, the Gii idex G k ca be approximately compute as follows: G k 1 b k k ρ(y, r y ), (21) 3 2k + 1 b k where ρ(y, r y ) is the correlatio coeciet betwee the reveue ad the rak r y. Remark 2.1 Note that for k 1, we have the result of Milaovic (1997), i.e. G 1 ρ(y,ry) ad for k 2, we have the result of Sadefo Kamdem et al. 3 (2010), i.e. G 2 2 ρ(y,ry) 15. Remark 2.2 Remarks that the Gii Idex G k approximatio depeds o k, the sig of b k ad ρ(y, r y ). Sice the Gii computatio is idepedet of parameters b i for i 0,..., k 1, we ca compute the Pe's Parade as follows: y b 0 + b k r k. Based o reveue data, it is simple to use a regressio to estimate the parameters b 0 ad b k. 6
3 Applicatio with Milaovic (1997) Data I relatio with the paramater k ad b k > 0, we have the followig table: Table 1: Computatio of some coeciet of variatio GV k. k 1 2 3 4 5 6 7 8 CV k 0,577 0,894 1,134 1,333 1,508 1,664 1,807 1,940 GC k 0.333 0.516 0.655 0.770 0.870 0.961 1.043 1.120 where CV k deotes the coeciet of variatio for polyomial Pe's Parade of order k ad GC k CV k / 3. Followig Milaovic's data (1997), we obtai the followig results: Table 2: Compariso the gii idexes of G k for k 1, 2, 3, 6 with true G est Coutry (year) ρ(y, r y ) G 1 G 2 G 3 G 6 G est Hugary (1993; aual) 22062 0.889 0.296 0.459 0.582 0.854 0.221 Polad (1993; aual) 52190 0.892 0.297 0.461 0.584 0.857 0.288 Romaia (1994; mothly) 8999 0.863 0.288 0.446 0.565 0.829 0.284 Bulgaria (1994; aual) 7195 0.889 0.296 0.459 0.582 0.854 0.308 Estoia(1995; quarterly) 8759 0.871 0.290 0.450 0.570 0.837 0.342 UK (1986; aual) 7178 0.815 0.272 0.421 0.534 0.783 0.320 Germay (1889; aual) 3940 0.744 0.248 0.384 0.487 0.715 0.305 US (1991; aual) 16052 0.892 0.297 0.461 0.589 0.857 0.391 Russia (1993-4; quarterly)) 16356 0.812 0.271 0.419 0.532 0.780 0.502 Kyrgyzsta (1993; quarterly) 9547 0.586 0.195 0.303 0.384 0.563 0.551 I the precedig table, G est deotes the estimatio of the true Gii idex by usig Milaovic data (1997). Remark 3.1 Based o the aalysis of the precedig table, we propose to cosider liear Pe's parade (k1) to compute the Gii idex of Polad (1993, aual), Romaia (1994, mothly), Bulgaria(1994; aual). For estoia (1995; quarterly) ad UK (1986; aual), we ca also choose k 1, but we cosider k 2 i the case where govermet policy prefers to overestimate iequality istead of uderestimate iequality. For Russia (1993-4; quarterly), we cosider k 3 ad for Kyrgyzsta (1993; quarterly) we choose k 6. 7
Remark 3.2 I our polyomial pe's parade, if r y 1, the y y 1 i1 b k. I practical applicatios, the parameters b k, estimated from the observed data usig multiple regressio, are chose so that the reveue y > 0 (i.e. b k > 0). It's very importat to ote that, the correlatio coeciet betwee y ad its rak r y deped o b k. Igorig this fact will arbitrarily restrict the Parade to pass through the origi ad may result i less accurate estimates of the Gii Idex. 4 Cocludig Remarks Followig Milaovic (1997), we have proposed aother simple way to calculate the Gii coeciet uder the assumptio of a geeral polyomial Pe's Parade of order k. By usig the data i Table 2, we coclude that the computatio of the Gii Idex of each reveues data eeds to d a specic iteger k, which is the order of a specic polyomial Pe's Parade. Our Gii computatio is useful i practical applicatios as soo as the limit expressio obtaied i (21)is a good approximatio of the Gii Idex for usual size. Two immediate ad practical applicatios ca be geerated from this ew Gii expressio. First, the possibility to address a sigicat test sice our Gii Idex (as well as Milaovic's) is based o Pearso's correlatio coeciet. Thereby, testig for the Gii Idex sigicace is equivalet to testig for the sigicace of Pearso's correlatio coeciet (up to the k costat 2k+1 ). This test relies o the well-kow studet statistics based o the hyperbolic taget trasformatio. Secod, estimatig the coeciets b i for i 1,..., k, e.g. with Yitzhaki's Gii regressio aalysis or a multiple regressio, eables a parametric Gii Idex to be obtaied. That procedure depeds o parameters reectig the curvature of Pe's Parade. This may be of iterest whe oe compares the shape of two icome distributios. 8
Refereces [1] Milaovic, B., "A simple way to calculate the Gii coeciet, ad some implicatios," Ecoomics Letters, 56, 45-49, 1997. [2] Lerma R.I., S. Yitzhaki, "A ote o the calculatio ad iterpretatio of the Gii idex," Ecoomics Letters, 15, 363-368, 1984. [3] Pe, J., Icome Distributio. (Alle Lae The Pegui Press, Lodo), 1971. [4] Pe, J., A parade of dwarfs (ad a few giats),i: A.B. Atkiso, ed., Wealth, Icome ad Iequality: Selected Readigs, (Pegui Books, Middlesex) 73-82, 1973. [5] Mussard, S., Sadefo Kamdem, J., Seyte, F., Terraza, M., "Quadratic Pe's parade ad the computatio of Gii Idex, Accepted for publicatio i Review of Icome ad Wealth, 2010. 9