Coefficient of variation and Power Pen s parade computation

Transcription

1 Coefficiet of variatio ad Power Pe s parade computatio Jules Sadefo Kamdem To cite this versio: Jules Sadefo Kamdem. Coefficiet of variatio ad Power Pe s parade computatio. 20. <hal > HAL Id: hal Submitted o 7 Apr 20 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.

2 Coeciet of variatio ad Power Pe's parade computatio Jules SADEFO KAMDEM Uiversité Motpellier I Lameta CNRS UMR 5474, FRANCE April 6, 20 Abstract Uder the the assumptio that icome y is a power fuctio of its rak amog idividuals, we approximate the coeciet of variatio ad gii idex as fuctios of the power degree of the Pe's parade. Reciprocally, for a give coeciet of variatio or gii idex, we propose the aalytic expressio of the degree of the power Pe's parade; we ca the compute the Pe's parade. Key-words ad phrases: Gii idex, Icome iequality, Raks, Harmoic Number, Pe's Parade. JEL Classicatio: D63, D3, C5. Correspodig author: Uiversité de Motpellier I, UFR Scieces Ecoomiques, Aveue Raymod Dugrad - Site Richter - CS 79606, F Motpellier Cedex 2, Frace; sadefo@lameta.uiv-motp.fr

3 Itroductio Iterest i the lik betwee icome ad its rak is kow i the icome distributio literature as Pe's parade followig Pe (97, 973). The precede has motivated research o the relatioship betwee Pes parade ad the Gii idex that is a very importat iequality measure. However, this research has so far focused o the liear Pes parade for which icome icreases by a costat amout as its rak icreases by oe uit (see Milaovic (997). Furthermore, a liear pe's parade does ot closely t may real world icome distributios Pes parade of which is covex i the absece of egative icomes (i.e., icome icreases by a greater amout as its rak icreases by each additioal uit). Mussard et al. (200) has recetly itroduced the computatio of Gii idex with covex quadratic Pes parade (or secod degree polyomial) parade for which icome is a quadratic fuctio of its rak (see also Ogwag (200)). This paper exteds the computatio of gii idex by usig a more geeral ad empirically more realistic case for which Pes parade is a power fuctio. Hece, the Gii idices for a liear Pe's parade ad for quadratic Pe's parade becomes special cases of our thus itroduced power Pes parade uder some costraits o parameters iducig covexity. Uder the assumptio that icome y is a power fuctio of its rak amog idividuals, we approximate the coeciet of variatio ad gii idex as fuctios of the power degree of the pe's parade. Reciprocally, for a give coeciet 2

4 of variatio or gii idex, we propose the aalytic expressio of the degree of the power pe's parade. It follows that, for a give coeciet of variatio or gii idex we ca compute the pe's parade. We use the Milaovic (997) data to illustrate the iterest of our results. The rest of the paper is orgaized as follows: I Sectio 2, the specicatio of a a power Pes parade is provided. I Sectio 3., the problem of ttig a power Pes parade to real world datasets of Milaovic (997) is discussed. The cocludig remarks are made i Sectio 4. 2 High degree Power 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 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 2 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 (3) as: G 2 σ y σ ry ρ(y, r y ) ȳ, (2) 3

5 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 (997) demostrates that for a sucietly large, the Gii idex ca be further expressed as: G σ y 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 (997, page 48) himself, "i almost all real world cases, Pe's parade is covex: icomes at rst rise very slowly, ad the their absolute icrease, ad ally eve the rate of icrease, accelerates". Thus, ρ(y, r y ) which measures liear correlatio will be less tha. Agai, from Milaovic (997), a covex Pe's parade may be derived from a liear Pe's parade throughout regressive trasfers (poor-torich icome trasfers). Ispired from Milaovic's dig, we demostrate i the sequel, without takig recourse to regressive trasfers, that the Gii idex ca be computed with a quite geeral oliear polyomial fuctio Pe's parade. The computatio of the gii idex usig a power fuctio of order 2 (i.e. quadratic fuctio) Pe's parade has bee itroduced i Mussard et al.(200). I this paper, we geeralize the precede to compute gii idex usig a power fuctio pe's parade. Note that a liear ad quadratic Pe's 4

6 parade are particular cases our power Pe parade. 2. Simple Gii Idex with oliear power Pe's Parade Cosider a power fuctio relatio betwee icomes ad raks: k y b i r α i + b k r α k. (4) with k N, α 0 0 ad α i R + for i,..., k, ad i0 α k > max{α i, i,..., k }. The covariace betwee y ad r α j for j N is give by: cov(y, r α j ) b i cov ( r α i, r α j ) (5) i i i ( b i [ b i rα i+α j + b i 2 r α i+α j + b i 2 r α i r α i r α i ) r α j ] The mea icome ȳ is the: y i0 [ b i r α i ] b i r α i i0 (6) 5

7 2.2 The coeciet of variatio ad Gii computatio Sice icomes y are positive, we use (3) by assumig that b k > 0 ad b j for j,..., k are chose such that y > 0. For istace, if k 2 the we ca use b j for j,..., k such that b 2 4b 2 b 0 < 0. We are ow able to compute the coeciet of variatio of icomes as follows: σ y y k i k j b ib j cov(r α i, r α j) k i b i r α i (7) k i b i k j b j cov(r α i, r α j) k i b i r α i (8) k j b j cov(y, r j y) k i b i r α i (9) k j b j k i [ b i rα i+α j + y [ ( k b i 2 r y rα i j] rα )]. i0 b ir α i where the variace of r α k is cov(r α k, r α k ) σ 2 r α k r 2α k ( 2 r k) α, (0) the covariace betwee r α j ad r α i for 0 < i, j k, is: cov (r α i, r α j ) r α i+α j 2 r α i r α j. () 6

8 After a double summatio ( ) cov b i r α i, b j r α j i j i i b i b j [ r α i+α j 2 r i r α j ]. (2) Lemma 2. Whe, for q R + ad r N, we have that r q q+ q +. (3) Based o the precedig lemma, the variace of y whe is equivalet to: ( cov r α i, i j r α j ) b2 k α k+α + + α k + α k + b2 k αk+ 2 α k + α k+ α k + ( αk bk k) 2 (2α k + )(α k + ). 2 Therefore whe the stadard deviatio of y is equivalet to (4) ( σ y α k bk k) 2 (2α k + )(α k + ) α k b k 2 (α k + ) 2α k + α k. (5) Whe, the mea of y is equivalet to y i0 b i r α i b k α k+ α k α k + b k α k + (6) 7

9 Thereby, as the coeciet of variatio is equivalet expressed as: σ y y α k b k (α k +) 2α k + α k b k α k α k + (7) the we have the followig limit which depeds o k ad the sig of b k : σ y lim y b k b k α k 2αk + sig(b k) 2αk +. (8) α k We have the proved the followig theorem: Theorem 2. Uder the assumptio of a power Pe's parade, i.e., y k i0 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 α k 2αk + (9) O the other had, followig Milaovic (997): lim 2 σ r y lim 2. (20) Remark 2. The followig theorem provide a iterestig result i practise. It tells us that, for a give icome data such as the coeciet of variatio, we ca compute the degree of the power Pe's parade. Therefore, we ca compute the Pe's parade. 8

10 Theorem 2.2 For b k > 0, If we kow the coeciet of variatio CV emp by usig icomes data, the we ca d the parameter s α k as a solutio of the followig equatio: s 2 s + CV emp. (2) We ca the use multiple regressio to d b i ad get the Pe's parade that correspods to icomes data. After a straightforward calculus, the positive solutio of the equatio (25) is : α k CV emp [CV emp + ] CVemp 2 +. (22) The product of (20), (8) ad ρ(y, r y ) etails the followig result: Theorem 2.3 Uder the assumptio of a power Pe's parade, i.e., y k i0 b i r α i, with b k 0, whe, the Gii idex G k ca be approximately computed as follows: G αk b k α k 3 2αk + ρ(y, r y), (23) b k where ρ(y, r y ) is correlatio coeciet betwee the icomes y ad their raks r y. Corollary 2. For b k > 0, the Gii idex is approximately equal to G αk b k α k 3 2αk + ρ(y, r y), (24) b k 9

11 where α k is the solutio of the followig equatio: s 2 s + CV emp. (25) where CV emp is the coeciet of variatio obtaied by usig icomes data y. Remark 2.2 Note that for α k, we have the result of Milaovic (997), i.e. G ρ(y,ry) 3 ad for α k 2, we have the result of Mussard et al. (200), i.e. G 2 2 ρ(y,ry) 5. Remark 2.3 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 do ot depeds o b i for i 0,..., k, we ca the compute the Pe's parade as follows: y b 0 + b k r α k. Based o reveues data, it is coveiet to use a simple regressio to estimate the parameters b 0 ad b k. Also we ca use the followig simplify power Pe's parade y b 0 +b r +b k r α k. If the reveues are ordered as follows: 0 y y 2... y, the the precede Pe's parade pass through the origi (, y ) implies that y b 0 + b + b k, y 2 b 0 + 2b + 2 α k bk ad y 3 b 0 + 3b + 3 α k bk where α k is give by 33. We ca the d b 0, b ad b k. 0

12 2.3 Correlatio betwee the icomes ad raks The covariace betwee icome y ad the rak r y is cov(y, r y ) b i [cov(r α i, r)] i0 i i b i [ r αi+ 2 r α i ] r [ b i H[, α i ] + ] H[, α i] (26) where H[, s] is a harmoic umber fuctio for s > ad N. I the simple case where, y b 0 + b r α, we have: r s [ cov(y, r y ) b H[, α ] + ] H[, α ] (27) The variace of the icome y is: cov(y, y) b 2 H[, α ] ( ) 2 H[, α ]. (28) The stadard deviatio of icome is: σ y cov(y, y) b H[, α ] (H[, α ]) 2. (29)

13 The stadard deviatio deviatio of idividuals raks is give by: σ ry r 2 ( ) 2 r (2 + )( + ) 6 ( + )2 4 (30) The correlatio coeciet betwee y ad r is: ρ(y, r y ) [ b rα + + ] rα b rα + ( ) 2 rα 2 2 [ b H[, α ] +H[, α ] ]. (3) b H[, α ] (H[, α ]) Theorem 2.4 If the icome y b 0 + b k r α k, with α > 0 ad b k 0 (i.e. b k > 0), the the Gii idex is approximately equal to G αk α k 3 2αk + H[, α k ] +H[, α k] H[, α k ] (H[, α k ]) (32) where α k CV emp ( CV emp + + CV emp ) (33) with CV emp beig the coeciet of variatio obtaied by usig icome data y. 2

14 3 Pe's parade of dieret coutries data ad their Gii idexes I this sectio, we use the Milaovic (997) coutries icomes data for the computatio of the Pe's parade ad gii idex of each coutry icome data. I relatio with the paramater k ad b k > 0, we have the followig table: Table : The Positive solutio s α k (see (22)) of equatio (25) i relatio with the empirical coeciet of variatio CV emp of each coutry. CV emp α k

15 Table 2: Coeciet of variatio ad Gii idex G αk for dieret degree α k for each coutry Pe's parade. Coutry (year) ρ(y, r y ) CV emp α k G αk Hugary (993; aual) Polad (993; aual) Romaia (993; aual) Bulgaria (994; mothly) Estoia(995; quarterly) UK (986; aual) Germay (889; aual) US (99; aual) Russia (993-4; quarterly)) Kyrgyzsta (993; quarterly) I the precede table, G αk deotes the estimatio of the Gii idex by usig Milaovic data (997). Remark 3. Based o the aalysis of the previous table, we ca propose to cosider power Pe's parade (α k 0.653) to compute the Gii idex for Hugary (993, aual), α k for Polad(993; aual), α k 0.98 for Romaia (994, mothly), α k.060 for Bulgaria(994; mothly), α k.285 for Estoia (995; quarterly), α k.285 for UK (986; aual), α k.375 for Germay (889; aual), α k.532 for US (99; aual), 4

16 α k 2.72 for Russia (993-4; quarterly)) ad α k for Kyrgyzsta (993; quarterly). Remark 3.2 I our power Pe's parade, if r y, the y y k i0 b k. I practical applicatios, the parameters b k which are estimated from the observed data usig multiple regressio (OLS) are such that the reveue y > 0 (i.e. b k > 0). It very importat to ote that, the correlatio coeciet betwee y ad its rak r y deped o b k. Igorig this fact will equivalet to arbitrarily restrictig the parade to pass through the origi ad may result i less accurate estimates of the Gii idex. 4 Cocludig Remarks I this paper, we proposed the power pe's parade as a alterative to the liear ad quadratic pe's parade. Followig Milaovic (997), we have proposed aother simple way to calculate the Gii coeciet uder the assumptio of a geeral power Pe's parade of order k. By usig the data of Table 2, we cocluded that the computatio of the Gii idex of each icome data coutry eed to d a specic α k R + which is the order of a specic power Pe's parade. We have therefore compute the Pe's parade ad the gii idex for each of the six coutries. It appeared that for a give coeciet of variatio of a coutry icomes data, we foud the associated degree of the power Pe's parade, hece eablig compute the pe's parade usig multiple regressio. It also provides 5

17 a coveiet ad straightforward way to compute the Gii idex of each coutry by usig the coeciet of correlatio betwee icomes ad raks. Oe immediate ad practical implicatios result from this ew Gii expressio. Estimatig the coeciets b i for i,..., k, e.g. with Yitzhaki's Gii regressio aalysis or a multiple regressio, eables a parametric Gii idex to be obtaied that depeds o parameters reectig the curvature of Pe's parade, which may be of iterest whe oe compares the shape of two icome distributios. 6

18 Refereces [] Milaovic, B., "A simple way to calculate the Gii coeciet, ad some implicatios," Ecoomics Letters, 56, 45-49, 997. [2] Lerma R.I., S. Yitzhaki, "A ote o the calculatio ad iterpretatio of the Gii idex," Ecoomics Letters, 5, , 984. [3] Pe, J., Icome Distributio. (Alle Lae The Pegui Press, Lodo), 97. [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, 973. [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, 200. [6] Ogwag, T., "The gii idex for a quadratic pr's parade, Workig paper,