Lorenz Curve and Gini Coefficient in Right Truncated Pareto s Income Distribution

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1 EUROPEAN ACADEMIC RESEARCH Vol. VI, Issue 2/ Mrch 29 ISSN Impct Fctor: (UIF) DRJI Vlue: 5.9 (B+) Lorenz Cure nd Gini Coefficient in Right Truncted Preto s Income Distribution EHTESHAM HUSSAIN Deprtment of Sttistics, Uniersity of Krchi, Pkistn MUHAMMAD AHSANUDDIN Deprtment of Economics, Uniersity of Krchi, Pkistn MASOOD UL HAQ Usmn Institue Technology, Krchi, Pkistn Abstrct Gini s coefficient of concentrtion hs been studied extensiely for the study of dispersion of the rible. Geometriclly, it is relted to the re enclosed by the Lorenz Cure nd the Distribution function of the uniform distribution. The study of the Lorenz Cure for some symmetric nd some skewed distributions hs been done lgebriclly nd grphiclly. The distribution which hs been selected is Right Truncted Preto s Income Distribution. The Gini s Index hs been obtined by clculting the relent re for different prmeters. Keywords: Lorenz Cure, Gini Coefficient, Right Truncted Preto s Income Distribution. INTRODUCTION The concept of spred, ribility or inequlity hs been discussed in the economics nd sttisticl literture in rious forms (Atkinson, 97; Allison, 978; Acemoglu, 23; Yitzhki & Schechtmn, 25; Willims & Doessel, 26). Anlyticlly to mesure the spred, ribility or inequlity in rndom ribles of interest (e.g. income) seerl mesures he been suggested, nmely, stndrd deition, men deition, Gini s 6963

2 men difference etc. (Fellmn, 22). To mesure reltie ribility their coefficients e.g. coefficient of rition, Gini coefficient etc. re lso used nd they re in terms of the prmeter(s) of the underlying distribution. In order to mesure ribility in the rible of interest geometriclly Lorenz cures re used (Arnold, 2; Fellmn, 28). In this pper we will discuss Lorenz cure, coefficient of rition nd Gini coefficient, (which is relted to Lorenz cure) ssocited with the truncted two prmeter Preto distribution (Arnold, 25). Usully left truncted Preto distribution is used s income distribution (Chernobi et l., 25), in this form the Lorenz cure nd Gini coefficient re independent of loction prmeter, but coefficient of rition depends on both prmeters. While in right truncted Preto distribution, these three mesures depend on both prmeters. It is suggested tht, to mesure the ribility, right trunction is importnt in Preto distribution becuse it is relted to rel situtions in the formultion of meningful indices. 2. DEFINITIONS Consider non-negtie rndom rible X with density function f(x), distribution function F(x) nd finite men. The moment distribution function of X, F(x) is defined by Sturt nd Ord (987). x p F( x) f ( x) dx <p< () o L ( p) F ( x) x f ( x) dx x (2) o E( X ) f ( x) dx EUROPEAN ACADEMIC RESEARCH - Vol. VI, Issue 2 / Mrch 29 (3) 6964

3 The grph of L(p) s function of p which we cll s the Lorenz function, is the popultion Lorenz cure on the unit interl [,],(see Fig.). If L(p)=p, then there is complete equlity nd if L(p)>p then disprity ppers i.e. ribility in the distribution of X exists. Twice the re between the Lorenz cure nd the line L(p) = p is clled the Gini coefficient nd define s: G 2 L p dp <G< (4).8.6 L(p) p Fig-. Lorenz cure 3. RIGHT TRUNCATION Corresponding to F(x), let Fk(x) denote the distribution function of the truncted distribution, truncted on the right t x=k, where k is n element of the support {o <F(x) <}. More specificlly we he right truncted distribution function where Fx(k)=Pr. (Xk) F x F ( k x ) ( ) F( k) x xk x k EUROPEAN ACADEMIC RESEARCH - Vol. VI, Issue 2 / Mrch

4 4. LORENZ CURVE IN TRUNCATED DISTRIBUTION The Lorenz cure Lk (p) of the truncted distribution is gien by Mooththu (985) L ( p F( k)) Lk ( p) L ( F ( k)) (5) For pplictions of Lorenz cure nd Gini coefficient see Goldie (977), Srbi (28), Mooththu (985b) nd references cited therein. Mooththu (985) lso contins references to the pplictions of the Preto distribution, Lorenz cure nd Gini coefficient. 5. PARETO DISTRIBUTION The Preto distribution with its density function lso known s income distribution is f x ( ) x x, where is loction prmeter nd is shpe or inequlity prmeter. Using (), (2), (3), (4) nd (5), the following results obtined Men : E( x), ( ) Gini coefficient : Stndrd deition : Lorenz Function : G( ) (2 ) L( p) { ( p), ( 2) },, > 2 In Preto distribution, mesures of ribility cnnot be used for <2, since does not exists in this cse. EUROPEAN ACADEMIC RESEARCH - Vol. VI, Issue 2 / Mrch

5 From the boe properties we see tht men, nd stndrd deition re functions of both prmeters (, ) but Lorenz cure nd Gini coefficient re only the function of prmeter nd no contribution of. 6. RIGHT TRUNCATED PARETO DISTRIBUTION The right truncted Preto distribution nd relted mesures re s follows: where f x F k x ( ), ( ) <x<k, > = elsewhere Men: Fk Stndrd deition: k, k Ek ( X ) Fk k k Fk 2 k 2 E X 2 Gini coefficient: 2 Gk (, ) k Lorenz cure: L p;, k k F( k) k F k p k / From boe properties we see tht men, stndrd deition, Lorenz cure nd Gini coefficient re function of both prmeters (, ). EUROPEAN ACADEMIC RESEARCH - Vol. VI, Issue 2 / Mrch

6 7. NUMERICAL ILLUSTRATION To see the effect of right trunction, comprison is mde by drwing Lorenz cures nd computing men, stndrd deition nd Gini coefficient. Mny combintions of prmeters (, ) he been studied nd it hs been found tht for compring ribility/ inequlity in income distribution Right truncted distribution should be effectie, since the loction prmeter lso hs its contribution while in Preto it is not. Due to lck of spce, results for selected prmeters (, ), gien in Tble, re shown in following figures nd tbles Tble I: Prmeters nd Truncted Points k Tble-I hs set of prmeters (, ) nd k. To drw Lorenz cures nd required computtions, Mthemtic System is used (Wolfrm, 99). Lorenz Cures for Preto s Income distribution nd Truncted to right Preto s Income distribution. Lorenz Cure Truncted Lorenz Cure Combine Lorenz Cure =5, =2.5 =5, =2.5, k= =5, =2.5, k= G=.25 G=.2422 G=.25 EUROPEAN ACADEMIC RESEARCH - Vol. VI, Issue 2 / Mrch

7 =25, =2.5 G=.25 =25, =2.5, k= G=.832 =25, =2.5, k= G>G k Tble II: Comprison, nd G with =5 nd =2.5 Prmeters Summry Results Preto distribution Right Truncted Preto distribution k= =5,= G Tble III: Comprison, nd G with =25 nd =2.5 Prmeters Summry Results Preto distribution Right Truncted Preto distribution k= =25,= G Similr results re obtined by computtion. Tble-II contins results computed for =5, =2.5 erses =25, =2.5 nd k= which show there is no much significnce difference in men, stndrd deition nd Gini coefficient. The results pper in Tble-III show tht if is incresed to =25 nd the remining prmeters re the sme, there is significnt difference in men, stndrd deition nd Gini coefficient, nd reels reduction in ribility. 8. CONCLUSION The study of the Lorenz Cure for some symmetric nd some skewed distributions re shown lgebriclly nd grphiclly in EUROPEAN ACADEMIC RESEARCH - Vol. VI, Issue 2 / Mrch

8 this study. It is found tht the distribution thus obtined by clculting the relent re for different prmeters is Right Truncted Preto s Income Distribution which is true for rel situtions in the formultion of meningful indices. REFERENCES. Acemoglu, D. (23). Cross country inequlity trends. The Economic Journl, 3(485), Allison, P. D. (978). Mesures of inequlity. Americn sociologicl reiew, Arnold, B. C. (2). The Lorenz cure: eergreen fter yers. In Adnces on income inequlity nd concentrtion mesures (pp ). Routledge. 4. Arnold, B. C. (25). Preto distributions. CRC press. 5.Atkinson, A. B. (97). On the mesurement of inequlity. Journl of economic theory, 2(3), Chernobi, A., Rche, S. T., &Fbozzi, F. J. (25). Composite goodness-of-fit tests for left-truncted loss smples. Hndbook of finncil econometrics nd sttistics, Fellmn, J. (22) Estimtion of Gini Coefficients Using Lorenz Cures. Journl of Sttisticl nd Econometric Methods,, Fellmn, J. (28). Income Inequlity Mesures. Theoreticl Economics Letters, 8, Goldie, C. M. (977). Conergence Theorems for Empiricl Lorenz Cures nd their Inerses, Adnces in Applied Probbility. 9, Mooththu, T. S. K. (985). Smpling distributions of Lorenz cure nd Gini index of the Preto distribution. Snkhyā: The Indin Journl of Sttistics, Series B, EUROPEAN ACADEMIC RESEARCH - Vol. VI, Issue 2 / Mrch

9 . Mooththu, T. S. K. (985b). Distribution of Mximum Likelihood Estimtors of Lorenz Cure nd Gini Index of Exponentil Distribution. Ann. Inst. Sttist. Mth., 37, Srbi, J. M. (28). Prmetric Lorenz cures: Models nd pplictions. In Modeling income distributions nd Lorenz cures (pp. 67-9). Springer, New York, NY. 3. Sturt, A. nd Ord, J. K. (987). Kendlls Adnced Theory of Sttistics, Vol. I, fifth edition, ChrlesGriffin, London. 4. Wolfrm, S. (99). Mthemtic: A System for Doing Mthemtics by Computer, 2nd ed.redwood City: C. A. Addison-Wesley. 5. Willims, R. F., & Doessel, D. P. (26). Mesuring inequlity: tools nd n illustrtion. Interntionl journl for equity in helth, 5(), Yitzhki, S., & Schechtmn, E. (25). The properties of the extended Gini mesures of ribility nd inequlity. EUROPEAN ACADEMIC RESEARCH - Vol. VI, Issue 2 / Mrch