Inequality measures for intersecting Lorenz curves: an alternative weak ordering

Transcription

1 h Inernaional Scienific Conference Financial managemen of Firms and Financial Insiuions Osrava VŠB-TU of Osrava, Faculy of Economics, Deparmen of Finance 7 h 8 h Sepember 25 Absrac Inequaliy measures for inersecing Loren curves: an alernaive weak ordering Tommaso Lando,, Lucio Beroli-Barsoi The Loren ordering is probably he mos logical ool for comparison of he dispersion of non negaive random variables. Indeed, many income inequaliy merics are order preserving (or isoonic) wih he aforemenioned preorder. However, in some siuaions Loren curves may inersec and he ordering is no fulfilled. Hence, some weaker crieria need o be o inroduced: we presen a new differen preorder, weaker han he Loren ordering, and propose a possible class of funcionals ha preserve i. This mehod could be especially useful under condiions of maximum uncerainy. Key words Loren ordering, inequaliy, dispariy, majoriaion, sochasic dominance JEL Classificaion: D3, D63, I32. Inroducion The Loren curve, which has been inroduced as a represenaion of inequaliy (e.g. income inequaliy), is generally used o rank probabiliy disribuions in erms of an order of preference, ha is, he Loren ordering (LO). In fac, he LO is conform o he idea ha he higher of wo non-inersecing Loren curves (as well as he corresponding disribuion) has o be preferred, in ha i shows less inequaliy compared o he lower one. As is well known, in an economic framework, he LO is coheren wih he Pigou-Dalon condiion ( principle of ransfers ), ha is, he higher of wo non-inersecing Loren curves can be obained from he lower one by an ieraion of income ransfers from richer o poorer individuals (he so called elemenary ransfers or T-ransforms, Marshall e al., 29, p. 32, also called progressive ransfers, Shorrocks and Foser, 987). For his reason, he coherence wih he LO represens a basic propery of many inequaliy (or concenraion) measures. Neverheless, i may happen ha Loren curves inersec or, equivalenly, he LO is no verified, which implies ha he Loren-preserving indices disagree: in his case we can rank he disribuions by relying on weaker orders of inequaliy. In he lieraure, his idea has been analyed in several works, relaed o he concep of hird-order sochasic dominance (see e.g. Akinson, 28), which emphasies he lef ail of he disribuion. Indeed, many auhors agree ha an elemenary ransfer should be more equaliing he lower i occurs in he disribuion (see e.g. Shorrocks and Foser, 987; Dardanoni and Lamber, 988; Akinson, 28). This concep has been defined as aversion o downside inequaliy or ADI (Davies and Hoy, 995). On he oher hand, one may be ineresed in wha happens in he righ ail of he disribuion. For insance, in an economic conex, a lo of aenion is recenly given o hose variaions occurring a he op of he income disribuion (Makdissi and Yabeck, 24). Logically, i is possible o define a preorder which akes ino accoun of his concep by emphasiing he Tommaso Lando, Universiy of Bergamo (Ialy) and VSB Technical Universiy of Osrava (Cech Republic), ommaso.lando@unibg.i Lucio Beroli-Barsoi, Universiy of Bergamo (Ialy), lucio.beroli-barsoi@unibg.i 656

2 h Inernaional Scienific Conference Financial managemen of Firms and Financial Insiuions Osrava VŠB-TU of Osrava, Faculy of Economics, Deparmen of Finance 7 h 8 h Sepember 25 righ ail of he disribuion: in paricular, he second-degree downward Loren dominance has been recenly inroduced by Aaberge (29). In his paper, we consider boh hese wo differen approaches and sudy he possibiliy of defining a class of inequaliy measures which is generally sensiive o ransfers occurring in one or boh of he ails of he disribuion (i.e. he lower or he higher, speaking in erms of income). More specifically, we presen wo differen preorders and combine hem in a new preorder. Then, we presen a class of funcionals ha are isoonic wih he new preorder. As for he noaion, we make use of he definiion of weak majoriaion (Marshall e al. 29). 2. Loren ordering and majoriaion In his firs secion, we define he Loren ordering and analye is relaion wih he majoriaion preorder. We recall ha a preorder is a binary relaion over a se S ha is reflexive and ransiive. In paricular, observe ha a preorder does no generally saisfy he anisymmery propery (ha is, a b and b a does no necessarily imply a = b) and i is generally no oal (ha is, each pair a, b in S is no necessarily relaed by ). Firs, we recall ha he (generalied) inverse of a disribuion funcion F is given by F (p) = inf{ [, ]: F() p}, p (,). If F has finie expecaion μ F, hen he Loren curve is defined as follows (Gaswirh, 97): L F (p) = p F ()d. μ F The Loren ordering L is a pre-order defined over he space F of non-negaive disribuions wih finie expecaions (F = {F: F() = for < and df() = μ F < }), and i is defined as follows. Definiion. Le F, G F : we wrie F L G if and only if L F (p) L G (p), p (,). On he oher hand, majoriaion is a preorder defined in he space of inegrable funcions, and i is aimed a comparing funcions in erms of diversiy beween heir values (Marshall e al., 29). Here we focus on coninuous majoriaion, hus we consider funcions ha are inegrable wih respec o he Lebesgue measure m on he se (,). Definiion. Le a, b L (,). We say ha a is majoried by b and wrie a b if and only if ) a (u) du b (u) du, (,), 2) a (u) du = b (u) du, where a (u) = (m a (x)) and m a (x) = m({u: a(u) > x}) (noe ha he funcion a is referred o as he decreasing re-arrangemen of a). When condiion 2) does no hold, we rely on weaker definiions of majoriaion. In paricular, in his paper we shall use he following one. Definiion. Le a, b L (,). We say ha a is weakly majoried by b from below and wrie a w b if and only if a (u) du b (u) du, (,). We say ha ha a is weakly majoried by b from above and wrie a w b if and only if a (u) du b (u) du, (,), where, similarly o a, a denoes he increasing re-arrangemen of a (see for insance Lando, Beroli-Barsoi, 24). 657

3 h Inernaional Scienific Conference Financial managemen of Firms and Financial Insiuions Osrava VŠB-TU of Osrava, Faculy of Economics, Deparmen of Finance 7 h 8 h Sepember 25 If we denoe he derivaive of he Loren curve by l F, ha is l F = (L F ) = F /μ F, we can express he relaion beween he LO and majoriaion as follows (Beroli-Barsoi, 2): if F L G hen l F l G. This equivalence relaion makes i possible o define several classes of funcionals ha are isoonic wih he LO, based on some well known resuls of majoriaion heory (Beroli- Barsoi, 2). 3. A new approach As explained in he inroducion, he aim of his paper is o propose a class of inequaliy measures ha are basically sensiive o ransfers occurring in he lower or he higher pars of he income disribuion. We firs consider wo differen weak preorders, defined as follows. Generally, F is preferable o G if i presens less inequaliy when we he comparison sars from he lef ails of he disribuions. By using he weak majoriaion definiion we can express his condiion wih: ) L F w L G, ha is: L F (p)dp L G (p)dp, for any in [,]. Observe ha condiion ) implies ha L F sars above L G and presens a larger (underlying) area, ha is, a lower value of he Gini index. Furhermore, noe ha L F w L G is equivalen o he second-degree upward Loren dominance of Aaberge (29). Similarly, one can also prefer F o G if i presens less inequaliy when he comparison sars from he righ ails of he disribuions. We can formulae his condiion as: 2) L G w L F, ha is: L G (p)dp L F (p)dp, for any in [,], (noe ha L F (p)dp = (L F ) (p)dp). According o Aaberge (29), he ordering L G w L F can be equivalenly referred o as he second-degree downward Loren dominance. Observe ha F L G implies L F w L G as well as L G w L F. In wha follows, we aemp o combine he preorders ) and 2) ino a single preorder, weaker han he LO, which emphasies inequaliy in boh he ails of he disribuion. Le us define: L F() = ( L F ()) = L F ( ) (noe ha if a() is a decreasing funcion in [,] hen a () = a( )). L F() can be inerpreed as a complemenary Loren curve (see Eliaar, 25). Acually, for a given percenage, L F () represens he percenage of oal possessed by he low % par of he disribuion, while L F() represens he percenage of oal corresponding o he op % par of he disribuion. Hence, as L F() L F () for any, he difference beween he Loren curves F () = L F() L F () expresses he dispariy beween he higher and he lower pars of he disribuion. In erms of income disribuions, F equals he difference beween he proporion of he sociey s overall wealh ha is held by he sociey s op (rich) %, and he proporion of he sociey s overall wealh ha is held by he sociey s low (poor) %. 658

4 h Inernaional Scienific Conference Financial managemen of Firms and Financial Insiuions Osrava VŠB-TU of Osrava, Faculy of Economics, Deparmen of Finance 7 h 8 h Sepember 25 I can be easily shown ha F is increasing for [,/2] and decreasing for (/2,], hus i aains is maximum a = /2. Moreover, for any in [,/2], F () = F ( ), herefore, in order o analye he behavior of F, i is sufficien o focus on he inerval [,/2]. Obviously, we wish ha F is (uniformly) as small as possible: his concep can be expressed in erms of weak majoriaion. Indeed, for any couple of disribuions F, G, we prefer F o G when: which is equivalen o F (p)dp G w F, G (p)dp for any in [,/2]. Observe ha he preorder G w F is weaker han he LO and combines boh he orderings ) and 2) presened in his secion. Moreover, G w F implies he condiion /2 /2 F (p)dp G (p)dp, which is equivalen o saying ha G presens a higher concenraion han F /2 according o he Gini index (indeed F (p)dp = (p L F (p))dp). Hence G w F expresses concenraion as well as inequaliy in he higher and lower pars of he disribuion. From hese consideraions, we propose o measure inequaliy wih a funcional ha preserves he ordering G w F. In paricular, we can rely on a well known resul of majoriaion heory and use he following measure of inequaliy: /2 Υ(F) = φ( F (p))dp where φ is increasing and concave (see e.g. Marshall e al., 29). Obviously, i is also easy o derive a normalied version of he inequaliy index Υ. Noe ha he formula for Υ deermines a class of inequaliy measures ha share cerain characerisics. Indeed, we obain ha Υ is isoonic wih he newly inroduced ordering, which is weaker han he LO. Therefore hese implicaions hold F L G G w F Υ(F) Υ(G). The usefulness of his new approach can be shown by a simple example. Example Consider he vecor X = (,,2,3,4,5,6,7,8,9) and he following wo vecors, where each of hem can be obained from X by wo elemenary ransfers, respecively in he ails and in he core of he disribuion: X = (5,5,2,3,4,5,6,7,85,85) X = (,,2,35,35,55,55,7,8,9). 659

5 h Inernaional Scienific Conference Financial managemen of Firms and Financial Insiuions Osrava VŠB-TU of Osrava, Faculy of Economics, Deparmen of Finance 7 h 8 h Sepember 25 I is easy o see ha X presens less inequaliy in he ails bu more inequaliy in he core, compared o X. Le us respecively denoe wih F, F and F he disribuions ha correspond o X, X and X. Since any inequaliy measure Υ is isoonic wih he LO, we know ha Υ(F ) Υ(F) and Υ(F ) Υ(F). However, we are ineresed in ranking F and F. Observe ha L F and L F inersec wo imes and moreover heir underling area is equal (i.e. F and F canno be ranked by he LO as well as he Gini index). The Loren curves in Figure also show ha L F sars and finishes above L F.. Figure : Loren curves L F (dashed) and L F (solid) On he oher hand, he curves F and F reveal ha F w F, as is shown in Figure 2. Figure 2: Dispariy curves F (dashed) and F (solid) Then, according o wha is saed above, any inequaliy measure Υ yields Υ(F ) Υ(F ) Υ(F). For his example we simply se φ(x) = x, hence /2 Υ(F) = ( F (p)) /2 dp which yields: Υ(F) =.29, Υ(F ) =.288, Υ(F ) =

6 h Inernaional Scienific Conference Financial managemen of Firms and Financial Insiuions Osrava VŠB-TU of Osrava, Faculy of Economics, Deparmen of Finance 7 h 8 h Sepember Conclusion We have proposed a new ordering of inequaliy/dispariy, and consequenly we inroduced a class of funcionals ha are isoonic wih his ordering. Fuure sudies will be aimed a he applicaion of his resul. Since he new ordering is weaker han he LO, i can be used o rank Loren curves when hey inersec. The raio of he ordering, as well as he corresponding index Υ, is ha ransfers should be more equaliing when hey occur in he ails of he disribuion. Acknowledgemen The research was suppored hrough he Cech Science Foundaion (GACR) under projec S and hrough SP25/5, an SGS research projec of VSB-TU Osrava, and furhermore by he European Social Fund in he framework of CZ..7/2.3./ References [] Beroli-Barsoi, L. (2). Some remarks on Loren ordering-preserving funcionals. Saisical mehods and applicaions : pp [2] Aaberge, R. (29). Ranking inersecing Loren curves. Soc. Choice Welf. 33, pp [3] Akinson, A.B. (28). More on he measuremen of inequaliy. J. Econ. Inequal. 6, pp [4] Dardanoni, V. and Lamber, P.J. (988). Welfare rankings of income disribuions: A role for he variance and some insighs for ax reforms. Soc. Choice Welf. 5, pp. -7. [5] Davies, J.B. and Hoy, M. (995). Making inequaliy comparisons when Loren curves inersec. Am. Econ. Rev. 85, pp [6] Eliaar, I. (25). The sociogeomery of inequaliy: Par. Physica A, 426, pp [7] Gaswirh, J.L. (97). A general definiion of he Loren curve. Economerica 39: pp [8] Lando, T. and Beroli-Barsoi, L. (24). Saisical Funcionals Consisen wih a Weak Relaive Majoriaion Ordering: Applicaions o he Minimum Divergence Esimaion. WSEAS Transacions on Mahemaics 3, Ar. #65, pp [9] Madkissi, P. and Yakbeck, M. (25). On he measuremen of pluonomy. Soc. Choice Welf. 44, pp [] Marshall, A.W., Olkin, I. and Arnold, B.C. (29). Inequaliies: heory of majoriaion and is applicaions. (second ediion), Springer. [] Shorrocks, A.F. and Foser J.E. (987). Transfer sensiive inequaliy measures. Rev. Econ. Sud., 4, pp