Inequality and the GB2 income distribution

Transcription

1 Working Pper Series Inequlity nd the GB2 income distribution Stephen P. Jenkins ECINEQ WP

2 ECINEC July Inequlity nd the GB2 income distribution Stephen P. Jenkins* University of Essex 27 My 2007 Abstrct The generlized entropy clss of inequlity indices is derived for Generlized Bet of the Second Kind (GB2) income distributions, thereby providing full rnge of topsensitive nd bottom-sensitive mesures. An exmintion of British income inequlity in 1994/95 nd 2004/05 illustrtes the nlysis. JEL Clssifiction: C16, C46, D31. Key words: inequlity, generlized entropy indices, generlized Bet of the second kind distribution, GB2 distribution, Singh-Mddl distribution, Dgum distribution * Address for correspondence: Jenkins: ISER, University of Essex, Wivenhoe Prk, Colchester CO4 3SQ, stephenj@essex.c.uk..

3 1 Introduction Prmetric functionl forms hve received considerble ttention in the literture on ernings nd income distribution. Although lrge number of functionl forms hve been proposed, the four-prmeter Generlized Bet of the Second Kind (GB2) model is now widely cknowledged to give n excellent description of income distributions, providing ne goodness-of- t with reltive prsimony, while lso including mny other models s specil or limiting cses. See, inter li, Bordley et l. (1999), Brchmnn et l. (1999), Butler nd McDonld (1989), McDonld (1984), nd McDonld nd Xu (1995). Feng et l. (2006) ddress issues of time-inconsistency in top-coded US Current Popultion Survey ernings dt by tting GB2 distributions tht ccount for top-coding, nd derive consistent time series of Gini coe cients from the estimtes. Prker s (1999) model of optimising rm behviour predicts tht the ernings distribution hs the GB2 shpe. Despite widespred use of the GB2 distribution, it is remrkble tht inequlity in the tted distribution hs been summrized in terms of the Gini coe cient lone. Although commonly used, the Gini is but one of mny mesures vilble, nd it incorportes prticulr ssumptions bout the wy in which income di erences in di erent prts of the distribution re summrized. (It is reltively sensitive to income di erences round the mode.) In other forms of income distribution reserch, generlized entropy (GE) nd Atkinson indices re widely used to ssess inequlity trends nd di erences these one-prmeter fmilies hve the dvntge tht vritions in inequlity version re strightforwrdly incorported. This pper provides formule for generlized entropy indices in the GB2 model, nd hence lso for the importnt specil cses of the three-prmeter Singh-Mddl nd Dgum models, thereby mking full rnge of top-sensitive nd bottom-sensitive mesures vilble to nlysts. The only GE index mentioned in Kleiber nd Kotz s (2003) encyclopedic survey of the GB2 nd relted distributions is the Theil index for the Singh- Mddl model. Cowell nd Flchire (2007) provide GE index formule for the Singh-Mddl model, but using di erent prmeteriztion from the stndrd one tht is employed by McDonld (1984) nd Kleiber nd Kotz (2003). There pper to be no extnt GE index formule for the Dgum distribution, which is surprising given Kleiber s (1996) rgument tht the Dgum distribution is likely to provide better t to income dt thn the Singh-Mddl distribution. This pper s derivtions for the GB2 model re illustrted with n exmintion of the chnge in income inequlity in Britin between 1994/95 nd 2004/05. 1

4 2 Generlized entropy indices Consider the distribution of rndom vrible y ( income ), which tkes strictly positive vlues. The generlized entropy (GE) clss of inequlity mesures, I(), is de ned s 1 I() = 1 ; 6= 0; 1 (1) ( 1) where Z = y df (y): (2) nd F (y) is the cumultive distribution function (cdf) for y. logrithmic devition (MLD) index is The men I(0) = lim!0 I() = log v 0 (3) where 0 = R log ydf (y) nd E(y) is the men of y. The Theil index is I(1) = limi() = log (4)!1 where 1 = R y log ydf (y). I(2) is hlf the squred coe cient of vrition. Prmeter 2 ( 1; 1) chrcterizes the sensitivity of I() to income di erences in di erent prts of the income distribution. The more positive tht is, the more sensitive is I() to income di erences t the top of the distribution; the more negtive tht is, the more sensitive is I() to income di erences t the bottom of the distribution. In empiricl work, the rnge of vlues for is typiclly restricted to [ 1; 2] becuse, otherwise, estimtes my be unduly in uenced by smll number of very smll incomes or very high incomes. For ech member of the Atkinson (1970) clss of inequlity indices, A(), there is n ordinlly equivlent member of the GE clss (but not vice vers). Speci clly, for inequlity version prmeter = 1, A() = 1 [ ( 1) I() + 1] 1 ; < 1; 6= 0 v 1 = 1 exp [ I(0)] ; = 0. (5) Since A() cn be computed from I(), this pper focuses on the derivtion of I() in the GB2 distribution cse. 1 On the chrcteriztion of the GE clss of inequlity indices, see Bourguignon (1980), Cowell (1980), nd Shorrocks (1980, 1984). 2

5 3 The GB2 distribution The GB2 distribution hs pdf f(y) = y p 1 b p B(p; q) [1 + (y=b) p+q ; y > 0 (6) ] where prmeters ; b; p; q, re ech positive, B(u; v) = (u) (v)= (u + v) is the Bet function, nd (:) is the Gmm function (McDonld 1984). Prmeter b is scle prmeter, nd ; p; nd q re ech shpe prmeters. The k th moment of the GB2 distribution is E(y k ) = bk (p + k ) (q k ) (p) (q) (7) nd exists only if p < k < q. The Singh-Mddl distribution is the specil cse of the GB2 distribution when p = 1; the Dgum distribution is the specil cse when q = 1. For discussion of other specil cses, see McDonld (1984) nd Kleiber nd Kotz (2003). 4 GE inequlity indices nd the GB2 distribution Expressions for ech GE index, I(), other thn for the cses = 0; 1, cn be derived by strightforwrd substitution, using the expressions for nd given by (2) nd (7). In prticulr, the bottom-sensitive index I( 1) is given by I( 1) = (p 1 ) (q + 1) (p + 1) (q (p) 2 (q) The top-sensitive index I(2) is given by ) : (8) I(2) = (p) (q) (p + 2 ) (q 2 ) 2 2 (p + 1 ) 2 (q 1 ) : (9) Expressions for the MLD nd Theil indices cn be derived noting tht the expression for I() cn be written s I() = g() h() (10) 3

6 where g() = 1, with = b (p + ) (q )= (p) (q) from (7), nd h() = ( 1). Hence, using L Hbopitl s rule, I(0) = g 0 (0) nd I(1) = g 0 (1), where g 0 () = ( ) 0 + ( ) 0. We therefore require expressions for ( ) 0 nd ( ) 0 evluted t the limits! 0, nd! 1. It cn be shown tht ( ) 0 = log (11) nd ( ) 0 = (p + ) given digmm function (z) = limits, (q ) + log b (12) 0 (z)= (z). Hence, pplying the pproprite nd I(0) = (p + 1 ) + (q 1 ) (p) (q) (p) + (q) (13) I(1) = (p + 1 ) (q 1 ) (p + 1 ) (q 1 ) + (p) + (q): (14) To derive the expression for I() in the specil cse of the Singh-Mddl model, set p = 1 nd note tht (1) = 1. For the Dgum model, set q = 1 insted. 5 Empiricl illustrtion: income inequlity in Britin, 1994/95 nd 2004/05 The derivtions re illustrted with nlysis of income inequlity in Britin in scl yers 1994/95 nd 2004/05. Estimtion is bsed on the unit record dt used to clculte the o cil income sttistics, derived from the Fmily Resources Surveys of 1994/95 nd 2004/05. Income is the distribution mong individuls of needs-djusted post-tx post-trnsfer household income, with ech individul ssumed to receive the income of the household to which s/he belongs. For further detils of the construction of the distributions, see Deprtment for Work nd Pensions (2006). Observtions with income equl to zero were excluded from the clcultions (182 observtions in the 1994/95 le nd 302 observtions in the 2004/05 le). 4

7 Estimtes of the GB2 prmeters for ech yer re shown in Tble 1, together with inequlity index estimtes implied by them. 2 According to probbility plots nd quntile plots (not shown), the GB2 distribution ts the dt well. The estimted GB2 shpe prmeters chnged mrkedly over the decde, with notble rise in combined with shrp fll in both p nd q. Put nother wy, the distribution ws well-chrcterized by Fisk distribution in 1994/95 (the GB2 cse when p = q = 1), but could not be described thus decde lter. These chnges contrst with the trend in GB2 prmeters for reported by Brchmnn et l. (1996) for household income in Germny, nd for for US white fmily income reported by Butler nd McDonld (1989). In both cses, there ws seculr decline in nd rise in p nd q. <Tble 1 ner here> The rise in combined with fll in p nd q implies tht neither distribution Lorenz-domintes the other one (Kleiber 1999), so conclusions bout whether inequlity incresed or decresed depend on the inequlity index used. As it hppens, the GB2 estimtes of the Gini coe cient nd ech of four GE indices incresed between 1994/95 nd 2004/05, nd the increse for the GE indices is greter the more positive tht is. However, of the ve indices, it is only for I(2) for which the estimted increse is some 28 per cent tht the increse is sttisticlly signi cnt. (In this cse the test sttistic for the relevnt t-test is 2.5, but it is mrkedly less thn 2 for the other four indices.) The signi cnt rise in top-sensitive index I(2) suggests tht the principl chnges over the decde in the British income distribution occured t the very top of the distribution. This is con rmed by the GB2 estimtes of the Lorenz curves (not shown), which indicte imperceptible chnges in income shres t the bottom of the income distribution but increses in income shres t the top. For exmple, the GB2 estimte of the income shre of the richest ve per cent incresed from 16.5 per cent to 17.3 per cent between 1994/95 nd 2004/05, nd the income shre of the richest one per cent from 5.6 per cent to 6.3 per cent. 2 A progrm for tting GB2 distribution to unit record dt using the sttisticl softwre Stt tm (SttCorp 2003), versions 8.2 nd lter, is provided by Jenkins (2007). Stt users cn instll the progrm directly by typing ssc instll gb2fit. The GE indices, nd ssocited stndrd errors computed using the delt method, cn be derived fter estimtion using the nlcom commnd. 5

8 If British inequlity trends over the decde hd been ssessed using the GB2-estimted Gini coe cient lone, number of importnt dimensions of the chnge would not hve been picked up. The bility to clculte rnge of indices incorporting di erent ssumptions bout ggregtion of income di erences in di erent income rnges is signi cnt extension to the utility of the GB2 model for nlysis of income nd ernings distributions. 6 Acknowledgements ISER s core-funding from the Economic nd Socil Reserch Council nd the University of Essex is grtefully cknowledged. Thnks to Emmnuel Flchire for suggesting the method to derive I(0) nd I(1), nd to Mrk Bryn, Shuizhng Feng, Lucind Pltt, nd Steve Pudney for comments. 7 References Atkinson, A.B., 1970, On the mesurement of inequlity, Journl of Economic Theory 2, Bordley, R.F., J.B. McDonld nd A. Mntrl, 1996, Something new, something old: prmetric models for the size distribution of income, Journl of Income Distribution 6, Bourguignon, F. 1989, Decomposble inequlity mesures, Econometric 47, Brchmnn, K., A. Stich nd M. Trede, 1996, Evluting prmetric income distribution models, Allegemeine Sttistiches Archiv 80, Butler, J.B. nd J.B. McDonld, 1989, Using incomplete moments to mesure inequlity, Journl of Econometrics 42, Cowell, F.A., 1980, On the structure of dditive inequlity mesures, Review of Economic Studies 47, Cowell, F.A. nd E. Flchire, 2007, Income distribution nd inequlity mesurement: the problem of extreme vlues, Journl of Econometrics, forthcoming. Deprtment for Work nd Pensions, 2006, Households Below Averge Income. An Anlysis of the Income Distribution 1994/5 2004/5. (Corporte Document Services, Leeds). Feng, S., R.V. Burkhuser nd J.S. Butler, 2006, Levels nd long-term trends in ernings inequlity: overcoming Current Popultion Survey censoring problems using the GB2 distribution, Journl of Business nd Economic Sttistics 24,

9 Jenkins, S.P., 2007, gb2fit: Stt module to t Generlized Bet of the Second Kind distribution by mximum likelihood, Sttisticl Softwre Components Archive S Kleiber, C., 1996, Dgum versus Singh-Mddl income distributions, Economics Letters 53, Kleiber, C., 1999, On the Lorenz order within prmetric fmilies of income distributions, Snkhy B 61, Kleiber, C. nd S. Kotz, 2003, Sttisticl Size Distributions in Economics nd Acturil Sciences. (John Wiley, Hoboken, NJ). McDonld, J.B., 1984, Some generlized functions for the size distribution of income, Econometric 52, McDonld, J.B. nd Y.J. Xu, 1995, A generliztion of the bet distribution with pplictions, Journl of Econometrics 66, Errtum: Journl of Econometrics 69, Prker, S.C., 1999, The generlized bet s model for the distribution of ernings, Economics Letters 62, Shorrocks, A.F., 1980, The clss of dditively decomposble inequlity indices, Econometric 48, Shorrocks, A.F., 1984, Inequlity decomposition by popultion subgroup, Econometric 52, SttCorp, 2003, Stt Sttisticl Softwre: Relese 8.0. (Stt Corportion, College Sttion TX). 7

10 1994/ /05 Prmeter estimtes (0.057) (0.179) b (4.602) (4.602) p (0.072) (0.042) q (0.058) (0.037) Inequlity indices (0.040) (0.039) Gini (0.003) (0.003) I( 1) (0.005) (0.005) I(0) (0.003) (0.004) I(1) (0.005) (0.006) I(2) (0.017) (0.030) Log-likelihood 196, ,850 N (households) 26,033 25,790 N (individuls) 62,055 59,804 Estimted stndrd errors shown in prentheses. Tble 1: Estimtes of GB2 prmeters nd inequlity indices, Britin, 1994/95 nd 2004/05. 8