Dominance Testing for Pro-Poor Growth with an Application to European Growth

Transcription

1 Domnance Testng for Pro-Poor Growth wth an Alcaton to Euroean Growth DANIEL SOTELSEK SALEM, ISMAEL AHAMDANECH ZARCO and JOHN. A BISHOP. IELAT (Latn Amercan Studes Insttute). Av/ Juan Carlos I, Alcalá de Henares, San (e-mal: smael.ahamdanech@uah.es). Deartment of Economcs, Unversty of Alcalá, Alcalá de Henares, San (e-mal: danel.sotelsek@uah.es). Economcs Deartment, East Carolna Unversty, USA. Abstract Ths aer ntroduces statstcal testng rocedures to evaluate ro-oor growth. Our measure of ro-oorness follows Kakwan (000), Kakwan and Perna (000), and Son (004), who decomose the generalzed Lorenz ordnates nto a growth effect and an nequalty effect. We derve an asymtotc dstrbuton-free covarance matrx for the decomosed generalzed Lorenz curves. Usng ths decomoston (and our standard errors) we test for ro-oor domnance n the growth rocess. We llustrate our test for ro-oor domnance by evaluatng the degree of ro-oor growth n fve Euroean countres. Key Words: Pro-oor Growth, Poverty, Stochastc domnance. JEL Classfcaton: D3, D63, I3

2 . INTRODUCTION My dream has always been to make the oor rcher, not to make the rch oorer...a rsng tde lfts all boats. John F. Kennedy Presdent Kennedy s statement was re-osed as a queston by Danzger and Gottschalk n ther 986 Amercan Economc Assocaton conference aer as: Do rsng tdes lft all boats? From the begnnng of the US War on Poverty economsts have noted that ncreased growth does not necessarly result n a declne n overty (see Anderson, 964 and Ahluwala, 974). More recently, ths queston has been recast to ask whch tyes of economc olces lead to ro-oor growth esecally n lght of the well-ublczed Mllennum Develoment Goals (MDG s). The ast decade has roduced a great deal of research relatng to how to measure ro-oorness. Essama- Nssah and Lambert (009) and Duclos (009) rovde thorough revews of ths lterature. The urose of ths aer s ntroduce statstcal testng rocedures to evaluate ro-oor growth. Our statstcal aroach s based on the semnal work of Beach and Davdson (983) who rovde asymtotc dstrbuton-free covarance matrces for Lorenz and generalzed Lorenz curves. Our theoretcal aroach to measurng rooorness follows Kakwan (000), Kakwan and Perna (000) and Son (004). The KPS aroach decomoses changes n the dfference n generalzed Lorenz ordnates nto a growth effect and an nequalty effect. Usng ths decomoston we can test for ro-oor domnance n the growth rocess. Furthermore, we mlement the KPS aroach drectly usng the tools of stochastc domnance, as oosed to the more restrctve ndex number aroach often used n ths analyss. We llustrate our test for ro-oor domnance by evaluatng the degree of rooor growth n fve Euroean countres. We consder two tme erods, 993 to 000 and 000 to 006. For the combned erod we fnd that San, France, and Italy enjoyed ro-oor growth, Germany s growth exerence was ant-oor, and the UK ro-oor rankng s ambguous. Only one researcher to our knowledge has studed ro-oor growth n Euroe (Henrch, 003). No researcher has emloyed domnance methods together wth formal nference tests.

3 . THEORETICAL FRAMEWORK. Second-order domnance (Generalsed Lorenz domnance) As Kakwan s method of evaluatng ro-oor growth has ts roots n second order stochastc domnance, we brefly ntroduce ths technque. Followng Atknson (970) we assume that the relatonsh between the dstrbuton of ncome and the standard of lvng s gven by a socal welfare functon, whch reresents the ethcal judgments regardng ncome dstrbutons. The well-known Atknson Lorenz Domnance Theorem as extended by Dasguta, Sen and Starrett (973) s restrcted to comarng dstrbutons wth the same mean. However, as Sen (973) onts out, the means of two dstrbutons wll rarely be equal. Shorrocks (983) addresses ths roblem by ntroducng generalsed Lorenz Domnance. From Gastwrth (97), the Lorenz curve can be defned as: L X X ( u ) : du, [] 0 and the generalsed Lorenz curve wll then be (Shorrocks, 983): GL X : X ( u) du L, [ 0, ] 0 X X [] Let WS be a S-concave and ncreasng welfare functon. Then we have the next theorem, demonstrated by Shorrocks (983): Theorem GL Domnance: w( X ) w( Y ), w WS ff GLX GLY for all wth at least one nequalty revalng. The mlcatons of ths aroach are straghtforward: assumng two wdely, though not unversally, acceted value judgments (the Pareto rncle and the Pgou- Dalton transferences rncle) we can rank the economc welfare assocated to two dfferent ncome dstrbutons. 3

4 . Truncated second-order domnance and overty Foster and Shorrocks (988) lnks second order domnance to overty. As t s well known, ncome-ga s the weghted sum of the ncome shortfalls of the oor, that s: P r z x ; [3] n( x) z ( x z ) Beng r the order statstc corresondng to the overty lne, z, and x the -th ndvdual s ncome. Ths crteron mles that ncome dstrbuton X domnates ncome dstrbuton Y, denoted by > Y, f, and only f, ( / n ) x > (/ n) y for all u to X z * * r and for any gven z. Then: GLX GLY ff X z < * * Y, z z Ths mles that f the dstrbuton s truncated at any arbtrary overty lne * z < z and X generalsed Lorenz domnates Y at and below that overty lne, then the ncome-ga overty n X cannot exceed overty n Y usng that overty lne, and ths s * the case for every overty lne z < z..3 Inequalty effect and growth effect: an stochastc domnance aroach Our aroach to measure ro-oorness follows Kakwan (000), Kakwan and Perna (000) and Son (004). The KPS aroach decomoses changes n the dfference n generalzed Lorenz ordnates nto a growth effect and an nequalty effect. One key ont n understandng the evaluaton of ro-oorness s to clearly dentfy the reference ont. The roosed method uses the underlyng ncome dstrbuton as reference ont. Bsho and Formby (994) evaluate the benefts of growth usng generalzed concentraton curves and lke the Kakwan method use the Lorenz curve as the reference ont. See Duclos (009) and Essama- Nssah and Lambert (009) for aroaches to measurng ro-oorness wth alternatve reference onts. 4

5 To evaluate changes n welfare due to economc growth we begn wth the generalzed Lorenz curve: GL L. Between two erods, ts varaton wll be gven as: GL ) GL GL L L ( ) [6] ( that s, the total change s due to both the change n the average ncome and the change n the ncome dstrbuton measured by the Lorenz curve. So the total effect can be decomosed n growth effect and nequalty effect. Followng Kakwan s (000) axomatc aroach, ths decomoston can be done as follows: GL I { L L L L } [7] GL g { L L L L } [8] and the sum of [7] and [8] equals the total change: GL GL GL [9] I g The nequalty effect, GL, shows the varaton of nequalty, measured by the I Lorenz curve, usng the ncome n both the begnnng and the fnal erod. The nterretaton of the growth effectgl s analogous. g As t s well known, the generalzed Lorenz curve s defned by the ar of coordnates { ; L( ) } account only the nequalty effect wll be:. Then, the generalzed Lorenz curve ordnates takng nto GL I GL { L L L L } [0] On the other hand, t s straghtforward that: 5

6 GL I GL g GL { L L L L } { L L L L } GL GL GL GL, [] Some nterestng nsghts arse from the exressons above. If the nequalty effect generalzed Lorenz curve gven by [0] domnates the ordnary generalzed Lorenz curve GL, the nequalty effect then reduces overty as measured by Atknson (987). We call ths result ro-oor domnance. The welfare mlcatons are straghtforward: ro-oor domnance mles an ncrease n economc welfare as measured the welfare functons ncluded n W S. The next theorem demonstrates ths result: Theorem Pro-Poor Domnance: If GL ( ) GL ( * I ), x < z, wth at least one nequalty holdng, then: X I X z * and X W * X I, where y s the ntal ncome dstrbuton and x the dstrbuton takng nto account only the nequalty effect. Ths theorem mles that f the generalzed Lorenz curve takng nto account only the nequalty effect domnates the frst erod generalzed Lorenz curve, the growth wll have been ro-oor, not only because the decrease of overty, but also because the ncrease of economc welfare of the oor, gven the assumtons made above. 3 The relatonsh between stochastc domnance and welfare can be extended to the truncated dstrbutons. If the nequalty effect generalzed Lorenz curve GL I domnates the ordnary GL curve, GL ( ), for all ncomes u to * * z (defnng z as the overty lne), then growth s overty-reducng for all the overty ndexes as defned n Atknson (987). 3 Note that our result s equvalent to Son s (004) overty growth curve. As noted by Son the generalzed Lorenz aroach s based on second order stochastc domnance whle Ravallon and Chen s (003) growth ncdence curve. Son also onts the advantages and dsadvantages of the two aroaches. Son recommends but does not rovde formal nference tests. 6

7 .4 Inference tests for Pro-Poor Domnance When comarng generalzed Lorenz curves there are three ossble, outcomes, equvalence, domnance, or crossng. Bsho, Formby and Thstle (989) recommend a ar wse statstcal nference test rocedure to evaluate generalsed Lorenz curves, gven these three alternatves. We adat ths aroach to test for ro-oor domnance. Usng a set of k sub-hyotheses to test for the overall hyothess of equvalence we have: H GL 0, : GL I and A I H, : GL GL [] If each of the sub-hyotheses s not rejected then the jont null hyotheses s not rejected, and we conclude that the growth rocess s nether ro or ant oor. On the other hand, f the any of the sub-hyotheses are rejected, then the followng are the ossble outcomes: - Weak Pro-Poor domnance: If for some quantles GL I > GL and for others GL I GL, then we conclude that growth s weakly ro-oor. If GL I > GL for all I then we have strong ro-oor growth. - If for some quantles GL I > GL and for others GL I < GL, then no unambguous rankng s ossble for all z (t wll be neccesary to analyze truncated domnance). The statstcal tests wll be: T GL GL ˆ I GL ˆ / I ϖ ˆ ˆ ϖ where,,,k. 4 [3] N I N Beach and Davdson (983) derve the statstcal dstrbuton for G ˆL. However, the dstrbuton for Ĝ LI s unknown needs to be derved. Equaton [0] mles: GL I GL { L L L L } [0] 4 The crtcal values for ths test are determned by the Student Maxmum Modulus dstrbuton. Tables can be obtaned from Stolne and Ury (979). 7

8 and can be wrtten as: GL I GL { GL GL GL GL } { GL GL GL GL },,,, [4] where: GL ) L ( ) [5], ( That s, GL ) s the generalzed Lorenz curve obtaned by scalng the ncome (, dstrbuton of the second erod by the mean ncome of the frst. In an analogous way, we have: GL ) L ( ) [6],( Takng ths nto account, the varance of GL I can be wrtten as: Var( GL, ) Var( GL ) Var( GL ) Var( GL,( )) Cov[ GL, ; GL ] Cov[ GL, ; GL ] Var( GLI ) [ ] [ ] [ ] [ ] [7] 4 Cov GL, ; GL, Cov GL ; GL Cov GL ; GL,( ) Cov GL ; GL, ndeendent: Some tems n [7] are equal to zero, snce the ntal and fnal dstrbutons are [ GL ); GL ] 0 Cov [8] ( [ GL ); GL ] Cov[ L ; L ] 0 Cov, snce t s the covarance of, ( the ntal and fnal dstrbuton scaled by the same quantty. Moreover: 8

9 [ GL, ; GL, ] Cov[ L ; L ] 0 [ GL ); GL ] Cov[ L ; L ] 0 Cov [9] Cov [0] (, There are stll two tems to be calculated. The frst one s: [ GL ); GL ] Cov[ L ; L ( )] Cov [], ( From a very well known roerty of the varance: Var [ L L ] Var[ ( ) L ] ( L ) Var( L ) Cov[ L ; L ] Var [] from where: [ L ; L ] Var( L ) Var( L ) Var[ ( ) L ( )] Cov [3] Analogously, we have: [ L ; L ] Var( L ) Var( L ) Var[ ( ) L ( )] Cov [4] Usng all these results n [7]: Var( GL, ) Var( GL ) Var( GL ) Var( GL,( )) Var( GLI ) 4 Cov[ GL ( ); ( )] [ ( ); ( )] [5], GL Cov GL GL, And after some manulatons, the next exresson s reached: Var( GLI ) { Var( L ) Var( L ) Var[ ( ) L ] Var[ ( ) L ]} [6] 4 9

10 As t can be seen n [6], the varance of the generalzed Lorenz curve that only has nto account the nequalty effect s comuted usng the varances of four dfferent ncome dstrbutons. Now, t s ossble to calculate both the varance of GL and ofgl s gven by Beach and Davdson (983). For exresson 5 : GL I. The varance GL I, t wll be gven, for j, by the * *,,,, { ϖ j ϖ j ϖ j ϖ } I Var( GL I ) ϖ j [7] 4 a, a beng ϖ the varance of the generalzed Lorenz curve of the dstrbutons seen n [6]. j 3. RECENT TRENDS IN EUROPEAN POVERTY AND INEQUALITY Accordng to the OECD (008) economc nequalty n Euroe s on the rse. For our samle of countres the Gn coeffcents n md-000s vary between 0.8 for France and 0.35 for Italy. Germany shows a Gn coeffcent of 0.3, San of 0.3 and Great Brtan of Whle the dfferences are not wde at all, t s nterestng to note that there has been changes n the nequalty trends among these countres n the last two decades. Two erods can be dstngushed n the evoluton of ncome dstrbuton, md- 980 s to md 990 s and the md-990 s to md 000 s. In the frst erod nequalty fell n San and France, whle t ncreased n Germany, Italy and UK. In the second erod, nequalty remaned steady n San and France, ncreased n Germany and n Italy, and decreased n the UK. From the md 990 s to md 000 s Germany s nequalty contnues to ncrease, France and San enjoy a decrease n nequalty, whle there was no sgnfcant change n nequalty n Italy and the UK. 6 If we turn our attenton to headcount overty evoluton n the last decade (wth 50% of medan ncome as overty lne) France, Italy and the UK exerenced decreases n overty whle overty n Germany and San ncreased. The analyss of relatve overty s nterestng, but also t s to know what has haened n absolute terms. One 5 The fnal exresson of the varances are develoed n the aendx. 6 These conclusons are based on the Eurostat database. 0

11 way to do ths s to fx a relatve overty lne for a erod t and, adjusted for nflaton, to use t to comare overty n, say, erod tn 7. Followng ths aroach, absolute overty decreased n all countres analyzed between md 990 s and md 000 s wth the exceton of Germany, where absolute overty ncreased. 4. DATA AND EMPIRICAL ANALYSIS 4. Data To dentfy the ro-oor asects of economc growth we need data of a suffcent tme san to cature the effects of dstrbutonal changes mbedded n the growth rocess. To ths am we have used data from two dfferent, although consstent, datasets: The Euroean Communty Household Panel (ECHP) and the Survey on Income and Lvng Condtons (SILC), both develoed by Eurostat. We have drawn data for the years 993 and 000 from the ECHP (waves frst and eght of the survey) whle for 006 we use data from the SILC (the 007 wave). Our ncome measure s er cata household dsosable ncome ncludng total market ncome, addng transfers, and deductng taxes and Socal Securty contrbutons, adjusted by the modfed OECD equvalence scale. The data are weghted usng the weghts gven by the samle desgn and the number of ndvduals n the households, so we convert household data nto ndvdual data, assumng the same ncome for each member of the household. Fnally, data have been deflated to 993 unts usng the HIPC from Eurostat. 4. Results In ths secton we aly formal nference tests to address two questons. Frst, dd welfare mrove (.e., generalzed Lorenz domnance) n each of the fve Euroean countes selected between 993 and 006? Secondly, and of more drect nterest to us, was economc growth durng ths erod ro-oor? In ths second case we comare the ordnary generalzed Lorenz curve of the ntal erod to the nequalty effect generalzed Lorenz curve of the second erod. 7 Ths aroach s comarable to the emrcal analyss carred out n ths aer.

12 Table shows the results of the GL domnance tests and the ro-oor tests for Germany, Columns and 3 rovde the 993 and 000 generalzed Lorenz ordnates whle the fourth column rovde the test statstcs for ordnary GL domnance. As each of the test statstcs s greater than the fve ercent SMM crtcal value of.80 we conclude 000 generalzed Lorenz domnates 993, mlyng mrovng welfare over tme. To test for ro-oor domnance we comare the GL ordnate for 993 (column ) wth the G I ordnate (column 5). All of the nne test statstcs are ostve, and eght of the nne test statstcs are statstcally sgnfcant at the fve ercent level. From these results we conclude that growth n Germany between 993 and 000 was unambguously ro-oor. Table examnes German growth over the erod and rovdes results that are qute dfferent from the erod Frst, we note that the ordnary GL curves cross Germany n 006 has a hgher mean ncome (GL 0 ) but the other nne GL ordnates are smaller n 006 as comared to 000. Fgure llustrates ths GL crossng (GL 00 vs. GL 06 ) and hghlghts the fact that crossng GL curves revent an unambguous welfare rankng. Whle no unambguous welfare concluson can be drawn from the German growth exerence of 000 to 006, we are able to rank ths tme erod as unfavourable to the oor. All of the nne test statstcs n column 6 are negatve and sgnfcant ndcatng that economc growth over ths erod s clearly ant-oor. These results are llustrated Fgure wth GL I lyng below GL 00,ndcatng that the German oor lost ground over ths tme erod. Table 3 rovdes the test results for GL and ro-oor domnance over the entre tme erod, 993 to 006. The generalzed Lorenz curve of 006 weak domnates to the one of 993 weak domnance mles some ostve and sgnfcant dfferences and no negatve and sgnfcant dfferences (see column 4). However, the falure of the lowest German ncomes to grow--gl decles and show no change n ncome for the oorest ersons suggests that the oor dd not fare well over ths tme erod. The falure of the oor to share the benefts of growth s verfed by testng for ro-oor domnance. Column 6 reorts the results of ths test and shows the tme erod s ranked as antoor. 8 8 In the case of Germany 993 to 006 f we defne the overty lne the 0 th ercentle (truncated domnance) we can conclude that the tme erod 993 to 006 was nether ro-oor or ant-oor (no sgnfcant dfference.

13 Table 4 rovdes a summary of the numercal and statstcal comarsons for all fve Euroe countres examned (see the Aendx for detaled tables for France, Italy, San and the UK). To understand how the table works consder the last comarsons made, Germany, 993 to 006, whch s shown n the last columns of the frst row. Under GL domnance we reort whch means that both the numercal and statstcal comarsons ranked 006 over 993. However, under ro-oor domnance we reort x - whch mles that the numercal crossng was relaced by an statstcal rankng of ant-oor. Ths table demonstrates the errors n rankngs that can occur n the absence of statstcal tests lke those develoed n the ths aer. In fact, sx out of 30 or 0 ercent of the comarsons n Table 4 are ms-ranked usng numercal comarsons. Whle Table 4 hghlghts the necessty of formal nference tests, t also summarzes the growth exerence of the fve Euroean countres. All fve countres show both GL domnance and ro-oor domnance for the 993 to 000 erod. For the erod all countres but Germany exerenced GL domnance; however, economc growth was ant-oor n Italy and the UK as well as n Germany. For the combned erod all countres show GL domnance, San, France and Italy are ro-oor, Germany s ant-oor, and the UK ro-oor rankng s ambguous. 5. CONCLUSIONS Ths aer ntroduces statstcal testng rocedures to evaluate ro-oor growth. Our theoretcal aroach to measurng ro-oorness follows Kakwan (000), Kakwan and Perna (000), and Son (004). The KPS aroach decomoses changes n the dfference n generalzed Lorenz ordnates nto a growth effect and an nequalty effect. Usng ths decomoston we can test for ro-oor domnance n the growth rocess. Furthermore, we test for ro-oor domnance drectly usng the tools of stochastc domnance, as oosed to the more restrctve ndex number aroach often used n ths analyss. We derve an asymtotc dstrbuton-free covarance matrx for the decomosed generalzed Lorenz curves on whch ro-oor domnance testng s based. We llustrate our test for ro-oor domnance by evaluatng the degree of rooor growth n fve Euroean countres. We consder two tme erods, 993 to 000 and 000 to 006. For the combned erod we fnd that San, France, and Italy enjoyed ro-oor growth, Germany s growth exerence was ant-oor, and the UK ro-oor rankng s ambguous. 3

14 4

15 Fgure : Generalzed Lorenz Crossng, Germany, GL GL 06 Fgure : Pro-Poor Domnance, Germany GL GL I 0 5

16 Table Generalzed Lorenz and Pro-Poor Domnance, Germany, Generalzed Lorenz Domnance Pro-Poor Domnance Decle () ĜL 93 () ĜL 00 (3) T GL (4) ĜL (5) T GL (6) * * * * * * * * * * * * * * * * * * * Fve ercent crtcal value for SMM.80 Table Generalzed Lorenz and Pro-Poor Domnance, Germany, Generalzed Lorenz Domnance Pro-Poor Domnance Decle () ĜL 00 () ĜL 06 (3) T GL (4) ĜL (5) T GL (6) * * * * * * * * * * * * * * * * * * Fve ercent crtcal value for SMM.80 Table 3 Generalzed Lorenz and Pro-Poor Domnance, Germany, Generalzed Lorenz Domnance Pro-Poor Domnance Decle () ĜL 93 () ĜL 06 (3) T GL (4) ĜL (5) T GL (6) * * * * * * * * * * * * * * * * Fve ercent crtcal value for SMM.80 6

17 Table 4 Summary of Generalzed Lorenz and Pro-Poor Domnance Country GL Domnance Pro-Poor GL Pro-Poor GL Domnance Domnance Domnance Domnance Pro-Poor Domnance Germany XX X France X X Italy X San X UK X XX Frst entry s numercal comarson, second entry s statstcal comarson. denotes domnance of nd year over frst, frst year over second, and X denotes a crossng. 7

18 REFERENCES. Ahluwala, M.S.(974): Income Inequalty: some dmensons of the roblem, n H. Chenery et. al., Redstrbuton wth Growth. Oxford: Oxford Unversty Press.. Anderson, W. (964): Trcklng Down: The Relatonsh Between Economc Growth and the Extent of Poverty Among Amercan Famles, Quarterly Journal of Economcs, 78: Atknson, A. B. (970): On the Measurement of nequalty, Journal of Economc Theory, : Atknson, A. B. (987): On the Measurement of Poverty, Econometrca, 55: Beach, C. and R. Davdson (983): Dstrbuton-Free Statstcal Inference wth Lorenz Curves and Income Shares, Revew of Economc Studes, 50: Bsho, J.A., J.P. Formby and P.D. Thstle (989), Statstcal nference, ncome Dstrbutons and socal welfare, en D.J. Slotje (ed.), Research on Economc Inequalty, Vol., Greenwch, CN: JAI Press. 7. Danzger, Sheldon and Gottschalk, Peter, 986. Do Rsng Tdes Lft All Boats? The Imact of Secular and Cyclcal Changes on Poverty, Amercan Economc Revew, Amercan Economc Assocaton, 76(): Dasguta, P., Sen, A., and Starrett, D. (973): Notes on the measurement of nequalty, Journal of Economc Theory, 6: Duclos, J.-Y. (009): What s Pro-oor, Socal Choce and Welfare, vol. 3, Essama-Nssah, B. and Lambert, Peter J. (009): Measurng Pro-Poorness : A Unfyng Aroach Wth New Results Revew of Income and Wealth, 55, 3:

19 . Foster, J. E., and A. F. Shorrocks (988): Poverty orderngs and welfare domnance, Socal Choce and Welfare, 5: Gatswrth, J.L. (97): A General Defnton of the Lorenz Curve, Econometrca, 39: Henrch, G. (003): More s not Necessarly Better: An Emrcal Analyss of the Inequalty-Growth Tradeoff usng the Luxembourg Income Study, LIS Workng Paer Nº Kakwan, N. and Pernía, M. (000): What s Pro-oor Growth?, Asan Develoment Revew, 8(): Kakwan, N. (000): On Measurng Growth and Inequalty Comonents of Poverty wth Alcaton to Thaland, Journal of Quanttatve Economcs, 6(): Kakwan, N. and Son, H. (008): Poverty Equvalent Growth Rate, Revew of Income and Wealth, 54, 4: OCDE (008): Growng Unequal? Income Dstrbuton and Poverty n OECD Countres, Pars. 8. Ravallon, M. and Chen, S. (003): Measurng Pro-Poor Growth, Economc Letters, 78: Sen, A.K. (973): On Economc Inequalty, Oxford Unversty Press. 0. Shorrocks, A. F. (983): Rankng ncome dstrbutons, Economca, 50: Son, H. (004): A note on Pro-Poor Growth, Economc Letters, 8: Stolne, M.R., and H.K. Ury (979): Tables of the Studentzed Maxmum Modulus Dstrbutons and an alcaton to multle comarsons among means, Technometrcs, : Wlks, S. S. (96): Mathematcal statstcs, Nueva York: John Wley & Sons. APPENDIX 9

20 The varance of the ordnates of the generalzed Lorenz curve takng nto account only the nequalty effect s as follows: Var( GL 4 I ) ϖ 4 * *,,,, { ϖ j ϖ j ϖ j ϖ j } { Var( L ) Var( L ) Var[ ( ) L ] Var[ ( ) L ]} I [A] To be able to comute ths exresson we need to calculate each of ts arts. To do ths we can begn wth Var ( L ( )). Ths s the varance of the generalzed Lorenz curve multlyng the Lorenz curve of the second erod by the average ncome of erod one. On the other hand, the varance of the generalzed Lorenz curve of the second erod s (Beach and Davdson, 983): ϖ j [ λ ( )( ξ γ ) ] [A] j where, snce j, s the quntle consdered (0., 0., ); j λ s the condtoned varance of each quntle, and ξ, γ are the maxmum and the condtoned average of each quntle. To calculate the generalzed Lorenz curve GL,, we scale the Lorenz curve of the second erod by the ncome mean of the frst erod, or begn from the generalzed Lorenz curve of the second erod and multly by and dvde by. Snce ths s only a change n scale, ths wll not affect the nequalty as measured by the Lorenz curve, snce ths s a relatve nequalty measure. Then, f [A] reresents the varance of the generalzed Lorenz curve of the second erod, the varance of Var ( L ( )) wll be:, ϖ ( ) j λ, j ξ, γ, [A3] 0

21 It s mortant to note that we are changng the scale of the dstrbuton of the second erod, so t wll be the samle of ths second erod that s relevant to comute the statstcal test. Reasonng n a smlar way, we have: ( ),,,, * j j γ ξ λ ϖ [A4] ( ),,,, * j j γ ξ λ ϖ [A5] Then the varance of each ordnate of the generalzed Lorenz curve that takes nto account only the nequalty effect wll be: ( )( ) [ ] ( ) ( ) ( ) ( ),,,,,,,,,,,, 4 ) ( j j j j I I GL Var γ ξ λ γ ξ λ γ ξ λ γ ξ λ ϖ [A6] And the statstcal test:

22 ( )( ) [ ] ( ) ( ) ( ) ( ) /,,,,,,,,,,,, ˆ / / / / 4 ˆ ˆ N N N N N GL GL T j j j j I GL ϖ γ ξ λ γ ξ λ γ ξ λ γ ξ λ ara,,,k. [A7] In ths exresson, all the varables re known or can be calculated.

23 ANNEX Table A. France: Decle G ˆL G ˆL T Sg. GL ĜL I TGL Sg. 37,0 548,45 -,69 *** 54,04-9,04 *** 045,46 355,04-5,50 *** 336,6 -,6 *** 3 865,43 35,7-7,9 *** 8,9-3,97 *** 4 80,8 3404,7-8,3 *** 3355, -3,8 *** 5 399,35 464,49-7,63 *** 4556,65-3,09 *** 6 569, 5985,6-6,58 *** 5898,39 -,87 *** 7 659,56 750,77-5,00 *** 7400, -0,33 *** 8 83,49 947,7-3,7 *** 909,9-8,57 *** 9 08,05 308,83-0,46 *** 39,39-6,05 *** 0 4, ,36 -,79 4,63 0,00 Table A.8 France: Decle G ˆL G ˆL T Sg. GL ĜL I TGL Sg. 5,97 548,45-3,7 ** 554,0-3,6 *** 308,89 355,04-3,56 *** 369,83-4, *** 3 60,3 35,7-3,4 ** 340,85-4,07 *** , 3404,7 -,5 3440,46-3,40 *** ,0 464,49 -, ,87 -, , ,6 0,0 6049,76 -, ,4 750,77 0, ,6-0, ,7 947,7, ,50-0, ,9 308,83,34 434,0 0, , ,36, ,57 0,00 Table A.7 France: Decle G ˆL G ˆL T Sg. GL ĜL I TGL Sg. 37,0 5,97-7,7 *** 5,09-3,36 *** 045,46 308,89-8,7 *** 77,30-4,69 *** 3 865,43 60,3-0,69 *** 05,33-6,40 *** 4 80,8 3353, -,5 *** 37,5-7,5 *** 5 399, ,0 -,77 *** 4480,5-7,53 *** 6 569, 5985,89 -,79 *** 5839,07-7,66 *** 7 659, ,4 -,60 *** 736,30-7,44 *** 8 83,49 935,7-0,6 *** 9090,65-6,53 *** 9 08,05 447,9-9,03 *** 58,9-5,07 *** 0 4, ,57 -,87 ** 4,63 0,00 3

24 Table A.9 Italy: Decle G ˆL G ˆL T Sg. GL ĜL I TGL Sg. 69,69 56,7-6,63 *** 36,89-0,85 *** 555,66 76,75-7,4 *** 658,76-0,0 *** 3 053,3 95,76-9, *** 88,94-9,89 *** 4 646,63 987,44-0, *** 8,08-9,83 *** 5 344,06 799,6 -,5 *** 565,0-0,05 *** 6 355,9 377,09 -,3 *** 343,4-9,85 *** 7 40, , -,94 *** 4365,83-8,63 *** 8 508,5 5959,7 -,9 *** 5449,78-6,9 *** 9 65, ,68-9,94 *** 670,8-4,67 *** 0 860, ,5-3,83 *** 860,04 0,00 Table A.0 Italy: Decle G ˆL G ˆL T Sg. GL ĜL I TGL Sg. 56,7 36,97 -,68 *** 37,00 3,3 *** 76,75 897,5-0,94 *** 67,46 4,3 *** 3 95,76 63,59-7,6 *** 6,84 5,58 *** 4 987,44 489,9-3,4 *** 865,98 6,33 *** 5 799,6 3505,80-35,5 *** 67,95 7,3 *** 6 377, ,9-38,80 *** 3499,9 7,87 *** , 6003,8-43,6 *** 4503,68 7,94 *** ,7 755,96-47,99 *** 5673,34 7,45 *** ,68 944,0-53, *** 7093,55 5,64 *** ,5 473, -5,08 *** 9434,5 0,00 Table A.3 Italy: Decle G ˆL G ˆL T Sg. GL ĜL I TGL Sg. 69,69 36,97-9,58 *** 9,77-0,95 *** 555,66 897,5-4,86 *** 633,50-8,34 *** 3 053,3 63,59-5,73 *** 35,87-6,59 *** 4 646,63 489,9-59,73 *** 734,74-5,44 *** 5 344, ,80-65,94 *** 437,0-4,60 *** 6 355,9 4668,9-70,75 *** 337,8-3,35 *** 7 40, ,8-73,38 *** 453,86 -, ,5 755,96-77,3 *** 55,4-0,9 9 65,78 944,0-80, *** 650,0 0, ,04 473, -70,70 *** 860,04 0,00 4

25 Table A.5 San: Decle G ˆL G ˆL T Sg. GL ĜL I TGL Sg. 8,9 3,85-9,55 *** 04,3-5,0 *** 57,0 607,86 -,9 *** 553,56-4,89 *** 3 94,04 090,93-4,43 *** 993,8-5,03 *** 4 443,75 660,5-6,0 *** 5,4-5, *** 5 07,44 37,65-6,88 *** 08,59-4,85 *** 6 70, ,0-7,96 *** 80,7-4,89 *** , ,5-7,87 *** 3597,58-4,38 *** , 4994,75-7,6 *** 4539,66-3,44 *** ,65 66,9-5,8 *** 5688,6 -, ,5 8347,7 -,49 *** 7579,5 0,00 Table A.6 San: Decle G ˆL G ˆL T Sg. GL ĜL I TGL Sg. 3,85 6,5-0,60 6,46,50 607,86 637,98-4,0 *** 60,9-0, ,93 5,44-5,99 *** 0,4-0, ,5 764,8-7,79 *** 688,09 -, ,65 483,06-9,63 *** 375,53 -,99 ** ,0 33,9 -,0 *** 370,3-3,83 *** ,5 466,83 -,97 *** 4083,09-4,3 *** , ,3 -,5 *** 547,87-4,44 *** 9 66,9 670,78 -, *** 6430,36-3,94 *** ,7 878,5-6,60 *** 8347,7 0,00 Table A. San. Perod Decl G ˆL G ˆL T Sg. GL ĜL I TGL Sg. 8,9 6,5 -,75 *** 97,86-4,8 *** 57,0 637,98-9,69 *** 556,66-6,5 *** 3 94,04 5,44-4,80 *** 005, -7,5 *** 4 443,75 764,8-8,73 *** 539,8-8,37 *** 5 07,44 483,06-3,06 *** 65,8-9,5 *** 6 70,35 33,9-35, *** 889,87-0,5 *** ,95 466,83-36,47 *** 37,40-0,55 *** , 5379,3-36,08 *** 4688,34-9,09 *** ,65 670,78-33,86 *** 585,40-6,33 *** ,5 878,5-5,45 *** 7579,5 0,00 5

26 Table A. UK: Decle G ˆL G ˆL T Sg. GL ĜL I TGL Sg. 87,85 333,39 -,0 *** 64,0-5,5 *** 568,86 875,35-3,08 *** 680,84 -,98 *** 3 066,60 538,85-4,86 *** 86,39-7, *** 4 657,95 3,43-5,65 *** 78,7-3,36 *** 5 346,55 38,60-57,8 *** 46,73-9,8 *** 6 340,00 44,97-43,5 *** 337,97-5,4 *** ,93 540,08-44,03 *** 4,60 -,88 ** 8 53,8 6768,90-44,0 *** 550, -0, , ,38-44,95 *** 6383,4, ,06 05,8-33,4 *** 838,06 0,00 Table A. UK: Decle G ˆL G ˆL T Sg. GL ĜL I TGL Sg. 333,39 553,98-3,5 *** 96,86 3,8 ** 875,35 473,8-4,68 *** 794,57 5,08 *** 3 538,85 66,76-57,44 *** 47,77 6,89 *** 4 3, ,0-69,90 *** 5,7 8,75 *** 5 38, ,6-78,03 *** 303,63 9, *** 6 44, ,69-77,00 *** 400,56 5,69 *** 7 540,08 94,87-8,4 *** 55,55 5,9 *** ,90 88,96-85,79 *** 6484,45 4,80 *** , ,43-88,94 *** 837,39 3,8 *** 0 05,8 9859, -64, *** 05,8 0,00 Table A.4 UK: Decle G ˆL G ˆL T Sg. GL ĜL I TGL Sg. 87,85 553,98-5, *** 65,7-7,76 *** 568,86 473,8-79,9 *** 657,94 -,0 *** 3 066,60 66,76-95,6 *** 30, -6,7 *** 4 657, ,0-08,6 *** 68,99 -, , ,6-6,7 *** 38,0, , ,69-5,7 *** 3070,97 3,3 *** ,93 94,87-3,9 *** 394,64 5,67 *** 8 53,8 88,96-37,3 *** 490,3 7,30 *** 9 643, ,43-40,0 *** 66,6 9,05 *** 0 838, , -99,4 *** 838,06 0,00 6