The measurement of multidimensional poverty

Transcription

1 The measuremet of multdmesoal poverty Fraços Bourgugo a ad Satya R. Chakravarty b Fal verso Jue 00 a Delta (ot research ut CNRS, EHESS, ENS), Pars, ad the World Bak. b Ida Statstcal Isttute, Calcutta

2 The measuremet of multdmesoal poverty Fraços Bourgugo ad Satya R. Chakravarty Abstract May authors have ssted o the ecessty of defg poverty as a multdmesoal cocept rather tha relyg o come or cosumpto epedtures per capta. Yet, ot much has actually bee doe to clude the varous dmesos of deprvato to the practcal defto ad measuremet of poverty. Estg attempts alog that drecto cosst of aggregatg varous attrbutes to a sgle de through some arbtrary fucto ad defg a poverty le ad assocated poverty measures o the bass of that de. Ths s merely redefg the more geeral cocept of poverty, whch the essetally remas a oe dmesoal cocept. The preset paper suggests that a alteratve way to take to accout the mult-dmesoalty of poverty s to specfy a poverty le for each dmeso of poverty ad to cosder that a perso s poor f he/she falls below at least oe of these varous les. The paper the eplores how to combe these varous poverty les ad assocated oe-dmesoal gaps to multdmesoal poverty measures. A applcato of these measures to the rural populato Bral s also gve wth poverty combg come ad educato. JEL Numbers: D63. Keywords: Poverty measure, multdmesoal. Authors address: F. Bourgugo, Delta, ENS, 48 Bd Jourda, 7504ars, Frace (Correspodg author) S. R.Chakravarty, Ida Statstcal Isttute, 03 Barrackpore Truk Road, Calcutta , Ida

3 3. Itroducto Poverty has bee estece for may years ad cotues to est a large umber of coutres. Therefore, targetg of poverty allevato remas a mportat ssue may coutres. I order to uderstad the threat that the problem of poverty poses, t s ecessary to kow ts dmeso ad the process through whch t seems to be deepeed. A atural questo that arses here s how to quatfy the etet of poverty. I a mportat cotrbuto, Se (976) vewed poverty measuremet problem as volvg two eercses: () the detfcato of the poor, ad () aggregato of the characterstcs of the poor to a overall dcator. I the lterature, the frst problem has bee solved mostly by the come (or cosumpto) method, whch requres the specfcato of a subsstece come level, referred to as the poverty le. A perso s sad to be poor f hs/her come falls below the poverty le. O the aggregato ssue, Se (976) crtcsed two crude poverty measures, the head cout rato (proporto of persos wth comes less tha the poverty le) ad the come gap rato (the gap betwee the poverty le ad average come of the poor, epressed as a proporto of the poverty le), because they are sestve to the redstrbuto of come amog the poor ad the former also remas ualtered f the posto of a poor worses. He also suggested a more sophstcated de of poverty usg a aomatc approach. However, the well-beg of a populato ad, hece ts poverty, whch s a mafestato of suffcet well-beg, deped o both moetary ad o-moetary varables. It s certaly true that wth a hgher come or cosumpto budget a perso may be able to mprove the posto of some of hs/her moetary ad o-moetary attrbutes. But at the same tme t may be the case that markets for some o-moetary attrbutes do ot est, for eample, wth some publc good. It may also happe that markets are hghly mperfect, for stace, the case of ratog. Therefore, come as the sole dcator of well-beg s approprate ad should be supplemeted by other attrbutes or varables, e.g. housg, lteracy, lfe epectacy, provso of publc goods ad so o. The eed for such a multdmesoal approach to the measuremet of equalty well-beg was already emphassed, amog others, by Kolm (977), Atkso ad Bourgugo (98), Maasoum (986) ad Tsu (995). Cocerg poverty, Ravallo (996) argued a recet paper that four sets of dcators ca be defeded as gredets for a sesble approach to poverty measuremet. These are: () real epedture per sgle adult o market goods, () Alteratves ad varatos of the Se de have bee suggested, amog others, by Takayama (979), Blackorby ad Doaldso (980), Kakwa (980), Clark, Hemmg ad Ulph (98), Foster, Greer ad Thorbecke (984), Chakravarty (990), ad Bourgugo ad Felds (997).

4 4 o-come dcators as access to o-market goods, () dcators of tra-household dstrbuto such as chld utrtoal status ad (v) dcators of persoal characterstcs whch mpose costrats o the ablty of a dvdual, such as physcal hadcap. I other words, a geue measure of poverty should deped o come dcators as well as o-come dcators that may help detfyg aspects of welfare ot captured by comes. We ca cte further ratoales for vewg the problem of measuremet of well-beg of a populato from a multdmesoal structure. For stace, the basc eeds approach advocated by developmet ecoomsts regards developmet as a mprovemet a array of huma eeds ad ot ust as growth of come -see Streete (98). There ests a debate about the mportace of low comes as a determat of uder-utrto -see Lpto ad Ravallo (995). Fally, well-beg s trscally multdmesoal from the vew pot of 'capabltes' ad 'fuctogs', where fuctogs deal wth what a perso ca ultmately do ad capabltes dcate the freedom that a perso eoys terms of fuctogs - Se (985, 99). I the capablty approach fuctogs are closely appromated by attrbutes such as lteracy, lfe epectacy, etc. ad ot by come per se. A eample of multdmesoal measure of well-beg terms of fuctog achevemets s the Huma Developmet Ide suggested by UNDP (990). It aggregates at the coutry level fuctog achevemets terms of the attrbutes lfe epectacy, per capta real GDP ad educatoal attamet rate. For reasos stated above we devate the preset paper from the sgle dmesoal come approach to the measuremet of poverty ad adopt a alteratve approach whch s of multdmesoal ature. I our multdmesoal framework stead of vsualsg poverty or deprvato usg come or cosumpto as the sole dcator of well-beg, we formalse t terms of fuctog falures, or, more precsely, terms of shortfalls from threshold levels of attrbutes themselves. We the eame varous aggregato rules whch permt to quatfy the overall magtude of poverty usg these shortfalls. It may be mportat to ote that the threshold levels are determed depedetly of the attrbute dstrbutos. I ths sese the cocept of poverty measuremet we eplore here s of absolute type. We beg the paper by dscussg the problem of detfyg the poor secto. Secto 3 the suggests reasoable propertes for a multdmesoal poverty de. Sce we vew poverty measuremet from a multdmesoal perspectve, a very mportat ssue that eeds to be dscussed s the trade off amog attrbutes. It s show that possblty/mpossblty of such trade-offs drops out as a mplcato of dfferet postulates for a multdmesoal measure of poverty. Ths s preseted secto 4 of the paper.

5 5 Secto 5 troduces some fuctoal forms for a multdmesoal poverty measure whereas secto 6 shows how they may be practcally mplemeted by cosderg the evoluto of come/educato poverty rural Bral. Secto 7 cocludes.. Idetfcato of the poor The purpose of ths secto s to determe the set of poor persos. We beg wth otatoal deftos. Wth a populato of se, perso possesses a m-row vector or of attrbutes, R+ m, where R + m s the o-egatve orthat of the Eucldea m-space R m. The vector s the th row of a m matr X M, where M s the set of all m matrces whose etres are o-egatve reals. The (, )th etry of X gves the quatty of attrbute possessed by perso. Therefore the th colum of X gves a dstrbuto of attrbute amog persos. Let M= M N, where N s the set of postve tegers. For ay X M, we wrte (X) - or, - for the correspodg populato se. It should be oted that quattatve specfcatos of dfferet attrbutes precludes the possblty that a varable ca be of qualtatve type -for stace, of the type whether a perso s ll or ot. A smple way of dealg wth the multdmesoalty of poverty s to assume that the varous attrbutes of a dvdual may be aggregated to a sgle cardal de of 'well-beg' ad that poverty may be defed terms of that de. I other words, a dvdual ca be sad poor f hs/her de of aggregate well-beg falls below some poverty le. However, such a approach would be severely restrctve ad would mostly amout to cosderg multdmesoal poverty as sgle dmesoal come poverty, wth some approprate geeralsato of the cocept of 'come'. Although there sometmes may be a good ustfcato for such a approach, ths s the case that we do ot wat to cosder here because t s coceptually strctly equvalet to the case of come poverty. The fudametal pot all what follows s that a multdmesoal approach to poverty defes poverty as a shortfall from a threshold o each dmeso of a dvdual's well beg. I other words, the ssue of the multdmesoalty of poverty arses because dvduals, socal observers or polcy makers wat to defe a poverty lmt o each dvdual attrbute: come, health, educato, etc... All the argumets preseted ths paper are based Tsu (00) provdes a aomatc ustfcato of such a approach. Note also that ths approach may go qute beyod aggregatg a few goods or fuctogs through usg approprate prces or weghts. For stace Pradha ad Ravallo (000) tred to tegrate to the aalyss uobserved welfare determats summared by reported subectve percepto of poverty.

6 6 o ths dea. 3 I agreemet wth ths basc prcple, a drect method to check whether a perso s poor the multdmesoal framework where he/she s charactersed by m attrbutes s to see whether he/she has the subsstece or threshold level of each attrbute. Let Z be a vector of thresholds, or 'mmally acceptable levels' - Se (99, p. 39)- for dfferet attrbutes 4, where Z s a subset of s ow to determe whether a perso,, s poor or ot o the bass of hs/her, ad the vector. R + m. The problem Oe uambguous way of coutg the umber of poor ths cotet s to detfy those for whom the levels of all attrbutes fall below the correspodg thresholds. But ths defto does ot ehaust the etre set of poor persos. For eample, a old beggar certaly caot be regarded as rch because of hs logevty, though the above oto ecludes hm from the set of poor. Therefore ths defto seems to be approprate. More geerally, perso may be called poor wth respect to attrbute f <. Perso s regarded as rch f for all. Aalogously, attrbute for perso s sad to be meagre or o-meagre accordg as < or. For ay X M, let S (X) (or S ) be the set of persos who are poor wth respect to attrbute. Oe may argue that the total umber of poor persos ca be obtaed by addg the umber of people S over. But ths procedure may lead to double coutg. To see ths, suppose that there are two attrbutes, ad. The subsstece levels ad are represeted by the les CD ad AB respectvely Fgure. U ad U are upper bouds o the quattes of the attrbutes. Clearly the total umber of poor ths two-attrbute case becomes the umber of persos for whom the attrbute quattes le sde the space (OABU + ODCU ). Ths shows that the umber of persos OAED s couted twce ths calculato. The double coutg may be avoded f we subtract OAED from (OABU + ODCU ). But wth a crease the umber of attrbutes the umber of sets o whch double coutg occurs wll crease. Cosequetly, gve that we should avod double coutg, determato of the total umber of poor usg S 's wll be very trcate. <Fgure aroud here> 3 Note that poverty lmts all dmesos are defed depedetly of the quatty of other attrbutes a dvdual may eoy. For a more geeral statemet see Duclos et al. (00) 4 Usg the same attrbutes as UNDP, emprcal eamples of these threshold quattes could be a come of $ (ppp corrected) a day, prmary educato, ad 50 year lfe epectacy.

7 7 A smpler way of defg poverty ad coutg the umber of poor s to eplctly accout for the possblty of beg poor ay poverty dmeso. A straghtforward way of dog so s to defe the poverty dcator varable : ρ( ; ) = f (,,..., m): < ad ρ( ; ) = 0, otherwse () The the umber of poor s smply gve by : = H ρ( ; ) () For further referece ad le wth the precedg argumets, t wll be coveet to adopt the followg deftos. The rego OAED fgure, where perso s poor wth respect to both attrbutes, wll be called the two dmesoal poverty rego (PR). I cotrast, the spaces AECU ad DEBU ca be called the oe-dmesoal poverty regos (PR) because the quatty of oe of the attrbutes s above the subsstece level these spaces. 3. Propertes for a multdmesoal poverty de I ths secto, we lay dow the postulates for a measure of multdmesoal poverty. A formal statemet of all these postulates s gve the Apped to ths paper. The followg dscusso s essetally verbal. A multdmesoal poverty de s a o-costat fucto P: M Z R. For ay X M, Z, P(X; ) gves the etet of poverty assocated wth the attrbute matr X ad thresholds. Thus, though we vew the poverty measuremet problem from a multdmesoal perspectve, we dcate the magtude of overall poverty by a real umber. The de P may be assumed to satsfy certa postulates. A frst set of postulates cludes the followg: STRONG FOCUS (SF), WEAK FOCUS (WF), SYMMETRY (SM), MONOTONICITY (MN), CONTINUITY (CN), PRINCIPLE OF POPULATION (PP), SCALE INVARIANCE (SI), ad SUBGROUP DECOMPOSABILITY (SD). These postulates are straght geeralsatos of the desderata suggested for a sgle dmesoal poverty

8 8 de 5. As such, most of them are lttle debatable. SF demads that for ay two attrbute matrces X ad Y, f Y s obtaed from X by chagg some o-poor attamet quattes so that the set of poor persos as well as ther attrbute levels below the relevat threshold rema the same, the the poverty levels assocated wth X ad Y must be equal. I other words, we say that the poverty de s depedet of the o-poor attrbute quattes. Therefore SF does ot allow the possblty that a perso ca gve up some amout of a o-meagre attrbute to mprove the posto of a meagre attrbute. If oe vews poverty terms of deprvato from thresholds, the SF s qute reasoable. I cotrast to SF, WF, the weak verso of the focus aom, says that the poverty de s depedet of the attrbute levels of the o-poor persos oly. SM states that ay characterstc of persos other tha the quattes of attrbutes used to defe poverty s umportat for measurg poverty. Accordg to MN f the posto of perso who s poor wth respect to attrbute mproves the overall poverty should ot crease. It may be oted that the mprovemet may make the beefcary o-poor wth respect to the attrbute uder cosderato. Cotuty (CN) requres P to vary cotuously wth s ad s essetally a techcal requremet. Cotuty esures partcular that the poverty de wll ot be oversestve to mor observatoal errors o quattes of attrbutes. PP s ecessary for cross populato comparsos of poverty. SI says that the poverty de should be varat uder scale trasformato of attrbutes ad thresholds. I other words, what matters for poverty measuremet s oly the relatve dstace at whch the quattes of all attrbutes are from ther poverty thresholds. SD shows that f the populato s parttoed to several subgroups wth respect to some homogeeous characterstc, say age, se, race, rego, etc., the the overall poverty s the populato share weghted average of subgroup poverty levels. Therefore SD eables us to calculate percetage cotrbutos of dfferet subgroups to total poverty ad hece to detfy the subgroups that are most afflcted by poverty. 6 We ow cosder postulates whch may less easly be geeraled to a mult-dmesoal framework or are specfc to t. We frst focus o redstrbuto crtera that volve a trasfer of a fed amout of some attrbute from oe perso to aother. We say that matr X s obtaed from Y by a Pgou-Dalto progressve trasfer of attrbute from oe poor perso to aother f the two matrces X ad Y are eactly the same ecept that the rcher poor wth respect to attrbute - has uts less of attrbute 5 For dscusso of propertes for a sgle dmesoal poverty de, see amog others, Foster (984), Doaldso ad Weymark (986), Cowell (988), Chakravarty (990), Foster ad Shorrocks (99) ad Zheg (997). 6 For further dscusso, see Tsu (00) ad Chakravarty, Mukheree ad Raade (998). Also t may be oted that that SD s ot the same as subgroup cosstecy dscussed Foster ad Shorrocks (99).

9 9 Y tha X whereas poorer poor t has uts more. Equvaletly, we say that Y results from X through a regressve Pïgou-Dalto trasfer attrbute. It s qute reasoable to argue that uder such a progressve (regressve) trasfer poverty should ot crease (decrease). Ths s what s demaded by the ONE DIMENSIONAL TRANSFER PRINCIPLE (OTP). A straghtforward eteso of that prcple that geeralses a smple maer the Pgou-Dalto trasfers prcple used come poverty measuremet, s a varat of the followg multdmesoal trasfers prcple troduced by Kolm (977). The Kolm property says that the dstrbuto of a set of attrbutes summarsed by some matr X s more equal tha aother matr Y (whose rows are ot detcal) f ad oly f X= BY, where B s some bstochastc matr 7 ad X caot be derved from Y by permutato of the rows of Y. Itutvely, multplcato of Y by B makes the resultg dstrbuto less cocetrated. I effect, ths trasformato s equvalet to replacg the orgal budles of attrbutes of ay par of dvduals by some cove combato of them. Followg Tsu (00), the aalogous property appled to the set of poor s the MULTIDIMENSIONAL TRANSFER PRINCIPLE (MTP). There s o more poverty wth X tha wth Y f X s obtaed from Y smply by redstrbutg the attrbutes of the poor accordg to the bstochastc trasformato. 8 Istead of the sgle dmesoal ad multdmesoal trasfer prcples OTP ad MTP, we ow cosder a redstrbutve crtero volvg two attrbutes, but wthout tyg dow the proportos whch they are echaged as MTP. For ths, suppose two persos, ad t, are the two-dmesoal poverty space assocated wth attrbutes ad k, ad has more of k but less of. Let us terchage the amouts of attrbute betwee the two persos. As perso who had more of k has ow more of too, there s a crease the correlato of the attrbutes wth the populato. It s reasoable to epect that such a swtch wll ot decrease or crease poverty accordg as the two attrbutes correspod to smlar or dfferet aspects of poverty. The NON-DECREASING POVERTY UNDER CORRELATION INCREASING SWITCH (NDCIS) postulate says that poverty caot decrease wth such correlato creasg swtches. The coverse property wll be deoted by NICIS. The eact meag of both postulates wll be dscussed more eplctly the et secto. 7 A square matr s called a bstochastc matr f each of ts etres s o-egatve ad each of ts rows ad colums sums to oe. Evdetly, a permutato matr s a bstochastc matr but the coverse s ot ecessarly true. 8 It s well kow that the oe-dmesoal Pgou-Dalto trasfer prcple s coected to Lore domace through the Hardy-Lttlewood-Polya theorem. No such theorem s avalable the mult-attrbute case.

10 0 4. Implcatos of propertes Ths secto dscusses some mplcatos of the propertes suggested the prevous secto. I the rest of ths paper we wll cosder mostly subgroup decomposable measures. A trval mplcato of SD s that a poverty de defed o M ca be wrtte as: P( X ; ) =. = p( ; ). I ths epresso p( ; ) may clearly be terpreted as the level of poverty assocated wth a sgle perso possessg attrbute vector. Most of our argumets ths secto are preseted terms of ths dvdual poverty fucto. Our frst proposto, whose proof s easy, makes a smple but etremely mportat observato about the shape of a sopoverty cotour a sgle dmesoal poverty rego. PROPOSITION : Uder SF, the sopoverty cotours of a dvdual a oe dmesoal poverty space are parallel to the as that shows the quattes of the attrbute wth respect to whch he/she s poor. Ths proposto s etremely mportat because t coveys the essece of multdmesoal poverty measuremet. If oe ssts o defg a poverty threshold depedetly for each attrbute, the at the same tme oe caot suppose that the poverty shortfall a gve attrbute may be compesated ad possbly elmated by creasg the quatty of aother attrbute deftely above ts threshold level. If I am poor because my come s below the poverty lmt, a very log lfe epectacy caot make my poverty dsappear. More precsely, proposto does ot allow trade off betwee meagre ad omeagre attrbute quattes of a perso. Thgs are slghtly dfferet whe usg WF rather tha SF. Sce WF assumes that the poverty de s depedet of attrbute levels of o-poor persos oly, t does ot rule out the possbltes of trade offs. WF gores formato o attrbutes of opoor persos but, ulke SF, takes to accout the o-poor attrbutes of a poor perso, that s, of a perso who has at least oe poor attrbute. Therefore, we ca o loger have straght le sopoverty cotours oe dmesoal poverty spaces f we assume WF.

11 I fact, f we assume covety of sopoverty cotours sgle dmesoal poverty regos 9, the the followg varat of Proposto emerges. PROPOSITION *: Uder WF, the cove sopoverty cotours sgle dmesoal poverty regos have vertcal ad horotal asymptotes. The reasog behd ths proposto s as follows. Although trade off s allowed uder WF oe dmesoal poverty spaces, poverty s ever elmated. That s, there s a postve lower boud of the poverty de alog ay vertcal or horotal as the poverty space. Ths meas that the cotour becomes a horotal or a vertcal le asymptotcally. However, ths property leads to aalytcally dffcult problems ad we shall be workg mostly wth SF what follows. Propostos ad * do ot gve ay dea about the estece or oestece of trade-offs the two dmesoal poverty space. The followg proposto descrbes the ature of trade offs that space. PROPOSITION (covety of sopoverty cotours): Suppose that m= ad that the poverty de satsfes MN, CN, SD ad OTP or MTP. The the sopoverty cotours the two dmesoal poverty rego are decreasg cove to the org. Proof: That the sopoverty cotour s decreasg s guarateed by MN. The covety makes use of OTP or MTP. s a lttle more delcate to prove. Deote the two attrbutes for whch cotours are to be eamed by ad. Sce we wll restrct our atteto to the two dmesoal space oly, let us suppose that < for =, ad for two persos ad. Let ther attrbutes (, ) ad (, ) be represeted by pots A ad A Fgure. Cosder a trasfer of attrbutes betwee these two persos whch makes ther budles detcal. Uder SD, the chage the overall poverty de s gve by: P= [. p(( + ) /,( + ) / ; ) p(, ; ) p(, ; )], (3) Both OTP ad MTP mply that ths epresso s o-postve. If I s the md-pot of the segmet A A Fgure, CN ad MN the mply that I les above the sopoverty cotour gog through the budle A or A, where dvdual poverty s mamum. If A ad A are o the same sopoverty cotour t follows that all budles o the segmet A A le o sopoverty cotours wth lower poverty. 9 Covety of the cotours mplctly assumes that MTP holds throughout the etre poverty space.

12 <Fgure aroud here> Ths proposto shows that o-creasgess of the margal rate of substtuto betwee two attrbutes for a perso the two dmesoal poverty rego s a mplcato of OTP or MTP. The oto of substtutablty betwee attrbutes somethg dfferet ad wll be take up below. It should be clear that uder SF, the poverty dfferece curves the oe dmesoal poverty regos wll be ether horotal or vertcal straght les depedg o whch as of the graph represets quattes of whch attrbute. Gve the shapes of the curves the respectve poverty spaces, we ca combe them to geerate sopoverty cotours for the etre doma. Cotuty eables us to coect the curves over the tervals [ - ε, ] ad [ - ε, ], by cotuous curves where ε>0 s ftesmally small. We show the combed graphs Fgure 3.,, ad 3 are three overall sopoverty curves. The poverty levels assocated wth s hgher tha that correspods to, ad represets more poverty tha 3. <Fgure 3 aroud here> I the precedg proposto, OTP ad MTP have a detcal role. It s clear, however, that requrg valdty of the trasfers prcple for all attrbutes s more demadg tha that for oe attrbute oly. Therefore the set of poverty dces satsfyg OTP must be more restrctve tha those satsfyg MTP. Our et proposto shows that deed the former cludes oly those dvdual poverty fuctos that are addtve across compoets. PROPOSITION 3 (addtvty): Suppose that a subgroup decomposable poverty de satsfyg OTP possesses frst order partal dervatves. The t s addtve across attrbutes, that s, P( X ; ) = m = = p (., ), (4) where p ( ) s the dvdual poverty fucto assocated wth attrbute. Proof: For smplcty let us cosder the two-perso, two-attrbute case. But oe may check that the result remas vald the geeral case too.

13 3 Cosder two dvduals ad wth attrbute levels (, ) ad (, ) respectvely. The for < OTP mples the followg: p( -ε, ) + p( +ε, ) - p(, ) - p(, ) 0 for all (,, ε>0) Lettg ε ted toward 0 ad takg lmts leads to: p (, ) - p (, ) 0 for all (,, ad < ), (5) where p ( ) s the partal dervatve of p( ) wth respect to ts frst argumet. Defe ow: g(t) = Ma p ( t, s) for s [0, [ ad h(t) = M p ( t, s) for s [0, [. (6) The (5) mples: h( ) g( ) 0 for all <. (7) But, by defto of g( ) ad h( ) (6), we have: h( ) g( ) 0 for all ad h( ) g( ) 0 for all. (8) Allowg to ted toward from below shows a cotradcto betwee (7) ad (8), uless h(t) = g(t) for all t. h(t) = g(t) mples that p (t, s) s depedet of s, whch tur shows that p (t, s) ca be wrtte as p (t)+ p (s). Usg (4) we ca determe the shares of dfferet attrbutes to total poverty. If a poverty de ehbts addtvty coucto wth SD, the we have a two-way poverty breakdow ad ca calculate the cotrbutos of alteratve subgroups to aggregate poverty wth respect to dfferet attrbutes. Cosequetly, detfcato of the subgroup-attrbute combatos that are more susceptble to poverty ca be made. Isolato of such subgroup-attrbute combatos becomes mportat desgg atpoverty polces whe a socety s lmted resource does ot eable t to elmate poverty for a etre subgroup or for a specfc attrbute. 0 We shall study later the practcal mplcatos of ths addtvty property ad see that they may ot always be coveet. We fally cosder the last trasfer propertes troduced the precedg secto, o-decrasg (ocreasg) poverty uder correlato creasg swtch. To uderstad ths ssue, defe substtutablty 0 For a umercal llustrato of ths two-way decomposablty formula, see Chakravarty, Mukheree ad Raade (998)

14 4 as promty the ature of attrbutes. A correlato creasg swtch meas that a perso who has hgher amout of oe attrbute gets hgher amout of the other through a rak reversg trasfer. If attrbutes are close to each other -.e. they are substtutes - such a trasfer should ot decrease poverty. The poorer perso caot compesate the lower quatty of oe attrbute by a hgher quatty of the other. A smlar argumet ca be provded for the complemetarty case. Atkso ad Bourgugo (98) argued rgorously that welfare should ot crease uder a correlato creasg swtch f the attrbutes volved the swtch are substtutes, where substtute attrbutes are such that the margal utlty of oe attrbute decreases whe the quatty of the other creases. The equvalet defto terms of the dvdual poverty fucto p(; ) -assumg that ths fucto s twce dfferetable s that two attrbutes ad k are substtutes wheever p k (; )>0 for all. I other words, poverty decreases less wth a crease attrbute for persos wth larger quattes of k. For stace, the drop poverty due to a ut crease come s less mportat for people who have a educatoal level close to the educato poverty threshold tha for persos wth very low educato, f come ad educato are cosdered as substtutes. O the cotrary, the drop poverty s larger for persos wth hgher educato f these two attrbutes are supposed to be complemets. Thus, the equvalet of the Atkso ad Bourgugo property the case of poverty s: PROPOSITION 4. Uder SD, o-decreasg (o-creasg) poverty uder creasg correlato swtch holds for attrbutes whch are substtutes (complemets) the dvdual poverty fucto. Of course, we observe that wth P( ) (4), attrbutes are ether substtutes or complemets. As epected OTP makes the propertes NDCIS or NICIS rrelevat. However, ths s ot the case wth MTP. There wll be dces satsfyg MTP ad NICIS ad others satsfyg MTP ad NDCIS. Tsu (00) argued that a poverty de should be uambguously odecreasg uder a correlato creasg swtch. But there s o a pror reaso for a perso to regard attrbutes as substtutes oly. Some of the attrbutes ca as well be complemets. 5. Some fuctoal forms for multdmesoal poverty dces Assumg that we may requre multdmesoal poverty dces to satsfy MN, FC, CN ad SD, the precedg secto led us to dstgush poverty dces satsfyg OTP from those satsfyg MTP. Further, amog the latter, there are dces that meet NDCIS (NICIS) but ot NICIS (NDCIS). I ths secto, we cosder smple fuctoal forms for poverty dces from these three sets, mposg

15 5 addto scale varace. We wll start from the two dmesoal case ad try to geeralse wheever ths s possble. - The set of addtve multdmesoal poverty dces As see above, poverty dces satsfyg OTP are addtve so that the geeral form of the dvdual poverty fucto the two dmesoal case s smply: ) ( ) ( ) ( ), ;, ( ) ( ad f f ad f f f p ad f f < < < + = < (9) where f ( ) are cotuous, decreasg ad cove fucto such that f (u)=0 for u. Note that homogeety wth respect to ad results from the SI property. (9) may also wrtte uder a more compact form as: (0), ). ( ). ( ), ;, ( S f S f p + = where S s the dcator fucto such that S = f S ad S = 0, otherwse. I the geeral case of m attrbutes ad dvduals, the epresso for the poverty de P correspodg to (0) becomes : (), ) ( ) ; ( = = m S f X P where X M, N, Z are arbtrary, f : [0, ) R s cotuous, o-creasg, cove ad f (t) = 0 for all t. To llustrate the precedg formula let us choose:

16 6 f ( t) = a ( t), 0 t <, () where > ad a (>0) may be terpreted as the weght gve to attrbute the overall poverty de. The the resultg measure s: P ( X ; ) = m = S a.( ). (3) Ths s a smple multdmesoal eteso of the Foster-Greer-Thorbecke (984) de. If = for all, the P becomes a weghted sum of poverty gaps all dmesos.. O the other had, f = for all, the m ( X ; ) = a. F.[ A + ( A ). V ], (4) = P where F s the populato se S as a fracto of, A s the average relatve poverty shortfall of persos S ad V s the coeffcet of varato of the dstrbuto of attrbute amog those S. It may be mportat to ote that though the use of S sets for determg the umber of poor leads to double coutg, ther use the costructo of a poverty de of the form () (ecludg the headcout rato) does ot volve ths problem. The reaso behd ths s that we are ot coutg the umber of poor but aggregatg ther poverty shortfalls the varous dmesos. However, as metoed earler, these measures are ot sestve to a correlato creasg swtch. - No-addtve poverty dces satsfyg MTP As see above, a more geeral famly of poverty dces s that satsfyg MTP rather tha OTP. It may be obtaed the two dmesoal case from sopoverty cotours whch are cove to the org. These poverty cotours may be geerated by takg o-decreasg ad quas-cocave trasformatos of the relatve shortfalls of the two attrbutes. The followg fuctoal form for the dvdual poverty fucto p(; ) s a compact way of represetg the so-poverty cotours show Fgure 3: p( ; ) = I Ma(, 0), Ma(, 0), (5)

17 7 where I(u, u ) s a creasg, cotuous, quas-cocave fucto wth I(0, 0) =0. The correspodg poverty de becomes: P( X ; ) = I Ma(,0), Ma( =,0). (6) Clearly, the addtve case aalysed above s a partcular case of (6) where I(u, u ) = f (u ) + f (u ). Dfferet forms of the poverty de may ow be geerated from alteratve specfcatos of I( ). A appealg specfcato may be derved from the CES form: I ( u, u ) = f [( a. u + a. u ) ], (7) where f( ) s a creasg ad cove fucto such that f(0) = 0, a ad a are postve weghts attached to the two attrbutes ad permts to parameterse the elastcty of substtuto betwee the shortfalls of the varous attrbutes. Note, however, that order to geerate sopoverty cotours cove to the org the two-dmesoal rego of the space of attrbutes, (7) must lead to sopoverty cotours that are cocave to the org the space of shortfalls. Ths s what s show fgure 3 whe sopoverty cotours are looked at from the org, O, or from the o-poverty pot, Ω. Ths cocavty requremet mposes that > (7). The full specfcato of poverty dces based o the dvdual poverty fucto (7) s obtaed by combg (6) ad (7). P( X ; ) =. = f.[ ( a Ma,0)] + a.[ Ma( Ths de seems a rather fleble fuctoal form cosstet wth MTP. Note, however, that t s ot clear,0)] a pror whether t satsfes NDCIS or NICIS. It s easy to see that MTP mples that >, whch tur mples that the cross secod dervatve of I( ) s egatve. However, the two shortfalls may stll be complemet determg poverty depedg o the shape of the fucto f( ).. (8) To see ths, ote that the cross secod dervatve of the dvdual poverty fucto p(, ;, ) wrtes wth obvous otatos : p = f'.i + f''.i.i. The codto > mples that I s egatve, but p may stll be

18 8 Three partcular cases of (8) are worth stressg. The frst case s whe teds toward fty so that the two shortfalls or equvaletly the substtutablty betwee the two attrbutes the defto of poverty teds toward ero. I that case, the sopoverty cotours become rectagular curves eve wth the two dmesoal poverty space. Ths s the shape show Fgure 4. It s terestg to ote that ths case the two attrbutes must ecessarly combe wth the two-dmesoal poverty space the same proportos as the threshold levels ad. The epresso for the poverty de the becomes: P( X ; ) where. f Ma Ma(,0), Ma(,0) =.. = = = I I= : M[,], I = : < M[,] f ( ) (9) These two sets may be called 'eclusve poverty sets' where two-dmesoal poverty s trasformed to oe-dmesoal poverty wth respect to the attrbute that s the farthest away from ts poverty le. Epresso (9) s aalogous to that for addtve poverty dces ecept that the poverty sets S are replaced by the sets I, ad the poverty fuctos are the same for the varous attrbutes. The etreme parsmoy of ths famly of poverty dces s to be oted. It actually requres o more tha the kowledge of the threshold levels ad a covetoal oe dmesoal poverty de f( ), for stace, the wellkow Foster-Greer-Thorbecke, P α de. Of course, these poverty dces satsfy MTP ad NICIS.. <Fgure 4 aroud here> The secod partcular case s at the other etreme whe the two attrbutes are perfect substtutes the two dmesoal poverty space. The sopoverty cotour s the a straght le that space whch coects the horotal ad vertcal straght les oe dmesoal poverty spaces, as Fgure 5. The geeral epresso of the correspodg poverty dces s: postve because of the secod term o the RHS. If ths were ote the case, a pot lke B Fgure 4 could be the summt of a rectagular sopoverty cotour, whch s obvously cotradctory sce poverty s ero for hgh values of a attrbute o the vertcal brach ad oero o the horotal brach.

19 9 P( X ; ) =. f a. Ma(, 0) + a. Ma( =, 0), (0) where, aga, f( ) may be ay oe-dmesoal poverty de, lke the Foster-Greer-Thorbecke P α de, ad, as before, the postve coeffcets a represet the weght gve to the attrbutes ad determe the slope of the so-poverty cotour the two dmesoal poverty space. Poverty dces of type (0) satsfy MTP ad NDCIS or NICIS depedg o whether the oe-dmesoal poverty fucto f( ) s cocave or cove. <Fgure 5 aroud here> A thrd partcular case of (8) s obtaed by usg the Foster-Greer-Thorbecke P α de for the fucto f ( ). Oe the obtas : α P ( X ; ) =. a.[ Ma(, 0)] + a.[ Ma( =, 0)] α, () where α s a postve parameter. The terpretato of that measure s straghtforward. The poverty shortfalls the two dmesos are frst aggregated to some 'average' shortfall through fucto I( ) wth a partcular value of ad the coeffcets a. Multdmesoal poverty s the defed as the average of that aggregate shortfall, rased to the power α, over the whole populato. Ths seems to be the measure the closest to oe dmesoal poverty measuremet cocepts ad the smplest geeralsato of these cocepts. Wth α=0, () yelds the multdmesoal headcout. Wth α=, α P becomes a multdmesoal poverty gap obtaed by some partcular averagg of the poverty gaps the two dmesos. Hgher values for α may be terpreted,as the oe dmesoal case,as hgher averso towards etreme poverty, A terestg property of that α P measure s that t satsfes NDCIS or NICIS depedg o whether α s greater or less tha. These three famles of poverty dces may easly be geeralsed to ay umber of attrbutes. However, dog so mples assumg the same elastcty of substtuto betwee attrbutes, ad therefore the resultg poverty dces are NDCIS or NICIS for all pars of attrbutes. Ths may ot be very satsfactory ad other more comple specfcatos have to be desged to avod ths.

20 0 Aother terestg geeralsato of the precedg measures cossts of assumg that the substtutablty betwee the poverty shortfalls the two attrbutes chages wth the etet of poverty. Whe someoe s very poor oe of the two dmesos, oe may be wllg to assume that the elastcty of substtuto betwee the two dmesos of poverty s of mor mportace. For stace, f a perso s 50 per cet below the poverty le terms of food, t s probably mmateral whether he/she s 0 or 0 per cet below the poverty le for educatoal attamet for evaluatg hs/her overall poverty. O the cotrary, f the food poverty gap s oly 0 per cet, the the etet of the poverty gap educato becomes a more mportat determat of overall poverty. The correspodg shape of the so-poverty cotours s show Fgure 6. But oe may also be wllg to assume the opposte, amely that the substtutablty betwee the two attrbutes decreases wth the etet of poverty. Aalytcally, a smple way of allowg for ths depedecy betwee the substtutablty of attrbutes ad the etet of poverty cossts of makg the parameter (8) a fucto of the level of poverty. Wth a P α framework, dvdual poverty s the defed mplctly by the followg equato: a( p) a( p) a( p) [ Ma(, 0)] + [ Ma(,0)] = p(,,, ) α where a(p) s a fucto that descrbes how attrbute substtutablty chages wth the etet of poverty. Obvous caddates for ths fucto are a(p) =/p ad a(p) = /(-p), assumg p s ormaled so as to le betwee 0 ad. Wth these fuctos, solvg umercally equato () s ot dffcult. It leads to poverty fuctos wth the same propertes as (), ecept for the fact that correlato creasg swtches may ow crease or decrease overall poverty depedg o whether they are performed amog very poor or moderately poor persos. We shall refer to these dces respectvely as (), p P α / ad /( p) α. P <Fgure 6 aroud here> It s worth stressg that all precedg mult-dmesoal poverty dces actually rely o the SF postulate. I effect, t may be show that the weak focus postulate (WF) rules out fuctoal forms of poverty dces that are addtve as well as the CES-lke substtutablty geeralsatos, α ad P p P / /( p) α α P measures, or eve ther varyg. As a matter of fact we have ot bee able to fd relatvely smple fuctos leadg to so-poverty cotours cosstet wth WF as show fgure 7, ad the other propertes of the dvdual poverty fucto p( ).

21 <Fgure 7 aroud here> 6. A eample of applcato To llustrate the use of the precedg measures as well as the cocepts behd them, we aalye here the evoluto of multdmesoal poverty rural Bral durg the 980s. Poverty cludes two dmesos : come o the oe had ad educatoal attamet o the other. The aalyss s performed o the rural populato oly, because ths s where Brala poverty teds to cocetrate. It s also restrcted to the adult populato, so as to avod the problem of mputg some fal educatoal level to chldre who are stll gog to school. Samples from the PNAD household surveys for the years 98 ad 987 are beg used. 3 The reaso for choosg these years s that they happe to correspod to a crease come poverty the rural populato. So, we felt t could be terestg to use the measures preseted the prevous secto to see whether ths crease come poverty had possbly bee compesated by a drop educatoal poverty. But, of course, ths ssue of the trade-off betwee these two partcular dmesos of poverty would also arse very dfferet cotets. For stace, desgg at-poverty polces may requre decdg whether t s better to reduce more come or educato poverty. Poverty s measured at the dvdual level. Each dvdual s gve the come per capta wth the household he/she belogs to. The come poverty threshold s $ a day, at 985 ppp corrected prces. 4 The educatoal poverty threshold s defed as the ed of prmary school, that s, 4 years of schoolg. The educatoal poverty shortfall s defed as the umber of years of schoolg short of that level. It may thus take oly 4 values. Yet, we treat t as a cotuous varable. The frst two colums of table show the level of poverty as measured by the covetoal P α measures separately for come ad educato. It may be see that come poverty creased from 98 to 987, whereas educato poverty fell. The "alpha = 0" rows show that there were 40.5 per cet of rural adults below the poverty le 98 whereas 74.4 per cet had ot completed prmary school. S years later these proportos were 4. ad 68 per cet respectvely, dcatg a crease come poverty ad a fall educato poverty. The poverty gap ("alpha = ") ad hgher levels of the P α measures show the 3 Irrespectvely of the fact that rural comes are kow to be mperfectly observed PNAD see for stace Elbers et al. (00). The calculatos below must therefore be take as mostly llustratve. 4 Survey data have bee made cosstet wth Natoal Accouts by a proportoal correcto of all come fgures.

22 same evoluto. <Table aroud here.> We ow cosder multdmesoal poverty measures of the P α type, wth α takg the same values as for the oe dmesoal poverty measures that we ust revewed ad takg the values, whch correspods to perfect substtutablty as (0) above, ad 5. We also use the varyg substtutablty measures, p P / α ad /( p) α P. The evaluato of multdmesoal poverty for 98 ad 987 accordg to these varous measures are reported table for two sets of weghts for the come ad educato attrbutes. The frst set gves equal weghts to the two dmesos whereas the secod gves more weght to come. Cosder frst the frst two rows whch correspod to headcout poverty measures. I the multdmesoal case, the headcout correspods to dvduals who are poor ether terms of come or terms of educato. Accordgly there were 79.7 per cet poor 98 versus 75.6 per cet 987. From these fgures ad the headcouts oe dmeso, t s easy to derve the proporto of people who were poor both dmesos. They were 35. per cet 98 ad 34.4 per cet 987. Readg dow the other rows, oe may check that the multdmesoal α P measures - as well as the measures wth varable substtutablty - are commesurate wth the oe dmesoal P α measures for come ad educato. There s othg surprsg here. As oted above, the multdmesoal α P measures are desged such a way that they may be terpreted as some partcular mea of oe dmesoal measures. Ths mea depeds o the weghg coeffcets, a ad a, but also o the substtutablty parameter,. So, multdmesoal measures s hgher whe more weght s gve to educato because oe dmesoal poverty s hgher for educato, as show the frst two colums of table. But multdmesoal poverty also teds to crease whe the substtutablty of the two attrbutes falls, or equvaletly the parameter creases. As suggested by the argumet leadg to (9) above, ths s because low substtutablty betwee the two attrbutes gves more weght for each observato to the attrbute wth the largest shortfall. Bold fgures table correspod to stuatos where poverty measures dcate more poverty 987 tha 98. We see that ths occurrece s more frequet whe the weght gve to the come dmeso s hgher. There s othg really surprsg here sce we have see that there was more

23 3 poverty, a oe dmeso sese, wth come tha wth educato. It s more terestg to otce that poverty appears to be hgher 987 tha 98 whe the poverty averso parameter, α, s hgh eough, although the value of that parameter for whch ths happes s ot systematcally show the table. Ths s true for each value of the substtutablty parameter,, as well as for both systems of weghts. Ths s true also wth the varable substtutablty measure, /( p) α P. A possble eplaato for ths patter would be that the worseg of the b-dmesoal come/educato dstrbuto rural Bral may have ts roots at the very bottom of the dstrbuto, where poverty s more severe. I other words, come losses may have bee more serous predomatly for people wth low come ad low educato. Regardg the correlato betwee the two dmesos of poverty, stll a more terestg feature table s the fact that poverty teds to be hgher 987 cases where the NICIS property holds. It was see the prevous secto that the α P measure was such that poverty would crease wth creasg correlato swtches whe α<. It happes table that cases where poverty s hgher 987 tha 98 occur oly whe the opposte s true.. Ths suggests that the crease oe-dmesoal come poverty was accompaed by a drop the correlato wth educatoal levels. The varyg substtutablty measures gve stll aother formato. Frst, t may be see that 987 ever ehbts more poverty tha 98 wth the p P / α measure. It does however wth the /( p) α P measure for hgh values of α whe both dmesos have equal weght ad much sooer whe more weght s put o the come dmeso. Ths evoluto s cosstet wth the dea that come losses were more proouced for poorer people wth more a large come tha educato shortfall. Wth the s lmted substtutablty for them ad the drop come could ot be compesated by a possble crease the educatoal level. /( p) α there P Ths terpretato of the fgures reported table would eed to be cofrmed by a more careful aalyss of the b-dmesoal dstrbuto of educato ad come rural Bral. Wth the preset framework, what matters s that measures drectly spred from the famlar oe dmesoal P α poverty dces ad elarged through a reduced set of parameters - parameters the case of α P ad a sgle oe the case of P / α p or multdmesoal perspectve. /( p) α - permt to descrbe adequately the etet of poverty a P

24 4 Cocluso We have eplctly argued ths paper why poverty should be regarded as the falure to reach mmally acceptable levels of dfferet moetary ad o-moetary attrbutes ecessary for a subsstece stadard of lvg. That s, poverty s essetally a multdmesoal pheomeo. The problems of coutg the umber of poor ths framework ad the combg the formato avalable o them to a statstc that summarses the etet of overall poverty have bee dscussed rgorously. Usg dfferet postulates for a measure of poverty, shapes of sopoverty cotours of a perso have bee derved alteratve dmesos. Ths tur establshes a perso s ature of trade off betwee attrbutes dfferet poverty spaces. We make a dstcto betwee addtve ad o-addtve poverty measures satsfyg the strog verso of the Focus Aom, whch demads depedece from o-poor attrbute quattes poverty measuremet. Oe problem wth addtve measures s that they are sestve to a correlato creasg swtch. A correlato creasg swtch requres gvg more of oe attrbute to a perso who has already more of aother. A fer subdvso amog oaddtve measures s possble depedg o whether a measure decreases or creases uder such a swtch. Specfc fuctoal forms have bee proposed that ft these varous propertes depedg o the value of a small umber of key parameters ad geeralg a easy way the famlar P α famly. As a llustrato, the resultg measures have bee used to evaluate the evoluto of come/educato poverty rural Bral the 980s.

25 5 Refereces Atkso, A. ad F. Bourgugo, The Comparso of Multdmesoed Dstrbutos of Ecoomc Status, Revew of Ecoomc Studes (98), 49, 83-0 Blackorby, C. ad D. Doaldso, Ethcal Idces for the Measuremet of Poverty, Ecoometrca (980), 48, Bourgugo, F. ad G.S. Felds, Dscotuous Losses from Poverty, Geeraled P α Measures, ad Optmal Trasfers to the Poor, Joural of Publc Ecoomcs (997), 63, Chakravarty, S. R. Ethcal Socal Ide Numbers, (990), Sprger-Verlag, Lodo Chakravarty, S.R., D. Mukheree ad R. Raade O the Famly of Subgroup ad Factor Decomposable Measures of Multdmesoal Poverty, Research o Ecoomc Iequalty (998),8, Clark, S., R. Hemmg ad D. Ulph, O Idces for the Measuremet of Poverty, Ecoomc Joural (98),9, Cowell, F. A., Poverty Measures, Iequalty ad Decomposablty, D. Bös, M. Rose ad C. Sedl (ed), Welfare ad Effcecy Publc Ecoomcs (988), Sprger-Verlag, Lodo Doaldso, D. ad J. A. Weymark Propertes of Fed Populato Poverty Idces, Iteratoal Ecoomc Revew (986), 7, Duclos, J-Y, D. Sah ad S. Youger, Robust Mult-Dmesoal Poverty Comparsos (00), Mmeo, Corell Uversty Elbers, C., J. Laouw, P. Laouw ad P. G. Lete, Poverty ad Iequalty Bral : New Estmates from Combed PPV-PNAD Data, (00), World Bak, DECRG, Mmeo Foster, J. E., O Ecoomc Poverty: A Survey of Aggregate Measures, R.L. Basma ad G.F. Rhodes (ed.), Advaces Ecoometrcs, (984),Vol. 3, JAI Press, Coectcut Foster, J., J. Greer ad E. Thorbecke, A Class of Decomposable Poverty Measures, Ecoometrca, (984), 5, Foster, J. ad A. F. Shorrocks, Subgroup Cosstet Poverty Idces, Ecoometrca, (99), 59, Kakwa, N.C., O a Class of Poverty Measures, Ecoometrca, (980), 48, Kolm, S.C., Multdmesoal Egaltarasms, uarterly Joural of Ecoomcs, (977),9, -3 Lpto, M. ad M. Ravallo, Poverty ad Polcy, J. Behrma ad T.N. Srvasa (eds.) Hadbook of Developmet Ecoomcs, (995), vol. 3, North-Hollad, Amsterdam Maasoum, E. The Measuremet ad Decomposto of Multdmesoal Iequalty, Ecoometrca, (986), 54,

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27 7 APPENDIX. FORMAL STATEMENT OF THE AXIOMS USED IN THE PAPER STRONG FOCUS (SF): For ay N, (X, Y) M, Z, {,,..., m}, f () for ay such that, y = + δ, where δ>0, () y t = t for all t, ad () y s = s for all s ad for all, the P(Y; ) = P(X; ). WEAK FOCUS (WF): For ay N, (X, Y) M, Z, f for some k k for all k ad () for ay {,,..., m}, y = + δ, where δ>0, () y t = t for all t, ad () y rs = rs for all r ad all s, the P(Y; ) = P(X; ). SYMMETRY (SM): For ay (X; ) M Z, P(X; ) = P(ΠX; ), where Π s ay permutato matr of approprate order. MONOTONICITY (MN): For ay N, X M, Z, {,,..., m}, f: () for ay, y = + δ, where <, δ>0, () y t = t for all t, ad () y s = s for all s ad for all, the P(Y; ) P(X; ). CONTINUITY (CN): For ay Z, P( ) s cotuous o M. PRINCIPLE OF POPULATION (PP): For ay (X; ) M Z, P(X k ; ) = P(X; ) where X k s the k-fold replcato of X. SCALE INVARIANCE (SI): For ay (X; ) M Z, P(X; ) = P(X ; ) where X = ΛX, = Λ, Λ beg the dagoal matr dag(λ,.λ m ), λ >0 for all. SUBGROUP DECOMPOSABILITY (SD): For ay X, X,, X K M ad Z: P(X, X,, X K ; ) = K = P(X ;), where s the populato se correspodg to X ad = Σ DEFINITION OF A PIGOU-DALTON PROGRESSIVE TRANSFER.. Matr X s sad to be obtaed from Y M by a Pgou-Dalto progressve trasfer of attrbute from oe poor perso to aother f for some persos, t : () y t < y <, () t y t = y >0, t, () r = y r for all r, t, ad (v) rk = y rk