Economics Letters 59 (1998) Jonathan Morduch* Received 12 January 1998; accepted 9 March 1998

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1 Economcs Letters 59 (998) Poverty, economc rowth, and averae ext tme Jonathan Morduch* Hoover Insttuton L27, Stanford Unversty, Stanford, CA , USA Receved 2 January 998; accepted 9 March 998 Abstract A smple transformaton of the Watts poverty ndex yelds a meannful measure wth appealn ordnal propertes and a natural nterpretaton n terms of the potental for economc rowth to allevate poverty. The ndex s llustrated wth data from Banladesh and Bolva. 998 Elsever Scence S.A. Keywords: Poverty measurement; Economc rowth; Watts poverty measure; Bolva; Banladesh JEL classfcaton: I32; O2. Introducton Economsts have proposed a wde varety of ndces of poverty, each wth competn attrbutes and justfcatons. For the most part, thouh, only two ndces are used reularly n poverty analyss: the headcount ndex and the poverty ap. The former smply measures the number of ndvduals below an ncome threshold, and the latter measures the total sze of transfers needed to brn poor ndvduals up to the threshold. Ther popularty remans stron despte coent crtues: n partcular, nether measure reflects chanes n the dstrbuton of ncome below the poverty lne (Sen, 976). In 968, Harold Watts proposed a smple alternatve measure that s dstrbutonally-senstve and decomposable as the wehted sum of poverty measures of populaton subroups. The ndex has appealn ordnal propertes, and below t s shown that a smple lnear transformaton of the ndex ves t cardnal propertes that can be useful as well. Ths modfed Watts measure s termed the averae ext tme. The ornal Watts measure s defned as W 5] O [ln(z) 2 ln( y )], N 5 where there are ndvduals n the populaton ndexed from to N n ascendn order of ncome and s the number of people wth ncome y below the poverty lne z. *Tel.: ; fax: ; e-mal: jmorduch@harvard.edu The attrbutes of the Watts ndex and a characterzaton are descrbed by Zhen (993) / 98/ $ Elsever Scence S.A. All rhts reserved. PII: S (98)

2 386 J. Morduch / Economcs Letters 59 (998) The averae ext tme, T, s smply W/,where.0 s a hypothetcal rowth rate of ncomes of the poor populaton. The ndex reflects the averae number of years that t would take the populaton to ext poverty f t were possble to ensure that all ncomes row at rate. The averae ext tme s derved below and appled to data from urban Bolva and rural Banladesh n Secton Ext tme A household s expected ext tme s defned as the tme t wll take t to reach a ven poverty lne va ncome rowth. If the ncome of household rows at a constant postve rate per year, the relatonshp of the poverty lne to current ncome can be wrtten t z 5 y ( ). Takn the loarthm and solvn for t yelds ln(z) 2 ln( y ) t ]]]], () for households below the poverty lne. For households above the poverty lne, t 50. The approxmaton ln() s used here. E. () holds exactly for exponental rowth such that z5y expht j. In Banladesh, for example, the medan poor rural household n the Household Expendture Survey spent Tk. 284 per month per capta relatve to the poverty lne of Tk. 370 (n 989, Tk. 32.5$). So, f ther expendtures rew at per year, t would take 8.8 years to reduce half of rural poverty throuh rowth alone. Whle rouh, ths sort of calculaton ves a sense of the contrbuton that rowth can make to poverty allevaton. Kanbur (987), for example, suests usn the ext tme for the averae poor person as a smple poverty measure: ln(z) 2 ln(m p) av t ]]]]], (2) where mp s the averae ncome of poor households. The measure has appeal, but t s not senstve to the dstrbuton of ncome below the poverty lne snce, lke the poverty ap, t s based on the averae ncome of the poor. The averae ext tme, n contrast, reflects dfferences between extreme and moderate poverty. The averae ext tme s smply t averaed over the entre populaton: N ln(z) 2 ln( y ) ] ] ]]]] N N T 5 O t 5 O 5 W/. (3) 5 5 Calculaton of the ndex s smple, nvolvn only an ordnal transform of the Watts measure.

3 3. Propertes and relatonshps to other measures J. Morduch / Economcs Letters 59 (998) The measure nests the Thel-L ndex of neualty. The Thel ndex s calculated as L 5] O [ln(m ) 2 ln( y )]. (4) p p 5 when appled just to the households below the poverty lne. The Thel ndex s bounded between 0 and, but by dvdn Lp by, the ndex can be put n unts of tme, yeldn L5L p /. Substtutn Es. (2) and (4) nto (3) yelds an euvalent form for the averae ext tme: av T 5 H(t L ). (5) When all ncomes below the poverty lne are eually dstrbuted, L50 and the averae ext tme s av smply Ht. Thus, the dstrbutonal-senstvty comes drectly from the nestn of the Thel ndex n much the same way as the Sen (976) ndex nests the Gn coeffcent. Because the measure s bult around a smple transformaton of the Thel ndex, t s easly checked that the measure satsfes the monotoncty and transfer axoms. The monotoncty axom states that a reducton n the ncome of a poor household must ncrease the poverty measure, holdn all else constant. The transfer axom states that a postve transfer from one poor household to any less poor household must ncrease the poverty measure. As a result of ts addtvely separable form, the measure s also decomposable nto the populatonwehted measures of subpopulatons of the poor, a feature that has proven useful for poverty analyss (Foster et al., 984). Thus the Watts ndex and averae ext tme have nearly dentcal propertes to the Foster Greer Thorbecke measures employed wdely n World Bank poverty assessments (Lpton and 3 Ravallon, 995) Applcaton to rural Banladesh and urban Bolva The concept of ext tme s llustrated below for a hypothetcal rowth rate of. The data come from three rounds of the Encuesta Interada de Hoares of Urban Bolva (989 92) and the rural 4 sample from the Household Expendture Survey of Banladesh. These are lare, representatve surveys, covern two low-ncome economes where poverty allevaton has been a contnun polcy oal. 2 When ncome below the poverty lne s dstrbuted eually, the Sen ndex reduces to H multpled by the poverty ap av measure. Ths s the analoue to Ht here. As wth the Watts measure and the Thel ndex, the averae ext tme s undefned when any household has zero ncome snce t ` as y 0. Zero ncome s most lkely due to under-reportn and the problem should be addressed n the data collecton process or by addn small amounts of ncome to zero values. 3 The Watts ndex has potental appeal relatve to the commonly-used suared poverty ap of Foster Greer Thorbecke measure by ben transfer senstve : reressve transfers between very poor households ncrease measured poverty more than reressve transfers between less poor households. A potental dsadvantae s that wehtn s not smooth at the poverty lne (Lpton and Ravallon, 995). Nether feature s lkely to affect poverty rankns much n practce, relatve to other propertes. 4 In 990, 3.2 Bs.5$. In 989, Tk. 32.5$ and 2.7 Bs.5$.

4 388 J. Morduch / Economcs Letters 59 (998) Whle the headcount ndex remaned farly constant at 55%, the averae ext tme n urban Bolva ncreased from 0.9 to 3.2 years durn the perod. The 20% ncrease reflects a declne n the averae ncome of poor households from Bs. to Bs., pushn the poverty ap up by 4%. Ths was a perod of worsenn dstrbuton: the ncome declne occurred whle averae ncomes of the 5 broader populaton rew by nearly 4%. At the same tme, ncome neualty of poor households, as measured by the Thel ndex, ncreased by over 20%. The averae ext tme provdes a way to put these peces toether to tell a coherent story about the chanes. Frst, snce the ncome declne of the poor averaed just under per year, by constructon av the ext tme of the averae poor household, t, should an just under a year for each of the 3.7 av years between rounds and 5 of the survey. Table shows ths exactly: t ncreased by 3.4 years av between March 989 and October 992. As shown n E. (5), T /H s the sum of t plus the rescaled Thel-L neualty ndex. Ineualty ncreases the averae ext tme of the poor populaton, av T /H, by nearly 20% over t, but the chane n neualty over the perod contrbutes only an extra 0.7 years to the chane n T /H. Thus, once measured neualty s put nto comparable terms, t can be seen drectly that ts contrbuton to the chane n poverty s mnor. These effects are summarzed by the averae ext tme, T. The ncrease from 0.9 to 3.2 years combnes these forces and testfes to the contnun need for economc rowth whch reaches down to the poorest households n urban Bolva. The stuaton n rural Banladesh shows the potental for rowth there as well. The poverty lne n rural Banladesh s much lower than n urban Bolva, $ versus $63 n 989, and ths explans the low averae ext tme of 4.9 years. The low averae ext tme also reflects a low deree of neualty amon poor households. The calculatons suest that a decade of steady rowth at can elmnate much of measured poverty. But t would take at least three eneratons of steady rowth at for the averae poor household n Banladesh to reach Bolva s poverty lne. Table Poverty measures and averae ext tme (assumn averae rowth per year). Urban Bolva and rural Banladesh Encuesta Interada de Hoares, urban Bolva Rural March 989 September 990 October 992 Banladesh Round Round 3 Round 5 HES Headcount ndex (H) Averae ext tme (T ) 0.9 years 2.0 years 3.2 years 4.9 years Ext tme of the av. 7.5 years 8.5 years 20.9 years 9.7 years av Poor household (t ) Ineualty of poor (Thel-L4) Poverty ap 69. Bs. 72 Bs Bs. Tk. 94 Poverty lne 69.9 Bs Bs Bs. Tk. 370 Sample sze Notes: Expendture data s per capta per month, calculated usn populaton wehts. Bolvan data s n September 990 Bolvanos. Banladesh data s n 989 takas. 5 Snce economc rowth pushes some households across the poverty lne, t s possble that all households have postve rowth rates but, at the same tme, the averae ncome of poor households may fall as the least poor households ext poverty, poorer households are left behnd, lowern the averae ncome of those remann below the poverty lne.

5 5. Meannful poverty measurement J. Morduch / Economcs Letters 59 (998) The headcount and the poverty ap are set apart from other poverty measures by ben meannful.e., there s partcular nterest n the cardnal representatons of the ndces themselves (Foster, 994). Suarn, cubn, or postvely transformn these ndces n any other way makes them much less nformatve measures. Ther popularty suests that for many purposes ths cardnal meannfulness outwehs other dsadvantaes of the two measures. Lke the poverty ap, the averae ext tme s meannful throuh descrbn an nterestn f then relatonshp. The poverty ap ndcates f there s perfectly tareted, costless redstrbuton then how much money wll be reured to completely elmnate poverty. These assumptons are clearly unrealstc redstrbuton wll never be perfectly tareted nor costless but the poverty ap nevertheless conveys useful nformaton. Smlarly, the averae ext tme s also based on best case assumptons. Even thouh rowth s unlkely to be constant or unform across households, the averae ext tme provdes a smple metrc of the potental for poverty allevaton throuh rowth. By usn best case assumptons rather than context-specfc assumptons, the averae ext tme s smple and comparable across reons and tme perods. Achevn broad-based economc rowth s frmly on polcy aendas, and the accumulatn evdence shows that achevn rowth and reducn poverty can be stronly complementary (e.., Denner and Sure, 996). Developn analytcal tools to consder poverty and rowth jontly s thus mportant. The usefulness of ths partcular measure depends on the relevance of the hypothetcal ueston. Whle the ndex s bult around the dmenson of tme, t remans a measure of statc condtons. It does not measure the actual duraton of poverty nor averae expected duraton based on ndvdualspecfc factors. Rasn the hypothetcal rowth rate here trvally reduces the averae ext tme, so comparsons across measures wth dfferent hypothetcal rowth rates shed lttle lht. The value of the ndex s n mappn an ncome dstrbuton nto a smple metrc that conveys nformaton about the nature of poverty. If an ndvdual earns $000 per year, t would take them 22 years to reach a poverty lne of $3000 f ther ncome were to row at a real rate of 5% annually. It would take just 8 years for someone startn at $2000 per year. These dfferences n ext tme reflect a potentally mportant dmenson by whch to aue what t means to have an annual ncome of $000 versus $2000. The averae ext tme presents one smple way to ben ncorporatn these dmensons nto poverty analyses. Acknowledements I have benefted from dscussons wth Tm Besley, Gary Felds, Peter Klbanoff, and Debraj Ray. Muel Urola and Wlson Jmenez provded crtcal nshts and assstance wth the Bolvan data. Fnancal support was provded by the Research Commttee at the Harvard Insttute for Internatonal Development. I am rateful for the hosptalty and data provded by the Undad de Analss des Poltcas Socales (UDAPSO), La Paz. Ths s a substantally revsed verson of a paper crculated prevously under the ttle A Postve Measure of Poverty (Harvard Insttute of Economc Research Dscusson Paper 669, January 994).

6 390 J. Morduch / Economcs Letters 59 (998) References Denner, K., Sure, L., 996. A new data set measurn ncome neualty. World Bank Economc Revew 0 (3), Foster, J., 994. Normatve measurement: Is theory relevant?. Amercan Economc Revew (AEA, Papers and Proceedns) 84, Foster, J., Greer, J., Thorbecke, E., 984. A class of decomposable poverty measures. Econometrca 52 (3), Kanbur, S.M.R., 987. Measurement and allevaton of poverty, wth an applcaton to the effects of macroeconomc adjustment. IMF Staff Papers 34 (), Lpton, M., Ravallon, M., 995. Poverty and polcy. In: Behrman, J., Srnvasan, T.N. (Eds.), Handbook of Development Economcs. North-Holland, Amsterdam, ch. 42. Sen, A.K., 976. Poverty: An ordnal approach to measurement. Econometrca 44, Zhen, B., 993. An axomatc characterzaton of the Watts poverty ndex. Economcs Letters 42 (), 8 86.