Poverty Intensity in Australia

Transcription

1 Unversty of Wollonon Research Onlne Faculty of Commerce - apers (Archve) Faculty of Busness 2000 overty Intensty n Australa Joan R. Roders Unversty of Wollonon, jrroder@uow.edu.au John L. Roders Unversty of Wollonon, john_roders@uow.edu.au ublcaton Detals Ths artcle was ornally publshed as Roders, JR and Roders, JL, overty Intensty n Australa, Australan Economc Revew, 33(3), 2000, Copyrht Blackwell Ornal journal avaalable here. Research Onlne s the open access nsttutonal repostory for the Unversty of Wollonon. For further nformaton contact the UOW Lbrary: research-pubs@uow.edu.au

2 overty Intensty n Australa Abstract Even thouh poverty ndces wth axomatcally sound propertes have been advocated for several decades, most emprcal studes of poverty n Australa and elsewhere contnue to use the crude, but easly understood, head-count rato. The dffculty of nterpretn the axomatcally more desrable ndces s a major reason why ther use has been ressted n appled poverty measurement. Ths paper demonstrates how the more sophstcated poverty ndces can be converted nto a form that s readly nterpreted as a measure of poverty ntensty of a roup, relatve to the populaton to whch the roup belons. The resultn poverty-ntensty ndex s easy to understand and t retans the axomatc propertes of the poverty ndex on whch t s based. We apply the method to Australan data. overty measures reported prevously n the lterature are converted nto measures of poverty ntensty and nterpreted accordnly. We also calculate and nterpret some new measures of poverty and poverty ntensty usn the Income and ousn Costs Survey, It s hoped our procedure wll lead to wder use of poverty ndces that are theoretcally superor to the head-count rato. Dscplnes Busness Socal and Behavoral Scences ublcaton Detals Ths artcle was ornally publshed as Roders, JR and Roders, JL, overty Intensty n Australa, Australan Economc Revew, 33(3), 2000, Copyrht Blackwell Ornal journal avaalable here. Ths journal artcle s avalable at Research Onlne:

3 overty Intensty n Australa by J. R. Roders and J. L. Roders Department of Economcs Unversty of Wollonon Wollonon, N.S.W., 2522, Australa h Fax: joan_roders@uow.edu.au john_roders@uow.edu.au January, 1998 Revsed March, 2000 Fnal Verson, Aprl, 2000 Summary Ths paper proposes a poverty-ntensty ndex that has desrable axomatc propertes and s easy to understand. The ndex s appled to recent Australan data.

4 overty Intensty n Australa Abstract Even thouh poverty ndces wth axomatcally sound propertes have been advocated for several decades, most emprcal studes of poverty n Australa and elsewhere contnue to use the crude, but easly understood, head-count rato. The dffculty of nterpretn the axomatcally more desrable ndces s a major reason why ther use has been ressted n appled poverty measurement. Ths paper demonstrates how the more sophstcated poverty ndces can be converted nto a form that s readly nterpreted as a measure of poverty ntensty of a roup, relatve to the populaton to whch the roup belons. The resultn poverty-ntensty ndex s easy to understand and t retans the axomatc propertes of the poverty ndex on whch t s based. We apply the method to Australan data. overty measures reported prevously n the lterature are converted nto measures of poverty ntensty and nterpreted accordnly. We also calculate and nterpret some new measures of poverty and poverty ntensty usn the Income and ousn Costs Survey, It s hoped our procedure wll lead to wder use of poverty ndces that are theoretcally superor to the head-count rato.

5 1. Introducton The measurement of poverty nvolves two dstnct sets of problems. The frst problem s to dentfy ndvdual ncome unts (people who share ncome) that are poor. 1 The second problem s to areate the poverty of ndvdual ncome unts nto an ndex of areate poverty for a roup of ncome unts. Ths paper focuses on the latter: on the measurement of areate poverty. overty n Australa has typcally been measured usn the head-count rato, whch s the proporton of the populaton that lves n ncome unts wth ncomes below some arbtrary poverty threshold. Exceptons are studes by Kakwan (1986), Johnson (1988, 1991 and 1996b) and Johnson and Dxon (1999). These authors and others (for example, Sen, 1976; Takayama, 1979; Kakwan (1980); Foster, Greer and Thorbecke, 1984) have reconsed the defcences of the head-count rato and, n response, have developed poverty ndces wth superor propertes. 2 In the emprcal lterature Kakwan (1986) reported, for example, that the value of hs ndex for all Australans n the year was 4.05 whle Johnson (1996b) reported that hs deprvaton-wehted poverty ndex for all Australan ncome unts n was Results such as these have lttle meann to most readers because the measures are functons, not only of the proporton of the populaton that s poor, but also of the mean ncome of the poor and the dstrbuton of ncome amon the poor. In contrast, head-count ratos such as those presented by Kakwan (1986) (7.02 percent of Australans were poor n ) and Johnson (1996b) (11.54 percent of Australans lved n poor ncome unts n ) are readly understood even thouh they consttute crude measures of poverty. 3 1

6 A major reason why the head-count rato contnues to domnate emprcal studes s ts ntutve appeal relatve to that of the theoretcally sound poverty ndces. Saunders and Whteford (1989, p.31) state that the head-count rato: s smple to derve, easy to understand and open to clear and obvous nterpretaton. The objectve of ths paper s to show how many of the new poverty ndces can be converted nto a form that s readly understandable yet preserves ther desrable propertes. In Secton 2 we present a poverty-ntensty ndex, 4 whch measures the poverty of a roup of ncome unts relatve to the populaton to whch the roup belons. The use of poverty-ntensty ndces should make the results of emprcal studes of poverty accessble to a wder audence. owever, economsts and other professonals who compute measures of poverty ntensty stll need to understand the propertes of poverty ndces n order to choose the poverty ndex on whch the poverty-ntensty ndex s to be based. In Secton 3 we draw upon the results of prevous research nto Australan poverty to demonstrate the applcaton of our poverty-ntensty ndex and n Secton 4 we present some new measures of poverty and poverty ntensty based on data from the Income and ousn Costs Survey, Australa. 2. overty Intensty Roders and Roders (1991, p.345) defne a poverty-ntensty () ndex for a roup,, of ncome unts n a populaton as: 2

7 = the proporton of populaton poverty contrbuted by roup [1] the proporton of populaton sze contrbuted by roup The numerator of [1] s equal to OV + / OV, where OV + s the contrbuton by roup to the poverty ndex value, OV, that apples to the entre populaton. The denomnator of [1] s equal to n / n, where n s the number of people n roup and n s number of people n the whole populaton. If OV s addtvely decomposable then t can be wrtten as a wehted averae of the poverty ndces for G mutually exclusve and collectvely exhaustve roups that comprse the populaton: OV = G = 1 n OV n [2] where OV s the level of poverty n roup, as measured by OV. In ths case OV + = n OV n and the ndex s smply the rato of the poverty ndex for roup to that of the entre populaton: OV = [3] OV Usn Equatons [2] and [3] t can be shown that / OV > 0 (Roders and Roders, 1991, p.347), that s, the poverty-ntensty ndex s a strctly ncreasn functon of the poverty ndex on whch t s based, provded that ndex s addtvely decomposable. Therefore, satsfes the same propertes as OV. The ndex has ntutve meann and so makes theoretcally sound measures of poverty accessble to a wde audence. If, for example, = 3 then the ntensty of poverty n roup s three tmes that of the populaton as a whole; = 0.3 ndcates that the ntensty of poverty n roup s 30 percent that of the 3

8 populaton as a whole. More enerally, f s less than, equal to, or reater than unty then ntensty of poverty n roup s respectvely less than, equal to, or reater than that of the whole populaton. The ndex addresses the queston: where s poverty most ntense? rather than how much poverty exsts? It s well suted to developed countres where even the poorest people receve enouh ncome for survval yet where analysts wsh to desn, mplement and evaluate alternatve polces amed at allevatn hardshp. Once the roups wth the hhest poverty ntensty are dentfed scarce publc funds can be allocated effcently to specfcally desned and tareted prorams. For example, vocatonal-trann prorams may be approprate for reducn youth poverty whereas subsdzed access to nursn homes may be effectve n reducn poverty amon the elderly. The ndex can be used to set prortes n desnn prorams to amelorate poverty and t can be used to track structural chanes n poverty over tme. To demonstrate the features of a ndex we consder two poverty ndces wth attractve propertes: the deprvaton-wehted ndex of Johnson(1996a and 1996b) and Johnson and Dxon (1999) and the poverty ndex of Foster, Greer and Thorbecke (1984). The Johnson-Dxon () ndex for an entre populaton comprsed of G mutually exclusve and collectvely exhaustve roups s equal to = 1 n G = 1 M Ø y h 1- Œº Ł z ł b ø œß 0 < β 1 [4] where: n s the number of people n the populaton; M s the set of poor ncome unts n roup (=1,2, G); 4

9 h s the number of people n ncome unt ; z s the poverty lne for ncome unt ; y s dsposable ncome of ncome unt, y < z ; and 0 < β 1, where β s a poverty averson parameter, whch determnes the rate at whch poverty ncreases as the dsposable ncome of the th ncome unt decreases, ceters parbus. 5 The ndex for a snle roup,, s: = 1 n M Ø y h - 1 Œº Ł z ł b ø œß [5] where: n s the number of people n roup and s addtvely decomposable and therefore can be wrtten as a wehted sum of poverty ndces for the G roups: G =1 n = n. = G = 1 n n [6] Substtutn Equaton [6] nto Equaton [1] we obtan the ndex for roup based upon the poverty ndex: n Ł n = łł ł = [7] n Ł n ł The numerator of Equaton [4]. s ven by Equaton [5] and the denomnator s ven by has the same propertes as the poverty ndex, namely, 5

10 addtve decomposablty, symmetry, focus, monotoncty, dstrbuton, transfer senstvty and substtuton senstvty (Johnson and Dxon, 1999, p ). The Foster, Greer and Thorbecke () ndex for a populaton comprsed of G mutually exclusve and collectvely exhaustve roups s: = 1 n a G y h 1- = 1 M Ł z ł α 2 [8] where: α s a poverty averson parameter, whch determnes the rate at whch poverty ncreases as the dsposable ncome of the th ncome unt decreases, ceters parbus. 6 The ndex for a snle roup,, s: = 1 n M h 1- Ł y z ł a [9] The ndex for roup based upon the poverty ndex s: = [10] has the same propertes as the poverty ndex, namely, addtve decomposablty, symmetry, focus, monotoncty, transfer and monotoncty senstvty (see Roders and Roders, 1991, p.345). A ndex based on the head-count rato s calculated for comparson purposes n Sectons 3 and 4 below. The head-count rato () for a populaton comprsed of G mutually exclusve and collectvely exhaustve roups s: = 1 n G = 1 M h [11] 6

11 The head-count rato for a snle roup,, s: = 1 n M h [12] and the ndex for roup based upon s: = [13] has the same defcences as the poverty ndex (see Roders and Roders, 1991, p.345). 3. overty Intensty n Australa, 1981 throuh 1990 Table 1 presents the ndex of poverty n Australa n , and and the correspondn ndces calculated usn Equatons [4], [5] and [7] wth β = 0.5. The values are easy to nterpret. In all three years, poverty amon couples was less ntensve, and poverty amon snles was more ntensve, than poverty n the populaton as a whole. Snles wth dependents were the most ntensely poor roup durn n 1980s. The latter were tmes as poor as the populaton as a whole n ; tmes as poor as the populaton as a whole n ; and tmes as poor as the populaton as a whole n Couples (wth no dependents) were the least ntensely poor roup durn n 1980s. The latter were tmes as poor as the populaton as a whole n ; tmes as poor as the populaton as a whole n ; and tmes as poor as the populaton as a whole n

12 A poverty-ntensty ndex movements throuh tme do not necessarly mmc those of the poverty ndex on whch t s based because poverty n the entre populaton may be chann. Between and , for example, poverty for snles wth dependents rose by 21 percent (from to ) but poverty amon all ncome unts rose even faster, by 30 percent (from to ), so the ndex for snles wth dependents fell by 9 percent from to Smlarly, between and , poverty for snles wth dependents fell (from to ) but poverty amon all ncome unts fell even faster (from to ), so the ndex for snles wth dependents rose by 24 percent from to Table 2 ves head-count ratos reported by Johnson (1996b) for , and The correspondn Equatons [11], [12] and [13] are also presented. Lke ndces calculated usn, ndcates that snles wth dependents are the most ntensely poor roup and couples wth no dependents are the least ntensely poor roup n the 1980s. A closer comparson of the ndces n Tables 1 and 2, however, demonstrates that Johnson s ndex can pant a dfferent pcture of poverty ntensty than the headcount rato. For example, n poverty amon snles wth dependents was tmes as ntense as poverty n the populaton as a whole accordn to (see Table 1) and tmes as ntense as poverty n the populaton as a whole accordn to (see Table 2). Smlarly, n and , ndcates reater poverty ntensty amon snles wth dependents than does. Both sets of statstcs are ntutvely understandable by a wde 8

13 audence but rato on whch t s based, s the better measure of poverty ntensty. Lke the head-count lacks axomatcally desrable propertes. Fnally, ndces, ben ratos of poverty ndces, have the practcal advantae of ben less senstve to errors and naccuraces n measurn ncome and less senstve to arbtrary choces of poverty lnes and equvalence scales than the poverty ndces on whch they are based. 8 Ths pont s demonstrated n Table 3, whch presents more ndces of Australan poverty for and Three sets of poverty lnes, z, (100 percent, 80 percent and 120 percent of the enderson poverty lnes) and three values of the poverty averson parameter, β, (0.25, 0.50 and 0.75) were used. Beneath the poverty ndces are lsted the poverty-ntensty values, whch were computed from poverty ndces. The fnal column of the table lsts the coeffcent of varaton of poverty or poverty ntensty for each ncome unt, computed across the fve combnatons of z and β that appear n the table. In each case, the coeffcent of varaton n poverty s more than three tmes as lare as the coeffcent of varaton n poverty ntensty. Ths demonstrates that the poverty-ntensty ndex s less senstve to the choce of poverty lnes and equvalence scales than the poverty ndces on whch t s based. 4. overty Intensty n Australa, Table 4 presents values of the, and ndces, and the ndces based upon them, for the year endn June The data used n the calculatons are ncome-unt-level data from the ABS Income and ousn Survey, Australa, The data are a sample of persons aed 15 years 9

14 and older lvn n prvate dwellns (houses, flats, home unts, caravans, tents and other structures used as places of resdence). An ncome-unt s classfed as poor f ts ncome net of tax, y, s less than the real value of the enderson poverty lne for an ncome-unt of ts sze and composton, z. (The poverty lnes used n our calculatons are ven n the Appendx to ths paper.) The value of h s set equal to the number of people n the ncome unt multpled by the weht attached to the ncome unt by ABS. For ncome unts wth neatve or zero dsposable ncome, we follow Johnson s (1996b, p 166) conventon of replacn dsposable ncome by an arbtrarly low value. We nflated Johnson s value of $100 n by the C to obtan $212 per annum n In , accordn to the ndex, snles wth no dependents were the most ntensely poor roup. They were tmes as poor as the populaton as a whole. Snles wth dependents were tmes as poor as the populaton as a whole. Couples wth no dependents were tmes as poor as the populaton as a whole. The least poor roup was couples wth dependents, who were tmes as poor as the populaton as a whole. Smlar poverty-ntensty values and the same rankn of the four roups are produced usn the ndex. The rankn of the four ncome-unt types by and and ther poverty-ntensty ndces s qute dfferent to that of the prevous decade (see Tables 1 and 2). In the 1980s snles wth dependents were more ntensely poor than snles wth no dependents and couples wth dependents were more ntensely poor than couples wth no dependents. These chanes n the relatve poverty status of ncome unts wth and wthout dependents deserve further nvestaton. It would be nterestn f t 10

15 were caused by structural chane n the ncome dstrbuton or by chanes over tme n the contrbuton of overnment cash transfers to ncome unts wth, and wthout, dependents. Stanton and Fuery (1995) descrbe the chanes n payments to famles wth chldren that occurred between 1983 and The head-count rato and ts poverty-ntensty ndex ndcate that the rankn of the four ncome-unt types n s unchaned from the 1980s. Thus we see how the choce of poverty ndex matters. Accordn to wth dependents were the most ntensely poor ( tmes as poor as the, snles populaton as a whole), followed by snles wth no dependents, couples wth dependents, and couples wth no dependents ( tmes as poor as the populaton as a whole). Unlke and, fals to take account of the mantude of the poverty ap for each of the four roups. The mean, normalzed poverty ap 10, for each of the four types of ncome unts s ven n Table 4. oor snle adults wth no dependents, on averae, have dsposable ncomes that are percent below the poverty lne; poor snles wth dependents have dsposable ncomes that are percent below the poverty lne. oor couples wth no dependents, on averae, have dsposable ncomes that are percent below the poverty lne; poor couples wth dependents have dsposable ncomes that are percent below the poverty lne. The conflctn results produced by compared wth and are consstent wth earler fndns that ndces based on axomatcally sound poverty ndces tend to be hhly correlated wth one another and less hhly correlated wth the axomatcally nferor Table 4). (Roders and Roders, 1991, 11

16 5. Concluson rorams to reduce poverty requre the measurement of poverty n varous roups n socety. Ths task s most commonly accomplshed usn the head-count rato even thouh t s known to be potentally msleadn. More sophstcated poverty ndces are avalable but are seldom used prmarly because ther values are dffcult to nterpret and complex to compute. Ths paper advocates the use of poverty-ntensty () ndces, whch are smple to nterpret and have the same propertes as the poverty ndces upon whch they are based. In the area of polcy desn and evaluaton, we beleve that poverty-ntensty ndces are a useful adjunct that ves ntutve meann to the results of emprcal studes of poverty. To demonstrate the use of ndces we have transformed measures of poverty n Australa n the 1980s nto measures and nterpreted the results. For example, snle adults wth dependents were the poorest roup, ben more than three tmes as poor as the populaton as a whole n The next poorest roup was snles wth no dependents, followed by couples wth the dependents. Couples wth no dependents were the least poorest roup, ben about half as poor as the populaton as a whole. We also performed some new computatons of poverty and poverty ntensty usn Australan unt-record data for An nterestn reversal of poverty-ntensty rankns between and was revealed by ndces based on the axomatcally sound poverty ndces of Johnson and Dxon (1999) and Foster, Greer and Thorbecke (1991). 12

17 By snle adults wth no dependents were the most ntensely poor roup. They were more than twce as poor as the populaton as a whole. Snle adults wth dependents were found to be approxmately 1.7 tmes as poor as the populaton as a whole and couples wth no dependents were approxmately 0.75 tmes as poor as the populaton as a whole. Couples wth dependents were the least ntensely poor roup. They were less than 40 percent as poor as the populaton as a whole. The observed chane n rankn of the four roups between and s not revealed by the head-count rato and ts poverty-ntensty ndex. Thus, the choce of poverty ndex matters. The avalablty of poverty-ntensty ndces that can be based upon sophstcated measures of poverty, yet have ntutve meann, should contrbute to effectve poverty measurement and polcy based upon t. 13

18 References Creedy, John. (1998) "Measurn overty: An Introducton." Australan Economc Revew, 31 (1): Foster, James, Joel Greer and Erc Thorbecke. (1984) A Class of Decomposable overty Indces. Econometrca, 52 (3): Johnson, Davd T. (1996a) "overty Lnes and the Measurement of overty." Australan Economc Revew, 29 (1): Johnson, Davd T. (1996b) overty, Inequalty and Socal Welfare n Australa. edelber: hysca-verla. Johnson, Davd T. (1991) The Measurement and Extent of overty Amon Immrants. Canberra: AGS. Johnson, Davd T. (1988) The Measurement of overty n Australa: and Australan Economc Revew, 22 (3): Johnson, Davd. (1987) The Calculaton and Use of overty Lnes n Australa. Australan Economc Revew, 21 (4): Johnson, Davd T. and eter B. Dxon. (1999) Australan overty Quantfed by a Famly-Based overty Index. Economc Record, 75(229): Kakwan, N. (1986) Analysn Redstrbuton olces: A Study Usn Australan Data. Cambrde: Cambrde Unversty ress. Kakwan, N. (1980) On a Class of overty Measures. Econometrca, 48 (2): Roders, John L. and Joan R. Roders. (1991) Measurn the Intensty of overty Amon Subpopulatons. The Journal of uman Resources, 26 (2):

19 Saunders, eter, and eter Whteford. (1989) Measurn overty: A Revew of the Issues. EAC Dscusson aper No. 89/11. Canberra: AGS. Sen, Amartya. (1976) overty: An Ordnal Approach to Measurement. Econometrca, 44 (2): Stanton, Davd, and Mchael Fuery. (1995) Developments n Famly ayments snce Socal Securty Journal, December: Takayama, Noryuk. (1979) overty, Income Inequalty and Ther Measures: rofessor Sen s Axomatc Approach Reconsdered. Econometrca, 47 (3):

20 erod Table 1: overty and overty-intensty Indces, Based on Johnson's Deprvaton Wehted overty Index, Australa, , , Couples Couples + dependents Income Unt Type Snles Snles + dependents All ncome unts OV n /n OV n /n n.a. n.a. n.a. n.a OV n /n Source: Johnson (1996b, Table 6.1, p.67). poverty ndces for and also appear n Johnson and Dxon (1999, Table 1). 16

21 erod Table 2: overty and overty-intensty Indces, Based on the ead-count Rato, Australa, , , Couples Couples + dependents Income Unt Type Snles Snles + dependents All ncome unts OV n /n OV n /n n.a n.a. n.a. n.a OV n /n Source: Johnson (1996b, Table 6.1, p.67). 17

22 Table 3: overty and overty-intensty Indces, Based on Johnson and Dxon's Index, Australa, 1982 and 1990 Senstvty to overty Lne and Dstrbuton Senstvty arameter overty Lne z 0.8z 1.2z z z Dstrbuton Senstvty arameter Coeff of Var overty Index, Couples Couples + dependents Snles Snles + dependents All ncome unts overty Index, Couples Couples + dependents Snles Snles + dependents All ncome unts overty-intensty Index, Couples Couples + dependents Snles Snles + dependents All ncome unts overty-intensty Index, Couples Couples + dependents Snles Snles + dependents All ncome unts Source: Johnson and Dxon (1999, Table 2). 18

23 erod Table 4: overty and overty-intensty Indces, Australa, Couples Couples + dependents Income Unt Type Snles Snles + dependents All ncome unts Based on Index wth β= OV Based on wth α = OV Based on head-count rato OV Mean Normalzed overty Gap GA roporton n the opulaton n /n Source: Computed by the authors usn data from the Income and ousn Cost Survey, Australa, (IDS96). 19

24 Type of Income Unt (1) Appendx Equvalence Scale Costs) ead n Workforce (2) (All ead not n Workforce (3) overty Lne ($ per annum) ead n Workforce (4) ead not n Workforce (5) Couple , , Couple plus , , Couple plus , , Couple plus , , Couple plus , , Couple plus , , Couple plus , , Couple plus , , Couple plus , , Couple plus , , Couple plus , , Snle person , , Snle arent plus , , Snle arent plus , , Snle arent plus , , Snle arent plus , , Snle arent plus , , Snle arent plus , , Snle arent plus , , Snle arent plus , , Snle arent plus , , Snle arent plus , , Note: overty lnes were calculated usn a benchmark ncome of $62.70 per week for a couple plus two dependents that apples to the September quarter, The benchmark ncome was nflated usn the consumer prce ndex to obtan ts equvalent real weekly ncome of $ n September, 1995; $ n December, 1995; $ n March, 1996; $ n June, These weekly ncomes areate to an annual total of $17, for overty lnes for other famly types (see Columns 4 and 5) are derved by multplyn $17, by the value assned to that famly type n the equvalence scales of Johnson (1987, Table 1), whch are reproduced n Columns 2 and 3. 20

25 Endnotes 1. See Johnson (1996a) for a dscusson of some of the ssues nvolved, partcularly n the Australan context. 2. See Roders and Roders (1991), Creedy (1998) and Johnson and Dxon (1999) for dscussons of the desrable propertes of a poverty ndex and for comparsons of the propertes of several of the better known poverty ndces. 3. Obvously a populaton, ten percent of whom have zero ncomes, s poorer than a populaton, ten percent of whom have ncomes one dollar below the poverty lne, ceters parbus. The head-count rato, however, makes no dstncton between the poverty of two such populatons. 4. The term poverty ntensty has been used (for example, Creedy, 1998) to refer to the extent to whch the ncome of a ven ncome unt falls below the poverty lne. We use the term poverty ntensty n a dfferent way. We are concerned wth the poverty of a roup of ncome unts rather than wth the poverty of an ndvdual ncome unt. 5. The role of β s revealed by an examnaton of the poverty of the th ncome unt, n whch z s constant. Frst, y = 0 mples y = 1 -, Ł z ł equals one, and y = z mples equals zero, for all 0 < β 1. For a ven y n the rane 0 < y < z, the closer s β to one, the larer s the poverty of the th ncome unt. Second, 0 < β 1 mples ncreases as y decreases. When β = 1 the rate of chane of wth respect to ncome s constant. When 0 < β < 1, ncreases at an ncreasn rate as y decreases. The closer s β to zero, the more convex s the poverty-ncome profle of the th ncome unt. That s, at small ncome levels, the smaller s β, the larer s the ncrease n resultn from a one-unt reducton n ncome. Conversely, at ncome levels close to the poverty lne, the smaller s β, the smaller s the ncrease n resultn from a one-unt reducton n ncome. See Johnson (1996b, p.44) for further dscusson of the poverty-averson parameter, β. 6. The role of α s revealed by an examnaton of the poverty of the th ncome unt, y = 1 -, Ł z ł n whch z s constant. Frst, y = 0 mples equals one, and y = z mples equals zero, for all α 2. For a ven y n the rane 0 < y < z, the larer s α, the smaller s the poverty of the th ncome unt. Second, α 2 mples ncreases as y decreases and ncreases at an ncreasn rate as y decreases. The larer s α, the more convex s the poverty-ncome profle of the th ncome unt. That s, at small ncome levels, the larer s α, the larer s the ncrease n resultn from a one-unt reducton n ncome. Conversely, at ncome levels close to the poverty lne, the larer s α, the smaller s the ncrease n resultn from a one-unt reducton n ncome. 7. To put these numbers n context, poverty-ntensty measures as hh as three are seldom found n the Unted States. In 1979, for example, Blacks were tmes as poor, and Blacks n rural farm areas were tmes as poor, as the U.S. populaton as a whole (Roders and Roders, 1991, Table 5). 8. Saunders and Whteford (1989, p. 33) contend that the head-count rato s preferred to the theoretcally superor poverty ndces because the latter are much more senstve to errors and naccuraces n the ncome data themselves. b a 21

26 9. Whle we were surprsed by the reversal we were astonshed by ts mantude. We note n passn that 46 percent of poor couples have a reference person 65 years or older and 55 percent of poor snles are 24 years or youner. Fnally, could the reversal be caused by an arbtrary chane, whch we have been unable to dentfy, n the way the data were collected? 10. The mean, normalzed poverty ap for roup s m = h 1- where m s the number of poor people n roup. 1 m M Ł y z ł 22