On Nonstable and Stable Population Momentum

Transcription

1 Demography DOI 1.17/s y On Nonsable and Sable Populaion Momenum Thomas J. Espenshade & Analia S. Olgiai & Simon A. Levin # Populaion Associaion of America 211 Absrac This aricle decomposes oal populaion momenum ino wo consiuen and muliplicaive pars: nonsable momenum and sable momenum. Nonsable momenum depends on deviaions beween a populaion s curren age disribuion and is implied sable age disribuion. Sable momenum is a funcion of deviaions beween a populaion s implied sable and saionary age disribuions. In general, he facorizaion of oal momenum ino he produc of nonsable and sable momenum is a very good approximaion. The facorizaion is exac, however, when he curren age disribuion is sable or when observed feriliy is already a replacemen. We provide numerical illusraions by calculaing nonsable, sable, and oal momenum for 176 counries, he world, and is major regions. In shor, he aricle brings ogeher disparae srands of he populaion momenum lieraure and shows how he various kinds of momenum fi ogeher ino a single unifying framework. Keywords Populaion momenum. Age disribuion. Decomposiion Inroducion The concep of populaion momenum refers o he fac ha a populaion ypically does no sop growing (or declining) he insan is feriliy reaches replacemen. Insead, in a closed populaion, growh or decline gradually slows unil a saionary populaion is aained, in much he same way ha a car gradually comes o a complee sop afer a driver s foo is lifed from he acceleraor pedal (Schoen and Kim 1991). The relaive T. J. Espenshade (*) Deparmen of Sociology and Office of Populaion Research, 249 Wallace Hall, Princeon Universiy, Princeon, NJ 8544, USA je@princeon.edu A. S. Olgiai Woodrow Wilson School of Public and Inernaional Affairs and Office of Populaion Research, Princeon Universiy, Princeon, NJ, USA S. A. Levin Deparmen of Ecology and Evoluionary Biology, Princeon Universiy, Princeon, NJ, USA

2 T.J. Espenshade e al. amoun of momenum is usually measured by he raio of he size of he long-run saionary populaion o ha of he populaion when replacemen feriliy is firs achieved. Momenum coefficiens for individual counries in 1935 ranged from 1.8 for Ausria, Belgium, and France o 1.6 for Puero Rico and Honduras (Vincen 1945). By 25, momenum coefficiens across all Unied Naions counries sreched from.81 in Bulgaria o 1.76 in Oman (auhors calculaions). In several cases, including Georgia, he Neherlands, and Poland, oal momenum was essenially zero. These examples illusrae ha populaion momenum can be posiive (coefficien greaer han 1) or negaive (coefficien less han 1), and ha i can play a large or small role in populaion dynamics (Knodel 1999; Preson and Guillo 1997). One reason for furher sudy of populaion momenum is ha i conribues imporanly o fuure populaion growh in developing counries. Bongaars (1994, 1999) has esimaed ha momenum accouns for nearly one-half of projeced fuure growh in he developing world over he nex cenury. For he period beween 2 and 21, momenum is he mos imporan facor in projeced fuure growh for he world and all major regions excep Europe and sub-saharan Africa. For every region in he developing world excep sub-saharan Africa, momenum is a more imporan conribuor o fuure populaion growh han all oher facors combined (Bongaars and Bulaao 1999). During he nex half-cenury, momenum is projeced o accoun for 58% of fuure populaion growh in he developing world (Bongaars 27). In addiion, one should no overlook he conribuion of negaive momenum o populaion decline in he developed world. Even wih an immediae feriliy rebound back o replacemen, Europe s populaion (ignoring migraion) is projeced o fall by 7% from curren numbers before leveling off. And, in addiion o Bulgaria, 11 oher counries (all of which, excep Japan, are in Europe) are projeced o decline by more han 1% owing o negaive momenum, even if feriliy were o rise insananeously and permanenly o replacemen. The purpose of his aricle is o convey a deeper undersanding of how age composiion conribues o long-erm populaion size. We decompose oal populaion momenum ino wo consiuen and muliplicaive pars: nonsable momenum and sable momenum. Nonsable momenum capures deviaions beween a populaion s curren age disribuion and he sable age disribuion implied by curren feriliy and moraliy. Sable momenum reflecs deviaions beween he sable age disribuion and he saionary age disribuion produced by replacemen feriliy. In addiion, by showing how oal, nonsable, and sable momenum fi ogeher ino a single heoreical and empirical framework, he aricle inegraes a number of disparae and seemingly unrelaed conceps in he populaion momenum lieraure. 1 Toal, Nonsable, and Sable Momenum The sylized ingrediens of a unified framework are shown in Fig. 1. Le P be he size of an arbirary iniial populaion arbirary wih respec o size, age srucure, 1 Previous work o decompose oal momenum has emphasized is age-specific componens (Preson 1986; Schoen and Kim 1991). Guillo (25) decomposed oal momenum ino wo muliplicaive facors: (1) he direc effec of improvemens in cohor survivorship and (2) flucuaions in annual numbers of birhs.

3 On Nonsable and Sable Populaion Momenum Fig. 1 Framework for undersanding oal, nonsable, and sable populaion momenum Populaion Size S 2 S 1 Q P P : Observed populaion size Q : Sable equivalen populaion size S 1, S 2 : Saionary populaion sizes R > 1 R = 1 Time feriliy, and moraliy. Assume only ha he populaion is closed o migraion and consiss enirely of females, some women are younger han he oldes age of childbearing, and feriliy and moraliy are bounded by he range of conemporary human experience. For he sake of illusraion, feriliy in Fig. 1 is above replacemen. Suppose ha birh raes are lowered insananeously o replacemen a ime = by dividing he observed feriliy schedule by he ne reproducion rae (R ) and ha feriliy and moraliy are hen held consan. As shown by he lower solid line, he populaion evenually converges o a saionary populaion wih size S 1.TheraioS 1 / P is he usual measure of oal populaion momenum. Suppose insead ha he observed populaion P is projeced holding curren feriliy and moraliy consan. I will ulimaely converge o a sable populaion as indicaed by he rajecory of he lower dashed line in Fig. 1. Once a sable sae has been aained, imagine using he sable growh rae o reverse projec he size of he sable populaion back o =. The reverse projecion follows an exponenial curve represened by he upper dashed line. The new populaion is he sable equivalen populaion. I has size Q, and is age disribuion is he sable age disribuion implied by he indefinie coninuaion of curren feriliy and moraliy. I is asympoically equivalen o he observed populaion in he sense ha if boh are projeced forward from = holding curren feriliy and moraliy consan, hey will evenually become indisinguishable wih respec o populaion size and age composiion. We measure nonsable momenum wih he raio Q / P. Finally, consider a projecion of he sable equivalen populaion using he same consan replacemen-level feriliy and moraliy used o produce he lower solid line in Fig. 1. The size of he populaion begins a Q and follows he upper solid curve before leveling off a a saionary populaion of size S 2. Sable momenum is measured by he raio S 2 / Q. The saionary populaions represened by he endpoins of he wo solid lines have he same proporionae age disribuions, bu hey do no necessarily have he same size. We may wrie oal momenum as he ideniy Toal Momenum ¼ S 1 P Q P S 2 Q S 1 S 2 :

4 T.J. Espenshade e al. In words, oal momenum is he produc of nonsable momenum, sable momenum, and an offse facor represened by he raio S 1 / S 2. If, however, S 1 = S 2,hen S 1 P ¼ Q P S 2 ð1þ Q _ and we would have accomplished an exac facorizaion of oal populaion momenum ino he produc of nonsable and sable momenum. Moreover, under hese condiions, he observed populaion P and is sable equivalen Q are asympoically equivalen no only wih respec o curren feriliy and moraliy bu also wih respec o replacemen feriliy and moraliy. Our ineres will cener on he relaion beween S 1 and S 2 and he condiions under which hey migh be equal. I is imporan firs o explain how he hree ypes of momenum are relaed o deviaions beween pairs of age disribuions. Toal Momenum The concep of he sable equivalen populaion plays a cenral and unifying role in our analysis. If an arbirarily chosen populaion a = has size P wih feriliy schedule m(a) and survival funcion p(a), hen he sable equivalen populaion is a sable populaion whose age disribuion and vial raes are deermined by m(a)andp(a). The size of he sable equivalen populaion a = is given by Q = n(x)v(x) dx, ð2þ b A r where n(x)dx is he number of females beween exac ages x and x + dx; b and A r are, respecively, he birh rae and he mean age of childbearing in he sable equivalen populaion; and " is he oldes age of childbearing. In Eq. 2, v(x) is Fisher s reproducive value funcion (Fisher 193:27 3) defined for a woman a exac age x as v(x) = 1 p(x) x e ra ( x) p(a)m(a) da. One may inerpre v(x) as he presen discouned value (using he sable growh rae r as he discoun rae) of he average expeced number of daughers remaining o be born per woman a age x. When he observed populaion and he sable equivalen populaion are projeced from = holding consan boh m(a) and p(a), hey will evenually converge and become indisinguishable. In Fig. 1, when he observed populaion P is projeced wih feriliy se a replacemen, i converges o a saionary populaion whose size is S 1. In oher words, S 1 is he size of he sable/saionary populaion ha is equivalen o P wih respec o replacemen feriliy. We can express his formally, using Eqs. 2 and 3, as S 1 = n(x) p(x) x ð3þ p(a)m (a) da dx /( b A ). ð4þ The subscrip is used o indicae replacemen values in he saionary populaion, where r = by definiion. Assume ha m (a) is obained by normalizing he feriliy schedule m(a) wih he ne reproducion rae (R ). We may rewrie n(x) asp c(x),

5 On Nonsable and Sable Populaion Momenum where c(x) is he observed proporionae age disribuion. And because he saionary populaion age disribuion c (x)=b p(x), Eq. 4 becomes S 1 = Pc(x) c (x) x p(a)m (a) dadx / A, from which i follows ha c(x) Toal Momenum = S 1 P = p(a)m (a)da dx / ð5þ c (x) An idenical expression for oal momenum appears in Preson and Guillo (1997:2 21), and i was anicipaed in somewha differen form by Keyfiz (1985: ). 2 Equaion 5 shows ha he oal momenum conained in a populaion s age srucure depends on he raio c(x) /c (x), which reflecs deviaions beween he proporionae curren and saionary age disribuions below he oldes age of childbearing. In paricular, oal momenum is a weighed average of hese deviaions, where he agespecific weigh is x p(a)m (a)da / A (Preson and Guillo 1997). These weighs sum o uniy, as can be seen by reversing he order of inegraion in he double inegral. Moreover, because he weighs are larges prior o he onse of childbearing and hen decline oward zero (Preson and Guillo 1997), deviaions beween c(x) andc (x) in he early par of life maer mos in deermining oal momenum. Finally, if he observed populaion is already saionary so ha here is no difference beween c(x) and c (x), hen all momenum has been wrung ou of he age disribuion and he oal momenum coefficien equals 1. (Preson and Guillo 1997). 3 Nonsable Momenum I may no be clear from Fig. 1 why he raio Q / P is a measure of populaion momenum, especially because mos discussions of momenum occur in he conex of replacemen feriliy. Suppose we separae he observed populaion growh rae a ime, r(), ino wo pars as r() = r +ρ(), where r ishesablegrowhraeproducedbym(a) andp(a), and ρ() is he residual. Then r reflecs he conribuion of feriliy and moraliy o r(), and ρ() measures he conribuion of he age disribuion a ime (Espenshade 1975). x A. 2 Vincen (1945) described he phenomenon of populaion momenum (wha he called he poenial increase of a populaion ) and inroduced a mehod o measure i. He also developed he heory behind he sable equivalen populaion and independenly invened Fisher s reproducive value wihou calling hem such. This work was laer exended and renamed by Keyfiz (1969, 1971). Anoher pah of developmen in he momenum lieraure examines he growh consequences of a gradual decline in feriliy o replacemen. Pioneering empirical research was underaken by Frejka (1973). Seminal work by Li and Tuljapurkar (1999, 2) has sparked a new line of formal analysis. 3 Oher work has approximaed oal momenum by comparing observed and saionary populaion age disribuions. Kim and Schoen (1993) expressed momenum as he raio of he proporion in he observed populaion o he proporion in he saionary populaion a a given age deermined by he crossover poin of heir respecive reproducive value funcions. Momenum has also been approximaed by he raio of he proporion under age 3 in he observed populaion o he proporion under age 3 in he saionary populaion (Kim and Schoen 1997; Kim e al. 1991).

6 T.J. Espenshade e al. Espenshade and Campbell (1977) showed ha Q T ρ d = lim e () ð6þ P Τ { }. The limi in Eq. 6 exiss because ρ() is he ransien par of r(), and ρ() ashe observed populaion converges o a sable populaion. Equaion 6 ells us how large P would evenually become if age composiion were he only source of fuure populaion growh or decline. In oher words, he raio Q / P is he long-erm relaive change in populaion size owing o nonsable momenum if curren feriliy and moraliy are held consan unil a sable populaion is aained. We call i nonsable momenum because is value depends on a flucuaing non-equilibrium age disribuion. Nonsable momenum can be expressed in a form similar o Eq. 5. To find he size of he sable equivalen populaion, Q, ha corresponds wih P, use Eqs. 2 and 3 in conjuncion wih m(a), p(a), and he observed age disribuion n(x). I follows ha Q = P n(x) P 1 be rx p(x) x e ra p(a)m(a) da dx / A r. By wriing n(x) / P as c(x) and recognizing ha he proporionae sable age disribuion c r (x) =be rx p(x), Eq. 7 becomes ð7þ Nonsable Momenum = Q P = c(x) c (x) r e ra x p(a)m(a) da dx / A r. ð8þ Equaion 8 says ha nonsable momenum is a funcion of deviaions beween he curren and implied sable proporionae age disribuions below he oldes age of childbearing. If he observed populaion is sable so ha here are no deviaions, hen c(x) =c r (x) a all ages, here is no nonsable momenum, and Q / P = 1 as can be verified by reversing he order of inegraion in Eq. 8. The same process shows ha nonsable momenum is a weighed average of dispariies beween c(x) andc r (x), where he weighs are e ra p(a)m(a)da / A x r and sum o 1. Equaion 8 gives equal weigh o deviaions up o he age when childbearing begins and monoonically declining weigh hereafer unil he weigh becomes zero a he highes age of childbearing. The raio Q / P appears a oher places in he momenum lieraure. Bourgeois-Picha (1971) inroduced he concep of he ineria of a populaion and developed he coefficien of ineria (Q / P) o measure i. He wen on o sugges a furher decomposiion of nonsable momenum ino he effecs of feriliy and moraliy on he one hand and hose of age disribuion on he oher. In reformulaing he concep of he sable equivalen populaion, Keyfiz (1969) calculaed Q / P for several empirical examples wihou relaing i o populaion momenum. The closes he came is referring o Q as a simple measure of he favorabiliy of he age disribuion o reproducion, given he curren regime of moraliy and feriliy (Keyfiz 1969:264). Schoen and Kim (1991:456) argued, The momenum concep need no be limied o cases in which he ulimae sable populaion has zero growh. More generally, momenum can be defined as he size of a populaion relaive o he size of is sable equivalen. Finally, Feeney (23:648) claimed, The momenum of he given age disribuion wih

7 On Nonsable and Sable Populaion Momenum respec o he given age schedules of feriliy and moraliy is he raio [Q / P] (firs emphasis in he original; second emphasis added). Sable Momenum The las kind of momenum in Fig. 1 is sable momenum, defined by he raio S 2 / Q. Because P and is proporionae age disribuion, c(x), can be chosen arbirarily, le he iniial observed populaion be sable wih size Q and age disribuion c r (x)=be rx p(x). Then se feriliy a replacemen and projec unil he populaion becomes saionary wih growh rae r = and fixed size S 2. The formula in Eq. 5 for oal momenum may be invoked and applied in his new siuaion o yield an expression for sable momenum as c r (x) Sable Momenum = S 2 Q = p(a)m (a) da dx / ð9þ c (x) We call i sable momenum because, jus like oher sable populaion conceps, i depends only on feriliy and moraliy and no on he curren age disribuion. Once again, he amoun of momenum depends on deviaions beween pairs of proporionae age disribuions. In he case of sable momenum, wha maers are he deviaions beween he sable and saionary age disribuions, weighed by he same age-specific weighs used in Eq. 5. These weighs are consan prior o he onse of childbearing and hen decline seadily o zero by age ". If feriliy is already a replacemen in he observed populaion, hen he sable age disribuion will be saionary, c r (x)=c (x) a all ages, S 2 / Q = 1, and here will be no sable momenum. Sable momenum is idenical o wha we migh call Keyfiz momenum. Keyfiz (1971) considered he long-erm size of an iniially sable populaion if feriliy raes are se immediaely o replacemen by normalizing he feriliy schedule, m(a), wih he ne reproducion rae (R ). To see he equivalence beween Eq. 9 and he Keyfiz formula, rewrie Eq. 9 as S 2 Q = be rx p(x) b p(x) x x p(a)m(a) da dx / ( A R ), A. ð1þ where b is he birh rae in he iniial sable populaion; b is he birh rae in he ulimae saionary populaion and equal o he reciprocal of life expecancy a birh. Simplifying Eq. 1 and reversing he order of inegraion, we have e o o S 2 Q = be A R a e rx p(a)m(a) dx da. ð11þ Bu because Z a Z a e rx dx ¼ 1 r e rx ð rþdx ¼ 1 r ð1 e ra Þ _ Eq. 11 becomes S 2 Q ¼ beo ðr 1Þ ra R _ which is exacly he formula in Keyfiz (1971:76) for momenum in a sable populaion.

8 T.J. Espenshade e al. In relaed work, Kim and Schoen (1993) derived momenum in an iniially sable populaion as a simple raio of proporions in he sable and saionary age disribuions a an age deermined by he crossover poin of reproducive value funcions. A furher decomposiion of momenum in a sable populaion is given in Kim and Schoen (1997). Much of he work on momenum in populaions wih gradually declining feriliy assumes iniial sabiliy. Schoen and Jonsson (23) decomposed his momenum ino a par ha reflecs he effec of growh coninuing a he original sable rae for half he period of decline and an offseing facor ha reduces momenum and he number of birhs in he long-run saionary populaion because of populaion aging (if feriliy is falling). Finally, Goldsein and Secklov (22) developed a simple analyic formula for esimaing populaion momenum in a sable populaion when feriliy declines gradually and linearly o replacemen. Their analysis suggess ha differences beween an observed populaion and is sable equivalen are sufficienly small ha hey can be ignored (Goldsein and Secklov 22: ). In oher words, heir analysis assumes ha nonsable momenum is inconsequenial o an undersanding of oal momenum and ha only sable momenum maers for all pracical purposes. Examples To add empirical conen o our analysis and anicipae laer resuls, we show projecions of he female populaions for wo counries in Fig. 2. The op porion illusraes he case of Indonesia, a populaion wih above-replacemen feriliy. Females numbered million in 25, and he size of he sable equivalen populaion is million. Fig. 2 Projecions of he female populaions of Indonesia and Japan, Populaion Size (in millions) Indonesia R = 1.9 R = 1 S 1 S Q P Q Japan 65.4 P R = 1 S 1 S 2 R =

9 On Nonsable and Sable Populaion Momenum When boh populaions are projeced assuming replacemen feriliy, i is no possible o deec any visual disance beween S 1 and S 2. They are boh very close o 15 million. A similar siuaion arises in he case of Japanese women shown in he boom porion of Fig. 2. Here he example is chosen o reflec feriliy below replacemen. Bu once again, i appears ha he observed populaion and is sable equivalen are asympoically equivalen wih respec o boh curren and replacemen feriliy. In his example, S 1 and S 2 are approximaely 57.5 million. For Indonesian and Japanese women in 25, he daa in Fig. 2 sugges ha oal momenum is exacly he produc of nonsable and sable momenum, or very nearly so. In he nex secion we invesigae analyically wheher S 1 = S 2 in all siuaions. A Unified Analyic Framework Because populaion projecions usually rely on discree formulaions of age and ime, i is convenien o develop an analyic soluion using marix algebra. Relevan inroducions are conained in Finkbeiner (196:23 43) and Keyfiz (1968:27 73). The General Case Le {P} be an n1 vecor for he age disribuion of an arbirarily chosen populaion a ime =. The elemens in he vecor represen he number of females in successive discree (say, five-year) age inervals. Suppose ha curren feriliy and moraliy are capured by a sandard nn Leslie marix L, which is assumed o be primiive. Le he Leslie marix L reflec replacemen feriliy, and assume ha L is obained from L by dividing feriliy raes by he ne reproducion rae. Then we may wrie fpg ¼ fqgþ fvg _ ð12þ where {Q} is he age disribuion of he sable equivalen populaion and {V} is he deviaion of {P} from{q}. When Eq. 12 is projeced forward infiniely wih L, he par associaed wih {V} is dominaed by he oher erm and hence can be negleced. Nex consider he infinie projecion of Eq. 12 using replacemen feriliy: { } { } { }. lim L P = lim L Q + lim L V ð13þ Bu because lim L P is he saionary populaion vecor {S 1 } and lim L Q is he saionary populaion vecor {S 2 } wih he same proporionae age disribuion as {S 1 }, we have from Eq. 13 he quaniy of ineres, which is { } { S } { S } =lim 1 2 { } { }. ð14þ Now change bases and rewrie {V} as a linear combinaion of he eigenvecors {Z j }ofl, specifically as fvg ¼ k 1 fz 1 gþ k 2 fz 2 gþþk n fz n g _

10 T.J. Espenshade e al. or more compacly as fvg ¼ k 1 fz 1 gþ fwg: ð15þ In Eq. 15, k 1 {Z 1 } is he saionary par of {V}. In paricular, k 1 {Z 1 } is he saionary equivalen populaion, where k 1 is he saionary populaion size and {Z 1 } is he proporionae saionary age disribuion. {Z 1 } is also he principal eigenvecor of L associaed wih he principal or dominan eigenvalue 1 1 =1. Finally, consider he infinie projecion of Eq. 15 using L, whereupon { } { } { }. lim L V = lim L k Z + lim L W ð16þ 1 1 The second erm on he righ-hand side of Eq. 16 can be ignored in he limi because {V} and k 1 {Z 1 } are asympoically equivalen under L.Moreover,L k 1 {Z 1 }=k 1 {Z 1 }. Tha his is rue is immediaely obvious from demography, because projecing an already saionary populaion using replacemen-level feriliy simply reproduces he original populaion. In addiion, however, because {Z 1 } is an eigenvecor of L, { }= { }= { }= { }. L { V}= k { Z }+ { } L k Z k L Z k λ Z k Z I follows from Eq. 16 ha lim 1 1 and, herefore, from Eq. 14, ha fs 1 g fs 2 g ¼ k 1 fz 1 g: ð17þ Because k 1 in general, i follows ha, in general, {S 1 } {S 2 }. We conclude ha when an observed populaion and is sable equivalen are projeced on he assumpion of replacemen-level feriliy, he saionary populaions o which hey converge are usually differen. In oher words, he facorizaion of oal populaion momenum ino he produc of nonsable and sable momenum ypically is no exac, bu only approximae. Special Cases There are, however, hree special cases in which he facorizaion is exac. Firs, suppose he observed age disribuion is sable. Then S 1 = S 2, as can be verified by subsiuing c(x)=c r (x) ino he main momenum formulas 5, 8, and 9. The leasdeveloped counries are mos likely o be described by his condiion. Second, suppose curren feriliy is a replacemen. To see ha he facorizaion is exac, subsiue r =,m(a) =m (a), and c r (x) =c (x) ino Eqs. 5, 8, and 9. Feriliy a or near replacemen is more likely o characerize developed counries han developing ones. Third, here is he degenerae case. If feriliy is a replacemen and he age disribuion is saionary, here is no momenum of any kind neiher oal, nor sable, nor nonsable. Each of he raios in Eq. 1 is 1.. Evaluaing he Approximaion Our analysis has shown ha apar from a few special cases, he sizes of he ulimae saionary populaions in Fig. 1, S 1 and S 2, are generally no idenical. Bu are hey

11 On Nonsable and Sable Populaion Momenum close in pracice? The beer is he agreemen beween S 1 and S 2, he more nearly oal momenum facors ino he produc of sable and nonsable momenum. In oher words, if S 1 and S 2 are very close, he decomposiion gives us a new way of hinking abou populaion momenum and undersanding differences among counries and regions of he world. We invesigae he relaionship beween S 1 and S 2 for each of 176 Unied Naions counries. Projecions are made using a sandard cohor-componen mehodology applied o recen Unied Naions (27) daa. The baseline populaion comes from an esimae for July 1, 25, for females, disaggregaed by five-year age groups from 4 up o 1 years and older. Feriliy raes and sex raios a birh are based on esimaes for he period 2 25, and projecions ha assume replacemen feriliy are consruced by dividing age-specific feriliy raes by he ne reproducion rae. Esimaes of deah raes by five-year age groups up o 1 years and older come from he World Healh Organizaion (28). All projecions are carried ou for 3 years assuming a closed populaion. 4 We graph on a logarihmic scale in Fig. 3 values of S 1 and S 2 for he female populaions of 176 U.N. counries. Values of S 1 range from 73 million for Tonga and 77 million for Grenada o 736 million for China and 777 million for India. Bu he mos sriking feaure of Fig. 3 is ha all he poins appear o lie on he 45-degree line. If S 1 and S 2 are no equal o each oher, hen he deviaion beween hem is very small. The simple correlaion coefficien beween S 1 and S 2 is The correlaion is idenical o four decimal places when populaion size is measured on a logarihmic scale in base 1. The daa in Fig. 3 reinforce an imporan conclusion: when he facorizaion of oal momenum ino he produc of sable and nonsable momenum is no exac, he approximaion is exremely good. Figure 4 examines he relaionship beween S 1 and S 2 in anoher way. Here we show he disribuion of he percenage deviaion of S 2 from S 1 for he same 176 counries. Mos of he deviaions cluser in a igh paern around zero, and only a small number fall ouside he range of ±.5%. Roughly wo ou of every five cases (39.2%) fall wihin.1% of he origin. In wo-hirds of he cases (64.8%), he deviaions are conained wihin.2%. And in hree-fourhs of all cases (74.4%), he relaive difference beween S 1 and S 2 lies wihin.3%. 5 The daa in Figs. 3 and 4 poin o one overarching conclusion. When S 1 and S 2 are compared, only one of wo oucomes is possible. Eiher S 1 = S 2 or S 1 S 2. This means ha oal momenum is eiher idenically equal o he produc of nonsable and sable momenum or very close o i. And we now know why his is he case: because for many of he world s counries, eiher curren feriliy is close o replacemen or he age disribuion is nearly sable. I is only in insances of a join deparure from replacemen feriliy and age disribuion sabiliy ha he exac facorizaion begins o dissolve ino an approximaion. 4 Nineeen U.N. counries ha are no WHO members have been excluded from he analysis. They range in size from Aruba (oal populaion of 13,, including men and women) o Hong Kong (wih a populaion of 7.1 million). The average populaion size of he excluded counries is approximaely 1.1 million oal persons. 5 The percenage deviaions in Fig. 4, including all 176 counries, have a sandard deviaion of.331, a mean value of.33, and a median value of.47.

12 T.J. Espenshade e al. Fig. 3 Relaionship beween S 1 and S 2 (N = 176 counries) 1B 1M S 2 1M 1M 1K 1K 1K 1M 1M 1M 1B S 1 Momenum in Global Perspecive We calculae values for oal, nonsable, and sable momenum for each of 176 Unied Naions counries, broad regional aggregaes, and he world. Resuls for he world and is major regions are repored in Table 1. Noice firs ha here is excellen agreemen beween he numbers in columns 1 and 4. Any differences are usually limied o he hird decimal place, which suggess ha he produc of nonsable and sable momenum is an unusually good approximaion o oal momenum. Moreover, our esimaes indicae ha world populaion would grow by an addiional Fig. 4 Disribuion of percenage deviaions beween S 1 and S 2 (N = 176 counries). Deviaions are calculaed as [(S 2 S 1 )/S 1 ] 1. Five counries fall ouside he inerval (.95,.95), ranging from Russia ( 1.14) o Erirea (1.168) 3 2 % Percenage Deviaion

13 On Nonsable and Sable Populaion Momenum Table 1 Toal, nonsable, and sable momenum for he world and major regions, 25 Region Toal Momenum Nonsable Momenum Sable Momenum Nonsable Sable World More-developed regions Less-developed regions Leas-developed counries Less-developed regions, excluding leas-developed counries Less-developed regions, excluding China Sub-Saharan Africa Africa Asia Europe Lain America and he Caribbean Norhern America Oceania Source: Auhors calculaions. 4% if global feriliy raes had moved insananeously o replacemen in 25. Nonsable and sable momenum conribue roughly equal shares o world populaion momenum. Taking naural logarihms shows ha nonsable momenum accouns for abou 53% of oal world momenum, and sable momenum conribues roughly 47%. Europe and he leas-developed counries represen wo exremes on he global momenum scale. The leas-developed counries possess he larges values for oal and sable momenum in Table 1 and have one of he lowes values for nonsable momenum. On he conrary, Europe has he lowes values for oal and sable momenum bu he larges value for nonsable momenum. Because oal momenum is a funcion of he raio beween proporions in he observed populaion and he saionary populaion a young ages, i will be influenced by he recen hisory of crude birh raes. High raes induce large values for oal momenum; low birh raes predic low values for oal momenum. Europe s crude birh rae was 1.2 per 1, for 2 25, in conras o a birh rae of 37.6 per 1, for he leas-developed counries during he same period (Unied Naions 27). Daa in Table 1 sugges ha if replacemen feriliy had been adoped in 25 and remained consan, he populaions of he leas-developed counries would evenually grow by more han one-half (51.3%) before becoming saionary. Even wih an increase in feriliy o replacemen in 25, Europe s populaion would ulimaely decline by 7%. The only oher region exhibiing negaive oal momenum is he group of more-developed counries. Several regions have posiive oal momenum coefficiens in he neighborhood of 1.5. Nonsable momenum depends largely on relaive proporions in he curren and sable age disribuions before he onse of childbearing. Populaions whose feriliy is subsanially below replacemen and whose age disribuions have no had ime o

14 T.J. Espenshade e al. adjus fully o he new feriliy regime will end o have high values for nonsable momenum. Wih a ne reproducion rae of.69 for 2 25, Europe has he larges value in Table 1 for nonsable momenum (1.383). On he oher hand, populaions wih high and relaively consan feriliy will have age disribuions ha are approximaely sable. This condiion produces nonsable momenum coefficiens near uniy. The leas-developed counries, sub-saharan Africa, and Africa as a whole all have nonsable momenum values close o 1.. And each has a hisory of high feriliy, wih modes declines occurring only recenly (Unied Naions 27). Values for sable momenum involve a comparison beween a populaion s sable and saionary age disribuions in he earlies par of life. When feriliy is subsanially above replacemen, he sable age disribuion will be young relaive o is saionary counerpar. Coefficiens of sable momenum should be large in his siuaion. Bu if feriliy is dramaically below replacemen, he opposie circumsance will arise and values for sable momenum will be less han 1. As seen in Table 1, sable momenum is greaes (a or above 1.45) in regions wih high feriliy. Europe s coefficien of sable momenum is jus.67 he lowes for any region. Noice finally in Table 1 ha he same value for oal momenum can be produced wih differen combinaions of nonsable and sable momenum. The oal momenum coefficien is eiher 1.49 or 1.5 for Lain America and he Caribbean, less-developed regions excluding China, and all of Africa. Nonsable momenum is larger han sable momenum in Lain America. These roles are reversed for lessdeveloped regions excluding China. Sable momenum accouns for pracically all of oal momenum in Africa. Table 1 suggess ha resuls depend on he level of developmen. To clarify his relaionship, Fig. 5 shows he disribuion of values for oal momenum for a group of 43 more-developed counries, 133 less-developed counries, and for all counries. For more-developed counries in panel (a), modal values fall beween.9 and 1., indicaing ha negaive populaion momenum is no uncommon in richer counries. Toal momenum values for less-developed counries in panel (b) cluser near 1.5. The disribuion in panel (c) for all counries exhibis a somewha bimodal shape, bu i is weighed oward values near 1.5 because of he greaer number 4 3 (a) (b) (c) World = % More-Developed Counries (N = 43) Less-Developed Counries (N = 133) Fig. 5 Disribuion of oal momenum by level of developmen, All Counries (N = 176)

15 On Nonsable and Sable Populaion Momenum of less-developed counries. For added perspecive, panel (c) also conains a verical line o indicae oal momenum for he enire world. 6 Finally, we consider nonsable and sable momenum values for individual counries. Figure 6 conains a scaerplo of poins whose coordinaes correspond o sable and nonsable momenum for 176 counries. Sable momenum ranges beween.48 and Counries wih he larges values include Timor-Lese (1.751), Guaemala (1.633), Yemen (1.58), Madagascar (1.574), and Guinea-Bissau (1.571). The five smalles values belong o Ukraine (.479), Czech Republic (.53), Bulgaria (.535), Belarus (.541), and Slovakia (.546). The level of he ne reproducion rae (R ) is an imporan deerminan of sable momenum. In he five counries where sable momenum is greaes, R values are 2.1 or higher. By conras, R values do no exceed.6 among he five counries wih he smalles coefficiens for sable momenum. Nonsable momenum varies beween.96 and Counries wih he smalles values include Sierra Leone (.961), Timor-Lese (.965), Guinea-Bissau (.969), Mozambique (.97), and Malawi (.974). A hisory of high and relaively consan feriliy is a good predicor of nonsable momenum near 1.. Each of hese counries has a oal feriliy rae above 5.5 for he period 2 25 (Unied Naions 27). On he oher hand, feriliy ha has fallen recenly o low levels is indicaive of large values for nonsable momenum. Counries wih he highes values include Republic of Korea (1.97), Armenia (1.918), Slovakia (1.824), Poland (1.758), and Azerbaijan (1.754). Feriliy experienced a recen collapse in each of hese cases (Unied Naions 27). 7 Toal momenum values of 1. and 1.5, respecively, are shown by poins along he wo hyperbolic curves in Fig. 6. I is clear from he graph ha muliple combinaions of sable and nonsable momenum are compaible wih he same oal momenum. Mos counries lie near or beween he wo curves, as panel (c) in Fig. 5 suggess hey should. Oman has he larges oal momenum (1.76), followed by Nicaragua (1.737), Guaemala (1.732), and Honduras (1.73). In each case, populaion would be expeced o grow by more han 7% if replacemen feriliy had been adoped in 25. The lowes values for oal momenum belong o Bulgaria (.811), Ukraine (.824), Germany (.844), and Ialy (.845). For hese counries, even if feriliy rebounded immediaely o replacemen, populaion losses beween 15% and 2% could be expeced. Counries in Fig. 6 are disinguished by heir level of developmen. There are some overlapping circles, squares, and crosses in he graph, and he respecive scaerplos for more-developed counries and less-developed counries are no oally disinc. Bu in general, he leas-developed counries are disribued around he 1.5 oal momenum curve, and more-developed counries are locaed near or somewha below he 1. curve. Tha here are so many poins beneah he 1. momenum curve emphasizes 6 Mean and median values, respecively, in Fig. 5 are as follows: in panel (a),.972 and.961; in panel (b), and 1.491; and in panel (c), and Low feriliy by iself is no enough o produce large values for nonsable momenum. Ialy s oal feriliy rae in 2 25 was 1.29 (Unied Naions 27), similar o ha in Korea (1.24), Armenia (1.35), Slovakia (1.22), and Poland (1.25). Bu Ialy s nonsable momenum coefficien is only because Ialian feriliy has been low for several decades, giving he younger par of is age disribuion ime o adjus o lower feriliy.

16 T.J. Espenshade e al. Fig. 6 Plo of nonsable and sable momenum for individual counries by level of developmen, 25 (N =176 counries). MDCs = moredeveloped counries; LDCs = less-developed counries Nonsable Momenum MDCs Leas DCs Oher LDCs World Sable Momenum how imporan negaive populaion momenum is among more-developed counries (Preson and Guillo 1997). Poins for oher less-developed counries are more dispersed beween he wo oal momenum curves. All of he more-developed counries lie above an imaginary 45-degree line drawn from he origin. For each of he more-developed counries, nonsable momenum is greaer han sable momenum. 8 Mos of he less-developed counries appear o lie below he 45-degree line, meaning ha for hem, sable momenum ouweighs nonsable momenum. The poin for he world is indicaed by a shaded riangle. This poin lies on or close o he 45-degree line, which suggess in anoher way ha on a global scale, nonsable and sable momenum are roughly equal in magniude. The akeaway message is his: as he level of developmen increases in Fig. 6, feriliy is generally lower, overall populaion momenum is less, sable momenum becomes weaker, and nonsable momenum becomes sronger. Discussion This aricle has described a decomposiion of overall populaion momenum ino wo consiuen and muliplicaive pars: nonsable and sable momenum. The value for nonsable momenum reflecs a counry s recen rend in feriliy. In a sable populaion where feriliy has been consan for a long ime, nonsable momenum is nonexisen. Bu counries ha have a hisory of feriliy decline, especially a recen and sharp decline, will have larger values for nonsable momenum. The value for sable momenum is dicaed by a populaion s curren level of feriliy in relaion o moraliy. A high (low) ne reproducion rae corresponds o a large (small) value for sable momenum. This 8 In addiion, each of he more-developed counries has a sable momenum value less han 1., which coincides wih below-replacemen feriliy. The lone excepion is Albania, whose ne reproducion rae is 1.5.

17 On Nonsable and Sable Populaion Momenum decomposiion gives us a new way of hinking abou he deerminans of overall populaion momenum. In addiion, i allows us o inegrae disparae srands of he populaion momenum lieraure and see how he various kinds of momenum ha researchers have considered fi ogeher ino a single analyic and empirical framework. Our empirical work shows ha he facorizaion of oal momenum ino he produc of sable and nonsable momenum is an exremely good approximaion much beer, in fac, han mos approximaions in demography. A a global level, sable and nonsable momenum are roughly equal in imporance. On a regional basis, however, a pariioning of he world by levels of developmen corresponds no only o differen values for overall momenum bu also o differen roles played by sable and nonsable momenum. Among more-developed counries, oal momenum is close o zero or slighly negaive. For hese counries, posiive nonsable momenum is offse by negaive sable momenum. By conras, in less-developed regions, and especially among he leasdeveloped counries, oal momenum values of 1.5 or greaer are common. For he poores counries, nonsable momenum is only slighly posiive. Sable momenum dominaes he overall projecion and gives by far he larger impeus o fuure populaion growh. In shor, among he 176 counries we have examined, as he level of developmen increases and feriliy subsides, values for oal momenum decline, sable momenum becomes weaker, and nonsable momenum becomes sronger. The research raises a number of quesions for furher work. Firs, we have idenified wo special cases in which he facorizaion of oal momenum is exac. Bu are here more? If so, hese addiional examples migh help o explain why he simple produc of nonsable and sable momenum is such a good approximaion o oal momenum. Second, he facorizaion of overall momenum is inexac whenever k 1 ineq.17. This corresponds o siuaions in Fig. 1 in which S 1 S 2. Wha are he condiions ha deermine he magniude and direcion of k 1? Third, work by Li and Tuljapurkar (1999, 2) has opened up new avenues of inquiry concerning momenum in populaions wih gradually declining feriliy (e.g., see Goldsein 22; Goldsein and Secklov 22; O Neill e al. 1999; Schoen and Jonsson 23; Schoen and Kim 1998). Migh i be useful in his conex o conemplae he roles of nonsable and sable momenum? Fourh, how does our undersanding of sable and nonsable momenum change if populaions are no longer assumed o be closed o migraion? Migrans modify a populaion s age disribuion (Guillo 25:293), bu hey can also affec levels of feriliy and moraliy. Fifh, our analysis has been largely saic, relying on esimaes of momenum in 25 for individual counries, he world, and major regions. There is a need o pu nonsable and sable momenum in a more dynamic conex and consider wha happens wihin populaions over ime. In Fig. 6, for example, European counries are clusered in he upper lef-hand corner of he diagram. Bu if one assumes ha feriliy and moraliy as measured in 25 are held consan, one can imagine he circles for European counries gradually drifing downward oward a nonsable momenum value close o 1. when momenum values are recalculaed periodically afer 25. More generally, i would be useful o race he arc of nonsable and sable momenum across he demographic ransiion. From his dynamic perspecive, one could conjecure ha nonsable momenum is somehing ha emerges during a ime

18 T.J. Espenshade e al. of feriliy ransiion and hen begins o disappear afer birh raes reach relaively low levels. In his sense, nonsable momenum is he ransiory par of oal populaion momenum. Finally, differen combinaions of sable and nonsable momenum are compaible wih he same amoun of overall populaion momenum. For example, in 25, oal momenum measured 1.46 in Chad and 1.45 in Mongolia. Boh counries could be expeced o grow by nearly 5% if feriliy had gone immediaely o replacemen. In Chad, sable momenum (1.49) ouweighed nonsable momenum (.98). Bu hese roles were reversed in Mongolia, where sable and nonsable momenum were esimaed a.96 and 1.51, respecively. Apar from implicaions ha heir respecive life expecancies have for long-run saionary age disribuions, does he paricular mix of sable and nonsable momenum in he wo counries maer for he pahs aken o saionariy? 9 Does he ime required o achieve a saionary populaion depend on he relaive srengh of nonsable and sable momenum? Wha abou he ripples ha are creaed in various age groups along he way? Carlson (28) showed for he Unied Saes no only ha successive generaions measured by 2-year birh cohors vary in size bu also ha here have been subsanial booms and buss in heir relaive sizes. These oscillaions have impacs hroughou he life course, affecing schooling, labor and housing markes, poliical aiudes and paricipaion, healh care, and pension sysems. For a given level of overall momenum, wha are he corresponding implicaions of varying combinaions of nonsable and sable momenum as cohors of flucuaing sizes pulse hrough insiuional areries? Acknowledgmens An earlier version of his aricle was presened a he annual meeings of he Populaion Associaion of America, Deroi, MI, April 3 May 2, 29. Parial suppor for his research came from he Eunice Kennedy Shriver Naional Insiue of Child Healh and Human Developmen (Gran #5R24HD47879). We are graeful o Adrian Banner, Laura Blue, Ronald Brookmeyer, Rober Calderbank, Chang Chung, Erhan Cinlar, Dennis Feehan, David Gabai, Diego Hofman, Igor Klebanov, Edward Nelson, John Palmer, David Poere, Germán Rodríguez, Lily Shen, and Shripad Tuljapurkar for useful commens and discussions. Suggesions from wo anonymous reviewers have helped o clarify he exposiion and highligh essenial feaures of he argumen. We hank Chang Young Chung for preparing he figures and Valerie Fizparick for adminisraive and echnical suppor. References Bongaars, J. (1994). Populaion policy opions in he developing world. Science, 263, Bongaars, J. (1999). Populaion momenum. In A. Mason, T. Merrick, & R. P. Shaw (Eds.), Populaion economics, demographic ransiion, and developmen: Research and policy implicaions (World Bank Working Paper, pp. 3 15). Washingon, DC: IBRD/World Bank. Bongaars, J. (27, November). Populaion growh and policy opions in he developing world. Paper presened a he Beijing Forum, Beijing. New York: The Populaion Council. Bongaars, J., & Bulaao, R. A. (1999). Compleing he demographic ransiion. Populaion and Developmen Review, 25, Bourgeois-Picha, J. (1971). Sable, semi-sable populaions and growh poenial. Populaion Sudies, 25, In 25, female life expecancy a birh was esimaed a 52. years in Chad compared wih 68.4 years in Mongolia. Chad had a oal feriliy rae of 6.54, and 18.6% of all persons were under age 5. By conras, he oal feriliy rae for Mongolia was 2.7, and jus 9.1% of he populaion were under age 5 (Unied Naions 27).

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