ECON 3710/4710 Demography of developing countries STABLE AND STATIONARY POPULATIONS. Lecture note. Nico Keilman
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1 ECON 37/47 Demogaphy of deveoping counies STAE AND STATIONARY POPUATIONS ecue noe Nico Keiman Requied eading: Sabe and saionay modes Chape 9 in D Rowand Demogaphic Mehods and Conceps Ofod Univesiy Pess 23, pp 3-36, Mode popuaions ae used o show vaious inks beween popuaion vaiabes, such as he numbes of bihs and deahs, he age disibuion, oa popuaion size, ec The inenion is o simpify eaiy, wihou osing main chaaceisics Saing poin: we conside a cosed popuaion, of which he membes ony epeience bih and deah In addiion: we beak he popuaion down by jus wo chaaceisics: age and se Definiion: a popuaion of his kind is caed sabe when boh is gowh ae and is eaive age disibuion do no change ove ime Hence he size of a sabe popuaion gows o diminishes a a consan ae, bu each age goup has a consan shae The size of each age goup gows/diminishes wih he same (consan ae as he size of he whoe popuaion Popey : The numbe of bihs and he numbe of deahs in a sabe popuaion change wih he same ae This ae is consan, and i equas he gowh ae of he popuaion Thus he cude bih ae (CR is aso consan, and so is he cude deah ae (CDR ahough he CR and he CDR ae usuay no equa o each ohe In ohe wods: gowh ae (= naua gowh ae = CR CDR = consan
2 Popey 2: Any wo ses of ime-consan age-specific bih aes and age-specific deah aes, when appied o a popuaion wih a ceain iniia age disibuion, wi in he ong un esu in a sabe popuaion, of which he gowh ae and he age disibuion ony depend of he bih aes and he deah aes The uimae (sabe age disibuion wi have a egua fom (cf beow This is he case even if he iniia age disibuion was no sabe fo insance was vey iegua We say ha a sabe popuaion has fogoen is hisoy This popey is known as he pincipe of song egodiciy (oka (9, Eue (76 The use of sabe popuaion heoy Sabe popuaion heoy is fequeny used when we ack ceain popuaion daa (fo eampe he CR whie we know some of he ohe daa (fo eampe he age disibuion wih ceainy As an eampe azi 96: fom he Census of 96 we know he age disibuion in five-yea age goups we assume ha he popuaion in azi in 96 is sabe Now we can compue ( esimae he CR, he CDR, he ife epecancy, he Goss Repoducion Rae, ec, by using he heoeica mahemaica eaionships ha eis in a sabe popuaion beween hese vaiabes and he age disibuion See Rowand Tabe 97 (p 336 Esimaion echniques of his kind ( Indiec esimaion mehods ae fequeny used fo deveoping counies, bu aso fo hisoica popuaions in deveoped counies Cf Appendi on he Nowegian popuaion census of 8, using Regiona Mode Sabe Popuaions o be discussed ae Using he song egodiciy pincipe Given a se of age-specific bih aes and one of age-specific deah aes, i is possibe o compue he annua gowh ae and he eaive age disibuion of he sabe popuaion ha hese aes impy The annua gowh ae summaizes, in one numbe, he age-specific bih and deah aes Definiion: he annua gowh ae of a sabe popuaion is caed he ininsic gowh ae 2
3 Noe: he ininsic gowh ae is geneay NOT he same as he gowh ae fo an acua popuaion These wo aes ae ony equa when he acua popuaion is sabe (and cosed fo migaion Ohewise he ininsic gowh ae is meey a heoeica noion Given wo ses of age-specific aes fo feiiy and moaiy, how do we compue he ininsic gowh ae? he sabe age disibuion? We imi ouseves o he femae pa of he popuaion Hence age-specific bih aes ae esiced o femae bihs Wie he ininsic gowh ae as Two appoaches wi be pesened: an appoimae one, and an eac one Appoimae cacuaion of he ininsic gowh ae and he sabe age disibuion We use he Ne Repoducion Rae (NRR, which can be compued on he basis of he age specific aes fo feiiy and moaiy See Rowand p 246 (no equied eading in his couse On page he wies he NRR as R NRR epesens he aveage numbe of daughes pe woman eow I wi deive an appoimae epession fo he ininsic gowh ae ha is somewha easie o undesand and simpe o compue han he one ha Rowand gives on page 36 (wih an eampe in Tabe 9 Usuay he wo fomuas give esus ha ae vey cose Assume ha he aveage geneaion disance beween mohes and daughes equas T yeas In ohe wods, mohes ae, on aveage, T yeas ode han hei daughes Then we know ha NRR epesens he eaive incease in he sabe popuaion ove a peiod of T yeas The annua gowh ae in he sabe popuaion equas Theefoe we can wie ( NRR = ( + T Now we can find, by soving epession (: T / T (2 NRR ( NRR 3
4 NRR foows fom he age-specific bih and deah aes u T? Conside a goup of women a age 5 They have finished hei epoducive caee They have given bih o vaious numbes of daughes a vaious ages When he mohes ae 5 yeas of age, he daughes ae beween and yeas od Define G as he numbe of daughes cueny of age of mohes aged 5, =,, 2,, Then he aveage disance beween he geneaion of mohes (a 5 and daughes (beween and equas G (5 G (5 G T G G G 5( G G G G G 5 G G G G G ( G G G G (5 G Thus he aveage geneaion disance T equas he diffeence beween wo ages: he age of he mohes (5 and he aveage age of he daughes Assume now ha his age diffeence equas he mean age of mohes when hey gave bih o hese daughes compued fom he ne feiiy aes (see page 6; see aso coumn E in Rowand s Tabe 9 The ae mean age is wien as µ (R /R in Tabe 9 We assume ha he aveage geneaion disance beween mohes and daughes is equa o he mean age of he mohes when hey give bih o he daughes, in ohe wods we assume ha T = µ This assumpion is ony pe cen coec in case =, in ohe wods when he sabe popuaion neihe inceases no diminishes (saionay popuaion, cf beow In ha case he numbe of women does no change ove ime, and he mean age compued based on absoue numbes (T equas he mean age compued fom age-specific aes (µ Fo ininsic gowh aes diffeen fom zeo, is T diffeen fom µ ( > : µ > T; < : µ < T Assuming ha T = µ, we use epession (2 o find an appoimae souion fo as (3 NRR 4
5 28 Fo eampe (angadesh 974 when µ = 28, and NRR = 23737, we find ha = 3, o 3 pe cen pe yea Afe he ininsic gowh ae has been cacuaed, we can find he sabe age disibuion fo he femae popuaion in wo sages Fis, compue an iniia shae of age goup (, + as C = ( + -(+½ ( / Hee and foow fom he ife abe ha is compued fom he age-specific deah aes: is he coumn of eposue imes, and is he adi The sum of hese quaniies C is no equa o one, as shoud be he case fo a pope shae Assume ha his sum equas C = C Ne, compue he fina shaes c = C /C Fo a poof, see Appendi 2 (no equied eading See aso Rowand pages 37-3 (no equied eading This fomua fo he age disibuion is eac, given 2 Eac cacuaion of he ininsic gowh ae and he sabe age disibuion In Appendi 2 I pove he foowing eac epession fo he ininsic gowh ae : f / (4 ( ( ½ Hee is epessed as a funcion of age-specific bih aes f (ony femae bihs and age-specific moaiy / (o be compued fom a ife abe, which isef is based on he age-specific deah aes m fo hese women This epession is caed oka s fundamena equaion of sabe popuaion heoy You do no have o emembe his epession by hea, bu you shoud be abe o inepe and use i How can we find he ininsic gowh ae, when we ae given age-specific feiiy (f and moaiy ( /? Epession (4 is oo compicaed fo a diec souion We find by ia and eo, as foows Choose a saing vaue fo he ininsic gowh ae Check whehe he ef-hand side of (4 equas wih his vaue (and given he age-specific feiiy and moaiy If no, choose a new 5
6 vaue: 2 Check (4 again ec, uni you have found a vaue fo he ininsic gowh ae ha esus in a vaue fo he ef-hand side of (4 ha is cose enough o Sysemaic seach: Choose NRR, in ohe wods, he appoimae souion Check he sum in he ef-hand side of (4 wih = : is sum( =? If no, choose a new vaue 2 fo : 2 = + [sum( ]/µ Check now he new sum agains one: is sum( 2 =? If no, epea he pevious sep as ofen as is necessay 3 = 2 + [sum( 2 ]/µ 4 = 3 + [sum( 3 ]/µ ec Usuay, his pocedue esus in ony a few seps in a vaue fo he ininsic gowh ae ha is accuae enough, in ohe wods ha poduces a vaue fo he ef-hand side of epession (4 which is cose enough o one Eampe: Peu 96 (Coae ; see websie ECON 37 PC-eecise Egyp 997, Noway 987 Rowand does no pesen his mehod fo he eac cacuaion of he ininsic gowh ae Commens Feiiy aes f ae zeo a a ages beow 5 o above 5 Thus in pacice, he sum in epession (4 is imied o ages = 5, 6, 7,, 49 2 The poduc (f / is caed he ne feiiy ae, ofen wien as φ We ecognize he sum φ as he Ne Repoducion Rae Indeed, when we assume no gowh in he sabe popuaion and inse = in epession (4, we find φ = O, he Ne Repoducion Rae equas one in his case! 3 When he age is given in five-yea inevas insead of one-yea inevas, epession (4 is wien as foows: f 5, / ( 2½ ( whee epesens inevas 5-9, 2-24,, As soon as we have compued he eac vaue of he ininsic gowh ae, we can aso compue he eac vaue of he geneaion disance T, as foows We know ha NRR = ( + T 6
7 Ne we can sove fo T : T = og(nrr/og( + 5 The fundamena equaion (4 is ofen wien as foows: e f ( p( d In ha case, age is a coninuous vaiabe no a discee vaiabe (as in ou case one-yea age goups, o five-yea age goups Theefoe - inega sign, insead of summaion - e -, insead of (+ -(+½ - p(, insead of / ; p( equas /, he individua suviva pobabiiy fom bih o eac age A he same ime we have ha NRR = e T ; insead of NRR = ( + T and = (/Tn(NRR insead of T NRR Whehe you use discee o coninuous age in a sabe popuaion mode has in genea vey ie effec on he numeica esus, uness you use oo wide age inevas Five-yea inevas give esus ha ae accuae enough fo ou pupose 6 Afe you have compued he eac vaue fo he ininsic gowh ae, you can compue he sabe age disibuion using he mehod descibed on page 5 Saionay popuaions A saionay popuaion is a sabe popuaion in which he ininsic gowh ae is zeo In ohe wods, none of he popuaion vaiabes in a saionay popuaion change ove ime: he annua numbe of bihs, he annua numbe of deahs, popuaion size, he size of a ceain age goup, ec ae a consan We can deive a numbe of simpe epessions fo hese vaiabes in a saionay popuaion Hee I wi menion hem, and y o give an inuiive jusificaion Appendi 2 conains a foma deivaion of hose epessions The epessions fo saionay popuaions in his secion ae equied eading, hose in Appendi 2 ae no Feiiy Given age-specific aes fo feiiy and moaiy fo women These aes ae combined ino ne feiiy aes φ = f ( / Since a saionay popuaion is a sabe popuaion wih zeo gowh ae, we can inse = in epession (4 and find ( ½ ( 7
8 We know ha he sum φ equas he Ne Repoducion Rae Thus we see ha a saionay popuaion has NRR equa o We find he same esu when we sa fom he epession NRR = (+ T, which hods fo evey sabe popuaion Inseing = esus in NRR = (+ T = Aso he ohe way ound: a cosed sabe popuaion ha has NRR equa o one is saionay Moaiy The ife epecancy a bih e equas he mean age a deah of he saionay popuaion In ohe wods, individuas die a age e on aveage The annua numbe of deahs in a saionay popuaion is consan Wie his numbe as D Popuaion size, wien as N, is aso consan Each yea, a shae /e in he popuaion dies Thus we find ha D = N/e This eads o he foowing epession fo he Cude Deah Rae (CDR in a saionay popuaion: CDR = D/MYP = D/N = /e, because he mid-yea popuaion MYP equas ½(N+N=N The ink beween feiiy and moaiy in a saionay popuaion In a sabe popuaion we have ha he gowh ae equas he diffeence beween cude bih ae and cude deah ae u in a saionay popuaion he gowh ae is zeo Thus we find fo a saionay popuaion ha (5 CDR = CR = /e Wie he annua numbe of bihs as Since CR = /N = /e, we find he foowing simpe epession fo oa popuaion size in a saionay popuaion: (6 N = e The age sucue of a saionay popuaion The ife epecancy e is defined in genea as e = T / = /, see ife abe heoy Inseing his epession fo e in epession (6, we find ha oa popuaion size in a saionay popuaion equas / Meanwhie one can pove ha he size of age goup (,+ in a saionay popuaion equas / and do no depend of age - ony vaies wih age This is he eason why (he -coumn in a ife abe is someimes caed "age sucue of he saionay popuaion" See Rowand page 37 8
9 Appendi Appicaion of sabe popuaion heoy o Noway s age disibuion in 8 In he eampe beow I anaysed he age sucue of Noway 8 by compaing i wih he age disibuions of a seies of mode sabe popuaions Afe I found a mode popuaion whose age pyamid esembed he empiica one cosey enough, I coud ead off he ife epecancy The Popuaion Census of 8 in Noway is geneay beieved o be of high quaiy (Dake 969 The age pyamid beow suggess ha he popuaion migh have been neay sabe Nowegian cude bih and deah aes wee ahe consan a 3-32 and pe housand especivey, in he peiod 7-8 (Dake 969, Tabe 36 Age goups beween 25 and 45 signa non-sabiiy, ahough some digi pefeence pobaby aso conibued o he ieguaiies y woking wih cumuaed age goups, his pobem wi have ony mino impac on he findings Age sucue, Now ay, Census of men w omen in housands I compaed he cumuaed age disibuions C(a fo men and women o hose of sabe popuaions based on Regiona ife Tabes Mode Noh of eves -5, ie ife epecancy vaues of fo women and fo men Fo successive vaues of a, I found he sabe gowh ae by inepoaion beween abuaed gowh aes If Noway s popuaion woud have been pefecy sabe, i shoud have he same inepoaed gowh ae fo each a In empiica appicaions, inepoaed aes vay by age Vaiaions acoss eves wee smaes fo eve 2, he sandad deviaion in he ae ove ages 5-7 being and 7 pe housand fo he wo sees The mean inepoaed gowh aes wee equa o 22 and 2 pe housand fo men and women, especivey This suggess a ife epecancy of aound 45 yeas befoe 8 Fo men aged -39 o 65-69, and women aged 3-34, 4-44, o 7-74, he esimaed gowh aes wee emakaby owe han hose fo ohe age goups Ecepionay ow bih aes and high deah aes in he 74s and 77s epain some of hese effecs (Dake 969 9
10 Appendi 2 Some mahemaics on sabe and saionay popuaions We anayse a sabe popuaion ha is cosed fo migaion, and esic he anaysis o women The aim is o find epessions fo he ininsic gowh ae and he sabe age sucue c, given ses of age-specific aes fo moaiy and feiiy The numbe of bihs inceases each yea by a consan faco (+ Wie he numbe of bihs in yea as y way of eampe, we noe ha 2 = 999 (+ = 998 (+ 2 = 997 (+ 3 = ec Geneay: hee ae ive bihs in yea - hese wee bon beween eac ime and eac ime +; a ime hee ae N pesons aged beween and +; Then we can wie ( 2 2 ( ( ( N N N Thus oa popuaion size a ime equas ( ( N One yea ae, a ime (+, popuaion size equas ( ( ( This impies ha he mid-yea popuaion in yea can be appoimaed by ( ½, ( ( MYP and he Cude ih Rae in he sabe popuaion in yea equas (A b MYP CR ½ ( (, which is independen of ime, see Popey in he main e
11 Insead of epession (A, we can aso wie (A2 ( ( b( ½ A he same ime we can wie fo he shae of age goup (,+ in he oa popuaion ( ( ( ( N c N The ems in he numeao and denominao cance, and he emainde of he denominao can be wien as /{b(+ ½ }, because of epession (A2 Hence we find fo he shae of age ( ½ (A3 c b( The Cude ih Rae b can be wien as a weighed aveage of he age-specific bih aes f, wih he age shaes c as weighs: b = f c Using (A3 his impies ha o b ( ½ f b(, (A4 f ( ½ ( Epession (A4 is known as oka's fundamena equaion of sabe popuaion heoy, o he oka equaion fo sho Given age-specific feiiy (f and age-specific moaiy ( /, he ininsic gowh ae is deemined by oka's equaion When we combine epessions (A and (A3 we find fo he age sucue c ( 2 ( ( ( 2, so ha, indeed, c =
12 A saionay popuaion is a specia case of a sabe popuaion - he gowh ae is zeo The oka equaion educes o f ( / = φ = NRR = 2 Cude ih Rae (b: Inse = in (A o find b, e ( / wih e he ife epecancy a bih 3 Cude Deah Rae (m: In a saionay popuaion, he Cude ih Rae equas he Cude Deah Rae Theefoe m e 4 Age disibuion (c : Use (A3 and inse =, o find c b e 5 Popuaion size (N: In genea, he CR equas /MYP In a saionay popuaion, he numbe of bihs is consan ( =, and so is popuaion size Hence MYP equas N fo a saionay popuaion Thus he Cude ih Rae is /N A he same ime, i is equa o /e In ohe wods, N = e 6 Size of age goup (,+: N N c e e 2
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