MODEL SCHEDULES IN MULTI STATE DEMOGRAPHIC ANALYSIS: THE CASE OF MIGRATION (1981)

8/3/2019 MODEL SCHEDULES IN MULTI STATE DEMOGRAPHIC ANALYSIS: THE CASE OF MIGRATION (1981)

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Working PaperMODEL SCHEDULES IN MULTISTATE DEMOGRAPHICANALYSIS: THE CASE OF MIGRATION

Andrei RogersLuis J. Castro

March 1981WP-81-22Prepared for presentation at theConference on Multidimensional' Demography, Washington, D. C. ,March 23-25, 1981

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International Institute for Applied Systems AnalysisA-2361 Laxenburg,Austria


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NOT FOR QUOTATIONWITHOUT PERMISSIONOF THE AUTHOR

MODEL SCHEDULES I N MULTISTATE DEMOGRAPHICANALYSIS: THE CASE OF MIGRATION

A n d r e i R o g e r sL u i s J . C a s t r oMarch 1981WP-81-22P r ep a r ed f o r p r e s e n t a t i o n a t theC o n f e r e n ce o n M u l t i d i m e n s i o n a lDemography, Washington , D . C . ,March 23-25, 1981

W o r k i n g Papers a re i n t e r i m r e p o r t s o n work o f t h eI n t e r n a t i o n a l I n s t i t u t e f o r A p pl i ed S y st e m s A n a l y s i sa n d h av e r e c e i v e d o n l y l i m i t e d review. V i e w s o ro p i n io n s e x p r e s s e d h e r e i n d o n o t n e c e s s a r i l y r e p r e -s e n t t h o s e o f t h e I n s t i t u t e o r o f i t s Nat iona l MemberO r g a n i z a t i o n s .INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSISA-2361 Laxe nbu rg , A u s t r i a


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PREFACE

I n t e r e s t i n human s e t t l e m e n t s y st em s a nd p o l i c i e s h a s b ee na c e n t r a l p a r t o f u r b a n - r e l a t e d work a t IIASA s i n c e i t s i n c e p t i o n .From 1 975 t h r o u g h 1 97 8 t h i s i n t e r e s t w a s m a n i fe s t ed i n t h e worko f the M i g r a t i o n a n d S e t t l e m e n t T as k , w h i ch w a s f o r m a l l y c o n c l u d e di n November 197 8. S i n c e t h e n , a t t e n t i o n h a s t u r n e d t o d i ss e mi n a-t i o n o f t h e T as k ' s r e s u l t s and t o t h e c o n c l u si o n o f i t s compara-t i v e s t u d y , w h ic h i s c a r r y i n g o u t a c o m p a r a t i v e q u a n t i t a t i v ea ss es s me nt of r e c e n t m i gr a t i o n p a t t e r n s a n d s p a t i a l p o p u la t i ondynamics i n a l l of I IASA's 17 NMO c o u n t r i e s .

T h i s p a p e r i s p a r t o f t h e Ta sk ' s d i ss e mi n at i on e f f o r t a ndi s t h e t h i r d o f s e v e r a l t o f o cu s o n t h e a g e p a t t e r n s o f migra-t i o n e x h i b i t e d i n t h e d a t a bank a s se mb le d f o r t h e c o mp ar at iv es t u d y . I t b e g i n s w i t h a c om pa ra ti ve a n a l y s i s o f o v e r 500 ob-s e r v ed m ig r a t i o n s c h e d u l e s a nd t h e n d e v e lo p s , o n t h e b a s i s oft h i s a n a l y s i s , a f a m i ly o f h y p o t h e t i c a l " s y n t h e t i c " s c h e d u l e sf o r u s e i n i n s t a n c e s where m i g r at i o n d a t a a re u n a v ai l a b l e o ri n a c c u r a t e .

R e p o r t s , su mm a ri zi ng p r e v i o u s work o n m i g r a t i o n a n d s e t t l e -ment a t IIASA, a r e l i s t e d a t t h e b ac k o f t h i s p a pe r . Theys ho uld b e c on s u l t e d f o r f u r t h e r d e t a i l s r e ga r di n g t h e d a t a b as et h a t u n d e r l i e s t h i s s tu dy . A t e c h n i c a l ap pe nd ix l i s t i n g t h ep a r a me t e rs a nd v a r i a b l e s of o v e r 600 m o d e l m i g r a t i o n s c h e d u l e si s a v a i l a b l e o n r e q u e s t .

Andre i RogersChairmanHuman Set t lementsa nd S e r v i c e s A r e a


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ACKNOWLEDGMENTS

The a u t h o r s a r e g r a t e f u l t o t h e many n a t i o n a l c o l l a b o r a t i n gsch o la r s who h ave p a r t i c ip a t e d i n IIASA's Co mp ara tive M ig ra t io nand S e t t l e m e n t S t ud y . T h i s p a pe r c o u ld n o t ha ve b e en w r i t t e nw i th o ut t h e d a t a bank produ ce d by t h e i r c o l l e c t i v e e f f o r t s .Thanks a l s o go t o R ic ha rd R a q u i l l e t f o r h i s c o n t r i b u t i o n s t ot h e e a r l y ph as es o f t h i s s t u d y and t o W a lt er K ogler f o r h i su n t i r i n g e f f o r t s on ou r b e ha l f i n f r o n t o f a c o n so l e i n IIASA'sco mp u te r cen te r .


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ABSTRACT

T h i s p ap e r d ra ws o n t h e f un da me nt al r e g u l a r i t y e x h i b i t e dby a g e p r o f i l e s o f m i g ra t io n a l l o ve r t h e wo rld t o d ev el op as y st em o f h y p o t h e t i c a l " s y n t h e t i c " model m i q r a t i o n s c h e d u l e st h a t c a n b e u sed t o c a r r y o u t m u l t i r e g i o n a l p o p u l a t i o n a n a l y s e si n c o u n t r i e s t h a t l a c k a d eq ua te m i g ra t io n d a t a .


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CONTENTS

1. INTRODUCTION2. MIGRATION AGE PATTERNS3. A COMPARATIVE ANALYSIS4. SYNTHETIC MODEL MIGRATION SCHEDULES: I.THE CORRELATIONAL PERSPECTIVE5. SYNTHETIC MODEL MIGRATION SCHEDULES: 11.THE RELATIONAL PERSPECTIVE6. CONCLUSIONAPPENDIXESAppendix AAppendix BREFERENCESRELATED PUBLICATIONS


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MODEL SCHEDULES I N MULTISTATE DEMOGRAPHICANALYSIS: THE CASE OF MIGRAT ION

1 . INTRODUCT IONThe a g e -- sp ec i fi c f e r t i l i t y a nd m o r t a l i t y s c h e du l e s of m ost

human p o p u l a t i o n s e x h i b i t re ma rk ab ly p e r s i s t e n t r e g u l a r i t i e s ;con seq uen t ly demographer s have found it p o s s i b l e t o summ arizeand c o d i f y su ch r e g u l a r i t i e s by means o f h y p o t h e t i c a l s c h e d u l e sc a l l e d m o d e l s c h e d u l e s . A lt ho ug h t h e d e ve l op m en t o f mod el f e r -t i l i t y a nd m o r t a l i t y s c he d ul e s h as r e c ei v e d c o n s i d e r a b l e a t t e n -t i o n , t h e c o n s t r u c t i o n o f model m i g ra t i o n s c h e d u l e s h a s n o t ,e v e n t ho ug h t h e t e c h n i q ue s t h a t h a ve be en s u c c e s s f u l l y a p p l i e dt o t r e a t tL1.e f or me r ca n r e a d i l y b e e xt e nd e d t o d e a l w i t h t h el a t t e r . The same may b e s a i d o f m ode l s c h e d u l e s o f l a b o r f o r c ee n t r y a n d e x i t , a nd o f m a r r i a g e , d i v o r c e , a nd r e m a r r i a g e. *

I n t h i s p a p e r w e c o n s i d e r t h e n o t i o n o f m odel m u l t i s t a t es c h e d u l es , f o cu s i ng i n p a r t i c u l a r o n t h e d e ve lo pm en t o f a f a m i l yo f m odel m i g ra t i on s c h ed u le s f o r u s e i n s i t u a t i o n s where t h ea v a i l a b l e m i g ra t io n d a t a a r e i na d eq u at e o r i n a c c u r a t e . W e b e g i nby examining r e g u l a r i t i e s i n a ge p r o f i l e t h a t a r e e x h i b i t e d bye m p i r i c a l s c h e d u l e s o f m i g r a t i o n r a t e s . E x p re s s in g t h i s r e gu -l a r i t y i n a m a t h em a t i ca l f or m c a l l e d a m od el m i g r a t i o n s c h e d u l e ,

*There are a few n o t a b l e e x c e p t i o n s , h ow ev er , s u c h a s t h ep ap e r o n mo de l d iv o r c e s ch ed u le s b y K r i sh n an an d K ay ani ( 1 9 7 3 ) .


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w e go on t o examine t h e p a t t e r n s o f v a r i a t i o n t h a t o c cu r i n al a r g e d a t a ban k o f s u c h s c h e d u l e s . Drawing o n t h i s c o m p a ra t i vea n a l y s i s , w e t h e n o u t l i n e two a l t e r n a t i v e a pp ro ac he s f o r gen-e r a t i n g f a m i l i e s o f h y p o t h e t i c a l " s y n t h e t i c " model m ig r a t i o ns c h e d u l e s an d c o nc l u d e t h a t f u r t h e r work i s n ee de d i f s u c h ap -p ro ac he s a r e t o b e o f p r a c t i c a l u s e i n m ig r a t i o n s t u d i e s c a r -r i e d o u t i n T h ir d World p o p u l a ti o n s e t t i n g s .


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2. MIGRATION AGE PATTERNSMigration measurement can usefully apply concepts borrowed

from both mortality and fertility analysis, modifying themwhere necessary to take into account aspects that are peculiarto spatial mobility. From mortality analysis, migration studiescan borrow thenotionof the life table, extending it to includeincrements as well as decrements, in order to reflect the mutualinteraction of several regional cohorts (Rogers, 1973a, b, and1975; Rogers and Ledent, 1976). From fertility analysis, migra-tion studies can borrow well-developed techniques for graduatingage-specific schedules (Rogers, Raquillet, and Castro, 1978).Fundamental to both "borrowings" is a workable definition of themigration rate.

2.1 Migration Rates and Migration Rate SchedulesDuring the course of a year, or some such fixed interval of

time, a nurnljer of individuals living in a particular communitychange their regular place of residence. Let us call such per-sons m o b i l e s to distinguish them from those individuals who havenot changed their place of residence, i.e., the n o n - m o b i l e s .Some of the mobiles will have moved to a new community of resi-dence; otherswill simply have transferred their household toanother residence within the same community. The former maybe called m o v e r s , the latter, r e l o c a t o r s . A few in each cate-gory will have died before the end of the unit time interval.

Assessing the situation with respect to the start and theend of the unit time interval, we may divide movers who s u r -v i v e d to the end of the interval into two groups: those livingin the same community of residence as at the start of the inter-val and those living elsewhere. The first group of movers willbe referred to as s u r v i v in g r e t u r n e e s , the second will be calleds u r v i v i n g m i g r a n t s . An analogous division may be made of moverswho died before the end of the interval to define n o n s u r v i v i n gr e t u r n e e s and n o n s u r v i v i n g m i g r a n t s .


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A move, then i s a n e ve n t : a s e pa ra t i on f rom a c om muni ty .A mover i s a n i n d i v i du a l who ha s made a move a t l e a s t onc ed u r in g a g i v en i n t e r v a l o f t i m e . A m i gra n t ( i . e . , a s u r v i v i n go r n o n su r vi v in g m i g r a n t ) , on t h e o t h e r h an d, i s a n i n d i v i d u a lwho a t t h e e nd o f a g i v e n t i m e i n t e r v a l no l o ng e r i n h a b i t s t h esame community o f re s i de nc e a s a t t h e s t a r t o f t h e i n t e r v a l .(The a c t o f s e p a r a t i o n f ro m o n e s t a t e i s l i n k e d w i t h a n a d d i-t i o n t o a n o th e r s t a t e . ) Thus p a r a d o x i c a l l y , a m u l t i p l e movermay b e a no n m ig r an t by o u r d e f i n i t i o n ; t h a t i s , i f a p a r t i c u l a rmover r e t u r n s t o t h e i n i t i a l p l a c e of r e s i d e n c e b e f o r e t h e endo f t h e u n i t t i m e i n t e r v a l , no " m i g ra t io n n i s s a i d t o h a ve t a ke np l a c e . *

The s i m p le s t and most common measure of mi gr a t io n i s t h ec r u d e m i g r a t i o n r a t e , d e f i n e d a s t h e r a t i o o f t h e number o fm ig ra nt s, l e a v i n g a p a r t i c u l a r p o pu l a t i o n l o c a t e d i n s p a c e an dt i m e , t o t h e a ve ra ge number of pe r son s (more e x a c t l y , t h e num-b e r o f p e r s o n - y e a r s ) e xp o se d t o t h e r i s k o f b ecom ing m i g r a n ts . * *

Be c a use m i g ra t i on i s h i g N y a g e s e l e c t i v e , w i th a l a r g ef r a c t i o n o f m i g r a n t s b e i n g t h e young, o u r u n d e r s t a n d i n g of m i -g r a t i o n p a t t e r n s a nd dy na mic s i s a i d e d by com put ing m i g ra t i o nr a t e s f o r e a c h s i n g l e y e a r o f a g e . Summing t h e s e r a t e s o v e ra l l a ge s o f l i f e g i v e s t h e g r os s m ig ra pr od uc tio n r a t e ( GM R ) , t h em i gr a ti o n a na lo g o f f e r t i l i t y ' s g r o s s r e p r od u c t i o n r a t e .

F i g u r e 2 . 1 i n d i c a t e s t h a t a g e- s pe c i f ic a nn ua l r a t e s of r e s i -d e n t i a l m o b i l i t y among w h i t e s a n d b l a c k s i n t h e U.S. d u r i n g1966-1971 e xh ib i t ed a common p ro f i l e . Mo bi l i ty r a t e s amongi n f a n t s a nd young c h i l d r e n m ir r or e d t h e r e l a t i v e l y h i gh r a t e so f t h e i r p a r e n t s , young a d u l t s i n t h e i r l a t e t w en t ie s . Them o b i l i t y of a d o l e s c e n t s w a s l ow e r b u t e x ce e de d t h a t o f y ou ng

* W e d e f i n e m i gr a t i o n t o be t h e t r a n s i t i o n b etween s t a t e se xpe r i e n c e d by a m i g ra n t .* *B ec au se d a t a o n n o n s u r v i v i n g m i g r a n t s a r e g e n e r a l l y un-a v a i l a b l e , t h e nu me ra to r i n t h i s r a t i o g e n e r a l l y e xc lu d esthem.


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I white- lackFigure 2.1 Observed annual migration rates by color and single

years of age: the United States, 1966-1971.


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t e e n s , w i th t h e l a t t e r showing a l o c a l l o w p o i n t around agef i f t e e n . T h e r ea f t e r m o b i l i t y r a t e s i n c re a s e d , a t t a i n i n g a h i g hpeak a t a b o u t a g e t w en ty two a nd t h e n d e c l i n i n g m o n o t o n i c a l l y

w i t h a ge t o t h e a g e s o f r e t i r e m e n t . The m o b i l i t y l e v e l s o fb o t h w h i t e s a nd b l a c k s were r o u g hl y s i m i l a r , w i t h w h i t e s show-i n g a g r o s s m i g r ap r od u ct io n r a t e o f a b ou t 1 4 moves and b la c k so n e o f ap p r o x im a te ly 1 5 o v e r a l i f e t i m e u n di s t ur b e d by m o r t a l i t yb e f o r e t h e e nd o f t h e m ob i l e a g e s .

Al though i t h a s bee n f r e q u e n t l y a s s e r t e d t h a t m i g ra t io ni s s t r on g l y s e x s e l e c t i v e , w i th males being more mobi le thanf em al es , r e c e n t re s e a rc h i n d i c a t e s t h a t s e x s e l e c t i v i t y i smuch l e s s p ro no un ce d t h a n a g e s e l e c t i v i t y and t h a t it i s l e s su n i f o r m a c r o s s t i m e and s p ace . Nev e r th e l e s s , b ecau s e m os tm od els a nd s t u d i e s o f p o p u l a t i o n d yn am ic s d i s t i n g u i s h be tw ee nt h e s e x e s , m o st m i g r a t i o n m ea su re s do a l s o .

F i g u r e 2 . 2 i l l u s t r a t e s t h e a ge p r o f i l e s o f m ale and f em al em ig ra t i on s c he d ul e s i n f o u r d i f f e r e n t c o u n t r i e s a t a b ou t t h esame p o i n t i n t i m e b etw ee n r o u g h l y co m pa r ab l e a r e a l u n i t s :communes i n t h e Net her l ands and Sweden, vo iv odsh ips i n Po land ,and c o u n t i e s i n t h e U . S . The mi gr a t io n l e v e l s f o r a l l b u tP ol an d a r e s i m i l a r , v a r y i n g be tw een 3 .5 and 5 .3 moves pe r l i f e -t i m e ; and t h e l e v e l s f o r m a le s and f e ma l es a r e r o u g hl y t h e s ame.The a g e p r o f i l e s , how ever, show a d i s t i n c t , a n d c o n s i s t e n t ,d i f f e r e n c e . The h i g h p ea k o f t h e f e m al e s c h e d u l e al w a y s p r e -c ed es t h a t o f t h e male s c h e d u l e by a n am ount t h a t a p p e a r s t oa pp ro xi ma te t h e d i f f e r e n c e betw ee n t h e a v e r a ge a g e s a t m a r r i a g eo f the two sexes.


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. .I aa .d\OQI *. 54 0 00- QI > *(I) 4 .d (I)91 ** 0 PIu a l Q > uL l 63 :5 a l uLn 0 ULnw b l CQ 0 6 .dQw al00 r .rU.d bl (I) Cad:: 4s

h a91 4J 3 Q ).c a lV 1 W Cw 4 uM -0 -0G a l C C. . u a.dp a 4Q S 4:C C-da al 4 uC + f ax u b l b laJ al M3 2.;h 6P z ha c a l ca! alu, apfa a8 P Ll" 2 . d Q


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Under n or ma l s t a t i s t i c a l c o n d i t i o n s , p o i n t - t o - p o i r ,t move-m ents a r e a g g r e g at e d i n t o s t r e a m s b etw ee n on e c i v i l d i v i s i o nand a n o t h e r ; c o n s e qu e n tl y , t h e l e v e l o f i n t e r r e g i o n a l m ig ra-t i o n depends on t h e s i z e o f t h e a r e a l u n i t s e l e c t e d . Thus , i ft h e a r e a l u n i t ch o sen i s a minor c i v i l d i v i s i o n s uc h a s ac o un t y o r a commune, a g r e a t e r p r o p o r t i o n o f r e s i d e n t i a l l o c a-t i o n w i l l b e i n c lu de d a s m i g r a t io n t ha n i f t h e a r e a l u n i tc h o se n i s a m ajo r c i v i l d i v i s i o n s uch a s a s t a t e o r a p ro vi nc e.

F i g u r e 2 . 3 p r e s e n t s t h e a ge p r o f i l e s o f f em al e m o b i l i t yand m i g r a t i o n s c h e d u l es a s m eas ured by d i f f e r e n t s i z e s o f a r e a lu n i t s : 1 ) a l l moves fro m o ne r e s i d e n c e t o a n o t h e r , 2 ) c h an g eso f r e s i d e n c e w i t h i n c o u n t y b o u n d a r i e s , 3 ) m i g r a t i o n b et we enc o u n t i e s , a n d 4 ) m i g r a t i o n b et we en s t a t e s . The r e s p e c t i v e f o u rg r o s s m i gr a pr o du c ti o n r a t e s ( GMR s ) a r e 1 4 . 3 , 9 . 3 , 5 .0 , a n d 2 . 5 ,r e s p e c t i v e l y . The f o u r a g e p r o f i l e s a p p e a r t o b e r em ar ka bl ys i m i l a r , i n d i c a t i n g t h a t t h e z e g u l a r i t y i n a g e p a t t e r n p e r s i s t sa c r o s s a r e a l d e l i n e a t i o n s o f d i f f e r e n t s i z e .

F i n a l l y , m i g r a t i o n o c c u r s o v e r t i m e a s w e l l a s a c r o s s s pa ce ;t h e r e f o r e , s t u d i e s o f i t s p a t t e r n s must t r a c e i t s o c c u r r e n c ew i th r e s p e c t t o a t i m e i n t e r v a l , a s w e l l a s o v e r a s y s te m o fg e og r ap h ic a l a r e a s . I n g e n e r a l , t h e l o n g e r t h e t i m e i n t e r v a l ,t h e l a r g e r w i l l b e t h e number o f r e t u r n m ov er s a n d n o n s u r v i v i n gm i g r a n t s an d; h e n c e , t h e more t h e c o u n t o n m ig r a n ts w i l l under -s t a t e t h e number o f i n t e r - a r e a movers ( a n d , o f c o u r s e , a l s o ofm o v e s ) . P h i l i p Rees, f o r examp le , a f t e r ex am in in g t h e r a t i o so f o n e -y ea r t o f i v e - y e a r m i g r a n t s b et we en t h e S t a n d a r d R e gi on so f G r ea t B r i t a i n , fo un d t h a t

t h e number o f m i g r a n t s r e c or d ed o v e r f i v e y e a r s i n a ni n t e r r e g i o n a l f l ow v a r i e s from f o u r t i m e s t o two t i m e st h e number o f m i g r a n t s r e c o r d e d o v e r o n e y e a r .(Rees, 1 9 7 7 , p . 2 4 7 ) .


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AGE- OTAL- ITHIN COUWTIES-- ETWEEN COUNTIES- ETWEEN STATESFigure 2 . 3 Observed female annual migration rates by levels ofareal aggregation and single years of age: the UnitedStates, 1966-1971.


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2.2 Model Migration Rate SchedulesIt appears that the most prominent regularity found in

empirical schedules of age-specific migration rates is the se-lectivity of migration with respect to age. Young adults intheir early twenties generally show the highest migration ratesand young teenagers the lowest. The migration rates of childrenmirror those of their parents; hence the migration rates of in-fants exceed those of adolescents. Finally, migration streamsdirected toward regions with warmer climates and into or out oflarge cities with relatively high levels of social services andcultural amenities often exhibit a "retirement peakv at ages inthe mid-sixties or beyond.

Figure 2.4 illustrates a typical observed age-specific mi-gration schedule (the jagged outline) and its graduation by amodel schedule (the superimposed smooth outline) defined as thesum of four components:

1) a single negative exponential curve of the pre-laborforce ages, with its rate of descent, a 1

2) a left-skewed unimodal curve of the labor force agespositioned at p2 on the age axis and exhibiting ratesof ascent, h2, and descent, a2.

3) an almost bell-shaped curve of the post-labor forceages positioned at p 3 on the age axis and exhibitingrates of ascent, A3, and descent, a3

4) a constant curve, c, the inclusion of which improvesthe quality of fit provided by the mathematical ex-pression of the schedule

The decomposition described above suggests the followingsimple sum of four curves (Rogers, Raquillet, and Castro, 1978):*

*Both the labor force and the post-labor force componentsin equation (1) are described by the "double exponential" curveformulated by Coale and McNeil (1972) for their studies of nup-tiality and fertility.


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a, = rate of descent of urc-labor-fcrce curve xp = :he low pointX 2 = rate of ascent of labor force curve xh = the high peaka2 = rate of descent of labor force curve xr = the retirement peakX , = rate of ascent o f post-labor-force curve X = the labor force shifta3 rate of descent of post-l abor-f ora curva A = the parental shiftMIGRATION RATE, M(x) , connam1 B =the jump

.020

,010 --- Cx X* !,, X!A Xr AGE, x

Figure 2 . 4 The model migration schedule.


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The " f u l l " model s c h e d u l e i n e q u a t i o n ( 1 ) h a s e l e v e n p a r a m e t e r s :a I ' a I t a 2 , 9 , a 2 , A 2 , a 3 t P 3 t a 3 , A 3 , and c . The p r o f i l e o ft h e f u l l model s c h e d u le i s d e f i n e d b y s ev e n of t h e e l e v e n p aram -e t e r s : n l , 1 1 2 , a 2 , A * , u 3 , a 3 , a n d A 3 . I t s l e v e l i s d e t e r m i n e dby t h e r e ma i ni n g f o u r p a r a m e t e r s : a 1 ' a 2 , a 3 , a n d c . A changei n t h e v a l u e o f t h e g r o s s m ig ra pr od uc t i on r a t e o f a p a r t i c u l a rmodel s c he du le a l t e r s p r o p o r t i o n a l l y t h e v a l ue s o f t h e l a t t e rb u t d o e s n o t a f f e c t t h e f o rm e r. However, a s w e s h a l l see i n t h en e xt s e c t i o n , c e r t a i n a s p e c t s o f t h e p r o f i l e a l s o depend on t h ea l l o c a t i o n of t h e s c h e d u l e ' s l e v e l among t h e l a b o r , p r e - l a b o r ,a nd p o s t - l a b o r f o r c e a g e co m po n en ts , a nd o n t h e s h a r e o f t h et o t a l l e v e l a c co un te d f o r by t h e c o n s t a n t t e r m , c . F i n a l l y ,m i g r a t i o n s c h e d u l e s w i t h o u t a r e t i r e m e n t p ea k may b e r e p r e s e n t e dby a " r e d u c e d " m od el w i t h s e v e n p a r a m e t e r s , b e c a u s e i n s u c h i n -s t a n c e s t h e t h i r d c omponent o f e q u a t i o n ( 1 ) i s o m i t t e d .

T ab le 2 . 1 s e t s o u t i l l u s t r a t i v e v a l u e s o f t h e b a s i c and d e r i v edm ea su re s p r e se n t e d i n F i g u r e 2 .4. The d a t a r e f e r t o 1 974 m i g ra t i o ns c h e d u l e s f o r a n e i q h t - r e g i o n d i s a g g r e g a t i o n o f Sweden ( An de rs so n a ndHo .lm berg ,l980).T he m ethod chos en f o r f i t t i n g t h e m odel s ch ed u l et o t h e d a t a i s a f u n c t i o n a l - m i n i m i z a t i o n p r o c e d u r e known a st h e m od i f i e d L evenber g - Mar qua rd t a lg o r i t hm . * Minimum ch i -s q u a r e e s t i m a t o r s a r e u se d i n s t e a d o f l e a s t s q ua r es e s t i m a t o r s .The d i f f e r e n c e s be tw ee n t h e two p a r a m e t r i c e s t i m a t e s t e n d t o b es m a l l , a nd b e ca u s e t h e f or me r g i v e m ore w e ig h t t o a g e g ro u p sw i th s m a l l e r r a t e s o f m i g r a t i o n , w e us e minimum ch i- sq ua re e s t i -m a to r s i n t h e r e ma in de r o f t h e p ap e r .

To a s s e s s t h e q u a l i t y o f f i t t h a t t h e model s ch ed u le pro -v ide s w hen i t i s a p p l i e d t o o b se rv ed d a t a , w e c a l c u l a t e t h e"mean a b s o l u t e e r r o r a s a p e r c e n t a g e o f t h e o b s e rv e d mean":

*See Appendix A an d Brown and Denn is (1972) , Levenberg ( 1944) ,and Mar qua r d t ( 1 963) .


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T a b l e 2 . 1 P a r a m e t e rs a nd v a r i a b l e s d e f i n i n g o b s e r v e d modelm i g r a t i o n s c h e d u l e s : S we di sh r e g i o n s , 1 9 7 4 .- - - - - ~~ -1. 2 . 3 . 4.

Param eters Ea st Middle- South Middle-and Stock holm Sweden Sweden So ut h

V a r i a b l e s M F M F M F M F

GMR*a1"1a2p2"2A 2a33

a3A 3C-n% (0-14)% (15-64)% (65+)&1c

632O12u2u 3X~

5,XrXAB

*The GMR, i t s p e r c e n ta g e d i s t r i b u t i o n a c r o s s t h e t h r e e ma jo r ag e c a t e --go r i es ( i . e . , 0 -14, 15-64 , 65+ ) , and t h e mean age , n , a l l a r e c a l c ul a te dwi th a model sched ule spanning an age range of 95 ye ar s .


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Table 2.1 Parameters and variables defining observed modelmigration schedules: Swedish regions, 1974 (cont.)-

5.Parameters

and WestVar iab le s M F

GMR 0.80 0.82

6. 7 . 8.North Middle- Lower North- Upper North -

Sweden Sweden SwedenM F M F M F


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T hi s me a su re i n d i c a t e s t h a t t h e f i t o f t h e model t o t h e S w e d i s hd a t a i s r e a s on a b l y goo d, t h e e i g h t i n d i c e s o f g o o d n e ss - o f - f i tb e i n g 6 . 8 7 , 6 . 4 1 , 1 2 . i 5 , 1 1 . 0 1 , 9 . 3 1 , 1 0 . 7 7 , 1 1 . 7 4 , a n d 1 C. 82 ,f o r m a l e s a nd 7 . 3 0 , 7 . 2 3 , 7 0 .7 1 , 8 . 7 8 , 9 . 3 1 , 1 1 . 6 1 , 1 1 . 3 8 ,a nd 1 3. 28 f o r f e m a l e s . F i g u r e 2 . 5 i l l u s t r a t e s g r a ,h ic a l lyt h i s g o o d n e s s - o f - f i t o f t h e model s c h e d u l e t o t h e o b se r ve d re -g i o n a l m i g r a t i o n d a t a f o r S w ed is h f em a le s .

Model m i g r a t i o n s c h e d u l e s o f t h e f o rm s p e c i f i e d i n e q u a t i o n( 1 ) may b e c l a s s i f i e d i n t o f a m i l i e s a c c o r d i n g t o t h e r a n g e s o fv a l u e s t a k e n o n by t h e i r p r i n c i p a l p a r a m et e r s . F o r e xa mp le , w emay o r d e r s c h e d u l e s a c c o r d i n g t o t h e i r m i g r a t i o n l e v e l s a sd e f i n e d by t h e v a l u e s o f t h e f o u r l e v e l p a r a m e t e rs i n e q u a t i o n( I ) , i . e . , a l l a 2 t a 3 t a n d c ( o r by t h e i r a s s o c i a t e d g r o s s mi g r a -

p r o d u c t i o n r a t e s ) . A l t e r n a t i v e l y , w e may d i s t i n g u i s h s c h e d u l e sw i t h a r e t i r e m e n t p e ak f ro m t h o s e w i t h o u t o n e , o r w e may r e f e r t os c h e d u l e s w i t h r e l a t i v e l y low o r h i g h v a l u e s f o r t h e r a t e o fa s c e n t A 2 o r t h e mean a g e n. I n many a p p l i c a t i o n s , it i s a l s om ea n i ng fu l t o c h a r a c t e r i z e m i g r a t i o n s c h ed u l e s i n t e r m s o f s e v e r a lo f t h e f un da me nt al me as ur es i l l u s t r a t e d i n F i g u r e 2 . 4 , .s uc h a s t h el o w p o i n t , x t h e h i g h peak , x h ,L' a nd t h e r e t i r e m e n t pe ak , xr .~ s s o k i a t e d i t h t h e f i r s t p a i r o f p o i n t s i s t h e l a b o r 0-rce s h i f t ,X I which i s d e f i n e d t o b e t h e d i f f e r e n c e i n y e a r s b etw ee n t h ea g e s o f t h e h i g h p e a k and t h e l o w p o i n t , i . e . , X = Xh - X 1 Thei n c r e a s e i n t h e m i g r a t i o n r a t e o f i n d i v i d u a l s ag ed xh o v e r t h o s ea g e d x e w i l l b e c a l l e d t h e jump, B .

T he c l o s e c o r r e s po n d e n c e b et w ee n t h e m i g r a t i o n r a t e s o fc h i l d r e n a nd t h o s e o f t h e i r p a r e n t s s u g g e s t s a n o t h e r i m p o r t a n ts h i f t i n o b s e r ve d m i g r a t i o n s c h e d u l e s . I f , f o r e a c h p o i n t xo n t h e po s t -h ig h- pe ak p a r t o f t h e m i g r a t i o n c u r v e , w e o b t a i n(where it e x i s t s ) by i n t e r p o l a t i o n t h e a ge , x - Ax s a y , w i t ht h e i d e n t i c a l r a t e o f m i g r a t i o n on t h e p re - l ow -p oi nt p a r t o ft h e m i g r a t i o n c u r v e , t h e n t h e a v e r ag e o f t h e v a l u e s o f Ax ,c a l c u l a t e d i n c r e m e n t a l l y f o r t h e number o f y e a r s b et we en z e r oa n d t h e l o w - p o in t xt, w i l l b e d e f i n e d t o b e t h e o b se r v e dp a r e n t a l s h i f C , A -


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F i g u r e 2 . 5 O b se rv ed a n d m od el m i g r a t i o n s c h e d u l e s : f e m a l e s ,S w e d i s h r e g i o n s , 1 9 7 4 .


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F i g u r e 2 . 5 O b se r ve d a n d m od el m i g r a t i o n s c h e d u l e s : f e m a l e s ,S w e d i s h r e g i o n s , 1 9 7 4 ( c o n t i n u e d ) .


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An observed (graduated) age-specific migration schedulemay be described in a number of useful ways. For example,references may be made to the heights at particular ages, tolocations of important peaks or troughs, to slopes along theschedule's age profile, to ratios between particular heights orslopes, to areas under parts of the curve, and to both horizontaland vertical distances between important heights and locations.The various descriptive measures characterizing an age-specificmodel migration schedule may be conveniently grouped into thefollowing categories and sub-categories:

B a s i c m e a s u r e s ( t h e 11 f u n d a m e n t a l p a r a m e t e r s a n d t h e i r r a t i o s )h e i g h t s : al a2, a31 cl o c a t i o n s : P2, L13s l o p e s : a l , a2, h2, a j I h3. G l c = al/cI &12 = al/a2,a t i o s . 632 = a a 2'

01 2 = a1/a2, o2 = 1 /a = i3/a32 2 ' 3D e r i v e d m e a s u r e s ( p r o p e r t i e s o f t h e m od e l s c h e d u Z e )

z r e a s : G M R , %(O-14), %(15-641, 5(65+)-l o c a t i o n s : n, x2, XhI Xr

d i s t a n c e s : X I A t B

A convenient approach for characterizing an observed m o d e lmigration schedule [i.e., an empirical schedule graduated byequation (I)] is to begin with the central labor force curve andthen to "add-on" the pre-labor and post-labor force componentsand the constant component. This approach is represented graph-ically in Figure 2.6.

One can imagine describing a decomposition of the model mi-gration schedule along the vertical and horizontal dimensions,e.g., allocating a fraction of its level to the constant compon-ent and then dividing the remainder among the other three (ortwo) components. The ratio 61c = al/c measures the former al-location, and 61 2 = al/a2 and Gj2 = a3/a2 reflect the latterdivision.


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The heights of the labor force and pre-labor force componentsare reflected in the parameters a2 and a respectively, therefore1 'the ratio a2/al indicates the degreeof "labor dominance", and itsreci~rocal, i1 2 = al/a2, the i n d e x of c h i l d d e p e n d e n c y , measuresthe l e v e l at which children migrate with their parents. Thusthe lower the value of 6 1 2 1 the lower is the degree of childdependency exhibited by a migration schedule and, correspondingly,the greater is its labor dominance. This suggests a dichotomousclassification of migration schedules into c h i l d d e p en d en t andl a b o r d o m i n a n t categories*

An analogous argument applies to the post-labor force curve,and 632 = a3/a2 suggests itself as the appropriate index. Howeverit will be sufficient for our purposes to rely simply on the valuetaken on by the parameter X with positive values pointing out3the presence of a retirement peak and a zero value indicatingits absence. High.values of X3 will be interpreted as identifyingr e t i r e m e n t d o m i n a n c e .

Labor dominance reflects the relative migration levels ofthose in the working ages relative to those of children andpensioners. Labor asymmetry refers to the shape of the skewedbell-shaped curve describing the profile of labor-force-age mi-gration. Inagine that a perpendicular line, connecting the highpeak with the base of the bell-shaped curve (i.e., the jump, B),divides the base into two segments X and Y as, for example, inthe schematic diagram:


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C l e a r l y , t h e r a t i o Y / X i s a n i n d i c a t o r o f t h e d e g r e e o f a sy mm etryo f t h e c u r v e . A more c o n v e n i e n t i n d e x , u s i n g o n l y t wo p a r a m e t e r so f t h e m od el s c h e d u l e i s t h e r a t i o o2 = A2/a2 , t h e i n d e x o f l a b ora s y m m e t r y , I t s movement i s h i g h l y c o r r e l a t e d w i t h t h a t o f Y / X ,b e c a u s e o f t h e a pp r o xi m a te r e l a t i o n :

w h e r e % d e n o t e s p r o p o r t i o n a l i t y . T hus o 2 may b e u s e d t o c l a s s i f ym i g r a t i o n s c h e d u l e s a c c o r d i n g t o t h e i r d e g r e e o f l a b o r a sy mm etry .

A g a in , a n a n a l o g o u s ar g u m en t a p p l i e s t o t h e p o s t - l a b o r f o r c ec u r v e , a nd o3 = A3/a3 may b e d e f i n e d t o b e t h e i n d e x o f r e t i r e m e n tasymmetry .

When " a d d in g - o n " a p r e - l a b o r f o r c e c u r v e o f a g i v e n l e v e l .t o t h e l a b o r f o r c e c om pon en t , i t i s a l s o i m p o r t an t t o i n d i c a t es o m e t h i n g o f i t s shape . F or ex am ple , i f t h e m i g r a t i o n r a t e s o fc h i l d r e n m i r r o r t h o s e o f t h e i r p a r e n t s , t h e n a , s h o u l d b e a p p r o x -i m a t e ly e q u a l t o a 2 , a n d B 1 2 = a 1 / a 2 , t h e i n d e x o f p a r e n t a l -s h i f t r e g u l a r i t y , s h o u l d b e c l o s e t o u n i t y .

The Sw ed is h r e g i o n a l m i g r a t i o n p a t t e r n s d e s c r i b e d i nF i g u r e 2 .5 a nd i n T a b l e 2. 1 may b e c h a r a c t e r i z e d i n terms o ft h e v a r i o u s b a s i c a n d d e r i v e d m ea su re s d e f i n e d a bo ve . W e be-g i n w i t h t h e o b s e r v a t i o n t h a t t h e o u t m i g r a t i on l e v e l s i n a l lo f t h e r e g i o n s a r e s i m i l a r , r a n g i n g fr om a low o f . 0 .80 f o r m a le si n Re gio n 5 t o a h i g h o f 1 .48 f o r f e ma l e s i n Reg io n 2 . T h i s s i m -i l a r i t y p e r m i t s a r e a so n a b l y a c c u r a t e v i s u a l a s se s sm e nt an d c h a r-a c t e r i z a t i o n o f t h e p r o f i l e s i n F i g u r es 2 .5 .

L ar ge d i f f e r e n c e s i n g r o s s m i g r ap ro du c t io n r a t e s g i v e r i s et o s l o p e s a nd v e r t i c a l r e l a t i o n s h i p s among s c h e d u l e s t h a t a r en o n- co m pa ra b le when e x am i ne d v i s u a l l y . R e c o u r s e t h e n m u s t b emade t o a s t a n d a r d i z a t i o n o f t h e a r e a s u n de r t h e m i g r a t i o n c u r v e s ,f o r e xa mp le , a g e n e r a l r e - s c a l i n g t o a GMR o f u n i t y . N ote t h a t


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t h i s d i f f i c u l t y d oe s n o t a r i s e i n t h e n um er i c a l d a t a i n T a b l e 2 . 1b e c au s e , a s w e p o in t ed o u t e a r l i e r , t h e p r i n c i p a l s l o p e and l o -c a t i o n p a ra m e te rs a nd r a t i o s u se d t o c h a r a c t e r i z e t h e s c he d u l e sa r e n o t a f f e c t e d by ch an ge s i n l e v e l s . Only h e i g h t s , a r e a s , a ndv e r t i c a l d i s t a n c e s , s uc h a s t h e jump, a r e le v e l- d ep e n de n t m ea su re s.

Among t h e e i g h t r e g i o n s ex am in ed , o n l y t h e f i r s t two e x h i b i ta d e f i n i t e r e t i r e m e n t p e a k , t h e m ale p ea k be i n g t h e more do mi na nto n e i n e a c h c a s e . The i n d e x o f c h i l d d ep en de nc y i s h i g h e s t i nR e gi on 1 a nd l o w e s t i n R e g io n 8 , d i s t i n g u i s h i n g t h e l a t t e r re-g i o n ' s l a b o r d om in an t p r o f i l e f r om St oc k ho lm ' s c h i l d d e pe n de n to u t m i g r a t i o n p a t t e r n . The i n d e x o f l a b o r a sym me try v a r i e s f roma low o f 2 .3 4, i n t h e c a s e o f m al es i n R eg io n 4 t o a h i g h o f4.95 f o r t h e f e m a le o u t m i g r a t i o n p r o f i l e o f R eg io n 8 . F i n a l l y ,w i t h t h e p o s s i b l e e x c e p t i o n o f m a l es i n Re gi on 1 a nd f e m a l e s i nR egion 6 , t h e m i g r a t i o n r a t e s o f c h i l d r e n i n Sweden d o i n d e e dseem t o m i r r o r t h o s e o f t h e i r p a r e n t s . The i n d e x o f p a r e n t a l -s h i f t r e g u l a r i t y i s 1 .2 6 i n t h e f o rm e r c a s e and - 73 0 i n t h el a t t e r ; f o r most o f t h e o t h e r s ch e du l e s it i s c l o s e t o u n i t y .


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3. A C O MPA R A T I V E ANALYSISThe p r e c e d in g s e c t i o n d e mo n st ra te d t h a t a g e - s p e c i f i c r a t e s

o f m i g r a t i o n e x h i b i t a f un da me nt al a g e p r o f i l e t h a t c an be e x-p r e s s e d i n m a th e m a ti c al fo rm c a l l e d a m odel m i g r a t i o n s c h e d u l e ,which i s d e f i n e d by a t o t a l of 1 1 p a r am e t er s . I n t h i s s e c t i o nw e s e ek t o e s t a b l i s h t h e r an g e s o f v a l u e s t y p i c a l l y assumed bye ac h o f t h e s e p a r am e t er s and t h e i r a s s o c i a t e d d e r i v e d v a r i a b l e s .T h is e x e r c i s e i s made p o s s i b l e by t h e a v a i l a b i l i t y o f a r e l a -t i v e l y l a r g e d a t a b a s e c o l l e c t e d by t h e C om pa ra ti ve M ig r at io nand S e t t l e m e n t S t ud y , r e c e n t l y c o nc lu de d a t t h e I n t e r n a t i o n a lI n s t i t u t e f o r A p p l ie d S ys te ms A n a l y s i s (IIA SA ) i n L ax en bu rg ,A u s t r i a ( R o g er s , 19 76 a, 1 9 76 b, 1 97 8; R o ge r s a nd ~ i l l e k e n s , 9 78 ,a nd W i l l e k e n s a nd R o g e rs , 1 9 7 8 ) . The m i g r a t i o n d a t a f o r ea c hc o un t ry i n c l ud e d i n t h i s s t u d y a r e s e t o u t i n t h e i n d i v i d u a ln a t i o n a l r e p o r t s .

3.1 D at a P r e p a r a t i o n , P a ra m et er ~ s t i m a t i o n , n d Summary S t a t i s t i c sThe a g e - s p e c i f i c m i g r a t io n r a t e s t h a t were used t o demon-

s t r a t e t h e f i t s of t h e model m ig r a t io n s c h ed u le i n t h e l a s t sec-t i o n w e r e s i n g le - y e a r r a t e s . Such d a t a a r e v e r y s c a r c e a t t h er e g i o n a l l e v e l an d, i n o u r c om p ar at iv e a n a l y s i s , a r e a v a i l a b l eo n l y f o r S wed en. A l l o t h e r r e g i o n -s p e c i f i c m i g r a t io n d a t a a r er e p o r t e d f o r f i v e - y e a r a g e g r ou p s o n l y a n d, t h e r e f o r e , m us t b ei n t e r p o l a t e d t o p ro v i de t h e n ec e s sa r y i n p u t d a t a by s i n g l e y e a r so f a g e. I n a l l s uc h i n s t a n c e s t h e r e g i o n -s p e c i f i c m i g r a t io ns c h e d u l e s were f i r s t s c a l e d t o a g r o s s m ig ra pr od uc ti on r a t e o fu n i t y (GMR = 1 ) b e f o r e b e i ng s u b je c t e d t o a c u b ic s p l i n e i n t e r -p o l a t i o n ( M c N e i l , T r u s s e l l , an d T u r n er , 1 9 7 7) .

S t a r t i n g w i t h a m i g r a t i o n s c h e d u le w i t h a GM R o f u n i t y a ndr a t e s by s i n g l e y e a r s o f a g e , t h e n o n l i n e a r pa ra m et er e s t i m a t i o na l go r i t h m u l t i m a t e l y y i e l d s a s e t o f e s t i m a t e s f o r t h e models c h e d u l e ' s p a r a m e t er s . * T a b le 2.1 i n S e c t i o n 2 p r e s e n t ed t h er e s u l t s t h a t w e r e o b t a i n e d u s i n g t h e d a t a f o r Sweden. S i n c et h e s e d a t a were a v a i l a b l e f o r s i n g l e y ea r s of a g e , t h e i n f l u e n c eo f t h e i n t e r p o l a t i o n p ro c e du r e c o u ld b e a s s e s s e d . T a b l e 3.1

*See Appendix A f o r d e t a i l s .


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c o n t r a s t s t h e e s t i m a t e s f o r f e ma le s c he d ul es i n T a b l e 2 . 1 w i t ht h o s e o b ta i ne d when t h e same d a t a a r e f i r s t a g g re ga te d t o f i v ey e a r a g e g ro up s and t h e n d is a g g r e ga t e d t o s i n g l e y e a r s o f a g eby a c u b i c s p l i n e i n t e r p o l a t i o n . A c o mp a ri so n o f t h e p a r a m e t e re s t i m a te s i n d i c a t e s t h a t t h e i n t e r p o l a t i o n p r oc ed ur e g i v e sg e n e r a l l y s a t i s f a c t o r y r e s u l t s .

T a b l e 3.1 r e f e r s t o r e s u l t s f o r r a t e s o f m i g r a t io n f rom e ac ho f e i g h t r e g i o n s t o t h e r e s t of Sweden. I f t h e s e r a t e s a r ed i s a g g r e g a t e d by r e g i o n o f d e s t i n a t i o n , t h e n 8 2 = 6 4 i n t e r -r e g i o n a l s c h e d u l e s ne ed t o b e e xa mine d f o r e a c h s e x , c o m p l i c a t i n gc om pa r i so ns a c r o s s s e v e r a l n a t i o n s . To r e s o l v e t h i s d i f f i c u l t y w es h a l l a s s o c i a t e a " t y p i c a l " s c h ed u le w i th e a ch c o l l e c t i o n o fn a t i o n a l r a t e s by c a l c u l a t i n g t h e mean o f e a c h p a ra m e te r a ndd e r i v e d v a r i a b l e . T a b le 3.2 i l l u s t r a t e s t h e r e s u l t s f o r t h eS w e d i s h d a t a .

To a v o i d t h e i n f l u e n c e o f u n r e p r e s e n t a t i v e " o u t l i e r " ob se rv a-t i o n s i n t h e c om pu ta ti o n o f a v e ra g e s d e f i n i n g t h e t y p i c a l n a t i o n a ls c h e d u l e , it was d e c i d e d t o d e l e t e a p p ro x im a t el y 10 p e r c e n t o f t h e" ex tr em e" s c h e d u l e s . S p e c i f i c a l l y , t h e p a r a m e t e rs a nd d e r i v e dv a r i a b l e s w e re o r d e r e d f r om low v a l u e t o h i g h v a l u e ; t h e l owes t5 p e r c e n t an d t h e h i g h e s t 5 p e r c e n t w er e d e f i n e d t o b e e x t r em ev a l u e s . S c h ed u le s w i t h t h e l a r g e s t number of l o w a n d h i g h e x t re m ev a l u e s were d i s c a r d e d , i n se qu en ce , u n t i l o n l y a b o ut 90 p e r c e n t o ft h e o r i g i n a l number o f s c h e d u l e s r e ma in e d. T h i s r e d u c e d s e t t h e ns e r v e d a s t h e p o pu l a t i o n o f s c h e d u le s f o r t h e c a l c u l a t i o n o f v a r -i o u s summary s t a t i s t i c s . T a b le 3.3 i l l u s t r a t e s t h e a v e r ag e pa-r a m e t e r v a l u e s o b t a i n e d w i t h t h e Sw edis h d a t a . S i n c e t h e m ed ia n,mode, s t a n d a r d d e v i a t io n - t o- m e a n r a t i o , a n d l o w e r a n d u p p e r b ou nd sa r e a l s o o f i n t e r e s t , t he y a r e in c l ud ed a s p a r t o f t h e more de-t a i l e d c om pu te r o u t p u t s r e pr o d uc e d i n Ap pe ndix B.


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Table 3.2 Mean values of parameters. efininq the full set of observed modelmigration schedules: Sweden, 8 regions*, 1974 observed data by single yearsof ave until 84 years and over.

SwedenMales k iemaWithout retirement With retirement Without retirement With retirementParameters peak (52 schedules) peak (11 scheduleq) peak (58 schedules) peak (5 schedules)

*Region 1 (Stockholm) is a single-commune region and hence there exists no "intraregional"schedule for it, leaving (812 - 1 = 63 schedules.


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The c o m p a r i s o n , i n T a b l e 3 .1 , o f e s t i m a t e s o b t a i n e d u s i n go ne -y ea r a nd f i ve - ye a r a g e i n t e r v a l s f o r t h e same S we di sh d a t ai n d i c a t e d t h a t t h e i n t e r p o l a t i o n p r o c ed ur e ga ve s a t i s f a c t o r y r e -s u l t s . H o w e v e r , i t a l s o s u g g e s t ed t h a t t h e p a ra m e te r X 2 w a s con-s i s t e n t l y u n d er e s t i m a t ed w i t h f i v e - y e a r d a t a . To c o n f i rm t h i s ,t h e r e s u l t s o f T a b le 3 . 3 w e r e r e p l i c a t e d w i t h t h e S w e d i s h d a t ab a s e , u s i n g a n a g g r e g a t i o n w i t h f i v e - y e a r a g e i n t e r v a l s . The re -s u l t s , s e t o u t i n T a b l e 3 .4 , show o n c e a g a i n t h a t A i s a l w a y s2u n d e r e s t i m a t e d b y t h e i n t e r p o l a t i o n p r o c e d u r e . A lt ho ug h t h e d e-g r e e o f u n d e r e s t i m a t i o n i s n o t l a r g e , t h i s t en d e n c y sh o u l d b en o t e d a n d k e p t i n mind .

I t i s a l s o i m p o r t a n t t o n o t e t h e e r r a t i c b e ha v i o r o f t h er e t i r e m e n t pe ak , a p p a r e n t l y d u e t o i t s e x t r e me s e n s i t i v i t y t ot h e l o s s o f i n f o r m a t i o n a r i s i n g o u t o f t h e a g g r e g a t i o n . T hu s,a l t h o u g h w e s h a l l c o n t i n u e t o p r e s e n t r e s u l t s r e l a t i n g t o t h ep o s t - l a b o r f o r c e a g e s , t h e y w i l l n o t b e a p a r t o f o u r s e a r c hf o r f a m i l i e s o f s c he d u l e s .

3 .2 N a t i o n a l C o n t r a s t sT a b l e s 3 . 3 a n d 3.4 o f t h e p r e c e d i n g s u b s e c t i o n s um ma ri ze d

a v e r a g e p a r a m e t e r v a l u e s f o r 57 m a l e a n d 57 f e m a l e S we d is h m od elm i g r a t i o n s c h e d u l e s . I n t h i s s u b s e c t i o n w e s h a l l e x p an d o u ra n a l y s i s t o i n c l u d e a much l a r g e r d a t a b a se , a d d i n g t o t h e 1 ' 1 4S w ed is h m od el s c h e d u l e s , a n o t h e r 164 s c h e d u l e s f ro m t h e U n i t e dK ingdom (T ab le 3 . 5 ) ; 11 4 f r o m J a p a n a n d 20 f r o m t h e N e t h e r l a n d s(T ab le 3 . 6 ) ; 58 f rom th e USSR, 8 f rom t h e USA, and 32 f ro m Hun-g a r y ( T a b l e 3 . 7 ) . Summary s t a t i s t i c s f o r t h e s e 510 s c h e d u l e sa r e s e t o u t i n A ppen dix B ; o f t h o s e , 206 a r e m al e s c h e d u l e s ,206 a r e f e m a le s c h e d u l e s , a n d 9 8 a r e f o r t h e c o m b i n a t io n o fb o t h s e x e s (males p l u s f e m a le s ) .

* T h i s t o t a l d o e s n o t i n c l u d e t h e 56 s c h e d u l e s ex c lu d e d a s" ex t r em e" s c h e d u l e s . D ur in g t h e p r o c e s s o f f i t t i n g t h e models c h e d u l e t o t h e s e more t h a n 500 i n t e r r e g i o n a l m i g r a t i o n s c h e d u l e sa f r e q u e n t l y e n c o u n t er e d p r ob le m w a s t h e o c c u r r e n c e o f a n e g a t i v ev a l u e f o r t h e c o n s t a n t c . I n a l l s uc h i n s t a n c e s t h e i n i t i a lv a l u e o f c was s e t e q u a l t o t h e l o w e s t o b se rv ed m i g r a t i o n r a t ea nd t h e n o n l i n e a r e s t i m a t i o n p r oc e d u re was s t a r t e d o n c e a g a i n .


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Table 3 . 3 Mean values of parameters defininq the reduced set of observed modelmigration schedules: Sweden, 8 regions, 1974, observed data by single years of ageuntil 8 4 years and over.*

Without retirement With retirement Without retirement With retirementParameters peak ( 4 8 schedules) peak- ( 9 schedules) peak ( 5 4 schedules) peak ( 3 schedules)

*Region 1 (Stockholm) is a single-commune region and hence there exists no intraregionalschedule for it, leaving (812 - 1 = 6 3 schedules, of which 6 were deleted.


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Table 3.4 Mean values of parameters defining the reduced set of observed model migrationschedules: Sweden, 8 regions, 1974, observed data by five years of age until80 years and over.

SwedenMales FemalesWithout retirement With retirement Without retirement With retirementParameters peak (49 schedules) peak (8 schedules) peak (54 schedules) peak (3 schedules)

X3 0.148 0.134*Region 1 (Stockholm) is a single-commune region and hence there exists no intraregional9schedule for it, leaving (81L - 1 = 63 schedules, of which 6 were deleted.


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A s i g n i f i c a n t number o f s c h e d u l e s e x h i b i t e d a p a t t e r n o fm i g ra t io n i n t h e p o s t - l ab o r f o r c e a ge s t h a t d i f f e r e d f rom t h a to f t h e 1 1 - p a r a m e te r m o de l migration s c he d ul e d e f i n e d i n eq u a t io n ( 1 ) .I n s t e a d of a r e t i r e m e n t p ea k, t h e ag e p r o f i l e t o o k o n t h e f ormo f a n "up ward s l o p e " . I n su c h i n s t a n c e s t h e f o l l o w i n g 9- pa ra m et erm o d i f i c a t i o n o f t h e b a s i c m odel m i g r a t i o n was i n t r o d u c e d

The r i g h t h a l f o f T a b l e 3 . 6, f o r e xa mp le , s e t s o u t t h e meanp a ra m e te r e s t i m a t e s o f t h i s m o d if i ed f or m o f t h e model m i g r a t i o ns c h e d ul e f o r t h e N e t he r la n ds .

T a b l e s 3. 3 t h r o u g h 3 .7 p r e s e n t a w e a l t h o f i n f o r m a t i o n a b o u tn a t i o n a l p a t t e r n s o f m i g r a t i o n by a g e. The p a r a m e t e r s , s e t o u t i nc ol um n s, d e f i n e a w i de r a n g e o f m od el m i g r a t i o n s c h e d u l e s .Four r e f e r o n l y t o m i g r a t i o n l e v e l : a a a 3 , a nd c . heir1 ' 2 'v a l u e s a r e f o r a GMR o f u n i t y ; t o o b t a i n co r re s po n di n g v a l u e sf o r o t h e r l e v e l s o f m i g r a t i o n , t h e s e f o u r numbers ne ed t o bem u l t i p l i e d by t h e d e s i r e d l e v e l o f GMR. F o r e x a m p le , t h e o b-s e r v e d GMR f o r f em a le m i g r a t i o n o u t o f t h e S tockh olm r e g i o n i n1974 was 1. 43 . M u l t i p l y i n g a l = 0 .0 29 by 1 . 4 3 g i v e s 0 . 0 4 1 , t h ea p p r o p r i a t e v a l u e o f a , w i t h w hich t o g e n e r a t e t h e m i g r a t i o ns c h e d u l e h a v i n g a GMR o f 1 . 4 3 .

The r e ma i ni n g model s c h e d u l e p a r a m et e r s r e f e r t o m i g r a t i o na g e p r o f i l e : a l I p 2 , a * I 1 2 , u 3 , a j I a nd X 3 . T h e i r v a l u e s re -main c o n s t a n t f o r a l l l e v e l s o f t h e GMR. T aken t o g e t h e r , t h e yd e f i n e t h e a g e p r o f i l e o f m i g r a t i o n f rom o n e r e g i o n t o a n o t h er .S c he du le s w i t h ou t a r e t i r e m e n t p eak y i e l d o n l y t h e f o u r p r o f i l e


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p a r a m e t e r s : a 1 P 2 , a 2 , a nd A 2 , a nd s c h e d u l e s w i t h a r e t i r e m e n ts l o p e have a n a d d i t i o n a l p r o f i l e p a r am e te r , a 3 .

A d e t a i l e d a n a l y s i s o f t h e p a r a m et er s d e f i n i n g t h e v a r i o u sc l a s s e s o f s c he d u l es i s beyond t h e s c op e o f t h i s r e p o r t , n ev er -t h e l e s s a few b a s i c c o n t r a s t s among n a t i o n a l a v e r a g e a g e p r o f i l e smay b e u s e f u l l y h i g h l i g h t e d .

L e t u s b eg in . w i t h a n e x a m in a t io n o f t h e l a b o r f o r c e compo-n e n t d e f i n e d by t h e f o u r p ar a me te r s a 2 , p 2 , a 2 , a nd A 2 . Then a t i o n a l a v e r ag e v a l u e s f o r t h e s e p a ra m e te r s g e n e r a l l y l i e w i t h -i n t h e f o l l ow i ng r a n g es :

I n a l l b u t two i n s t a n c e s , t h e f e m al e v a l u e s f o r a 2 , a 2 , andh 2 a r e l a r g e r t h a n t h o s e f o r m a le s . The r e v e r s e i s t h e c a s e f o rp 2 , w i t h two e x c e p t i o n s , t h e m ost i m p o r t a n t o f w hic h i s e x h i b i t e dby J a p a n ' s f e m a l e s who c o n s i s t e n t l y s how a h i g h p ea k t h a t i so l d e r t h a n t h a t o f m a le s.

The two p a r a m e t e r s d e f i n i n g t h e p r e - l a b o r f o r c e c om p on en t,a and a l , g e n e r a l l y l i e w i t h i n t h e r a n g e s o f 0 .0 1 t o 0 .0 3 a nd10 .0 8 t o 0 .1 2, r e s p e c t i v e l y . The e x c e p t i o n s a r e t h e S o v i e t U niona nd H un gary , w hi ch e x h i b i t u n u s u a l l y h i g h v a l u e s f o r a l . U n l ik et h e c a s e o f t h e l a b o r f o r c e c omponen t, c o n s i s t e n t s e x d i f f e r e n -t i a l s a r e d i f f i c u l t t o i d e n t i f y .

Average n a t i o n a l m i g r a t i o n a g e p r o f i l e s , l i k e mo st a g gr e-g a t i o n s , h i d e more t h a n th e y r e v e a l . Some i n s i g h t i n t o t h e r a n g eso f v a r i a t i o n s t h a t a r e av e ra g ed o u t may b e fo un d by c o n s u l t i n g t h el o w e r a n d u p p e r b o un ds a nd s t a n d a r d - d e v ia t i o n - t o -m e a n r a t i o sl i s t e d i n Appendix B f o r e ac h s e t o f n a t i o n a l s c h ed u l es . Addi-t i o n a l d e t a i l s a r e a v a i l a b l e i n t h e t e c h n i c a l a pp e n d i x t o t h i sr e p o r t * . F i n a l l y , T ab le 3 .8 , i l l u s t r a t e s how p a r am e t e rs v a r y i n

*The t e c h n i c a l a p p e n d ix e n t i t l e d " 63 8 Model M i g r a t i o n S ch e-d u l e s : A T e c h n i c a l A p p e n d i x " i s a v a i l a b l e on r e q u e s t .


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s e v e r a l unaveraged n a t i o n a l s c h e d u l e s , b y way o f ex a mp l e. Themodel s c h e d u l e s p r e s e n t e d t h e r e d e s c r i b e m i g r a t i o n f l o w s o u t o fand i n t o t h e c a p i t a l r e g i on s o f e ach o f s i x c o u n t r i e s : F i n la n d ,Hungary , Jap an , t h e Ne the r la nd s , Sweden, and th e Uni t ed Kingdom.The former s ch ed u l es d e s c r i b e c a p i t a l o u t f l o w p r o f i l e s , t h e l a t -t e r d e f i n e c a p i t a l i n f l o w p r o f i l e s . A l l a r e i l l u s t r a t e d i nF i g u r e 3 . 1.

The m ost a p p a r e n t d i f f e r e n c e b etw ee n t h e a g e p r o f i l e s o ft h e c a p i t a l o u t f l o w and i n f l o w m i g r a t io n s c h e d u l e s i s t h e d o m i -n an ce o f young l a b o r f o r c e m i gr an t s i n t h e l a t t e r , t h a t i s , p r o -p o r t i o n a t e l y more m i g r a n t s i n t h e young l a b o r f o r c e a g e s a p p e a ri n c a p i t a l i n f l o w s c h e d u l es . I n d i c a t i n g t h i s l a b o r dom inancea r e t h e l a r g e r v a l u e s o f t h e p r o d u c t a2A2 i n t h e i n f l o w s ch e-d u l e s and o f t h e r a t i o 6 = a 1 /a 2 i n t h e o u t fl o w s c h ed u l es .12

A seco n d p r o f i l e a t t r i b u t e i s t h e d e g r e e o f a s ym me tr y i nt h e l a b o r f o r c e com ponent o f t h e m i g r a t i o n s c h e d u l e , i . e . , t h er a t i o o f t h e r a t e of a s c e n t X 2 , t o t h e r a t e o f d e s ce n t a 2 , de-f i n e d a s c 2 i n S e c t i on 2 . I n a l l b u t t h e J ap an es e c a s e , t h el a b o r f o r c e c u r ve o f t h e c a p i t a l o ut f l ow p r o f i l e i s more asym-metr ic t h a n t h a t o f t h e c or r es p on d in g i n f l o w p r o f i l e . W e r e f e rt o t h i s c h a r a c t e r i s t i c a s l a b o r asymmetry.

E xa min ing t h e o b s e rv e d r a t e s o f d e s c e n t o f t h e l a b o r an dp re - la b or f o r c e c u r v e s , a 2 a nd a r e s p e c t i v e l y , w e f i n d , f o r1 'exam ple , t h a t t h e y a r e c l o s e t o be in g e q u a l i n t h e o u t f l o w s ch e-d u l e s o f H e l s i n k i a nd S to ck ho lm a nd a r e h i g h l y u n eq u al i n t h ec a s e s o f B u d a pe s t ,T o k y o, a n d Amsterdam . I n f o u r o f t h e s i xc a p i t a l i n f l o w p r o f i l e s a > a l .2 P r o f i l e s w i t h s i g n i f i c a n t l yd i f f e r e n t v a l u e s f o r a 2 and a l , a r e s a i d t o b e i r r e g u l a r .

I n c o n c lu s i on , t h e e m p i r i ca l m i g r a t io n d a t a o f s i x i n -d u s t r i a l i z e d n a t i o n s s u g g e s t t h e f o l l ow i ng h y p o th e s is . Them ig r a t i o n p r o f i l e o f a t y p i c a l c a p i t a l i n f l o w s ch ed ul e i s , i ng e n e r a l , m or e l a b o r d o mi n a n t and m or e l a b o r s y m m e t r i c t h a n t h em i g r a t i o n p r o f i l e o f t h e c or re sp on d in g c a p i t a l o u t f l o w s c h e d u l e .No comparab le hy po th es i s ca n be made re ga rd in g i t s a n t i c i p a t e dd e g re e o f i r r e g u l a r i t y .


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Table 3.5 Mean values of parameters defining the reduced set of observedmodel migration schedules: United ~ i n g d o m , 0 regions, 1970.*

United KingdomMales FemalesWithout retirement With retirement Without retirement With retirementParameters peak (59 schedules) peak (23 schedules) peak (61 schedules) peak (21 schedules)

X 3 0.309 0.403*No intraregional migration data were availa ble; hence (10) - 10 = 90 schedules were analyzedand 8 were deleted.


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Table 3 . 6 Mean values of parameters defining the reduced set of observed modelmigration schedules: Japan, 8 regions, 1970 and the Netherlands, 12 regions, 1974.*

Japan NetherlandsMales Females Males FemalesWith~ut etirement Without retirement With retirement with retirementParameters ~ e a k 57 schedules) Deak (57 schedules) s l o ~ e 10 schedules) s l o ~ e 10 schedules)

*Region 1 in Japan (Hokkaido) is a single-prefecture region and hence there exists no intraregionalschedule for it, leaving ( 8 1 ~ 1 = 6 3 schedules, of which 6 were deleted. The only migrationschedules available for the Netherlands were the migration rates out of each region without regardto destination; hence only 12 schedules were used, of which 2 were deleted.


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T a b l e 3 . 7 Mean v a l u e s o f p a r a m e t e r s d e f i n i n g t h e re d u c e d s e t o f o b s e r v e d mo de l m i g r a t i o ns c h e d u l e s : USSR, 8 r e g i o n s , 1 9 7 4 , USA, 4 r e g i o n s , 3 9 7 0 ~ 7 1 , a n d H u n g a ry , 6 r e g i o n s ,1 9 7 4 . *

I 1 S S R US A Hunaarv---. ~ ..-., - - AT o t a l (M a 1 e s p l u s F e ma l es ) T o t a l ( M a 1 e s p l u s F e m a l e s ) T o t a l ( M a 1 e s p l u s F e m a le s ) T o t a l ( M a 1 e s p l u s F e m a le s )W i t h ou t r e t i r e m e n t p e ak W i th r e t i r e m e n t p e ak W i t ho u t r e t i r e m e n t s l o p e W i t h r e t i r e m e n t s l o p e( 5 8 s c h e d u l e s ) ( 8 s c h e d u l e s ) ( 7 s c h e d u l e s ) ( 2 5 s c h e d u l e s )

A 3 0 . 1 1 9* I n t r a r e g i o n a l m i g r a t io n w a s i n c l u d e d i n t h e USSR a nd ~ u n g a r i a n a t a b u t n o t i n t h e USA d a t a ; h e n ce t h e r e w e re( 3 1 ~ = 6 4 s c h e d u l e s f o r t h e USSR, o f w h i c h 6 were d e l e t e d , ( 6 12 = 3 6 s c h e d u l e s f o r H u n ga r y. o f w h i c h 4 w e re d e l e t e d .

La n d ( 4 ) - 4 = 1 2 s c h e d u l e s f o r t h e USA, o f w h i c h 2 were d e l e t e d b e ca u s e t h e y l a c k e d a r e t i r e m e n t p e ak a nd a n o t h e r2 w e r e d e l e t e d b e c a u s e o f t h e i r e x t re m e v a l u e s .


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E Ii i i i i i . . a l~ a ~ ~ ~ C : p : .

Il l


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Table 3.8 Parameters defining observed total (males plus females) model migration sc hedulesfor capital-region flows: Finland, 1974, Hungary, 1974, Japan, 1970, Netherlands,1974, Sweden, 1974, United Kin gdom, 1970.

Finland Hungary Japanparameters From Helsinki T o Helsinki From Budapest TO Budapest From Tokyo To Tokyo


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e 3 . 8 P a r a m e t e r s d e f i n i n g o b s e r v e d t o t a l ( m a l e s p l u s f e m a l e s ) m o de l m i g r a t i o n s c h e d u l e sf o r c a p i t a l - r e g i o n f l o w s : F i n l a n d , 1 9 7 4; H u ng a ry , 1 9 74 ; J a p a n , 1 9 7 0 ; N e t h e r l a n d s ,1 9 7 4 ; S w ed e n, 1 9 7 4 ; U n i t e d K in g do m , 1 9 7 0 ( c o n t i n u e d ) .

Ne the rla nds Sweden Unit ed KingdomPa ra m et er s From Amsterdam To Amsterdam From Sto ckh olm To Sto ckh olm From London To London


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3 . 3 F a m i l i e s o f S c h e d u le sT h r e e s e t s o f m ode l m i g r a t i o n s c h e d u l e s h a v e b e e n d e f i n e d

i n t h i s p ap e r : t h e 1 1- pa ra me te r s c he d u l e w i t h a r e t i r e m e n t p ea k ,t h e a l t e r n a t i v e 9 -p ar am et er s c h e d u le w i t h a r e t i r e m e n t s l o p e , a ndt h e s i m p l e 7- pa ra m et er s c h e d u le w i t h n e i t h e r p ea k n o r s l o p e .Thus w e have a t l e a s t t h r e e b r oa d f a m i l i e s o f s c h e d u l e s .

A d d i t io n a l d i me ns io ns f o r c l a s s i f y i n g s c he d u le s i n t o f a m i l i e sa r e s u g g e s te d by t h e a b ov e co m p ar a t i v e a n a l y s i s o f n a t i o n a l m ig ra -t i o n a g e p r o f i l e s and t h e b a s i c m ea su re s a nd d e r i v e d v a r i a b l e sd e f i n ed i n S e c t i o n 2 . T he se d i me ns io ns r e f l e c t d i f f e r e n t l o c a -t i o n s on t h e h o r i z o n t a l and v e r t i c a l a xe s o f t h e s c h e d u l e , a sw e l l a s d i f f e r e n t r a t i o s o f s l o p e s and h e i g h t s .

Of t h e 528 model m i g r a t i o n s c h e d u l e s s t u d i e d i n t h i s S e c t i o n ,416 a r e s e x - s p e c i f i c a n d, o f t h e s e , o n l y 336 e x h i b i t n e i t h e r ar e t i r e m e n t p ea k n o r a r e t i r e m e n t s l o p e . B ec au se t h e p a r a m e t e re s t i m a t e s d e s c r i b i n g t h e ag e p r o f i l e o f p o s t - l a b o r f o r c e m ig ra -t i o n a r e u n r e l i a b l e , w e s h a l l r e s t r i c t o u r s e a r c h f o r f a m i l i e so f s c h e d u l e s t o t h e s e 164 m a le a nd 1 72 f e m a l e m od el s c h e d u l e s ,summary s t a t i s t i c s f o r w hic h a r e s e t o u t i n T a b l e s 3 . 9 a nd 3 .1 0 .

An e xa m i n a t io n o f t h e p a r a m e t r i c v a l u e s e x h i b i t e d by t h e 336m i g r a t i o n s c h e d u l e s sum ma riz ed i n T a b l e s 3 . 9 a n d 3 .1 0 s u g g e s t st h a t a l a r g e f r a c t i o n o f t h e v a r i a t i o n e x h ib i t ed by t h e s e s ch e -d u l e s i s a c o ns eq ue nc e o f c ha n ge s i n t h e v a l u e s o f t h e f o l l o w i n gf o u r p a r a m e t er s and d e r i v e d v a r i a b l e s : p 2 , 6 1 2 , a 2 , and B 1 2 .

M i g r a t io n s c h e d u l e s may be e a r l y o r l a t e p e a k in g , d e pe n d in gon t h e l o c a t i o n o f p 2 o n t h e h o r i z o n t a l ( a g e ) a x i s . A lt ho ug ht h i s p a ra m et er g e n e r a l l y t a k e s on a v a l u e c l o s e t o 2 0, r ou gh ly3 o u t o f 4 o b s e r va t i o ns f a l l w i t h i n t h e r an ge o f 17 t o 25 - W es h a l l c a l l t h o s e below a g e 19 a s e a r l y p e a ki ng s c h e d u l e s a n dt h o s e ab o ve 2 2 a s l a t e p ea k i ng s c h e d u l e s .

The r a t i o o f t h e two b a s i c v e r t i c a l p a r a me t e rs , a l a nd a 2 ,i s a me as ur e o f t h e r e l a t i v e i mp or ta nc e o f t h e m i g r a t i o n o f c h i l -d r e n i n a model m i g r a t i o n s c h e d u l e . The i n d e x o f c h i l d d ep en -d en cy , = a , / a 2 , t e n d s t o e x h i b i t mean v a l u e o f a b o u t a t h i r dw i t h 80 p e r c e n t o f t h e v a l u e s f a l l i n g b et we en o n e - f i f t h a nd


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four-fifths. Schedules with an index of one-fifth or less willbe said to be l a bo r d o m i n a n t ; those above two-fifths will becalled c h i d d e p e n d e n t .

Migration schedules with labor force components that takethe form of a relatively symmetrical bell-shape will be said tobe labor symmetrical. These schedules will tend to exhibit anindex of labor asymmetry, o = h2/ct2, that is less than 2.2L a b o r a s y m m e t r i c schedules, on the other hand, will usuallyassume values for o2 of 5 or more. The average migration sche-dule will tend to show a 02-value of about 4, with approximately5 out of 6 schedules exhibiting ao2within the range of 1 to 8.

Finally, the index of parental-shift regularity in manyschedules is close tounity,with approximately 70 percent ofthe values lying between one-third and four-thirds. Valuesof B 1 2 = al/a2 that are lower than four-fifths or higher thansix-fifths will be called i r r e g u l a r .

~ h u s e may image a 3 by 4 cross-classification of migra-tion schedules that defines a dozen "average families."

Introducing a low and a high value for each parameter givesrise to 16 additional families for each of the three classes ofschedules. Thus we may conceive of a minimum set of 60 families,equally divided among schedules with a retirement peak, scheduleswith a retirement slope, and schedules with neither a retirementpeak nor a retirement slope.

peakingp2=20

Dominance612=1/3

+

+

+

RetirementPeakRetirementSlope

ReducedForm

+

+

+

Symmetry02=4+

+

+

~egularityB 1 ,=1+

+

+


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U -0 r=m aJm aa a,3m V)a,d !-Is oa lei

a .cUm r n *k mO C a ,+J O da,+ 3E + J aafua,k k.cF Ua.4Ew 3

k0lei


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e ~ N ~ a Q b m N ~ 8 a O + m N d Q + b ~ N J W Q P Q e N QQ m m ~ P N W ~ . W m Q m Q O S b Q W a a G b P I 6 P I v 1 a N + de Q Q N Q O m M Q 9 3 3 ~ m N a * O e w O * ~ a m ~ ~ 3 O ~D m W ~ ~ + m * O W E G Q Q m * ? m G Q ~ C d 9 6 d m G Q ~ * ?d Q = a d m Q d * Q Q Q Q 3 9 e Q * M W Q a N O m * m m m Q.................... w e . . . L . . . . . . a , . . c......t B - m ~ m m m m Q B m Q Q ~ m Q O Q P m m m p I Q = N C ) + ~ Q1 - 1 IU I P INSd dN (U


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To com plem ent t h e a bo ve d i s c u s s i o n w i t h a f ew v i s u a l i l l u s -t r a t i o n s , w e p r e s e n t i n F i g ur e 3.2A s i x l a b o r do mi na nt p r o f i l e s ,w i t h 6 1 c f i x e d a t 22 . The t a l l e s t t h r e e e x h i b i t a s t e e p r a t e o fd e s c e n t a 2 = . 3 ; t h e s h o r t e s t t h r e e s how a much m ore m o d e r a t es l o p e o f a 2 = . 0 6 . W i th in e ac h f a m i l y o f t h r e e c u r v e s , o n e f i n d sv a r i a t i o n s i n u and i n t h e r a t e o f a s c e n t , X 2 . I n c r ea s i n g t h e2fo rm er s h i f t s t h e c u rv e t o t h e r i g h t a lo ng t h e h o r i z o n t a l a x i s ;i n c r e as i n g t h e l a t t e r p a r a m et e r r a i s e s t h e r e l a t i v e h e i g h t o f t h eh i g h p e a k .

The s i x s c h e d u l e s i n F i g u r e 3.2B d e p i c t t h e c o r r e s p o nd i n gtwo f a m i l i e s o f c h i l d d ep en de nt p r o f i l e s . The r e s u l t s a r e g en -e r a l l y s i m i l a r t o t h o s e i n F i g ur e 3 .2At w i t h t h e ex c ep t io n t h a tt h e r e l a t i v e i m po rt an ce o f m i g r a t i on i n t h e p r e- la bo r f o r c e ag eg r o u p s i s i n c r ea s e d c o n s i de r a b ly . The p r i n c i p a l e f f e c t s o f t h echange i n a r e : ( 1 a r a i s i n g o f t h e i n t e r c e p t a l + c a l o n gt h e v e r t i c a l a x i s , a nd ( 2 ) a s i m ul ta ne ou s r e d u c t i o n i n t h e h e i g h to f t h e l a b o r f o r c e c omponent i n o r d e r t o m a i n t a i n a c o n s t a n t a r e ao f u n i t y u n d e r e a c h c u r v e .

F i n a l l y , t h e d o z en s c h e d u l e s i n F i g u r e s 3 .2C a n d 3.2D d e-s c r i b e s i m i l a r f a m i l i e s o f m i g r a t i on c u r v es , b u t i n t h e s e p ro -f i l e s t h e r e l a t i v e c o n t r i b u t i o n o f t h e c o n s t a n t component t o t h eu n i t GMR h a s b ee n i n c r e a s e d s i g n i f i c a n t l y ( i t le = 2 . 6 ) . I ti s i m p or t an t t o n o t e t h a t s uc h " p ur e" m ea su re s o f p r o f i l e a s x R ,xh, X , and A r e m ai n u n a f f e c t e d by t h i s c h a n g e , w he r ea s " i m pu r e"-p r o f i l e m e a s ur e s , s u c h a s t h e mean a g e o f m i g r a t i o n , n , now t a k eon a d i f f e r e n t s e t o f v a l u e s .


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. .l-l k7 d da (a aa, UC .4 4JU c, Cm a, a,C k4J c, a,C 0 urd a'uC h-4- 4 C\I 3:aE *0 IIa c\( N


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3 .4 S e n s i t i v i t y A n a ly s i sT he p r e c e d i n g s u b s e c t i o n s h a ve f o c u s e d o n a c o m p a r is o n o f

t h e f u n d am e n ta l p a r a m e t e r s d e f i n i n g t h e m od el m i g r a t i o n a g e pr o -f i l e s o f a number of n a t i o n s . The c om p ar i so n y i e l d e d r a n g e s o fv a l u e s w i t h i n w hi ch e a c h p a ra m e te r may b e e x p e c t e d t o f a l l a n ds u g g e s t e d a c l a s s i f i c a t i o n o f s c h e d u l e s i n t o f a m i l i e s . W e nowt u r n t o a n a n a l y t i c e x a m in a t i on o f how c ha n g e s i n s e v e r a l o f t h em ore i m p o r t a n t p a r a m e t e r s become m a n i f e s t e d i n t h e a g e p r o f i l eo f t h e m odel s c h e d u le . F o r a n a l y t i c a l c o n ve n ie n ce w e b e g i n byf o c u s i ng o n t h e p r o p e r t i e s o f t h e d o u b l e e x p o n e n t i a l c u r v e t h a td e s c r i b e s t h e l a b o r f o r c e component :

W e b e g i n b y o b s e r v i n g t h a t i f a i s s e t e q u a l t o h 2 i n t h e2a bo ve e x p r e s s i o n , t h e n t h e l a b o r f o r c e com po nen t a ss um es t h es h a pe o f a well-known e x t r e m e v a l u e d i s t r i b u t i o n u s e d i n t h es t u d y o f f l o o d f l o w s (G um be l, 1 9 4 1 ; K i m b al l , 1 9 4 6 ) . I n s u c h acase t h e f u n c t i o n f 2 ( x ) i s s y m m e t r i c a l a r o u n d i t s mean xh = p 2a nd r e a c h e s i t s maximum, y h, a t t h a t p o i n t . To a n a l y z e t h e moreg e n e r a l case w h e r e a 2 # h 2 , w e may d e r i v e a n a l y t i c a l e x p r e s s i o n sf o r b o t h o f t h e s e v a r i a b l e s by d i f f e r e n t i a t i n g e q u a t i o n ( 4 ) w i t hr e s p e c t t o x , s e t t i n g t h e r e s u l t e q u a l t o z e r o , a nd t h e n s o l v i n gt o f i n d

a n e x p r e s s i o n t h a t d o e s n o t i n v o l v e a and2 '

a n e x p r e s s i o n t h a t d o e s n o t i n v o l v e p 2Note t h a t i f X 2 > a 2 , w h ic h i s almost a lw ay s t h e c a s e , t h e n

xh > p 2 . And o b s e r v e t h a t i f a 2 = h 2 , t h e n t h e a bo ve t w o e q u a t i o n s


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s i m p l i f y t o

and

S i n c e p 2 a f f e c t s x h o n l y a s a d i s p la c em e n t, w e may focuson t h e v a r i a t i o n o f xh a s a f u n c t i o n o f a 2 an d X 2 . A p l o t o fxh a g a i n s t a 2 , f o r a f i x e d X 2 , shows t h a t i n c r e a s e s i n a 2 l e a dt o d e c r e as e s i n x

h 'A n al og o us ly , i n c r e a s e s i n X f o r a f i x e d2 'a 2 , p r o d uc e i n c r e a s e s i n xh b u t a t a r a t e t h a t d e c r e as e s r a p i d l y

a s t h e l a t t e r v a r i a b l e a p p r o a c h e s i t s a sy m p t o t e .Th e b e h a v i o r o f y h i s i n d e p e n d e n t o f p 2 an d v a r i e s p r op or -

t i o n a t e l y w i th a 2 . Hence i t s v a r i a t i o n a l s o d e pe nd s fundamen-t a l l y o n l y on t h e two v a r i a b l e s a and A 2 . A p l o t o f yh a g a i n s t2a 2 , f o r a f i x e d X 2 , g i v e s r i s e t o a U-shaped c u r v e t h a t r e a c h e si t s minimum a t a = X 2 . I n c r e a s i n g X w id en s t h e s h ap e o f t h e2 2u .

The i n t r o d u c t i o n o f t h e p r e - la b o r f o r c e com ponent i n t o t h ep r o f i l e g e n e r a l l y moves xh t o a s l i g h t l y y ou ng er a ge and r a i s e s

-a Xyh b y a b o u t a l e h , u s u a l l y a n e g l i g i b l e q u a n t i t y . The a d d i -t i o n o f t h e c o n s ta n t t e r m c , o f c o u r s e , a f f e c t s o n ly yh , r a i s i n gi t b y t h e amount o f t h e c o n s t a n t . Thus t h e m i g r a t i o n r a t e a ta g e x may b e e x p r e s se d a sh

A v a r i a b l e t h a t i n t e r r e l a t e s t h e p r e - l a b o r and l a b o r f o r c ecomponents i s t h e p a r e n t a l s h i f t , A . To s i m p l i f y o u r a n a l y s i so f i t s d e pe nd e nc e o n t h e f u n d a m e n ta l p a r a m e t e r s , it i s c o n v e n i e n tt o a ssume t h a t a a nd a 2 a r e a p p ro x im a te l y e q u a l . I n s u ch i n -1s t a n c e s , f o r a ge s i mm e d ia t el y f o l l ow i n g t h e h ig h pe ak x h, t h el a b o r f o r c e com po nen t o f t h e m odel m i g r a t i o n s c h e d u l e i s c l o s e l ya p p ro x im a t ed by t h e f u n c t i o n


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Recalling that the pre-labor force curve is given by

when a l = a 2 , we may equate the two functions to solve for thedifference in ages that we have called the parental.shift:

This equation shows that the parental shift will increasewith increasing values of u and will decrease with increasing2values of a and 6 1 2 .2 Table 3 .11 compares the values of thisanalytically defined "theoretical" parental shift with the cor-responding observed parental shifts presented earlier in Table 2 .1for Swedish males and females. The two definitions appear to pro-duce similar numerical values, but the analytical definition hasthe advantage of being simpler to calculate and analyze.

Consider the rural-to-urban migration age profile definedby the parameters in Table 3 . 1 2 . In this profile the values ofa 2 and A 2 are almost equal making it a suitable illustration ofseveral points raised in the above discussion.

First, calculating xh with equation ( 5 ) gives

as against the xh = 2 1 . 5 9 set out in Table 3 . 1 1 . Deriving yhwith equation (6) gives

where a2/A2 = 0 . 2 3 7 / 0 . 2 7 0 = 0 . 8 7 8 . Thus M(21 . 5 9 ) is approxi-mately equal to yh + c = 0 . 0 69 + 0 . 0 0 4 = 0 . 0 7 3 . The value givenby the model migration schedule equation is also 0 . 0 7 3 .


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T a b l e 3 . 1 1 O bs er ve d a nd t h e o r e t i c a l v a l u e s o f t h e p a r e n t a l s h i f t : S we di sh r e g i o n s , 1 9 7 4 .( a S o u r c e : T a b l e 2 . 1 ; b ~ o u r c e : R o ge r s, R a q u i l l e t , a nd C a s t r o L 1 9 7 8 1 , p . 4 9 7 . )

L

?

REGIONS OF SWEDEN--1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 .T he P a r e n t a l S h i f t E a s t S o u t h N o r th U pp er U pp erS t o ck h o l m Mi d d l e - Mi d d l e - . So u t h West Mi d d l e - N o r t h - N o r t h -Sweden Sweden Sweden Sweden Sweden

aO b s e r v e d , m a l e s 2 7 . 8 7 2 9 . 8 6 2 9 . 9 1 2 9 .8 9 2 9 . 5 7 2 9 . 9 2 3 0 . 1 5 3 1 . 6 1T h e o r e t i c a l , b m a l e s 2 6 . 6 7 2 8 . 9 7 2 9 . 6 3 2 9 . 7 4 2 8 . 8 4 2 9 . 4 3 2 9 . 7 4 3 0 . 5 9

aO b s e r v e d , f e m a l e s 2 5 . 4 7 2 7 . 2 1 2 7 . 2 6 2 7 . 8 7 2 7 . 4 2 2 7 . 0 9 2 6 . 9 4 2 8 . 3 6T h e o r e t i c a l , b f e m a l e s 2 4 . 4 9 2 6 . 3 3 2 7 . 5 1 2 8 . 2 1 2 7 . 1 9 2 7 . 6 9 2 7 . 5 3 2 8 . 5 9


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Since a, # a2, we cannot adequately test the accuracy ofequation ( 7 ) as an estimator of A. Nevertheless, it can be usedto help account for the unusually large value of the parentalshift. Substituting in the values for p a*, and 612 into2'equation (7) we find

And although this is an underestimate of 45.13, it does suggestthat the principal cause for the unusually high value of A isthe unusually low value of 612' Had this latter parameter thevalue found for Stockholm's males, for example, the parentalshift would exhibit the much lower value of 22.52.


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T a b l e 3 .1 2 P a r a m e t er s a nd v a r i a b l e s d e f i n i n g o b s e r v e du r b a n / r u r a l model m i g r a t i o n s c h e d u l e s f o ru r b a n / r u r a l f l ow s : USSR, 1974.

V ar i a b l e s USSRa nd T o t a l ( M al es p l u s F e m a le s )P a r a m e t e r s U rb an t o R u r a l R u r a l t o U rb anGMR


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4. SYNTHETIC MODEL MIGRATION SCHEDULES: I. THE CORRELATIONALPERSPECTIVEA z y n t h e t i c model schedule is a collection of age-specific

rates that is based on patterns observed in various populationsother than the one being studied and some incomplete data on thelatter. The justification for such an approach is that age pro-files of fertility, mortality, a n d m i g ra t i o n vary within predeter-mined limits for most human populations. Birth, death, and migra-tion rates for one age group are highly correlated with the corre-sponding rates for other age groups, and expressions of such inter-relationships form the basis of model schedule construction. Theuse of these regularities to develop synthetic (hypothetical)schedules that are deemed to be close approximations of the unob-served schedules of populations lacking accurate vital and mobil-ity registration statistics has been a rapidly growing area ofcontemporary demographic research.

4 . 1 Introduction: Alternative PerspectivesThe earliest efforts in the development of model schedules

were based on only one parameter and hence had very little flex-ibility (United Nations, 1 9 5 5 ) . Demographers soon discoveredthat variations in the mortality and fertility regimes of dif-ferent populations required more complex formulations. Inmortality studies greater flexibility was introduced by provid-ing families of schedules (Coale and Demeny, 1 9 6 6 ) or by enlarg-ing the number of parameters used to describe the age pattern(Brass, 1 9 7 5 ) . The latter strategy was also adopted in thecreation of improved model fertility schedules and was augmentedby the use of analytical descriptions of age profiles (Coale andTrussell, 1 9 7 4 ) .

Since the age patterns of migration normally exhibit a greaterdegree of variability across regions than do mortality and fertil-ity schedules, it is to be expected that the development of anadequate set of model migration schedules will require a greaternumber both of families and of parameters. Although many alter-native methods could be devised to summarize regularities in the


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form of families of model schedules defined by several parameters,three have received the widest popularity and dissemination:

1. the regression approach of the Coale-Demeny model lifetables (Coale and Demeny 1966

2. the logit system of Brass ( ~ r a s s , 971), and3. the double-exponential graduation of Coale, M c ~ e i l nd

Trussell (Coale, 1977, Co.ale and McNeil, 1972, and Coaleand Trussell, 1974)

The regression approach embodies a c o r r e l a t i o n a l perspectivethat associates rates at different ages to an index of level,where the particular associations may differ from one "family" ofschedules to another. For example, in the Coale-Demeny model lifetables the index of level is the expectation of remaining life atage 10, and a different set of regression equations is establishedfor each of four "regions" of the world.*

Brass's logit system reflects a r e l a t i o n a l perspective inwhich rates at different ages are given by a standard schedulewhere shape and level may be suitably modified to be appropriatefor a particular population.

The Coale-Trussell model fertility schedules are relational inperspective (they use a Swedish standard first-marriage schedule),but they also introduce an analytic description of the age profileby adopting a double exponential curve that defines the shape ofthe age-specific first-marriage function.

In this section and the next we mix the above three approachesto define two alternative perspectives for creating syntheticmodel migration schedules to be used in situations where onlyinadequate or defective data on internal (origin-destination)migration flows are available. Both perspectives rely on theanalytic (double plus single exponential) graduation defined bythe basic mod

MODEL SCHEDULES IN MULTI STATE DEMOGRAPHIC ANALYSIS: THE CASE OF MIGRATION (1981)

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