More Computer Programs for Spatial Demographic Analysis

-MORE COMPUTER PROGRAMS F O R SPATIAL DEMOGRAPHIC ANALYSIS

F r a n s W i l l e k e n s A n d r e i R o g e r s

J u n e 1 9 7 7

Research Memoranda are interim reports on research being conducted by the International Institute for Applied Systems Analysis, and as such receive only limited scientific review. Views or opinions contained herein do not necessarily represent those o f the Institute or o f the National Member Organizations supporting the Institute.

Preface

Interest in human settlement systems and policies has been a critical part of urban-related work at IIASA since its incep- tion. Recently this interest has given rise to a concentrated research effort focusing on migration dynamics and settlement patterns. Four sub-tasks form the core of this research effort:

I. the study of spatial population dynamics;

11. the definition and elaboration of a new research area called demometrics and its application to migration analysis and spatial population fore- casting;

111. the analysis and design of migration and settle- ment policy;

IV. a comparative study of national migration and settlement patterns and policies.

As part of the comparative study of migration and settle- ment, IIASA ,is developing a set of computer programs for spatial demographic analysis. A first set of programs has already been published (RM-76-58). This paper presents another set - - - one

focusing on 'the analysis of stationary and stable multiregional populations.

Related papers in the comparative studies series, and other publications of the migration and settlement study, are listed on the back page of this report.

Andrei Rogers Chairman

Human Settlement & Services Area

May 1977

iii

A b s t r a c t

T h i s r e p o r t p r e s e n t s t h e a l g o r i t h m s a n d l i s t s t h e FORTRAN I V c o d e s o f computer programs f o r t h e a n a l y s i s o f m u l t i r e g i o n a l p o p u l a t i o n s y s t e m s . I t i s a c o n t i n u a t i o n o f t h e IIASA r e p o r t RM-76-58. The f o l l o w i n g t o p i c s a r e i n - c l u d e d : m o b i l i t y and f e r t i l i t y a n a l y s e s o f l i f e t a b l e a n d s t a b l e p o p u l a t i o n s ; methodology a n d a p p l i c a t i o n s o f t h e s p a t i a l r e p r o d u c t i v e v a l u e ; and t h e s t u d y o f t h e s p a t i a l d e m o g r a p h i c i m p a c t s o f f e r t i l i t y r e d u c e d t o r e - p l a c e m e n t l e v e l . T h i s r e p o r t f o c u s e s on t h e i n t e r p r e t a t i o n o f t h e o u t p u t o f t h e computer p r o g r a m s .

Acknowledgements

The numerous r e a c t i o n s t o o u r f i r s t s e t o f computer p r o - grams (RM-76-58)

were

e x t r e m e l y h e l p f u l f o r t h e p r e p a r a t i o n o f t h i s r e p o r t . I n p a r t i c u l a r , w e acknowledge t h e d e t a i l e d comments o f Tom C a r r o l , R i c h a r d R a q u i l l e t a n d P h i l i p R e e s . We a l s o hope t h a t t h i s r e p o r t w i l l p r o v o k e r e a c t i o n s a n d s u g g e s t i o n s t h a t m i g h t i m p r o v e t h e u s e r - o r i e n t a t i o n o f t h e computer p r o g r a m s .

D u r i n g t h e d e v e l o p m e n t o f t h e p r o g r a m s , w e have b e n e f i t e d from t h e a s s i s t a n c e o f IIASA's Computer S e r v i c e s . We a r e

e s p e c i a l l y i n d e b t e d t o James C u r r y a n d Mark P e a r s o n f o r t h e i r a d v i c e a n d f o r s o l v i n g o u r s o f t w a r e p r o b l e m s .

W e

a l s o a r e g r a t e f u l t o J a c q u e s L e d e n t a n d R i c h a r d R a q u i l l e t who r e a d a n e a r l i e r v e r s i o n o f t h i s r e p o r t a n d s u g g e s t e d s e v e r a l

improvemente.

The b u r d e n o f t y p i n g t h e s u c c e s s i v e d r a f t s o f t h i s r e p o r t

war

b o r n e by L i n d a Samide, E l i s a b e t h G r a n d v i l l e , M a r i n a Hornasek a n d 8 o n j a Belwyn, i n c h r o n o l o g i c a l o r d e r .

W e

a p p r e c i a t e w i t h

many

t h a n k r t h e

skills

and e f f o r t 6 t h e y d e v o t e d t o t h i s r e p o r t ,

i n

p a r t i c u l a r t h e c o n t r i b u t i o n o f S o n j a who t y p e d t h e f i n a l v e r r i o n .

E R R A T A

Willekens F., and Rogers A.,

Computer Programs for Spatial Demographic Analysis RM-76-58, July 1976

1. p. 21, eq. 12:

a

(10) = (5) +

10a12

(5)

2. p. 21, eq. 13:

a

⁽¹⁰⁾ =

10a2

( 5 )

P~~

(5) + ^{(5) p12 (5)} 10 2

3. p. 28, eq. 28: e(x)

-

^{= T(x)}

-

^p(x)

- -

^(o)]

,

where I(x) is the diagonal matrix with elements

i - -

^{1 1}

' a (x) ,

^or io" (x)

.

7

4. p. 28, bottom: e(10)

-

^{= T(10)}

-

^[~(10)

- -

^Q-~(O)]-~

5. p. 3 0 , top:

1 0.897131 ₌

[

^55.264362 0.799071 L.96707

o. :go7J

7.702079 63.154427 7.448485 56.251442

where the matrix inverse is [~(10)

- -

^(o)]-~.

vii

T a b l e o f C o n t e n t s

P a g e P r e f a c e

. . .

i i i.

A b s t r a c t a n d A c k n o w l e d g e m e n t s

...

^v

E r r a t a

...

^{v i i}

1. THE POPULATION DISTRIBU'I'ION 13Y AGE ANT) 2

REGION

. . .

1 . 1 T h e L,ife T a b l e P o p u l a t i o n

. . .

³

1 . 2 '].'he S t a b l e P o p ~ l l a t i o n . . . 8 2 . " ' E R T T L I r r Y A N A L , ' , S T S

. . .

'I 0 2 . 1 ' l ' h e C e n e l - a l i z e t l N e t MaLer11 i t y 1;'11not- ~ ( J I I

. . .

¹²

2 . 2 'Ilhe W e i . q h t e d C t ~ ~ ? ~ - ? r a l . i z c c : l N e t M a . l c l r l . l i l \ ,

F ' u l l c t - i o n

. . .

3 . MORIT,:TT'Y ANA1,Y::lS

. . .

3 . 1 'T'he G e n c r a : L i z e , l N e t P.lolt i l i l..y I ~ ' L I I I C : I: i ^{O I I} . . .

3 . 2 ' I t h e blei,cjlrk.ed ^(I;, ! n e r a l i . z r , d N t h ( b 1 0 t . 1 ~ 1 ^I L.11 ~ " I I I I C : ~ I ( J ~ I

- . . .

I;'E:HIL'I f,T1!'Y ANAT,"S 1s: ' I I N I L

'rhe 'I'lic_.ory of ^I h e S p a t i \ t ' l l ? t - ~ ! , ~ ~ o ( l ~ ~ c : l : i

V a l - u e

. . .

'l'lie ( . ' o n \ ~ ) ~ . ~ l l a t i o ~ ~ o f t h e S p ~ l t . i . , ~ ~ l l < c ? l , r o ( ~ l l ~ c . : l I V C ,

V a l u e

. . .

E'URII'IIER S1l'AU.Lb.: E'C)PUL,Arl' 1 C)N AN/\ L<:'ST S

. . .

T h e U l t i r n a t ~ e ' I ' c a j e c t o r l . o f B.i t.t-11s a ~ ~ d [ ~ o [ , u l ^{d - - -} t i o n s

. . . . . .

T h e S t a b l e E q u i v a l e n t s a n d I n t r i n s i c Rates

SPATIAL ZERO POPULATION GROWTH

...

T h e N u m e r i c a l A ~ p r o a c h

...

Page

...

6.2 The Analytical Approach

7

.

PROGRAM DESCRIPTION

...

...

7.1 The General Purpose Subroutines

...

7 . 2 The S p c i a l Purpose Subroutines

. ...

7 2 Tne ?lain Program

7.4 The Input Data

...

APPENDICES :

1

I

.

Glossary of Mathematical Symbol:; and FORTRAN Names of Demographic Variables

...

...

2

.

^List~.ng of Computer Programs

REFERENCES

More Computer Programs for SDatial Demoara~hic Analvsis

One of the objectives of the Migration and Settlement Study at IIASA is to develop a package of computer programs for spatial demographic analysis. The reasoning has been that a basic re- quirement for an effective policy regarding the growth and the distribution of the population is a well-developed understanding of spatial population dynamics. Such an understanding is enhanced if the analyst and the policy-maker are provided with a ready tool for analysis, one which encompasses both the existing methodolo- gical knowledge and the computational procedures necessary to implement the methodology. This tool is a set of computer pro- grams.

A first set of programs for spatial demographic analysis has already been published (Willekens and Rogers, 1976)

.

^They

include the computation of the multiregional life table and the projection of a multiregional population system forward in tkme until it stabilizes. This paper focuses on the analysis of stable populations.' It consists of seven sections. The first section focuses on the basic input for the analysis: the age and regional distribution of the population. Demographers use three types of population distributions: the observed population distribution, the stationary population distribution, as expressed by the life table, and the stable population distribution. For each type of distribution fertility, mortality and mobility analyses may be performed. This is the task of sections two to five. Sections two and four deal with fertility analysis;

section three treats mobility; and section five derives inter- esting stable population characteristics. The sixth section studies the spatial consequences of a sudden drop of fertility

' A stable population is a population in steady-state equili-

brium. It is a zero-growth population only if the stable rate of growth is zero.

t o r e p l a c e m e n t l e v e l . The c h a r a c t e r i s t i c s o f s p a t i a l ZPG-popu- l a t i o n s a r e d e r i v e d n u m e r i c a l l y and a n a l y t i c a l l y . The l a s t s e c t i o n o f t h e p a p e r p r e s e n t s a u s e r - o r i e n t e d d e s c r i p t i o n o f t h e computer p r o g r a m s . The a c t u a l l i s t i n g s o f t h e programs a r e c o n t a i n e d i n t h e Appendix.

T h i s p a p e r f o c u s e s o n t h e i n t e r p r e t a t i o n o f t h e o u t p u t o f t h e computer programs. A l l n u m e r i c a l i l l u s t r a t i o n s r e f e r t o t h e same r e a l t w o - r e g i o n s y s t e m : S l o v e n i a and t h e R e s t o f

Y u g o s l a v i a . The demographic d a t a on which t h e c o m p u t a t i o n s a r e b a s e d r e f e r t o t h e y e a r 1961 and a r e g i v e n i n Rogers ( 1 975a)

.

The same example h a s been u s e d t o i l l u s t r a t e t h e p r e v i o u s p r o - grams ( W i l l e k e n s and R o g e r s , 1 9 7 6 ) . The m u l t i r e g i o n a l l i f e t a b l e and t h e s t a b l e p o p u l a t i o n computed t h e r e a r e u s e d a s

i n p u t i n f o r m a t i o n i n t h i s p a p e r .

1 . THE POPULATION DISTRIBUTION BY AGE AND REGION

The dynamics o f a m u l t i r e g i o n a l p o p u l a t i o n s y s t e m a r e g o v e r n e d by f e r t i l i t y , m o r t a l i t y and m i g r a t i o n . A g e - s p e c i f i c r a t e s o f f e r t i l i t y , m o r t a l i t y and m i g r a t i o n a r e t h e f u n d a m e n t a l components o f d e m o g r a p h i c a n a l y s i s ( R o g e r s and W i l l e k e n s

,

1976c)

.

They d e t e r m i n e n o t o n l y t h e growth o f t h e p o p u l a t i o n , b u t a l s o ( i n t h e l o n g r u n ) i t s a g e c o m p o s i t i o n , s p a t i a l d i s t r i b u t i o n , and c r u d e r a t e s .

The o b s e r v a t i o n t h a t a u n i q u e c o m b i n a t i o n o f a g e - s p e c i f i c r a t e s r e s u l t s i n a p a r t i c u l a r a g e and r e g i o n a l c o m p o s i t i o n

h a s i n d u c e d demographers t o r e a d i n e a c h p o p u l a t i o n d i s t r i b u t i o n a p a r t i c u l a r s e q u e n c e o f v i t a l r a t e s . "The d e m o g r a p h i c h i s t o r y o f a p o p u l a t i o n i s i n s c r i b e d i n i t s a g e d i s t r i b u t i o n " ( K e y f i t z , e t a l . , 1967, p. 862; see a l s o Namboodiri, 1 9 6 9 ) . F o r example, a n o b s e r v e d p o p u l a t i o n d i s t r i b u t i o n ( p o p u l a t i o n p y r a m i d ) re-

f l e c t s p e r i o d s o f h i g h f e r t i l i t y (baby boom) and h i g h m o r t a l i t y ( w a r s ) . A p a r t i c u l a r l y u s e f u l way f o r u n d e r s t a n d i n g how t h e a g e and r e g i o n a l s t r u c t u r e o f a p o p u l a t i o n i s d e t e r m i n e d , i s t o i m a g i n e a p a r t i c u l a r d i s t r i b u t i o n a s d e s c r i b i n g a p o p u l a t i o n which h a s been s u b j e c t e d t o c o n s t a n t f e r t i l i t y , m o r t a l i t y and m i g r a t i o n s c h e d u l e s f o r a p r o l o n g e d p e r i o d o f t i m e . The

population that develops under such circumstances is called a stable multiregional population.

We may now reverse the procedure and derive the population distribution that would evolve if the actual observed schedules would remain unchanged for a prolonged period of time. This is the stable population associated with the observed demographic behavior. It is obvious that the age-specific rates do not

remain constant and therefore that a stable population will never be realized. However, the stable population is a concept that enables us to look behind observed rates to explore what is

hidden in the current fertility, mortality, and migration behav- ior. It shows where the system is heading,.in the long run, under the current demographic forces. Keyfitz (1972, p. 347) compares stable population analyses with "microscopic examina- tions" because they magnify the effects of differences in current rates and therefore show more clearly their true meaning.

Rogers (1971, p. 426) and Coale (1972, p. 52) compare these to ."speedometer readings" to emphasize their monitoring function and hypothetical nature.

In addition to the observed population distribution and the stable population distribution associated with the observed

fertility, mortality and migration schedules, demographers usually consider a third population distribution, namely the distribution of the life table population. The multiregional life table is a device for exhibiting the mortality and mobil- ity history of an arbitrary birth cohort or radix. The repre- sentation and interpretation of life table and stable

populations will now be discussed in some more detail.

1.1 The Life Table Po~ulation

The population distribution that results from applying given mortality and migration schedules to regional radices is repre-

sented by R

-

(x) and L (x) of the life table (Rogers, 1975a)

- .

^The

matrix R(x) represents the distribution of the population of

-

exact age x, whereas L(x) denotes the distribution of the popu-

-

lation in age group x to x

+

h, with h being 5 (age intervals

o f 5 y e a r s ) . The m a t r i x R ( x ) w i l l b e u s e d i n t h e c o n t i n u o u s

-

m o d e l s , a n d L ( x ) f o r t h e d i s c r e t e app::oximations

-

o f t h e con- t i n u o u s m o d e l s . F o r e x a m p l e , f o r a t w o - r e g i o n s y s t e m ,

F o r u n i t r e g i o n a l r a d i c e s , a n e l e m e n t L . ( x ) d e n o t e s i 0 7

t h e number o f p e o p l e o f r e g i o n j i n a g e g r o u p x t o x

+

4 , who

9

w e r e b o r n i n r e g i o n i , p e r u n i t b i r t h i n i . L F o r a r b i t r a r y r a d i c e s

t h e number a f p e o p l e i n r e g i o n j b e t w e e n a g e s x a n d x

+

5 a n d b o r n i n i i s L

.

( x ) Qi a

,

a n d i n g e n e r a l L ( x ) { a a ) . L ( x ) a n d

i 0 7

... ... -

i t s e l e m e n t s a r e computed f o r u n i t r a d i c e s . The a b s o l u t e number o f p e o p l e i n e a c h a g e g r o u p and r e g i o n i s f o u n d by m u l t i p l y i n g L ( x ) by t h e g i v e n v e c t o r o f r a d i c e s

...

10

-

a

1 .

N o t e t h a t L ( x ) r e p r e s e n t s t h e r e l a t i v e p o p u l a t i o n d i s t r i -

-

b u t i o n by p l a c e o f r e s i d e n c e a n d p l a c e o f b i r t h . I n s t e a d o f b e i n g e x p r e s s e d i n p e r c e n t a g e s ( f r a c t i o n s o f t h e t o t a l ) , o r i n some o t h e r m a n n e r , t h e p o p u l a t i o n i s g i v e n i n u n i t b i r t h s * T h i s i s a l o g i c a l p r o c e d u r e i n demography s i n c e i t s e p a r a t e s t h e f e r t i l i t y component f r o m t h e s u r v i v o r s h i p ( m o r t a l i t y and m i g r a t i o n ) component. I t w i l l become c l e a r l a t e r t h a t t h i s i s

' ~ n e q u i v a l e n t i n t e r p r e t a t i o n , w h i c h i s more s u i t e d f o r l i f e t a b l e c : o n s t r u c t i o n i s t h e " p e r s o n - y e a r s l i v e d " i n t e r p r e t a - t i o n . I n t h i s s e n s e i o L j ( x ) i s t h e number o f y e a r s e x p e c t e d t o b e l i v e d i n r e g i o n j b e t w e e n a g e s x t o x

+

5 by a p e r s o n b o r n i n r e g i o n i .

a l s o a v e r y c o n v e n i e n t way o f " n o r m i n g " i n s p a t i a l p o p u l a t i o n a n a l y s i s .

T a b l e 1 g i v e s t h e d i s t r i b u t i o n o f t h e o b s e r v e d , l i f e t a b l e and s t a b l e p o p u l a t i o n s o f t h e o n e - s e x ( f e m a l e )

,

t w o - r e g i o n

s y s t e m : S l o v e n i a - R e s t o f Y u g o s l a v i a , 1961. The o b s e r v e d popu- l a t i o n i s g i v e n by p l a c e o f r e s i d e n c e . The l i f e t a b l e p o p u l a t i o n i s computed by a p p l y i n g t h e 1961 s c h e d u l e s o f m o r t a l i t y a n d

m i g r a t i o n t o u n i t r a d i c e s . The c o m p u t a t i o n i s p a r t o f t h e c o n s t r u c t i o n o f m u l t i r e g i o n a l l i f e t a b l e s . ( T a b l e I b i s i d e n t i c a l t o T a b l e 8 o f W i l l e k e n s and R o g e r s ( 1 9 7 6 , p . 2 5 ) 3

.

To d e r i v e t h e p o p u l a t i o n by p l a c e o f r e s i d e n c e , a n d t h e a g g r e - g a t e p o p u l a t i o n , o n e must i n t r o d u c e t h e r a d i c e s

C Q -

a

1 .

T a b l e 1 . p o p u l a t i o n d i s t ' r l b u t i o n b y a a e a n d r e g i o n

* * * * * * * * * * * * * * * * * * * * * * * 3 f * * * * ~ f * * ~ f * t * * * * *

T a b l e l a , o b s e r v e d p o p u l a t i o n ( b y p l a c e o f r e s i d e n c e )

. . .

. . .

s l o v e n i a r . y u g o s .

t.o t a l 832800. 8670200.

3 ~ h e m u l t i r e g i o n a l l i f e t a b l e i s computed u s i n g t h e R o g e r s - L e d e n t Method (see W i l l e k e n s and R o g e r s , 1976, p p . 3 3 - 3 6 ) . A s a c o n s e q u e n c e , t h e n u m e r i c a l r e s u l t s shown i n t h i s p a p e r d e v i a t e s l i g h t l y f r o m t h o s e o f R o g e r s ( 1 9 7 5 ) a n d R o g e r s a n d W i l l e k e n s

( 1 9 7 6 b )

,

which u s e d t h e s o - c a l l e d O p t i o n 1 method.

Table _{I b .} l i f e t a b l e p o p u l a t i o n

. . . . . .

i n i t i a l r e g i o n o f c o h o r t s l o v e n i a

...

t 0 t . a . - s l c v e n i a r . y u g o s .

i n i t i a l r e g i o n o f c o h o r t r . y u g o s . t o t . a l s l o v e n i a r . y u g o s .

t o t a l

66.245674 C.8'0323 65.435349

Table I c . s t a b l e p o p u l a t i o n ( g r o w t h r a t e = ^0.006099 )

- - - - - -

i n i t i a l r e g i o n o f c o h o r t s l o v e n i a

...

t o t a l s l o v e n i a r . y u g o s .

t o t a l

57.856662 52.227596

5.619005

i n i t i a l r e g i o n o f z o h o r t r . y u g o s .

...

t o t a l s l o v e n i a r . y u g o s .

1 . 2 The S c a b l e P o p u l a t i o n

The s t a b l e p o p u l a t i o n b y p l a c e o f r e s i d e n c e a n d p l a c e o f b i r t h , p e r u n i t r a d i c e s , i s g i v e n by

a n d

L ( r ) (x)

2 0 ~ ; r )

' 1

^(x)

w h e r e r i s t h e a n n u a l q r o w t h r a t e o f t h e s t a b l e p o p u l a t i o n , i . e . t h e i n t r i n s i c g r o w t h r a t e . The r a t e r o n l y d e p e n d s o n t h e

o b s e r v e d s c h e d u l e s a n d i s i n d e p e n d e n t o f t h e o b s e r v e d p o p u l a t i o n d i s t r i b u t i o n . 1t i s computed a s f o l l o w s :

w i t h h b e i n g t h e a g e i n t e r v a l ( 5 y e a r s ) , a n d

X

t h e e i g e n v a l u e o f t h e p o p u l a t i o n g r o w t h m a t r i x . The v a l u e o f

X

i s c o m p u t e d

by t h e s u b r o u t i n e PROJECT i n w i l l e k e n s a n d R o g e r s ( 1 9 7 6

,

p . 5 0 ) . The a b s o l u t e number o f p e o p l e i n e a c h a g e g r o u p b y p l a c e o f r e s i d e n c e i s

w h e r e { Q ) i s t h e s t a b l e d i s t r i b u t i o n o f b i r t h s a n d w i l l b e

w

d e t e r m i n e d i n s e c t i o n f i v e o f t h i s r e p o r t . E x p r e s s i o n ( 1 . 5 ) i s

t h e n u m e r a l e v a l u a t i o n o f t h e c o n t i n u o u s $ o r m u l a ( R o g e r s a n d b l i l l e k e n s , 1 9 7 6 b , p . 2 2 ) .

A t t h i s p o i n t i t i s u s e f u l t o s t r e s s t h a t :

i. The l i f e t a b l e p o p u l a t i o n d i s t r i b u t i o n i s a s p e c i a l c a s e o f ( 1 . 3 ) w i t h r = 0 .

i i . Any s t a t i o n a r y p o p u l a t i o n , i . e . s t a b l e p o p u l a t i o n w i t h z e r o g r o w t h r a t e , i s d i s t r i b u t e d a c c o r d i n g t o a l i f e t a b l e . p o p u l a t i o n . I t s r e l a t i v e d i s t r i b u t i o n ( i n t e r m s o f u n i t b i r t h s ) i s t h e r e f o r e i n d e p e n d e n t o f how f e r t i l i t y i s r e d u c e d t o r e p l a c e - ment l e v e l .

iii. The column t o t a l s i n T a b l e 1b a r e t h e number o f p e o p l e i n t h e l i f e t a b l e p o p u l a t i o n p e r baby b o r n . A d o p t i n g t h e " p e r s o n - y e a r s l i v e d " i n t e r p r e t a t i o n o f L ( x ) , t h e t o t a l s w o u l d b e t h e

-

l i f e e x p e c t a n c i e s a t b i r t h by p l a c e o f b i r t h a n d p l a c e o f r e s i d e n c e ,

F o r e x a m p l e , t h e t o t a l l i f e e x p e c t a n c y o f a baby g i r l b o r n i n S l o v e n i a i s 7 2 . 4 8 y e a r s . A t o t a l o f 6 4 . 9 0 y e a r s a r e e x p e c t e d t o b e l i v e d i n S l o v e n i a a n d 7 . 5 7 y e a r s i n t h e R e s t o f Y u g o s l a v i a .

i v . The column t o t a l s i n T a b l e 1 c a r e t h e number o f p e o p l e i n t h e s t a b l e p o p u l a t i o n p e r b a b y b o r n . I f t h e g r o w t h r a t e r i s p o s i t i v e , t h e n t h e s t a b l e p o p u l a t i o n i s g r o w i n g a n d t h e s h a r e o f t h e b i r t h s i n t h e t o t a l p o p u l a t i o n i s g r e a t e r t h a n i n t h e s t a t i o n a r y p o p u l a t i o n . T h e r e f o r e , f o r r > 0

F o r e x a m p l e , f o r e a c h b a b y b o r n i n S l o v e n i a , t h e r e a r e 5 7 . 8 6 p e r s o n s l i v i n g i n Y u g o s l a v i a who w e r e b o r n i n S l o v e n i a . Of t h e s e 5 2 . 2 3 a r e l i v i n g i n S l o v e n i a a n d 5 . 6 3 i n t h e R e s t o f Y u g o s l a v i a . A n a l o g o u s t o t h e e x p e c t a t i o n o f l i f e a t b i r t h - i n t e r p r e t a t i o n o f e ( 0 )

- ,

t h e m a t r i x e ( r )

-

( 0 ) may b e c o n s i d e r e d a s t h e d i s c o u n t e d l i f e e x p e c t a n c y m a t r i x , w i t h

-

r b e i n g t h e r a t e o f d i s c o u n : ( W i l l e k e n s , 1 9 7 7 ) . The m e a n i n g a n d r e l e v a n c e o f t h i s a p p r o a c h w i l l b e d i s c u s s e d i n s e z t i o n f o u r .

The t h r e e t y p e s o f a g e d i s t r i b u t i o n a r e t h e c o r n e r s t o n e s f o r f u r t h e r s t u d y . F e r t i l i t y a n a l y s i s i s p e r f o r m e d - b y a p p l y - i n g a g e - s p e c i f i c f e r t i l i t y r a t e s t o t h e a g e d i s t r i b u t i o n s . ^' I n . m o b i l i t y a n a l y s i s , a g e - s p e c i f i c o u t m i g r a t i o n r a t e s a r e u s e d i n s t e a d . The n e x t two s e c t i o n s d e a l w i t h t h e s e t o p i c s i n g r e a t e r d e t a i l .

2 . FERTILITY ANALYSIS

The f e r t i l i t y a n a l y s i s p r o c e e d s b y a p p l y i n g t h e

f e r t i l i t y s c h e d u l e t o t h e t h r e e t y p e s o f a g e d i s t r i b u t i o n s . L e t t h e d i c g o n a l m a t r i x m ( x ) c o n t a i n t h e a n n u a l r e g i o n a l

-

f e r t i l i t y r a t e s o f t h e women a t e x a c t a g e x , a n d l e t F ( x ) b e -..

t h e d i a g o n a l m a t r i x o f a n n u a l r e g i o n a l f e r t i l i t y r a t e s o f a g e g r o u p x t o x

+

^{4 ,} e . g .

The i n t e g r a t i o n o f t h e m a t r i c e s o f a y e - s p e c i f i c f e r t i l i t y r a t e s o v e r a l l a g e s i s t h e g r o s s r e p r o d u c t i o n r a t e m a t r i x , i . e .

GRR _.. = jW m ( x ) d x 5

1

F ( x )

.

0

- -

X

T h e GRR-matrix i s a d i a g o n a l m a t r i x w i . t h t h e r e g i o n a l g r o s s r a t e s o f r e p r o d u c t i o n a s i t s e l e m e n t s . A g e - s p e c i f i c f e r t i l i t y r a t e s f o r S l o v e n i a a n d t h e R e s t o f Y u g o s l a v i a a r e g i v e n i n

T a b l e 2 . The c o l u m n t o t a l s d e n o t e t h e r e g i o n a l g r o s s r e p r o d u c t i o n r a t e s .

*

'-9

*

(D *7

*

u? 4 t'. *I-

*

t-'.

*

Cr *Y

*

*w *3 *P, *I- *Y *cn

*

I-'. *cn

The r e g i o n a l c r u d e b i r t h r a t e s may b e d e r i v e d by m u l t i p l y - i n g t h e a g e - s p e c i f i c f e r t i l i t y z a t e s by t h e o b s e r v e d p o p u l a t i o n d i s t r i b u t i o n , i n f r a c t i o n s o f t h e t o t a l , a n d summing o v e r a l l a g e g r o u p s . D e n o t i n g t h e r e g i o n a l d i s t r i b u t i o n o f t h e p e o p l e a g e d x t o x

+

4 b y t h e d i a g o n a l m a t r i x K ( x ) , t h e r e g i o n a l

-

₀

c r u d e b i r t h r a t e s a r e g i v e n by t h e v e c t o r { b

- 1 :

The p r o d u c t F ( x ) K ( x )

- -

i s o f c o u r s e th.2 o b s e r v e d r e g i o n a l number o f b i r t h s t o a m o t h e r a g e d x t o x

+

4 .

The a p p l i c a t i o n o f t h e a g e - s p e c i f i c f e r t i l i t y r a t e s t o t h e l i f e t a b l e p o p u l a t i o n a n d t o t h e s t a b l e p o p u l a t i o n h a s r e c e i v e d much a t t e n t i o n i n t h e d e m o g r a p h i c l i t e r a t u r e .

2 . 1 The G e n e r a l i z e d N e t M a t e r n i t y F u n c t i o n

The g e n e r a l i z e d n e t maternity ( G N M ) f u n c t i o n i s d e f i n e d a s t h e p r o d u c t ( R o g e r s , 1 9 7 5 a , p . 9 3 )

w h e r e

An element $.(x) denotes the expected number of children to i _I

be born during a unit time interval in region j to a woman of exact age x , who was born in region i, and who is part of a stationary (life table) population. The fertility rates applied to this stationary population are the observed fertility rates.

Since the actu.al population data are usually given for five-year age groups, one normally evaluates (2.3) with the numerical approximation

in which the integral

1

^{M (x}

⁺

^t)

^e

^(x

⁺

t) dt is replaced by

0

- -

the product of F(x) and L(x). The numerical evaluations or

- -

the integrals of the generalized net maternity function are given in Table 3. They are obtained by multi~lying the fertil- ity rates of Table 2 by the age composition of the life table population (Table Ib)

.

For example, -

-

$ (20) is:

The GNM function gives the number of offspring by age of a population which is distributed according to the life table

(stationary) population, and which is subjected to the observed regional fertility schedules. The total number of offspring per unit birth is

N R R

-

⁼

1

^$(x)

- .

X

An e l e m e n t

d e n o t e s t h e t o t a l number o f c h i l d r e n * z x p e c t e d t o b e b o r n i n r e g i o n j t o a woman who was b o r n i n r e g i o n i , and who

i s a member o f a l i f e t a b l e p o p u l a t i o : ~ . 4 The m a t r i x NRR _,.. i s t h e n e t r e p r o d u c t i o n r a t e m a t r i x , and i s t h e m u l t i r e g i o n a l g e n e r a l - - i z a t i o n o f t h e N e t R e p r o d u c t i o n R a t e ( N R R ) ( R o g e r s , 1975a, p . 1 0 6 ) . The e l e m e n t s o f NRR _* a r e t h e t o t a l s i n t a b l e 3 .

The m a t r i x NRR

-

g i v e s t h e r e g i o n a l d i s t r i b u t i o n o f t h e o f f - s p r i n g p e r u n i t b i r t h i n e a c h r e g i o n . I t h a s b e e n computed u s i n g u n i t r a d i c e s . From t h e d i s c u s s i o n o f t h e l i f e t a b l e i n t h e

p r e v i o u s s e c t i o n i t i s c l e a r t h a t a b i r t h c o h o r t o f { Q , )

-

would l e a d t o a r e g i o n a l number o f o f f s p r i n g , a f t e r a g e n e r a t i o n , o f

The GEM f u n c t i o n c o n t a i n s a d d i t i o n a l u s e f u l i n f o r m a t i o n f o r f e r t i l i t y a n a l y s i s . D e f i n e t h e n - t h moment o f t h e GNM f u n c t i o n

( 2 . 3 ) a s ( R o g e r s , 1 9 7 5 a , p . 1 0 6 )

w h e r e a a n d B a r e t h e l o w e s t a n d h i g h e s t r e p r o d u c t i v e a g e s r e s p e c t i v e l y , a n d , f o r e x a m p l e ,

4 ~ e c a l l t h a t a l i f e t a b l e p o p u l a t . i o n i s a s t a t i o n a r y p o p u l a t i o n t h a t would r e s u l t i f t h e m o r t a l i t y and m i g r a t i o n s c h e d u l e s were a p p l i e d t o a r b i t r a r y r e g i o n a l r a d i c e s .

Table 3 . i n t e g r a l s of g e n e r a l i z e d n e t m a t e r n i t y f u n c t i o n

. . . . . .

i n i t i a l r e g i o n of c o h o r t s l o v e n i a

...

age 0 5 10 15 20 25 30

s l o v e n i a r . y u g o s . 0,000000 0,000000 0.000000 0

.oooooo

0.000333 O.OOOOO?

0.073095 0.005626 0.314869 0.030727 0.272029 0.035961 0.172122 0.025579 0.093916 0.014786 0.031564 0.007811 0.002824 0.001403 0.001

133

0.000462 0.000000 0,000000 0.000000 0,000000 0.000000 0,000000 0.000000 0.000000 0.000000 0.000000 0,000000 0.000000 0.000000 0.000000

i n i t i a l r e g i o n of c o h o r t r . y u g o s .

...

age 0

s l o v e n i a r . y u g o s . 0.000000 0.000000 0.000000 0.000000 0.000001 0.000297 0.000324 0.116755 0.002626 0.384742 0.003247 0.321421 0.002429 0.189757 0.001464 0.099737 0.000525 0.050403 0.000049 0.008840 0.000020 0.002859 C.000000 0.000000 0,000000 0.000000 0.000000 0.000000 0,000000 0,000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

t o t a l 0.010687 1.174812

The numerical approximation of (2.7) is

Observe that the 0-th moment, R(O), is identical to NRR.

..,

-

The 0-th, first and second moments of the GNM function of the two-region system Slovenia-Rest of Yugoslavia are given in Table 4. The column totals of R(0) represent the total

-

number of offspring per woman born in a certain region, e.g.

The row totals of R(0) give the total number of children born

..,

in a certain region during one generation per woman born in that region. It is the number of daughters by which a girl child in a region is replaced. Noting that R(0) = NRR, the

..,

-

total number of children born in region j during one generation, is

and the row total of the j-th region is

The value of R.(O) depends on the radix ratio O l i

/

Q l j of the

3

life table population. Since we have assumed unit radices in all regions, the row totals in Table 4, i.e. Rj (0)

,

^{are the}

sum of the elements in the row. Other radices would give R.(O) l and the row totals of the subsequent moments other values.

T a b l e 4 . m o m e n t s o f i n t e g r a l f u n c t i o n

. . . . . .

0 moment

---

t o t a l s l o v e n i a r . y u g o s . s l o v e n i a 0 . 9 7 2 5 6 3 0 . 9 6 1 8 7 6 0 . 0 1 0 6 3 7 r . y u g o s . 1 . 2 9 7 1 7 6 0 . 1 2 2 3 6 4 1 . 1 7 4 8 1 2 t o t a 1 1 . 0 8 4 2 4 0 1 . 1 8 5 4 9 9

1 moment

---

t o t a l s l o v e n i a r . y u g o s . s l o v e n i a 2 6 . 8 1 3 1 0 1 2 6 . 4 9 9 4 3 9 0 . 7 1 3 6 6 2 r . v u g o s . 3 5 . 7 4 9 5 9 2 3 . 5 V 4 9 4 3 2 . 1 ~ ~ 7 0 9 8

t o t a l 3 0 . 0 8 5 9 3 3 3 2 . 4 7 5 7 6 1

2 m o m e n t

---

t o t . a l s l o v e n i a r . y u p o s . s l o v e n i a 7 7 7 . 5 6 5 4 3 0 7 6 7 . 9 4 0 0 0 2 9 . 6 2 5 4 5 6 r . y u g o s . 1 0 4 4 . 0 3 9 4 2 9 1 1 0 . 8 3 3 0 8 4 9 3 3 . 2 0 6 2 9 9 t o t a l 8 7 8 . 7 7 3 0 7 1 9 4 2 . F 3 1 7 2 6

T a b l e 5 r e p e a t s R ( 0 ) o r NRR and g i v e s t h e d o m i n a n t e i g e n -

w

-

v a l u e a n d a s s o c i a t e d e i g e n v e c t o r s o f R ( 0 ) . The e i g e n v a l u e o f

w

R ( 0 )

,

X ( R ( 0 ) )

,

g i v e s a n i n d i c a t i o n o f t h e n e t r e p r o d u c t i o n r a t e

-

¹

o f t h e whole s y s t e m o r c o u n t r y ( R o g e r s a n d W i l l e k e n s , 1 9 7 6 c , p . 2 8 ) . A l i f e t a b l e r a d i x r a t i o t h a t y i e l d s a g l o b a l NRR e q u a l t o X 1 ( R

-

( 0 ) ) i s g i v e n by t h e r i g h t e i g e n v e c t o r o f R

-

( 0 )

.

The g l o b a l NRR r e s u l t i n g f r o m a r a d i x r a t i o a s s p e c i f i e d by t h e u s e r , 1:1 s a y , i s a l s o g i v e n i n T a b l e 5 . I t i s e q u a l t o 1.224257. The n e t r e p r o d u c t i o n a l l o c a t i o n i p j d e n o t e s t h e f r a c t i o n o f t h e o f f s p r i n g o f t h e i - b o r n women, t h a t a r e b o r n i n r e g i o n j ( R o g e r s , 1975b. p. 2 . ) . F o r example.

i . e . 1 1 . 2 9 % o f t h e d a u g h t e r s o f S l o v e n i a - b o r n women, a r e b o r n i n t h e R e s t . o f Y u g o s l a v i a .

The moments o f t h e GNM f u n c t i o n g i v e r i s e t o o t h e r demo- g r a p h i c a l l y m e a n i n g f u l s t a t i s t i c s : t h e mean and t h e v a r i a n c e o f t h e GNM f u n c t i o n . I n t h e s i n g l e r e g i o n c a s e , t h e mean o f t h e n e t m a t e r n i t y f u n c t i o n i s d e f i n e d a s ( K e y f i t z , 1968, p. 1 0 2 )

I t r e p r e s e n t s t h e mean a g e o f c h i l d b e a r i n g o f t h e l i f e t a b l e p o p u l a t i o n ( g i v e n t h e o b s e r v e d f e r t i l i t y s c h e d u l e ) . The v a r i a n c e o f t h e n e t m a t e r n i t y f u n c t i o n i s

5 T h e a r r a n g e m e n t s o f t h e e l e m e n t s i n T a b l e 5 i s t h e t r a n s p o s e o f T a b l e 2 i n R o g e r s ( 1 9 7 5 b , p . 5 ) .

Table 5 . . s p a t i a l f e r t i l i t y e x p e c t a n c i e s

. . .

n e t r e p r o d u c t i o n r a t e

t o t a l s l o v e n i a r . y u g o s . s l o v e n i a 0 . 9 7 2 5 6 3 0 . 9 6 1 8 7 0 0.01068'7 r . y u g o s . 1 . 2 9 7 1 7 6 0 . 1 2 2 3 6 4 1 . 1 7 4 8 1 2

t o t a l 1 . 0 8 4 2 4 0 1 . 1 8 5 4 9 9

. e i g e n v a l u e 1 . 1 8 0 7 8 6 e i g e n v e c t o r

-

r i g h t 1 . 0 0 0 0 0 0 2 0 . 4 8 3 9 3 8

-

^{l e f t} 1 . 0 0 0 0 0 0 1 . 7 8 9 0 2 5

n e t r e p r o d u c t i o n a l l o c a t i o n s

...

t o t a l s l o v e n i a r . y u g o s . s l o v e n i a 0 . 8 9 6 1 5 8 0 . 8 8 7 1 4 3 0 . 0 0 9 0 1 5 r . y u g o s . 1 . 1 0 3 5 4 2 0 . 1 1 2 8 5 7 0 . 9 9 0 9 9 5

g l o b a l n r r = 1 . 2 2 4 2 5 7

and represents the variance of the mean age of childbearing.

Multiregional generalizations of (2.12) and (2.13) are (Rogers, 1975a, p. 106) :

and

respectively.

The matrix of mean ages of childbearing of the life table popu- lation is given in Table 6 as Alternative 1. For example, the mean age of childbearing among Slovenia-born women who are living in the Rest of Yugoslavia is 29.32 years. The mean age of the women living in Slovenia is lower, namely 27.55 years. This is consistent with the observation that mothers who have migrated are normally older.

The single-region measures (2.12) and (2.13) may be

generalized to a multiregional system in a different way, one which is analogous to the extension of the single-region survivorship proportion to the multiregional survivorship matrix in the life table. The mean age of childbearing matrix in this case is

and the variance matrix is

These matrices are given in Table 6 as Alternative 2. The average age at childbearing of a woman who conceived in Slovenia is 27.795 years. Of this total 27.548 have been lived in Slovenia and 0.247 in the Rest of Yugoslavia.

T a b l e 6 - m 2 t r i c e s o f mean a g e s a n d v a r i a n c e s

. . . . . .

* *

a l t e r n a t i v e 1

* *

t * * * t * # * * * * * * * * * * * *

m e a n s

---

t o t a l s l o v e n i a r . y u g o s . s l o v e n i a 2 8 . 4 4 9 9 6 3 2 7 . 5 4 9 7 4 0 2 9 . 3 5 0 1 8 5 r

.

y u g o s . 2 8 . 3 4 7 3 5 5 2 9 . 3 1 8 3 2 7 2 7 . 3 7 6 3 7 9 t o t a l 2 8 . 4 3 4 0 3 4 2 3 . 3 6 3 2 8 3

v a r i a n c e s

---

s l o v e n i a r . y u g o s . s l o v e n i a 3 9 . 3 8 8 9 7 7 3 9 . 2 4 6 2 7 7 r

.

y u g o s . 4 6 . 2 0 4 7 7 3 4 4 . 8 7 9 1 5 0

" *

a l t e r n a t i v e 2

* "

* f * * * * * * t f * * * * * B * * *

m e a n s

---

t o t . a l s l o v e n i a r . y u g o s . s l o v e n i a 2 7 . 5 6 4 0 5 1 2 7 . 5 4 7 6 5 3 0 . 0 1 6 3 9 7 r

.

y u g o s . 2 7 . 6 2 1 4 5 8 0 . 2 4 7 3 2 8 2 7 . 3 7 4 1 3 0 t o t a l 2 7 . 7 9 4 9 8 1 2 7 . 3 9 0 5 2 5

v a r i a n c e s

---

t o t a l s l o v e n i a r

.

y u g o s . s l o v e n i a 3 9 . 4 1 2 5 4 4 3 9 . 3 8 1 4 0 9 0 . 0 3 1 1 3 7 r

.

y u g o s . 4 5 . 4 7 6 3 8 7 0 . 6 0 7 3 0 6 4 4 . 8 6 9 0 8 0 t o t a l 3 9 . 9 8 8 7 1 6 4 4 . 9 0 0 2 1 5

2 . 2 The W e i g h t e d G e n e r a l i z e d N e t M a t e r n i t y F u n c t i o n

T h u s f a r w e l i m i t e d o u r s e l v e s t o t h e f e r t i l i t y a n a l y s i s o f a p o p u l a t i o n , d i s t r i b u t e d a s i n t h e m u l t i r e g i o n a l l i f e t a b l e . I t i s a s t a t i o n a r y p o p u l a t i o n t h a t i s g e n e r a t e d by t h e o b s e r v e d m o r t a l i t y and m i g r a t i o n s c h e d u l e s . The l i f e t a b l e p o p u l a t i o n was a u g m e n t e d by t h e o b s e r v e d f e r t i l i t y s c h e d u l e s t o g i v e t h e GNM f u n c t i o n a n d t h e d e r i v e d s t a t i s t i c s d i s c u s s e d a b o v e . W e now r e p l a c e t h e l i f e t a b l e p o p u l a t i o n b y t h e s t a b l e p o p u l a t i o n , g i v e n i n T a b l e I c , a n d p e r f o r m a n a n a l o g o u s a n a l y s i s . The r e g i o n a l r a d i c e s , u s e d i n t h e l i f e t a b l e , a r e now r e p l a c e d b y t h e r e g i o n a l b i r t h s i n t h e s t a b l e p o p u l a t i o n . A s b e f o r e w e a s s u m e u n i t b i r t h c o h o r t s .

C o m p u t a t i o n a l l y , t h e f e r t i l i t y a n a l y s i s i n t h e s t a b l e p o p u l a t i o n i s c o m p l e t e l y a n a l o g o u s t o t h e o n e d e s c r i b e d a b o v e . The o n l y d i f f e r e n c e i s t h a t

-

R ( x ) i s r e p l a c e d by

a n d L ( x ) by

-

D e f i n e t h e W e i s h t e d G e n e r a l i z e d N e t M a t e r n i t y (MGNM) F u n c t i o n a s t h e p r o d u c t

-

r x

The w e i g h t a p p l i e d i s e

.

S i n c e t h i s may be c o n s i d e r e d a s a d i s c o u n t i n g t o b i r t h , w i t h r b e i n g t h e r a t e o f d i s c o u n t , w e may d e n o t e t h e WGNM f u n c t i o n a s a GMM f u n c t i o n w i t h d i s c o u n t i n g . T h e u s e f u l n e s s o f t h e n o t i o n o f d i s c o u n t i n g f o r d e m o g r a p h i c

a n a l y s i s becomes c l e a r i n t h e t r e a t m e n t o f t h e r e p r o d u c t i v e v a l u e ( R o g e r s a n d W i l l e k e n s , 1 9 7 6 b

.

An e l e m e n t _{i j}

6

^{( r ) ( x )}

d e n o t e s t h e e x p e c t e d number o f c h i l d r e n t o be b o r n i n r e g i o n j

t o a n i - b o r n woman o f e x a c t a g e x who i s p a r t o f t h e s t a b l e popu- l a t i o n . I t may a l s o be c o n s i d e r e d a s t h e number o f c h i l d r e n d i s c o u n t e d b a c k t o t h e t i m e o f b i r t h o f t h e m o t h e r .

The n u m e r i c a l a p p r o x i m a t i o n o f ( 2 . 1 8 ) i s

m _-

^{( r )}

( X I

^{= F}

-

^{( x ) L ( I )}

-

^{( x )}

,

a n d t h e r e s u l t i s g i v e n i n T a b l e 7 . T a b l e 7 i s o b t a i n e d by m u l t i p l y i n g t h e f e r t i l i t y r a t e s o f T a b l e 2 b y t h e a g e composi- t i o n o f t h e s t a b l e p o p u l a t i o n ( T a b l e I c ) . F o r e x a m p l e ,

The WGNM f u n c t i o n g i v e s t h e number o f o f f s p r i n g by a g e o f a u n i t b i r t h i n t h e s t a b l e p o p u l a t i o n . Summing o v e r a l l a g e g r o u p s w e g e t

The m a t r i x

-

Y ( r ) i s t h e c h a r a c t e r i s t i c m a t r i x o f t h e m u l t i r e g i o n a l p o p u l a t i o n s y s t e m ( R o g e r s , 1 9 7 5 a , p . 9 3 ) . An e l e m e n t Y . ( r )

i

I

d e n o t e s t h e t o t a l number o f c h i l d r e n e x p e c t e d t o b e b o r n i n r e g i o n j t o a woman who was b o r n i n r e g i o n i , a n d who i s

a member o f t h e s t a b l e p o p u l a t i o n . The c h a r a c t e r i s t i c s m a t r i x i s t h e s t a b l e a n a l o g u e o f t h e NRR

-

m a t r i x . I t g i v e s t h e r e g i o n a l d i s t r i b u t i o n o f t h e o f f s p r i n g p e r u n i t b i r t h i n e a c h r e g i o n o f t h e s t a b l e p o p u l a t i o n . F o r e x a m p l e , T a b l e 7 shows t h a t a woman b o r n i n t h e s t a b l e p o p u l a t i o n i n S l o v e n i a g i v e s b i r t h t o a t o t a l o f 0.916100 c h i l d r e n on t h e a v e r a g e . Of t h e m , 0.813686 a r e b o r n i n S l o v e n i a a n d 0 . 1 0 2 4 1 4 i n t h e R e s t o f Y u g o s l a v i a .

T a b l e 7 . i r l t . e y r a l s o f w e i g h t e d E e n e r a l i z e d n e t P a t e r n i t y f u n c t l o n

. . .

. . . i n i t i a l r e g i o n o f c o h o r t s l o v e n i a

...

a g e 0 5 1 0 1 5 2 0 2 5 3 0

3

5 4 0 4 5

15

3 5 5 5 0 6 5 7C 7 5 8 0

85

s l o v e n i a 0 . 0 0 0 0 C 0 0 . 0 0 0 0 0 0 0 . 0 0 0 3 0 9 0 . 0 6 5 6 9 6 0 . 2 7 4 4 9 1 0 . 2 3 0 0 2 2 0 . 1 4 1 1 7 1 0 . 0 7 4 7 15 0 . 0 2 4 3 5 6 0 . 0 0 2 1 1 4 0 . 0 0 0 ? , 2 3 0 . 0 0 0 0 0 C 0 . 0 0 3 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 5 . 0 0 G 0 0 0 0 . 0 0 0 9 0 0 0.0130000

r . y u g o s . 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 8 0 . 0 0 5 0 5 6 0 . 0 2 6 7 8 6 0 . 0 3 C 4 0 8 G.02C979 0 . 0 1 1 7 6 3 0 . 0 0 6 0 2 7 0 . 0 0 1 0 5 0 0 . 0 0 0 3 3 5 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0

. o o o o o o

C . 0 0 0 0 0 0 o . n o 0 0 0 0 0 . 0 0 0 0 0 0 0

. o o o o o o

t o t a l 0 . 8 1 3 6 8 6 0 . 1 0 2 4 1 4

i n i t i a l r e g i o n o f c o h o r t r . y u a o s .

s l o v e n i a 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 C 0 0 0 1 0 . 0 0 0 2 9 2 c ) . 0 0 2 2 6 ? 0 . 0 0 2 7 4 6 0 . 0 0 1 9 ? 2 0 . 0 0 1 '165 0 . 0 0 0 4 0 5 0 . 0 0 0 0 3 7 0 . 0 0 0 0 1 5 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 3 . 0 0 0 0 0 0 0.000C,00 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0

t o t a l 0 . 0 0 5 9 4 2 0 . 9 9 4 9 5 6

If the stable distribution of births is {Q'}, _s

-

then the distribution of offspring is also

I Q -

(Rogers, 1975a, p. 93):

Equation (2.21) is the multiregional characteristic equation.

It can be seen from (2.21) that the relative distribution of births is given by the right eigenvector of Y(r). In our

-

numerical example,

where the subscript denotes "arbitrary norming." Since the eigenvector of a matrix is fixed up to a scalar, we may choose the norming of the eigenvector freely. The result (2.22) implies that 4.58% of the births occur in Slovenia and 95.42% in the

Rest of Yugoslavia (in the observed population it was 6.91%

and 93.09%, respectively)

.

As with the GNM function, we define the n-th moment of the

WGNM function (2.18) as (Rogers, 1975a, p. 112)

and evaluate 5t numerically as follows:

The moments a r e g i v e n i n T a b l e 8 . N o t e t h a t t h e 0 - t h moment o f t h e WGNM f u n c t i o n c o i n c i d e s w i t h Y ' ( r ) .

-

The column t o t a l s o f Y(r)

-

r e p r e s e n t t h e t o t a l number o f o f f s p r i n g i n t h e s t a b l e p o p u l a t i o n p e r woman by h e r p l a c e o f b i r t h , e . g .

The row t o t a l s g i v e t h e t o t a l number o f d a u g h t e r s by w h i c h a f e m a l e baby i s r e p l a c e d i n h e r r e g i o n o f b i r t h i n t h e s t a b l e p o p u l a t i o n . I t d e p e n d s o f c o u r s e o n t h e s t a b l e r a t i o o f b i r t h s :

w h e r e i s a n e l e m e n t o f t h e r i g h t e i g e n v e c t o r o f Y

-

^{( r ) .}

T a b l e 9 r e p e a t s t h e - , Y .. ( r ) m a t r i x . I n a d d i t i o n , it shows ( r )

,

w i t h

t h e n e t r e p r o d u c t i o n a l l o c a t i o n s

p j

F o r e x a m p l e ,

i . e . 1 1 . 1 8 % o f t h e d a u g h t e r s b o r n t o S l o v e n i a - b o r n women, a r e b o r n i n t h e R e s t o f Y u g o s l a v i a .

The mean and t h e v a r i a n c e o f t h e WGNM f u n c t i o n a r e g i v e n i n T a b l e 1 0 . A g a i n , two a l t e r n a t i v e e x p r e s s i o n s a r e d i s t i n - g u i s h e d .

T a b l e 8 . moments o f i n t e g r a l f . l n c t i o n

. . . . . .

0 moment

---

t o t a l

s l o v e n i a r . y d g o s .

s l o v e n i a 0.822628 0.813686 0.008342

r

. y u g o s . 1 . 0 9 7 3 8 0 0.102414 0.994966

t o t a l 0.916100 1.003908

1 moment

---

t o t a l s l o v e n i a

r

. y u g o u .

s l o v e n i a 22.483915 22.223598 0.260318

r .

yilgou

.

^{29.944412 2.974082 26.970331}

t o t a l 25.197681 27.230650

2 moment

---

t o t a l a l o v e n i a r . y u g o s . s l o v e n i a 6 4 6 . 2 1 2 0 9 7 638.288757 7.923365

r . y u g o s . 865.373718 90.987785 774.385925

t o t a l 729.276550 782.309265

T a b l e 9 . s p a t i a l f e r t i l i t y e x p e c t a n c i e s

. . .

n e t r e p r o d : ~ c t i o n r a t e

...

t o t a l s l o v e n i a r . y ~ g o s .

Y l o v e n i a 0 . 8 2 2 6 2 8 0 . 8 1 3 6 8 6 0 . 0 0 8 9 4 2

r

.

y ~ ~ p o s . 1 . 0 9 7 3 8 0 0 . 1 0 2 4 1 4 0 . 9 9 4 9 6 6

t c t a l 0 . 9 1 6 1 0 0 1 . 0 0 3 9 0 8

e i g e n v a l s e 6 0 . 9 9 9 8 8 4 e i g e n v e c t o r

-

r i g h t 1 . 0 0 0 0 0 0 2 0 . 8 2 3 6 6 2

-

l e f t 1 . 0 0 0 0 0 0 1 . 8 1 P 1 1 6

n e t r e p r o d ~ ~ c t i o n a l l o c a t i o n s

...

t o t a l s l o v e n i a r . y d g o s . s l o v e n i a

r

.

y t ~ g o s . t o t a l

6 ~ h e e i g e n v a l u e s h o u l d be e q u a l t o o n e . D e v i a t i o n i s d u e t o r o u n d i n g o f t h e i n t r i n s i c g r o w t h r a t e r t o s i x d e c i m a l p l a c e s . The g r o w t h r a t e h a s b e e n computed by p r o j e c t i n g t h e p o p u l a t i o n g r o w t h m a t r i x u n t i l s t a b i l i t y .

A i i e r n a t i v e i (Rogers, 1975a, p. 113) :

The matrix of mean ages of childbearing in the stable population, A, has elements:

-

and the variance a 2

-

with elements

A l t e r n a t i v e 2 :

T a b l e 1 0 . m a t r i c e s o f mean a g e s a n d v a r i a n c e s - - - . - - -

- - - . - - -

* *

a l t e r n a t i v e 1

* *

* * * * * * * * * # * I * * * * * * *

means

---

t o t a l s l o v e n i a r . y u g o s . s l o v e n i a 2 8 . 2 1 2 5 8 2 2 7 . 3 1 2 2 5 4 2 9 . 1 1 2 9 0 6 r

.

y i l g o s

.

2 8 . 0 7 3 2 9 6 2 9 . 0 3 9 8 0 8 2 7 . 1 0 6 7 8 3 t o t a l 2 8 . 1 7 6 0 3 1 2 8 . 1 0 9 8 4 4

v a r i a n c e s

---

s l o v e n i a r . y : ~ p o s . s l o v e n i a 3 8 . 4 8 1 8 7 3 3 8 . 5 5 5 4 8 1 r . y d g o s . 4 5 . 1 2 0 7 8 9 4 3 . 5 2 6 0 6 2

* Y a l t e r n a t i v e 2

* *

f * Y * * * * * t Y * Y * * * * * * *

t o t a l s l o v e n i a r . y u g c s . s l o v e n i a 2 7 . 3 2 6 4 1 4 2 7 . 3 1 0 2 1 3 0 . 0 1 6 2 0 1 r

.

y : ~ g o s . 2 7 . 3 4 8 1 6 7 0 . 2 4 3 5 7 4 2 7 . 1 0 4 5 9 3 t o t a l 2 7 . 5 5 3 7 8 7 2 7 . 1 2 0 7 9 4

v a r i a n c e s

---

t o t a l s l o v e n i a r . y u g o s . s l o v e n i a 3 8 . 5 0 7 4 7 3 3 8 . 4 7 4 2 4 3 0 . 0 3 3 2 3 0 r . y d p o s . 4 4 . 1 3 8 8 7 4 0 . 6 2 2 7 6 0 4 3 . 5 1 6 1 1 3 t o t a l 3 9 . 0 9 7 0 0 4 4 3 . 5 4 9 3 4 3

3. MOBILITY ANALYSIS

There are two alternative approaches to expressing the level of migration in a multiregional system (Rogers, 1975b).

The first expresses the migration level in terms of expected durations

,

i.e. the fraction of an individual's lifetime that is spent in a particular region. The expectation of life at

birth by place of residence is computed in the multiregional life table. The life expectancy matrix

for the system Slovenia-- Rest of Yugoslavia is given in Table 1 1 - The total life expectancy of a girl born in Slovenia is 72.48 years, of which 64.90 years are expected to be lived in Slovenia (lel(0)) an2 7.57 years in the Rest of Yugoslavia (l e2 (0) )

.

Expressing these expectancies as fractions of the total life- time yields the migration levels i 8 j :

The second approach adopts a fertility perspective to mi- gration analysis. Unlike death, migration is a recurrent event, analogous to birth. Like fertility, its level can be measured by counting the events, i.e. the number of moves an average person

7 I

makes during his lifetime

.

Such i n d i c ~ s have been developed

by Wilber (1963) and Long (1973) for a population aggregated at the national level. Rogers (1975b) combines Wilber's and Long's ideas of "expected moves" with the approach generalizing the expected number of children (NRR) to a multiregional system (NRR).

-

7 ~ h e number of moves is defined here as the number of times a person is in another region at the end of the unit time interval.

Back and forth moves during a unit interval are not counted (a. similar assumption has been adopted by Wilber (1 963) and Long (1 973) )

.

T a b l e 11. s p a t i a l m i g r a t i o n e x p e c t a n c i e s

* * * * * * * * * * * * * * * * * * * * * * * t * * * * Z *

e x p e c t a t i o n s o f l i f e

...

t o t a l s l o v e n i a r . y u g o s . s l o v e n i a 6 5 . 7 1 2 9 9 7 6 4 . 9 0 2 6 7 2 0 . 8 1 0 3 2 3 r . y u g o s . 7 3 . 0 0 9 1 4 8 7 . 5 7 3 5 0 1 6 5 . 4 3 5 3 4 9 t o t a l 7 2 . 4 7 6 4 7 1 6 6 . 7 4 5 5 7 4 e i g e n v a l u e 6 7 . 6 6 0 6 2 9

e i g e n v e c t o r

-

r i g h t 1 . 0 0 0 0 0 0 1 . 4 0 3 5 2 5

-

^{l e f t} 1 . 0 0 0 0 0 0 0 . 3 6 4 1 4 4

~ i g r a t i o n l e v e l s

---

t o t a l s l o v e n i a r . y u g o s . s l o v e n i a 0 . 9 0 7 7 3 2 0 . 8 9 5 5 0 0 0 . 0 1 2 2 3 2 r . y u g o s . 1 . 0 9 2 2 6 8 0 . 1 0 4 5 0 C 0 . g o 7 7 6 9

t o t a l 1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0

As before, let R(x) be the distribution of the life table

-

population of exact age x, and let L(x) be the stationary life table

-

population aged x to x

+

4, by place of birth and residence.

Define m0 as the diagonal matrix of annual regional outmigration

-

rates of people at exact age x, and MO(X) as the diagonal matrix

-

of outmigration rates of people in age group x to x

+

^{4, e.g.}

with M . 0 ( x ) =

1

^{Mij (x)}

^,

^Mij (x) being the age specific migration

1 j#i

rate from region i to region j. Integration of the matrices of dye-specific outmigration rates over all ages gives the gross miqra-wroduction rate matrix:

The origin-destination migration rates of the two-region system Slovenia

-

Rest of Yugoslavia are given in Table 3 of Willekens and Rogers (1976a, p. 9). Table 12 shows the age-specific re- gional total outmigration rates. Since the system under con- sideration contains only two regions, M . 0 (x) -

- ( x ) for i # j.

1

The column totals denote the regional gross migra-production rates.

The application of the age-specific outmigration rates to

the life table and to the stable populations yields, respectively, the generalized and the weighted generalized net mobility functions.

T a b l e 1 2 . m i g r a t i o n a n a l y s i s

* * * * * * * * * * * + * * * * * *

o b s e r v e d r a t e s

- - - - - -

a g e s l o v e n i a r . y u g o s .

3 . 1 The G e n e r a l i z e d N e t M o b i l i t v F u n c t i o n

The g e n e r a l i z e d n e t m o b i l i t y ( G M ) f u n c t i o n i s t h e p r o d u c t

An e l e m e n t y . ( x ) d e n o t e s t h e e x p e c t e d number o f m i g r a t i o n s i 3

o u t o f r e g i o n j , made d u r i n g a u n i t t i m e i n t e r v a l f o l l o w i n g a g e x , by a woman b o r n i n r e . g i 0 n . i . S i n c e t h e s y s t e m o n l y c o n s i s t s o f two r e g i o n s , y . ( x ) m e a s u r e s t h e r e t u r n m i g r a t i o n o f t h e

i 3 x - y e a r o l d .

The n u m e r i c a l e v a l u a t i o n of e q u a t i o n ( 3 . 4 ) i s

The v a l u e s o f y ( x )

-

a r e g i v e n i n T a b l e 1 3 . The c o m p u t a t i o n a l

p r o c e d u r e i s c o m p l e t e l y a n a l o g o u s t o t h e o n e u s e d i n t h e f e r t i l i t y a n a l y s i s . The o n l y d i f f e r e n c e

_-

i s t h a t F

-

(x) o f ( 2 . 4 ) i s r e p l a c e d by p1°

-

( x )

.

F o r e x a m p l e , y ( 2 0 )

-

i s

The e x p e c t e d number o f m i g r a t i o n s a n i n d i v i d u a l makes d u r i n g h i s l i f e t i m e i s g i v e n by t h e summation o f y ( x ) o v e r a l l x. The

*

r e s u l t i s t h e n e t m i g r a - p r o d u c t i o n m a t r i x ( R o q e r s , 1 9 7 5 b , p . 8 ) : NMR -- =

1

y ( x )

. -

X

w h e r e

NMR =

1

Table 13. i n t e g r a l s o f g e n e r a l i z e d n e t m o b i l i t , y f u n c t i o n

. . . . . .

i n i t i a l r e g i o n o f c o h o r t s l o v e n i a

...

a g e 0 5 1 C!

15 20 25 3 0

3"

4

C 45

5Q 55

60 65 70 75

so

9.;

s l o v e n i a r . y u g o s . 0.01 3849 0.000009 C.010892 0.000015 0.006972 0.000021 0.023773 0.000144 0.031954 0.000327 9.023813 0.000245 0.0 15729 0.000202 0.007251 0.000142 0.004101 0.000119 0.00216C 0.000061 0.002573 0.000084 0.002354 0,000130 0.003130 O.OC0123 C.003040 0.000087 0.002335 0.000037 Q.00218? 0.000036 0.000517 0.000047 0.000000 0.000000

t o t a l 0.156926 0.001830

i n i t i a l r e g i o n o f c o h o r t r . y u g o s .

...

a y e 0 5 10 15 20 25 30

? C ,

49

4

5 50 55 50 65 70 75 50 85

s l o v e n i a r . y u g o s . 0.000009 0.001287 0.000019 0.000739 0.000017 0.000697 0.000106 0.002996 0.000267 0.004098 0.00028~? 0.002190 C1.000222 0.001500 O.OOOll3 0.000958 0.000068 0.000765 0.000032 0.000386 0.000046 0.000521 0.00004~ 0.000788 0.000052 0.000730 0.000063 0.000502 0.000050 0.000208 0.009047 0.000197 0.009018 0.009250 0.000000 0.000000

t o t a l 0.001473 0.018513

The column sum iNMR denotes the total expected number of migrations to be made by a person born in region i. Some of these, i.e.

NMR. migrations are made out of region j. In other words, iNMR

i

I

j

is the number of times a person born in region

i

is expected to leave region j. The total number of migrations expected to be made by the current birth cohorts out of region j is of course

or in matrix notation

The moments of the GM-function are completely analogous to those of the GNM

-

function. The n-th moment of the GM-function is defined as:

where w is the highest age of the population. The numerical approximation of (3.8) is:

with 2 being the highest age in the discrete case and 2-5 the starting age of the highest age group.