In Sections 4 and 5 of this report, we perforved some intro- ductory analyses of the fertility and migration characteristics of stationary populations. In this section, stable population analysis is advanced by means of the notions of spatial reproduc- tive value that was developed in the previous section.
If age-specific birth, death and migration rates remain fixed, then a population exposed to these rates ultimately will evolve into a stable population whose principal characteristics are:
unchanging regional age compositions and regional shares; constant regional annual rates of birth, death, and migration; and a fixed multiregional annual rate of growth that also is the annual rate of population growth in each and every region (Rogers and Willekens,
1976c, p. 12). The constant growth rate implies that births
and
population increase at the same rate and follow an exponential growth path. This trajectory may be expressed in terms of observed population characteristics. This is the topic of the first part of this section. The second part focuses on the calculation of the intrinsic rates of birth, death, out- and inmigration.
where r is the stable urowth rate, V is the total reproductive value of the whole population system, Iv(O)}' and {Q1} are, re-
- -
spectively, the left and right eigenvectors of Y(r), associated
?The superscript of {Q
-
s 1 is dropped for convenience.w i t h t h e d o m i n a n t e i g e n v a l u e , a n d K
-
i s t h e m a t r i x o f mean a y e s o f c h i l d b e a 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 , d e f i n e d i n ( 4 . 3 0 ) a s :The e x p r e s s i o n
I v ( O ) I ' K C Q - -
-1I
i s a n o r m a l i z i n g f a c t o r . W r i t i n gy i e l d s t h e s i m p l e e x p r e s s i o n f o r t h e u l t i m a t e b i r t h t r a j e c t o r y :
I f { Q
-
1 1 i s c h o s e n s u c h t h a t i t s e l e m e n t s sum u p t o u n i t y , t h e n t h e u l t i m a t e t o t a l number o f b i r t h s i s p r o p o r t i o n a l t o t h e t o t a l r e p r o d u c t i v e v a l u e . The t o t a l number o f b i r t h s i s t h e n a l l o c a t e d t o t h e d i f f e r e n t r e g i o n s a c c o r d i n g t o { Q 1 l .-
S u b s t i t u t i n g V o f( 6 . 1 5 ) i n t o ( 7 . 4 ) shows t h a t t h e s t a b l e number of b i r t h s i n e a c h r e g i o n I Q ( ~ ) 1 may a l s o b e e x p r e s s e d a s a l i n e a r c o m b i n a t i o n o f t h e d i s c o u n t e d number o f o f f s p r i n g by r e g i o n o f b i r t h ( s e e W i l l e k e n s ,
1 9 7 7 , p p . 3 2 - 3 3 ) . The v e c t o r o f s t a b l e e q u i v a l e n t b i r t h s i s :
R e c a l l o u r n u m e r i c a l i l l u s t r a t i o n . The m a t r i x o f mean a g e s o f c h i l d b e a r i n g i s g i v e n i n T a b l e 4 . 9 . S i n c e t h e g r o w t h r a t e r i s 0 . 0 0 6 0 9 9 . t h e n o r m a l i z i n q f a c t o r K i s 1054.256 ( T a b l e 7 . 2 ) . The t o t a l r e p r o d u c t i v e v a l u e V h a s b e e n computed t o b e 9 , 7 3 8 , 4 6 6 ; h e n c e t h e s t a b l e e q u i v a l e n t o f b i r t h s i s by ( 7 . 5 ) :
The total number of births is 2 0 1 , 5 9 2 . Of this number of babies 4 . 5 8 % will be born in Slovenia and 9 5 . 4 2 % in the Best of Yuqo- slavia. t
The stable equivalent population in each age group x to x
+
4 is easily obtained by the formula ( 3 . 2 2 ) :The stable equivalent of the total population is:
Defining
as the matrix of discounted life expectancies at birth, reduces equation ( 7 . 7 ) to
The numerical values of Yugoslavia's stable equivalent popu- lation are given in Table 7 . 1 . Note that those values are very close to the ones given in Table 3 . 3 , which was obtained by pro- jecting the observed population-tt
Equations ( 3 . 2 2 ) and ( 7 . 7 ) demonstrate that for population analysis it is more convenient to express the relative age comgosi- tion of the population in unit births instead of in fractions or percentages of the total population. The values of
*Compare this allocation with the observed number of births ( 2 0 5 , 0 1 0 ) and the regional distribution: 6 . 9 0 % in Slovenia and 9 3 . 1 0 % in the Rest of Yugoslavia.
ttMinor d-eviations are due to rounding error.
T a b l e 7.1. S t a b l e e q u i v a l e n t o f t o t a l p o p u l a t i o n . t o t a l S l o v e n i a r . y u g o s .
t o t a l 10717010.
p e r c e n t a g e d i s t r i b u t i o n
...
t o t a l S l o v e n i a r . y u g o s .
t o t a l 100.030 100.000 100.000
s h a r e 100.000 5.578 9 4 . 4 2 2
are given in Table 3.4.
7.2 Stable Equivalents and Intrinsic Bates
The fertility, mortality and migration characteristics of a stable population may be described by a small number of param- eters, e.g., the intrinsic rates (Rogers, 1975a, pp. 109-115).
The intrinsic rates are directly related to the stable equivalents of births, deaths, and migrants. Therefore, we treat both rates and equivalents simultaneously.
Applying the fixed age-specific schedules of fertility, mortality and migration to the stable equivalent of the population gives the stable equivalent of births, deaths and migrants. The stable equivalent of births has already been computed. Applying the fertility schedule to the population distribution of (3.22) and summing over all age groups yields, of course, the character- istic equation:
{ Q }
-
=1
F-
(x) { ~ ( x )- 1
X
The intrinsic birth rate of region i is the ratio between Qi and the stable equivalent population Yi, which may be written as (Rogers, 1975a, p. 115) :
The v e c t o r o f i n t r i n s i c b i r t h r a t e s i s
w h e r e Y
-
i s t h e d i a g o n a l m a t r i x o f s t a b l e e q u i v a l e n t t o t a l p o p u l a t i o n s , i . e .The v e c t o r { b ] a l s o may b e e x p r e s s e d a s
-
w h e r e { C ( x ) } d e n o t e s t h e a g e c o m p o s i t i o n o f t h e p o p u l a t i o n a s
-
f r a c t i o n s o f t h e t o t a l , i . e .
The p r o p o r t i o n o f t h e r e g i o n a l p o p u l a t i o n , w h i c h i s a g e d x t o x
+
4 , may b e e x p r e s s e d a ss i n c e by ( 7 . 1 0 ) Y-' i s e q u a l t o "-'b,
- -
w h e r e b o t h Q a n d b a r e d i a g o n a l m a t r i c e s . D e f i n i n g C ( x ) a s-
g i v e s
{ C ( x ) 1 = Q-I
-
C ( x ) { Q ]- .
To c o m p u t e t h e s t a b l e e q u i v a l e n t s o f d e a t h s , o u t m i g r a n t s and i n m i g r a n t s , we m u s t r e c o n s i d e r t h e a g e - s p e c i f i c d e a t h a n d m i g r a - t i o n r a t e s ( L e d e n t , 1 9 7 7 ) . The d e a t h s a n d o u t m i g r a n t s i n a g e
group x to x
+
4 in a life table population are given by (Rogers and Ledent, 1976, p. 289).A
where (I(x) represents the distribution of the life table popula-
-
tion at exact age x by place of birth and place of residence (in terns of unit born),
L(x) is given in (2.10) and represents the distribution of
-
the life table population aged x to x + 4 by place of- r x "
( x ) = e 9. ( x )
..
Hencew i t h II
- -
.
5i ('I
( x )-
( r ) ( ~ + 5 )= I
u ( ( x + t ) e - r ( x + t ) ;-
( x+
t ) d t,
0
I n t e g r a t i o n by p a r t s y i e l d s
i
( r ) ( x )-
i ( r ) ( ~ + 5 ) = e - r x A ~ ( x )-
e - r ( ~ + 5 ) i ( ~-
+ 5 )The a g e - s p e c i f i c d e a t h a n d o u t m i g r a t i o n r a t e s i n t h e s t a b l e p o p u l a t i o n a r e g i v e n by t h e m a t r i x
d r ) -
( x ) =.
( x )- i -
( r ) ( x+
5 ) 1 [ L ( ~ ) ( x ) 1 - lwhich after substitution yieldst
For the last age group z , the rates are:
The outmigration rates ~.l!~) (x) are contained in the off-
11
diagonal elements of ll(rl
.
(x). The death rates M):: (x) are equal to the diagonal elements minus the outmigration rates, i.e. plus the off-diagonal elements in the same column.To facilitate further analysis, define the diagonal matrix Gbl(r)
-
(x) of regional death rates, and the diagonal matrix O ? I ( ~ )-
(x) of total regional outmigration rates, i.e.Ohljr) (x) =
I
?!!.P (x).
j+i '7
Let O O ~ ( r ) (x) be the matrix of outmigration rates, i.e.
tCompare this with the formula for the life table ( = observed) rates. Solving equation (2.25)
5
-
1 5P(x)
-
= [I-
+ 7I [I -
7 !(XII
for M(x) yields-
0
. . . .
PI;;) (x)Once consistent age-specific death and migration rates are derived, we may proceed with the computation of the stable eguiva- lents of deaths and out- and inmigrants, and of the associated intrinsic rates. The stable equivalent of deaths is
The intrinsic death rates follow immediately:
The stable equivalent of the outmigrants from region i to region j is
where
~ : f )
(x) is the age-specific migration rate andxi
(x) is the stable population of region i aged x to x+
4. In general, wemay w r i t e t h e o r i g i n d e s t i n a t i o n f l o w o f s t a b l e e q u i v a l e n t m i g r a - t i o n s a s
w h e r e O O ~ I ( ~ )
-
( x ) i s d e f i n e d i n ( 7 . 2 3 ) a n d K ( x ) i s a d i a g o n a l m a t r i x-
o f s t a b l e r e g i o n a l p o p u l a t i o n s o f a g e s x t o x
+
4 . The o u t m i g r a - t i o n r a t e s a r e s i m p l yw h e r e z ( x )
-
= K ( x ) Y-I- ,
i . e . E ( x ) { l l-
= { C ( x ) I- -
The s t a b l e e q u i v a l e n t o f t h e t o t a l number o f o u t m i g r a n t s i s
a n d t h e t o t a l o u t m i g r a t i o n r a t e s a r e :
A n o t h e r e x p r e s s i o n f o r ( 7 . 3 1 ) i s :
- - x
1
{ O ~ ( r ) ( x ) j ' K ( x )- ,
where I O M ( ~ ) (x)
l
is the vector of total outmigration rates, de- fined in (7.22).The stable equivalent of the total number of inmigrants by region is
and the inmigration rates are
= Y-l
-
0- - I l l .
The matrix h
-
= Y-'-
0-
contains inmigration rates by region of origin and region of destination. An element h i j describes the migrants from region i to j as a fraction of the population in j .There exists a unique relationship between inmigration rates and outmigration rates. Since by (7.29)
we have
and the total inmigration rates
-
1Iil
-
= i- - I l l
= Y o y- - - -
I 1 1.
The stable equivalents of births, deaths and outmigrants and inmigrants for Yugoslavia are given in Table 7.2, together with the intrinsic rates. Note that the intrinsic rates obey the following definitional relationship:
r = b.
-
d .-
o .+
i i - (7.35)~ h u s equation (7.38) provides an independent check of the results.