SPATIAL ZERO POPULATION GROWTH - Spatial Population Analysis: Methods and Computer Programs

death, and migration rates, and a unique and constant qrowth rate

The growing public concern about rapid population increase has generated a vast literature on the social and economic im- pacts of high fertility and has focused attention on fertility decline as a means for relieving socio-economic problems. An immediate drop of fertility to replacement level, however, would not stop population growth. Children outnumber parents in a growing population. Consequently, the number of potential par- ents in the next generation will be larqer than at present.

This built-in tendency for continued growth causes the number of people to increase for some time before the population becomes stationary (i-e. stable, but with zero growth). The ratio by which the ultimate stationary population exceeds a current popu- lation undisturbed by migration has been studied by Keyfitz (1971).

Although population growth is an important concern, where people choose to live in the future presents issues and. p r o b l e m that are potentially as serious as those posed by the number of children they choose to have. A drop in fertility, for example, not only causes the population to continue to grow for a while, but, in the presence of migration, also affects the regional distribution of this population. The spatial impact of fertility reduction has been studied by Rogers and Willekens (1976a, 1976b).

The spatial momentum of zero population growth may he com- puted numerically and, if the initial population is stable, analytically. In the first section, the numerical approach is discussed. The analytical approach is examined in the following section.

8 . 1 The N u m e r i c a l Approach

e x p r e s s e s t h e b i r t h s i n o n e g e n e r a t i o n a s a f u n c t i o n o f b i r t h s

whence

Therefore, the cohort replacement alternative yields the replace-

ment fertility rates F (x)

-

where F(x) is the diagonal matrix of observed regional fertility

-

rates of age group x to x

+

^4, ^and^y

-

is the diagonal matrix with the elements of { y l

-

in the diagonal.

Recall our numerical illustration: the two-region system of Slovenia and the Rest of Yugoslavia. The matrix of fertility adjustment factors is given in Table 8 . 1 . Since the women of both regions originally had a net reproduction rate greater than unity, the fertility adjustment factors are less than one, causing a fertility drop in both regions. In Slovenia, fertility rates drop to 9 3 . 2 4 % of their original values, whereas in the Rest of Yugoslavia they decline to 8 4 . 2 7 % of their previous levels. The difference is caused by differences in the initial fertility levels.

The new fertility rates F(x) are also given in Table

-

8 . 1 . Mote that the gross rates of reproduction must decrease in the same proportion as the age-specific fertility rates. fertility rates of each region are reduced by the same proportion.

The fertility adjustment factor is identical for each region and is equal to

Table 8 . 1 . Zero p o p u l a t i o n growth a l t e r n a t i v e 1 .

matrix of f e r t i l i t y adjustment f a c t o r s

...

slovenia r.yugos.

s l o v e n i a 0.932431 0.000303 r.yugos. 0.000000 3.842719 total 0.932431 0.842719

f e r t i l i t y a o a l y s i s

* * * * Y * * Y t * l l * * * * * *

ade-spec i f ic rates

- - - - - -

ade s l o v e n i a r.yu&os.

Table 8.2. Integrals of generalized net maternity function.

Table 8.3. Moments o f integral function.

0 moment

---

slovenia r.yugos.

slovenia 0 . 8 9 6 8 8 2 0 . 0 0 9 9 6 5 r.yugos. 0 . 1 0 3 1 1 8 0 . 9 9 0 0 3 5 total 1 . 0 0 0 0 0 0 1.000000

1 moment

---

slovenia r.yugos.

slovenia 2 4 . 7 0 8 9 2 1 0 . 2 9 2 4 6 9 r.yugos. 3.023252 2 7 . 1 0 3 6 0 7 total 2 7 . 7 3 2 1 7 4 27.396076

2 moment

---

slovenia r.yugos.

slovenia 7 1 6 . 0 5 3 8 8.9751 r. yugos. 9 3 . 4 0 1 2 786.4307 total 8 0 9 . 4 5 5 0 795.4058

-153-

Table 8.4. Spatial fertility expectancies.

net reproduction rate

...

slovenia r.yugos.

slovenia 0.896882 0.009965 r.yugos. 0.103118 0.993035 total 1.000000 1.000003 eigenvalue 1.000030 eigenvector

-

^right ^1.000000 ^10.348280

-

^left ^1.000000 ^1.000005

net reproduction allocations

...

slovenia r.yugos.

slovenia 0.896882 0.009965 r.yugos. 0 . 1 0 3 1 1 8 0.930035 total 1.000000 1.000030

Table 8 . 5 . M a t r i c e s of mean a g e s and v a r i a n c e s .

Table 8.6. Discounted number o f offspring per person of exact

T a b l e 8 . 7 . S p a t i a l r e p r o d u c t i v e v a l u e p e r p e r s o n o f e x a c t a g e x.

s l o v e n i a

T a b l e 8 . 8 . Discounted number o f offspring per person in age

T a b l e 8 . 9 . S p a t i a l r e p r o d u c t i v e v a l u e p e r p e r s o n i n a g e g r o u p

Table 8 . 1 0 . T o t a l d i s c o u n t e d number o f o f f s p r i n g of observed 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 . s l o v e n i a 389998. 358522. 3 1 4 7 7 . r . y u a o s . 4741045. 25023. 4 7 1 6 0 2 2 . t o t a l 5 1 3 1 0 4 3 . 3 8 3 5 4 5 . 4 7 4 7 4 9 9 .

Table 8 . 1 1 . Reproductive v a l u e o f t h e t o t a l p o p u l a t i o n .

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

t

5 1 3 1 0 7 0 . 100.00

?The s m a l l d e v i a t i o n from t h e t o t a l d i s c o u n t e d number of o f f s p r i n g of t h e observed p o p u l a t i o n i s due t o rounding e r r o r .

Table 8.12. Stable equivalent o f total population.

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

t o t a l 12496551. 1208238. 1 1 2 8 8 3 1 3 .

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.yugos.

t o t a l 1 0 0 . 0 0 0 100.000 100.000 s h a r e 100.000 9.669 93.331

o C U - 0 Y m m m c m aru =

0 L N O 0

4 3 0 0

J 5 0 0

. . .

0 0 0 ." 0

E C

4 . .

.

w 0 9 m

3 ~ m IC E N m 0 3 m ~ u C

The matrix of fertility adjustment factors for Yugoslavia is given in Table 8.14 together with the new fertility rates:

where y

-

= y:.

This reduction scheme produces a different stationary popu- lation. A haby girl born in Slovenia is replaced by only 0.918 daughters on the average, while a girl born in the Rest of Yugoslavia replaces herself with 1.004 daughters. Further re- sults of this replacement alternative are given in Tables 8.15 to 8.26.

8.2 The Analytical Approach

If the initial population is stable, the momentum of spatial zero population growth may be expressed as a simple analytical formula. The ultimate number of stationary equivalent births is by (7.1)

where the caret designates a stationary population. The total

reproductive value V is

with {k(x)} being the vector defining the regional distribution

-

of people at exact age x. If the distribution {k(x)I is stable, then by (3.23)

T a b l e 8 . 1 4 . Zero p o p u l a t i o n growth a l t e r n a t i v e 2 .

Table 8.15. Integrals of generalized net maternity function.

-1 6 5 -

T a b l e 8 . 1 6 . Moments of i n t e g r a l f u n c t i o n .

Slovenia r.yugos.

slovenia 0.814607 0.009051 r.yugos. 0.103629 0.994941 total 0.918236 1.003992

1 moment

---

Slovenia r.yugos.

slovenia 22.442253 0.265639 r .yugos. 3.038233 27.237909 total 25.480436 27.503548

total 744.2309 798.4795

Table 8.17. Spatial fertility expectancies.

net reproduction rate

...

slovenia r.yugos.

slovenia 0 . 8 1 4 6 0 7 0.009051 r.yugos. 0.103629 0 . 9 9 4 9 4 1 total 0.918236 1 . 0 0 3 9 9 2 eigenvalue 1.000000 eigenvector

-

^right 1 . 0 0 0 0 0 0 20.483990

-

^left ^1.000000 ^1.789017

net reproduction allocations

...

slovenia r.yugos.

slovenia 0 . 8 8 7 1 4 3 0.009015 r

.

Y U ~ O S . 0.11 2 8 5 7 0.990985 total 1.000093 1.000000

Table 8.18. Matrices of mean ages and variances.

Table 8.19. Discounted number of offspring per person of exact a g e x.

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

...

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

r e g i o n o f r e s i d e n c e r . y u g o s .

...

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

T a b l e 8 . 2 0 . S p a t i a l r e p r o d u c t i v e v a l u e p e r p e r s o n of e x a c t a g e x.

slovenia

T a b l e 8 . 2 1 . D i s c o u n t e d number o f o f f s p r i n g p e r p e r s o n i n a g e

Table 8.22. Spatial reproductive value per person in age group x to x

+

^4.

s l o v e n i a 1.00979 1 1.013931 1.006159 0.95765 1 0.749220 0.449374 0.222677 9.085883 0.020747 0.002726 0.000619 0.003003 0.0003i10 0.000000 0.000000 9.000000 0.000000

Table 8.23. Total discounted number of offspring of observed population.

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

s l o v e n i a 354222. 325533. 2 8 5 8 9 .

r . Y U i 3 O S . 4 7 6 4 5 3 8 . 2 5 1 4 7 . 4 7 3 9 3 9 0 . t o t a l 5 1 1 8 7 6 0 . 3 5 0 7 8 0 . 4 7 6 7 9 8 0 .

Table 8.24. Reproductive value of the total population.

t o t a l p e r c e n t a g e s l o v e n i a 3 7 0 6 2 1 . 4.17 r . y u g o s . 8 5 9 7 4 4 0 . 9 5 . 8 3 t o t a l 8 8 7 8 0 6 1 . 100.00

Table 8 . 2 5 . S t a b l e e q u i v a l e n t of t o t a l p o p u l a t i o n .

U NO. ru

4 n - 0

c m c -

0 L = L 3 O

-

^{C O}⁰

4 0 0 0

. . .

O D 0 m

. . .

U C C 3 2 :

9 0 0 - m i l

E * r n O . L V

3 m.0 = m

C C

- -

E 8^-. W 3 Cn 0 C L .3

where

I Q I -

represents the regional distribution of births before the drop in fertility. Substituting {k

-

(x)

I

in (8.8) into (7.1) and simplifying gives (Rogers and Willekens, 1976b, p. 22) :

where

u

= {G(o)}'

-

- -

inl), with

-

= p

-

= y R(1) !-'(0)

- -

1-l being the matrix of mean ages of childbearing in the stationary popu- lation after the decline in fertility. The matrices R(0) and Y(r) and the vector of stable equivalent births refer to the

-

stable population before the drop in fertility. The matrix of fertility adjustment factors is y.

-

It can be shown that equation (8.9) is equivalent to

so -

IQ)

- ,

^say.

The stationary births are therefore a linear combination of the stable births, before the drop in fertility. The conversion matrix is SO.

-

A numerical evaluation is given in Table 8.27.

The ultimate stationary population is

and the total reproductive value is

Let Y

-

be the diagonal matrix of the total population before the drop in fertility, then

w h e r e e

-

( r l ( 0 ) h a s b e e n l a b e l e d t h e m a t r i x o f d i s c o u n t e d l i f e e x p e c t a n c i e s . R e c a l l i n g t h e c h a r a c t e r i s t i c e q u a t i o n , ( 8 . 1 3 ) a l s o may b e w r i t t e n a s

whence

The s p a t i a l momentum o f z e r o p o p u l a t i o n g r o w t h i s t h e n

( 8 . 1 7 ) w h e r e { b }

-

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

t h e d r o p i n f e r t i l i t y . A p p l y i n g ( 8 . 1 0 ) t h e momentum becomes

I n t r o d u c i n g ( 8 . 1 5 ) i n t o ( 8 . 1 6 ) g i v e s y e t a n o t h e r e x p r e s s i o n f o r t h e momentum

( 8 . 1 9 ) The a n a l y t i c a l a p p r o a c h i s i l l u s t r a t e d i n T a b l e 8 . 2 7 . I t i s a s s u m e d t h a t t h e i n i t i a l p o p u l a t i o n c o i n c i d e s w i t h t h e s t a b l e e q u i v a l e n t p o p u l a t i o n o f 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 . Hence t h e r e g i o n a l b i r t h s a r e c o n t a i n e d i n t h e v e c t o r

and the population by age-group and region is given in Table 7.1.

Table 8.27 reveals that, given a population of 597,806 in Slovenia and 10,119,204 in the Rest of Yugoslavia, an immediate drop of fertility to replacement level would result in an ulti- mate population increase of 15.745 in Slovenia and of 10.66% in the Rest of Yugoslavia. The momentum is a consequence of the growth potential in the initial age and regional distribution of the popu1ation.t

?Note that the stationary population distribution in unit births was given in Tables 2.4 and 3.5.

T a b l e 8 . 2 7 . S p a t i a l momentum o f z e r o p o p u l a t i o n growth.

m a t r i x c o n v e r t i n g s t a b l e t o s t a t i o n a r y b i r t h s

...

s l o v e n i a r . y u g o s . s l o v e n i a 0.916844 -0.000054 r . y u g o s . -0.000574 0 . 9 1 6 7 8 6

s t a b l e an3 s t a t i o n a r y e q u i v a l e n t s

...

b i r t h s p o p u l a t i o n p o p u l a t i o n

s t a b l e s t a t i o n a r y s t a b l e s t a t i o n a r y aomentum

s l o v e n i a 9 2 3 7 . 8 4 5 9 . 5 9 7 8 0 6 . 69 1925. 1.1574

r . yugos. 192355. 176343. 10119204. 11603081. 1.1465

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

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