The Concentration of Reproduction: A Global Perspective

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Working Paper

The Concentration of Reproductiom A Global Perspective

Wo&ang Lutz

June 1987 WP-87-51

International Institute for Applied Systems Analysis

A-2361 Laxenburg, Austria

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The Concentration of Reproduction:

A

Global Perspective

June

1987 W - 8 7 - 5 1

Working Papers are interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and have r e c e i v e d only limited review. Views or opinions e x p r e s s e d h e r e i n d o not necessarily r e p r e s e n t those of t h e Institute or of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

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Foreword

Standard techniques f o r demographic analysis deal mostly in averages, and s o implicitly t a k e populations as homogeneous. The standard life table tells us only t h a t at a given age, f o r example, .99 (say) of t h e population will survive to t h e next age. Births a r e expressed as children p e r woman or s o m e o t h e r average. The original r a w data in t h e case of deaths are incapable of saying much about distributions, since they c a n tell us only t h a t M r . A died at a g e z. Women can have m o r e than one birth, and so i t i s possible t o know t h e distribution of childbearing among women, but in f a c t in t h e usual form of compilation t h e d a t a are stripped of information on distributions.

While confining t h e analysis to averages i s convenient f o r many purposes, i t also leaves out much. Birth rates a r e t h e subject of policy, in s o m e countries to r a i s e below-replacement fertility, in o t h e r countries to hold increase down to what the economy and ecology can stand. But if m o s t of t h e children are borne by a (relatively small) fraction of women, then those devising of policies will want to t a k e account of this heterogeneity.

IIASA has produced a s e r i e s of papers, authored by Anatoli Yashin and James Vaupel, dealing with t h e implications of heterogeneity in various fields. The present p a p e r by Wolfgang Lutz takes an empirical turn; i t assembles such d a t a on childbearing a s shows distributions, particularly thcse collected in the World Fer- tility Survey. The present assembly of d a t a and its analysis is a f u r t h e r s t e p for- ward in filling what has been a major gap in demographic analysis.

Nathan Keyfitz

Leader, Population Program

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Abstract

This p a p e r s t u d i e s t h e heterogeneity of women with r e s p e c t to achieved fertility under a global p e r s p e c t i v e . Besides t h e g r a p h i c a l analysis of Lorenz c u r v e s , t h e p r o p o r t i o n of women t h a t h a s half t h e number of a l l children i s used as a quan- t i t a t i v e indicator of concentration. The empirical basis f o r t h i s p a p e r are mainly t h e World Fertility S u r v e y d a t a f o r 41 less developed c o u n t r i e s and 11 industrialized nations. This i s supplemented by o t h e r s o u r c e s including t h e Chinese one p e r thousand f e r t i l i t y s u r v e y .

The study shows t h a t during t h e c o u r s e of demographic transition concentration i n c r e a s e s while t h e a v e r a g e level of f e r t i l i t y d e c r e a s e s . This s t r o n g negative association holds f o r t i m e s e r i e s of t h e h i s t o r i c a l f e r t i l i t y decline in Germany and Austria and also v e r y c l e a r l y f o r t h e cross-section of 41 LDC's. The r e c e n t fertili- t y decline in China p r e s e n t s a major exception from t h i s g e n e r a l p a t t e r n s i n c e i t w a s not associated with increasing concentration. In modern European s o c i e t i e s concentration h a s been diminishing mainly because of a n approximation of r e a l p a r i t y distributions to t h e relatively homogeneous e x p e c t e d family sizes. The fu- t u r e t r e n d of concentration in childbearing in l o w f e r t i l i t y c o u n t r i e s will mainly depend on t h e e x t e n t of childlessness.

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The a u t h o r wants to thank James Vaupel and Dianne Goodwin f o r stimulating discussions on concentration in human reproduction, and Nathan Keyfitz and Doug- las Wolf f o r helpful comments on t h e manuscript. Thanks are extended t o Andreas Bakany f o r his a b l e r e s e a r c h assistance and to Susanne Stock f o r skillful textpro- cessing and editing. Parts of t h e work presented in t h i s study were s u p p o r t e d by G r a n t No. P6006 of t h e Austrian Science Foundation.

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vii

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Contents

Page

1

.

THE MEASUREMENT OF CONCENTRATION IN DEMOGRAPHY

...

¹

1.1. S t a t i c and Dynamic Concentration

...

1

1.2. The Absolute Concentration of Fertility

...

²

1.3. The Relative Concentration of Fertility

...

0

2

.

RELATIVE CONCENTRATION OF FERTILITY ALONG VARIOUS DEMOGRAPHIC DIMENSIONS

...

¹⁶

2.1. Time

...

¹⁶

2.2. Age

...

¹⁹

2.3. Marital Duration

...

²²

2.4. Space

...

²³

2.5. Individual Women

...

²⁶

3

.

DEMOGRAPHIC TRANSITION AND CONCENTRATION

...

³⁰

3.1. Marital Fertility in Germany and Austria from t h e Late 1800's t o 1939

...

30

3.2. Time S e r i e s f o r Selected Less Developed Countries

...

³⁵

...

3.3. A Cross-Sectional View on 41 Less Developed W F S Countries 38 4

.

THE CONCENTRATION OF PERIOD FERTILITY IN CHINA 1955-1981

...

⁴⁹

5

.

CONCENTRATION OF MARITAL FERTILITY IN EUROPE AND THE USA IN THE 1970s: A WFS PERSPECTIVE

...

⁵⁷

5.1. Socio-Economic Differential Concentration

...

57

5.2. Birth Control and Concentration

...

64

5.2.1 Concentration and Current U s e of Contraception in Spain and Portugal

...

⁶⁴

5.2.2 Ever-Use of Contraception in Eight European Countries

...

66

5.3. The Distribution of Expected Family Size

...

⁶⁷

6

.

THE CONCENTRATION OF OVERALL FERTILITY IN AUSTRIA

...

⁷¹

7

.

CONCLUSIONS

...

⁷⁴

REFERENCES

...

⁷⁶

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The Concentration of Reproduction:

A

Global Perspective

1.

THE HEA.SUBEId[ENT OF CONCENTRATION IN DEYOGBAPEIY

1.1.

Static and Dynamic Concentration

E v e r y real population i s heterogeneous with r e s p e c t to various a s p e c t s : individual survival, r e p r o d u c t i o n , migration, m a r r i a g e , d i v o r c e , etc. T h e r e are always some subgroups of t h e population t h a t h a v e a h i g h e r r i s k of d e a t h , marriage, o r divorce at c e r t a i n a g e s . T h e r e i s also no population where all women b e a r t h e same number of children. If w e d o not t a k e t h e individual b u t a t o t a l (e.g. national) population as t h e unit of observation, we also find considerable d i v e r s i t y among t h e populations in t h e world, which in some cases i s g r e a t e r , in o t h e r s smaller, t h a n t h e diversity within t h e populations.

Dispersion and c o n c e n t r a t i o n are two notions t h a t are closely r e l a t e d . Without dispersion in t h e distribution t h e r e i s n o concentration a n d vice v e r s a . The questions behind those two notions a r e , however, somewhat different: t h e i n d i c a t o r s of dispersion t e l l us how strongly t h e units of observation d i f f e r from e a c h o t h e r with r e s p e c t to t h e i r o u t p u t , w h e r e a s indicators of concentration tell u s how t h e total o u t p u t i s a t t r i b u t e d to individual units.

Our speaking of concentration usually c o v e r s t w o quite d i f f e r e n t a s p e c t s . W e may r e f e r to dynamic concentration, i.e. t h e p r o c e s s of a distribution becoming m o r e c o n c e n t r a t e d , or to s t a t i c concentration. S t a t i c concentration o b s e r v e s t h e s t a t u s of t h e distribution at a given point in time. This second meaning seems to b e much wider s p r e a d , at least in t h e economic l i t e r a t u r e ( s e e Bruckmann 1981). The analysis of concentration as a p r o c e s s i s t h e n usually done by a comparative stat- i c s a p p r o a c h . This implies t h e calculation of s t a t i c concentration measures at dif- f e r e n t points in time a n d t h e i r comparative analysis.

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For demographic applications this s t a t i c understanding of t h e word concen- t r a t i o n a l s o seems t o be a p p r o p r i a t e . Demographers may study t h e distribution at one point in time and t h e n compare i t s changes o v e r time. This seems e a s i e r t o define unambiguously t h a n t h e dynamic meaning which only would make s e n s e under a s p a t i a l p e r s p e c t i v e , e.g. concentrating t r o o p s in one place. The question of c l e a r definition i s c r u c i a l to t h e demographic application of concentration analysis. Un- like f o r many economic questions, in population science i t i s less clear what i s t h e output and what a r e t h e producing units. A s mentioned above, t h e units may be all individuals, individuals with c e r t a i n c h a r a c t e r i s t i c s (e.g women of a c e r t a i n age group) o r populations. The o u t p u t may b e virtually everything t h a t i s measured in demography. Only nonrepeatable events t h a t o c c u r to everybody, such as d e a t h , are not a p p r o p r i a t e f o r conaentration analysis, because t h e r e cannot be any concentration. But o t h e r a s p e c t s of mortality, s u c h as t h e concentration of child mortality among families, might well be studied. In t h i s context, however, we will only focus on t h e concentration of fertility. The b i r t h of a child is, on a global p e r s p e c - tive, i s t h e most often r e p e a t e d demographic e v e n t ( e x c e p t p e r h a p s f o r local mi- grations) and i t s distribution h a s important consequences f o r t h e mother, t h e child, a n d t h e population as a whole: population reproduction seems t o b e t h e most a p p r o p r i a t e demographic field f o r concentration analysis.

Before discussing specific indicators of concentration w e s t i l l must make a n o t h e r important distinction: concentration may b e understood in absolute o r relative terms. Absolute concentration focuses on t h e o u t p u t of a s m a l l absolute number of t h e highest producing units, whereas r e l a t i v e concentration looks at t h e proportion of t h e units t h a t produce t h a t o u t p u t . The essential empirical differ- e n c e i s t h a t if a g r e a t number of additional units with little o u t p u t were added t o t h e t o t a l , t h e measure of absolute concentration should remain essentially un- changed, whereas r e l a t i v e concentration should i n c r e a s e significantly. This will b e discussed in g r e a t e r detail below.

1.2. T h e A h l u t e C o n c e n t r a t i o n o f F e r t i l i t y

Because of t h e n a t u r e of demographic phenomena t h e study of absolute con- c e n t r a t i o n i s only r e a s o n a b l e and informative f o r a v e r y r e s t r i c t e d number of questions. Generally t h e c o n c e p t of r e l a t i v e concentration will b e t h e a p p r o p r i a t e one because demographic information i s mostly r e l a t e d to t h e t o t a l of t h e population o r c e r t a i n subgroups (age groups). One g r e a t advantage of measures of abso-

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lute concentration i s t h a t t h e y may b e applied to distributions t h a t are truncated at t h e lower end (e.g. in t a x r e c o r d s , o r lists of companies t h a t start at a c e r t a i n size). In population statistics, however, this kind of d a t a problem usually does not exist. Furthermore, in demography t h e r e i s a natural limit to concentration. The number of children women c a n b e a r during t h e i r lifetime i s biologically limited and i t i s v e r y small in relation to t h e total number of children b o r n e in a society.

Hence a small absolute number of even extremely f e r t i l e women would n e v e r account f o r a sizable proportion of t h e total number of b i r t h s in a society.

With a l l t h e s e r e s e r v a t i o n s , t h e r e a r e a few questions in demography where t h e study of absolute concentration i s informative if t h e units of observation a r e a g g r e g a t e populations instead of individuals. I t i s a meaningful statement to s a y t h a t in 1950-1955 more t h a n one q u a r t e r of a l l children in t h e world where born in China, and t h a t China, India, and t h e Soviet Union accounted f o r almost half of a l l b i r t h s (see Table 1). Similar calculations c a n b e made f o r o t h e r periods and t h e s t a t i c statements on absolute concentration may t h e n b e compared o v e r time.

How c a n absolute concentration b e measured adequately? Bruckmann (1969) specified f o u r conditions which e v e r y indicator of concentration should meet; t h e f i r s t t w o hold for absolute and relative concentration as well, t h e t h i r d and fourth a r e differentiating c r i t e r i a between absolute and r e l a t i v e concentration:

1. Indicators of absolute and relative concentration must b e insensitive to proportional changes in t h e o u t p u t p e r unit; instead they should only depend on t h e proportion of t h e t o t a l o u t p u t produced by e a c h unit.

2. An i n c r e a s e of t h e proportion p ( A ) at t h e expense of p ( B ) by a c e r t a i n amount e should--given p ( A )

>

p @)-lead t o a n i n c r e a s e in t h e indicators of absolute and r e l a t i v e concentration (and vice versa). F o r maximum concentration t h e indicator should b e equal to unity, and otherwise be g r e a t e r or equal to zero.

3. An i n c r e a s e in t h e number of producing units by drawing from t h e s a m e distribution should a f f e c t t h e indicator of absolute concentration but not t h a t of r e l a t i v e concentration. In o t h e r words, only t h e indicator of absolute concen- t r a t i o n should depend o n t h e sample size.

4. The inclusion of a g r e a t e r number of producing units with v e r y little o u t p u t should a f f e c t t h e indicator of absolute concentration only marginally, whereas t h e indicator of r e l a t i v e concentration should respond markedly to such a change.

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Table l a . Absolute concentration of a v e r a g e annual numbers of b i r t h s by coun- t r i e s in 1950-1955.

Proportion

of total Conoentration

Rank Number of births ratio

order Country births ~ ( i ) Z P ( ~ )

I China 26675 0.2647 0.2647

2 India 17024 0.1689 0.4336

3 USSR 4947 0.0491 0.4027

4 USA 3946 0.0392 0.5128

5 Indonesia 3573 0.0355 0.5573

6 Brazil 2569 0.0255 0.5828

7 Pakistan 2091 0.0207 0.6035

8 Japan 2052 0.0204 0.6239

9 Nigeria 1784 0.0177 0.6416

10 Bangladesh 1769 0.0176 0.6591

I 1 Mexloo 1378 0.0137 0.6728

12 Vietnam 1206 O.OU-0 0.6848

13 Korea 1121 O . O l l 1 0.6959

14 Philippines 1094 0.0109 0.7067

15 Thailand 1016 0.0101 0.7168

16 Egypt 1014 0.0101 0.7269

17 Turkey 1003 0.0100 0.7368

18 Iran 920 0.0091 0.7460

19 Ethiopia 885 0.0088 0.7547

20 Italy 870 0.0086 0.7634

21 Franoe 829 0.0082 0.7716

22 Great Britain 808 0.0080 0.7796

23 GFR 807 0.0080 0.7876

24 Poland 785 0.0078 0.7954

25 Rep. of Korea 773 0.0077 0.8031

Herftndahl-Index: 0.1079

These c r i t e r i a will b e used as guidelines in t h e following evaluation of abso- l u t e (this section) and r e l a t i v e (next section) measures of concentration.

The most often used indicator of absolute concentration, t h e concentration r a t i o , i s defined as follows: l e t a ( i ) be t h e absolute amount of o u t p u t produced by unit i , and m a c e r t a i n small integer. Then

F o r this r a t i o t h e producing units a r e ranked by o u t p u t size and t h e m g r e a t e s t

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p r o d u c e r s are compared to t h e rest of t h e units. The selection of m is a r b i t r a r y . The r a t i o tells us what p r o p o r t i o n of t h e total output i s produced by t h e g r e a t e s t 3, 5, 10, 25, or m o r e production units.

Table lb. Absolute c o n c e n t r a t i o n of a v e r a g e annual numbers of b i r t h s by coun- t r i e s in 1975-1980.

Proportion

of total Conoentration

Rank Number of births ratio

order Country births ~ ( 2 )

c

^{~ ( i )}

1 India 23583 0.1893 0.1893

2 China 21313 0.1711 0.3603

3 Indonesia 5220 0.0419 0.4022

4 USSR 4745 0.0381 0.4403

5 Bangladesh 3889 0.0312 0.4715

6 Nigeria 3751 0.0301 0.5016

7 Brazil 3671 0.0295 0.5311

8 USA 3621 0.0291 0.5601

9 Pakistan 3569 0.0286 0.5888

10 Mexioo 2433 0.0195 0.6083

11 Vietnam 1995 0.0160 0.6243

12 Japan 17!57 0.0141 0.6384

13 Egypt 1583 0.0128 0.6512

14 Phlllppines 1540 0.0124 0.6636

15 Iran 1526 0.0122 0.6758

16 Ethiopia 1507 0.0121 0.6879

17 Korea 1478 O.Oll9 0.6998

18 Turkey 1475 0.0118 0.7116

19 Thailand 1380 0.0111 0 . W

20 Burma 1264 0.0101 0.7328

21 Zaire 1225 0.0098 0.7427

22 SouthAfrica 1033 0.0083 0.7510

23 Rep. of Korea 930 0.0075 0.7584

24 Tanzania 885 0.0071 0.7655

25 Kenya 855 0.0069 0.7724

Herfindahl-Index: 0.0758

In case of t h e global distribution of b i r t h s t h e concentration r a t i o s c a n b e s e e n in Table 1 f o r m ranging from 1 to 25. F o r t h e a v e r a g e annual number of b i r t h s in t h e p e r i o d 1950-1955, t h e t o p 25 c o u n t r i e s accounted f o r m o r e t h a n 80%

of a l l children b o r n in t h e 156 c o u n t r i e s considered by t h e UN population statistics. This value declines to 77% in 1975-1980 and i s p r o j e c t e d to decline to 74% by 1995-2000. These f i g u r e s indicate a d e c r e a s e in absolute concentration of b i r t h s

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among a l l c o u n t r i e s in t h e world. A s w e c a n see from t h e t a b l e t h i s decline i s mainly due to t h e s u c c e s s of t h e b i r t h c o n t r o l program in China, while in o t h e r c o u n t r i e s t h e number of b i r t h s i s s t i l l increasing. The proportion of all b i r t h s t h a t t a k e place in China h a s been declining from 26.5% in 1950-1955 to 17.1% in 1975-1980; i t i s e x p e c t e d to decline f u r t h e r to 15.3% in 1995-2000. The five countries with t h e highest numbers of k i r t h s , i.e. t h e concentration r a t i o f o r m

=

5 , accounted f o r 55.7% in 1950-1955, b u t only f o r 47.2% in 1975-1980, and t h e i r contribution i s ex- p e c t e d to decline to 43.0% in 1995-2000.

This decline in absolute concentration can a l s o be d e s c r i b e d by giving t h e number of c o u n t r i e s t h a t a c c o u n t f o r about half of all children b o r n in t h e world.

These f i g u r e s cannot b e e x a c t because c o u n t r i e s may not be interpolated. In 1950- 1955 t h r e e c o u n t r i e s accounted f o r almost half of a l l b i r t h s in t h e world, in 1975- 1980 i t needed s i x c o u n t r i e s to produce half t h e children and in 1995-2000 i t will be s e v e n t o e i g h t countries. Two t h i r d s of t h e b i r t h s w e r e born in 11 c o u n t r i e s in 1950-1955, in 15 c o u n t r i e s in 1975-1980, and will b e b o r n in 18 c o u n t r i e s in 1995- 2000.

Hence, in o u r case any selection of m o b s e r v e d o v e r time r e s u l t s in t h e same finding: absolute concentration of b i r t h s among c o u n t r i e s diminishes considerably during t h e second half of t h e twentieth century. But not always t h e selection of dif- f e r e n t m yields equivalent r e s u l t s . For some distributions a c e r t a i n m indicates a n i n c r e a s e in concentration whereas a n o t h e r m may indicate a d e c r e a s e . This pro- p e r t y of t h e concentration r a t i o i s c l e a r l y not d e s i r a b l e . In o u r c a s e , even though all m indicated t h e same d i r e c t i o n of change, t h e e x t e n t of change measured still depended on t h e a r b i t r a r y selection of m

.

This deficiency in t h e consistency of t h e concentration r a t i o may be overcome by considering a l l information given by t h e distribution and not only t h a t of those at t h e t o p of t h e list. Herfindahl (1950) f i r s t provided a consistent indicator f o r absolute concentration. The Herfindahl index is defined by

where p ( i ) again i s t h e p r o p o r t i o n of output produced by unit i . In t h e case of maximum concentration, i.e. a l l output being produced by one unit, t h i s index r e a c h e s unity. In t h e case of no concentration, i.e. a n even distribution, t h e Her-

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findahl index will b e 1 / N , N being t h e number of units observed. The Herfindahl index meets all t h e conditions given above. Its minimum value of I/ N as compared to a possible value of z e r o i s not a disadvantage but even a necessity in o r d e r t o m e e t condition 3, i.e. t h e sensitivity of t h e index to changes in t h e sample size.

Table lc. Absolute concentration of a v e r a g e annual numbers of b i r t h s by coun- t r i e s in 1995-2000.

Rank Number of

order Country births

Proportion of total

births

~ ( i )

China India Nigeria USSR Bangladesh Indonesia Pakistan Brazil USA Mexi o o Ethiopia Zaire Vietnam Iran Kenya Egypt Tanzania Philippines Turkey Japan Burma South Afrioa Korea Thailand Sudan

Herfindahl-Index: 0.0603

Conoentration ratio

The Herfindahl index f o r t h e t h r e e periods is a l s o given in Table 1. I t con- firms t h e above finding t h a t absolute concentration declines significantly from 1950-1955 to 1995-2000. I t also becomes clear t h a t t h e decline w a s markedly s t r o n g e r during t h e f i r s t 2 5 y e a r s than t h e expected decline o v e r t h e next q u a r t e r century. The main reasons f o r this lie in t h e dramatic Chinese fertility declines

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between 1965 and 1980 t h a t h a s a l r e a d y r e a c h e d s u c h low levels, t h a t i t cannot decline much f u r t h e r . Simultaneously to t h e Chinese decline t h e number of b i r t h s in a l a r g e f r a c t i o n of less developed countries, most prominently in India, h a s in- c r e a s e d substantially, a f a c t t h a t brought about more evenness among t h e coun- t r i e s with t h e highest numbers of births, i.e. less concentration. For t h e period up to t h e end of t h e c e n t u r y , absolute numbers of b i r t h s are e x p e c t e d to decline in s e v e r a l l a r g e Asian countries, whereas t h e y are p r o j e c t e d t o i n c r e a s e in p a r t s of Latin America a n d especially s t r o n g in Africa. This will f u r t h e r diminish t h e dif- f e r e n t i a l s between t h e l a r g e s t countries in terms of t h e f u t u r e s i z e s of t h e i r youth cohorts.

1-3. The Eeiative Concentration of Fertility

The major c r i t e r i o n t h a t distinguishes r e l a t i v e concentration from absolute concentration i s t h a t measures of r e l a t i v e concentration should b e a l s o sensitive to t h e addition of units at t h e v e r y end of t h e list, ranking units by output. Sup- pose two populations, A and B, had e x a c t l y t h e same numbers of f e r t i l e women and identical p a r i t y distributions b u t only population B had, in addition t o t h e f e r t i l e women, a sizable number of childless women; t h e n t h e r e l a t i v e concentration of f e r t i l i t y in B should b e h i g h e r t h a n t h a t in A although t h e absolute numbers of chil- d r e n born and t h e i r distribution are identical.

The Lorenz c u m e provides a n a t u r a l tool to d e s c r i b e r e l a t i v e concentration.

Introduced by Lorenz (1905), i t d e p i c t s t h e relationship between cumulated producing units and cumulated output units as f r a c t i o n s of t h e total of producing units and t h e t o t a l output. Because of this basic s e t u p t h e Lorenz c u r v e becomes t h e c e n t r a l a p p r o a c h ( s e e Piesch 1975) t o t h e analysis of r e l a t i v e concentration. Al- most a l l f u r t h e r generalizations in t h e measurement of r e l a t i v e concentration t a k e t h i s a p p r o a c h and focus on specific f e a t u r e s , e.g. t h e slope of t h e Lorenz c u r v e at d i f f e r e n t points.

The Lorenz c u r v e c l e a r l y r e f e r s t o r e l a t i v e concentration because on both a x e s t h e r e are cumulated proportions: t h e x-axis shows t h e cumulated proportion of producing units, on t h e y-axis t h a t of output units. If t h e ranking of producing units by output i s done in a way t h a t puts t h e most productive units t o t h e left, t h e n t h e c u r v e comes to Lie above t h e diagonal, otherwise below. Lorenz (1905) origi- nally g a v e c u r v e s t h a t lie above t h e diagonal but especially in economic concentration analysis t h e c u r v e s often lie below t h e diagonal. Whether t h e ranking goes

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f r o m lowest t o highest or highest t o lowest does not affect t h e mathematics o r t h e general setup of t h e curve. It only r e f l e c t s t h e focus of t h e specific r e s e a r c h , whether one i s more interested in t h e lower end (such as poverty analysis in economics) or in t h e upper end of t h e distribution. In t h e life sciences and animal ecology where questions of dominance play an important r o l e (see Goodwin and Vaupel1985), r e s e a r c h e r s mostly put t h e m o s t productive at t h e beginning.

Figure 1. Densities of children e v e r born in Jordan, Portugal, and Cameroon t o ever-married women aged 40-49 (WFS).

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 C H I L D R E N E V E R BORN

JORDAN A PORTUGAL 0 CAMEROON

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Figures 1 to 5 show how t h e Lorenz c u r v e may b e built up by a graduation scheme (see Piesch 1975). Figure 1 gives t h r e e densities, indicating how women are distributed o v e r 1 6 possible completed p a r i t i e s (0 t o 15 children) in t h r e e WFS countries with v e r y d i f f e r e n t reproductive p a t t e r n s : Jordan, Cameroon, and Portu- gal. Women aged 40-49 in J o r d a n at t h e time of t h e s u r v e y had t h e highest f e r t i l i t y of a l l countries considered with t h e mode at p a r i t y nine. In Cameroon t h e p a t t e r n i s quite d i f f e r e n t mainly because of a high p r o p o r t i o n being childless (15%). The distribution of women with more t h a n one child p e a k s at p a r i t y seven in Cameroon.

Portugal i s a n example of a country t h a t has essentially completed t h e fertility transition. In P o r t u g a l completed marital fertility c l e a r l y peaks at p a r i t y two with more t h a n 15% of t h e women e a c h at p a r i t i e s one and t h r e e and around 10% at parity four.

Figure 2 gives t h e distribution functions of t h e t h r e e densities f o r Jordan, Cameroon, and Portugal. This i s simply done by cumulating t h e densities o v e r a l l p a r i t i e s from lowest t o highest. Because of t h e concentration of women at lower p a r i t i e s in P o r t u g a l t h e distribution function t h e r e i n c r e a s e s s h a r p l y . In Portugal women with 0 to 3 children a l r e a d y make up more t h a n 70% of a l l women. In Jordan t h e corresponding f i g u r e i s under 10%. Figure 3 n e x t gives t h e inverted distribution function with t h e cumulated proportion of women now on t h e x-axis and t h e distribution of children e v e r b o t n on t h e y-axis.

Figure 4 shows a modification of t h e i n v e r t e d distribution function where t h e x-axis i s normed and adjusted t o t h e f a c t t h a t t h e c a t e g o r i e s of children e v e r born contribute differently t o t h e t o t a l number of children born. Hence y from Figure

y . c

3 became y -, in Figure 4 , where c i s t h e density shown in Figure 1.

mean of y

Since Figure 4 may a l s o b e s e e n as a density with r e s p e c t t o t h e cumulated p r o p o r t i o n of women, t h e distribution function of t h i s density may b e calculated by cumulating again along t h e x-axis. This transition to cumulated p r o p o r t i o n s of children on t h e y-axis yields t h e Lorenz c u r v e given in Figure 5. The f u r t h e r t h e Lorenz c u r v e lies from t h e diagonal, t h e h i g h e r t h e concentration. In o u r example t h e distributions in Cameroon and P o r t u g a l are c l e a r l y more c o n c e n t r a t e d than t h a t in Jordan.

The Lorenz c u r v e i s a v e r y intuitive way to d e s c r i b e concentration and i t gives a complete p i c t u r e of t h e relationship between cumulated p r o p o r t i o n s of producing units and output units. In some instances however, i t seems d e s i r a b l e to have t h e concentration of a c u r v e not only given by a g r a p h b u t a l s o calculate a

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Figure 2. Distribution functions f o r t h e t h r e e densities from Figure 1.

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 C H I L D R E N EVER BORN

JORDAN A PORTUGAL 0 CAMEROON

single quantitative i n d i c a t o r of concentration. T h e r e are a l s o cases where t h e Lorenz c u r v e s cross o v e r e a c h o t h e r and w e need some additional c r i t e r i a to de- cide which distribution i s h i g h e r c o n c e n t r a t e d .

Numerous indicators of r e l a t i v e concentration have b e e n suggested in t h e l i t e r a t u r e . A t t h i s point we will not give a s u r v e y of them. Since r e l a t i v e concen- t r a t i o n i s d i r e c t l y a function of r e l a t i v e variance, many i n d i c a t o r s of d i s p a r i t y , dissimilarity, and unevenness may a l s o b e t a k e n as indicators of t h e d e g r e e of con- c e n t r a t i o n . O t h e r c o n c e n t r a t i o n coefficients are d i r e c t l y based on t h e Lorenz c u r v e . P e r h a p s t h e b e s t known i n d i c a t o r of t h i s kind i s t h e G i n i coeppicient. In terms of t h e Lorenz c u r v e a p p r o a c h , t h e C i n i coeppicient i s twice t h e area

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between t h e concentration c u r v e and t h e diagonal (Foster 1985).

The Gini coefficient h a s t h e advantage of summarizing t h e complete information given by t h e Lorenz c u r v e . I t may b e used t o compare t h e d e g r e e of concen- t r a t i o n of two d i f f e r e n t distributions; b u t i t i s not e a s y t o i n t e r p r e t i t in terms of t h e original d a t a . What does a Gini coefficient of .7 in t h e distribution of b i r t h s mean in terms of c e r t a i n f r a c t i o n s of women having c e r t a i n proportions of all chil- d r e n b o r n ?

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Figure 4. Adjusted inverted distribution function ( s e e text).

0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0.5 0.6 0.7 0 . 8 0.9 1 . 0 CUMULATED P R O P O R T I O N OF WOMEN

JORDAN A PORTUGAL 0 _CAMEROON

In t h i s context i t seems d e s i r a b l e to use a n analogon t o t h e concentration ratio, c,

,

which we introduced f o r t h e study of absolute concentration. The concen- t r a t i o n r a t i o tells us what p r o p o r t i o n of t h e total output w a s produced by a c e r t a i n absolute number of p r o d u c e r s . For r e l a t i v e concentration w e would a s k f o r t h e proportion of c h i l d r e n produced by a c e r t a i n proportion of women. These p r o p o r - tions may b e called f r a c t i l e s . If women a r e o r d e r e d by t h e i r family size we may ask what proportion of women gave b i r t h to 10% ( . I fractile), 50% (.5 f r a c t i l e ) , o r 90%

(.9 f r a c t i l e ) of all children. These values may be readily s e e n from t h e Lorenz c u r v e .

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Figure 5. Lorenz c u r v e s resulting from cumulation of adjusted inverted distribution functions.

CUMULATED PROPORTION OF WOMEN JORDAN A PORTUGAL 0 CAMEROON

Being v e r y intuitive and easy t o i n t e r p r e t as quantitative indicators of con- c e n t r a t i o n , t h e f r a c t i l e s have t h e shortcoming of not reflecting all t h e information given in t h e c u r v e . The .1 f r a c t i l e t e l l s us what p e r c e n t a g e of highest-fertility women gave b i r t h t o 10% of all children. I t mainly r e f l e c t s t h e distribution among v e r y high-fertility women and does not tell much a b o u t t h e d e g r e e of concentration at t h e lower end of t h e distribution. The .9 f r a c t i l e would s u f f e r from t h e opposite shortcoming. The l e a s t amount of information would b e lost when using t h e .5 f r a c - tile because i t i s at t h e c e n t e r of t h e distribution a n d r e f l e c t s both t h e s h a p e of t h e c u r v e b e f o r e and a f t e r t h e .5 f r a c t i l e .

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This may b e illustrated with t h e help of Figure 5. Since in t h i s f i g u r e women were ranked from low f e r t i l i t y ( a t t h e l e f t ) t o high f e r t i l i t y , t h e c u r v e lies below t h e diagonal. But because in this study t h e f r a c t i l e s will consistently b e defined as p e r c e n t a g e s of a distribution o r d e r e d from highest to lowest. In Figure 5 , t h e x- f r a c t i l e s should b e r e a d as l-x. Although t h e Gini coefficients are at similar mag- nitude f o r P o r t u g a l and Cameroon, t h e c u r v e s look quite d i f f e r e n t . In Cameroon t h e c u r v e i s s t e e p e r f o r high-parity women and less s t e e p f o r l ~ w - ~ a r i t ~ women, mainly because of high proportions of childless women. Because of t h i s cross-over in t h e concentration c u r v e s , t h e f r a c t i l e s from highest f e r t i l i t y down to about t h e c e n t e r of t h e c u r v e indicate h i g h e r concentration f o r P o r t u g a l and t h e r e a f t e r h i g h e r concentration f o r Cameroon. In t h i s specific case t h e .5 f r a c t i l e indicates s t i l l somewhat h i g h e r concentration f o r P o r t u g a l s i n c e i t i s above t h e cross-over of t h e two c u r v e s . But generally t h e c o r r e l a t i o n between t h e Gini coefficient and t h e -5 f r a c t i l e should b e h i g h e r than f o r o t h e r f r a c t i l e s . Empirical studies (Goodwin, Lutz, and Vaupel1906) on 4 1 less developed c o u n t r i e s r e s u l t in c o r r e l a - tion coefficients between t h e Gini index and t h e .5 f r a c t i l e of above .9.

P o r t u g a l and Cameroon are e x t r e m e cases in a way t h a t t h e i r normalized pari- t y distributions (Figure 4 ) have v e r y d i f f e r e n t s h a p e s . I t is interesting to f u r t h e r note t h e d i f f e r e n c e s in t h e c u r v e s f o r Portugal and Cameroon. The c u r v e f o r Por- tugal i s initially s t e e p e r b u t then i t crosses t h a t of Cameroon and f l a t t e n s out.

Thus t h e c u r v e f o r P o r t u g a l initially shows more concentration t h a n t h a t f o r Cam- e r o o n but beyond t h e c r o s s o v e r point this situation i s r e v e r s e d . The f l a t end on t h e c u r v e f o r Cameroon indicates an e x t r a o r d i n a r i l y high proportion of married women with completed p a r i t y z e r o (15.4%) and p a r i t y one (9.6%). F o r most o t h e r c o u n t r i e s in t h e world t h e Lorenz c u r v e s are of similar s h a p e (varying only in t h e i r distance from t h e diagonal) and hardly c r o s s .

Vaupel and Goodwin (1985) call t h e f r a c t i l e s w e defined h e r e Have-statistics because t h e y indicate what proportion of women has a c e r t a i n p r o p o r t i o n of chil- d r e n . The .5 f r a c t i l e in this terminology becomes t h e have ha^, a name t h a t c l e a r - ly indicates t h e meaning of t h e coefficient. Because of i t s e a s y i n t e r p r e t a t i o n and t h e f a c t t h a t i t i s a consistent s t a t i s t i c a l measure (Goodwin and Vaupel 1985) t h a t i s sensitive to not-proportional changes in any of t h e values of t h e underlying f r e - quency distribution, w e will use t h e .5 f r a c t i l e (interchangeably called Haveham as o u r basic measure of r e l a t i v e concentration in t h e following analysis.

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2.

RELATIVE

CONCENTRATION OF FERTILITY AUING VABIOUS DEMOGRAPHIC DIMENSIONS

2.1. Time

Concentration analysis always r e f e r s to t h e distribution of o u t p u t units among producing units. The o u t p u t units are always b i r t h s in t h i s study; t h e pro- ducing units may b e individual women b u t a l s o a g g r e g a t e s of women defined with r e s p e c t to d i f f e r e n t demographic dimensions. With r e s p e c t to h i s t o r i c a l time w e may define a given time unit ( y e a r , month, day) as t h e producing unit and study t h e distribution of b i r t h s o v e r those time units. With r e s p e c t to a g e , i.e. subjective time, we may define c e r t a i n y e a r s of a g e as producing units and o b s e r v e t h e distribution of b i r t h s o v e r a g e s . An equivalent a p p r o a c h can b e t a k e n when we study t h e distribution of b i r t h s o v e r marital durations. F'inally, not only temporal a g g r e g a t e s but also s p a t i a l a g g r e g a t e s may b e r e g a r d e d as producing units. W e may f o r instance look at t h e distribution of b i r t h s o v e r all c o u n t r i e s in t h e world or o v e r all km2 of land on o u r planet.

In t h i s section we will focus on t h e distribution of b i r t h s o v e r h i s t o r i c a l time.

Table 2 and Figure 6 show t h e concentration of b i r t h s on t h e world o v e r t e n thousands of y e a r s from 8,000 B.C. to 2,000 A.D. The estimates f o r t h e period be- f o r e modern demographic records became available are based on population sizes and a v e r a g e annual i n c r e a s e s suggested in United Nations (1973), assuming a c r u d e b i r t h rate of 5 0 p e r thousand up to 1900 and later empirical figures. F o r t h e period 1950-2000 A.D., United Nations (1985) estimates and projections h a v e been used.

Table 2. Crude estimates of t h e distribution of t o t a l b i r t h s between 8,000 B.C. and 2,000 A.D.

Total of Births 8,000 B.C. t o 2,000 A.D.: about 80 billion

Fraotile Birth Time Period

.5 40 billion BOO-1,000 A.D. t o 2,000 A.D.

.25 20 billion around 1760 A.D. to 2,000 A.D.

.10 8 billion 1931 A.D. to 2,000 A.D.

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The estimates suggest t h a t half of all children e v e r b o r n within t h e s e 10,000 y e a r s have been b o r n within t h e l a s t 1,000 y e a r s , i.e. one t e n t h of t h e time con- s i d e r e d . Since world population growth h a s speeded up especially during t h e last c e n t u r y t h e most r e c e n t p e r c e n t i l e s show a n even g r e a t e r d i s c r e p a n c y between cumulated b i r t h a n d cumulated y e a r s : 10% of all b i r t h s w e r e and will b e b o r n during only 7 0 y e a r s from 1 9 3 1 t o 2,000 A.D. These are only .07% of t h e total time con- s i d e r e d . If w e had not only t r i e d t o estimate t h e distribution of b i r t h s o v e r time b u t also t h a t of people alive at any point in time, t h e c o n c e n t r a t i o n would b e even g r e a t e r because of significant improvements in live expectancy.

If w e want t o b a s e o u r s t u d y on r e l i a b l e records r a t h e r t h a n t h e c r u d e estimates given above, w e are r e s t r i c t e d to t h e analysis of much s h o r t e r time periods.

Like Sweden, Finland h a s r e l i a b l e annual f i g u r e s on f e r t i l i t y s i n c e 1722. The con- c e n t r a t i o n of b i r t h s o v e r t h e 264 y e a r s of Finnish population h i s t o r y up t o 1 9 8 5 are a l s o plotted in Figure 1. I t t u r n s out t h a t 32% of all y e a r s considered (i.e. 8 5 y e a r s ) produced half of t h e Finns b o r n o v e r t h e last 264 y e a r s . The 8 5 most f e r - t i l e y e a r s d o not include t h e most r e c e n t y e a r s s i n c e 1968; t h e y include y e a r s from t h e middle of t h e nineteenth c e n t u r y t o t h e post-war baby boom. Even though t h e total f e r t i l i t y rate w a s highest in t h e eighteenth c e n t u r y , t h e s e e a r l y p e r i o d s con- t r i b u t e d only r e l a t i v e l y l i t t l e t o t h e t o t a l number of b i r t h s because t h e s i z e of t h e population in 1 7 5 0 w a s less t h a n one t e n t h of today's population size. But s t i l l t h e 264 y e a r s of Finnish f e r t i l i t y h i s t o r y are much less c o n c e n t r a t e d t h a n t h e 10,000

y e a r s of world history.

Temporal variations of b i r t h s d o not only o c c u r between y e a r s , t h e y may a l s o b e o b s e r v e d within a y e a r . In most c o u n t r i e s b i r t h s show some distinctive seasonal p a t t e r n and a l s o some variation between d i f f e r e n t days of t h e week. The seasonali-

t y of b i r t h s seems t o b e especially s t r o n g in a N o r t h e r n climate (see Rantakallio 1971) and in c o u n t r i e s where t h e m a r r i a g e p a t t e r n i s s t r o n g l y seasonal. F o r t h i s r e a s o n w e again s e l e c t e d Finland t o demonstrate t h e concentration with r e s p e c t t o t h e month of b i r t h . In 1976-a y e a r with h i g h e r seasonal variation t h a n more re- c e n t years-53% of all b i r t h s o c c u r r e d within 50% of all months. In this calculation w e adjusted f o r t h e f a c t t h a t months are of d i f f e r e n t length. By f a r t h e g r e a t e s t numbers of children w e r e b o r n in t h e months of March and April, nine months a f t e r midsummer. A second smaller peak o c c u r r e d in September, nine months a f t e r Christmas.

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Figure 6. Lorenz c u r v e s f o r t h e temporal concentration of fertility: o v e r t h e period 8,000

B.C.

to 2,000 A.D. f o r t h e world as a whole, f o r 1722-1985 in Finland, and f o r monthly variations within t h e y e a r 1976 in Finland.

CUMULATED PROPRTION OF T I M E

O F I N 1 9 7 6 A F I N 1 7 2 2 - 1 9 8 5 OWORLD 8000 BC

-

²⁰⁰⁰^AD

Very short-term variations in b i r t h s , namely variations by weekdays, a r e not likely to depend on t h e time of conception but r a t h e r on t h e working h o u r s in hos- pitals around t h e time of b i r t h . Table 3 gives t h e adjusted distribution of b i r t h s o v e r weekdays in Austria. The p a t t e r n r e v e a l s t h a t on Sundays only 85% of t h e b i r t h s o c c u r r e d t h a t usually o c c u r on a n a v e r a g e working day. On Saturdays t h i s r a t i o i s 91%. I t i s h a r d to calculate a .5 f r a c t i l e in t h e c a s e of seven weekdays; b u t assuming a n even distribution t h a t 52.1% of a l l b i r t h s a r e born on t h e 3.5 days t h a t have t h e highest b i r t h frequencies. The Lorenz c u r v e of t h e concentration of f e r - tility among weekdays h a s almost exactly t h e same s h a p e as t h a t f o r t h e seasonal

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Table 3. Distribution of b i r t h s o v e r weekdays: Austria, 1984.

Absolute number* In peroent

Monday 12.684 14.5

Tuesday 13.318 15.3

Wednesday 12.941 14.8

Thursday ^12.625 14.5

Friday 12.868 14.8

Saturday 11.702 13.4

Sunday 11.065 12.7

*adjusted for the aase that all weekdays ooour 52 times a year

concentration.

In summary w e may s a y t h a t w e found h i g h e r d e g r e e s of r e l a t i v e variation with r e s p e c t t o temporal changes when w e o b s e r v e longer periods of time t h a n within s h o r t e r time units. The distribution of b i r t h s o v e r 10,000 y e a r s of global history i s c l e a r l y one of t h e highest c o n c e n t r a t e d c u r v e s t h a t c a n b e found in t h e field of f e r t i l i t y analysis. Monthly and daily variations are minimal by t h e s e s t a n d a r d s .

2.2. Age

The concentration of f e r t i l i t y with r e s p e c t t o subjective time, i.e. a g e , i s v e r y d i f f e r e n t from t h a t o v e r historical time. We d o not evaluate t h e concentration within a period of time t h a t includes many demographic changes, but w e o b s e r v e t h e distribution of b i r t h s o v e r t h e a v e r a g e life c o u r s e of a r e a l o r synthetic c o h o r t . This distribution c a n b e made s u b j e c t to t h e analysis of s t a t i c concentration. Comparing t h e s t a t i c measures o v e r different c o h o r t s in d i f f e r e n t historical periods o r d i f f e r e n t c o u n t r i e s makes t h e analysis one of comparative s t a t i c s .

When studying t h e a g e concentration of fertility w e look at t h e distribution of b i r t h s o v e r t h e potentially f r a c t i l e a g e s (15-49), r a n k single-year a g e g r o u p s according to t h e i r f e r t i l i t y and calculate what proportion of a g e s produces (single- y e a r a g e g r o u p s being t h e producing unit) a c e r t a i n fraction of a l l b i r t h s . Table 4 shows f o r selected l e s s developed c o u n t r i e s participating in WFS how many single-year a g e g r o u p s produce half t h e number of children b o r n t o all women in- terviewed in t h e s u r v e y . The f i r s t t h r e e columns give t h e f i g u r e s t h a t r e f l e c t t h e a c t u a l a g e distribution of women, column 4 gives age-standardized r e s u l t s , i.e. t h e r e s u l t s f o r a synthetic c o h o r t disregarding mortality. Since w e consider 35 a g e groups, 6 a g e g r o u p s mean a .5 f r a c t i l e of .17 while eleven a g e g r o u p s indicate a

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value of -34 f o r t h e havehaw.

Table 4. Number of single-year age g r o u p s t h a t produced 50% of a l l children in t h e y e a r b e f o r e t h e s u r v e y f o r selected WFS countries.

Total Rural women Urban women Total agestandardized

Costa Rioa 9 5 7 8

Kenya 9 7 9 ii

Pakistan 9 6 9 9

P e n 9

Sri Lanka 8

The table indicates t h a t t h e a g e concentration of f e r t i l i t y i s h i g h e r among urban women t h a n among r u r a l women. The total i s usually close to t h e r u r a l value because of high r u r a l p e r c e n t a g e s in t h e populations considered. Age- standardization seems to make t h e distribution o v e r a l l a g e s in m o s t c a s e s more even.

The difference between t h e a g e concentration of Finnish f e r t i l i t y in 1776 and 1983 plotted in Figure 2 l i e s a l m o s t p a r a l l e l to t h e u r b a n / r u r a l differential ob- s e r v e d in Table 4. In both c a s e s a lower level of f e r t i l i t y i s associated with h i g h e r a g e concentration. An explanation f o r t h i s may b e found in t h e concepts of n a t u r a l f e r t i l i t y and controlled fertility. A s defined by Henry (1961) w e may s p e a k of na- t u r a l f e r t i l i t y if t h e b i r t h of a child does not depend on t h e number of children al- r e a d y borne by t h e mother. Since p a r i t y and a g e strongly c o r r e l a t e t h e existence of n a t u r a l f e r t i l i t y i t may a l s o b e i n f e r r e d from t h e a g e p a t t e r n of f e r t i l i t y if i t i s concave, i.e. shows a relatively slow decline a f t e r t h e a g e of peak fertility. In c o n t r a s t t o this, t h e p a t t e r n of controlled fertility produces a r a t h e r pronounced peak with f e r t i l i t y s h a r p l y declining a f t e r t h e peak ages. Because of t h i s a n a t u r a l f e r t i l i t y p a t t e r n like t h a t in Finland in 1776 or in many r u r a l less developed populations today i s e x p e c t e d to show only a weak age concentration as compared t o populations with widespread u s e of b i r t h c o n t r o l methods.

In Figure 7 t h e a g e concentration c u r v e s f o r t h e two high f e r t i l i t y populations-Kenya in t h e 1970s and Finland in 1776--contrast against t h e p a t t e r n of much h i g h e r concentration f o r t h e low f e r t i l i t y of Finland in 1983. F o r Finland where we have annual time-series of age-specific fertility r a t e s f o r more than 200

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Figure 7. Lorenz c u r v e s f o r t h e a g e concentration of b i r t h s (considering a g e s 15-49) in Kenya (WE'S), Finland 1776, and Finland 1983.

Cum. P r o p . o f A g e e

C l K e n l a A F l n I a n d 1 9 8 3 O F l n l a n d 1 7 7 6

y e a r s , we may follow this development and identify t h e p e r i o d during which t h e transition from low to high a g e concentration took place. Figure 8 plots t h e t o t a l f e r t i l i t y rate against t h e .5 f r a c t i l e of ages f o r 67 three-year periods in I?inland1 f o r 1776-1976. W e c a n see t h a t until a r o u n d 1930 t h e a g e concentration did not show any lasting i n c r e a s e d e s p i t e a s t r o n g decline in t h e total f e r t i l i t y rate from w e l l above 5 to under 2.5. For t h e pre-industrial period w e see some short-term fluctuations t h a t had no lasting e f f e c t . The s e c u l a r i n c r e a s e in a g e concentration 'since only five-year age groups ere given, we had to interpolate to single-year age groups.

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s t a r t e d about 20 y e a r s a f t e r t h e onset of t h e g r e a t f e r t i l i t y decline. I t is a l s o in- t e r e s t i n g t o notice t h a t t h e post-war baby boom t h a t peaked around 1947 in Finland w a s associated with h i g h e r t o t a l fertility rates but not with a r e t u r n to a p r e - modem s h a p e of t h e c u r v e of age-specific fertility rates ( s e e Lutz 1984). Since 1960 t h e p a t t e r n of age-specific rates h a s become even more focused around 25, t h e c u r r e n t a g e of peak f e r t i l i t y in Finland. The increasing concentration of childbearing within a relatively s h o r t a g e s p a n i s typical f o r m o s t industrialized coun- t r i e s ( s e e Lutz and Yashin 1987).

2.3. Marital Duration

Another a s p e c t of t h e subjective time dimension t h a t i s of high r e l e v a n c e f o r f e r t i l i t y variations o v e r t h e life cycle i s marital duration. W e know t h a t a g e a n d marital duration e x e r t independent influences on f e r t i l i t y ( s e e e.g. P a g e 1977 a n d Hobcraft and Casterline 1983) and t h a t t h e r e i s some interaction between those t w o variables. When women m a r r y late--as in t h e case of t h e GFR in Table 5-and t h e o v e r a l l level of f e r t i l i t y i s low, t h e n t h e bulk of children tends t o be b o r n within t h e f i r s t few y e a r s of marriage. The c a s e of Nepal shows t h e o t h e r e x t r e m e where women m a r r y at a v e r y young a g e (often b e f o r e s e x u a l maturity) and almost no voluntary family limitation t a k e s place o v e r t h e life course. In s u c h a case childbearing i s c o n c e n t r a t e d around some c e n t r a l marital duration of 10-14 y e a r s .

F o r P e r u t h e distribution o v e r marital durations i s somewhat less concentrat- e d t h a n in Nepal, mainly because of a h i g h e r a g e at marriage which r e s u l t s in h i g h e r b i r t h f r e q u e n c i e s during t h e f i r s t 0-4 y e a r s of marriage. Marital f e r t i l i t y in Portugal shows t h e m o s t even distribution of t h e f o u r countries considered h e r e : with a substantially l o w e r a g e at marriage t h a n in Germany and some d e g r e e of voluntary childspacing combined with a r a t h e r s t r o n g heterogeneity of t h e population with r e s p e c t to t h e quantum and timing of b i r t h s . Portugal h a s t h e Lorenz c u r v e t h a t lies closest t o t h e diagonal.

In Figure 9 t h e German distribution of b i r t h s o v e r marital duration in 1983 i s by f a r t h e most c o n c e n t r a t e d one: in t h e German Federal Republic more t h a n 57%

of all marital b i r t h s o c c u r during t h e f i r s t f o u r y e a r s of marriage. A s has been hinted at above, t h i s i s not only because of t h e high a g e at marriage (27.2 f o r wom- e n in 1983) but also because of t h e f a c t t h a t many couples have only one child, and if they have two they often t r y t o s p a c e them close to e a c h o t h e r . All in all, w e c a n s a y t h a t marital duration c o n c e n t r a t i o n s of f e r t i l i t y v a r y within t h e s a m e r a n g e as

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Figure 8. Changes in t h e relationship between t h e a g e concentration of f e r t i l i t y and t h e total f e r t i l i t y rate in Finland 1776-1976 (in t h r e e - y e a r steps).

I

I I I I I I I I I

.50 2 . 0 0 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.

TOTAL F E R T I L I T Y RATE

a g e concentration: both depend crucially on t h e f e r t i l i t y and nuptiality p a t t e r n of t h e population concerned.

2.4. Space

W e may not only consider temporal a g g r e g a t e s as units t h a t produce children but w e might as w e l l think of s p a t i a l entities as bringing a b o u t children. In t h e section on absolute concentration w e a l r e a d y considered nations as producing un- its. The same a p p r o a c h may a l s o b e t a k e n f o r t h e analysis of r e l a t i v e concentration.

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Table 5. Fraction of children b o r n to c e r t a i n marital durations of all children b o r n up to duration 25-29.

0-4 3-9 10-14 15-19 20-24 2529

GFR .573 2 9 1 .lo6 .025 .004 .001

Nepal .OBI 216 .250 -216 .I50 .086

Figure 1 0 plots t h e Lorenz c u r v e s f o r t h e 100 c o u n t r i e s t h a t produce most b i r t h s . The f i g u r e shows c u r v e s f o r 1950-1955 and 1995-2000. The Lorenz c u r v e f o r 1975-1980 comes to between those two c u r v e s . In a l l c a s e s t h e People's Repub- lic of China i s to t h e v e r y l e f t because i t c o n t r i b u t e s by f a r t h e l a r g e s t proportion of b i r t h s . W e s e e t h a t in 1995-2000 China i s supposed to c o n t r i b u t e a smaller f r a c - tion to t h e total number of children b o r n on e a r t h t h a n in 1950-1955. This i s a ma- j o r r e a s o n f o r t h e f a c t t h a t t h e 1995-2000 c u r v e lies c l o s e r t o t h e diagonal a n d hence i s l e s s concentrated.

Countries are of v e r y d i f f e r e n t size and i t i s not at a l l s u r p r i s i n g t h a t l a r g e countries c o n t r i b u t e a h i g h e r fraction of a l l b i r t h s . For t h i s r e a s o n i t s e e m s d e s i r a b l e from a s p a t i a l point of view t o t a k e a country's area into account. We then may a s k what p r o p o r t i o n of all km2 of land on e a r t h p r o d u c e s what p r o p o r t i o n of children. This i s done by calculating a b i r t h d e n s i t y f o r e a c h country (i.e.

b i r t h p e r km2) and ranking t h e c o u n t r i e s according to t h a t density. After ranking, t h e c o u n t r i e s must b e weighted again by t h e land area t h e y c o v e r in o r d e r to calculate fractiles.

From Table 6 w e see t h a t only l e s s than 1% of land area p r o d u c e s 10% of a l l children; exactly 10% of a l l land p r o d u c e s half t h e children, a n d less t h a n half of t h e land p r o d u c e s 90% of a l l children. This indicates a l r e a d y a v e r y high concen- t r a t i o n of b i r t h s with r e s p e c t to s p a t i a l distribution. In o r d e r to calculate t h i s we had to make one v e r y r e s t r i c t i v e and unrealistic assumption, namely t h a t within e a c h nation b i r t h s are evenly distributed o v e r a l l km2. Especially f o r c o u n t r i e s like t h e Soviet Union o r nations with high f r a c t i o n s of t h e i r t e r r i t o r y being d e s e r t o r r a i n f o r e s t , t h i s i s v e r y misleading. In r e a l i t y t h e s p a t i a l concentration of b i r t h s i s much h i g h e r , especially if we consider t h e g r e a t u r b a n agglomerations s e p a r a t e l y .

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Figure 9. Lorenz c u r v e s f o r t h e concentration of b i r t h s along s i x marital duration c a t e g o r i e s (see Table 4) f o r t h e German F e d e r a l Republic in 1983, and t h e W F S d a t a f o r Portugal, P e r u , and Nepal.

V. V V L I I I I I

0.00 0.20 0.40 0.60 0.80 1 .OO

CUM. PROP. OF MARITAL DURATIONS 0 GFR A PORTUGAL 0 PERU X NEPAL

Without going into t h e vast geographical l i t e r a t u r e on s p a t i a l distribution, i t s h a l l b e pointed o u t t h a t t h e distribution of b i r t h s in some r e s p e c t i s more impor- t a n t t h a n t h e distribution of living people. The s p a t i a l distribution of b i r t h s i s c r u - c i a l f o r f u t u r e food supply, industry, and education-all f a c t o r s t h a t a f f e c t t h e s t a n d a r d of living and t h e level of development.

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Figure 10. Lorenz c u r v e f o r t h e 100 countries t h a t produce t h e most b i r t h s , 1950-1955 and 1995-2000.

2.5. Individual Women

So f a r w e had looked at temporal and s p a t i a l a g g r e g a t e s t h a t produced children. These w e r e all a g g r e g a t e s of women t h a t gave b i r t h to children at d i f f e r e n t times, d i f f e r e n t a g e s , d i f f e r e n t marital durations, and in d i f f e r e n t places. These a g g r e g a t e s a l l showed specific macro-aspects of t h e distribution of b i r t h s . To study t h e concentration of reproduction in a specific g r o u p of women at a specific time a n d place, however, w e have t o g o back t o t h e micro level and consider t h e original p r o d u c i n g unit: t h e individual woman.