Growth and Structure of Human Populations: A Mathematical Investigation

Growth and Structure of Human Populations: A Mathematical Investigation

by Ansley Johnson Coale


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ISBN-13: 9780691619811
Publisher: Princeton University Press
Publication date: 03/08/2015
Series: Office of Population Research
Pages: 246
Product dimensions: 6.90(w) x 9.90(h) x 1.80(d)

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The Growth And Structure of Human Populations

A Mathematical Investigation

By Ansley Johnson Coale


Copyright © 1972 Princeton University Press
All rights reserved.
ISBN: 978-0-691-09357-4


Fertility, Mortality, and Age Distributions: Introduction

The age composition of a population that neither gains nor loses by migration is determined by the recent sequence of fertility and mortality risks at each age to which it has been subject. Its overall birth rate, death rate, and rate of increase at each moment are determined by the current age composition and the current age schedules of fertility and mortality. In principle, then, both age composition and vital rates can be determined from knowledge of the history and present value of fertility and mortality schedules. Consider a female population in which the annual death rate of persons at age a and time t is μ,(a, t), and the annual rate of bearing a female child at age a and time t is m(a, l). If these schedules are specified for a sufficient time interval — in practice no more than a century — the age composition, birth rate, death rate, and rate of increase can all be calculated. If an age distribution in the past is among the available data, the calculation of current age composition and vital rates can be made by standard methods of population projection. If no past age distribution is stipulated, it appears at first that knowledge of past fertility and mortality schedules is not sufficient to calculate the current age distribution. However, Alvaro Lopez has proven a conjecture made by the author in 1957: that two arbitrarily chosen age distributions no matter how different, subject to identical sequences — whether varying or constant — of fertility and mortality, ultimately generate populations with the same age composition (Coale [2], Lopez [8, 9], also McFarland [10]). Age distributions gradually "forget" the past. It is therefore possible to reproduce the age composition of a given population by projecting an arbitrarily selected initial population from many years in the past, employing the age schedules of fertility and mortality that the population actually experienced in the interim. If the period of projection is long enough, the effect of the arbitrary initial age composition wholly disappears, and the current age composition (c (a, t0)) is seen to be entirely a function of μ(a, t) and m(a, t) during a substantial interval before t0. Moreover, if (c(a, t)da) is the proportion of the female population between age a and a + da, b(t) is the birth rate and d(t) the death rate, and ωis the highest age attained,


The rate of increase of the population at time t — r(t) — is the difference between h(t) and d(t), and the proportionate increase in numbers 2 r (t)dt



The principal purpose of this book is to analyze the relation of levels, age patterns, and time patterns of fertility and mortality to the growth and age composition of populations. If our minds could readily visualize the outcome of the large number of multiplications and additions that constitute a population projection, the book could be terminated right here, or at most would need to include a brief description of how a population projection is made and a recapitulation of Lopez' proof. But in view of our incapacity to visualize elaborate calculations, what has been presented is enough to enable a demographer merely to calculate an age distribution, a birth rate, a death rate, and a rate of increase from a long sequence of fertility and mortality schedules without providing him the basis for understanding what features of mortality, fertility, and their changes account for particular characteristics of the age structure, for changes in the birth and death rates, etc. In trying to explain how age structures are formed, and vital rates determined, we shall consider the age distributions produced by schedules of fertility and mortality that do not change with the passage of time (Chapters 2 and 3); and then the age distributions produced by changing fertility and mortality (Chapters 4 to 7).

Fertility and Mortality Schedules in Human Populations


The variation of the rate of childbearing with age depends partly on biological factors. The capacity to conceive and bear children normally begins at about age 15 (following menarche), attains a broad maximum beginning at about age 20, falls slightly by age 30, and then follows an accelerated pace of decline after age 35. A typical mean age at the birth of the last child among married women not practicing birth control is about 40 years; only a small minority can bear children after age 45 and practically none after age 50.

The actual curve of childbearing as a function of age of course depends not only on the varying capacity to conceive and bear live issue, but on variations with age in exposure to intercourse with a fertile partner, and on whether or not measures are taken in different degree at different ages to prevent conception or to cause an early termination of pregnancy. The rise of fertility with age is strongly affected by laws and customs that determine when women enter fertile unions. Among societies in which marriage is ordinarily a prerequisite for fruitful intercourse, there are wide variations in mean age at marriage, from less than 15 to nearly 30 years. The decline of fertility with age is influenced by the rising incidence of widowhood, divorce, and perhaps of abstinence, as well as by declining fecundability. The practice of contraception and abortion could in principle reduce fertility at any age in the fertile span and thus produce an essentially arbitrary modification of the age structure of fertility; but, in fact, voluntary control seems always to cause a greater proportionate reduction of fertility among older women (i.e., women 30 to 45) than among younger.

Figure 1.1 shows schematically how the principal factors that determine the shape of fertility schedules operate. In panel A the variations in the shape of age-specific fertility schedules of cohabiting women that occur with and without contraception are shown. The schedules have been adjusted so that the maximum of each is 100; otherwise the schedule of women practicing contraception would ordinarily have a lower maximum. In panel B can be seen variations in the proportion of women cohabiting, again with the maximum proportions set at 100. With very early marriage — as in India, Pakistan, and much of Africa — the highest proportions are reached in the early twenties, while in nineteenth century northern Europe, the peak occurs well in the thirties. In each panel, the curves are based on recorded values; almost all recorded experience would be matched by these curves, or by intermediate ones.

The combined effect of variations in fecundability, cohabitation rates, pregnancy wastage, and contraception with age is to produce age schedules of childbearing that vary quite substantially, both in overall level (as indicated by total fertility, or by the gross reproduction rate if referring to female births) and in the shape of the fertility schedule, in the sense of the proportion of childbearing that occurs in different segments of the fertile span. Nevertheless, age schedules of fertility in human populations have a number of general features in common. All rise smoothly from zero at an age in the teens to a single peak in the twenties or thirties, and then fall continuously to near zero in the forties and to zero not much above age 50. The lowest mean age of fertility I have found is a little less than 26 years (in recent years in Hungary); the highest a little over 33 years (in nineteenth century Sweden). In all but a few fertility schedules, 75% of total fertility occurs within a span of 16 years, and in every case I have examined within a span of 20.

The area under the fertility schedule — total fertility if the schedule refers to all live births, and the gross reproduction rate if to female births only — could range from zero (in a celibate or sterile population) to a maximum that would be attained by constant exposure to intercourse from menarche to menopause, the prohibition of contraception, abortion, and breastfeeding, by medical treatment for the sterile, etc. However, no actual population on record approaches such a maximum, and certainly no large population closed to migration has ever had zero fertility.

Variations in the gross reproduction rate can be described in terms of the two components of age-specific fertility shown in panels A and B of Figure I. I . The highest fertility that can be realistically imagined would result from a combination of early and continued cohabitation — the left curve in panel B modified to remain near its peak in the early twenties — and the absence of contraception or abortion — the right curve in panel A. This combination would provide the shape, or age structure, of a very high fertility schedule. In Figure 1.1 the maximum vertical ordinate of each curve has been set at 100, but in fact the maximum differs from one population to another. There are large differences in fertility rates at the same ages even among different groups of cohabiting women who do not practice "controlled fertility" as defined by Louis Henry, the highest rates being at least 50% above the lowest at almost every age. The existence of control in this sense can lower the whole schedule of fertility, including its maximum, well below the level that would otherwise exist. The proportion cohabiting sometimes approaches a peak of 100%. In many populations virtually every woman by her mid-twenties has been married, or has been a partner in a consensual union of some sort, but widowhood, divorce, and separation usually reduce the proportion currently cohabiting at any given age noticeably below 100%, and to progressively lower levels in the thirties and forties. In societies characterized by late marriage, there is often a substantial proportion (as high as 30%) who remain single at age 50 (Hajnal [5]).

The highest gross reproduction rate based on complete and reliable records is 4.17 for the women in the Cocos-Keeling Islands (Smith [11]). The high fertility in this population results from the combination of a favorable age structure of fertility (because cohabitation proportions follow the left curve in panel B of Figure 1.1) and the absence of fertility control as defined earlier. However, the fertility of the married women in this population is surpassed by nearly 50% at every age by marital fertility among the Hutterites (Eaton and Mayer [4]) and among the French population of Canada in the eighteenth century (Henripin [6]); and therefore a population with the age pattern of co-habitation characteristic of the Cocos-Keeling Islands and the fertility at each age of married Hutterites or eighteenth century French Canadians would have a gross reproduction rate of over six.

The lowest gross reproduction rates for a national population were recorded in Europe during the 1930s — a level of about 0.80 in Austria and Sweden. Very much lower fertility has occurred within smaller geographical units — 0.30 in Vienna at the time the Austrian GRR was 0.80, for example.


In almost every accurately recorded schedule of death rates by age, mortality declines sharply during the first year from a high value immediately after birth, falls more moderately after age 1 to a minimum between age 10 and 15, increases gradually until about age 50, then increases ever more steeply until the highest age for which a rate is given. There are exceptions to this pattern: When tuberculosis is a major cause of death there may be a broad peak of mortality in the twenties and thirties followed by a slight decline centered on age 35 or 40 (before the accelerated rise with age beginning at 45 or 50); when diarrhea and enteritis are major causes of deaths under age 5 there may be a minor rise in mortality at about age 6 months, breaking the usual pattern of rapid decline from age 0 to 2 or 3.

There is a remarkable general similarity in the age schedules of mortality among today's highly industrialized countries. In all such countries (Australia, New Zealand, Canada, the United States, the Soviet Union, almost all of the countries of Europe, and Japan) current mortality schedules would produce an average duration of life for women of more than 70 years. Infant mortality among females varies from about 12 to about 40 per thousand live births; the death rate falls to an extremely low minimum between ages 10 and 15 of from .2 to .9 per thousand; then increases gradually until the fifties; and rises sharply by age 65 to 70, lying between 14 and 27 per thousand in this age interval. There is also a broadly similar age pattern of mortality among populations with high death rates. When the expectation of life at birth is 30 years or less, mortality in the first year of life is more than 200 per thousand, falls to a minimum at or soon after age 10 of 5 to 10 per thousand, and by 65 to 70 is over 70 per thousand, and rising rapidly with age.

These generally similar patterns found in high and low mortality schedules are a manifestation of an interrelation that exists among the mortality risks experienced by different age segments of the population: If mortality at a particular age in one population is 10 times as high as in another, one normally expects higher mortality at every other age as well. Such an interrelationship is caused by the fact that persons of different ages in the same population share conditions that affect the health of all: the state of medical technology and the organization of medical practice and public health services, environmental sanitation, standards of diet and housing, etc. At the Office of Population Research, after examining all of the national mortality tables based on high quality data we could assemble, we found four characteristic sets of age patterns of mortality. Within each set the intercorrelations among mortality rates at different ages are extremely high — over .90 from age 0 to age 70 in each set, and over .95 in most instances. In other words, if the mortality rate is known at one age within one of these sets of schedules, the rate at other ages can be accurately estimated. One of these age patterns characterizes the mortality experienced in Norway, Sweden, and Iceland; another the mortality schedules of central and parts of eastern Europe; a third the schedules of Spain, Portugal, and southern Italy; and a fourth encompasses mortality in western Europe, northern America, Oceania, Japan, and Taiwan. Twenty-four model life tables have been calculated for each of these age patterns of mortality at levels of mortality ranging from a life expectation of 20 years to one of 77.5 (Coale and Demeny [3]). Each set of model life tables provides age schedules of mortality at every level of mortality from close to the lowest to close to the highest of peacetime, nonepidemic human experience — age schedules conforming in pattern to the recorded experience of a particular group of populations. These model tables will be used frequently in this book to illustrate the effect of differences in, or changes in, mortality.

Figure 1.2 shows two typical schedules of female mortality — one at a high overall level (e00 = 30 years), and the other at a low level (e00 = 65 years).

Age Distribution and Birth and Death Rates

The birth and death rates in a population are the result of the interaction of the age schedules of fertility and mortality on the one hand and of the age composition on the other, as indicated by Equations (1.1) and (1.2). Sometimes the same schedule offertility or mortality can produce quite different birth or death rates in populations with differing age composition. Because age composition has an important influence, birth and death rates are not satisfactory summary indexes of age schedules of fertility and mortality. For example, Austria had lower mortality rates at every age in 1961 than Mexico (less than half the Mexican rates at all ages less than 50) but had a higher death rate; and the Japanese birth rate of 1956 was higher than the birth rate in the United Kingdom in 1961, although Japanese fertility rates at every age were lower. In each instance, of course, the source of the anomaly is the difference in age distribution in the populations compared: Austria (when compared to Mexico) had nearly four time as large a proportion over 65, a part of life with high mortality rates in every schedule, and Japan had a higher birth rate with a lower fertility schedule because of larger proportions at the childbearing ages than in the United Kingdom.


Excerpted from The Growth And Structure of Human Populations by Ansley Johnson Coale. Copyright © 1972 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents

  • Frontmatter, pg. i
  • Preface, pg. v
  • Contents, pg. vii
  • List of Figures. List of Tables, pg. xi
  • CHAPTER 1. Fertility, Mortality, and Age Distributions: Introduction, pg. 1
  • CHAPTER 2. The Stable Population, pg. 16
  • CHAPTER 3. Convergence of a Population to the Stable Form, pg. 61
  • CHAPTER 4. Population with Fertility that Changes at a Constant Rate, pg. 117
  • CHAPTER 5. Birth Sequences and Age Distributions with Changing Mortality, pg. 152
  • CHAPTER 6. The Birth Sequence and the Age Distribution That Occur When Fertility Is Subject to Repetitive Fluctuations, pg. 165
  • CHAPTER 7. The Relation Between the Birth Sequence and Sequence of Fertility Schedules in Any Time Pattern Derived by Fourier Analysis, pg. 194
  • CHAPTER 8. Conclusion, pg. 206
  • INDEX, pg. 225

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