Keyfitz N., Caswell H. Applied Mathematical Demography

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x Preface to the Third Edition

tics Department at the University of Auckland. I owe a special thanks to

Sinauer Associates for their permission to use material from Matrix Popu-

lation Models, where a much more detailed treatment of these models may

be found.

Finally, my deepest thanks go to Nathan Keyﬁtz, for his contributions to

mathematical demography over a remarkably long and productive career,

and for the privilege of working with him on this book.

Woods Hole, Massachusetts, USA Hal Caswell

Preface to the Second Edition (1985)

What follows is a new edition of the second in a series of three books

providing an account of the mathematical development of demography.

The ﬁrst, Introduction to the Mathematics of Population (Addison-Wesley,

1968), gave the mathematical background. The second, the original of the

present volume, was concerned with demography itself. The third in the

sequence, Mathematics Through Problems (with John Beekman; Springer-

Verlag, 1982), supplemented the ﬁrst two with an ordered sequence of

problems and answers.

Readers interested in the mathematics may consult the earlier book, re-

published with revisions by Addison-Wesley in 1977 and still in print. There

is no overlap in subject matter between Applied Mathematical Demography

and the Introduction to the Mathematics of Population. Three new chapters

have been added, dealing with matters that have come recently into the

demographic limelight: multi-state calculations, family demography, and

heterogeneity.

Nathan Keyfitz

Preface to the First Edition (1977)

This book is concerned with commonsense questions about, for instance,

the eﬀect of a lowered death rate on the proportion of old people or the

eﬀect of abortions on the birth rate. The answers that it reaches are not

always commonsense, and we will meet instances in which intuition has to

be adjusted to accord with what the mathematics shows to be the case.

Even when the intuitive answer gives the right direction of an eﬀect, tech-

nical analysis is still needed to estimate its amount. We may see intuitively

that the drop from an increasing to a stationary population will slow the

promotion for the average person in a factory or oﬃce, but nothing short

of an integral equation can show that each drop of 1 percent in the rate of

increase will delay promotion to middle-level positions by 2.3 years.

The aim has been to ﬁnd answers that will be serviceable to those working

on population and related matters, whether or not they care to go deeply

into the mathematics behind the answers. My earlier book, Introduction

to the Mathematics of Population, had the opposite purpose of developing

the theory, and mentioned applications mostly to illuminate the theory.

Because of their diﬀerent objectives there is virtually no overlap between

the two books.

Population theory has developed at a suﬃciently fast rate and in enough

directions that no book of reasonable size can include all of its applications.

A full development of theory ought to recognize not only age-speciﬁc rates

of birth and death but also two sexes and two or more species. Age-speciﬁc

rates can vary through time, and the theory can be stochastic in allow-

ing to each individual member of the population his own separate risk, or

deterministic in supposing that whatever probability applies to each in-

xiv Preface to the First Edition

dividual is also the fraction of the population that succumbs to the risk.

Thus population theory can be classiﬁed into the 16 categories shown in

the accompanying table. By far the largest part of what is taken up in this

book falls into the upper left category: it deals with one sex, usually female,

and one species, man; it takes the age-speciﬁc rates of birth and death as

ﬁxed through time; it is deterministic rather than stochastic. This upper

left-hand cell is conceptually the simplest of the 16, and it is mathemati-

cally the most tractable. But are these decisive arguments for its emphasis,

given that real populations include two sexes; human populations interact

with other species; birth and death rates change through time; and all life

is stochastic?

The art of theory construction is to start with simple assumptions

and then to introduce greater realism, which means more complexity, as

required. On the path from simplicity to realism one must stop at a com-

promise point. My taste may not always be that of my readers; they may

often say that a particular model I use is too simple, that they need to take

into account factors that I neglect. This line of criticism is welcome, even

though it leads to further and more diﬃcult mathematics.

Classiﬁcation of population theory

Fixed rate Changing rate

Deterministic Stochastic Deterministic Stochastic

One sex

One species *

Two or more species

Two sexes

One species

Two or more species

During 10 or more years of work on this book I have incurred more

obligations than I can acknowledge or even remember. Students pointed

out errors and obscurities; they helped in some cases by conspicuously

failing to understand what I was saying and compelling me to think the

matter through afresh. Colleagues looked at drafts and were generous with

comments. Editors and referees of journals were helpful, especially Paul

Demeny. No one is responsible for errors that remain but me.

Among these colleagues, students and correspondents who have been

a source of ideas and a means of correcting errors, I recall especially

William Alonso, Barbara Anderson, Brian Arthur, John C. Barrett, Ans-

ley J. Coale, William Cochrane, Joel E. Cohen, Prithwis Das Gupta, Paul

Demeny, Lloyd Demetrius, James Dobbins, Barry Edmonston, Jamie Eng,

Thomas Espenshade, Noral Federici, Griﬃth Feeney, Gustav Feichtinger,

Jair Fereira-Santos, James Frauenthal, A. G. Fredrickson, Robert Gardiner,

Campbell Gibson, Noreen Goldman, Antonio Golini, David Goodman, Leo

A. Goodman, Louis Henry, Jan Hoem, Barbara Keyﬁtz, S. Krishnamoorthy,

Paul Kwong, Juan Carlos Lerda, John Lew, Gary Littman, Robert Lundy,

Preface to the First Edition xv

James G. March, Robert Mare, George Masnick, John McDonald, David

McFarland, Geoﬀrey McNicoll, Paul Meier, Jane Menken, Walter Meyer,

George C. Myers, Frank Oechsli, Beresford Parlett, James Pick, Robert G.

Potter, Jr., Samuel H. Preston, Thomas Pullum, Robert Retherford, Roger

Revelle, Andrei Rogers, Normal Ryder, Paul Samuelson, Robert Sembir-

ing, David P. Smith, Leroy O. Stone, Richard Stone, Michael Stoto, Michael

Teitelbaum, Harold A. Thomas, Robert Traxler, James Trussell, Etienne

Van de Walle, Kenneth Wachter, Frans Willekens and Harrison White,

every one of whom has had some eﬀect on this book. To be singled out

especially for their help in the ﬁnal stages are Noreen Goldman, David P.

Smith, Gary Littman, S. Krishnamoorthy, and Michael Stoto, who read

much of the manuscript and all of the page proof.

The most important acknowledgment is to my wife, who edited and typed

the manuscript, some parts of it many times.

Cambridge, Massachusetts, USA Nathan Keyfitz

February 1977

Acknowledgments

An early version of Section 1.4 appeared as “How many people have lived

on the earth?” Demography 3: 581–582 (1966).

Section 1.10 was published as “A general condition for stability in

demographic processes,” Canadian Studies in Population 1: 29–35

(1974).

The life table method of Chapter 2 ﬁrst appeared as “An improved life table

method,” Biometrics 31: 889–899 (1975) (with James Frauenthal).

Material in Chapter 3 is extracted from Chapters 2 and 3 of Matrix

Population Models: Construction, Analysis, and Interpretation,2dedi-

tion, Sinauer Associates, Sunderland, Massachusetts, USA. Used by

permission.

Much of Chapter 4 was included in “Mortality comparisons: The male-

female ratio,” Genus 31: 1–34 (1975) (with Antonio Golini).

Section 5.8 is a condensed version of “Individual mobility in a stationary

population,” Population Studies 27: 335–352 (1973).

Chapter 7 contains material extracted from Chapter 4 of Matrix Population

Models: Construction, Analysis, and Interpretation, 2d edition, Sinauer

Associates, Sunderland, Massachusetts, USA. Used by permission.

Some of Section 8.5 appeared in “Migration as a means of population

control,” Population Studies 25: 63–72 (1971).

Section 8.6 is an abridged version of “On the momentum of population

growth,” Demography 8: 71–80 (1971).

xviii Acknowledgments

Parts of Section 8.8 were published as “Age distribution and the stable

equivalent,” Demography 6: 261–269 (1969).

Chapter 11 contains material from Chapter 5 of Matrix Population Mod-

els: Construction, Analysis, and Interpretation, 2d edition, Sinauer

Associates, Sunderland, Massachusetts, USA. Used by permission.

Some parts of Chapter 12, as well as Section 14.3, were included in “On

future population,” Journal of the American Statistical Association 67:

347–363 (1972).

Parts of Chapter 13 come from Chapters 9 and 10 of Matrix Population

Models: Construction, Analysis, and Interpretation, 2d edition, Sinauer

Associates, Sunderland, Massachusetts, USA. Used by permission.

Section 14.5 is an abridged version of “Backward population projection

by a generalized inverse,” Theoretical Population Biology 6: 135–142

(1974) (with T. N. E. Greville).

The substance of Sections 14.10 and 14.11 was included in “Popula-

tion waves,” in Population Dynamics (edited by T. N. E. Greville),

Academic Press, 1972, pp. 1–38.

Parts of Chapter 15 are from “Family formation and the frequency of

various kinship relationships,” Theoretical Population Biology 5:1–

27 (1974) (with Leo A. Goodman and Thomas W. Pullum), and

“Addendum: Family formation and the frequency of various kinship

relationships,” Theoretical Population Biology 8: 376–381 (1975) (with

Leo A. Goodman and Thomas W. Pullum).

An early statement of Section 16.1 appeared as “How birth control aﬀects

births,” Social Biology 18: 109–121 (1971).

Chapter 20 is reprinted with some modiﬁcations from “How do we know the

facts of demography?” Population and Development Review 1: 267–288

(1975).

Contents

Preface to the Third Edition vii

Preface to the Second Edition xi

Preface to the First Edition xiii

Acknowledgments xvii

A Word About Notation xxv

1 Introduction: Population Without Age 1

1.1 Deﬁnitions of Rate of Increase ............... 2

1.2 DoublingTimeandHalf-Life................ 3

1.3 One-Sex Versus Two-Sex Models: Descendants of the

PilgrimFathers ....................... 9

1.4 HowManyPeopleHaveLivedontheEarth?....... 11

1.5 A Mixture of Populations Having Diﬀerent Rates of

Increase ........................... 13

1.6 Rate of Increase Changing over Time ........... 18

1.7 Logistic Increase and Explosion .............. 20

1.8 TheStalledDemographicTransition ........... 23

1.9 Diﬀerential Fertility Due to the Demographic

Transition .......................... 25

1.10 MatricesinDemography .................. 26