Demographic
Research
Monographs

David P. Smith
Nathan Keyfitz

Mathematical
Demography
Selected Papers
Edited by
Kenneth W. Wachterand Hervé Le Bras
Second, revised edition

123

Demographic Research Monographs
A Series of the Max Planck Institutefor Demographic Research

Editor-in-chief
James W. Vaupel
Max Planck Institute for Demographic Research,
Rostock, Germany

For further volumes:
http://www.springer.com/series/5521

David P. Smith • Nathan Keyfitz

Mathematical Demography
Selected Papers
Second, revised edition

Edited by Kenneth W. Wachter and Hervé Le Bras

123

David P. Smith
Harvard University
Cambridge, USA

Nathan Keyfitz
Department of Sociology
Harvard University
Cambridge, USA

Editors:
Kenneth W. Wachter
University of California
Berkeley, USA
Hervé Le Bras
School for Advanced Studies in the Social Sciences
Paris, France

Printed with the financial support of the Max Planck Society
ISSN 1613-5520
ISBN 978-3-642-35857-9
ISBN 978-3-642-35858-6 (eBook)
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Foreword to New Edition of Smith and Keyfitz

The test of a classic is whether its echo can be heard today. We at the Max Planck
Institute for Demographic Research are very pleased to be publishing this new
edition of Smith and Keyfitz’ collection of demographic classics. We hope that thisnew edition will amplify the echoes of the great demographic discoveries of the lastcenturies.
This new edition comes from both practical and intellectual needs of demographers. On a practical side, Smith and Keyfitz’ 1977 edition is out of print andunavailable to anyone wishing to have it on their shelves. With this re-publication,
we are not only making hardcopies once again available but also, as is our tradition
with the Springer Demographic Research Monographs, ensuring that the entire
publication is accessible on-line as open access at www.demogr.mpg.de/books/drm.
It will be easier than ever for modern scholars and students to obtain the classics
and their accompanying commentary.
On the intellectual side, the classics of the field represent essential nourishment
for the modern demographer, giving a sense of the long history of demographic
science as well as to see the insights of classic discoveries as their discoverers
framed them. We are thankful to Kenneth W. Wachter and Hervé Le Bras, the editorsof this new edition, for updating Smith and Keyfitz’ collection, selecting the mostenduring works and adding new commentaries where needed. As they explain in
their editors’ preface, the inspiration for this new edition had its source in Rostockat the Institute’s 2009 celebration of Demography’s birthday on February 27th, theday when John Graunt first read his ”Observations” at the Royal Society in London.
The publication of this new edition coincides with two important anniversaries– 2012 is the 350th anniversary of John Graunt’s foundational work, and 2013 is thecentenary of Nathan Keyfitz’ birth – a propitious time for the ring of demography’sclassics to be heard once again.
Rostock
December, 2012

Joshua R. Goldstein
James W. Vaupelv

Preface to the Second Edition

Mathematical demography has a distinguished early history, intertwined withdiscoveries by the great figures of mathematics. Our generation of demographerslearned this history from the volume which we now in large part republish. David
Smith and Nathan Keyfitz in their collection of 1977 made the founding works
available to all. But this collection has long been out of print, making it far moredifficult for the present generation of demographers to locate their own works within
the tradition of their forerunners.
The idea of republishing “Smith and Keyfitz” arose during a symposium at
the Max Planck Institute for Demographic Research in Rostock, Germany, on 27
February 2009. The symposium, held in the Süssmilch Auditorium, was convened
by James Vaupel and Joshua Goldstein to inaugurate several years of celebrationof the 350th birthday of demography. The birthday follows on the February 1662
appearance of the Natural and Political Observations ... Made Upon the Bills of
Mortality... published under the name of “John Graunt, Citizen of London”. The
event in Rostock also celebrated the contributions of Peter Laslett. Peter inspired
and guided the fusion of mathematical demography with historical sociology andstudies of aging which marks the discipline of population studies today. He alsoorganized and edited the series of full facsimile reprintings by Gregg International
Publishers of the classics of demography which have served over the intervening
decades as indispensable companions to the Smith and Keyfitz volume.
The past cannot just be archived. New steps forward bring into the light newmeanings for previous formulations and give new salience to what may have been
obscure. Likewise, the genealogy of an idea has power to reshape its evolving
future form. We hope that readers of this volume will feel encouraged to submit
to current demographic journals studies of other works important to the history ofmathematical demography and write about the meaning of those works in their own
time and ours.
The lasting influence of Smith and Keyfitz is found not only in the works they
made accessible, but even more in the commentaries they wrote about these works.
These commentaries explain the terminology, notation, and intention of the originalvii

Preface to the Second Edition

authors and go on to situate their results in the framework of Twentieth-Century
mathematical demography. The commentaries combine historical scholarship with
scientific insight.
The 514 pages of the 1977 volume exceed the size desirable for a reprinting. A
selection of papers to retain was made at the outset of this project. In their originalcollection, Smith and Keyfitz included works as late as the 1970s which are now
readily available on the world-wide web and hardly need reprinting. Not all the
choices of works that were recent in 1977 have stood the test of time and some thenrecent works not chosen have come into the limelight now. A decision was made toomit from this reprinting the nearly contemporary works and concentrate on older
already classic contributions.
A decision was also made not to reprint the section from the 1977 volume
devoted to “Branching Processes and Other Stochastic Processes”. This subject
remains central to mathematical demography, but our understanding of its history
was subsequently transformed by the rediscovery of Bienyamé’s contributions. The
story is told by Heyde and Seneta [18] and the original works are readily accessible.
We retain the full commentaries by Smith and Keyfitz including discussion ofworks omitted from this reprinting. New editorial notes are inserted within bold
curly brackets, marked “Ed. Note:”, to alert the reader to anything omitted and to
set the commentaries into the context of our own Twentyfirst Century.
The date of original publication of this collection, at the end of the 1970s, marked
a watershed in the discipline. Three central themes dominate the volume – lifetables,
stable populations, and measures of fertility and its component processes. These
themes all go back to the early days of demography. Advances continue to bemade in these areas, but by 1977 the established core was coming to completion.
Stable population theory was being rounded out and turned into a versatile part of
the repertory, particularly by Ansley Coale and Nathan Keyfitz himself. Digitalcomputers were ending the need for ingenuity in numerical computations that hadgone hand-in-hand with the study of lifetables and projections. The spread of sample
surveys into developing countries eased the challenges for indirect measures offertility championed by William Brass.
Today, a third of a century later, mathematical demography is enlivened by
new directions, which have taken their place beside the themes featured bySmith and Keyfitz. We think of non-stable theory, demographic feedback models,heterogeneity and frailty, spatial demography, and the mathematical innovationsof biodemography. These endeavors trace some of their ancestry to the founders
represented in this volume, but also to other forerunners whose roles we hope future
historical studies will bring to center stage.
As this reprinting was in preparation, Nathan Keyfitz passed away at the ageof 96. A few months before his death, speaking with him by telephone, one of us(KWW) told him about plans for this project. He was delighted. He still took an
active interest in mathematical demography, and in the whole range of demographic
thinking, and he looked forward to having this cherished work back in print. In
preparing it we have benefited from encouragement and advice from BarbaraviiiPreface to the Second Edition

Keyfitz. We also salute David R. Smith, who took the lead in crafting the original
1977 volume. We dedicate this reprinting to Nathan. His own works are classics in
our field.
Berkeley, California, June 2013
Paris, June 2013

Kenneth W. Wachter
Hervé Le Bras

ix

Preface to the First Edition

This volume is an effort to bring together important contributions to the mathematical development of demography and to suggest briefly their historicalcontext. We have tried to find who first thought of the several concepts anddevices commonly used by demographers, what sort of problem he was facing
to which the device or concept seemed the solution, and how his invention
developed subsequently in the hands of others.
Historically, the book starts with a Roman table of life expectancies from thethird century a.d. about which we know little, and with John Graunt's explorations in an area that was still popularly suspect when he wrote in 1662. These
are followed by the astronomer Halley, who looked into the field long enoughto invent the life table and to notice that Their Majesties would take a sizeable
loss on the annuity scheme they had just launched; and by Euler, who was firstto devise the formulas of stable population theory and to apply them to filling
gaps in data To these we add the handful of further contributions in the 19th
century and many pieces from the explosion of contributions that began in thiscentury with Lotka. We doubt that we have managed to trace everything backto its ultimate beginning, and suspect that our nominees in some cases havebeen anticipated by predecessors who will be turned up by other students.
The works we include form a living heritage in demography: Graunt; Halley;
Euler; Lotka; Milne, who formalized life table construction; Lexis, who was
preoccupied with the way members of a population are situated simultaneously
in age and in time, and showed how a plane chart, now known as a Lexis diagram, can help analysis. Much less alive, and largely excluded here, are such
notions as that of George King, that graduation of data for a life table was moreaccurate from pivotal death rates calculated at five-year intervals; John Graunt's
belief that the right way to describe the dynamics of a population was as the ratioof births to deaths, without considering age; and devices that once reduced thelabor of numerical calculation but are obsolete in a computer age. These andmany other ideas that have proved to be dead ends and are now of merely antiquarian interest we tried to distinguish from those that were part of a chain ofdevelopment that is still advancing. As far as we could discriminate our excerpts
are confined to the latter.
To determine which works most deserve attention among the large number
written has not been easy, and we have undoubtedly made mistakes both ofinclusion and of omission. We were far from insisting on subtle mathematicalxiPreface to the First Edition

ideas, but did look for the effective uses of mathematics that have come to be
assimilated into population work. Articles that profess to deal with population
but whose main interest was mathematics we tried to avoid, and we avoided
them doubly if they were a mere import from some other subject that seemed
unlikely ever to be naturalized in population analysis. Some ideas and techniques
have a kind of droit de Ia cite in contemporary population study, and we hopethese are the ones th_at predominate in our selections.
To find passages that were self-contained and suitable for contemporary
reading was occasionally difficult. Writers often used symbols well known totheir place and time, and their immediate readership had no need for definitions
we would now miss. To this the earlier works add key formulas with no hint
as to how they are derived. Where we expect readers to have trouble as a result;because we did, we include a brief explanation of what is being done.
The choice of excerpts from the classic articles and books rather than complete reprints in all cases was dictated partly by economy of publication, but
this was not the only constraint. Benjamin Gompertz fairly compactly introduces
his Law of mortalitv. but spends above fifty pages fitting it to life tables andworking out its implications for annuity payments. Harro Bernardelli publishedthe first article on the use of matrices in population projection in the Journalof the Burma Research Society, which is not a source that most of us would
come across in our ordinary reading. He has top priority for inclusion, but hedeals partly with problems of the Burmese economy under British colonial ruleand with speculations on cyclic events that do not carry much interest forreaders today. Leslie, whose reading in a sickbed had taken him deep into themathematical properties of matrices, went into cogredient and contragredienttransformations that are unlikely to have demographic application. We saw
no need to burden the reader with these only to have him discover at the end
that he would never need them.
In editing we did not strip down our authors to the point of losing the context of their contribution to our subject. We learned much of an incidental
character in our reading and have tried to retain that richness. Where substantive
omissions are made we note these for the reader's benefit.
Several topics that fall in the province of demography are not included,
among them treatments of human spatial ecology, urbanization, and migration.
Omission is partly due to space limitations, and partly to lack of confidence in
our ability to decide what is basic in fields whose mathematical explorations
are recent and expanding rapidly.
We expect from the reader at least some background in calculus and matrix
algebra, and several papers will require an understanding of stochastic processes.
The reader lacking a background in elementary mathematics will find the greaterpart of the book difficult.
Secondary accounts of much of what we present can be found in Keyfitz(1968), and stochastic processes are well handled in Feller (1968) and Chiang(1968). Our chief sources for the early histories given here are Hendriks (1852,
1853), Westergaard (1969), and Lorimer (195~).
Cambridge, MA 1977
1977

David Smith
Nathan KeyfitzxiiNathan Keyfitz: A Biographical Sketch

Nathan Keyfitz was the world’s leading figure in mathematical demography in thesecond half of the Twentieth Century. His books Introduction to the Mathematics ofPopulation in 1968 and Applied Mathematical Demography in 1977, with expanded
editions in 1985 and 2005, defined the subject for generations of students andscientists. They were accompanied by other major volumes, the collection World
Population: An Analysis of Vital Data from 1968 with Wilhelm Flieger, with successor volumes in 1972 and 1991, as well as Causes of Death: Life Tables for National
Populations with Samuel Preston and Robert Schoen, and the collection from 1977
with David Smith, Mathematical Demography: Selected Papers, reprinted here.
These works tied together practical demography with mathematical analysis andcontinue in use today.
Nathan Keyfitz was born on 29 June 1913 in Montreal, Canada. His family had
emigrated from Mogilev, now in Belarus, around 1900. After taking a degree inMathematics at McGill University in 1934, he looked for an academic position,
but there were few if any opportunities at that time of great depression. For somemonths he worked in life insurance societies. In 1936 he was hired as a clerk at
the Dominion Bureau of Statistics in Ottawa (later Statistics Canada) where he
had to correct the inconsistencies in the individual census results, a monotonousand boring activity. Taking an interest in the data themselves, comparing resultsfrom the Canadian census with those from the American and English censuses,
led to admonishment by his immediate chief. But it attracted the attention of R.H.
Coats, head of the Bureau, and Coats encouraged Keyfitz’ vocation as a researcher
alongside his mundane duties.
John Robbins, head of education statistics at the Bureau, encouraged Keyfitz
to consider studying for a Ph.D. and directed him toward an opportunity at the
University of Chicago. William Ogburn, head of the Sociology Department at
Chicago offered him a Carnegie Foundation Fellowship, and in 1942 Keyfitz began
graduate study there. After a year, Keyfitz returned to the Dominion Bureau of
Statistics but continued work on a dissertation and received a Ph.D. in Sociology
from the University of Chicago in 1952. In Chicago, he attended lectures by LouisexiiiNathan Keyfitz: A Biographical Sketch

Wirth and Ernest Burgess, leaders of what became known as the Chicago School ofurban sociology.
In 1946, Keyfitz was sent to Raleigh, North Carolina, to a six week session ofcourses in statistics taught by Ronald Fischer, Abraham Wald, William Cochran,and Frank Yates. It was the beginning of what he called “chance meetings that
establish firm friendships” which he enjoyed throughout his long life. A year later
he was sent to Geneva to represent Canada at a United Nations subcommission on
sampling, meeting such great statisticians as Fisher, Prasanta Chandra Mahalanobis
and Georges Darmois.
Having climbed a number of steps in the hierarchy of the Dominion Bureauof Statistics, Keyfitz was taken as an expert to advise South Asian statistical andcensus offices: in 1951 in Burma where he became friends with travel writer
Norman Lewis; in Indonesia in 1952–1953 where he he met his life-long friend,
the French mathematician and academician Marc-Paul Schützenberger. At that timeSchützenberger was serving as a medical expert for the World Health Organization.
All his life Keyfitz remained a lover of Indonesia and he returned in 1964, 1979,
1984, and 1989 for long stays.
Keyfitz served for eighteen months as technical director for the Colombo Plan for
Cooperative Development. Stationed in Ceylon, now Sri Lanka, his duties took him
to Laos, Vietnam, Singapore and India. At the invitation of Mahalanobis he resided
some months at the Statistical Institute in Calcutta together with the geneticist J.B.S.
Haldane. During these years of statistical practice, he did not abandon his desire to
do research.
In 1959 he took up his first academic position at the University of Toronto in the
Department of Economics, Political Science, and Sociology. He described leaving
the Dominion Bureau of Statistics with more hopes than regrets. He was 45 years
old, but on the threshold of what would be a prodigiously long career in academic
demography.
Keyfitz remained on the faculty at Toronto until 1962, teaching the works ofWeber and Durkheim among others. The next decade brought a series of moves
from university to university. In 1962, figuratively speaking, he climbed Montreal’sMont Royal to teach for a year at the French Université de Montréal. It was only
a kilometer from his undergraduate university, McGill, but for a person born andeducated in the English-speaking part of the city, it was a venture into unknown
territory. In 1963 he returned to the University of Chicago as Professor of Sociology.
From that time on, he could concentrate on mathematical demography, a field that hepractically invented as he drew together into a unified whole piecemeal work frommany disciplines. In 1968 he headed west to the University of California, Berkeley.
On the urging of Fred Mosteller, founder of the Statistics Department at Harvard
University, Keyfitz set about writing what would become his influential Introduction
to the Mathematics of Population. The book galvanized the field and established his
reputation. He served as president of the Population Association of America in 1970
and was elected to the American Academy of Arts and Sciences in 1971 and the
National Academy of Sciences, USA, in 1977. It was rapid recognition, although
not quite as swift as his forerunner in demography, John Graunt, who after signingxivNathan Keyfitz: A Biographical Sketch

the Natural and Political Observations ... in 1661 was elected a member of the Royal
Society in England the same year.
The move to U.C. Berkeley came at the invitation of Kingsley Davis and Judith
Blake who were leading a new Department of Demography there. The four years
at Berkeley can be considered the acme of Keyfitz’ scientific production. He wrote
seminal papers on topics ranging from the evaluation of demographic projections
and the structure of causes of mortality around the world to the number of people
who have lived on the earth.
The Berkeley years were also, as he later described them, the most intense periodof his broader academic life. The entire university system was under pressure fromprotests and demonstrations. Keyfitz felt close to students involved in condemnationof the war in Viet Nam. The department became politically polarized, especiallyafter an arson fire struck a building in which Kingsley Davis had an office. There
were tensions between Judith Blake and others, some surrounding a joint courseof lectures which Keyfitz delivered on contraception and its history with John T.
Noonan at the U.C. Berkeley School of Law, later a U.S. Appeals Court judge.
In the wake of turmoil at Berkeley, Keyfitz accepted an invitation by Roger
Revelle to come to Harvard. Other demography faculty also departed, SamuelPreston to the University of Washington and Etienne van de Walle to the Universityof Pennsylvania. The Berkeley Department of Demography was abolished. It was
refounded in 1978 as a Graduate Group and later again as a Department under theleadership of Eugene A. Hammel and has been known for carrying forward the
tradition of mathematical demography which Keyfitz initiated.
At Harvard, Keyfitz was based partly at the Center for Population Studies led
by Roger Revelle in a quaint building on Bow Street in Cambridge, Massachusetts,
partly in the Department of Sociology, and partly across the Charles River at theHarvard School of Public Health. One of the editors of the present volume, Kenneth
Wachter, shared an office with him at Bow Street. These were the years of theconsolidation of Stable Population Theory at Keyfitz’ hands and the years when
his celebrated book Applied Mathematical Demography took shape.
Harvard was not Keyfitz’ final harbor. He taught there from 1972 to 1983. He
had a part-time position at the Ohio State University between 1981 and 1983,
and a little later spent three months at Stanford working with Marc Feldman,
Brian Arthur, and Paul Ehrlich. In 1984 he took on leadership of the population
program at the International Institute for Applied Systems Analysis (IIASA) in
Laxenburg outside Vienna, Austria. IIASA had been founded to promote East-Westscientific cooperation under the shadow of the Cold War. Keyfitz made it a centerfor creative mathematical demography, fostering the work of a new generation ofleaders including James W. Vaupel. While based at IIASA up to 1993, Keyfitz
travelled throughout eastern Europe, although he never obtained the opportunityof visiting Mogilev, the cradle of his ancestors.
After 1995, Keyfitz returned to live by the Charles River in Cambridge, Massachusetts. He remained intellectually active. There was hardly a week without avisit to his former department at Harvard and phone-calls and exchanges with axvNathan Keyfitz: A Biographical Sketchwide scientific circle. Each year a reflective and thought-provoking Christmas letterfrom the Keyfitz family was eagerly awaited by their many friends.
Keyfitz married Beatrice Orkin in 1939. They were a devoted couple throughout
their life, until her death in 2009. His achievements would have been unthinkable
without her love, support, and intellectual stimulation, and the welcome she gave tothe many people whose lives they both touched. Their daughter Barbara was born in1944 and their son Robert in 1947. Barbara went on to be Professor of Mathematics
at Ohio State University and Robert to be Senior Economist at the World Bank andat the International Monetary Fund.
By grasping opportunities offered partly by chance, Keyfitz found himself able
to combine the very different aspects of being a demographer. As a sociologist,
he wrote on the role of the middle class in the developed as well as in thedeveloping world. As a statistician, he had 23 years of practice in a national
statistical office. As a mathematician, he gave a new unity to the mathematical studyof populations. Interested in biology, he was one of the founders of the journal
Theoretical Population Biology. He was engaged with economics and wrote on
family allowances and on retirement schemes.
In this variety, Keyfitz was helped by the diversity of his friends around the world,by his ability to collaborate, and by his sense of balance. He could converse on the
one hand with Paul Ehrlich, who at one time advocated a maximum populationof a billion humans, and on the other with Roger Revelle who did not exclude
the possibility of feeding forty billion humans. He was on friendly terms withspokespersons for the theory of evolution and with critics of it, with Stephen Jay
Gould and with Marc-Paul Schützenberger. It was too much, perhaps, to hope at
Berkeley to reconcile Judith Blake with John T. Noonan. But for him what was
important was not to reach the same opinion but to engage in free discussion.
Keyfitz had a strong sense of history. In collaboration with David Smith, hehearkened back to founding works of demographic theory and method, gave themcareful scholarly readings, and thought hard about their relationship to the progressof his own time. The collection which is reprinted here represents the fruits of thiscollaboration and this endeavor. It stands as a tribute to Nathan Keyfitz’ legacy.

(Principal sources for Nathan Keyfitz’ life are the Memoir that he wrote, available
on the Keyfitz family website at www.keyfitz.org, and the bibliography posted by
Harvard University at oasis.lib.harvard.edu. The editors have also drawn on theirown reminiscences for this brief sketch.)xvi

David P. Smith: A Biographical Sketch

David P. Smith became well-known in the demographic world already asa Harvard graduate student in 1977 when he edited, together with Nathan
Keyfitz, the collection reprinted here. He went on to a notable career inapplied demography at the World Fertility Survey and the University of Texas
at Houston. Retiring in 2005, he remains active in the field.
David Paul Smith was born on 26 September 1944 in Detroit, Michigan.
He matriculated at Albion College in 1962, where he took classes and came toknow John A. Ross, who was a professor at Albion at the time. In 1964, Smithtransferred to the University of California, Berkeley and completed a B.A. inthe study of developing nations in 1966. He joined the Peace Corps for two
years, serving in Turkey and renewing his friendship with the Ross family,
who were stationed in Ankara with the Population Council. After his Peace
Corps service, Smith spent a semester at the School of Public Health at the
University of Michigan. He then arranged to work for the Population Council
in Seoul, working on family planning surveys and data analysis with John
Ross in the Korean Family Planning Program. Mathematical Demography:
Selected papers would later be dedicated to John Ross.
In 1971, after two years in Korea, Smith applied for graduate study at
Berkeley, but the Demography Department was being closed down. Nathan
Keyfitz had moved to Harvard. John Ross recommended Smith to Keyfitz,
who invited Smith to visit. Smith immediately drove from Michigan to Massachusetts, and after an hour’s conversation with Keyfitz was admitted to
Harvard’s graduate program in Sociology. His classes included Keyfitz’ memorable introduction to mathematical demography. As Smith recently recalled,
The book got written because Nathan mentioned in class one day that he had acollection of the pioneering papers in mathematical demography he was hoping to
put into a book. I went to his office to find out who the pioneers were, thinking it was
one of the things a graduate student in demography ought to know. After a shortxvii

David P. Smith: A Biographical Sketch
conversation l left with a stack of xeroxes and a vague commitment to help him on
the book. By the next summer I was starting to edit papers to get them down toessential content, filling in some holes in the collection, tracking down the originals
in Widener Library to be photographed, and starting on the chapter introductions.

Smith’s dissertation, completed in 1978 under Keyfitz’ supervision, tested
empirical evidence for the theory put forward by Richard Easterlin to explain
cycles in U.S. fertility. Papers based on the dissertation in Population Studies
and Theoretical Population Biology played an important role in empiricalcritiques of Easterlin’s theory. At the time, mathematical understanding of
the population dynamics implied by Easterlin-style systems of demographic
feedback was limited, but would flourish in the following decade.
While at Harvard, Smith married Joan M. Smith and moved to Hartford
a couple of years before his Ph.D. was awarded in 1979. Stepsons Kenneth
Clair Smith and Russell L. Smith had been born in 1965 and 1968. From
Hartford, David Smith commuted periodically to Cambridge, Massachusetts
to assist in classes Keyfitz was teaching. Then Smith was hired by the World
Fertility Survey and moved to London, where he implemented methods forcomputing fertility measures, family planning indices, and life table estimatesfrom computerized files of survey responses.
In 1981 Smith returned to the United States and joined the faculty of
the University of Texas at the Health Science Center in Houston as Associate Professor of Demography. He served at Texas throughout the next 24
years of his career. Early research at Texas included studies of duration ofbreastfeeding in Sri Lanka and in the United States and patterns of prevalence of ischemic heart disease. From 1989 to 1994 he served as Director andPrincipal Investigator of the TexNet Project funded by the Hogg Foundation. The TexNet Project was devoted to improving the quality of evaluation
research in the analysis of teen service programs in Texas relating to pregnancy. Guides for assisting programs in analyzing data and appreciating thelimitations that small non-random samples place on statistical analysis were
published in 1992 and 1994.
Over the next decade Smith continued his contributions to evaluation research, including analysis of the Episcopal Health Charities Healthy Communities Project with funding from the Center for Health Policy Studies at
the University of Texas, Houston, analysis of substance abuse prevention programs in San Antonio directed at pregnant and parenting teens, and a major
assessment of perinatal health needs at the turn of the millenium.
In 1992, Smith returned to his early interests in mathematical demography, publishing the volume Formal Demography with Plenum Press. Toward
the end of the decade, in collaboration with Hélène Rossert, he completed atranslation of Alfred Lotka’s Théorie Analytique des Associations Biologique,a volume in which Lotka summed up his discoveries and a valuable supplement to the papers by Lotka reprinted in the present volume.

xviii

David P. Smith: A Biographical Sketch

In the following years, in collaboration with Benjamin Bradshaw, Smith
undertook an extensive project on the estimation of cause-specific death rates
for 72 causes of death, by Hispanic and non-Hispanic ethnicity, race, and age.
This work, commissioned by the National Center for Health Statistics, led
Smith to focus on the role of inconsistencies in self-identification of ethnicity
between Census returns and death certificates. It culminated in an important
paper by Smith and Bradshaw in 2006, “Rethinking the Hispanic Paradox”,
in the American Journal of Public Health. The extent to which apparently
favorable health and survival outcomes among U.S. Hispanics reflect artifactsof the estimation process, a question much in the public eye, continues to bea focus of Smith’s research.

(The authors are grateful to David Smith for supplying information andbibliographic data on which this biographical sketch is based.)xix

Contents

Part I
1

The Life Table“Tables of Annuity Values Which Were Sanctionedby the Roman Law for the Purposes of the Lex Falcidia”,
C. F. Trenerry (1926) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .

72“Natural and Political Observations Mentioned in a
Following Index, and Made Upon the Bills of Mortality”,
John Graunt (1964 (1662)).. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 11

3

“An Estimate of the Degrees of the Mortality of Mankind”,
Edmund Halley (1693) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .

21

4“A Treatise on the Valuation of Annuities and Assuranceson Lives and Survivors”, Joshua Milne (1815) . . . . . .. . . . . . . . . . . . . . . . . . . . . 27

5

“Statistical Applications of the Mortality Table”, George
King (1902) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 3 5

6“Formal Treatment of Aggregate Mortality Data”,
Wilhelm Lexis (1875) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 39

7

“A Short Method for Constructing an Abridged Life
Table”, Lowell J. Reed and Margaret Merrell (1939) . . . . . . . . . . . . . . . . . . . 43

8“Short methods of Constructing Abridged Life Tables”,
T. N. E. Greville (1943) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 53

9“Life Tables for Natural Populations of Animals”, Edward
S. Deevey, Jr. (1947) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 61

xxi

Contents

Part II

Stable Population Theory

10 “An Illustration of Population Growth”, Johann PeterSüssmilch (1761) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 79
11 “A General Investigation into the Mortality andMultiplication of the Human Species”, Leonard Euler(1970 (1760)).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 83
12 “Relation Between Birth Rates and Death Rates”, Alfred
J. Lotka (1907) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 93
13 “A Problem in Age-Distribution”, F. R. Sharpe and Alfred
J. Lotka (1911) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 97
14 “The Stability of the Normal Age Distribution”, Alfred J.
Lotka (1922) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 101
15 “Resolving a Historical Confusion in Population
Analysis”, Paul A. Samuelson (1976) . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 109
16 “On the Integral Equation of Renewal Theory”, William
Feller (1941) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 131
17 “A New Method for Calculating Lotka’s r — The Intrinsic
Rate of Growth in a Stable Population”, Ansley J. Coale (1957) .. . . . . . 157
18 “The Fundamental Theorem of Natural Selection”,
R. A. Fisher (1958 (1930)).. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 161
19 “How the Age Distribution of a Human Population is
Determined”, Ansley J. Coale (1957) .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 167
20 “On the Reproduction of Organisms with Overlapping
Generations”, W. R. Thompson (1931) .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 173
21 “The Population Consequences of Life History
Phenomena”, Lamont C. Cole (1954) . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 183
Part III

Attempts at Prediction and the Theory They
Stimulated

22 “The Probability of a Cessation of the Growth ofPopulation in England and Wales during the Next
Century”, Edwin Cannan (1895).. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 197
23 “Population Waves”, Harro Bernardelli (1941) . . . . .. . . . . . . . . . . . . . . . . . . . . 207
24 “On the Use of Matrices in Certain Population
Mathematics”, P. H. Leslie (1945).. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 213

xxii

Contents

Part IV

Parameterization and Curve Fitting

25 “On the Nature of the Function Expressive of the Law of
Human Mortality”, Benjamin Gompertz (1825) .. . .. . . . . . . . . . . . . . . . . . . . . 231
26 “On the Law of Mortality”, William M. Makeham (1867).. . . . . . . . . . . . . 235
27 “Calculation of Model Tables”, Ansley J. Coale and Paul
Demeny (1966) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 241
28 “Methods of Analysis and Estimation”, William Brassand Ansley J. Coale (1968).. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 253
29 “Nuptiality, Fertility, and Reproductivity”, S. D. Wicksell (1931) . . . . . 261
30 “Model Fertility Tables: Variations in the Age Structureof Childbearing in Human Populations”, Ansley J. Coaleand T. James Trussell (1974) . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 269
31 “A Note on the Law of Population Growth”,
Pierre-François Verhulst (1838).. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 279
32 “On the Rate of Growth of the Population of the UnitedStates since 1790 and its Mathematical Representation,
Raymond Pearl and Lowell J. Reed (1920) . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 287
Part V

Probability Models of Conception and Birth

33 “First Investigations on the Fecundability of a Woman”,
Corrado Gini (1924) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 299
34 “Theoretical Basis of Measures of Natural Fertility”,
Louis Henry (1972 (1953)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 305
35 “On the Time Required for Conception”, Mindel C. Sheps (1964) . . . . 315
36 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 325
Additional Bibliographic References . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 331
Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 333

xxiii

Part IThe Life Table

Mathematical demography has its modern beginnings in the gradual developmentof correct procedures for forming life tables, and in a single remarkable paper by
Leonard Euler (1760), paper 11 in Part II, that introduced stable age distributions.
One far older work of at least some quality survives: a third century A.D. tableof life expectancies attributed to Ulpian (paper 1) that remained in use in northern
Italy through the 18th century. The table and an accompanying discussion have beentaken from C. F. Trenerry (1926).
The mathematician Girolamo Cardano took up the problem briefly in 1570, but
without substantive results. Cardano made the assumption that a man who took greatıcare in all things would have a certain life expectancy ˛, so that e x D ˛  x for
all ages x, and then asked what part of this might be forfeited by a relaxation of1
prudence. He proposed letting life expectancy fall by 40
of its value during each year
in which a man was reasonably careful but not fastidious: by the nature of the life
expectancies, a man might be born with the prospect of living say 260 years and yet
die at age 80, having every year thrown away by inattention a part of what remained
to him (Cardano 1570, pp. 204—211). A modern interpretation would be that acohort carries its life table with it. The result was not generalized to populations.
John Graunt’s Natural and Political Observations Upon the Bills Mortality(1662) is the first substantive demographic work to have been written. The bookis occasionally curious but most often impressive, even from a perspective of
three hundred years: Graunt culled a remarkable amount of information from thechristening and death lists begun in the later plague period and usually understoodits implications. Parts of the treatise are included here as paper 2. f Ed. Note: This
founding work of demography is surrounded by some mystery. A reconstruction of
the algorithm for the 1662 lifetable and an investigation into questions of authorshipmay be found in [22] g
A second work of great importance followed upon Graunt’s, Edmund Halley’s(1693) presentation of the Breslau (Wroclaw Poland) life table (paper 3). Halley
had made an effort to obtain the Breslau lists in order to see what might be donewith them, after learning of their apparent quality.

I The Life Table

The methods of calculation Halley used in his life table were partly informal, as
in his remarks on stationarity and in his unorthodox subtraction (where `x is thenumber of survivors to exact age x in the life table from among `0 births and Lx
represents the number between ages x and x C 1)
890ŒD `1   198 D 692ŒD L6 ;
explained by the oblique statement: “198 do die in the Five Years between 1 and 6
compleat, taken at a Medium.”
The terminology has created confusion down to the present century. Raymond
Pearl (1922, p. 83), apparently reading the Lx terms that make up Halley’s table as`x , calculated life expectancy at birth as 33:5 years by the table instead of the correct
27:5. The mistake is carried over in Dublin, Lotka and Spiegelman (1949, p. 34).
Johan DeWit (1671) preceded Halley in the correct calculation of annuities,
using exact (`x ) as against Halley’s approximate (Lx ) denominators, and most of
Halley’s other Uses can be answered differently, but the quality of Halley’s table anddiscussion much surpasses the few earlier works and several of the subsequent ones.
His table is graphed below alongside Ulpian’s, Graunt’s, DeWit’s, and as references
the middle level table for Crulai c. 1700 (Gautier and Henry 1958, pp. 163, 190) and
one of the Coale and Demeny (1966) model life tables.
After Halley, the next impressive contribution was the series of life tables for annuitants and monastic orders by Antoine Deparcieux, printed in 1746. The accuracyof Deparcieux’s data was sufficient for him to show that adult life expectancies hadıbeen increasing over the previous half century. Deparcieux calculated his e x values
by the simple but adequate formula
!
P
ı

ex D

i Dx

!
P

.`i  `i C1 /.i C 0:5  x/
D

`x

i Dx

`x

`i
 0:5 :Of their utility he writes (1760, pp. 58—59):
ı“Les vies moyennes [i.e., e x ] sont ce qui m’a paru de plus commode pour faire
promptement & sans aucun calcul, la comparaison des different ordres de mortalité
qu’on a établis ... [Life expectancies are what have appeared to me most convenient
for making promptly and without any calculation a comparison of different ordersof mortality that one has established... ].”Two later efforts merit attention here: Daniel Bernoulli (1766) introduced
continuous analysis and suggested the force of mortality [ .x/ ] in an applicationof differential calculus to the analysis of smallpox rates. Later Emmanuel Etienne
Duvillard (1806), in an article that also introduced the Tx column (defined as!
P
Tx DLi ), applied Bernoulli’s method to estimate the increase in life expectancyxthat would follow if smallpox were eliminated by Edward Jenner’s vaccine. The
calculus, which these and most modern work employ, dates to a seventy year period2

I The Life Table(1665—1736) between Isaac Newton’s first investigations and the publication ofhis principal works. Westergaard (1969, pp. 92—93) comments however that it was
not until the late nineteenth century that continuous analysis was widely enoughunderstood for Bernoulli’s work to be appreciated.
Joshua Milne in his excellent Treatise on the Valuation of Annuities (1815), which
includes a careful analysis of life tables made prior to his, was first to suggest aformula by which `x values could be calculated for real populations. His is the wellknown expression
d x D `xDx
Px C 12 Dx!

where dx is the number dying between ages x and x C 1 in the life table population,
Dx represents calendar year deaths between these ages in an observed population,
and Px C 12 Dx constructs an initial population analogous to `x except for scale byadding half of the yearly deaths to the observed midyear population Px .
Excerpts from Milne’s discussions of the life table and of age-specific fertility
rates (due to Henrie Nicander (1800, pp. 323—324)) are given in paper 4. Milne’s
method for graduating data from grouped to single ages, not the best, has been
omitted. His footnoted criticism of Thomas Simpson’s (1742) work opens an areaof discussion that is easily missed: Milne’s life table was misread by William Sutton(1884) – whose clarification in 1874 of the construction of Richard Price’s 1771
Northampton Table is a more competent work – but was immediately reestablishedby George King (1884). Like other fields, demography does not only move forward.
The fifth article in this section excerpts from George King (1902), whose notationis contemporary, his explanations of terms of the life table. From the middle ofthe 19th century William Farr standardized much of the life table, but he did not
put his work in a form at all comparable to King’s excellent textbook. A recent
addition to the life table, from C. L. Chiang (1960a), is the term n ax . This is King’s
unremembered “average amount of existence between ages x and x C n, belonging
L `
to those who die between these ages,” i.e.: n ax D n x n dx xCn .
Out of sequence, the Lexis (1875) diagram is introduced in paper 6. For mostof a century it has been a standby of all analysis attempting to relate age and time.
Among contemporary works, those of Roland Pressat (1969, 1972) exploit it mostfully. f Ed. Note: Wilhelm Lexis’ own version of the diagram is not the one in use
today. Several of Lexis’ contemporaries played a role in the invention of diagrams ofthis kind and controversies around priority occurred in the later 1800s, as discussed
in [37]. g
The important contributions to the life table in this [20-th] century have been
competent abridgement techniques for generating tables by five or ten year age
groupings in place of single years of age. The Lowell Reed and Margaret Merrell(1939) article, paper 7, did much to establish the validity of abridgement techniquesby its introduction of an attractive expression:

3

I The Life Table
32n qx D 1  expŒn n mx  an n mx for estimating n qx from n mx values where wide age groupings are used. In theexpression, n mx is the age-specific death rate in the life table population ages x
to x C n (that is, n mx D n dx =n Lx and n qx the probability of dying within theinterval for a person of exact age x, ( n qx D n dx =`x ). By empirical examination
the authors found that the constant a required by the expression could be the samefor all ages above infancy. Reed and Merrell examine two other approximations to
n qx , the first of which:n qx Dn n mx
1 C n2 n mx

can be derived from Milne’s formula for dx , the other due apparently to Farr (1864,
pp. xxiii—xxiv), and evident earlier to Gompertz (1825, paper 25 in Part III.
n qx D 1  expŒn n mx :Following their article we include derivations for both expressions.
T. N. E. Greville (1943) was able through ingenious expansions to derive each ofthese equations by working with the definitions of n mx and n qx , and to show that
the Reed-Merrell formula incorporates Gompertz’ Law that the force of mortality is
an exponential function of age. The assumption is appropriate at older ages andinappropriate for infancy, and thus defines the age range over which the ReedMerrell formula is applicable. In the same article Greville discusses approximations
to n Lx values where, as before, age groupings are wide (paper 8).
The methods used by Greville can be generalized to take advantage of theobserved age structure of a population as well as its mortality schedule. In thesimplest case this gives rise to the formula, due to Nathan Keyfitz and James
Frauenthal (1975),n qx D 1  exp n n Mx Cn.n PxCn  n Pxn /.n MxCn  n Mxn /48 nPxwith as before the caveat that infancy requires separate consideration.
The chapter concludes with excerpts from the well known article by Edward
Deevey (1947) in which he evaluates efforts that had been made up to that time todevelop life tables for animal populations.
f Ed. Note: The methods for abridged lifetables featured in papers 7 and 8
are now of mainly historical interest. It has become standard practice to calculate
abridged lifetables directly from tables with single-year intervals, selecting therequired `x values and cumulating the 1 Lx values for the table. Exposition is foundin the documentation for the Human Mortality Database developed by John Wilmothand Vladimir Shkolnikov on the World Wide Web, the leading resource for highquality estimates of mortality rates and lifetables.

4

I The Life TableThe subject of lifetables is taken up again in Part IV of this volume, which includes parametric mortality models and model lifetables. By far the most important
has come to be the 1825 model of Benjamin Gompertz [16] discussed there.
In the first decade of the Twentyfirst Century, the subject of lifetables has
been dominated by applications of proportional hazard models capitalizing on an
extension of statistical regression techniques to survival analysis from 1972 by
Sir David Cox [10]. A proportional hazards approach to unobserved heterogeneity,
the frailty modeling introduced by James Vaupel, Kenneth Manton, and Eric Stallard[38] has come into widespread use.
Estimation of non-human lifetables has gone far beyond the works discussed by
Deevey. Progress is described, for instance, in [29] for animals and [35] for plants.
Our knowledge of life-history parameters for populations of individuals from otherspecies continues to grow rapidly, stimulated by progress in biodemography. An
introduction to the field of biodemography is found in the collection by Wachterand Finch [42]. g

5

6

1. Tables of Annuity Values Which Were Sanctioned by the Roman Law for the Purposes of the
Lex Falcidia
C. F. TRENERRY (1926)
From The Origin and Early Historv of Insurance, pp. 151--152. London: P.S. King & Son.

On the question of whether Ulpian's table contains an interest factor Hendriks(1852, p. 224) writes: " ... as it was the special duty of the jurist to look to a sufficient fund being created for the discharge of the obligations under such annuities, he is not likely in those times to have taken much account of interest,
which at best could only have been continuously obtained with great difficulty."
The table values should be recognized as excessive if an interest rate much above
1 % is incorporated, and we believe with Hendriks that none is. The table isreproduced below:
Age

Provision0-1920·······24
25-29
30-34
35-39

30 years
25
22
20

40
41
42
43
44
45
46
47
48
49

19
18
17
16
15
14
13
12
11
10

50-54
55-59
60+

9
7
5

28

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_1, © Springer-Verlag Berlin Heidelberg 2013

7

ALTHOUGH there is no trace of any attempt on:ot:-:
the part of the Romans to construct a tableof premiums, of mortality or rates of subscriptions payable byb~.:R~:!ts. members of burial clubs, yet there are records of
two tables which were sanctioned by the civil
law for the capitalization of annuities. The necessity
for such tables arose from the provisions of the~1cidia. Lex Falcidia, 1 by which it was ruled that theheir or heirs to an estate should receive not lessthan one~quarter of the total property left by the testator.
As it frequently happened that the will was made when thetestator was in better circumstances than at the time of hisdeath, it became necessary in such cases for the values ofthe various legacies to be proportionately reduced in orderthat the total value should not exceed three-quarters of theestate. In the case of the ordinary legacy, this reduction
presented no difficulty, but when the testator had left a life
annuity to one or more legatees, the question arose as to thebasis of reduction of the values of such annuities.
Two tablesof annuity In order to meet this requirement, it was necesvalues
sary to capitalize the annuities, and, in order toauthorized.
permit of this being done, a table of annuity
values, known subsequently as "Macer's Table," wasused. This table was1 as will be seen from the quotation below, authorized in a responsum of the jurisconsult
Macer, who, at the same time, sanctioned the use of anotherand more correct table, the authorship of which heattributed to Ulpian. 1 ' In the case of the first table,

1

Passed 40 B.C. See Digest, XXXV, ii, 1.

1 ' Digest, XXXV, ii, 68.
lEmilius Macer, Lib. 2, ad legem vicesimam
hereditatium. " Computationi in alimentis faciendre hanc formam esse
Ulpianus scribit, ut a prima retate usque ad annum vicesimum quantitas
alimentorum triginta annorum computetur, ejusque quantitatis Falcidia
prrestetur : ab annis vero viginti usque ad annum vicesimum-quintum
annorum viginti octo : ab annis vigintiquinque usque ad annos triginta,
annorum vigintiquinque : ab annis triginta usque ad annos trigintaquinque,annorum vigintiduo : ab annis trigintaquinque usque ad annos quadraginta, annorum viginti : ab annis quadraginta usque ad annos quinquaginta, (tot) annorum computatiofit, quotretatiejus ad annum sexagesimum
deerit, remisso uno anno : ab anno vero quinquagesimo usque ad annum
quinquagesimum quintum, annorum novem : ab annis quinquagentaquinque usque ad annum sexagesimum annorum septem : ab annis
sexaginta cujuscunque retatis sit, annorum quinque. . . . Solitum est
tamen a prima retate usque ad annum trigesimum computationem
annorum triginta fieri, ab annis vero triginta, tot annorum computationem
inire, quot ad annum sexagesimum deesse videntur, nunquam ergo ampliusquam triginta annorum computatio initur."8

Constructionit is clear on inspection that the values are
lf~acer'.s arbitrary, and in no way based on a mortality
ar~i~~te experience. Ulpian's Table, however, shows~b~an~s ed distinct signs of an attempt to represent more
o! U:e as nearly the actual values of annuities.
experience.
It has been held by various writers 8 that thevalues given by Ulpian are those of the expectation of life
Table ofrather than those of annuities, and the reason put
Ulpianforward for this assumption is that the Romans
f!~~a~}Y were unlikely to have used the factor of interest
values ofwhen compiling the table. There seems, however,a,., not of e., to be no particular justification for this assumption.
It is generally admitted that the table of Ulpian shows signsof having been calculated with direct reference to either amortality or a subscription experience, but, so far, no satisfactory suggestion has been made of the way in which thetable was constructed.

a Cf. Hendriks, J.I.A., VI, p. 313.

9

2. Natural and Political Observations Mentioned
in a Fallowing Index, and Made Upon the Bills of
Mortality
JOHN GRAUNT (1964 (1662))

From Journal of the Institute of Actuaries 90. Excerpts are from pages 15, 19-23, 35-38. 44-47.

In extracting from Graunt's observations we have tried to give the reader a feel
both for Graunt's imaginativeness and tenacity and for the materials he had
to work with. We include both of his estimates for London's population: thefirst (460,000) he found by attributing to the city one-fourteenth of a national
population estimate of 6,440,000 not well derived. The second (384,000), fromhis more careful knowledge about London, is probably of the correct order ofmagnitude, and suggests a reasonable total population figure (5 to 5.5 million)(cf. Wrigley 1967, pp. 44-45). Graunt's life table entries and his comments on
London's doubling time are not of this quality.
We have mostly avoided Graunt's discussion of specific diseases, and haveomitted his remarks on the expansion of London beyond the old city walls,
on the healthfulness of the city and countryside, and on the country bills.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_2, © Springer-Verlag Berlin Heidelberg 201311

CHAPTER I
OF THE BILLS OF MORTALITY, THEIR BEGINNING, AND PROGRESS

THE first of the continued weekly Bills of Mortality extant at the Parishclerks Hall, begins the 29th of December, 1603, being the first year of KingJames his reign; since when, a weekly account hath been kept there ofburials and christenings. It is true, there were Bills before, viz. for theyears 1592,-93,-94, but so interrupted since, that I could not depend upon
the sufficiency of them, rather relying upon those accounts which havebeen kept since, in order, as to all the uses I shall make of them.
2. I believe that the rise of keeping these accounts, was taken from the
Plague: for the said Bills (for aught appears) first began in the said year
1592, being a time of great mortality; and after some disuse, were resumed
again in the year 1603, after the great Plague then happening likewise ....
10. We have hitherto described the several steps, whereby the Bills of
Mortality are come up to· their present state; we come next to shew how
they are made and composed, which is in this manner, viz. when anyone
dies, then, either by tolling or ringing of a bell, or by bespeaking of a graveof the Sexton, the same is known to the searchers, corresponding with thesaid Sexton.
11. The Searchers hereupon (who are ancient matrons, sworn to their
Office) repair to the place where the dead corpse lies, and by view of thesame, and by other enquiries, they examine by what disease or casualty thecorpse died. Hereupon they make their Report to the Parish-clerk and he,every Tuesday night, carries in an account of all the burials and christinings, happening that week, to the clerk of the Hall. On Wednesday thegeneral account is made up and printed, and on Thursdays published, anddispersed to the several families, who will pay four shillings per annum forthem.
12. Memorandum. That although the general yearly Bills have been set
out in the several varieties afore-mentioned, yet the original entries in theHall-books were as exact in the very first year as to all particulars, as now;and the specifying of casualties and diseases, was probably more.

CHAPTER If
GENERAL OBSERVATIONS UPON THE CASUALTIESIN my discourses upon these Bills I shall first speak of the casualties, thengive my observations with reference to the places and parishes comprehended in the Bills; and next of the years, and seasons.

121. There seems to be good reason why the Magistrate should himself
take notice of the numbers of burials and christenings, viz. to see whether
the City increase or decrease in people; whether it increase proportionately
with the rest of the Nation; whether it be grown big enough, or too big, etc.
But why the same should be made known to the People, otherwise than toplease them as with a curiosity, I see not.
2. Nor could I ever yet learn (from the many I have asked, and those notof the least sagacity) to what purpose the distinction between males andfemales is inserted, or at all taken notice of; or why that of marriages was
not equally given in? Nor is it obvious to everybody, why the account ofcasualties (whereof we are now speaking) is made? The reason, which seems
most obvious for this latter, is, that the state of health in the City may at all
times appear.
3. Now it may be objected that the same depends most upon the accounts
of epidemic diseases, and upon the chief of them all, the Plague; wherefore
the mention of the rest seems only a matter of curiosity.
4. But to this we answer, that the knowledge even of the numbers which
die of the Plague, is not sufficiently deduced from the mere report of the
Searchers, which only the Bills afford; but from other ratiocinations, andcomparings of the Plague with some other casualties.
5. For we shall make it probable that in years of Plague a quarter partmore dies of that disease than are set down; the same we shall also proveby the other casualties. Wherefore, if it be necessary to impart to theworld a good account of some few casualties, which since it cannot well
be done without giving an account of them all, then is our common
practice of so doing very apt, and rational.
6. Now, to make these corrections upon the perhaps, ignorant, and careless Searchers' Reports, I considered first of what authority they were inthemselves, that is, whether any credit at all were to be given to their distinguishments: and finding that many of the casualties were but matter ofsense, as whether a child were Abortive, or Stillborn; whether men wereAged, that is to say, above sixty years old, or thereabouts, when they died,without any curious determination whether such aged persons died purelyof Age, as for that the innate heat was quite extinct, or the radical moisturequite dried up (for I have heard some candid physicians complain of thedarkness which themselves were in hereupon) I say, that these distinguishments being but matter of sense, I concluded the Searchers' Report might
be sufficient in the case.
7. As for Consumptions, if the Searchers do but truly report (as they may)whether the dead corpse were very lean and worn away, it matters not tomany of our purposes whether the disease were exactly the sam~ . asphysicians define it in their books. Moreover, in case a man of75 years olddied of a cough (of which had he been free, he might have possibly lived toninety) I esteem it little error (as to many of our purposes) if this person bein the table of casualties, reckoned among the Aged and not placed under
the title of Coughs.

13

8. In the matter of Infants I would desire but to know clearly, what the
Searchers mean by Infants, as whether children that cannot speak, as theword Infant seems to signify, or children under two or three years old,
although I should not be satisfied, whether the Infant died of Wind, or of
Teeth, or of the Convulsion, etc. or were choked with Phelgm, or else of
Teeth, Convulsion, and Scowring, apart, or together, which, they say, dooften cause one another: for, I say, it is somewhat, to know how many die
usually before they can speak, or how many live past any assigned numberof years.
9. I say it is enough if we know from the Searchers but the most predominant symptoms; as that one died of the Head-Ache, who was sorely
tormented with it, though the physicians were of opinion that the disease
was in the stomach. Again, if one died suddenly, the matter is not great,
whether it be reported in the Bills, Suddenly, Apoplexy, or Planet-strucken,
etc.
10. To conclude, in many of these cases the Searchers are able to report
the opinion of the physician who was with the patient, as they receive thesame from the friends of the defunct, and in very many cases, such as
Drowning, Scalding, Bleeding, Vomiting, Making-away themselves, Lunaticks, Sores, Small-Pox, etc. their own senses are sufficient, and the generality of the world, are able pretty well to distinguish the Gowt, Stone,
Dropsy, Falling-sickness, Palsy, Agues, Pleurisy, Rickets, etc. one fromanother.
11. But now as for those casualties which are aptest to be confounded,
and mistaken, I shall in the ensuing discourse presume to touch upon themso far as the learning of these Bills hath enabled me.
12. Having premised these general advertisements, our first observation
upon the casualties shall be, that in twenty years there dying of all diseases
and casualties, 229,250, that 71,124 died of the Thrush, Convulsion, Rickets,
Teeth, and Worms; and as Abortives, Chrysoms, Infants, Liver-grown, andOverlaid; that is to say, that about one-third of the whole died of thosediseases, which we guess did all light upon children under four or five years
old.
13. There died also of the Small-Pox, Swine-Pox, and Measles, and ofWorms without Convulsions, 12,210, of which number we suppose likewise,that about one-half might be children under six years old. Now, if we
consider that 16 of the said 229 thousand died of that extraordinary andgrand casualty the Plague, we shall find that about 36 per centum of all
quick conceptions died before six years old.
14. The second observation is, that of the said 229,250 dying of all
diseases, there died of acute diseases (the Plague excepted) but about
50,000 or 2/9 parts. The which proportion doth give a measure of the stateand disposition of this climate and air, as to health, these acute andepidemic diseases happening suddenly and vehemently, upon the like corruptions and alterations in the air.
14

15. The third observation is, that of the said 229 thousand about 70 diedof chronic diseases, which shews (as I conceive) the state and disposition of
the country (including as well its food, as air) in reference to health, orrather to longevity: for as the proportion of acute and epidemic diseases
shews the aptness of the air to sudden and vehement impressions, so thechronic diseases shew the ordinary temper of the place, so that upon the
proportion of chronic diseases seems to hang the judgment of the fitness ofthe country for long life. For, I conceive, that in countries subject to great
epidemic sweeps men may live very long, but where the proportion of thechronic distempers is great, it is not likely to be so; because men being longsick, and always sickly, cannot live to any great age, as we see in several
sorts of Metal-men, who although they are less subject to acute diseases
than others, yet seldom live to be old, that is, not to reach unto those years
which David says is the age of man ....
17. In the next place, whereas many persons live in great fear and apprehension of some of the more formidable and notorious diseases following;
I shall only set down how many died of each: that the respective numbers,
being compared with the total 229,250, those persons may the better
understand the hazard they are in.
Table of notorious diseases
Apoplexy
1,306
Cut of the Stone
38
Falling Sickness
14
243
Dead in the streets
Gowt
134
Head-Ache
51
Jaundice
998
Lethargy
61
Leprosy
6
Lunatick
158
Overlaid, and Starved 529
Palsy
423
Rupture201
Stone and Strangury, 863
Sciatica
5
Sodainly
454

Table of casualties
Bleeding
69
Burnt, and Scalded
125
Drowned829
Excessive drinking
2
Frighted
22
Grief
279
Hanged themselves
222
Killed by several
1,021
accidents
Murdered86
14
Poisoned
Smothered
26
Shot
7
Starved
51
Vomiting
136

18. In the foregoing observations we ventured to make a standard of thehealthfulness of the air from the proportion of acute and epidemic diseases,
and of the wholesomeness of the food from that of the chronic. Yet forasmuch as neither of them alone do shew the longevity of the inhabitants, weshall in the next place come to the more absolute standard, and correctionof both, which is the proportion of the aged, viz. 15,757 to the total
229,250. That is of about 1 to 15 or 7 per cent. Only the question is, what

15

number of years the Searchers call Aged, which I conceive must be thesame, that David calls so, viz. 70. For no man can be said to die properly ofAge, who is much less: it follows from hence, that if in any other country
more than seven of the 100 live beyond 70 such country is to be esteemed
more healthful than this of our City....

CHAPTER VII
OF THE DIFFERENCE BETWEEN BURIALS AND CHRISTENINGS

THE next observation is, that in the sai4 Bills there are far more burials
than christenings. This is plain, depending only upon arithmetical computation; for, in 40 years, from the year 1603 to the year 1644, exclusiveof both years, there have been set down (as happening within the sameground, space, or parishes) although differently numbered, and divided,
363,935 burials, and but 330,747 christenings within the 97, 16, and 10 outparishes, those of Westminster, Lambeth, Newington, Redriff, Stepney,
Hackney, and Islington, not being included.
2. From this single observation it will follow that London should havedecreased in its people, the contrary whereof we see by its daily increaseof buildings upon new foundations, and by the turning of great palacious
houses into small tenements. It is therefore certain that London is suppliedwith people from out of the country, whereby not only to repair the overplus difference of burials above-mentioned, but likewise to increase its
inhabitants, according to the said increase of housing. . ..
4. But, if we consider what I have upon exact enquiry found true, viz.
that in the country, within ninety years, there have been 6,339 christenings
and but 5,280 burials, the increase of London will be salved without
inferring the decrease of the people in the country; and withal, in case all
England have but fourteen times more people than London, it will appear,
how the said increase of the country may increase the people, both of
London and itself; for if there be in the 97, 16, 10, and 7 parishes, usuallycomprehended within our Bills, but 460,000 souls as hereafter we shallshew, then there are in all England and Wales, 6,440,000 persons, out of
which subtract 460,000, for those in and about London, there remains
5,980,000 in the country, the which increasing about 1/7 part in 40 years,
as we shall hereafter prove doth happen in the country, the whole increaseof the country will be about 854,000 in the said time, out of which number,
if but about 250,000 be sent up to London in the said 40 years, viz. about
6,000 per annum, the said missions will make good the alterations which
we find to have been in and about London, between the years 1603 and1644 above-mentioned. But that 250,000 will do the same, I prove thus,
viz. in the 8 years, from 1603 to 1612, the burials in all the parishes, and of16

all diseases, the Plague included, were at a medium 9,750 per annum. And
between 1635 and 1644 were 18,000, the difference whereof is 8,250, whichis the total of the increase of the burials in 40 years, that is about 206 perannum. Now, to make the burials increase 206 per annum, there must be
added to the City thirty times as many (according to the proportion of 3
dying out of 11 families) viz. 6,180 advenae, the which number multiplied
again by the 40 years, makes the product 247,200, which is less thanthe 250,000 above propounded; so as there remains above 600,000 ofincrease in the country within the said 40 years, either to render it more
populous, or send forth into other colonies, or wars. But that England hathfourteen times more people is not improbable, for the reasons following.
1. London is observed to bear about the fifteenth proportion of thewhole tax.
2. There is in England and Wales about 39,000 square miles of land,
and we have computed that in one of the greatest parishes in Hampshire, being also a market town and containing twelve square miles,
there are 220 souls in every square mile, out of which I abate 1/4 forthe overplus of people more in that parish, than in other wild
counties. So as the 3/4 parts of the said 220, multiplied by the totalof square miles, produces 6,400,000 souls in all, London included.
3. There are about 10,000 parishes in England and Wales, the which,although they should not contain the 1/3 part of the land nor the
1/4 of the people of that country parish which we have examined, yet
may be supposed to contain about 600 people, one with another,according to which account there will be six millions of people in
the Nation. I might add, that there are in England and Wales about
five and twenty millions of acres at 16t foot to the perch; and if
there be six millions of people, then there is about four acres forevery head, which how well it agrees to the rules of plantation, I
leave unto others, not only as a means to examine my assertion,
but as an hint to their enquiry concerning the fundamental trade,which is husbandry and plantation. . ..

CHAPTER VIII
OF THE DIFFERENCE BETWEEN THE NUMBERS OF MALES AND FEMALES

THE next observation is, that there be more males than females.
There have been buried from the year 1628 to the year 1662, exclusive,
209,436 males and but 190,474 females: but it will be objected, that in
London it may indeed be so, though otherwise elsewhere; because London
is the great stage and shop of business, wherein the masculine sex bears
17the greatest part. But we answer, that there have been also christened
within the same time, 139,782 males and but 130,866 females, and that thecountry accounts are consonant enough to those of London upon thismatter.
2. What the causes hereof are, we shall not trouble ourselves to conjecture, as in other cases, only we shall desire that travellers would enquire
whether it be the -same in other countries. . ..

CHAPTER XIOF THE NUMBER OF INHABITANTSI HAVE been several times in company wiJh men of great experience inthis City, and have heard them talk seldom under millions of people to be
in London, all which I was apt enough to believe until, on a certain day,one of eminent reputation was upon occasion asserting that there was in
the year 1661 two millions of people more than in the year 1625, before thegreat Plague; I must confess that, until this provocation, I had beenfrighted with that misunderstood example of David, from attempting any
computation of the people of this populous place; but hereupon I both
examined the lawfulness of making such enquiries and, being satisfied
thereof, went about the work itself in this manner: V'iz.
2. First, I imagined that, if the conjecture of the worthy person aforementioned had any truth in it, there must needs be about six or sevenmillions of people in London now; but repairing to my Bills I found that
not above 15,000 per annum were buried, and consequently, that notabove one in four hundred must die per annum if the total were but six
millions.
3. Next considering, that it is esteemed an even lay whether any man
lives ten years longer, I supposed it was the same, that one of any 10 might
die within one year. But when I considered, that of the 15,000 aforementioned about 5,000 were Abortive, and Stillborn, or died of Teeth,
Convulsion, Rickets, or as Infants, and Chrysoms, and Aged. I concluded
that of men and women, between ten and sixty, there scarce died 10,000
per annum in London, which number being multiplied by 10, there must be
100,000 in all, that is not the one-sixtieth part of what the Alderman
imagined. These were but sudden thoughts on both sides, and both far
from truth, I thereupon endeavoured to get a little nearer, thus: viz.
4. I considered, that the number of child-bearing women might be double
to the births: forasmuch as such women, one with another, have scarcemore than one child in two years. The number of births I found, by those
years wherein the registries were well kept, to have been somewhat lessthan the burials. The burials in these late years at a medium are about18

13,000 and consequently the christenings not above 12,000. I therefore
esteemed the number of teeming women to be 24,000: then I imagined,that there might be twice as many families as of such women; for that there
might be twice as many women aged between 16 and 76, as between 16 and40, or between 20 and 44; and that there were about eight persons in a
family, one with another, viz. the man and his wife, three children andthree servants, or lodger-s: now 8 times 48,000 makes 384,000.
5. Secondly, I find by telling the number of families in some parishes
within the Walls, that 3 out of 11 families per annum have died: wherefore,
13,000 having died in the whole, it should follow there were 48,000 families
according to the last mentioned account.
6. Thirdly, the account which I made of the trained bands and auxiliary
soldiers, doth enough justify this account.
7. And lastly I took the map of London set out in the year 1658 by
Richard Newcourt, drawn by a scale of yards. Now I guessed that in 100
yards square there might be about 54 families, supposing every house to be
20 foot in the front: for on two sides of the said square there will be 100
yards of housing in each, and in the two other sides 80 each; in all 360
yards: that is 54 families in each square, of which there are 220 within theWalls, making in all 11,880 families within the Walls. But forasmuch as
there die within the walls about 3,200 per annum, and in the whole about
13,000; it follows that the housing within the Walls is 1/4 part of the whole,and consequently, that there are 47,520 families in and about London,
which agrees well enough with all my former computations: the worst
whereof doth sufficiently demonstrate that there are no millions of people
in London, which nevertheless most men do believe, as they do, that there
be three women for one man, whereas there are fourteen men for thirteen
women, as elsewhere hath been said.

8. We have (though perhaps too much at random) determined thenumber of the inhabitants of London to be about 384,000: the which being
granted, we assert that 199,112 are males and 184,886 females.
9. Whereas we have found that of 100 quick conceptions about 36 ofthem die before they be six years old, and that perhaps but one surviveth76, we, having seven decades between six and 76, we sought six mean
proportional numbers between 64; the remainder living at six years, andthe one which survives 76, and find that the numbers following arepractically near enough to the truth; for men do not die in exact proportions, nor in fractions: from whence arises this Table following:Viz. of 100 there dies
within the first six years 36
The next ten years, ordecade24
The second decade
15
The third decade9

The fourth
The next
The next
The next
The next
19

6
4
3
2
1

10. From whence it follows, that of the said 100 conceived there remains
alive at six years end 64.
At sixteen years end
At twenty-six
At thirty-six
At forty-six

40
25
16
10

At fifty-six
At sixty-six
At seventy-six
At eighty

63
1
0

11. It follows also, that of all which have been conceived, there are now
alive 40 per cent above sixteen years old, 25 above twenty-six years old,& sic deniceps, as in the above Table: there are therefore of aged between
16 and 56, the number of 40, less by six, viz. 34; of between 26 and 66, thenumber of25less by three, viz. 22: & sic deniceps.
Wherefore, supposing there be 199,112 males, and the number between
16 and 56 being 34. It follows, there are 34 per cent of all those males
fighting men in London, that is 67,694, viz. near 70,000: the truth whereof
I leave to examination, only the 1/5 of 67,694, viz. 13,539, is to be added
for Westminster, Stepney, Lambeth, and the other distant parishes, making
in all 81,233 fighting men.
12. The next enquiry shall be, in how long time the City of London shall,by the ordinary proportion of breeding and dying, double its breeding
people. I answer in about seven years, and (Plagues considered) eight.
Wherefore since there be 24,000 pair of breeders, that is one-eighth of thewhole, it follows that in eight times eight years the whole people of theCity shall double without the access of foreigners: the which contradicts
not our account of its growing from two to five in 56 years with suchaccesses.
13 According to this proportion, one couple viz. Adam and Eve, doubling themselves every 64 years of the 5,610 years, which is the age of theworld according to the Scriptures, shall produce far more people than are
now in it. Wherefore the world is not above 100 thousand years old as somevainly imagine, nor above what the Scripture makes it.

20

3. An Estimate of the Degrees of the Mortality of
MankindEDMUND HALLEY (1693)

From Philosophical Transactions XVII. Excerpts are from pages 596-604, 610, 655-656.

Halley's text may be clearer if the reader notes that he intends: /0 = 1238;
L 0 =1000; / 1 =890; L 1 =855. According to Westergaard (1969, p. 34) the Breslau
lists showed 1218 births in 1691, among whom 992 were alive on January 1,
1692. Rounding the figure, Halley set L 0 = 1000 for 1238 births.
In his Use 5 Halley refers to "years purchase", the price charged for an annuity paying one dollar per year. Parliament granted William and Mary permission in 1691 to borrow money through an annuity scheme; the rate Halleyrefers to was set the following year.
We have omitted Halley's discussion of joint and group annuities.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_3, © Springer-Verlag Berlin Heidelberg 2013

21

The Contemplation of the Mortality of Mankind, has besides the Moral, its
Physical and Political Uses, both which have been some years since most judiciously considered by the curious Sir William Petty, in his Natural and Political
Observations on the Bills of Mortality of London, owned by Captain John Graunt.
And since in a like Treatise on the Bills of Mortality of Dublin. But the Deduction
from those Bills of Mortality seemed even to their Authors to be defective:
First, In that the Number of the People was wanting. Secondly, That the Ages of
the People dying was not to be had. And Lastly, That both London and Dublin
by reason of the great and casual Accession of Strangers who die therein, (asappeared in both, by the great Excess of the Funerals above the Births) rendred
them incapable of being Standards for this purpose; which requires, if it werepossible, that the People we treat of should not at all be changed, but die where
they were born, without any Adventitious Increase from Abroad, or Decay by
Migration elsewhere.
This Defect seems in a great measure to be satisfied by the late curious Tablesof the Bills of Mortality at the City of Breslaw, lately communicated to this Honourable Society by Mr. Justell, wherein both the Ages and Sexes of all that die aremonthly delivered, and compared with the number of the Births, for Five Years
last past, viz. 1687, 88, 89, 90, 91, seeming to be done with all the Exactness andSincerity possible.
This City of Breslaw is the Capital City of the Province of Si/esia; or, as theGermans call it, Schlesia, and is scituated on the Western Bank of the River Oder,anciently called Viadrus; near the Consines of Germany and Poland, and verynigh the Latitude of London. It is very far from the Sea, and as much a M editerranean Place as can be desired, whence the Confluence of Strangers is but small, andthe Manufacture of Linnen employs chiefly the poor People of the place, as well
as of the Country round about; whence comes that sort of Linnen we usually callyour Sclesie Linnen; which is the chief, if not the only Merchandize of the place.
For these Reasons the People of this City seem most proper for a Standard;
and the rather, for that the Births do, a small matter, exceed the Funerals. The only
thing wanting is the Number of the whole People, which in some measure I haveendeavoured to supply by comparison of the Mortality of the People of all Ages,
which I shall from the said Bills trace out with all the Acuracy possible.
It appears that in the Five Years mentioned, viz. from 87 to 91 inclusive,
there were born 6193 Persons, and buried 5869; that is, born per Annum 1238,
and buried 1174; whence an Encrease of the People may be argued of 64 per
Annum, or of about a 20th part, which may perhaps be ballanced by the Levies
for the Emperor's Service in his Wars. But this being contingent, and the Births
certain, I will suppose the People of Breslaw to be encreased by 1238 Births
annually. Of these it appears by the same Tables, that 348 do die yearly in thefirst Year of their Age, and that but 890 do arrive at a full Years Age; and likewise,that 198 do die in the Five Years between 1 and 6 compleat, taken at a Medium;so that but 692 of the Persons born do survive Six whole Years. From this Age
the Infants being arrived at some degree of Firmness, grow less and less Mortal;
and it appears that of the whole People of Breslaw there die yearly, as in thefollowing Table, wherein the upper Line shews the Age, and the next underit the Number of Persons of that Age dying yearly.
22

7 . 8 9 . . 14
11 . 11 . 6 . 5! . 2. 18 . 21 . 27 . 28 . . 35 .
3! 5 6 4! 6! 9 . 8 . 7 . 7

36
42 .
45
49 54 . 55 . 56
8 . 9! 8 . 9 . 7 . 7 . 10 11 . 9 . 963
10 . 1270 71 . 72
77
81
84
90 91
9! 14 9 . 11 9! 6 . 7 .3 .4 .2 . 1 . 1 . 1 .
98 . 99 . 100 .
0 . t. ~And where no Figure is placed over, it is to be understood of those that die
between the Ages of the preceding and consequent Column.
From this Table it is evident, that from the Age of 9 to about 25 there does notdie above 6 per Annum of each Age, which is much about one per Cent. of thosethat are of those Ages: And whereas in the 14, 15, 16, 17 Years there appear to diemuch fewer, as 2 and 3!, yet that seems rather to be attributed to Chance, as
are the other Irregularities in the Series of Ages, which would rectifie themselves,
were the number of Years much more considerable, as 20 instead of 5. And by
our own Experience in Christ-Church Hospital, I am informed there die of theYoung Lads, much about one per Cent. per Annum, they being of the foresaid Ages.
From 25 to 50 there seem to die from 7 to 8 and 9 per Annum of each Age; and afterthat to 70, they growing more crasie, though the number be much diminished,
yet the Mortality en creases, and there are found to die 10 or 11 of each Age per
Annum: From thence the number of the Living being grown very small, they
gradually decline till there be none left to die; as may be seen at one View in the
Table.
From these Considerations I have formed the adjoyned Table, whose Uses are
manifold, and give a more just Idea of the State and Condition of Mankind, than
any thing yet extant that I know of. It exhibits the Number of People in the City ofBreslaw of all Ages, from the Birth to extream Old Age, and thereby shews the
Chances of Mortality at all Ages, and likewise how to make a certain Estimateof the value of Annuities for Lives, which hitherto has been only done by animaginary Valuation: Also the Chances that there are that a Person of any Age
proposed does live to any other Age given; with many more, as I shall hereafter
shew. This Table does shew the number of Persons that are living in the Agecurrent annexed thereto.
Thus it appears, that the whole People of Breslaw does consist of 34000 Souls,
being the Sum Total of the Persons of all Ages in the Table: The first use hereof
is to shew the Proportion of Men able to bear Arms in any Multitude, which arethose between 18 and 56, rather than 16 and 60; the one being generally too weak
to bear the Fatigues of War and the Weight of Arms, and the other too crasie andinfirm from Age, notwithstanding particular Instances to the contrary. Under 18
from the Table, are found in this City 11997 Persons, and 3950 above 56, which
together make 15947. So that the Residue to 34000 being 18053 are Persons
between those Ages. At least one half thereof are Males, or 9027: So that thewhole Force this City can raise of Fencible Men, as the Scotch call them, is about
23

Age
Curt.

Persons

Age
Curt.

Persons

Age
Curt.

Persons

Age
Curt.

Persons

Age
Curt.

Persons

Age
Curt

Persons

Age

Persons

2
3
4
5
6
7

1000
855
798
760
732
710
692

8
9
10
11
12
13
14

680
670
661
653
646
640
634

15
16
17
18
19
20
21

628
622
616
610
604
598
592

22
23
24
25
26
27
28

586
579
573
567
560
553
546

29
30
31
32
33
34
35

539
531
523
515
507
499
490

36
37
38
39
40
41
42

481
472
463
454
445
436
427

Age
Curt.

Persons

Age
Curt.

Persons

Age
Curt.

Persons

Age
Curt.

Persons

Age
Curt.

Persons

Age
Curt.

Persons

43
44
45
46
47
48
49

417
407
397
387
377
367
357

50
51
52
53
54
55
56

346
335
324
313
302
292
282

57
58
59
60
61
62
63

272
262
252
242
232

64
65
66
67
68
69
70

202
192
182
172
162
152
142

71
72
73
74
75
76
77

131
120
109
98
88
78
68

78
79
80
81
82
83
84

58
49
41
34
28
23
20

7
14
21
28
35
42
49
56
63
70
77
84
100

5547
4584
4270
3964
3604
3178
2709
2194
1694
1204
692
253
107

222

212

34000
Sum Total

9000, or 394 , or somewhat more than a quarter of the Number of Souls, which may
perhaps pass for a Rule for all other places.
The Second Use of this Table is to shew the differing degrees of Mortality,
or rather Vitality in all Ages; for if the number of Persons of any Age remaining
after one year, be divided by the difference between that and the number of theAge proposed, it shews the odds that there is, that a Person of that Age does not die
in a Year. As for Instance, a Person of 25 Years of Age has the odds of 560 to 7 or
80 to 1, that he does not die in a Year: Because that of 567, living of 25 years ofAge, there do die no more than 7 in a Year, leaving 560 of 26 Years old.
So likewise for the odds, that any Person does not die before he attain any
proposed Age: Take the number of the remaining Persons of the Age proposed,and divide it by the difference between it and the number of those of the Age of
the Party proposed; and that shews the odds there is between the Chances of theParty's living or dying. As for Instance; What is the odds that a Man of 40 lives
7 Years: Take the number of Persons of 47 years, which in the Table is 377, andsubstract it from the number of Persons of 40 years, which is 445, and the differenceis 68: Which shews that the Persons dying in that 7 years are 68, and that it is377 to 68 or 5-! to 1, that a Man of 40 does live 7 Years. And the like for any othernumber of Years.
Use III. But if it be enquired at what number of Years, it is an even Lay that a
Person of any Age shall die, this Table readily performs it: For if the numberof Persons liviny of the Age proposed be halfed, it will be found by the Table atwhat Year the said number is reduced to half by Mortality; and that is the Age, towhich it is an even Wager, that a Person of the Age proposed shall arrive before
he die. As for Instance; A Person of 30 Years of Age is proposed, the numberof that Age is 531, the half thereof is 265, which number I find to be between 57

24and 58 Years; so that a Man of 30 may reasonably expect to live between 27
and 28 Years.
Use IV. By what has been said, the Price of Insurance upon Lives ought to be
regulated, and the difference is discovered between the price of ensuring theLife of a Man of 20 and 50, for Example: it being 100 to 1 that a Man of 20 dies
not in a year, and but 38 to 1 for a Man of 50 Years of Age.
Use V. On this depends the Valuation of Annuities upon Lives; for it is plainthat the Purchaser ought to pay for only such a part of the value of the Annuity, as
he has Chances that he is living; and this ought to be computed yearly, and the
Sum of all those yearly Values being added together, will amount to the value ofthe Annuity for the Life of the Person proposed. Now the present value of Money
payable after a term of years, at any given rate of Interest, either may be had from
Tables already computed; or almost as compendiously, by the Table of Logarithms:
For the Arithmetical Complement of the Logarithm of Unity and its yearlyInterest (that is, of 1.06 for Six per Cent. being 9.974694.) being multiplied by the
number of years proposed, gives the present value of One Pound payable after theend of so many years. Then by the foregoing Proposition, it will be as the numberof Persons living after that term of years, to the number dead; so are the Odds
that any one Person is Alive or Dead. And by consequence, as the Sum of bothor the number of Persons living of the Age first proposed, to the number remaining
after so many years, (both given by the Table) so the present value of the yearly
Sum payable after the term proposed, to the Sum which ought to be paid for the
Chance the person has to enjoy such an Annuity after so many Years. And thisbeing repeated for every year of the persons Life, the Sum of all the present Valuesof those Chances is the true Value of the Annuity. This will without doubt appear
to be a most laborious Calculation, but it being one of the principal Uses of thisSpeculation, and having found some Compendia for the Work, I took the pains tocompute the following Table, being the short Result of a not ordinary numberof Arithmetical Operations; It shews the Value of Annuities for every Fifth Yearof Age, to the Seventieth, as follows.

Age

Years PurchaseAge

Years PurchaseAge

Years Purchase

1
5
10
15
20

10.28
13.40
13.44
13.33
12.78

25
30
35
40
45

12.27
11.72
11.12
10.57
9.91

50
55
60
65
70

9.21
8.51
7.60
6.54
5.32

This shews the great Advantage of putting Money into the present Fundlately granted to their Majesties, giving 14 per Cent. per Annum, or at the rate of 7
years purchase for a Life; when young Lives, at the usual rate of Interest, are worth
above 13 years Purchase. It shews likewise the Advantage of young Lives overthose in Years; a Life of Ten Years being almost worth 13! years purchase,whereas one of 36 is worth but 11. . ..
25

It may be objected, that the different Salubrity of places does hinder this
Proposal from being universal; nor can it be denied. But by the number that die,being 1174 per Annum in 34000, it does appear that about a 30th part die yearly, as
Sir William Petty has computed for London; and the number that die in Infancy,
is a good Argument that the Air is but indifferently salubrious. So that by what I
can learn, there cannot perhaps be one better place proposed for a Standard. At
least 'tis desired that in imitation hereof the Curious in other Cities would attempt
something of the same nature, than which nothing perhaps can be more useful. ...
A second Observation I make upon the said Table, is that the Growth andEncrease of Mankind is not so much stinted by any thing in the Nature of theSpecies, as it is from the cautious difficulty most People make to adventure
on the state of Marriage, from the prospect of the Trouble and Charge of providing
for a Family. Nor are the poorer sort of People herein to be blamed, since theirdifficulty of subsisting is occasion'd by the unequal Distribution of Possessions,
all being necessarily fed from the Earth, of which yet so few are Masters. So that besides themselves and Families, they are yet to work for those who own the Groundthat feeds them: And of such does by very much the greater part of Mankind
consist; otherwise it is plain, that there might well be four times as many Births
as we now find. For by computation from the Table, I find that there are nearly
15000 Persons above 16 and under 45, of which at least 7000 are Women capable
to bear Children. Of these notwithstanding there are but 1238 born yearly, which
is but little more than a sixth part: So that about one in six of these Women do
breed yearly; whereas were they all married, it would not appear strange or
unlikely, that four of six should bring a Child every year. The Political Consequences hereof I shall not insist on, only the Strength and Glory of a King
being in the multitude of his Subjects, I shall only hint, that above all things,
Celibacy ought to be discouraged, as, by extraordinary Taxing and MilitaryService: And those who have numerous Families of Children to be countenanced
and encouraged by such Laws as the Jus trium Liberorum among the Romans.
But especially, by an effectual Care to provide for the Subsistence of the Poor,by finding them Employments, whereby they may earn their Bread, without
being chargeable to the Publick.

26

4. A Treatise on the Valuation of Annuities andAssurances on Lives and Survivors
JOSHUA MILNE (1815)

London. Excerpts are from pages vi-xii, 89-91, 97-100, 487-489, 582.

Milne's notation and principal equations translate into modern form as:"~=In=survivors to exact age n from among /0 births at observed
age-specific death rates
x=nn

L = Pn = observed population at ages n to n + 1
n

D = Dn = annual deaths to persons ages n to n + 1
b = dn = deaths to persons ages n to n + 1 in the life tablero

L lx-t/nx=n/n= en= life expectancy at age n

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_4, © Springer-Verlag Berlin Heidelberg 2013

27

The general theorems throughout the work are adapted to any law of mortality
whatever, and even to different laws, for all the different lives that may be involved;
but they cannot be applied to practical purposes, unless one or more tables of mortality, adapted to the lives proposed, be given. The construction of such tables,
although one of the most important parts of the subject, has remained the longest
imperfect; principally from the want of data, and partly from its having occupied
but little of the attention of mathematicians.
This is the objeCt of the third chapter.
It is first assumed that the population remains stationary without being
affected by migration, the method of constructing a table of mortality from theregisters of burials in these circumstances is then shown, and the principal properties of such tables are demonstrated (143-165).
These might with more propriety be called tables of vitality than of mortality,
as their principal use is, to show the mean duration of life, and the probabilityof its continuance; were it not, that the measure of vitality is often materially different from that of the continuance of life. For one individual may, and certainly
often does, to every purpose of utility, enjoyment, or suffering, live much morethan another during the same time. So that the absolute quantity of vitality
depends upon its intensity, as well as its duration.
They are also, when properly constructed according to the general formof article 162, tables of population as well as mortality, since they show the
proportion of the people of each sex in every interval of age, as well as the proportions of them that annually enter upon and die in each of these intervals,
when the population is stationary, (art. 163 and 840).
Considered in this point of view, they are of importance in the science of
Political Economy, and as the subject is treated here, in a manner which, at thesame time that it is general and comprehensive, is perfectly elementary, this partof the work may, perhaps, be useful to those who cultivate that science, buthave made little or no progress in mathematics.
When the state of the population is variable, and affected by migration, it isvery difficult to ascertain the law of mortality, or even to approach near to it bymeans of the registers of burials alone. Mr. Simpson 1 endeavoured to make suchallowances for the influence of the continual influx of new settlers upon the
London bills, as to deduce the law of mortality from them, and from his great
judgment and accuracy there is reason to believe that his success was considerable, but he did not explain distinctly how he proceeded.
Dr. Price afterwards, in his "Essay on the Method of forming Tables ofObservations 2 ," showed how the number of the inhabitants of a town, at every
age, might be determined from the bills of mortality, when the number at eachage was maintained stationary by the influx of new settlers, although the deathsof every year exceeded the births; provided that the annual number of settlersof each age could be ascertained, and remained always the same: But besides
that the practical applications of that method are extremely limited by the
hypotheses, the requisite data can hardly in any instance be obtained.
The principle of the method is here demonstrated in article 168, is extended
in article 169 to places wherein the annual births exceed the deaths, while thepopulation is kept stationary by emigration; and in article 170, a general theorem28

is given, that includes both these. But it is shown in article 172, that such a table
could not exhibit the true law of mortality, nor enable us to determine the probabilities of life.
For that purpose, wherever the population is variable, that is, strictly speaking,in all cases, it is necessary that the number of the living, as well as that of theannual deaths at each age, should be known; and the method of constructing an
accurate table of mortality from these data, is next given (174-183).
Almost all who have hitherto treated the subject of this chapter, have neglected
the use of symbols, and have been content to illustrate their reasoning by examples
in numbers. But in this, and all other inquiries that admit of the application ofmathematical reasoning, the employment of symbols is attended with many
advantages, since they enable us to treat of generals with almost the same facility
as particulars; they relieve the mind from the fatigue and obscurity that attend
circumlocution and ambiguity of language, and increase its powers much morethan mechanical engines do those of the body.
It has generally been assumed that the number of the people when not affectedby migration, increases in geometrical progression; a few theorems are given
which determine the relations between the time elapsed, the rate of augmentation,and its amount, upon that hypothesis (184-187). Then it is shown how, upon thesame hypothesis, a table of mortality may be constructed from the registersof deaths alone, when the annual rate of increase is given (188-191).
1 See his excellent little tract, entitled The Doctrine of Annuities and Rerersions, etc. (8 vo. London,
1742). and his supplement to it, printed first in his Select Exercises for youny proficients in the mathematics. (8vo. London. 1752). but since separately.
In that supplement, the author complained with some bitterness of M. Deparcieux having criticised.
with more severity than judgment. the alterations which he had made in Mr. Smart's table of mortality
for London.
But Mr. Simpson's information of what M. Deparcieux had advanced in his Essai sur les Prohahilites
de Ia duree de Ia rie humaine. was derived from the account of the work given in the History of theRoyal Academy of Sciences at Paris (An. 1746, p. 45.). What Mr. Simpson objected to, and was really injurious. appears to have been given by the writer of that account. rather as his own sentiments thanthose of M. Deparcieux, which he does not appear to have understood, any more than Mr. Simpson's.
I consider the remarks which M. Deparcieux did make on Mr. Simpson's table. to be both candid andjudicious. and. in justice to his memory. will here insert what is most material in them.
After observing upon Dr. Halley's table for Breslaw. and that of Mr. Smart for London. which
Mr. Simpson corrected. he proceeds thus. "II est bien difficile, pour ne pas dire impossible. qu'on
puisse etablir un ordre de mortalite approchant du vrai, par le moyen des regitres d'une ville commecelle de Londres. a cause de Ia quantile prodigieuse d'etrangers qui vont s'y etablir et mourir. Aussi
M. Simpson a-t-il juge a propos d'y faire quelques corrections. sans trop dire comment. On verra
dans Ia suite par Ia comparaison qu'on fera de cette table. avec quelques aut res. si on peut beaucoup ycompter. malgre Ia correction." (p. 38.).
He then shows several of the difficulties that attend the construction of tables of mortality forlarge towns. from their mortuary registers. and adds. "II suit de toutes ces raisons. que Ia table duDocteur Hallei doit etre preferee a celle de M. Simpson. II est vrai que ce dernier semble ne vouloir
donner Ia sienne que pour les habitans de Londres. ce qui pourroit etre approchant du vrai s'il n'entendoit
parler seulement que de ceux qui naissent dans cette ville; ce qui ne peut servir de regie pour aucun
autre endroit qu'on ne J'ait examine." (p. 41.).
2 See his valuable work, entitled Observations on Reversionary Payments, vol. ii. p. 73, of the 7th
and last edition (2 vols. 8vo. London, 1812), which is that always quoted in this work.

29

In the Memoirs of the Academy of Sciences at Berlin for the year 1760, there
is a paper by the celebrated EULER, entitled Recherches generales sur Ia mortalite
et Ia multiplication du genre humain, wherein the subject is treated algebraically. He
assumes that the population is not affected by migration, and that both the annual
births and deaths are always as the contemporaneous population; consequently,that the number of the people increases or decreases in geometrical progression.
Then he gives several theorems, exhibiting the relations that would obtain between
the annual births and deaths, and the population; and determines the law ofmortality upon these hypotheses, but does not show how it may be deduced
from actual observations independent of hypotheses; neither does he undertakethe construction of any table of mortality, but, by way of example, gives that of
M. Kerseboom, with the changes in the numbers which became necessary inconsequence of his altering the radix from 1400 annual births to 10003 .
M. Euler was perfectly aware how much the application of his theory was
limited by the hypotheses it was derived from, he allows that his conclusions
will not apply where migration has place, nor in case of any extraordinary increaseof the people, such, he says, as takes place in new colonies; but he appears tohave considered the increase in geometrical progression to prevail generally,
though I believe it seldom does, either in countries that have been long or very
recently settled (192-197)....
162. In a society, none of the members of which enter it but by birth, nor go out
of it but by death, when~ the number of the members remains always the same, andone uniform law of mortality obtains; if the number of deaths at every age, in any
one year be given, a table may be constructed which will exhibit. the numbers of
the living in that society at all ages, and the law of mortality, according to which
the members continually pass out of it by death, while their places are supplied byothers that are continually rising from birth towards the greater ages, and whichshall also show the expectation of life at every age-Thus:
Let there be five columns, in the first of which insert the agesO, 1, 2, 3, ... (w-1);
then, against every age, insert in the fifth column the given number that died in theyear between that and the next greater age; which being done, the numbers to beinserted in the third and fourth columns may be easily determined, by beginning at
the greatest age in the table, and proceeding regularly, year by year, to the leastThus:
To the number against any age in the fourth column, add that against thenext less age in the fifth, and the sum will be the number to be inserted againstthat next less age in the fourth column (150).
To the sum of the numbers in the third and fourth columns, against any age,
add half the number in the fifth column, against the next less age; and the sum
last obtained will be the number to be inserted against that next less age in thethird column (161).
3 He assumes unity for the number of the born, consequently, the number completing each yearof age is expressed by the fraction which measures the probability that a child just born will attainto that age. Perhaps it is from this property that these have, by some writers, been called tables of theprobabilities ~I" lif'e: but although they show those probabilities directly at birth, when the radix is
any power of 10; they only furnish the means of determining them at all ages after that.

30

Lastly, divide the number against any age in the third column, by the number
against the same age in the fourth; the quotient will be the expectation of lifeat that age {154 and 157) to be inserted in the second column:
And the general form of the table will be this:No. of Col.
Age

2

3

4

5

Expectationof life at thatage

No. of the living at that age
and upwards

No. thatannuallycompletethat yearof theirage

No. that dieannually intheir next succeeding year

0

0

snp

lnp

0-1

11

Greatest age

w-1

Limiting agew

0S"(; -~"e

"(;

1(·J-

I~

w-Ig

0

-!w-tg20163. From the third and fourth columns of this table, it will be easy to determine
the numbers both of the living, and of the annual deaths, between any two agesthat may be assigned; for we have only to subtract the number in each column
against the greater age, from that against the less; the remainder will, in eachcase, be the number sought. ...
174. But whether the population be stationary, or increasing, or decreasing;
and whether such changes be produced by procreation, mortality, or migration,or by the joint operation of any two or more of those causes; provided that themode of their operation be uniform, or nearly so, and not by sudden starts, thelaw of mortality may be approached near enough for any useful purpose, by
actual enumeration and the bills of mortality-Thus:
175. Let the number of persons, in each year of their age, that are resident
in a place at any one time be taken; and let an accurate register be kept of thenumber that die annually in each year of their age, during a term of eight or ten
years at the least, whereof the first half may precede, and the second follow,
the time of the enumeration.
Then, if the number of inhabitants of every age either increase or decrease
uniformly during that term, the mean number of annual deaths in every year of age,thus registered, will be the same as if the population of the place had continued
throughout that term the same as when the enumeration was.made.
176. In consequence of the several causes of change mentioned in art. 174, bywhich the population we are considering may be affected, the proportion between
the numbers that annually complete any two ages therein may differ materiallyfrom that which would obtain in a society not affected by migration, but subject
to the same law of mortality at every age, and maintaining its population stationary,by a constant equality between the annual numbers of the births and deaths.
31

And therefore the probabilities of life in that place cannot be determined
immediately from the proportions furnished by the enumeration.
But from the enumeration, and the register together, the law of mortality,
and consequently those probabilities, may be determined in the following manner.
n177. Let L denote the number of the living in the place, at the time of theenumeration, of the age of n years, that is, in the (n + 1)th year of their age; andnD, the mean number that died annually in the same year of their age, during theterm for which the register was kept (175); while·~ represents the number, in thesociety mentioned in the last article, that annually complete the nth year of theirage; and o= ·~- "+ 1 ~, the number of the same society that die annually in theirnext succeeding year, as in art. 149:nn

Then will L: D:: t("~ +"+ 1 ~): n~ _n+ 1 ~:: ·~-to: 0, and

n

n

n

n~ D-toD=oL;n"~Dwhence - n - - n = 0.
L+tD
n

n

But Land Dare always given by the enumeration and the register; therefore
When n~ is given, 0, and thence n +I~= n~- 0, may be easily Obtained.
In this manner, assuming at pleasure 0 ~ or ~. the number of annual births, andproceeding year by year, to determine successively the number that die in eachyear of their age, and, consequently, the number that complete that year, thenumbers against every age in the fourth and fifth columns of the table in article 162
may be inserted; from whence the numbers for the second and third columns
thereof may also be supplied, as directed in that article.
n178. From about the age of seven or eight years, till forty or fifty, t D isngenerally so small in comparison with L, that it might be neglected in the denominator of the fraction in the formula just given, without producing any
considerable error in the value of o, whereby that formula would become
n

"~D

-n- =

o, and this would facilitate the calculation a little.

L

179. This approximate value of o would also result from the supposition
that all who enter upon any year of their age continue alive throughout that year;
and, that such of them as do not enter upon the next expire at its commencement;
which hypothesis has generally been assumed in constructing tables of this kind.
But as the accuracy of such tables is of great importance, and the abridgmentof the labour of constructing them in this way is but small, it will be better always
to use the correct formula given in article 177....
788. In K. V. Ac. Handl. 1800, s. 323. Mr. Nicander has given statementsof the mean number of women in all Sweden and Finland, with the annual average
number of deliveries, and the proportion delivered annually in each of the under32mentioned intervals of age; during sixteen years ending with 1795, which arepresented at one view in the following table 4 :Mean
Number of
Females
livingAnnualaverage
Number of
Deliveries

One
Woman of

Proportionof
1000
Deliveries

15 & 20
20 & 25
25 & 3030 & 35
35 & 40
40 & 4545 & 50
Above 50

134,548
129,748121.707111.373
97,54390,85278,89769.2683,298
16,50726,32925,61818,093
8,518
1,694
39

40.8
7.8
4.6
4.3
5.4
10.6
46.5
1776.0

33
165
263
256
181
85
17
0.4

Above 15

833,936

100,0968.3

1000

Between theages of~-··-------------

According to the proportion of legitimate and illegitimate children, Mr.
Nicander found, that of 100,096 5 women delivered annually, 96,124 were married,and 3972 unmarried.
Upon comparing the legitimate births with the married women, and theillegitimate with the unmarried above fifteen years of age, he also found that 10
married women out of 54, and 10 unmarried out of 918, were annually brought
to bed.
Some other results given by Mr. Nicander are presented under table XI.,
which table differs but little from that given above. What difference there is,arises from hence, that I have deduced the mean number of women in eachinterval of age from that gentleman's table marked Q, in K. V. Ac. Hand!. 1801,1
Qu., and have calculated the two last columns from that and the column ofdeliveries.
It should be observed, that as the annual numbers here given are the average of
sixteen years' observations, the number of deliveries they were derived fromwas 1,601,536.
789. The proportion of males born to females, appears to be the result of a
uniform law of nature, and is probably always the same under the same circumstances, though it seems to vary with the situation of the parents, and I
should rather suspect that variation to depend principally upon their age.
~ It will be observed that this table only shows the actual fecundity, which is always kept below
the natural aptitude or physical power, by the obstacles to marriage; as appears by the table itself,
for the proportion of women delivered annually between the ages of 30 and 35. is almost twice as
great as between 20 and 25; which can only arise from the proportion of the married in the first ofthese intervals of age being at least so much greater than in the second.
5 This number is printed 100,098 by Mr. Nicander, and I believe it should be so, as it occurs several
times. hut if that be right. some other number in the column of deliveries must be wrong, and as I haveno! the means of determining which it is. I have thought it belter to state the amount of them as theystand. because the error bears a less proportion to that than any other. it is indeed of no kind of consequence.

33

The following table exhibits the proportion with considerable precision,under a few varieties in the circumstances.
Place

Term.
endingwith

Males

FemalesNo. of Male
Births forevery 10,000
Females

22
r8
4
9

1800
1810
1763
1795
1802
1800
1810
1800
1810
1800
1796
1792
1791
1795
1795
1763

3,285,188
1,468,677
400,086
1,006,420
110,3122,997,842
1,391,977
177,401
76,70067,3532,400
12,91993,701
117,928
24,227
12,015

3,150,922
1,410,229
387,702
965,000
105,2872,879,011
1,338,998
166,593
71,231
62,6362,271
12,14591,404
115,191
23,787
11,706

10,426
10,415
10,436
10,42910,477
10,413
10,396
10,64910,76810,753
10,568
10,637
10,251
10,238
10,185
10,264

20
21

1795
1792

37,7001,373

37,060
1,362

10,173
10,081

Years

England and Wales
Sweden and Finland
FranceEngland

r
r
10
9
20
3

10
29
10
29
18
21

Wales
ScotlandCarlisle
Montpellier
Diocess of Borgo in Finland
Stockholm
Illegitimate
In Sweden and Finland
In Montpellier

Number of BirthsofTable XI. Showing the Fecundity of Women at the different Periods of Life, in Sweden and Finland,
from 1780 to 1795, both Years inclusive. (Art. 788).
Absolute Fecundity

Intensity of Fecundity

Between theAges of

Mean Number ofFemales living

Annual average
Number of Deliveries

That is, of
10,000 living

Or one of

15 & 20
20 & 25
25 & 3030 & 35
35 &4040 &4545 &5050 & 55

132,765
131,377
121,650
112,250
98,71089,25974,002
69,0353298
16,50726,32925,61818,093
8518
1694
39

248
1257
2164
2282
1833
954
229
5.65

40.256
7.959
4.620
4.382
5.456
10.47943.6861770.1

20 &40
15 & 55

463,987
829,048

86,547100,096

1865
1207

5.361
8.283

Double Births, 1 of58
Triple,1 of 3,365.
Quadruple,
1 of 143,000.
One Birth in 104 produced the death of the mother.
And the number of deaths from Childbirth, was to the whole number ofdeaths of females between the ages of 15 and 55, as 2 to 19, or as 1 to 10 nearly.
34

5. Statistical Applications of the Mortality Table
GEORGE KING (1902)

From Institute of Actuaries' Textbook, Part II, Second Edition, pages 56-58, 63. London: Charles
and Edward Layton.

We omit several numerical examples given by King of the uses of the life table,
which concern populations at various ages, annuity payments, and the effects
of immigration on observed death rates. The article introduces the life tablein modern notation.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_5, © Springer-Verlag Berlin Heidelberg 201335

4. The fundamental column of the Life Table is the column of l3:.
The first value in that column, l0 , called the radix, is the number ofannual births in the imaginary population; and the succeeding numbers
show how many persons, out of l0 born alive, complete each year of age.
In the table the number of annual births is 127,283; and we observethat 100,000 live to complete the tenth year of their age; 89,685 live
to complete the thirtieth ; and so on. We also see that only 4 live tocomplete the century ; and that, although 1 survives 101 years, all die
before reaching the age 102. Age 102 is therefore the limiting age ofthe table, being the year of age on which some lives enter, but which
none complete. To the limiting age, the Greek letter (.1) is assigned
for symbol; and therefore l(I)=O. Also, the difference between thelimiting age, and the present age, is called the complement of life;
so that, at age x, the complement of life is (I)-X; in the case of ourtable, 102-x.
5. The column of d3: contains the differences between the numbers in
the column of l3:; and shows how many, out of l0 persons born alive, diein each year of their age. Thus, by the table, out of 127,283 persons
born alive, 14,358 die before completing their first year; 691 survive to
age 30, but die before reaching 31; and so on. The number, then, incolumn d opposite any age, x, is the number who complete that year ofage, but die before completing the next; that is, the number in column
d opposite age x, is the number who die in the (x+ 1)th year of age.
As all born must die, it follows that the sum of all the numbers incolumn dis equal to l0 : also, the sum of the numbers in column d, fromage x to the oldest age, is equal to l3:.
6. To find how many die aged between x and x+n, we may take thesum of the numbers in column d, from d3: to dx+n- 1 , inclusive ; but
unless n be very small, it will be easier to obtain the result by means of
the l column; because lx-lx+n=dx+dx+l+&c.+dx+n- 1 • Thus, by
the table, the number of persons, out of 127,283 born alive, who die
between ages 20 and 30, is l~-l30 =6376.
7. Passing now to the column L, we have the population living in astationary community. Such a community, sustained by l0 annual
births, will, on the supposition of uniform distribution of births anddeaths, always contain i(l0 +li)=L0 children in the first year of their36

age; i(l1+l2)=L1 in the second year of their age; and so on. Thus, by
the table, a population sustained by 127,283 annual births, will always
contain 95,787 young persons aged between 20 and 21. The total
population at all ages will be the sum of all the numbers in column L ;and that is given in column T. By the table, the total population thatwould be supported by 127,283 annual births, is 6,082,0,!3l=T0 • The
column T bears exactly the same relation to the column L, that thecolumn l bears to the column il: that is, Tz is the sum of the numbers incolumn L, from age x to the oldest age: therefore Tz is the total population, aged x and upwards, in the community. In the community of
T 0 inhabitants, there must be 10 deaths annually; because there are 10
births; and, the population being stationary, the deaths must be equalin number to the births. Similarly, there must be lz deaths annually of
persons aged x and upwards; and lz-lz+n deaths annually, of persons
aged between x and x+n. Also, the number of inhabitants aged
between x and x+n, is Tz-Tz+n; and, therefore, the proportion ofdeaths to population, between ages :c and x+n, is ~z-~~+n ; and, for thez - z+nwhole community, the proportion of deaths to population is ~:.

When n• umty,·il:e = m:e, t h e centra1 death rat e at age x.
1s
Tl:e-l:e+n
T= Lz - :e+n:eBy

the table, the proportion of deaths to the population for the wholecommunity is 127 ' 283 = ·020928 or about 21 per thousand. The
6,082,031.
127,283-72,795
proportion for the population aged less than 50, 1s 6 ,082 ,031 _ 1 ,475 ,603
54•488 = ·011829, or not quite 12 per thousand, while the pro4,606,428. aged 50 and upwards, . 72,795
portion for the population
IS 1 ,475 , 603·049332 '_or over 49 per thousand. These figures illustrate the remarks madein Chapter iii, Art. 13. They show that if from any cause there isan unusual proportion of young persons in a community, the ratio of
deaths to population will be diminished; but, as previously remarked, it
does not follow that therefore the members of that community enjoy
unusuallongevity....
37

We have seen that, of l:1: persons who attain the precise age tt, l:1:+1 willcomplete a year of life in the first year; and d:1: will live on the average halfa year each ; therefore the quantity of existence in the first year due tothe l:1: persons will be l:1:+ 1 + i-d:1:= L:1:. Similarly for future years ; therefore lL:1:=T:1: will be the total future existence due to the l:1: persons;giving ~= =e:1: years to each; and the average age at death of the l<»persons will be te+e:D years.

The existence within the next n years due

1.'o
T:D-T:D+n. .
. T ~~:- T ~~:+n; givmg
t o the l ~~:persons, IS
each .
years 10r
= lne:el~~:

Of these years, n X l~~:+n are due to those who complete age te+n; leaving
T~~:-T~~:+n-nl~~:+n for those who die between age te and age te+n. Butl~~:-l:e+n persons die between these ages; therefore the average amountof existence between ages te and te + n, belonging to those who die.
• T~~:-T~~:+n-nlx+n, and therr average age at deathllbetween these ages, IS:JJ- :e+n
if
Tx-T:D+n-nla:+n•te=20 and n=10, we find
• For example,ll
IS te +~~:- a:+n
Tro-T30olro =9·680. Also, the existence within the 10 years, due tohoe2o=
all those who reach age 20, is T 20 -T30 =929,902; the existence inthe period due to those who survive it is 10l30 = 896,850 ; leaving
33,052 years to the l20 -l30=6,376 persons who die in the 10 years;
or 5·184 to each. The average age at death of those who die between
20 and 30, is therefore 25·184. Similarly the average age at death of. To- T20-20l20.
= 3·734.
lo-lrothose who die below 20 IS
13. We have seen, Art. 7, that the ratio of deaths to population
in a stationary community is ~: : also (Chap. iii, Art. 16), that thecomplete expectation of life is ~:.

It therefore follows that the ratio

of deaths to population is equal to the reciprocal of the complete
expectation of life.

38

6. Formal Treatment of Aggregate Mortality Data
WILHELM LEXIS (1875)

From Einleitung in die Tl1eorie der Bevolkerungs-Statistik, pp. 5-7. Strasbourg: Trubner. Translated
by Nathan Keyfitz.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_6, © Springer-Verlag Berlin Heidelberg 201339

As an example of a population with a single source of variation, let us consider
that of a fixed territory subject only to death. (One could consider birth as thealteration of condition, but it seems more natural to take the births as the absolute starting point of the elements). The proposed treatment is valid also forthe presentation of any other kind of alteration of condition of those born.
The development of our example yields the formal theory of mortality statistics, which covers not only mortality, but also its correlate, survivorship.
Provisionally disregarding out- and in-migration, we consider a number ofindividuals who are distinguished for us only by being born at different times,
and dying at different times and ages. Thus the population can only be decomposed according to these variables. The decomposition to reveal the inner
structure of the population can be seen most simply in a graphical scheme.
Let (Fig. 1) N be an axis of coordinates, on which time is measured from anarbitrary fixed origin 0, and let the birth times of individuals be marked on ON.
These birth points occur discontinuously and more or less densely. The numberof points that lie on equal intervals of the line 0 N would serve to compare thedensity of births at different times.
Two kinds of birth points are to be distinguished, those corresponding tolive births, and those corresponding to stillbirths. We consider only the first.
Starting from the birth points, we now measure the ages of individuals vertically to the birth axis, and parallel to the axis of ages OQ. That age at which
the event under consideration, namely death, occurs to an individual is represented by a point.

X

Fig. 10'

A'

0

In that way the plane QO N is strewn with scattered death points, not shownin the diagram. The diagram is closed by the line Q X, which runs parallel to
0 N at a distance equal to 0 Q = w, which corresponds to the highest age reached.
The figure can be thought of as extending indefinitely to the right.
40

On every life-line there now lies only one death point, but with a high densityof births there would tend to be in each small interval of births dying pointsof all the ages from 0 to w.
The mass of points so presented can be divided by parallels to the two axes.
If the birth times of the abcissa are in general expressed by 11, the equationn = 0 P represents the vertical line P n. This divides the dying times of theindividuals who are born before the time 0 P from those born after.
If the ages, represented by the ordinates, are designated by a, then a= OQ
is the equation of the line QA, which distinguishes the dying points of those
who die before reaching the age OQ from those who die after.
Finally, there is a third division of the plane, such that the cases of deathoccurring before a certain absolute time are divided from those that lie beyond
this time. If z is the general designation of the time of observation, then theequationn+a=zapplies for the points that fall exactly at time z.
Let us now suppose that z takes a fixed value OZ' as measured on the birthaxis, corresponding to the age 0, and let z go through all values from 0 to w,
then n goes backwards through all values from z (or OZ') to z-w. The above
equation thus represents the straight line Z' n, which on further extension would
thus cut the axis OQ at a distance z from point 0 and makes an angle of 45owith either axis.
Consider now an age at death that belongs to an arbitrary birth point in
the interval z- cv to z (or P Z'). If it is below that determined by the above equation, (that is, lower than z -n) then the death point of the individual concernedlies below line Z' n. But if the age at death is higher than would be determinedby the n from the above equation, then the death point lies above that line.
The line Z' n divides those who die before the time z from those whose deathoccurs after this time. For values of 11 that exceed z it is obvious that the corresponding cases of death can only occur after the time z. It is also plain that persons
born before the time z-w (or OP) necessarily die before the time z, for they
otherwise would exceed the maximum age cv.

41

7. A Short Method for Constructing an AbridgedLife Table
LOWELL J. REED and MARGARET MERRELL (1939)
From American Journal o( Hygiene 30. Excerpts are from pages 33~38, 51.

For infancy Reed and Merrell revise their qx formula to correct underenumeration, an approach that is not general. This has been omitted, as have their equations for Lx and Tx values by approximate integration. The article is followed
by a note on other nqx formulas.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_7, © Springer-Verlag Berlin Heidelberg 2013

43

The life table is so valuable a description of the age variation in danger ofdying that it is desirable to have rapid
methods of obtaining it from the recorded
vital statistics. The principal value ofthe life table lies in the form in which agespecific risks of dying are stated. They
are expressed as the probability of dying
within a designated interval subsequent
to an exact age, or as the probability ofsurviving from one exact age to another.
Such probabilities are not only very descriptive, but they lend themselves
readily to the treatment of mortality ina wide variety of problems.
The fundamental element in the construction of a life table for a particulartime is thus to convert the observed agespecific death rates for that time into
the probabilities of dying within stated
age intervals. There is nothing in thedefinition of the probability of dying
which fixes the length of these intervals.
They may be chosen to suit the needs ofthe problem. The term "complete life
table" has by custom come to designatea table in which the interval is 1 year,
and the probability is stated for every
year of age. It should be remembered,however, that this is pure convention,
since a table computed for monthly
intervals would be more complete andone for weekly intervals still more so.
The term "abridged life table" is lessstabilized in its usage, since there arevarious ways in which a life table may
depart from the form called "complete."
One form of abridged table is illustrated
by Foudray's 1920 Tables for the UnitedStates (1923) and by Dublin and Lotka'sAppendix Tables in Length of Life (1936),
in which the probabilities of dying are
given for single years of age, but thesevalues are tabulated at intervals of 5 or
10 years rather than for every year.
They constitute, therefore, selected probabilities from a complete table, theintermediate values being omitted.
Another form of abridgement, whichis the one considered in this paper, increases the interval over which theprobabilities are stated, usually to 5 or
10 years. These probabilities cover theinterval between the stated ages andtherefore form a table which is not incomplete except in the defined technicalsense. They differ from the values in acomplete life table only in the fact thatthe interval chosen for study is greaterthan 1 year. The abridgement in thiscase, being through a condensation of thecomplete table rather than through omission, has certain advantages, since thesurvivorship values can be computed
directly from the probabilities of dying,
and the probabilities may be used asthey stand, without interpolation, in any
analysis where an interval greater than
1 year gives sufficient detail.
The method proposed is a short procedure for obtaining these probabilities
of dying directly from the observed rates
without going through the elaboration of
constructing a complete table. The
principal function to be derived is therefore .,.q:r:, the probability of dying within
n years after age x, and the other life
table functions flow directly from thesevalues by procedures which are described.
Direct observation shows that there
is a very high degree of association
between the observed age-specific death
rate over an age interval and the probability of dying within that interval.
... The basis of this relationship may beseen from the two following equations
expressing, respectively, the age-specificdeath rate, .,.m:z:, in a stationary popula1 Paper no. 212 from the Department of
Biostatistics, School of Hygiene and Public tion, and the probability of dying, ..q:z:,
Health, The Johns Hopkins University, Balti- as functions of l:r:, the number surviving
more, Md.
to age x out of a given number born alive.

44

average annual death rate becomes
smaller, the probability of dying within
n years approaches n times the value per
year. With regard to the upper limit(2) imposed on ,.q,., the catalytic with its
limit of 1 is more rational than the
These equations show -that ,.q,. is ex- hyperbola with its limit of 2, since thepressible in terms of ,.m,. and that if the probability of dying cannot exceed unity.
equation for l,. were known, the function For intervals as small as 1 year, it isrelating the two types of rates could be hard to say whether the hyperbola (3)stated explicitly. The assumption that or the catalytic (4) is to be preferred asl,. is expressible as a straight line over an expression of the relationship between
the interval n leads to the well-known m,. and q,., but neither proves to be satisfactory for longer intervals.
relationship 3
Other attempts to state an equation
2n,.m,.
(3) describing the l,. curve over longer inter,.q,. = 2 + n ,.m,. 'vals have led to rather involved equations of association. Furthermore, forwhich is the equation of a hyperbola,
practical applications it would be necespassing through the origin with a slope
sary to adjust such equations for then, and having an upper asymptote,difference between observed and life,.q,. = 2 ....
table values of ,.m,.. Therefore, theAs an alternative, we might assume relationship between the observed ,.m,.
that l,. is an exponential curve over an and ,.q"' was studied directly without aninterval n, since it is as reasonable to explicit assumption with regard to the lz
break up the arithlog graph of l,. into equation. For this purpose Glover's
linear segments as it is to treat the arith- 1910 life tables (1921) were examined as
metic graph in this way. This assump- to the relation of the observed 5-yeartion gives for the relationship between and 10-year death rates to the derived
the age-specific rate of the life table and probabilities of dying over the samethe probability of dying over the same intervals.
age interval, the catalytic equation 4
Figure 1 presents for a 5-year interval
the association between the observed,.q,. = 1 - e-n nmz.
(4)6m., and the derived 6qz for the 33 tables
This equation, like the hyperbola (3), in Glover's 1910 series, the x values
passes through the origin with a slope taking all the multiples of 5 through ageof n, but has an upper limit ,.q,. = 1, 80. It is apparent from this graph that
where the hyperbola has a limiting value the observations fall exceedingly closeof 2. Both of these curves have, there- to a smooth curve. This is especiallyfore, a rational position at the origin, impressive since the life tables are not at
the probability of dying being 0 when the all similar in their mortality schedule.
death rate is 0. Furthermore, they have They represent tables for the different
a rational slope at the origin since, as the sexes and colors, for urban and rural
areas, and for various combinations ofthese factors, with the variation inpatterns that comes from such diverse
1 For the derivation of equation (3), seegroups. It would thus appear that the
appendix 1.
'For the derivation of equation (4), see relationship of sm,. and 5q,. is one which
need not be stated for a particular age orappendix 1.
(1)45

- - sqx =1- e -s,mx -too8)'.5) sm.
J

2 (.5)smx- - - - 5fx= 2+.5 5 171~-------- sqx= 1- e -.s "'cl/Jx

100 110 JJO 130 HO 150
FIGURE 1.

Correlation between the observed age specific death rate, ,ms, and the probability ofdying, illz1 stated per 1000, from Glover's 1910 life tables.

size of death rate, but is independent ofthese factors. The statement of thisrelationship in the form of an equation
would clearly permit passing directlyfrom the age-specific death rate to theprobability of dying.
The observed relationship is of somewhat the same general form as the hyperbola (3) or the catalytic (4), as may beseen in figure 1. Neither of these theoretically derived equations, however,
could be considered an adequate description of the relationship. The catalytic begins to fall below the points
when &m, is approximately 20 per 1000
(about age 50) and remains below thepoints throughout the rest of the range.
The hyperbola differs from the points
46

less consistently than the catalytic, butdoes not have the proper curvature.
For the ages beyond 80 which are notshown on this graph, it is in generalfarther above the points than the catalytic is below them, since the l., curve in
extreme old age is better described by an
exponential than a linear function.
Thus, the catalytic and the hyperbola
deviate from the points in different form
but neither stays with them. This isdue to the fact that the basic assumptions from which they were derived do
not hold for intervals much longer than
1 year.
An equation was therefore sought
which would describe the entire range of
observations and one which would also

be suitable for expressing the relationshipof the rates for intervals other than 5
years. This requirement proved to besatisfactorily met by the equation

the catalytic depart from the trend of
the points in the same way as they did
for the 5-year rates, but to a greater
degree.
The empirical curve, derived above,with the same value of the arbitrary
in which a is an arbitrary £onstant. Its constant, is seen to give a very satisvalue determined empirically for the factory description of the relationship,
method presented in figure 1 is a = .008. and the points hang closely to the central
This equation is seen to be similar in curve. The values beyond age 80 areform to (4), passing through the origin also well described by the equation andwith a slope of n and having an upper thus for a grouping as broad as 10 years,
asymptote ,.q., = 1, but it has a cor- the equation effects a satisfactory transrective term in the exponent which has formation of the observed age-specificthe effect of lifting the latter part of the death rates into probabilities of dying
curve. Thus it is a rational curve in its from age 5 to the end of life.
Although the observations have been
position and slope at the origin and inits limiting value, and from figure 1, it presented only for an interval of 5 andis apparent that it fits the points well, of 10 years, it should not be inferredrather remarkably so, in view of the fact that the equation is limited to thesethat it contains only one arbitrary con-· values of n. The equation derived
stant. It should be stated that the depends implicitly on the area andcurve fits the points beyond age 80, ordinate relationships in the Z., curve andwhich are not shown on this graph, as is sufficiently< elaborate to summarize
well as those presented, and the equation these relationships with a high degree 0f
may therefore be used from age 5 to the approximation for an interval as large
end of life, to derive the life table prob- as 10 years. Consequently it holds
abilities from the observed rates.
equally well for smaller intervals, proThe complexity of the l., curve pre- vided the age-specific rates over thesevents the use of such a simple equation intervals can be obtained, either byover very broad intervals. The group- direct observation or adjustment of theing to which it may, with fair accuracy, observations, with the same accuracy as
be extended is indicated by the relation- those observed for an n of 5 or 10 years.
ship for 10-year values. Figure 2 shows For an interval as small as 1 year, thisthe scatter of points for all the 10-year curve, the catalytic (3), and the hyperrates from Glover's 1910 tables, at 5-year bola (4) give virtually the same resultsintervals from age 5 to 80. Here again, over the entire age range.
The procedure for summarizing thethe points from the various life tables
fall into the same pattern, none of the relationship between the observed ,.m.,tables showing any :n1arked or con- and ,.q., statistically in the form of asistent divergence from the others. regression equation does not take accountThis scatter of points is compared of the indivtdual variation about thegraphically with the three equations just central curve. However, the individual
considered. The assumption that l., variation is, in general, well within the
could be expressed as a series of linear range of error of observation, and foror exponential equations would naturally areas such as states or smaller units, it is
be more violent for an interval of 10 even within the variation of simple
than of 5 years. It is not surprising, sampling. Thus the refinement of ,.q.,therefore, that both the hyperbola and values beyond the value predicted by47

Relafion5nip br?tween 06~erved JolllJi and Derived1o t'fx, Glover:Y 1910 Life.Tab!e~700'.--r~--~-r~r-,-~--~-r-,--~~--r-,-~--~-r~r-~~--

600

200

100

FIGURE 2.

Correlation between the observed age specific death rate, , 0mz, and the probabilityof dying, lo!lz, stated per 1000, from Glover's 1910 life tables.

the regression curve from the observed
specific rate is, for the usual case, carrying the statistical analysis beyond thepoint justified by the accuracy of thematerial. ...

Then
nm·x

=

a

+ bx - a - b(x + n)
Ix+n (a + bx)dx-bbn'

a+ bx + 2

Appendix 1- bn
nqx =a+ bx.

(b) Derivation of equation (3) in text:
Assume that over the interval n,lx = a+ bx.
By definition

Eliminating a + bx between
these two equations gives
2n nmz(3)nqx =2 + nnmx(c) Derivation of equation (4) in text:
Assume that over the interval n

(1)

and
(2)

lz = ea+bz.

48

By substitution in equations (1)and (2),.m.,=

andnq., =-bea+bz(1 - ebn)
ea+bz·= 1 - ebn.

Eliminating b between these
two equations gives..q., = 1- e-nnm:..
(4)

REFERENCES

Foudray, E.
1923 United States Life Tables, 1919-20. Washington (Bureau of the Census).
Dublin, L. I. and A. J. Lotka.
1936 Length of Life. New York. Ronald Press Company.
Glover, J. W.
1921 United States Life Tables, 1890, 1901, 1910, and 1901-1910. Washington (Bureauof the Census).

49

Editors' Note: Approximations to nqx
The equations for nqx Reed and Merrell examine apart from their own derivefrom Milne and Farr. Their equation (3)-equation (1) in the article by Grevillethat follows-is found by setting Mx = mx and generalizing from 1 year to nyear intervals Milne's equation (1815, p. 99)
px
D.~tUx+ fx+ I)
fx-/x+ I

to yield

On dividing by lx this becomes

Reed and Merrell arrive at the same result by setting lx =a+ b x, which of course
Milne assumed. The equation can also be identified as DeMoivre's (1725) hypothesis, discussed in chapter 4.
Reed and Merrell's equation (4)-Greville's equation (2)-derives from
Farr's (1864, pp. xxiii-xxiv) English Life Table No.3. Farr introduced a better
form, which would generalize for abridged tables as

where

Here, r is "the rate at which the rate of mortality increases or decreases." Thederivation of the formula is explained by Wolfenden (1954, pp. 130-131). Assuming the force of mortality J.lx to be approximately nmx-n/Z and applying
Gompertz' Law that the force of mortality increases exponentially with age,
we have50

This permits the substitution
nqx=1-exp[ -j.u(x+t)dt]=1-exp[ -nmxr;n12

1r/ dt]=1-exp[ -nmxr;n12 (~-1)/lnrJ.
In application Farr used single year rates and dropped the minor term r;n12 •If the force of mortality is assumed constant (i.e., r = 1) the equation reduces toThis yields the lx formula given in the appendix to Reed and Merrell:

Graphed with the simple exponential and the Reed-Merrell formula (seetheir Figure 2) the Farr equation would be the least accurate for the low nmx
rates at younger ages and intermediate for the higher rates that come later inlife, reflecting its close dependence on Gompertz' Law. Greville's article completes this analysis by uncovering a derivation for the Reed-Merrell formula,which employs Gompertz' Law more effectively as a correction term.

51

8. Short Methods of Constructing Abridged Life
TablesT. N. E. GREVILLE (1943)
From Record o( the American InstitUte of Actuaries, Part I 32. Excerpts are from pages 29, 34-40.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_8, © Springer-Verlag Berlin Heidelberg 201353

I. INTRODUCTION

In recent years sociologists, public health officers, and students
of population problems have been taking an increasing interest in
the life table as a description of the age-variation in the chances ofdeath and survival. As a result of this interest in the subject, attention has been focused on rapid methods of constructing abridged life
tables. Some of the most interesting and useful methods have beendevised by nonactuaries and, probably for that reason, have not received the attention they deserve from the actuarial profession. It
is the purpose of this paper to bring some of these methods to thenotice of actuaries and also to show that certain formulas whichhave been put forward by their originators on purely empiricalgrounds actually have a valid mathematical basis ....
III. RELATION BETWEEN ,.m:e AND ,.q:eThe basic feature of the type of methods of life-table construction
under consideration here is the mathematical relation assumed tohold between ,.m:e and ,.qz. There are a number of such relations
which may be used. Perhaps the simplest is obtained by assumingthat l:e can be regarded as a linear function in the age-interval. This
leads to the equation

a formula which, for the particular case n = I, is well known to
actuaries [5, p. s]. Another possible assumption is that l:e can be
represented by an exponential function. This leads to the equationa relation which is also familiar to actuaries in the special case ofn = I (5, p. I6).
More precise equations of the same general form as equations(I) and (2) will now be derived. In the life table

Integrating,54

Now, applying the Euler-Maclaurin summation formula, we have

Differentiating and substituting from equation (4) gives

Making use of equations (3) and (5), we getor,

2n. nm:enq:e = --------(;----'---:----;----2
nm~ - nm:e) + ••••1
2 + n • nm:e + ~:Xan equation of the same general form as equation (I).
In order to derive an equation of the form of equation (2), we
write
2 2

2
d mz ) + · · · ·n • nmz + -n6 ( nm z2 - dxn+ n . nm:e + n6 nm~ - dxd nm:e ) + .... '2(

whencecolog., nP:e =log.,[I +; nm:e + :: (nm~- :x nm:e) + · · · ·][I - ;: nm:e + :: (nm; - :x nm:e) + · · · ·J

- log.,

55

This gives, upon expansion and simplification,
colog. nPx = n · nmxn3d+ -I2 nmx d-X "m"' + · · · · ,(8)a formula of the same general form as equation (2).
The first question that arises in the actual application of formulas (7) and (8) concerns the evaluation of the derivative of nmz. It
turns out that great exactness is not necessary, as the terms into
which this derivative enters are in the nature of minor adjustments
which do not have a great effect on the resulting nqz values. If it isassumed that the function nmz can be represented by a polynomial,
it is found thator

according to whether a polynomial of the second or the fourth degree is assumed. These formulas, however, are not applicable whenneighboring age-groups do not contain the same number of years.
As an alternative, it may be noted that the well-known formulafor the derivative of nPz [s, p. I 7] givesddx colog. nPx = P.x+n - P.x .

Therefore, differentiating equation (8) givesId- nmz = - (P.z+n - P.z)dnx

approximately. Substituting this approximation m equation (8)gives, finally,colo g. nPx = n . nmz[I + ~
56(P.x+n -P.z)]

This is the formula given (for the case of n = 5) by the editors of
JIA [I, p. 30I; 2, pp. I2I-22]. However, in the practical application of this formula it was found necessary to approximate the expression in parentheses in terms of nmz values. This makes the formula equivalent to equation (8), used in conjunction with the approximation (9).
Since nmz is approximately equal to the force of mortality at the
middle of the period, a more logical assumption is that nmz is an exponential function, in accordance with Gompertz' Law. If

then

where
k = log~ e.

Substituting this value in equations (6) and (8) gives us
(6a)

and

As a moderate variation in the value of k has little effect on thevalue of nqz except at the older ages, where most mortality tables
follow Gompertz' Law fairly closely, and at the very young ages,
which are generally dealt with by a different method, k may be safely taken as a constant throughout the table. Henderson states [6,p. 90] that for practically all mortality experiences log1o c lies between .035 and .045. Therefore, k =loge c would fall between .o8oand .I04.
It is interesting to observe that Reed and Merrell, after finding
equations (I) and (2) unsatisfactory at the older ages, suggested on
empirical grounds the equation(II)57

a relation exactly equivalent to equation 8(a) if a is written for
k/12. By fitting this curve to the thirty-three tables in Glover's
1910 series, they arrived at the value .oo8 for a. This would correspond to k = .og6, a value well within the range given by Henderson. Reed and Merrell have published extensive tables of the function (II) with a= .oo8, both for n = 5 and for n = 10 and also forn = 3 over a limited range, intended for use in the age-interval
two to five. With the aid of these tables a good computer can construct an entire abridged life table in less than two hours.
Although experiment would indicate that equation (6a) generally
gives nearly as good results as equation (Sa), the latter has certain
theoretical advantages. In both cases nqs = o when nms = o.
However, in equation (Sa) nqs approaches unity as it should do,when nms increases without limit, while in equation (6a) there is apoint beyond which nqs ceases to increase with nms, and its limit
is actually zero. . ..
Reed and Merrell seem to have considered that, were it not forunderenumeration, the expression (II) (with a= .oo8) should beapplicable to ages zero and one as well as to the remainder of thelife-span [3, p. 6go]. Theoretical considerations would indicate thatin a period of decreasing mortality k = log. c (and therefore a in the
Reed-Merrell formula) should be negative, and rough calculations
based on recent United States Life Tables indicate for age zero avalue of about - ·3 for a instead of .oo8. The fact is, however, thatfor an age-interval of only one year, this adjustment is not of muchconsequence, and for such an interval equation (II) would be closeenough for practical purposes. However, equation (x) or equation(2) would be slightly preferable, as the additional adjustment contained in equation (II) is in the wrong direction.
V. LIFE-TABLE POPULATIONThe calculation of the values of ls and nds = ls- ls;-n from thoseof nqs requires no explanation or comment. However, the computation of the nLs column introduces some questions. Two distinct
methods of obtaining nLs have been suggested. The first [7] is basedon the assumption that nm:.: has the same value in the actual population and in the life-table population and employs the relationn

L : c_- nds •
nms
58

(12)

which means, in practice, using an approximate integration formula,such asThis approximate method, although less direct and in theory lessexact, generally produces superior results in actual practice. This isbecause the values of lz obtained by the abridged process containinaccuracies and irregularities, and it turns out that the effect on thevalue of .,Lz in equation (1,3.) of slight errors in lz is less than theeffect in equation (12) of the corresponding error in .,d~ and .,mz. A
mathematical demonstration of this fact may be of interest.
From equation (13) we can write by the Theorem of Mean Value

where 8 is a number beween o and n. Therefore, if d(.,Lz) denotes
the error in the value of .,Lz, we have

From this it appears that a given percentage of error in the lz values
will tend to produce about the same percentage of error in .,Lz. On
the other hand, equation (12) gives

This indicates that, when equation (12) is used, a given percentage
of error in either .,dx or .,mz tends to produce about the same percentage of error in .,Lz. However, it is obvious that errors of a given
size in the values of t~ would tend to produce errors of the same absolute magnitude (and therefore much greater percentage errors)in .,dz. Likewise, the values of .,mz are likely to contain greater relative errors than the lz values. This explains why in practice the approximate integration formula gives better results.

59

BIBLIOGRAPHY
r. EDITORS of J I A. Note appended to the article "On a Short Methodof Constructing an Abridged Mortality Table, by GEORGE KING, JIA,XLVIII, JOI-J.
2. WoLFENDEN, H. H. Population Statistics and Their Compilation.
("Actuarial Studies," No. J.) New York: Actuarial Society of America,
1925.
3· REED, L. J., and MERRELL, M. "A Short Method for Constructing
an Abridged Life Table," American Journal of Hygiene, Vol. XXX,
No. 2. Reprinted by the U.S. Bureau of the Census as Vital StatisticsSpecial Reports, Vol. IX, No. 54· All page references are to the Census
publication.
4· DUBLIN, L. I., and LOTKA, A. J. Length of Life. New York: Ronald Press Co., 1936.
5· SPURGEON, E. F. Life Contingencies. London: Macmillan & Co.,
Ltd., I9J8.
6. HENDERSON, R. Mathematical Theory of Graduation. ("Actuarial
Studies," No.4.) New York: Actuarial Society of America, 1938.
7· DoERING, C. R., and FoRBES, A. L. "A Skeleton Life Table," Proceedings of the National Academy of Sciences, XXIV, 400-405.
8. GLOVER, J. W. United States Life Tables, r8go, rgoo, rgro, andrgor-rgw. Washington: Government Printing Office, 1921.

60

9. Life Tables for Natural Populations of Animals
EDWARDS. DEEVEY, Jr. (1947)

From Quarterly Review of' Biology 22. Excerpts are from pages 283-284, 287-292, 294, 296-302,
312.

In selecting from Deevey's extensive review, we have tried to emphasize the different types of problems that arise in working with animal populations; ourmost serious omission is a detailed study of barnacles that examines crowding
effects and mortality. Deevey begins his article with discussions of the life tableand of different general survival patterns, which we also omit.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_9, © Springer-Verlag Berlin Heidelberg 201361

Having gained some idea of the limits circumscribing his own mortality, man
has turned to look at the other animals. In 1935 Pearl and Miner, in their discussion of the comparative mortality of lower organisms, attempted to formulate
a general theory of mortality. They quickly gave up the attempt upon realizingthat the environmental determinants of life duration can not, at least as yet, bedisentangled from such biological determinants as genetic constitution and rateof living. They ended with a plea for "more observational data, carefully andcritically collected fof different species of animals and plants, that will follow
through the life history from birth to death of each individual in a cohort ofstatistically respectable magnitude." Thus by implication Pearl and Miner appealed to the ecologists, who for the most part have been busy elsewhere. Accounts
ofthe conceptions and methodology of life tables have not yet found their way into
textbooks of ecology, and while field naturalists have devoted increasing attention
to the dynamics of natural populations most of them have been content to leave
the construction of life tables to the statisticians and laboratory ecologists.
This article, which is designed as an introduction to the subject rather thanas a formal review, brings together from the ecological literature a mass of information bearing on the survival of animals in nature. This information has notheretofore been considered relevant by biometricians working with human populations, nor has it ever been considered in its context by ecologists. In collecting
the material it was immediately obvious that it is still too early to formulate
general theories. Serious deficiencies are only too apparent in the data. But thedifficulties differ from case to case, and are therefore not insurmountable. Moreover, the bibliography will show that virtually all of this knowledge has beenacquired in the twelve years since the appearance of the review by Pearl andMiner. By taking stock now, and by calling attention to gaps in our information,
it is hoped that some guidance can be given to ecologists and others in the gathering of new material ....

Ecological Life Tables
The field ecologist deals with populations which are by no means so elementary
as those inside Drosophila bottles. Even the total size of the population of a species
cannot be easily ascertained for an area large enough to be representative, andcalculations of the birth rate and death rate are uncertain at best, largely owing
to immigration and emigration. It is seldom indeed that the ecologist knows
anything of the age structure of a natural population. In a few cases, growthrings on the scales or otoliths (fish) or horns (ungulates) make it possible todetermine the age of an animal. Moore (1935) has shown that annual growthrings occur in the genital plates of sea-urchin tests, as they do in the shells ofsome molluscs. Moore checked the validity of the age determination by reference
to the size-frequency distribution in his catches, and the separation of modal sizeclasses in a population often affords a clue to age, particularly for younger age
groups. The age of adult females can be determined in the case of certain mammals(whales, Wheeler, 1934, Laurie, 1937; seals, Bertram, 1940) by counting the corpora
62

Jutea in the ovaries. But for most animals it is possible to find out the ages ofindividuals only by marking them in some way.
Even when the age of a member of a natural population is known, it is nota simple matter to obtain accurate vital statistics. The source of greatest confusion lies in the impracticability of keeping the individuals under continuous
observation. Migratory birds, for example, are easy to band as nestlings, butnearly impossible to find between fledging and the time they leave for winterquarters. Often they can not be found at all unless they return to the same area
to breed, when they can be trapped in nest boxes. Their mortality between fledging
and breeding can be calculated, but the calculation is rendered uncertain by thetendency of young birds not to return to their birthplaces as breeding adults.
As sources of data for the construction of life tables, the ecological information
falls into three groups: (1) cases where the age at death (dx) is directly observed
for a large and reasonably random sample of the population; (2) cases where
the survival Uxl of a large cohort (born more or Jess simultaneously) is followed
at fairly close intervals throughout its existence; (3) cases where the age structure
is obtained from a sample, assumed to be a random sample of the population,and d, is inferred from the shrinkage between successive age classes. It should
be noticed that only the second sort of information is statistically respectable,
since in so far as the breeding can safely be assumed to be simultaneous, it iscomparable to that obtained from a Drosophila bottle. The first and third types
can be used only if one is prepared to assume that the population is stable in time,so that the actual age distribution and the life table age distribution are identical.
This assumption would certainly not be true of a human population; it may
be approximately true for many natural populations of animals ....

Age at Death Directly Observed
In the course of his careful investigation of the wolves of Mt. McKinley, Murie(1944) picked up the skulls of 608 Dall mountain sheep (Ovis d. dalli) which had
died at some time previous to his visit, and an additional 221 skulls of sheep
deceased during the four years he spent in the Park. The age of these sheep atdeath was determinable from the annual rings on the horns. 'Time, which antiquates antiquities, and hath an art to make dust of all things, hath yet sparedthese minor monuments" (Sir Thomas Browne, Urn Burial). Most of the deaths
presumably occurred directly as a result of predation by wolves. Many skulls
showed evidence of a necrotic bone disease, but it is not possible to say whether
death was due solely to the disease or whether the disease merely ensured deathby predation.
The mean longevity of the later sample is significantly greater (7.83 years)
that that of the earlier (7.09 years), but the interpretation of this fact is not clear.
The form of the distribution of deaths is sensibly the same in the two samples.
As the survival of the members of this population is astonishingly great. it seems
best to be conservative, and attention has been focussed on the larger, earlier
sample. Except for the "lamb" and "yearling" classes, which are doubtless underrepresented in the data owing to the perishability of their skulls, there is no63

reason to suppose that either group is anything but a fair sample of the total
population, i.e., the probability of finding a skull is not likely to be affected by
the age of its owner. A life table for the 608 sheep has accordingly been prepared(Table 1). The survivorship curve, plotted logarithmically in Fig. 2, is remarkably
Table 1. Life table for the Dall Mountain Sheep (Ovis d. dalli) based on the known age at death of608
sheep dying before 1937 (both sexes combined)* Mean length of life 7.09 years. Data from Muric (1944)X

Age(years)

x'

dx

1,

1000 qx

Age as%Deviationfrom Mean
Length of Life

NumberDying in Age
Interval out of1000 Born

NumberSurviving at
Beginning of
Age Interval
out of1000 Born

Mortality Rate Expectation of
per Thousand Life, or Mean
Life-Time
Alive atRemaining to
Beginning ofThose Attaining
Age Interval
Age Interval

e,(years)

0-0.5
0.5-1
1-2
2--3
3-4
4-5
5-6
6-7
7-8
8-9
9-10
10-11
11-12
12-13
13-14

100
-93.0
-85.9
-71.8
-57.7
-43.5
-29.5
-15.4
- 1.1
+13.0
+27.0
+41.0
+55.0
+69.0
+84.0

54
145
12
13
12
30
46
48
69
132
187
156
90
3

3

1000
946
801
789
776
764
734
688
640
571
439
252
96
6
3

54.0
153.0
15.0
16.5
15.5
39.3
62.6
69.9
108.0
231.0
426.0
619.0
937.0
500.0
1000

7.06
7.7
6.8
5.9
5.0
4.2
3.4
2.6
1.9
1.3
0.9
0.6
1.2
0.7

* A small number of skulls without horns, but judged by their osteology to belong to sheep nine
years old or older. have been apportioned pro rata among the older age classes.
"human" in showing two periods of relatively heavy mortality, very early andvery late, with high and nearly constant survival ratios at intermediate ages.
The adult sheep have two principal methods of defense against wolves, theirchief enemies: flight to higher elevations, where wolves can not pursue; andgroup action or herding. It is clear that these recourses confer a relative immunity
to death by predation and that only the very young, which have not learned byexperience, and the very old, which are too feeble to escape, suffer heavy losses ....
The second case to be discussed is that of an aquatic invertebrate, the sessile
rotifer Floscularia conijera. This species has been 1>tudied by Edmondson (1945)
under conditions which are fully as natural as those enjoyed by Murie's mountainsheep. Floscularia lives attached to water plants, especially Utricu/aria, surrounded by a tube constructed by itself out of pellets of detritus. The tube isadded to at the top continuously throughout life, and Edmondson was able toidentify all the members of a population living in a pond by dusting the U tricularia
plant with a suspension of powdered carmine. On subsequent visits the Floscu/aria

64present at the time of dusting were conspicuously marked by bands of carminestained pellets in the walls of their tubes, each band being surmounted by new
construction of varying widths. Thus in one operation the stage was set for ananalysis of growth, age, birth-plus-immigration, and death in a natural population.
Among other spectacular results, Edmondson found that the expectation of lifeof solitary individuals was only half as great as that of members of colonies of
two or more, and he presented separate life tables for each component of the
population, calculated from the age at death. To facilitate comparison with otherspecies, however, solitary and colonial individuals have been lumped together(for Edmondson's "Experiment f') in the life table of Table 2.
Table 2. Life table for the sessile rotifer Floscularia conifera based on the known age at death of 50
rotifers, both solitary and colonial. Mean length of life 4. 74 days. From Edmondson (1945), Experiment 1
X

x'

dx

/x

1000 q.,ex

Age(days)

Age as%Deviationfrom Mean
Length of Life

NumberDying in Age
Interval outof 1000
AttachingNumberSurviving at
Beginning of
Age Interval
out of 1000
AttachingMortality
Rate perThousandAlive at
Beginning of
Age Interval

Expectationof Life. or MeanLife Time
Remaining toThose Attaining
Age Interval(days)

0-1
1-2
2-3
3--4
4-5
5-6
6-7
7-8
8-9
9-10
10-11

-100
-78.9
-57.8
-36.7
-15.6
+ 5.4
+26.7
+47.7
+68.8
+90.0
+111.0

20
200
60
0
300
140
60
140
40
20
20

1000
980
780
720
720
420
280
220
80
40
20

20
204

4.76
3.78
3.70
2.98
1.97
2.02
1.79
1.14
1.25
1.00
0.50

77

0
416
333
214
636
500
500
1000

The survivorship curve (Fig. 2). like that of the Dall sheep, shows unexpectedly
good survival. As Edmondson has pointed out, it is not so good as that of otherrotifers reared in the laboratory under standard conditions ( Proales decipiens,
P. sordid a, Lecane inermis), but it is only a little less good ....
The case of Floscularia is almost above reproach as an example of a life tableobtained under natural conditions. It is, of course, open to the objection that
only the age at death is known, and the age structure of the living animals must
be assumed to be constant. Apart from this deficiency, it should also be realizedthat the origin of the life table is not at birth. The pelagic larval life of the rotifer,like the larval life of barnacles and insects, is omitted from consideration insuch a table ....
In his delightful book, The Life of the Robin (1943a) and in two admirable
papers, Lack (1943 b. c) has investigated the age at death of certain British birds,
65

Doll sheepe>-------o Herring Gullo ................ o Floscularia

500--""'(}...-. . . . . . . ----o..._. . __

c

0
.0

q_

"D

c
0

~
0

'D.._ __..............

••••••

50

CJ... ..............

....

..c

''0,

Q...

',

',

··...

',

.....

0

0

\
\\

\

\\

:

\
\
\

\

\\

\

\
\

\\

;

__
+100

1L-----------~-----------&

-100

0

\

\

\
\

\

_ l l_ _ _ _ _ _ _ _L __ _ _ _ _ _ _ _ _ __Lil---~

+2dO

+300

Percent devJOtion from mean length of Ii fe
Fig. 2. Survivorship (/J Curves for the Dall Mountain Sheep, the Sessile Rotifer Floscularia conifera,
and the Herring Gull, Age being Expressed as Percentage Deviation from the Mean Length of Life

1000~---------------------------------------------,. ·~.

\ \ ··:"'!\,·...,. ·..,

500\~\., "~

·.,

~

::J
'0

0

'0

c
d

l/1

·'tl.,_

..,
\

50

·,.

\\

·.. \
\ .....
\___
.....

\

0..

0
>

\

\

r,_

..........

',

·~

[) ',

\

(;;

"'-.6

.,

v..

-E

.,.

·<l.\

\

::J
0

l/1

''A..

·...._ ...~,
v, ... ,·. \3;

100• Blackbird
o Song Thrush
7 British Robino American Robin" Lapwing+ Starlingll....,\·. u..
....

....

·.

'

...

b..

2:

::J

\

',

\

\

····o

...

.......

'b.,.

·,.

\...

\

'

...

\

''A..

·~

+' ',

+\
\·.

Ul

, .)'.

+ ',\'o\... .. ....;.,:\'-··-··"':·-··

10......

~

\

\

0.
\\

I.

\I

o \'

\ \

i

\\

\.

'

\

\

\

1 L __ _ _ _~_ _ _ __ L_ _ _ _~_ __rr_ _~~L__L_ _ _ _-J~--~

1

3

5

7

9

Age in years

11

13

Fig. 3. Survivorship (/..) Curves for the British Robin, Song Thrush. Blackbird, Starling, Lapwing
and American Robin, Age being Expressed in Years66

as obtained by recoveries of individuals banded as nestlings. Because banded
nestlings are likely to be picked up near the banding stations or not at all, it is
impossible to estimate the whole of the mortality in the first year of life withany accuracy. and Lack begins his life tables on August 1. The proportion of birds
banded which are later recovered is small, ranging from 1.0 per cent for the robin
to 18.4 per cent for the cormorant; but after August 1 of the first year it is considered that the ages at death of birds recovered are not likely to differ from theages at death among the total population. The samples are small and of course
become progressively smaller with increasing age, so that Lack does not regard
the mortality rates and expectation of life as reliable beyond the fourth or fifth
year.
Several of Lack's life tables are reproduced in Fig. 3 ....
The striking feature of these survivorship curves is their diagonal form. The
mortality in the first year varies from 380 per thousand for the lapwing to 723 perthousand for the robin, but for a given species the mortality remains approximately
constant throughout life, or at least for as long as the data are reliable ....

Survivorship Directly Observed
The cases now to be discussed differ from the preceding in the character of theoriginal observations. Instead of a fairly large sample of individuals about whichlittle or nothing can be told except their age at death, we have a group of individuals known to have been born at a particular time and to have been present
or absent at some later time. Their presence gives their survivorship, their absence
implies death in the interval since they were last observed. This is the best sortof information to have, since it does not require the assumption that the age
composition of the population is stable in time. Provided only that the seasonof birth is a small fraction of the age interval between successive observations,so that births can be assumed to be simultaneous, as in a Drosophila bottle, ahorizontal life table can be directly constructed from the survivorship data. Unfortunately, most of the species which have been studied in this way have shortspans of natural life, and when census data are obtained only once a year thenumber of points on the survivorship curve is too small to be satisfactory.
The best example of such observed survivorship comes from Hatton's work
(1938) with the barnacle, Balanus balanoides . ... The case is very nearly ideal.
The barnacle settles on rocks during a short time (two to six weeks) in early
spring. Test areas were scraped clean one winter, and after new populations had
settled, the survival of their members was followed at intervals of one to fourmonths for three years. Barnacles which disappeared from the areas between
observations were certainly dead, for emigration does not complicate the problem.
Immigration, however, does present difficulties, though since it is confined to theattachment seasons of subsequent years it should be possible to control it insubsequent work. There is one further disadvantage in that the life tables necessarily start at metamorphosis, leaving out of account mortality during pelagic
larval stages. A life table for a typical population of barnacles is presented inTable 6.
67

Table 6. Life table for a typical population of Balanus balanoides, based on the observed survival of
adult barnacles settling on a cleaned rock surface in the spring of 1930. The population is that at
Cite, (St. Malo, France), a moderately sheltered location, at Level III, at half-tide level. The initial
settling density (2200 per 100 cm 2 ) is taken as the maximum density attained on May 15. Mean lengthof life 12.1 months. Data from Hatton (1938)
d,

IX

X

X

*Age

NumberNumber*Age as%Deviation frofn Dying in Age Surviving to
Beginning of
Mean Length Interval of
1OOOAttaching Age Intervalof Lifeout of 1000
Attaching(months)

0-2
2-4
4-6
6-8
8-10
10-12
12-14
14-16
16-18
18-20
20-22
22-24
24-26
26-28
28-30

-100
-83.5
-67.0
-50.4
-33.9
-17.4
- 0.9
+16.0
+32.2
+49.0
+65.4
+82.0
+98.8
+115.0+132.0

90
100
50
60
80
160
80
100
50
40
100
60
20
8
2

1000
910
810
760
700
620
460
380
280
230
190
90
30
10
2

1000 qxex

Mortality Rate Expectation
per Thousand of Further Life(months)
Alive at
Beginning of
Age Interval

90
110
62
79
114
258
174
263
179
174
526
667
667
800
1000

12.1
11.3
10.5
9.1
7.8
6.7
6.7
5.9
5.7
4.7
2.4
1.9
1.8
1.4
1.0

* Survivorship data given graphically by Hatton were smoothed by eye, and values at every othermonth were then read from the curve. The original observations were made at irregular intervals
during three years.

The remaining examples suffer from more serious defects, and the data do
not justify extended treatment. Green and Evans (1940) in their important studyof the snowshoe rabbit (Lepus americanus) in Minnesota, followed the survivalof marked individuals of several year classes, the total population present on thearea and the number in each age-class being obtained by the mark-and-recapture
method-also known as the "Lincoln index" (Jackson, 1939). Marking was done
during most of the winter, and the annual census was made in February. It isperhaps unnecessarily, and certainly uncharitable, to point out two sources oferror in this excellent and ingenious work. In the first place, when marked individuals are released into a population and later recaptured, the calculation of
the total population from the fraction
when recapturingof sample
size_:__ _____:: _ x number marked__:__________
number recaptured
depends on two assumptions, neither of which is likely to be true in this case:that there is no mortality between marking and recapturing; and that the marked

68individuals disperse at random through the whole population. Secondly, the flowof vital events in this population was so rapid, very few rabbits more than three
years old ever having been found, that observations made annually can give only
a very rough idea of the life table.
The latter objection applies with equal force to the study of a pheasant population made by Leopold eta!. (1943) in Wisconsin. The former objection, though
doubtless it could be urged, has less validity here, since the population, as ascertained by trapping, was checked by census drives.
Nice's thoroughgoing work (1937) on the song sparrow (Melospiza melodia)
included a consideration of the survival of banded birds from year to year. The
number of individuals which could be kept under continuous observation wasnecessarily small, and to find a sample large enough to use as the basis of a life
table, it is necessary to take the 144 males banded in the breeding season between
1928 and 1935. Unfortunately, some of these males were of unknown age when
first banded. Even if one assumes, (and the assumption is not far from the truth)that all new males appearing are first-year birds born elsewhere, the survival ratiosfrom year to year will be too low if any adult males were still alive but failed toreturn to the area. Evidently such emigration is of minor importance with adultmale song sparrows. With adult females, however, it is so serious that Nice did
not think it worth while to publish the data on their return. Clearly, work on
the survival of migratory birds is full of uncertainties, though the same may be

1000.---------------------------------------------------~

500• Snowshoe Rabbit (from birth)o Pheasant ( from matching )o Song Sparrow (from fledging)

E

D.

0
..0

Song Sparrow ( from egg )

= 100

\"-

Ll

c

:;i 50:;J

0.L~~~'· ............. _·A··......., ..... ................

',....., '"'-.. ........
..................
',,

',o,,

~

0

~ 10
:;J

'q\·····.....

.........

...

~'·,·...........

'',,,

',

(/)

5

...........

..... ~ ............. .

',,'o..,·q....·········...

...........,.

.................. .
....

CJ, ••••• •. ··D·•···· ... {]~~....\,;.-.

.......

""t'.,,

1~~---L~~--L-~~-L--~L-~~---Lllil~~L_~~-L--~-....~

+BOO

-100

+1400

Percent deviation from mean length of life

Fig. 5. Synthetic Survivorship (f_J Curves for the Snowshoe Rabbit. the Pheasant. and the Song
Sparrow. the Latter Calculated from Two Different Biological Ages. Data for different parts of thelife spans are derived in different ways, as explained in the text.

69

said of resident species such as the wren-tit (Erickson, 1938) and the robin (Lack,
1943a).
All of these cases, snowshoe rabbit, pheasant, and song sparrow have one
defect in common. This is the necessity of calculating the survival between birthand the first year of adult life from other data than those given by banding. For
the snowshoe rabbit, the initial strength of the year-class is calculated from theestimated breeding population present and its known fertility. Leopold et al.,lacking observations of their own on the pheasant mortality between birth andthe first census period, used the estimates given by Errington for pheasants inanother state. Nice calculates the survival of fledged young song sparrows to theirfirst breeding season, by assuming a stable population and combining the estimatedmean length of life of adults with their average nesting success. These procedures,
while perfectly defensible as approaches to the problem, are inadequate substitutes for direct observation.
The three sets of data, with all their uncertainties, have been used as bases forsynthetic life tables, and the survivorship curves are presented in Fig. 5 ....

Age Structure Directly Observed
Ecological information of a third sort is available for a number of natural populations, principally of fishes and birds. In these cases the investigator has beenable to determine how many individuals of each age are living in the population,and the age at death, though calculable from the shrinkage between successive
age classes, has not been directly observed. This kind of information lends itself
just as well as either of the others to the computation of life tables. As in thegroup where only the age at death is known, of course, it is necessary to assume
the age composition to be unchanged with time. When this assumption is unreasonable, as it often is for fish populations, with their outrageous fluctuationin strength of year-classes, average age compositions obtained from several years'
work can often be used. As it happens, however, all the life tables which fall inthis third group are incomplete for one reason or another, and the data do notbear close comparison with such examples of natural life tables as those of theDall sheep and the barnacle.
Kortlandt (1942) has recently given a very elaborate analysis of the Netherlands population of cormorants ( Phalacrocorax carbo sinensis). Birds banded
as nestlings were later observed in their breeding and playing colonies, the numbers on the bands being read with the aid of a telescope. The age distribution of
the banded birds being known in 1940 and 1941, it should be possible to infer
the age distribution of the total population and from this to compute the annual
mortality suffered by each year-class. A number of complicating conditions arepresent in this case, however, making direct calculation unreliable and necessitating a more circuitous approach: (1) the size of the Dutch cormorant population
is not constant, but has been increasing by about 10 per cent per year, as estimatedby counts of nests at the breeding colonies; (2) differences between the observed
sex ratio among sexually mature birds and the sex ratio predicted on the basisof estimated mortality by sex and age class show clearly that there is some band70

mortality; that is, some birds either lose bands or die because of the band, making
estimates of natural mortality too high by a factor of about 2; (3) it is not possible
to infer the complete age structure from observations made at breeding colonies,
since the one- and two-year-old birds occupy "colonies" elsewhere, returning totheir birthplaces to breed no sooner than their third year.
In view of these difficulties, and others which need not be discussed here,
Kortlandt's results must be regarded as schematic and preliminary only, andscarcely warrant the construction of a life table. His computations suggest that
cormorants suffer mortality somewhat as follows: 17 per cent between fledging
and the first May 1; 8 per cent in the first year; 6 per cent in the second year;
and about 4 per cent per year in the third to twelfth years. These are astonishingly
low figures for a natural population, but it must be remembered that the population is increasing rapidly ....
The literature of fisheries biology is full of attempts to estimate the mortalityof fishes, to distinguish natural mortality from rate of exploitation, and to determine the rate of exploitation which, given certain mortality rates and certain
relationships between age and size, will steadily yield an optimum catch. These
complex questions are fully discussed in the important works of Russell (1942),
Thompson and Bell (1934), and Ricker (1940, 1944), and by various authors in
the Rapports et Proces- Verbaux of the International Council for che Study qj' theSea, Volume 110, 1939. Little of this information can be directly used for our
purpose. The explanation is as simple as it is regrettable: although the age of acaught fish can be ascertained with more or less complete confidence, fishes of
all ages can not be caught with equal facility. Inevitably the methods so skilfully
developed for catching fish of desirable sizes will fail to catch fish of undesirabl~
sizes. It is true that on various occasions the whole fish population of a lake has
been removed by poisoning or drainage. The estimates given by Eschmeyer (1939)
for the abundance of large-mouth black bass (Huro salmoides) in Howe Lake,
Michigan, at the time of its poisoning in 1937, may be cited as a example:
Age00 (cannibals)
I
II
III
IV
V and older
Total

Number

18,374
229
25
10
105
7
9
18,759The implication of enormously greater mortality in the first year of life is plainfrom these figures, but such data can not be taken as they stand, partly becauseof very variable annual recruitment, and partly because young of the year were
removed from the lake at various times for hatchery purposes ....
This section may logically be concluded with a brief reference to the data for
the fin whale, in which, as recent investigations have shown, the age of the female
can be determined from the number of old corpora lutea in the ovary. By this71

method Wheeler (1934) arrived at the following as the age structure, observed over
five seasons, 1926-1931, of the catch of female fin whales in the Antarctic:Age
3-4 years
5-6 years
7-8 years
9-10years
11-12 years
13-14 years
15-16 years
17-18 years
19-20 years
21-22 years

Number caught
130
95
72

53
37
28
10
4
1
1

The data imply (subject to the usual qualifications) a biennial mortality of about
26 per cent, increasing beyond the 15th year to much higher values. The author
considers that the increased rate of loss with age is not real, but is due to failureof the older whales to return from their winter quarters in the north. This belief
may or may not be well founded, but one suspects it to be predicated on theidea that mortality. at least when it is primarily due to exploitation, is constantamong animals with respect to age. Edser (to whom the statistical analysis iscredited) assumed, for the purpose of a rough calculation of the necessary rateof replacement, that the mortality between birth and breeding age is also 26 pe.r
cent. The improbability of this assumption may be surmised by reference to thelife table for the Dall sheep (T able 1). Edser' s calculation has the great merit ofyielding a minimal estimate of the alarming exploitation being conducted by thewhaling industry in the Antarctic. More realistic assumptions would darken the
picture even more. In any case the data can not yet be cast into a life table ....

Comparisons and Conclusions
Both in nature and in the laboratory, animals differ characteristically in theirorder of dying. When the mortality rate at all ages is constant, the survivorship(/) curve is diagonal on semi-logarithmic graph paper. Such a curve is found formany birds from adults stages onward: the mortality of adult birds is about
320 per thousand per hundred centiles of mean life span. If the constant agespecific mortality rate observed for the first few years of adult life is really maintained throughout life, the oldest bird in a cohort of 1000 lives 6.6 times as longas the average bird. Not all animals resemble birds in this respect, however,
although many (e. g., fish) are assumed to do so. The Dall sheep, the rotifer, andpossibly the barnacle are more like civilized man in that they seem to have evolved
a mechanism for stretching the mean life span toward the maximum, so that thesurvivorship curve is convex. In these cases the maximum life span (among asample of 1000) is only two or three times the mean. On the other hand there are72

undoubtedly species in which juvenile mortality is very heavy, but the few survivors to advanced ages die at reduced rates. This J-shaped or concave survivorship line, with the maximum longevity perhaps 15 or more times the mean, is
presumed to characterize the oyster and other species, but it has not yet beenformally recognized either in the laboratory or in nature. The closest approach
to it, so far, is found when the survival of song sparrows is reckoned from theegg stage; but the mackerel will almost certainly provide an even better example.
Detailed comparisons between species cannot yet be made, partly because of
the diverse statistical foundations of the life tables and partly because the data begin
at different biological ages (birth, hatching, metamorphosis, sexual maturity, etc.)
in the different cases. In all cases it is the youngest ages about which we know
least, and ecologists should therefore concentrate their efforts on this segmentof the life span of animals in nature.

List of Literature
Bertram. G. C. L. 1940. The biology of the Weddell and Crabeater Seals, with a study of the comparative
behaviour of the Pinnipedia. Brit. Mus. (Nat. Hist.): British Graham Land Expedition 1934-37.
Sci. Rep .• 1 (1): 139 pp., pls.1-10.
Edmondson. W. T. 1945. Ecological studies of sessile Rotatoria, Part II. Dynamics of populations andsocial structures. Ecol. Mon .• 15: 141-172.
Erickson, M. M. 1938. Territory, annual cycle, and numbers in a population of wren-tits (Chamaea.fitsciata). Unir. Ca[U: Puh. Zoo/., 42 (5): 247-334, pis. 9-14.
Errington, P. L. 1945. Some contributions of a fifteen-year local study of the northern bobwhite
to a knowledge of population phenomena. Ecol. Mon., 15:1-34.
Eschmeyer. R. W. 1939. Analysis of the complete fish population from Howe Lake, Crawford County,
Michigan. Papers Mich. Acad. Sci. Arts Lett., 24 (II): 117-137.
Green, R.G., and C.A. Evans. 1940. Studies on a population cycle of snowshoe hares on the Lake
Alexander area. I. Gross annual censuses. 1932-1939. J. Wild/. Man., 4: 220-238. II. Mortality
according to age groups and seasons. Ibid., 4: 267-278. III. Effect of reproduction and mortalityof young hares on the cycle. Ibid., 4: 347-358.
Hatton, H. 1938. Essais de bionomie explicative sur quelques especes intercotidales d'algues etd'animaux. Ann. Inst. Ocf>anogr., 17: 241-348.
Jackson, C. H. N. 1939. The analysis of an animal population. J. Anim. Ecol., 8: 238-246.
Kortlandt, A. 1942. Levensloop, samenstelling en structuur der Nederlandse aalscholver bevolking.
Ardea, 31: 175-280.
Lack. D. 1943a. The life of the rohin. 200 pp. H. F. & G. Witherby, London.
Lack, D. 1943b. The age of the blackbird. Brit. Birds, 36: 166-175.
Lack, D. 1943c. The age of some more British birds. Brit. Birds, 36: 193-197, 214-221.
Laurie, A. H. 1937. The age of female blue whales and the effect of whaling on the stock. Discor. Ret'·· 15:
223-284.
Leopold, A., T. M. Sperry. W. S. Feeney, and J. S. Catenhusen. 1943. Population turnover on a Wisconsin
pheasant refuge. J. Wi/c/1. Man., 7: 383-394.
Moore, H. B. 1935. A comparison of the biology of Echinus esculentus in different habitats. Part II.
J. Mar. Bioi. Ass., n.s., 20:109-128.
Murie, A. 1944. The wolt·es of' Mount McKinley. (Fauna of the National Parks of the U.S., Fauna
Series No.5, xx+238 pp.) U.S. Dept. Int., Nat. Park Service, Washington.
Nice. M. M. 1937. Studies on the life history of the song sparrow. Vol. I. A population study of thesong sparrow. Ti·an~. Linn. Soc. N.Y.. 4: vi+ 247 pp.
Pearl, R. and J.R. Miner. 1935. Experimental studies on the duration of life. XIV. The comparative
mortality of certain lower organisms. Quart. Rev. Bioi., 10: 60-79.

73

Ricker, W. E. 1940. Relation of "catch per unit effort' to abundance and rate of exploitation. J. Fish.
Res. Bel. Ccmac/a, 5:43-70.
Ricker, W. E. 1944. Further notes on fishing mortality and effort. Copeia, 1944: 23-44.
Russell, E. S. 1942. Tile ow:fisl1iny problem. viii + 130 pp. Cambridge Univ. Press, Cambridge.
Thompson, W. F., and F. H. Bell. 1934. Biological statistics of the Pacific halibut fishery. (2) Effectof changes in intensity upon total yield and yield per unit of gear. Rep. Intern. Fish. C omm., 8: 49 pp.
Wheeler, J. F.G. 1934. On the stock of whales at South Georgia. Discor. Rep., 9: 351-372.

74

Part II

Stable Population Theory

In 1760 Leonard Euler introduced the concept of a stable age structure in which
proportions in all age categories would remain fixed if mortality were constantand births increased exponentially over time. In his analysis, Euler also included
a series of problems to demonstrate the usefulness of stable population theory
for filling in gaps in population information. His paper was familiar to Milne(1815: his comments on Euler are included in paper 4 above), who recognized its
theoretical interest but emphasized its restricted applicability. Milne’s comments
were appropriate for his time: the very limited empirical data that existed for Europe
did not suggest stability, and did not indicate how abusive of the facts the assumption
might be. [Some years later Milne applied Euler’s theory in a communication
to Robert Malthus (1970 (1830), pp. 228—229) demonstrating that three stable
populations, one with a 25 year doubling time, one with a 50 year doubling time,and one stationary, would have substantially different crude death rates if theyshared a common mortality schedule. Malthus used the analysis to show that crude
death rates could not adequately measure the health of a population.] Euler’s article
and an earlier note to Johann Peter Süssmilch. who may have brought population
mathematics to his attention, are presented here as papers 10 and 11.
The link between stable theory and real populations was very largely uncoveredby Alfred Lotka (1907, 1922) and F. R. Sharpe and Lotka (1911), in works thatform a singular achievement in demography. The three papers included here will
indicate how Lotka’s insight expanded from his realization that population could
be represented as a renewal process displaying some stability, to his discovery
that populations would nearly always stabilize, by predictable paths, if fertility andmortality were held constant.
The papers introduce the renewal equation
Zt
B.t  x/p.x/m.x/ dx

B.t/ D G.t/ C
0II Stable Population Theory

where B.t/ are births at time t, and are composed of births to the population alive attime zero [ G.t/ ] and births to those born since; B.t  x/ being persons born t  xyears ago and, subject to their survival probability p.x/, currently age x; m.x/ being
their chance of giving birth in the interval x to x C dx. For G.r/ D 0, substitutionof an exponential birth function B.t/ D B0 e rt provides the characteristic equation
Z1
.r/ D

e rx p.x/ m.x/ dx; D 1

0whose single real root is the intrinsic growth rate of the population and whose otherroots define the rate of convergence toward the stable age structure. [An examinationof high order roots is provided in Coale (1972).]
The conjecture that populations stabilize was attributed by R. R. Kuczynski (1931b, 1935) to L. von Bortkiewicz (1911), but this seems not to be the meaning of therelevant text. Bortkiewicz essentially followed Euler in examining relationships under stability, with the distinctions that he employed density functions corresponding
to rectangular sections of the Lexis diagram, and provided applications to resolve
current disputes in the literature. Except for a brief note on Euler, the approach tostability is not discussed. Lotka’s debate with Kuczynski. which touches on otherearly contributions to stable theory, is the subject of paper 15 by Paul A. Samuelson.
Lotka’s arguments were also subject to mathematical criticism and did not gain
universal acceptance until a rigorous mathematical proof that in the main they held
for both discrete and continuous cases was given by William Feller (1941), whose
article is included here as paper 16. f Ed. Note: The mathematical object that playsa central role is known as the “renewal function”. Building on the paper by Feller
reprinted here, general conditions for convergence of the renewal function and sofor the convergence of population age distributions to stable age distributions were
established in Blackwell’s Renewal Theorem proved in [5] along with work byFeller, Erdös, and Pollard described in [12]. g
Widespread application of the theory has come more recently, with the introduction of projection matrices and of modern computing equipment to facilitate theextraction of characteristic roots and vectors.
A more direct and considerably simpler method for calculating the intrinsic rateof growth than that developed by Lotka has been given by Ansley Coale (1957b) andis included here as paper 17. It is followed by a selection from R. A. Fisher (1958(1930)) in which he develops the concept of Reproductive Value, complementing
stable theory. It is found in Lotka [1939, equation (214)] as the term P , and inFeller [1941, equation (3.2)] as .s/, but without the intuitive meaning Fisher givesit.
In paper 19 we present an application of stable population theory by Coale(1957a). In the work he discusses the effects that changes in birth and death rates
have on age distributions of real populations, among other points correcting thewidespread misconception that a lowering of death rates necessarily results in an

76

II Stable Population Theory

older population. The article also introduces the concept of weak ergodicity, an
important contribution to stable theory that will be taken up in Part III in conjunction
with projection matrices.
We conclude the chapter with two articles that attack a problem commonly
overlooked in stable theory: the treatment of reproduction as a discontinuous
process in work with very small populations. W.R. Thompson’s (1931) pioneering
analysis is included here as paper 20. To take into account chance fluctuations, astochastic approach becomes more relevant to Thompson’s small populations thanthe deterministic solution he gives, and readers should be aware that this limitation
exists. It does not detract from Thompson’s contribution: as he indicates, both thenature of the problem and of the solution change radically when numbers are small.
(We present in the introduction to Thompson’s paper a note on its matrix formulation
which will relate it to the discussion of matrices in Part III. f Ed. Note: In the 1977
edition, the authors went on to say “The stochastic treatment of birth processes istaken up in chapters 6 and 7”; these parts are omitted in this edition. g The closing
article by Lamont Cole (1954) clarifies Thompson and ties his work to stable theory.
It includes as well a fascinating note on Fibonacci numbers and their relation to theThompson problem.
f Ed. Note: From the 1970s onward, stable population theory expanded into
a subject with a rich variety of applications, many developed in the first, 1977
edition of Applied Mathematical Demography by Nathan Keyfitz, republished with
substantial additions in a third edition by Keyfitz and Caswell [19]. Among theprominent applications are measures of population momentum under the “Keyfitz
Scenario” in which fertility rates within a previously stable population suddenly
drop to replacement levels at all ages by the same fixed factor with no change
in mortality rates (see [19], pages 196–199). A decomposition of momentum into
stable and non-stable contributions is given by Espenshade, Oligati, and Levin [11].
Biological applications extend ideas from stable population theory to populations
structured both by age and life-cycle stage, as in [9].
Stable population theory applies to settings in which demographic rates do
not depend on the size or age structure of the population. When fertility andmortality rates do respond to population through processes of demographic feedback, sustained cycles in births and age distributions over time may take theplace of a stable state. Inspired by ideas of Richard Easterlin about mechanisms
contributing to the baby booms that followed World War II, inherently non-linear
demographic feedback models were developed by Lee [24] and Wachter [41] andothers. Analysis of complex systems of feedbacks between population, environment,
and economic varibles has gradually become feasible, largely employing computerbased simulations.
Further developments building on Stable Population Theory through partial
differential equations are discussed in Part III. g

77

10. An Illustration of Population Growth
JOHANN PETER SOSSMILCH (1761)

From Die gottliche Ordnung, Vol. 1, pp. 291-299. Berlin. Translated by Nathan Keyfitz.

The extract which follows assembles several fragments of Euler's work in stable
theory that complement, and apparently precede, his better known 1760 article(paper 11 below). Euler here prepares for Siissmilch a population projection thatbegins with one couple of age 20 who give birth to an additional couple at ages
22, 24 and 26, and who die at age 40, later generations repeating the same fertility
and mortality schedules. Euler demonstrates that after 300 years the population
will be growing approximately geometrically and recognizes this as the limit ofa recurrent series; i.e., he is working with a very general result.
The recurrent series is in this case a net maternity function expressed as adifference equation. We have:B(t) = B(t- 22) + B(t- 24) + B(t- 26);of which the solution is

A.= 1.04696
where B(t) represents births at time t, B 0 is a constant and A.=er, the annual
intrinsic growth rate. From earlier work (1748) Euler recognized that the ratio
between successive terms B(t), B(t +a) in series of this form would approach aconstant limit determined by the intrinsic rate of growth as t--+ oo; and both heand Joseph Louis Lagrange (1759) had worked with the conversion of the equations to polynomials and extraction of roots, learned in part from Abraham
DeMoivre's Miscellanea Analytica de Seriebus et Quadraturis (1730, 72-83). A
hint of this insight is contained in the closing paragraph of the present article.
(Euler's note that the birth series can be generated by dividing out an algebraic
fraction is also from DeMoivre. It reappears in Thompson (paper 20 below), in
whose notation births at ages 22, 24, and 26 would be representable by the"Generation Law" G = T 22 + T 24 + T 26 . The complete birth series is produced
by dividing out the fraction 1/(1- G).)
Euler's mathematical contribution to Stissmilch goes no further and is very
reminiscent of Fibonacci's rabbits (discussed in Cole, paper 21 below). It is as
well the point at which Euler's important 1760 paper begins: from the net rnaD.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_10, © Springer-Verlag Berlin Heidelberg 201379

ternity function an intrinsic rate of growth has been found; this rate, together
with the complete mortality table, provides the stable age distribution and the
intrinsic birth and death rates of that paper. E. J. Gumbel (1917) was first to
see the full implication of the two papers.
SUssmilch's opening paragraph refers to an earlier table of population daublings (i.e., powers of 2) with imagined doubling times inserted; calculated, withsome multiplication errors, by himself. It gives a total population after 300 yearsof 2 23 = 8,338,608 from an initial cohort of 2, a figure twice as great as in the
Euler projection and one which corresponds to an annual growth rate of 5.3 %.
The Euler projection runs to several pages and is shown here only in part.

80

To show that nothing impossible is contained in the preceding table I present
another due to Professor Euler and prepared some years ago at my request. It
is too elegant for me to omit ....
Professor Euler assumes (1) at the outset there exists a married couple aged 20,(2) their descendents also always marry at the age of 20, (3) 6 children are born toeach marriage. (This could certainly happen if couples of differing ages wereprecluded from marrying, if like marry like, and if all are able to marry at thecorrect time.) Also (4) variations must never occur; therefore twins will always
be born, the first pair to each marriage coming in the 22nd, the next in the 24thand the third in the 26th year. It will be assumed (5) that all children survive,
marry, and remain living until reaching age 40. (This is the average expectation oflife on the whole: lives were of much longer duration and the average muchhigher before the Flood. If the fertility of the marriages appears too high it canbe revised throughout.)
On these assumptions there will be only two people initially, 4 after 2 years,
6 after 4 years, 8 after 8 years. After this time no changes occur until the first two
Table of Growth According to the Conditions Assumed
Year

Numberof Births

Birthsto DateDeathsto DateNumber
Alive

0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34

0
2
2
2
0
0
0
0
0
0
0
0
2
4
6
4
2
0

2
4
6
8
8
8
8
8
8
8
8
8
10
14
20
24
26
26

0
0
0
0
0
0
0
0
0
0
2
2
2
2
2
2
2
2

2
4
6
8
8
8
8
8
8
8
6
6
8
12
18
22
24
24

280
282
284
286
288
290
292
294
296
298
300

71632
122112
178036260362342310403268426034404348346570273 8842143701679344
1791456
1969492
222985425721642975432
3401466380581441523844426268
464063828048432861.0379908428068467934497 348517 874534572555274589 506
646684

1 398 860
1462846
1589 584
1801 786
210423024780842883 592
32712423597110
3 836 752
3 993954

81

children reach their 22nd year, which takes place after 24 years, when their firsttwo children come into the world. Two years later this couple will produce 2 more,
but the couple born in the 4th year will also produce 2 children; in the 28th year
6 children will arrive; in the 30th again only 4, and so forth. In order to grasp
the nature of this initially irregular growth, the attached table shows in its second
column the births of even numbered years. The third column shows the sum of
all births to date, which would be the same as the number living if no one
were to die.
But at age 40 all die, from which are derived the numbers of column 4. If
these deaths are subtracted from the number born, the result is the number living
each year, found in the 5th column.
One can see from this that after 24 years there is a trebling of the number
alive, from which after 1000 or more years an astonishing increase must ensue.
After 300 years the number living already runs to 4 million; if the trebling timeis set at 25 years there will be 324 million people after 400 years, and after 450
years fully 3000 million, more than now live on the entire earth. The increase
before the Flood, while not so great as this, was surely not an insignificant progression; so the earth must at that time have been more heavily populated
than now ....
Notice that although great unorderliness seems to rule in Euler's table, thenumbers of births belong to a progression called a recurrent series [ einem Geschlecht von ProgreBionen, welche man Series recurrentes nennet] and which
can be produced by dividing out an algebraic fraction. While these progressions
initially appear irregular, if they are continued they finally change into a geometric progression; the initially perceived irregularities decrease with time until
they finally almost entirely disappear.

82

11. A General Investigation into the Mortality andMultiplication of the Human Species
LEONARD EULER (1970 (1760))

Theoretical Population Biology 1: 307-314. Translated by Nathan and Beatrice Keyfitz.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_11, © Springer-Verlag Berlin Heidelberg 2013

83

TRANSLATOR's NoTE

Euler's virtually unknown article, published by the Belgian Academie Royale desSciences et Belles-Lettres in 1760, anticipates important parts of modern stable population
theory for a one-sex population closed to migration. Its ideas have been published manytimes during the subsequent two centuries by writers who independently rediscoveredthem.
The application to which Euler oriented his argument was inference from incomplete
data. Life tables were already in use for working out probabilities and expected values inrelation to individuals; Euler saw them as a means of studying populations, provided theassumption of stability was appropriate. He follows a cohort of individuals born "enmeme terns" (sic) by means of the probability of survival from birth to age x that weknow as l,. ; he supposed that is available from some source other than the population
under consideration, as contemporary users of model life tables do. The equation for the
unknown rate of increase A in paragraph 18, as well as the proposed method of solution,
bear a resemblance to procedures now in use for inferring the rate of increase of .a
population.
If total births and deaths are known (but not by age}, then Euler s\tows how assuming
a life table permits the number of the population to be inferred, as well as the age distribution. Euler turned his stable population theory to filling the gaps in information Of his
own day: Western Europe had baptism and burial records before it had censuses. His
model, used in only a slightly different fashion, would enable a present-day underdeveloped
country with censuses and no vital statistics to infer its birth rate.
Euler's "hypothesis of mortality" means "life table," and is taken as unchanging intime and operating deterministically. The ratio A of population ihcrease per year is thesame as er, if r is Latka's intrinsic rate of natural increase. Though the cohort with which
Euler deals is a group of children born at one moment, and hence projected by the life
table function l,. , the alteration needed to work with cohorts evenly spread over the year
is slight: We replace the probability of survival l., by L.,/L 0 , where L., is the integral
L., =l.,+t dt.
Euler's annual births B occurring at one moment, while his deaths Dare spread throughthe subsequent year, give a simple but unfamiliar expression for the ratio of increase:A = (P- D)/(P- B), where P is the population just after the annual births haveoccurred, and D the deaths in the succeeding twelve months.

z.

J!

The following list of the problems in stable population theory solved by Euler willserve as a table of contents of the important paragraphs of his work.
Paragraph

Given

Find

18
19

P,B,l.,
A, l.,
P,B,l.,
P,B,l.,
B,D,l.,
B,D,l.,
P,B,D.,

A
P/B
P.,
D
A,P
D.,
l.,

20
21
22
24
26

where Pis total population; P., population by age; Dis total deaths;!)., deaths by age;
B is births for the given sex; l., is the probability of surviving to age x; A is the ratio of
annual increase.

84Euler uses "hommes" for the population and "enfants" for the births, which could
be interpreted as meaning that the sexes are combined in his model. As he nowhere deals
with age at childbearing, his argument would apply to the two sexes together, with aconsolidated life table. Essentially a one-sex model seems to be intended, with males as
the illustration, and this is the rendering of the translation that follows.
N.K.

1. The registrations of births and of deaths at each age which are
published in various places every year give rise to many different questions on
the mortality and the multiplication of mankind.
2. All of the general questions depend on two hypotheses. I shall callthe first the hypothesis or law of mortality (i.e., life table), by which can be
determined how many out of a certain number of men born at one time will be
still alive at the end of each period of years. Here population increase does notenter the matter at all, and we go on to the second hypothesis, which I shallcall that of multiplication: by how many the number of men is increased ordiminished in the course of one year. The second hypothesis depends, of course,on the number of marriages and upon fertility, whereas the first is based on thevitality or life potential characteristic of man.

HYPOTHESIS OF MoRTALITY

3. For the first hypothesis, let us suppose a number B of children bornat the same time; the number of these still alive at the end of one year is B11 ,at the end of two years B12 , of three years B13 , •••• These show the decrease of
the number of men born at one time; for each region and way of life they will
have particular values. The numbers indicated by 11 , 12 , ••• , constitute a
decreasing progression of fractio~s of which the greatest, 11 , is less than unity,and beyond 100 they disappear almost completely. For if of 100 million men
not one reaches the age of 125, this means that 1125 must be less than
1/100,000,000.

4. Having established for a certain place, by a sufficiently large numberof observations, the values of the fractions 11 , 12 , ••• , one can resolve several
questions which are commonly posed on the probability of human life. First,
if the number of children born at the same time is equal to B, the expected
number dying each year will be
85

From AgeTo AgeNumber of Deaths

0
1
2

1
2
3

B-Bl1
BlcBl2
BlrBl 3And since of B children born Blx will be expected to be still alive after x years,
the number of deaths before the end of x years must be equal to B - Blx .

5. Given a number of men of the same age, find how many of them will
be expected to be alive after a certain number of years.
Suppose there are P men, all aged x years, and we want to know how many
we can expect to be living after n years. We put P = Blx to obtain B = Pflx,
where B is the number of male births, of whom P are still alive after x years.
Of this number Blx+n will be the expected number still living x + n years
after their birth, and hence n years after the time in question. Then the number
sought is equal to P(lx+nflx); that is, after n years we can expect there to be thatmany survivors of P men who are presently all aged x years.
Then of the P men all aged x years, the expected fraction who will die before
the end of n years is 1 - lx+nflx .

6-14. (These paragraphs derive life table probabilities for individuals,
along with the formula for a life annuity purchasable for the amount of money a.
The annuity payable at the end of each year purchasable for a is shown to be

per year, where i is the rate of interest. If the annuitant is just born, and theannuity is to be deferred to start at age x, then the annual value purchasable forthe amount a isaEuler then gives the lx column by single years of age of a life table due toKeerseboom, and goes on to discuss births.)

86

HYPOTHESIS OF MuLTIPLICATION

15. Just as I assume that the regime of mortality remains ever the same,
I shall assume a like constancy of fertility, so that the number of children borneach year will always be proportional to the total number living. Let B be thenumber of children born in the course of one year, and >JJ the number ofchildren born in the following year. Insofar as the ratio which ch:anges B to >..B
continues to hold from any Year to the next the number of births increases in
the ratio 1 to >... Consequently the births of the third year will be >..2B, of the
fourth >..3B, .... Either the number of annual births will constitute a geometricprogression, increasing or decreasing, or it will remain constant, according as>.. > 1, or >.. < 1, or >.. = 1.
16. Suppose then that in a town or province the number of (boy)children born this year is equal to B, and the number born next year is equal
to >JJ, and so on according to this progression:
Number of Births

B

This year
After 1 year
After 2 years>. 2B

After 99 years
After 100 years).loon

MJ).99 BIf we suppose that after 100 years none of the men alive at this moment will
still be living, all the men in existence after 100 years will be the survivors of
the above births. Then, bringing the hypothesis of mortality to bear, we could
determine the total number of men who will be alive after 100 years. Thus,
since the number born in that year will be >.._100B, we will have the ratio of births
to the total number living.
17. To make this cle~rer, let us see how many men will be alive after
one hundred years from the births of the preceding years:

Number of
BirthsNumber Living
After 100 Years

At present
After 1 year
After 2 years

B
>.B
).2B

Bl1oo
>.Bl99
A2Bl9sAfter 99 years
After 100 years).DDB).9DBh>,.1oo B).lOOB87

Thus the total number living after 100 years will be

AlOOB(1 + i1_A + ~A2 + ... + A1oo •/100 )

18. The terms of this series disappear finally, by virtue of the hypothesisof mortality, and since the total number living has a certain relationship to thenumber of births during the course of one year, the multiplication from one
year to the next, which we have assumed to be A, will reveal this relationship.
For if (at the end of 100 years) the total number living is equal to P, and thenumber of children produced in the course of one year is B, we will havep = B

(t + !1_A + A212 + ... + A1oo
110o )·

Thus, if we know the ratio PfB as well as the hypothesis of mortality, i.e. thevalues of the fractions 11 , 12 , ... , 1100 , this equation will determine the ratio of
multiplication A from any year to the next. For each life table, if we calculate
the quantity 1 11 /A
12/A2
for several values of A, and set up a tableof them, it will be easy to ascertain for the given ratio PfB, which expresses
fertility, the annual increase A in the number living.

+

+

+ ...

19. Having this equation, it is indifferent whether we know the fertility
PfB, or the multiplication A, the one being determined by the other throughthe life table.
20. Given the hypotheses of mortality and fertility and the total number
living, find how many there are at each age.
From P, B, and the life table we calculate the ratio of annual multiplication A.
From ,\ and paragraph 17 we see that there will be among the number P
B,
Bllf,\,
B12f,\2,B1af,\3,children JUSt born;
aged one year;
aged 2 years;
aged 3 years;

and in generalaged a years.
The sum of all these numbers is equal to P.
21. The same things being given, find the number of persons who will
die in one year.
88Call P the number of persons living at present, including the number ofchildren B born this year. The quotient PfB will determine the annual increase.\on a given life table. Then next year the number living will be .\P, amongwhom the number just born will be .\B; the others, of whom the number is.\P - .\B, are the survivors from the P persons of the previous year; from which
it follows by subtraction that (1 - .\) P + .\B of them have died. Thus if thenumber now living is P,.then D = (1 - .\) P + .\B ofthem will die in the courseof one year.
22. Knowing both the number of births and of burials occurring in thecourse of one year, find the total number living, and their annual increase, fora given life table.
The preceding paragraph gave us
D = (1 - .\) P + .\Bor.\B -D
P= .\-1.

From the hypothesis of mortality
p = B(l + .!!_.\ + !!_+ ... + ,\100
l1oo ) •.\2

Substituting P = (.\B- D)/(.\- 1) (and subtracting B from both sides), we
obtain
B-D
B(l1
12
11oo)
.\ - 1 =T + .\2 + ... + .\100

from which .\ may be calculated. (Then from .\, B, and D the total population P
can be found.)
24. Given the number of births and of burials in one year, find thenumber of each age among the deaths.
Call B the number of children born during a year, and D the number ofdeaths; from the preceding problem we have the number living P, along withthe ratio of increase .\. Consider how many persons will be living at each age,
this year as well as next year:

Just born
Aged I year
Aged 2 years
Aged 3 years:

Aged 100 years

This Year

Next Year

B
Bl1/A
Bl1 /A1
Bla/A8BABl1
Bl1 /A
Bla!A1Bl1oo/A100

Bl1oo/A11:

89

Thus the deaths in the course of the year must number:
Number of Deaths
Under one year
From 1 to 2 years
From 2 to 3 years

B(1 - l1)
B(/1 - 12)//..
B(lz - la)/1.. 2

Fmm 99 to 100 years
Over 100 years

B(/99 - l1oo)/l..99
Bl1oo/A 100

26. Knowing the total number living P but not their ages, the numberof births B, the number of deaths D, and the number of ·deaths at each age
during the course of one year, find the law of mortality (i.e., calculate the life
table).
We first find the annual multiplication.\= (P- D)f(P- B). Next, for thisyear the number of deaths, from the preceding problem, must be
Do= B(l -11 )D 1 = B(/1 - 12 )/1..
D2 = B(/2 - la)/1..2

Under one year
From 1 to 2 years
From 2 to 3 years

and hence we will find the fractions 11 , 12 , 13 , ••• , which contain the law ofmortality:

11 = 1- D0 /B,
1
l 2 = l 1 - ill1.IB = 1 - Do +B >.D ,
1a = 12 - ).2D2/B = 1 - Do + ~ + ).2D2 ,
1, = 1a- >.aDafB = 1 - Do+ >.D1 + ).2D2 +>.aDa ,B
28. I should point out once more that in the calculations which I havedeveloped here I have assumed that the total number living in one place remains
the same, or that it increases or decreases uniformly, so that I have had to
exclude such extraordinary devastations as plague, wars, famine, as well as
extraordinary increases like new colonies. It is well to choose a place where all
those borri remain in the region, and where strangers do not come in to live ordie, which would upset the principles upon which I have based the preceding90calculations. For places subject to such irregularities, it would be necessary tokeep exact registrations of all those living and dying, and then, by following
the principles which I have established here, one would be in a position to
apply the same calculation. It always comes back to these two principles, thatof mortality and that of fertility, which, once they have been established for acertain place, make it easy to resolve all the questions which one could propose
on this subject, of which I am satisfied to have given an account of the principal
ones.

91

12. Relation Between Birth Rates and Death
Rates
ALFRED J. LOTKA (1907)
Science, N.S. 26: 21-22.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_12, © Springer-Verlag Berlin Heidelberg 201393

A short notice appeared on page 641 of Science, 1907, of a paper read by
C. E. Woodruff before the American Association for the Advancement of Science,on the relation between birth rates and death rates, etc.
In this connection, it may be of interest to note that a mathematical expression
can be obtained for the relation between the birth rate per head b and the death
rate per head d, for the case where the general conditions in the community are
constant, and the influence of emigration and immigration is negligible.
Comparison with some figures taken from actual observation shows thatthese at times approach very nearly the relation deduced on the assumptions
indicated above.
I give here the development of the formula, and some figures obtained by
calculation by its aid, together with the observed values, for comparison.
Let c(a) be such a coefficient that out of the total number N 1 of individuals
in the community at time t, the number whose age lies between the values a and(a+da) is given by N 1 c(a)da.
Now the N 1 c(a)da individuals whose age at timet lies between the values a and(a+da), are the survivors of the individuals born in time da at time (t-a).
If we denote by B(t-aJ the total birth rate at time (t- a), and by p(a) the probability at its birth, that any individual will reach age a, then the number of theabove-mentioned survivors is evidently B(t-aJp(a)da.
Hence:
N 1 c(a)da = B(t-a)p(a)dac(a) = B~-a) p(a)I

Now if general conditions in the community are constant, c(a) will tend toassume a fixed form. A little reflection shows that then both N and B will increasein geometric progression with time, 1 at the same rate r = (b- d). We may, therefore,
write:
Brc(a) = N e-ra p(a)I= be-'"p(a){1)

Now from the nature of the coefficient c(a) it follows that'X>

Jc(a)da=10

Substituting this in (1) we have:
1
-=

b

1

'X>

Je-'"p(a)da(2)

0Compare M. Block. "Traite theorique et pratique de statistique:· 1886. p. 209.

94

Equation (1) then gives the fixed age-distribution, while equation (2) (which
may be expanded into a series if desired), gives the relation between b, the birthrate per head, and r, the rate of natural increase per head, and hence between b andd, since r=b-d.
Applying these formulae to material furnished by the Reports of the RegistrarGeneral of Births, etc., in England and Wales, the following results were obtained:
England and Wales 1871-80 (Mean)bcl(b-cl)=r

Birth-rate per head
Death-rate per head
ExcessObserved 2
0.03546
0.021390.01407

Calculated
0.03520.0211(0.0141)p(a) from Supplement to 45th Ann. Rep. Reg. Gen. Births, etc., England and Wales, pp. vii and viii,
assuming ratio:

male births
female births- - - - = 1.04.

Age Sca/e.-1,000 individuals, in age-groups of 5 and 10 years
at al

0-- 5
5-10
10-15
15-20
20--25
25-35
35-45
45-55
55-65
65-75
75-oo

1000 "'Jc(a)da
136
120
107
97
89
147
113
86
59
33
13

"'

138
116
106
97
87
148
116
87
59
33
13

It will be seen that in the above example the values calculated for the agescale and especially forb and d, show a good agreement with the observed values.
The above development admits of further extension. But this, as well as further
numerical tests, must be reserved for a future occasion. In view of the recent noteof the work by Major Woodruff, it appeared desirable to the writer to publish
this preliminary note.

2

Mean band cl from 46th Ann. Rep. Reg. Gen. Births, etc., England and Wales, p. xxxi.

9513. A Problem in Age-Distribution
F. R. SHARPE and ALFRED J. LOTKA (1911)
Philosophical Magazine, Series 6, Volume 21: 435-438.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_13, © Springer-Verlag Berlin Heidelberg 2013

97

The age-distribution in a population is more or less variable. Its possible
fluctuations are not, however, unlimited. Certain age-distributions will practicallynever occur; and even if we were by arbitrary interference to impress some extremely unusual form upon the age-distribution of an isolated population, intime the "irregularities" would no doubt become smoothed over. It seems therefore
that there must be a limiting "stable" type about which the actual distribution
varies. and towards which it tends to return if through any agency disturbed
therefrom. It was shown on a former occasion 1 how to calculate the "fixed"
age-distribution, which. if once established, will (under constant conditions)maintain itself.
It remains to be determined whether this "fixed" form is also the "stable"
distribution: that is to say, whether a given (isolated) population will spontaneously return to this "fixed" age-distribution after a small displacement
therefrom.
To answer this question we will proceed first of all to establish the equations
for a more general problem, which may be stated as follows:"Given the age-distribution in an isolated population at any instant of time.
the 'life curve' (life table), the rate of procreation at every age in life, and the ratioof male to female births, to find the age-distribution at any subsequent instant."
1. Let the number of males whose ages at time t lie between the limits a anda+da be F(u.z)du. where F is an unknown function of a and t.
Let p(a) denote the probability 2 at birth that a male shall reach the age a, ·so that p(O) = 1.
Further, let the male birth-rate (i.e. the total number of males born per unitof time) at time t be B(t).
Now the F(a,t)da males whose age at timet lies between a and a+du are thesurvivors of the B(t- a) d a males born a units of time previously, during an intervalof time d u. Hence
F(a, t)da = B(t -a)p(a)duF(a.t)=p(a)B(t-a).

( 1)

2. Let the number of male births per unit time at time t due to the F(a, t) d a
males whose age lies between a and a+da be F(a,t){j(a)da.
If;· is the age at which male reproduction ends, then evidently
B(t)=JF(a.t)f!(a)da0= JB(t- a)p(a) f3(a)da.

(2)0

Now in the quite general case /1(a) will be a function of the age-distribution
both of the males and females in the population, and also of the ratio of malebirths to female births.
1

2

A.J. Lotka. Am. Journ. Science, 1907, xxiv. pp.199, 375; 'Science,' 1907, xxvi. p. 21.
As read from the life table.

98We are, however, primarily concerned with comparatively small displacementsfrom the "fixed" age-distribution, and for such small displacements we may
regard f3(a) and the ratio of male births to female births as independent of theage-distribution.
The integral equation (2) is then of the type dealt with by Hertz (Mat h. Ann.
vol. lxv. p. 86). To solve it we must know the value of B(t) from t=O to t=}', or,what is the same thing, the number of males at every age between 0 and~· at time~·.
We may leave out of consideration the males above age ~· at time /', as they willsoon die out. We then have by Hertz, lac. cit.,
h=x ()(~

B(t)=

L

i'

a

0

0

J{B(a)- Jf3(a )p(a )B(a_ja )dadQ(;;"da
1

h=t

1

1

Jaf3(a)p(a)Q(;;"da(3)0where ()( 1, ()( 2 , ••• are the roots of the equation for ()(;
7

1 = Jf3(a)p(a)Q(-"da.

(4)0

The formula (3) gives the value of B(t) for t > ~·, and the age-distribution thenfollows from
F(a, t) = p(a) B(t- a).

(1)

4. From the nature of the problem p(a) and f3(a) are never negative. It follows
that (4) has one and only one real root r, which is~ 1, according as

Jf3(a)p(a)da~1.

(5)0

Any other root must have its real part less than r. For if r 1 (cos8+isin0) is aroot of (4),;· f3(a) ap(a) cos a Od a.
1= J
o

(6)

rt

It follows that for large values of t the term with the real root r outweighs allother terms in (3) and B(t) approaches the value(7)

B(t)=Ar1 •The ultimate age-distribution is therefore given by(8)

F(a,t) =Ap(a)r1 - "(9)=Ap(a)er'(t-al.

99

Formula (9) ~!xpresses the "absolute" frequency of the several ages. To findthe "relative" frequency c(a, t) we must divide by the total number of male individuals.
F(a,t)c(a, t) = 00-___:___.:~-

JF(a,t)da0

Ap(a)er'(t-a)p(a)e-r'a00

Je-r'a p(a)da00

J

Ae''' e-r'ap(a)da00= be-r'a p(a),where

The expression (10) no longer contains t, showing that the ultimate distribution
is of "fixed" form. But it is also "stable;" for if we suppose any small displacementfrom this "fixed" distribution brought about in any way, say by temporary disturbance of the otherwise constant conditions, then we can regard the new distribution
as an "initial" distribution to which the above development applies: that is to say,the population will ultimately return to the "fixed" age-distribution.
It may be noted that of course similar considerations apply to the females in
the population. The appended table shows the age-distribution calculated according to formula (10) for England and Wales 1871-1880. The requisite data(including the life table) were taken from the Supplement to the 45th Annual
Report of the Registrar General of Births, etc. The mean value of r' (mixed sexes)
for that period was 0.01401, while the ratio of male births to female births was
1.0382.
It will be seen that at this period the observed age-distribution in Englandconformed quite closely to the calculated "stable" form.
Table
Age
(Years)Males

Females

Persons

Calc.

Obs.

Calc.

Obs.

Calc.

Obs.

0-5
5-10
10-15
15-20
20-25

139
118
107
97
88

139
123
110
99
87

136
115
104
95
87

132
117
104
95
91

138
116
106
96
87

136
120
107
97
89

25-35
35-45
45-55
55-65
65-75

150
116
86
57
30

144
112
84
59
31

148
116
88
62
35

149
87
61
35

149
116
87
59
33

147
113
86
59
33

75-x

11

12

15

15

13

13

3

11}5

Compare Am. Journ. Science, xxiv. 1907, p. 201.

100

14. The Stability of the Normal Age Distribution
ALFRED J. LOTKA (1922)

Proceedings of the National Academy of Sciences 8: 339-345.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_14, © Springer-Verlag Berlin Heidelberg 2013

101

There is a unique age distribution which, in certain circumstances, 2
has the property of perpetuating itself when once set up in a population.
This fact is easily established, 3 as is also the analytical form of this uniquefixed or normal age distribution.
More difficult is the demonstration that this age distribution is stable,that a population will spontaneously revert to it after displacement therefrom, or will converge to it from an arbitrary initial age distribution. Sucha demonstration has hitherto been offered only for the case of small displacements, 4 by a method making use of integral equations. The purpose of the present communication is
to offer a proof of stability which employs only elementary analytical operations, and which is readily extended tocover also the case of large displacements. This method presents the further advantage that it is molded in
more immediate and clearly recognizable relation to the physical causes that
operate to bring about the normal age
distribution.
Consider a population which, at time
t has a given age distribution such as
athat represented by the heavily drawn
Fig. 1curve in fig. 1, in which the abscissae
represent ages a (in years, say), while the ordinates y are such thatthe area comprised between two ordinates erected at a1 and a2, respectively, represents the number of individuals between the ages a 1
and a2.
If we denote by N(t) the total population at timet, and if the ordinatesof our curve are
y = N(t) c(t,a)(1)

we have, evidently,(2)
where N(t,a1,a2) denotes the number of individuals living at time t andcomprised within the age limits a1 and a2. We may speak of c(t,a) as
the coefficient of age distribution. It is, of course, in general a functionof t, only in the special case of the fixed or self-perpetuating age distribution is c(a) independent of the time.
Now, without assuming anything regarding the stability of the selfperpetuating age distribution, it is easy to show 2 that its form must be
102

c(a)

(3)

foooe-rap(a)dawhere r is the real root of the equation

1 = i>-rap(a)f3(a)da(4)

In this equation r is the natural rate of increase of the population, i.e.,the difference r = b - d between the birthrate per head b and the deathrate per head d, and p(a) is the probability, at birth, that a random individual will live to age a (in other words, it is the principal function tabulated in life tables, and there commonly denoted by lx). The limits a;
and ai of the integral are the lower and upper age limits of the reproductive
period. The factor {3(a), which might be termed the procreation factor,
or more briefly the birth factor, is the average number of births contributed per annum by a parent of age a. (In a population of mixed sexes
it is, of course, immaterial, numerically, to what parent each birth iscredited. It will simplify the reasoning, however, if we think of eachbirth as credited to the female parent only.)
The factor {3(a) will in general itself depend on the prevailing age distribution. This is most easily seen in the case of extremes, as for example
in a population which should consist exclusively of males under one yearof age and females over 45. But, except in such extreme cases, {3(a) will
not vary greatly with changes in the age constitution of the population,
and we shall first develop our argument on the supposition that {3(a)is independent of the age distribution. We shall then extend our reasoning to the more general case of {3(a) variable with c(t,a).
Referring now again to fig. 1, let two auxiliary curves be drawn, a minor
tangent curve and a major tangent curve
(5)

(6)

the constants Kh K2 being so chosen that the minor tangent curve lies
wholly beneath the given arbitrary curve, except where it is tangent
thereto, while the major tangent curve lies wholly above the given curve,
except where it is tangent thereto.
The given arbitrary curve representing the age constitution of the population at time t then lies wholly within the strip ot area enclosed between
the minor and the major tangent curves.
Now consider the state of affairs at some subsequent instant t'. Had
the population at time t consisted solely of the individuals represented by
the lightly shaded area in fig. 1, i.e. the area under the minor tangent
curve, then at time t' the population would be represented by the lower
curve of fig. 2, whose equation is103

Y't = K'te-rap(a)= Kter(t'-e>e-rap(a)

(7)
(8)

For the age distribution (5) is of the :fixed form (3), and therefore persistsin (7); on the other hand, given such:fixed age distribution, the population asa whole increases in. geometric progression, 3 so that K '1 = Kte'"<''- e>. In pointof fact, we have left out of reckoningthat portion of the population which in:fig . 1 is represented by the dotted area.
Hence, in addition to the population under the lower curve of :fig . 2, there will,at time t', be living a body of population which for our present purposes it
is not necessary to determine numerically. We need onlyknowthatitissome
positive number, so that the curve representing the actual population at timet' must lie wholly above or in contact
with the curve (8).
By precisely similar reasoning it is readily shown that at time t' the actual curvelies wholly beneath or in contact withthe curvey'2 = K~r(t'-t>e-rap(a)(9)
Fig. 2Hence at time t' the actual curve lies The curves shown are m· t ended t owholly within the strip comprised be- be interpreted only in a qualitative
tween the two curves (8) and (9).
sense; so, for example, the increase inConsider now an elementary strip, of the ordinates in passing from fig. 1 towidth da of the original population fig. 2 is very much exaggerated, to. :fig. l) , wh'tch render it obvious to the eye.
' .1y sh a ded m(sh own h eav1
at time t is in contact with the minor tangent curve. Let this contact occur at age at, so that y1(a1) = K~e-ra•p(a 1). At time t' the survivors of the individuals comprised in this elementary strip will be ofage (a1 + t' - t), so that they will then be represented by a strip ofwidth da and of altitude
Ye•(al + t' - t) = K'te-r111 p(at + t'- t)= Kter(l'-t>e-r<a• +t'-t>p(at+ t'- t)From this it is seen that the elementary strip of population which at time
t' contacts with the minor tangent curve (8) is built up of the survivors104

of the strip which at time t contacted with the minor tangent curve (5).
In other words, if we follow up, by identity of individuals, the elementof the population which at any instant contacts with the minor tangent
curve, this element (so long as any part of it survives) continues in contact
with that curve. (It must be remembered, however, that the tangent
curve itself changes with time according to (5), (8).)
Similarly it follows that the element of population which at any instantcontacts with the major tangent curve continues in this contact so longas any part of it survives.
And again, considering any element of the population which does notcontact with the minor or the major tangent curves, but has its upper
extremity at some point within the area enclosed between these curves,
it can be shown by precisely similar reasoning that such element continues
in such intermediate position.
Turning now from the consideration of the survivors of the original
population, and taking in view the new population added by births since
the time t = t, we note that if the original population had been that represented by the shaded area in fig. 1, i.e., by the area under the minor tangent
curve, then the birthrate would at all times have been such as continuously
to reproduce a population represented by the minor tangent curve (5),(8).
In point of fact, provided that contact with the minor tangent curve
is not continuous over a range of ages equal to or greater than ai - a,, itis easily seen that the total birthrate

B = N i>(a)fi(a)dais always greater (equality is here excluded) than that which would result
from and in turn reproduce the age distribution represented by the minor
tangent curve.
Similarly the total birthrate is always less than that which would result
from a population and age distribution represented by the major tangent
curve.
From this it is clear that, after the original population has once diedout, the representative curve can never again contact with the original
minor and major tangent curves, but must henceforth be separated fromthem everywhere by a finite margin (except, of course, where the several
curves terminate upon the axis of a).
We may then begin afresh by drawing a new pair of tangent curves,
lying within the original pair, and so on indefinitely; until the minor andmajor tangent curves coincide, and with them also coincides the actual
curve of age distribution, which is then of the form

N(t) c(a) = Ke71 e-rap(a)105which we recognize as the fixed age distribution (3), with
N(t) = Kert

fooo e-rap(a)daIn this argument we have expressly excluded the case that the original
age distribution curve contacts continuously with one of its tangent curves
over a range greater than the reproductive period (aj- ai) = A. If
such continuous contact occurs in the Oth generation over a range nA,
where aj/ A> n > 1, then a simple reflection shows that in the next generation(the first) contact will occur over a range (n- l)A, in the second generation over (n - 2)A, etc. A time is therefore reached (in practice verysoon), when contact is over a period less than the period of reproduction
A. After that the argument set forth above applies.
If the curve representing the original age distribution contacts with
one of the tangent curves all the way from a = 0 to a = ai, so that
aj/A ~ n, then, of course, the fixed age distribution is practically E-stablished ab initio, or at any rate from the moment the original population
above reproducing age aj has died out.
It remains to consider the effect of variability in the form of {3(a) withchanges in c(t,a). Some such variability undoubtedly exists owing to
the influence of the ages of the male and female constituents of the population upon the frequency of matings. We may, nevertheless, in thiscase also, define a minor and a major tangent curve (5), (6) in terms ofthe value of r given by equation (4); in order, however, to make this value
determinate, it is now necessary to make some definite disposition regarding the form of {3(a), which is now variable. Merely for purposesof defining r, we shall suppose that the function {3(a) under the integral
sign has that particular form which corresponds to the fixed age distribution.!\We cannot, however, now reason, as before, that the portion of the population represented by the shaded area in fig. 1 will, by itself, reproduce
its own form of age distribution. For its constitution as to sex will in
general differ from that of the "normal" population with self-perpetuating
age distribution. We must therefore consider three different possibilities:
1. The shaded area alone will produce a population exceeding at allages the normal or fixed type continuation (8) of the original shaded area.
Should this be the case, then the argument presented with regard to thecase of invariable {3(a) holds a fortiori, so far as the minor tangent curveis concerned.
2. The shaded area alone will produce a population deficient at someor all ages, as compared with the normal type continuation (8) of the original shaded area. In that case two alternatives present themselves:106

a. The deficiency is more than counterbalanced by the additional
population produced by that portion of the original population which isrepresented by the dotted area in fig. 1. In this case also the original
argument, so far as the minor tangent curve is concerned, applies essentially as before.
b. After the contributions from all parts of the population havebeen taken into account, there remains an unbalanced deficiency short ofthe population defined by (8), in the population resulting from that originally present. In such case the argument presented on the assumptionof invariable {3(a) fails, and the population may move away from, nottowards the age distribution(3). Stability of the fixed age distribution
may not extend to such displacements as this.
Similar reflections apply, mutatis mutandis, as regards the major tangent
curve.
If conditions (1) or (2) prevail with respect to the minor, and corresponding conditions with respect to the major tangent curve, then we canargue as in the case of invariable {3(a), that after expiration of the existing
generation a new pair of tangent curves can be drawn, which will lie within
the pair defined by (8), (9). And if conditions (1) or (2) still persist withreference to the new tangent curves, the same process of dosing in thesecurves at the expiration of the current generation can be repeated, andso on, as in the argument first presented. Now conditions (1) or (2) willthus continue to prevail for each new pair of tangents, if, as the minor andmajor tangents dose up, {3(a) approaches a limiting form. In such case,therefore, the age distribution defined by (3), (4) is stable even for displacements of any magnitude, provided always that conditions (1) or (2)prevail, as indicated.
1 Papers from the Department of Biometry and Vital Statistics, School of Hygieneand Public Health, Johns Hopkins University, No. 71.
2 These circumstances are: (a) an invariable life curve (life table); (b) an invariableratio of male to female births; (c) an invariable rate per head of procreation at eachyear of age, for any given sex-and-age distribution in the population.
a Lotka, A mer. ]. Sci., 26, 1907 (21); Ibid., 24 (199); ]. Wash. A cad. Sci., 3 1 1913
(241, 289).
4 Sharpe, F. R., and Lotka, A. ]., Phil. Mag., April 1911, 435.
s A certain ambiguity is introduced here by the fact that, with {3(a) of variable form,
equation (4) might have more than one real root for r. In practice, however, in a humanpopulation at any rate, probably only one such root exists, or has any effect upon thecourse of events.

10715. Resolving a Historical Confusion in Population
Analysis*
PAUL A. SAMUELSON (1976)

This essay was written for this volume, and also appears in Human Biology 48, 559-580 (1976).

Introduction
As Robert K. Merton (1973) has so well discussed, creative scientists are notimmune to preoccupation with priorities: their priorities. Alfred J. Lotka (1880
to 1949), the father of self-renewal models in linear population analysis, wasleast of all an exception in this regard. A lone pioneer throughout much of his
career, 1 with no cadre of graduate students and colleagues, he could naturally
be expected to be prickly over failures to recognize and acknowledge his important
contributions to demography.
The present brief note attempts to sort out a controversy that arose some
40 years ago between Lotka and another able self-made demographer, R. R.
Kuczynski (1876-1947). The matter has an interest at two levels. Substantively,a reader of their polemics has still been left in ignorance of the true merits of thepoints being argued-whether Bortkiewicz (1911) and other writers had already established the asymptotic approach to a stable exponential equilibriumof a population subject to invariant (one-sex) age-specific mortality and fertility,
an accomplishment properly attributed today primarily to Lotka. Psychologically, as an exercise in how new science gets itself done, the matter is also ofsome interest. Once emotions entered in, Lotka ceased to be his own best advocate and came gratuitously to denigrate legitimate theorems in demographythat are of interest for their own sake. I shall briefly sort out the truths and misunderstandings of the discussion.

Act I
The story begins with Lotka (1929), a review of Kuczyl}ski (1928), a book withan honorable role in the history of demography for its forceful pointing out animpending decline in European population levels, a decline shown to be concealedby swollen numbers of people of fertile age inherited from earlier generations ofhigher fertility. Kuczynski could have benefitted from the fundamental findingsof Sharpe and Lotka (1911) and Lotka (1913, 1922). Perhaps he did not thenknow of the 1911 integral-equation finding; and perhaps he had neither themathematical equipment nor the inclination to master it even were it called toD.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_15, © Springer-Verlag Berlin Heidelberg 2013109

his attention. For the purpose of discerning a shift from a growing to an ultimately declining population, Kuczynski could rely on a common-sense criterion, 2
the net reproduction rate (i.e., the expected number of female babies that will
be born to a representative female baby who throughout her life will be subject
to current age-specific mortality and fertility rates).
Lotka's review is favorable on Kuczynski's substantive thesis, but accuses
him of not acknowledging that his analysis is taken from Dublin and Lotka(1925), a work said to receive only cursory acknowledgment by Kuczynski.
Kuczynski (1930) replies sturdily that the finding about swollen fertile-age
numbers is in fact an ancient staple in the literature, even having been dealt with
by Kuczynski in an 1897 book (when Lotka was only 17 and Kuczynski only 23);
moreover, the concept of the net reproduction rate that Kuczynski relied on,
Kuczynski points out, was put forth by his mentor, Richard Bockh, as far backas 1886. 3
In his rebuttal to Kuczynski, Lotka (1930) had to scale down his accusations.
But he does assert the incompleteness of the early work Kuczynski cited. And
he makes the valid point that, although the net reproductive rate is an accurate
indicator of whether the intrinsic rate of population growth is positive or negative,
it only gives the rate of growth per ambiguous "generation", and not per year,
so that it falls short of the full analysis of Sharpe and Lotka and of Dublin and
Lotka. He concludes with the barb that, if he had known that the discussionof Dublin and Lotka (1925) on the net reproduction rate had been anticipatedearlier, he, Lotka, would have felt obliged to acknowledge it-whatever
Kuczynski's scholarly code might be.
This ends Act I of the drama. The antagonists are now sensitized to eachother. In the years between 1930 and 1937, Lotka apparently became increasingly of the opinion that Kuczynski's publications were spreading misleading
accounts of the true priorities. This finally culminated in two publications,Lotka (1937a, 1937b), which attempt to set the record straight and clarify thetruth. To understand them, I first review some now familiar fundamentals.
And then I formulate a number of distinguishahie propositions or theorems
that are needed to judge the various allegations.

Review of Fundamentals: Mortality Relations
Let N(a, t) be the number of people (females presumably) of age a at time t, withtotal population number at t given by
oo

(1)

n

N(t)=JN(a,t)da=JN(a,t)da,0

a<n<oo0where n is the maximum length of life. Let p(a) be the fraction of new-born
females surviving to age a. Then with births at t given as a prescribed functionof time, N (0, t) = B( t),
110

(2)

N(a,t)=p(a)B(t-a)

=

p(a)

p(a-0)

p(O)= 1,

N( -0 t-0)a''p(n)=O,O<O<a,·

p'(a)~O,(3)

i3N(a, t)ji3a + i3N(a, t)ji3t = [p'(a)jp(a)] N(a, t),(4)N(t) = fp(a)B(t-a)da,

n

0

n

(5)

N'(t)=B(t)+ fp'(a)B(t-a)da.
0

A solution to (5), and under suitable specifications to (4), can be given by
the "renewal function" so useful in industrial-equipment as well as actuarial
analysis, ll(t), as discussed in Feller (1941) and Lotka (1933, 1939a}, namely byt(6)

S JI(t-a)N'(a)da

B(t)=N'(t)+-wwhere JI(t-a) is the Volterra resolvent kernel to p'(t-a), defined as the solution
to
t

(7)

JI(t)= -p'(t)- fp'(t-a)Jl(a)da0t= -p'(t)- SJl(t-a)p'(a)da.
0

One way of solving for JI(t) is via its Laplace transform
'lJ

(8)

fl(w) = S Jl(t)e-wr dt
0r.r_:-p'(w) = S -p'(t)e-'"' dt,
0fi(w)= -p'(w)-p'(w)n(w)-p'(w)

1 +p'(w) ·Alternatively, writing symbolically,t.f(t)= -p'(t),.f(t)·g(t) = S.f(t-a)g(a)da0= g(t) ·.f(t)'(9)Jl(t)= f(t)+ f(t)·f(t)+.f(t)·.f(t)·f(t)+ ...

111

which ts a rapidly converging explicit solution, known to have the property
n

(10)

n

lim fl(t)= c0 = 1/J -ap'(a)da= 1/J p(a)da >0.
o

r-oo

0

Also, with n finite, fl(t) can be written as an infinite series of exponentials,
of Hertz-Herglotz-Lotka type:
ro

(11)

II(tr=c 0 + L e"'i 1 cj-oowhere mj are the infinite number of complex roots of the transcendental equation(12)nJ-p'(a)e-'"ada=1.
0

The sole real root is m0 = 0, and the real parts of all the conjugate complex roots
are demonstrably negative

U=1,2, ... ).

(13)

If any mj root is multiple, the coefficients cj and c _ j will be polynomials in t

rather than constants.
It is worth noting that the non-real roots of (12) are also the roots of

(12')

n

Jp(a)e-mada=O0

as an integration by parts will verify.

Review: Fertility and Mortality Relations
So far nothing has been said about fertility rates. Writing the number of female
births at time t to mothers of age a as B(a, t), we define an invariant age-specificfertility function, m(a), by
(14)

m(a)

=B(a, t)/N(a, t):?. 0,tm(a)=O,

O<a<a,

m(a)>O,a~a~f3<n,m(a)=O,f3 <a.
112

A population self-propelled by invariant age-specific mortality and fertility
functions, p(a) and m(a), and starting out from an initial non-negative age distribution, N(O,a), will have births and numbers that forever after satisfyp(15)

B(t)= Jm(a)p(a)B(t-a)da,t>n>f3>rx>0,

Jm(t-a)p(t-a)B(a)da+ Jm(a)-(--N(a-t,O)da,
o
pa-t)
1

=

p~

n

O<t<n,

1I= Jcp(t-a)B(a)da+G(t),

(16)

t>O,

0

where
(17)

cp(a)

=m(a)p(a),j

G(t) = m(a)p(p(a) ) N(a- t,O)da:?.O,
a-t
1

=0,

n<t

0< t<n0

Just as II(t-a) was the resolvent kernel of -p'(t-a) in (6)-(9), so is therea useful resolvent kernel to cp(t-a), which I write as cp*(t-a), and which provides
a solution to B(t) in (16) and has the other listed verifiable properties:

J
I

(18)

B(t)=G(t)+ cp*(t-a)G(a)da,

t>O,

0

I

(19)

J

cp*(t)=cfJ(t)+ cf>(t-a)cp*(a)da0
IJ=c/J(t)+ cf>*(t-a)cf>(a)da0=c/J(t)+c/J(t)ocp(t)+

000

00

=Qoerol+

L Qieril,

Qo>O,

-oo

where r 0 is the sole real root and ri are the complex roots, necessarily infinite
in number when f3 is a finite positive number, of the transcendental equation

Jcp(a)e-'"da-1,
00

(20)

t/J(r) =0(21)

r 0 >real coefficients of r ±i

(j = 1, 2, o)000The (Q±) are constants or polynomials in t, and like Q0 can be determined fromthe initial condition N(a,O)o113

To determine whether r 0 ~0, we can employ the useful Bockh (1886) net
reproduction rate as a criterion, namely
n

(22)

n

r 0 ~0+-4Jm(a)p(a)da=0

Jc/J(a)da~1.

0

As was done in (8), the Laplace transform may be used to solve for c/J*(t) interms of known c/J(t). totka (1928) gave the interpretation of c/J*(t) as the progenyof a Dirac pulse of new births at initial time zero:(23)

N(a,0)=.:5(a-0),
n

n

0

0

JN(a,O)da = J.:5(a-O)da= 1.

From (19) and (20), or from more general analysis of Feller, one can prove
(24)
(25)
(26)

(27)

lim c/J*(t)e-ror = Q0 = -1/tf/(r0 )>0,

r-oo

lim B(t}e-rot =b 0 >0,r-oolimN(a,t)e-ror=b 0 p(a)e-roa,t-oo( } -roa .
. N(a, t)
1Im--=paer-oo N(O,t)

These two-dozen-odd equations are now standard in the demographic literature, as discussed in Lopez (1961 ), Keyfitz (1968), Coale (1972), J. H. Pollard(1973), and elsewhere. All of these relations owe much to Lotka; in particular
there can be no doubt that equations (15)-(20), (25)-(27) are primarily due tohim, so that any account which failed to indicate this fact is open to criticism
by an impersonal jury, to say nothing of criticism by the injured scientist himself.

A Bouquet of Theorems
To illuminate the disputed issues, let me write down a number of relevant theorems. The list could be amplified, abbreviated, or arranged differently.
The first three theorems involve essentially nothing more than the actuarial
assumption of age-specific mortality.
Theorem lA. Suppose that, for a finite time interval, t0 - .:5 < t < t 0 + .:5(i) p(a) applies, so that from (3)o[ln N(a, t}] oa + o[ln N(a, t}]jot= d[lnp(a)]/da114

(ii) the age distribution is stable, in the sense that
N(a, t)/N(O, t) = c(a):Then N(O,t)=B(t)=B(O)e"', t 0 -r'5<t<t 0 +r'5, u~ON(a,t)/N(O,t)=p (a)e-"",00

b= B(t)/N(t)= 1/Jp(a)e-uada,=-[I
0

d= D(t)/N(t)p'(a)e-ua da

J/Ip(a)e-"" da,u=b-d.

Bortkiewicz (1911) essentially states and proves Theorem 1A; however, it
would be surprising if a thorough search did not turn it up in the earlier actuarial
literature. With charity, and some charity is needed, Lotka (1907a, 1907b), in
his maiden demographic papers, can be construed to have glimpsed the truthof 1A.
To prove 1A, using my notation rather than that of Bortkiewicz (1911),
combine (i) and (ii) to derive(28)

O=oln[N(a, t)/N(O,t)]/ot

o=dIn [p(a)]/da- In [ N(a, t)]jca- B'(t)/ B(t),(29)c(a)=p(a)e-"",u = B'(t)/ B(t),a constant of any sign.

No doubt Bortkiewicz (1911) and Lotka (1907 a, 1907 b) thought of r'5 as oo or
as a large number; however, so long as r'5 > 0, the theorem holds.

Theorem 1 B. Suppose, for all positive time, t;:::: 0,(i) p(a) applies, and(ii) B(t)=B(O)e"', u~O, t;:c:O.
Then
N(a, t) = p(a) B(t- a),=B(O)[p(a)e-"" ]e"',N(t)=B(O)b- 1 e"',t::C:n,ex.

b- 1 = Jp(a)e-uada.
0This almost trivial theorem, whose proof is direct from substitution, is in
Euler (1760), paragraph 17; in perusing this celebrated early paper, I was sur115

prised to find that Euler seems to have not gone beyond its fertility analysis,
despite his earlier promising remarks that the multiplication of a population"depends, of course, on the number of marriages and upon fertility ... ". However, in Euler's private communic ation to Siissmilch, reported in Siissmilch(1761) and discussed in my footnote 2, Euler did anticipate a case of the SharpeLotka Theorem 2A about to be discussed.

Theorem 1C. Suppose, for all t ~ 0,(i) p(a) applies, and(ii) N(t) is of exponential growth
N(t)= N(O)e"',

t~O,

u~O:

Then
( ) -ua ,. N(a, t) pae
l lm--=

r-oo N(O,t)

limB(t)e-" '=bN(O),b- 1 = Jp(a)e-""d a>O.
Q

f-tOO

If u were zero, and N(a, 0) were Dirac's b(a- 0), this would be the renewal
equation for ll(t) B(t). However, the theorem is true whatever the admissible
specifications of N(a, 0), and it is far from absurd or trivial.
To prove the theorem, utilize (5) and (6). Thus, for N(t)=e"', u> ll±I•

=

(30)

N'(t)=ue"'=B(t)+ Jp'(t)B(t- a)da-H(t) ,0H(t)N(a,t)da~O,=,J_p'((a))
pa

O<t<n

n p'(a)= J -N(a-t,O )da,' p(a)

H(t)=:O,n<t.

Any general solution to (30) is known from the principle of superposition to be
the sum of the special exponential solution proportion al to e"' plus Y(t), thegeneral solution to the following homogeneous integral equation:n(31)

Y(t)=Jp(a )Y(t-a)da000=

L hiemi1,

where mi is as in (12').

-oo

116

When the h's are tailored to admissible initial H(t), the fact that all !li are negative
guarantees that as t grows large, the non-exponential Y(t) is damped down to
zero. Q.E.D.
Remark: The fact that numbers in any age group can never be negative puts
a restriction on how fast total N(t) can fall exponentially: thus, if u in e"1 weremore negative than the r_eal part of some mi in (13), fli, that would not be anadmissible observed situation unless B(t) was already exactly proportional to e"'.
This serves as a reminder that one could strengthen the hypothesis in all
three theorems to require their postulated conditions to hold for all time,- oo < t < oo. In that case, Theorems 1A and 1B have conclusions that hold for
all time. And the conclusion of Theorem 1C holds not merely asymptotically but
for all t, as in the following:

Theorem lC': Suppose, for - oo <t< oo,(i) p(a) applies, -_(ii) N(t)=N(O)e"', -XJ<t<oo,(iii) N(a, t) must, of course, always be non-negative.
Then
N(a,t)/N(O,t)=p(a)e-"',nB(t)=N(O)e"'jfp(a)e-uada,t0

For proof, recall that any initial deviation from the stable distribution would,
when projected backward in the fashion of Samuelson (1976), generate negative
numbers from the backward-anti-damped ( !) oscillatory components implied to
be present. This shows that there can be no such initial deviation.
The condition of non-negativity, (iii), is important, even if usually left implicit. Thus, Lotka (1931, 1939b) seems a bit casual in assuming that a particular-x;,formal solution for B(t) is unique in fp(a)B(t-a)da= a prescribed N(t) function.
0

We can add to such a special solution terms like Lk±iemjr and have new formal
solutions: however, with JLi all negative in (12')'s m ±i = !li ± v;, these "appendages" would make B(t) negative as t->- oo; and it is this property that Lotka
should utilize in a cogent treatment.
All theorems up to now, 1A, 1B, 1C, or 1C', have involved only age-specificmortality data as contained in the p(a) survival function. They have not involved
age-specific fertility data from an m(a) function, all e"' growth functions having
been postulated or deduced from postulates not involving m(a). They all belong
to the pre-Sharpe-and-Lotka era of Euler, Bortkiewicz, and Lotka (1907a, 1907b).
117

The next set of theorems depend on both m(a) and p(a) belonging to the post-1911
age of Lotka.
The first theorem is the basic one of Sharpe and Lotka (1911 ), with Bernardelli(1941) and Leslie (1945) equivalences holding for discrete-time, discrete-age
models.

Theorem 2A: Suppose, for t~O,(i) p(a) applies,(ii) age-specific fertility, m(a), applies{J

B(t) = Jm(a)N(a,t)da,t~O(iii) initial N(a, 0) is an arbitrarily given, non-negative, integrable function withsome females not yet beyond the fertile ages
f3

JN(a,O)da>O:0

Then
lim B(t)e-'"' = c 0 >0,
t~'"D

n

lim N(t)e-'"' = c 0 Jp(a)e-roa da,
0(--t'.J). N(a, t)
hm - - = p(a)e-rou'
N(O,t)

Hoc

lim N(a, t)e-rot = p(a)e-'"'' c 0 ,r-x

fJlj;(r 0 )= Jm(a)p(a)e-•oada-1,

r0 real,{J

Je-roa G(a)da
0

Co= P

G(t) as defined in (17).

Ja¢(a)e-rada

Warning: My lj;(r) is often written in the demographic literature as lj;(r) -1.
Two corollaries may be stated.
f3

Corollary 2A. Depending upon whether the net reproduction rate, Jm(a)p(a)da,is greater than, less than, or equal to unity, the population will ultimately grow,decay, or approach a constant level.
118

Corollary 2AA. The algebraic sign of r 0 , the asymptotic or intrinsic rate ofnatural self-propelled increase, is determined by the algebraic sign ofpJm(a)p(a)da-1 =1/J(O), which represents the rate of growth per (ambiguous)2"generation".
For lr0 1 not too large, good approximations are given to r 0 by r~. r~. r';:r~ =- t/J(O)/t/J'(O)=(R 0 -1)/R 1 =(R 0 -1)/J.I.Ro,p

Ri= Jaicp(a)da,average age of becoming a mother,

Jl=R 1/R 0 ,2

a 2 =(R 2 /R 0 ) - Jl 2 =variance of net fertility,
,

ro =

jl-

VJl -2a lnR
2

2

(J

2

0
•

r'; = r~- 1/J(r~)t/J'(r~).

Corollary 2A is essentially due to Bockh (1886) who intuitively inferred it,
and from whom Kuczynski (1928, 1932, 1935) derived his understanding.
Corollary 2AA was popularized by Dublin and Lotka (1925), but its essentials
had already been stated in Lotka (1913). The first approximation given above
can be derived from various series expansions of 1/J(r) and related functions, or
as a Newton-Raphson approximation using an initial r0 = 0; the final approximation given above involves a second Newton-Raphson whirl; the intermediateapproximation has alternative derivations-from certain ratios of power expansions involving cumulants, or from fitting a Gaussian function to m(a)p(a)(with its bizarre implication of some mothers who are not themselves conceived!).
S.D. Wicksell, and others, have given alternative non-Gaussian graduations.
Lotka's formal proofs of 2A were made rigorous by Feller (1941); however,
Lopez (1961) showed that the finiteness postulated for f3 guaranteed that the rootsof 1/J(r)=O were infinite in number and sufficed to provide an arbitrary B(t) on
00

the interval (- {3,0) with its Fourier-like expansion,

L cie'i1•-oo

A different theorem from 2A is provided by the following:
Theorem 2B: Suppose that, for all time, - oo < t< + oo,(i) p(a) applies,(ii) m(a) applies,(iii) N'(t)/N(t)::aconstant, -oo<t<oo.
Then, the asymptotic stable state of 2A holds already, and holds all the time
N(a,O)/N(O,O):=p(a)e-roa=N(a,t)/N(O,t),t

B(t)= B(O)e'"1 ,- x <t< t·.JJ,

N(t)=[B(O)Ip(a)e-roa da]e

01 ,

119

N(a, t)=[B(O)p(a) e-' e'o',tf;(ro) =

0 "]

fJ

Jm(a)p(a)e-roada-1 =0.
,

Theorem 28 was stated very loosely in Lotka (1937 a), and a purported proof ofits more careful restatement was offered in Lotka (1937b).
A final theorem may be distinguished from 2A and 28.
Theorem 2C: Suppose that, for a time period as long as the length of life, namely
for 0< f<'/?.n,(i) p(a) applies,(ii) m(a) applies,(iii) N'(t)/ N(t) =a constant.
Then
N(a, 0) = B(O)p(a) e -roa,
N(a, t)tf;(r 0 ) = 0,=[B(O)p(a) e -roa daJe'"',N(a,t)/N(O,t)=p(a)e-roa,
B(t)=B(O)e'"',t ?_ 0,t?.O,

N(t)=[B(O) ip(a)e-'''"da}'"',

t?.O.

Remark: If the time interval in the hypothesis of 2C, /., were permitted to be lessthan n, the conclusion that a stable age distribution already holds from the
beginning could be shown by numerous counter-examples to be definitely false,
contrary to what some of the remarks in Lotka (1937 a) seem to me to suggest
and thereby undermining his contention.
A proof of 2C would show that N'(t)/N(t) would undergo a transient oscillation if any complex-root harmonic e'i' were present: since this is ruled out by
hypothesis, the only admissible initial N(a,O) is already in the stable configurationp(a) e -r,~• B(O).

Arbitrating the Quarrels
We are now armed to judge the litigants. Lotka (1937a) begins by alleging that
Kuczynski's writings "probably" caused a recent monograph, German Statistical
Office (1935), to credit to 8ortkiewicz (1911) the first formulation of the SharpeLotka results of my Theorem 2A and my Equation (20). The jury must agree
that what 8ortkiewicz did accomplish, namely Theorem 1A, cannot possibly be
identified with Theorem 2A 4 . So Latka's complaint, if accurate, is a serious one.
In support of his charge against Kuczynski, Lotka (1937a, p. 104) quotes
the following passage from Kuczynski (1935, p. 226):
120

Bortkiewicz had come to the conclusion that a population constantly subject to the same mortality and with a constant rate ofincrease must ultimately become stable, that is to say has a stableage composition, a stable birth rate and a stable death rate.
The reader will find this odd, since Kuczynski's quoted words can be construed as attributing to Bortkiewicz not Theorem 2A, but rather my Theorem
1C, which seems to be a perfectly valid theorem, albeit a<> far as I know not appearing explicitly in the pre-1976 literature.
Lotka goes on to remark that there is no m(a) fertility function in Bortkiewicz,
saying 5 in Lotka (1937a, p. 104, n. 2) that even "Kuczynski (1932, p. 43) admits
this in his otherwise misleading history of the 'stable' population." Lotka goeson to argue that (1) Kuczynski has not correctly stated what Bortkiewicz (1911)
did do-which Lotka correctly states was to begin with a stable age distribution
and to deduce from it a balanced exponential growth rate proportional to mye"' [not to be confused with my eror since u is not deduced from an m(a)p(a) integral], and then to find a numerical estimate for u that made the p(a)e-""
theoretical distribution fit tolerably well the turn-of-the-century German data.
The jury must agree with Lotka's contention under (1), which is that
8ortkiewicz (1911) proves my Theorem 1A. Kuczynski, like Homer, nodded.
Lotka (1937a, p. 106) however goes a bit far in writing: "All this was old in theliterature (which Kuczynski fails to point out) ... ", having already been done in
Lotka (1907a, 1907b) with data for England and Wales. First, we might deem
1907 not to be long before 1911. Second, Lotka's 1907 utterances of Theorem 1A
was, as noted already, loose at best and no semblance of an adequate proof wasprovided. Third, 8ortkiewicz's work is thorough to the point of being tedious,
much of it going back to the 1890's, while Lotka's maiden publications are briefand suggestive notes.
(2) Lotka (1937a, pp. 106-7) goes on seemingly to argue that what Kuczynski
wrongly attributes to 8ortkiewicz is in any case not worth doing, being "inherently absurd" in the light of Lotka's claim that, exponential growth in total
self-propelled population already implies realization of the stable age distribution-as in Theorem 28 here. It escapes me why the truth of 28 should makeit absurd for 8ortkiewicz (if only he had done it!) to have formulated Theorem 2A(or, Kuczynski aside, for 8ortkiewicz to have formulated Theorem 1C). Indeed,taken literally, Lotka seems to be cutting his own throat and that of Sharpe!
Obscurely, Lotka (1937a, p. 106) argues that if 8ortkiewicz had done what heis credited with doing, there would have been no need "for Kuczynski to go
through the agonizing labor of testing the approach ... to the stable distribution
in 15 pages of closely printed figures" of Kuczynski (1932c, p. 65).
This Lotka point seems dubious. The 1911 Sharpe-Lotka proof, or the 1941
proof of Feller of Theorem 2A is admittedly better than Kuczynski's numerical
exercise, which (a) starts from an m(a)p(a) such thatnJm(a)p(a)da=1 and the0

population is in stationary, stable equilibrium, (b) suddenly lowers m(a) by 10
per cent, and then (c) by laborious numerical projection depicts the 70 years oftransient approach to a new exponential equilibrium with negative r0 . Still
121

numerical exercises are frequently performed, and for a non-mathematician like
Kuczynski this was a particularly valuable and insightful thing to do.
Lotka (1937a) concludes with his imperfect statement of the truth of Theorem 2B-perhaps not being clear in his own mind that, for the age distributionof a self-propelled system to necessarily already be in its stable form, the posited
exponential growth for N(t) would have had to have held forever, or at least as
in my 2C for n~100 years. Lotka (1937b), when it comes to provide a more
careful statement of 2B and a proof, evidently assumes that his hypotheses holdfor all time, - oo < t < oo; and even then I cannot follow the cogency of his
proof. 6
From my literal account of Latka's 1937 papers, one can perhaps conclude
that the plaintiff has not optimally pressed his case. But the jury cannot conclude
from this that the defendant, Kuczynski, is without fault in Kuczynski (1935) or
Kuczynski (1932). This requires special investigation, the results of which do
not clear Kuczynski of fault in the matter.
Let me audit Kuczynski (1935), a much better-known work than Kuczynski(1932), one which purports to be a basic exposition. Around pp. 6-7, the author
concentrates on the qualitative problem of intrinsic growth of population, or
decay, properly pointing out that Bockh's net reproduction rate provides anappropriate answer. Fair enough. He goes on to document at length, p. 207, n. 2,
how Bockh, Kuczynski, Hirschberg, Rahts had computed in the 1884--1912
period dozens of fertility tables for Germany, Sweden, Denmark, and France-sothat Dublin and Lotka (1925) was something of a Johnny-come-lately. Fair
enough as documentation of the Kuczynski (1930) reply to the Lotka (1929)
review.
But it was quite misleading for Kuczynski to gloss over the difference between
the rate of growth per annum, r0 , and the rate of growth per generation: Kuczynski(1935, p. 207) misleadingly says that, from his 1884 calculation of a Berlin fertility
table, Bockh " ... concluded that the real rate of increase of the Berlin population
.
'
2,172
.
m 1879 was -2- - - 1 = 6 per cent. ' The growth rate per year or decade IS of,053
course not 0.06, and Bockh could not give the correct number.
To conserve time and reduce tedium, I shall reproduce some further misleading passages from Kuczynski (1935, p. 224) with my bracketed editorialcomments.
"Will a population constantly subject to the same fertility andthe same mortality ultimately become stable?
The mathematical elements of this problem have for a long time
[how long? and when before Sharpe and Lotka of 1911 ?] attracted
the attention of both European [name one!] and American mathematicians.
They [Lotka and who else?] have come to the conclusion that a
population with a constant fertility and mortality will in fact ultimately
become stable [yes, Lotka, Feller, Leslie, Lopez]. A comparatively
easy approach to the computations necessary for ascertaining theage composition and the birth and death rates of the stable popula122

tions is to be found in the report represented by Bortkiewicz to the1911 congress of the International Statistical Institute [literally
correct as written, but Kuczynski (1932, p. 41, n. 1) had alreadynoted that Lotka (1907b) had already done this; and, in any case,this literally true assertion is a non sequitur in its seeming implicationthat Theorem 1A or 1C is Theorem 2A].
There soon follows the p. 226 passage on Bortkiewicz that Lotka quoted inprotest, and which I showed to be ambiguous. Kuczynski (1935, p. 226) goeson to say:... But one of his [Bortkiewicz's] assumptions, the stable rate ofincrease, was not and could not be based on the actual conditions
presented by some specific statistical example [being not based onany m(a) data, as Lotka pointed out was devastating to Kuczynski's
apparent link up of Bortkiewicz with Theorem 2A], his findings,
interesting as they were from a theoretical standpoint [misleading inthat "theoretical" versus "statistical" is being confused with the
Theorem 1A versus Theorem 2A issue], did not attract the attentionof demographers.
The attentions of demographers [Kuczynski at least] was indeed
only aroused when 14 years later the American mathematician,Lotka, who for a long time had studied the theoretical propertiesof the stable population [which one? the 1A case? or the 2A case?]published with Dublin ... Their approach is highly mathematicaland we shall confine ourselves here to showing how through Lotka's
formulae the stable yearly rate of increase (r) may be derived fromthe net reproduction rate [and the first and second moments of m(a)].
A cross that mathematical pioneers in a subject must always bear is to havetheir pearls dismissed as theoretical and vaguely impractical; later, after capitulation, they receive their revenge.
It is true that in the rarer item, Kuczynski (1932), the author deals at greater
length with Lotka's contributions, quoting from him copiously and correcting
his numerical errors (but still writing the same misleading sentence about
Bortkiewicz !). At one point, Kuczynski (1932, Appendix, pp. 62- 3) makes thisexplicit acknowledgment:
It goes without saying that we would not have devoted so muchspace [more than 17 pages!] to the presentation of the trend of Dublin
and Lotka's argument and to the translation of their mathematicaloperations into simple arithmetic if we were not convinced that someof the methods which they apply are of great scientific value and if
we did not feel the strong desire that those methods be applied in
the future also by such statisticians as are unfamiliar with higher
mathematics. We wish even to state expressly that the computationof the exact rate of increase in the stable population, as presented
in this study of Dublin and Lotka, in our opinion, marks the only
great progress that has been made in the methodology of measuring
123net reproduction since Boecke in 1886 puplished his first table offertility. But just because we so emphatically recommend the application of those methods we feel obliged to show that the manner inwhich they themselves applied them to statistical data is inadequate.
This is merited if qualified praise, and all the greater the pity that a reader
like myself can find it only in the Appendix of a work so rare that, in the end,
I had to rely on a xerox from a copy in the Princeton Office of Population
library, a copy which I judge from some marginal caligraphy, in comparison
with some samples of Latka's handwriting in my own possession, to have comefrom Latka's own copy and which must have been among his books bequeathedto Princeton.
Before concluding, I ought to venture an opinion as to whether Kuczynski
was being deliberately ungenerous to Lotka. At the conscious level, I think not.
Kuczinski was always a plain spoken scholar, and such people never hesitate
to point out the motes that they see in the eyes of others. In this regard he was
not extreme: certainly, Bortkiewicz customarily meted out more trenchantcriticisms than did Kuczynski, and R.A. Fisher's quill was dipped in stronger
acid than Kuczynski's. Moreover, as I read and reread Kuczynski, both a thirdof a century ago when I developed a mild interest in the matter and recently inthe preparation of this article, I sensed that he may really never have fully understood the magic of Latka's Theorem 2A. He sensed that, if you knew r0 you
could derive the stable age distribution, and then from that you could computefrom p(a) alone the death rate; he may also have sensed the element of simultaneity involved, because only if you had happened to guess the right r0 would
your resulting death rate, when subtracted from the birth rate computable fromm(a), be consistent with the originally assumed r 0 . But he shows signs of being
unclear on the essential logic involved? Nevertheless, I ought to point out
explicitly that there is a way of reading his train of thought which makes his
individual sentences about Bortkiewicz literally true, even though misleading intheir context. Thus, Kuczynski, in the end seems to have considered the following
procedure as optimal: First compute r0 from the first few moments of the fertility
table m(a)p(a) (a result he ought to have clearly excluded Bortkiewicz from infavor of Lotka); second, use Bortkiewicz (or pre-Sharpe Lotka) to get the actuary's stable age distribution and the implied death rate (and, as a residual,
the birth rate). This will be faster and digitally more accurate, Kuczynski decided,
than computing out a Monte Carlo version of Latka's Theorem 2A. But why
did he not make this clear?
Perhaps at the unconscious level, Kuczynski was a bit grudging in his treatment of Lotka, writing passages, (1932, p. 65), likeA good mathematician may be a poor statistician; a good statistician may be a poor mathematician. And since the author of thisbook, if anything, is a poor mathematician ...
A Gestalt psychologist trained in Vienna would expect the reader to complete
the chiasmus by regarding Lotka as the poor statistician.
124

A sociologist of science like Merton would not be surprised to observe that
controversy sours both contestants. Lotka (1925, p. 112, n. 7) praised Euler's
early anticipations, writing, "An exceedingly interesting effort of early date to
demonstrate the ultimate approach to geometric increase of the birth rate,
independently of the initial conditions (e. g., starting with a single pair of parents)
is to be found in L. Euler (1760)." This is generous praise-as I have argued
even over-generous, if Lotka did not in 1925 know of Euler's private communication to Siissmilch (1761 ), and seemingly involving an error like that of Kuczynski's
mistake in attributing to Bortkiewicz Theorem 2A rather than 1A-but understandable praise of a great scholar. However, once he has become alarmed for
his own property rights, Lotka (1937a, p. 107) takes a shriller tone in defending
his own originality, now writing: " ... This must be abundantly clear to anyone
who takes the trouble to examine the pertinent publications, from the first crude
approach by Euler (based on highly unrealistic assumptions, and quite inapplicable to actual statistical data), ... ".
As a final word, I ought to emphasize that the reason for now discussing
this historical contretemps in detail has to do primarily with the need to clar(fythe substance of the matter. By no means was this controversy of unprecedented
virulence: neither scholar ever stood in an extreme position with respect totemperament or emotion; both always conducted themselves with honor anddignity.
Moreover, in a sense Lotka has been the ultimate victor. It is he who is accorded full scholarly homage today. The danger is that, if anything, it is Kuczynski's
commendable role in the development of the subject that will be lost. Thus,
Lotka appears with a full page of references in the excellent Keyfitz (1968,
pp. 424-5) bibliography; the works of Kuczynski escape notice. The one noticetaken of Kuczynski in J. H. Pollard (1973, p. 82) is only in connection with his1932 computation of a male NRR for France after World War I that exceeds
unity whereas the female NRR falls short of it, a dramatic consequence of war
casualties that, ironically, Lotka had adverted to in his original cited review ofKuczynski. Richard Bockh receives no citations in either text. Again, a Merton
would understand how the brighter light drowns out the earlier light. By a process somewhat the opposite of Gresham's law the more polished mathematics
seems to drive out the less certain experimenting with numerical data.
* I owe thanks to the National Institutes of Health for financial aid on demographic research, NIH Grant # 1-R01 HD-09081-01, and to Vicki Elms for help
in preparing the manuscript. Wilma Winters, librarian at the Harvard Center
for Population Studies, provided me with appreciated help in locating rare
items. And I owe gratitude to Professor Ansley Coale for providing me with a
Princeton library English translation of Bortkiewicz (1911), and with xerox
pages from Kuczynski (1932), which might be Lotka's own copy bequeathedwith his other demographic collection to the Princeton Office of Population
Research. More than a third of a century ago, I benefitted from some correspondence with Dr. A.J. Lotka on aspects of these questions; however, from so
cursory a dialogue, I was not able to sort out then the present formulation of theissues. Professor Nathan Keyfitz's invitation to include this material in theSmith and Keyfitz collection of historical items provided the final stimulus for125the present effort, and I owe thanks to David P. Smith for translation of the
Slissmilch (1761) account of Euler's important private communication. All interpretations must, of course, be on my own responsibility.
1. See Spengler (1968) on Lotka in The International Encyclopedia of the Social
Sciences; also, the obituary notices of Dublin (1950) and Notestein (1950).

2. Edwin Cannan, a no-new-fangled-nonsense economist if there ever was one,showed by arithmetic projections of the absolute number of English births that
ultimate U.K. population decline was likely. See Cannan (1895).
3. Cf. Kuczynski (1935, p. 207, n.2) and his citation of Bockh (1886).
4. On the attainment of stability Bortkiewicz says only (1911, pp. 63, 69-70;
the translator is unnamed):Three qualifying hypotheses underlie the following statements.
It is assumed: 1) that in the population an unchanging order ofdeaths prevails; 2) that the current age distribution likewise is invariable, and 3) that no immigration or emigration takes place.
Accordingly the "stationary" and the "stable" population appear
in a sense, as the terms will be used here, almost as ideal types, towhich reality never exactly corresponds, but to which it comes all
the closer, the less significant the actual changes of the death orderand the age distribution, and, relatively speaking, the smaller theimmigration and emigration.
The above hypotheses can be considered as the three characteristics
which are held in common with the two concepts of the stationary
and stable population. In addition, there comes a fourth characteristic: with the stationary population there is a continual constancy,
whereas with the stable population a continual accretion in the total
number of persons living.
. . . [T]he geometric progression as a standard norm for thegrowth of the population was established by L. Euler. He proceededfrom one human couple, and let it as well as its offspring propagate
from generation to generation according to certain invariable conditions. These conditions referred to age of marriage, the number ofchildren begotten by each couple, and the number dying. Hence it
resulted that the number of living at the end of every calendar year
forms a line which, in its further course, approaches more and morea geometric progression.
The key second paragraph reads in the original German: ,Demnach erscheinen die ,stationare" und die ,progressive" Bevolkerung in dem Sinne, wie dieseTermini bier gebraucht werden, gleichsam als Idealtypen, denen die Wirklichkeit niemals genau entspricht, denen sie aber urn so naher kommt, je unerheblicher
die tatsachlich vor sich gehenden Wandlungen der Absterbeordnung und der126

Altersverteilung und je gennger, relativ genommen, die Zahlen der Ein- und
Auswandernden sind."
The statement suggests that real populations may resemble the stable ideal
type, and not that they approach stability. Kuczynski may have given it thesecond meaning, taking the comment that the geometric is a standard norm as
supportive evidence. (The illustration by Euler is from a personal communication
to J.P. Si.issmilch (1761, Vol. 1, pp. 291-299). Euler's example treats 1 female
and 1 male as a "couple", and in effect postulates B(t)=B(t-22)+B(t-24)+
B(t- 26), D(t) = B(t -40), a Bernardelli-Leslie case that is "cyclic" because of an
unfortunate choice of even numbers only: along with dominant (A.*)' terms go
also dominant (-A*)' terms and no strict approach to a stable age distribution
at both odd and even ages.)
5. I follow Lotka (1937a) in giving the date 1932 to Kuczynski's Fertility andReproduction of 1931; likewise Kuczynski (1928) gives 1886 for Bockh (1884),
and I follow him in this (actually, the computation might date from 1886).
6. I am unable to discern the cogency of Lotka (19 37 b)'s purported proof that
if N(t) is observed to be in exponential growth (over some consecutive time
periods? over all time-periods?-which is another thing), N(a, t) is already inthe stable configuration. The following discrete-time example meets the only
hypotheses he purports to use in his demonstration-yet it is not "already inthe stable distribution", and after (t = 0, 1, 2) its N(t) ceases to grow like N(0)(2)'!
The example is based on [p(O),p(1),p(2),p(3)]=[1,r 1,2- 2 ,2 3 ], B(0)=1,032,[m(O), m(1), m(2), m(3)] = [2,0,0,0], with initial (and asymptotic!) increase ofN(t) like (1 + 1)', but lacking such exact exponential growth in the near future.
Period, t0123Births, N(O, t)
1,032
2,064
4,128
8,256

N(1,t)

N(2, t)N(t, 3)

88
516
1,032
2,064

192
44
258
516

48
96
22
129

Total N(t)1,360
2 X 1,360
22 X 1,360
85 + (2 3 X 1,360)

The example was fabricated by perturbing an exact exponential solution to theself-propelled Sharpe-Lotka system through adding to a stable initial state,[N(j,t)]=[1,024, 256,64, 32], a non-exponential solution, [8, -168,128, 16], of
higher harmonics fitted to the initial conditions that total N(t) of the add-on
initially vanish as for t = (0, 1, 2). Only if Lotka postulates initial exponential
growth of N(t) over a long enough initial interval-the whole length of life,n> P-will it become impossible for me to find such a perturbing add-on. No
hint of this appears in his purported proof. The demonstration goes off the
tracks because he seems to confuse a functional-equation requirement of the type.f(a, t)/ g(a, t)

independent of t for all awith his actual type[f,{t)+.f2 {t)]/[g 1 (t)+g 2 (t)]independent oft.
127

I must make clear that these last interpretations of mine have to be regarded as
only tentative.
7. This surmise is corroborated by Kuczynski (1931, pp. 20, 32, 166): writing apparently simultaneously with his Fertility and Reproduction, the author eschews
relations like those in (22) and Corollary 2AA, even though he already knows
them, in favor of stable-state relations liker=N'(t)/N(t)::B(t)/N(t)-D(t)/N(t),pj m(a)p(a)e-rada
r = """~-------nj -p'(a)e-rada
0n= O(r) = I/J(r)+1 + rp(r)-1
p(r)

p(r)

1/J(r)=-+rp(r)where real r is put in the indicated Laplace Transforms and where we solve forthe unique r0 root. Keyfitz (1968, p. 176) calculates r0 iteratively fromr~= O(r~).

r;;' = O(r~) • ....

Even more rapid convergence would occur for the Newton-Raphson variant
r~ = r~- O(r~)/O'(r~).

References
Bernardelli, H.: Population Waves. Journal of the Burma Research Society 31, Part I, 1-18 (1941 ).
Bockh, R.: Statistik des Jahres 1884. Statistisches Jahrbuch der Stadt Berlin, Volume 12 (Berlin)(1886).
Bortkiewicz, L. V.: Die Sterbeziffer und der FraueniiberschuB in der stationliren und in der progressiven Beviilkerung. Bulletin de I'Institut International de Statistique 19, 63-183 (1911 ).
Cannan, E.: The probability of a Cessation of the Growth of Population in England and Wales
during the Next Century. Economic Journal 5, 505-15 (1895).
Coale, A.J.: The Growth and Structure of Human Populations. A Mathematical Investigation.
Princeton, N.J.: Princeton University Press 1972.
Dublin, L.I.: Alfred James Lotka, 1880-1949. J. Amer. Statistical Assoc. 45, 138-9 (1950).
Dublin, L. I., Lotka, A.J.: On the True Rate of Natural Increase of a Population. J. A mer. Statistical
Assoc. 20, 305-39 (1925).
Euler, L.: Recherches generales sur Ia mortalite et Ia ml.\ltiplication. Memoires de I'Academie Royale
des Sciences et Belles Lettres 16, 144-64 (1760).
Feller, W.: On the Integral Equation of Renewal Theory. The Annals of Mathematical Statistics 12.
243-67 (1941).

128

German Statistical Office: Neue Beitrage zum deutschen Bev6lkerungsproblem (Berlin) (1935).
Keyfitz, N.: Introduction to the Mathematics of Population. Reading, Mass.: Addison-Wesley
Publishing Co. 1968.
Kuczynski, R.R.: The Balance of Births and Deaths, Volume I, Western and Northern Europe.
New York: Macmillan Co. for The Brookings Institution 1928.
The Balance of Births and Deaths, Volume II, Eastern and Southern Europe. Washington, D.C.:
The Brookings Institution 1931.
Kuczynski. R. R.: A Reply to Dr. Lotka's Review of 'The Balance of Births and Deaths.' J. A mer.
Statistical Assoc. 25, 84-5 (! 930).
Kuczynski, R.R.: Fertility and Reproduction. New York: Falcon Press 1931/32.
Kuczynski, R.R.: The Measurement of Population Growth. London: Sedgewick & Jackson. Ltd.
1935; reproduced by Gordon and Breach. New York: Science Publishers 1969.
Leslie, P.H.: On the Use of Matrices in Certain Population Mathematics. Biometrika 33, 183-212(1945).
Lopez. A.: Problems in Stable Population Theory. Princeton, N.J.: Office of Population Research
1961.
Lotka,A.J.: Relation between Birth Rates and Death Rates. Science, N.S. 26,21-22 (1907a).
Lotka, A.J.: Studies on the Mode of Growth of Material Aggregates. Amer. J. Science 24, 199-216(1970b).
Lotka. A. .I.: A Natural Population Norm. J. Washington Acad. Sciences 3, 241-48, 289-93 (1913).
Lotka, A. .I.: The Stability of the Normal Age Distribution, Proc. Nat'! Acad. Sciences 7, 339-45(1922).
Lotka, A.J.: Elements of Physical Biology. Baltimore: Williams & Wilkins 1925; reproduced in
posthumous edition with bibliography as: Elements of Mathematical Biology. New York: Dover
Publications, Inc. 1956.
Lotka. A.J.: The Progeny of a Population Element. Amer. J. Hygiene 8, 875-901 (1928).
Lotka, A.J.: Review of 'The Balance of Births and Deaths, Vol. I, Western and Northern Europe'by R. Kuczynski. New York: Macmillan, 1928; J. Amer. Statistical Assoc. 24, 332-3 (1929).
Lotka. A. J.: Rejoinder to 'A Reply to Dr. Lotka's Review of The Balance of Births and Deaths by
R. Kuczynski.' J. Amer. Statistical Assoc. 25, 85-6 (1930).
Lotka, A.J.: The Structure of a Growing Population. Human Biology 3, 459-93 (1931).
Lotka, A.J.: Notes: A Historical Error Corrected. Human Biology 9, 104-7 (1937a).
Lotka, A.J.: Population Analysis: A Theorem Regarding the Stable Age Distribution. J. Washington
Acad. Sciences 27, 299-303 (1937 b).
Lotka, A.J.: A Contribution to the Theory of Self-Renewing Aggregates, with Special Reference toIndustrial Replacement. Annals of Mathematical Statistics 10, 1-25 (1939 a).
Merton, R.K.: The Sociology of Science. Theoretical and Empirical Investigations. Chicago: The
University of Chicago Press 1973.
Notestein, F. W.: Alfred James Lotka: 1880-1949. Population Index 16, 22-9 (1950), containing
a Lotka bibliography of 114 items.
Pollard, J. H.: Mathematical Models for the Growth of Human Populations. Cambridge: Cambridge
University Press 1973.
Samuelson, P.A.: Time Symmetry and Asymmetry in Population and Deterministic Dynamic Systems.
J. Theoretical Population Biology 9, 82-122 (1976).
Sharpe, F.R., Lotka, A.J.: A Problem in Age-Distribution. Philosophical Magazine 21, 435-8(1911).
Spengler, J.J.: Alfred J. Lotka. International Encyclopedia of the Social Sciences 9. New York:
Crowell Collier Macmillan 475-6 (1968).
Siissmilch, J.P.: Die g6ttliche Ordnung, Volume 1, Second Edition. Berlin: 1761.

129

16. On the Integral Equation of Renewal Theory
WILLIAM FELLER (1941)

From Annals of Mathematical Statistics 12. Excerpts are from pages 243-261, 263-266.

Feller's paper is a rigorous treatment of renewal theory, and to assist the reader
his principal results are summarized below in demographic form and notation.
Equation (1.1) is the renewal equation(1.1 a)J'

B(t)= G(t)+ B(t-a)p(a)m(a)da,0' B(t-a)cfJ(a)da=G(t)+ J(1.1 b)0where B(t) is births at time t and is composed of births G(t) to the parent population alive at time 0 plus, inside the integral, births to those born since; B(t-a)
being the number born a years ago, p(a) their probability of surviving to age a,and m(a) the probability of their giving birth to offspring in the interval a to
a+da. In equation (1.1 b), cfJ(a)=p(a)m(a). Stieltjes integrals, which translate
for discrete distributions as summations, are allowed by a comment following
equation (1.3). The equation generalizes from births at time t to births in theinterval 0 to t.
For Feller's example ii, equation (2.1) does not apply. We have instead(2.1 a)G(t) =fH

p(t+a)op(a)J k(a)--m(t+a)da,{t<f3t "?; f30

where k(a) is the parent population at ages a to a+ da at time 0, and other terms
are as above. For t"?;[3 the parent population is no longer in the reproductive
ages and equation (1.1) takes the homogeneous form
p(1.1 c)B(t) = JB(t-a)p(a)m(a)da,

where rx is the youngest age at which m(a)#O. Equation (2.3) is the net reproduction rate
(2.3 a)"-'Jf(t)dt =p

Jp(a)m(a)da = R 0 .

0D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_16, © Springer-Verlag Berlin Heidelberg 2013131

For Theorems 1 and 2 we have:t(3.1 a)F(t) = fp(a)m(a)da,0

which for t?;.{J becomes R 0 as above. Feller's G(t) will be the integral of equation(2.1 a), hence it is total births to the initial population over the interval 0 to t.
Both are finite and are cumulative functions, hence non-decreasing. The Laplace
Transforms (3.2) give their characteristic functions:p(3.2a)

cp(s)=cf>*(r) [or t/J(r)] = fe-' 1p(t)m(t)dt,(3.2b))'(s) =G*(r) [or V(r)] =P p-rp(t+x)

f f e-' k(x)-(-)-m(t+x)dxdt.
1

o o

px

The first of these is the characteristic equation for r, the intrinsic growth rate;
the second is the total reproductive value introduced by R.A. Fisher (1930,
paper 18 below). Both integrals are convergent for t?;,./3 since m(t) then becomes 0.
Hence, by Theorems 1 and 2 the renewal equation has a unique solution.
Theorem 3 establishes stable properties of mean births B(t). We make thesubstitutions:(4.2a)

a= R 0 ,the net reproduction rate.

(4.2b)

b = B,

total births to the initial population.

(4.4a)

m = Jl,the mean age at childbearing in the stationary
population.

(4.7 a)m1 = -t/l'(r)=A,,

the mean age at childbearing in the stable
population.

All of these terms are finite, and therefore:(i) For R 0 = 1 the population is stationary and mean births approach thelimiting value B/J1;(ii) For R 0 < 1 the total number of births accruing to the population after time
0 will be B/(1-R 0 );(iii) For R0 > 1 mean births when deflated by their intrinsic increase over time
approach the limiting value B/A,.
Theorem 4 establishes conditions under which births B(t) as well as meanbirths B(t) stabilize. The theorem states that for R 0 = 1 and B finite as'before,
B(t) stabilizes if: by equation (5.3) the fertile age distribution has a finite mean
and variance; and by equation (5.4) births to the initial population eventuallytaper out, as by the population dying or ageing out of the reproductive years.
132

The Remark extends Theorem 4 to all R 0 , establishing that under the conditions given by the theorem births at time t stabilize toward the limiting value
B(t)= Ve''IA,, whereas above (4.7a),m1=m'1 =A,

b' = V = total reproductive value

as above (3.2 b).

The result has an intuitive meaning: VIA. is the number of births at t=O thatwould give rise over time to the same population as the parent generation's
observed birth distribution. It is found as the backward projection of their births
to time 0 at their intrinsic growth rate, adjusted for the delay to childbearing.
[In the Remark, note that the characteristic function 1/J(r) converges in theinterval 0 > r > - oo and that for the case R 0 < 1 the dominant root r1 is also
negative; hence the Lemma of Section 4 is applicable. For the case R 0 = 1,
VI A,= BlJl· In the discrete case, which Feller does not complete, Theorem 4
holds, to an approximation for finite population size, if the fertility distribution<J>(a) can be expressed as an irreducible and primitive projection matrix. These
requirements are discussed in Parlett (1970, paper 29 below).]
Lotka's solution to the renewal equation enters in Theorem 6. He first substituted B 0 e'' for B(t) and B 0 e'<r-a) for B(t-a) in the homogeneous form of theequation, (1.1c), by which it reduces topJe-•ap(a)m(a)da=1.
Given an infinite number of simple roots r 1 , r 2 , ... he was able to assert that
there existed constants Q such thatp(6.2a)

B(t) = JB(t-a)p(a)m(a)da = L Q;e''';ai

and to identify the Q; terms as Q;= V(r;)I-1/J'(r;), where these are the total reproductive value and mean age at childbearing associated with the i'th root of
equations (3.2 b) and (4.7 a) (Lotka 1939, 64-67, 85-87).
Feller shows that the use of the Laplace Transform is both simpler than
Lotka's solution and more general since it also applies where the characteristic
equation has only a finite number of roots. By taking the transforms of the threeterms of equation (1.1 b)t

B(t)= G{t)+ JB(t-a)<J>(a)da,0

and noting that the last term is a convolution whose transform is the productof the transforms of the two functions, Feller comes down toB*(r) = G*(r)+ B*(r) </>*(r).

133

The solution becomes simply(6.6a)* _ G*(r) _ V(r)B (r) - 1 - c/J*(r) - 1 - t/J(r)'where on the right we use the notation of equations (3.2 a) and (3.2 b).
By Theorem 6, when equation (6.6a) admits the expansion(6.6b)and L IV(rJ/- t/J'(r;)l is convergent, it has the inverse(6.2b)
given by Lotka for the case of infinite simple roots. In demographic work theconditions of the Theorem are always met.
Equation (6.10) extends the analysis to multiple roots, which introduce nongeometric components to growth and are not found in demographic work.
Section 7 provides further comments on discrete solutions.

134

1. Introduction. In this paper we consider the behavior of the solutions of
the integral equationu(t) = g(t)(1.1)

+ { u(t- x)f(x) dx,

where f(t) and g(t) are given non-negative functions. 1 This equation appears,
under different forms, in population theory, the theory of industrial replacement
and in the general theory of self-renewing aggregates, and a great number of
papers have been written on the subject. 2 Unfortunately most of this literature
is of a heuristic nature so that the precise conditions for the validity of different
methods or statements are seldom known. This literature is, moreover, abundant in controversies and different conjectures which are sometimes supported
or disproved by unnecessarily complicated examples. All this renders an orientation exceedingly difficult, and it may therefore be of interest to give arigorous presentation of the theory. It will be seen that some of the previously
announced results need modifications to become correct.
The existence of a solution u(t) of (1.1) could be deduced directly from a wellknown result of Paley and Wiener [21] on general integral equations of form
(1.1).1 However, the case of non-negative functions f(t) and g(t), with which
we are here concerned, is much too simple to justify the deep methods used by
Paley and Wiener in the general case. Under the present conditions, the existence of a solution can be proved in a simple way using properties of completely
monotone functions, and this method has also the distinct advantage of showing
some properties of the solutions, which otherwise would have to be proved
separately. It will be seen in section 3 that the existence proof becomes mostnatural if equation (1.1) is slightly generalized. Introducing the summatory
functions(1.2)

U(t) = {u(x) dx,F(t) = { f(x) dx,

G(t) = { g(x) dx,

equation (1.1) can be rewritten in the form
(1.3)

U(t) = G(t)+f

U(t- x) dF(x).

However, (1.3) has a meaning even if F(t) and G(t) are not integrals, provided
F(t) is of bounded total variation and the integral is interpreted as a Stieltjes
integral. Now for many practical applications (and even for numerical calculations) this generalized form of the integral equation seems to be the most

For the interpretation of the equation cf. section 2.
Lotka's paper [8] contains a bibliography of 74 papers on our subject published before
1939. Yet it is stated that even this list "is not the result of an exhaustive search." At
the end of the present paper the reader will find a list of 16 papers on (1.1) which haveappeared during the two years since the publication of Lotka's paper.
a This has been remarked also by Hadwiger [3].
1

1

135

appropriate one and, as a matter of fact, it has sometimes been used in a more orless hidden form (e.g., if all individuals of the parent population are of the sameage). Our existence theorem refers to this generalized equation.
We then tum to one of the main problems of the theory, namely the asymptotic
behavior of u(t) as t -+ <Xl. It is generally supposed that the solution u(t)"in general" either behaves like an exponential function, or that it approaches

in an oscillating manner ~ finite limit q; the latter case should arise if

1.

f( t) dt = 1,

thus in particular in the cases of a stable population and of industrial replacement. However, special examples have been constructed to show that this is
not always so.' In order to simplify the problem and to get more general conditions, we shall first (section 4) consider only the question of convergence in mean,that is to say, we shall study the asymptotic behavior not of u(t) itself but oftthe mean value u*(t) = 1t .lo u(x) dx.

The question can be solved completely

using only the simplest Tauberian theorems for Laplace integrals. Of course,
if u(t) -+ q then also u*(t) -+ q, but not conversely. The investigation of the
precise asymptotic behavior of u(t) is more delicate and requires more refinedtools (section 5).
Most of section 6 is devoted to a study of Lotka's well-known method of
expanding u(t) into a series of oscillatory components, and it is hoped that thisstudy will help clarify the true nature of this expansion. It will be seen that
Lotka's method can be justified (with some necessary modifications) even insome cases for which it was not intended, e.g., if the characteristic equation has
multiple or negative real roots, or if it has only a finite number of roots. On
the other hand limitations of the method will also become apparent: thus it
can occur in special cases that a formal application of the method will lead to afunction u(t) which apparently solves the given equation, whereas in reality itis the solution of quite a different equation.
Of course, most of the difficulties mentioned above arise only when the function f(t) has an infinite tail. However, it is known that even computational
considerations sometimes require the use of such curves, and, as matter of fact,
exponential and Pearsonian curves have been used most frequently in connection with (1.1). It will be seen that even in these special cases customary
methods may lead to incorrect results. Besides, our considerations show how
much the solution u(t) is influenced by the values of f(t) for t -+ CXl, and, accordingly, that extreme caution is needed in practice. The last section contains
some simple remarks on the practical computation of the solution.

• Cf. Hadwiger [2] and also Hadwiger, "Zur Berechnung der Erneuerungsfunktion nacheiner Forme} von V. A. Kostitzin," Mitt. Verein. schweizerischer Versich.-Math., Vol. 34(1937), pp. 37-43.

136

2. Generalities on equations (1.1) and (1.3}. This section contains a few
remarks on the meaning of our integral equation and on an alternative form
under which it is encountered in the literature. A reader interested only in the
abstract theory may pass immediately to section 3.
Equation (1.1) can be interpreted in various ways; the most important among
them are the following two:(i) In the theory of industrial replacement (as outlined in particular by Lotka),
it is assumed that each individual dropping out is immediately replaced by a
new member of zero age. f(t) denotes the density of the probability at the
moment of installment that an individual will drop out at age t. The function
g(t) is defined by(2.1)

g(t) =

f.,(x)j(t- x) dx,

where .,(x) represents the age distribution of the population at the momentt = 0 (so that the number of individuals of an age between x and x
8x is

+

.,(x)8x + o(8x)). Obviously g(t) then represents the rate of dropping out attimet of individuals belonging to the parent population. Finally, u(t) denotes
the rate of dropping out at timet of individuals of the total population. Now
each individual dropping out at timet belongs either to the parent population,or it came to the population by the process of replacement at some momentt - x (0 < x < t), and hence u(t) satisfies (1.1). It is worthwhile to note thatin this case

(2.2)

f'

j(t) dt = 1,

sincef(t) represents a density of probability.
(ii) In population theory u(t) measures the rate of female births at time t > 0.
The function j(t) now represents the reproduction rate of females at age t (thatis to say, the average number of female descendants born during (t, t + 8t)
from a female of age tis j(t)8t + o(8t)). If .,(x) again stands for the age distribution of the parent population at t = 0, the function g(t) of (2.1) will obviously
measure the rate of production of females at time t by members of the parent
population. Thus we are again led to (1.1), with the difference, however, that
this time either of the inequalities(2.3)

may occur; the value of this integral shows the tendenc)f of increase or decreasein the total population.
Theoretically speaking, j(t) and g(t) are two arbitrary non-negative functions.
It is true that g(t) is connected with f(t) by (2.1); but, since the age distribution.,(x) is arbitrary, g(t) can also be considered as an arbitrarily prescribed function.
It is hardly necessary to interpret the more general equation (1.3) in detail:it is the straightforward generalization of (1.1) to the case where the increase or
decrease of the population is not necessarily a continuous process. This form137

of the equation is frequently better adapted to practical needs. Indeed, thefunctions f(t) and g(t) are usually determined from observations, so that only
their mean values over some time units (years) are known. In such cases it issometimes simpler to treat f(t) and g(t) as discontinuous functions, using
equation (1.3) instead of (1.1). For some advantages of such a proceduresee section 7. It may also be mentioned that the most frequently (if not theonly) special case_ of (1.1) studied is that where g(t) = f(t). Now it is apparent
from (2.1) that this 'tneans that all members of the parent population are ofzero age: in this case, however, there is no continuous age-distribution 71(x).
Instead we have to use a discontinuous function 71(x) and write (2.1) in the formof a Stieltjes integral. Thus discontinuous functions and Stieltjes integrals
present themselves automatically, though in a somewhat disguised form, evenin the simplest cases.
At this point a remark may be inserted which Will prove useful for a better
understanding later on (section 6). In the current literature we are frequently
confronted not with (1.1) but with
(2.4)

u(t) = {" u(t- x)f(x) dx,

together with the explanation that it is asked to find a solution of (2.4) which
reduces, fort < 0, to a prescribed function h(t). Now such a function, as is
known, exists only under very exceptional conditions, and (2.4) is by no means
equivalent to (1.1). The current argument can be boiled down to the following.
Suppose first that the function g(t) of (1.1) is given in the special form
(2.5)

g(t) = /,.., h(t- x)f(x) dx,

where h(x) is a non-negative function defined for x < 0. Since the solution
u(t) of (1.1) has a meaning only fort > 0, we are free to define that u( -t) =h( -t) fort > 0. This arbitrary definition, then, formally reduces (1.1) to (2.4).
It should be noted, however, that this function u(t) does not, in general, satisfy(2.4) fort < 0, for h(t) was prescribed arbitrarily. Thus we are not, after all,concerned with (2.4) but with (1.1), which form of the equation is, by the way,
the more general one for our purposes. If there really existed a solution of(2.4) which reduced to h(t) for t < 0, we could of course define g(t) by (2.5) andtransform (2.4) into (1.1) by splitting the interval (0, oo) into the subintervals(0, t) and (t, oo ). However, as was already mentioned, a solution of the required
kind does not exist in general. It will also be seen (section 6) that the true
nature of the different methods and the limits of their applicability can be understood only when the considerations are based on the proper equation (1.1) andnot on (2.4).

138

3. Existence of solutions.
THEOREM 1. Let F(t) and G(t) be two finite non-decreasing functiom which
are continuous to the right5• Suppose that
F(O) = G(O) = 0,

(3.1)

and that the Laplace integrals6(3.2)'Y(s) = {" e_., dG(t)'P(s) = { ' e-•' dF(t),converge at least for s > u ~ 01 •In case that lim 'P(s) > 1, let u' > u be the root•

·-o+o
·-o+o
Under these conditions there exists fort> 0 one and only one finite non-decreasingof the characteristic equatwn 'P(s) = 1; in case lim 'P(s) $ 1, put u' = ,,functwn U(t) satisfying (1.3).

With this function the Laplace ~ntegral(3.3)

w(s) =

L.

e_,'dU(t)converges for s > u', and
(3.4)

'Y(s)
w(s) = 1 - 'P(s)'PRooF: A trivial computation shows that for any finite non-decreasing solution U(t) of (1.3) and any T > 0 we have

6 It is needless to emphasize that this restriction is imposed only to avoid trivial ambiguities.
1 The integrals (3.2) should be interpreted as Lebesgue-Stieltjes integrals over opm
intervals; thus

which implies that <P(8) -+ 0 as 8 -+ oo. Alternatively it can be supposed that F(t) andG(t) have no discontinuities at t = 0. Continuity of F(t) at t = 0 means that there is no

reproduction at zero age. This assumption is most natural for our problem, but is by nomeans necessary. In order to investigate the case where F(t) has a saltus c > 0 at t = 0,
one should take the integrals (3.2) over the closed set [0, oo ], so that
<P(8) = c

+ lim

J..

·-+O .

e-''dF(t).

It is readily seen that Theorem 1 and its proof remain valid if 0 < c < 1.

However, if

c > 1, then (1.3) plainly has no solution U(t). The continuity of G(t) at t = 0 is of noimportance and is not used in the sequel.
7 The condition is formulated in this general way in view of later applications (cf., e.g.,
the lemma of section 4). In all cases of practical interest IT = 0.
s <P(8) is, of course, monotonic for 8 >IT and tends to zero as 8-+ oo. In order to ensure
the existence of a root of <P(8) = 1, it is sufficient to suppose that the saltus c of F(t) at t = 0
is less than 1 (cf. footnote 6).

139

herein all terms are non-negative and hence by (3.2)

LT e-at dU(t) ~ "Y(s) + <P{s) LT e-•l dU(t).
Now <P(s) < 1 for s > u', and hence it is seen that the integral (3.3) exists fors > u' and satisfies (3.4). On the other hand it is well-known that the values ofw(s) for s > u' determine the corresponding function U(t) uniquely, except foran additive constant, at all points of continuity. However, from (1.3) and (3.1)
it follows that U(O) = 0 and, since by (1.3) U(t) is continuous to the right, themonotone solution U(t) of (1.3), if it exists, is determined uniquely.
To prove the existence of U(t) consider a function w(s) defined for s > u' by(3.4). It is clear from (3.2) that <P(s) and 'Y(s) are completely monotone functions, that is to say that <P(s) and 'Y(s) have, for s > 11, derivatives of all orders
and that ( -l)n'I'Cn>(s) ~ 0 and ( -l)n'Y(nl(s) ~ 0. We can therefore differentiate(3.4) any number of times, and it is seen that w<n>(s) is continuous for s > u'.
Now a simple inductive argument shows that ( -l)nw<n>(s) is a product of{1 - 'I'(s) } -< n+ll by a finite number of completely monotone functions. It
follows that ( -l)nw<n>(s) ~ 0, so that w(s) is a completely monotone function,
at least for s > u'. Hence it follows from a well-known theorem of S. Bernsteinand D. V. Widder9 that there exists a non-decreasing function U(t) such that
(3.3) holds for s > u'. Moreover, this-function can obviously be so defined that
U(O) = 0 and that it is continuous to the right. Using U(t) let us form a new
functionV(t) =(3.5){

U(t- x) dF(x).

V(t) is clearly non-negative and non-decreasing. It is readily verified (and, ofcourse, well-known) that
1/t(s)= f' e-•l dV(t) = w(s)<P(s).

It follows, therefore, from (3.4) that 1/t(s) = w(s) - 'Y(s), and this implies, by theuniqueness theorem for Laplace transforms, that V(t) = U(t) - G(t). Combining this result with (3.5) it is seen that U(t) is a solution of (1.3).
Suppose that f(t) arul g(t) are measurable, non-negative and
THEOREM 2.
bounded in every finite interval 0 ~ t ~ T. Let the integrals(3.6)<P(s) = [ ' e-• 1/(t) dt,

converge for s > u.

Then there exists one arul only one non-negative solution u(t)g This theorem has been repeatedly proved by several authors; for a recent proof cf.
Feller [19].

140

of (1.1) which is bounded in every finite interval 10•
w(s) =(3.7)

f'With this function the integral

e-' 1 u(t) dt

converges at least for s > u', where u' = u if lim <p(s) ~ 1, and otherwise u' > u
11-11+0

is defined as the root of the characteristic equation <p(s) = 1. For s > u' equation(3.4) holds.
If f(t) is continuous except, perhaps, at a finite number of points then u(t) - g(t)is continuous.
PROOF: Define F(t) and G(t) by (1.2). Under the present conditions thesefunctions satisfy the conditions of Theorem 1, and hence (1.3) has a non-decreasing solution U(t). Consider, then, an arbitrary interval 0 ~ t ~ T and supposethat in this interval f(t) < M and g(t) < M. If 0 ~ t < t + h ~ T we haveby (1.3)

0 ~ ~ {U(t + h) - U(t)}= ~ {G(t +h) - G(t)} + ~ it+h U(t + h- x)f(x) dx

~ M

Ml'

11'

+ ii 0 {U(t + h- x)- U(t- x)Jf(x)dx+ MU(T) + h 0 { U(t + h- x)- U(t- x)} dx

Mlh= M + MU(T) + hMft+hI
U(y)dy- h 0 U(y)dy< M + 2MU(T).
Thus U(t) has bounded difference ratios and is .therefore an integral. The
derivative U'(t) exists for almost all t and 0 ~ U'(t) ~ M. Accordingly we candifferentiate (1.3) formally, and since U(O) = 0 it follows that u(t) = U'(t)satisfies (1.1) for almost all t. However, changing u(t) on a set of measure zero
does not affect the integral in (1.1), and since g(t) is defined for all tit is seen thatu(t) can be defined, in a unique way, so as to satisfy (1.1) and obtain (1.3).
Since the solution of (1.3) was uniquely detBrmined it follows that the solution
u(t) is also unique. Obviously equations (3.7) and (3.3) define the same function
w(s), so that (3.4) holds, and (3.7) converges for s > u'.
Finally, if f(t) has only a finite number of jumps, the continuity of u(t) - g(t)becomes evident upon writing (1.1) in the formu(t) - g(t) = { u(x)f(t- x) dx.

10 Without the assumptions of positiveness and boundedness this theorem reduces to a
special case of a theorem by Paley and Wiener [21]; cf. section 1, p. 243.

141

4. Asymptotic properties. In this section we shall be concerned with theasymptotic behavior as t-+ Q() not of u(t) itself but of the mean value u*(t) =1'!t 0 u(T)dT. If u(t) tends to the (not necessarily finite) limitC, then obviously

also u*(t) -+ C, whereas the converse is not necessarily true.
the theorem we shall need the following obvious but useful
LEMMA: If u(t) ~ 0 is a solution of (1.1) and if

ux(t) = e"'u(t),(4.1)

ft(t) = e"'J(t),then Ut(t) is a solution ofux(t) = Ut(t) +1'

For the proof ofUt(t) = e"'g(t),Ut(t - x)fx(x) dx.

THEOREM 3: Suppose that using the functions defined in Theorem 2 the integrals

f"

(4.2)

1"" g(t) dt = b,f(t) dt = a,are finite.
(i) In order that

11' U(T) dT-+ 0

u*(t) = t

(4.3)

0

as t -+ Q() , where C is a positive constant, it is necessary and sufficient that a = 1,
and that the moment,

1"" t f(t)d~ = m.

(4.4)

be finite. In this case
b
0= -.
m

(4.5)

(ii) If a < 1 we have

(4.6)

1.

b
u(t) dt = 1 _ a.

(iii) If a > 1 let q' be the positive root of the characteristic equation ~(s) = 1(cf. (3.2)) and put11
(4.7)

Then
(4.8)

11

1•

e""'''tj(t) dt = m~o

11' e-.·~ u()dlimc-.. tT(4.2) implies the finiteness of m1.

142

T

= -b.

mx

REMARK: The case a = 1 corresponds in demography to a population of
stationary size. In the theory of industrial replacement only the case a = 1
occurs; the moment m is the average lifetime of an individual. The case a > 1
corresponds in demography to a population in which the fertility is greater thanthe mortality. As is seen from (4.8), in this case the mean value of u(t) increases
exponentially. It is of special interest to note that in a population with a < 1
the integral (4.6) always eonyerges.
PRooF: By (4.2) and (3.7)

lim .p(a) = a,

(4.9)

lim -y(8) = b.

•-+O

·-+O
If a < 1, it follows from (3.4) that lim w(8) = b/ (1 - a) is finite.

Since u(t) ~ 0

this obviously implies that (4.6) holds. This proves (ii).
If a = 1 and m is finite, it is readily seen thatlim 1 -and hence by (3.4)

·-+O

~(8)
8

=m,

lim 8w(8) = lim -y(8) lim 1

•-+O

•-+O

8 ( ) = ~.

•-+O - IP 8m

By a well-known Tauberian theorem for Laplace integrals of non-negative
functions 12 it follows that u*(t) --+ ~. Conversely, if (4.3) holds it is readilymseen thae3

lim 8w(8) = 0,a-+0which in tum implies by (3.4) and (4.9) thatlim 1 - ~(8) = ~.
•-+o

0

8

This obviously means that the moment (4.4) exists and equals b/0. This
proves (i).
Finally case (iii) reduces immediately to (ii) using the above lemma withk = -u'. This finishes the proof.
Cf. e.g. Doetsch [18], p. 208 or 210.
Indeed, if (4.3) holds and if U(t) is defined by (1.2), then there is aM= M{e) such that
I U(t) - Ct I < M + et. Now
12
11<P(a) = aand hencea<P(s) - C = s1orls<P(s) - C I ~ s1

L"' e-•t U dt,(t)

L"' e-•'(U(t) - Ct) dt,

L"' e-•'(M + et) dt = sM + •·143

It may be remarked that the finiteness of the integrals (4.2) is by no means
necessary for (4.3) ....
5. Closer study of asymptotic properties. In this section we shall deal almost
exclusively with the most important special case, namely where

f'

(5.1)

f(t) dt = 1.

The question has been much discussed whether in this case necessarily u(t)- C
as t - oo, which statement, if true, would be a refinement of (4.3). Hadwiger[2] has constructed a rather complicated example to show that u(t) does· notnecessarily approach a limit. Now this can also be seen directly and without
any computations. Indeed, if u(t) - C and if (5.1) holds, then obviously
lim l' u(t - x)f(x) dx = C,t-oo0

and hence it follows from (1.1) thatg(t) -o. Inorderthatu(t) -citistherefore
necessary that g(t) - 0, and this proves the assertion. In Hadwiger's example
lim sup g(t) = oo, which makes his computations unnecessary.
It can be shown in a similar manner that not even the condition g(t) - 0 is
sufficient to ensure that u(t)- C. Some restriction as to the total variation off(t) seems both necessary and natural (conditions on the existence of derivatives
are not sufficient). In the following theorem we shall prove the convergence ofu(t) under a condition which is, though not strictly necessary, sufficiently wide
to cover all cases of any possible practical interest.
ThEOREM 4: Suppose that with the functions f(t) and g(t) of Theorem 2

1
00

(5.2)

1
00

f(t) dt = 1,

g(t) dt = b < 00.

Suppose moreover that there exists an integer n ~ 2 BUCh that the moments

m~: = 1oo t" f(t) dt,(5.3)

k = 1, 2, •.. , n,

are finite, and that the functions f(t), tf(t), t2f(t), • • • , t"-2f(t) are of bounded total
variation over (0, oo ). Suppose finally that

(5.4)

lim t-2g(t) = 0

and

t~~

lim t"-2
t~~

1
00

'

g(x) dx = 0.

Thenlim u(t) = .!!_

(5.5)

mt

t-oo

and

(5.6)

lim tn-2 {u(t) -

t-oo

144

.!!_}
= 0.
mt

REMARK:As it was shown in section 4, the case wheref'f(t) dt > 1

can readily be reduced to the above theorem by applying the lemma of section 4
with k = u', where u' is the positive root of 1,0(s) = 1: it is only necessary tosuppose that e-u''j(t) is of bounded total variation and that e-u'tg(t) ~ 0. Obviously all moments of e-u''j(t) exist, so that the above theorem shows thatu 1(t) = e-u'tu(t) tends to the finite limit b' /m~, whereb' =1"" e-u't g(t) dt,

Thus in this case and under the above assumptions u(t) ,....., .!!:_, ev't, so that the
m1

renewal function increases exponentially as could be expected.

If however

1"" j(t) dt < 1,
u(t) will in general not show an exponential character.

If f(t) is of boundedvariation and has a finite moment of second order, and if g(t) ~ 0, then it can beshown that u(t) ~ 0. However, the lemma of section 4 can be applied only if
the integral defining 1,0(s) converges in some negatives-interval containing a value
s' such that 1,0(s') = 1, and this is in general not the case.
PROOF: The proof of Theorem 4 will be based on a Ta.uberian theorem due toHaar16 • With some specializations and obvious changes this theorem can beformulated as follows.
Suppose that l(t) is, for t ;;::: 0, non-negative and continuous, and that the
Laplace integral(5.7)

A(s) =

1"" e-•t l(t) dt

converges for s > 0. Consider >.(s) as a function of the complex variables=x + iy and suppose that the following conditions are fulfilled:(i) For y -¢ 0 the function >.(s) (which is always regular for x > 0) has continuous boundary values >.(iy) as x ~ +O, for x ;;::: 0 andy -¢ 0
(5.8)

c

X(s) = - + 1/J(s),8where 1/l(iy) has finite derivatives 1/l'(iy), ••• , 1/J<•>(iy) and 1/J<•>(iy) is boundedin every finite interval;
(ii)

+oo

1"" eitll>.(x + iy) dy

converges for some fixed x > 0 uniformly with respect tot ;;::: T > 0;(iii) >.(x + iy) ~ 0 as y ~ ± oo, uniformly with respect to x;;::: 0;(iv) >.'(iy), >."(iy), ·. · , >.<•>(iy) tend to zero as y ~ ± oo;
145

1:

(v) The integrals

1i~ e''")..<r>(iy)dy

e1111 )..<r>(iy)dy and(where Y1 < 0 and Y• > 0 are fixed) converge uniformly with respect tot ~ T > 0.
Under these conditions

lim t'{l(t) - C} = 0.

(5.9)

Now the hypotheses of this theorem are too restrictive to be applied to thesolution u(t) of (1.1). We shall therefore replace (1.1) by the more special
equation(5.10)

v(t) = h(t)f+v(t- x)f(z) dz,

where(5.11)

h(t) = { f(t - z)/(z) dz.

Plainly Theorem 2 can be applied to (5.10).
and non-negative and that (by (5.1))

L.

(5.12)

It is also plain that h(t) is boundedh(t) dt = 1,

(5.13)
Accordingly we have by Theorem 2(5.14)

r(s)= 1. e-a v(t) dt = 1 t/(s).
- <p(8)1

0

We shall first verify that t(s) satisfies the conditions of Haar's theorem with
For this purpose we writer = n - 2.

(5.15)

j(t) = f1(t) - ft(t),where !1(t) and fs(t) are non-decreasing and non-negative functions which are,by assumption, bounded:(5.16)

0 ~ /1(t) < M,

0 ~ fs(t) < M.

(a) We show that v(t) is continuous. Now by Theorem 2 the solution v(t)of (5.10) is certainly continuous if h(t) is continuous; however, that h(t) is continuous follows directly from (5.11) and the fact that the functions{ /I(t - z)f(z) dz and { fs(t - z)f(z) dz

n Haar [20] or Doetsch [18], p. 269.

146

are continuous.
(b) In view of (5.1) the function <p(s) exists for X = m(s) ~ 0. Obviously
I <p(x + iy) I < 1 for x > 0. Now
1 - <p(iy) ==1"" (1 - e

4111)f(t) dti"" (1 - cos yt)f(t) dt + i 1"" sin yt.f(t) dt,

and, since 1 - cos yt ~ 0 and f(t) ~ 0, the equality <p(iy) = 1 for y ¢ 0 would
imply that f(t) = 0 except on a set of measure zero. It is therefore seen that<p(x + iy) ¢ 1 for all x > 0 and for x = 0, y ¢ 0.
It follows furthermore from (5.3) that fork = 1, · · · , nand x ~ 0 the derivatives

exist and that
Finally, it is readily seen that in the neighborhood of y = 0 we have<p(iy) =(5.17)!'"

e4 " 1f(t) dt= 1 - m1iy + 1nt(iy) 2 - + ...
2+ (-1)"- (nm~-~)! (iy)"-1 + O(j Y 1").
1(c) From what was said under (b) it follows by (5.14) that t(s) is regular for
1 S(n)(s) approach continUOUS boundary values
as s = x + iy approaches a point of the imaginary axis other than the origin.
Now put
X> 0, and that s(s), s'(s), • • •(5.18)
so that by (5.14)(5.19)For x > 0 and x = 0, y ¢ 0 the function 1/;(x + iy) is obviously continuous;
the derivatives 1/;'(iy), · · · , 1/;<">(iy) exist. To investigate the behavior of
1/;(iy) in the neighborhood of y = 0 put(5.20)P( )Y = m1 -21nt ( ~y· ) + -· •• -(- 1).. (nrn,._l(. )"- 1)! ~y•-1

By (5.17), (5.18) and (5.20)(5.21)

1/;(iy) = [{1 -;r~(y)}2- ~Ji~147

+ O(jyj"-2).

2Now the expression in brackets represents an analytic function of y which
vanishes at y = 0. Hence 1/;(iy) = $(y)0(1 y 1"-2), where $(y) denotes a
power series. It follows that the derivatives !J;'(iy), ... , 1/;<n-2l(iy) exist for all
real y (including y = 0) and are bounded for sufficiently small I y I : since they
are continuous functions they are bounded in every finite interval.
(d). Next we show that there exists a constant A > 0 such that for sufficiently
large I y I+

I~J?(X + iy) I < 1:1(5.22)

uniformly in x ;::: 0.

By (5.15)~J?(s) = 1"" {cos yt- i sin yt}e-" {/l(t) - /2(0} dt.

(5.23)

1

Now !1(t) is non-decreasing and accordingly by the second mean-value theorem
we have for any T > 0 andy

1r0

cos yt-/l(t) dt = J1(T)jr cos ytdt = /I(T) sin Ty - sin ry ,YTwhere Tis some value between 0 and T (depending, of course, on y; at points of
discontinuity, !1(T) should be replaced by lim /I(t)). Hence by (5.16)
t-T-0

1

f'

cos yt·e-"1·J~co dt < f~.
1

Treating the other terms in (5.23) in a like manner, (5.22) follows.
Combining (5.22) with (5.14) it is seen that for sufficiently large I y I
l!(s) I< 2A2

y2

uniformly in x ~ 0. This shows that the assumptions (ii) and (iii) of Haar's
theorem are satisfied for A(s) = !(s). In order to prove that also conditions(iv) and (v) are satisfied it suffices to notice that the proof of (5.22) used only
the fact thatj(t) is of bounded total variation. Now IP<">(s) is the Laplace transform of ( -t)"f(t), and, since t"f(t) is of bounded total variation fork :$; n - 2,
it follows thatk = 1, 2, ... , n - 2,
for sufficiently large I y I, uniformly in x ~ 0.
also seen that

Differentiating (5.14) k times it is

I !<">(s) I = O(l y r\

k = 1, 2, ... , n - 2,

as y --+ + oo, uniformly with respect to x ~ 0.
This enumeration shows that v(s) = l(t) and A.(s) = !(s) satisfy all hypothesesof Haar's theorem with r = n- 2 and C = 1/ml. Hence(5.24)..!_}

lim t"-2 {v(t) = 0.
e-..
m1
148

Returning now to (5.14) we getw(s) = -y(s) + 'Y(S)lP(s) + 'Y(s)t(s),or, by the uniqueness property of Laplace integrals,u(t) = g(t) +(5.25)

f

g(x)f(t - x) dx +fg(x)v(t - x) dx= g(t) + u1(t) + U2(t)(which relation can also be checked directly using (5.10)).
the last term. We have by (5.2)

U2(t) -.!!_ =

m1

and hencet"-2 1U2(t)- .!!_I~ 2n-2
m1

J'

~~

1'
o

Let us begin withg(t - x) {v(x) - _!_}dx,m1g(t- x)xn-21v(x)- _!_ldx

J'

m1_!_I dy.
+ tn-2 112 g(y) Iv(t - y) - m1
If tis sufficiently large we have by (5.24) in the first integral xn-2 1v(x) -~1 ~ <

E

In the second integral v(t- y)- _!_is bounded, and hence by (5.4)m1

U2(t) lim t-2 1, .... ..,

= 0.
.!!_I
m1

The same argument applies (even with some simplifications) also to the second
term in (5.24); it follows that
lim t"-2u1(t) = 0,
whilst tn-2g(t)-+ 0 by assumption (5.4). Now the assertion (5.6) of our theorem
follows in view of (5.25) if the last three relationships are added. This finishes
the proof of Theorem 4.
It seems that the solution u(t) is generally supposed to oscillate around its
limit b/m1 as t -+ oo. It goes without saying that such a behavior is a priori
more likely than a monotone character. It should, however, be noticed that
there is no reason whatsoever to suppose that u(t) always; oscillates around its
limit. Again no computation is necessary to see this, as shown by the following
EXAMPLE: Differentiating (1.1) formally we getu'(t) = g'(t) + g(O)f(t) +fu'(t - x)f(x) dx,

which shows that, if g(t) andf(t) are sufficiently regular, u'(t) satisfies an integral
equation of the same type as u(t). Thus ifg'(t) + g(O)f(t) 2: 0
149

for all t, we shall have u'(t) ~ 0, and u(t) is a monotone function. In particular,
if g'(t) + g(O)f(t) = 0, then u'(t) = 0 and u(t) = const. For example, letf(t) =g(t) = e- 1• Then <;(s) = 'Y(s) = 1/(s + 1) and hence w(s) = 1/s, which is the
Laplace transform of u(t) = 1. It is also seen directly that u(t)1 is thesolution. We have however the following
THEOREM 516 : If the functions f(t) and g(t) of Theorem 4 vanish identicaUy fort ~ T > 0, then the solution u(t) of (1.1) oscillates around its limit b/mast---+ oo,=

PRooF: For t ~ T equation (1.1) reduces toand since

1
1

1-f'

u(t) = J,~f' u(t - x)f(x) dx,
f(x) dx = 1 it follows that the maxima of u(t) in the intervals nT <t < (n + 1)T form, for sufficiently large integers n, a non-increasing sequence.
Similarly the corresponding minima do not decrease. Since u(t) - b/m1, by
Theorem 4, it follows that the minima do not exceed b/m1 and the maxima are
not smaller than b/mi.

6. On Lotka's method. Probably the most widely used method for treating
equation (1.1) in connection with problems of the renewal theory is Lotka's
method. As a matter of fact this method consists of two independent parts.
The first step aims at obtaining the exact solution of (1.1) in the form of a seriesof exponential terms (this is achieved by an adaptation of a method which wasused by P. Herz and Herglotz for other purposes. The second part of Lotka's
theory consists of devices for a convenient approximative computation of thefirst few terms of the series. While restricting ourselves formally to Lotka's
theory, it will be seen that some of the following remarks apply equally to othermethods.
Lotka's method rests essentially on the fundamental assumption that thecharacteristic equation(6.1)<;(s) = 1

has infinitely many distinct simple17 roots So , s1 , • • • , and that the solution u(t)of (1.1) can be expanded into a series(6.2)

u(t) = :E A~oe'k 1"

where the A~o are complex constants. The argument usually rests on an assumed
completeness-property of the roots. Thus, starting from (2.4) it is required that

11 Under some slight additional hypotheses and with quite different methods this theorem
was proved by Richter [16].
11 Hadwiger [3] objected to the assumption that all roots of (6.1) be simple. The modifications which are necessary to cover the case of multiple roots alEto will be indicated below.

150(6.2) reduces to h(t) for t < 0; in other words, that an arbitrarily prescribedfunction h(x) be, for x < 0, respresentable in the form(x < 0).
h(x) = L: Ake•k'"(6.3)
k
In practice we are, of course, usually not concerned with h(t) but with g(t) (cf.
(2.5)), and according to Lotka's theory the coefficients Ak of the solution (6.2)
of (1.1) can be computed directly from g(t) in a way similar to the computationof the Fourier coefficients.
Lotka's method is known to lead to correct results in many cases and also tohave distinct computational merits. On the other hand it seems to require asafer justification, since its fundamental assumptions are rarely realized. Thus
clearly an arbitrary function h(x) cannot be represented in the form (6.3): tosee this it suffices to note that (6.1) frequently has only a finite number of roots(cf. also below). It should also be noted that, the series (6.3) having regularity
properties as are assumed in Lotka's theory, any function representable in theform (6.3) is necessarily a solution of the integral equation (2.4), whereas thetheory requires us to construct a solution u(t) which reduces to an arbitrarilyprescribed function h(t) for t < 0, (which frequently is an empirical function,
determined by observations). Nevertheless, it is possible to give sound foundations to Lotka's method so that it can be used (with some essential limitations
and modifications) sometimes even in cases for which it originally was notintended. For this purpose it turns out to be necessary that all considerations
be based on the more general equation (1.1), instead of (2.4) (cf. alsosection2).
Before proceeding it is necessary to make clear what is really meant by a root of(6.1). The function !f'(s) is defined by (3.2), and the integral will in generalconverge only for s-values situated in the half-plane ffi(s) > IT. Usually onlyroots situated in this half-plane are considered . It is also argued that !f'(s)is, for real s, a monotone function, so that (6.1) has at most one real root: accordingly the terms of (6.2) are called "oscillatory components." However,the function !f'(s) can usually be defined by analytic continuation even outside
the half-plane 9?(s) > 11, and, if this is done, (6.1) will in general also have rootsin the half-plane 9?(s) < 11. It will be seen in the sequel that these roots playexactly the same role for the solution u(t) as the other ones, and that the applicability of Lotka's method depends on the behavior of !f'(s) in the entire
complex s-plane. . ..
From now on we shall consistently denote by !f'(s) the function defined by the
integral (3.4) and by the usual process of analytic continuation; accordjngly weshall take into consideration all roots of (6.1). The main limitation of Lotka's
theory can then be formulated in the following way: Lotka's method depends
only on the function g(t) and on the roots of (6.1). Now two different functions
f(t) can lead to characteristic equations having the same roots. Lotka's method
would be applicable to both only if the corresponding two integral equations(1.1) had the same solution u(t). This, however, is not necessarily the case.
Thus, if Lotka's method is applied, and if all computations are correctly performed, and if the resulting series for u(t) converges uniformly, there is no
possibility of telling which equation is really satisfied by the resulting u(t):151it can happen that one has unwittingly solved some unknown equation of type(1.1) which, by chance, leads to a characteristic equation having the same roots
as the characteristic equation of the integral equation with which one was reallyconcerned....
These preparatory remarks enable us to formulate rigorous conditions for the
existence of an expansion of type (6.2). The following theorem shows the limitsof Lotka's method,- but at the same time it also represents an extension of it.
In the formulation of the theorem we have considered only the case of absoluteconvergence of (6.2). This was done to avoid complications lacking any practical significance whatsoever. The conditions can, of course, be relaxed alongcustomary lines.
THEOREM 6: In order that the solution u(t) of Theorem 2 be repreBentable inform (6.2), where the series converges absolutely fort ;;:: 0 and where the sk denote theroots of the characteristic equation21 (6.1), it is necessary and sufficient that the Laplace transform w(s) admit an expansion(6.6)

w(s) =-y(s)1-tp(S)and that 2:1 A" I corwerges absolutely.
(6.7)

Ak =

=

:E ~

S-8~;

The coefficients Ak are determined by- 'Y (s")

tp'(Sk) •In particular, it is necessary that w(s) be a one-valued function.
PRooF: All roots SJ: of (6.1) satisfy the inequality Sl(s") ~ u', where u' wasdefined in Theorem 2. It is therefore readily seen that in case 2: I AJ: I converges, the Laplace transform of (6.2) can be computed for sufficiently large
positives-values by termwise integration so that (6.6) certainly holds for sufficiently large positive s. Now with 2: I A" I converging, (6.6) defines w(s)uniquely for all complex s (with singularities at the points s,. and the points ofaccumulation of sk , if any). Since the analytic continuation is unique, it follows
that (6.6) holds for all s. The series 2: I Ak I must, of course, converge if (6.2)
is to converge absolutely for t = 0, and this proves the necessity of our condition. Conversely, if w(s) = 1 ~(s~(s) is given by (6.6), and if 2: I A" I converges, then w(s) is the Laplace transform of a function u(t) defined by (6.2).
Since the Laplace transform is unique, u(t) is the solution of (1.1) by Theorem 2.
The serie.'3 (6.2) converges absolutely for t ;;:: 0 since I A"e""' I -~ ! A,. le"''.
Finally (6.7) follows directly from (6.6).
It is interesting to compare (6.7) with formulas (50) and (56) of Lotka's
paper [8]. Lotka considers the special case g(Z) = f(t); in this case -y(s~:) =tp(St) = 1, and (6.7) reduces to A" = - '(1 ) . If s,. lies in the domain of conIPSk

21 The number of roots may be finite or infinite.
It should also be noted that it is not
required that Bk--> oo. If the Bk have a point of accumulation, c.~(s) will have an essential
singularity. That this actually can happen can be shown by examples.

152

vergence of the integral ~(s) =

f'

~A: =

(6.8)

e-''f(t) dt, that is, if m(sA:) ;;:: tT thenf'e-at tf(t) dt,

in accordance with Lotka's result. However, (6.8) becomes meaningless for theroots with m(sA:) < tT, whereas (6.7) is applicable in all cases.
Theorem 6 can easily be generalized to the case where the characteristic equa·tion ~ muUiP.le roots. The expansion (6.6) (which reduces to the customary
expansion into partial fractions whenever w(s) is meromorphic) is to be replaced by(6.9)

where m,. is the multiplicity of the root s,. • This leads us formally to an
expansion(6.10)

u(t) = "'e' ..'{A (1) +A (2) !:_~+ ... +A Cm,~:)tm..-1\"" 1!
k(mk - 1) !/'which now replaces (6.2). Generalizing Theorem 6 it is easy to formulate some
simple conditions under which (6.1) will really represent a solution of (1.1).
Other conditions which ensure that (6.9) is the transform of (6.10) are known
from the general theory of Laplace· transforms; such conditions usually use only
function-theoretical properties of (6.9) a.nd are applicable in particular whenw(s) is meromorphic. We mention in particular a theorem of Churchill [17]which can be used for our purposes.

7. On the practical computation of the solution. There are at hand two main
methods for the practical computation of the solution of (1.1). One of themhas been developed by Lotka and consists of an approximate computation of afew coefficients in the series (6.2). The other method uses an expansion
00

u(t) = ~ u,.(t),

(7.1)

n-o

where u,.(t) represents the contribution of the nth "generation" and is defined
by X

(7.2)

Uo(t) = g(t),u,..rt(t) = [u,.(t- x)f(x) dx.

Now the Laplace transform of u,.+1(t) is -y(s)~"(s), and hence (7.2) corresponds
to the expansion(7.3)

w(s) = 1 -y(s)( ) = -y(s)-~ 8f: ~"(s).

n-oIn practice the functions g(t) and f(t) are usually not known exactly. Frequently their values are obtained from some statistical material, so that only
their integrals over some time units, e.g. years, are actually known or, in other153

words, only the values
1f(n+l)6j(t) dt,f,. = ~ .. a

(7.4)

1f(n+l)&g(t) dt,g,. = "8 .. aare given, where 8 > 0 is a given constant. Ordinarily in such cases sometheoretical forms (e.g. Pearson curves) are fitted to the empirical data andequation (1.1) is solved with these theoretical functions. Now such a procedure
is sometimes not only very troublesome, but also somewhat arbitrary. Consider for example the limit of u(t) as t ---+ oo; this asymptotic value is the main
point of interest of the theory and all practical computations. However, as has
been shown above, this limit depends only on the moments of the first twoorders of f(t) and g(t), and, unless the fitting is done by the method of moments,
the resulting value will depend on the special procedure of fitting. Accordingly
it will sometimes happen that it is of advantage to use the empirical material
as it is, and this can, at least in principle, always be done.
If only the values (7.4) are used it is natural to consider f(t) and g(t) as stepfunctions defined byj(t) = J,.,

(7.5)g(t) = g,.'lforn8 ::::; t < (n+ 1)6.

In practice only a finite number among the f,. and g,. will be different from zero:accordingly the Laplace transforms -y(s) and IP(s) reduce to trigonometrical polynomials, so that the analytic study of w(s) = 1 :<s~(s) becomes particularly
simple. Lotka's method can be applied directly in this case.
For a convenient computation of (7.1) it is better to return to the more generalequation (1.3), instead of (1.1). The summatory functions F(t) and G(t) should
not be defined by (1.2) in this case, but simply by
F(t) =(7.6)[~]

L j,.,

G(t) =

n=O[~]

L g,..

n-o.

It is readily seen that the solution U(t) of (1.3) can be written in the formU(t) =

L: U,.(t), wheren...O

U0(t) = G(t),U,._,.,(t) = [U,.(t- x) dF(x);in our case U,.(t) will again be a step-function with jumps at the points ko, thecorresponding saltus being
(k)

k
""

(k-r)J,

Un+l = L.... U,.
r=Or•

Thus we arrive at exactly the same result as would have been obtained if theintegrals (7.2) had been computed, starting from (7.4), by the ordinary methods154

for numerical integration of tabulated functions. It is of interest to note that
this method of approximate evaluation of the integrals (7.2) leads to the exact
rJalue8 of the renewal function of a population where all changes occur in a discontinuous way at the end of time intervals of length o in such a way that eachchange equals the mean value of the changes of the given population over thecorresponding time interval.
REFERENCES

I. Papers on the integral equation o.f renewal theory
Note: Lotka's paper [8] contains a list of 74 papers on the subject published before 1939.
The following list is to bring Lotka's list up to June 1941; however no claims to completenessare made.
[1J A. W. BaowN, A note on the use of a Pearson type III function in renewal theory,"
Annals of Math. Stat. Vol. 11 (1940), pp. 448--453.
(2] H. HADWIGEB, "Zur Frage des Beharrungszustandes bei kontinuierlich sich erneuernden Gesamtheiten," Archiv f. mathem. Wirtschafts- und Sozialforschung, Vol.
5 (1939), pp. 32-34.
[3] H. HADWIGER, "fiber die Integralgleichung der Bevolkerungstheorie," Mitteilungen
Verein. schweizer Yersicherungsmathematiker (Bull. Assoc. Actuaires suisses),Vol. 38 (1939), pp. 1-14.
[41 H. HADWIGER, "Eine analytische Reproduktionsfunktion fiir biologische Gesamtheiten," Skand. Aktuarietidskrift (1940), pp. 101-113.
[5] H. HADWIGER, "Natiirliche Ausscheidefunktionen fiir Gesamtheiten und die Losung
der Erneuerungsgleichung," Mitteilungen Verein. Schweiz. Versich.-Math.,Vol. 40 (1940), pp. 31-39.
[6] H. HADWIGER and W. RuCHTI, "fiber eine spezielle Klasse analytischer Geburtenfunktionen," Metron, Vol. 13 (1939), No.4, pp.17-26.
(7] A. LINDER, "Die Vermehrungsrate der stabilen Bevolkerung," Archiv f. mathem.
Wirtschafts- und Sozialforschung, Vol. 4 (1938), pp. 136-156.
iS) A. LoTKA, "A contribution to the theory of self-renewing aggregates, with specialreference to industrial replacement," Annals of Math. Stat., Vol. 10 (1939),
pp. 1-25.
[9) A. LoTKA, "On an integral equation in population analysis," Annals of Math. Stat.,Vol. 10 (1939), pp. 144-161.
[10) A. LoTKA, "Theorie analytique des associations biologiques II," Actualites Scientifiques No. 780, Paris, 19a9.
(11] A. LoTKA, "The theory of industrial replacement," Skand. Aktuarietidskrift (1940),
pp. 1-14.
[12) A. LOTKA, "Sur une equation integrale de !'analyse demographique et industrielle,"
Mitt. Yerein. Schweiz. Yersich.-Math., Vol. 40 (1940), pp. 1-16.
[13) H. MuNZNER, "Die Erneuerung von Gesamtheiten," Archiv f. math. Wirtschafts- u.
Sozialforschung, Vol. 4 (1938).
[14) G. A. D. PREINREICH, "The theory of industrial replacement," Skand. Aktuarietidskrift(1939), pp. 1-19.
[15) E. C. RHODEs, "Population mathematics, I, II, III," Roy. Stat. Soc. Jour., Vol. 103(1940),pp.61-89,218-245,362-387.
[16) H. RICHTER, "Die Konvergenz der Erneuerungsfunktion," Blatter f. Versicherungamathcmatik, Vol. 5 (1940), pp. 21-35.
[16a] H. HADWIGER, "Uber eine Funktionalgleichung der Bevolkerungstheorie und einespezielle Klassc analytischer Losungen," Bl. f. Versicherung8mathematik, Vol. 5(1941), pp. 181-188.
[16b] G. A. D. PRIENREICH, "The present status of renewal theory," Waverly PrellS, Baltimore (1940).

155

II. Other papers quoted[17] R. V. CHURCHILL, "The inversion of the Laplace transformation by a direct expansion
in series and its application to boundary-value problems," Math. Zeits., Vol. 42(1937), pp. 567-579.
[18] G. DoETSCH, Theorit und Anwendung der Laplace Transformation. J. Springer, Berlin,
1937.
[19] W. FELLER, "Completely monotone functions and sequences," Duke Math. Jour.,Vol. 5 (1939), pp. 661-674.
[20] A. HAAR, "Vber asymptotische Entwicklungen von Funktionen," Math. Ann., Vol. 96(1927), pp. 69--107.
[21] R. E. A. C. PALEY and N. WIENER, "Notes on the theory and application of Fouriertransforms, VII. On the Volterra equation," Amer. Math. Soc. Trans., Vol. 35(1933), pp. 785-791.

156

17. A New Method for Calculating Lotka's rThe Intrinsic Rate of Growth in a Stable Population
ANSLEY J. COALE (1957)
Population Studies 11: 92-94.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_17, © Springer-Verlag Berlin Heidelberg 2013

157

The determination of the intrinsic rate of increase in a stable population involves
finding the real root r0 of the following integral equation:"'

f-rap(a)m(a)da = I(I)0where r is the intrinsic rate of growth, p(a) is the probability of surviving toage a, and m(a) is the probability of giving birth to a female child at age a.
Lotka uses the first three terms of a Taylor expansion to obtain a quadratic
equation in r yielding a very close approximation to the exact root.
The equation is :2
I~ f3 r 2+ rJ.r- loge Ro=o

(z.)

Rl
where rJ.=Ro/3=rJ.2- ~Ro

and Rn"'fan p(a)m(a)da0

The calculation required is fairly laborious. Typically one calculates R0 ,R 1, and R 2 by dividing the childbearing age interval into five-year groups andapproximating the integral expression for Rn by a finite sum. Three such sums
must be calculated, and then the quadratic equation (z.) solved.
Lotka points out that the correctness of his solution can be verified by substituting the value of r in Equation (I), and approXimating the integral byforming a summation by five-year age groups. The summation should, ofcourse, have a value of unity. This method of verification suggests a methodof finding a root for (I) that is somewhat less laborious and at least as precise.
In brief, the simpler computation involves substituting a rough approximation
for r in Equation (1 ), ascertaining how much the resulting integral diverges fromunity, and using the amount of divergence to determine how the approximate
value of r must be adjusted.
The rough approximation can be obtained from the relation

Ro=e"T(where Tis the mean length of generation), and the fact that T normally takes
on only a limited range of values, with an average of about 2.9 years. An assumption that the value of T is 2.9 years gives a first approximation of r-as Cog e~
2.9

1 The author proposed in an earlier note a still simpler method of calculating approximate intrinsic
rates. This earlier method produced an approximation that was adequate for most purposes but notnearly as close as that produced by the procedure here outlined. Cf. Coale, A. J., "The Calculationof Approximate Intrinsic Rates", Population Index, vol. xxi, April 1955, pp. 94"""97·• Dublin, Louis I., and Lotka, Alfred J., "On the True Rate of Natural Increase", Journal of theAmeri&an Statistical .Ar.rociation, vol. xx, no. IF, September 1925.

158

that may be designated r 1 • Substituting r 1 in (I) yields an integral slightly larger
or smaller than unity-smaller if r 1 is greater than the correct value of r.
w

je-r,a p(a)m(a)da=I +8(4)0

The problem is to infer from 8-the divergence of the integral from unitythe difference between r 1 and the root r 0 of (I). Again following Lotka, letw

y=Je-ra p(a)m(a)da, then
0

!= -f
w

e-ra p(a)m(a)da0=- Ay, where A is the average age of childbearing
in the stable population.
But sincey=I when r=r0 (the root of Equation (I)),

(dy)
dr

r=r0

=-A

(6)

Hence for values of r differing only slightly from r 0 , the integral in (I) will
differ from unity by an amount- A D,r, where D,r is the error in r. Conversely,
the indicated adjustment in r in order to yield a value of unity for the integral is8D,r=-AThe mean age of childbearing (A) differs from the mean length of generation(T) by an amount that depends on r, 1 but that in no case exceeds o · 6 of a year,
or about 2% ofT. Hence 29 years is also a good approximation for A.
Thus r can be estimated by adding~ (with due regard for the sign of 8) to29
the first approximation, r 1 • Or, by taking note of the fact that the error inassuming that A equals 29 years is nearly the same as in assuming Tis 29 years,r'· error 1n· t h e latter lS· T
D. T ._· ·and that t h erelattve= - D,r a more exact determmatton
is obtained by the following expression :8

r=ri+ --8-

(8)

29--

ri

and Tis (neglecting higher powers of r) ~ {J r, with {J defined as(~) • - ~·2.
RoRo
(cj. page 92.). {J lies between about 35 and 50 years, with a typical value of 42. If r is as great as 3 %, T
and A will differ by about o · 6 years. (See Dublin and Lotka, op. cit.)
1 The difference between A

159

As a check of the validity of the technique here described, it was used to recompute a number of values of r already computed by Lotka's procedure. The
agreement is excellent. In fact, if the mean length of generation is within abouta year of 29 years (as it almost always is), (8) should usually give a closer approximation than the algebraic solution of (2). However, when the intrinsic rateof increase for mid-nineteenth century Sweden is computed, Lotka's procedureyields a value of 9 · 65 per thousand and (8) gives a value of 9 · 68. In this instance,
the mean length of generation is about 32 · 3 years, which is unusually long. The
value of r that brings the integral in (I) within one part in ten thousand of unity
is 9 · 65 5 per thousand. But even in this instance (where r 1 is a relatively poor
first approximation) the technique outlined here gives an error of only 2 • 5 per
Joo,ooo in r, or of less than three-tenths of I% of the correct value.
This method of computation has two virtues to recommend it-it is lesslaborious than the solution of the quadratic equation while achieving about
the same precision, and it has the pedagogical advantage of staying close to
fundamental ideas of stable population theory.
Equation (I) follows from the fact that the stable age distribution is(9)c (a)da=be-ra p(a)dawhere c(a)da is the fraction of the population in the interval between a anda+d a, and b is the intrinsic birth rate; and the further fact that the number ofbirths per unit of time must equal the product of the number of women at eachchildbearing age and the age-specific fertility rate for each age ; or(JJ

b= ~=f (a)m(a)da

(Io)

0

Equation (3) relates the rate of increase per generation (the net reproductionrate) in the stable population to the annual rate of increase. No further concepts(such as Thielean semi-invariants) need be introduced.

160

18. The Fundamental Theorem of Nat ural
Selection
R.A. FISHER (1958 (1930))
From The Genetical Theory of Natural Selection, pp. 25-30. New York: Dover.

The total reproductive value, due to Fisher, is of great importance in stable
population theory. It is defined asV=

00

JP(x) v(x)dx0where P(x) is the observed population between ages x and x + dx at time 0,v(x) is the reproductive value, and standard notation r, m(x) is used in place ofFisher's m and bx for the intrinsic growth rate and probability of giving birthin the age interval x to x + dx, respectively. The fraction l(t + x)//(x) is theprobability that an individual age x at time 0 survives t years to his (t + x)th
birthday. In words, V is the backward projection of fertility accruing to an observed population to find the size of a birth cohort that would be reproductively
equivalent to it. Its application is shown in Feller (1941, paper 16 above).

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
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161

The Malthusian parameter of population increase
If we combine the two tables giving the rates of death and reproduction, we may, still speaking in terms of human populations, atonce calculate the expectation of offspring of the newly-born child.
For the expectation of offspring in each element of age dx is lxbxdx,
and the sum of these elements over the whole of life will be thetotal expectation of offspring. In mathematical terms this is00

Jlxbxdx,0

where the integral is extended from zero, at birth, to infinity, to cover
every possible age at which reproduction might conceivably takeplace. If at any age reproduction ceases absolutely, bx will thereafter
be zero and so give automatically the effect of a terminating integral.
The expectation of offspring determines whether in the population
concerned the reproductive rates are more or less than sufficient tobalance the existing death rates. If its value is less than unity the
reproductive rates are insufficient to maintain a stationary population, in the sense that any population which constantly maintained
the death and reproduction rates in question would, apart from
temporary fluctuations, certainly ultimately decline in numbers at a
calculable rate. Equally, if it is greater than unity, the population
biologically speaking is more than holding its own, although theactual number of heads to be counted may be temporarily decreasing.
This consequence will appear most clearly in its quantitative aspect
if we note that corresponding to any system of rates of death andreproduction, there is only one possible constitution of the populationin respect of age, which will remain unchanged under the action ofthis system. For if the age distribution remains unchanged therelative rate of increase or decrease of numbers at all ages must be
the same ; let us represent the relative rate of increase by m; which
will also represent a decrease if m is negative. Then, owing to theconstant rates of reproduction, the rate at which births are occurring
at any epoch will increase proportionately to emt. At any particularepoch, for which we may take t=O, the rate at which births were
occurring x years ago will be proportional to e-mx, and this is the rate
at which births were occurring at the time persons now of age x werebeing born. The number of persons in the infinitesimal age interval
162

dx will therefore be e-=z:iix, for of those born only the fraction lx
survive to this age. The age distribution is therefore determinate if
the number m is uniquely determined. But knowing the numbers
living at each age, and the reproductive rates at each age, the rate at
which births are now occurring can be calculated, and this can beequated to the known.rate of births appropriate to t=O. In fact, thecontribution to the total rate, of persons in the age interval dx, mustbe e-mxzxbxdx, and the aggregate for all ages must be
CX)

Je-mxlxbxdx,
0which, when equated to unity, supplies an equation form, of which
one and only one real solution exists. Since e-mx is less than unity for
all values of x, if m is positive, and is greater than unity for all valuesof x, if m is negative, it is evident that the value of m, which reduces
the integral above expressed to unity, must be positive if the expectation of offspring exceeds unity, and must be negative if it falls shortof unity.
The number m which satisfies this equation is thus implicit in any
given system of rates of death and reproduction, and measures therelative rate of increase or decrease of a population when in the steadystate appropriate to any such system. In view of the emphasis laidby Malthus upon the 'law of geometric increase' m may appropriately
be termed the Malthusian parameter of population increase. It
evidently supplies in its negative values an equally good measure of
population decrease, and so covers cases to which, in respect of mankind, Malthus paid too little attention.
In view of the close analogy between the growth of a population
supposed to follow the law of geometric increase, and the growth ofcapital invested at compound interest, it is worth noting that if weregard the birth of a child as the loaning to him of a life, and the birthof his offspring as a subsequent repayment of the debt, the method bywhich m is calculated shows that it is equivalent to answering thequestion-At what rate of interest are the repayments the justequivalent of the loan~ For the unit investment has an expectationof a return l;cb:ilx in the time interval dx, and the present value of thisrepayment, if m is the rate of interest, is e-=zxb:ilx; consequently the
Malthusian parameter of population increase is the rate of interest at163which the present value of the births of offspring to be expected isequal to unity at the date of birth of their parent. The actual valuesof the parameter of population increase, even in sparsely populated
dominions, do not, however, seem to approach in magnitude the rates
of interest earned by money, and negative rates of interest are, I
suppose, unknown to commerce.

Reproductive value
The analogy with money does, however, make clear the argument
for another simple application of the combined death and reproduction
rates. We may ask, not only about the newly born, but about personsof any chosen age, what is the present value of their future offspring ;and if present value is calculated at the rate determined as before, thequestion has the definite meaning-To what extent will persons of
this age, on the average, contribute to the ancestry of future generations ~ The question is one of some interest, since the direct action of
Natural Selection must be proportional to this contribution. There
will also, no doubt, be indirect effects in cases in which an animal
favours or impedes the survival or reproduction of its relatives ; as asuckling mother assists the survival of her child, as in mankind amother past bearing may greatly promote the reproduction of herchildren, as a foetus and in less measure a sucking child inhibits
conception, and most strikingly of all as in the services of neuter
insects to their queen. Nevertheless such indirect effects will in very
many cases be unimportant compared to the effects of personal reproduction, and by the analogy of compound interest the present valueof the future offspring of persons aged x is easily seen to beVxz:= emx Je-mtztbtdt.
00

X

Each age group may in this way be assigned its appropriate
reproductive value. Fig. 2 shows the reproductive value of womenaccording to age as calculated from the rates of death and reproduction current in the Commonwealth of Australia about 1911. The
Malthusian parameter was at that time positive, and as judged fromfemale rates was nearly equivalent to I! per cent. compound interest;
the rate would be lower for the men, and for both sexes taken together,owing to the excess of men in immigration. The reproductive value,1643~------~---------r--------.---------r--------.

w

>

1-

u

::::>

a
0

a:

a.. I
w

a:

0~--------~10~--------2~0----------3LO----------L---~~--5~0

AGE

IN

YEARS

Fm. 2. Reproductive value of Australian women.
The reproductive value for female persons calculated from the birth- and deathrates current in the Commonwealth of Australia about 1911. The Malthusianparameter is +0·01231 per annum.

which of course is not to be confused with the reproductive rate,
reaches its maximum at about IS!, in spite of the delay in reproduction caused by civilized marriage customs; indeed it would havebeen as early as 16, were it not that a positive rate of interest gives
higher value to the immediate prospect of progeny of an older woman,compared to the more remote children of a young girl. ...
A property that well illustrates the significance of the method ofvaluation, by which, instead of counting all individuals as of equalvalue in respect of future population, persons of each age are assigned
an appropriate value vx, is that, whatever may be the age constitutionof a population, its total reproductive value will increase or decrease
according to the correct Malthusian rate m, whereas counting all165heads as equal this is only true in the theoretical case in which the
population is in its steady state. For suppose the number of persons
in the age interval dx is nxdx ; the value of each element of the population will be nxvxdx; in respect of each such group there will be a gainin value by reproduction at the rate of nxbxv0 dx, a loss by death of
nxf.LxVxdx, and a loss by depreciation of -nxdvx, or in all
nx{ (bxvo- f.LxVx) dx +dvx},
but by differentiating the equation by which vx is defined, it appears
thator that
Consequently the rate of increase in the total value of the population
is m times its actual total value, irrespective of its constitution in
respect of age. A comparison of the total values of the population at
two census epochs thus shows, after allowance for migration, thegenuine biological increase or decrease of the population, which may
be entirely obscured or reversed by the crude comparison of thenumber of heads. The population of Great Britain, for example, musthave commenced to decrease biologically at some date obscured by
the war, between 1911 and 1921, but the census of 1921 showed anominal increase of some millions, and that of 1931 will, doubtless inless degree, certainly indicate a further spurious period of increase,
due to the accumulation of persons at ages at which their reproductive value is negligible.

16619. How the Age Distribution of a Human Population is Determined
ANSLEY J. COALE (1957)

Cold Spring Harbor Symposia on Quantitatit·e Biology 22: 83-88.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_19, © Springer-Verlag Berlin Heidelberg 2013167

Mortality and the stable age distribution'!-'his analysis of age distribution will be restncted to closed populations of human females·
restricted to closed populations because to con~sider ~he effects o~ migration would be unduly
eomphcated; restncted to human populations
beP-ause the author is a demographer rather thana biologist; and restricted to females because
differences in age composition and in age atparenthood between the sexes make a combined
treatment awkward. If data were equally availa.ble, a completely similar analysis could be madefor males, with only one ma)'or reservation: The
special effects of war mortality on males of military age would require special analysis.
I. STABLE PoPuLATIONS
The forces affecting the shape of the age distribution are most readily visualized in the special
case of an unchanging distribution. Lotka (Dublin
and Lotka, 1925; Lotka, 1939) has shown that
constant mortality and fertility schedules will
ultimately produce a constant age distribution
and a constant rate of growth. This constant ag~
distribution can be calculated as the product of afactor representing relative number of survivors
to each age from birth (the life table) and a factor
reflecting the continuously growing (or shrinking)number of births. Specifically:c(a) = be-'"p(a)(1)

where c(a) is the fraction at any given age a, b is
the birth rate (which Lotka shows is constantunder the specified conditions) r is the growthrate, and p(a) is the fraction who survive frombirth to age a. No matter what the initial age
distribution (provided the childbearing ages are
amply represented) a constant age schedule offertility (m(a)) and probability of surviving (p(a))
eventually establish the distribution given inequation (1). Actually, a period of 50 to 100 years
is adequate to produce a very close approximation
to the stable population.
The basis for equation (1) can be simply explained: The birth rate (b) is the proportion atage zero (set a = 0, and the other factors areeach unity). The term e-'• relates the size at birthof the cohort now aged a to the current birthcohort, and p(a) allows for the attrition of mortality.
Since b="1

J, e-'"p(a) da,(where w is thehighest age attained) the stable age distribution iswholly determined by the growth rate r and thesurvivorship function p(a). The value of'r in turncan be calculated from the second fundamental
equation of stable population theory.
(2)fe-'•p(a)m(a) da = 1

The real root of this integral equation is thestable growth rate, while the complex roots determine how the stable population is approachedfrom arbitrary initial conditions.
We now turn to the role of fertility and mortality in determining the shape of the stable age
distribution. The role of mortality will be described first.

Two rather surprising conclusions emerge whenthe effect of mortality schedules on the stable age
distribution is examined:(a) The. effect. of ~~;lternative mortality
sched';lles. IS ~elat1vely mmor. Roughly similar
age d1stn_butw~s res_ult fr_om a given fertility
schedule m conJunctiOn With a very high mortality l_ife t~~;ble on the one hand, or a very low
mortality hfe table on the other. Figure 1
illustrates this point. The life table of Sweden
for 1860, with a life expectancy of 45.4 years,
and that for 1946 to 50, with a life expectancy
of 71.6 years produce about the same stableage distributions when combined with thesame fertility schedule.
(b) Among life tables reflecting experience so
far recorded, it is nearly universally true that amore favorable mortality schedule-with ahigher life expectancy-will yield a stable
population with a higher proportion under 15.
In a vast majority of contrasting life tables
the lower mortality life table will produce ~distribution with a lower average age; and at
least half the time, lower mortality will produce
a smaller fraction over 65.
These conclusions are partly analytical andpartly empirical. A full analysis of the contrasting
stable age distributions associated with different
mortality schedules is somewhat laborious (Coale,1956). A simple qualitative argument is enoughto dispel the common belief that lower mortality
inevitably means an older population. If in one
life table the probability of surviving for a year
at each age exceeded the corresponding probability in another life table by a fixed proportion(for example, 1 %) the two life tables would produce precisely the same age distributions. The
tendency toward more survivors with advancing
age would be exactly offset by a faster rate of
growth that tends to make each cohort smaller
than the next younger.
This point can be proved by assuming a sudden
shift to a life table with a one per cent higher
probability of surviving at each age. The year following this change, there would be one per centmore one-year olds surviving from birth, one percent more two-year olds, etc. Improved survivorship would produce one per cent more persons at
every age above 0. But since the increase wouldyiel~ one per cent more persons at every childbearmg age, with constant fertility there would
be one per cent more births-persons at age {}--aswell. The whole population would be one per centlarger; and the age distribution would be unaffected.
The common view that lower mortality means
an older population takes account of only part ofthe effect of lower death rates. It is immediately
clear that lower death rates produce more oldpeople. However, lower death rates also produce
more parents, more births, and more children.
'Y!Je~her. the dominant effect on the stable age
d1stnbutwn of a lower level of mortality is to enl~~;rge the upper en_d of the a~e distribution throughh1gher survivorship, or to tilt the age distribution
more steeply_ through more rapid growth depend~on the relatiVe age pattern of mortality in the
two life tables (Coale, 1956).

168

STABLE AGE DISTRIBUTIONS, SWEDEN"'

FEMALEPOPULATION3MORTALITY FERTILITY- - - 1946-50••••••••• 1851 • 70

1950
1860·611950
1860·61oL-----~IO;----i2o~---,3bo,----4i,0,----,50~---,~----,k----,J,I
AGE

FIGURE 1.

A particularly illuminating way of comparmg
the age pattern of mortality is to compute theproportionate difference in the probability of surviving for one year at each age. Let the probabilityof surviving from age a to a + 1 be r'(a) in the
better life table, and r(a) in the higher mortality
table, and consider the ratio r' - r(a). We ha,·e1r'lr' -

1r

already shown that a con~tant value of - - ( a )1r

implies the same age distribution~. We ean subtract the minimum value of r' ,..
- ,.. from the \·alue

at other ages-the subtracted portion is Pqui\·alent to no difference in mortality-and consider'lr'-ronly the residual. A large excess of --,..- above
the minimum at ages below childbearing means
there is a large difference in the growth rates of thetwo stable distributions. Moreover, since a large
I
excess r - ,.. at the youngest ages implies more
1r

survivors at all subsequent ages, the proportion ofsurvivors above-say-age 50 in the better life
table would not be much larger than in the poorer
one. In short, an excess of 1r' - ,.. above the minir

mum at young ages means that the better mortality schedule tends to have a younger age
distribution.

On the other hand, an above-the-minimum,..

r' - ""(a) at ages over 50 implies no difference inthe long term growth rate (since the reproductive
ages are not involved) but does mean more old
age survivors in the better life table. The net
result is a higher fraction at ages above 50.
The effects of three special instances of percent
difference in the probability of surviving havebeen described in a non-rigorous fashion. These
particular patterns have been emphasized because
differences among the great majority of recorded
life tables can be approximated as the sum of
three components-a minimum per cent difference in the probability of surviving from age 5 to50, above minimum differences below age 5, andabove minimum differences above age 50 (see Fig.
'fr'-,..
2). The general pattern of -1r-(a) is roughly
U-shaped, decliniag from a high at age zero tonear its minimum by age 5. It is relatively constant until age 50 or 60 where it frequently but
not universally rises. The central portion (from 5
to 50) contributes nothing to the differences instable age distribution, serving only to diminish
the effect of the differences represented by the twolegs of the U, and to raise the growth rate of the
population.
The two legs of the Ucshaped pattern work
more or less in opposition in causing differences in

169

%INCREASE
lN'II'(o)

0

10

20

30

40

50

AGE

FIGURE 2.

age distribution. If there were no right leg, the age
distribution with the better life table could be approximated at all ages except those under 5 by asort of pivoting of the higher mortality distribution on its average age, with increased fractionsat less than the average age, and reduced fractions
above. If there were no left leg, rising per centdifferences in survivorship over age 50 would
mean higher fractions above 50, and lower fractions below. When the two legs operate together,
the net effect is to reduce the differences eachwould cause alone. If the left leg is large, and theright leg is small, the effect of growth rates will
dominate; and the low mortality population may
have smaller proportions at all ages above theaverage. This domination is typical when a life
table of high mortality is compared with a moderate or low mortality table. However, when mortality of 20 years ago in the most advanced areas
is compared with current mortality, the rightleg-improved survivorship at the older agestakes on more importance. There is not muchroom for a left leg in a comparison of future life
tables with the best current ones. A two or three
per cent further improvement in the probabilityof surviving to age 5 in Sweden would raise thisprobability to one.
There is no logical necessity that mortality differences should produce only minor differences instable age distributions. This result arises from theage structure of differences among life tables observed to date. However, a contributing factor in
the typical U-shaped age structure of differencesin the probability of surviving is that large differences can last only when there is room for them.
Differences of the order of those observed for age
0 could not occur at those ages where even withhigh general mortality levels the risk of death ismoderate.
Fertility levels and the stable age distributimt
The role of fertility in shaping the stable age
distribution is at once simpler than that of mortality, and quantitatively much more important,when areount is taken of the range of mortality
and fertility observed in the world. The simplicity
and the quantitative importance have a commonsour<·e: the fact that differences in fertility operate
in a single direction in affecting the relative shapeof the age distribution. While higher probabilities
of sur\"iving simultaneously flatten out p(a) inequation (1) and make e-"4 steeper, higher probabilities of giving birth affect only the growthrate-making it larger, of co11rse, and making the
exponential factor in equation (1) contribute to amore rapid taper in the distribution.
If a high fertility stable age distribution is compared with a low fertility distribution with thesame mortality, the two will be found to intersect
at the mean of the average ages. The higher fertility population will have higher proportions atages below the average, of course.
Figure 1 makes it clear that fertility differences
can produce profoundly different stable age distributions. This fact together with the relatively
slight influence of mortality on the stable age distribution means that a schedule of fertility by age
is sufficient to give at least a fair approximation
to the stable age distribution even if mortality
rates by age are not known. One would simply use
whatever female life table was lying around in
calculating the stable population. It must be admitted that there is a much better chance of aclose fit if some hint about infant mortality isvisible.
One final comment on a common sense basis forunderstanding the powerful influence of fertilityon the stable age distribution. The general fertility rate establishes the ratio of the area in theage distribution from ages 17 to 44 to the zeroordinate of the distribution. If fertility is twice as
high, this ratio must be cut in half irrespective ofmortality. The exact inverse relation of this ratio
to fertility is a consequence of the fact that thezero ordinate is proportional to births, and thearea from 17 to 44 to the number of women ofchildbearing age. Thus the fertility level clamps avice on the relation of the beginning of the distribution to an area near the middle.
Several straightforward conclusions emerge
from this consideration of stable age distributions:(1) Sustained high fertility (average completed size of family, 6 or more children) produces a young population with a median age
well below 25 years, more than 40 per cent of
the population under 15, and no more than
three or four per cent over 65.
(2) Sustained low fertility (average completed size of family below 2.5 children) pro-170

duces an old population with a median age
above 35, no more than 20 per cent of the population under 14, and at least 15 per cent over 65.
(:3) In general, the approximate form of theage distribution is determined hy the level offertility. The level of mortality has more or leHssecond order effects on the distribution. The
general quality of these effects has been to datethat low mortality yields a slightly larger fraction at ages up to at least 15 but as high as. theaverage age, and somewhat smaJ.let fraet10ns
either at all higher ages or until age 50 or higher.
More often than not, sustained low mortality
yields a slightly lower average age.

II. VARYING FERTILITY AND MOitTALITY
AND THE AGE DISTRIBUTION

We tum now to a brief consideration of how an
age distribution is determined ~nder a regim~. ofcontinuously changing mortahty and fert1hty
rates.
The general equations corresponding to equation (1) are:(3) n(a, t) = B(t - a)p(a, t) where B(t - a)is the number of births a years before time t,n(a, t) is the number of persons at age a at timet, and p(a, t) is the proportion of those born at(t - a) who survive to achieve age a at timet,andn(a, t)(4) c(a, t)

1~ n(a, t) da .

The equation corresponding to (2) is:(5) B(t) =fn(a, t)m(a, t) da where m(a, t)

is the probability of bearing a female child atage a and time t.
These equations of course do not lead to Lotka's
tidy solutions since we here permit mortality andfertility to vary with time. In fact, they do noteven give much clue to the role of fertility andmortality in determining age distributions, sinceequation {3) requires us to know B(t - a), andequation (5) tells us that B(t - a) depends on theage distribution at time (t - a) as well as on fertility rates at time (t - a). It would appear thatto account for the present age distribution, one
needs to know:(a) an age distribution at some past date, and(b) schedules of mortality and fertility sineethat date.
However, as the date of the past age distribution is made more remote, its form makes less andless difference to the shape of the current age distribution. It seems intuitively plausible, in fart,that if the course of fertility and mortality were
known since t - "', the proportionate age distribution would be wholly determined at time t no
matter what the distribution at t - oo, assumingof course that the initial distribution was not
one-for example with no one under 50-headed
for extinction. Lotka (1939) shows this statement
to hold in the special case of endlessly un<"hanging
fertility and mortality. But the same factors thatcause the transient efferts of a particular initial
age distribution to disappear from the stable
population would also operate for any observed
time path of fertility and mortality rate. Mter aa suitably long period the effect of an initial age
distribution is swamped by the cumulative effectof the time pattern of vital rates. To put the point
more concretely, any age distribution with persons at every age could be assumed for 100 years
ago in place of the actual distribution. If such anassumed population were projected to the present
with observed fertility and mortality rates, theproportionate age distribution would differ negligibly from the actual. In short, the age distribution of a closed population is determined by themortality and fertility rates of recent history.
Another result from stable population analysis
is suggestive for our general case-the conclusionthat mortality differences have only second ordereffects on the age distribution. An increase or decrease in mortality tends to decrease or increase
all cohorts, implying a small effect of mortality on
the immediate age distribution as well as thestable. The short run transient age distribution
effect of a mortality change may differ somewhat
in form from the long run effects as expressed bystable age distributions, but the magnitude Hhould
be small in both instances.
We shall proceed with the provisional hypothesis that changes over time in mortality schedules
do not have major effects on the age distribution.
If this hypothesis is valid, a current age distribution could be closely approximated by ealculating
what the distribution would have been had observed fertility rates and unchanging mortality
rates prevailed for the last 80 or 90 years. I have
tried such a calculation using Swedish data. The
results are shown in Figure 3. The proportionate
distribution based on mortality unchanged for 90
years indeed does come close to the actual distribution. Moreover, the differences between thecensus population and the hypothetical are nearly
identical to the differences between two stable
populations with the same fertility, one based on
an 1860 Swedish life table, and the other on a life
table for 1946 to 50 (compare Fig. 1). The principal
effect of projecting with unchanged mortality is to
produce smaller fractions at ages under 25 andover 70. The reason is that since 1860 there havebeen disproportionate improvements in the probability of surviving in infancy and in the older
ages. If the projection from 1860 is calculatedwith mortality unchanging at 1946 to 50 levels,
the result is a distribution virtually indistinguishable from the census distribution. If in the 90
years before 1950 mortality had remained as highas in 1860 the result would have been a muchsmaller female population with an age distribution similar to the actual; if in these 90 years mortality had always been as low as in 1950, the result
would have been a much larger female population
with an age distribution nearly identical to theactual.
Figure 3 also shows the distribution that would
have resulted with observed mortality risks during
the 90 years before 1950, and with fertility assumed constant at 1860 levels. This figure makes
clear what is the major determinant of an age
distribution-the course of fertility. I wish I were
now able to present a short, simple explanation of
the effects of fertility on the age distribution. It isclear that fertility determines the age distribution, but an attempt to explain the relationshipprecisely soon runs afoul of major complications.
The basic difficulty is that when high fertility
produces a brief series of unusually large cohorts,
for example, these cohorts not only exceed their

171

...

FEMALE POPULATION, SWEDEN

FEMALEPOPULATIONPER CENT DISTRIBUTION

3r-------------------------------------------~ACTUAL POPULATION, 1950 - -FERTILITY UNCHANGED SINCE 1860 · - · - ·MORTALITY UrfCHANGEO SINCE 1860 ----~MORTALITY CONSTANT AT 1946-50 LEVELS ·-····AGE

FIGURE 3.

neighbors through their lifetime (tlris is a simple
effect), but also when they pass through the childbearing ages they produce a diffused "echo" oflarger birth cohorts. It is easy to give instructions
about how to compute these consequences, but
impossible to describe them simply in terms of theresultant age distribution.
We can make these observations about theeffects of fertility on age distribution:(l) Current and recent high fertility produce
a younger population than would low fertility,and vice versa.
(2) Transitory waves of unusually high or
low fertility create humps and hollows thatmove out through the age distribution as thecohorts move through life. In fact all of thenotches and knobs in the Swedish female age
distribution of 1950 can 1-)e traced back to unusual birth crops.
(:3) A long period of high fertility, or a periodof rising fertility creates a section of the age
distribution that tapers rapidly with age. Conversely a long period of low fertility, or a periodof falling fertility creates a relatively flat (oreven rising) section of the age distribution.
(4) When a cohort of unusual size reaches thechildbearing ages, it sets up an attenuated andflattened out "echo" in the number of births. It
is this second generation effect that makes theanalysis of the relation of the age distribution
to the course of fertility so complex.
Finally I turn to one or two practical or at least
worldly observations arising from this anslysis.

First, the rising fraction of the aged in westerncountries has not resulted from lowered death
rates but almost wholly from a long history of declining fertility. However, future improvements
in mortality may make this statement obsolete.
Second, improved mortality will not reduce the"burden of dependency" imposed on low income
areas because of their age distributions. To thecontrary, mortality reduction can be expected tolead to a somewhat higher fraction in dependentages. If the burden is to be reduced, fertility must
be lowered. Third, the remarkable variation in
fertility in the U. S. during the past 20 years has
produced a very irregular age distribution. The
succession of variously sized cohorts will doubtlesshave interesting implications in the next half century. Thus in the next 20 years the number of persons 20 to 24 is due to increase by at least 75 per
cent, while the number 40 to 44 can be expected to
decline by about three per cent.
REFERENCES

CoALE, ANSLEY, J., 1956, The effects of changes in mortality and fertility on age composition. Milbank
Mem. Fund Quart. 34: 79-114.
DuBLIN, Loms I., AND LoTKA, ALFRED J., 1925 On thetrue rate of natural increase. J. Amer. Stati~. Assoc.
£0: 30&-327.

LoRIMER, FRANK, 1951, Dynamics of age structure in a
po~;>ulation wi~h initially high fertility and mortality. PopulatiOn Bull. U.N. 1:31-41.
LoTKA, ALFRED J., 1939, Theorie analytique des associations biologiques. Deuxieme partie: Analyse demographique avec application particuliere a l'espece
humaine. Paris, Hermann et Cie.

172

20. On the Reproduction of Organisms with
Overlapping Generations
W.R. THOMPSON (1931)
From Bulletin of Entomological Research 22. Excerpts are from pages 147-152, 154-160, 163,168-170.

We omit the final section of Thompson's paper, in which he shows the effects
on population size of(a) lengthening the pre-reproductive interval, (b) lengthening
the period of reproduction, (c) increasing the number of offspring produced
per day, and (d) compacting the reproductive period while leaving the number
of offspring constant. The effects can be summarized in the approximate equality,
due to Lotka (1939, pp. 70-71)

where r is the intrinsic rate of growth, R 0 the net reproduction rate, and J.1 and(J 2 the mean and variance of the net maternity functjon. Where the net maternity'
function takes the form of a normal distribution the equalities are exact.
Much of Thompson's work can be translated to matrix form, where his
generation law G becomes the first row of a projection matrix whose sub-diagonal terms (survival) are unity. Thus, the generation law: G =(aT 1 + b T 2 +cT 3 +dT4 ), becomes the projection matrix M:

yz

T3

b

c

0

0

1
0

0

0d
0
0

l rB(T0 )'l = rB(T
B(T )~
0

0
0

1
0)

0
0

M
which we show post multiplied by the initial forebears B(T 0 ) to yields births
1) and total population B< 0 at time 1. (Mortality can be subsumed in thegeneration law or in the subdiagonals, hence the form shown here is quite general.) Matrix formulation facilitates the calculation of characteristic roots andvectors but was not yet adopted when Thompson wrote. It does not lend itself
as conveniently to the separation of births by generation or to an appreciationof generation overlap, which constitute important contributions of Thompson's
work.

B(T

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_20, © Springer-Verlag Berlin Heidelberg 2013

173

INTRODUCTION.

In the great majority of organisms, the reproducing individual does not engender
Between the beginning and the end of the reproductive period there is usually a certain interval of time, during which the productionof offspring continues in a manner depending, on the one hand, upon the specificcharacteristics of the organism, and on the other, upon the conditions under which
it lives. From this simple fact follow some interesting consequences.
all of its progeny simultaneously.

If the successive generations are separated by a long interval of time and conditions
during this interval are of such a nature that the organism is inactive and in the condition of obligatory dormancy, known as diapause, then, in spite of the increase of
the population, the generations will remain separate and distinct, and a comparisonof the total populations in the successive reproductive periods will enable us to determine the rate of increase.
On the other hand, if the interval between generations is short and favourable
to activity and the organism does not pass through a period of obligatory dormancy
at this time, then, as the species increases, the successive generations will come tooverlap, and the amount of overlapping will increase progressively as time goes on.
This is primarily due to the fact that when there is reproduction over a periodof time-intervals, in a typical individual of the species, and conditions remain constant,then the number of time-intervals during which births occur will increase from generation to generation. In this way there will be produced a.lengthening of there~
ductive period for the whole mass of individuals ; and this will cause a diminutionof the intervals between stages in the same generations, and between generations,so that individuals belonging to different stages of development, and to different
generations, will be found simultaneously in the field.
On account of these facts the calculation of the number of progeny and of thenumber of individuals in successive stages, existing at any given moment, presents
serious difficulties, which make both the practical and the theoretical treatment of
the subject laborious. Some years ago I began to study the matter and succeededin getting the process of calculation into a form in which the required data could beobtained, but the method of calculation was extremely tedious. The results obtained
were accordingly submitted, through the kindness of Professor M. Greenwood, ofthe Department of Epidemiology and Vital Statistics of the London School of Hygiene
and Tropical Medicine, to Mr. H. E. Soper. Within a few days I received from
Mr. Soper a communication, in which he explained and developed in detail an
e$emely elegant method of dealing with the problem. It is this method which
I have utilised in this paper. I have added such explanations as seem to me necessary in order to make the process of reasoning clear to those who, like myself, havelittle knowledge of mathematics, and have worked out a series of examples to show
how the method may be employed in entomological problems. The credit for thesolution utilised is entirely due to Mr. Soper, and I am glad to have this occasion of
thanking him for the kind and patient interest he has shown in the problems submitted to him. . ..
The first part of the paper is devoted to an exposition of the problem ; the second
part. to ~he methods which can be used in dea~ng with it ; and the third part to an
apphcation of these methods. The mathematical theory of the method is given by
Mr. Soper in the Appendix.
. Th~ object of the paper is, in. the firs~ place, to enable the entomologist to
mvestigate the natural control of ~ns~cts. With reproduction of this type ; and, inthe second place, to show how vanatlon m the length of the pre-reproductive andr~productive periods, and in .the production of offspring, affect the increase in population. When the length of life and of the pre-reproductive and reproductive periods

174have been determined, and the average daily production of progeny is known, we
can calculate from the formulae given the daily births or the daily population ofindividuals of all ages, when no causes of mortality intervene. By comparing thefigures with those obtained in an actual experiment, we can then determine whencauses of mortality begin to operate and get some idea as to the nature and effectof the controlling factors which come into play at any given date. Organisms withoverlapping generations are particularly suitable for work of this kind, because of
their rapidity of reproduction, but the complex way in which their populations increase
has, hitherto, constituted a ve'ry serious obstacle to their study. The formulaedeveloped by Mr. Soper enormously simplify the problem and constitute, I believe,a very important contribution to our methods of study of the process of naturalcontrol.

I.

Let us suppose, to begin with, that we have an initial organism or stem-mother,producing d offspring per day during a certain period, which we Iilay call k days
for the moment ; and let us suppose that the reproducing individual dies on the day
it has produced its last offspring. Then, in the first generation, if we put k=4, forthe sake of simplicity, we shall have births occurring as fo1lows :Day •. .
1
2
3
4
5
6
7
8
9
10
d
d
d
d
G.I
Now suppose that both the organisms produced in the first generation and theenvironmental conditions under which they live remain identical. The group oforganisms produced on the second day will reproduce a day later than those producedon Day 1 ; the group produced on the third day, two days later than those producedon Day 1, and so on. The second generation, produced by the reproduction of thegroups occurring on the successive days, will thus be as follows, since each individualof the group of d produces d offspring :DAY

G.II
G.II
G.II
G.II

1

2

3

4

5

6

7

1d2 · uz · 3d"

4d2

3d2

U2

ld2

8

9

10

1 d2
2
3
4

Total

In the next generation the process will obviously continue in the same way ;during the first reproductive period ld2 individuals will produce d offspring per day ;during the second period 2d2 will produce d each per day ; during the third period
3d2 will produce d per day, and so on-the result being as follows :DAY

G.III

1 dS

2
3

4

5
6
7
Total

1

2

3

4

5

dS

dS

dS

3dS

3dS
4dS

us us us

us

3d3
4d3
3dS

6

7

3dS
4d3
3d3

4dS
3d3

us us

dS

8

9

3d3
2dS
d3

us

dS

IdS . 3dS . 6d3 • lOdS · JUS · JUS · lOdS · 6d3 . 3dS .

175

10

d3

IdS

When we compare the series. of births in the successive generations the first thingthat we notice is that the number of days (or time-intervals) on which births occur
becomes greater in each generation. Thus, as an organism increases in numbers,
the length of time during which births occur in the field will also increase, from generation to generation, and so will the length of time during which the successive
developmental stages are found, without there being, necessarily, any deviation
whatever from the specific reproductive habits, or any change in the environmental
conditions; or, in other words, any real irregularity in the course of events....
Furthermore, since the number of time-intervals during which births occur
increase from generation to generation, it is also evident that, as time goes on, theinterval between the end of one generation and the beginning of another will become
smaller and smaller, and then disappear, after which the successive generations will
overlap. The overlap will become greater and greater as the generations succeed
one another. Finally, the number of generations overlapping in a given timeinterval will also become greater, though the rate of increase is not a smooth andregular one.
Thus, suppose the pre-reproductive period of a species is of two days (including
day of birth) and the reproductive period also two days, each individual producing
d births per day, the succession of events will be as follows : The first and second generations do not overlap, but the second and third overlap
on one day, the third and fourth on two days, the fourth and fifth on three days,
the fifth and sixth on four days, the nth and (n+I)th on (n-1) days. On the first
four days individuals of only one of the two generations produced are found on any
one day; on the fifth day individuals of G.II and G.III co-exist; on the seventhand eighth days individuals of G.III and G.IV, on the ninth and tenth days individualsof G.IV and G.V; on the eleventh day individuals of G.IV, V, and VI, and on theseventeenth day individuals of G.VI, VII, VIII, and IX.

II.
that the direct calculation of the progeny
preceded
has
what
from
evidentbe
It will
produced day by day in the successive generations, to say nothing of the total
number of progeny produced by all generations taken together, or, to take the mostdifficult case of all, of the total population, including both the newly born and theolder individuals, is a very complex matter. In order to deal with the problem it is,therefore, necessary to simplify the operation.
It is, of course, possible to represent the increase in population by using the compound interest formula, according to which the number of individuals existing at
any given moment is given by the equation
Nl=NoeKt
Where N 1 is the population at the moment considered, N 0 the initial population,t the number of time-intervals (hours, days, years, etc.), e the incommensurable
number 2·71828 . . . , and K a constant depending upon the particular case considered. This formula is, however, not satisfactory for experimental work. In
the first place, the constant K cannot be determined until the growth of the population
under certain definite conditions has been studied during a considerable period ; inthe second place, no intelligible significance can be attached to the constant afterits value has been determined; in the third place, the growth of the population isconsidered in this formula to be at every moment proportional to the size of the
population, which is not true except with large numbers and over long periods, andcannot be safely taken as a basis for the examination of experimental data. Thus,
using the figures for births in the last example given in this paper, we obtain values
from K ranging from 0-652 to 0 ·693. The former value is valid when the population
reaches a high level, but under such conditions accurate analytical experimental
work is not possible. The number of individuals at the end of five time-intervals
might be anything from 208 to 255. It is therefore necessary to deal with thereproductive process as a discontinuous phenomenon.

176

Suppose the ancestor or stem-mother is born on ro, and that we represent thesuccessive days or " time-intervals " on which reproduction takes place by
T 1, T2, rs, T 4 , • • • meaning the first, second, third, fourth, etc., days, and suppose
one offspring is produced each day. Suppose the reproductive period is one of fourdays, and the pre-reproductive period is of one day, including the day of birth, thenthe first generation, = G1 may be writtenG1=1 T1+1 T~>-+1 TB+1 T'.
Now the one individual born on Tl will begin to reproduce on T2 and from thence•
forward during four days ; the one individual born on T2 will begin to reproduce on
T3 and from thenceforward during four days ; and similarly, for the individuals
born on rs and T', which begin reproducing on T' and T5 respectively. Thus,
writing the series of births for G~>- we haveG2=1 T~>-+1 TB+1 T'+1 T5
1 TB+1 T'+1 T5+1 T&
1 T'+1 T5+1 T&+1 T7
1 T5+I T6+1T7+1 rs

which is-=(Tl+ T~>-+ TB+ T4)2.
In GB, the one individual produced on T2 will reproduce on four succeeding days,
and similarly, for the two produced on rs, the three produced on T 4 and so on ;and if these series of births are written out and added as before, we obtainG8=1 TB+3T'+6T5+IOT6+12T7+12TS+IO:fii+6Tl0+3Tll+l Tl~>­=(Tl+ TZ+ TB+ T')B.
It is clear that, in general, we haveG.,={ Tl+ T~>-+ TB+ T')",and that by multiplying this out, or, as the mathematicians say, " expanding it,"we shall obtain a series of T's multipled by numbers, or coefficients in which theindex of T designates the day and the coefficient gives the number of births on thatday in the generation considered.
From this simple case all others may readily be derived without any alterationin the reasoning.
Thus, suppose the pre-reproductive period is of three days, including the day ofbirth, and the reproductive life is of four days, as before. If the original ancestor
or stem-mother was born on ro {an assumption made in the first case), then its
first offspring would be produced, not on Tl but on rs, and the first generation would
be writtenG1=(1 TB+1 T'+1 T5+1 T&);
then the individual produced on rs would begin reproducing on T 6 , and we should
obtain, by proceeding as before, a series for the second generation equivalent to{1 TB+ 1 T'+ 1 T5+ 1 T6)2.
Finally, if, instead of one birth on each day, we assume d on each day, we should
obtainG1=d(Tl+T2+TB+T');
G 2 =d~>-(T1+ T2+ rs+ T')~>-;
G,.=d •( Tl+ T2+ TB+ T'),.;
while if the number of births varied from day to day, so that we ha.dGl=(aTl+bT2+cTB+dT'); we shall have177

Suppose that on one or more days in the reproductive period no progeny are
produced. This is allowed for simply by omitting these days. Thus, in the lastseries mentioned, let b=O. The formula for the nth generation will then be :G,.=(aTl+c TS+dT4)". .. .
What has preceded: covers fully the series of births (or production of offspring)in each generation considered separately ; but it does not give us any information
as to the overlapping of the generations, nor does it enable us to calculate directly
the total number of births produced by the individuals of all the different generations
which happen to be reptooucing on any given day ; and that individuals belonging
to different generations will be producing simultaneously we have already seen....
If, taking the successive generations in the example already studied, we write
them down and add, we haveG1+ G2 + G3 =(T1+ TS+ Ta+ T4)+(T1+ T2+ TS+ T4)Z
· + ( Tl+ TS+ TS+ T4)3.
It is obvious, from what has preceded, that by expanding these expressions andadding up all the Tl, Til, TS ......... T12 we shall get the number of births producedby all the individuals of all the generations existing simultaneously on any given day,up to and including the third generation.
If we designateTl+ TS+ TS+ T4 by G
so that we have

we can write

G,.=G",

G1+G 1 +G 8=G+GZ+GS
G(I-GS)=1-G

by the usual rule for the summation of a geometrical series.
But this formula does not represent the total number of individuals produced
up to the day on which G3 ends (TlZ), because by that time generations G4 to G11
inclusive will already have begun.
Let us, therefore, take the sum of G1+G 2 +G 8 + ............to n generations.
We have, including the original ancestor or stem-mother1+G1+G1+G8+ ......... +G,. =1+G+G2 +G•+ ......... +~(1-G")= (1-G).
We have also G= T1+ TS+ TS+ T4, which may be writtenT-T5

1-T

as division will prove.
Substituting, we have--{Tl-T5}"
I-T
I+G 1 + ......... +G,. = -{Tl-T5}
I

1

1-T

(I-T) (1-{ Tl+ TS+ TB+ T4 ~ ")1-2T+T5(1-7) (I-T'! -higher powers of T)
I.-2T+T5(I-T -T"-higher powers of 7).
=
I-2T+T5=

178

Now here, exactly as in the case of-{ T-T~l_I-T J"

we have only to divide the numerator by the denomination to get a series in which
the index of T will designate the day and the coefficient of T the total number ofbirths for that day.
If we suppose n is infinitely large, then T" and the powers of T following it
may be neglected, and the eXpression becomes(I-T)(I-2T+T5)
which is the same thing as
I+G+GZ+Ga+ ......... +G"
summed to infinity as a diminishing series
1=I-G"
When we substitute{ T-T5}
I-T
for G.
The same method of procedure, up to this point, may obviously be used whatever
the initial value ofT, whatever the number of terms in the series for G1 , and whatever
the daily birth rate, provided it is constant and regular.
Thus we haveG1=(TS+ T'+ To+ TG+ T?) ; this can be written(I-T6) TS-T8.
G1 =TS l-T
I-T 'so that Gl+ GB+ GS+ ........... .
l-T
l-T-TB+TB
Ifwehave-

G1=(dTS+dT'+dTo +dT6+dT7)= d(~~J/)
the formula will bel-T
I+G1+G 1+Ga+ ......... etc. (l-T-dTS+dT8)
If the number of births on the successive days of the period of reproduction is
not the same, suppose these births to be a, b, c, d, as before. We then haveG =aTl+bTZ+cTS+d T'
G1=(aTl+bTZ+cTS+d T')2=G2 etc., and1
I+G1+G 1 + ......... =l-G
1

1-(aTl+bTI!+cTS+ dT')
which will be valid even when there are no births on one or more days of the reproductive period, or, in other words, when one or more of the coefficients of T are madeequal to 0....
It is evident that if the organism continues to live after it has ceased to reproduce,
this will still be valid, so that if the life is of l days, the series of births must be multiplied by-179to get the population series. Thus, if the formula of the births series is-

I-T

1-2T+T5
the formula for the population series will be-

1-P

l-2T+T5
The rule for finding the successive members of the series is precisely the same as before,
after the term Tl has been obtained.
.
The series given by the formulae includes, of course, only the individuals which
are capable of reproducing themselves, i.e., only the females; to obtain the total
number of individuals, each member of the series must be multiplied by a figure
representing the ratio of total population to females, e.g., by 2, if the numbers ofmales and females in the progeny are equal, on the average ; by 4, if there are threetimes as many males as females, etc.
Provided we know the length of the reproductive period we can readily obtainfrom observations giving either the series of daily births, or the series giving thetotal number of individuals produced up to and including each day, the formulafor individual reproduction.
It has already been shown that if the generation formula is represented by G,
the series of daily births is given by the formula!

1-G~

Thus, if the series be represented by S, we have1
S = l-G' whence
1
G=

1-s·

Thus, suppose the series of daily births, due to an individual reproductive period offour days, is1+1 T1+31'2+8TB+21 T~+ ................................. ,we obtain for the generation formula, by dividing the above series into 1 and subtracting the result from 1 :G=l T1+21'2+3TS+4T~.
APPENDIX.
PROBLEMS OF SIMPLE PROPAGATION WHOSE SOLUTION MAY BE FACILITATED BY
THE USE OF OBJECTIVE SYMBOLS.

By H. E. SOPER,
Division of Epidemiology and Vital Statistics, London School of Hygiene andTropical M edicime.
The problem of finding the resultant at any time, from given independent incoming
births, or outgoing, i.e., destroyed, births of the operation of a given law of reproduction and the converse problems of how to find the independent birth happenings,
knowing the resultant births and the law, and how to find the law knowing theindependent birth happenings and resultant births, and many such problems, aresusceptible of analysis by a process, akin to that known as " generating functions "(akin also to chemical formulation), but which is more aptly described as one that
employs objective symbols to indicate and array the differing characters of the things
enumerated.

180

In the present instance we epumerate " events " ; and the " time of occurrence "is the characteristic that we desire to set against any event. Let then T be thesymbol, whose power or index has the duty of indicating the number of units oftime (which unit we shall call a day for convenience although any other period may
be chosen), from any zero time, at which the event happens. Hence T 0 (or 1),
Tl (or T), T2, ... 1", ... indicate zero time and times 1, 2, ... t, ... days after zeroand an array or formula as,
F o=2+5T+11 T2+25TS+ ... ,... (i)will be :read as " 2 events at zero time, 5 events one day after, 11 events two days
after, 25 events three days after zero .... "Suppose the event is a birth and suppose each event gives rise to other like events,that is births, at known intervals, then the law of reproduction can also be expressedin symbols and, for instance, the law,
G=5 TS+ 10 T~>+3 Ts,.. . (ii)
would indicate 5 births 3 days after, 10 births 5 days after, and 3 births 8 days afterzero, which can be the birth of the reproducing organism.
In this case it will be clear that if F 0 stand for a births formula as (i) and G fora generation formula as (ii), then
Fl=F 0 x G... (iii)
will formulate, in the time symbol T, the births F 1 in the first generation arisingfrom the original births F 0 .
Here it may be pointed out that T is an objective symbol only, without thequantitative connotation of such a symbol as t in algebra, and that other large
letters as F and G stand for arrays of the symbol T such as(~) or (ii). We interpret
11 T 2 in statistics in much the same way as we should interpret 11 H2 in chemistry,
but in statistics the convention leads to a useful calculus, based upon such algebraic
renderings as (iii).
The following elementary propositions may now be enunciated.
Proposition I.-The independent or original incoming births being F 0 and the generation

law being G to find the births in the first and each succeeding generation and thetotal births.

The original births are
The 1st generations births are
The 2nd
The r-1th

..
..

..
..

The total births to the r-lth generation are
The total births to unlimited generations are
Proposition II.-To find the population under these conditions when each birth st~r­
vives s days or unit intervals.
A birth at zero time gives rise to a population 1 in the succeeding intervaland 1 in each succeeding interval to s intervals. This succession may be
represented by
1-T•

1+T+T2+ ... +T•-1= 1 -T.

• We say that Gr is negligible when ,.=co, not because G <I, but because we do not concern
ourselves with the exact enumeration of events at infinite time.

181

Hence from births F formulated as occurring at successive instants oftime 0 1 2 ...... there results a population in the succeeding intervals of time, twhich we may call the 0 1 2 ... intervals, formulated as,

pi-Ts
1-T
Instants at which births occur

Tl

TO

Intervals in which population is enumerated

I

TO

I

TS

Tl

I

T"

Proposition III.- To find the independent incoming births F 0 that must be postulated
to give rise to the observed total births F under the knqwn generation law G.
Since, by Proposition I, F
it follows that,

Fo

=I=G,

F 0 = F (1-G).

Proposition IV.-To find the generation law, given the independent incoming or originat-

ing births F 0 and the observed total births F.
FoSince, by Proposition I, F

it follows that,= l-G'

Fo
G =1-y.

Proposition V.-A series of events being expressed as an array F in the symbol T, tofind the slope of the curve and its inflexson and to find the successive sums or total$,these also being expressed in a"ay form.
The slope or backward first difference is (1-1) F.
The in:flexion or backward 2nd difference is (1-T)•F.
The sum to (and including) the instant is 1F T'
The second, third, etc., sums are (i F1) 2-, (I ~1) 8 , etc.

The succeeding items follow from the first, and the first is obvious since ifthen

F=a+bT+cT2+ ......

(l-1)F=a+(b-a)T+(c-b)T2+ ......

Many other propositions could be stated, but the process iS so elementary .thatthe student will be able, with the help of the above examples and using similar
algebraical renditions and transformations, to evolve many desired results.

t It would perhaps have accorded better with our common conceptions had populations
been put at instants ana births in intervening intervals of time.

182

21. The Population Consequences of Life History
Phenomena
LAMONT C. COLE (1954)

From Quarterly Reriew of Biolow· 29. Excerpts are from pages 107-115.

The part of Cole's article included here outlines the relationship of Thompson's
work (1931, paper 20 above) to stable population theory. Omitted sections
discuss the relationships between the rate of increase, numbers and spacing ofoffspring, and age structure, with their implications for species survival.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_21, © Springer-Verlag Berlin Heidelberg 2013183

Quite recently a number of ecologists haverecognized the importance of a knowledge of thevalue of r for non-human populations and havecomputed its value for various species by employing empirical values of age-specific birth rates
and survivorship (Leslie and Ranson, 1940;
Birch, 1948; Leslie and Park, 1949; Mendes, 1949;
Evans and Smith, 1952). While Chapman's term
"biotic potential" would seem to have ecological
merit as the name for this parameter r it has beenvariously called by Lotka the "true," the "incipient," the "inherent," and the "intrinsic" rateof increase, and by Fisher (1930) the "Malthusianparameter" of population increase. Probably forthe sake of stlrbilizing nomenclature it is advisable
to follow the majority of recent writers and refer
tor as "the intrinsic rate of natural increase."
In the works of Dublin and Lotka (1925),
Kuczynski (1932), and Rhodes (1940) on humanpopulations and in the papers mentioned above
dealing with other species, the value of r has
typically been determined by some application of
three fundamental equations developed by Lotka(1907a, b; Sharpe and Lotka, 1911). He showedthat if the age-specific fecundity (b,.) and survivorship (l,) remained constant, the population wouldin time assume a fixed or "stable" age distribution
such that in any interval of age from x to x + dx
there would be a fixed proportion (c,.) of the population. Once this stable age distribution is established the population would grow exponentiallyaccording to our formula (1') and with a birthrate per head, (3. Then the following equations
relate these quantities:

1"' e-n~l. b, dx 1
1"' e-nl,.dx 1/fJ=

(4)

{Je-•"' l., = c,..

(5)

=

and

(3)

While the use of formulas (3), (4), and (5) tocompute the value of r often presents practicaldifficulties owing to the difficulty of approximating
the functions l., and b., by a mathematical function,
and also because the equations usually must be
solved by iterative methods, it may fairly bestated that Lotka's pioneer work establishing these
relationships provided the methods for interpreting
the relationships between life history features andtheir population consequences.

However, the exceedingly important ecologicalquestions of what potential advantages might berealized if a species were to alter its life history
features have remained largely unexplored.
Doubtless, as already noted, this is largely to be
explained by a certain suspicion felt by biologists
toward analyses such as those of Lotka, which
seem to involve assumptions very remote from therealities of life histories as observed in the field
and laboratories. A particularly pertinent statementof this point of view is that of Thompson (1931),
who recognized the great practical need for methods of computing the rate of increase of natural
populations of insects adhering to particular life
history pattems but who insisted that the reproductive process must be dealt with as a discontinuous phenomenon rather than as a compoundinterest phenomenon such as that of formula (1').
His methods of computation were designed to givethe exact number of individuals living in any
particular time period and, while he recognizedthat the population growth can be expressed in an
exponential form such as (1'), he rejected its useon these grounds:"In the first place, the constant (r) cannot be determined until the growth of the population under certain
definite conditions has been studied during a considerable period; in the second place, no intelligible
significance can be attached to the constant after its
value has been determined; in the third place, thegrowth of the population is considered in this formula
to be at every moment proportional to the size of the
population, which is not true except with large numbers and over long periods and cannot be safely taken
as a basis for the examination of experimental data."
In the following sections of the present paper an
effort will be made to reconcile these two divergent
points of view and to show under what conditions
Thompson's "discontinuous" approach and thecontinuous methods lead to identical results.
Practical methods of computation can be foundedon either scheme, and there are circumstanceswhere one or the other offers distinct advantages.
It is hoped that a theoretical approach to population phenomena proceeding from exact computational methods will clarify the meaning of someof the approximations made in deriving equations
such as (3), (4), and (5) by continuous methods,
and will stimulate students of ecology to a greaterinterest in the population consequences of life
history phenomena.
Before proceeding to a discussion of potential
population growth, one point which has sometimes

184caused confusion should be mentioned. This concerns the sex ratio and the relative proportions ofdifferent age classes in the growing population.
Once stated, it is obvious that if a population isalways growing, as are the populations in themodels used for determining potential population
growth, then each age and sex class must ultimately come to grow at exactly the same rate as
every other class. If this were not the case thedisproportion between any two classes would cometo exceed all bounds; the fastest growing classwould continue indefinitely to make up a larger
and larger proportion of the total population. It
is thus intuitively recognizable that with fixed life
history features there must ultimately be a fixed
sex ratio and a stabfe age distribution. In discussing potential population growth it is often
convenient to confine our attention to females oreven to a restricted age class, such as the annual
births, while recognizing that the ultimate growthrate for such a restricted population segment mustbe identical to the rate for the entire population.
SIMPLEST CASES OF POPULATION GROWTH

Non-overlapping generations
The simplest possible cases of population growthfrom the mathematical point of view are those inwhich reproduction takes place once in a lifetimeand the parent organisms disappear by the time
the new generation comes on the scene, so that
there is no overlapping of generations. This situation occurs in the many plants and animals which
are annuals, in those bacteria, unicellular algae,
and protozoa where reproduction takes place byfission of one individual to form two or moredaughter individuals, and in certain other forms.
Thus in the century plants (Agave) the plant dies
upon producing seeds at an age of four years or
more, the Pacific salmon (Oncorhynchus) dies after
spawning, which occurs at an age of two to eight
years (two years in the pink salmon 0. gorbuscha),
and cicadas breed at the end of a long developmental period which lasts from two years (Tibicen)
to 17 years in M agicicada. For many other insects
with prolonged developmental stages such asneuropterans and mayflies potential population
growth may be considered on the assumption thatgenerations do not overlap.
In these cases, perhaps most typically illustrated
in the case of annuals, the population living in
any year or other time interval is simply the

number of births which occurred at the beginningof that interval. Starting with one individual whichis replaced by b offspring each of which repeats thelife history pattern of the parent, the population
will grow in successive time intervals according to
the series: 1, b, lfl, 1;3, b4 , • • • b~. Hence the numberof "births," say B,., at the beginning of any timeinterval, T .,, is simply b"' which is identical withthe population, P ,., in that interval of time. If the
population starts from an initial number Po we
have:
(6)

which is obviously identical with the exponential
formula (1'), P., = Aer"', where the constant A is
precisely Po, the initial population size, and r =In b; the intrinsic rate of increase is equal to thenatural logarithm of the litter size.
If litter size varies among the reproductive
individuals, with each litter size being characteristic of a fixed proportion of each generation, it isprecisely correct to use the average litter size, sayb, in the computations, so that we have r = In b.
Furthermore, if not all of the offspring are viable,
but only some proportion, say lt, survive to reproduce, we shall have exactly r = In fill. Thus,
mortality and variations in litter size do not complicate the interpretation of population growth incases where the generations do not overlap. On
the other hand, even in species which reproduce
only once, if the generation length is not the samefor all individuals, this will lead to overlapping
generations, and the simple considerations which
led to formula (6) will no longer apply. In otherwords, we can use an average figure for the litter
size b but not for the generation length x. It will
be shown in the next section, however, that themore general formula (1') is still applicable.
In these simplest cases the assumption of ageometric progression as the potential form of
population growth is obviously correct, and numerous authors have computed the fantastic numbers of offspring which could potentially result
from such reproduction. For example, according
to Thompson (1942), Linnaeus (1740?) pointed
out that if only two seeds of an annual plant grew
to maturity per year, a single individual could
give rise to a million offspring in 20 years. (In all
editions available to the present writer this interesting essay of Linnaeus' is dated 1743, and thenumber of offspring at the end of twenty years isstated by the curious and erroneous figure 91,296.)

185

That is, P 20 = 220 = & 0 In2 = 1,048,576. Additional
examples are given by Chapman (1935, p. 148).
Formulas (1') or (6) may, of course, also be usedin an inverse manner to obtain the rate of multiplication when the rate of population growth is
known. For ~e example given by Molisch (1938,
p. 25), referring to diatoms reproducing by binary:fission where the average population was observed
to increase by a factor of 1.2 per day, we have
1.2 = e"' In 2, where xis the number of generations
per day. Solving for 1/x, the length of a generation,
ln 2.69315
we obtain 1/x = - - = - 8 - = 3.8 days.
ln 1.2.1 232

Overlapping generations
Interest in computing the number of offspringwhich would be produced by a species adhering toa constant reproductive schedule dates back at
least to Leonardo Pisano (=Fibonacci) who, in theyear 1202, attempted to reintroduce into Europe
the study of algebra, which had been neglected
since the fall of Rome. One of the problems in his
Liber Abbaci (pp. 283-84 of the 1857 edition)
concerns a man who placed a pair of rabbits in an
enclosure in order to discover how many pairs of
rabbits would be produced in a year. Assumingthat each pair of rabbits produces another pair in
the first month and then reproduces once more,giving rise to a second pair of offspring in thesecond month, and assuming no mortality,
Fibonacci showed that the number of pairs in eachmonth would correspond to the series
1, 2, 3, 5, 8, 13, 21, 34, 55, etc.,
where each number is the sum of the two preceding
numbers. These "Fibonacci numbers" have arather celebrated history in mathematics, biology,
and art(Archibold, 1918; Thompson, 1942; Pierce,1951) but our present concern with them is merely
as a very early attempt to compute potential
population growth.
Fibonacci derived his series simply by following
through in words all of the population changes
occurring from month to month. One with sufficient patience could, of course, apply the sameprocedure to more complicated cases and could
introduce additional variables such as deductions
for mortality. In fact, Sadler (1830, Book III) did
make such computation~ for human populations.
He was interested in discovering at what ages
persons would have to marry and how often theywould have to reproduce to give some of the rates
of population doubling which had been postulatedby Malthus (1798). To accomplish this, Sadler
apparently employed the amazingly tedious procedure of constructing numerous tables corresponding to different assumptions until he found
one which approximated the desired rate of
doubling.
Although we must admire Sadler's diligence,anyone who undertakes such computations willfind that it is not difficult to devise various waysof systematizing the procedure which will greatly
reduce the labor of computation. By far the bestof these methods known to the present writer isthat of Thompson (1931), which was originallysuggested to him by H. E. Soper.
In the Soper-Thompson approach a "generationlaw" (G) is written embodying the fixed life history
features which it is desired to consider. The symbol
T'' stands for the xth interval of time, and a generation law such as G = 2T1 + 2T2 would be readas "two offspring produced in the first time interval
and two offspring produced in the second timeinterval." This particular generation law might,
for example, be roughly applicable to some bird
such as a cliff swallow, where a female produces
about four eggs per year. Concentrating our attention on the female part of the population, we mightwish to compute the rate of population growthwhich would result if each female had two female
offspring upon attaining the age of one year andhad two more female offspring at the age of two
years. The fundamental feature of the Thompson
method is the fact that the expression:
1

1-G

(7)

is a generating function which gives the series ofbirths occurring in successive time intervals. In
the algebraic division the indices of the terms
Tl, T2, etc., are treated as ordinary exponents andthe number of births occurring in any time interval
T"' is simply the coefficient of T"' in the expansionof expression (7). Thus, for our example whereG = 2T1 + 2T2 we obtain:
1= 1 + 2Tt + 6T2 +1- 2T 1 - 2T2

16P + 44T• + 120T6 + 328T6 + ··· ,showing that one original female birth gives riseto 328 female offspring in the sixth year. The

186

generation law G = 2Tl + 2T2 + 21". If we were
further interested in the case where not all of theoffspring survive for three years, the coefficientsin the generation law need only be multiplied by
the corresponding survivorship values. For example, if one-half of the individuals die between
the ages of one and two, and one half of the remainder die before reaching the age of three we= 1, l 2 = ~' 13 = X, and thewould have
above generation law would be revised to G =2Tl + T2 + ~1". The future births per originalindividual would then beseries could be continued indefinitely to obtain thenumber of births any number of years hence.
However, in practice it is not necessary to continue the division. In the above series the coefficientof each term is simply twice the sum of the coefficients of the two preceding terms; hence the generation law gives us the rule for extending theseries. G = 2T1 + 2P. instructs us to obtain eachnew term of the series by taking twice the preceding term plus twice the second term back. In
the case of the Fibonacci numbers we would have
G = Tl + P., telling us at once that each newterm is the sum of the two preceding it.
From the birth series we can easily obtain theseries enumerating the total population. If eachindividual lives for A years, the total populationin T"' will be the sum of A consecutive terms in the
expansion of the generating function. Multiplying
formula (7) by the length of life expressed in theform 1 + T 1 + T2 + T3 + · · · + TX-t will givethe population series. In our above example if we
assume that each individual lives for three years,
although, as before, it only reproduces in the firsttwo, we obtain for the population

z,

- 1-

1-G

=

1 + 2T' + 5T2 + 25/2T3 + 31P +151/2T~+ ···.

Very generally, if the first reproduction for a
species occurs at some age a and the last reproduction occurs at some age w, and letting b. andl"' represent respectively the age-specific fecundity
and survivorship, we may write the generation
law as:G = lab, T"' + la+l ba+l T<Hl + •••1 + T' + T2 = 1 + 3T' + 9P. +1- 2P- 2T2(8)

24P + 66T• + 180T~ + 492T6 + ··· ,a series which still obeys the rule G = 2T1 + 2P..
Thompson's method for obtaining the exact
number of births and members of the populationin successive time intervals is very general. As in
the case of non-overlapping generations, the coefficients in the generation law may refer to averagevalues for the age-specific fecundity. Also thelength of the time intervals upon which thecomputations are based can be made arbitrarily
short, so that it is easy to take into accountvariations in the age at which reproduction occurs.
For the above example, time could have beenmeasured in six-month periods rather than years
so that the generation law would become G =2T2 + 2T4, with the same results already obtained.
Furthermore, the factor of mortality can easily
be included in the computations. For example,
suppose that we wish to determine the rate of
population growth for a species where the females
have two female offspring when they reach theage of one, two more when they reach the age oftwo, and two more when they reach the age ofthree. Neglecting mortality, this would give us the

Therefore, in the Thompson method we have acompact system of computation for obtaining theexact number of births and the exact population
size at any future time, assuming that the significant life history features (a, w, l., and b.) do notchange.
Not all of the possible applications of Thompson's method have been indicated above. For
example, formula (7) may be used in an inverse
manner so that it is theoretically possible to workback from a tabulation of births or population
counts made in successive time intervals and discover the underlying generation law. Formulas (7)and (8), together with the procedure of multiplying
the birth series by the length of life expressed as asum of T• values, provide the nucleus of the systemand offer the possibility of analyzing the potential
population consequences of essentially any lifehistory phenomena. The system has the merit oftreating the biological units and events as discontinuous variates, which, in fact, they almost
always are. The members of populations are
typically discrete units, and an event such as
reproduction typically occurs at a point in time187with no spreading out or overlapping between
successive litters. While survivorship, l.,, as apopulation quantity, is most realistically regarded
as continuously changing in time, the product l.,b,.
which enters our computations by way of formula(8) is typically discontinuous because of the discontinuous nature of b.,.
It is quite obvious that eq'\18-tions of continuous
variation such as (1') are often much more convenient for purposes of computation than theseries of values obtained by expanding (7). This isespecially true in dealing with the life histories of
species whicl. have long reproductive lives. In
writing a generation law for man by (8) we shouldhave to take a at least as small as 15 years and oo
at least as great as 40 years, since for the population as a whole reproduction occurs well outside ofthese extremes and it would certainly be unrealistic
to regard b., as negligibly small anywhere between
these limits. Thus there would be at least 25 terms
in our generation law, and the computations would
be extremely tedious. By selecting special cases
for study it is sometimes possible greatly tosimplify the procedures. For example, if one isinterested in the case where there is no mortality
during the reproductive span of life and where thelitter size is a constant, say b, the expression for
the generation law (8) can be simplified to:bT"- bT•-+1
G - bT" + bT"+1 + ··· bT"' - --:1 ---T=--··
Since one can also write the length of life as
1- T"

1+T1 +Tt+ .. ·+T"-1 ... 1 _T'
the generating function for the total population
simplifies to
1- T"
1 - T - bT" + b'P'+' .

This last formula is much more convenient forcomputations than one containing 25 terms or soin the denominator, but it applies only to a very
special case and is much less convenient thanformula (1 '). Consequently, great interest attaches to these questions: can (1') be used as a
substitute for (7)? (i.e., does Thompson's method
lead to a geometric progression?) and, if it is soused, can the constants, particularly r, be interpreted in terms of life-history features?

...,.

-1000

-...
.,.

'5100

J...

..

00

121•$671

Time Intervals

FIG. 1. ExACT VALUES OF PoPULATION GROWTH IN
TERMs OF Bmms PEl!. UNIT Tnm UNDER SEVERAL GENERATION LAws, WHEN EACH FEMALE
HAS A TOTAL 01!' FOUR FEMALE Ol!'l!'SPB.ING

In each case it is assumed that a single female exists
at time zero and produces her four progeny on or before
her fourth birthday. The plotted points represent exact
values as determined by Thompson's method. To theextent that the points for any generation law fall on astraight line in this logarithmic plot, they can be represented by the exponential growth formula (1'), and theslope of each line is a measure of the intrinsic rate ofnatural increase (r).

THE GENERALIZATION OF THOMPSON'S METHOD

Fig. 1 shows the exact values, as determined byThompson's method, of the birth series arisingfrom several generation laws (life-history patterns)
which have in common the feature that in eachcase every female produces a total of four female
offspring in her lifetime and completes her reproductive life by the age of four "years." The numberof births is plotted on a logarithmic scale, hence
if it can be represented by formula (1'), P = Ae•'"or, logarithmically, In P - In A + rx, the points
should fall on a straight line with slope proportional to r. It is apparent from Fig. 1 that after
the first few time intervals the points in each caseare well represented by a straight line. Therefore,
except in the very early stages, formula (1 ') does188

give a good representation of potential population
growth. The question remains, however, as towhether we can meet Thompson's objection to (1 ')and attach any intelligible significance to the constants of the formula. From Fig. 1 it is obviousthat the lines do not, if projected back to time 0,
indicate exactly the single individual with which
we started. Thus, in these cases the constant A
cannot be precisely Po as was the case with nonoverlapping generations.
Before proceeding to interpret the constants offormula (1') for the case of overlapping generations it will be well to notice one feature of Fig. 1
which is of biological interest. In the literature ofnatural history one frequently encounters references to the number of offspring which a female
can produce per lifetime, with the implicationthat this is a significant life-history feature. The
same implication is common in the literature
dealing with various aspects of human biology,
where great emphasis is placed on the analysis of
total family size. From Fig. 1 it will be seen that
this datum may be less significant from the standpoint of contributions to future population thanis the age schedule upon which these offspring areproduced. Each life history shown in Fig. 1 represents a total production of four offspring withinfour years of birth, but the resulting rates ofpotential population growth are very different
for the different schedules. It is clear that the casesof most rapid population growth are associatedwith a greater concentration of reproduction into
the early life of the mother. This is intuitively
reasonable because we are here dealing with acompound interest phenomenon and should expect
greater yield in cases where "interest" begins toaccumulate early. However, the writer feels that
this phenomenon is too frequently overlooked inbiological studies, possibly because of the difficulty of interpreting the phenomenon quantitatively.
In seeking to reconcile the continuous and discontinuous approaches to potential population
growth, let us first note that Thompson's discontinuous method corresponds to an equation offinite differences. We have seen above that thegeneration law gives us a rule for indefinitely
extending the series representing the population
size or the number of births in successive timeintervals by adding together some of the precedingterms multiplied by appropriate constants. If we

let f<z) represent the coefficient of T"' in the expansion of the generating function (7) and, forbrevity, write in (8) V,. = l,.b., then our population series obeys the rule:

J(z) = VaJ(~) + Va+1/(z..-.a-1) + · · •+ v,.f(:!>-<4),(9)which may be written in the alternative form,f(z+<A) -V af(z+O>--<Z) - V a+d(:o~-1) - · · •- V.,J(z)J = 0.
(10)

Thus for our "cliff swallow" example, where we had
G= 2T1 + 2T2 we havef(z) = 2f(z-l)+ 2f(-2) or,f(z+2) -2f<•+1) -2J(z) = 0.

Formula (10) represents the simplest and best
understood type of difference equation, a homogeneous linear difference equation with constantcoefficients. It is outside the scope of the present
paper to discuss the theory of such equations,which has been given, for example, by Jordan(1950). By the nature of our problem as summarized in formula (9), all of our V,. values areeither equal to zero or are positive real numbers
and all of the signs of the coefficients in (9) are
positive: features which considerably simplify
generalizations. By virtue of these facts it can beshown that there is always a "charactenstic"
algebraic equation corresponding to (10). This isobtained by writing p"' for f<zl and dividing throughby the p value of smallest index. This givesv., = 0(11)

an algebraic equation which has the roots p1,
P2, etc.
The general solution of the corresponding difference equation (10) is
f<zl = CtPi + C2P~ + · · · + C,.p!

(12)

where the C's are constants to be determined by
the initial conditions of the problem. Formula (12)is precisely equivalent to Thompson's method andis a general expression for the number of births or
the population size in any future time interval.
As an example we may consider the case whereG = 2Tl + 2T2. The difference equation, as already noted, is f<z+2l - 2f<z+t) - 2f<z) = 0 andthe characteristic algebraic equation is p2 - 2p 2 = 0 which is a quadratic equation with the roots
P1 = 1 + vf3, and P2 = 1 Hence the general

189

y3.

solutionis!<=> = Ct(1 + Vl)"' + C2(1 - Vl)"'. To
determine the constants cl and c2 we look at the
beginning a of the seriesnd note that we havef<o> == 1 and f(ll = 2. Substituting these values in

V:0

1 and
the general solution we obtain Ct =1
C2 = 2
Therefore, the general expression

V3VJ .

for the number of births in time interval T"' isf<=> "'

0+1_/'1.,

2Vl (1 + v 3) V3 - 1"'+ 2VJ (1- VJ)

which can be simplified to f<zl =Ptn-t - P2:rf-t

y'3=

p~ + p~IP2 + • , • + p;.

In order to have the difference equation (12)correspond to the equation of exponential growth(1'), the ratio between populations in successive
time intervals must assume a constant value giving
f<=+t> = e•
f<z>
•

(13)

By the nature of our problem, as already noted, thepotential population growth is always positive, sothat any limit approached by the ratiof~z+1 l must
be a positive real number.
<,.>
It is beyond the scope of the present paper todiscuss the conditions, for difference equations ingeneral, under which this ratio does approach as alimit the largest real root of the characteristic
algebraic equation. (See, for example, MilneThompson, 1933, chap. 17). Dunkel (1925) refers
to the homogeneous equation with real constantcoeflicients corresponding to our formulas (10)and {11). The algebraic equation (11) has a single
positive root which cannot be exceeded in absolutevalue by any other root, real or complex. Using (12)to express the ratio between successive terms,
we havef<=+tlC,pr+1 + C:p~+t + ... + C,.p';.+t__f<:e) .., C,p,"' + c"'!PI+ .•• + c,.p,."' .

(14)

If we let Pl represent the root of (11) of greatest
absolute value and divide both numerator anddenominator of {14) by CJPt"' we obtainf<=+tlf<z> == Pt[

The expressions in parentheses are all less thanunity, on the assumption that Pl is the largest
root, and the entire expression in brackets approaches unity as x increases. Consequently wehave, for x large(16)This then explains the shape of the potential
birth and population series as illustrated in Fig. 1.
In the very early stages population growth isirregular, because the expressions in (12) and {15)involving the negative and complex roots of (11)are still large enough to exert an appreciable influence. As x increases, the influence of these otherroots becomes negligible and the population grows
exponentially, conforming to (16). In considering
potential population growth we are concerned withthe ultimate influence of life-history features, andthe equation of geometric progression or compoundinterest does actually represent the form of potential population growth. We are interested onlyin the single positive root of (11) for the purposeof determining the constant r, and this can readily
be computed with any desired degree of preclsion
by elementary algebraic methods.
Having established the relationship of formula(13) or {16), it is easy to reconcile Thompson's
discontinuous approach to population growth with
Lotka's continuous approach, as exemplified byformulas (3), {4), and (5).
Employing formula (9) we may write the ratio
between populations in successive time intervals as
cf=+t> = v ./<=-«+~> + v..+/<=-a>+
f<=>
f<=>
f<:e)

... + V./<z--1)/(z)

Substituting the relationship given by {13), thisbecomese• = V,.~f-r(cr-1) + V«+te-r« + · · · + V.,e-r<--1), or
1 = V,.e-r« + V«+te-•<«+ll + · · · + V.,e-rw.

(p,.)z+']C,. (p,.)"
·
c,

1 +C2
- (Ps)"'+l+Ca- (Pt)"'+l+ ..• +C,.
- C, Pt
C, Pt
C, Pt

~)" + Ca (~)"' + .•• +
c, Pl
c, PI

1 + Ca

190

Pl

(15)

LIST OF LITERATURE

Replacing V. by its equivalent, l:b:r:, this is.

1=:L: e-•=z"'b"'.
z-a

(17)

Formula (17) is the precise equivalent in terms offinite time intervals of Lotka's equation (3) for
infinitesimal time intervals. In Lotka's equation,
as in (17), the limits of integration in practice area and w since bx is zero outside of these limits.
Formula (17) was in fact employed by Birch (1948)
as an approximation to (3) in his method of determining r for an insect population. The only
approximation involved in our derivation of (17)is the excellent one expressed by formula (13);otherwise the formula corresponds to Thompson's
exact computational methods. It is hoped thatrecognition of this fact will make some of theapproaches of population mathematics appear
more realistic from the biological point of view.
Formulas (4) and (5), originally due to Lotka,
are also immediately derivable from the relationship (13). In any time interval, T:r, we may saythat the population members aged 0 to 1 aresimply the births in that interval, say Bz. The
population members aged 1 to 2 are the survivorsof the births in the previous interval, that isl1B.,-t, or employing (13), ltB:e-•. Quite generally,
the population members aged between z and z + 1
are the survivors from the birth z intervals previous, or lr:Bxe-". If )1. is the extreme length of life
for any population members (~+ 1 = O) we havefor the total population

P: = Bz(l + lte-• + l,e- +>.

+ l>.e-r = B: L e-r:r: l:.
2,_oThe birth rate per individual, (:J, is B:/P., therefore,(18)
which is the equivalent in finite time intervals of
Lotka's equation (4). Also the proportion, c., of
the population in the age range z to z + 1 isl.B.,e-••. . .
- - - which iS Simply,
P,.
(5)

ARCHIBOLD, R, C. 1918. A Fibonacci series. A mer .
math. Monthly, 25: 235-238.
BIRCH, L. C. 1948. The intrinsic rate of natural increase of an insect population. J. Anim. &ol.,
17: 15-26.
CHAPMAN, R. N. 1938. Animal &ology. McGraw·
Hill, New York.
DuBLIN, L. I., and A. J. LoTKA. 1925. On the true
rate of natural increase as exemplified by the
population of the United States, 1920. J. Amer.
statist. Ass., 20: 305-339.
DuNKEL, 0. 1925. Solutions of a probability difference equation. Amer. math. Monthly, 32: 354370.
EVANS, F. C., and F. E. SMITH. 1952. The intrinsic
rate of natural increase for the human louse,
Pediculus humanus L. Amer. Nat., 86: 299-310.
FIBONACCI, L. (1857). Liber Abbaci di Leonardo
Pisano publicati da Baldasarre Boncompagni.
Tipografia delle Scienze mathematiche e fisiche,
Roma.
FISHER, R. A. 1930. The Genetical Theory of Natural
Selection. Oxford Univ. Press, London.
JORDAN, C. 1950. Calculus of Finite Differences,
2nd ed. Chelsea, New York.
KucZYNSKI, R. R. 1932. Fertility and Reproduction.
Falcon Press, New York.
LESLIE, P. H., and R. M. RANSON. 1940. The
mortality, fertility, and rate of natural increaseof the vole (Microtus agrestis) as observed in thelaboratory. J. Anim. &ol., 9: 27-52.
- - , and T. PARK. 1949. The intrinsic rate ofnatural increase of Tribolium castaneum Herbst.
&ology, 30: 469-477.
LINNAEus, C. 1743. Oratio de telluris habitabilis.
In Amoenitates Academicae seu Dissertationes
Variae.... Editio tertia curante J. C. D. Schrebero, 1787, Vol. 2:430--457. J. J. Palm, Erlangae.
LOTKA, A. J. 1907a. Relation between birth rates
and death rates. Science, 26: 21-22.
- - . 1907b. Studies on the mode of growth of material aggregates. A mer. J. Sci., 24: 119-216.
MALTHUS, T. R. 1798. An Essay on the Principle ofPopula-tion ds it Affects the Future Improvement ofSociety, witlt Remarks on the Speculations of Mr.
Godwin, M. Condorcet, and Other Writers. Printed
for J. Johnson in St. Paul's Churchyard, London.
MENDES, L. 0. 1949. Determina!;ao do potencial
biotico da "broca do cafe''-Hypothenemus hampei(Ferr.)-e considerac6es sabre o crescimento desua popula!;ao. Ann. A cad. bras. Sci., 21:275-290.
MILNE-THOMPSON, L. M. 1933. The Calculus ofFinite Differences. Macmillan, London.

191

MousCH, H. 1938. The Longevity of Plants. E. H.
Fulling, Lancaster Press, Lancaster, Pa.
PIERCE, J. C. 1951. The Fibonacci series. Sci.
Monthly, 73: 224-228.
RHODES, E.

C. 1940. Population mathematics. J.
R. statist. Soc., 103: 61-89, 218-245, 362-387.

M. T. 1830. The Law of Population: A
Treatise, in Six Books; i11 Disproof of the Superfecundity of Human Beings, and Developing theReal Principle of their Increase. Murray, London.

SADLER,SHARPE, F. R., and A. J. LoTKA. 1911. A problem
in age distribution. Phil. Mag., 21: 435-438.
THOMPSON, D'ARcY W.

1942. On Growth and Form,
new ed. Cambridge Univ. Press, Cambridge.
-THOMPSON, W. R. 1931. On the reproduction oforganisms with overlapping generations. Bull.
ent. Res., 22: 147-172.

192

Part III

Attempts at Prediction and the Theory
They Stimulated

The tools needed for population projections, age-specific fertility and mortality rates,
were in existence by the early part of the 19th century. They remained unused
almost to the 20th. Edwin Cannan’s (1895) article, paper 22, which begins Part
III, both established life table rates as the necessary base for competent projections
and demonstrated that if care were taken the rates could reasonably be adjusted in
anticipation of future changes.
Independently of Cannan’s work and later contributions by A. L. Bowley (1924,
1926), P. K. Whelpton (1936) also developed component population projections.
Whelpton in his projections for the United States divided total population into ethnic
groupings recognized by the census, handling each separately, and considered immigration as an independent component: apart from fertility he followed procedures
he and W. S. Thompson had previously worked out (Whelpton 1928: Thompson
and Whelpton 1933). The estimates were almost immediately overtaken by events,
a war that was not widely expected in 1936 and a postwar baby boom not expectedeven when it hit. Yet for the few years that the future was orderly it accorded with
Whelpton’s calculations, and the clear logic of separating age groups has never since
been at issue. f Ed. Note: Portions of Whelpton’s 1936 article were included in
the 1977 edition but omitted here. g Mathematical treatment of component-based
projection followed shortly afterward with the introduction of matrix methods.
The use of matrices in population projections was first suggested by Harro
Bernardelli (1941), paper 23, who noted from the sizes of Burmese birth cohortsthat birth peaks occurred at intervals of a generation and that these might carry
important economic and social implications, a question examined more recently
by Richard Easterlin (1966) for its possible relationship to Kuznets cycles. Froma projection matrix formed by holding fertility and mortality constant Bernardelliisolated the characteristic roots of the net maternity function for the discrete case,identifying the dominant root as the intrinsic growth rate of the population andrelating the other roots to oscillations in the birth rate. By mischance, Bernardelliutilized a matrix concentrating fertility into a single age category. This gives rise to
the exceptional result, which he took to be a general case and drew several spurious

III

Attempts at Prediction and the Theory They Stimulated

conclusions from, that oscillations in the birth rate and hence in the age structure
might be maintained or amplified over time rather than becoming damped.
In a paper that followed Bernardelli’s address by three months, E. G. Lewis(1942) presented a better but still essentially preliminary exploration of the projection matrix. This was followed by P. H. Leslie’s (1945) definitive article, paper
24. Leslie develops the use of matrices fully, with attention to the several typesof information they make accessible and their relationship to the earlier theory
developed by Lotka. The present widespread use of projection matrices is anextension of his work. A generalized inverse for the Leslie matrix to permit limited
backward projection is given by Greville and Keyfitz (1974).
f Ed. Note: Results from the use of generalized inverses are, however, usuallyquite unstable. When estimates of aggregate counts of births and deaths over earlier
historical years are available and certain conditions are satisfied, the quite different
technique of “inverse projection” is now available for working out estimates of age
structure backward in time. See Ronald Lee’s original paper [25], the collection [3],and mathematical results on ergodicity by Wachter [39]. g
We follow Leslie’s article with an extension due to Leon Tabah (1968), in which
he shows how matrices may be applied to migration and labor force projections aswell as to overall population change. f Ed. Note: The paper by Tabah is omitted
from the present edition. g The method is also explained in Rogers (1968, pp. 6–15).
f Ed. Note: In 1977 Smith and Keyfitz concluded their introduction to Part III
with the following paragraph. As explained below, our knowledge of the history has
since changed. g
From the development of the projection matrix we turn to the concepts of strong andweak ergodicity, whose initial proofs were developed through matrix formulation
but which otherwise are a continuation of the discussion in Part II. A paper f in
the 1977 edition, omitted here, g extracts from Alvaro Lopez (1961) a statement ofergodicity. The concluding f omitted g article by Beresford Parlett (1970) compactly
summarizes matrix development of ergodicity and general stable population theory.
The reader might return from Parlett to the opening article of the chapter toappreciate the impressive achievements in discrete stable theory and determinationof characteristic roots and vectors that have followed from Cannan’s introduction of
simple but correct principles for population projection.
f Ed. Note: In demography ergodic theorems are theorems that assert that the age
structure of a population projected into the future becomes independent of the initialage structure. An ergodic theorem for population processes with time-varying rates
was actually proved by Henry T. J. Norton [30] and published in the Proceedingsof the London Mathematical Society in 1928. Unaware of this result, at the urging
of Ansley Coale, Alvaro Lopez reproved a version of Norton’s theorem. Golubitsky,
Keeler, and Rothschild [15] pointed out that the theorem follows from a famouslemma of Birkhoff [4], pages 381–390. A compact elementary proof of this kind isgiven by Le Bras [23], Chapter 8.
The term “weak ergodicity” is often used by demographers for cases where rates
vary over time, with “strong ergodicity” used for cases where rates are assumed

194

III

Attempts at Prediction and the Theory They Stimulated

constant over time. This terminology is less than ideal, since the weak ergodic
theorem implies the strong ergodic theorem.
When rates are constant over time, the age structure of a population generally
converges to a stable state independent of the starting age structure, as featured inPart II. Seen from the perspective of renewal theory, this convergence involves aprocess of averaging, set forth in section 7 of Feller’s (1941) paper, paper 16, and in
the textbook by Feller [12], pages 181 to 189. A version for demographers is given
by Arthur [1].
The age-specific rates that go into population projections may also be taken
to vary randomly over time in accordance with some stochastic process. Ergodic
theorems in such a setting are proved in a series of papers by Joel E. Cohen (see[8]) and by Le Bras [21]. More wide-ranging theory for time-varying environments
is found in Tuljapurkar [36].
Matrix projection methods for demographers are covered extensively in the third
edition of Keyfitz’ Applied Mathematical Demography by Keyfitz and Caswell [19].
When time and age are treated as continuous variables and age-specific rates
of fertility and mortality for a closed one-sex population are allowed to vary overtime, population dynamics can be described by a partial differential equation whichgoes back at least to a 1926 paper of McKendrick [27], a pioneer of epidemiology.
Boundary conditions along the time axis are supplied by the Renewal Equation.
Population as a function of age and time is represented by a surface over the Lexis
Plane called a Lexis Surface. The partial derivative with respect to time of thelogarithm of the height of this surface is the continuous counterpart of an agegroup growth rate. In 1982 Preston and Coale [32] developed theory for age-group
growth rates, often called “variable r”, leading to an identity reminiscent of Lotka’s
Equation, as well as to applications exploiting the robustness of age-group growth
rates in the face of common kinds of census coverage errors. Arthur and Vaupel[2] elucidated connections to the properties of general paths on the Lexis Surface.
Applications to population momentum were explored and clarified by Preston [31],Wachter [40], and Schoen [34]. Further developments relating both to prediction
and to parametrization are mentioned in Part IV. g

195

22. The Probability of a Cessation of the Growthof Population in England and Wales during the
Next Century
EDWIN CANNAN (1895)

From The Economic Journal 5. Excerpts are from pages 505-514.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_22, © Springer-Verlag Berlin Heidelberg 2013197

Many things about which .we are habitually obliged to form
estimates are of a much more speculative character than thegrowth of population, and the estimates which we do form about
them in many cases actually depend on our estimate of theprobable growth of population. There is not a builder nor atown ~ouncil in the country that is not obliged to prophesy every
month what the growth of population in a particular district islikely to be, and it was the speculations of the Metropolitan Water
Commission as to the population of London in 1931 that inspired
me to make this contribution to the literature of the subject.
The real question is not whether we shall abstain altogether fromestimating the future growth of population, but whether we shallbe content with estimates which have been formed without
adequate consideration of all the data available, and can be shown
to be founded on a wrong principle.
The generally accepted principle is that of ' as the increase
has been in the past so it will be in the future.' This is susceptible of more than one interpretation. All we really knowof the increase in the past is contained in the totals obtainedby the censuses. A schoolboy whose arithmetic is describedas 'v. G.' would probably boldly disregard all the intermediate censuses and divide the total increase of 20,109,989
between 1801 and 1891 by ninety years. The quotient of
223,444 he would call 'the average annual increase,' and saythat in order to get the probable increase of population in acertain number of years after 1891 you must simply multiply
223,444 by that number of years ....
A person with a slight smattering of statistics will probably
say that the schoolboy's method is quite wrong, and that inarguing from the past to the future, you ought to consider not
the number which has been added to the population, but thefactor by which the population has been multiplied. You ought
to consider not the absolute amount, but the rate of the increase.
But which rate? For there is a different rate in every one of thenine decades. Are we to take the rate from 1801 to 1891, disregarding the intermediate censuses, or are we to somehow deduce some1 This article contains the substance of a paper read before Section F of the
British Association at Ipswich.

198

thing which may be called an average rate from the nine different
rates? The Metropolitan W a.ter Commission groped for an answer
to this question and found none. A very able mathematician towhom I gave the figures of the ten censuses tells me that they areso irregular that no law of increase of the smallest value can be
deduced from them.· The Registrar-General's method in forming
what are called the ' official estimates ' cuts the knot by disregarding all the decades except the last. The only conceivable
argument in favour of this course is the allegation that theimmediate past being nearer the future than the long past,affords a. better basis for estimating future probability. By
itself this would justify the 'red'U,ctio ad absu,rdum, 'if the last ten
years are better than the last ninety, the last year must be better
than the last ten,' but 'temporary fluctuations' have of course to
be considered, and it may perhaps be asserted that ten years is
sufficient, while twenty years is more than sufficient, to give an
average unaffected by such fluctuations.
Which rate of multiplication be taken, however, does not verymuch matter, since any rate that can possibly be deduced fromthe census figures gives ridiculous results. Carried back into thepast the assumption of a. uniform rate of increase equal to thmt
which prevailed in the decade 1881 to 1891 necessitates the
supposition that in A.D. 525 there were scarcely two persons in
England, and carried forward into the future it gives a. populationof a. thousand millions in less than 300 years. After 350 years
more there would be only just room for the people to lie down on
the ground. Yet the rate of increase in the decade 1881-90 is
the lowest of the nine.
These periods, it may be urged, are long. But applied toshort periods the assumption of a uniform rate of increase equal
to that shown by the last decade leads to just as absurd results
as when it is applied to long periods. As is well known, theRegistrar-General, whose annual reports reveal no acquaintancewith the very accurate statistics of immigration and emigration
now collected by the Board of Trade, ignores his own figuresof births and deaths, on the ground that in estimating a. population immigration and emigration cannot be disregarded. The
official estimates of the population in the middle of each of thefollowing ten years which he forms after each census on the199assumption that the rate of increase shown by it will continue, areallowed to stand in each of his annual reports till the next census,
and would, we must suppose, be allowed to stand if a plague swept
away half the people. When the next census is taken they are, ofcourse, found to be wrong. [At the last census the error amounted
to 701,843 at the end of the decade, while Dr. Longstaff had in
the pages of this Journal estimated the population within
10,251. 1 ]They are then corrected on the assumption thatthroughout the decade population annually increased at the rate
observed during the whole decade. Now whenever the rate ofincrease in a decade is less than in the preceding decade, these'corrected' figures, on which the 'corrected' birth, marriage,
and death-rates are calculated, present a very odd and most
unnatural series of increases. Thus they represent the increasefor 1871-2 to have been 307,901, and make the annual increase
grow gradually till in 1879-80 it amounted to 342,799. Then
comes the census year, which stands by itself, as three-quartersof it lie in one decade and the other quarter in another ; it isallotted an increase of 331,854. After this the new period begins
in earnest with an increase of only 288,800 in 1881-2; and therising process starts again and continues till the increase of 1890-1
amounts to 318,912. In short, the increase grows steadily till it
reaches the census of 1881, then it comes down with a crash only
to begin immediately to rise again. It must always happen under
this system that the addition to the population in the last half ofa decade is greater than the addition in the first half, even when
the addition in each decade is less than the addition in thepreceding decade.
The truth is that every estimate of population, past, present, and to come, ought to be founded on aconsideration of the factors on which the growth or decline o.
population is dependent-births, deaths, immigration, and emigration. The number of births and deaths, and of immigrants andemigrants, is now so well known, that if two government departments, the Board of Trade and the General Registry Office,would only recognise each other's existence, the population at
the present time, or at any point in recent years, could be given
within ten thousand of the actual number.
1 ECONOMIC JOURNAL, VOl. i., p. 382.

200

30
0
401------l---+---+---+-----r--+---+--+-+-----l

The population in each census year is measured down from the top, the figures on the left indicating
the number of millions. The lines sloping downwards from left to right divide the population ateach census into persons over 80 ; persons between 70 and SO ; between 60 and 70 ; and so on down tohetween 0 and 10. The ages are shown by the figures on the right. The lines sloping upwards fromleft to right divide the persons born in each decade from those born before and after it. They thus
show graphically how each generation becomes smaller as it passes from under ten years of age at one<'ensus to over ten but under twenty at the next, and then to over twenty but under thirty, and so on.

In estimating future population the most important data
we have to rely upon are the ages of the people as taken atrecent censuses. These are shown measured down from the
top on the first five perpendicular lines of the diagram opposite. The lines sloping downwards from left to right divide
the total population according to each ten years of age fromeighty downwards, so that the top layer of persons includes allover eighty years of age, the next all between seventy and eighty,
201

the next all between sixty and seventy, and so on down to thechildren under ten. Now as the number of immigrants at any
particular decade of age is probably always exceeded by the
number of emigrants of that age, the people who are between ten
and twenty at one census are (with the exception that some immigrants are substituted for some emigrants) the survivors inEnglandof those who were under ten at the preceding census, and similarly
with each ten years of age. So if we call the persons born
between one census and the next a generation, the lines sloping
upwards from left to right divide each generation from the next,
and show how each generation becomes smaller and smaller, till
it finally becomes extinct on the death of some centenarian.
Now, provided that the rate of mortality and loss by migration
remain the same at each age period, it is possible to calculate withabsolute accuracy from the observed decrease of a generation
between any two censuses, how much the next generation will
decrease when it arrives at the same age. For example, if it be
known that everyone over 100 in 1881 died before 1891, it may
be confidently assumed that everyone over 100 in 1891 will die
before 1901. If it be known that the people between fifty andsixty in 1881 lost about 27 per cent. of their numbers before
1891, it may be taken for granted that the people between
fifty and sixty in 1891 will lose the same proportion between1891
and 1901. Nothing is requisite to get the total population overten at the next census except to work out by proportion sums
the population in each ten years of age and add up the results.
Before, however, we can apply this method, it is necessary toinquire whether we are justified in assuming that the rate ofmortality and loss by migration will not change. As to the futureof mortality and loss by migration every man has a right to his
own opinion, but no one can expect to prove anything. We may
therefore fairly treat it as an even chance whether the rate of loss
is likely to increase or diminish. In the past a decrease of therate of loss by mortality has been counteracted by an increase in
the rate of loss by migration, so that estimates in which thismethod has been used have been surprisingly accurate ....
The question is, therefore, what is the probable number ofbirths ? Here we border on considerations of a more speculative
character, but we have still something statistical to rely upon.
202

The fact that it is the custom to calculate birth-rates as a rate
per thousand on the whole population makes it natural to saythat the number of births depends in the first place upon the
population, so that if the population increases rapidly the births
show
The continuous lines of the diaaram
will increase rapidly.
0•that the populatiOn over certain ages will increasA rapidly formany decades, so that we might consequently expect a considerable increase in the number of births. But of course, as a matterof fact the increase of population over and under certain ages
has obviously no tendency to increase the number of births.
The number of old women and children may be doubled without
making it the least more likely that the births will increase until
the children have grown to marriageable age. The number ofbirths is more likely indeed to be somewhat repressed, since the oldpeople and children form a burden upon the shoulders of thosein the prime of life. It is consequently much more true to saythat the number of births depends in the first place upon thenumber of men and women between certain ages. For practical
purposes the ages of twenty and forty are sufficiently near themark, and they are much more convenient than the ages for eachsex which would have to be taken if perfect accuracy were required.
Now from 1853 to 1876 the number of births, after allowing forsome deficiency in registration, increased rapidly and was almost
uniformly just about 12 per cent. on the number of persons
between twenty and forty. From 1876 onwards the number ofbirths has been almost stationary, and the rate per cent. on the
persons between twenty and forty has consequently been rapidly
declining. In ten years it had fallen to 11 per cent. ; by 1890
it had further fallen to 10 per cent. ; in 1891 it went up to about
10·4 per cent.; in the next two years it was 10 per cent.; in 1894
it descended to 9·8 per cent.
Now if the future births were estimated at this rate on thenumber of persons between twenty and forty we should have nogrounds for expecting a cessation of the growth of population,
though the growth would be at a much less rate than heretofore.
But the statistics make it probable that the birth-rate on persons
between twenty and forty will continue to fall. It must be remembered that the effect of the births having been nearlystationary for twenty years will be to change considerably the203age distribution of the group of people between twenty and forty.
A larger proportion of them will be at the higher ages. It seems
at first sight paradoxical to say that the persons between twenty
and forty can be older at one time than another, but it is really quite
simple. The persons between 0 and 100 years old obviously do notaverage fifty years, and in just the same way the persons between
twenty and forty do not average thirty. What the exact average
age will be depends chiefly on the variation in the number ofbirths between twenty and forty years before. As the number of
persons born rose rapidly year by year before 1876, the number of
persons becoming twenty years of age in each year has been
rising rapidly and will continue· to rise rapidly till1896. After that
it will still continue to rise because there has been a reduction ofjuvenile mortality, but the rise will be far less rapid. The consequence will be that while in 1891 30 per cent. of the twenty toforty group were under twenty-five, in 1911 not more than 27i
per cent. will be so.
This increase in the average age of persons in the prime oflife, or, to put the same thing in other words, this diminution in
the increase of the number of persons reaching marriageable
age in each year, must tend somewhat to reduce the birth-rate.
How much effect should be attributed to it would not be very
difficult to calculate if the necessary statistics of the ages ofparents were forthcoming, but unfortunately none such exist.
If, however, we take into account not only the diminution to
be expected from this cause, but also the diminution to be
expected from the working of the enormously strong economic
and social forces which have brought about the diminution of thelast twenty years, it seems a very moderate hypothesis to supposethat the rate of births on the number of persons between twenty
and forty may fall to a little below 9 per cent. by 1901 and to theneighbourhood of 8 per cent. by 1911. This, with a further
slight diminution to a little below 8 per cent. by 1921 is all thatis necessary in order to keep the number of births stationary at
the level of 1881-90. Adopting then this hypothesis, I havecontinued the lines of the diagram so as to show what the
future of the population will be if the rate of mortality and loss
by migration at each age, and also the absolute number of births,
remain the same as in 1881-90. It will be seen that the increase204

of population, large at first, becomes less and less, till it is triflingin 1941-51. It would continue, but always growing less and less,till about 1995, when the last survivor of the period before 1891
would disappear, and the population would then stand at its
maximum of 37,376,000.
I have no desire to stake my reputation as a prophet on thegrowth of population following exactly the line shown in the
diagram, and ceasing to increase in 1991. I am only prepared
to assert confidently that the line shown is a much more probable_
one than that which might be laid down by the ' official ' method,
and which would shoot through the bottom of the diagram
between 1921-31 and encircle the globe before the diagram waswidened very many yards.. Whether the cessation of the growthof population is reached, as I personally should expect, before
1991, or afterwards, it must be reached at last, and if it is reachedwithout any violent changes in mortality, migration, or natality, it
will necessarily be reached by a curve of increase closely approximating to that laid down in the diagram. The value of the
diagram lies not in its prediction of a maximum population of
thirty-seven millions, but in the fact that it shows how a cessationof growth may be reached within no very long period without any
violent or unnatural changes.

20523. Population Waves
HARRO BERNARDELLI (1941)

From Journal of the Burma Research Society 31, Part 1. Excerpts are from pages 1-4, 6-7.

We omit Bernardelli's discussions of economic cycles and his further comments
on cycles in human and animal populations, which are of uneven quality. The
appendix referred to in the text is also omitted: it introduces a projection matrix
and indicates the derivation of its characteristic roots.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_23, © Springer-Verlag Berlin Heidelberg 2013207

1. In thinking of population growth usually the notion is applied-consciouslyor unconsciously-that such growth takes place along a smooth curve: either inthe form of Malthus' geometric progression, the interest law, or according toVerhulst- Pearl's logistic equation, or along such curves as have been worked out
recently by Dr. Kuczynski and his collaborators in order to trace the decline of
the European race. Practical statisticians, no doubt, realise that populations
actually show considerable deviations from such models, but they have tended
on the whole to consider.such "irregularities" as "random variations" which are
best smoothed out.
Having worked on a variety of problems which all involve assumptions about
the manner in which the numbers of a population change, it appears to me now
rather doubtful that the preference given to smooth curves is justified even as
a first approximation. Smoothing always means introducing a definite hypothesis
about the nature of the causes that determine the value of the variable underconsideration; in eliminating the irregular oscillations typiCal for a populationgraph, we assume, in fact, that there are two sets of causes operating: a fundamental, basic set of causes which, left alone, would produce the picture of oursmooth curve, and a second set-more irregular and haphazard in occurrence
and intensity-which accounts for the superimposed aberrations.
The justification of the procedure of smoothing thus depends essentially on
the validity of the hypothesis that the basic causes will manifest themselves
smoothly. If it can be shown that this is not the case, that the basic causes themselves are bound, in fact, to generate oscillations, then the operation of smoothing
will lose its rationale. Its application will merely blur the picture, and possibly
even block the way for a correct understanding of the causal effects which wewish to elucidate.
2. The oscillations of which I am talking can easily be demonstrated in any
Census Report. The following table, taken from the Burma Census 1931, gives
the age-distribution of 10,000 persons of each sex for the indigenous population

Age

Males

Females

Group

1931

1921

1911

1901

1931

1921

1911

1901

0-5
5-10
10-15
15-20
20-25
25-30
30-35
35-40
40-45
45-50
50-55
55-60
60-65
65-70

1397
1282
1186
916
886
838
776
604
532
412
361
283
239
130

1265
1271
1223
1002
862
777
722
600
571
446
410
252
274
133

1354
1355
1236
899
775
760
767
656
566
409
373
241
278
133

1433
1301
1143
877
828
828
799
626
539
407
383
244

1414
1236
1134
1002
997
848
737
560
504
397
358
278
229
130

1286
1260
1146
1082
941
801
687
539
558
406
418
251
264
126

1372
1344

1442
1277
1046
961
917
850
756
552
521
369
395
241

208

1140

966
864
792
727
584
552
374
384
242
278
128

of this country 1 • The figures have been compiled for the indigenous inhabitants
only, because they reveal best the pure effects of population growth, and are not,in any way, affected by migration.
The table shows a remarkable instability of the age-structure. Its characteristic features can be studied most effectively by averaging for the four censuses
the frequencies in each age-group and recording for each census the deviationsof the actual frequencies from their mean. The result is shown in Fig. 1.
It will be observed that there was in 1901 an abnormally high frequency ofchildren in the age-group 0-5 which subsequently produced a wave of survivors
in the later census records. Similarly the low frequency for the age-groups 10-20
in 1901 can be traced as a depression in the waves for the following three decades.
The waves for females (shown in the Census Report 1931 page 73ff.) run exactly
parallel. It is obvious that the new wave crest of children which is seen to develop
in 1931 originates from the survivors of the wave of female children in 1901.
These waves have a sufficiently large amplitude to exert a significant influenceon the economic and social conditions of the country. This became painfully
evident during the Great Depression, in 1929 and the succeeding years. There
was during those years an unusually large proportion of indigenous males in the
age-group 15-30 which forms, of course, the most active element of a populationfrom which the new entries into the labour market are drawn (See Section ABCof the 1921 and DEF of the 1931 wave). With agriculture disorganised by the
impact of the unprecedented depression, a large number of these persons-who
normally would have found accommodation in agricultural pursuits were fm:ced
to look for openings in the industrial labour market which up to that time had
been the undisputed domain of the Indian immigrant 2 •.••
This unfortunate constellation of a large wave of indigenous males looking
for employment, and of an industrial labour market which offered no chances of
employment, undoubtedly contributed to the series of unprecedented riots, andthe intense growth of racial animosity, which-busily fanned by interested
parties-has never since subsided, and which is likely now to produce the usual
crop of obnoxious restrictionism, although-among other things-the labour
market is long again in perfect adjustment, and although at the present, andduring the next years, the entry of indigenous people into the labour market willbe by no means brisk (See the depression GHD of the 1931 wave which is relevant for the present phase).
3. Naturally the question arises, how do such waves originate? A little reflection
shows that population cycles easily can be produced synthetically. Take a species,
say a beetle, which lives three years only, and which propagates in its third yearof life. Let the survival rate of the first age-group be t, of the second l, andassume that each female in the age 2-3 produces, in the average, 6 new living
females. (These values are, of course, entirely arbitrary.) Then a population ofthese beetles which, we may assume, has in a certain year 1 a thousand females
1 Census of India, 1931, Vol. XI, Burma, Part L Rept. by J.J. Bennison, Rangoon
1933, pp. 73ff.
See also Census of India, 1921. Vol. X, Burma, Part I, Rept. by S.G. Grantham, Rangoon 1923, p. 121fT.
2 "The economic depression"--says Bennison I.e. p. 77-"has caused agreat deal of unemployment,
and this has apparently been aggravated by the unusually large proportion of males between 20 and 30."209Age Groups

05

10 15 20 25 30 35 40 45 50 55 60 65 70

0
+75

-50
-75
0
-2S

-so

+50

-2S

-so~~~~~~~~~~;=~~~~~~~
0

5

10 15 20 25 30 35 40 45

so 55 60 65 70

Age Groups
Age Distribution of indigenous Males
since1901

Fig. 1210

in each age-group-males are irrelevant in matters of reproduction-will produce
waves in the following manner:II

Age Group

0-1
1-2
2-3
Total

1QOO
1000
1000
3000

2

3

4

6000
500
333.3
6833.3

1999.9
3000
166.6
5166

1000
1000
1000
3000

So long as the rates of fertility and mortality remain constant, this development will, of course, continue indefinitely in regular three-year cycles. The population will oscillate round a stationary level, as the Net Reproduction Rate R
is here assumed to be = 1.
What determines the amplitude of these waves? Without going into the
Mathematics of the process, it can be seen by means of simple Arithmetic, thatthe amplitudes depend on the assumptions made with regard to the initial distribution of females in the age-groups. If one takes in the year 1 the numbers
1000, 600, 200 respectively, it is easily seen that the amplitudes of the resulting
waves become much smaller: and with the numbers 1000, 500, 167 instead, thewaves disappear altogether.
For a mathematical analysis interested readers are referred to the Appendix
where a system of equations is given from which all the relevant features of theprocess can be deduced with greatest ease. Here only the following observations
will be used:
I. For any given set offertility and mortality conditions there exists always one
characteristic age distribution (1000:500:167 in the above example) which does
not give rise to disequilibrium waves; but whenever the population deviates fromthis equilibrium structure, waves will be set up with amplitudes that depend
entirely on the degree of these deviations.
II. It is not necessary that the population should be stationary, as in the above
example; waves will also occur, if the Net Reproduction Rate R> 1, indicatinga growing population, or if R < 1, so that the population eventually dies out.
III. It is not necessary that the waves repeat themselves year after year with thesame amplitude. The occurrence of dampening, or of magnifying factors depends
entirely on the given fertility and mortality rates.
IV. Similarly the wave length depends entirely on those rates, so that any changeof the fertility and mortality conditions will produce waves of a different character. Wave length as well as the degree of dampening, and the amplitudes will
be modified, if the situation changes.

21124. On the Use of Matrices in Certain Population
Mathematics
P. H. LESLIE (1945)
From Biometrika 33. Excerpts are from pages 183-185, 187-193, 199-202.

Leslie's work, rather than that of his predecessors Bernardelli and Lewis, is mostcommonly cited in the widespread literature using matrices, largely for the reason
that Leslie worked out the mathematics and the application with great thorough~ness. Some of his elaboration was designed to save arithmetic-for example histransformation of the projection matrix into an equivalent form with unity inthe subdiagonal positions. Such devices, like a considerable part of classicalnumerical analysis, are unnecessary in a computer era.
The case of double roots seemed to require attention, for Leslie could notthen know that no case of double roots would ever arise with real data. In thepresent excerpt we omit this, as well as the spectral decomposition of the matrix,
which has not found extensive application.
Among the latent roots or eigen values and the corresponding stable vectors,
as Leslie points out, demographic and biological interest is confined to three.
The first, of largest absolute value, is positive and represents the ratio of population at the end of a cycle to that at the beginning, when the process has been
operating for a considerable period; it provides the component of geometricincrease. The second and third roots produce waves of diminishing amplitude
having the length of the generation, usually 25 to 30 years. These waves measure
the echo effect-after a baby boom they provide for a smaller boom a generation later-on the condition that the age-specific rates of birth and death remain
constant.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_24, © Springer-Verlag Berlin Heidelberg 2013

213

CONTENTS
PAGE

PAGE

1. Introduction

2. Derivation of the matrix elements
3. Numerical example
4. Properties of the basic matrix
5. Transformation of the co-ordinate
system
6. Relation between the canonical form
B and the L~m~ column
7. The stable age distribution
8. Properties of the stable vectors .
9. The spectral set of operators
10. Reduction of B to classical canonicalform
11. The relation between rp and 1fr vectors
12. Case of repeated latent roots

13. The approach to the stable age dis·tribution
14. Special case of the matrix with only asingle non-zero F • element
15. Numerical comparison with the usual
methods of computation
16. Further practical applications

183
184
185
187
188
190
191
192
193
194
195
197

Appendix: ( 1) The tables of mortality andfertility(2) Calculation of the rate ofincrease(3) Numerical values of thematrix elements

199
200
201
207
209
210
212
212

References

1. INTRODUCTION

IT we are given the age distribution of a population on a certain date, we may require toknow the age distribution of the survivors and descendants of the original population at
successive intervals of time, supposing that these individuals are subject to some given agespecific rates offertility and mortality. In order to simplify the problem as much as possible,
it will be assumed that the age-specific rates remain constant over a period of time, and thefemale population alone will be considered. The initial age dii!!tribution may be entirely
a.rbitrary; thus, for instance, it might consist of a group of females confined to only one of
the age classes.
The method of computing the female population in one unit's time, given any arbitrary
age distribution at time t, may be expressed in the form of m + 1 linear equations, wherem to m + 1 is the last age group considered in the complete life table distribution, and when
the same unit of age is adopted as that of time. Ifn., = the number of females alive in the age group x to x + I at time t,
P, = the probability that a female aged x to x + 1 at time t will be alive in the age groupx + I to x + 2 at time t + 1,
F., = the number of daughters born in the interval t to t + I per female alive aged x to x + 1
at timet, who will be alive in the age group 0--1 at timet+ 1,
then, working from an origin of time, the age distribution at the end of one unit's intervalmwill be given by= nol~ F.,nzO
z-0

P0 n 00= n 11

P1n1o

= n21
= n31

.Pzn2opm-1 nm-1,0 = nml

or, employing matrix notation, Mn0 = nv where n 0 and n 1 are column vectors giving theage distribution at t = 0 and 1 respectively, and the matrix
Fk Fk+lF0 F1 F2

r.

Po

.

.

~

p2

M

~

l: : : : : : : : :~ : ;: : : : : :;.~ : :214

O<P.,< 1; F.,~ o.

This matrix is square and consists of m + 1 rows and m + 1 columns. All the elements arezero, except those in the first row and in the subdiagonal immediately below the principal
diagonal. The P, figures all lie between 0 and 1, while the Fx figures are by definition necessarily positive quantities. Some of the latter, however, may be zero, their number andposition depending on the reproductive biology of the species we happen to be consideringin any particular case, and on the relative span of the pre- and post-reproductive ages. If
Fm = 0, the matrix M is singular, since the determinant I M I = 0.
Since Mn0 = n 1 , and Mn 1 M~ = n 2 , etc., the age distribution at timet may be found by
pre-multiplying the column vecto~ {n 00 n 10 n 20 ••• nm 0}, i.e. the age distribution at t = 0, by
the matrix M 1• Moreover, it will be seen that with the help of the jth column of M 1 the age
distribution and number of the survivors and descendants of the n1_ 1,0 individuals, who werealive at t = 0, can readily be calculated. Thus, n 1_ 1,0 times the sum of the elements in the jthcolumn of M 1 gives the number of living individuals contributed to the total population attime t by this particular age group.

=

2.

DERIVATION OF THE MATRIX ELEMENTS

The basic data, from which the numerical elements of this matrix may be derived, are given
usually in the form of a life table and a table of age specific fertility rates. To take the P"
figures first; if at t = 0 there are nxa females alive in the age group x to x + 1, the survivors ofthese will form the x + 1 to x + 2 age group in one unit's time, and thus Px n,0 = nx+1,1 • Then
it is usually assumed (e.g. Charles, 1938, p. 79; Glass, 1940, p. 464) thatp = Lx+1
X

where

L, =

LX'

J l,dx,x+1
xor the number alive in the age group x to x + 1 in the stationary or life table age distribution.
This method of computing the survivors in one unit's time would be exact if the distributionof those alive within a particular age group was the same as in the life-table distribution.
The F, figures are more troublesome, and in the numerical example which will be given
later they were obtained from the basic maternal frequency figures (m.r =the number oflive daughters born per unit of time to a female aged x to x +I) by an argument which ran
as follows. Consider the nxo females alive at t = 0 in the age group x to x +I, ani! let us suppose that they are concentrated at the midpoint of the group, x + t. During the interval oftime 0-1 some of these individuals are dying off, and at t = 1 the nx+l,l survivors can be
regarded as concentrated at the age x+ It. Although these deaths are taking place continuously, we may assume them all to occur around t = t, so that at this latter time thenumber of females alive in the age group we are considering changes abruptly from nxo ton,o+ 1,1 = P,nxa. Then during the time interval O-t these nxo females will have been exposedto the risk of bearing daughters, and the number of the latter they will have given birth toper female alive will be given by the maternal frequency figure for the ages x + t to x + 1.
This figure may be obtained by interpolating in the integral curve of them., values, and thus
expressing the latter in t units of age throughout the reproductive span instead of in single
units. The daughters born during the interval of time O-t will be aged !-1 at t = 1, thenumber of them surviving at this time being determined approximately by multiplying theJ:appropriate m.-.+i figure by the factor 2

lxdx according to the given life table. Similarly,

each of the P...,nxa females during the interval of time 1--1 give birth to mx+l-x+ll daughters,
the survivors of which form part of the O-! age group at t = 1. The survivorship factor isin this case taken to be 2s:lxdx.

215

Combining these two steps together we obtain a series of Fx figures, which may be definedas the number of daughters alive in the age group 0-l at t = l per female alive in the age groupx to x+ l at t = 0. Putting

then
m

and

I; F., n.,o = no1'

x=O

the total number of daughters alive aged 0-l at t = l.

4. PROPERTIES OF THE BASIC MATRIXThe matrix M is square and of order m+ l; it is not necessary, however, in what follows to
consider this matrix as a whole. For, if x = k is the last age group within which reproduction
occurs, Fk is the last Fx figure which is not equal to zero. Then, if the matrix be partitioned
symmetrically at this point,

The submatrix A is square; B is of order (m-k) x (k+ l); 0 again is square consisting of
m- k rows and columns, the only numerical elements being in the subdiagonal immediately
below the principal diagonal. The remaining submatrix is of order (k+ 1) x (m-k) andconsists only of zero elements. Then in forming the series of matrices M 2 , M 3 , M', etc.,r

A'
M'=U(ABO)
The submatrix 0 is, however, of such a type that om-k = 0, so that M', t;;; m- k, will haveall its last m- k columns consisting of zero elements. This is merely an expression of theobvious fact that individuals alive in the post-reproductive ages contribute nothing to the
population after they themselves are dead. It is the submatrix A which is principally ofinterest, and in the mathematical discussion which follows, attention isfocusedalmostentirely
on it and on age distributions confined to the prereproductive and reproductive age groups.
The matrix A is of order (k + 1) x (k + 1}, where x = k is the last age group in which reproduction occurs, and written in full,

F0 F1 F2 F3
Po

Fk_ 1 Fk

...

pk-1

This matrix is non-singular, since the determinant I A I = ( -I)k+2(P0 P1 P2 ... Pk_ 1 Fk).
There exists, therefore, a reciprocal matrix of the form
p 01

A-t=

1

P11

.......................................................................................... I.

.

.

.

P;1

.
F"k1

.
- (PoFk)-1 Fo·- (P1 Fk)-1 Fl- (P2Fk)-l F2

·

.

···

P"k!.1- (Pk-1 Fk)-1 Fk-t

Thus, given an initial age distribution nxa (x = 0, 1, 2, 3, ... , k) at t = 0, in addition to theforward series of operations An0 , A2n0 , A3n0 , ••• ,etc., there is also a backward series A - 1 n 0 ,216

A-Zn0 , A-3n 0 , ••• , etc. There is, however, a fundamental difference between these; for,whereas the forward series can be carried on for as long as we like, given any initial age distribution, the backward series can only be performed so long as n.,1 remains ;;::; 0, since a
negative number of individuals in an age group is meaningless. Apart from this limitation,
it is possible to foresee that the reciprocal matrix might be of some use in the solution ofcertain types of problem.

5.

TRANSFORMATION OF THE CO-ORDINATE SYSTEM

Hitherto an age distribution nx1 has been regarded as a matrix consisting of a single columnof elements. For simplicity in notation, this column vector will now be termed the vectoretc.). We may picture
sand different fs will be distinguished by different subscripts (sa,an age distribution as a vector having a certain magnitude and related to a definite direction
in a vector space, the space of the fs. The different age distributions which may arise in thecase of any particular population will be assumed to be fs all radiating from a common origin.
The numerical elements of a s vector are thus taken to be the co-ordinates of a point in
multi-dimensional space referred to a general Cartesian co-ordinate system, in which thereference axes may make any angles with one another. At this point in the argument anothertype of vector will be introduced, which in matrix notation will be written as a row vector,
and which will be termed the vector 7J. There is an intimate relationship between this new
there is a uniquely determim~d vector
type and the old, for, associated with each vector
1J,., and vice versa. The inner or scalar product, 1JaSa, is the square of the length of the vector
Either we may picture each of these vectors as associated with a different kind of vector
space, the space of the fs and the dual space of the 7J's, which are not entirely disconnected
but related in a special way; or, alternatively, we may regard them as two different kinds ofvector associated with the same vector space. The relationship between 1J and sis precisely
the same as that between covariant and contravariant vectors in differential geometry.
If we pass from our original co-ordinate system to a new frame of reference, and thevariables 1J and s undergo the non-singular linear transformations,

sx,

sa,

s,..

s

= H- 1'1jf, I HI 9=0,
1J = cpH,
it can be seen that since the variables are contragredient, 1JS = cp'ljf, so that the square of thelength of a vector remains invariant. Moreover, since the result of operating on a vectors"
with the matrix A is, in general, another vector Sb, where Sa and Sb are both referred to theoriginal co-ordinate system, it follows that in the new frame of reference which is defined by
the linear transformations given above, the relationship

Asa

= Sb
HAH-Va = 'ljfb,
orBlfra = lfrb·
Thus, in the new frame of reference the matrix B = HAH- 1 operating on the vector lfra isequivalent to the matrix A operating on the vector Sa in the original frame.
It is convenient, for the purposes of studying the matrix A and of performing any numericalcomputations with it, to transform the variables 1J and sin the above way, choosing thematrix H so as to make B = HAH-1 as simple as possible. For Bt = (HAH- 1)' = HAIH-I,
and since A is non-singular, by the reversal law, (HAH-1 )-1 = HA- 1H-1 • Thus, if f(A) isa rational integral function of A, f(B) =f(HAH-1 ) = Hf(A)H- 1; and the properties ofmatrix functions /(A) can be studied by means of the simpler forms f(B). Moreover, thematrices A and B have the same characteristic equation and, therefore, the same latentroots. For B- ii.I = H(A- ii.I) H-1 and, forming the determinants of both sides,

becomes

I B-ii.II = IHIIA-ii.IIIHI-1,

217

so that the characteristic equation is

lA-AII =I B-All= 0.
If, in the present case, the transforming matrix is taken to be(Po~ F. ··· Pk-1>

H=(JlJJlJ ··· Pk-1)..................... '; ......................................................................... .
(Pk-Bpk-1)

1

in which, it is to be noted, the only numerical elements lie in the principal diagonal and arederived eB.tirely from the life table, thenF0 P0 F1 P0 P1 F1 P0 P1 P1 F8
11

B=HAH-1 =1
1
1Comparing this matrix B with the original form A, it can be seen that the latter has been
simplified to the extent that the original P., figures in the principal subdiagonal are now replaced by a series of units, and the matrix A has been reduced to the rational canonical form
B = HAH-1 (see Turnbull & Aitken, 1932, chap. v). In this way any computations with thematrix A are made easier, and we may work henceforward in terms of ifJ and 1fr vectors
together with the matrix B, instead of with the original?] and 6 vectors, and the matrix A.
Any results obtained in this new system of co-ordinates may be transformed back again
to the original system whenever necessary. It is evident that by suitably enlarging H theoriginal matrix M may be transformed in a similar way.
This linear transformation of the original co-ordinate system is equivalent biologicallyto the transformation of the original population we were considering into a new and completely imaginary type which, although intimately connected with the old, has certain quite
different properties. Thus, it can be seen from the transformed matrix B that the individuals
in this new population, instead of dying off according to age as the original ones did, live
until the whole span of life is completed, when they all die simultaneously. This is indicatedby the P., figures being now all equal to unity; an individual alive in the age group x to x + 1
at t = 0 is certain of being alive at t = 1, excepting in the last age group of all where none of
the individuals will be alive in one unit's time. Accompanying this somewhat radical change
in the life table, there is a compensatory adjustment made in the rates of fertility so that the
new population has the same inherent power of natural increase (r) as that of the old. This
follows from the fact that the latent roots of the matrices A and B are the same, and, as
will be shown later, the dominant latent root is closely related to the value of r obtained by
the usual methods of computation. Insomuch as the transformation is reversible and
A= H-1BH, it can be seen that by changing H we could transform the canonical form B,
if we wished, into another matrix in which the P., subdiagonal might be a specified set of
figures derived from some ot}ler form oflife table. But, for our present purposes, the canonicalform B, in which all the P., figures are units, offers advantages over any other matrix of asimilar type owing to the greater ease with which it can be handled. • •.

218

7. THE STABLE AGE DISTRIBUTION
The result of operating on an age distributon 1/r., with the matrix B is, in general, a different
distribution 1/r11 • But, in the special case when the relation between the two distributions is

Bl/r" = AVra•such that

where A is an algebraic number, then 1/ra may be said to be a stable age distribution appropriate to the matrix B. For the sake of brevity it will be referred to as a stable 1/r. SimilarlytPaB = AtPa•
for initial row vectors, ifthen 9~ is said to be a stable t/J.
The matrix equation defining a stable 1/r may be written as k + I linear equations, of which
the ith isk+t
I: btini-Ant = 0,
i-1
where nt (i = 1, 2, ... ,k+ 1) are the co-ordinates of the stable y, and b1i the element in the
ith row and jth column of B. Elinlinating the n 1 from this system of equations, we obtainthe characteristic equation of B, namely,
IB-AII=O;
and, expanding this determinant in powers of A, we have in the present case,

Ak+l_ FoAr.- PoFt Ar.-t_ PoPt FaAr.-z_ ... - <PoPt ... P,._z) Fr.-t A- <Po~ ... P,._t) Fk = 0.
The k + 1 roots Aa of this equation are the latent roots of B, and corresponding to each distinct Aa there is a pair of stable vectors, tPa and Vra• determined except for an arbitrary
scalar factor.
Once a latent root Aa has been determined, it is a comparatively simple matter to find theappropriate stable 1/ra and tPa vectors. Thus, it is easily shown that the stable 1/rais the column
vector {A!A!- 1 A!-• ... Aal}. A short method of estimating tPa is the following. Suppose, totake a simple case, thatand let y, (x = 1, 2, 3, 4) be the elements of the stable tPa appropriate to the root i\a. ThentPaB = [ayl +Y2 byl +Ya cyl +Y4= [ AaYt

AuYz

AaYa

dytJ
AaY4].

By equating similar elements and putting y 1 = I, y4 = diA"' y3 =c;"y4 , etc., it is easy tosee how the required row vector can be built up. Having in this way obtained the stable
1/r and rp vectors for the matrix B, they may be transformed to the appropriate stableand 1J for the matrix A by means of the relationsss

1J = tjJH,= H-11/r.
The characteristic equation of the matrix B, when expanded, is of degree k + 1 in A, andonce B has been obtained this equation can immediately be writtlen down, since the numericalcoefficients of Ak, Ak-l, Ak- 2, etc., are merely the elements of the first row taken with a negativesign. Since there is only one change of sign in this equation, only one of the latent roots
will be real and positive. Excluding the rather special case when the first row of B has onlya single non -zero element, and taking the more usual type of matrix which will be met with,
namely, that for a species breeding continuously over a large proportion of its total life span,
it will be found that the modulus of this root (A1) is greater than any of the others,

I At I> IAzl >I Aal >... >IAk+ll•
219

the remaining roots being either negative or complex.
This dominant latent root i\1 , which will be ~ 1 according as to whether the sum of theelements in the first row of B is ~ 1, is the one which is principally of interest. Since it isreal and positive, it is the only root which will give rise to a stable Y, or ~ vector consistingof real and positive elements. It is this stable ~1 associated with the dominant root A1 which
is ordinarily referred to as the stable age distribution appropriate to the given age specificrates of fertility and mortality. Since
it can be seen that the latent root i\1 of the matrix .A and the value of r obtained in the usual
way frome-nz.,m.,dx = 1,J:

are related bylog."-1 = r . ..•8.

PROPERTIES OF THE STABLE VECTORS

Before proceeding further it is necessary to mention briefly the reasons why the methods
given above for the computation of the stable Y, and tf> vectors were adopted, apart from theirsimplicity in practice. If the k + 1 distinct roots of the characteristic equation are known,
we may form a set of k + 1 matricesl("-a) by inserting in turn the numerical vq.lue of each rootin the matrix [B- "-a 1]. The adjoint of 1<"-a> is
F(Aa)= II[B-1\,1] and 1<"-a)F(Aa)=O.
b;joa
It may be shown that the stable Y,aappropriate to the root "-a can be taken proportional toany column, and the stable tf>a proportional to any row of the matrix F(Aa) (see e.g. Frazer,
Duncan & Collar, 1938, chap. m). Moreover, F(Aa) is a matrix product of the type 1/Ftf>,
where the 1/F vector is given·by the first column and the tf> vector by the last row of F(Aa),each divided "by the square root of the element in the bottom left-hand corner; and thetrace of the matrix is equal to the scalar product tf>Y,. Now {B- "-all is a square matrix oforder k + 1 with only zero elements below and to the left of the principal subdiagonal, whichitself consists of units. The product of k such matrices, which gives F(,\a), will have therefore
a unit in the bottom left-hand corner. Since the stable tf>a and !/Fa vectors obtained by themethods suggested in § 7 have respectively their first and last elements = 1, it follows that

1/FatPa = F(,\a), tPa 1/Fa = trace F(,\..).
The stable vectors may now be normalized. If the scalar product, tf>ai/Fa = z2, say, thentPa "'" = 1·
rzrrzrFrom now on it will be assumed that the stable vectors appropriate to each of the latentroots have been normalized in this way.
These vectors have the following important properties:(1) The k+ 1 stable Y, are linearly independent. There is thus no such relationship, with
non-zero coefficients c, as

c1Y,1+ca!/Fa+cs!/Fs+···+ck+lfk+l = 0.
(2) The scalar product of a stable Y,, Y,a with the associated vector of another stableY,, 1/Fb is zero, i.e.
tPbi/Fa = 0 (a,Pb).
The normalized stable Y, thus form a set of k + 1 independent and mutually orthogonal
vectors of unit length.
(3) Any arbitrary Y,-1/F., say-can be expanded in terms of the stable Y,, thus

1/F., = C1 !/F1 +ca!/Fa +ca !/Fa+···+ ck+l 1/Fk+l•

220

where the coefficients c may be either real or complex. Similarly an arbitrary vector <fox canbe expanded in terms of the stable <fo. , ••13.

THE APPROACH TO THE STABLE AGE DISTRIBUTION

A stable age distribution appropriate to the matrix B has been defined mathematically by
the equation
Blfr = A.lfr,
and it has already been shown that since only one latent root of B is real and positive, only
one of the stable 1fr will consist of real and positive elements. But, in addition to this Malthusian age distribution, it is also of some interest to inquire whether any significance canbe attached to the remaining stable 1/r associated with the negative and complex roots of thecharacteristic equation.
Any age distribution 1fr.,, the elements of which are necessarily ;;; 0, may be expressed asa vector of deviates from the stable 1/r1 associated with the dominant latent root, and we may
therefore write the expansion of 1/r., in terms of the stable 1/r as(1/rz-cllfrl) = c21fr2+calfra+···+ck+1Vrk+l = 1/rr~.,
where the coefficients care given by the vector c = U 1fr.,. Thus, the way in which a particular
type of age distribution will approach the stable form may be studied by means of thevector Vrr~.·
Among the terms occurring in the right-hand side of this expression there will be, corresponding to each negative root, a single term ca 1fra which will consist of real elements alternately positive and negative in sign. (Even if the normalized 1/ra is imaginary this term will'consist of real numbers, since in this case ca will also become im~inary.) Moreover, corresponding to every pair of complex roots there will be a pair of terms (em 1/rm+ c, 1/rnl whichtaken together will also give a single vector with real elements. This follows from the factthat em is the conjugate complex of c, owing to the way in which the matrix U is constructed;
Then, apart from the scalar c1 which must necessarily be > 0, some of the remaining coefficients c2 , c3, ••• , ck+l in the expansion of 1/rd may be zero. The first and most obvious case iswhen they are all zero, and the age distribution 1frx is therefore already of the stable form.
But, if either,,, = c ,,,'Y da 'f' a'where 1fra corresponds to a negative latent root, orVra = cmlfrm+c,.!Jr,,where 1/rm and 1frn are associated with a conjugate pair of complex roots, then it follows thatthe age distribution lfrx will, as time goes on, approach the stable form in a particular way
defined by either
Bllfra = caAtlfra or Bllfra = cmNlfrm+c,.XI!Jrn,in which N for a pair of complex roots u + iv with modulus r may be written in the form otr( cos !:It± i sin Itt). Thus, the negative and complex latent roots of B serve to determine a
number of age distributions which are of some interest owing to the fact that they willapproach the Malthusian form in what may be termed a stablE~ fashion.
Since IA. 1 I > I"- 2 1> I "-al > ... > I Ak+l I, the vector of deviates 1/ra will tend towards zero
as t-..rx:; whenever ,\1 < l. Thus, in the case of a stationary population, any 1/r., will converge
to the stable form of age distribution. But if A. 1 > 1, there is a possibility of one or more of
the remaining roots having a modulus ;;; 1, e.g. I A. 2 l;;; 1. In the latter case there may becertain age distributions with c2 9o 0 for which the amplitude of the deviations from the stableform tend either to increase (I A. 2 1> 1), or to remain constant (I A.2 1= 1). From the practicalpoint of view, however, we may still say that a population with such an age distribution
approaches or becomes approximately equal to the stable population, since A.i is much greaterthan ,\~ when t is large.

221

14.

SPECIAL CASE OF THE MATRIX WITH ONLY A SINGLE NON-ZERO Fx ELEMENT

The interesting case of the matrix A having only a single non-zero element in the first row has
been illustrated in a numerical example by Bernardelli (1941}.* This author has also useda matrix notation in the mathematical appendix to his paper, and the form of his basic
matrix is the same as that referred to here as MorA. It is not clear, however, from thedefinitions which he gives whether he regards the elements in the first row of his matrix as
being constituted by the maternal frequency figures (m,) themselves, or by a series of values
similar to those defined here as the F"' figures. He refers to the_m merely as the specificfertility rates for female births.
In discussing the causes of population waves, Bernardelli describes a hypothetical species,
such as a beetle, which lives for only three years and which propagates in the third year of
life. He a~:~sumes, for the sake of argument, that--to employ the terminology used hereP0 = -~ and P1 = !, and that 'each female in the age 2-3 produces, on the average, 6 newliving females'. Assuming for the moment that he is here defining a Fx figure, we may write
this system of mortality and fertility rates as

0 0
A= [ t 00

0 0
B = HAH-1 = [ 1 o

l

0

1

The characteristic equation expanded in terms of A is A3 -1 = 0; and the latent roots are
therefore 1, -! ±~ i, all three being of equal modulus. The matrix A has the interesting

properties
so that any initial age distribution repeats itself regularly every three years. Thus, as
Bernardelli shows, a population of 3000 females distributed equally among the three agegroups becomes a total population of 6833 at t = l ; of 5166 at t = 2; and again 3000 distributed equally among the age groups at t = 3. Unless a population has already an initial
age distribution in the ratio of {6: 3: l }, no approach will be made to the stable form associated with the real latent root, and the vector of deviates will continue to oscillate with astable amplitude, which will in part depend on the form of the initial distribution. Although
this numerical example refers specifically to a stationary population, it is evident that asimilar type of argument may be developed in the case when I AI> land A 3 = A3 / .
We have assumed here that his definition of the fertility rate refers to a F, figure. But, if
we were to interpret the words quoted above as referring to a maternal frequency figure,namely that every female alive between the ages 2-3 produces on the average 6 daughters
per annum, then the results become entirely different. For, deriving the appropriate Fx
figures by the method described in § 2, the matrix is now

saA=[~ l~ ~0]·0!

B = HAH- 1 = [ ~

~

0

t]~

'

and the latent roots are l, -! ± !i. The modulus of the pair of complex roots is Ij.J'i, whichis < 1, so that every age distribution will now converge to: the stable form associated with the• At this point I should like to acknowledge the gift of a reprint of this paper, which was received by t-he
Bureau of Animal Population at a time when I was in the middle of this work, and when I was just beginning
to appreciate the interesting results which could be obtained from the use of matrices and vectors: also a personalcommunication from Dr Bcrnardelli, received early in 1942, at a time when it was difficult to reply owing to thedevelopments of the war situation in Burma. Although the problems we were immediately interested in differedsomewhat, this paper did much to stimulate the theoretical development given here, and it is with great pleasurethat I acknowledge the debt which I owe to him.

222

real root. Thus, to take the same example as before, 3000 females distributed equally among
the age groups will tend towards a total population of 4000 distributed in the ratio of{6: 3: 1}, and it was found that this age distribution would be achieved at approximatelyt = 23. During the approach to this stable form periodic waves are apparent both in the age
distribution and in the total number of individuals, but these oscillations are now damped,in contrast with the results obtained with the first type of matrix.
This simple illustration serves to emphasize the importance which must be attached to
the way in which the basic data are defined and to the marked difference which exists between
what are termed here the m"' and F,. figures. Nevertheless, apart from the question of the
precise way in which the definition of the fertility rates is to be interpreted in this example,
the first type of matrix with only a single element in the first row does correspond to thereproductive biology of certain species. Thus, in the case of many insect types the individuals
pass the major portion of their life span in various immature phases and end their lives in ashort and highly concentrated spell of breeding. The properties of this matrix suggest thatany stability of age structure will be exceptional in a population of this type, and that even
if the matrix remains constant we should expect quite violent oscillations to occur in the
total number of individuals.
15. NUMERICAL COMPARISON WITH THE USUAL METHODS OF COMPUTATION:From the practical point of view it will not always be necessary to estimate the actual valuesof all the stable vectors and of the associated matrices which are based on them. Naturally,much will depend on the type of information which is required in any particular case. In
order to compute, for instance, the matrices U, Q and G, it is necessary first of all to determine all the latent roots of the basic matrix. The ease with which these may be found depends
very greatly upon the order of the matrix. Thus, in the numerical example for the brown
rat used previously in § 3, the unit of age and time is one month and the resulting squarematrix A is of order 21. To determine all the 21 roots of the characteristic equation would bea formidable undertaking. It might be sufficient in this case to estimate the positive realroot and the stable vector associated with it. On the other hand, it is possible to reduce thesize of the matrix by taking a larger unit of age, and in some types of problem, where extreme
accuracy is not essential, a unit say three times as great might be equally satisfactory, which
would reduce the matrix for the rat population to the order of 7 x 7. It is not too difficult tofind all the roots of a seventh degree equation by means of the root-squaring method (Whittaker & Robinson, 1932, p. 106). But tP.e reduction of the matrix in this way will generally
lead to a value of the positive real root which is not the same as that obtained from the larger
matrix, and it is therefore necessary to see by how much these values may differ owing to
the adoption of a larger unit of time.
Another important point which must be considered is the following. By expressing theage specific fertility and mortality rates in the form of a matrix and regarding an age distribution as a vector, an element of discontinuity is introduced into what is ordinarily taken
to be a continuous system. Instead of the differential and integral calculus, matrix algebrais used, a step which leads to a great economy in the use of symbols and consequently toequations which are more easily handled. Moreover, many quite complicated arithmeticalprobleins can be solved with great ease by manipulating the matrix which represents thegiven system of age specific rates. But the question then arises whether these advantages
may not be offset by a greater degree of inaccuracy in the results as compared with thoseobtained from the previous methods of computation. It is not easy, however, to settle thispoint satisfactorily. In the way the usual equations of population mathematics are solved,
a similar element of discontinuity is introduced by the use of age grouping. Thus, in the caseof a human population, if we were estimating the inherent rate of increase in the ordinaryway, we should not expect to obtain the same value of r from the data grouped in five year223

intervals of age as that from the data grouped in one year intervals. The estimates of theseminvariants would not be precisely the same in both cases. Nevertheless, the estimate
from the data grouped in five year intervals is usually considered to be sufficiently accurate
for all ordinary purposes, and there is little doubt that if we merely require the inherent rateof increase and the stable age distribution, these methods of computation are perfP.ctly
satisfactory when applied to human data.

REFERENCESBERNABDELLI, H. (1941). Population waves. J. Burma Bes. Soc. 31, no. 1, 1-18•.
CHAm.Es, ENID (1938). The effect of present trends in fertility and mortality upon the future populationof Great Britain and upon its age compoSition. Chap. II in Political Arithmdic, ed. Lancelot
Hogben. London: Allen and Unwin.
FBAzEB, R. A., DuNcAN, W. J. & CoLLAR, A. R. (1938). Elementary Matrices. Camb. Univ. Press.
GLASs, D. V. (1940). Population Policies and Movements in Europe. Oxford: Clarendon Press.
TuRNBULL, H. W. & AITKEN, A. C. (1932). An Introduction to the Theory of Canonical Matrices. London
and Glasgow: Blackie and Son.
WHITTAKER, E. T. & RoBINSON, G. (1932). The Oalcul'UB of Observations, 2nd ed. London and Glasgow:
Blackie and Son.

224

Part IV

Parameterization and Curve Fitting

An early attempt to describe demographic observations in a mathematical formula
was contributed by Abraham DeMoivre, whose hypothesis (1725. p.4) “consists insupposing that the number of lives existing at any age is proportional to the numberof years intercepted between the age given and the extremity of old age,” i.e.
x;`x D `0 1 !
from which we have for the force of mortality
x D

1
:
! x

DeMoivre chose the upper limit ! D 86 for its quite good fit to Halley’s (1693) life
table, and qualified his analysis by setting a lower age limit of 12. His assumptionthat `x , is a linear function of age was used for short age intervals by Joshua
Milne (1815) to develop his qx formula and it remains in use in that context. Asan approximation to the full life table it has long been obsolete. DeMoivre (1733)
was also first to develop the normal distribution. discussed below in connection withfertility.
We begin this chapter with an article by Benjamin Gompertz (1825), paper 25,
who used the better mortality statistics of his time to suggest the expressions
x D Bc x ;`x D `0 g .c 1/ ;x

where B, c, and g are constants; in effect a law that mortality increases exponentiallywith age. Gompertz was innovative in that he reasoned from a law of mortality
in finding `x , where a century earlier DeMoivre had not attempted to justify his
implied mortality function. Gompertz is as much remembered for having reasoned
well: his Law is a better description of the upper ages than DeMoivre’s hypothesis

IVParameterization and Curve Fittingthat populations collapse towards a specific age limit. From Gompertz are derived
Farr’s (1864) and indirectly Reed and Merrell’s (1939) excellent approximations to
n qx .
An improvement on Gompertz’ Law was contributed by William Makeham(1860, 1867), whose 1867 article is included here as paper 26. In a careful
examination of causes of death (as they were then listed), Makeham found that
overall mortality levels could be better represented if a constant term were added to
x to account for causes of mortality not dependent on age, a possibility Gompertz
also had noted. This givesx D A C B c x ;`x D `0 s x g .c 1/ :x

For ages beyond childhood and youth Makeham’s Law combines intuitive plausibility and a close fit to observed mortality.
Other equations have been suggested that give a slightly better fit to mortality
rates overall but have spurious inflection points or are otherwise not intuitive.
For these the reader may consult Hugh M. Wolfenden (1942, pp. 79—85; 1954,
pp. 164—167). A further contribution to the interpretation of Gompertz’ andMakeham’s Laws and their generalization will be found in Brillinger (1961).
The lack of a satisfactory analytic expression for mortality that includes bothinfancy and adulthood has driven workers in the field to the alternative approachprovided by model life tables, which average observed mortality schedules byregression, usually on life expectancy at birth or at age ten where the strong impact
of infant mortality is not felt. An early attempt was published by the United Nations(1955); but by far the best known is that of Ansley Coale and Paul Demeny (1966).
From their examination of historical mortality patterns in countries having reliable
data, Coale and Demeny were able to isolate four general patterns and to constructregressions of the forms
ın qx D a C b e 10 ;ı

ln n qx D a C b e 10
for the separate age groups within each of the four families of tables. Their
blending of the regressions and general methodology are given as paper 27. How
the model tables are applied to countries with poor demographic data is discussed ina paper from United Nations Manual IV (1967) which Coale and Demeny preparedf omitted here g. An application of factor analysis to the problem of sorting out
existing life tables and making the models will be found in Ledermann and Breas(1959).
William Brass (Brass and Coale 1968) has suggested an ingenious combinationof analytic curves and model tables that provides greater flexibility in table use,which we include here as paper 28. Brass takes a selected model table as the

226

IV

Parameterization and Curve Fitting.s/standard, say `x and adjusts it to an observed set of rates by a regression of theform.s/lnŒ.1  `x /=`x / D ˛ C ˇ lnŒ.1  `.s/x =`x :The equation preserves the extreme values `0 D 1, `! D 0, and allows model tables
to be used in cases where observed rates are incomplete and not quite of the (usuallyEuropean) pattern the models reflect. The Brass technique also has much potential
value for forecasting, since in many instances the constants ˛ and ˇ show fairly
clear trends over time (Brass 1974, pp. 546—551).
The early attention and degree of success achieved in mapping and analyzing
mortality patterns contrasts sharply with the very meager efforts that were madeto understand marriage and fertility. Most of the 19th century elapsed between
Nicander’s publication of fertility rates and their use by Milne (1815), and RichardBöckh’s (1886, p. 30) introduction of the Net Reproduction Rate, defined as theintegralZˇ
R0 D

p.a/ m.a/ da;˛

where p.a/ is the probability that an individual survives from birth to age a andm.a/ is the expected number of offspring of the same sex born to him (her) in
the interval a to a C da. The integration is across all fertile ages and hence gives
expected progeny to an individual just born. [Richard R. Kuczynski, at one timea student of Böckh’s and an assistant in the Berlin statistical office, in several of
his works drew attention to the implications of the rate for population survivaland growth; e.g. (Kuczynski 1928, pp. 41—42): “The pertinent question is not:is there an excess of births over deaths? but rather: are natality and mortality
such that a generation which would be permanently subject to them would during
its lifetimes, that is until it has died out, produce sufficient children to replacethat generation? If, for instance, 1,000 newly born produce in the course of theirlives exactly 1,000 children, the population after the death of the older 1,000 willremain unaltered ... and if natality and mortality remain permanently the same,the population will always exactly hold its own. If more than 1,000 children are
produced by a generation of 1,000 newly born, the population will increase: if lessthan 1,000 are produced, the population will decrease and finally die out.”] Neither
the net reproduction rate nor the simpler age-specific fertility rates were in use in
England when Cannan carried out his population projections in 1895.
With Lotka’s contributions to stable population theory good approximations to
the net maternity function and related measures became immediately valuable, andseveral attempts to fit equations to them were made. The earliest employ Pearson
distributions (Elderton and Johnson 1969, pp. 35—109; Kendall and Stuart 1969,
Vol. 1, pp. 148—154), which are solutions to the differential equation

227

IV

Parameterization and Curve Fitting.x  a/d ln f .x/
D;dx
b0 C b1 x C b2 x 2
most but not all having a single mode at x D a and making smooth contact withthe x-axis at the extremities. They include the normal distribution, investigated by
Dublin and Lotka (1925); the Pearson Type I by S. J. Pretorius (1930) and Lotka(1933); and Pearson Types I and III by S.D. Wicksell (1931). (For the normal, b1 D
b2 D 0; for the Type III, b2 D 0.)
The normal, Type III and an exponential introduced by Hadwiger (1940) are
analyzed in detail in Keyfitz (1968, pp. 141—169). We include here as paper 29
Wicksell’s remarks about the normal and Pearson curves.
None of the equations we mention has been defined to express either marriage
patterns or birth interval and family size distributions, and the most recent work in
the field has been directed toward providing this essential base. A large measureof success has been achieved by Ansley Coale (Coale 1971; Coale and McNeil
1972; Coale and Trussell 1974), in work directed toward the development of model
nuptiality and fertility tables. The work combines four elements. Drawing on
evidence from a number of populations, Coale (1971) found that the age-specificrisk of marriage among those who ever marry could be closely approximated by adouble exponential (Gompertz) distribution, differing between populations in origin(age at which marriages essentially begin) and the relative intensity of the marriage
process once underway. For fertility by duration of marriage, Coale suggested taking
as a standard that of a natural fertility population (i.e., one in which birth limitation is
not intentionally practiced), reducible by an exponential decay function to represent
fertility patterns imposed by family planning practice.
In later work extending Griffith Feeney’s (1972) suggestion that the age-specificprobability of marriage could be represented as the convolution of a normal distribution to represent age at eligibility with an exponential distribution representing
delays between eligibility and marriage, Coale and McNeil found that as further
exponentials are introduced the distribution takes as its limiting value a Makeham
curve of the form (Coale and McNeil 1972, p. 744)
g.x/
N
D


expf˛.x  /  expŒ.x  /g; .˛=/0 .˛=/where  is the gamma function,  D a C N
.˛=/ , and a is the mean of g.x/.
The distribution has the important properties that it closely matches the probability distribution g.x/ generated by the Coale risk function and can be approximatedto a high degree of accuracy using 1 to 3 exponentials. The distribution is thus
both empirical and intuitive. The process of fitting the curve is greatly simplified
by taking advantage of the similarities between different populations to establish astandard distribution. This distribution, fitted by two constants, and the distribution
for fertility by duration of marriage, fitted by one constant, form the base for
the Coale and Trussell (1974) model fertility tables. Paper 30 is from this article.

228

IV

Parameterization and Curve Fitting[For another model expressing age at first marriage. also with an intuitive base,see Hernes (1972). The various models about equally reflect observed marriage
patterns.]
The future growth of populations was first treated mathematically by PierreFrançois Verhulst (1838), in response to Malthus’ argument that populations would
tend to grow exponentially until checked by resource limits. By implication,
long range projections might be formed without detailed reference to current age
structure, fertility or mortality, an assumption compatible with the lack of fertility
information in Verhulst’s time. His suggestion, paper 31, was to represent population
growth by the logisticPt;rt D b 1 
k
Pt D

k
;
1 C e abt

which makes the intrinsic growth rate rt a linearly decreasing function of population
size Pt , k being the asymptotic population. The equation was independently applied
to U.S. population growth by Raymond Pearl and Lowell Reed (1920), whose article
here follows Verhulst’s paper 31. Pearl and Reed’s satisfaction with the logistic was
not warranted when they wrote; in particular they failed to notice that the good fitof their equation neglected extraordinary contributions of technology, immigrationand territorial expansion to U.S. population growth, making the correspondence ofrt and Pt rather more fortuitous than imperative. (As it turned out, the curve was
0.3% under the 1930 census but over by 3.5% at the next count. The higher figuretranslates as a 57% overestimate of intercensal growth. It may be compared with
Whelpton’s fine componant projection for 1940, 0.2% above the census, and his
6.0% and 49% underestimates for 1950 population and 1940—1950 intercensal
growth respectively. In this case the better method was not a better guaranteeof success.) Pearl and Reed toward the end of the paper take care in suggesting
limitations of the logistic, and the reader will find there some of the reasons why
they should not have used it; others were provided by Lancelot Hogben (1931, pp.
176—184) and William Feller (1940).
In another work that deserves mention, Thomas Edmonds (1852) applied
Gompertz’ Law to population growth by defining the rate of population increaseas:dP
D B ct ;dt
with the distinction that c < 1 in order that the rate of growth would decline rather
than increase over time. (In Verhulst, this corresponds to the damping factor .p/.)
Edmonds’ equations are those later reapplied to mortality by Farr (1864), and he isalmost certainly Farr’s immediate source. The equations have not been rememberedin population projections.

229

IV

Parameterization and Curve Fitting

An examination by Otis Dudley Duncan (1958) f omitted here g treats human spatial measurement and efforts that have been made to fit curves to city
size distributions, using either Felix Auerbach’s (1913) rank-size rule or Mario
Saibante’s (1928) power function. These analyses confront measurement problems
more complex than we have treated elsewhere in the book. f Ed. Note: A treatmentof spatial demography, rank-size rules, and power laws from the perspective of early
Twentyfirst Century demography is found in [23], chapters 13 to 15. g
Readers should be aware that the equations presented here are fitted to real data in
different ways: DeMoivre’s and those given by Duncan are solved by regression; theearly fertility functions by moments; and Gompertz and Verhulst used simultaneous
equations with selected data points. [Here, note that we might not follow thesame approach. Curves in which constants enter non-linearly are accessible to leastsquares fitting by computer, using iterative methods; neither moment nor selectedpoint fittings are of equivalent quality.] f Ed. Note: Maximum likelihood estimation
has replaced these specialized fitting procedures over the course of time. g
The method of least squares was first suggested by Carl Friedrich Gauss inpersonal communications and by Adrien Marie Legendre (1805) in print; fitting
by moments is due principally to Pearson (1893, 1948).
f Ed. Note: Parametric models for lifetables like the one in paper 28 generally
known as the Brass Relational Logit model are useful for population forecasting.
Trends in parameter values can be extrapolated into the future using econometrictime-series methods which yield measures of uncertainty as well as of central
tendency. Ronald D. Lee and Larry Carter [26] introduced methods for creatingsuch a family of models from a matrix of historical rates of mortality arrayed byage and time using the matrix technique known as the singular value decomposition.
The factor varying with time supplies a parameter which can often be treated for the
purpose of forecasting as mimicking a random walk with drift. Lee-Carter forecasts
have given impetus to the larger field of stochastic forecasting, in which Bayesianstatistical approaches, as in [13], also play a prominent role. g

230

25. On the Nature of the Function Expressive of
the Law of Human Mortality
BENJAMIN GOMPERTZ (1825)

From Philosophical Transactions 27. Excerpts are from pages 513·-519.

We omit Gompertz' method of fitting, which the reader will find in Makeham,below. Also omitted are Gompertz' mortality tables, and his discussion of life
expectancy and annuities under his Law. The hyperbolic logarithms in Art. 5 are
the natural logs; we are not able to follow completely his integration, which is by
Newton's method of fluxions.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_25, © Springer-Verlag Berlin Heidelberg 2013231

Article 1. In continuation of Art. 2. of my paper on the valuation of life contingencies, published in the Philosophical Transactions of this learned Society,in which I observed the near agreement with a geometrical series for a shortperiod of time, which must pervade the series which expresses the number ofliving at ages in arithmetical progression, proceeding by small intervals of time,
whatever the law of mortality may be, provided the intervals be not greater thancertain limits: I now call the reader's attention to a law observable in the tablesof mortality, for equal intervals of long periods; and adopting the notation of my ·former paper, considering Lx to express the number of living at the age x, andusing A. for the characteristic of the common logarithm; that is, denoting by
A.(Lx) the common logarithm of the number of persons living at the age of x,
whatever x may be, I observe that if A.(Ln)-A.(Ln+m), A.(Ln+m)-A.(Ln+2m)).(Ln+ 2m)-),(Ln+Jm), etc. be all the same; that is to say, if the differences of thelogarithms of the living at the ages n,n+m; n+m, n+2m; n+2m, n+3m; etc.
be constant, then will the numbers of living corresponding to those ages form ageometrical progression; this being the fundamental principle of logarithms.
Art. 2. This law of geometrical progression pervades, in an approximate
degree, large portions of different tables of mortality; during which portions thenumber of persons living at a series of ages in arithmetical progression, will benearly in geometrical progression; thus, if we refer to the mortality of Deparcieux,
in Mr. Baily's life annuities, we shall have the logarithm of the living at the ages
15, 25, 35, 45, and 55 respectively, 2.9285; 2.88874; 2.84136; 2.79379; 2.72099, fori.(L 15 ); i.(L 25 ); i.(L 35 ); etc. and we find i.(L 25 )- i.(L 35 ) = .04738, i.(L 35 )- i.(L 45 ) =.04757, and consequently these being nearly equal (and considering that forsmall portions of time the geometrical progression takes place very nearly) we
observe that in those tables the numbers of living in each yearly increase of age
are from 25 to 45 nearly, in geometrical progression. If we refer to Mr. Milne'stable of Carlisle, we shall find that according to that table of mortality, the numberof living at each successive year, from 92 up to 99, forms very nearly a geometricalprogression, whose common ratio is i; thus setting out with 75 for the numberof living at 92, and diminishing continually by we have to the nearest integer
75, 56, 42, 32, 24, 18, 13, 10, for the living at the respective ages 92, 93, 94, 95,
96, 97, 98, 99, which in no part differs from the table by 317 th part of the living
at 92....
Such a law of mortality would indeed make it appear that there was no positive
limit to a person's age; but it would be easy, even in the case of the hypothesis,
to show that a very limited age might be assumed to which it would be extremely
improbable that any one should have been known to attain.
For if the mortality were, from the age of 92, such that t of the persons living
at the commencement of each year were to die during that year, which I haveobserved is nearly the mortality given in the Carlisle tables between the ages 92
and 99, 1 it would be above one million to one that out of three millions of persons,whom history might name to have reached the age of 92, not one would havet.

1 If from the Northampton tables we take the numbers of living at the age of 88 to be 83. and diminish
continually by± for the living. at each successive age. we should have at the ages 88. 89. 90. 91. 92.
the number of living 83; 61.3; 45.9; 34.4; 25.8; almost the same as in the Northampton table.

232attained to the age of 192, notwithstanding the value of life annuities of all ages
above 92 would be of the same value. And though the limit to the possible durationof life is a subject not likely ever to be determined, even should it exist, still it
appears interesting to dwell on a consequence which would follow, should themortality of old age be as above described. For, it would follow that the nonappearance on the page of history of a single circumstance of a person having
arrived at a certain limited age, would not be the least proof of a limit of theage of man; and further, that neither profane history nor modern experience
could contradict the possibility of the great age of the patriarchs of the scripture.
And that if any argument can be adduced to prove the necessary terminationof life, it does not appear likely that the materials for such can in strict logic begathered from the relation of history, not even should we be enabled to prove(which is extremely likely to be the state of nature) that beyond a certain period
the life of man is continually becoming worse.
Art. 4. It is possible that death may be the consequence of two generally
co-existing causes; the one, chance, without previous disposition to death ordeterioration; the other, a deterioration, or an increased inability to withstanddestruction. If, for instance, there be a number of diseases to which the young
and old were equally liable, and likewise which should be equally destructive
whether the patient be young or old, it is evident that the deaths among theyoung and old by such diseases would be exactly in proportion of the number ofyoung to the old; provided those numbers were sufficiently great for chance tohave its play; and the intensity of mortality might then be said to be constant; andwere there no other diseases but such as those, life of all ages would be of equalvalue, and the number of living and dying from a certain number living at a given
earlier age, would decrease in geometrical progression, as the age increased byequal intervals of time; but if mankind be continually gaining seeds of indisposition,or in other words, an increased liability to death (which appears not to be an unlikely supposition with respect to a great part of life, though the contrary appears
to take place at certain periods) it would follow that the number of living out
of a given number of persons at a given age, at equal successive increments ofage, would decrease in a greater ratio than the geometrical progression, and thenthe chances against the knowledge of any one having arrived to certain definedterms of old age might increase in a much faster progression, notwithstanding
there might still be no limit to the age of man.
Art. 5. If the average exhaustions of a man's power to avoid death were suchthat at the end of equal infinitely small intervals of time, he lost equal portions of his
remaining power to oppose destruction which he had at the commencement of
those intervals, then at the age x his power to avoid death, or the intensity of his
mortality might be denoted by aqx, a and q being constant quantities; and if
Lx be the number ofliving at the age x, we shall have a Lx x qx ·.X for the fluxion of theL
LX

number of deaths = - (Lx)";. ·.a b qx = - ----=-, .·.a b qx = - hyp.log of b x hyp.log
of Lx, and putting the common logarithm of ~ x square of the hyperbolic logarithm of 10=c, we have c·qx =common logarithm of233i;d being a constantquantity, and therefore Lx or the number of persons living at the age of x = d · gw~;
g being put for the number whose common logarithm is c. The reader should
be aware that I mean gqx to represent g raised to the power qx and not gq raised
to the x power; which latter I should have expressed by (gqy, and which would
evidently be equal to gqx. I take this opportunity to make this observation, as
algebraists are sometimes not sufficiently precise in their notation of exponentials.
This equation between the number of the living, and the age, becomes deservingof attention, not in consequence of its hypothetical deduction, which in fact iscongruous with many natural effects, as for instance, the exhaustions of thereceiver of an air pump by strokes repeated at equal intervals of time, but it isdeserving of attention, because it appears corroborated during a long portion oflife by experience; as I derive the same equation from various published tables ofmortality during a long period of man's life, which experience therefore proves thatthe hypothesis approximates to the law of mortality during the same portion oflife; and in fact the hypothesis itself was derived from an analysis of the experience
here alluded to.

234

26. On the Law of Mortality
WILLIAM M. MAKEHAM (1867)

From Journal of the Institute of Actuaries 13. Excerpts are from pages 335-340, 348.

We omit Makeham's opening comments on life table functions, some of themortality tables he examined, and his appendix, in which he develops some of his
formulas. Makeham's Fx is the force of mortality.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_26, © Springer-Verlag Berlin Heidelberg 2013

235

The formula for Fx, according to Mr. Gompertz's theory, is Bqx. For this I
propose to substitute A+ BqX, where A is the sum of certain partial forces which
we assume to be, in the aggregate, of equal amount at all ages. The quantity
Bqx may also consist of the aggregate of several forces of a similar nature. So
that we may put
Fx=(a+a' +a"+ ... )+(b+b' +b" + ... )qx,

where a+a'+a"+ ... =A, and b+b'+b"+ ... =B.
I do not profess to be able to separate the whole category of diseases into thetwo classes specified-viz., diseases depending for their intensity solely upon thegradual diminution of the vital power, and those which depend upon othercauses, the nature of which we do not at present understand. I apprehend that
medical science is not sufficiently advanced to render such a desideratum possibleof attainment at present. I propose only at present to show that there are certain
diseases-and those too of a well-defined and strictly homogeneous characterwhich follow Mr. Gompertz's law far more closely than the aggregate mortality
from all diseases taken together. I shall then have given sufficient reason for thesubstitution of the form Bqx + c/J(x) for the force of mortality in lieu of Bqx: theproof that the terms included in c/J(x) form, in the aggregate, a constant quantity,
I shall leave until we come to the examination of data more satisfactory than the
returns of population and the public registers of deaths.
The two following tables are taken from the supplement before referred to.
They give, first, the number of annual deaths (from all causes) to 1,000,000 living;and secondly, the number of annual deaths from certain specified causes to thesame number living. The causes of death, as well as the ages for which they are
given, have of course been selected as the most favourable exponents of the law ofgeometrical progression; but is will be observed that the former embrace all theprincipal vital organs of the body, and the latter include the whole of the period
from early manhood to the confines of extreme old age.
The column headed "total force of mortality" should form a geometricalprogression if Gompertz's law were applicable thereto. That it does not, however,form such a progression, is evident by inspection; the rate of increase in the earlier
terms being less than 50 per cent., and gradually increasing until it exceeds 100 per
cent. A similar result is found in all the known tables when the law is applied
to the total force of mortality, the remedy for which (in constructing mortality
tables by Mr. Gompertz's formula) is usually sought in a change of the constantsof the formula after certain intervals. It is this gradual but constant variation of therate of increase in one direction, and the fact of its being uniformly found in all
tables, that show unmistakeably that if the law itself be true, its application
stands imperatively in need of some modification.
The modification which I have suggested, viz., there are certain partial forcesof mortality (how many I do not pretend to say) which increase in intensity withthe age in a constant geometrical ratio, while there are also certain other partial
forces which do not so increase, may be tested by an examination of the six
columns which follow that of the total force above referred to. The tendency toa geometrical progression is more or less apparent in all of them; the average rate236

Male Life, 1851-60

Ages

25-34
35-44
45-54
55-64
65-74

Partial Forces of Mortality

Total
Force
of
Mortality

Lungs

Heart

Kidneys

Stomach
and
Liver

Brain

Sum of five
precedingColumns

9,574
12,481
17,95630,85565,332

772
1,524
3,092
6,616
13,416

514
1,002
1,898
4,130
8,714

174
292
471
937
2,453

464
890
1,664
3,032
4,837

638
1,180
1,990
4,097
9,831

2,562
4,888
9,115
18,812
39,251

Female L!fe, 1851-60

Ages

25-34
35-44
45-54
55-64
65-74

Total
Force
of
Mortality

Partial Forces of Mortality
Lungs

Heart

Kidneys

Stomach
and
Liver

Brain

Sum of five
precedingColumns

9,925
12,147
15,19827,00758,656

582
1,049
2,062
5,027
11,016

603
1,118
2,064
4,558
8,916

109
151
212
317
485

570
937
1,608
2,967
4,692

532
872
1,681
3,818
8,905

2,395
4,127
7,627
16,68734,014of increase being such that the force of mortality somewhat more than doubles
itself in 10 years.
It should be observed that, in addition to the diseases of the particular organs
specified, other diseases of a kindred nature are also included under each of theabove five partial forces. Possibly if more detailed information were accessible,
we might be able to trace the geometrical character during a still more extended
period of life. This, at least, I find to be the case in reference to one particulardisease, viz., bronchitis, which in the preceding tables is included in the class of"lung diseases." Now it so happens that the deaths from bronchitis alone, for along series of years, are given in the 26th Annual Report of the Registrar-General,from which the materials for the following table are taken. The number living is
supposed to be 100,000, instead of 1,000,000 as in the two preceding tables.
In the preceding examination of the results of the Registrar-General's returnsof deaths, I have confined myself to the object of proving that Gompertz's law is
traced much more distinctly in the deaths arising from certain specified diseases,
than in the deaths arising from all causes together. If I have succeeded in thisobject (and I think it can scarcely be denied that I /wee succeeded), I have justified
the introduction of an additional term in the formula representing the total force ofmortality; but I have as yet brought forward nothing to show that such additionalterm is a constant in respect of the age, and varying only with the peculiar
characteristics which distinguish different sets of observations from each other.
237

Ages

15-25
25-35
35-45
45-55
55-65
65-75
75-85

1855 to 1857
(3 Years)

1848 to 1854
(7 Years)

1858 to 1863
(6 Years)

Males

Females

Males

Females

Males

Females

8
17
42
107
259
589
1,027

9
16
34
85
218
525
906

9
21
55
133
333
801
1,463

9
22
45
112
316
697
1,325

9
22
59
151
379
876
1,614

9
21
50
126
351
834
1,479

The several observations, however, which I now proceed to examine, if they
do not enable us (like the former) to test particular terms of the function referred to,yet they will nevertheless afford a very satisfactory criterion of the complete
expression. Not only, therefore, do they form by themselves (on account of theirunquestionable accuracy and trustworthiness) ample evidence of the truth of the
supposed law of mortality, but they also supply the deficiency, above adverted to,in the preceding investigation, as regards the requisite proof of the constancyof the term representing the aggregate of the remaining partial forces of mortality.
Commencing with the very valuable observations on the "Peerage Families"(both sexes), I find, by dividing the entire period of life into intervals of 14 yearsneglecting, however, the first-the following results:log L 14 = 0.99034 _ 0.05068
log L 28 :0.9396~ _ 0 .05784 ::::0.00716
log L42 - 0.8818 __ 0 .08343 0.02559
logL 56 =0.79839 _ 0 .19738 -0.11395
logL 70 =~.60101_ 0 . 61011 -0.41273

log L 84 = 1.99090

The tendency to a geometrical progression in the four terms of the second
order of differences is sufficiently apparent. In order, however, to show this more
distinctly, I have devised the following method of correcting the series log Lx
so that the four terms in question shall form a perfect geometrical progression.
If the series consist of jive terms, and consequently the second order ofdifferences of three, the latter may be converted into a pure geometrical progression
by substituting for the original series another of the following form, viz.,log L 0 + p, log L.- p, log L 2 n + p, log L 3 n - p, log L4 n + p,
where p is derived from the equation' The differences are those of the function log L,.

238

This method, it is true, changes the value of the radix of the table, but I see nonecessity for making a distinction between that and other terms of the series; forin comparing the terms of the altered with those of the original series, the object
is to ascertain their bearing with respect to the original series generally, and not
to any one term in particular. Secondly, by the method adopted, the first differences(which are the logarithms of the probabilities of living n years) are increased ordiminished by an uniform quantity; whereas by omitting the correction in log L0 ,the first term of the first order of differences would be increased or diminishedby one-half of the quantity introduced into the remaining terms. Lastly, theequation for p would be of the second order, instead of the simple one given above.
Again, if the series consist of six terms-in which case there will be four terms
in the second order of differences-the required effect may be produced bysubstituting for log Lx the series
logL 0 +(v-w), logL,.-(v-w), logL 2 ,.+v,logL 3 ,.-v,
logL 4 ,.+(v+w), logL 5 ,.-(v+w),
r and w being determined from the equations

and

A'C'-B' 2
8 v = A'+ 2 B' + C''

B'=~~,

C'=~~,.-4w.

AC-B 22 w = A+ 2 B + C
where
and
A'=~~+4w,

Here again, by involving the corrections symmetrically, we obtain for the
unknown quantities simple instead of complicated quadratic equations.
Applying these formulae to the [preceding series], we have 2w =0.002651,
and 8 r = 0.007725. The transformed series therefore becomeslog L:14 = 0.989980 _ 0 .049960
logL~ 8 =0.940020 _ 0.057234 -0.007274
logL~ 2 =0.882786 _ 0.085362 -0.028128
logL;; 6 :0.797424 _ 0.194122 ::::0.10876~
log L;o -Q.603302 _ 0.614694 0.42057log L 84 = 1.988608

log= ~.86177 0.58737
log= ~.44914 058733
log= ~.03647 0 58737
log= 1.62384 ·The logarithms of the third series, and their differences, show that thetransformed series fulfils the required conditions.
I have now to show that this result has been attained without a greater alterationof the original series than is warranted by the probable errors of the latter. In thefollowing table the first column contains the age, the second the natural numbers
corresponding to the original series log L,, the third gives the decrement (deduced
from the original data) of the year immediately following, while the fourth and239

fifth contain respectively the transformed series (denoted by L~) 2 and the amountby which it differs from the original series in the second column.

X

LX

D,

L~

Lx-L~

14
28
42
56
70
84

9780.08702.87617.66286.23990.3
979.3

41.4
69.8
78.3
150.3
220.3
152.2

9771.98710.07634.66272.34011.5
974.1

+ 8.1
- 7.2
-17.0
+13.9
-21.2
+ 5.2

712.3

72.6

Comparing columns 3 and 5 together, term by term, we find that in one
instance only (viz. at age 42) does the alteration made in the numbers living
exceed one-fifth [0.2171] of the corresponding yearly decrement; while from thesums of the same columns it appears that the average alteration is little morethan one-tenth of the average decrement. We may, therefore, say the limit of thevariation of the two series (cols. 2 and 4) is about one-fifth of a year ....
We have seen that the expression for the number living at any given age in anormally-constituted increasing population-in which the yearly births as well as
the yearly deaths (and consequently also the excess of the former over the latter)are proportional to the existing population-is of the same form as that representing the numbers living at the given age in a stationary population, and also thenumbers living in a table of mortality. But the function Lxvx is also of the same
form as the latter, for Lx contains a factor sx which combines with that introducedby the interest of money. Hence it follows that to determine the number living
between given ages in a population normally constituted- whether increasing or
stationary-as well as the expectation of life and the value of annuities, thesummation of a function of one form only (viz., dgq"' sx) is required.

2 Hitherto the accent has been used to distinguish the "partial" from the "total"' forces of mortality,
but as we have now done with this branch of the subject. no confusion will be caused by using it todenote (as it will be used henceforth) the corrected values of the function to which it is applied.

240

27. Calculation of Model Tables
ANSLEY J. COALE and PAUL DEMENY (1966)

From Regional Model L(fe Tables and Stable Populations, Chapter 2, pp. 11-14,20,23-26. Princeton:
Princeton University Press.

We have omitted sections of the chapter on the correlation of nqx to nqy and elO•which was high in all cases, and on the calculation of stable populations. The
regression COefficientS for the nqx On elO and the Separation faCtOrS k,rJ. are also
left out here, since these are less important than the methods underlying theirderivation.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_27, © Springer-Verlag Berlin Heidelberg 2013241

CALCULATION OF FOUR FAMILIESOF MODEL LIFE TABLES
The model tables presented here are based on a tendency that we
noted for the life tables based on accurate data to cluster aroundfour different lines,-. representing distinct age patterns of mortality
in certain geographical regions.
The four families of life tables were isolated as the result ofworking with a preliminary one-parameter family of model tables
designed to represent the entire collection of 326 life tables. The
following method was used to construct the preliminary modeltables: all of the q:r values were ordered, from highest to lowest, ateach age. The values were then plotted as a function of order foreach age, and occasional erratic fluctuations remov~d by handsmoothing. Preliminary model tables were then formed by putting
together mortality rates with the same rank. This system of construction has the virtue of simplicity, and of symmetrical treatmentof mortality rates at all ages. That is to say, no particular rate or
rates are singled out as the basis for estimating others. The usefulness of the preliminary set of tables is questionable, however, because the 326 tables are so widely scattered about the "line" thesemodel tables formed, as we could see by examining two-dimensional
scatter diagrams.
The next step was to examine deviations of individual life tablesfrom the age pattern of mortality in the preliminary model tables.
The pattern of deviations was measured by the difference between
q.r in the given table and q:r in a model life table with the samegeneral level of mortality. The "comparison" model table wasselected by first noting the expectation of life at birth in the model
life tables with the same nq:r as the given life table, at x = 0, 1, 5,. . . , 75. The median model table (the table with the median e0 °)WaS the "comparison" table. The graph showing ,.q;r - ( nq ;r) morl~ 1
displays, for ready visual comprehension, the age pattern of mortality relative to the age pattern in the model tables, in a way that
does not depend on the over-all level of mortality. Graphs of thissort were constructed for all 326 life tables. The following generalobservations emerged from an examination of the patterns:242

1. Some life tables have large variations from the model patterns.
Large deviations were especially frequent under age 10 and overage 60.
2. The biggest deviations were found in life tables where thequality of the underlying data is suspect: life tables for Western
Europe prior to 1850; certain Eastern European life tables-for
Russia in 1897 and 1926, for Bulgaria and Greece; and life tables
in underdeveloped countries in Asia, Africa, and Latin America.
In most of these tables, age-reporting in the censuses, and doubtlessin the mortality register, is inaccurate. The completeness of deathregistration-especially for infants-is uncertain.
3. Where data are known to be accurate, deviations are usually
moderate. Statistics that are comparatively free of age misstatement
and omission are found in some European countries since 1870, in
20th-century Canada, the United States, Australia, and New
Zealand. Unfortunately, these areas have a rather narrow rangeof cultural diversity and doubtless represent a narrow sample ofage patterns of mortality. It is of special interest to note, then, thatthe life tables of Taiwan and the more recent Japanese life tables-both based on accurate data-conform as well as do European
tables to the preliminary model tables.
Two features of the deviations found within life tables basedon accurate data led to the construction of four separate families
of model tables: (a) The fact that the pattern of deviations isoften similar among life tables expressing the mortality of the same
population at different times, and (b) the fact that several groupsof geographically linked populations exhibited similar patternsof deviations.
Figure 1 shows patterns of deviation in certain Scandinavian life
tables; in tables from Germany, Austria, Czechoslovakia, Poland,and North Italy; and in tables from South Italy, Spain, andPortugal. Within each group the similarity pf deviations is easily
visible, although from group to group the patterns differ.
On the basis of an examination of the patterns of deviations in
the 326 life tables for each sex, correlation matrices were calculated
for various groups of life tables-zero-order correlations among 19
variables: log ,,qJ. at various ages (x = 0, 1, 5, 10, ... , 75), andexpectation of life at age 0 and age 10. We experimented with243

nine principal sets of correlations for each sex: ( 1) tables before
1870; ( 2) tables for Russia and certain Balkan areas; ( 3) tables
for selected Central European areas; ( 4) tables for Scandinavian
countries; ( 5) tables for Spain, Portugal, and Southern Italy; ( 6)tables for Switzerland; ( 7) tables for countries with reliable data
not included in ( 3), ( 4), ( 5), or ( 6); ( 8) tables reflecting mortality when there is an unusually high incidence of tuberculosis;and ( 9) modern tables based on relatively inaccurate data-primarily from Asia, Africa, and Latin America.
The nature of the life tables underlying each of the "regional"
model tables is as follows:
1. Tables underlying "East" model tables. The life tables ofAustria, Germany (including tables in 1878 and the 1890's for
Bavaria and Prussia), Czechoslovakia, North and Central Italy,
Hungary, and Poland show deviations from the preliminary model
life tables characterized by high mortality rates in infancy, andincreasingly high rates over age 50 (F igure 1, central panel).
Switzerland's life tables show deviations very similar to this groupuntil 1920, although the early Swiss life tables have a less conspicuous positive deviation in infancy. After 1920, the Swiss life
tables show zero or negative deviations in infant mortality. Hungarian life tables exhibit substantial deviations in an age patternindicating an extraordinary incidence of tuberculosis. Inclusionof the Hungarian life tables lowers the correlation coefficients fromage 5 to 35, but has little other effect, and they were omitted fromthe calculation of the "East" model tables, as were the Swiss life
tables. The tables in the "East" group include 13 from Germany(of which 3 are from Prussia or Bavaria), 5 from Austria, 3 from
Poland, 4 from Czechoslovakia, and 6 from North or Central Italy.
2. Tables underlying "North" model tables. The life tables of
Norway, Sweden until 1920, and Iceland deviate from the preliminary model tables in having low infant mortality rates, andrates that are lower than the model rates by an increasing margin
at ages beyond 45 or 50. Later Swedish life tables do not have thischaracteristic pattern ( F igure 1, left panel). In the life tables of all
three "North" countries from 1890 or 1900 to 1940, there are deviations in the mortality rates from age 5 to 35 or 40 indicating theeffect of an unusual incidence of tuberculosis. Model tables in244

245

DEVIATIONS OF

"cr"

-~

\

\

1~040

40110

70

- - PORTUGAL 1919· 1922

0

10

20

30

40

50

--SOUTH ITALY 1950-1953- - · · SPAIN 1940..-.. - SOUTH ITALY 1921-1922......... SPAIN 1920----PORTUGAL 19ll9-19<,2

60

70

-.oao

-070

-.060

-050

-040

-.010

-020

·DID

.000

.010

020

-030

0

10

'20

30

40

110

60

...... CZECHOSLOVAKIA 1949-1951•-·-• POLAND 1931·1932--NORTH ITALY 1921-1922......... AUSTRIA 1900·1901.. _ .. _ AUSTRIA 1949•1951---- GERMANY 1949-1951- - GERMANY 1881 -1890

70 AGE

SELECTED LIFE TABLES.

FIGURE I. Deviations of mortality rates in. three regional collections of life tables ( .q.) from mortality rates in a preliminary model life table (.q.'). Selected tables for females.

0

30

"070

20

\\I
.
\ ~"07010I.

\\\l·-080, i l,oeo

-.080

I I

_ _ SWEDEN 1881-181D\ \ 1-.050
I I
I.

\ i

'I

"030

-020

---- NORWAY 1871-1880........ NORWAY 1946-1950____ ICELAND 1941-1950

\

.

.

............l-.010\\ I\I

80

'

\\

~

~

..

··· \

II

.030 .1

1 .o4o!

.050! ! .040

i~

;

~i

.o6o
.050

!1.070.080!
i I .o6o;

070'".0110

"050

-.040

-.oso

"

·D20~

I

I
-,010~.0401-

.0!50

.060

.070

.080

FROM PRELIMINARY MODEL LIFE TABLE VALUES,"'Cfx. ~nq~------=-------.

corporating this experience would be suitable only for populations
with a high endemicity of tuberculosis. Consequently, the "North"model tables are based on Swedish mortality from 1851 to 1890
( 4 tables), Norwegian mortality from 1856 to 1880 and from 1946
to 1955 ( 4 tables), and the Iceland life table for 1941-1950.

3. Tables underbjing "SoutJi' model tables. The life tables of
Spain, Portugal, and Southern Italy have high mortality under age
5, low mortality from about age 40 to 60, and high mortality overage 65, relative to the preliminary model tables (Figure 1, rightpanel). Early tables-prior to 1912-for all Italy had these samecharacteristics. The "South" model life tables were based on 5
tables for all Italy ( 1876 to 1910), 8 tables for Portugal ( 1919 to
1958), 1 table for Sicily ( 1951), 3 for South Italy ( 1921 to 1957),
and 5 for Spain ( 1910 to 1940).
4. Tables underlying "West" model life tables. The "West"model life tables were based on mortality experience recorded in
populations known to have relatively good vital statistics, and notshowing a persistent systematic pattern of deviations from thepreliminary model tables. In other words, the tables underlying
the "West" models are a residual collection after the "East," "South,"and "North" tables have been removed. (The Swiss tables were also
omitted, having a recent pattern resembling the "West" up to age
45 or 50, and the "East" above age 50.) The tables underlying the'West" models include 7 from Australia ( 1881 to 1955), 4 from
Belgium ( 1880 to 1949), 7 from Canada ( 1926 to 1959), 11 from
Denmark ( 1895 to 1955), 11 from England and Wales ( 1871 to1959), 1 from Estonia (1933), 4 from Finland (1881 to 1955), 16
from France ( 1871 to 1959), 5 from Ireland ( 1925 to 1952), 3 from
Israel ( 1949 to 1959), 6 from Japan ( 1949 to 1960), 1 from Latvia( 1936), 1 from Luxemburg ( 1947), 10 from the Netherlands( 1870 to 1955), 12 from New Zealand ( 1906 to 1959), 2 from
Northern Ireland (1925 to 1959), 7 from Scotland (1891 to 1959),
5 from Sweden ( 1931 to 1959), 3 from Taiwan ( 1921 to 1959), 4
from the Union of South Africa white population ( 1920 to 1947),
and 10 from the United States ( 1901 to 1958). Tables before 1870
were eliminated because most of those examined (from France,
the Netherlands, and England and Wales) had irregular patterns
that appeared to arise from faulty data. . ..
246

CONSTRUCIION OF FOUR SETS OF MODEL LIFE TABLES

The principal steps in the calculation of the four sets of model
life tables were as follows:
1. Intercorrelation matrices for nqz and log1o(nqz) were calculated for the "North," "South," "East," and ''West" data. [eo 0 and
e1o 0 were left untransformed (no logarithms taken) in the second
sets of intercorrelations.]
2. Least-square linear regressions of nqz and of log nq:c on e 10°
were fitted for both sexes in all four "regions." The regression coefficients are given in Table XI.
3. The values of nq 111 estimated from the logarithmic regression
are always above those from the regression of untransformed
mortality rates at the high and low extremes of observed life expectancies, and the logarithmic regression values are always lower inthe middle range. In other words, the two regression lines always intersect twice within the range of observations. In constmcting themodel life tables, nq:c values were taken from the simple regression
at all points to the left (i.e., at points with lower life expectancy)of the first intersection of the regression lines; and to the right ofthe second intersection, nq :c values were taken from the logarithmic
regression. Between the two intersections, the mean of the nq:cvalues from the two regressions was used.
4. From various values of the independent variable ( e10 ° ), 11q:c's
at ages 0, 1, 5, 10, . . . , 75 were calculated. From each such set of
nq:c's, l1, ls, l1o, ... , lao were computed with lo taken as 100,000.
5. nL111 and e111° were estimated on the use of the following
formulae:
1Lo = kolo + (1 - ko)ll
4L1 = k1l1 + (4 - k1)l5
5Lz = 2.5(/z + lz+5), X = 5, 10, • • • , 75

e80° = 3.725 + 0.0000625/so
Tso = eso0 lso

l Lz +
75

Tz =

Tso

X

Cz

o

=

Tz
-r;
247

6. Age-specific mortality rates ( nmx) were calculated from theformula nmill = ndillf.nLill where ndill = l:r -lill+n
7. Five-year survival rates for projecting five-year age groups( 5 Pill)
were calculated by the formula sPill = sLill+s/ sLill, x = 0,
5, ... ' 70.
The first survival ~ate is the proportion surviving to the end of afive-year time interval of persons born during the interval, estimated as sLo I 5l0 • The last survival rate is of persons over 75 at
the beginning of an interval (and over 80 at the end), estimatedas Tso/T7s·
8. Both male and female tables are calculated by regression ofnqill on e 10 °. The values of e 10 ° that were used as the independent
variable in constructing the female tables were chosen so as togive even 2.5-year intervals of e0 ° from 20 to 77.5 years. The valuesof e 10 ° for males were chosen so as to correspond with the female
e10 °'s in a way that preserves the typical relation of e 10° for males
and females at each level of mortality within each family of life
tables. The relationship posited was as follows:

where <Tm and u 1 are the standard deviations of expectation of life
at age 10 for males and females. This expression is the equation forthe straight line with a slope intermediate between the regressionof ( e1o 0 ) m on ( e1o 0 ) 1, and the inverse of the regression of ( e10 ° ) r on
(et0 °)m. The correlation between e 10 ° for the two sexes is more than
.99 in all instances, so that the two regression lines are almost
identical. ...
Tim USE OF TWO REGRESSIONS IN CALCULATING nqill IN MODEL TABLES

There are two considerations that make questionable the use ofregression equations in constructing model life tables: ( 1) A
regression of a particular functional form (linear, quadratic,
logarithmic, etc. ) often fits the data better over some parts of therange of observations than others, and often represents the data
poorly at extremes, and may in particular provide an implausible
extension or extrapolation of relationships beyond the range of ob248

servations; ( 2) "least squares" regression equations of any givenfunctional form establish two relations between any pair ofvariables, as each is considered the independent variable, and theother dependent. In an estimation of one variable from a known
value of another, the use of one variable as dependent and theother as independent is clearly appropriate and there is a logicalreason for employing one regression rather than the other. But
there is no logical basis for selecting one mortality rate or anotheras independent when trying to estimate a set of interrelationships
among mortality rates at different ages. In the original United
Nations model life tables, the selection of 1 q 0 as the sole purely independent variable, and the calculation of a chain of regressions,with each mortality rate first being a dependent and then an independent variable, created a tendency (because of the wellknown "regression toward the mean") to incorporate less of theobserved range of variability at higher ages than at lower ages.
A regression chain starting at the highest ages and ending with
1 q 0 would have been equally plausible, and would have produced
somewhat different model tables.
The use of two linear regressions (logarithmic and non-logarithmic) and the selection of e10 ° as the index of mortality (the independent variable in calculating mortality rates for every age
group) result in a procedure that lacks elegance, but takes accountof the considerations stated in the preceding paragraph.
The expectation of life at age 10 is an approximate general index
of the level of mortality, an index that is not strongly dependent
on the mortality rates for any one age group. (In contrast, e0 ° ' isstrongly influenced by the value of 1 q0 . ) Correlations with e111 ° arenear to unity at most ages in all "regions." The choice of e 10 ° as thebasis for regression estimates is not an arbitrary fixing of any one
mortality rate in the resultant model life tables. All mortality
rates are separately subject to "regression toward the mean." Hence
if an e1o 0 two standard deviations above the mean is chosen as thebasis for estimating a model life table, all of the mortality rates will
be somewhat less than two standard deviations below their meanvalue; but the sequence of mortality rates is not ever more compressed toward the mean as age advances. . ..
249

The facts that led us to combine estimates from regressions of
nqx and log nqx on e1o 0 were as follows:
1. The two sets of correlation coefficients were comparable inmagnitude. There were 48 instances where the non-logarithmiccorrelation exceeded the logarithmic by at least .003; 55 where
the logarithmic coefficients were larger by this margin; and 33
instances where the difference was less than .003. (Table XII).
However, there is a different relative effect on the two kinds ofcorrelation coefficients of "scatter" in mortality rates at high levelsof mortality on the one hand, and low levels of mortality on theother....
2. The logarithmic regression line at the highest observed expectations of life is closer to the observations, and represents a moreplausible extension of the observations, than the non-logarithmic
regression line. This closer fit is caused partly by the sensitivity of
the logarithmic regression to relative rather than absolute deviations,and partly by the fact that declining linear exponential functions arenecessarily asymptotic to zero, and never give negative values. In
contrast, the non-logarithmic regression line falls below the observed mortality rates for the higher values of e10 °, and sometimes
indicates negative mortality estimates near the upper limits of observed values of e 10 °-a patent absurdity.
3. At the other end-high mortality, low expectation of life-of
the range of observed values, the non-logarithmic regression line isusually closer to empirical mortality rates, and, in this range, also
represents a more plausible extension of observed relations. The
exponential necessarily becomes steeper at lower expectations oflife, and would yield absurdly high estimated mortality rates if
e1 0 ° were assumed to have a low value.
Through the middle range-between the two intersections of thelines-sometimes one line is closer, and sometimes the other to theobserved mortality rates. To have a uniform rule, and minimize
discontinuity in the sequence of model tables, we estimated mortality rates from the non-logarithmic regression up to the firstintersection, used the mean of the two regression estimates between
the two intersections, and used the estimates from the logarithmic
250

regression at expectations of life at age 10 above-to the right ofthe second intersection.
ESTIMATION OF LIFE TABLE VALUES OTHER THAN

nqx

The weights ( ko and k 1 , the "separation factors") that relate 1Lo tolo and l1, and 4 L 1 to Z1 and l5 , can be shown to equal the age atdeath of those members of the life table population who die underage 1 (for ko), and the age at death minus 1 of those who die
between ages 1 to 4 (for k 1 ). The values of the factor ko were therefore determined by examining the average age at death under 1
year in the records of the populations whose mortality experience
was the basis for the four regional model life tables. No consistentvariation in average age of infant deaths was found at infantmortality levels above 0.100, nor any consistent differences amongthe "regions," except that ko for "East" infants was consistently lessthan in the other "regions," and ko for males was generally slightly
less than for females. When 1qo is very low-in the range from
0,015 to 0.100-there is a clearly apparent tendency for ko to vary,because at very low levels of infant mortality infant deaths aremuch more concentrated immediately after birth. This tendency
was represented by allowing ko to rise linearly from a typical levelat the lowest observed infant mortalities until at 1qo = 0.100 itreaches the plateau typical of higher infant mortality.
The value of k1 in the expression 4 L 1 = k1Z1
4 - k1) Z5 wasdetermined from the estimates of 12 , l3 , and Z4 given in Table XV atthe end of Chapter 3. In fact 4 L 1 was calculated on the assumptionthat lx can be considered linear in each single-year age intervalfrom 1 to 5; thus k1 = 0.5 a~a3
a 4 , when l2 = a2Z1
( 1 - a 2 ) Z5, etc. Each ai was based on the relation of li to l1 and Zu
observed in the life tables upon which each family of model tables
was based. As with ko, it was noted that when 1 q 0 > 0.100, thevalues of each ai showed no tendency to va:ry as a function of thelevel of mortality, although at levels of mortality below 1 q 0 = 0.100
there is a tendency for the deaths from age 1 to age 5 to be more
evenly distributed, and for l:c in this range to be more nearly linear.
Hence at low levels of mortality, each aL was allowed to fall as
1qo increases in a manner analogous to the rise in ko. The valuesof a2, aa, and a4 are somewhat different in each region. It is reassur-

+(

+ + +

251

+

ing to note that when 1 qo > 0.100, the a/s observed in the life
tables underlying each regional set of model tables clustered about
an average typical of the set, and differing from the averagevalue of ai for the other "regions." In other words, each region
has a distinctive pattern of mortality from 1 to 5, as well as atother ages.
The formula for expectation of life at age 80 is based on a line
fitted to observations in 70 life tables ( 35 for males and 35 forfemales) from the Netherlands, Norway, Sweden, Switzerland,
and Japan. These tables were selected because age reporting in thecensus and mortality records in these countries for persons over
80 appeared especially trustworthy.

252

28. Methods of Analysis and Estimation
WILLIAM BRASS and ANSLEY J. COALE (1968)
From William Brass, et. al.: The Demography of Tropical .1/i"ica, pp. 127-132. Princeton: Princeton
University Press.

In this extract Brass fits logit transformations of observed and model life tablesurvival rates to a power function that preserves the extreme values p(O) = 1,p(w)=O, where p(x) is the probability of surviving from birth to age x. The
work may be clearer if his equations (9) and (10) are rewritten slightly asq(a)jp(a) =A [ qs(a)/Ps(a)]P u(a),In [q(a)jp(a)] =In A+ j3ln [qs(a)/Ps(a)] + lnu(a)

(10a)

(9a)

with the multiplicative disturbance term u(a), whose expected value in (9a)becomes E(lnu)=O. From (9a) the estimators A.,/3 are found by linear regression.
The equation to generate the new survival estimates jJ(a) from the fitted regressionis/J(a) = { 1 +A [qs(a)fps(a)]P}- 1 .

Loosely, A is associated with life expectancy and j3 with the distribution of mortality
between younger and older ages, relative to the standard table.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_28, © Springer-Verlag Berlin Heidelberg 2013

253

The Brass Model Life Tables and Stable Populations
The Brass method of constructing model life tables is to subject thesurvivor function in a life table chosen as a "standard" to the so-called
logit transformation, and then to consider the life tables generated byassuming that their logits are linearly related to the logit of the standard
table. In this way, a two-parameter set of model life tables can be constructed.
The logit function is as follows:1
1 -x
logit (x) = :zlog. - x -(8)For x in equation (8), Brass substitutes p(a) (or la/10 in life-table notation). He then selects a standard table, p8 (a), and constructs his model
life tables by assigning different values to a and {3 inlogit p(a) = a + 13 logit p.(a)(9)

This transformation implies that:q(a) = A ( q_,(a))tJp(a)\j,(a)(10)

where A = e2"'.
The Nature of the Linear Logit Transformationof a Standard Life Table

The logit transformation can generate a survivor function p (a) thatpasses through arbitrarily preassigned values at any two ages a1 and a2,
by the selection of a and (3. 19 The life table thus generated has the preassigned values of p(a) at the selected ages and shares the shape, in a
generalized sense, of the "standard" life table.
Figure 3.4 illustrates the effect of different values of a and {3 on thestandard life table that Brass has constructed for Africa. The survival
curves include: (a) the standard (a= 0, {3 = 1.00); (b) a curve thatdiffers from the standard by having a {3 of 1.20 instead of 1.00; (c) aGiven any preassigned p(a,) and p(a.), we obtain two equations:
11 + fJ logit p.(a,) = logit p(a,)" + fJ logit p.(a.) = logit p(a.)
The known values of p,(a,) and p,(a2 ) together with the preassigned values ofp(a,) and p(a2 ) provide a unique determination of " and {3, and hence a complete life table.
10

254

plal
1.0,.---------------------.9h'"i ··~,.\ ··.:--,··........ ...
··· ..... ~ .......... ......
\~·1.20)'· ..... ...
\
\
.?
··.
. ex··... ··..
'·'·
··.
'·

.8

I ·.,,

1.61•••.... ...

'· .......

.6e•t.oo ' ·'·IX=o.3o
.5

..·.

••••

··..··.,'·'·'·'·, ..··...
···...
.,
••

.,

··.

''·..........,·.·,····.··...

.4···...
' . .......

,.3·, ....

·,.... ''.,<<.. \\j.2"·\:. ,I
.I

0

5

10

15

20

25

30

35

40

AGE

45

50

55

60

65

70

75

3.4. Proportion surviving to age a, p(a), obtained from thetransformation p(a) =a+ r>togit p,(a) for various valuesof a andr..

curve that has the same value of p(2) as the standard, but a {3 of 1.20,
and (d) a curve that differs from the standard by having a value of aof 0.30 instead of zero. Note that the curve with a = 0 and f3 = 1.20
intersects the standard at an age where p 8 (a) = 0.50. This is a generalrelation: all curves with the same a and different f3's intersect at ii de255

termined by p 8 (li) = 0.500, because logit 0.5 = 0. Hence variationof f3 produces a sub-family of survival curves, fixed at age 0 and w, andwith a node at li (about age 51 in the standard table). At ages below thisnode or intersection, the life table based on the larger f3 has higher valuesof p (a), and at higher ages the opposite relation holds. As a result thearea under these two curves (which in each instance is the expectationof life at birth) is very nearly the same. 20 However, this near constancyof e 0 for different values of f3 is not a general relationship, but is the result of the selection of a standard life table with p (51 ) = 0.50. If ahigher-mortality standard life table had been chosen, tables with thesame a would intersect at a younger age (say 25) and higher values of
f3 would lower the expectation of life at birth.
Note that the logit transformation permits the acceptance of a trustworthy estimate of childhood survival-e.g., an estimate of p(2)-andthen the selection of some overall estimate of adult mortality, in the formof a choice of p(30) or p(50) given the accepted value of p(2). The
transformation of the standard table (with a and f3 chosen in this manner) then provides a whole model life table.
0

The Importance of the Standard Life Tablein the Linear Logit Transformation

The logit transformation will generate life tables having any preassigned level of childhood mortality-expressed, for example, by a valueof q(2)-and of adult mortality-expressed, for example, by 48q2, orp(50)/p(2). How well do such life tables conform to the observed age
pattern of mortality in empirical populations? The Coale-Demeny families of model tables provide an interesting test. Each family of thesetables expresses the particular age pattern of mortality found in the wellrecorded experience of a group of populations. The groups of life tables
underlying each family were assembled because of .noticeable similarities in age patterns. Figure 3.5 shows the typical result of trying to express a model table within one of the families (a) as the linear logit
transformation of another table in the same family, and (b) as the logit
transformation of a table from a different family. Note that logit p(a)°For the standard life table, e=43.6, and for {3 = 1.2, e 0 0 = 44.1. There0
is a kind of symmetry among curves with different values of {3, and a. = 0. The
difference in p for each young age such that p,(a')>0.5 is matched by an equal(and opposite) difference in p for the older age a" such that p,(a") = 1 - p,(a').
Thus when p,(a) = 0.50 occurs near the middle of the range zero to IIJ, life tables
with a.
0 and different values of {3 have nearly the same e 0 0 •
2

0

=

256

L200

,.

1.00

Jls (a.) WEST FEMALES.,.., ......
(=50 YEARS

.8
.6

en
a:

tt.""

~.4

,.
0

_,"'

...... ..-,'"'"'

p 0 (a) SOUTH FEMALES,

It, • 50 YEARS

.

": .2

...,

•o

~

...;;

o.•. 2

9-.4
·.6

-.8
-I

·2.6

·2.4

-22

-2.0

-1.8

-1.6

-1.4

-12

LOG IT p5(a ),

-1.0
·.8
·.6
e,~ 20 YEARS

·.4

·.2

0

.2

.4

3.5. Logit p(a) _plotted against Jogit p,(a), when p(a) isSouth model life table, females, e', =·20 years, and p,(a) isalternatively West model11Dd South model table, e'. =SO
years.

for the South model life table (females) with e o = 20 has a very nearlylinear relation to logit p(a) for the South model table where e 0 = 50
years, but that the relation to logit p(a) for the West model table withe 0 = 50 is by no means linear. Figure 3.6 shows the result of attempting to approximate the South model table with eo 0 = 30 years by alogit transformation of the South model table with eo 0 = 50. and theWest model table with eo o = 50. Calculations of the same sort with theother Coale-Demeny families of model tables led to a similar result:linear logit transformations reproduce the model life tables within a family very closely, but between families not nearly so well. Table 3.10
shows the values of a and /3 needed to transform the female table witheo o = 50 into the tables with e 0 of 20, 30, 40, 60, and 70 in each family. Note that in all families except South (where {3 is quite large at highmortality levels) the parameters needed to transform from one mortality level to another are almost identical.
0

0

0

0

257

p(a)

10--------------------------------------.9--p.(a)•WEST MOOEL,o\'0 •50.8l- - - p• (a): SOUTH MODEL~•50•••••••• SOUTH MODEL, 80 ,30

O·L---~5~--~~o~~~~5---2~0~~2~5--~3~0~~3~5---4~o~~4~5--~5~o~~5~5--~6~o~-6~5~~7~u--~?5AGE

3.6. South model life table, females, ~·, = 30 years com·pared with life tables obtained by linear logit transformati<>nof South and West model tables, e', =SO years.

258

+

Table 3.10. Values of a and {3 in logit p(a) = 11 {3logit p,(a) for North, South,
East, and West model life tables (female) , various levels of e•., life tablewith e"0 =50 serving as P..(a)

e 00
North

West

South

East

= 20 ec0 = 30 e 00 = 40 e 00 = SO e 00 = 60 e 00 = 70

a

1.008

0.612

0.295

o.ooo

-0.316-0.734

a.

1.296

1.154

1.064

1.000

0.950

0.900

a

1.036

0.632

0.307

0.000

-0.339-0.819

B

1.306

1.161

1.068

1.000

o. 944o. 893

a

1.057

0.650

0.318

o.ooo

-0.356-0.803

B

1.533

1.318

1.155

1.000

0.810

0.599

a

1.000

0.618

0.302

o.ooo

0.338

0.813

B

1.299

1.165

1.073

1.000

0.929

0.852

These examples show that the selection of a standard life table affects
the detailed form of the survival curve in a life table fitted to the estimated mortality experience of a population by a linear logit transformation. The standard life table to be used in Africa, in other words, should
ideally incorporate typical features (if such exist) of African age patterns of mortality.

259

29. Nuptiality, Fertility, and Reproductivity
S.D. WICKSELL (1931)
From Skandinavisk Aktuarietidskrift, pp. 149-157.

We omit Wicksell's discussion of Swedish nuptiality and fertility, and his distinction
between net reproduction rates based on legitimate and illegitimate births. For
his equation (31), a Pearson Type I distribution, the fitted constants should be
Case1: b 1 =1.857,

b2 =2.518

Case 2: b 1 = 1.689,

b 2 = 2.593.

Formulas for fitting the Type I distribution are given in Elderton and Johnson(1969, pp. 51-52). The fitting requires the mean and variance (JJ.,JJ. 2 ) and two
next central moments (JJ. 3 ,JJ. 4 ) of the net maternity function, given by
az

Jlo =

I p(x)m(x)dx=R 0

az

Jl =

I xp(x)m(x)dx/Jlo

az

Jl; =I (x-JJ.)ip(x)m(x)dx/JJ.oi=2,3,4 ...

where p(x) is the probability of survival from birth to age x and m(x) the probabilityof giving birth in the interval (x,x+dx). The lower and upper limits of the fertile
age distribution are a 1 and a 2 , respectively.
In Wicksell /(x) f(x), or r(x), replaces p(x)m(x), and we have for the frequency function

with r the familiar gamma function. The equation is most easily solved from themoments by proceeding in steps, through the intermediate values c 1 ,c 2 ,c 3 :D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_29, © Springer-Verlag Berlin Heidelberg 2013261

If age boundaries are preset, b 1 and b 2 can be found using only the first two
moments of the distribution. Here we would have, on rearranging terms in theequations above,b2-- b I az -Jl
Jl-al

A fitting of the distribution using preset age limits has been made by Mitra (1967)
for a large number of populations.
Wicksell's equation (21) uses the more easily fitted Type III distribution,
for which a 1 =0, a 2 =co, andy = Jl!Jlz

f3 = Jlz/Jlz.

26210. Puttingr(x)=f(x)l(x),the function r(x)-being proportional to the distribution of the ages at which ageneration of girls bear their daughters-plays a rather important part in themathematical theory of population.
In Table 7 the function r(~) is given, integrated over quinquennial age-groups(in the columns headed "total").
Table 7. Number of daughters that will be born to 1000 new-born girls if they have the maternity of
Sweden in 1926, the mortality of Sweden in 1921-1925 and the nuptiality referred to as Case 1 andCase 2.
Age ofmotherCase 1

Case2

Number of daughters

Number of daughters

Legitimate

Illegitimate

Total

Legitimate

Illegitimate

Total

15-20
20--25
25-30
30-35
35-40
40-45
45-50

13
130
209
195
151
76
8

27
54
27
13
8
4

40
184
236
208
159
80
8

27
230
315
263
192
95
10

27
44
15
5
3
2

54
274
330
268
195
97
10

Sum

782

133

915

1132

96

1228

Both these series can be described with sufficiently good approximation bymeans of frequency functions of Pearson's type III, starting at x=O. If we put
JJ

(21)

r(x)=R · _Y_ xP-l e-yx
T(/3)

and define the moments in the usual way, viz.
v0 =
v'1 =

00

Jdxr(x),000

Jdxxr(x),000v2 = Jdx(x-v'1 ) 2 r(x),0263

we getR = v~;

Thus, by means of the method of moments, we find
In Case 1: R=0.915,/3=19.6,

y=0.65.

In Case 2: R= 1.228,/3=19.5,

y=0.65.

The curves are thus very nearly of the same form-differing only by the factor R-and we adopt for both curves the values of the parameters: f3= 19.5 and y=0.65. ·The expression (21) for r(x) is a very convenient one. In the mathematicaltheory of population, as developed chiefly by Lotka, the integral
R(t) =(22)"'Jdxr(x)e-rx,
0plays a very important part. In particular, the equation(23)

R(t) = 1

is of fundamental interest, and several rather complicated and laborious methods
have been devised for solving it. But when r(x) is given in the form (21)-and it
seems that r(x) can generally be expressed in this way-(22) and (23) assume
very simple forms. We actually have (putting w = oo)(23*)

R(t) = (

R

t

)fl,

1+y

when the real part oft> - y and the solution of (23) will simply be

We getCase 1 : t = - 0.0029
Case 2: t = + 0.0070.
The values of the there calculated simply give the rates of increase per annum,
corresponding to the net reproductivity rates given in Section 9. They are, what
Lotka terms the "true rates of natural increase". The following table, calculatedon the assumption that we have /3=19.5 and y=0.65, gives the value of the true
rate of natural increase, t, for different values of the net reproductivity rate, R.
The corresponding time for the population being doubled or halved is added.
264

This time is, as well known, given by the formula

In Fig. 5 a graph oft as a function of R has been given (/3 = 19.5 and y = 0.65).
To save space also the negati"l(e values oft have been plotted to the positive side of
the axis.
Table 8.
Net
reproductivity
rate
R

2.50
2.00
1.80
1.60
1.40
1.30
1.20
1.15
1.10
1.05

Population
growth inpromille per
annum
1000t
31.2
23.5
19.9
15.9
11.3
8.8
6.1
4.7
3.2
1.6

Period of
doubling or
halving the
population in
years
y

Net
Population
reproductivity growth inrate
promille perR
annum
1000 t

22
29
35
44
61
79
113
148
218
426

1.00
0.95
0.90
0.85
0.80
0.70
0.60
0.40
0.20
0.10

0.0
- 1.7
- 3.5
- 5.4
- 7.3
-11.8
-16.8
-29.5
-51.5
-72.4

Period of
doubling or
halving the
population in
years
y
00

410
197
128
94
59
41
23
13
10

11. In a recent paper (Metron. Vol. VIII, 1930) Dublin and Lotka give tablesof the function r(x) for the U.S. (22 states) in the year 1925 and following years.
The central ordinate is given in quinquennial age groups. Multiplying by 5 we get
the following series for 1925:
Table 9. Number of daughters born at different ages to a generation of 1000 girls. (Mortality according
to the U.S. life table 1925. maternity of U.S. (22 states) in 1925 and actual civil status distribution in
the same registration area, 1925.)
Age

10-15

15-20 20-25

25-30

30-35

35-40

40-45

45-50

50-55

Sum

Number ofdaughters

0.6

100.6

291.0

218.9

149.3

53.0

5.6

0.0

1128

309.0

The moments are: R = 1.128; v'1 = 28.33; v2 = 45.88, and we get f3 = 17.49;
y =0.6175, and equation (24) gives
t= +0.00427.

265

t

t

-31

I

-30
-29
-28

0 ;-.REASE

I
I

INCREAS

-27

+31
+30
+29
+28
+27

-26

+26

-25

+25

-21.

+21.

-23

+23

-2:!

+22

-n

+21

-20

+20

-19

+19

-18

+18

-17

+17

-16

+16

-15

+15

-14

+14

-13

+13

-12

+12

-11

+11

-10

+10

-!1

+9

-!I

+8

-7

+7

-6

+6

-5

+5

-I.

+I.

-3

+3

-2

+2

-1

+1

o.o 0.1 o.z o.a 0.4 0.5 0.6 0.7 0.8 0.9 l.o 1.1 1.~ l.a 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.s 2.4 2.5 R
Fig. 5. Net reproductivity, R, and true rate of natural increase, r.

Dublin and Lotka give the same value of t, which value has, however, been
computed by a quite different method.
In fact, Dublin and Lotka compute the value oft from the equation(26)

which is obtained by assuming r(x) to be described by the normal, LaplaceGauss'ian frequency function.

266

As the solution of equation (26) can, in our notation be written in the form
(27)

t=y(t<+>V1-~log.,R),

it is easily seen that when R -1 is a small quantity (24) and (26) give the samevalue of the root. In fact (24) is the first term in an expansion of (27) in powers of(R1111 -1).

The reason why our equation (24) gives the same values oft to three decimal
places as obtained by Dublin and Lotka 1 is thus only that the values of R used,
happen to deviate very little from unity.
Dublin and Lotka's equation for t, (26), is, as already mentioned, the cQnsequence of assuming r(x) to be given by the ordinary, normal frequency function.
A glance at the asymmetry of table 9 shows, however, that the normal curve
here gives rather a poor fit. Thus, when R deviates substantially from unity,
equation (26) will not give a good value of t. To meet this difficulty Lotka has, inanother paper, extended equation (26), so as to take account also of the higher
moments. Equations of the third or fourth degree are the results. This moregeneral method is obtained by neglecting the higher terms in the series(28)

which series follows from the semi-invariant theory of Thiele (the A.k being the socalled semi-invariants).
Notwithstanding the fact that the rate of convergence of this series is, generally
spoken, open to question, the method is at times certainly rather unwieldy fromthe numerical point of view. It therefore seems to me to be of practical importancethat the formula (24), which is a skew curve, gives a better fit than the normal
curve and that consequently formula (24) has a wider range of applicability thanformula (26).
12. In concluding, the following additional remarks may be made. The curve(21) for r(x) is a Pearson Type III curve with the starting point in x=O. This
choice of the starting point causes the computations to depend only on the
moments of the first and second order. It is clear, however, that a still better fit
may at times be obtained by using the third order moment to determine thestarting point. Putting
II

(29)

r(x ) = R -')' - ( x-a)fl-1 e - y(x-a)r([J)

we getRe-ar(30)

R(t) = (

t)li.

1+')'

1 This takes place not only in the example just cited but also in all the other examples given in thepaper referred to.

267

The parameters are now obtained from the equations

andThe factor e-ar here makes the solution of the equation R(t) = 1 more difficult,and this is the reason why we have preferred to put a= 0, as a fairly good fit wasnevertheless obtained. Cases may occur, however, where a substantial improvement
is obtained by leaving the starting point a free, and using the third order moment to
fix it. The equation R(t) = 1 may then be solved numerically by a graphical method.
A still better fit may be obtained by using Pearson's type I, i.e. by putting(31)
Using the ordinary moment-method of determining the parameters we find for the
Swedish material (Table 7):
Case1

b 1 =0.857

b2 =1.518

a 1 =17.02

a 2 =48.50

Case2

b 1 =0.689

b 2 =1.593

a 1 =17.65

a 2 =48.83.

The curves thus obtained give a nearly ideal fit in the cases in question, but ofcourse here the solution of the equation R(t) = 1 is still more complicated.
The advantage of using the formula (21) ofr(x) with the corresponding formula(23*) of R(t), instead of the expression (28) used by Lotka, is more clearly realized
when we consider that in the mathematical theory of population growth not
only the real root of the equation R(t) = 1 is required, but also a number of thecomplex roots.

268

30. Model Fertility Tables: Variations in the Age
Structure of Childbearing in Human Populations
ANSLEY J. COALE and T. JAMES TRUSSELL (1974)
From Population Index 40. Excerpts are from pages 186-192.

The model tables and the authors' discussions of their uses and methods of fitting
are omitted.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_30, © Springer-Verlag Berlin Heidelberg 2013

269

The Basis for the Model Schedules of Fertility

The basic assumption upon which the model schedules are calculated is that
fertility conforms to the structure by age created by multiplying together twomodel subschedules: a sequence of model proportions ever married at each age anda model schedule of marital fertility. Thus, if the proportion ever marrie<i at age ain the model schedule of nuptiality is G(a), and the proportion of married
women at age a experi~ncing a live birth in the model schedule of marital fertilityis r(a), age-specific fertility is j(a) = G(a) • r(a). This construction applies exactly toa hypothetical population in which there is no fertility outside marriage, and no
dissolution of marriage before the end of the childbearing span of ages. But it also
duplicates quite adequately the age structure of fertility in actual populations
through the selection of a G(a) that differs slightly from the proportion evermarried in the actual population, and of an r(a) that differs slightly from theactual marital fertility schedules.
The representation j(a) = G(a) • r(a) makes possible the calculation of model
fertility schedules from three specified parameters-two parameters required to
specify a model schedule of proportions ever married, and one parameter required to
specify a model schedule of marital fertility.
Age Structure of the Proportion Ever Married, G(a), Specified by Two Parameters

First-marriage frequencies, defmed as the number of first marriages in a short
age interval divided by the number of persons in that interval, have been shown toconform to a curve of the same shape in different populations (or more precisely
in different cohorts). What differs from population to population is the age at
which first marriage begins, the duration of the age span within which the
majority of the marriages occur, and the proportion of the survivors in the cohort
who, at advanced ages, have been married at some time. The similarity in structureof the age distribution of first marriages in different cohorts is analagous to thecommon shape characterizing different normal (Gaussian) distributions, which arealike only when the mean (location), standard deviation (horizontal scale), andvertical scale (number of cases, or size of population) are specified.
If the effect of differential mortality by marital status on the proportion evermarried is neglected, the existence of a standard distribution of first marriage
frequencies implies a standard curve describing the proportion ever married indifferent cohorts. The form of the curve is standard, but there are differences, ofcourse, in the starting age of a tangible proportion ever married, in the pace atwhich the curve rises and in the ultimate proportion experiencing marriage-the
proportion ever married by the age at which first marriage rates have fallen
essentially to zero. If the standard proportion ever married x years after firstmarriages begin is Gs(x), in any cohort G(a) = c. Gs((a- a0 )/k), where C is afactor determined by the ultimate proportion ever married, a0 is the age at which
first marriages begin, and k is the scale factor expressing the number of years ofnuptiality in the given population equivalent to one year in the standard population. If k is 1.0, first marriages occur at the same pace as in the nineteenthcentury Swedish population that served as the basis of the standard; if k is 0.5, or
one-half, first marriages occur at twice the pace of the standard. Specifically,270according to the standard schedule half of the population that will ever marry has
experienced first marriage ten years after the earliest age at which a consequential
number of first marriages occur; if k is equal to 0.5, one-half the cohort has
experienced first marriage five years after a0 •The standard proportions ever married were published in an earlier article(Coale 1971), but for computational convenience, we have calculated G(a) from aclosed-form analytical expre~ion for first marriage frequencies developed by
Donald R. McNeil (Coale and McNeil 1972). This expression is:(1) g(a) ={0.19465/k) exp {{-0.174/k)(a-a0 -6.06k)- exp [{-0.2881/k){a-a0 -6.06k)]}

No analytical expression for G(a) has been found, but G(a) can be calculated bynumerical integration of g(a), since G(a) = f 0° g(x)dx. This representation of G(a),0with appropriate estimates of a0 and k, provides an approximation of the proportion ever married in a cohort, if multiplied by a scale factor to allow for theparticular proportion ultimately experiencing marriage. However, since the standard schedules of fertility that we have constructed represent only the age patternof fertility and not the level, the proportion ultimately marrying is omitted here.
Only the age of initiation and the pace of first marriages affect the structure of
fertility; the proportion remaining celibate influences the level but not the age
pattern of fertility.

The Age Structure of Marital Fertility, r(a), Specified By a Single Parameter.

Louis Henry found that there is a characteristic pattern of marital fertility inpopulations in which there is little or no voluntary control of births. He defined
voluntary control as behavior affecting fertility that is modified as parity increases,
and the absence of control-natural fertility-as behavior, whether affecting fertility or not, that is the same no matter how many children have been born (Henry
1961). The regularity in marital fertility that makes possible a single-parameter setof schedules is this: marital fertility either follows natural fertility (if deliberatebirth control is not practiced), or departs from natural fertility in a way that
increases with age according to a typical pattern. In a population in which fertility
is voluntarily controlled, the ratio of marital fertility at each age, r(a), to aschedule of natural fertility, n(a), is given by:(2)r(a)/n(a) = M exp (m • v(a))

The factor M is a scale factor expressing the ratio r(a)/n(a) at some arbitrarily
chosen age. Since we are concerned only with the age pattern of fertility (not its
level), the value of M {like the value of the factor C in the model schedule ofproportion ever married) is of no significance for the construction of our fertility
schedules. The function v(a) expresses the tendency for older women in populations practicing contraception or abortion to effect particularly large reductions offertility below the natural level.
271

Model schedules of r(a) are required at single years of 'age over the full range at
which there is found both 1) a non-zero proportion cohabiting, and 2) non-zero
marital fertility. The two functions n(a) and v(a), assumed to be invariant, must
therefore be estimated by single years of age; the requisite family of modelschedules is then obtained by assigning values tom, from zero, in which case l{a)equals n(a), to a maximum expressing the greatest likely departure of fertility
from the age pattern of natural fertility resulting from a very high degree ofvoluntary control of births.
The functions n(a) and v(a) were derived from empirical data. There were two
steps in the derivation: first, the estimation of approximate values of n(a) and v(a)by five-year age intervals above age 20, and second, determination of single-year
values by freehand interpolation above age 20 plus extension to ages below 20 on
somewhat arbitrary common sense principles.
Seven values of n(a) at ages 20-24 through 45-49 were derived by calculating
the arithmetical average of schedules designated by Henry as natural (Henry 1961).
Henry's schedules begin at 20 because premarital conceptions have a large andirregular effect on teenage marital fertility. Ten schedules of natural fertility wereaveraged after discarding schedules known to be based on surveys in which age
misreporting was especially prevelant and might have distorted the pattern of
fertility. The effect of this selection (compared to the acceptance of all schedules
listed by Henry) is minor, since the age pattern of all of those listed is broadly
similar.
Seven values of v(a), at ages 20-24 through 45-49, were obtained by calculations employing the marital fertility schedules listed in the United Nations Demographic Yearbook for 1965 (United Nations 1966). Again, schedules known orsuspected to be distorted by age misreporting or other forms of faulty data werediscarded. Each of the forty-three schedules not eliminated on this basis were
provisionally accepted as embodying, each in its own degree, the typical pattern ofdeparture from natural fertility.
For the ;th schedule an individual v;(a) can be calculated by setting m = 1.0 in
equation (2). For the ;th schedule we find(3)v;(a) =log [r;(a)/(M • n(a))]

M is chosen so that v;(a) is zero for the age interval 20-24. The arithmetical averageof the forty-three values of v;(a) in each of the seven age intervals was thendefmed as v(a) for each interval. The values of n(a) and v(a) are as follows:
20-24

25-29

30-34

35-39

40-44

45-49

n(a) 0.460v(a) 0.000

0.431-0.3160.396-0.8140.321-1.0480.167-1.4240.024-1.667

The function v(a) calculated in this way can be validated by substituting thetabulated values in equation (2) and seeing how well the result fits each marital
fertility schedule. A value of M is chosen that equates M. n(a) with r(a) at ages
20-24. One way of getting a visual impression of how well v(a) fits a given marital272

fertility schedule is to calculate a separate value of m for each age interval. If
equation (1) were fully valid, and v(a) appropriately estimated, the separately
determined values of m for age intervals 25-29 through 4549 would all be thesame. The sequence of m's calculated for the forty-three empirical marital fertility
schedules is not in every instance highly uniform. However, the set of m's formost marital fertility schedules falls on a reasonably level plateau, and thedifference in level of m between different populations is quite evident (see Figure
1).

2.4,...CZECHOSL,1961-HUNGARY, 1963

2.0

1.6

1.2

.8

.4

=--- ·-~ ----~,GUADELOUPE,I961

\

0

\

EL SALVADOR,I962

25-29 30-34 35-39 40-44 45-49
AGE INTERVAL

Fig. 1. Values of m, where m = log[r(a)/(M • n(a)) I /v(a), for selected marital fertility schedules

.5

.12
-v(a)

.8
.4

.I

Fig. 2. Values of n(a) (natural fertility). and v(a) (logarithmic departure from n(a))

273Single-year values of n(a) and v(a) are shown in Figure 2, and tabulated as partof the FORTRAN program in Appendix A. The hand-fitted values of n(a) aboveage 20 approximately match, in average value for each five-year interval, the values
at five-year intervals listed earlier. The extension of n(a) back to age 12 is basedon general biomedical information that full reproductive capacity is reached a few
years after menarche, and that the mean age at menarche varies from about 12 to
16 years in different populations. The particular choice of rates to represent n(a)below age 20 is not of major importance because of the dominant role of G(a) in
determining the rise of age-specific fertility with age.
Values of v(a) at single ages were chosen so that their sum over five-year age
intervals matched (above age 25) the values at five-year intervals given earlier. To
avoid a sharp change in the neighborhood of age 25, non-zero values were assumed
to begin at age 20.
With single-year values of our three functions, we have the means of calculating
a full range of fertility schedules for hypothetical populations in which there is no
illegitimacy and no marital dissolution, and in which marriage begins at various
initial ages and occurs over various age spans, and in which marital fertility ranges
from the gradual decline with age characteristic of natural fertility to the muchsteeper decline characteristic of populations in which there is extensive control of
fertility within marriage. The age pattern is given by equation (4):(4)f(a) = G(a)n(a)em•v(a)where !(a) is age-specific fertility, G(a) is the proportion ever married (in apopulation where first marriage occurs according to a schedule characterized by
selected values of the parameters a0 and k), n(a) is natural fertility, v(a) is thecharacteristic pattern of departure from natural fertility, and m is the extent ofthat departure.
Model Schedules of Age-Specific Fertility, and Their Similarity to the Age Patternof Fertility in Actual Populations

In actual populations, of course, births occur outside of marriage as well as
within, and the proportion of the population currently married differs from the
proportion ever married because of the presence of the widowed and divorced.
However, the structure of fertility in an actual population may closely resemble
that in a hypothetical population with no marital dissolution or extramarital
fertility if the latter population has slightly different parameters of nuptiality andmarital fertility from those found in the actual population. The effect of illegitimate births and of premarital conceptions on the age structure of fertility isequivalent to a schedule of first marriages that is slightly different from theobserved one at early ages; the effect of illegitimate births at the older ages is
equivalent to a slight increase in marital fertility at those ages. The proportion of
the ever married population that is widowed and divorced rises monotonically withage, thus reducing fertility toward the end of childbearing in a way that is
topographically similar to the effect of v(a) on marital fertility. In other words, it
is probable that the standard schedule of first-marriage frequencies, with a suitable274

choice of initial age and pace of occurrence of firsf marriages, can serve as a usable
surrogate for the age of entry into sexual union (including unions that do not infact involve marriage), and that modification of natural fertility by the properchoice of m by which to multiply v(a) can serve to approximate the effect both ofmarital dissolution in reducing the fraction married at higher ages and of control
of fertility on marital fertility. On the provisional assumption that such is the case,
we have calculated a large array of model fertility schedules by single years of age;each schedule is composed of the product of an estimated proportion ever marriedand of marital fertility in each single-year age interval. The starting age ofnuptiality was allowed to range from 12.5 to 18 years; the pace of marriage from
56 percent of the pace (k = 1. 7) to five times the pace (k = 0.2) in the Swedish
standard nuptiality schedule. The value of m was permitted to range from zero(natural fertility) to 3.9, on a scale in which 1.0 is the average value forforty-three schedules in the 1965 Demographic Yearbook. A total of 795 modelschedules was tabulated. Each schedule has been normalized so that the sum of thefertility rates at all ages is 1.0; the schedules embody only an age pattern of fertility
and carry no implication with respect to total fertility.
The tabulated schedules have been selected to produce mean ages at integral
values from 24 to 34 years and values of standard deviation (achievable within the
stipulated limits of the three underlying parameters) at intervals of half a year.
The range of standard deviation is from 4.0 to 7.5, but some combinations (e.g.
standard deviations of 7.0 or 7.5 with a mean age of 25) could not be attained
within the limits of the three controlling parameters.
When a0 was 15.0 or more, the single-year rates under age 20 were modified toconform to an observed feature of reliably recorded single-year schedules; non-zero
fertility rates typically begin at about age 15 even when marriage begins relatively
late. Positive fertility rates at ages 15 and 16 in such populations are probably the
result primarily of extramarital conceptions that occur to a small number ofadolescents. The requisite modification was achieved as follows: the value of
fertility at exact age 20 and the cumulated value of fertility up to age 20 were
accepted as initially calculated from equation (3). Values of n and R were foundsuch that !(a) equals R(a- 15)n matches the calculated value at age 20, and suchthat R J?~(a- 1srda matches cumulated fertility (as calculated) up to age 20.
A crucial question is whether this family of model fertility schedules provides aclose fit to the fertility of actual populations. We have tried to determine how
well the model schedules operate by fmding a schedule (through interpolation
among the printed values) that matches each of a number of recorded schedules interms of the mean age and the standard deviation and the ratio of the averagevalue of fertility in the interval from ages 15 to 20 to the average value from ages
20 to 25. Figure 3 shows the goodness of fit for three selected fertility schedules
recorded by single years of age.
The schedules were chosen because they had the lowest and highest mean ages
(Hungary, 1970, and Sweden, 1891-1900), and the lowest standard deviation(Japan, 1964) among the single-year fertility schedules that we examined; in spiteof the fact that the schedules fitted are extreme, the fit in every case is quite
close. In fact the absolute value of the area between the model schedule and the
275

.24

• Recorded• Model>- .20

I-

;!
~ .16
w

IL

1i: .12uw:li .08Iw<t .04(!)

AGE
Fig. 3. Age-specific fertility rates of three populations fitted by rnotlel fertility schedules

recorded rates is in each instance less than 2.5 percent of the total area undereither curve. We have fitted a number of other recorded fertility schedules withequal success.
Figure 4 shows the structure of fertility that results when entry into cohabitation is early and rapid or late and gradual, combined with natural fertility, andwith fertility that is highly controlled. In interpreting Figure 4 the reader must
keep in mind the normalization of each schedule so as to produce an arbitrary
total fertility of 1.0. The figure illustrates the distribution of fertility by age, notdifferences in level of fertility associated with age patterns .
.090
.080
.070

!'\
I
I

I
I
I
I

m• 2.4a 0 •13.5 ,k = 0.4'/\

a 0 =18.0,k•l.2

\

\
\
\

-.060
0

u.: .050t::.
0:;:- .040
.030

.020

AGE

35

40

45

49

Fig. 4. Model fertility schedules, total fertility = 1.0. Combinations of early marriage with various
degrees of fertility control and late marriage with various degrees of control

276

REFERENCES

Coale, Ansley J. 1971. Age pattern of marriage. Population Studies 25(2):193-214.
Coale, Ansley J., and McNeil, Donald R. 1972. The distribution by age of the frequency offirst marriage in a female cohort. Journal of the American Statistical Association
67(340):743-749.
Henry, Louis. 1961. Some data on natilral fertility. Eugenics Quarterly 8(2):81-91.
United Nations. Department of Economic and Social Affairs. 1966. Demographic Yearbook
1965. Sales No.: 66.XIII.l. New York.

277

31. A Note on the Law of Population Growth
PIERRE-FRANc;:Ois VERHULST (1838)

Correspondence Mathematique et Physique Publiee par A. Quetelet (Brussels) 10, pp. 113-117.
Translated by David Smith.

Of Verhulst's tables we have included only that for France, which best fits his
formula and has an asymptotic population of approximately 40 million. Readers
should note that errors in Verhulst's estimates of annual population change areserious and directional with respect to the official figures, and that the proportional
error figures he gives are not the ones we most care about.
In a paper published some years later Verhulst (1845) gave the name logistiqueto his curve and discussed its properties in more detail. This article included
his method of fitting the curve, which we reproduce in a note following his paper.
He also considered more carefully than in the present work the quality of available
population figures.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_31, © Springer-Verlag Berlin Heidelberg 2013279

As is well known, the celebrated Malthus has established as a principle that thehuman population tends to increase in geometric progression so as to double
after a certain period, for example every twenty-five years. This proposition isincontestable, if one neglects the ever increasing difficulty of procuring provisions
after the population has acquired a certain degree of agglomeration; or [if heneglects] the resources that the population draws on in its growth, even while thesociety is still in its infancy-such as a greater division of labor, the existence ofregular government and the means of defense which assure the public tranquility,etc.
In fact, all other things equal, if one thousand souls have become two thousand
after twenty-five years, these two thousand will become four thousand after
the same lapse of time.
In our old European societies, where suitable lands have long been cultivated,
the work applied to improve a field already being farmed can add to its products
only ever decreasing quantities. Admitting that in the first twenty-five years one
could have doubled the product of the soil, in the second period one could barely
succeed in perhaps making it produce a further third. The effective growth of the
population thus finds a limit in the extent and fertility of the country, and the
population, in consequence, tends more and more to become stationary.
It is not the same in certain cases, purely exceptional to be sure; for example,
when a civilized people cultivates a fertile territory until then uninhabited, orwhen it exercises an ingenuity that gives large temporary benefits. A large family
then becomes a source of wealth and the second generation finds it easier toestablish itself than the first, because it need not like the former struggle against
the obstacles that the untamed land offered to the first colonists.
To judge the speed with which the population grows in a given country, it isnecessary to divide the increase of the population in each year by the population
which furnished it. The relationship, being independent of the absolute size of the
population, can be regarded as the measure of this speed. If it is constant the
population increases in a geometric progression; if it is increasing the progression
is greater than geometric, and less than geometric if it is decreasing.
One can make diverse assumptions about the resistance or the sum of the
obstacles opposed to the indefinite expansion of the population. Mr. Quetelet
supposes it proportional to the square of the speed with which the population tends
to increase (Adolphe Quetelet: Essai de Physique Sociale, Vol. 1, p. 277. [Paris1835]).
This is to liken the movement of the population to that of a body falling througha resistant medium. The results of this comparison accord, in a satisfying manner,with statistical data and with those that I obtained by my own formulas, when
one supposes in the layers of the medium it passes through a density increasing
indefinitely.
The growth of the population necessarily has a limit, if only in the extent of
the soil indispensible for the lodging of this population. When a nation has
consumed all the fruits of its fields, it can to be sure procure supplies from abroad
through the exchange of its other products, and support in this way a new increaseof population. But it is evident that these importations must have limits and stopa very long time before the entire of the country is converted to cities. All the280formulas by which one attempts to represent the law of population must thus
satisfy the condition of admitting a maximum which is attained only in an epoch
infinitely distant. This maximum will be the number of the population when it
becomes stationary.
I tried for a long time to determine by analysis the probable law of population;
but I abandoned this type of inquiry because data from observations are too
scarce for the formulas to be verified in such a way as not to leave doubt about
their exactness. Still, as the steps that I followed appear to me to lead necessarily
to knowledge of the true law when the data become sufficient, and as the results I
have arrived at may be of interest, at least as an object of speculation, I thought
I ought to accede to the invitation of Mr. Quetelet and bring them to the public.
Let p be the population. and let us represent by dp the infinitely small increasethat it receives during an infinitely short time dt. If the population were growing
in a geometric progression, we would have the equation ~~ = mp. But as therate of growth of the population is retarded by the augmentation of the number of
inhabitants in itself, we must deduct from mp an unknown function of p; so thatthe formula to integrate becomesdp
dt = mp-¢(p) ·The simplest hypothesis that one can make as to the form of the function (pis tosuppose ¢(p) = n p 2 . One then finds for the integral of the equation above1
t =- [lnp-ln(m-np)] +constant,
l1l

and three observations will suffice for determining the two constant coefficients m
and n and the arbitrary constant.
On solving the last equation for p, it becomes(1)

p = n p' em' + m - n p' 'designating by p' the population that corresponds to t = 0, and by e the base ofNaperian logarithms. If one sets t= oo, one sees that the corresponding value ofmp is P = -. That is thus the upper limit of the population.
nInstead of supposing ¢(p)=np 2 , one may take cjJ(p)=npa, a being [anynumber] whatever, or ¢(p)=nlnp. All these assumptions satisfy the observed
facts equally well, but they give very different values for the upper limit of the
population.
I supposed successively

281

and the differences between the calculated populations and the numbers
observation furnished were appreciably the same.
When the population increases by a progression greater than geometric,
the term - c/J(p) becomes + c/J(p); the differential equation then integrates
as in the preceding cases, but one conceives that there can no longer be a maximum
population.
I calculated the ta,bles which follow by formula (1). The figures for France,
Belgium, and the coun.ty of Essex were drawn from official documents. Those
which relate to Russia are found in the work by Dr. Sadler, Law of Population[London 1830], and I cannot guarantee their authenticity, being ignorant by
what manner they have been developed. I could have extended the tables forFrance and Belgium as far as 1837, by means of the Annuaires published in these

Table 1. Progress of the Population of France from 1817 to 1831, According to the Annuairefor 1834.
Year

According to theregistrar

According to theformula1817

29,981,336
195,902

29,981,336
208,2810.00007.4768490

1818

30,177,238
161,948

30,189,500
204,500+0.0004

7.4798565

1819

30,339,186
199,863

30,394,000
200,500+0.0018

7.4827875

1820

30,539,049
188,227+0.0018

7.4856461

1821

30,727,276
212,144

30,594,500
197,300
30,791,800
192,700+0.00217.4884310

1822

30,939,420
198,634

30,984,500
189,500+0.00147.4911453

1823

31,138,054
221,286

31,174,000
185,223+0.00127.4937907

1824

31,359,340
220,546

31,359,340
182,7770.00007.4963719

1825

31,579,886
175,974

31,542,000
178,000-0.00127.4988859

1826

31,755,860
157,533

31,720,000
175,000-0.00117.5013366

1827

31,913,393
189,071

31,895,000
172,000-0.0005

7.5037257

1828

32,102,464
139,402-0.00117.5060547

1829

32,241,866
161,074

32,067,000
168,000
32,235,000
164,500-0.00027.5083251

1830

32,402,940
157,9940.00007.5105385

1831

32,560,934*

0.00007.5126965

32,399,500
161,434
32,560,934* Census figure (January 1)

282

Proportionalerror

Logarithmof the calculated
population

two countries since 1833, and in this way verify my formula; but my pursuits
have not left me the leisure. My work was terminated in 1833, and I have nottouched it again since.
I will point out in passing that the table which relates to France appears toannounce that the formula is especially exact, since the observations [for France]
bear on the largest numbers and have been made with greater care. For the rest,
the future alone can unveil for us the true mode of action of the retarding force
which we have represented by cf>(p).

283

Editors' Note: Fitting the Logistic
The form in which Verhulst presents the logistic makes its solution somewhat
cumbersome. In a more standard form the equation becomesr1 =b(1- :')k
Pr = 1 +exp[a-bt]"

Given the three population estimates P10 ,1'r 0 +.~,Pro+H (which we may index
PpP2,P3), the constants a,b and asymptotic population k are found by theformulas (from Verhulst 1845, pp. 12-13; Croxton, Cowden and Klein 1967,
pp. 274-275)
k = 2P1 P2 P3 -P~(~1 +P3 )PlP3-P2

b = (1/<5)ln[P 2 (k-P 1)/P 1 (k-P 2 )]a= ln[(k-P 1 )/P 1 ]+bt~
where t~ is a computational origin. The terms t~,b vanish under the linear transformation t--+t*, defined by t*=(t-t~)/<5.
To reconcile this form with Verhulst's equation (1) we write, using Verhulst's
notation,
mp'emtp = ----:--c:=------::t
np' em1 +m-np'

m/n

k = m/nb = m

Pearl and Reed, whose paper follows, also use a non-standard form for thelogistic. Their equation and the conversion to standard form are
B

Yr = ----=-----=--

exp[ -A(t-t~)]+C
B/C

1 +exp [(At~ -lnC)-A t]
k = B/C

b=A

a= At~-lnC.
284

The inflection point, given by Pearl and Reed as (cx,ya) has as its coordinates(a/b,k/2) in the standard form. An ex not the same is given in their equation xvi.
Further treatment of the logistic, with its implied birth and death rates, will
be found in Latka (1939, pp. 48-62). Latka centers the curve about its inflection
point for convenience in developing its theoretical properties, by setting

t6 =albk
1+eP,. =P,_,~ = -~.-;b"'*r•.

285

32. On the Rate of Growth of the Population of
the United States since 1790 and its Mathematical
Representation
RAYMOND PEARL and LOWELL J. REED (1920)
From Proceedings of the National Academy of Science 6. Excerpts are from pages 280-288.

Pearl and Reed begin their paper with a discussion of exponential and parabolic
growth formulas for past U.S. population growth, which we have omitted.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_32, © Springer-Verlag Berlin Heidelberg 2013287

It is quite clear on a priori grounds, as was first pointed out by Malthus
in non-mathematical terms, that in any restricted area, such as the UnitedStates, a time must eventually come when population will press so closely upon
subsistence that its rate of increase per unit of time must be reduced to the vanishing point. In other words, a population curve may start, with a convex face to thebase, but presently it must develop a point of inflection, and from that point on
present a concave fa,ce to the x axis, and finally become asymptotic, the asymptote
representing the maximum number of people which can be supported on thegiven fixed area. . ..
It would be the height of presumption to attempt to predict accurately the
population a thousand years hence. But any real law of population growthought to give some general and approximate indication of the number of people
who would be living at that time within the present area of the United States,
provided no cataclysmic alteration of circumstances has in the meantime intervened.
It has seemed worth while to attempt to develop such a law, first by formulating a hypothesis which rigorously meets the logical requirements, and then by
seeing whether in fact the hypothesis fits the known facts. The general biologicalhypothesis which we shall here test embodies as an essential feature the ideathat the rate of population increase in a limited area at any instant of time is
proportional (a) to the magnitude of the population existing at that instant(amount of increase already attained) and (b) to the still unutilized potentialities of
population support existing in the limited area.
The following conditions should be fulfilled by any equation which is todescribe adequately the growth of population in an area of fixed limits.
1. Asymptotic to a line y=k when x= + oo.
2. Asymptotic to a line y = 0 when x = - oo.
3. A point of inflection at some point x = il( and y = {3.
4. Concave upwards to left of X=ll( and concave downward to right of x=ll(.
5. No horizontal slope except at x= ± oo.
6. Values of y varying continuously from 0 to k as x varies from - oo to + oo.
In these expressions y denotes population, and x denotes time.
An equation which fulfils these requirements is(ix)
when a, b and c have positive values.
In this equation the following relations hold:

x=+oo

b
y=c

(x)

x=-oo

y=O

(xi)

Relations (x) and (xi) define the asymptotes.
288

The point of inflection is given by 1-ce"x=O, or1

x= --loge
a

b
2c

(xii)

y=-

The slope at the point of inflection is ~~ ....
b
Y= e-bx + c

Fig. 2. General form of curve given by equation (ix).

The general form of the curve is shown in figure 2.
The question now is how well does (ix) represent the known historical facts
as to the growth in population of the United States, and to what legitimate
deductions as to the future course of population in this country does it lead?
It is obvious that equation (ix) as it stands cannot be fitted to observational
data by the method of least squares. It is possible to write momental equations
and fit by the method of moments, but at this time we do not care to developthat method because, as will presently appear, we do not regard equation (ix)as the final development of this type of equation for representing population,
and we have no desire to encumber the literature with a mathematical discussion
which we expect later to discard.
For present purposes it will be sufficient to fit (ix) to the observations bypassing it through three points. Given three equally spaced ordinates, y 1 ,y 2 andy 3, the necessary equations are:bc

2YtY2Y3-y~(Yt+Y3)(xiv)

YtY3-Y~289

(xv)where his the abscissa! distance in years between y 1 and y 2 , or Y2 and y 3.

1 (Ytc = Y2- Yt

eaa -

Y2 )

(xvi)

ea<a+hl

where oc is the abscissa! distance in years from the origin to y 1 •Putting x 1 at 1790, x 2 at 1850, and x 3 at 1910, and taking origin at 1780
we have

a=10
h=60
J3=91,972
and taking (ix) in the formb(xvii)

we find these values for the constants:

2,930.3009

y = ---;~;-;-;-r....-:::---::-::-:-:-=-::-:c
e 0.0313395x+0.014854(xviii)

The closeness of fit of this curve is shown graphically in figure 3.
Though empirically arrived at this is a fairly good fit of theory toobservations. . ..
The significance of the result lies in this consideration. A curve which ona priori grounds meets the conditions which must be satisfied by a true law of
population growth, actually describes with a substantial degree of accuracy
what is now known of the population history of this country.
Let us examine some further consequences which flow from equation (xviii).
The first question which interests one is this: when did or will the population
curve of this country pass the point of inflection, and exhibit a progressively
diminishing instead of increasing rate of growth? From (xii) it is easily determinedthat this point occurred about April I, 1914, on the assumption that the numericalvalues of (xviii) reliably represent the law of population growth in this country.
In other words, so far as we may rely upon present numerical values, the United
9

Omitting 000 here and in the subsequent calculations till the end.

290

Population of united states

100000000/__

2g3Q3009

y = e -0313395x +014854c

60000000

0

0
:::J

a.
0

Q._

40000000

Yv

200000000r--,....---'

v

v

80000000

H1

I
i/

v

1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900 1910
Year
Fig. 3. Showing result of fitting equation (xviii) to population data.

States has already passed its period of most rapid population growth, unlessthere comes into play some factor not now known and which has never operated
during the past history of the country to make the rate of growth more rapid.
This latter contingency is improbable. While prophecy is a dangerous pastime,
we believe, from the fragmentary results already announced, that the 1920 census
will confirm the result indicated by our curve, that the period of most rapid
population growth was passed somewhere in the last decade. The population
at the point of inflection works out to have been 98,637,000, which was in fact
about the population of the country in 1914.
The upper asymptote given by (xviii) has the value 197,274,000 roughly.
This means that according to equation (xviii) the maximum population which
continental United States, as now areally limited, will ever have will be roughly
twice the present population. We fear that some will condemn at once the whole
theory because the magnitude of this number is not sufficiently imposing. It is soeasy, and most writers on population have been so prone, to extrapolate population
by geometric series, or by a parabola or some such purely empirical curve, andarrive at stupendous figures, that calm consideration of real probabilities is mostdifficult to obtain. While, as will appear from the next section of this paper,
we have no desire to defend the numerical results of this section, and indeed
ourselves regard them only as a rough first approximation, it remains a factthat if anyone will soberly think of every city, every village, every town in thiscountry having its present population multiplied by 2, and will further think291

of twice as many persons on the land in agricultural pursuits, he will be bound, we
think, to conclude that the country would be fairly densely populated. It would
have about 66 persons per square mile of land area.
It will at once be pointed out that many European countries have a muchgreater density of population than 66 persons to the square mile, as for example
Belgium with 673, Netherlands with 499, etc. But it must not be forgotten thatthes~ countries are far from self-supporting in respect of physical means ofsubsistence. They are economically self-supporting, which is a very different
thing, because by their industrial development at home and in their colonies
they produce money enough to buy physical means of subsistence from lessdensely populated portions of the world. We can, of course, do the same thing,provided that by the time our population gets so dense as to make it necessary
there still remain portions of the globe where food, clothing material, and fuel are
produced in excess of the needs of their home population. But in this, and in any
other scientific discussion of population, it is necessary to limit sharply the area
one is to talk about. This paper deals with population, and by direct implication
the production of physical means of subsistence, within the present area ofcontinental United States.
Now 197,000,000 people will require, on the basis of our present food habits, 10
about 260,000,000,000,000 calories per annum. The United States, during theseven years 1911-1918, produced as an annual average, in the form of humanfood, both primary and secondary (i.e., broadly vegetable and animal), only
137,163,606,000,000 calories per year. 11 So that unless our food habits radicallychange, and a man is able to do with less than 3000 to 3500 calories per day,or unless our agricultural production radically increases, 12 it will be necessary
when our modest figure for the asymptotic population is reached, to import nearlyor quite one-half of the calories necessary for that population. It seems improbablethat the population will go on increasing at any very rapid rate after such a
condition is reached. And is it at all reasonable to suppose that at such time,with all the competition for means of subsistence which the already densely
populated countries of Europe will be putting up, there can be found any portionof the globe producing food in excess of its own needs to an extent to make itpossible for us to find the calories we shall need to import?
Altogether, we believe it will be the part of wisdom for anyone disposed to
criticise our asymptotic value of a hundred and ninety-seven and a quarter
millions because it is thought too small, to look further. into all the relevant facts.
III

With the above numerical results in hand it is desirable to discuss a little
further the general theory of population growth set forth in the preceding section.
At the outstart let it be said that we are convinced that equation (ix) represents° Cf. Pearl, R., Tile Nation's Food, Philadelphia (W. B. Saunders Company), 1920 (247).

1

Pearl, R., /oc. cit., p. 76.
As a matter of fact East, in his able presidential address on "Population."' before the American
Society of Naturalists has shown that the United States has already entered upon the era of diminishingreturns in agriculture in this country.
11

12

292

no more than a first approximation to a true law of population growth. There
are several characteristics of this curve which are too rigid and inelastic to meetthe requirements of such a law. In (ix) the point of inflection must of necessity
lie exactly half-way between the two asymptotes. Furthermore the half of thecurve lying to the right of the point of inflection is an exact reversal of the half lying
to the left of that point. This implies that the forces which during the latter partof the population history of an area act to inhibit the rate of population growth areequal in magnitude, and exactly similarly distributed in time, to the forces whichin the first half of the history operate to accelerate growth. We do not believethat such rigid and inelastic postulates as these are, in fact, realized in population
growth ....
We attach no particular significance to the numerical results of the preceding
section. They obviously can give only the roughest approximation to probablefuture values of the population of the United States. Our only purpose in presenting them at all at this time is to demonstrate that the hypothesis here advanced
as to the law of population growth, even when fitted by a rough and inadequate
method, so closely describes the known facts regarding the past history of thatgrowth, as to make it potentially profitable to continue the mathematicaldevelopment and refinement of this hypothesis further. There is much thatappeals to the reason in the hypothesis that growth of population is fundamentallya phenomenon like autocatalysis. In a new and thinly populated country the
population already existing there, being impressed with the apparently boundless
opportunities, tends to reproduce freely, to urge friends to come from older
countries, and by the example of their well-being, actual or potential, to induce
strangers to immigrate. As· the population becomes more dense and passes
into a phase where the still unutilized potentialties of subsistence, measured interms of population, are measurably smaller than those which have already beenutilized, all of these forces tending to the increase of population will become
reduced.

293

Part V

Probability Models of Conception and
BirthPregnancy and birth distributions are mathematically unlike mortality in being
repeated events, separated by intervals of non-risk and admitting as a matter ofcourse a high degree of individual control. Corrado Gini (1924) was first to explore
these distinctions and their implications, by his suggestion that birth intervals betreated as waiting time problems dependent on fecundability. This he defined as theprobability that a woman at risk would conceive in a given monthly cycle.
Gini’s article, paper 33, limits itself to first pregnancies and births under constantfecundability. The obvious possibilities for extending the work were not considered
for about 30 years, when Louis Henry (1953) published the first of a series ofinvestigations in the mathematical treatment of birth intervals. Henry’s initial article,
presented here in paper 34 in a translation by Mindel C. Sheps and Evelyne LapierreAdamcyk (1972) indicates how Gini’s analysis is applied to birth orders beyond thefirst.
An equation of Henry’s that figures in much later work is the proportion of
women conceiving each month in a homogeneous population, given that at any point
in time some women are already pregnant or in amenorrhea and hence not at risk.
Henry shows that if the monthly probability of pregnancy to non pregnant womenis p and the non susceptible period associated with pregnancy is s months, bothconstants, then the mean birth interval i approaches the stable value
i D

1
Cs
p

The inverse 1= i is the proportion conceiving each month. The result was extended
by Tietze (1962) to show the effects of contraception and by Potter (1963) tocompare rates of abortion that would arise if it were used exclusively or used inconjunction with contraception, assuming in both cases that all pregnancies that
occur are aborted. Contraception of effectiveness e changes the probability ofpregnancy from p to p.1  e/, while abortion affects the sterile period s. The results
show in general that if abortion is used alone as a means of controlling fertility it
is required frequently. If s 0 < s is the sterile period associated with abortion, theVProbability Models of Conception and Birthintervals being compared are1
C s0
p.1  e/
and
1
C s0:
p
With highly effective contraception, the first interval can be of great length, while
the second is less than the normal interval between births.
In the paper that follows Henry’s, D. Basu (1955) f omitted here g derives themonthly proportion conceiving independently, by means of difference equations.
The solution is also attractively reached through matrix formulation, for which thereader may consult Keyfitz (1968, pp. 390—392). Henry’s work is expanded in later
papers to cover general fertility histories; several are available in English in Henry(1972).
Important contributions have been added by Robert G. Potter and M. P. Parker(1964), and Mindel C. Sheps (1964), whose related articles appeared together.
Sheps, whose paper is included here, as paper 35 finds generating functions forthe distribution of conception delays under Henry’s assumption that fecundability
differs among couples but is constant over time. Potter and Parker specify birthprobabilities as a Pearson Type I distribution and find waiting times for this specificcase.
The analysis of waiting times and conception probabilities has been extended
very considerably beyond the level treated here, including allowance for pregnancy
wasteage, for declining fecundability with age, and for contraceptive use. Much of
Sheps’ later work will be found in Sheps and Menken (1973). Sheps, Menken andRadick (1969) provide an introduction to other contributions.
f Ed. Note: The models of Sheps and Menken have formal connections withthe renewal processes discussed in Part II. Methods for estimating the relative
contributions of a full set of “proximate determinants” of fertility came into use
with Bongaarts and Potter’s work [7]. The lines of inquiry initiated by Louis Henry
led to advances in historical demography exploiting data from parish registers,
partly through aggregate analysis and partly through the individual-level approachof family reconstitution, epitomized by Wrigley and Schofield’s volumes [43] and[44], building on Peter Laslett’s classic work [20]. Appreciation of relationships
between period and cohort indices of fertility from the perspective of Norman Ryder[33] have led to tempo-adjusted indices of total fertility defined by Bongaarts andFeeney [6].
Over the years the field of probability models for conception and birth has
broadened to include modeling of other individual-level demographic life-history
processes, many in the domain of kinship and family demography. Demographic
computer microsimulation allows treatment of interacting individual-level and296

V Probability Models of Conception and Birthpopulation-level processes. Early landmarks are found in work by Menken [28] andby Hammel, Wachter, and McDaniel [17]. Such microsimulation approaches havespread throughout the social sciences under the name of “agent-based modelling”.
An illustration of their power may be found, for instance, in the application byGoldstein [14]. g

297

33. First Investigations on the Fecundability of a
Woman
CORRADO GINI (1924)
Proceedings q{the International Mathematical Congress (Toronto) 2: 889-892. Translated by David

Smith.

For the Gini Index, Gini's "approximate measure of the mean fecundability ofprimipares who had their child in the months of marriage from x + 9 to x + y + 9,"
we may write
F _ nx+9-nx+y+lo _
G-

-

x+y+9

I

x+ 9

n;

nl-ny+l
y

In;

'

I

where n; are births in the i'th, and n; births in the (i+9)'th month of exposure.
If fecundity differs among women but for each woman is constant over time, theIndex has as its limiting value the arithmetic mean fecundability of the nonsterile population N*. This is seen by noting that as y approaches infinity, ny+ 1 -+0yandIn;-+ N*.
I

The Pear/Index (Pear11933), defined as the number of pregnancies experienced
divided by person-months of exposure, is often more useful. It iswhere N is the population size. If analysis is again restricted to the non-sterile
population this has as its limiting value the harmonic mean fecundability, whose
inverse is the mean waiting time to pregnancy: (~in) IN*. The expression isformally a life expectancy, correct to the measurement error of the exposure
time i.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_33, © Springer-Verlag Berlin Heidelberg 2013299

I call fecundability of a woman the probability that a married woman becomes
pregnant during the month, neglecting any Malthusian or neo-Malthusianpractice intended to limit procreation.
It is unnecessary to stress the theoretical and practical importance of a measurefor fecundability. We would have, in particular, the means to decide what fractionof the differences we observe between the coefficients of natality of different
countries, different· soe.ial classes, or different times arises from physiologicalcauses and what fraction arises from voluntary causes.
It is also unnecessary to demonstrate that fecundability cannot be measured
directly. Quite clearly, we are not in a position to observe the consequences ofMalthusian and neo-Malthusian practices on natality. Nor do we know thefrequency of ovulatory abortions that occur in the initial months of pregnancy.
We can, however, formulate an indirect measure of fecundability, based on theconsiderations that follow.
Let us suppose that married women able to bear children are n in number
during the whole period being considered, and that all have the same fecundability
p. On these assumptions the number of women who would conceive for the firsttime during their first month of marriage will be pn; the number who would
conceive for the first time in the second month will be (1- p) p11; the number
who would conceive for the first time in the third month will be (1- p) 2 pn, and soforth. The numbers obtained in this way form a geometric progression whose
ratio is 1- p.
If we assume that the percentage of abortions (or of abortions and stillbirths)
and likewise the percentage of pregnancies longer or shorter than 9 months
are the same for the products of conceptions that have taken place in the successivemonths of marriage, we can substitute for the ratio between first conceptions
taking place in the month x + 1 of marriage and first conceptions taking place inthe preceding month x, the ratio between the firstborn (or first live births) coming
into the world in the month x + 10 of marriage, and the firstborn (or first live
births) coming into the world in the month x + 9.
It is evident that the value of p arrived at by this route is independent of thenumber, large or small, of women who are not in the childbearing population,either because of the sterility of the marriage or because of Malthusian or neoMalthusian practices.
It is also evident that it is independent of the frequency of abortions andstill births.
The value of p thus determined measures, on the assumptions mentioned
above, the fecundability of women able to conceive.
Among these assumptions there is one which clearly does not agree withreality however: it is that all fertile married women have the same fecundability p.
This circumstance, which cannot be neglected, does not prevent our arriving
at the measure of fecundability; though the route to follow is a bit longer.
Let s ~ 11 be the number of groups into which the 11 married women can beclassed according to their fecundability, p;l; = 1. 2 , .. . ,s will be the fecundabilityof women in group i and n;x the number of women of group i who, in the month x
after marriage, have not yet conceived.
300

The number of women expected to conceive for the first time in the x'th monthof marriage will be

2: p;n;x; and the number expected to conceive for the firsti::::: ltime in the next month x + 1 of marriage will be

L (1- p;)p;n;x· The expectedvalue of the ratio between the second and first numbers (if we agree to give toeach of the possible values of the ratio a weight proportional to the probability
that this value is realized and to the number which forms its denominator) will bes~i=~~~-------- =s

L P;11;x

i=l

But

itt

pf n;x1

L pfnix

i=ls

L P;l1;xi= 1

I;~ P;n;x is precisely the expected value of the mean fecund-ability of women who conceive for the first time in the x' th month of marriage.
Thus we will be able to deduce the mean fecundability of women who conceive
for the first time in the month x from the complement of the ratio between women
who conceive for the first time in the month x + 1 and women who conceive
for the first time in the month x.
This calculation rests on the following assumptions, which can be acceptedwithout difficulty:(a) The number of women who were eligible for first conceptions in month xof marriage, and did not conceive in this month, will equal the number of women
eligible for first conceptions in month x + 1 of marriage. This is a way of sayingthat we can neglect the effects from one month to the next of mortality, migrations,and passages from the category of married women not able to conceive to thecategory of married women eligible for first conceptions (for example by cessationof contraceptive practices or by elimination of the cause of sterility), as well as
passages from the second category to the first (for example, due to unexpectedsterility or the adoption of contraception);(b) in each of the s groups of women eligible for first conceptions after marriage,
fecundability remains unchanged from month x to month x + 1.
Supposing further (assumption c) that the probability of abortion (or abortionand stillbirth) and the probability of a gestation period greater or less than 9months are the same for the products conceived in month x and for those conceived
in month x + 1, we will be able to substitute for the ratio between women who
conceive for the first time in month x + 1 and women who conceive for the firsttime in month x, the ratio between the firstborn (or first live births) of the marriage
who came into the world in month x + 10 and the firstborn (or first live births)of the marriage who came into the world in month x + 9.
We are thus able to obtain a measure of the mean fecundability of primipares
who, after marriage, conceived for the first time in month x. This fecundability
can in turn be regarded as essentially equal to the fecundability of primipares who
had their child in month x + 9.

301

The complement of the ratio between the firstborn (or first live births) of themarriage coming into the world in month x + 10 and the firstborn (or first live
births) of the marriage coming into the world in month x + 9 will give us, byassumptions (a), (b), and (c), an approximate measure of the mean fecundabilityof ptimipares who had their child in month x + 9 of marriage.
If we calculate the mean of the ratios between the firstborn of the months ofmarriage x+10 and x,+9, x+11 and x+10, ... ,x+y+10 and x+y+9, giving
each ratio a weight proportional to its denominator; and if we take the complement of this mea:n, we obtain by assumptions (a), (b), and (c) an approximate
measure of the mean fecundability ofprimipares who had their child in the monthsof marriage from x+9 to x+ y+9.
Here are the results obtained for several areas. They are based on data publishedby official sources, except for Italy, for which the data are the fruit of special
inquiries, using the registers of vital events of 24 communes.

Area

Years

Mean fecundability of primipares who had
their child during the months following
marriage:
10-17
11-17
10-23
11-23

BerlinAustralian Confederation
Western Australia

1894-1895
1917-1921
1895-1915
1893-1905}
and
1916-1921

26.1
19.4
23.9

24.4
20.6
22.7

23.0
18.5
21.9

20.5
19.2
20.2

20.6

22.2

18.9

19.7

21.0

24.4

23.8

21.0

21.4

18.6

New South Wales
Victoria
Tasmania
Italy (24 Communes)

1898~1900}

1905-1906
1900-1921

The uniformity of the results obtained is remarkable, above all when we
exclude from the calculations the firstborn coming into the world during thetenth month of marriage, corresponding to conceptions of the first month ofmarriage. For the ratio between conceptions of the first and those of the second
month of marriage, we cannot in fact accept the validity of assumption (a),fecundability being, in the first month, less than in the second due to the frequentvirginity of the wife; nor that of assumption (c), ovulatory abortions being
especially frequent for the products conceived in the first month because ofhoneymoon trips and perhaps other circumstances.
The obstacle to fertilization represented by virginity naturally increases inimportance with the age of the wife, but beyond the first month it does not appear
that the ability to conceive is weaker for older than for younger wives. That is aremarkable finding, and was brought to light by the data of the Australian Confederation for the period 1907-1914 and those of New South Wales for theperiod 1893-1898.
Another important finding is that the diminution of natality from 1901--02
to 1911-12 which has been verified in the Kingdom of Saxony and has reached302

50%, does not appear to have been accompanied by a diminution of the abilityof primipares to conceive.
The methods and results summarized in this note are set forth together
with complementary details in two papers presented at the Instituto Venetodi Scienze, Lettere ed Arti on July 1, 1924, and in the article "Decline in the birthrate and 'fecundability' of women," in The Eugenics Review, January 1926.
Further investigations are in process which can perhaps permit us to assemble
other interesting results in the new area that the method proposed appears
to open to statistical research.

303

34. Theoretical Basis of Measures of Nat ural
Fertility
Lours HENRY (1972 (1953))
From On the Measurement of Human Fertility: Selected Writings of Louis Henry, translated and editedby Mindel C. Sheps and Evelyne Lapierre-Adamcyk. New York: Elsevier. Excerpts are from pages 2,
4-9. 15-20.

In his remarks on Gini, Henry notes that the mean fecundability of women who
conceive in the first month of marriage (in Gini's notation, paper 40 above:
I pf nil/ I Pi ni 1) may also be written p(1 + c2 ), with p the arithmetic mean
fecundability of the non-sterile population and c the coefficient of variation.
asThis permits calculation of the variance of fecundabilityv;,where p=nrfN*; n 1 and n 2 being the number of women conceiving in the firstand second months of exposure respectively, and N* the total non-sterile population(cf. Sheps and Menken 1973, pp. 129-130, 398-399).
We omit sections of the paper discussing sterility, natural fertility andfecundability, and the fitting of observed data to the models, and some remarkson the works of Gini and Pearl.

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_34, © Springer-Verlag Berlin Heidelberg 2013305

Introduction
One of the most interesting and difficult problems in demography is that ofestimating natural fertility, i.e. the fertility of a human population that makes
no deliberate effort to limit births. Natural fertility, which depends essentially on
biological factors, is primarily a biological phenomenon, and a "natural" phenomenon has a particular attraction. It shares in the prestige of the natural sciences,
overshadowing that of the social sciences which are still in their infancy. In
addition, efforts to appraise the effectiveness of contraceptive practice lead tocomparisons between actual fertility and the hypothetical fertility that a given
population would have if it did not use any form of birth control.
The difficulties encountered in the study of natural fertility result from itsvery definition. Natural fertility is, for easily observable populations, hypothetical.
No doubt one can find populations living under conditions of quasi-natural
fertility; but, for the most part, little is known about them. For European populations, before the spread of contraceptive practice, only scanty data are available.
Thus, it is not surprising that indirect approaches are made to circumvent thedifficulty. But in following indirect routes we risk losing sight of the goal: An
investigation of the theoretical basis of measures of natural fertility is indispensable.
We will start with certain assumptions regarding the biological basis offertility. These assumptions are not completely arbitrary; observed results
are used to construct a mathematical model that is as close to reality as possible
and yet simple enough to be easily manageable.

Fundamental Assumptions
It has long since been ascertained that in large families of a fixed size, the interval
between successive births is more or less independent of birth order, except
perhaps for the last few births. Now, a priori, the interval between births depends
on the duration of pregnancy (independent of order), on the duration of thenonsusceptible period that follows confinement, and on the natural fecundityof the couple. If the mean interval between successive births does not vary, it isnatural to think that these various factors also do not vary for a given couple aslong as they are not sterile.
We were led, therefore, to characterize each fecund couple (couple able toproduce living children, at the time or later):
1. by the duration, g, of the nonsusceptible period associated with pregnancy;
g is the time that elapses between a conception and the first ovulation after
delivery.
2. by fecundability or the probability of conception per unit time outsideof the previously mentioned nonsusceptible periods.
Since conception is possible only at the time of ovulation, one ought, rigorously,
to treat fecundability as discrete and define it as the probability of conceiving per306

menstrual cycle. In this case we designate it by p. But since it is often more convenient to treat time as continuous, we also introduce the probability cpdx ofconceiving during the infinitely small time interval dx. The term fecundability
is also used for¢.
The quantity g is the sum of two terms: the duration of pregnancy, g 0 , and theduration, g- g 0 , of the period of nonsusceptibility following delivery. The latter
varies appreciably between women, because, while it is very short in the absence ofbreastfeeding, it can be very long in the case of prolonged lactation. Moreover,fecundability certainly varies between women, whether because ofthe physiologicalcharacteristics of the couple or of the frequency of their sexual relations. To
take these variations into account, we introduce two distributions: h(g), theprobability density function (p.d.f.) of the duration of nonsusceptibility, andj(¢), the p.d.f. of fecundability, or f(p) in the discrete case.
Before going on, let us examine the foregoing. To characterize each couple by apair (cp,g), is to assume that these quantities are invariant in time as long as thecouple is not definitely sterile. It is certain that, in reality, ¢ and g vary overtime. For g, this is obvious. Illness of the mother or death of the child may interrupt
lactation; furthermore, if one considers conceptions, the duration of pregnancy
also varies. 1
It is more difficult to affirm, a priori, that ¢ varies. Doubtless our sense ofcontinuity makes us think that women do not become completely sterile suddenly,
without a preceding progressive decline in fecundability. But this feeling forcontinuity, though undeniable, does not constitute an argument. One could objec;tto the assumption of a constant ¢ on the grounds of the existence of nonsusceptibl<lperiods outside of those associated with pregnancy, e. g. those due to separation
or illness. But to the extent that these separations and illnesses are distributed
uniformly in time, they intervene by modifying the probability of conception;
they do not affect its assumed invariance.

On the other hand, we must admit that the distribution of these separations
and illnesses is certainly not uniform. Long established marriages accept temporary separations more easily than do young married couples; also, the frequencyof illness increases with age and therefore with duration of marriage. Thus,
separations and illnesses tend to lower ¢ with increasing duration of marriage.
The relatively small variation in observed mean intervals between successive
births shows, however, that variations in ¢ and y depend little on the duration of
marriage, age or parity. With respect to g, this suggests that if this function varies
for a given couple, the variation is independent of t<he duration of marriage.
On the average, g will be the same at any point in a marriage. We shall see that
under these conditions, the results are the same as if we assume a constant g
for each couple.
1 In the case of spontaneous abortion. pregnancy is shorter. For convenience. we often operate as if
there were a fixed time between conception and birth and pass from one to the other at our choice.
We are not unaware, however, that the existence of spontaneous abortions and stillbirths complicatesmatters. But these are secondary difficulties which it appears unnecessary to expound here.

307

Reproductive History of a Group of Couples
We begin by considering the reproductive history of a group of couples with thesame characteristics (¢,g) in the absence of permanent sterility and of dissolutionof the marriage by separation, divorce or death.
It is natural to take marriage as the time origin, under the assumption of nopremarital sexual relations and, hence, no premarital conceptions. But, on thisassumption, the period from marriage to the first live birth (if, as we are doing,
one considers live births only) differs from subsequent intervals because g 0 , theduration of pregnancy, rather than g, intervenes in this first interval. One could,
obviously, retain this difference, but only at the cost of complicating the notation.
It is simpler to change the origin, placing it at g- g 0 before marriage.
Let us designate the elapsedtime from this translated origin by x. For x<g,
all couples are childess; for x > g, the expected number of couples without births,
taking the initial sample size as unity, is equal to e-<t><x-ol and the number of firstbirths in the interval (x,x+dx) is ¢e-<t><x-oldx.
For x < 2g, there are no second order births; it can easily be shown that,for x 2 2g, the expected number in the interval (x,x+dx) is equal to¢ 2 (x- 2g)e-<t><x-Zg) dx.

It is then easy to show by recursion that, for x 211 g, the expected numberof births of order n in the interval (x,x +d x) is equal to 2

(x-ngt- 1¢" ---=--- e- </>\x -ng) dx.

( 1)(n-1)!

The expected number of births B(x)dx in the interval (x,x+dx), regardlessof order, is equal to the sum of (1) for values of n from 1 tom such that mg~x.
B(x) is given by:
X!

B(x) = S B(x-t)¢e-<t><t-oldt(2)which is of the same form as equations studied by Lotka. Its solution is of theform I Qsersx. The Qs are coefficients and the rs are roots of the equationoc= S ¢e-rt-</>(t-gldt.

(3)

g

If one had considered couples with fecundability rf> and g0 , ... g._ 1 for the duration of nonsusceptability corresponding to each successive ,birth order, one would have obtained for the number ofbirths of order n in the interval (x,x+dx):

2

c/Jn ( X -go -gl- · · · -gn-I )"- 1 e-,P(x-g0 -g 1 - ... -g.,-ddx(n -1)!
. 'for x::?:g 0 + g 1 + ... +g._ 1 , given that g. is not dependent upon the duration of marriage. For a set of
couples in which g 0 +g 1 + ... +g._ 1 =ng, we again arrive at the earlier result. There is no need,
therefore, to introduce variations of g independent of marital duration.

308

The real root is zero and the real component of the imaginary roots is negative.
Consequently, B(x) is the sum of a constant and of damped periodic functions.
With increasing duration of marriage, this sum approaches the constant part of
B(x), which we designate by ¢' (asymptotic fertility rate). This is equal to 1/i,
where i is the mean interval between births. Now,
i =

.

00

~

g

g

J¢xe-<f>(x-gldx = g + Jcf>(x-g)e-<f>(x-gldx(4)1

whence 1 = g + -;J>. Hence¢'=-1-=_¢__
g+ 11¢(5)

1 +g¢

Often in demography, rates are only intermediate results in the attempt toderive the mean number of events per capita over a long period: here, the mean
number of births per marriage. Hence, we are led to study cumulative fertility as afunction of time, that is, the expected number of children born in x years ofmarriage. Let us designate it by E(x). We have:E(x) =

X

X

0

s 0

JB(()d(=cf>'x +I JQ,e"·'d(.

(6)

E(x) is the sum of a linear function of x and of damped periodic functions. After anumber of oscillations, that is, after a given duration of marriage, E(x) approaches¢' x + E 0 , where E 0 depends on ¢ and g. Given ¢ and g, E(x) can be calculated
from appropriate tables, e.g. X2 tables [since Eq. (1) is equivalent to a x2 (chisquare) distribution with 2n degrees of freedom]. From this, one deduces E 0 •But it is not necessary to consider all pairs (¢,g). Let us assume two values, g andi.q. The maximum value of n in timex given y is the same as the maximum value intime i.x given i.q. If ¢ is the fecundability associated with y, let us consider thepair (cf>/i.Jy). We then have:E(xlcf>,y)=I S¢"(~
1ng-ny)"-l e-<P(~-ng)d~(n-1)!

(7)
(8)

Let

it follows that E(i.xi¢/Jc,i.y)=E(xlcf>,g). The two terms in x are(=i.~:

equal to _¢_ x for the pair (¢,g), and to

1+y¢
,

¢

),(1 +g¢)

,

leX

309

for the pair(c/J/A,Ag). They are therefore equal and we have finally,
E 0 (c/J,g)=E 0 (c/J/),,).g). Hence, when gcp is a constant, E 0 is fixed. Values of E 0
for selected gcp were calculated, and a regression fitted as:

0.115
E 0 = - 0.5 + ----;;:--- .

(9)

Y'f'

Table 1 shows calculated and estimated values.
Table 1. E 0 According to gcj>

gcj>

0.5
-0.275E 0 calculated
E 0 estimatedfrom Eq. (9)-0.3752-0.4454
3-0.470 -0.480

5-0.485

6-0.490-0.500-0.270 [ -0.385]-0.443-0.462-0.477-0.481-0.5001-0.471C()

We now pass to the more realistic case of a heterogeneous group where thecouples have all possible characteristics (c/J,g). Assume that there is no correlation
between ¢ and g: the group is then characterized by two probability densities,j(¢) and h(g). This group, by virtue of the heterogeneity of¢ and g, has diverse
values for¢'; the distribution of¢' has a probability density k(c/J').
In such a group, the expected number of births in the interval (x,x+dx) is
equal to the sum of the births of the homogeneous subgroups (c/J,g) which constitute the entire group. If x is large, the expected number in the subgroup (c/J,g)
departs little from ¢' dx; the expected number of births for the total, therefore, isequal to dxf¢'k(c/J')dc/J'=$'dx, and the cumulative fertility of the group isapproximately equal toX

Jc/J' k(c/J')dc/J' +Sf E 0 f(c/J)h(g)dc/Jdg(10)or x$' + E0 , where E0 is the mean of the values of E 0 corresponding to thevarious combinations (c/J,g).
If we substitute Eq. (9), the approximation for E0 referring to a sufficiently
long time, into Eq. (10), the result is:where (ijh and gh are the harmonic means of¢ and of g. . ..

The Discrete Case: Gini's Method
To study the fecundability of newlyweds, it is preferable to abandon the continuous
notation used until now. Let p be the probability of conception in the course of a310

menstrual cycle, where the cycle is assumed to be of a fixed duration equal to one
month. Let us consider, then, a homogeneous group with fecundability p in which
the couples were married on the same date and had no premarital conceptions.
If the original number is put equal to unity, there are p conceptions expected
the first month, p(1- p) the second, p(1- p) 2 the third, and so on.
In heterogeneous groups characterized by a density f(p), the expected numberof conceptions in the first-month is p and in the second month it is p- p2 (1 + c2 ),where c is the coefficient of variation of p [ c is equal to the standard deviation
divided by the mean]. The complement of the ratio of the second month's con2
2
ceptions to those of the first is expected to be 1 - [p-p ~1 +c )] = p(1 +c 2 ).
This is the same as the mean fecundability of women conceiving the first month,which may be written as:
Jp -pf(p)- - dp=p(1 +c 2 ).

(22)

p

More generally, the complement of the ratio of the expected conceptions ofthe(n + 1)th month to those of the nth month is equal to the mean fecundabilityof women conceiving in the nth month. The mean fecundability of women conceiving in the course of the first n months of marriage is equal to a weighted
mean of the preceding quantities, the weights being the number of women conceiving the first, second, ... , nth month, i.e. the denominators of the preceding
quantities. It is written:(23)

As 11 increases indefinitely, this quantity approaches p, which is thus the mean
fecundability of women who conceive at least once.
The essentials of this theory are due to C. Gini ....

Intervals Between Births
We have already seen that in the case of a homogeneous group (</>,g), the meaninterval between births is g + 1/4>; by a change of origin we have equated marriage
to a birth; if one returns to the real origin at marriage, the mean interval between
marriage and the first birth is reduced by g -g 0 and becomes g 0 + 1/4>.
In discrete notation, the mean number of ovulations (or of months) between
marriage and the first conception is equal to p + 2 p( 1 - p) + ... + np( 1 - p)n- 1 ,that is, on setting 1 - p = q,d[ L: 1~ d[q]

pdq

00

i=l

q

=p-dq 1-q

1
=-

311

p.

(24)

If k is the mean interval, counted in menstrual cycles (or months), between marriage
and the first ovulation, the mean interval between marriage and first conception isequal to (k-1) + _!__ If we assume k equal to 1/2, the mean interval is 1/p-1/2.
p

Division by 13 (or by 12) gives the mean interval in years; on adding g0 we havethe interval between marriage and the first live birth.
In a heterogeneous group the mean interval, I, between births is given, in thecontinuous case, by the relation
I=

JJ(g+1/<P)f(¢)h(g)d¢dg=g+1/(/jh.

(25)

One can also write, since i = 1I¢',l

1k(¢')d¢'
¢'
- ¢~

=J

(26)

Since ¢~<¢',it follows that 1(1<(/j', 1>1/(/j'. Analogous relations hold in thediscrete case. In particular, the mean interval between marriage and the first
conception is equal to k-1+(1/ph). Evidently, ph<p; but, the inverse of theis also greater than Ph. InPh
1 -(1-k)phpractice the difference between Ph and p is generally expected to be large; theinverse of the mean interval must therefore be less than p.
The foregoing holds in the absence of sterility; its presence may modify themean intervals between births. Let n(x)dx be the births of a given order thatwould occur in (x. x + dx) after the preceding births in the absence of sterility,
and let F'(x) be the proportion of couples still fecund x years after the precedingbirth, when all were, by definition, fecund. The mean interval between births
becomesmean interval to conception, equal to_, Jxn(x)F'(x)dx1(27)= Jn(x)F'(x)dx ·This is a new weighted mean of x with larger weights for low values of x andsmaller weights for high values; T' is therefore smaller than T; its inverse is consequently larger than that ofT. One then wishes to determine whether 1(1' canequal or surpass (/j'. For a homogeneous group of medium or high fecundability.
the reduction in the mean interval by sterility at younger ages, and even up to
about 40 years, is almost negligible. The reduction at older ages, on the contrary,
may be very great with low and very low values of fecundability.
From the little we know of the distribution of fecundability, its mean is sufficiently high so that sterility will not affect the interval between births until
older ages. However, the dispersion of fecundability is apparently great. Hence,
those with low fecundability, although probably small in number, may have asufficient effect to reduce the mean interval appreciably.
From available information on the mean interval between marriage and thefirst birth by the woman's age, there does not appear to be a reduction with312

increasing age. No doubt these data relate to populations that already practicecontraception and should be verified. We think, however, we may assume that,except at older ages, the reduction in the mean interval to the first birth caused by
the onset of sterility is not very important. It follows that l'-:::; I, and therefore, theinverse of the mean interval between marriage and the first birth, 1[1'1 will be lessthan ¢,given that age at marriage is still low. If!' is the mean interval between
two births calculated for relatively low ages (the first and the second birth forexample), 1[1' should, for the same reasons, be less than ¢'.
Instead of the mean interval between two births of a given order, we could.
calculate the mean interval between all births. Let us examine what would happen
in this case. Consider women who marry young; let x be the duration of marriage
and dS the risk of sterility between x and x+dx. When age at marriage is low,dS
· neg 1·1g1"bl e when xIS. Iow; when dx
dS IS. not neg1·1g1"bl e, x IS
. Iarge enough sodx ISthat cumulative fertility is approximately equal to tf>' x for a homogeneous groupof fertility tf>'.
Let us pass to the calculation of intervals. For women who will becomesterile at x, let ( be the duration of marriage at a given birth. In the interval((,( +d(), there are 4>' d( births (except when (is low, which is unimportant forwhat follows). If (is between 0 and x-g, then tf>'e-<f>(x-{-g)d( of these children
are last births. If (is between x-g and x, all 4>' d( births are last births. The mean
duration of marriage at the last childbirth is, therefore,(28)

that is,
,[ (

g)

x-g

4> g X - 2 + T

1- 4>2 +

e-<f>(x-g)J

4>2

,

(29)

or, ignoring the term in e-<f>(x-u),(30)

The duration of marriage at first birth is equal toX

S (tj>e-<f>({-go)dC,

9owhich is(31)

which, for sufficiently large x, reduces to g 0 + 1/4>. Then, the time between firstand last births reduces to x-g-g 0 -1/tf>+tf>'(g 2 /2-1f4> 2 ). The number of313

births of order 2 and higher is, on the other hand, equal to(32)

being, for all practical purposes, ¢' x -1. For a heterogeneous group, the time
between first and last births is equal to x- k, wherek = ss[g+go + ~-¢'(; ;z)].r(¢)h(g)d¢dg.
2

-

(33)

The total number of births is ¢' x -1. For all values of x, one then has
J(¢'x-1)dS=¢'JxdS-JdS births and, for the durations: JxdS-kJdS.
In practice, only large values of x enter the picture. Therefore, the ratio of births
to durations is in practice reduced to¢'. We emphasize, however, that this result is
valid only for the assumptions made; in particular that the population is noncontracepting and that the distribution of fecundability is such that the frequencyof low values (those for which e-cJ>x and xe-4>x are not negligible) is negligible.

314

35. On the Time Required for Conception
MINDEL C. SHEPS (1964)

From Population Studies 18. Excerpts are from pages 85--92.

We omit the appendix, and sections of the paper discussing correlations between
two successive conception delays and numerical results. (The maximum likelihood
estimates for the mean and variance of fecundability, developed in the appendix,
are the p and VP given in our introduction to Henry, paper 41 above.)

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6_35, © Springer-Verlag Berlin Heidelberg 2013315

In trod ucti on
The purpose of this paper is to investigate some characteristics of rates of conception in a mixed or heterogeneous population. Consider that one has data
showing, for a sample of couples, the number of women conceiving in eachconsecutive month of observation. These data on the monthly incidence ofconception may be used to characterize the fecundability (monthly chance ofconception) of the couples or to estimate the effectiveness of a contraceptive
method in use by them. The data may also be regarded as showing the distributionof the interval to conception. This interval constitutes one component of theinterval between successive births and hence is a subject of interest in the study offertility. 1
If we accept the reasonable hypothesis that the couples in the sample vary
among themselves in their fecundability, what can we expect our data to show?
How will the monthly incidence of conception vary? What is the expected distribution of the intervals to conception? Given a set of data as described above, can wearrive at any conclusions about the underlying distribution of fecundability?
The results to be given below, in the attempt to answer these questions, are generalin the sense that they do not depend on any assumptions regarding the specificform of the underlying distribution of fecundability.
The discussion is related to part of the subject matter considered by Potter
and Parker in the preceding paper. 2 As is well illustrated in that paper among
others, 3 a reasonable model for the distribution of the intervals to conception
assists both in interpreting observed data and in considering reasonable expectations for a group of women. A number of investigators have considered thisproblem and have developed theoretical formulations for it, as many of theseformulations have either: (1) involved the admittedly unrealistic assumptionthat all the couples in a population have an equal probability of conception permonth, (2) utilized numerical examples to illustrate what might result if the
population consisted of some mixtures of couples having two or three different
levels of fecundability, or (3) assumed that fecundability has a continuous frequency distribution of a specified form.
The most recent example of the last procedure is given in the preceding paper
by Potter and Parker, who assume that the frequency distribution of fecundability
1 The importance of birth intervals as an approach to fertility
analysis and their components are
discussed for example by: L. Henry. 'Intervals between confinements in the absence of birth controL
Eugenics Quarterly, 5, 1958, pp. 200-11; R.G. Potter, Jr., 'Birth intervals: Structure and change',
Population Studies, 17, 2,1963, pp. 155-166; and M.C. Sheps and E. B. Perrin, in 'Changes in birthrates as a function of contraceptive effectiveness: some applications of a stochastic modeL American
Journal 4 Public Health, 53, 1963, pp. 1031-46, and in 'The distribution of birth intervals under aclass of stochastic fertility models', Population Studies. 17, 3, 1964, 321-31.
2 R.G. Potter, Jr., and M.P. Parker, 'Predicting the time required to conceive', Population Studies(this issue).
3 See, for example: C. Tietze, 'Differential fecundity and effectiveness of contraception'. The Eu~Jenics
Reciew, 50, 1959, pp. 231-4; R.G. Potter, Jr., loc cir .. and 'Length of the observation period as afactor affecting the contraceptive failure rate: Milbank Memorial Fund Quarterly, 38, 1960, 142-4
and other references cited by Potter and Parker.

316

is a unimodal Type I (Beta type) distribution. They derive a number of importantand interesting results, including expressions for the mean and variance of theconception delay and the correlation between two successive delays in the same
woman. Further, they estimate the parameters of Type I distributions from two
sets of data and illustrate their results numerically by using the estimated values.
Many of their inferences have general applications.
Gini and Henry inves·tigated some properties of the expected results for moregeneral distributions of fecundability. 4 The present paper will extend this investigation, considering some of the same problems as are discussed by Potter
and Parker but the assumption of a specified frequency distribution of fecundability is relaxed to permit the distribution to assume practically any shape. 5
It can, for example, be a bimodal distribution in which an appreciable proportionof couples use highly effective contraceptives (with resultant low or zero fecundability) while the fecundability of the remainder depends on the interaction
between inherent fecundity, patterns of sexual relations and somewhat carelesscontraceptive use. Accordingly, some of the conclusions constitute a generalizationof earlier results; specifically they support and generalize some of the conclusions
in the preceding paper. In addition, parameters that characterize a distributionthat may underline a set of observations are suggested. Estimators of theseparameters are derived for situations where only some of the women have conceived when the data were collected, as well as from complete observations.

Assumptions
The assumptions made here agree with those of Potter and Parker, with the
exception already indicated. They are:
1. Conception is a chance (random) occurrence.
2. The fecundability of each couple in a population remains constant during theperiod of observation.
3. Fecundability varies between couples in the population in an unspecified way.
Conceptions, regardless of the outcome of the pregnancy, are considered here.
In conformity with the preceding paper, let the fecundability of a particular
couple be denoted as p. Postulate an arbitrary distribution of fecundability <P(p),1where 0 s; p::::; 1 andJ<P(p) dp = 1. A group of couples with identical values of p
60will be referred to as a homogeneous population, and a group with a variety ofvalues, distributed as <P(p), will be referred to as a heterogeneous population. A
C Gini. 'Premieres recherches sur Ia fecondabilite de Ia femme·. Proc. of' the International Mathematics
Con{;ress. 1924, 889-92. and L. Henry, 'F on dements theoriques des mesures de Ia fecondite naturelle',
Rer. de flnst. Int. de Statistique. 21, 3. 1953, 135-51. Among others. the results in expressions (1), (2)4and (5) below were given by Henry.
5 With some restrictions. as defined below.
" Since pis bounded and non-negative all moments of p and of 1-p are finite and non-negative. See
H. Cramer. Mathematical Methods of' Statistics, Princeton. 1946. p. 175.

317

heterogeneous population can be considered as an aggregate of homogeneous
sub-populations.

The Proportion Conceiving Each Month
In a homogeneous population, under the assumptions given, the expected
proportion of women who conceive in the first month of exposure is p. In thesecond month it is p(1- p) which is equivalent to a proportion p of those who
failed to conceive the first month. Generally, the expected proportion conceiving
monthly decreases geometrically at a ratio of (1- p). At every month, the expected
number conceiving is p times the number still at risk, i.e. the conditional probabilityof conceiving is constant.
Under the same assumptions, in a heterogeneous population the expected
proportion conceiving during the first month of exposure is1

c1 = Sp<P(p)dp=f>

( 1)

0

which is equal to the arithmetic mean fecundability of the population. During
the second month the expected proportion conceiving is:1

C2 = Sp(1-p)<P(p)dp=E(p)-E(p 2 )=p-p 2 -a/(2)0

where E(u) is the expected value of u and (Ji is the variance of pin the population.
Hence, a smaller proportion of the total group may be expected to conceive in thesecond month than would be the case for a homogeneous population with p = p.
The expected number conceiving in the second month, considered as a proportionof those who failed to conceive during the first month (conditional probability ofconceiving in the second month), is:1

sp(1-p)<P(p)dp -0

1

1- Sv<P(p)dp(J;=p-(1-p)'(3)0a smaller quantity than p.
As shown in the appendix, similar results hold for all subsequent months,
i.e. the proportion expected to conceive out of those still at risk (conditional
probability) decreases monthly. Accordingly, we may conclude that in aheterogeneous population where the assumption of constant fecundability
for any couple holds reasonably well, the proportion of conceptions-as in a'contraceptive failure rate'-will tend to fall during each successive month of
exposure.

318

This conclusion, which was previously reached by Tietze and Potter using aless general approach, 7 indicates that it may be misleading to estimate contraceptive effectiveness by calculating the number of conceptions per 100 'person
years' of use. As just indicated, 100 'person years' may have very different expectations of conception rates, depending on whether they represent two years
for each of 50 couples, a half year for each of 200 couples, or a mixture of longand short periods of observation. Instead, a method utilizing a life table approach,such as was recently suggested by Potter, 8 would provide a more appropriate
description.
Furthermore, as is shown in the appendix and illustrated numerically, expressions (1) and (2) and their analogues lead to a method of describing the
distribution of fecundability (or of contraceptive effectiveness) from knowledgeof the proportion (of a sample) conceiving each month. Inferences about the
distribution of fecundability, such as estimates of its mean, variance and third
central moment, can be made from observations on the first three months only.
They do not require information on the conception delays of all the womenin the sample; nor is it necessary to exclude sterile couples for whom p=O.

Distribution of Waiting Times
The probability that conception occurs in a given month after the initial exposure
is simultaneously the probability that the time required for conception ('waiting
time') will be that number of months. In a homogeneous population, the timerequired for conception is, on the defined assumptions, distributed geometrically.
Thus the mean, the variance and other moments are known functions of p, andmore specifically of the powers of (1/p).
A heterogeneous population, as previously defined, may be considered togenerate a compound geometric distribution of waiting times. In such a population
the expected values of the moments of the waiting times can be obtained by
calculating the expected values of the appropriate functions of p. This procedure
will be demonstrated after a review of the moments of the simple, homogeneousgeometric distribution.
(a) Distribution of Waiting Times of a Homogeneous Population

In the literature there is some variation in the definitions of the geometric
distribution describing the process of conception in a homogeneous population,and therefore variation in the formulae derived for the mean conception delay.
The apparent disagreement originates simply in two different ways of counting themonths of exposure. Thus one may say that, given a <tonstant value of p, theprobability of conception during the xth month is p(1 - py- 1 • One may also saythat the probability that the delay to conception will be exactly x months (i.e.
that conception will occur during month (x + 1) is p(1- py. The first formulation
See the papers cited in footnotes 2 and 3.
R. G. Potter, 'Additional measures of use-effectiveness in contraception·, Milbank Memorial Fund
Quarterly 41, 1963, pp. 400-18.
7

8

319

might be said to refer to the number of trials necessary (each cycle being defined
as a single trial), whereas the second refers to the conception delay. The twoalternatives may be shown as follows:

Cycle (i.e. trial) during which
conception occurs

Delay (in cycles)

Probability

1st
2nd

0

p
p(1-p)xth

x-1

p(1-py-'If one uses the values in the first column, counting the month when conception
occurs, the probability of conceiving after zero trials is zero, and the mean numberof trials needed is _.!._. If one uses the values in the second column, not countingpthe month when conception occurs, the probability of a delay of zero months is p,
.

(1-p)

and the mean delay IS equal to - - months.
pDepending on the choice made, the expression relating to month x is, as
shown before, p(1- pt- 1 or p(1- py. Letting q = 1- p, the moment generating
functions 9 of these two distributions are:(a) for the number of trials: 10
C(s) = xi"' pes (qe")x-1 = ~

1-qe

x=1

(4A)

and (b) for the conception delay:D(s) =x= X!

p

I p(qeT = -s ·
x=O
1-qe

(4B)

From (4A) or (4B), the kth moment about zero of the waiting time, i.e. E(xk),
is obtained by differentiating the generating function k times with respect to s andthen evaluating the derivative at s =0. This gives, for the two homogeneous
distributions, the first three moments as shown in Table 1. 11
M.G. Kendall and A. Stuart, The Adranced Theory of Statistics. Vol. 1. London. 1958.
Since the probability of a conception after zero trials is zero in this case. summation from x = 1 isequivalent to summation from x =0.
1 1 The third moments are derived to exemplify the general approach and because these moments
may be needed for studies of the distribution of the number of births to a group of women in a specified
interval following marriage. See E. B. Perrin and M. C. Sheps, 'Human Reproduction. A Stochastic
Process'. Biometrics, 20, 1, 1964, 28-45.
9

10

320

(b) Distribution of the Waiting Times of a Heterogeneous Population
For the discussion that follows, only fecund women may be included (i.e.
0 < p :$; 1) and it is assumed that conception times of the whole fertile population
are known. The kth moment of x in a heterogeneous population is the expectedvalue of the components of the kth moment of x in a homogeneous population.
Table 1. Moments of the homogeneous geometric distributions.
Number of trials

Delay

(4A)

(4B)

1
q
= - -1pp

E(x) (mean)

-

q

2q 2
p2

2

3

1
-+-=---+
p
p

6

6

1

p2

6

12

7

p3

p2

p

---+--1- -pl+ -p
p3
E(x 2 ) - [ E(x)] 2 (variance)

1

3

2

- -pl+ -pp3

E[x- E(x)] 3 (third central moment)

Thus, for example, the second moment of the delay (distribution 4 B) will be

E (:2 -%+ 1)=

E (:2 )-

E(%) + 1. We accordingly need to derive theexpected values of powers of _!_ in a compound geometric distribution. Fromp

[Jf(u)du]-1
cIear that
1t
, ..
-·-u1

. mean as...
o f t he h armomc
t he usua I de fmitlonIS

0

E (}) is the reciprocal of the harmonic mean. Let the harmonic mean be
designated p' and its reciprocal as R. Then:J}
1

E (}) =

cP(p)dp =~ = R.

(5)0

Further, let the mean square of the deviations of _!_ about this reciprocal bepdenoted Q. Q is thus a measure of dispersion, a kind of 'variance', of the reciprocalsof p about the harmonic mean. In order that the results to be given be meaningful
it is necessary that the distribution cP(p) be such that Q is finiteY
12

An analogous point is made by Potter and Parker with reference to the Beta distribution.

321

Now

(6)
whence(7)
In terms of the Type I distribution (Beta distribution) discussed by Potter anda+b-1
b(a+b-1)
Parker, R = a _ 1 and Q = (a _ 1)2 (a_ 2) . Therefore, in this type of distri-1

bution, given a fixed value of p', Q = (a _ ~~ _ 2). It increases with increasing b,
and decreases as a increases.
Similarly, we may define the third moment of _!_ about its mean (a measurepof absolute skewness) as:

where a finite value of L1 is assumed. Then(8)

We may now apply the result of expressions (5)--{8) to the values in Table 1
to obtain the moments of the heterogeneous distribution. For example, as alreadyshown, the second moment (M! 2 >) of the conception delay (distribution 4B) is£(: -£(%)
2)+1. From (5) and (7) this is equal to 2(Q+R 2 )-3R+1.

The variance of the delay is then M! 2 >-M 2 which is equal to 2Q+R(R-1).
More complete results are displayed in Table 2.
In the case of a homogeneous distribution Q= 0, L1 = 0, and the results inTable 2 reduce to those in Table 1. The mean delay of a heterogeneous populationis the same as for a homogeneous population with p = p'. Since the harmonicmean of any distribution is smaller than its arithmetic mean, the mean delay in aheterogeneous population is accordingly longer than might be expected from thearithmetic mean of the distribution or the proportion conceiving in the firstmonth of exposure. In other words, R = 1/p' is always greater than 1/P, where pis the (arithmetic) average fecundability in the population.
Further, the variance of the waiting times in a heterogeneous population
exceed.s that of a homogeneous population with the same mean delay by twice the322Table 2. Moments of the heterogenous (compound) geometric distributions
Number of trials

Delay

(4A)

(4B)

Mean number of months: (M)EG)=R=1/p'

EG) -1=R-1=(1-p')jp'

Second moment: M121

2Q+2R 2 -R

2Q+2R 2 -3R+1

Third moment: M< 31

6L1+18QR-6Q+6R 3 -6R 2 +R

6L1+18QR-12Q+6R 3 -12R 2 +7R-1

Variance: Mm-M 2

2Q+R(R-1)

Third central moment:
M< 31 -3(M)M< 21 +2M 3

6L1 + 12QR -6Q+R(R-1)(2R-1)

fG-~y<P(p)dp
fG1

where Q =01

and L1 =;,y <P(p)dp.

quantity Q. This difference is always positive, since Q, the mean of the squared

1 1)2 , . a Iways greater than zero m. a miXe. d or compoun'd. .
( p - p'd ev1atwns

IS

distribution.

323

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IndexA
Abortion, 295
Agent-based modelling, 297
Arthur, W. Brian, xv, 195

BBasu, D., 296
Bayesian forecasting, 230
Bernardelli, Harro, 193
Bienyamé, I.J., viii
Birkhoff, Garrett, 194
Birth interval, 295
Blackwell, David, 76Blake, Judith, xv
Böckh, Richard, 227
Bongaarts, John, 296
Bowley, A.L., 193
Brass, William, viii, 226
Breslau, 1
Brillinger, David, 226
Burgess, Ernest, xiv

CCannan, Edwin, 193, 197, 227
Cardano, Girolamo, 1
Carter, Lawrence, 230
Chiang, Chin Long, 3
Coale, Ansley, viii, 2, 76, 195, 226, 228
Coates, R.H., xiii
Cohen, Joel E., 195
Cole, Lamont, 77
Component projections, 193
Contraception, 295
Cox, Sir David, 5D
Darmois, Georges, xiv
Davis, Kingsley, xv
Deevey, Edward, 4
Demeny, Paul, 2, 226
DeMoivre, Abraham, 225
Deparcieux, Antoine, 2
DeWit, Johan, 2Duvillard, Emmanuel, 2E
Easterlin, Richard, 77, 193
Edmonds, Thomas, 229
Ehrlich, Paul, xv
Ergodic theorems, 195
Espenshade, Thomas, 77
Euler, Leonard, 1, 75
F
Family reconstitution, 296
Farr, William, 3, 226, 229
Feedback, 77
Feeney, Griffth, 228
Feldman, Marc, xv
Feller, William, 76, 195, 229
Finch, Caleb, 5Fisher, Sir Ronald, xiv, 76
Flieger, Wilhelm, xiii
Frauenthal, James, 4G
Gauss, Carl Friedrich, 230
Generalized inverse, 194
Gini, Corrado, 295, 299
Goldstein, Joshua R., vii, 297

D.P. Smith and N. Keyfitz, Mathematical Demography: Selected Papers, Demographic Research
Monographs, DOI 10.1007/978-3-642-35858-6, © Springer-Verlag Berlin Heidelberg 2013333

IndexGompertz, Benjamin, 4, 225, 231
Graunt, John, vii, xiv, 1, 11
Greville, Thomas N.E., 4, 53

H
Halley, Edmund, 1, 21, 225
Hammel, Eugene A., xv, 297
Henry, Louis, 2, 295
Human Mortality Database, 4

I
IIASA, xv
Intrinsic growth rate, 76
Inverse projection, 194

J
Jenner, Edward, 2K
Keyfitz, Nathan, vii, viii, xiii–xvi, 4, 77,
194, 296
King, George, 3
Kuczynski, Richard, 76, 227
Kuznets, Simon, 193

L
Laslett, Peter, vii, 296
Le Bras, Hervé, ix, 194, 195
Lee, Ronald D., 77, 194, 230
Legendre, Adrien Marie, 230
Leslie, Patrick H., 194
Lewis, E.G., 194
Lexis Plane, 195
Lexis Surface, 195
Lexis, Wilhelm, 3Limit cycles, 77
Lopez, Alvaro, 194
Lotka, Alfred J., 28, 75
Lotka’s Equation, 195

M
Mahalanobis, P.C., xiv
Makeham, William, 226
Malthus, T.R., 75, 229
Max Planck Institute, vii
McKendrick, Anderson, 195
McNeil, Donald, 228
Menken, Jane, 296

Merrell, Margaret, 3Microsimulation, 297
Milne, Joshua, 3, 75, 225
Model lifetables, 226
Model marriage schedules, 228–229
Momentum, 77, 195
Mosteller, Fred, xiv

NNet reproduction rate, 225
Newton, Isaac, 3
Nicander, Henrie, 3Noonan, John T, xv
Northampton, 3
Norton, Henry T.J., 194

O
Ogburn, William, xiii
Orkin, Beatrice, xvi

P
Parker, M.P., 296
Parlett, Beresford, 194
Pearl, Raymond, 2, 229
Pearson, Karl, 230
Populatin momentum, 77, 195
Potter, Robert G., 295, 296
Pressat, Roland, 3
Preston, Samuel, xiii, xv, 195
Price, Richard, 3
Proximate determinants, 296

R
Reed, Lowell, 3
Renewal theory, 75–76, 195
Reproductive value, 76
Revelle, Roger, xv
Robbins, John, xiii
Rogers, Andrei, 194
Roots of Lotka’s equation, 75
Rostock, Germany, vii

S
Samuelson, Paul, 76
Schoen, Robert, xiii, 195
Schofield, Roger, 296
Schützenberger, Marc-Paul, xiv
Sharpe, F.R., 75
Sheps, Mindel, 295

334

Index
Shkolnikov, Vladimir, 4Simpson, Thomas, 3Simulation, 77, 297
Smith, David R., vii, xiii
Stable age structure, 75
Stochastic forecasting, 230
Strong ergodicity, 194Süssmilch, Johann Peter, vii, 75
Sutton, William, 3TTabah, Leon, 194
Tempo, 296
Thompson, W.R., 77, 193
Tietze, Christopher, 295
Trenerry, C.F., 1
Trussell, James, 228
Tuljapurkar, Shripad, 195

U
Ulpian, 1V
Van de Walle, Etienne, xv
Variable r; 195
Vaupel, James W., vii, xv, 195
Verhulst, P.-F., 229
von Bortkiewicz, L., 76

W
Wachter, Kenneth W., xv, 5, 77, 194, 195
Wald, Abraham, xiv
Weak ergodicity, 77, 194
Whelpton, P. K., 193
Wilmoth, John, 4Wirth, Louise, xiv
Wolfenden, Hugh, 226
Wrigley, Sir E. Anthony, 296

Y
Yates, Frank, xiv

335