DEMOGRAPHIC MODELS AND ACTUARIAL SCIENCE
Robert Schoen, Hoffman Professor of Family Sociology and Demography, Pennsylvania State
University, USA
Keywords
Dynamic models, life table, marriage squeeze, multistate population, population momentum,
population projection, stable population, timing effects, “two-sex” population models
Contents
1. Introduction
2. Life Table Models
2.1 Life Table Structure and Functions
2.2 Actuarial Science
2.3 Analytical Representations of Mortality
3. Stable Populations
3.1 The Stable Population Model in Continuous Form
3.2 The Stable Population Model in Discrete Form
3.3 Population Momentum
3.4 Analytical Representations of Fertility and Net Maternity
4. Multistate Population Models
5. “Two-Sex” Population Models
5.1 The “Two-Sex” Problem
5.2 Analyzing the Marriage Squeeze
6. Population Models with Changing Rates
6.1 Population Projections
6.2 Dynamic Mortality Models
6.3 Timing Effects with Changing Fertility
6.4 Dynamic Birth-Death and Multistate Models
Glossary
Cohort: In demographic usage, a closed group formed on the basis of an initial shared
characteristic and followed over time (e.g. a birth cohort, a marriage cohort).
Dynamic models: In demographic usage, population models with changing vital rates.
Life table: An analytical representation of the survivorship of a hypothetical group of people,
typically from birth to the death of the last survivor.
Markov assumption: In demographic work, the assumption that a person’s risk of transfer
depends only on the person’s present characteristics, independent of past history.
Marriage squeeze: An imbalance between the number of males and females at the primary
marriage ages.
Multistate population models: Demographic models that recognize persons in more than one
living state. For example, a multistate life table describing marital status could have four distinct
states for persons Never Married, Presently Married, Widowed, and Divorced.
Period: In demographic usage, a cross-section of time, often (but not always) one year.
Population momentum: The extent that a population will grow after it attains replacement level
fertility.
Stable population: The result when a population experiences constant age-specific rates of birth
and death for a long period of time. A stable population grows exponentially at a constant rate and
has a fixed age composition, both determined by the unchanging rates of birth and death.
Timing effects: In demographic usage, changes in the level (or quantum) of a demographic
measure produced by changes in the timing (or tempo) of another demographic measure.
“Two-sex” models: Models that examine marriage or fertility, simultaneously reflecting the
behavior of both males and females.
Summary
Population models typically describe patterns of mortality, fertility, marriage, divorce, and/or
migration, and depict how those behaviors change the size and age structure of populations over
time. Demographic behavior in fixed rate, one-sex populations is quite well understood by means
of existing demographic and actuarial models, principally the stationary population (life table),
stable population, and multistate models. Progress has been made on “two-sex” marriage and
fertility models, though no consensus has been achieved. Recent work has led to a better
understanding of timing effects and the relationship between period and cohort measures over time.
Generalizations of the stable model that allow changing vital rates are now available, but such
dynamic modeling remains in a relatively early stage.
1. Introduction
Mathematical models of population date back at least to 1662, when John Graunt used parish
records of birth and death to describe the demography of London, England and present the first
example of a life table. Fundamentally, population models are elaborations of the basic differential
equation
P’ = m P
(1)
where P represents the number in the population of interest; P’ is the time derivative of P, i.e. how
the size of the population changes over time; and m is a force of transition, or an instantaneous rate
(or probability) of change with respect to the demographic behavior of interest. The most common
demographic behaviors are birth, death, marriage, divorce, and migration, though models have
frequently been applied to many other topics, including educational enrollment, labor force status,
and disability status.
Population models have the great advantage of being logically closed, that is population size and
composition (or stock) at one time can be found from the population stock at an earlier time and the
demographic events (or flows) that occurred between those time points. The classic expression of
that closure is the so-called Vital Statistics Balancing Equation which, ignoring age and sex, can
be written
P(t+1) = P(t) + B(t) -D(t) + I(t) -O(t)
(2)
where P(t) is the size of the population of interest at the beginning of year t, B(t) is the number of
births during year t, D(t) is the number of deaths during year t, I(t) is the number of inmigrants
during year t, and O(t) is the number of outmigrants during year t. Fertility, mortality, and
migration constitute the core demographic processes.
Most demographic models employ two significant simplifications. The first is that the models are
deterministic, i.e. that they ignore chance fluctuations and focus on expected values of population
behavior. For example, the same rates of death always have the same impact on survivorship, with
no chance (or stochastic) variability. The second simplification is that within the categories
specified, population heterogeneity is ignored. Thus in a model that only recognizes age and sex,
all persons of the same age and sex are at equal risk of experiencing an event (i.e. of “transferring”).
Population models are rooted in Markov processes, and embody the Markovian assumption that
a person’s risk of transfer depends only on current characteristics, with past experiences having no
effect. While those assumptions are generally counterfactual, their practical importance varies
greatly.
2. Life Table Models
2.1 Life Table Structure and Functions
The basic life table model follows a birth cohort to the death of its last member. More formally,
there is only one living state, and the only recognized transfers are exits (decrements) from that
state. At any exact age x, basic differential equation (1) can be written in the form
m(x) = -P’(x)/P(x)
(3)
which defines m(x), the force of mortality (or decrement) at exact age x, to be minus the derivative
of the natural logarithm of the change in the number of persons at exact age x. Since the size of the
life table cohort decreases monotonically, it is customary to use a minus sign so that the force
[often written as µ(x)] is positive. Let the initial size of the life table cohort, referred to as its radix,
be designated by l(0). Then equation (3) can be integrated to show that l(x), the number of
survivors to exact age x, is given by
x
l(x) = l(0) exp[ - m(y)dy]
(4)
0
Equation (4) is the basic solution for a life table, as it transforms rates of decrement m into
probabilities of survival l(x)/ l(0). As a practical matter, equation (4) can be implemented by
choosing a simple functional form for m(x). For single year age intervals, assuming that m(x) is
constant within intervals is generally acceptable.
An alternative approach, which has been termed the “General Algorithm” for life table
construction, sets forth three sets of equations. The first set, the flow equations, describe the
permissible flows into and out of states. In the basic life table, there is one flow equation which can
be written
l(x+n) = l(x) - d(x,n)
(5)
where d(x,n) is the number of deaths (or decrements) between exact ages x and x+n. Equation (5)
simply states that the number of survivors to exact age x+n is the number of survivors to age x less
the number who exit the table between exact ages x and x+n.
The second set of equations are the orientation equations, which relate the observed rates of
transfer to the model rates. It is generally convenient and desirable to equate those sets of rates,
hence the basic orientation equation is
M(x,n) = m(x,n)
(6)
where M(x,n) and m(x,n) refer, respectively, to the observed and model rates of transfer prevailing
between exact ages x and x+n.
The third set of equations are the person-year equations, which relate the number of survivors to
each exact age to the number of person-years lived in an interval (where one person year is one year
lived by one person). The exact relationship is
n
L(x,n) =  l(x+u) du
(7)
0
where L(x,n) is the number of person-years lived between exact ages x and x+n. The General
Algorithm shifts the task of making a life table from integrating the force of decrement function in
equation (3) and then integrating the survivorship (l) function in equation (7), to simply integrating
the survivorship function. The General Algorithm also readily generalizes to specify more
complex models (e.g. multistate models).
Constructing a life table from a set of data involves, implicitly or explicitly, making two choices.
On is an assumption about how the observed rates relate to the life table rates; typically they are
assumed to be the same. The second concerns the functional form of either the force of decrement
or the survivorship curve. Many choices are possible. The simplest choice for the survivorship
curve is a linear relationship, that is
L(x,n) = ½ n [ l(x) + l(x+n) ]
(8)
which is generally adequate for single years of age and for 5 year age intervals when the force of
decrement is rising within the interval. Many other choices are possible and often desirable,
especially for intervals longer than one year or when the force of decrement is large. The details
of life table construction are discussed in the works cited in the bibliography. While there is no
“ideal” solution, a number of adequate methods are available.
[Table 1. Life Table for California Females, 1970]
Table 1 provides an example of a life table, based on the experience of California Females during
the year 1970, and shows the principal life table functions. The table is “abridged”, as age is shown
for roughly every fifth year up to age 85. Age 1 is also shown as mortality in the first year of life
is typically much higher than in the immediately following years. In the past, age 85 was the
highest age commonly shown in life tables, though with improved survivorship and better data for
the high ages, many tables now show survival to age 90. The next column is the survivorship
column, which begins with the frequently used radix l(0)=100 000. Of that number, 32 338
females survive to exact age 85. Next is the d(x,n) column, calculable from equation (5), which
shows the age distribution of decrements. By definition, l(85)=d(85,).
The probability of dying between exact ages x and x+n, denoted q(x,n), is shown in the fourth
column, and is found from the relationship
q(x,n) = d(x,n) / l(x)
(9)
The fifth column shows the life table decrement rates, m(x,n). In terms of life table functions, they
satisfy the relationship
m(x,n) = d(x,n) / L(x,n)
(10)
The sixth column is often not presented, but can be quite useful. It shows Chiang’s a(x,n), the
average number of years lived between exact ages x and x+n by those decrementing (or dying)
during the interval. Mathematically, it must satisfy the equation
L(x,n) = n l(x+n) + a(x,n) d(x,n)
(11)
as the total number of person-years lived in an interval is the sum of the years lived by those who
survive the interval and those who do not.
The seventh column shows the number of person-years lived in every age interval. The last entry,
L(85,)= 213 964, gives the total number of person-years lived above age 85. The eighth column,
T(x), shows the number of person-years lived at and above each exact age x, and is the sum of the
L(x,n) column from age x through the highest age group in the table. The last column, e(x), shows
the expectation of life at age x, or the average number of years a person exact age x is expected to
live. In terms of life table symbols
e(x) = T(x) / l(x) = Li / l(x)
(12)
where Li indicates the sum of the person-years lived at and above exact age x. For the highest age
interval, e(85)=a(85,). The most commonly used life table summary measure is e(0), the
expectation of life at birth, here 75.66 years.
To this point, the life table has been viewed as following the experience of a birth cohort, and
showing the implications of a set of decrement rates on survivorship. A second perspective needs
to be recognized, a period perspective that gives rise to a stationary population. Assume that age
ω is the highest age attained in the life table (i.e. no one survives to attain exact age ω+1). Now
assume that the rates of decrement remain constant over time, and that for at least ω+1 consecutive
years there are annual birth cohorts of l(0) persons. The result is a stationary population, that is a
population of constant size (specifically of T(0) persons) and of unchanging age composition. The
number of persons between the ages of x and x+n, at any time, is given by L(x,n). The experience
of the stationary population in one year captures that of the life table cohort over its entire lifespan.
Each year in the stationary population l(x) persons attain exact age x, there are d(x,n) deaths
between the ages of x and x+n with a total of l(0) deaths, and there are L(x,n) person years lived
between the ages of x and x+n giving a total of T(0) person-years lived.
Life tables have been widely used to reflect mortality patterns, and are now routinely produced by
most national statistical organizations. The basic model can easily be generalized to recognize
more than one cause of decrement. Cause-of-death life tables, nuptiality-mortality life tables, and
numerous other types of multiple decrement models have been constructed, and describe the
implications of competing risks. Cause eliminated life tables (also known as associated single
decrement life tables) have also been calculated. They can address the hypothetical question of
how survivorship would change if one (or more) observed causes disappeared while the forces of
decrement from the remaining causes remained unchanged.
Various means have been used to deal with problems related to population heterogeneity. Separate
life tables are routinely prepared for males and females, and frequently recognize other
characteristics such as geographical area or race/ethnicity. That approach, however, can quickly
exhaust the data available for even a large population. Proportional hazard (or Cox) models have
been used to incorporate more covariates, though usually with the assumption that risks differ by
a constant factor over age. More sophisticated efforts have sought to explicitly model individual
“frailty” (or susceptibility to death). They have been hampered by the complexity of the problem
and the very limited empirical evidence available on the distribution of frailty, both between
persons and over the life course.
2.2 Actuarial Science
Perhaps the leading use of the life table has been in the insurance and pension industries. Actuarial
science uses the life table, and other models reflecting life contingencies, to determine insurance
and pension risks, premiums, and benefits. In essence, actuarial methods combine the life table
with functions related to an assumed fixed rate of interest.
Let ι denote the annual rate of interest such that $1 at time t will increase to 1+ι dollars in exactly
one year. It is convenient to denote the present value of a dollar by v, where v represents the
amount one must have at time t (under prevailing interest rate ι) in order to have $1 one year in the
future. In symbols,
v = 1/ (1+ι)
It immediately follows that vn is the present value of $1 n years in the future.
(13)
In its simplest form, a life annuity is a periodic series of payments, with the first payment of $1 due
in one year, and the payments continuing annually as long as the recipient is alive. Consider the
present value of a life annuity payable to a person now exact age x. The present value of the
payment due in one year is v l(x+1)/l(x). The present value of the payment due in two years is
v2l(x+2)/l(x). It follows that ax, the present value of a life annuity to a person exact age x is
ax = [ j vj l(x+j) ] /l(x)
(14)
where summation index j ranges from 1 to the highest age in the life table. [Here age is indicated
by a subscript. That is conventional in actuarial usage, and here serves to distinguish the life
annuity from Chiang’s a function, defined in equation (11).]
In its simplest form, a life insurance for a person exact age x is a payment of $1 made at the end of
the year in which that person dies. Combining an appropriate rate of interest and life table, and
following the same logic as used in deriving equation (14), Ax, the present value of a life insurance
for a person exact age x is
Ax = [ j vj+1 d(x+j) ] /l(x)
(15)
In practice, of course, equations (13) - (15) are modified to deal with shorter intervals of time, more
complex benefit options, and a variety of expense and other “loading” factors. Actuaries often
make calculations using a range of interest rates, andextend the applicability of life tables through
the use of “setbacks”. When a life table is set back n years, the risks for a person age x are
evaluated as if they applied to a person age x–n.
Insurance companies are quite conscious of “selection” factors, which motivate atypical persons to
seek coverage. An extreme but not unlikely example is a person recently diagnosed with a fatal
disease who seeks to buy life insurance. To gauge the size of those selection/population
heterogeneity effects, actuaries have used “select and ultimate” life tables. For example, in the
United States, the Society of Actuaries has produced life tables with a 15 year select period. That
is, for the first 15 years of coverage, the risk of death is seen as depending on policy duration as
well as age (and usually other characteristics such as sex). After 15 years, when the selection effect
is deemed negligible, mortality rates depend on age alone. Tables that impose a second time
varying parameter of that sort are known as semi-Markov models. They are common in actuarial
work, but infrequently encountered in demographic analyses.
The life table, as a stationary population, is the basic model underlying benefit calculations. The
classic article by C.L. Trowbridge on pension funding viewed the participants in a “mature”
pension plan as constituting a stationary population. The “service” table commonly used by
actuaries in benefit calculations is a multiple decrement life table where a steady stream of entrants
to a benefit plan are followed over time subject to risks of death, withdrawal from employment,
disability retirement, and “normal” retirement. Defined benefit pension plans, especially those
where the benefit is based on the employee’s final salary, are more complex and involve additional
economic assumptions. Defined contribution plans, which are becoming more common, offer
employees more choice and portability, while not assuring them of any specific pension amount.
When an employer “buys” an annuity from an insurance company, or an employee selects an
option to annuitize a pension account, the benefit level follows directly from the life table and
interest assumptions used.
2.3 Analytical Representations of Mortality
Actuarial analyses of mortality have often used the classic nineteenth century “Laws” of mortality
propounded by Gompertz and Makeham. The persistence of those formulations is possible
because the age pattern of mortality varies relatively little over a wide range of life expectancies.
The risk of death is moderately high in the first year of life, declines to a minimum around ages
10-12, and then rises steadily with age. As the level of mortality changes, the whole curve tends to
rise or fall.
Those regularities in the age pattern of the force of mortality can be reflected by a function with
three terms, specifically
μ(x) = Fg-x + A + Bcx
(16)
with g, c>0. The rightmost term in equation (16) reflects Gompertz’ Law, and describes mortality
as rising exponentially with age. The middle term, A, is Makeham’s constant, and its inclusion
typically leads to a substantial improvement in fit over that of a Gompertz curve. Even so,
Makeham’s Law does not reflect the declining mortality of the pre-teen ages and almost certainly
overstates mortality rates at high ages. The first term on the right side of equation (16), introduced
more recently by Siler, deals with the discrepancy at the young ages. It adds a negative exponential
term to reflect the relatively high mortality in infancy that rapidly declines after age 1.
Without practical pressures to simplify calculations by mathematical functions, several alternative
methods of representing mortality patterns have been pursued. One uses collections of “model”
life tables to identify a set of age-specific patterns of mortality and how they vary with the level of
mortality. The leading such collection is that of Coale and Demeny, whose model life tables
recognize four distinct age patterns related to those observed in North, South, East, and West
(including overseas) Europe. Separate model life tables are provided for males and females, with
female life expectancies at birth ranging from 20 years to 80 years. As life expectancy at birth in
a few countries now exceeds 80 years, model tables with higher life expectancies have been used
in population projections.
A second approach uses relational models that involve a “standard” mortality function and a rule
for applying that standard to an observed population. The most widely used relational model is the
Brass logit transformation (see Chapter 18).
3. Stable Populations
Stable population theory is closely associated with the twentieth century work of Alfred J. Lotka,
but many of the underlying ideas can be found in a 1760 paper by Leonhard Euler.
3.1 The Stable Population Model in Continuous Form
Assume that a population closed to migration experiences constant age-specific rates of birth and
death for a long period of time. That population becomes “stable”, i.e. it grows exponentially at a
constant rate (generally denoted by r) and has an age structure that does not change over time. In
other words, change at every age x can be described by the differential equation
P’(x) = r P(x)
(17)
In continuous form, the structure of the stable model is captured by its characteristic (or renewal)
equation. Let us assume that there is one birth in the stable population during our reference year.
Then, with the integral ranging over all ages,
1 =  e-rx p(x) f(x) dx
(18)
where p(x)=l(x)/l(0) represents the probability that a newborn will survive to exact age x, and f(x)
represents the fertility rate at exact age x. Equation (18) can be interpreted as showing that the unit
birth in the reference year represents the effect of constant fertility rates [f(x)] acting on the
survivors [p(x)] of previous births [e-rx], where the number of births has been increasing
exponentially at rate r. When mortality and fertility are known, equation (18) is the basis for
calculating “intrinsic” growth rate r. There are a number of ways of doing that, one being
functional iteration on the discrete form of the equation, beginning with the estimate r=0.
The classic stable population is a one-sex model. The great majority of stable populations refer to
females, because of the more limited female reproductive age span and the greater availability of
fertility data by age of mother. Thus, unless otherwise noted, fertility in a stable population refers
only to the birth of daughters. [Data on total births are often adjusted to female births by using a
factor of 100/205, which reflects the typical sex ratio at birth of 105 males to 100 females.]
The crude birth rate in a stable population, b, often termed the “intrinsic” birth rate, is given by
b = 1 /  e-rx p(x) dx
(19)
or as the unit birth divided by the total number of persons in the stable population in the reference
year. To specify the unchanging age composition of the stable population, let c(x) represent the
proportion of the population attaining exact age x in any year. We then have
c(x) = e-rx p(x) /  e-ry p(y) dy = b e-rx p(x)
(20)
which satisfies the necessary constraint that  c(x) dx = 1.
The function p(x)f(x) is known as the net maternity function, and is often denoted by φ(x). When
net maternity is summed over all ages, an important demographic measure known as the Net
Reproduction Rate (NRR or R0) results. Mathematically,
R0 =  p(x) f(x) dx =  φ(x) dx
(21)
and the NRR can be interpreted as the average number of daughters born per woman under a given
set of age-specific fertility and mortality rates. An NRR of 1 is referred to as replacement level
fertility, as under that regime each woman, on average, has one daughter.
In the stationary population of the life table (which can be seen as a stable population with r=0),
period and cohort experience is identical. That is not the case in stable populations because of
differential cohort size. For example, the average age at death for each cohort in a stable
population is the e(0) implied by its schedule of mortality. However, the average age at death in
the stable population during any year will generally be less (if r>0) or more (if r<0) than e(0)
because the period population will be younger (if r>0) or older (if r<0) than the stationary
population.
In the stable population, the rates (flows) clearly determine the size and structure of the population
(the stocks). Although the stable composition only emerges after a number of generations, some
features are quite robust, and it is often possible to use stable population relationships to estimate
demographic parameters. For example, equation (20) can be written in the form
ln[ c(x)/p(x) ] = ln b - rx
(22)
Given a census age distribution and an appropriate life table, linear regression can be used to
estimate the birth and growth rates. Moreover, the stable population provides an analytic
framework for examining a broad range of demographic issues, including such topics as
population aging, population turnover (“metabolism”) and, as we examine shortly, population
momentum.
3.2 The Stable Population Model in Discrete Form
For many purposes it is useful to view stable populations in terms of discrete age groups. Five year
age groups are the standard in demographic work, though larger (and smaller) age groups have
been useful in analytical work. The so-called Leslie matrix is frequently used to array discrete
fertility and mortality values. For example, assuming that age 45 is the highest age of reproduction,
a stable population up to age 45 can be represented by the 3x3 Leslie matrix, A, where
A=
┌
┐
│ b1 b2 b3 │
│ s1 0 0 │
│ 0 s2 0 │
└
┘
(23)
The elements in the first row of A are related to fertility. Specifically, element bi reflects the
contribution of a person in the ith age group at the beginning of a (here) 15 year time interval to the
population in the first age group at the end of that time interval. The si are survivorship proportions.
Specifically, si represents the proportion of persons in the ith age group at the beginning of an
interval who survive to be present in the i+1st age group at the end of the interval. A Leslie matrix
is a population projection matrix since, if the column vector Pt, here with elements P1t, P2t, and P3t,
represents the time t population in age groups 1-3 (here at ages 0-14, 15-29, and 30-44), then the
definition of A yields the projection relationship
Pt+15 = A Pt
(24)
Equation (24) holds for Leslie matrices of any number of rows and columns, with projection
interval n always equal to the length of the age intervals used. Leslie matrices with five year age
intervals have frequently been used in carrying out population projections.
It is useful to examine the eigenstructure of a Leslie matrix, because it relates directly to the stable
population implied by the mortality and fertility underlying that matrix. Let the eigenvalue matrix
of Leslie matrix A be diagonal matrix Λ, and let its dominant eigenvalue (or latent root, or
characteristic value) be λ1. From the Theorem of Perron and Frobenius, as long as the Leslie
matrix meets certain commonly occurring criteria (satisfied, for example, if two consecutive age
groups have nonzero fertility), then λ1 is unique, real, and greater than the real part of any complex
eigenvalue. Relative to the continuous model, λ1=exp[nr], where n is the length of the
age/projection interval and r is the stable population growth rate.
Let the right eigenvector matrix be denoted by U, with first row elements equal to 1 and its leftmost
(dominant component) column vector denoted by u. Then the relative size of the elements of u
reflect the age structure of the stable population implied by the mortality and fertility of the Leslie
matrix.
With U-1=V, any Leslie matrix A can be written in eigenstructure form as
A=UΛV
(25)
Projecting a population across two age intervals is accomplished by left multiplying the initial
population vector by AA=A2, and across m intervals by Am. Thus with n year age intervals, we
have
Pt+nm = Am Pt = U Λm V Pt
(26)
Because λ1 is the dominant root, after some number of intervals M, all of the other eigenvalues,
raised to the Mth power are either essentially zero (as is almost always the case for observed human
populations) or insignificant relative to λ1M. Then equation (26) reduces to
Pt+nM = λ1M u v’ Pt
where v’ is the first row of V=U-1, and v’u=1.
(27)
Equations (26) and (27) reflect the process of convergence to stability under a fixed regime of
mortality and fertility. In general, convergence starts at the lowest ages. The time to convergence
depends on the square of the relative size of the first two eigenvalues, and is influenced by initial
population composition. However, an empirical rule of thumb is that most populations are
approximately stable up to age x after 60+x years of exposure to a set of constant vital rates.
3.3 Population Momentum
Consider a population that, at time t, experiences a decline in fertility to replacement level. With
that zero growth regime of fertility and mortality continuing indefinitely, the population will
eventually become stationary. In the classical, or Keyfitz form, population momentum refers to the
amount of growth that the population will experience after initial time t. Denoting momentum by
Ω, we can write
Ω = P / P t
(28)
where Pt denotes the total number of persons in the population at time t, and time “” refers to the
ultimate stationary population.
An analytical expression for momentum can be developed from equation (27). If the ultimate
population is stationary, then λ1=1 (and r=0). Thus the ultimate population vector is
P = u (v’ Pt) = uQ
(29)
The size of the ultimate stationary population is thus the sum of the elements of vector u multiplied
by the scalar Q=v’ Pt. The elements of vector v’, the dominant left eigenvector of the Leslie matrix,
can be seen as a function of the “present value” of future births under the given mortality and
fertility regime, with Lotka’s (intrinsic) r as the rate of interest [see equation (13)]. Here, the ith
element of v’ is the contribution of a person in the ith age group at time t to the size of the first age
group of the ultimate population. Scalar Q thus represents the ultimate size of the stationary
population’s birth cohort (or the “stable equivalent number of births”). With BR denoting the
crude birth rate of the initial population, we can write
Ω = BR e(0) Q
(30)
Population momentum is often on the order of 1.5 to 1.7 in contemporary developing populations,
indicating the substantial growth potential of their young age structures. [On the other hand, some
“old”, low fertility, developed populations have Ω<1, indicating a population decrease if fertility
were to rise to replacement value.] Population forecasts indicate that most world population
growth after the year 2000 will be attributable to population momentum.
In comparing the initial population with the ultimate stationary population, both analytical and
numerical work has shown that the number of persons under age 30 (about one generation) remains
essentially constant. The population growth typically associated with momentum occurs at ages
over 30, especially at ages over 60. Population momentum can thus be seen as an inherent part of
the process of population aging that is associated with long term fertility decline.
3.4 Analytical Representations of Fertility and Net Maternity
The net maternity function φ(x)=p(x)f(x) of stable population characteristic equation (18) is
unimodal with a bit of a skew to the right. Nonetheless, it has been successfully approximated by
the symmetric normal curve for such purposes as the calculation of Lotka’s r. Other work has
represented φ by a gamma distribution (Wicksell) or a series of exponential curves (Hadwiger).
Recent efforts to model fertility have involved relational models. Henry considered “natural
fertility” to be the fertility of populations that did not practice deliberate birth control, and he
described a characteristic age pattern of “natural” marital fertility. If w(x) represents an observed
marital fertility rate at age x, Coale and Trussell proposed the relational equation
w(x) = M n(x) exp[ m y(x) ]
(31)
where M reflects the level of marital fertility, n(x) is the age schedule of marital fertility in a natural
fertility population, y(x) represents a standard age schedule of departure from natural fertility, and
m is the extent of departure from natural fertility. A simple approach to estimating M and m is to
rewrite equation (33) as
ln[ w(x)/n(x) ] = ln M + m y(x)
(32)
and use ordinary least squares regression. A major application of the approach has been to
determine whether a population’s marital fertility has been affected by the use of contraception.
4. Multistate Population Models
Multistate models have two or more living states that intercommunicate, i.e. where a decrement
from one state is an increment to another. Those models are discussed in detail in Chapter 16.
5. “Two-Sex” Population Models
5.1 The “Two-Sex” Problem
Demographic behavior varies markedly by age and, in most instances, by sex as well. Accordingly,
age-sex specific rates are widely used to control for the effects of the age-sex composition of the
population, and to identify the underlying behavior. However, fertility and marriage inherently
involve the joint behavior of males and females. Every birth has one mother and one father; every
marriage involves one bride and one groom. Thus the use of sex-specific rates of marriage and
fertility ignores the essential role played by members of the other sex.
The crux of the “two-sex” problem is that one-sex marriage and fertility rates depend on the
age-sex composition of the entire population. To keep the number of fathers equal to the number
of mothers, and to keep the number of brides equal to the number of grooms, the one-sex rates must
change as the number of males and females change. Ignoring that fact can lead to absurd results.
For example, an observed population can have male fertility and mortality rates implying a positive
intrinsic growth rate, and female vital rates implying a negative intrinsic growth rate. Calculating
separate male and female stable population models based on those observed rates would show the
number of males increasing to infinity while the number of females fell to zero. Demographers
became aware of the inconsistencies resulting from ignoring the dependence of one-sex rates on
population composition in the 1940s, and a great deal of work has been done on the problem since
then. It has been difficult to achieve closure because there is no obvious method for adjusting the
one-sex rates, nor any clear way to bring empirical evidence to bear.
A number of proposed solutions have sought to create “two-sex” rates by incorporating both male
and female populations at risk in the denominator. For example, a geometric mean two-sex
marriage rate between males age x and females age y could be written as
GM(x,y) = C(x,y) / [M(x) F(y) ]½
(33)
and a corresponding harmonic mean rate as
HM(x,y) = C(x,y) [M(x)+F(y)] / [M(x) F(y)]
(34)
where C(x,y) is the number of marriages between males age x and females age y, M(x) is the
number of males age x, and F(y) is the number of females age y.
McFarland has proposed a solution, related to the geometric mean, that uses a “bordered” marriage
matrix, that is a matrix that arrays marriages by age of bride and age of groom and includes the
number in each age-sex group that does not marry. The row and column sums of the bordered
marriage matrix yield the total number of males and females in each age group. If the age-sex
composition of the population changes, the new population figures become the row and column
sums. The elements of the array are then adjusted to those new figures by Iterative Proportional
Fitting (IPF, also known as the Deming-Stephan or RAS method). The procedure preserves the
cross-product ratios of the original marriage matrix while adjusting the array of marriages to
preserve consistency between the sexes.
Schoen sought to define a “two-sex force” (or “propensity”), and argued that the sum of the male
and female age-specific forces of marriage reflected the marginal change in the number of (x,y)
marriages given a marginal change in the number of (x,y) couples. That approach led to a
harmonic mean solution, which related the propensity, or the “magnitude of marriage attraction”
for (x,y) marriages to the observed number of marriages divided by the average number of (x,y)
pairs per person. As compositional changes would raise the marriage rate for one sex and lower it
for the other, the harmonic mean solution sees the sum of those two rates as being constant.
A number of other specific solutions have been advanced, including one by Henry that
hypothesized “panmictic” mating within population subgroups. Pollak took a different approach
that emphasized axioms or “mating rules” that solutions must satisfy. While clarifying the nature
of an acceptable solution, that has not resolved the problem because a number of possible solutions,
with different quantitative implications, satisfy the proposed requirements. An important
requirement is that the solution reflect the competitive context of the marriage (or fertility)
“market”. Thus if the number of males age z increases, and there are marriages between males age
z and females age y, then there should be fewer (x,y) marriages because of the increased
competition from males age z. In any instant, neither the geometric nor harmonic mean approaches
satisfy the competition requirement. However, over any interval of time, both do reflect
competition from other age-sex groups, as a change in the number of males age z leads to an
immediate increase in (z,y) marriages that reverberates through the age-sex composition of the
population and influences the future number of (x,y) marriages.
5.2 Analyzing the Marriage Squeeze
The principal manifestation of two-sex population dynamics is the “marriage squeeze”. That
evocative but imprecise term refers to an imbalance between the number of marriageable males
and females that effects the number and the characteristics of marriages. Most commonly, a
marriage squeeze is the result of a rapid rise (or fall) in the annual number of births. In most
countries, husbands are several years older than their wives. Hence a rapid rise in births means,
some 20 or so years later, that men born in relatively small cohorts would potentially be marrying
women born in relatively large cohorts. The women, facing a possible shortage of marriage
partners, would be in a “squeeze”. Similarly, a rapid fall in births would lead to a squeeze against
men. Marriage squeezes, especially in local areas, can also result from differential migration or
sex-selective mortality.
Some observers have argued that marriage squeezes have had profound social consequences. A
marriage squeeze against women in the 1960s in many Western countries was seen as playing an
important role in the emergence of the Women’s Movement. Arguments have been made that a
“surplus” of women has been associated with sexual libertarianism and a devaluation of marriage
and family life. Certain groups, such as well educated women in most countries and Black women
in the United States, have been seen as facing particularly unfavorable marriage markets.
Much applied work has used population sex ratios, e.g. the number of males per 100 females at the
primary marriage ages. While suggestive of an imbalance, such sex ratios do not capture the
essential feature of a marriage squeeze—i.e. that male and female marriage rates are directly
affected by changes in the age-sex composition of a population.
Work using the harmonic mean approach has examined differences between male and female
age-specific marriage rates in a number of countries, based on the idea that in the absence of a
marriage squeeze the sum of the male age-specific marriage rates would equal the sum of the
female age-specific marriage rates. In the West, that work has found that the marriage squeeze has
had relatively little impact on the proportion ever marrying, and noted that in the 1910s and 1920s
there were large (relative to the 1960s) marriage squeezes against women that had little observable
effect and that passed by with little comment. On the other hand, it documented marriage squeeze
effects on the average ages at marriage, on the average age difference between brides and grooms,
and on the variance of the distribution of marriages. For example, the age difference between
brides and grooms moved from two to three years. As might be expected, marriage squeezes
against women led to men having earlier marriages and less variability in their marriage ages,
while causing women to have later marriages and more variability in their marriage ages. However,
the substantial declines in marriage observed in a number of Western countries in the later decades
of the twentieth century were primarily produced by declines in the propensity to marry, not by
compositional effects related to the marriage squeeze.
While data availability has meant that most analyses examined more developed societies, there is
good reason to expect that marriage squeeze effects may actually be greater in developing countries.
In particular, because marriage squeezes against women are associated with population growth,
and a number of developing countries have a history of birth cohorts of increasing size, there could
well be significant effects on the age difference between brides and grooms, on the availability of
second wives in societies that practice polygamy and, indirectly, on fertility rates (a “birth
squeeze”). Similarly, subgroups in different populations are likely to experience substantial effects
as possibilities for finding partners with desirable or similar characteristics are restricted by
compositional factors. Two-sex dynamics are inherently a part of partner choice, and are likely to
play a significant role in many situations.
6. Dynamic Population Models
Fixed rate models of demographic behavior have proven to be of great value, but are inadequate to
capture the complex dynamics of observed populations. For the general case, dynamic models, i.e.
models with rates that change over time, are necessary. Such models follow from the differential
equation
Pt’ = μt Pt
(35)
where both the matrix of rates and the vector of population stocks change over time.
6.1 Population Projections
Estimates of the future size and composition of a population are needed for many administrative,
commercial, and policy purposes, and have been produced by demographers and others for many
years. In addition to long term, national level projections, statistical time series methods have been
used for short run forecasts, local area projections (emphasizing migration) have been made for
different localities, and structural equation models have been applied to examine social, economic,
and demographic interactions.
The mechanics of projecting a population, essentially by means of Leslie matrices with adaptations
to incorporate migration, are fairly straightforward. Predicting future demographic behavior is far
more complicated, and has been done with at best partial success. For most contemporary
populations, mortality estimates for a generation or so can be made with reasonable accuracy,
though there is some controversy about the extent of future mortality declines at the high ages. The
largest projection errors have stemmed from estimates of future fertility and, to an almost equal
extent, from estimates of future migration. In many projections, future fertility is seen as reflecting
past fertility, as demographic theory provides no firm basis for predicting changes in trends.
Future international migration has an even weaker theoretical underpinning, and is often estimated
in terms of a fixed number of (net) migrants each year, or is assumed to decline to zero. That is
especially problematic in an era of globalization.
Projections are usually made employing “High”, “Medium”, and “Low” estimates of future
fertility, mortality, and migration. The Medium projection is implicitly seen as the most likely,
with the High and Low values often taken as approximate confidence intervals. Recent work by
Lee and Carter and by Lee and Tuljapurkar have pioneered stochastic population projections,
which offer statistically-based confidence intervals. Although world population growth has been
slowing in recent years, long term global projections (e.g. the 2000 U.S. National Academy of
Sciences report Beyond Six Billion) still see the world’s population increasing from about 6 billion
in the year 2000 to about 9 billion in the year 2050.
6.2 Dynamic Mortality Models
The profound mortality declines that occurred during the twentieth century nearly doubled human
longevity. Death rates at younger ages, especially infancy, declined greatly, and in many countries
the latter decades of the century saw continuing declines in death rates at the higher ages. The near
elimination of infant and child mortality in some countries makes it likely that future mortality
changes will be qualitatively different from past changes, and there is no evidence to suggest that
mortality declines cannot continue into the future. New models emerged as research sought to
explore changing mortality.
To identify a new dynamic perspective, consider a model population with a constant (unit) size
birth cohort and mortality free to vary over age and time. At any point in time, that model
population consists of the survivors of past birth cohorts, which experienced the changing
mortality risks that characterized past ages and times. The population is composed of multiple
cohorts, and is not an ordinary period population because it does not reflect the implications of a
single set of age-specific rates. It has been termed a “wedge-period” population, as it reflects the
effects of a “wedge” of age-time-specific mortality rates. The total size of the time t wedge-period
population has been designated by CAL(t), with “CAL” taken from “cross-sectional average length
of life”. If mortality remains constant for a long period of time, the wedge-period population will
grow in size by a factor of e(0,t)/CAL(t), where e(0,t) is the life expectancy at birth under the rates
prevailing at time t.
Bongaarts and Feeney noted that in a wedge-period model with Gompertz form mortality, a shift in
the age schedule of death rates (i.e. a lateral movement of the entire schedule) produces a shift in
the age-specific composition of the wedge-period population. That relationship between changes
in rates and changes in composition provides a basis for dynamic modeling. In a shifting Gompertz
mortality model, CAL(t) is always the time t wedge-period population average age at death. When
mortality changes over time in a linear fashion, there are further simplifications in the shifting
Gompertz model. Linear relationships describe changes over time in life expectancy and
population size, and also characterize the gaps and lags between values of period and cohort
longevity. At time t, CAL(t) also equals the life expectancy of the cohort born CAL(t) years earlier.
6.3 Timing Effects with Changing Fertility
The Total Fertility Rate (TFR) is a widely used measure of fertility. It is defined as the sum of a
given schedule of age-specific fertility rates, and can be interpreted as the number of children an
average woman would have, in the absence of mortality, if she experienced those rates over the
course of her life. The period TFR for time t, PTFR(t), is the average number of children a woman
would have under the fertility rates of year t. While the TFR always applies to a hypothetical
cohort, it can also be calculated from data for an actual cohort, i.e. for women born in a given year.
Period fertility determines the size of annual birth cohorts, and there is considerable empirical
evidence showing that period changes account for most of the over-time variability in fertility.
However, cohorts reflect the behavior of real groups of women sharing a common year of birth and
thus experiencing period events at the same age. Theories of fertility are generally expressed in
cohort terms (e.g. completed family size), and cohort TFRs have been found to be more stable over
time. Hence both period and cohort perspectives are important in fertility analysis.
Bongaarts and Feeney examined a dynamic fertility model where a fixed-shape age schedule of
fertility shifted, to a higher or lower age range, by a constant amount each year. If each year’s shift
is ±k years, 0k<1, then the cohort TFR (CTFR) is also constant over time and
CTFR = PTFR / (1-k)
(36)
In that Bongaarts-Feeney shifting fertility model, it is possible, with k>0, to have PTFR below
replacement level while CTFR is above replacement level. In that case, cohorts of steadily
diminishing size more than replace themselves.
The classic formulation of a fertility timing effect is when the timing (or tempo) of cohort fertility
changes and thus alters the level (or quantum) of period fertility. For example, assume that a
number of women in a particular birth cohort postpone their childbearing by having a child in year
t+1 instead of in year t. The CTFR of that cohort would not change, but PTFR(t) would be lowered
and PTFR(t+1) would be raised.
Timing effects can make period fertility a poor indicator of cohort behavior. The matter took on
particular significance in the latter decades of the twentieth century when period fertility dropped
well below replacement levels (i.e. a TFR of a bit over 2), while there appeared to be delays in
childbearing (i.e. the average age at reproduction rose substantially). A number of observers
rightly questioned whether those PTFR figures accurately reflected the completed family size of
the reproducing cohorts.
Developed around 1970, independently by Ryder and by Butz and Ward, a measure called the
Average Cohort Fertility (ACF) provides a readily interpretable way to quantify the average
fertility of cohorts active in a given year. Essentially, ACF(t), the ACF for year t, provides a
weighted average of the fertility of the cohorts reproducing during year t, with each cohort’s
ultimate TFR weighted by the fraction of the cohort’s fertility that occurred during year t.
Calculating ACF(t) requires fertility data for approximately 30 years before and after year t. For
recent years, that necessitates some estimation, though the ACF appears to be relatively unaffected
by a range of plausible assumptions regarding future fertility. Analyses using the ACF with
fertility data for the USA showed that average cohort fertility often deviated markedly from the
PTFR. During the “baby boom” of 1946-64, the ACF was substantially lower than the PTFR, as
the boom was accentuated by timing effects. During the “birth dearth” of the 1970s, the ACF was
substantially higher. For example, in 1976, the ACF was 2.1, while the PTFR was only 1.7.
Especially in times when fertility is becoming later (or earlier), it is important to interpret a PTFR
as reflecting the implications of the fertility rates of a single year, and not see it as reflecting the
behavior of any actual cohort or group of cohorts.
Similar timing effects have been found in other demographic behaviors. Analyses using
modifications of the ACF have found substantial differences between period measures and average
cohort measures of both marriage and divorce. For example, in 2000 the probability of ever
marrying in England and Wales was 0.69 based on period first marriage rates, while the average
probability for the active cohorts was about 0.77.
6.4 Dynamic Birth-Death and Multistate Models
A variety of population models have sought to accommodate changing vital rates. The difficulty
in doing so stems from the inability to readily express stocks in terms of flows when rates change
over time. Differential equation (35), or equation (41) with a constant rate matrix, can readily be
solved to express population size and structure in terms of those fixed rates, i.e.
Pt = [exp(μt)] P0
(37)
However, no general closed form solution is possible when μ varies over time. The general
solution can only be expressed as a product integral, which must be evaluated numerically. As a
result, a number of efforts have been made to find special cases where the population structures
generated by changing rates can be determined analytically.
Early efforts by Coale to examine changing rates produced “quasi” stable models that showed the
effects of linear declines in mortality and fertility rates. Later, the “variable r” approach of Preston
and Coale provided a formal structure that allowed any observed population to be expressed by
equations that paralleled those of stable populations.
Cyclically stable populations, where a given sequence of rates re-occurs repeatedly, is a natural
extension of stability. The idea that demographic behavior is cyclical pre-dates Malthus, who
considered it obvious that populations would oscillate in size. More recently, Easterlin has argued
that birth cohort size has profound social and economic influences. With small cohorts having
better opportunities and more children, and large cohorts having the opposite experiences,
self-generating cycles of 40-60 years in length could be produced and maintained. The dynamics
of “Easterlin” cycles and related repetitive behaviors have been explored in depth by Tuljapurkar,
Lee and others.
Given the widespread occurrence of cyclical behavior, it is worth noting that there are many
instances where a stationary population model could usefully be replaced by the more flexible
cyclically stationary model, which allows cohort size to vary over time in the absence of any long
term growth. When a cyclically stationary model is applied to a population with simple, age-based
intergenerational transfers, the potential for large equity disparities based on cycle position can be
seen.
Interacting populations have been another field of sustained inquiry. Models of the spread of
epidemics have been formulated and applied. A general linear model encompassing competition,
predation, scavenging, and symbiosis has been studied. The classic Lotka-Volterra predator-prey
equations provide a soluble nonlinear model of population interaction, and lead to limit cycle (or
cyclically stable) solutions. The basic model is
P1(t)’ = a P1(t) - b P1(t) P2(t)
P2(t)’ = -c P2(t) + d P1(t) P2(t)
(38)
where subscripts 1 and 2 denote the prey and predator, respectively, with parameters a, b, c, d>0.
Such patterns have actually been observed in nature, the foremost example being that of the lynx
and the snowshoe hare. When the number of hares declines, the lynx (for whom hares are the
primary food source) must also diminish in number. With fewer lynx, the number of hares rises,
but as the hares become numerous, the lynx population recovers. The lynx then reduce the number
of hares, and the cycle repeats itself.
Schoen and Kim sought to generalize the birth-death or multi-age stable population model by
exploring changing birth and death rates in the context of “hyperstable” models. Under the
generally reasonable assumption of a fixed distribution of births by age of mother, such models can
relate a given birth sequence to consistent underlying schedules of age-specific net maternity rates.
They also explored models that found the birth sequences produced by a trajectory of net
maternity (NRR) values, assuming a constant length of generation. Changes in net reproduction
that followed an exponentiated polynomial of degree n led to a birth trajectory that changed
according to an exponentiated polynomial of degree n+1. Trigonometric changes in net
reproduction led to trigonometric changes in births. In one special case, an Easterlin cycle model
was found that was formally equivalent to a predator-prey model.
Work by Schoen, Kim, and Jonsson produced a “metastable” population, fully defined by closed
form equations, that generalizes the stable model by allowing fertility to change monotonically at
any specified rate. To specify a metastable model, assume an initial (time 0) schedule of net
maternity, φ(x,0), which, by equation (18), implies a stable growth rate r. Now let net maternity
change over age and time according to
φ(x,t) = φ(x,0) exp[hxt]
(39)
where parameter h reflects the chosen rate of change. Note that net maternity changes by h over
both age and time. Then B(t), the number of births in the metastable population at time t, is given
by
B(t) = B(0) exp[(r+s)t +ht2 /2]
(40)
where parameter s is defined by the characteristic equation
1 =  exp[-x(r+s)+hx2 /2] φ(x,0) dx
(41)
The exponential change in net maternity described by equation (39) yields a quadratically
exponential birth trajectory with a relatively simple form. When h>0, s>0, as s reflects the
additional growth that results from having a metastable, as opposed to a stable, age composition.
When h=0, s=0, and the metastable model reduces to stability.
The metastable model can facilitate analyses of transitions from one set of vital rates to another.
As a result, it can be used to estimate population momentum under a gradual decline to
replacement level fertility. To a first approximation, with momentum from an immediate decline
to replacement given by equation (30) and the length of the gradual decline being Y years, the
extended momentum, ΩY, is roughly
ΩY  Ω exp[rY/2]
(42)
where r is the predecline rate of stable growth. However, in a metastable model, the extent of
population growth is very closely approximated by
Ω*Y = Ω exp[rY/2] exp[sY]
(49)
where the second exponential term reflects the additional growth produced by the changing age
composition.
A second generalization of the stable model is provided by Intrinsically Dynamic Models (IDMs),
which assume that the subordinate eigenvalues of the sequence of population projection matrices
remain constant over time. IDMs allow any pattern of change in fertility levels over time, but
involve convergent infinite series and require a pattern of age-specific fertility change that is likely
to be unrealistic when age intervals of less than 15 years are used.
Dynamic modeling can be extended to allow multiple living states. Such models have a broader
analytical scope, but are substantially more complex because changes in state are no longer linked
to time. Hyperstable, metastable, and IDM approaches can be used, but only under stronger
assumptions. Dynamic multistate cohort analyses are possible if it can be assumed that all of the
age-specific transfer rate matrices are scalar multiples of one another. That assumption appears
tenable in some cases, e.g. parity status or marriage/divorce/remarriage models, but is often
inappropriate.
Finding a sequence of demographically reasonable transition matrices that reproduces a given
sequence of population values is a more difficult problem in the multistate case than in the
birth-death case. Essentially there are only n constraints on some n2 transitions. Two solutions are
available, and they generate very similar results from quite different assumptions. One is Iterative
Proportional Fitting, previously mentioned with regard to the Geometric Mean solution to the
“two-sex” problem, which is equivalent to entropy maximization (or finding the pattern of flows
that can be achieved in the largest number of ways). The second is the Relative State Attraction
(RSA) method, which assumes that every state becomes more (or less) “attractive” over time. The
estimated flow from state i to state j is then estimated from a base transfer rate which is multiplied
by a calculated attractiveness factor for state j and divided by the corresponding attractiveness
factor for state i. Both approaches generally yield plausible and nearly identical values, but both
depend on a set of standard values and both perform better when an increase in the transfer rate
from state i to state j is accompanied by a decrease in the transfer rate from state j to state i.
Efforts have just begun to examine dynamic models with multiple ages and states. Hyperstable
and metastable approaches can be applied, most easily using generalized Leslie matrices, but
additional restrictions are needed to obtain the population trajectories in terms of the changing
rates. Given the broad scope of such models, they have great potential for extending present
knowledge of population dynamics.
Although regularities in dynamic populations are only beginning to be explored in depth,
considerable progress has been made in understanding how population structure responds at the
margin to changing rates. Schoen and Kim found that, at any moment, every population moves
toward the stable population implied by its currently prevailing rates, with the extent of that
movement determined by the covariance between its age-specific growth and its age-specific log
momentum.
Acknowledgements
Assistance from Stefan Jonsson is gratefully acknowledged.
Bibliography
Bongaarts, J. and R.A. Bulatao (Eds.) (2000). Beyond Six Billion: Forecasting the World’s
Population. Washington DC: National Academy Press. [The volume discusses and examines
recent world population projections.]
Caswell, H. (2001). Matrix Population Models, (2d ed). Sunderland MA: Sinauer. [This book
provides an excellent overview of population models and their mathematical bases with an
emphasis on biological applications]
Coale, A.J. (1972). The Growth and Structure of Human Populations: A Mathematical
Investigation, Princeton: Princeton University Press. [This monograph is a classic, but still an
important treatment of populations with changing rates]
Coale, A.J. and P.Demeny. (1983). Regional Model Life Tables and Stable Populations (2d Ed).
New York: Academic Press. [The extremely useful reference volume containing model life tables,
their method of construction, and associated stable populations that bracket most of human
demographic experience.]
Halli, S.S. and K.V. Rao. (1992). Advanced Techniques of Population Analysis, New York:
Plenum. [A rather comprehensive look at demographic models, through multistate models, from
a fairly statistical perspective.]
Jordan, C.W. Jr. (1975). Life Contingencies, Chicago: Society of Actuaries. [The indispensable
book on actuarial mathematics]
Keyfitz, N. (1977). Introduction to the Mathematics of Population (with revisions), Reading MA:
Addison-Wesley. [ A comprehensive book on basic population mathematics]
Keyfitz, N. and H. Caswell. (2005) Applied Mathematical Demography (3d ed). New York:
Springer. [Interesting analyses on a range of demographic topics and how they apply to actual
populations.]
Keyfitz, N. and W. Flieger. (1990). World Population Growth and Aging: Demographic Trends in
the Late Twentieth Century, Chicago: Univ. of Chicago. [ A detailed exposition of discrete
population models with applications that provide data on many contemporary populations.]
Land, K.C. and A. Rogers (Eds). (1982). Multidimensional Mathematical Demography, New York:
Academic Press. [A collection of basic papers on multistate modeling]
Pollard, J.H. (1973). Mathematical Models for the Growth of Human Populations, Cambridge:
Cambridge University Press. [A concise treatment of stable populations and related topics]
Preston, S.H., P. Heuveline, and M. Guillot. (2001). Demography: Measuring and Modeling
Population Processes, Oxford: Blackwell Publishers. [An excellent, current demographic methods
text with an introduction to fixed rate demographic models and the basic patterns of demographic
behavior]
Preston, S.H., N. Keyfitz, and R. Schoen. (1972). Causes of Death: Life Tables for National
Populations. New York: Seminar Press. [A valuable resource that includes 180 populations over
the years 1861-1964, providing data on 12 causes of death]
Schoen, R. (1988). Modeling Multigroup Populations, New York: Plenum. [A comprehensive
treatment of fixed rate population models, with details on multistate life table construction and an
emphasis on two-sex models]
Schoen, R. (2006). Dynamic Population Models. New York: Springer. [An in depth examination
of population models with changing rates. Topics include population momentum, population
change at the margin, dynamic mortality models, fertility timing, and dynamic multi-age and
multistate models.]
United Nations, Dept of International Economic and Social Affairs. (1983). Manual X: Indirect
Techniques for Demographic Estimation by K. Hill, H. Zlotnik, and J. Trussell. Population Studies,
No. 81. New York: United Nations. [The authoritative compendium of techniques for estimating
demographic measures from incomplete data.]