Keyfitz N., Caswell H. Applied Mathematical Demography

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10.3. Age at Marriage 235

Let D(t) be the curve of divorces with time, and M(t)thecurveof

marriages. Again let the (unknown) continuous duration-speciﬁc force of

divorce be µ

(x) for marriages that have lasted x years, and call the prob-

ability of survival of the marriage for x years against the contingencies of

both death and divorce p(x). Then the current number of divorces D(t)

must be

D(t)=



∞

M(t − x)p(x)µ

(x) dx, (10.3.14)

and under a ﬁxed rate of increase r the current number of marriages is

M(t)=M

. (10.3.15)

Therefore, on entering M(t − x)=M

r(t−x)

in the expression (10.3.14)

for D(t), we have

D(t)=



∞

−rx

p(x)µ

(x) dx. (10.3.16)

From (10.3.15) and (10.3.16)

D(t)

M(t)



∞

−rx

p(x)µ

(x) dx. (10.3.17)

How much does this ratio depend on the rate of increase of marriages r?

Taking the logarithm and then diﬀerentiating both sides of (10.3.17) gives

thesimpleresult

log



D(t)

M(t)

(

= −A

, (10.3.18)

where A

is the mean number of years of marriage at the time of divorce,

calculated for the part of the population that ultimately does divorce. One

could conﬁne the argument to ﬁrst marriage and ﬁrst divorce, or else to

ﬁrst marriage and any divorce, but to take all marriages and all divorces

accords with the data most commonly available.

Integrating (10.3.18) gives

log



D(t)

M(t)



=log



(t)



− rA

where D

(t)/M

(t) is the ratio in the stationary population. Taking

exponentials provides

D(t)

M(t)

(t)

−rA

, (10.3.19)

which would have been obtained more directly as an approximation to

(10.3.17).

For populations that have not experienced sudden changes in the mar-

riage rate (10.3.19) enables us to infer the ratio in the stationary population,

(t)/M

(t), which is the same as the probability that a marriage will end

236 10. Understanding Population Characteristics

in divorce. The mean number of years married for those who divorce was

constant through the 1960s at about 7 (Statistical Abstract for the United

States, 1972, p. 3). Suppose a rate of increase of about 1 percent per year.

Then Preston’s result (10.3.19) can be used to calculate the probability

that a random marriage will end in divorce, assuming values for r and A

and applying them to the observed D(t)/M (t):

(t)

D(t)

M(t)

D(t)

M(t)

(0.001)(7)

=1.073

D(t)

M(t)

. (10.3.20)

This result is directly applicable only if birth, marriage, divorce, and

death rates have all been ﬁxed in the past. But even under instability in

these rates (10.3.20) is useful, for the birth rates really do not matter;

what is necessary is that marriages have been increasing in the past at a

reasonably steady and known rate.

10.4 The Foreign-Born and Internal Migrants

The 1970 census gave 9,619,000 foreign-born in the United States, a decline

from previous censuses and considerably less than the 11,595,000 reported

in 1940. If the ﬂow of immigrants (net of those leaving) had been constant

and at constant ages, improvements in mortality would have led to a steady

increase in the number alive to be counted in the census; the reason for the

decline must have been a greater decline in entrants than the increase in

fraction surviving. But the preceding decade’s immigration records had

shown 3,322,000 entrants, the peak to that time in a steady rise from a low

of 528,000 immigrants in 1931–40. The number of immigrants entering the

United States rose sharply in the postwar period; why, then, were fewer

and fewer foreign-born counted in successive censuses?

The answer requires a longer view backward than the postwar period.

The 40 years from 1931 to 1970 showed entrants numbering 7,400,000

against 22,326,000 for the preceding 40 years. This contrast is what was

reﬂected in the declining immigrant population; the years since the 1930s

had witnessed the dying oﬀ of the immigrants from a time earlier in the

century. Immigrants are typically young adults who have on the average 40

or more years to live after their arrival.

Any statistical reconciliation is rendered more diﬃcult yet by the con-

siderable volume of illegal immigrants, who are certainly not included in

the annual oﬃcial numbers of entrants; some unknown fraction are picked

up by the census.

10.4.1 A Matrix Analysis

The next stage in analysis, beyond the accounting for present numbers of

foreign-born in terms of the preceding stream of immigration, is to project

10.4. The Foreign-Born and Internal Migrants 237

the consequences for future population of the present rates of immigration

or of some hypothetical change in them. This is easily done using matrix

population models (Chapter 7) that classify individuals by location instead

of (or in addition to) age. Such models go beyond description and recon-

ciliation of reports into at least a primitive kind of mechanism: the eﬀect

of ﬁxed (or changing) rates on the pattern of residents from abroad, where

“abroad” may mean another country, state, or county.

Rogers (1968; see also 1995) provides an exposition of the theory and

examples of the migration process for the United States. As an example he

considers 1955–60 migration and natural increase between California and

the rest of the United States with the matrix

A =



1.0215 0.0127

0.0627 1.0667



He starts the process with the initial 1955 vector in thousands:

n(1955) =



12,988

152,082



and ﬁnds a 1960 distribution between California and the rest of the United

States equal to

n(1960) =



1.0215 0.0127

0.0627 1.0667



12,988

152,082





15,199

163,040



Continuation of the process, or use of eigenvalues, gives the ultimate ratio

of California to United States population if these rates continue.

10.4.2 Migration and Age

Among other simpliﬁcations the models of the preceding paragraphs suﬀer

from the omission of age. Rogers goes on to show (1968, p. 10) that one

canmakeeachofthea

itself a matrix, with nonzero elements in the ﬁrst

row and subdiagonal, and so he combines the migration and age eﬀects in

a single large matrix. With 50 states and 18 ages for females the matrix

would be 900 × 900.

Arrangement of the elements within A and n(t) is arbitrary, as long as

the same order is used in both. One could put the states in alphabeti-

cal order or in order of their 1970 populations, and even the ages could

be rearranged with no loss other than inconvenience in remembering the

sequence. Moreover the ages need not be a subdivision of the states; the

states could just as well be a subdivision of the ages. In fact Feeney (1970)

shows the advantage of considering the ages separately, making a block of

the interstate migration for ages 0 to 4, then another block for ages 5 to 9,

and so on. The block matrix for ages is then largely empty, and its proper-

ties are readily worked out. More recently, Hunter and Caswell (2004) have

238 10. Understanding Population Characteristics

shown how to easily construct multiregional models with either classiﬁca-

tion, using an operator called the vec-permutation matrix. Their approach

maintains a useful block-diagonal structure for the projection matrices.

The literature on migration and settlement is too extensive to refer to

here in any complete way, let alone to summarize. Classic papers on the

mathematics are due to Blumen, Kogan, and McCarthy (1955), Goodman

(1961b), and McGinnis (1968). Alonso (1978), Lowry (1964), and Stone

(1971) provide theory on rural–urban movements, and an annotated bib-

liography is due to Price and Sikes (1975). The economics of migration

is treated by David (1974) and Todaro (1969), and geographical aspects

by Berry (1973) and Zelinsky (1971). Rogers (1984) explores important

aspects of the age-dependence of migration.

Of interest as potentially unifying migration theory and integrating it

with multiple decrement life tables, tables of the work force, and tables of

marital status, is the work of Rogers (1975, 1995) and Rogers and Willekens

(1976). Their matrix formulation simpliﬁes and generalizes the work of

Schoen (1975). For other approaches using matrix formulations, see Lebre-

ton and Gonzalez-Davila (1993), Lebreton (1996), and Hunter and Caswell

(2004). Alonso (1978; see de Vries et al. 2000) has developed a general

framework that holds promise of being able to handle nonlinearities that

are important in the real world.

10.5 Human Stocks and Flows

The study of population involves relating stocks to ﬂows, as Richard Stone

(1972) reminds us. Censuses report the stocks, according to the categories

of age, marital status, education, occupation, and other characteristics,

while vital statistics, numbers of graduates provided by schools, and similar

current data report the ﬂow. In contrast to economic data, which are more

plentiful with respect to ﬂows, social statistics are richer and higher in

quality with respect to stocks. Relating the two is important in all ﬁelds.

Underlying all social statistics are individuals passing through the sev-

eral states of their life cycles. They begin by being born, as reported in a

ﬂow series, spend some years at home, go into the educational system at

the elementary level, sooner or later move to the intermediate and higher

levels, eventually enter the labor force, where they may stay brieﬂy or long,

perhaps go back to school, and ultimately retire, possibly after several en-

tries into and withdrawals from the labor force. At the same time they

pass from a household of orientation, are married and form a household

of procreation, have children, divorce, remarry, become widowed, and go

through other domestic transitions. If it were useful, we could consider each

economic activity state classiﬁed by the several domestic states, but it is

10.5. Human Stocks and Flows 239

less demanding of data to avoid this and to use two models in which the

two collections of states are separately treated.

The two sequences outlined above through which individuals go are

referred to by Stone as the active sequence, relating the individual to

production and the market, including schooling and retirement, and the

passive sequence, including the successive types of household of which he

is part, the housing conditions, and his neighborhood and location. A fur-

ther sequence is that of health and medical care, as the individual passes

through the stages of being well and sick, encounters successive medical

practitioners, is hospitalized and released. Some individuals go also through

a sequence of delinquency states, starting with aberrant behavior and its

consequences as they pass through the hands of police, courts, and prisons.

These can all be treated as cases of stage-classiﬁcation (Chapter 3), and

approached via matrix models.

For each of the states in each of the four sequences mentioned above,

a precise deﬁnitional boundary is required for purposes of measurement.

For education it happens to be relatively easy to draw a sharp boundary

around formal full-time schooling, and this is where the boundary is usually

placed for statistical purposes; the important component of home and part-

time learning is neglected. A conﬂict between precision of measurement and

relevance of concept exists in any empirical science. As another example,

it is sharper to restrict statistically economic production to that which

goes through the market, locating self-transport to place of work and other

do-it-yourself contributions outside the boundary of production.

To introduce the notation and ideas with an uncomplicated example,

consider just two states, alive and dead, and the corresponding ﬂows of

birth and death. The statement is an adaptation of the Leslie (1945) model

to a stationary population. Suppose a matrix S in which the subdiagonal

elements are the probabilities of surviving from one year of age to the next,

so that the ﬁrst is, in the notation of Chapter 2, L

, the second L

and so on; all elements but the subdiagonal are zero. The new population

that enters each year through birth or migration is a vertical vector b.

Suppose also that b consists of the number L

in its ﬁrst position and

zeros elsewhere. Then, if the population at time t is (t), also a vertical

vector, we have

n(1) = Sn(0) + b

n(2) = Sn(1) + b = S

n(0) + Sb + b

After the process has been going on for a long time, the term in S

n(0)

will have vanished; in fact all powers of S beyond the (m −1)th, if it is an

m × m matrix, will consist entirely of zeros. Similarly, if t is m or greater,

such terms as S

b will be zero, so we can add them in with impunity. This

240 10. Understanding Population Characteristics

givesustheseries

n(t)=b + Sb + S

b + ··· t>m (10.5.1)

=[I + S + S

+ ···]b, (10.5.2)

which can be thought of as inﬁnite. The sum of a geometric series of ma-

trices is obtained in the same way as the sum of a series of scalars of which

the common ratio is less than unity. Let the sum be N:

N = I + S + S

+ ···.

Multiplying N by S gives the same series, except that the I is missing;

hence we have

N = NS + I,

or, on subtracting NS from both sides and factoring out N,

N[I − S]=I,

and, on multiplying on the right by the inverse [I − S]

−1

N =[I − S]

−1

Entering this result in (10.5.2) gives for the population distribution by

age at time t

n(t)=[I − S]

−1

b. (10.5.3)

The factor multiplying b, which is called the fundamental matrix of the

absorbing Markov chain implied by the matrix S, will appear again in

Chapter 11.

For our particular matrix S giving survivorships as the transitions be-

tween successive ages, the determinant of [I −S] is readily seen to be unity,

and the inverse is the transpose of the cofactors of the elements of I − S;

that is to say, for the jth element of the ith row of the inverse we enter the

determinant obtained by deleting the ith column and jth row of I−S.This

procedure provides the fundamental matrix of the survivorship matrix:

[I − S]

−1

⎛

⎜

⎝

1000···

100···

10···

1 ···

⎞

⎟

⎠

. (10.5.4)

The elements of the ﬁrst column provide the probability of attaining the

successive ages, starting at age zero; the elements of the second column

10.6. The Demography of Organizations 241

give the same probabilities, starting at age one, and so on for the several

columns. The totals of the columns are the expectations of further life,

starting in the ﬁrst age, the second age, and so on. Expressed formally, if e

is the column vector consisting of ones then the ith element of e

[I −S]

−1

is the expectation of further life for a person who has attained the ith

age. (In contrast to most demographic and actuarial work, this way of

developing the subject takes individuals in midstate rather than completed

ages. Unless the states were age intervals much shorter than 1 year, calling

the midstates

,... intervals would not make the expectations quite

equal numerically to those calculated in the usual life table. The diﬀerence

is of no consequence for the present purpose.)

Merely to arrive at a somewhat inaccurate version of the life table, the

matrix analysis above is cumbersome as well as superﬂuous. Its advantage

is that it gives stocks (of people) in terms of ﬂows, not only for death but

also for many other conditions, and that it permits a connection to powerful

results on absorbing Markov chains (Chapter 11).

We were able conveniently to sum the powers of S because S vanishes

beyond a certain power. That does not happen in general, but in many situ-

ations the elements of the powers become smaller and smaller in something

like a diminishing geometric progression. They do this when the matrix S

is part of an absorbing chain; our matrix S above is part of an absorbing

chain in the sense that everyone is absorbed by death sooner or later in

the course of his progress through the several ages. See Chapter 11 for a

complete description of these methods.

10.6 The Demography of Organizations

This book deals for the most part with large populations. It presents theory

suited to the understanding of national populations, like those of the United

States or Mexico. But this is an inappropriate restriction; every city ward,

corporation, school, hospital, or seaside resort has its demography. The

operations of communities and institutions depend on a ﬂow of people,

whose entries are analogous to births and their departures to deaths. For

institutions that have relatively well-determined posts, diﬀerentiated by age

and sex like a national population and in addition by skills and other role

requirements, a thoroughgoing demography would be an extensive subject,

and the pages that follow are barely an introduction to it. They draw from

contributions due to Bartholomew (1982), Coleman (1973), March (1975),

and Waugh (1971). Their claim to a place in this chapter is the help they

oﬀer in understanding distributions of income and similar variables.

We tackle the sheer size aspect ﬁrst in political, then in economic,

organizations.

242 10. Understanding Population Characteristics

10.6.1 Loss of Power

One aspect of the number of persons in an organization has been called

by Coleman “loss of power,” the diminishing inﬂuence of the individual

as the organization grows in size. This appears clearly in voting, where

the individual has a deciding voice only when his associates are evenly

divided. The probability of such even division can be shown to be inversely

proportional to the square root of the size of the voting group. If there are

100,000 individuals aside from Ego, the chance that they will split evenly

is approximately one-tenth as great as if there were 1000 individuals. This

inverse square root law is easily derived.

With 2n + 1 members voting, the chance that Ego’s vote will be decisive

is the same as the chance that the other 2n individuals will split n for and

n against whatever motion is under debate. The number of ways in which

this can happen is the number of ways of putting 2n objects equally into

two boxes, that is, (2n)!/n!n!. The total number of combinations of the 2n

votesis2

, since each could vote “yes” or “no.” With equal chances or

each vote the probability that the 2n votes will be evenly split is

p =





(2n)!

n!n!

The solution happens to be formally identical with that for the fraction

unmarried in (10.3.7).

Entering as before the Stirling ﬁrst approximation to the factorial n!=

−n

√

2πn, we ﬁnd that everything cancels except 1/

√

πn. With n =3,

√

πn is 0.3257 against the exact 0.3125; with n =5,1/

√

πn is 0.2523

against the exact 0.2461. For higher values of n the error of the approxi-

mation is negligible, as Table 10.3 shows. This establishes the proposition

that a person’s power goes down as the square root of the size of the group

within which he is operating provided he is the last one to vote; in the

symmetrical condition before the voting starts each member can have no

more and no less than 1/n of the power.

10.6.2 Organizing Political Success

As Coleman points out, one means by which an individual can apply social

skill to attain power is by restructuring the electorate. In real life an elec-

torate, whether it be a local boys’ club, a town, a nation, or the Congress

of the United States, is made up of cliques, factions, committees, parties,

and other subgroups. In the smallest cell the probability 1/

√

πn provides

an appreciable chance to the individual. Ego may well gain a sense of power

through having the decisive vote in the clique with relative frequency, even

though the decisions concern small matters such as how the clique as a

whole will vote in a larger contest. If the voting is to be by groups, like an

electoral college, Ego’s chance of being the deciding voter is little diﬀerent

10.6. The Demography of Organizations 243

from that in individual voting. But evidently a person gains power nearly in

the ratio of

√

2=1.41 by being a member of a coalition containing half the

members plus one. He gains if he becomes a member of a coalition of any

size that votes together while others vote individually. Coleman proves that

the size of the coalition that maximizes the person’s power is considerably

less than the 51 percent that gives absolute control.

10.6.3 Economic Hierarchies

Economic organizations often fall naturally into hierarchies by virtue of

the technical requirements of production and decision making. To simplify

a typical organization, assume that there is one head of it, he has s persons

reporting directly to him, and each of these has s persons reporting to him

in turn, where s can be a number like 5 and is called the span of control

(Cramer 1972). If there are n levels then counted from the top downward

the personnel of the organization will be distributed as follows:

Level 123 ··· n

Number of employees 1 ss

··· s

n−1

and the total personnel, say at time t, will be N(t)=(s

−1)/(s−1). Then,

if a retirement takes place at the ith level, the chance of promotion for each

individual at the (i+1)th level is 1/s

, which is a very small number for the

lower echelons of the hierarchy. The theory here does not have any simple

relation to that relating promotion and age of Sections 5.8 and 14.10.

Note that, as long as s is greater than 1, there are more persons in the

lowest level than in all higher levels together. If seniority in promotion

is maintained, the halfway point between bottom and top will always be

within the lowest level. A person in an organization with s = 5 will be 80

percent of the way to the top by the time he is promoted out of the lowest

level, when the scale is in terms of individuals. But when the scale is in

terms of levels, he will be only 1/n of the way to the top.

Suppose that the organization starts with N

employees, in k layers; then

=1+s + s

+ ···+ s

k−1

− 1

s − 1

;

and if it is growing at rate r, its size at time t can be expressed as N(t)=

. Equating this to the above expression N(t)=(s

−1)/(s −1) gives

− 1

s − 1



− 1

s − 1



. (10.6.1)

Even for a small population the error in substituting s

for s

− 1and

for s

− 1 is negligible. The substitution permits solving (10.6.1) for n:

= s

244 10. Understanding Population Characteristics

so that

n = k +

log s

, (10.6.2)

where k, the initial number of levels, depends only on the initial population

and the span of control. The conclusion is that the creation of new levels

goes on at the steady rate of r/(log s) per unit of time or, equivalently, that

each (log s)/r units of time a new level is created.

A typical entrant who makes a career in a large organization virtually

ensures that he will not rise far above the bottom. With a span s of 5, even

spectacular success, rising above 0.999 of fellow employees, will put him

only at 4.4 levels from the bottom, or in the ﬁfth layer. This is of course

modiﬁed for a two-class organization, like the traditional British army, in

which one entered either as an enlisted man, with zero chance of rising

above sergeant, or as an oﬃcer, with appreciable chance of rising to the

highest ranks.

The boundaries of an organization are not always well deﬁned; in fact,

they may be determined by the viewpoint of the individual. An orderly

in a hospital is near the bottom of the paid hierarchy; but if he thinks

of patients as part of the hospital, he adds a substantial layer below him.

A teaching assistant in a college is at the bottom or not, depending on

whether students are included in the hierarchy. A faculty member may be

high in her department, but low in the collection of departments in the

country that make up her professional peer group.

Suppose that the rules of an organization call for absolute seniority, so

that everyone has a turn at the top, if only for a short period. In a stationary

condition the fraction of his service that he could have at the top is

s − 1

− 1

If s =5,n = 9, he could be boss for 4/(5

− 1) = 1/488,000 of his career,

which works out to 11 oﬃce minutes in the 45 years from age 20 to age 65.

But if the situation is hopeless for the mass of people in an organization,

by the same logic the person who has an exceptional endowment of luck,

brains, or inﬂuence can very quickly rise to the top. If the organization has

488,000 employees, and everyone in it above the bottom level has 5 people

reporting to him, there are only 9 layers. A person starting at the bottom

need be promoted only once every 3 years to become head of General

Motors before he is 50.