Keyfitz N., Caswell H. Applied Mathematical Demography

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10.3. Age at Marriage 225

Then the sex ratio in the age group is

∗

−(2

+x)r

−(2

+x)r

∗

in which everything has canceled but s and the life table survivors.

In application to the United States in 1967, we have s =1.050,

329,111, and

∗

= 225,346, so the stable model gives the sex ratio at age

70 as

(1.050)(225,346)

329,111

=0.72,

compared with 0.76 observed. The diﬀerence arises because the current life

tables have a higher ratio of male to female mortality than the cohort in

question experienced; immigration and other factors also operate.

10.3 Age at Marriage

A curve to describe proportion married with age, backed by a convincing

rationale, is provided by Hernes (1972). In such an age-graded population

it hardly occurs to individuals to marry when most of their contemporaries

are unmarried; but as marriage gets under way after about age 18, each

sees the number of his or her unmarried friends diminishing and begins to

feel left out—experiences, indeed, an increasing pressure to marry. Hence,

argues Hernes, we can take the rate of transfer into the married state as

proportional to the fraction married. But it must also be proportional to

the fraction not yet married; the marrying can come only from the stock

of the unmarried. The result is a logistic curve of which Savage (1973)

provides a generalization.

10.3.1 A Sum of Random Intervals Model

Coale (1971) found by experimenting with a number of observed distribu-

tions of age at marriage that all ﬁtted a standard curve, provided only that

the curve was adjusted for (a) the age at which girls become marriage-

able, (b) the proportion ever marrying by the end of life, and (c) horizontal

scale. If the distributions are presented as risk functions, that is, age-speciﬁc

marriage rates for the still-single population but excluding those who never

marry, they take the form for ﬁrst marriages

r(x)=0.174e

−4.411e

−0.309x

, (10.3.1)

where x is the age of the person, measured from the origin for the particular

population. Calculation yields the following values:

226 10. Understanding Population Characteristics

xr(x)

0 0.0021

1 0.0068

2 0.0161

3 0.0304

4 0.0483

5 0.0679

10 0.1424

20 0.1724

This zero-parameter or three-parameter curve, depending on how one looks

at it, gives close ﬁt to data but, as Coale points out, lacks an obvious

behavioral rationale.

Feeney (personal communication) suggested that the marriage curve may

be composed of a random age of entry followed by a random delay. Coale

and McNeil (1972) carried this thought further and developed a distribution

involving three delays.

If the probability that X falls between x and x + dx is f

(x) dx,and

the probability that Y falls between y and y + dy is f

(y) dy; the joint

probability is f

(x)f

(y) dx dy. To ﬁnd the distribution f(z)ofZ = X +Y

we need to integrate this over all x and y such that x + y = z,thatis,

(z)=



∞

−∞

(x)f

(z − x) dx. (10.3.2)

This expression for f

(z) is called a convolution, like (7.5.1), because the

sum of the arguments of the functions in its integrand does not involve x,

the variable of integration. It is readily applied to the sum of two negative

exponential distributions, f

(x)=r

−r

and f

(y)=r

−r

, to obtain

the distribution of Z = X + Y as

(z)=

− r

−r

− e

−r

). (10.3.3)

It may also be applied to more than two exponentials and to a normal

curve followed by three exponentials, as long as independence holds.

This is how Coale and McNeil (1972) proceed to account for the dis-

tribution of marriage ages. They suppose a normal distribution of ages at

which girls become marriageable, followed by three delays, each exponen-

tially distributed. Fitting to data for French couples, they ﬁnd the mean

age of entry into a state of marriageability as 16.6 years, the mean inter-

val between then and meeting the future husband as 4.02 years, the mean

interval from acquaintance to engagement as 1.53 years, and the mean in-

terval from engagement to marriage as 0.93 year. The closeness of ﬁt of the

convolution to the observed data for this and other populations conﬁrms

their behavioral model.

10.3. Age at Marriage 227

10.3.2 Small Marriage Circles

So much for marriage partners in a large population, where random varia-

tion in the time of individual marriage is recognized, but random variation

in the number of men and women of marrying age can have only a minor

eﬀect. If in fact each individual, as he or she comes of age, seeks a marriage

partner from a relatively small circle, some part of nonmarriage can be due

to random diﬀerences in the numbers of men and women in these circles.

Henry (1969) has addressed himself to this aspect of the problem.

He asks us to think of a circle containing candidates for marriage, whether

or not they see themselves in that light. A circle may be an oﬃce or other

workplace, a social club, or a neighborhood. Suppose that it contains 2n

members, all of such age and disposition as to be candidates for marriage to

one another. Let them be randomly distributed by sex; that is, the proba-

bility of a member being male is 1/2, and of being female 1/2. The expected

number of male candidates would be n, and of females likewise n,sothat

in any realization of our hypothetical circle in which these expected values

held all 2n candidates would marry. The task is to ﬁnd the expected num-

ber of candidates that fail to ﬁnd mates over all realizations of the circle.

The following solution of Henry’s problem is due to McFarland (1970).

If the chance of each individual being male is 1/2, the chance that of the

2n members the numbers of males is n + k andoffemalesn − k is



n + k





. (10.3.4)

In this situation the nonmarrying males will number 2k = n+k−(n−k), k>

0. Hence the expected number of males not marrying must be n+k−(n−k)

times probability 10.3.4, added through all possible constitutions of the

circle that have males in excess of females:



[(n + k) − (n − k)]

(2n)!

(n + k)!(n − k)!









(2n)!





(n + k − 1)!(n − k)!

−

n−1



(n + k)!(n − k − 1)!



. (10.3.5)

We have made the upper limit in the right-hand summation k = n − 1,

because k = n gives a zero term owing to the factor n − k. Now note that

the term for k = 2 in the left summation cancels with that for k =1on

the right, both being equal to 1/(n + 1)!(n − 2)!. All other terms similarly

cancel, except that for k = 1 on the left, which is equal to 1/n!(n − 1)!.

Hence the required expected number of males left over is





(2n)!

(n)!(n − 1)!

= n





(2n)!

n!n!

. (10.3.6)

228 10. Understanding Population Characteristics

The expected number of females left over must be the same by symmetry,

and hence for both sexes we have an expected





(2n)!

n!n!

persons unmarried or, as a fraction of the 2n individuals in the circle,





(2n)!

n!n!

. (10.3.7)

McDonald (1965) ﬁnds the same result (10.3.7) for the matching of male

and female worms in the spread of certain infestations. For no readily under-

standable reason, the expected fraction of marriage candidates (or worms)

unmatched is the same as the probability that the group is equally divided

between males and females.

Stirling’s approximation to the factorial is n!=n

−n

√

2πn [1+(1/12n)],

omitting terms from 1/n

onward. Entering this in (10.3.7) gives





(2n)

−2n

√

4πn

1+(1/24n)

−n

√

2πn

1+(1/12n)

-7

which after cancellation becomes

√

πn

1+1/(24n)

1+1/(12n)

≈

1 − 1/(8n)

√

πn

. (10.3.8)

Thus, if a marriage circle containing 2n individuals is considered, the

expected fraction of excess of one sex or the other, that is to say, the average

fraction who cannot marry within the circle, is slightly under 1/

√

πn.For

a group of 20 individuals this is 1/

√

πn =0.178, reduced to 0.176 by the

factor 1 − 1/(8n) (Table 10.3 and Figure 10.2).

This solution may be extended to a circle having a disequilibrium of the

sexes, in the sense that the probability of a random individual being male

is not 1/2 but some other number; in this case, another device is required

to sum the series.

10.3.3 How Many Households Are Implied by Birth, Death,

and Marriage Rates?

Age-speciﬁc rates of birth and death, along with the fraction of women

married at each age, imply an average size of nuclear family (Goodman,

Keyﬁtz, and Pullum 1974).

For the general case, call the age distribution of women p(a), not nec-

essarily stable, so that



p(a) da = N

, and the fraction married at age

a is q(a); here the total number of married women is



p(a)q(a) da.Itis

reasonable to attach a separate family to each married woman, where for

this purpose only those currently living with a spouse will be deﬁned as

10.3. Age at Marriage 229

Table 10.3. Values of (2n)!/(2

n!n!) and Stirling approximations

Size of

group

2n (2n)!/(2

n!n!) 1/

√

πn



1 −





√

πn

2 0.5 0.5642 0.4937

4 0.375 0.3989 0.3740

6 0.3125 0.3257 0.3122

8 0.2734 0.2821 0.2733

10 0.2461 0.2523 0.2460

12 0.2256 0.2303 0.2255

14 0.2095 0.2132 0.2094

16 0.1964 0.1995 0.1964

18 0.1855 0.1881 0.1855

20 0.1762 0.1784 0.1762

40 0.1254 0.1262 0.1254

80 0.08893 0.08921 0.08893

120 0.07268 0.07284 0.07268

160 0.06298 0.06308 0.06298

200 0.05635 0.05642 0.05635

0.1

0.2

0.3

0.4

0.5

246810121416

Size of circle 2n

Expected fraction for whom there is no spouse

Figure 10.2. Diminishing fraction of expected number without spouse as marriage

circle increases.

married. Then the total number of families is also



p(a)q(a) da, and the

mean number of persons per family is



p(a)q(a) da

(10.3.9)

230 10. Understanding Population Characteristics

if the numbers of males and females in the population are equal.

To go from this to the mean size of family implied by age-speciﬁc rates

in a stable population, we replace p(a)/N

by be

−ra

l(a), as in Section 5.1.

Then we have for the mean number of persons per family



−ra

l(a)q(a) da

. (10.3.10)

This is diﬀerent in concept from the average we would derive from the

census itself by counting persons in families and dividing by the number

of families. Our (10.3.10) neglects persons living not in families but in

prisons and other institutions, and implicitly somehow allocates illegitimate

children to married women, but this is a minor divergence from the census.

The main diﬀerence is that (10.3.10) gives what is implied by the current

rates rather than by rates that have existed over the lifetimes of the people

counted in a census. To reconcile these two kinds of average would be a

diﬃcult but worthwhile piece of research.

What about the size of the extended family, supposing that a separate

extended family is formed by a married woman if her mother is dead?

We need merely enter in (10.3.10) the probability that a woman aged a

does not have a living mother, a subject to be developed in Chapter 15.

If the probability that a woman aged a has a living mother is M

(a)[an

expression for this is given later as (15.1.4)], the chance that she has not is

1 − M

(a), and from (10.3.9) extended families will average



[1 − M

(a)] p(a)q(a) da

persons, (10.3.11)

or in the stable case



[1 − M

(a)] be

−ra

l(a)q(a) da

persons. (10.3.12)

10.3.4 Intrinsic Rates of Natural Increase: Age, Parity, and

Nuptiality

The intrinsic rate of growth, as deﬁned by Lotka and discussed in Sections

6.1 and 7.2, is that which ultimately results from the continuance of ob-

served age-speciﬁc rates of birth and death. The intrinsic rate depends in

no way on the observed age distribution, but is a means of interpreting the

age-speciﬁc rates. And yet the choice of age-speciﬁc rates was in a sense ar-

bitrary; we could have used other classiﬁcations—say rural–urban-speciﬁc

rates, classiﬁed by age or not. Any characteristic that shows diﬀerentials of

birth and death rates is a candidate for providing an intrinsic rate, although

not all characteristics are equally logical candidates.

10.3. Age at Marriage 231

Oechsli (1971) has calculated rates intrinsic for age, parity, and nuptial-

ity, and for certain combinations of these. The age–parity rate, for instance,

tells what the increase of the population would be once its age–parity dis-

tribution came to be that resulting without disturbance from the observed

set of rates in the several age–parity groups.

To see the meaning of various directions of adjustment consider the

following ﬁve rates (per 1000 population) for United States females in 1960:

Crude Crude rate of natural increase 14.7

A Age intrinsic rate 20.8

A–P Age–parity intrinsic rate 23.0

A–N Age–nuptiality intrinsic rate 18.4

A–N–P Age–nuptiality–parity intrinsic rate 19.6

Therisewhenwegofromthecruderateintheﬁrstlinetotheage

intrinsic rate in the second means that the observed age distribution was

unfavorable to increase—that there was a smaller proportion of women

of childbearing age than are present in the stable condition at 1960 age-

speciﬁc rates of birth and death. This is a well-known result of the small

cohorts born in the 1930s.

When we go from the age intrinsic to the age–parity intrinsic rate,

we ﬁnd a further increase, which can mean only that within age groups

the distribution of observed parities was unfavorable to reproduction. The

highest-bearing parities are the low ones; apparently the stable condition

on 1960 age-parity speciﬁc rates would have relatively more individuals of

low parity than the actual situation of 1960.

On the other hand, when we go from the age intrinsic rate to the age–

nuptiality intrinsic rate (from 20.8 to 18.4), we ﬁnd a decline; this means

that the proportion of married women in the population of 1960 was greater

(in the main ages of childbearing) than the proportion of married women in

the stable condition. If the marriage rates, the age-speciﬁc nuptial fertility

rates, and the age-speciﬁc mortality rates of 1960 were allowed to work

themselves out, they would result in a lower overall birth rate than would

the age-speciﬁc fertility and mortality rates of 1960, disregarding marriage.

The fall from the age–parity to the age–nuptiality–parity says that sep-

aration of the married women in the age–parity analysis brings the rate

down, and the eﬀect is so drastic that we ﬁnd A–N –P below the simple

age intrinsic rate A. In short, the eﬀect of nuptiality more than oﬀsets the

eﬀect of simple parity. Karmel (1950) suggested in his comment on Whelp-

ton (1946) that the parity correction by itself could be deceptive. Suppose

from one year to the next a rise in marriages, while age–parity rates at

each marriage duration remained the same. The age–nuptiality–parity in-

trinsic rate would be unchanged, whereas both the age intrinsic and the

age–parity intrinsic rates would rise. But the latter would rise more, for

232 10. Understanding Population Characteristics

the higher ﬁrst births would be divided by zero-parity women rather than,

as in the age-speciﬁc rate, being diluted by division by all women.

Therisefromthe18.4inA–N to the 19.6 in A–N–P means that within

the married group, age by age, the parity distribution of the observed pop-

ulation was less favorable to fertility than the parity distribution of the

stable condition.

All of this can be shown on a diagram. The lozenge form brings out the

contrast between the eﬀects of parity and of nuptiality (A = age; P =

parity; N = nuptiality):

Crude

14.7

20.8

A – N – P

19.6

A – P

23.0

A – N

18.4

10.3.5 The Life Cycle

According to Glick’s calculation (1955), the following changes in median

ages of various events in American families took place between 1890 and

1950:

Median age of Median age of

husband wife

Event 1950 1890 1950 1890

First marriage 22.8 26.1 20.1 22.0

Birth of last child 28.8 36.0 26.1 31.9

Marriage of last child 50.3 59.4 47.6 55.3

Death of one spouse 64.1 57.4 61.4 53.3

Death of other spouse 71.6 66.4 77.2 67.7

Whereas in 1890 a minority of couples survived until the last child was

married, by 1950 most survived, on the average by about 14 years. A new

demographic element came into existence: husband and wife living together

after their last child had left home—the empty nest. This corresponds to

a new pattern of spatial movement as well; a couple may start out in an

apartment in the city, move to a suburb when their ﬁrst child is born, and

remain there until their last child has left home. If the husband is middle

class, his income has usually continued to climb, and his earnings are higher

than when the children were at home; moreover, his wife may well take up

10.3. Age at Marriage 233

some new kind of paid work or resume an earlier career. The couple no

longer need their suburban home and schools and are in a position to move

into town and pay a substantial rent. Collectively such couples have aﬀected

the skylines of many American cities as new high-rise, high-rent apartments

spring up to accommodate the new demographic element.

The change in life cycle is due to improved mortality, especially for

women, along with earlier marriage in both the older and the younger

generation—a 1-year fall in the age of marriage subtracts 2 years from the

age at which the last child is married, everything else remaining the same.

Moreover the birth rate has fallen so that children are fewer; and, espe-

cially if the wife is eager to resume her career, the children that couples do

have are more closely spaced. To all this is added an increased tendency

for children to leave home before marriage.

We thus know to what factors the changes are due, but we do not know

how much of a particular change is due to each factor. Yet it need not be

diﬃcult to break down any part of the diﬀerence. In 1890, for husbands, the

ﬁrst death of spouse occurred 2 years before the last child was married; in

1950 it was 14 years after. One would like to know how much of the change

was due to improved mortality, how much to earlier marriage, and so on.

These eﬀects would be ascertained by the technique exhibited in Table 10.2.

The ﬁrst step would be to set up a model, including death rates, marriage

rates, home-leaving rates, birth rates, all by age. Then the model with

the rates for 1890 would be used to generate the 1890 column of Glick’s

table, or an approximation to it. The same would be done for the 1950

column. Then the various elements of the models would be interchanged;

for example, the marriage rates for 1950 would be combined with the other

elements for 1890 to obtain an estimate of the eﬀect of changed marriage

ages. The result would be somewhat more elaborate to interpret than Table

10.2, since it would have several orders of interaction, but no new principle

would be required. See Chapter 13 for other ways to decompose eﬀects of

factors and their interactions on the rate of increase.

This demographic theory of the life cycle needs to be further developed,

and then integrated with the theory of savings and consumption over the life

cycle that has become a standard part of macroeconomics. Stage-classiﬁed

matrix population models would certainly be applicable to it.

10.3.6 Married and Divorced

The numbers of married and divorced persons counted in a census, like

those of the several ages and sexes, are related to the country’s previous

history of marriage and divorce. It is necessary to consider the time series

of marriages of preceding years, along with other series on the dissolution

of marriages by death of one spouse or divorce. The present is in principle

explainable by and reconcilable with the past. If the reconciliation and

explanation are in a small local area, one would have to take account of

234 10. Understanding Population Characteristics

immigration and emigration as well as marriages, deaths, and divorces, but

this complication will be disregarded here. The discussion that follows is

due to Preston (1975b).

Still for simplicity, disregard the age of the individuals concerned, and

take account only of duration of marriage. (First marriages fall in a rela-

tively narrow band of ages, and the theory will apply to them more strictly

than to all marriages.) Call µ

(x) dx the chance that a marriage that has

been in existence x years will break up during the time x to x + dx through

the death of the husband, µ

(x) that it will break up through the death of

the wife, and µ

(x) that it will break up through the divorce of the couple.

Then the argument that started in Section 1.6, of which a special case was

the survivorship of an individual in the face of death (Section 2.1), is here

applicable to the survivorship of a couple in the face of the death of either

member or of divorce. It is as though a group of marriages, initially N

number, had to run the gauntlets of death and divorce, and only a certain

fraction survived in each time period. The surviving couples at the end of

t years would be

N(t)=N

exp

−





(x)+µ

(x)



(

. (10.3.13)

The expected number of divorces between time t and t + dt would be

N(t)µ

(t) dt, assuming independence, and disregarding remarriage.

These causes of dissolution compete with each other. Higher mortality

lowers the divorce rate, everything else remaining the same. The reason for

this is to be seen in the expression for the number of marriages that will

eventually end in divorce:

D =



∞

N(t)µ

(t) dt.

This, divided by the number N

of marriages that started the cohort oﬀ,

gives the fraction expected to end in divorce. If, as between two populations

otherwise similar, mortality is higher at any age in one, N(t) will be lower in

that one for all subsequent ages, so D will also be lower. This is analogous to

the way that deaths from cancer in the multiple-decrement table of Section

2.6 depend on deaths from other causes.

The Current Divorce–Marriage Ratio. The ratio of current divorces to cur-

rent marriages, say for the year 1973, is a convenient measure of the extent

of divorce. This measure is often used, and almost as often criticized on the

ground that current divorces come from the marriages of a period going

back a number of years, and hence the denominator of the ratio ought to be

these marriages. The ideal is a schedule of divorce rates according to dura-

tion of marriage, so that duration-speciﬁc divorce rates can be calculated.

But when these are not available and we resort to the current divorce–

marriage ratio, we can apply an interpretation of this ratio that has been

provided by Samuel Preston.