Keyfitz N., Caswell H. Applied Mathematical Demography

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5.8. Promotion Within Organizations 125

will receive 10

dollars. Each individual may be thought of as an ancestor

of those whose names are below his on the list, and the payments are made

by descendants to ancestors.

There is no fallacy in the operation of a chain letter, provided that re-

cipients follow instructions and new people continue to be available to be

brought into the scheme. Once the available population has been brought

in, so that names have to be repeated, however, the scheme breaks down.

And this is certain to happen long before 10

names are in the process.

Payments by descendants to ancestors, say sons to great-grandfathers,

in analogy to the chain letter with four names, have a very diﬀerent way of

cumulating than payments by ancestors to descendants, as in the ordinary

passing of property from father to son. With a growing population, equal

inheritance will fractionate property and give successively less to each gen-

eration. Large families and growth of population are a drawback in a regime

of inherited private property, just as clearly as they are an advantage in

the chain letter or promotion. Norman Ryder has emphasized this point in

his lectures.

Whether the institutions of a community are such that beneﬁts ﬂow

upward, as in promotion or the chain letter, or ﬂow downward, as in in-

heritance, is crucial to attitudes on collective growth and to the size of

individual families. It is arguable that the peasants’ preoccupation with

holding property intact in nineteenth century France was associated with

low fertility; a peasant obtained security in his old age by having few chil-

dren and thereby avoiding division of his land. In Java, on the other hand,

wage labor is a main means of livelihood, and the landless laborer wants

many children, whose combined contributions will permit him to subsist in

old age. This supposes appropriate discipline in the children, just as does

the chain letter.

The American pay-as-you-go social security system resembles that of

Java in that the working generation supports the retired one, with the

diﬀerence that in the United States the unit is no longer the family but the

entire country. The U.S. system works well when there are many children

to support few old people, as was the case a generation back; it runs into

trouble when many old people have to be supported by few children, as

will be the case early in the twenty-ﬁrst century. Table 14.4 suggests the

consequences for pay-as-you-go social security of the present decline in U. S.

births.

It would be useful to inventory the situations in which something—

money, supervisory positions, prestige—ﬂows up from the younger gen-

eration to the older, and examine empirically corresponding attitudes on

reproduction; the hypothesis of this section is that they would be found to

be pronatalist. On the other hand, in situations where the ﬂow is from the

older to the younger, of which inheritance of property is the most familiar,

restrictive attitudes toward reproduction would be expected.

126 5. Fixed Regime of Mortality and Fertility

Growth is popular in comparison to mortality: having children is prefer-

able to dying. Moreover, a given amount of growth helps as much as a

given amount of mortality; and since growth varies 3 times as much as

mortality among observed populations, it can accelerate promotion about

3 times as much as even the survivors can expect in a high-death commu-

nity. Cessation of growth has costs and is inevitable, and the question is

what generation ought to incur the cost.

Birth and Population Increase from

the Life Table

The stable age distribution of Chapter 5 supposed births growing at ﬁxed

rate r and subject to probability l(a) of survivorship to age a.Various

conclusions were reached by noting how the stable age distribution was

aﬀected by r, treated as an arbitrarily disposable parameter. Now we regard

rate r not as arbitrary but as determined by the joint action of birth and

death rates.

The argument rests on birth rates m(a), where m(a) da is the probability

that a woman who has reached age a will give birth to a girl child before

age a + da, or alternatively that each woman will produce exactly m(a) da

of a child in interval da. A model in which exactly m(a) da of a child is

produced by each woman in any small interval of time da departs consid-

erably from the reality of human reproduction, where children are born,

not continuously, but in units of one child each, with occasional multiple

births, and random variation is conspicuous. Moreover, we will deal with

the female part only of the population, and disregard the contribution of

males to reproduction. (Or, less often, with the male part, but not with

both together.) Such ﬁctional treatment permits some useful conclusions

to be drawn and ought to be exploited to the limit before resorting to more

realistic (and necessarily more complex) models. Although unusable for

families and other small populations, the continuous deterministic model

provides useful answers to many questions concerning mean values in large

populations.

128 6. Birth and Population Increase from the Life Table

6.1 The Characteristic Equation

To ﬁnd an equation for the ultimate rate of increase r by a continuation

of the means developed in Chapter 5, suppose that for a long time the

chance of living to age a is l(a), and that births increase at ﬁxed rate r.

Then the age distribution becomes stable, a proposition proved many times

(Sharpe and Lotka 1911, Lotka 1939, Parlett 1970, Lopez 1961, McFarland

1969) and illustrated in Section 1.10. When stability has been attained,

the fraction of population at ages a to a + da is be

−ra

l(a) da,asshownin

the argument leading to (5.1.1). Since births are calculated by multiplying

exposed population by m(a), the fraction be

−ra

l(a) da of the population

must give rise to be

−ra

l(a)m(a) da births per unit of total population. The

integral of this last expression, from the youngest fertile age α to the highest

β,is



−ra

l(a)m(a) da, the overall birth rate.

But b is the overall birth rate in the population with age distribution

−ra

l(a) da, and it may therefore be equated to the integral derived in the

preceding paragraph. This provides the equation



−ra

l(a)m(a) da = b, (6.1.1)

or, on dividing by b,



−ra

l(a)m(a) da =1, (6.1.2)

obtained in more general fashion by Lotka (1939, p. 65; see Section 7.5 of

the present chapter) and called the characteristic equation for the unknown

r. Equation 6.1.2 tells us the ultimate rate of increase implied by l(a)and

m(a) after they have been acting long enough for stability to be attained.

For a general net maternity function l(a)m(a), equation (6.1.2) has an

inﬁnite number of roots, of which only the real root answers the questions

treated in this chapter. The uniqueness of the real root follows from the

fact that



−ra

l(a)m(a) da,sayψ(r), has a negative ﬁrst derivative [prove

this] and so is a monotonically decreasing function of r, which takes the

value ∞ for r = −∞ and the value 0 for r = ∞ (Figure 6.1). Thus ψ(r)

can only once cross any given horizontal line in the half-plane ψ(r) > 0,

including the line one unit above the r axis, and hence ψ(r)=1canhave

only one real root. The function ψ(r) crosses the vertical axis at

ψ(0) =



l(a)m(a) da = R

the net reproductive rate.

6.1. The Characteristic Equation 129

0 r

␺(r) = 1

␺(r)

Figure 6.1. Values of ψ(r), showing ψ(0) = R

, and root r

of ψ(r)=1.

6.1.1 Why Stress the Female Model?

Although the argument leading to (6.1.2) is applicable to either sex, it is

most often applied to females because:

1. Data on age of parent at the birth of a child are more often collected

by the registration authorities for women than for men.

2. Even when the authorities present births by age of father as well as by

age of mother, more cases of unstated age of father occur, especially

for illegitimate births.

3. Women have their children within a narrower and more sharply de-

ﬁned range of ages than do men, say 15 to 50 as against 15 to

80.

4. Both the spacing and number of children are less subject to variation

among women; a woman can have children only at intervals of 1 or

2 years, so she is physiologically limited to a score or so of children,

whereas a man can have hundreds. Under monogamy this asymmetry

of the sexes is of no consequence, but every society allows some de-

partures from strict monogamy, even if only to permit the remarriage

of widowers.

The reasons for the emphasis on females apply over a wide range of

population analysis. Both understanding and prediction must be based on

whatever constancies we can ﬁnd; that potential variation is less in births

to women than in births to men makes the female model more useful.

130 6. Birth and Population Increase from the Life Table

On the other hand, when the age distributions and life tables for males as

well as for females are available, the rate of increase can be estimated from

either sex, and two estimates provide more information than one. When

only one is to be had, we must rest on the fact that usually in the short

run, and always in the long run, the males and females of any population

increase at nearly the same rate.

6.1.2 An Iterative Method for Calculating r

Means of calculating this root are given in several places (Lotka 1939,

p. 70; Coale 1957; Keyﬁtz 1968, p. 108) and need not be repeated here.

But an iterative method is worth sketching because it is suggestive for

other equations.

If T is an arbitrary age chosen near the mean age of childbearing, multiply

both sides of (6.1.2) by e



−(a−T )r

l(a)m(a) da = e

, (6.1.3)

then take logarithms of both sides, and divide by T to obtain

∗

log



−(a−T )r

l(a)m(a) da. (6.1.4)

The left-hand side of (6.1.4) has been marked with an asterisk to suggest

the use of the equation for iteration. The unknown r enters on the right,

but in a way that does not greatly inﬂuence the integral. The extent to

which r inﬂuences the right-hand side of (6.1.4) determines the speed of

convergence, and is measured by the absolute value of the derivative dr

∗

/dr,

a fact demonstrated formally in books on numerical analysis, such as that

of Scarborough (1958, p. 209). If |dr

∗

/dr| is less than 1 in the neighborhood

of the solution, convergence will occur; if it is much less than 1, convergence

will be rapid. From (6.1.4) dr

∗

/dr is seen to have a factor 1 − κ/T ,where

κ is the mean age of childbearing in the stable population:

κ =



−ra

l(a)m(a) da.

(Since r satisﬁes (6.1.2), the denominator of the expression for mean age

κ is unity.) The derivative dr

∗

/dr seems to be telling us that if we can

put the arbitrary T equal to some number near κ, rapid convergence will

result. In fact we do not know κ before we ﬁnd r, but κ varies little among

populations and T can be put at 27.5 years.

No direct use can be made of a continuous form like (6.1.4)—it must be

converted to the discrete form for calculations. If T is chosen as 27.5, then

with the customary approximation



x+5

−ra

l(a)m(a) da = e

−(x+2

6.1. The Characteristic Equation 131

Table 6.1. Second iteration of intrinsic rate of natural increase for the United

States, females, 1968

1000F



−(x+2

−27

)(0.0056)

15 4.867 67.1 0.327 1.058

20 4.851 167.4 0.812 1.028

25 4.831 140.3 0.678 1.000

30 4.807 74.9 0.360 0.972

35 4.770 35.6 0.170 0.946

40 4.714 10.2 0.048

0.919

Total 2.395

Source:For

, Vital Statistics of the United States (1968, Vol. 2, Section 5,

pp. 5–7). For F



, Statistical Abstract of the United States (1973, p. 52).

where



x+5

l(a) da,andF

is the observed age-speciﬁc fertility rate

for ages x to x + 4 at last birthday. We translate (6.1.4) into calculable

terms as

∗

27.5

log(e

10r

+ e

−5r

+ e

−10r

+ e

−15r

(6.1.5)

The expression in parenthesis contains six terms if we include births under

15 with the 15 to 19 group and those over 45 with the 40 to 44 group.

Data for United States females in 1968 are shown in Table 6.1, where F



the birth rate of children of both sexes, is converted to F

, the female

birth rate, by multiplying by the fraction of births that were girls in 1968,

that is, 0.487. The calculation of (6.1.5) starting with r = 0 gives r

∗

log(2.395 × 0.487)/27.5=0.00560 and r

∗∗

=0.00587 in two successive

iterations.

The method is applicable to complex as well as to real roots. With com-

plex roots we must check to ensure that the root being approached is the

one sought—the real root being unique, no such precaution is needed. If all

complex roots are sought, one can sweep out each as it is found, after the

ith, by dividing out the polynomial in e

−5r

; this will prevent the iteration

from converging to any root more than once.

6.1.3 The Intrinsic Rate for Various Kinds of Data

The usual application of (6.1.2) is to the births and deaths of a particular

period, 1 or 3 or 5 calendar years, as when the Statistical Abstract of the

United States (1973) tells us that the intrinsic rate of increase was 0.6

percent for the United States in 1968, with 0.4 percent for White and 1.6

percent for Nonwhite components.

Male Period Intrinsic Rates. The above numbers apply to females; a corre-

sponding calculation can be made by imputing male births to fathers, and

132 6. Birth and Population Increase from the Life Table

Table 6.2. Male and female intrinsic rates during the 1960s

1000r

Country and year Male Female

United States

1966 12.82 9.70

1964 17.48 15.70

1963 18.68 17.10

1962 19.81 18.83

1959–61 20.92 20.70

Chile, 1964 23.55 21.05

Hungary, 1964 −2.95 −7.28

England and Wales, 1960–62 9.55 9.77

Source: Keyﬁtz and Flieger (1968).

in general it produces a diﬀerent answer. Sometimes the answer is substan-

tially diﬀerent, as for the United States, 1964, data, where females show

1000r as 15.70, and males as 17.48. The higher male rate is largely due to

fathers being older than mothers on the whole, while numbers of persons

in the main childbearing ages during the 1960s reﬂected the rising births

during the late 1930s and 1940s. If the men of an age to be fathers come

from earlier smaller cohorts, their rates will be higher for the given number

of births that actually took place, and it is largely this that is reported by

the diﬀerence between the male and female intrinsic rates.

Table 6.2 shows some male and female rates; the diﬀerences are consid-

erable and larger than can be explained by misstatement of age. For the

United States in particular the steady increase in the gap between male

and female rates seems due to the relative shortage of men of the age to be

fathers in the wake of the baby boom.

Cohort Intrinsic Rate. The same equation (6.1.2) can help interpret the

death and birth rates in a cohort. By following deaths and births to the

individuals born in a given year or 5 years, we obtain a cohort life table

and a set of age-speciﬁc birth rates that together determine r. Applied to

the women who were 0 to 4 years of age in 1920 in the United States, and

who therefore were 5 to 9 years of age in 1925, and so forth, they yield

an intrinsic rate of 1.5 per thousand (Keyﬁtz and Flieger, 1968, p. 601).

This means that the age-speciﬁc rates of birth and death were such that, if

they persisted, the cohorts (and incidentally the period cross section) would

ultimately increase at 1.5 per thousand. This value of 1000r is low because

the group in question passed through its principal ages of childbearing in

the late 1930s and early 1940s. Period and cohort inﬂuences are confounded

in the concrete historical record.

Intrinsic Rate for One Family. The concept of intrinsic rate can be applied

to a particular family. If a girl child is born at age a

of the mother and

6.2. A Variant Form of the Characteristic Equation 133

another girl at age a

, r is found from

−ra

l(a

)+e

−ra

l(a

)=1, (6.1.6)

which is easily solved for r by the method of (6.1.4) or otherwise. The

meaning of r is that a population in which all women married and bore girl

children at ages a

and a

would increase at this rate; the r so calculated

is an interpretation of the fertility behavior of the particular family. Non-

marriage or sterility within marriage can be accommodated by using l(a)

as the probability of surviving and being married and fertile by age a.

Whatever the group to which the characteristic equation 6.1.2 applies,

one of its uses is to ﬁnd how the rate of increase r depends on various

features of the net maternity function. We will see how this is done for an

increment in the birth rate m(a)atasingleagex. More frequently referred

to, however, are the relations between the intrinsic rate and the several

moments of the net maternity function. These relations are easily found

once we express the characteristic equation in terms of the moments.

6.2 A Variant Form of the Characteristic Equation

It happens that ψ(r), the left-hand side of (6.1.2), when divided by R

the moment-generating function of the normalized net maternity function

l(a)m(a)/R

ψ(r)



−ra

l(a)m(a)

da.

The sense in which ψ(r)/R

generatesthemomentsisthatitisequivalent

to the inﬁnite series

ψ(r)

=1−

−···,

where R



l(a)m(a) da,sothatR

is the ith moment about zero

of the distribution l(a)m(a)/R

. But the moments about zero are large; and

even though the series converges, an impractical number of terms would be

required for a good approximation. A rapidly converging series is obtained

by taking the logarithm of ψ(r)/R

, and so generating the functions of the

moments called cumulants (Kendall and Stuart 1958, Vol. 1, p. 70) that

we have already encountered in Section 5.3:

log

ψ(r)

= −rµ +

−

−···. (6.2.1)

The cumulants expressed in terms of the R

areasfollows:

µ =

134 6. Birth and Population Increase from the Life Table

−





− 3





and in terms of the moments about the mean,



(a − µ)

l(a)m(a) da

they are

= µ = µ

= σ

= µ

− 3µ

= µ

− 10µ

... .

Now the characteristic equation is ψ(r) = 1; that is, log[ψ(r)/R

−log R

; or, written out in terms of the cumulants, from (6.2.1) the

equivalent to (6.1.2) is

log R

= rµ −

−

+ ···. (6.2.2)

The successive terms diminish rapidly in value, even where the cumu-

lants increase. For the United States in 1967 (1000r =7.13), we have the

following:

Cumulant Term of (6.2.2)

1.205 0.186

µ 26.28 0.187

34.98 −0.000889

127.5 0.0000077

−120.7 0.000000013

The rate of convergence is slower for populations of higher r,andmuch

slower for complex roots of (6.2.2).