Keyfitz N., Caswell H. Applied Mathematical Demography

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5.6. Several Ways of Using the Age Distribution 115

Further work remains to be done on robustness, however, in the face of

inaccurate data and inappropriate assumptions.

5.6.1 Incomplete Population and Deaths

William Brass has suggested (personal communication) that one can make

use of an incomplete census, along with incomplete death statistics, to

ascertain r, the rate of increase of a population, provided only that the

omissions at the several ages are nearly constant and the population is more

or less stable. Although such assumptions may seem strong, the calculation

itself checks them. Several values of r may be produced by the calculation;

and if they are not in agreement, we know that the assumptions are not

met and reject the method for the case in question.

The Brass suggestion requires an expression for c(x), the fraction of

persons at age x, in terms of the population and deaths cumulated from

age x to the end of life,

∞

and

∞

, respectively. An integration by parts

in the expression for deaths over age x as a fraction of total population will

establish the required relation. The deaths from age x to the end of life as

a fraction of total population are as follows:

∞



−ra

l(a)µ(a) da

= −b



−ra

dl(a)

= −b



−ra

l(a)|

+ r



−ra

l(a) da



= be

−rx

l(x) −

∞

= c(x) −

∞

Transposing gives the required relation of c(x) to the cumulative population

and deaths:

c(x)=

∞

. (5.6.4)

This is interpretable as a ﬂow equation, in which the entrants c(x)into

the age group x and over are equal to the net natural increase r

∞

/N of

that age group plus the deaths

∞

/N , all in terms of ratios to the total

population.

Equation (5.6.4) may be used to calculate r, given the age distribution

and the deaths, but it involves a gratuitous error insofar as the deaths suﬀer

from more (or less) incompleteness than the population. Brass avoids this

by multiplying and dividing the term on the right,

∞

/N ,byd = D/N,

116 5. Fixed Regime of Mortality and Fertility

the death rate, and then treating d as unknown:

c(x)=

∞

. (5.6.5)

Now the fact that the population and deaths are incomplete by some un-

known fraction at all ages makes no diﬀerence. Any two ages, say x and y,

provide a pair of equations solvable for r and d. With more than two ages

one can ﬁt by least squares, and use the magnitude of departures from

the ﬁtted r and d to check the assumptions of stability and of constant

proportional completeness in population and deaths.

The logic of populations permits modiﬁcation of this without further

algebra—for instance, if the population and deaths under age 5 are thought

to be especially uncertain; in that case one would reinterpret all the ele-

ments of (5.6.5) on the basis of the population and deaths at 5 years of

age and over: c(x) would be the ratio of those aged x to the population 5

and over, D would be the deaths for ages 5 and over, and N would be the

population aged 5 and over. To ﬁnd the rate of increase of a long past time

one might consider only the part of the population 60 and over, interpret

c(x) as the ratio of those aged x to the number aged 60 and over, and D

and N as deaths and population, respectively, over 60, and put x equal to

60 and to 70 to obtain two equations.

Taking Mexico, 1966, females as an example, we have as the equation for

x = 0, using all ages,

900

22,029

= r + d,

and, for x =60

81.8

22,029

= r

1161

22,029

+ d

58,923

198,893

Solving these together gives r =0.0345, d =0.0064, and b = d+r =0.0409.

Now suppose that we disregard the ﬁrst 60 years of life, and use x = 60;

then

81.8

1161

= r + d,

and, for x = 70, still considering population 60 and over,

43.4

1161

= r

448

1161

+ d

41,148

58,923

which solve to r =0.0378 and d =0.0326. In this formulation the death

rate d does not have the usual meaning, since its denominator is population

60 and over, but r is an estimate of the usual rate of increase.

5.6. Several Ways of Using the Age Distribution 117

5.6.2 Estimates from Two Censuses

When two censuses, say 10 years apart, are available, much more can

be done; not only can assumptions be weakened, but also increased

redundancy permits a better estimate of error.

If the population is closed, and whether or not it is stable, survivorship

rates can be found by comparing the number in a cohort as given by the

two censuses. For instance, the chance of surviving 10 years for those aged

10 to 14 is the ratio of the number 20 to 24 at the later census to that 10

to 14 at the earlier one; if

(t)

is the population aged 10 to 14 at time t,

then

(t+10)

(t)

Having a set of survivorships for pairs of ages 10 years apart, one could

interpolate to construct the life table. Unfortunately the countries less de-

veloped statistically, for which reliable mortality registrations are lacking,

are not likely to have accurate censuses, and inaccuracies in the census

reporting of ages produce a very irregular table. Therefore we must seek a

way to use the survivorships that is less sensitive to their irregularities.

One such way is to match the survivorships to model life tables. Each of

the 10-year survivorships would permit picking a member of the series of

life tables. With perfect data and a series of model tables appropriate to the

underlying mortality of a closed population, the identical life table would

be picked by the several ages. In practice, however, diﬀerent life tables will

be picked, and if they do not diﬀer too greatly they can be averaged.

Once the life table is available, chosen by the observations rather than

imported more or less arbitrarily as with a single census, one can use either

census for estimating r by the methods outlined above. Again the degree

of agreement among the several estimates of r will inform us as to how far

to trust the results.

A further and obvious resource for calculating r is the intercensus in-

crease. In some cases the accuracy of enumeration is inadequate to provide

survivorships between the censuses for the several cohorts, but is suﬃcient

to show how fast the total population is increasing; r would be obtained

from the ratio of census totals as r =

log

(t+10)

(t)

.Thevalueofr

so obtained, taken in conjunction with either age distribution, can provide

a life table. Arriaga (1968) used an observed age distribution along with

a value of r in his study of mortality in Latin America; from (5.1.1) he

obtained l(x)ase

c(x)/c(0).

118 5. Fixed Regime of Mortality and Fertility

5.7 Sensitivity Analysis

The stable model was used in Sections 5.3 to 5.6 to estimate rates of

increase of populations lacking in vital statistics. It can also provide an-

swers to substantive questions concerned, not with data, but with the way

populations—or at least their parameters—behave.

5.7.1 Mean Age as a Function of Rate of Increase

We have prepared the way for one example with (5.3.2), showing how the

mean age of a population is related to its rate of increase:

¯x ≈

− σ

Insofar as this applies, it tells what a change in r will do to the population

mean age ¯x. If in our population r were higher by ∆r,then¯x would be lower

by σ

∆r, on the understanding that the life table remained unchanged.

Essentially the same fact can be expressed by the derivative:

d¯x

≈−σ

, (5.7.1)

and derivatives are useful in cases where the device by which (5.3.2) was

established is not applicable.

For Canada, 1968, σ

, the variance of women’s ages in the life table

population, was 573.7. Hence we can say from (5.7.1) that a fall of 0.010

in r would raise the mean age by 5.737 years, the life table remaining

unchanged. This touches genuine causation, although it does not say what

would happen in the real world, where many other changes would occur

at the same time. Like any other causal law, it is capable only of telling

what will happen conditionally, provided that its action is not overlaid, or

reversed altogether, by other changes such as migration.

One added qualiﬁcation is that the eﬀect does not immediately follow

the cause in time. In this section and others presenting sensitivity analysis

it is two ultimate stable conditions that are compared, a device known in

economics as comparative statics.

5.7.2 Pension Cost

Consider a pension scheme in which those over exact age 65 are paid a

pension equal to salary, and salary is unity and the same for all individuals

and at all ages between 20 and 65. If N is the number of persons of all

ages in the population, the total payments by the fund each year will be



−rx

l(x) dx, and the total receipts of the fund each year will be



−rx

l(x) dx,whereg is the premium as a fraction of salary. For

5.7. Sensitivity Analysis 119

the fund to remain in balance these two quantities must be equal, so we

have for the premium

g =



−rx

l(x) dx



−rx

l(x) dx

, (5.7.2)

still as a fraction of the uniform annual salary of individuals aged 20 to 65.

Adapting this to salaries and pensions that vary with age, to salaries that

vary at each given age, to cases where increasing numbers of persons are

unable to work as they become older, and to other practical circumstances

complicates the formula somewhat but entails no new principle.

A formula enables us to see how demographic conditions aﬀect pension

arrangements. For the scheme whose premium is given by (5.7.2) we might

ask what diﬀerence it makes if mortality improves beyond age 65, if the

age at pension is 60 rather than 65, and if the population increases more

slowly or becomes stationary. The stable model permits pencil-and-paper

experiments to answer such questions; let us consider the last one brieﬂy.

To see what diﬀerence it makes if r changes we calculate dg/dr from

(5.7.2). Taking logarithms and using the fact that

d log g

gives immediately, with ﬁxed life table,

d log g

= −



−rx

l(x) dx



−rx

l(x) dx



−rx

l(x) dx



−rx

l(x) dx

The ﬁrst term on the right is minus the mean age of the pensioners, say M

and the second is the mean age of the contributors, say m. Hence

= −(M − m). (5.7.3)

According to (5.7.3) a small ﬁnite change ∆r in r ultimately causes a

relative change ∆g/g in the premium, i.e.,

∆g

≈−(M − m)∆r. (5.7.4)

In words, the fractional change in the premium equals minus the diﬀer-

ence of mean ages in the pensioned and working groups times the absolute

change in rate of increase. If the mean M is about 70 years and m about

40 years, M − m is about 30, and we conclude that a decrease in r by 1

percentage point, say from 0.020 to 0.010, increases the fraction of salary to

be paid as premium by −(30)(−0.01) = 0.30. Practically without data we

120 5. Fixed Regime of Mortality and Fertility

have found that a 30 percent increase in premiums for a contributory non-

reserve pension scheme is associated with a fall in the population growth

rate of 0.01. We could sharpen the 30 percent by using observed numbers

for M − m, but 30 years is close enough to suggest how much such ﬁnan-

cial calculations depend on the rate of increase of populations. As Thomas

Espenshade pointed out to me, the lower birth rate as r becomes smaller

releases women into the labor force and so, if jobs are available, helps to

pay the premium for the increased fraction of aged persons. With some

additional data the eﬀect of this factor and others may be calculated.

5.7.3 Fraction of Old People

The objective of the preceding pages can be attained more precisely by

employing derivatives and then treating the Taylor expansion as a diﬀer-

ential equation. Suppose we would like to know how the fraction of the

population that is old depends on the rate of increase, and assume that

death rates do not change. The fraction f(r) over age 65 is

f(r)=



−ra

l(a) da



−ra

l(a) da

. (5.7.5)

Taking logarithms and diﬀerentiating both sides gives

f(r)

df (r)

= κ

− k

, (5.7.6)

where κ

is the mean age of the entire population and k

of the part of it

65 years of age and over. Integrating (5.7.6) and taking exponentials gives

f(r)=f

(κ

−k

where f

is the life table proportion 65 and over.

The result is obtainable from the term in r of the diﬀerence of two cu-

mulant generating functions, one for the distribution of those 65 and over,

the other for all ages:

log f(r)=logf



−rk

−

+ ···



−



−rκ

−

+ ···



where the k’s are the cumulants of the persons 65 and over and the κ’s of

everyone. Taking exponentials up to the term in r

gives

f(r)=f

exp



r(κ

− k

) −

(κ

− k

)



. (5.7.7)

5.8. Promotion Within Organizations 121

Table 5.7. First and second approximations to the percent over 65 years of age

from (5.7.7), along with the total as taken from Table 5.1

Approximation from (4.7.7)

As totaled

from

r To term in r To term in r

Table 4.1

−0.01 19.9 19.6 19.46

0 13.8 13.8 13.80

0.01 9.5 9.3 9.31

0.02 6.5 6.1 6.02

0.03 4.5 3.8 3.74

0.04 3.1 2.2 2.25

Experimenting with Table 5.1, and using round numbers for the means

= 72, κ

= 35, so that κ

− k

= −37, and for the variances k

= 30,

= 450, so that κ

− k

= 420, we have the fraction 65 and over as

0.138e

−37r

to a ﬁrst approximation, and as

f(r)=0.138e

−37r−210r

to a second approximation. Table 5.7 shows that the second approximation

at least is very close. Such discrepancies as exist between the last column

and the preceding column are due to the crude estimates of the ﬁrst and

second cumulants and to the omission of later cumulants.

It bears repeating that (5.7.7) gives only the ultimate diﬀerence between

the original situation and the one with incremented r; it says nothing about

the path by which the second is reached from the ﬁrst. When the birth rate

declines with unchanged l(x), it requires more than one generation under

the new conditions for the age distribution to stabilize. The projection

techniques of Chapter 12 are capable of showing the actual path of change in

a quantity such as our premium g when birth rates change in any speciﬁed

manner. The results are inevitably more complicated than those of stable

theory.

5.8 The Degree to Which Promotion Within

Organizations Depends on Population Increase

In a stationary population the progress of individuals through whatever

hierarchy they belong to, that is, their social and economic mobility, will on

the average be slower than in an increasing population. To show the eﬀect

in its pure form we will disregard the diﬀerences in merit, luck, inheritance,

and inﬂuence among individuals whereby some move up faster and others

more slowly; our interest is in average rates of promotion. In other words, we

seek the rate of promotion insofar as it is aﬀected by one rate of population

increase as opposed to another (Waugh 1971, Bartholomew 1967).

122 5. Fixed Regime of Mortality and Fertility

The concept of promotion stands out most clearly in a hierarchical or-

ganization that observes some degree of seniority in promotion, and that

permits a certain fraction of its employees to be foremen and above, or

colonels and above, or associate professors and above. Most organizations

do not have exact ratios for the number of those above to those below these

ranks, but the nature of their work and budgets is such that de facto limits

exist on the numbers at higher ranks. We will take as ﬁxed and given the

ratio k of those above the given rank to those below, and we want to ﬁnd

the age x at which a person on the average reaches the status expressed by

k; we seek x as a function of r, the rate of increase. Suppose as in (5.1.1)

that the population is stable, so that the fraction of individuals between

ages a and a+da is be

−ra

l(a) da. Then the fraction of the population above

the rank in question is the integral of this from x to β, and the fraction

of the population below is the integral from α to x,whereα is the age of

recruitment and β the age of compulsory retirement.

The condition is described by an equation in which k is identiﬁed with

aratioofintegrals:

k =



−ra

l(a) da



−ra

l(a) da

. (5.8.1)

Once the life table, along with k, α,andβ, is known, (5.8.1) implicitly

gives the age of promotion x as a function of r.

Equation (5.8.1) cannot be solved in closed form for x, but the theory

of implicit functions enables us to approximate the derivative of x with

respect to r. If the ratio of integrals in (5.8.1) is called u/v, then (5.8.1)

is the same as φ = u − kv = 0. With a few steps of elementary calculus

(details are given in Keyﬁtz 1973) we ﬁnd

= −

∂φ/∂r

∂φ/∂x

= −

1+k

⎡

⎢

⎣



−ra

l(a) da

−rx

l(x)

⎤

⎥

⎦

(M − m), (5.8.2)

where M is the mean age of the group between x and β,andm of the

group between α and x. Evidently dx/dr is negative; as r increases, the

age of promotion declines.

To obtain an idea of how much it declines, suppose the age of recruitment

α to be 20 and the age of retirement β to be 65. Then M − m cannot be

very diﬀerent from half of the interval from 20 to 65, that is, 22.5. Consider

a rank somewhat above the middle, where the person holding it has one

employee above him for each two below; that is, k =

. With a small

positive rate of increase r the expression in the square brackets of (5.8.2)

5.8. Promotion Within Organizations 123

Table 5.8. Age x of passing through position k for k =1uptok =0.2, for

r =0.00 to r =0.04, based on male life table for the United States, 1968

Value of Value of r

k 0.00 0.01 0.02 0.03 0.04

1.0 40.86 38.55 36.39 34.43 32.71

0.8 43.26 40.91 38.64 36.54 34.66

0.6 46.31 43.97 41.64 39.41 37.35

0.4 50.32 48.14 45.86 43.56 41.33

0.2 55.93 54.27 52.36 50.27 48.08

will be about 15. Then

≈−

(15)(22.5) = −225.

This means that, when r falls from 0.02 to stationarity, the age of promotion

to a rank two-thirds of the way up the hierarchy rises by −225 ×(−0.02) =

4.5years.

For a more exact result we can solve (5.8.1) numerically for x with r =

0.02 and r =0.00. This requires interpolation in the 5-year intervals for

which such data are given, including solution of a cubic equation (Keyﬁtz

1973). Some of the results are shown as Table 5.8 for l(x)fromtheUnited

States, 1968, male life table. With k =0.6 we ﬁnd x =46.31 for r =0.00

and x =41.64 for r =0.02, a diﬀerence of 4.67 years, or very nearly what

the derivative showed in the preceding paragraph. With k = 1 the diﬀerence

from r =0.04 to r =0.00 is 40.86 − 32.71 = 8.15 years.

The same calculation has been made on the male life table of Sweden,

1783–87, with

=33.6 years, representing about the highest observed

mortality for which a life table is available. Such high mortality ought at

least to have the advantage of speeding the promotion of the survivors, and

indeed, at a given r, they show promotion to middle positions about 2 to

3 years younger than the ages in the United States, 1968, table. Thus the

possible eﬀect of high mortality in favoring promotion is 2 to 3 years; the

possible eﬀect of rapid population growth is 8 years. The reason for this is

that the range of recorded human population growth is from −

percent

through stationary to 4 percent per year. The range of human mortality

averaged over ages 20 to 65 is much narrower, say

to 2 percent (Table

5.9) or one-third as much.

A Simpliﬁcation. The theory can be made simpler with only a small sac-

riﬁce of realism by supposing constant mortality between ages α and β

(McFarland, personal communication). Mortality rises over the age inter-

val, but we are encouraged to suppose constancy by the small eﬀect that

mortality seems to have in any case. If the death rate is ﬁxed at µ,the

124 5. Fixed Regime of Mortality and Fertility

Table 5.9. Average mortality µ for males between ages 20 and 65, where

−45µ

= l

, compared with intrinsic rate on female-dominant model

1000µ 1000r

Country (males) (females)

Austria, 1966–68 8.1 7.8

Canada, 1966–68 7.3 7.4

Ceylon, 1967 6.9 24.6

France, 1967 8.2 8.3

Malaysia (West), 1966 9.5 30.9

Mauritius, 1966 11.3 30.5

Mexico, 1966 10.1 34.6

Norway, 1967 5.7 10.4

Spain, 1967 6.5 10.1

Sweden, 1783–87 20.2 1.4

United States, 1968 9.2 5.9

Venezuela, 1965 8.8 37.6

probability of surviving to age a is l(a)=e

−µa

, and the equation becomes

k =



−ra

−µa



−ra

−µa

Now the right-hand side of this expression lends itself to integrations in

closed form, and we have

k =

−(r+µ)x

− e

−(r+µ)β

−(r+µ)α

− e

−(r+µ)x

which may be solved for x to give

x = −

r + µ

log



−(r+µ)α

+ e

−(r+µ)β

k +1



, (5.8.3)

an expression that is exactly the same function of µ as it is of r. Hence dx/dr

is equal to dx/dµ; population increase and high mortality help promotion

to exactly the same degree.

5.8.1 The Chain Letter Principle

The analogy of the promotion process to a chain letter is easily drawn. In

a chain letter (say of four names in each transmittal) the recipient (Ego)

sends the person at the top of the list he has received a sum, say 1 dollar,

crosses this name oﬀ the list, adds his own name at the bottom, reproduces

the letter, and sends it on to four other persons. If four copies are sent on

at each stage, in the short time required for ﬁve successive letters to be

mailed and delivered Ego will receive 4

= 256 dollars. With 10 names he