Keyfitz N., Caswell H. Applied Mathematical Demography

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6.3. Perturbation Analysis of the Intrinsic Rate 135

6.3 Perturbation Analysis of the Intrinsic Rate

A main use of the foregoing theory is to compare populations that diﬀer

in some birth or death parameter, and note by how much they diﬀer in in-

trinsic rate. For the corresponding calculations for stage-structured matrix

models, see Section 13.1.

6.3.1 How the Intrinsic Rate Varies with the Moments

We are now in a position to ﬁnd how the intrinsic rate varies with the

several moments. To ﬁnd, for example, how r in (6.2.2) varies with σ

might ﬁrst think of solving explicitly for r and noting how σ

appears in

the solution. Since an explicit solution is impossible, we are fortunate that

it is not necessary: the theory of implicit functions will again provide the

derivative of r with respect to σ

.Ifσ

is subject to a small increment, say

∆σ

, and the corresponding change in r is ∆r, then, writing φ(r, σ

)for

the right-hand side of (6.2.2), we have in terms of partial derivatives

∆φ(r, σ

∂φ

∂r

∆r +

∂φ

∂σ

∆σ

Insofar as φ(r, σ

) is a constant (namely, log R

) for values of r satisfying

the characteristic equation, its change ∆φ(r, σ

) must be zero; therefore

∂φ

∂r

∆r +

∂φ

∂σ

∆σ

=0,

and solving for ∆r/∆σ

results in

∆r

∆σ

= −

(∂φ/∂σ

)

∂φ/∂r

. (6.3.1)

In the limit as the increment tends to zero this gives the derivative of r

with respect to σ

in terms of the two partials, with all other moments

constant.

The partials are readily calculated from (6.2.2) as

∂φ

∂r

= µ − rσ

−··· (6.3.2)

∂φ

∂σ

= −

(6.3.3)

Therefore

dσ

≈

2(µ − rσ

)

, (6.3.4)

if we truncate after the variance, equivalent to ﬁtting a normal curve to

the net maternity function l(a)m(a)/R

136 6. Birth and Population Increase from the Life Table

This tells us that if two increasing populations are identical in all mo-

ments, except that one has a larger variance in age of childbearing than

the other, the one with the larger variance will have the higher rate of

increase. If the ages of childbearing are more spread out, apparently the

gain through some children being born earlier more than oﬀsets the loss

through those born later. [What if R

is allowed to vary?]

Easier to understand is the relation between r and µ, the mean age of

childbearing. The same technique as before gives for the required derivative

dµ

−r

µ − rσ

. (6.3.5)

The term rσ

in the denominator is relatively small; therefore it seems

that for an increasing population dr/dµ is negative, and the larger µ is the

smaller r is, again with everything else the same. Larger µ implies slower

turnover.

At one time comparison of the United States and Canada showed a

contrast between larger families in Canada, and thus a higher R

,and

younger marriage and childbearing in the United States, and thus a smaller

µ. The smaller µ reversed the eﬀect of the higher R

, and the net outcome

was a higher r for the United States.

Finally, and most obviously, the intrinsic rate is positively related to R

Neglecting all but the ﬁrst two terms on the right-hand side of (6.2.2), we

have

= −

(∂ψ/∂R

)

∂ψ/∂r

(µ − rσ

)

If r is small, the intrinsic rate rises with R

as long as µ − rσ

is positive,

that is to say, always.

One kind of change can be compared with another and a set of equiva-

lencies found. What increase in R

exactly oﬀsets an increase of ∆µ in µ

and leaves the rate of increase unchanged? Should eﬀort, say in India, go

into raising the age of marriage or into disseminating birth control within

marriage. India should of course do both, but the question still remains

of where the marginal eﬀort should go. With R

=1.77, and µ = 25, if

raising the average age of childbearing by 3 years is easier than lowering

age-speciﬁc birth rates by 5.9 percent, the eﬀort should concentrate on age

at marriage. In general if a change from R

to R

∗

is to be equivalent to

one from µ to µ

∗

then if σ

does not change much, the equation

log R

∗

≈

log R

∗

6.3. Perturbation Analysis of the Intrinsic Rate 137

holds to a close approximation. But this is true only if we can be sure

that a higher rate of childbearing will not take place within the delayed

marriages, and that illegitimacy will not be substantially increased.

Such statements can be made for any of the cumulants. The ﬁrst three

derivatives from (6.2.2), along with their values for the case of R

=2,

µ = 27, and σ

= 40, so that r =0.02618, disregarding terms beyond

/2, are as follows:

(µ − rσ

)

=0.01927

dµ

−r

µ − rσ

= −0.001009 (6.3.6)

dσ

2(µ − rσ

)

=0.0000132.

Thus a small increase ∆R

increases r by 0.01927∆R

and so forth. In

proportions the derivatives are

dµ

dσ

: −r :

The inﬂuence of successive cumulants of the net maternity function alter-

nates in sign and decreases with the powers of r. As a matter of curiosity

we can extend the series, and ﬁnd the ratios

dκ

: ···= −

: −

: ···.

6.3.2 Change in Births at One Age

Other calculations of the relation of rate of increase to aspects of the net

maternity function depend on the form (6.1.2) of the characteristic equa-

tion. For example, we would like to know whether the use of contraception

by young women will have more eﬀect on the rate of increase than its use

by older women. Suppose that for the 1-year interval around x the value

of m(x) is changed to m(x)+∆m(x); our problem is to ﬁnd the value ∆r

by which this modiﬁes r.

The new value r +∆r is obtained from the characteristic equation 6.1.2

in the form



−(r+∆r)a

l(a)[m(a)+∆m(x)] da =1, (6.3.7)

wherewehavemodiﬁedm(a) by adding to it the quantity ∆m(x) for the

1yearx. [This unorthodox notation will not cause any trouble if we think

of ∆m(x) as a function of a, deﬁned to be zero everywhere except in the

interval x −

to x +

, where it is ∆m(x).] The left-hand side of (6.3.7)

138 6. Birth and Population Increase from the Life Table

consists of two additive parts, one an integral, and the other approximately

−(r+∆r)x

l(x)∆m(x). If for e

−(r+∆r)a

we write e

−ra

(1 − a∆r) within the

integral, two integrals emerge, of which



−ra

l(a)m(a) da equals unity

by our original equation (6.1.2), and



−ra

l(a)m(a) da (without denom-

inator) equals κ, the mean age of childbearing in the stable population. If

∆m(x) is small enough that the term involving ∆r∆m(x) may be ignored,

we obtain

∆r ≈

−rx

l(x)

∆m(x). (6.3.8)

Thus the intrinsic rate r is changed by e

−rx

l(x)/κ times the change

in the age-speciﬁc birth rate m(x). Note that the coeﬃcient of ∆m(x)is

proportional to the number of women in the stable age distribution. Thus

the relative eﬀect of changes in rates at the several ages is proportional to

the number of women at these ages. Such sensitivity analysis is a way of

exploiting models to obtain conditional statements of cause and eﬀect in

other instances where the result is less obvious.

An equivalent method of working out the eﬀect of change in a birth or

death rate involves the use of implicit functions. The interested reader can

work out other examples. Here we proceed to a more general case in which

not single ages but groups of ages are considered.

6.4 Arbitrary Pattern of Birth Rate Decline

As the birth rate declines in the United States or any other country, it falls

more rapidly at some ages than at others. For modernizing populations

the initial fall has been greatest at the oldest ages of childbearing, as has

been noted alike for the United States and for Taiwan. The ages that drop

are partially related to the means of population control used: sterilization

applies mostly to the older ages of childbearing; the pill, to younger ages

(at least while it is a novelty); abortion, to all ages. The intrauterine device

(IUD) is not much used by women until they have had a child, suggesting an

aggregate impact on ages intermediate between the pill and sterilization.

Possible cuts g(a) that might be taken out of the birth function m(a)

by these three methods of birth control are suggested in Figure 6.2. The

fertility function that remains is m(a) − kg(a), where k is a constant.

We will ﬁrst consider an arbitrary function g(a) and see the eﬀect of re-

moving it times some constant k from the birth function m(a). Our analysis

will concentrate on two special cases:

1. The eﬀect on r of deducting kg(a)fromm(a), where k is small, as

though one were trying to examine the direction and pace of the ﬁrst

move toward fertility reduction.

6.4. Arbitrary Pattern of Birth Rate Decline 139

Births per woman

Pill

IUD

Sterilization

Age a

m(a)

m(a) – kg (a)

kg(a)

Figure 6.2. Area kg(a) removed from net maternity function by the several

methods of birth control.

2. The ﬁnal condition of stationarity that will result from subtraction of

kg(a)fromm(a), where now k is large enough to produce stationarity.

Theﬁrstcaseissimplebecausek is small, and the second is simple

because the ﬁnal condition is stationary; intermediate values of k are more

diﬃcult to handle and will not be discussed here. From the ﬁrst case we

will know in what direction the system starts to move, and from the second

where it ends; by comparing these two we obtain at least a suggestion of

how the movement is modiﬁed as k goes through intermediate ﬁnite values.

6.4.1 Eﬀect of Small Arbitrary Change in Birth Function

Suppose an arbitrary function g(a), positive or negative, such as unity for 1

year of age and zero elsewhere, or equal to m(a), the birth function, or some

other variant. We consider deductions from m(a) equal to kg(a), that is to

say, kg(a) is a bite of arbitrary shape taken out of m(a). For purposes of this

part of the argument k is small enough that we can neglect second-order

terms like (∆r)

and k ∆r,where∆r is the diﬀerence that the deduction

of kg(a)fromm(a) makes to r.

To ﬁnd the value of ∆r that corresponds to kg(a) we have to solve the

equation



−(r+∆r)a

l(a)[m(a) − kg(a)] da =1, (6.4.1)

where r is deﬁned by (6.1.2). Expanding the exponential e

−∆ra

≈ 1 −∆ra

in (6.4.1), ignoring the term involving k ∆r, and solving for ∆r gives

∆r ≈−



−ra

l(a)g(a) da

k, (6.4.2)

where κ is the mean age of childbearing in the stable population.

140 6. Birth and Population Increase from the Life Table

If g(a)=m(a) for all a, (6.4.2) becomes

∆r = −

or k = −κ ∆r.

The question might be what fractional change in age-speciﬁc rates will

bring the rate of increase down by ∆r; the answer here given is that a

subtraction of 100κ ∆r percent at each age does it. A variety of results,

including (6.3.8), can be obtained as special cases of (6.4.2). The method

is still that of comparative statics: birth rates are taken as m(a) for one

population and as m(a) −kg(a) for another, and the two stable conditions

compared.

6.4.2 Amount of Change Needed for Drop to Bare

Replacement

For bare replacement the net reproduction rate, the expected number of

girl children by which a girl child born now will be replaced,



l(a)m(a) da,

must equal unity. If R

is to equal unity on the age-speciﬁc birth rates

m(a) − kg(a), we must have



l(a)[m(a) − kg(a)] da =1,

where the solution for k will tell just how much of the change shaped like

g(a) in the age pattern is required for the reduction to replacement. The

answer is evidently

k =

− 1



l(a)g(a) da

For g(a)=m(a)thisisk =(R

− 1)/R

, as accords with intuition. For

applications of this logic in epidemiology, where the reproductive rate of

concern is that of the pathogen, see p. 194.

6.4.3 Eﬀect of Uniformly Lower Death Rates

Suppose two populations of which one has the force of mortality µ(a)and

the other µ

∗

(a)=µ(a)+δ,whereδ is the same for all ages. Then the

population with µ(a)+δ will have survivorship

∗

(x)=exp

−



[µ(a)+δ] da

(

= e

−δx

l(x).

6.5. Drop in Births Required to Oﬀset a Drop in Deaths 141

Its characteristic equation will be



−r

∗

(a)m(a) da =1



−(r

∗

+δ)a

l(a)m(a) da =1.

But this is identical with the characteristic equation for l(a)m(a), as given

in (6.1.2), except that it has r

∗

+ δ rather than r. And, since the char-

acteristic equation has a unique real root, it follows that r

∗

+ δ = r,or

∗

= r − δ. In words, the solution with the incremented mortality is the

original solution less the increment of mortality.

From this it follows that age distribution is unaﬀected by a constant

increment of mortality. This is so because the age distribution is propor-

tional to e

−rx

l(x) dx; and if we change l(x)tol

∗

(x)=e

−δx

l(x), so that r

goes to r

∗

= r − δ,thene

−rx

l(x) dx goes to e

−(r−δ)x

−δx

l(x), that is, it is

unchanged on cancellation of e

−δx

That a change in death rates that is the same at all ages has no eﬀect

on age distribution is true more generally. Suppose an age distribution,

however irregular, and a decline of 0.001 in mortality rates at all ages.

Then exactly one person in a thousand who would have died on the former

regime now survives, age by age. This increases the number at every age

by exactly 0.001, that is, multiplies it by 1.001, and multiplying every age

by 1.001 can have no eﬀect on the age distribution (Coale 1956).

6.5 Drop in Births Required to Oﬀset a Drop

in Deaths

Let us ﬁnd the fraction by which existing fertility must decline so as to

oﬀset an absolute decrease in mortality equal to k at every separate age.

In symbols, suppose that for women who have reached age a the chance of

dying between ages a and a + da drops from µ(a) da to [µ(a) − k] da,and

this applies with the same k at all ages; by what uniform fraction f at all

ages would women have to lower their birth rates in order for the intrinsic

rate r to remain the same? If the drop in births takes place uniformly at all

ages, the old probability of having a child between ages a and a + da being

m(a) da, and the new probability being (1 − f)m(a) da, we can determine

the unknown f.

The eﬀect of lowering the death rate by the amount k at each age is equiv-

alent to increasing the survivorship l(a) by the factor e

, for by deﬁnition

142 6. Birth and Population Increase from the Life Table

the new l

∗

(a)mustbe,asbefore,

∗

(a)=exp

−



[µ(t) − k] dt

(

, (6.5.1)

and carrying out the integration gives l

∗

(a)=e

l(a). Alongside the old

characteristic equation 6.1.2 we have a new one:



−ra

l(a)(1 − f)m(a) da =1. (6.5.2)

We will replace e

by 1 + ka, permissible if k is small, take advantage of

(6.1.2) to cancel



−ra

l(a)m(a) da on the left of (6.5.2) against the 1 on

the right, and solve for f as

f =

kκ

1+kκ

, (6.5.3)

where κ is again the mean age of childbearing in the stable population.

Approximately, the fraction that the births would have to be reduced to

oﬀset a fall of k in the mortality of every age is kκ. For a population whose

κ is 27 years, an absolute fall of 0.001 in the death rate at each separate age

would require a fall of the fraction (0.001)(27)/[1 + (0.001)(27)] = 0.0263

in the birth rates at each age to oﬀset it.

A variant of the same question, left to the reader, asks how much must

be cut oﬀ the m(a) curve at its upper end to oﬀset a drop k in the age-

speciﬁc death rates. We go on to ﬁnd the drop in fertility that would oﬀset

the largest possible decrease in mortality—its complete elimination.

6.5.1 The Drop in Fertility That Would Oﬀset a Drop in

Mortality to Zero

The question has been raised how a drop in mortality to zero could be

oﬀset by a change in fertility at all ages of women. In the simplest form,

what value of f , the fractional fall in fertility at all ages, is associated with

an intrinsic rate r that would remain unchanged if mortality to the end of

reproduction were to drop to zero? Putting l(a) = 1 and changing m(a)to

m(a)(1 − f) in (6.1.2) provides the equation that f must satisfy:



−ra

m(a)(1 − f) da =1, (6.5.4)

while r is still subject to (6.1.2). The value of f from (6.5.4) is

f =1−



−ra

m(a) da

. (6.5.5)

A fertility drop by the fraction f at all ages would preserve r, the ultimate

rate of increase, against the hypothetical drop to zero of mortality up to

6.5. Drop in Births Required to Oﬀset a Drop in Deaths 143

age β. A very slightly diﬀerent problem is to ﬁnd the fraction f that would

preserve the generation ratio of increase R

, the net reproduction rate,

deﬁned as



l(a)m(a) da.

The required fraction is the solution in f of the equation



m(a)(1 − f) da = R

, (6.5.6)

f =1−



m(a) da

=1−

, (6.5.7)

the integral in the denominator being the gross reproduction rate G

(Coale

1973b).

Applying this to developed countries shows that a 3 or 4 percent decline

in births would oﬀset the drop to zero mortality. In 1967 the United States

was 1.26 and the R

was 1.21 (Keyﬁtz and Flieger 1971, p. 361). For less

developed countries where mortality is somewhat higher, a 5 to 15 percent

drop in births would suﬃce. No one need be concerned that a further fall in

the Mexican death rate, for example, will add seriously to the demographic

problem of that country. Mexico’s G

in 1966 was 3.17, its R

2.71, and

its birth rate 1000b =43.96 (Keyﬁtz and Flieger 1971, p. 345). A drop to

1000b =37.58 would oﬀset a transition to zero mortality.

The conclusion that further decline in mortality need not provoke any

substantial rise in the rate of population growth probably applies to the

world as a whole. Insofar as mortality up to the end of childbearing is

already very low, further diminution in it can have only a small eﬀect

on the rate of increase, and this small eﬀect is likely to be oﬀset for most

countries by the decline in fertility during the 1970s. The parts of the world

where this conclusion does not yet apply are mostly in tropical Africa.

6.5.2 Diseases of Infancy Versus Heart Disease: Their Eﬀects

on Population Increase

The death rate from cardiovascular renal diseases for females in the United

States in 1964 was 448 per 100,000 population, and that from certain dis-

eases of infancy (as deﬁned) was 26 (Preston, Keyﬁtz, and Schoen, 1972,

p. 770). Standardized, the numbers were 383 and 26, still a ratio of nearly

15 to 1. Eliminating certain diseases of infancy adds only 0.930 year to

, whereas eliminating CVR adds 17.068 years, or 18 times as much. All

indications are that CVR has at least 15 times the eﬀect on mortality of

certain diseases of infancy.

144 6. Birth and Population Increase from the Life Table

But no such proportion appears in the eﬀects on overall population in-

crease of eliminating these diseases. Consider ﬁrst the eﬀect on the net

reproduction rate. Using the life table calculated as though CVR were

eliminated raises the net reproduction rate by 0.002. Eliminating certain

diseases of infancy without altering the other causes raises the net reproduc-

tion rate by 0.018. Heart disease may be 15 times as prevalent as diseases

of infancy, but its long-term eﬀect on increase is only about one-ninth as

great.

6.6 Moments of the Dying Population in Terms

of Those of the Living, and Conversely

If we know the age distribution of the living population of the life table,

we ought to be able to ﬁnd the age distribution of the dying. Stated in life

table symbols, the problem is simply that of ﬁnding the distribution d

terms of l

, and the answer is d

= l

− l

x+1

Translating moments of the living into those of the dying is straight-

forward; given the moments of l(x), we proceed to ﬁnd the moments of

l(x)µ(x), where µ(x)=−

l(x)

dl(x)

. The result will enable us to ﬁnd the

mean age, variance, and other parameters of the living from those of the

dying, and vice versa.

One device is to relate the cumulant-generating function of the ages of the

dying to that of the ages of the living. For a general distribution function

f(x), deﬁne ψ(r) as the transform

ψ(r)=log





−rx

f(x) dx



and deﬁne the cumulants of f(x) as the coeﬃcients of the powers of r in

the expansion

ψ(r)=−rκ

−

+ ···. (6.6.1)

For the special case where f(x) is proportional to l(x), the transform is

log

⎡

⎢

⎣



−rx

l(x) dx



l(x) dx

⎤

⎥

⎦

which will be called ψ

(r); the κ

are the cumulants of the distribution

l(x)/



l(x) dx. In reference to our distribution of population the mean

age is µ

,thevarianceσ

, the skewness κ

, and the kurtosis κ

.For

deaths the mean is κ

,thevarianceκ

= σ

, the skewness κ

,and

the kurtosis κ