Keyfitz N., Caswell H. Applied Mathematical Demography
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xx Contents
2 The Life Table 29
2.1 DefinitionofLifeTableFunctions............. 29
2.2 Life Tables Based on Data ................. 31
2.3 FurtherSmallCorrections ................. 39
2.4 PeriodandCohortTables ................. 40
2.5 FinancialCalculations ................... 41
2.6 Cause-DeletedTablesandMultipleDecrement...... 42
2.7 The Life Table as a Unifying Technique in
Demography......................... 46
3 The Matrix Model Framework 48
3.1 TheLeslieMatrix...................... 49
3.2 Projection:TheSimplestFormofAnalysis........ 51
3.3 TheLeslieMatrixandtheLifeTable ........... 55
3.4 Assumptions:ProjectionVersusForecasting ....... 62
3.5 State Variables and Alternatives to
Age-Classification...................... 64
3.6 Age as a State Variable: When Does it Fail? ....... 65
3.7 TheLifeCycleGraph.................... 67
3.8 TheMatrixModel...................... 69
4 Mortality Comparisons; The Male-Female Ratio 71
4.1 TheMultiplicityofIndexNumbers ............ 73
4.2 Should We Index Death Rates or Survivorships? ..... 76
4.3 Effect on
o
e
0
of Change in µ(x)............... 78
4.4 EverybodyDiesPrematurely................ 88
5 Fixed Regime of Mortality and Fertility 93
5.1 StableTheory ........................ 94
5.2 PopulationGrowthEstimatedfromOneCensus..... 97
5.3 MeanAgeintheStablePopulation ............ 103
5.4 Rate of Increase from the Fraction Under 25 ....... 107
5.5 Birth Rate and Rate of Increase Estimated for a
StablePopulation...................... 109
5.6 SeveralWaysofUsingtheAgeDistribution ....... 111
5.7 Sensitivity Analysis ..................... 118
5.8 PromotionWithinOrganizations ............. 121
6 Birth and Population Increase from the Life Table 127
6.1 TheCharacteristicEquation................ 128
6.2 AVariantFormoftheCharacteristicEquation ..... 133
6.3 Perturbation Analysis of the Intrinsic Rate ........ 135
6.4 Arbitrary Pattern of Birth Rate Decline ......... 138
6.5 DropinBirthsRequiredtoOffsetaDropinDeaths. . . 141
6.6 Moments of Dying and Living Populations ........ 144
Contents xxi
7 Birth and Population Increase from Matrix Population
Models 148
7.1 SolutionoftheProjectionEquation............ 148
7.2 TheStrongErgodicTheorem ............... 155
7.3 TransientDynamicsandConvergence........... 165
7.4 Computation of Eigenvalues and Eigenvectors ...... 175
7.5 MathematicalFormulations ................ 176
8 Reproductive Value from the Life Table 183
8.1 ConceptofReproductiveValue .............. 184
8.2 Ultimate Effect of Small Out-Migration Occurring in a
GivenYear.......................... 190
8.3 Effect of Continuing Birth Control and Sterilization . . . 191
8.4 LargeChangeinRegime .................. 193
8.5 EmigrationasaPolicyAppliedYearAfterYear..... 194
8.6 TheMomentumofPopulationGrowth .......... 196
8.7 EliminatingHeartDisease ................. 199
8.8 TheStableEquivalent ................... 200
8.9 Reproductive Value as a Contribution to
FutureBirths ........................ 206
9 Reproductive Value from Matrix Models 208
9.1 ReproductiveValueasanEigenvector........... 208
9.2 TheStableEquivalentPopulation............. 213
10 Understanding Population Characteristics 216
10.1 AccountingforAgeDistribution.............. 217
10.2 MoreWomenThanMen .................. 223
10.3 AgeatMarriage....................... 225
10.4 TheForeign-BornandInternalMigrants ......... 236
10.5 HumanStocksandFlows.................. 238
10.6 TheDemographyofOrganizations............. 241
11 Markov Chains for Individual Life Histories 245
11.1 A = T + F .......................... 245
11.2 Lifetime Event Probabilities ................ 251
11.3 Age-Specific Traits From Stage-Specific Models ..... 253
12 Projection and Forecasting 268
12.1 UnavoidableandImpossible ................ 268
12.2 TheTechniqueofProjection................ 272
12.3 ApplicationsofProjection ................. 278
12.4 TheSearchforConstancies................. 282
12.5 FeaturesofForecastingandForecastingError ...... 287
12.6 The Components of Forecasting Error Ex Ante ..... 292
xxii Contents
12.7 Ex Post EvaluationofPointEstimates .......... 296
12.8 A Division of Labor ..................... 299
12.9 IntervalEstimatesasCurrentlyProvided......... 301
13 Perturbation Analysis of Matrix Models 303
13.1 Eigenvalue Sensitivity .................... 305
13.2 ElasticityAnalysis...................... 317
13.3 Sensitivities of Eigenvectors ................ 321
13.4 LifeTableResponseExperiments ............. 325
13.5 Fixed Designs ........................ 327
13.6 RandomDesignsandVarianceDecomposition ...... 329
13.7 Regression Designs ..................... 332
13.8 ProspectiveandRetrospectiveAnalyses.......... 333
14 Some Types of Instability 335
14.1 Absolute Change in Mortality the Same at All Ages . . . 335
14.2 ProportionalChangeinMortality............. 338
14.3 Changing Birth Rates .................... 341
14.4 Announced Period Birth Rate Too High ......... 343
14.5 BackwardPopulationProjection.............. 348
14.6 The Time to Stability .................... 352
14.7 RetirementPensions .................... 358
14.8 The Demography of Educational Organization ...... 361
14.9 TwoLevelsofStudentsandTeachers ........... 363
14.10 Mobility in an Unstable Population ............ 365
14.11TheEasterlinEffect..................... 366
15 The Demographic Theory of Kinship 370
15.1 Probability of Living Ancestors .............. 372
15.2 Descendants ......................... 379
15.3 SistersandAunts ...................... 381
15.4 MeanandVarianceofAges ................ 386
15.5 Changing Rates of Birth and Death ............ 387
15.6 Sensitivity Analysis ..................... 388
15.7 Deriving Rates from Genealogies .............. 394
15.8 Incest Taboo and Rate of Increase ............. 396
15.9 BiasImposedbyAgeDifference .............. 397
16 Microdemography 399
16.1 BirthsAvertedbyContraception ............. 399
16.2 Measurement of Fecundity ................. 410
16.3 Three-Child Families Constitute a Population
Explosion .......................... 425
16.4 AFamily-BuildingtoAvoidExtinction.......... 427
16.5 Sex Preference and the Birth Rate ............. 430
Contents xxiii
16.6 Parental Control over SexofChildren ........... 433
16.7 Mean Family Size from Order-of-Birth Distribution ... 439
16.8Parity Progression ...................... 440
16.9 Lower Mortality and Increase ............... 441
17 TheMulti-StateModel 444
17.1 Single Decrement and Increment-Decrement ....... 446
17.2 The
Kolmogorov Equation ................. 449
17.3 Expected Time in the SeveralStates ........... 452
17.4Projection .......................... 455
17.5 Transition Versus InstantaneousProbabilityof
Moving ............................ 456
17.6 StablePopulation ...................... 459
18 Family Demography 461
18.1 Definitions
.......................... 462
18.2 Kinship ............................ 464
18.3 The Life Cycle ........................ 469
18.4 Household Size Distribution ................ 471
18.5 Economic, Political,andBiologicalTheory........ 474
18.6 Family Policy ........................ 475
19 Heterogeneityand Selection inPopulat
ion Analysis 477
19.1 Conditioning and the Interpretation of Statistical
Data ............................. 479
19.2 Heterogeneity and Selection ................ 481
19.3 ApplicationtoMortality .................. 482
19.4 Modelling Heterogeneity .................. 486
19.5 Experimentation ....................... 492
20 Epilogue: How Do We Know theFactsof Demography? 49
4
20.1 Proportions ofOldPeople ................. 496
20.2Promotion in Organizations ................ 501
20.3 No Model, No Understanding ............... 503
20.4 Too ManyModels ...................... 504
20.5 Development and Population Increase ........... 504
20.6 How Nature Covers Her Tracks .............. 508
20.7 The Psychologyof Research ................ 510
Bibliography 513
Index 551
A Word About Notation
Throughout the book, matrices are denoted by boldface capital letters (A)
and their elements by lower-case letters (a
ij
). Vectors are denoted by bold-
face lower-case letters (w) and their elements by lower-case letters (w
i
).
Vectors are column vectors by default. Occasionally, it is necessary to refer
to the elements of subscripted matrices; in such cases, e.g., a
(k)
ij
is the (i, j)
element of A
k
. The transpose of A is A
T
. The complex conjugate trans-
pose of A is A
∗
. The determinant of A is denoted det(A)or|A|. Unless
otherwise specified, all logarithms are natural logarithms.
It has not been possible to develop a completely uniform set of symbols.
Population size, in numbers of individuals, is usually denoted by N (as a
scalar) or n (as a vector). The age distribution as a density function is
usually denoted p(a), so that, for example, the total population would be
given by N =
∞
0
p(a)da. Double subscripts (e.g.,
5
q
x
) are used to denote
quantities defined by an age (the right subscript) and an age interval (the
left subscript). The example just given is the probability that an individual
aged x will die in the next 5-year age interval; i.e., between ages x and x+5
(see p. 30). There are exceptions, but symbols are always defined in the
context of their use.
Equations are numbered within sections, so (12.2.3) or “equation 12.2.3”
refers to the third numbered equation in Section 12.2.
1
Introduction: Population Without Age
Age is a characteristic variable of population analysis; much of the theory
of this volume will be concerned with the effect of age distribution, either
fixed or changing. But age is not the only characteristic variable, and part
of this book will deal with methods capable of describing the effects of
other variables besides, or in addition to, age. Abstraction is necessary in
demographic as in other theory; is it possible to abstract even from age
and still obtain results of value?
To represent a population as a number varying in time, and in disregard
even of its age composition, is like treating the earth as a point in space—
though too abstract for most purposes, it is useful for some. Partly as an
introduction to the rest of the book, the nine sections of this chapter take
up questions to which answers can be obtained without reference to age.
These topics vary in depth and subtlety. The first deals only with the
use of logarithms in base 2, that is, the number of doublings to go from
one total to a larger total. Such logarithms show dramatically how close
birth rates must have been to death rates throughout human history. Next
comes an aspect of bisexual reproduction, and some of the difficulty it
causes; consideration of one sex at a time is a powerful simplifying device
for understanding demographic mechanisms. Another question concerns a
population made up of a number of subpopulations, each increasing at
a fixed rate; we will see at what increasing rate the total population is
expanding. Attention will be given also to the migration problem, from
which the one-sex population recognizing age will emerge as a special case;
that a proposition concerning age follows directly from one on migration
illustrates the advantage of a formal treatment.
2 1. Introduction: Population Without Age
1.1 Definitions of Rate of Increase
Population change is expressed in terms of rate of increase, a notion differ-
ent in an important respect from the rate of speed of a physical object such
as an automobile. If a country stands at 1,000,000 population at the begin-
ning of the year, and at 1,020,000 at the end of the year, then by analogy
with the automobile its rate would be 20,000 persons per year. To become
a demographic rate this has to be divided by the population, say at the
start of the year; the population is growing at a rate of 20,000/1,000,000
or 0.02 per year. More commonly it is said to be growing at 2 percent per
year, or 20 per thousand initial population per year.
The two kinds of rate are readily distinguished in symbols. The analogue
of the physical rate x
in terms of population at time t and t +1, N
t
and
N
t+1
,is
x
= N
t+1
− N
t
or N
t+1
= N
t
+ x
,
while the demographic rate x is
x =
N
t+1
− N
t
N
t
or N
t+1
= N
t
(1 + x).
Just as x
is similar to a rate of velocity, so x is similar to a rate of compound
interest on a loan. Both are conveniently expressed in terms of a short time
period, rather than 1 year, and in the limit as the time period goes to zero
they become derivatives:
x
=
dN(t)
dt
and x =
1
N(t)
dN(t)
dt
.
The rates x
and x give very different results if projected into the future.
A population continuing to grow by 20,000 persons per year would have
grown by 40,000 persons at the end of 2 years, and by 60,000 persons at
the end of 3 years. One increasing at a rate of 0.02 would be in the ratio
1.02 at the end of 1 year, in the ratio 1.02
2
at the end of 2 years, and so on.
The latter is called geometric increase, while the fixed increment is called
arithmetic increase.
Thus the very notion of a rate of increase, as it is defined in the study of
population, seems to imply geometric increase. In fact, it implies nothing
for the future, being merely descriptive of the present and the past; rates
need not be positive and can be zero or negative. Moreover, being positive
today does not preclude being negative next year: ups and downs are the
most familiar feature of the population record.
The future enters demographic work in a special way—as a means of
understanding the present. To say that the population is increasing at
2 percent per year is to say that, if the rate continued, the total would
amount to 1.02
10
=1.219 times the present population in 10 years, and
1.02
1000
= 398,000,000 times the present population in 1000 years. If this
1.2. Doubling Time and Half-Life 3
tells anything, it says that the rate cannot continue, a statement about the
future of a kind that will be studied in Chapter 8.
Still, the conditional growth rate whose hypothetical continuance is help-
ing us to understand the present can be defined in many ways. In particular,
in the example above, it could be an increase of 20,000 persons per year
just as well as of 0.02 per year. Arithmetical and geometrical projections
are equally easy to make; why should the latter be preferred?
The reason for preferring the model of geometric increase is simple:
constancy of the elements of growth translates into geometric increase. If
successive groups of women coming to maturity have children at the same
ages, and if deaths likewise take place at the same ages, and if in- and
out-migration patterns do not change, then the population will increase
(or decrease) geometrically. Any fixed set of rates that continues over time,
whether defined in terms of individuals, families, or age groups, ultimately
results in increase at a constant ratio. That any fixed pattern of childbear-
ing, along with a fixed age schedule of mortality, implies long-run geometric
increase is what makes this kind of increase central in demography. (The
stationary population and geometric decrease can be seen as special cases,
in which the ratios are unity and less than unity, respectively.)
1.2 Doubling Time and Half-Life
The expression for geometric increase gives as the projection to time t,
N
t
= N
0
(1 + x)
t
, (1.2.1)
where x is now the fraction of increase per unit of time. The unit of time
maybeamonth,ayear,oradecade,solongasx and t are expressed
in the same unit. Any one of N
0
, N
t
, x,ort may be ascertained if the
other three are given. When the quantity x is negative, the population is
decreasing; and when t is negative, the formula projects backward in time.
If x is the increase per year as a decimal fraction, then 100x is the increase
as percentage and 1000x the increase per thousand population.
We are told that the population of a certain country is increasing at
100x percent per year, and would like to determine by mental arithmetic
the population to which this rate of increase would lead if it persisted over
a long interval. Translating the rate into doubling time is a convenience in
grasping it demographically as well as arithmetically.
Since annual compounding looks easier to handle than compounding by
any other period, we will try it first. We will see that it leads to an un-
necessarily involved expression, whose complication we will then seek to
remove.
If at the end of 1 year the population is 1 + x times as great as it was
at the beginning of the year, at the end of 2 years (1 + x)
2
,..., and at the
4 1. Introduction: Population Without Age
end of n years (1 + x)
n
times,thenthedoublingtimeisthevalueofn that
satisfies the relation
(1 + x)
n
=2. (1.2.2)
To solve for n we take natural logarithms and divide both sides by log(1+x):
n =
log 2
log(1 + x)
=
0.693
log(1 + x)
. (1.2.3)
The Taylor series for the natural logarithm log(1 + x)is
log(1 + x)=x −
x
2
2
+
x
3
3
−···. (1.2.4)
Entering the series for log(1 + x) in (1.2.3) gives for doubling time
n =
0.693
log(1 + x)
=
0.693
x − (x
2
/2) + (x
3
/3) −···
. (1.2.5)
The right-hand expression can be simplified by disregarding terms beyond
the first in the denominator. We could write n =0.693/x, but for values of x
between 0 and 0.04, which include the great majority of human populations,
arithmetic experiment with (1.2.5) when x is compounded annually shows
that it is, on the whole, slightly more precise to write n =0.70/x or, in
terms of x expressed as a percentage (i.e., 100x), to write n =70/100x.
The expression
n =
70
100x
(1.2.6)
is a simple and accurate approximation to (1.2.3). It tells us that a popula-
tion increasing at 1 percent doubles in 70 years, and so forth. For Ecuador
in 1965, 100x was estimated at 3.2 percent, so doubling time would be
n =70/3.2 = 22 years. The 1965 population was 5,109,000, and if it dou-
bled in 22 years the 1987 population would be 10,218,000; this compares
well with the more exact (5,109,000)(1.032)
22
=10,216,000. Similarly a
further doubling in the next 22 years would give a population in the year
2009 of 20,436,000, as compared with (5,109,000)(1.032)
44
=20,429,000.
The formula n =70/100x is plainly good enough for such hypothetical
calculations.
The Period of Compounding. All this is based on a definition of x by which
the ratio of the population at the end of the year to that at the beginning of
theyearis1+x. Interest calculations are said to be compounded annually
on such a definition. A different definition of rate of increase simplifies this
problem and others and without any approximation gets rid of the awkward
series in the denominator of (1.2.5).
Suppose that x is compounded j timesperyear;thenattheendof1year
the population will have grown in the ratio (1 + x/j)
j
. Is there a value j