Demography

Abstract

Demography is the analysis of population, including both techniques and substance. It is applied most often to human populations, and includes the gathering of data, the construction of models, interpretation of population changes, policy recommendations. The data used by demographers are partly cross-sectional in the form of censuses and sample surveys, partly flow data consisting of time series of births and deaths. Models that express the relation between the flow series of births, deaths and migration on the one side and the cross sections on the other are a main tradition of demography, running through the work of Lotka, Leslie and many others. Interpretation includes tracing causes of changes, and assessing their future consequences. Policy recommendations aim at lowering birth rates in countries of rapid growth, and raising it in countries below replacement.

Demography is the analysis of population, including both techniques and substance. It is applied most often to human populations, and includes the gathering of data, the construction of models, interpretation of population changes, policy recommendations. The data used by demographers are partly cross-sectional in the form of censuses and sample surveys, partly flow data consisting of time series of births and deaths. Models that express the relation between the flow series of births, deaths and migration on the one side and the cross sections on the other are a main tradition of demography, running through the work of Lotka, Leslie and many others. Interpretation includes tracing causes of changes, and assessing their future consequences. Policy recommendations aim at lowering birth rates in countries of rapid growth, and raising it in countries below replacement.

Demography on the whole belongs to social science, though part of it (some of the analysis of mortality, was well as questions of fecundity) falls within the field of biology. It draws from and overlaps with other social sciences, especially sociology and economics. Reliability engineers deal with the life and demise of equipment and face problems analogous to those of human mortality; the mathematics they use is in many respects the same as that of demography, with superficial differences of notation. Epidemiology deals with some of the same problems as demography, though it too has developed a different tradition of exposition and notation. In so far as demographers collect and interpret data they necessarily borrow the techniques of statistics, including probability and stochastic processes. Ecology, a branch of biology, makes use of demographic techniques and results (Sauvy 1954; Scudo 1984).

For the more numerically minded demographer the subject begins with John Graunt (1662), who published his Observations on the Bills of Mortality more than three centuries ago. Yet Graunt’s close study of the primitive death certificates of his day is not often referred to by working demographers now More often mentioned as a predecessor is Lotka, who applied the renewal equation, developed in mathematical physics about the beginning of the century, to the renewal of a human population. The part of his long career, with publications dating all the way from 1907 to 1948, that is most remembered was devoted to developing the consequences of that one equation. Those who see demography as emphasizing forecasting are likely to think of Cannan (1895), Bowley (1924), and Whelpton (1936), whose components method was put into convenient matrix form by Leslie (1945).

Data

The most fundamental of all demographic data is the Census. Census taking is by no means novel. Ten cases of enumeration of the whole people (the earliest under Moses (Exodus xxxviii) and the last under Ezra (Ezra ii, 64)) are reported in the Old Testament, and one very famous occasion by the Romans is reported in the New Testament (Luke ii, 2). For a time the Romans took a census every five years. Classical Chinese literature contains innumerable references to counts in one part of the country or another. Premodern censuses were taken primarily to establish obligations on payment of taxes and military service, and they were correspondingly subject to evasion.

Modern censuses have been associated with the national state, as were other kinds of statistics: the word statistics itself itself reminds us of the association. Among the early acts of the revolutionary government of France was legislation providing for collection of data, including the taking of censuses. This was anticipated by Sweden, whose series of censuses goes back to the 18th century. Depending on the definition, the first census of modern times was taken in Sweden, Canada, or Virginia.

The association of the census with the national state has been seen in many of the new countries established after World War II. Countries seized on censuses to legitimate their nationhood, just as did France two centuries ago.

What characterizes modern censuses is (a) that they take place periodically, (b) that the enumeration is name-by-name, (c) that they seek to include all the persons belonging in a given area, (d) that they ask questions on age, sex, activity, etc., some of the questions often being on a sample, (e) that they recognize the problem of error and omission.

Geographic preparation is a major part of the effort to attain accuracy and completeness. The country is divided into enumeration areas on maps, with boundaries indentifiable on the ground, and each such area is assigned to an enumerator to be held responsible for its coverage. This principle of a division, first on maps and then on the ground, into an exhaustive set of non- overlapping areas is the essential principle of censustaking. It was apparently Morris H. Hansen who first applied the fact that every such area need not be covered for surveys (for example, population surveys taken between censuses). In area sampling the identification of individuals with a point on the map constitutes an implicit listing; the sample is specified in such a way that all individuals, including those unknown to the sample designers, have a prescribed chance of inclusion.

Equally valued with censuses for demographic calculations, though much less widely available, are accurate vital statistics. Partial records of births and deaths are to be found in many places and in many historical epochs, but effectively complete registration was largely a 19th-century innovation; the Swedish series going back to the 1700s is virtually unique.

Only under modern conditions do citizens need passports and other identification that depend on birth registration, and the citizen co-operation that is a condition for good vital statistics comes only with modernization. Censuses have now been taken in most countries of the world, but accurate vital statistics, covering current births and deaths, are to be had for countries including no more than about 30 per cent of the world’s population. If we had to wait for the general awakening of public statistical consciousness that is required for a complete vital statistics system the population problem of the world would be solved before it could be measured.

Comparison

One of the oldest demographic problems is the simple comparison of mortality level as between two populations, or one population between two points of time. US advances in longevity were slow and uncertain in the 1950s and 1960s; it is a statistically delicate question whether mortality was lower in the United States in 1980 than it was in 1950 and by how much. A first attempt to answer it is comparison of crude rates, and we find that for white females the crude rate, deaths D divided by population P, was the same in both years. But this is not a pure comparison of mortality. If the populations number p¹_x and p²_x at age x, and their death rates are μ¹_x and μ²_x, then the comparison of crude rates is D¹/P¹ versus D²/P² or

$$ \frac{\sum {p}_x^1{\mu}_x^1}{\sum {p}_x^1}\;\mathrm{versus}\ \frac{\sum {p}_x^2{\mu}_x^2}{\sum {p}_x^2} $$

whose sole advantage as an index is that it may be calculated from the number of deaths and the number of exposed population at each of the two times, without any breakdown of the data by age. The p¹_x and the p²_x confound the comparison, and if they are systematically different then the comparison of crude rates tells little about relative mortality. In particular one population having a larger proportion of old people than the other badly distorts the comparison.

To meet this difficulty, basic information was collected by age as far back as the 18th century in Sweden. To eliminate the different age weighting of the two populations from the comparison, it is common to use the directly standardized index with fixed p¹_x.

$$ \frac{\sum {p}_x^1{\mu}_x^2}{\sum {p}_x^1{\mu}_x^1}, $$

whose analogue in economics is the base-weighted aggregative price index. (The μs are similar to prices, and the ps to quantities used.) This formula gives for white females 6.5 in 1950 and 4.1 in 1980, a major difference from the crude rates, that were unchanged. Other formulas, for instance that obtained by replacing p¹_x by p²_x, give different answers, and the choice among them is difficult to make on logical grounds. Thus the famous price index number problem carries over to demographic comparison, though not the difficulty that rising or falling prices by themselves affect the amounts purchased. (Kitagawa and Hauser 1973).

Demography has a resource not available to the study of price changes: the life table model. If the death rates of this year, including all ages at which anyone is living, can be seen as the successive ages in the life of an individual, then the individual subject to those rates would have a certain expectation of life. No real person will have such an expectation, but the model provides what is the most common means of interpreting a current pattern of mortality.

If μ(x) is the age-specific death rate at age μ to μ + dx then the chance of a baby living to age α is $l\left(\alpha\ \right)=\exp \left[-{\int}_0^a\mu (x)\mathrm{d}x\right]$, this being the solution of the differential equation defining the death rate,

$$ {\mu}_x=\frac{1}{l(x)}\frac{\mathrm{d}l(x)}{\mathrm{d}x}. $$

The expectation of life at age x is then

$$ {e}_x^0=\frac{\int_x^{\omega }l(a)\mathrm{d}a}{l_x}, $$

where ω is the highest age to which anyone lives. US white females showed e⁰_x equal to 72 years in 1950, 79 years in 1980. Elandt-Johnson et al. (1980) apply the expectations comparisons in clinical follow-up studies.

Mortality and its Changes

To classify mortality according to the single parameter of life expectancy captures a good part of the variation in age incidence from one population to another, but not all. For instance a population may have high infant mortality and low mortality in later life while, in another, mortality may be low for infants and high in later life, with the two populations having the same overall expectations. Two dimensions differentiate among patterns much better than one. Coale and Demeny (1983) show four families of model tables. The United Nations (1985) show a Latin American, a Chilean, a South Asian, and a Far Eastern pattern. A particularly effective set of tables is due to William Brass (1971), who regresses the l_x column of a given table on that of a standard table, after both have been transformed by logits, and the regression of the one on the other turns out to be close to a straight line. Given the standard table, Brass’s is a two constant system.

As mortality improves along the path that we have seen in advanced countries over the past generation the age specific rates at all ages go down, most being reduced by half in each generation. Because the span of life has changed little, a given per cent fall in age specific rates now has a much smaller effect on life expectancy than an equal percentage fall 50 years ago. In fact a drop of 1 per cent now in all age specific rates causes a rise of only about 0.10 to 0.15 per cent in life expectancy; 50 years ago it caused a rise of 0.30 per cent. This number, the derivative of the life expectancy with respect to the age specific rates, has been called H:

$$ H=\frac{\int_x^{\omega }l(x)\ln \left[l(x)\right]\ \mathrm{d}x}{\int_x^{\omega }l(x)\mathrm{d}x}. $$

On the present course it is becoming smaller and smaller, as we proceed to a time when everyone dies at about the same age. Demetrius (1974) has carried this analysis further.

Note that the progress against mortality need not go this route. We can imagine a slowing of the ageing process by which the l_x curve moves out to the right, rather than merely moving up to a horizontal line with a fixed boundary on the right. A slowing of the ageing process by 50 per cent would mean an extension of average life not of 50H per cent, or about 7 years, but a full doubling of life expectancy. One of the questions that physicians, pension officials and demographers ask one another is which of the two courses will be taken in the future by mortality improvements, especially at the oldest ages which count more and more for this as mortality under age 70 becomes small.

The life table with one exit–death–can be extended to several exits, representing the several causes of death, and on from these to several increments, taking place not only at age zero, but at arbitrary ages.

Fertility Measures

Children are born to women only at a restricted range of ages, so comparison for births are a somewhat different problem than for deaths. If we divide the number of births B by the whole population P to obtain a crude birth rate then we are subject to the irrelevant variation of the young and old people in the denominator; it is better to divide by the number of women in the childbearing ages. Some further small gain in precision of comparison is obtained by working with age-specific rates, the births ₅B₁₅ to women 15–19 years at last birthday divided by the number of women ₅P₁₅ in the population of that age at mid-period, and similarly for the six other ages under 50. With single years of age, if B_x is average girl births during a year to women aged x, then the rates are f_x = B_x/P_x, and these over the childbearing ages may be added to obtain the Gross Reproduction Rate (GRR):

$$ \mathrm{GRR}=\sum {f}_x=\sum \left({B}_x/{P}_x\right) $$

including boy and girl births in the numerator B_x gives the total fertility rate (TFR), approximately double the GRR.

For measuring the natural increase of a population survivorship l_x is incorporated in the formula to give the net reproduction ratio (NRR), R₀ = Σl_xf_x, where now f_x is again the girl birth rate. R₀ is the number of girl children expected to be born to a girl child on a particular set of rates of birth and death. By virtue of that definition it is the ratio of the number of persons in one generation to the number in the preceding, taken in abstraction from any irregularities in the age distribution, and disregarding the length of time over which one generation is replaced by another.

Estimating the effect of abortion and contraception raises some further issues. Since one abortion of a conception leading to a live birth reduces the number of live births in the population by 1, it might be thought that 1000 abortions would reduce the number of births by 1000, but this is not so. If the probability of a conception that leads to a live birth in a given month is p, and the sterile period following conception is s months, then there will be a birth on the average every (1/p) + s months. If the sterile period following conception when abortion occurs is α, then there will be an abortion on the average every (1/p) + α months. Hence the number of abortions that avoid one birth is

$$ \frac{\frac{1}{p}+s}{\frac{1}{p}+\alpha }. $$

This can come out above 2 if no contraception is used, but is only slightly over 1 if the abortion is a backstop to more or less efficient contraception (Potter 1972).

Momentum

With an NRR equal to unity a population will just replace itself over the long run; population in this condition of bare replacement will ultimately become stationary. If it drops to bare replacement after a history of rapid increase, then because of its young age distribution, with many women in the childbearing ages, it will continue to increase for one or two generations, until it attains a number that may be as much as 70 per cent higher than when its NRR dropped to unity, a phenomenon called population momentum. If the population has been increasing uniformly over a considerable period of time the ratio of the ultimate population to that at the onset of bare replacement is simply expressed as

$$ \mathrm{Ratio}=\left(\frac{b}{r}\right)\frac{e_0^0}{\mu}\left(\frac{R_0-1}{R_0}\right), $$

where b is the birth rate, r the rate of natural increase, μ the mean age in the stationary population (Keyfitz 1985, p. 156).

This result is exact under the assumptions stated, and is one of numerous inferences from stable population theory.

Pension Cost as a Function of the Rate of Increase

Stable population theory also tells us the relation between certain variables when other circumstances are held constant. A pension of unity to all members of the population over age 65 will cost those aged 20 to 64 at last birthday the annual premium

$$ p(r)=\frac{\int_{65}^{\omega }{e}^{- rx}l(x)\mathrm{d}x}{\int_{20}^{65}{e}^{- rx}l(x)\mathrm{d}x}, $$

and this cost can be approximated as

$$ p(r)={p}_0\;\exp \left[r\left({m}_1-{m}_2\right)-\frac{r^2}{2}\left({\sigma}_1^2-{\sigma}_2^2\right)\right] $$

where m₁ and m₂ are the mean ages of the 20–64 and the 65 and over respectively, and σ²₁ and σ²₂ their variances. Since m₁ < m₂ and the term in r² is small, the premium is necessarily a decreasing function of the rate of increase of the population (Keyfitz 1985, p. 106).

Kinship

If the population can be assumed to be stable and some assumptions of continuity are made then kin relations become determinate. Knowing the age specific rates of birth and death, and supposing the various demographic events are independent, we can find exact expressions for the probability that a person aged α has a living mother, living grandmother, as well as the expected aunts, cousins etc. (Goodman et al. 1974).

Lotka (1931) gives the probability that a girl aged α has a living mother. His answer is obtained in two steps: (1) with the condition that at the girl’s birth the mother was x years old the probability is simple: l_{x +α}/l_x ; (2) removal of the condition by averaging over all ages of mothers at childbearing gives, on the stable assumption:

$$ {M}_1\left(\alpha \right){\int}_{\alpha}^{\beta}\frac{l_{x+\alpha }}{l_x}{e}^{- rx}l(x)f(x) dx. $$

From this it follows that the probability of a living grandmother is

$$ {M}_2\left(\alpha \right){\int}_{\alpha}^{\beta }{M}_1\left(x+\alpha \right){e}^{- rx}l(x)f(x)\mathrm{d}x, $$

and so on. Other expressions are obtainable for sisters, aunts, cousins (Le Bras 1973). Noreen Goldman (1978) has applied the formulas for younger sisters and older sisters, equating the ratio as given in theory to the ratio observed in a sample of a population, and solving for the intrinsic rate. Her method for finding the rate of increase has the advantage of requiring no knowledge of age on the part of respondents.

Notice that the preceding formulas, like others based on stable theory, are essentially comparative statics, and give a result very different in meaning from one based on observed age data. They answer questions like ‘What happens to the premium for old age pensions in the stable condition with the given parameters?’ The formula for M₁(α) gives the fraction of girls aged α who have a living mother given the life table and birth rates, and disregarding all else. The observed fraction of girls aged α who have a living mother takes account of all other elements affecting the real population.

Inferring Vital Rates by Indirect Methods

In the absence of complete vital statistics much effort has had to be devoted to inferring vital rates from censuses, and one early method was based on the stable age distribution. In a fast growing population the preponderance of numbers is shifted to the younger ages, and this fact makes it possible to infer the rate of growth from examination of the age distribution. If birth rates and death rates are constant and the population closed, then as we saw the number of persons aged x per current birth is e^−rxl_x. If the l_x can be taken from a reference or model table, and a census gives c_x persons at age x and c_y persons at age y > x, then the equation

$$ \frac{c_x}{c_y}=\frac{e^{- rx}{l}_x}{{e^{- ry}}_y} $$

can be solved to find

$$ r=\frac{1}{y-x}\ln \left(\frac{c_x/{l}_x}{c_y/{l}_y}\right) $$

(Bourgeois-Pichat 1966).

The matter is not that simple in practice, since growth is irregular, censuses are subject to error, and one does not know what life table to apply. In general any pair of ages combined with a life table gives an estimate, and one can try to use ages that are less vulnerable to reporting error. The theory is readily extended to populations in which mortality is falling (Coale 1963). More recently methods have been developed that do not depend on the assumption of stability (Brass 1975; Preston and Coale 1982; Coale 1984; United Nations 1985).

Periods and Cohorts

Demography moves back and forth between consideration of a population existing at a given moment or period of time, and a cohort that is a group of individuals followed from birth or some other event. Comparison of mortality can be made between periods or between cohorts. The same formulas apply to both, for standardization as well as the life table. In fact, the usual life table is referred to as a synthetic cohort: it treats a set of age-specific rates referring to a particular moment as though they were applicable to individuals and extended over time. Cohorts are in a sense more real than periods, but being only calculations after the last individual member has died, they can never be up-to-date (Ryder 1964).

The cohort – a number of individuals observed from a given starting point – is a demographic unit appropriate to fields other than mortality; one can assemble death and divorce statistics from individual data by following the marriages occurring in a particular year to the time where the couple divorces or one member of the couple dies (Henry 1957a, b; Pressat 1961).

Multi-Dimensional Demography

The above questions and techniques have been largely concerned with counts of people, and in disregard of characteristics other than age and sex. But for many purposes we need to examine marital status, or labour force status, or place of residence within a country. We need to take account of the transitions of individuals, for instance between the states of married and single, between school and labour force, etc. Combinations of sequences are numerous in any of these matters, and in order to bring the number down the Markov assumption is usually introduced, whereby the probability of a person moving into the several states in each period depends only on the last previous state the person was in, and not at all on the path by which he or she arrived at that state.

It fortunately happens that the ordinary life table can be extended to the multi-dimensional case, with matrix analogues for the most common formulas. If μ_ij(x) is the rate at which people aged x are moving from the jth to the ith state, then the probability of going from the jth state at the beginning of a period to the ith state at the end of the period, is the ijth element of P_x, where M_x, is the matrix of the μs

$$ {\mathbf{P}}_x={\left(1+{\mathbf{M}}_x/2\right)}^{-1}\ \left(1-{\mathbf{M}}_x/2\right) $$

and so on through all the usual life table formulas (Rogers 1975). This way of handling the arithmetic has the convenience of simple formulas, easily implemented on a computer. An equivalent method that dispenses with matrices is due to Robert Schoen (1975) and Leo A. Goodman (1961, 1969).

Mixtures and Heterogeneity

Everything said so far supposes that the several members of the population in any one category have the same probabilities – of dying, of giving birth, or of migrating – an assumption that cannot be correct. The usual demographic models recognize age, sex, and a few other sources of variation among individuals; they make no allowance for statistically unobserved heterogeneity.

Yet we know that some people are in vigorous condition, while others of the same age, sex, etc. are moribund. Among a group of individuals who are not all in the same condition the less vigorous die sooner, leaving the remainder with more favourable mortality than an unselected group would average. This process goes on through life, and the observed death rates, arising as they do from a population selected by differential mortality towards the more robust, are too low to represent an individual who at the start is of average frailty.

If we each had a mark on us indicating our degree of frailty then in estimating our own chances of survival we would use the experience of a group with the same mark as ourselves. We could avoid the unsatisfactory procedure of applying to ourselves the experience of a collection of people among whom average robustness was steadily increasing. Not knowing our condition, we must choose one of two ways of expressing our ignorance and deriving a probability. We can take ourselves as average at the start, and then we must accept that we will have an expectation lower than the published tables show; or else we can take ourselves as the average of the surviving population throughout the whole course, in which case we are supposing that we as individuals are steadily improving in robustness (Vaupel and Yashin 1985).

The recognition of heterogeneity can explain some of the crossovers that are otherwise puzzling, for instance the fact that in the United States blacks show higher mortality than whites at ages up to 70, and beyond that they have lower mortality. Selection by the higher mortality at the younger ages is a way of explaining this; another explanation is defective data.

The curious paradoxes that arise through mixed distributions have been explored by reliability engineers (Mann et al. 1974). In application to demography, the familiar rise in the proportion of divorces with duration of marriage, reaching a peak at five to ten years, could be due to married couples being of two kinds – one group that has a low and constant probability of divorce, not changing with duration of marriage, and another group that has a steadily rising probability with duration of marriage. First the overall rate, following this latter group, rises, but as these divorce and so drop out of the exposed population the rate falls towards that of the lower group. Neither of the component groups has a peak in rates at any time, yet the mix shows such a peak and subsequent fall because those more prone are eliminated from the exposed population.

The point is particularly important in respect of pregnancy. If we follow a group of fertile women through time, and note when they become pregnant we have the same problem of a changing mix, as those that are more fertile drop out, leaving less and less fertile ones behind. That may be a matter of fecundity, the biological ability to have a child, or it may be skill in using birth control, and both of these vary among women (Potter 1972; Potter and Parker 1964). It was Gini (1924) who showed that only in the first month can the rate refer to an unselected group. Goodman (1961) provides methods for the corresponding problem in migration, that had earlier been introduced by Blumen et al. (1955).

The order of magnitude of the effect can be very large in respect of susceptibility to pregnancy, or in respect of divorce; for mortality it cannot be so large because the event in question can only occur once to each member of the population. If a population were divided into three groups, one with an expectation of life of 65 years, one with 73 years, and one with 80 years, then the expected lifetime for the mixed population would be about one year greater than the expected lifetime of the middle group, that we take as the prospect for an individual who is initially of average frailty. About the only general statement that can be made is that expectation as given in published life tables is anything up to one year higher than the initially average person can expect to live.

Forecasting

The activity of demographers that is most often noticed by the public is forecasting: estimating the future population of a country or other area (Brass 1974). The forecasting problem is essentially unsolvable, just as is extrapolating from previous stops to estimate where the wheel will next stop in a casino. There is somewhat more continuity in the demographic than in the casino serials, but to know in advance the major turning points, especially in births, is at least for the present impossible.

While the public may think of demography as principally concerned with the forecasting of population, yet the literature of demography does not give a great deal of attention to this subject, and the best-known demographers have in recent years turned their attention to other problems; explaining the past is providing difficult enough, and until one can say why past events have occurred there is not much prospect of foretelling future ones.

Demographic forecasts are bound to be subject to especially large error for two reasons: they concern the long-term future, and they are self-contained within the narrow set of demographic variables. Forecasting a year ahead would be extremely useful in regard to the unemployment rate or housing construction, not to mention the stock market, while for a year ahead the population is so close to that of today that the forecast is of no interest. Demographic forecasts are typically for 10, 25, and more years into the future.

Since what the population will do depends on many variables outside of demography, it has often been suggested that demographers take into account these non-demographic variables. But that would require knowing future attitudes towards work and the family, and other matters more resistant to forecasting than population itself. Beyond that problem, even if we knew all of these independent variables for the next 25 years, the nature of the functional relation between them and population is beyond present knowledge.

During the present century death rates have been decreasing in most parts of the world, and extrapolations have been moderately successful. The increase in life expectancy has typically been almost three years per decade in developed countries, and has often reached five years per decade elsewhere.

What affects forecasts most is the birth rates assumed, and here is where the biggest failures have been. There was no way to forecast the postwar rise in births shown by developed countries, and equally little understood is the decline of births in the 1960s, and why birth rates continue to be so low. It was during the prosperity of the 1960s that the birth rates started to fall, and during the depressed late 1970s and 1980s that they fall even lower, so we do not know whether births depend directly or inversely on income. A theory that has strong logic on its side, that of Richard Easterlin (1980), by which the small cohort finds itself prosperous and produces a large cohort in its turn has not so far seemed precise enough either in timing or in quantity of the effect to be used by practising forecasters.

Migration is even more difficult for those few countries in which it is substantial. We do not know the amount of immigration into the United States now, let alone the amount that will occur during the 21st century.

Once the future mortality, fertility and migration are assumed, the forecast is easily made. In the usual projection by age and sex one starts with females, sets up a vertical vector P₀ consisting of the numbers at each age, premultiplies that vector by a matrix whose first row is the age-specific fertility rates for girl children, and whose subdiagonal is the survivorship rates. if M is the matrix with fertility rates m_1j in its first row, and surviviorships m_{j+1, j}, j = 1,…, n – 1, in the sub-diagonal, then the age vector at time 1 is

$$ {\mathbf{P}}_1=\mathbf{M}{\mathbf{P}}_0. $$

and at time t is

$$ {\mathbf{P}}_t={\mathbf{M}}^t\ {\mathbf{P}}_0, $$

if the rates are assumed constant over time (Leslie 1945), If the rates change, the matrix being M₁ in the first period, M₂ in the second period, then

$$ {\mathbf{P}}_t={\mathbf{M}}_t,\dots, {\mathbf{M}}_2{\mathbf{M}}_1{\mathbf{P}}_0. $$

The assumed migrants would be added in each time period.

Experiments have shown that extrapolating birth and death rates does better, though not by much, than supposing that birth and death rates will continue unchanged at their level at the jumping-off point.

Even simpler than projecting with fixed birth rates is using fixed absolute numbers of births into the future. This method, that might be called instant stationarity, also gives results not much inferior to the usual assumption of changing future rates. A rationale for the fixed absolute numbers is provided by the Easterlin hypothesis, by which birth rates are higher for small parental cohorts.

Forecasting Error

Badly needed are probability methods. Some have been proposed (e.g. Pollard 1966) for ex ante computation of error, but so far these have had little influence on forecasting practice.

Ex post the problem is simpler. The assessment of earlier projections, leading to an estimate of the intrinsic error of the process, demands first of all a metric that will be comparable between different points of time for a given population, and between large and small populations, growing and declining populations, long- and short-range projections. Such a metric has been found to be the difference between the forecast rate of growth of the population in question and the (subsequently known) realized rate:

$$ \mathrm{Error}=\sqrt{\sum {\left[{\left(\frac{{\widehat{p}}_t}{{\widehat{p}}_0}\right)}^{1/t}-{\left(\frac{p_t}{p_0}\right)}^{1/t}\right]}^2} $$

where ${\widehat{p}}_t$ is the forecast population at time t, p_t the realized population, t being the time interval between when the forecast was made and the date to which the projection applies. For some 300 forecasts applying to 15 developed countries, error as so measured turns out to be about 0.003, or 0.3 percentage points.

To interpret this result, consider an estimate for the United States of 268,000,000 for the year 2000, when we are now (1984) at 236,000,000. This is a projected annual rate of increase of 0.8 per cent, so odds are 2 to 1 of the true outcome falling within the range 0.8 ± 0.3 or 0.5 – 1.1; one can bet 2 to 1 odds that the population in the year 2000 will be in the range (236)(1.005)¹⁶ to (236)(1.011)¹⁶, or 256 to 281 millions. This supposes that the present forecast is no better and no worse than the 300 similar forecasts on which this estimate of error has been based (Keyfitz 1981).

Exponential and Logistic Growth

There may have been situations in the past when populations were growing uniformly and it was possible to make some kind of credible prediction by supposing constant increase for the future. By definition of the rate of increase,

$$ r=\frac{1}{P_t}\frac{\mathrm{d}{P}_t}{\mathrm{d}t}, $$

so that the population at time t is

$$ {P}_t={P}_0{e}^{rt}. $$

It is hard to think of cases where such exponential growth persists over more than a very short period.

The patent defect of the exponential that nothing can grow uniformly for very long suggested a further factor in the differential equation to produce the curve known as logistic:

$$ {r}_t=\frac{1}{P_t}\left(1-\frac{P_t}{A}\right)\frac{\mathrm{d}{p}_t}{\mathrm{d}t}, $$

where A is the asymptotic population at which growth stops. The rate of increase r_t is no longer constant, and the solution of the equation is

$$ {P}_t=\frac{A}{1+b{e}^{- ct}}, $$

where b and c are constants.

The logistic seemed to have merit when births were slowing and total population growth tapering off. It reached the height of its popularity when the Americas could be seen as empty, and as they filled would move towards a population ceiling. Unfortunately the ceiling keeps changing with changing society and technology.

One might take a different line in support of the logistic: not the logic of the model but goodness of fit to the historical series. That does not work either; an inverse tangent, or a cumulative normal fit just as well as a logistic, and an impossible curve, a hyperbola moving off to infinity in a near future, is not much inferior to any of the three in fitting the past (Cohen 1984).

For animal populations the story is different; real niches filling under constant conditions do appear, and in ecological studies the logistic has on occasion provided a useful representation of the process.

Difficult Matters

Some demographic results are perfectly explicable: when Romania suddenly banned abortion, the birth rate, which presumably included a proportion of unwanted children, rose sharply; after the public adapted to the ban the birth rate settled back to where it was. Others remain puzzles even after much study: why does West Germany stay at the lowest recorded fertility of all time, much lower than neighbouring France? The effectiveness of determined population policy in East Germany is partly explained by the large expenditure on it, but not Hungary’s extremely low fertility after the war, and the subsequent partial recovery.

Similarly, there is much to explain in poor countries; some countries have seen their fertility fall drastically, while others remain high. Cultural inheritance is apparently a factor. Islamic populations have higher fertility than non-Islamic that are otherwise similar; thus for 1980–85 the UN (1985) estimates Pakistan’s TFR (Total Fertility Rate) at 5.84 and Bangladesh’s at 6.15 against India’s 4.41. What feature of Islam is the cause of the differential remains to be discovered.

A key question in contemporary demography is whether and how quickly the countries whose death rates have fallen can follow through with declines in birth rates that will bring them to zero growth. No one knows for sure whether the fall of deaths – for instance and especially of infant mortality – in and by itself brings about a decline of births; the literature contains proofs that it does and proofs that it does not. Even if we knew for sure that the demographic transition to a stationary condition will take place everywhere, forecasting for the years ahead is impeded by our ignorance of how quickly it will come. And professional opinion on the effectiveness of family planning programmes is by no means unanimous.

See Also

• Demographic Transition

• Historical Demography

• Life Tables

• Stable Population Theory