The analysis of linkages in demographic theory

Many seemingly different questions that arise in the analysis of population change can be phrased as the same technical question: How, within a given demographic model, would variable y change if the age- or time-specific function f were to change arbitrarily in shape and intensity? At present demography lacks the machinery to answer this question in analytical and general form. This paper suggests a method based on modern functional calculus for deriving closed-form expressions for the sensitivity of demographic variables to changes in input functions or schedules. It uses this “linkage method” to obtain closed-form expressions for the response of the intrinsic growth rate, birth rate, and age composition of a stable population to arbitrary marginal changes in its age patterns of fertility and mortality. It uses it also to obtain expressions for the transient response of the age composition of a nonstable population to time-varying changes in the birth sequence, and to age-specific fertility and mortality patterns that change over time. The problem of “bias” in period vital rates is also looked at.     

Age composition in the absence of mortality and in other odd circumstances

The age composition of populations experiencing no mortality and various kinds of fertility is derived. It is then shown that, if successive mortality schedules can be expressed as the sum of a component that varies with age and a component that varies with time, only the former has any effect on the age structure of the population. One implication is that a population in which mortality varies with time but not with age has the same composition as a population with no mortality at all.     

Calculation of age-specific fertility schedules from tabulations of parity in two censuses

The mathematics of stable populations recently has been generalized to cover populations with time-varying fertility and mortality by a modification incorporating the sum of age-varying growth rates in place of the fixed growth rate of a stable population. Equations that characterize nonstable populations apply to any cohort-like phenomenon with a measurable property that cumulates gains or losses through time. In particular, the equations fit the relation between a population's average parity at a given age and age-specific fertility rates previously experienced at lower ages. Techniques devised to derive an intercensal life table from single-year age distributions in two censuses are adapted to estimate accurate intercensal fertility schedules from distributions of parity by age of woman in two censuses. Birth-order specific fertility schedules are also estimated.     

Population dynamics in Germany: the role of immigration and population momentum

The effects of changes in rates of mortality, fertility, and migration depend not only on the age-specific patterns and levels of these rates, but on the age structure of the population. In order to remove the influences of the age structure and concentrate on the effects of the demographic rates themselves, a common practice is to analyze the influences of the rates for a standard age structure. This paper analyzes current and future population changes in Germany, using a stationary population equivalent model (SPE) that shows long-term effects of current fertility, mortality, and international migration patterns. Results indicate that the German population will eventually decline because of below replacement fertility, if net immigration does not counteract this decrease. This means, for instance, that the long-term stationary population levels for Germany will decrease by approximately 6.5 million during a decade in which current fertility, mortality, and international migration levels prevail. The paper also reports how various other assumptions for mortality, fertility, and international migration affect the SPE model for Germany.     

Immigration and the stable population model

This paper reports on work aimed at extending stable population theory to include immigration. Its central finding is that, as long as fertility is below replacement, a constant number and age distribution of immigrants (with fixed fertility and mortality schedules) lead to a stationary population. Neither the level of the net reproduction rate nor the size of the annual immigration affects this conclusion; a stationary population eventually emerges. How this stationary population is created is studied, as is the generational distribution of the constant annual stream of births and of the total population. It is also shown that immigrants and their early descendants may have fertility well above replacement (as long as later generations adopt and maintain fertility below replacement), and the outcome will still be a long-run stationary population.     

Compensating changes in fertility and mortality

This paper illustrates a method of studying changes in vital rate schedules which have no effect on the intrinsic rate of population growth. These changes are described as compensating changes in fertility and mortality. The analysis proceeds from the discrete perspective of Leslie matrices, wherein the central idea is to establish the set of all compensating changes by identifying that class of Leslie matrices which possess the same positive eigenvalue, λ₁. A root-squaring technique is adapted for the purpose of estimating λ₁. Finally, a variety of compensating fertility and mortality changes is illustrated using data from Japan.     

Estimation of local demographic variation in a flexible framework for population projections

A cohort component projection of local populations based on sex and single year of age offers great value for planning local services, but demands data beyond the detail available. Local fertility, mortality and migration schedules by age and sex must be estimated sensitively to local variation if the results are to be of greater value than simpler methods of projection. Two approaches are compared, using data for the recent past: (a) direct estimation of local area age-specific schedules of fertility, mortality and migration based on data available to the national statistical agency; (b) graduation of national schedules using only local area population estimates by age, total numbers of births, and total numbers of deaths; age-specific migration is indirectly estimated from successive population estimates. These two approaches are compared with a projection using the same rates for each area. The three projections have been implemented for electoral wards in the Fife local government area of Scotland, using the flexible framework provided by POPGROUP software. Persuasive local population projections based on standard data for standard areas are feasible without the regular publication of migration flows.     

Asymptotic implications of fluctuating nuptiality and fertility considering both sexes together

In this paper we have suggested a procedure of measuring population change which takes into account fluctuating sequences of nuptiality and fertility schedules as they reflect a population’s response pattern to its changing socioeconomic conditions. Through numerical experiments, the two-sex population model of cyclical change, which considers the interaction between sexes through marriage, is seen to converge to an asymptotic stability. The advantage of such a convergence is to enable comparative investigations, in terms of a set of asymptotic parameters, of rather complex series of nuptiality and fertility changes and their implications for short-run oscillation in population structure as well as for long-run population growth.     

Model fertility schedules: variations in the age structure of childbearing in human populations

The nature of the roots for a set of fertility functions were explored in this study, resulting in tables from creation of a family of model fertility schedules. These model fertility schedules accurately represent the full range of age structures of fertility in large populations; they have close fit to various accurately recorded fertility schedules of very different form. The text for the tables includes discussions of: 1)the basis for the fertility schedules, 2)the age structure of the proportion ever married (G(a) specified by 2 parameters, 3)single parameter specified age structure of marital fertility, 4)the similarity of model schedules of age specific fertility to the age pattern of fertility in actual populations, 5)model fertility schedules' suitability during changing nuptuality. Possible uses of the schedules and their application to different countries (England, Wales, Peru) are also described.     

Age structure,growth, attrition and accession: A new synthesis

This paper shows that each equation describing relationships among demographic parameters in a stable population is a special case of a similar and equally simple equation that applies to any closed population and demonstrates some implications of these new equations for demographic theory and practice. Much of formal demography deals with functions that pertain to individuals passing through life, or to a stationary population in which births of individuals are evenly distributed over time. These functions include life expectancy, probabilities of survival, net and gross reproduction rates, expected years spent in various states and the probability that certain events will occur in the course of life. The stable population model permits the translation of population structure or processes in a more general type of population, with constant growth rates, back into equivalent populations for a stationary population. The method for translation developed in this paper, requiring only a set of age-specific growth rates is even more general, applying to any population. Age specific growth rates may also be useful for performing reverse translations, between a population's life table and its birth rate or its age distribution. Tables of numbers of females by single years of age in Sweden are used to illustrate applications. Tables summarize the basic relations among certain functions in a stationary population, a stable population and any population. Applications of new equations, particularly to demographic estimation of mortality, fertility and migration, from incomplete data, are described. Some other applications include; the 2 sex problem, increment decrement tables, convergence of population to its stable form, and cyclical changes in vital rates. Stable population models will continue to demonstrate long term implications of changes in mortality and fertility. However, in demographic estimation and measurement, new procedures will support most of those based on stable assumptions.     

Stationary and stable population]

The properties of stationary and stable population models are discussed. In particular, the changes in population size and age structure that will occur under assumptions of constant fertility and mortality rates and no migration are explored.     

Effect of mortality change on stable population parameters

The stable population model is used to establish formulas expressing the effects of mortality change on population growth rates, birth rates, and age composition. The change in the intrinsic growth rate is shown to be quite accurately approximated by the average decline in age-specific death rates between age zero and the mean age at childbearing in the stable population. This change is essentially independent of the initial level of fertility in the population. Changes in birth rates and age composition are shown to be simple functions of the age pattern of cumulative changes in mortality rates relative to an appropriately defined “neutral” standard.     

Stable age by region distributions

If the pattern of fertility, mortality and interregional migration exhibited by the United States population during 1950–60 were to continue in the future, the proportions of persons in the various age groups and regions would fluctuate from decade to decade. These fluctuations would become less marked with time, however, and eventually all the proportions would stabilize at certain fixed values. This collection of values may be called astable age by region composition corresponding to the given schedule of fertility, mortality and migration. The same phenomenon may be observed when individuals move between socioeconomic categories as, for example, socioeconomic status or educational attainment levels. The substantial differences between these various situations conceal remarkable similarities. In each case the continued operation of schedules of fertility, mortality and mobility between categories may result in a stable composition. The purpose of this paper is to shed some light on the nature of these stable compositions, on the interrelation between their various components, and on their relation to the patterns of fertility, mortality and mobility which generate them.     

Patterns underlying fertility schedules: A decomposition by both age and marriage duration

Summary Model fertility schedules based on the proportions married and the age pattern of marital fertility are unsatisfactory to the extent that marital fertility does not depend on age alone but also on other factors. Most notably, models based just on age fail to allow for differences between populations in their composition by marriage duration. Examination of the major series of fertility rates specific by both age and duration of marriage (Sweden 1911-70, England and Wales, 1941-70) reveals striking underlying regularities. The marital fertility rates observed at any given point in time can be factored into three independent components - an overall level, a vector of age effects common to all marriage durations and a vector of duration effects common to all age groups. A simple product of these three components is shown to approximate the data very closely over the entire series, despite major changes in the aggregate levels of fertility and nuptiality during the periods concerned. Not only are the data tightly structured, conforming very closely to this simple multiplicative model, but the vectors of age and duration effects themselves are shown to exhibit clear and meaningful regularities.     

Empirical analysis of the contribution of age composition to population growth

The method of decomposition is applied to rates of natural increase in order to elucidate the role played by age composition in the growth of populations. A population’s age distribution and fertility schedule are contrasted to those in a "stationary" population having the same mortality rates and having a fertility schedule equal to that of the observed population divided by its net reproduction rate. In this manner it is shown that about one-quarter to one-third of the growth of most current high-growth populations can be attributed to non-stationarity of their age distributions. This fraction will rise, as it has in most industrialized countries, if fertility is reduced and age distributions become middle-heavy. In projections of the 1963 Venezuelan female population with fertility rates declining by 20/0 and 1% annually, more than half of the growth (in numbers) that occurs prior to zero-growth attainment is contributed by non-stationarity of its intervening age distributions.

     

Demographic change and the needs-based planning of government services: projecting small area populations using spatial microsimulation

Changing patterns of longevity, fertility and migration in Australia have driven substantial changes in population age structure and household size and composition. Of the various dimensions of population change, population ageing is expected to present major challenges to the financing and sustainability of welfare state programs in industrialized countries. One key issue for many of these countries will be assessing where particular services will be required in the future. This paper outlines the application of new forecasting techniques that age a spatial microdataset to 2027. Two illustrative examples are provided to highlight the potential capacities of the new modelling approach for government service delivery planners. For many older people, ageing in place is important, but is more difficult when the person is single: and so the first illustrative application focuses on where aged single people will be living in 2027. The second application examines where future childcare places will be required given the projected growth in the number of children aged 3–4 years living in families where all parents are working. This information will be important for Government planners in deciding the best location for childcare places. The creation of synthetic small-area household microdata for future years offers great potential for a number of purposes, such as analysis of the likely future sociodemographic characteristics of individuals and families at the local level and assessment of the future geographic effect of alternative scenarios such as changes in labour force participation or fertility rates.     

Impact of developmental variance on behavior of models for insect populations I. Models for populations with unrestricted growth

Summary The concept of developmental variance is discussed with reference to its use in models for insect populations. When included in a model, developmental variance is typically used to describe the variation of developmental periods among individuals. However, its presence in a model can also have indirect impact on survival and fertility schedules. This impact can lead to significant changes in population growth rates and generation times. These relationships between developmental variance and population growth in models are quantified and discussed.     

The development and use of demographic models

In this review, we first examine two classical demographic models - conventional life tables and stable populations - and a modern generalization of stable population theory; we then discuss mathematical models of conception and birth. These models involve purely mathematical relations in formal demography as opposed to empirical regularities. Next we consider model age schedules of mortality, nuptialitiy, marital fertility, fertility, and migration that are explicitly based on such empirical patterns. We close this empirical section with a discussion of model stable populations, which are based on model life tables. We next examine the use of demographic models in forecasting future mortality, nuptiality, and fertility and in population projection. Following a discussion of microsimulation models, which gives us the opportunity to mention model age schedules of post partum amenorrhoea and of sterility, we close with observations about the purposes and uses of demographic models.     

Reflections on the demographic conditions needed to establish a U.S. stationary population growth

Abstract It is assumed that in the long run U.S. population growth will have to cease, as otherwise life will become physically impossible. Various hypothetical possibilities of achieving such a type of development are investigated. Alternatives of reproduction rate trends are considered in terms of alternatives of interactions of assumed age-specific fertility and mortality trends and these are computed and evaluated. The various computations then indicate the nature of childbearing attitudes and behaviour, which the 'average population' would have to adopt in order to achieve the desired stationary population growth after a certain period of time. On the other hand the results presented in the paper also indicate that a certain population growth (of the order of 30-50%) and change in age structure is inevitable in the coming 5-10 decades given the initial childbearing behaviour.     

Parameterizing age patterns of demographic rates with the multiexponential model schedule

"For nearly 200 years actuaries, statisticians, and demographers have sought to summarize the age pattern of mortality rates by means of a limited number of parameters. Such 'model schedules' have also been useful in representing schedules of rates other than mortality....This paper illustrates a particular general functional form for such model schedules: the multiexponential function. It discusses the changing behavior of this function as its parameters take on different values and examines the quality of the fits of this function to observed data on mortality, fertility, and migration." This is a revised version of a paper originally presented at the 1993 Annual Meeting of the Population Association of America. (SUMMARY IN FRE)