Growth of U.S. population, 1940–1971, in the light of an interactive two-sex model

Since it is logically impossible to hold constant both male and female age-specific fertility rates, the intrinsic growth rates or the net reproduction rates for males and females, based on that assumption, are internally inconsistent. The interactive two-sex model presented in this paper holds constant a set of bivariate age-specific fertility rates by age of men and women and allows the male and female age-specific fertility rates to adjust themselves to achieve stability. The model gives the same intrinsic growth rate for both sexes and generates intrinsic age-specific fertility rates and intrinsic net reproduction rates for males and females which are consistent and can operate simultaneously on a population. The model is applied to the U.S. data for 1940–1971, and the results are compared with those obtained from the one-sex models.     

Effect of adjustment for sex composition in the measurement of fertility on intrinsic rates

The inconsistencies inherent in the one-sex models created a need for the construction of what are known as marriage functions, especially for the measurement of fertility. But attempts to develop marriage functions have been frustrated by the inability of the proposed functions to meet certain consistency conditions and also by difficulties in empiracally determining function parameters. Among several functions proposed so far, Das Gupta's (1972) "effective population" deserves special mention. He uses both sexes in its formulation, and has shown that the constancy of fertility rates based on such a function together with the constancy of survivorship probabilities of both sexes would result in a stable population. It is suggested in this article that the major source of the problem in a two-sex model seems to be the requirement that the model has to be specific both for sex and age. The ideal of incorporating the relative composition of one sex in the age-specific rates of the other, thereby creating a function dependent on both sexes, is advanced in this article. Such functions, defined explicitly for births, can be easily translated into age-specific birth rates. In addition to simplicity in the definition and form, the conditions leading toward stability can also be established. Interestingly enough, the intrinsic rate for this two-sex model lies in the interval determined by the rates obtained from the two one-sex models.     

Populations with constant immigration

"We consider a Leslie-type model of a one-sex (female) population of natives with constant immigration. The fertility and mortality schedule of the natives may be below or above replacement level. Immigrants retain their fertility and mortality, their children adopt the fertility and mortality of the natives. It is shown how this model may be written in a homogeneous form (without additive term) with a Leslie-type matrix. Reproductive values of individuals in each age group are discussed in terms of a left eigenvector of this matrix. The homogeneous form of our projection model permits the transformation into a Markov chain with transient and recurrent states. The Markov chain is the basis for the definition of genealogies, which incorporate immigration. It is shown that genealogies describe the life histories of individuals in a population with immigration. We calculate absorption times of the Markov chain and relate them to genealogies. This extends the theory originally designed for closed populations to populations with immigration." (SUMMARY IN FRE)     

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.     

The stable equivalent population,age composition,and Fisher’s reproductive value function

This paper deals with a further analysis of the stable equivalent population for human populations. It first shows that the arithmetic difference between the size of the stable equivalent population and the actual population captures the total prospective contribution of age composition to the growth of the present population, if fertility and mortality rates are held constant at their current levels. The second part of the paper examines Fisher’s reproductive value function and investigates the location of maxima in relation to alternative values of the intrinsic growth rate. One conclusion which follows from this analysis is that, contrary to what has been discovered by others in specific examples, the reproductive value function can possess a global maximum in the first one or two years of life, provided the intrinsic rate of population growth is sufficiently negative. This principle is illustrated by reference to reproductive value functions calculated from the recent experience of U.S. females.     

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.     

An economist's non-linear model of self-generated fertility waves

Summary Standard one-sex linear models of Lotka or Bernardelli always approach asymptotically an exponential growth mode with stable age distribution. Realistic non-linear models need not possess this property. The present analysis uncovers a possibly realistic ease where an existent mode of balanced growth is 'unstable', giving way when slightly perturbed to an asymptotic every-other generation limit cycle of determinable amplitude, and which is stable. The nonlinear model utilizes the hypothesis of R. A. Easterlin that age-specific fertility will tend to be lower for age classes that are relatively swollen in total number. By virtue of the law of diminishing returns, wages and feeling of security will tend to be low for such swollen groups. A possible rebound in fertility in the 1980s is implicit in the Easterlin hypothesis.     

A demographic perspective on the Muslim world

A demographic perspective is relevant to understanding the position of Muslims in today’s world. This paper examines the size and growth of Muslim populations, and whether most Muslims live in overwhelmingly Muslim countries. It also examines indices of poverty and human development for Muslimmajority countries, and the growth of the youth population; finally, it examines the key components of population growth: mortality and fertility. Mortality has declined sharply over the past 15 years in many Muslim countries, though not in all, and Muslim countries are no longer prominent among the ‘outliers’ with higher mortality than expected on the basis of their income levels. Fertility rates are also declining sharply in a number of major Muslim-majority countries, raising interesting issues about attitudes of different schools of Islamic jurisprudence, village-level religious leaders and ordinary Muslims towards contraception and abortion, as well as the role of socio-economic development and family planning programs in fertility declines. Despite these declines, past high fertility in many Muslim-majority countries leaves as a legacy a rapidly growing adolescent population and a burgeoning, inadequately educated labour force.     

A two-sex nuptiality-mortality life table

The “problem of the sexes” has been one of trying to reconcile inconsistent male and female demographic rates. The present paper deals with that question in the context of a two-sex nuptiality-mortality life table. A “rectangular” population, with equal numbers of persons in each age-sex group, is introduced as a standard, and a standardization relationship expressed in equation (9) relates changes in rectangular population rates to changes in age-sex composition. The standardization relationship is shown to satisfy a number of desirable properties and produce a realistic two-sex model. The standardization approach is then applied to data from Sweden for 1973, and the results and their implications are discussed. In particular, it is seen that the total number of marriages in a two-sex population neither is nor should be bounded by the total numbers of marriages in the associated male and female one-sex nuptiality-mortality tables.     

非洲的人口动态与分布

非洲的人口动态长期以来具有高出生率、高死亡率的特点,20世纪90年代以后,非洲的人口动态由高出生高死亡模式向高出生中死亡模式转变．死亡率的持续下降在很大程度上是由于数种过去危害最严重的急性传染病基本上得到有效控制的结果。正是死亡率的下降和持续的高出生率导致非洲人口迅速增长。在非洲人口增长的过程中．人口分布是极不平衡的。非洲人口分布的变化与经济因素的人口定期迁移是密切相关的,大致可分为三种情况．这种独特的迁移模式均与经济活动和生产方式直接相关。     

A new method for estimating the level of natural fertility in populations practicing birth control

The natural fertility schedule of a population is the schedule of age-specific marital fertility we would observe if no birth control were being practiced. In natural fertility (no birth control) populations we can observe the natural fertility schedule directly, but in populations practicing birth control the natural fertility schedule is disguised by the marital fertility rates in those age intervals in which control is exercised, the marital fertility rates being below the natural rates. This paper elaborates a method for estimating the natural fertility schedules of populations practicing birth control. Two alternative models are presented, one nonlinear and one linear. The use of these models is then illustrated with reference to the Old Order Amish population of Lancaster County, Pennsylvania. It is shown that for Amish women born between 1860 and 1900 both the nonlinear and linear models yield an estimate of .843 as the ratio between Amish and Hutterite natural fertility schedules.     

Aids and population growth in sub-Saharan Africa: Assessing the sensitivity of projections

Despite different models to project the course of the AIDS pandemic and a scarcity of data to provide standard input parameters for those models, a limited consensus emerges from distinct sets of population projections. In sub-Saharan Africa, population growth rates are projected to remain positive in spite of the pandemic over the next few decades. To investigate this conclusion, alternative projections of an HIV/AIDS epidemic and its related mortality are first produced from different sets of input parameters and assumptions. Their incorporation into the population projections of a fast-growing country illustrates the robustness of projected population growth rates under very different scenarios of the future epidemic but with the common assumption that it will not affect the mortality of the uninfected population, fertility nor migration. This paper then shows that the projected growth rates are much less robust when interactions between the epidemic and the demographic regime are allowed and identifies several potential mechanisms for such interactions. In particular, it suggests that improving our confidence in the medium-term projections of the demographic impact of AIDS in the region requires less a refinement of the projections of the epidemic than a better understanding of its impact on the timing of the postulated fertility decline.     

Bridewealth and Birth Control: Low Fertility in the Indonesian Archipelago, 1500–1900

Before the onset of the present demographic transition, population growth in Indonesia had reached unprecedentedly high levels. This article demonstrates that such high levels were a recent phenomenon. Prior to 1900 rates of natural population increase were low to very low in most areas in Indonesia. This runs counter to expectations based on Hajnal's “Eastern marriage pattern,” which could imply high growth levels in extended family areas, such as most Indonesian regions outside Java in the past. Usually, the low population growth rates in Southeast Asia are attributed to high mortality owing to high levels of violent conflict. It is argued that other factors contributing to such high levels of mortality should receive more attention. In this article it is also argued that low fertility rates, too, played a role in generating low rates of natural increase. The article discusses the influence of marriage patterns, household structure, methods of birth control, adoption, and slavery on fertility.     

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.     

On the dependence of age structure on asequence of mortality and fertility schedules: An exposition of a cyclical model of population change

In this paper a method to investigate the dependence of age structure and growth rate on a given sequence of fertility and mortality schedules under the conditions of unchanging mortality and absence of migration is discussed. The method consists in projecting an arbitrary population classified by age to the ends of successive periods assuming that a given age pattern of mortality will remain without change and that a given sequence of fertility schedules will repeatedly operate on the population in a cyclical fashion. It is shown that after a sufficiently large number of repetitions of the cycle, the shifts in age structure between the ends of successive periods and the changes in the growth of the different age groups from one period to the next show a cyclical pattern. Formulas are derived expressing the above changes in terms of a sequence of k growth multipliers, k being the number of schedules in the fertility sequence, and the survival rates in the mortality schedule. A numerical illustration of the theory is given using fertility data from Finland.     

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.     

Mortality level,desired family size and population increase: Further variations on a basic model

A series of computer-simulation models relating mortality level to fertility behavior and to rates of natural increase assuming that couples made use of a perfect means of birth control, that they wanted to be highly certain of having at least one son survive to the father’s 65th birthday, and that all women were biologically capable of having the same number of children were reported earlier. A model identical to one presented earlier but assuming women to be of variable fecundity is presented here; the results are quite similar. Also, the results of models assuming that parents want to be highly certain of at least two surviving children regardless of sex are compared with models assuming parental desire to be highly certain of at least one surviving son. At high levels of mortality the rates of natural increase are quite similar. When mortality is at intermediate to medium-low levels, the two-surviving-children model shows a lower rate of natural increase than the surviving-son model. At very low levels of mortality, such as now experienced by the most advanced nations, the two-surviving-children model manifests a higher rate of natural increase than the surviving-son model.     

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.     

Crossovers that link populations with the same vital rates

"In this paper, we consider crossovers of demographic density distributions from...populations that have the same fertility and mortality rates. We focus on observed populations and their associated stationary and stable models, and on proportional distributions of persons, births, deaths and reproductive values....Three different populations were selected to represent a range of demographic behavior. Those populations are Japan 1963, a low mortality, low fertility population; Togo 1961, a high mortality, high fertility population; and the United States 1919-1921, a population whose fertility and mortality are intermediate."     

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.