The effect of variability in the fertility schedule on numbers of kin

"The kinship formulas developed by Goodman, Keyfitz, and Pullum (1974) are extended to encompass populations in which fertility varies among women. An expression is derived to determine the amount by which the number of sisters in a heterogeneous population exceeds that in a population with homogeneous fertility. This expression, which is a function of the variance in the gross reproduction rate of the population, can readily be applied to numbers of other kin, such as aunts and cousins. Several trial calculations indicate that calculations of average numbers of sisters based on an assumption of uniform fertility could result in a underestimate of about 13 percent." (SUMMARY IN FRE)     

Estimating the intrinsic rate of increase of population from the average numbers of younger and older sisters

Based on stable population theory, a mathematical relationship is developed between the intrinsic rate of increase (r) of a population and the ratio (Z) of the average number of younger sisters ever born to the average number of older sisters ever born, for a random sample of women in the population. This mathematical formula is then converted into a technique for estimating r from data on numbers of sisters. The extent to which the technique may be generalizable to actual populations is discussed.     

基于符号互动论视角下的个体生育态度探究

近年来随着人口老龄化进程的明显加快,以及不同地区、城市出现的生育率持续走低等情况,中国未来人口数量的变化成为了一个十分重要的议题。近年来,国家先后制定了“双独生二胎”、“单独生二胎”以及全面放开二胎等鼓励民众生育的相关政策;相关学者也从人口学和经济学等视角对生育问题展开了诸多研究。本研究则是从符号互动论这一微观社会学视角出发,试图了解民众（主要集中在1970年至1995年出生的群体）生育态度是否会受到社会心理因素,尤其是从小生长的家庭环境中,兄弟姊妹等手足人数、手足关系、父母对待子女公平与否和出生排行的影响;以及当前生活环境,如社会治安状况、经济发展前景和政治稳定性等方面的主观判断对生育态度的影响;最后研究还将结婚意愿、性别等因素纳入到影响个体生育态度的分析模型之中。研究结果发现,兄弟姊妹等手足越多者,未来越倾向要生孩子;手足间关系越好者,未来也越倾向于要生孩子;排行中间的比排行老大的生孩子意愿高;无论父母对自己和手足公平情况怎样,都不会影响其生育态度。此外在所有外在环境中,经济状况是影响人们生育态度的主要因素,有结婚意愿者更倾向于生育孩子,男性也显著比女性更倾向于生育孩子。     

Population growth with variable family size

The Sharpe‐Lotka continuous time deterministic model of population growth is developed to take account of some possible forms of mother‐daughter fertility association, characterised here by a bivariable measure, A. This leads to a linear double integral equation for which, subject to certain conditions, a finite time solution can be found by Laplace transform methods and thus also model specific results relating the intergenerational fertility effect to the long term population growth rate and magnitude are established. The quantitative implications of the theory are illustrated by a consideration of a general bilinear form of A and in this context numerical results illustrating the finite time growth and also the long term distribution of fertility levels in the stable female population are obtained. In particular, it is shown that different fertility specific subpopulations can coexist indefinitely.     

Population momentum: A wider definition

Summary

Keyfitz has derived an elegant formula for estimating the ultimate size of an initially stable, growing population that abruptly reduces its fertility to replacement level. Reduction of fertility is achieved by the rather unrealistic device of dividing the original age schedule nffertility rates by the net reproduction rate. Only the inertia of the age distribution is thus accounted for, but not that of the fertility schedule. The key idea of an abrupt imposition of a fixed regimen capable in the long run of generating zero population growth may be retained, but the regimen made more realistic. By elaborating the population setting, such disparate ZPG regimens as reduction of marital fertility by contraception, delayed and/or less universal marriage, raised mortality risks, or permanent net out-migration may be formulated. Convergence of the populaton to stationarity becomes a two-phase process: a primary adjustment period of changing fertility rates followed by a period of age adjustment.

The present paper treats what happens when a fixed ZPG sterilization regimen, defined by a minimum age of sterilization γ and constant continuous risk φ of sterilization among unsterilized wives aged γ to β, is imposed abruptly (or else progressively over an interval T) upon an initially stable, growing population. Additional sources of residual growth are: (1) the nine-month lag in sterilization effect owing to pregnancy: (2) the more youthful pattern of child-bearing under sterilization: (3) the extra adjustment period (of length β-γ-0.75) of changing fertility rates; and (4) any delays in exposing elements of the population to the sterilization regimen.

Two questions are pursued. First, how important are the additional sources of residual growth? Secondly, how do their relative sizes vary as a function of the characteristics of the initial population?     

How responsive is U.S. population growth to immigration? A situational sensitivity analysis

There has been public concern about the effect of immigration on population growth in the U.S. But how responsive is population growth to immigration? This paper examines the sensitivity of intrinsic population growth to immigration and situates such sensitivity in fertility and survival changes. The application of second derivatives on a modified Leslie matrix facilitates the analysis of situational sensitivity of U.S. population growth to immigration. The results show that the sensitivity to immigration is not as influential as the sensitivity to fertility, and that the sensitivity to immigration further depends on changes in fertility and survival.     

Estimating the goodman,keyfitz, pullum kinship equations: An alternative procedure∗

As is often the case in demography, Goodman, Keyfitz and Pullum (1974) developed their theory of the interrelationships of fertility, mortality and kinship numbers by means of continuous mathematics [integrals], but resorted to ad hoc finite approximations for calculating results in concrete empirical cases. Their reason: ‘Ordinarily, we cannot evaluate the l(x) and m(x) functions for arbitrary values of x, since the data are usually collected for five‐year age intervals’ [p. 24]. Recent developments in computer software now provide an alternative, two‐step procedure that avoids extensive programming of finite approximation algorithms: 1) using a popular scientific curve‐fitting package, functions are found to represent particular sets of discrete data on fertility and mortality, 2) the resulting functions and parameter estimates are then inserted directly into the kinship equations, and the integrals evaluated numerically using readily available mathematics software. This procedure has potentially wide application in other areas of population mathematics where theory is given by integrals and other continuous expressions, but data are for discrete age groups.     

Religious Commitment and Social Relationships: Their Relative Contributions to Self-Esteem of Catholic Sisters in Later Life

SUMMARY

In this paper we examine religious commitment and social relationships of Catholic sisters and the relative contributions of these and other variables to their self-esteem in later life. Using a sample of 377 Catholic sisters with an average age of 63.5, we conducted a series of a hierarchical recession analyses to examine the relative contributions of blocks of socio-demographic variables, religious commitment variables, personal relationship variables, and psychological variables to self-esteem. In the overall model, the extent to which relationships were rewarding, perceptions of themselves as women, coping strategies, and perceived self-control were significant and thus predictive of the self-esteem of Catholic sisters.     

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.

     

Pre‐procreative ages in population stability and cyclicity

Age‐specific models of population renewal (with and without feedback) which imply convergence to a stable state for some levels of fertility or feedback may imply the presence of long‐term cycling around a constant or exponentially changing equilibrium for other levels of fertility or feedback. The switch from one regime to the other is a “bifurcation.”; The conditions for bifurcation involve the roots of an analogue of Lotka's Equation.

Typically bifurcation is induced by raising the strength of feedback or the level of fertility. It has been known since the early 1980s, however, that this is sometimes impossible. It is sometimes impossible even with the linear renewal equation itself and with the most basic of non‐linear models, Lee's cohort feedback model.

Here it is proved that this typical route to bifurcation does not fail for these basic models in the presence of a condition which always holds for realistic applications with higher organisms: the existence of a span of ages before the onset of fertility.

Specifically, a strictly positive lower bound on ages of procreation is proved to be sufficient to guarantee the existence of a rescaling of Lotka's Equation for which the real part of some complex root vanishes. This result holds for absolutely Lebesgue‐integrable (signed) net maternity functions on the positive real line and for absolutely summable (signed) net maternities on the positive integers.

It follows that Coale's rescaling device for the analysis of approach to stability in stable population theory can be implemented for all realistic human net maternity schedules. It also follows that the many special cases of the cohort feedback model throughout population biology will all generate persistent cycling instead of stability if feedback is sufficiently strong.     

The effects of gender control on fertility and children's consumption

Effects of sex preference on investments in children‘s human capital, bequests and fertility are studied, with and without sex selection, in a model based on parental altruism. Both pure sex preference, a feature of the parental utility function, and indirect preference, which arises from gender-related differences in earnings opportunities, are examined. When there is no gender control the impact of pure sex preference is seen in smaller consumption for daughters than for sons. However, when gender control is exerted, sex preference raises the sex ratio and it is possible that sisters may, on average, consume no less than their more numerous brothers. In an example of the model with specific functional forms, parents who practise gender control have larger families than if sex selection techniques were unavailable. The effect is magnified if sons‘ earnings opportunities are better than daughters‘. JEL classification: D11, J13, J16 Received August 31, 1995 / Accepted May 2, 1996     

On the distribution of completed parities when fertility is heritable

Over the last one hundred years, there has been, in many developed countries, a demographic convergence towards the two child family. The possible implications for population growth of such a tendency are considered in this paper in terms of both family limitation and also the intergenerational transmission of fertility. These two effects interact so that as the proportion of two‐child families increases, the possible influence of mother‐daughter fertility associations on population growth decreases, though even now it could override otherwise significant changes in either or both of the birth and death intensities. In particular, it is shown that according as to how fertility is transmitted through generations, it is still possible to have zero growth rates consistently with a widely dispersed stable distribution of family size as well as a typical mortality regime.     

The multistate stable population model with immigration

This paper outlines the discrete‐time and continuous‐time formulations of the stable population model with immigration, showing their commonality. It then illustrates how the model can be extended to include multiple interacting populations, and goes on to consider a multistate version of reproductive value that further illuminates the evolutionary dynamics of an “open”; model of multistate population growth and redistribution. Attention is restricted to results arising from a fertility regime that is below replacement level.     

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 immigratioa     

Statistical modeling of selected aspects of the childbearing process with application to world fertility survey countries∗

The childbearing process should be monitored in developing countries experiencing high population growth rates and high levels of maternal and infant mortality. A mathematical model for estimation of certain aspects of the childbearing process, which requires only data on age‐specific fertility rates, is developed. Synthetic maternal childbearing indices, namely, mean ages at first and last birth, length of reproductive life span, inter‐birth spacing, and proportion of childless women, in addition to the well‐known mean age at childbearing, for the WFS countries are obtained using the proposed model. The indices are free from age truncation effects, and, under certain assumptions, provide information about a cohort's completed fertility before the women stop reproducing. The effects of women's residence and education on fertility are also examined.     

Books and publications received

Efforts to control rampant population growth in sub-Saharan Africa have been stymied by confusion between the potential causes and consequences of high fertility in the region. A controversy has surfaced over the causal direction of the fundamental relationship between human fertility and size of landholdings. Members of one school of thought claim that farm couples modify their fertility behaviour according to the amount of land they own or operate. Yet others argue that the size of landholdings varies as a function of family size (an indicator of the availability of family labour). In the present study we use a two-stage least-squares regression on data from a 1988 survey of 747 farm households in Rwanda to disaggregate and compare the strengths of these two possible paths of influence. The results show that landholdings exert a positive influence on human reproduction, but not the reverse. Moreover, this influence is slightly stronger for couples who own all the land they operated, probably because they have larger incomes from equity in the land. The size of the farm is unrelated to the size of the family's potential farm labour force (measured as the number of household members aged 15–65) or to the husband's total desired number of children. These findings suggest that farm size boosts the number of living children not by creating a demand for more children but by increasing the supply of children through higher natural fertility and child survival.     

Comparison of microcomputer programs for making population projections: An update

There are many programs for making population projections now available for use with microcomputers. This article reviews six of approximately 15 microcomputer population projection programs. Each program is compared to a standard set of criteria relating to such items as hardware and software requirements, input data requirements and specification of assumptions, methodology and documentation, and summary output indicators. Numerical results from projections of six test data sets reflecting different assumptions about mortality, fertility, and migration are compared. Qualitative comments are included for describing special features and for making an overall assessment of each program.     

Estimation of vital rates by means of monte carlo simulation

Monte Carlo simulation has been used to estimate age-specific fertility and mortality rates for a small population,the French-derived isolate of Northside on St. Thomas, U. S. Virgin Islands. Estimates were based on data collected in a household census and genealogical survey and on birth, death, and marriage records for the years 1916to 1966. During this 50-year period (in which the population size increased from 202 to 657), the numbers of births and deaths were too, small to estimate age-specific rates directly, and in addition, death registration was incomplete. Mortality rates were estimated using a simulation program in which mortality was the only stochastic variable. A model mortality schedule was chosen which most accurately reproduced the growth pattern of the population over the 50-year period. To estimate fertility rates, a more complex simulation model was used in which fertility, nuptiality, and mortality were random variables with probability distributions. Preliminary estimates of fertility were made from the birth records and used as input to this simulation program. Birth probabilities were adjusted empirically from one set of simulation runs to the next, until population growth rates, as well as other demographic characteristics, were similar in the real and simulated populations. The birth rates which produced the best fit to the real population data were taken as the estimated age-specific fertility schedule. To reproduce the real population age structure more closely, secular changes in birth probabilities were applied.     

Human capital,technological progress and the demographic transition*

We emphasize the importance to consider components of population growth — fertility and mortality ‐ separately, when modeling the mutual interaction between population and economic growth. Our model implies that two countries with the same population growth will not converge towards the same level of per capita income. The country with the lower level of birth and death rates will be better off in the long run. Introducing a spill over effect of average human capital on total productivity our model implies multiple equilibria as illustrated in Becker el al. (1990) and Strulik (1999). Besides the existence of a low and high level equilibrium ‐ as characterized by low and high levels of per capita output respectively ‐ we show the existence of multiple low level (Malthusian) equilibria. Initial conditions and parameters of technological progress and human capital investment determine whether an economy is capable to escape the low level equilibrium trap and to enjoy sustained economic growth.