Determining the birth function for an age structured population

This paper deals with an inverse problem in age‐structured population dynamics, namely the recovery of the unknown birth function from the additional or overposed data consisting of the total population over a time interval equal to the maximum life span of the species. Conditions on the data are given to guarantee the existence and uniqueness of a solution, and the question of continuous dependence of the birth function on the data is addressed. Some numerical simulations are presented to indicate that one can in fact use the methods of the paper to reconstruct the birth function.     

Determining the initial age distribution for an age structured population

"This paper deals with an inverse problem in age-structured population dynamics; the recovery of an unknown initial age distribution. We attempt to recover this function from overposed data which consists of either the total population over a time interval equal to the maximum life span of the species or the age structure of the population at a fixed later time. Existence, uniqueness and continuous dependence of the initial distribution function on the data are addressed. Some numerical simulations are presented to illustrate the feasibility of recovery using the methods of the paper."     

Individual and aggregate influences on the age at first birth

This study examines the role of both individual adolescent expectations of market and nonmarket income and aggregate influences on first birth timing for Blacks and Whites and for three birth cohorts of American women. Using two panels of the National Longitudinal surveys, results suggest that between-race differences in age at first birth result from differences in individual expectations about market and nonmarket income. Cohort differences in age at first birth result from relationship differences in both individual and aggregate influences, with aggregate influences differentially altering the role of individual expectations on first birth timing. These results suggest that employment policies that reduce poverty and increase wages would effectively delay childbearing. For Blacks, since early childbearing results from lower expectations about future economic success, similar fertility timing patterns between the races would result if Blacks had White employment and wage levels. Over time, these employment and wage policies will not only alter individual expectations about economic success but also will alter the impact of expectations on fertility timing.     

Consumption and population age structure

In this paper, the effects on aggregate consumption of changes in the age distribution of the population are analysed empirically. Economic theories predict that age influences individuals’ saving and consumption behaviour. Despite this, age structure effects are rarely controlled for in empirical consumption functions. Our findings suggest that they should. By analysing Norwegian quarterly time series data, we find that changes in the age distribution of the population have significant and life-cycle-consistent effects on aggregate consumption. Furthermore, controlling for age structure effects stabilises the other parameters of the consumption function and reveals significant real interest rate effects.      

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.     

Computer display of population age structure

The standard human population pyramid contains only current information; it is often inadequate when making comparisons of different countries or years. A graphic computer program superimposes current population, life table survivor ships, stable population, and past population projected up to the present. Fertility, health conditions, migration, and demographic transition stage are better revealed by this expanded graph form. A time series of the modified pyramids reveals detailed age-specific trends for a country or region. With 'minor modifications the computer program is applicable to all machines with a Fortran IV compiler.     

Intergenerational transmission of age at first birth in the Netherlands for birth cohorts born between 1935 and 1984: evidence from municipal registers

Whereas most research on the intergenerational transmission of fertility behaviour has focused on transmission of the number of children, this paper studies the transmission of the timing of first births. Specific attention is paid to changes in the strength of transmission across cohorts. Theoretically, it is unclear whether the strength of intergenerational transmission of entry into parenthood can be expected to increase or to decrease across cohorts. Event history analyses of data in Dutch registers show a substantial degree of intergenerational transmission in the age at which people have their first child. The degree of transmission from mothers to children increases for successive cohorts. Intergenerational transmission becomes weaker the longer children postpone entry into parenthood. At young ages transmission from mothers to children is stronger than from fathers to children.     

Some aspects of the use of birth expectations data from sample surveys for population projections

The value of national sample survey data relating to birth expectations for projecting births is reassessed in the light of data limitations pointed out by Ryder and Westoff among others and of the methods of projection used by the United States Bureau of the Census. The annual level of fertility under the cohort-fertility projection method depends on the assumptions regarding completed fertility, about which the available survey data are fairly informative, and on the assumptions regarding the timing of births, about which the survey data tell us very little. Test calculations suggest that Ryder and Westoff have overstated the significance of timing relative to completed family size for the level of future births. We believe that the fall in the annual total fertility rate in the first half of the sixties is to be explained only in part by a general delay in childbearing; a moderate to substantial decrease in completed family size has also occurred. Analysis of the latest set of fertility projections of the Census Bureau also suggests that the assumptions about completed fertility are a much more important determinant of the level of future births than timing, both in the short and long term. Although the available expectations data cannot help in predicting short-term annual changes in fertility, they appear useful for making long-term projections of annual fertility. Expansion of the size, frequency, and content of the sample surveys and incorporation of parity and birth interval into the projection method may improve projections.     

The outlook of China's population after the stabilization of birth policy

A professor of the Institute of Population Research of the People's University of China attempts to project the future population development of China so that stabilization of the birth policy can be assessed. He divides China into the economically developed and population-controlled area (29% of the population), the economically subdeveloped and population fairly-controlled areas (59% of the population), and the economically less-developed areas where fertility is high (12% of the population). China's population is expected to increase because of the baby boom to 1.25-1.3 billion by the year 2000. Between 1996 and 2000, the growth rate is expected to decelerate and reach zero growth. After 2010, if growth is held at the replacement rate of 2.1, the population will still continue to grow slowly. Around 2100, China's population should be around 1.45-1.59 billion. This would cause a decrease of 27% of arable land. With a decline in fertility rate comes a rise in the amount of the aged population (4.9% in 1982 vs. 5.5% in 1987). The proportion of aged citizens is expected to rise with the stabilization policy until around 2040 where it can be held at about 18%. China's GNP by the year 2000 is expected to be US$1183.8 billion with the per capita GNP about US$934 (providing the population is controlled). Compare this figure with the per capita GNP of the world (US$30,100) and of developed countries (US$10,700) in 1988, and one can see that China is far behind the rest of the world in economic growth.     

Summary indices of the age distribution of a population

In this paper a method for characterizing the age distribution of a population by means of an ‘index word’ is outlined. It is shown that the method can be applied to yield approximations to population forecasts.     

Investigation on birth desires of couples of childbearing age in Hongsheng and three other villages conforming to consideration conditions for the second birth]

The opinions of inhabitants of outlying rural areas regarding second births are explored in this report. By means of household interviews, it is revealed that 22 households out of 153 (i.e., 14.4%) that are eligible for a second birth usually do not opt for the second birth. In a study that took place in four villages within one county, findings showed that desire for a second birth is determined in large part by the sex of the firstborn, males being preferred to females. Moreover, this desire decreases in proportion to time in that the elegible married couple often realizes the economic benefit of having only one child after the first child's birth and chooses not to have a second one. Similarly, many low-income couples would like to have a second child but find that it is not within their economic means. Mid-range income couples often want a second child as well. However, it is generally the couples with higher income that choose to remain with only one child. The desire for a male child is quite strong in the People's Republic of China, especially in rural areas. The fact that the male carries on the family name is very important in most people's opinion, as is the fact that a son has the potential to increase the family's income through manual labor. Thus, the inhabitants of rural areas traditionally favor male children.     

An evaluation of population projections by age

A number of studies have evaluated the accuracy of projections of the size of the total population, but few have considered the accuracy of projections by age group. For many purposes, however, the relevant variable is the population of a particular age group, rather than the population as a whole. We investigated the precision and bias of a variety of age-group projections at the national and state levels in the United States and for counties in Florida. We also compared the accuracy of state and county projections that were derived from full-blown applications of the cohort-component method with the accuracy of projections that were derived from a simpler, less data-intensive version of the method. We found that age-group error patterns are different for national projections than for subnational projections; that errors are substantially larger for some age groups than for others; that differences in errors among age groups decline as the projection horizon becomes longer; and that differences in methodological complexity have no consistent impact on the precision and bias of age-group projections.     

Optimal age of retirement and population growth

The purpose of this paper is to study intergenerational optimal resources sharing when the social planer can choose the retirement age in addition to consumptions and investment. We use the extension of the Diamond analysis by Hu [1979] that incorporates endogenous retirement age. We found that the optimal retirement age is an increasing function of the population growth rate if the elasticity of substitution of old agents' labor for young agents' labor is lower than one. In the millian case, when the size of a population does not matter, and when the elasticity of substitution of old agents' labor for young agents' labor is strictly higher than one, the optimal retirement age is a decreasing function of the population growth rate. In the benthamite case, the change in the optimal retirement age is indeterminate. Received: 19 February 1999/Accepted: 27 February 2001 All correspondence to Bertrand Crettez. We would like to thank Jean-Pierre Vidal for very helpful comments on an earlier draft. An anonymous referee provided insightful comments on a previous version of this paper. Responsible editor: Alessandro Cigno.     

On the relationship between household composition and population age distribution

Summary Composition of households by age of head and by age of other household members has recently been presented in a convenient algebraic expression, the household composition matrix. It has been shown that this matrix operates as a linear transformation from the vector of household distribution by age of head to the vector of population age distribution. A further analysis will show that the first row of the matrix may be interpreted as representing a vector of average household fertility rates. If the linear relationship between household and population distributions is fully implemented, then a relationship between household fertility and the size of the youngest age group can be derived. If w is the population age distribution and w (1) is the number of persons in the youngest age group, then: where α is the first row of the household composition matrix with its first element eliminated, C is the household composition matrix with its first row and first column eliminated, and Ψ is the vector w with its first element, w (1) eliminated. Extension of this result will enable simultaneous projection of population and households, suitable for computer application to conventional five-year age groups.     

An alternative formulation of the birth function in a two-sex model

The inconsistency in Lotka's stable population model (two different intrinsic growth rates for the two sexes) arises from the fact that he considers two equations (for male and female births), and not because his equation for one sex does not involve the other. Many authors in the past have erroneously put emphasis on the latter point and modified Lotka's equations for male and female births. Since sex ratio at birth is constant, two independent equations for male and female births cannot exist. The correct approach is to attempt to form an equation for all births. The author followed this approach in his earlier works on the problem, but his birth functions were formulated from axiomatic considerations. The present paper provides a new birth function which has an intuitively appealing physical interpretation, and for which the interaction between the sexes is empirically determined from the data.     

Social determinants of low birth weight in a high-risk population

Over ninety low-weight infants were born per thousand live births in South Carolina, based on 96,000 birth records from 1975 and 1979. Higher incidence of low birth weight for black infants cannot be explained away as a result of black/white differences in age or education of mothers, prenatal care, parity or length of birth intervals. Though all these factors are important predictors of birth weight, an increasing propensity to have low-weight babies persists among black mothers even after all these factors are controlled.     

A survey of China's birth control among women of child-bearing age

To further implement China's family planning policy of "prevention first, birth control first," a study of the current family planning situation was conducted. A survey of the birth control methods employed by women of childbearing age and by men was based on a nationwide randomized sampling of 1/1000. In the different age groups, ranging from 15-49 years old, IUD users accounted for over 50%, tubal sterilization 25%, and vasectomy 10%. The main IUD users were women in the 20-24 age group. Tubal sterilization was more prevalent among the women in the 35-39 age group. The use of oral contraceptives (OCs) was more common among younger women but accounted for less than 10% of the total. The survey was based on the replies to questionnaires from 172,788 married women of childbearing age; 120,022 of them practiced contraceptive methods for a birth control rate of 69.46%. The breakdown was as follows: IUD, 34.84%; tubal sterilization, 17.63%; vasectomy, 6.94%; OCs, 5.86%; condom users, 1.39%; and other methods (including chemical suppositories, rhythm, or safe period method and withdrawal before ejaculation), 2.78%. There was a higher percentage of OC users in urban areas, and a marked preference for IUDs in the rural communities. The rural birth control rate was 68.58%; the urban rate was 74.17%. The use of the IUD has priority in all the areas; its percentage approaches the national average level. The use of vasectomy as a birth control method varies considerably according to area as does the use of OCs, condom, and tubal sterilization. Rural minority groups prefer the IUD and OCs; tubal sterilization, the condom, and vasectomy are preferred by the Han nationality. The birth control rate differed according to the different occoupation groups: 77.85%, workers; 76.01%, farmers; 85.15%, cadres; 59.52%, housewives; and 66.67%, others. The birth control rate was higher among those who received a college education than the illiterates, but statistics did not show a significant difference in the rate of those with a high school education and the illiterates. Mothers of 0-1 children generally preferred OCs; tubal sterilization was preferred by mothers with 2-3 children. The nonusers of contraception accounted for 30.54% among married women of childbearing age. A breakdown gives the following figures: menopause and infertility, 6.17%; divorced and widowed, 1.64%; planned parenthood, 10.51%; nonusers who should have practiced contraception, 12.22%. On a national level, the estimated number of nonusers of contraception among those who should be practicing contraception comes to about 20,000,000 women.