On the theory of stable populations: A new and elementary proof of the theorems under weaker assumptions

Expositions and elementary proofs are given for the basic theorems of stable population theory: That a population subjected to vital rates (not necessarily constant over time) satisfying certain postulates will eventually "forget" its original age distribution and take on one (not necessarily constant over time) which depends only on its history of agespecific vital rates, a process called "weak ergodicity." That consequently the subsequent birth, death, and growth rates (none of these necessarily constant over time) depend only on the history of age-specific vital rates and not on the original age distribution. And, in particular, that these results apply to the special case, herein called "classic" stable population theory, in which the age-specific vital rates are constant over time, and in which after the "forgetting" takes place the subsequent age distribution and birth, death, and growth rates all become constant. This formulation of the theory differs from previous ones in two respects: First, the postulates required are weaker, and hence the theorems more general, than previously; in particular, this formulation permits the highest age of childbearing to change from cohort to cohort, which is important for populations practicing contraception. Second, none of the advanced mathematics used in previous formulations is needed; only the manipulation of sums and inequalities from high school algebra and the concept of "limit" from freshman calculus are required.     

Malthusian Dynamics in a Diverging Europe: Northern Italy, 1650–1881

Recent empirical research questions the validity of using Malthusian theory in preindustrial England. Using real wage and vital rate data for the years 1650–1881, I provide empirical estimates for a different region: Northern Italy. The empirical methodology is theoretically underpinned by a simple Malthusian model, in which population, real wages, and vital rates are determined endogenously. My findings strongly support the existence of a Malthusian economy wherein population growth decreased living standards, which in turn influenced vital rates. However, these results also demonstrate how the system is best characterized as one of weak homeostasis. Furthermore, there is no evidence of Boserupian effects given that increases in population failed to spur any sustained technological progress.     

The solution of time‐dependent population models

We analyze the dynamics of age‐structured population renewal when vital rates make a transition in a finite time interval from arbitrary initial values to any specified final values. The general solution to the renewal equation in such cases is obtained. This solution describes the birth sequence explicitly, and also leads to a general formula for population momentum. We show that the duration of the transition determines the complexity of the solution for the birth sequence. For transitions that are completed in a time smaller than the maximum age of reproduction, we show that the classical Lotka solution found in every textbook also applies, with a small modification, to the time‐dependent case. Our results substantially extend previous work that has often focused on instantaneous transitions or on slow and infinitely persistent change in vital rates.     

The effects of demographic factors on age composition and the implications for per capita income

Multiple regression analysis techniques are used to measure the impact of variations in fertility and mortality rates on the population of labor-force age. The results of this analysis suggest that the impact of reductions in mortality on age composition are dependent on the level of mortality already attained, whereas reductions in fertility, as expected, increase the population in the nondependent age groups. The direct effect of vital rates on potential per capita income is assessed using the results of the regression equations, and such results suggest that variations in per capita income as large as 20 percent could be accounted for by variations in vital rates.     

Cyclically stable populations∗

The cyclically stable population relaxes the stable population assumption of fixed vital rates and replaces it with the assumption of a recurring sequence of schedules of vital rates. From any point (or stage) in one cycle of the sequence to the same stage in the next cycle, the cyclically stable population grows at a constant rate (λ). While the age composition of the cyclically stable population is different at different stages of the same cycle, it always has the same age composition at the same stage of every cycle. The essential dynamics of the cyclically stable model are captured by its birth projection matrix (BPM). The dominant eigenvalue of the BPM is growth rate A, and the right eigenvector associated with λ gives the within cycle‐birth sequence.

An important special case occurs when λ = 1, and a cyclically stationary population arises. Such populations challenge simplistic ideas about “Zero Population Growth.”; A population projection based on the sets of rates observed in the United States, 1970–90, shows a cyclically stationary population arising in less than 100 years. While it experiences no long term growth, that cyclically stationary population exhibits fluctuations in total size and considerable variability in age structure.     

Atechnique for estimating life expectancy with crude vital rates

This paper describes a method of estimating life expectancy at birth on the basis of crude vital rates. The method is derived from stable population theory and it furnishes good estimates insofar as the current crude vital rates of a population are close to its intrinsic rates. This condition is generally met in closed populations which have not experienced sharp movements in fertility. The method is useful for estimating life expectancy in developing nations with good sample registration systems but for which information on age is of poor quality. It is also useful for estimating the movement of life expectancy in certain European nations in the period prior to regular census taking. There are a number of nations and regions in Europe for which long series of birth and death rates are available but for which census age counts are widely spaced.     

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.     

The female population of Chile, 1855-1964: a microcomputer balance sheet method

This article summarizes the essential features of the inverse projection method and applies it to data on the female population of Chile for the period 1855-1964. Changes in age distribution, vital rates, life expectancy, fertility, and gross and net reproduction rates over time are described.     

Stabilization,birth waves,and the surge in the elderly

This study examines the transition of a population to stability following a shift to a new fixed set of vital rates. Specifically, the authors develop a simple discrete population model and use it to derive an explicit solution for the birth trajectory. "The new vital rates interact with the population's initial age composition and generate birth waves whose amplitude and attenuation depend on the ratio of ultimate to initial growth and on the new pattern of stable net maternity. A greater change in growth and a later stable net maternity pattern produce larger fluctuations in the number of births. Stabilization begins at the youngest ages and proceeds upward. Sixty years after the shift, the birth waves have largely disappeared and the proportion under age 15 approximates the stable level implied by the new rates. Those patterns are manifest in the stabilization of both observed and Coale-Demeny model stable populations." (SUMMARY IN FRE)     

Vital rates in India 1961-71 estimated from 1971 census data

Abstract In the last decade the increase in the population of India, while, of course, very large, was smaller than predicted by official forecasts. With the use of recent census and sample registration data - in the absence of age-specific rates and adequate vital statistics - this paper provides estimates of fertility and mortality through the reverse-survival and forward-projection methods. Birth rates are estimated as 40·5-42, death rates as 18-20, and life expectancy at birth as 45-46 years. Mortality decline had been smaller than forecast but more than during any comparable period in the past, even though current mortality levels, particularly infant mortality, are still high. Males continue to have a longer life expectation than females, with a difference that has widened in the past decade. The decline of between seven and ten per cent in the crude birth rate is largely due to changes in marital fertility and to some extent to changes in age and marital composition. Because of greater decline in death rates than birth rates, the 1961-71 decade shows a higher rate of population growth than previous periods.     

Mortality among elderly hispanics in the United States: Past evidence and new results

We used vital records and census data and Medicare and NUMIDENT records to estimate age- and sex-specific death rates for elderly non-Hispanic whites and Hispanics, including five Hispanic subgroups: persons born in Cuba, Mexico, Puerto Rico, other foreign countries, and the United States. We found that corrections for data errors in vital and census records lead to substantial changes in death rates for Hispanics and that conventionally constructed Hispanic death rates are lower than rates based on Medicare-NUMIDENT records. Both sources revealed a Hispanic mortality advantage relative to non-Hispanic whites that holds for most Hispanic subgroups. We also present a new methodology for inferring Hispanic origin from a combination of surname, given name, and county of residence.     

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.     

Age-distribution dynamics and aging indexes

I analyze the dynamics of the age distribution as some vital rates change. When the fertility or mortality rate declines, the age distribution often manifests a dynamic stochastic-dominance relationship. I also propose some alternative indexes for measuring population aging. These indexes are closely connected with the age-distribution dynamics and contain more refined information about the distribution of age among the old.     

A study of the correlation bias of unrecorded events by two independent enumeration systems

In this paper the allegation that, when using the Chandrasekar-Deming technique with two independent enumeration systems to record the incidence of vital events and migrations, each individual system misses the same type of event is investigated. If this correlation bias were to occur frequently enough, it could result in a serious underestimation of a given type of event. Using vital events and migrations data derived from a longitudinal survey conducted in Liberia between 1969 and 1973, the estimated number of recorded events is tabulated into homogeneous groups that are demonstrated to affect omission rates. The proportion of missed eyents is computed separately for each enumeration system, and the presence or absence of a significant correlation mathematically determined. The results of this study suggest that no significant correlation could be demonstrated in the type of birth, death or infant death that was commonly missed by each individual enumeration system. This indicates that the omission of a birth, death or infant death was apparently a random occurrence as far as the individual casefinding systems were concerned. The authors conclude that any underestimation of vital events from this type of correlation bias may not be as serious as originally imagined.

In contrast to this observation, the omission of particular types of migrations by each enumeration system was apparently not a random occurrence, but was associated with the direction of migration. This implies that selected migration rates in the liberian survey were probably understated and this could happen in other longitudinal surveys, unless specific precautions are taken to avoid it.     

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.     

Demographic Conditions Responsible for Population Aging

This article develops and applies two expressions for the rate of change of a population's mean age. In one, aging is shown to be negatively related to contemporary birth rates and death rates. In a general sense, aging occurs when vital rates are too low, as illustrated through applications to the United States, the Netherlands, and Japan. The other expression relates the rate of aging to a population's demographic history, in particular to changes in mortality, migration, and the annual number of births. Applications to the United States and Sweden show that the dominant factor in current aging in these countries is a history of declining mortality. Migration also contributes significantly but in opposite directions in the two countries. The two approaches are integrated after recognizing that the rate of change in the mean age is equal to the covariance between age and age-specific growth rates. A decomposition of this covariance shows that the two seemingly unrelated expressions contain exactly the same information about the age pattern of growth rates.     

The Momentum of Real Population Under Linear Fertility Transition

Population momentum is the ratio of a population's ultimate size after a demographic transition to its initial size before the transition. For stable population and instantaneous drop to replacement fertility, Keyfitz found a simple formula for the momentum. Although Keyfitz's formula has been extended to cover gradual demographic transitions, the stable initial population, however, is still an obstacle to approaching reality. Using the solution of the Lotka equation with time-varying vital rates and extending it to negative values of time, exact and accurate formulae for the momentum of any initial population that undergoes a linear fertility transition are expressed. Examples using the data from the United Nations indicate that the accurate formula works well for the age structures that are far from stable such as the more and less developed regions, as well as that close to stable such as the least developed region.     

The population census of Northern Nigeria 1952: Problems and results

A brief survey of censuses previously taken in Northern Nigeria makes clear by comparison the importance of the 1952 Census in the acquisition of demographic data. Some of the problems of organising and conducting the census are considered.

The analysis of the census results pays special attention to the pattern of population density but consideration is also given to age and sex structure, occupations, tribal data, literacy and religions. Shortcomings and omissions in the census, particularly the absence of any information on migration, are noted.

The conclusion stresses the importance of the 1952 Census as a basis for future demographic studies ; the evidence for an increasing population but the absence of vital statistics to explain this ; the need for much further detailed analysis to be undertaken of the data in the 1952 Census.     

A study of the correlation bias of unrecorded events by two independent enumeration systems

Summary In this paper the allegation that, when using the Chandrasekar-Deming technique with two independent enumeration systems to record the incidence of vital events and migrations, each individual system misses the same type of event is investigated. If this correlation bias were to occur frequently enough, it could result in a serious underestimation of a given type of event. Using vital events and migrations data derived from a longitudinal survey conducted in Liberia between 1969 and 1973, the estimated number of recorded events is tabulated into homogeneous groups that are demonstrated to affect omission rates. The proportion of missed eyents is computed separately for each enumeration system, and the presence or absence of a significant correlation mathematically determined. The results of this study suggest that no significant correlation could be demonstrated in the type of birth, death or infant death that was commonly missed by each individual enumeration system. This indicates that the omission of a birth, death or infant death was apparently a random occurrence as far as the individual casefinding systems were concerned. The authors conclude that any underestimation of vital events from this type of correlation bias may not be as serious as originally imagined. In contrast to this observation, the omission of particular types of migrations by each enumeration system was apparently not a random occurrence, but was associated with the direction of migration. This implies that selected migration rates in the liberian survey were probably understated and this could happen in other longitudinal surveys, unless specific precautions are taken to avoid it.     

Demographic data collection in less developed countries 1946-1996

This evolution over the last 50 years of data collection systems in less developed countries is assessed. The progress made by civil registration systems has been extremely disappointing. Except in Central and South America, their role in providing vital rate estimates is still very limited. In contrast, the promulgation of regular population censuses has been a success, particularly in Africa. The relative merits and demerits of different types of demographic surveys are described. To some extent multi-round designs have given way to single-round surveys, such as the Demographic and Health Surveys (DHS). DHS-style enquiries are particularly suitable for evaluation of interventions but are less appropriate if the main aim is to measure vital rates.