A stochastic version of the Malthusian Trap Model: consequences for the empirical relationship between economic growth and population growth in LDC's

A stochastic version of the Malthusian trap model relating the growth rate of income per capita to the population growth rate of a given country is described. This model is applied to the a priori evaluation of the cross-sectional correlation between these 2 growth rates under 2 additional assumptions: 1) the relations in the model at national levels include country-specific and time-invariant random components, and 2) these growth rates are measured with a certain degree of temporal aggregation. It is shown that these 2 assumptions can explain near-0 correlations between the 2 growth rates even if there exist a strongly negative effect of population growth on economic growth. However, it is not clear whether these assumptions fully explain such insignificant correlations. Indeed, the implementation of the model is complicated by the structural shifts which are likely to occur in the equations over the course of the demographic transition.     

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.     

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.     

Induced population growth and induced technological progress: Their interaction in the accelerating stage

A simple model of Malthusian population growth combined with population‐induced technological progress generates accelerating growth. The model may be relevant for a first stage of growth in which natural resource limitations can be overcome through technological progress; it is not applicable to a later stage in which resource constraints are more resistant. Parameter values are roughly inferred from historical experience. Exogenously more rapid population growth initially depresses income, perhaps for up to several centuries, then raises it without limit. More rapid population growth is desirable only when the social discount rate is less than the ratio of the parameters for induced technical progress and static diminishing returns. Imposed population fluctuations cause inverse movements in incomes, so that induced progress is very difficult to detect empirically even for population fluctuations up to 500 years.     

A stochastic computer model for simulating population growth

Summary A model is described for investigating the interactions of age-specific birth and death rates, age distribution and density-governing factors determining the growth form of single-species populations. It employs Monte Carlo techniques to simulate the births and deaths of individuals while density-governing factors are represented by simple algebraic equations relating survival and fecundity to population density. In all respects the model’s behavior agrees with the results of more conventional mathematical approaches, including the logistic model andLotka’s Law, which predicts a relationship betwen age-specific rates, rate of increase and age distribution. Situations involving exponential growth, three different age-independent density functions affecting survival, three affecting fecundity and their nine combinations were tested. The one function meeting the assumptions of the logistic model produced a logistic growth curve embodying the correct values orr _m andK. The others generated sigmoid curves to which arbitrary logistic curves could be fitted with varying success. Because of populational time lags, two of the functions affecting fecundity produced overshoots and damped oscillations during the initial approach to the steady state. The general behavior of age-dependent density functions is briefly explored and a complex example is described that produces population fluctuations by an egg cannibalism mechanism similar to that found in the flour beetleTribolium. The model is free of inherent time lags found in other discrete time models yet these may be easily introduced. Because it manipulates separate individuals, the model may be combined readily with the Monte Carlo simulation models of population genetics to study eco-genetic phenomena.     

The Census Bureau on Prospects for US Population Growth in the Twenty‐First Century

The U.S. Census Bureau periodically releases projections of the US resident population, detailed by age, sex, race, and Hispanic origin. The most recent of these, issued 13 January 2000, for the first time extend to the year 2100 and also include information on the foreign‐bom population. (Earlier projections were carried up to 2080.) The extensive tabulations presenting the new set, and detailed explanation of the methodology and the assumptions underlying the projections, are accessible at the Census Bureau's web site: http://www.census.gov . A brief summary of some of the main results of these projections is reproduced below from United States Department of Commerce News, Washington, DC 20230. (The Census Bureau is an agency of the Department of Commerce.) Uncertainties as to future trends in fertility, mortality, and net migration over a period of some 100 years are very great, as is illustrated by the massive difference in the projected size of the population for 2100 in the three variants produced. The “middle” projected population figure of 571 million (which represents a growth of some 109 percent over its current level) is bracketed by “lowest” and “highest” alternative projections of 283 million and 1.18 billion, respectively. With somewhat lesser force, the point also applies to the 50‐year time span considered in the well‐known country‐by‐country projections of the United Nations. These projections are also detailed in three variants: low, middle, and high. The UN projections (last revised in 1998) envisage less rapid growth in the United States during the first part of the twenty‐first century than do the Census Bureau's. The projected population figures for 2050 in the three variants (low, middle, and high) are as follows (in millions):

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.     

Conditions determining the local stability of populations

The conditions that determine the local stability classification of an equilibrium population configuration are analyzed. The population investigated is age‐structured and density‐dependent, where density is determined by an age‐weighted population size. Two demographic parameters are introduced: the marginal birth rate and marginal death rate, which describe the marginal density‐dependence of the birth and death rates of the equilibrium population. Certain necessary and/or sufficient conditions determining stability are developed, most of them involving the net reproduction rate of the population, and examples illustrating these conditions are presented.     

The Relation Between Actual and Intrinsic Growth Rates

This paper demonstrates that, to a close approximation, the intrinsic growth rate of a population is equal to the mean of age-specific growth rates below age T, the mean length of a generation. This mean is normally close to the growth rate of the entire population block below age T. Therefore, when a disparity exists between the intrinsic growth rate and the actual growth rate of a population (whether or not net migration is included in both rates), it must be attributable to an unusual growth rate of the population block above age T. One implication is that the ‘momentum of population growth’ is entirely confined to the age span above T, approximately age 28 in developing countries to-day.     

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.     

On demographic transition,structural change,and economic growth and stagnation

The paper analyzes an economy with an agrarian and an industrial sector. Demand is determined by Engel's Law. Population growth follows a non‐linear income dependent path according to the theory of demographic transition. In case of decreasing returns to scale in the agrarian sector the existence of a stable low‐income equilibrium with high population growth can be shown. If this equilibrium is globally unstable, the system evolves towards a steady‐state of perpetual economic growth and low population growth. The path of demographic transition coincides with a path of structural change from an economy specialized in agriculture to a fully industrialized economy. The introduction of an income dependent savings rate allows the interpretation of the low‐income equilibrium as a limit cycle and, therefore, the explanation of high fluctuations in population growth and per capita income in least developed economies.     

Stabilization,birth waves,and the surge in the elderly*

Substantial regularities characterize the transition to stability that follows a shift from one set of vital rates to another. 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.

When fertility falls, the new stable population has a larger fraction at all ages above (approximately) 30, with greater changes characterizing the extremes of life. Fifteen years after the fall, there is a trough in the number at ages 0–14. Sixty years after the fall, when the largest pre‐decline cohort is age 60–74 and the smallest post‐decline cohort is age 45–59, there is a surge in the relative size of the elderly population. Thus after two generations, the birth waves produced by a rapid decline in fertility accentuate the effects of population aging.     

Stable growth in native‐dependent multistate population dynamics

Users of multistate life tables and projections have recognized that the Markovian assumptions underlying such models are unduly restrictive and should be relaxed whenever data permit. Efforts to include the influences of previous occupancies have included the incorporation of place‐of‐birth dependence. This paper addresses the stable growth properties of such generalized multistate models. It shows how place‐of‐birth‐specific stable growth measures can be calculated without projection simply by solving the characteristic equation. An example using Canadian data illustrates the argument.     

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.     

Mathematical investigations of the escape from the Malthusian trap

We present a simulation model that synthesizes Malthusian and Boserupian notions of the way population growth and economic development were intertwined. The non‐linear stochastic model consists of a system of equations whose dynamics culminate in an industrial revolution after hundreds of iterations. The Industrial Revolution can thus be conceptualized as a permanent “escape”; from the Malthusian trap that occurs once the economy is capable of permanently sustaining an ever growing population. We investigate the conditions for such an escape and their sensitivity to the parameters of the model. This is done in an attempt to understand why some economies might have had difficulties escaping from the Malthusian trap (in contrast to the European experience in the eighteenth and nineteenth centuries). Our results show that the likelihood of an escape is sensitive to the savings rate and to the output elasticities of the two sectors of the economy. When not in a subsistence crisis, the chances that an escape will occur increase for larger values of the ratio of the savings rate to the growth rate of the population. The chances of an escape also increase substantially for larger values of the output elasticities of labor.     

A population dynamic model of Batesian mimicry

Summary A population dynamic model of Batesian mimicry, in which populations of both model and mimetic species were considered, was analyzed. The probability of a predator catching prey on each encouter was assumed to depend on the frequency of the mimic. The change in population size of each species was considered to have two components, growth at the intrinsic growth rate and carrying capacity, and reduction by predation. For simplicity in the analyses, three assumptions were made concerning the carrying capacities of each population: (1) with no density effects on the mimic population growth rate; (2) with no density effects on the model species; and (3) with density effects on both species. The first and second cases were solved analytically, whereas the last was, for the most part, investigated numerically. Under assumption (1), two stable equilibria are possible, in which both species either coexist or go to extinction. Under assumption (2), there are also two stable equilibria possible, in which either only the mimic persists or both go to extinction. These results explain the field records of butterflies (Pachliopta aristolochiae and its mimicPapilio polytes) in the Ryukyu Islands, Japan.     

Ethnic population projections for the UK, 2001–2051

This paper reports on projections of the United Kingdom’s ethnic group populations for 2001–2051. For the years 2001–2007 we estimate fertility rates, survival probabilities, internal migration probabilities and international migration flows for 16 ethnic groups and 355 UK areas. We make assumptions about future component rates, probabilities and flows and feed these into our projection model. This model is a cohort-component model specified for single years of age to 100+. To handle this large state space, we employed a bi-regional model. We implement four projections: (1) a benchmark projection that uses the component inputs for 2001; (2) a trend projection where assumptions beyond 2007 are adjusted to those in the UK 2008-based National Population Projections (NPP); (3) a projection that modifies the NPP assumptions and (4) a projection that uses a different emigration assumption. The projected UK population ranges between a low of 63 millions in 2051 under the first projection to a high of 79 million in the third projection. Under all projections ethnic composition continues to change: the White British, White Irish and Black Caribbean groups experience the slowest growth and lose population share; the Other White and Mixed groups to experience relative increases in share; South Asian groups grow strongly as do the Chinese and Other Ethnic groups. The ethnic minority share of the population increases from 13% (2001) to 25% in the trend projection but to only 20% under our modified emigration projection. However, what is certain is that the UK can look forward to be becoming a more diverse nation by mid-century.     

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.     

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.     

Population aging, health care, and growth: a comment on the effects of capital accumulation

In a recent paper, Hashimoto and Tabata (J Popul Econ 23:571–593, 2010) present a theoretical model in which the increase in the rate of dependence due to aging of the population leads to a reallocation of labor from non-health to health production and, as a consequence, to a decline in economic growth. We argue that these results rely heavily on assumptions of a “small economy” and perfect capital mobility, which tie down the amount of capital. In this paper, we proceed by analyzing the case of an economy in which the availability of capital is endogenously determined by domestic savings. We find that the new “capital accumulation effect” is opposite to the previous “dependency rate effect,” leaving the effect on economic growth ambiguous. In particular, if the former prevailed, population aging would foster economic growth, a result that finds support in recent empirical work.