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1993 - VOLUME 4, NUMBER 1

94.61.01 - English - Young J. KIM and ROBERT SCHOEN, Department of Population Dynamics, The Johns Hopkins University, Baltimore, MD 21205 (U.S.A.)

Crossovers That Link Populations with the Same Vital Rates (p. 1-19)

This paper looks at relationships that exist among an observed population and its associated model populations by examining "crossover" points. Crossovers are mean ages where proportional distributions from two populations with the same vital rates intersect, for example curves representing the age distributions of an observed population and its stable counterpart. Crossovers conceptually unify a number of diverse threads in mathematical demography, including generation length, population momentum, and the relationship between a population's stable growth rate and its observed age-specific growth rates. More specifically, Lotka's length of generation T is the stable-stationary birth crossover age, and intrinsic growth rate r is equal to the average growth rate of an observed population up to the observed-stable birth crossover age. The momentum of a population is the ratio of observed to stationary population proportions at the reproductive value function crossover age. Crossovers thus provide useful insights into demographic relationships.

94.61.02 - English - Peter EKAMPER, Netherlands Interdisciplinary Demographic Institute (NIDI), P.O. Box 11650, NL-2502 AR The Hague (The Netherlands), and Nico KEILMAN, Central Bureau of Statistics, Section for Research on Demography and Living Conditions, P.O. Box 8131 Dep., N-0033 Oslo I (Norway)

Sensitivity Analysis in a Multidimensional Demographic Projection Model with a Two-sex Algorithm (p. 21-36)

Formulas are derived for the effect that a slight change in the occurrence/exposure rate of the multidimensional projection model has on the elements of the population vector. The projection model classifies the population by sex, age, and marital status. The model includes a two-sex algorithm in order to ensure consistency between numbers of male and female marriages, number of divorces for the two sexes, and new widows (widowers) and deceased spouses. The sensitivity functions and elasticities are applied to data from the Netherlands for the period 1980-1984. The results indicate that marriage market mechanisms, in particular competition and substitution effects, are reasonably well modelled.

94.61.03 - English - Dov FRIEDLANDER, Moshe POLLAK and Jona SCHELLEKENS, The Hebrew University of'Jerusalem, Faculty of Social Sciences, Mount Scopus, Jerusalem 91905 (Israel)

A Method of Estimating the Time of Marital Fertility Decline and Associated Parameters (p. 37-49)

This paper presents a new method for estimating the time of the onset of marital fertility decline. The proposed method produces a maximum likelihood least squares estimate for the point of change in a sequence of marital fertility indexes. It is suggested that the proposed method has certain advantages over previous attempts to estimate the time of the onset of marital fertility decline.

94.61.04 - English - Noreen GOLDMAN, Office of Population Research, Princeton University, 21 Prospect Avenue, Princeton, NJ 08540 (U.S.A.), Graham LORD, Mathtech (Mathematica), Inc., Princeton, NJ 08540 (U.S.A.), and Yuanreng HU, WESTAT, Inc., Rockville, MD 20850 (U.S.A.)

Marriage Selection and Age Patterns of Mortality: A Mathematical Investigation (p. 51-73)

Researchers have acknowledged that the greater longevity of married as compared with unmarried persons may result in part from the mate selection process, namely, more frequent rejection of unhealthy men and women as potential spouses. Can one make inferences about the importance of such selection processes by examining the age pattern of mortality differences by marital status? Several investigators have hypothesized that, if selection were the driving force behind the excess mortality of the single population, the relative mortality ratio (RMR) would decline steadily beyond the upper age at marriage. In this paper, we use simulation models to demonstrate the complexity of the relationship between the marriage selection process and the resulting RMRs. In particular, we show that marriage selection alone can produce a relative mortality ratio which remains large and relatively constant at ages far beyond the marriage span.

1993 - VOLUME 4, NUMBER 2

94.61.05 - English - Young J. KIM and Robert SCHOEN, Department of Population Dynamics, The Johns Hopkins University, Baltimore, MD 21205 (U.S.A.)

On the Intrinsic Force of convergence to Stability (p. 89-102)

Observed populations differ greatly in the speed with which they approach the stable form, but what determines rates of convergence is not fully understood. The present paper shows that the force of convergence does not approach a fixed value, but oscillates indefinitely around an "intrinsic" level. That level, h*, is determined by the square of the ratio of the 2 largest eigenvalues of the Leslie matrix. The value of h* can be closely approximated by a simple function that changes directly with the square of the coefficient of variation and inversely with the mean of the stable net maternity function. Population entropy, another measure of dispersion relative to the mean, is also highly correlated with h*. (MATHEMATICAL DEMOGRAPHY, STABLE POPULATION)

94.61.06 - English - Elizabeth A. ZENGER, Office of Population Research, Princeton University, 21 Prospect Ave., Princeton, NJ 08544 (U.S.A.)

Infant Mortality, Birth order, and Sibship Size: The Role of Heterogeneous Risk and the Previous-death Effect (p. 103-116)

This paper derives an analytic model to study biases in infant mortality estimates by birth order and sibship size, which occur when the death of an infant tends to shorten the next birth interval and mortality risk varies among families. We find that order-specific and sibship-size-specific estimates are biased by a selection for high-risk women across birth orders, since women with higher risk will tend to have shorter intervals, and more births, within a given period of time. Sibship-size-specific estimates are, in addition, biased by a selection of women who have experienced deaths, even if there is no heterogeneity in risk. Numerical examples based on data from Matlab, Bangladesh, are used to illustrate the possible magnitude of these biases. The results resemble patterns of infant mortality by birth order and sibship size which are often observed empirically. (MATHEMATICAL DEMOGRAPHY, INFANT MORTALITY, BIRTH ORDER)

94.61.07 - English - John H. POLLARD, School of Economic and Financial Studies, Macquarie University, Sydney, NSW 2109 (Australia)

Heterogeneity, Dependence among Causes of Death and Gompertz (p. 117-132)

Heterogeneity in a population with respect to mortality, or variation in "frailty" among members of that population, which has been discussed extensively in the literature over the last decade and a half is essential to any realistic model of dependence among causes of death. The main problem then is the development of a mortality model incorporating heterogeneity and cause of death which is both realistic and of manageable proportions. In a recent paper (J.H. Pollard, 1991), it has been shown that many life table results are remarkably insensitive to the strict shape of the mortality curve, at least for more developed populations, and that accurate approxmations can in many cases be obtained knowing only the mortality rates at two representative ages (e.g. 50 and 70). These results and the Gompertz "law' of mortality can be used to develop manageable approximate formulae for the expectation of life under heterogeneity and correlation among the causes of death. The formulae are confirmed by simulation. Numerical results indicate, somewhat surprisingly, that the effects of correlation among causes of death, even at quite high levels, on expectation of life and changes on expectation of life when particular causes of death are reduced or eliminated are relatively minor. (MATHEMATICAL DEMOGRAPHY, CAUSES OF DEATH, MODELS)

94.61.08 - English - Kenneth G. MANTON, Gene LOWRIMORE, and Anatoli YASHIN, Center for Demographic Studies, Duke University, 2117 Campus Drive, Durham, Nc 27706 (U.S.A.)

Quelques méthodes utilisant des données auxiliaires dans des modèles stochastiques à compartiments utillisés pour la mortalité par cancer: une généralisation des modèles d'hétérogénéité (Methods for Combining Ancillary Data in Stochastic Compartment Models of Cancer Mortality: Generalization of Heterogeneity Models) (p. 133-147)

In analysing mortality data there may be available information from survey and other sources that describe the marginal distribution of risk factors. We present a mortality model where nationally representative survey data on risk factor distributions are combined with data on cohort mortality rates to increase information, i.e., a fixed marginal risk factor distribution is combined with a cohort model representing unobserved individual risk heterogeneity. The model is applied to lung cancer mortality in nine U.S. white male cohorts aged 30 to 70 in 1950 and followed 38 years. Estimates of the cohort specific proportions of smokers were made from the National Health Interview Survey. Comparisons are made for models with different patterns of changes with age of individual heterogeneity. (MATHEMATICAL DEMOGRAPHY, STOCHASTIC MODEL, MORTALITY, CANCER)

1993 - VOLUME 4, NUMBER 3

94.61.09 - English - Jun ZHU, Office of Population Research, Princeton University, 21 Prospect Avenue, Princeton, NJ 08544 (U.S.A.)

A Model of the Age Patterns of Births by Parity in Natural Fertility Populations (p. 153-173)

This research develops a convolution model to express the age patterns of fertility at each birth order in natural fertility populations in terms of six parameters, directly representing the proximate determinants of fertility, and a series of parity level indicators. The parity level indicators at each birth order are simply the proportions of women in a cohort who will eventually have births at each birth order if the age-related fecundity decline is controlled. The Coale-McNeil nuptiality model is adopted to represent the age pattern of first marriage rates and the natural fertility schedule employed in the Coale-Trussell fertility model is incorporated to adjust age effects. The fast Fourier transform is used in solving the model numerically. It proves that the model is able to provide excellent fits to fertility for rural Chinese women in the 1950s. (MATHEMATICAL DEMOGRAPHY, FERTILITY, MODELS, PARITY)

94.61.10 - English - Andrei ROGERS and Jani S. LITTLE, University of Colorado, Boulder, CO 80309-0484 (U.S.A.)

Parameterizing Age Patterns of Demographic Rates with the Multiexponential Model Schedule (p. 175-195)

For nearly 200 years actuaries, statisticians, and demographers have sought to summarize the age pattern of mortality rates by means of a limited number of parameters. Such "model schedules" have also been useful in representing schedules of rates other than mortality. Model schedules may be used to summarize, analyze, compare, and fill in gaps in observed data. They may be used in forecasting and they also facilitate indirect estimation made on the basis of incomplete data. This paper illustrates a particular general functional form for such model schedules: the multiexponential function. It discusses the changing behavior of this function as its parameters take on different values and examines the quality of the fits of this function to observed data on mortality, fertility, and migration. (MATHEMATICAL DEMOGRAPHY, AGE-SPECIFIC RATE)

94.61.11 - English - Evert VAN IMHOFF, NIDI, P.O. Box 11650, 2502 AR The Hague (Netherlands)

A Consistency Algorithm Based on Information Theory (p. 197-203)

This paper provides a geometric-mean solution to the consistency problem of multidimensional demographic projection models, based on the constrained minimization of an entropy function. A comparison with the existing harmonic-mean solution yields many similarities and almost no differences: both solutions satisfy the properties of availability, monotonicity, homogeneity, competition, and symmetry, and, for both solutions, there is a convenient one-to-one relationship between adjustments in aggregate numbers of events, on the one hand, and age-specific numbers of events, on the other hand. However, one major advantage of the geometric mean is that its corresponding distance function is firmly based on (information) theory. (MATHEMATICAL DEMOGRAPHY, MODELS, POPULATION PROJECTIONS)

94.61.12 - English - Rainer WINKELMANN and Klaus F. ZIMMERMANN, SELAPO, University of Munich, Ludwigstr. 28 RG, 80539 Munich (Germany)

Count Data Models for Demographic Data (p. 205-221)

Key demographic variables, such as the number of children and the number of marriages or divorces, can only take integer values. This papers deals with the estimation of single equation models in which the counts are regressed on a set of observed individual characteristics such as age, gender, or nationality. Most empirical work in population economics has neglected the fact that the dependent variable is a non-negative integer. In the few cases where this feature was recognized, the authors advocated the use of the Poisson regression model. The Poisson model imposes, however, the equality of conditional mean and variance, a restriction which is often rejected by the data. We propose a generalized event count model to simultaneously allow for a wide class of count data models and account for over- and underdispersion. This model is successfully applied to German data on fertility, divorces and mobility. (MATHEMATICAL DEMOGRAPHY, DEMOGRAPHIC MODELS, REGRESSION ANALYSIS)

1994 - VOLUME 4, NUMBER 4

94.61.13 - English - Lorenzo MORENO, Mathematica Policy Research, Inc., P.O. Box 2393, Princeton, NJ 08543-2393 (U.S.A.)

Frailty Selection in Bivariate Survival Models: A Cautionary Note (p. 225-233)

One problem that researchers face in analysing the survival times of groups of related individuals is selecting how the distribution of frailty - an unobserved (or not adequately observed) random factor - should be specified. Several distributions have received attention - for instance, the gamma distribution and a nonparametric N-point, discrete probability distribution. Researchers have selected these distributions more for mathematical convenience than for their ability to represent biological, social, or economic reality, and the implications of choosing one functional representation of frailty over alternative choices have not been studied extensively. In particular, researchers have paid little attention to the type of association that exists among survival times of individuals in a group or between those of a pair under specific frailty distributions. This research paper explores the association among survival times under gamma, inverse Gaussian, nonparametric N-point, and Poisson distributions. It shows that the pattern and strength of this association depends on how the distribution of frailty is specified. (MATHEMATICAL DEMOGRAPHY, MODELS, MORTALITY)

94.61.14 - English - Kao-Lee LAW, Department of Geography, 1280 Main Street West, McMaster University, Hamilton, Ontario (Canada), and Hiroshi KAWABE, Department of Geography, Senshu University, 1-1, Higashimita 2-chome, Tama-ku, Kawasaki-shi, Kanagawa-ken 214 (Japan)

The Dependence of Marriage Migrations in Japan on Personal Factors and Ecological Variables (p. 235-258)

Marriage is an important migration-inducing life-cycle event. This paper uses a nested logit model to explain the interprefectural migration behaviors at marriage by personal factors and prefectural attributes, based on the micro data of the 1986 national migration survey of Japan. Before marriage, each person is considered a potential migrant making a two-level decision: (1) to stay or depart and (2) to choose a destination. The main findings are as follows. Destination choice propensities were affected by such attributes of potential destination as income (+), employment growth (+), distance (-), contiguity (+), and linguistic similarity (+). Non-natives appeared to be less sensitive to the attraction of economic opportunities. Personal factors were less important than prefectural attributes in affecting destination choice propensities. Departure propensities were affected by not only such attributes of origin prefecture as income employment growth (-), and population density (+) but also the "inclusive variable" (+), which reflected the attractiveness of the rest of the system. Despite being strongly emphasized in the literature, sibling status was less important than gender, nativity and education in affecting departure propensities. Personal factors were much more important than prefectural attributes in determining the departure propensities. (JAPAN, MATHEMATICAL DEMOGRAPHY, MIGRATION DETERMINANTS)

94.61.15 - English - Karin AARSSEN and Laurens DE HAAN, Econometric Institute, Erasmus University, P.O. Box 1738, 3000 DR Rotterdam (Netherlands)

On the Maximal Life Span of Humans (p. 259-281)

Mortality data from the Netherlands are analyzed using recently developed statistical methods in the field of extreme value theory. It is shown that there is a finite age limit. A 95% confidence interval for the age limit is 113-124 years. The results suggest differences between men and women. The suggested hypotheses could be tested on a larger data set. (NETHERLANDS, MATHEMATICAL MODELS, LIFE SPAN)

94.61.16 - English - Robert SCHOEN and Young KIM, Department of Population Dynamics, Johns Hopkins University, Baltimore, MD 21205 (U.S.A.)

Cyclically Stable Populations (p. 283-295)

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 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 and the right eigenvector associated with gives the within cycle-birth sequence. An important special case occurs when 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 DEMOGRAPHY, STABLE POPULATION)

1994 - VOLUME 5, NUMBER 1

94.61.17 - English - Joel COHEN, Rockefeller University, 1230 York Avenue, New York, NY 10021-6399 (U.S.A.)

Fertility Incentives and Participation in Localities with Limited Means: A Dynamic Model of Per Capita Resources (p. 3-24)

Several countries have attempted to change human fertility through economic incentives. This paper presents simple mathematical models of the participation of couples in a locally funded program of economic incentives. The models take as a springboard China's one-child program. Localities with a low per capita incentives attract few couples to the program, while localities with high incentives attract many couples at first but the value of the benefits is then watered down. The models show that participation in the program may persistently oscillate or may decay to a stationary level. Which behavior occurs is determined by whether there are decreasing, constant, or increasing returns in the rates of participation in response to successive equal increments in the incentive offered, and by the extent to which prospective parents learn from experience with past oscillations in the incentives. The models raise many empirical questions about the dynamics of incentive programs. (MATHEMATICAL DEMOGRAPHY, MATHEMATICAL MODELS, PRONATALIST POLICY)

94.61.18 - English - Kenneth W WACHTER, Department of Demography and Statistics, University of California, 2232 Piedmont Avenue, Berkeley, CA 94720 (U.S.A.)

The Cohort Feedback Model with Symmetric Net Maternity (p. 25-44)

Demographic cohort feedback models were introduced by Ronald D. Lee in 1974. In these non-linear models, a cohort's net reproduction ratio responds inversely to cohort size at birth, while the shape of the net maternity function remains constant. For certain response strengths such a model leads to sustained cycles in the trajectory of births. Suppose that the net maternity function is symmetric under reflection around some mean age of childbearing. I prove that the model then has cyclic solutions with period exactly equal to twice the mean age of childbearing not merely in a neighborhood of equilibrium but for a range of parameter values which is unbounded in a certain suitable sense. Sobolev space methods are introduced for the theorem's proof. This "global" bifurcation theorem for the cohort feedback model with symmetric net maternity provides a benchmark case for understanding the characteristics of non-linear population waves of realistic size. MATHEMTICAL DEMOGRAPHY, DYNAMIC MODELS)

94.61.19 - English - Robert CHUNG, Department of Demography, University of California, Berkeley, Berkeley, CA 94720 (U.S.A.)

Cycles in the Two-sex Problem: An Investigation of a Nonlinear Demographic Model (p. 45-73)

A fundamental shortcoming of classic stable population theory is its failure to handle populations differ entiated by sex. The classic theory is linear while the two-sex problem is inherently nonlinear. Previous two-sex investigations have focused on equilibrium conditions rather than dynamics, and ignored competition between age groups for marriage partners. This study makes a start at analysing dynamics models that incorporate competition, which can play an important role in any realistic marriage mode and can turn a model with a stable equilibrium sex ratio into one with a cycling equilibrium. As competition for mates increases is the cycle period stable or does it respond more sensitively? Over a wide rang of demographic parameters the period at bifurcation is a good predictor of the cycle period for highe levels of competition; however, other cycle characteristics are insufficiently predicted by linear method and nonlinear methods are needed to complete the picture. (MATHEMATICAL DEMOGRAPHY, STABLE POPULATION, MATE SELECTION)

94.61.20 - English - Gustav FEICHTINGER, Institute for Econometrics, Operations Research and Systems Theory, Vienna University of Technology, Argentinierstr. 8/119, a-1040 Vienna (Austria), and Andy J. NOVAK, Department of Statistics, Operations Research and Computational Methods, University of Vienna, Universitätsstr. 5/9, A-1010 Vienna (Austria)

How Stock Dependent Flow Rates May Imply Chaos in Educational Planning (p. 75-85)

The aim of the present paper is to illustrate how extremely complex patterns may be generated in a simple model of educational planning. In particular, we will show that certain dependencies of the flow rates on the teacher/student ratio imply nonlinearities which are substantial enough to generate erratic behaviour of the time paths. The main message is that chaos in educational planning may result from assumptions which are indeed qualitatively realistic but which are quantitatively exaggerated. (MATHEMATICAL DEMOGRAPHY, EDUCATIONAL PLANNING)

94.61.21 - English - Alexia PRSKAWETZ, Institute for Demography, Austrian Academy of Sciences, Vienna (Austria), Gustav FEICHTINGER and Franz WIRL, Institute for Econometrics, Operations Research and Systems Theory, Vienna University of Technology, Argentinierstr. 8/119, a-1040 Vienna (Austria)

Endogenous Population Growth and the Exploitation of Renewable Resources (p. 87-106)

We consider a demo-economic model where the economy consists of two sectors ('hunting and farming' and 'industry'), and both sectors depend directly or indirectly on the exploitation of a renewable resource. The primary sector harvests a renewable resource (fish, corn or wood) which is used as the input into industrial production, the secondary sector of our economy. Labour is divided up between these two sectors under the assumption of competitive labour markets. A system of two nonlinear differential equations for the resources and the population is studied by phase space analysis. Using the Hopf bifurcation theorem, we obtain two different routes to limit cycles and prove numerically the existence of a stable Malthusian limit cycle. (MATHEMATICAL DEMOGRAPHY, MATHEMATICAL MODELS, ECONOMIC MODELS, HUMAN ECOLOGY)

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