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.     

Future mortality: a bumpy road to Shangri-La?

Americans are getting fatter, and it is known that increased obesity may increase the risk of death. Olshansky et al. have argued that this increase in obesity will likely slow, or even reverse, increases in life expectancy in the United States and perhaps save U.S. Social Security as a result. We discuss historical changes in the mortality rate and the reasons why other analyses argue that life expectancies will continue to increase. We also discuss the limitations of using single risk factors such as obesity as predictors of mortality risk. Finally, we explore the relation between risk factors and the long-term historical increase in human life expectancy.     

Disaggregation in population forecasting: do we need it? And how to do it simply

"We have described a method for reducing the dimensionality of the forecasting problem by parsimoniously modeling the evolution over time of the age schedules of vital rates. This method steers a middle course between forecasting aggregates and forecasting individual age specific rates: we reduce the problem to forecasting a single parameter for fertility and another one for mortality. We have described a number of refinements and extensions of those basic methods, which preserve their underlying structure and simplicity. In particular, we show how one can fit the model more simply, incorporate lower bounds to the forecasts of rates, disaggregate by sex or race, and prepare integrated forecasts of rates for a collection of regions. We also discuss alternate approaches to forecasting the estimated indices of fertility and mortality, including state-space methods. These many versions of the basic method have yielded remarkably similar results." (SUMMARY IN FRE)     

Death and Taxes: Longer life, consumption, and social security

We analyze in three steps the influence of the projected mortality decline on the long run finances of the Social Security System. First, on a theoretical level, mortality decline adds person years of life which are distributed across the life cycle. The interaction of this distribution with the age distribution of labor earnings minus consumption, or of taxes minus benefits, partially determines the corresponding steady state financial consequences of mortality decline. The effect of mortality decline on population growth rates also matters, but is negligible in low mortality populations. Second, examination of past mortality trends in the United States and of international trends in low mortality populations, suggests that mortality will decline faster than foreseen by the Social Security Administration s forecasts. Third, we combine the work of the first two parts in dynamic simulations to examine the implications of mortality decline and of alternative forecasts of mortality for the finances of the social security system. Also, we use stochastic population forecasts to assess the influence of uncertainty about mortality decline on uncertainty about finances; we find that uncertainty about fertility still has more important implications than uncertainty about mortality, contrary to sensitivity tests in the official forecasts.     

Divergence in Age Patterns of Mortality Change Drives International Divergence in Lifespan Inequality

In the past six decades, lifespan inequality has varied greatly within and among countries even while life expectancy has continued to increase. How and why does mortality change generate this diversity? We derive a precise link between changes in age-specific mortality and lifespan inequality, measured as the variance of age at death. Key to this relationship is a young–old threshold age, below and above which mortality decline respectively decreases and increases lifespan inequality. First, we show for Sweden that shifts in the threshold’s location have modified the correlation between changes in life expectancy and lifespan inequality over the last two centuries. Second, we analyze the post–World War II (WWII) trajectories of lifespan inequality in a set of developed countries—Japan, Canada, and the United States—where thresholds centered on retirement age. Our method reveals how divergence in the age pattern of mortality change drives international divergence in lifespan inequality. Most strikingly, early in the 1980s, mortality increases in young U.S. males led to a continuation of high lifespan inequality in the United States; in Canada, however, the decline of inequality continued. In general, our wider international comparisons show that mortality change varied most at young working ages after WWII, particularly for males. We conclude that if mortality continues to stagnate at young ages yet declines steadily at old ages, increases in lifespan inequality will become a common feature of future demographic change.     

Stochastic population forecasts for the United States: beyond high,medium, and low

"This article presents and implements a new method for making stochastic population forecasts that provide consistent probability intervals. We blend mathematical demography and statistical time series methods to estimate stochastic models of fertility and mortality based on U.S. data back to 1900 and then use the theory of random-matrix products to forecast various demographic measures and their associated probability intervals to the year 2065. Our expected total population sizes agree quite closely with the Census medium projections, and our 95 percent probability intervals are close to the Census high and low scenarios. But Census intervals in 2065 for ages 65+ are nearly three times as broad as ours, and for 85+ are nearly twice as broad. In contrast, our intervals for the total dependency and youth dependency ratios are more than twice as broad as theirs, and our ratio for the elderly dependency ratio is 12 times as great as theirs. These items have major implications for policy, and these contrasting indications of uncertainty clearly show the limitations of the conventional scenario-based methods."     

Inequality in Life Spans and a New Perspective on Mortality Convergence Across Industrialized Countries

The second half of the twentieth century witnessed substantial convergence in life expectancy around the world. We examine differences in the age pattern of mortality in industrialized countries over time to show that inequality in adult life spans, which we measure with the standard deviation of life table ages at death above age 10 years, S₁₀, is increasingly responsible for the remaining divergence in mortality. We report striking differences in level and trend of S₁₀ across industrialized countries since 1960, which cannot be explained by aggregate socioeconomic inequality or differential external‐cause mortality. Rather, S₁₀ reflects both within‐ and between‐group inequalities in life spans and conveys new information about their combined magnitudes and trends. These findings suggest that the challenge for health policies in this century is to reduce inequality, not just lengthen life.