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

Abstract

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.

Appendix: Materials and Methods

We downloaded our data from the Human Mortality Database (HMD) in January 2013 (Human Mortality Database 2012).³ For international comparisons, we used the same data series as Vaupel et al. (2011). Table S1 (Online Resource 1) shows the range of years between 1947 and 2011 that is covered by these data. We conducted all calculations in the R environment (R Development Core Team (2012)). Online Resource 1 gives the derivation of each formula presented. For our age decomposition of change in V(15), we adapted Eq. (2) to

$$ \frac{ dV(15)}{ dt}=-2{\int}_{15}^{\infty}\frac{a(x)\upmu (x)}{l(15)}{\int}_x^{\infty }l(z)\left[z-M(15)\right] dzdx, $$

(9)

which gives the change in V(15) over time interval dt. The observed proportional changes in age-specific mortality, a(x), between years t and t + 1 were computed as

$$ a(x)=\frac{\upmu \left(x,t+1\right)-\upmu \left(x,t\right)}{\upmu \left(x,t\right)}. $$

(10)

To convert time in these formulae to discrete one-year intervals, we discretized the probability density function of age at death, considered survivorship to the midpoint of each age interval, and substituted instantaneous mortality for central death rates. We counted age in one-year intervals starting from 0.5 years. Where we present mortality change over wider age intervals, we computed a weighted average of the one-year central death rates in each interval, using the probability of survival to each age as weights (Ahmad et al. 2001).