Sensitivity Analysis of Longevity and Life Disparity

Abstract

The population growth rate (λ or r) analyzed in Chap. 3 is a population-level consequence of the individual-level vital rates. A similarly basic outcome, at the individual or cohort level, is longevity: the length of individual life. The most commonly encountered description of longevity is its expectation, the life expectancy. However, longevity is a random variable, differing among individuals (even when those individuals are subject to the same rates and hazards) because of the random vagaries of mortality and survival. Therefore, it is important to also consider its variance and higher moments. This chapter introduces the sensitivity analysis of longevity, which will be explored in more detail in Chaps. 5, 11, and 12. As in Chap. 3, we will begin by reviewing a classic formula for the sensitivity of life expectancy in age-classified models. The we will use matrix calculus to derive more general formulas for the moments of longevity, the distribution of age or stage at death, and the life disparity, applicable to age-or stage-classified populations. 4.2 Life Expectancy in Age-Classified Populations Notation It is customary to denote life expectancy by symbols like e o x or e(x), but in general the symbol e plays too many roles in mathematics to be helpful for our purposes. So, when we make the transition to matrix formulations, I will use the symbol η, in various vector and scalar manifestations, to indicate longevity. Perturbation analysis of longevity has been pursued mostly within the framework of age-classified life cycles (e.g.,