The age separating early deaths from late deaths

Averting deaths may either increase or reduce life disparity, as measured by e†. The sign of the effect depends on the age at which the deaths are averted. We show that there may, depending on the entropy of the life table, be an age such that averting deaths before that age reduces disparity, but averting deaths after that age increases disparity. We say that this age separates “early” from “late” deaths (in terms of their effect on disparity) and investigate the factors determining this threshold age. If life table entropy is less than one, a unique threshold age separating early and late deaths always exists. If life table entropy is greater than one, averting deaths at any age increases disparity. If entropy equals one, averting deaths at age zero has no effect, but averting a death at any age after zero increases disparity. 1. Relationship We measure life disparity by life expectancy lost due to death, e† = ∫∞ 0 e(a) d(a) da, where e(a) = (∫∞ a `(x) dx ) /`(a) is remaining life expectancy at age a and d(a) = `(a) μ(a) is the life table distribution of deaths. Note that `(a) = exp(− ∫ a 0 μ(x) dx), with `(0) = 1, gives the probability of survival to age a, that μ(a) denotes the agespecific hazard of death, and that H(a) = ∫ a 0 μ(x) dx is the cumulative hazard function with H(0) = 0. Consider a perturbation that alters the cumulative hazard function by a step function with a single negative step of size s. Let es be the corresponding value of life disparity with the perturbation. To study the impact on e† of a concentrated decrease in mortality at age a, it is useful to consider the function (1) k(a) = 1 `(a) des ds ∣∣∣∣ s=0 = e†(a) + e(a) ( H(a)− 1), 1 Max Planck Institute for Demographic Research. E-mail: [email protected] 2 Max Planck Institute for Demographic Research. E-mail: [email protected] 3 We thank Kenneth Wachter, Hal Caswell, Trifon Missov, Peter Wagner and two anonymous reviewers. http://www.demographic-research.org 721 Zhang & Vaupel: The age separating early deaths from late deaths where e†(a) = (∫∞ a e(x) d(x) dx ) /`(a) is life expectancy lost due to death among people surviving to age a. Mortality reductions at ages with k < 0 decrease life disparity, while those at ages with k > 0 increase life disparity. We prove below that if e†/e(0) < 1, then there exists an age a† greater than zero such that k > 0 for a greater than a† and k < 0 for values of a less than a†. We also prove that if e†/e(0) = 1, then averting a death at age zero does not change life disparity, but averting a death at any higher age increases disparity. Finally we prove that if e†/e(0) > 1, then averting a death at any age increases life disparity. In the proof we make use of the fact that k(0) = e†−e(0). Note that the “entropy of the life table” equals e†/e(0) (Keyfitz 1977; Mitra 1978; Goldman and Lord 1986; Vaupel 1986), which differs from the standard statistical definition by a term equal to ln e(0). If life table entropy is less than one, then k(0) < 0; if entropy is one, then k(0) = 0; and if entropy is greater than one, then k(0) > 0.