On Equality and Inequality in Stationary Populations

Although it is an analytic construct important in its own right, a stationary population is an integral component of a life table. Using this perspective, we discuss well-known and not-so-well known equalities that are found a stationary population as well as a set of inequalities. There are two parts to the set of inequalities we discuss. The first (theorem 1) is that at any given age x, the sum of mean years lived and mean years remaining exceeds life expectancy at birth when x is greater than zero and less than the maximum lifespan (When x = zero or x =maximum lifespan, then the sum of mean years lived and mean years remaining is equal to life expectancy at birth). The second inequality (theorem 2) is a generalization of the first, namely that for the entire population, the sum of mean years lived and mean years remaining exceeds life expectancy at birth. It may be that the inequality we identify as Theorem 1 is common knowledge in some circles. However, we have found no formal description of it and believe that Theorem 1 represents a contribution to the literature. Similarly, it may be the case that one would expect that Theorem 2 would hold, given Theorem 1, but we also have not found a formal description of this in the literature and believe that it also represents a contribution. Finally, we note we have not found any discussion of an equality we found embedded in Theorem 1 (when age = 0 and when age = ?, then ?x+ex= e0) and believe that the identification of this equality represents a contribution. We provide illustrations of the two inequalities and discuss them as well as selected equalities.

Keywords: Carey’s Equality Theorem, Two Inequality Theorems, Mean years lived, mean years remaining, life expectancy at birth, sum or mean years lived and mean years remaining, mean age at death, variance in age at death