The threshold age of Keyfitz' entropy

BACKGROUND Indicators of relative inequality of lifespans are important because they capture the dimensionless shape of aging. They are markers of inequality at the population level and express the uncertainty at the time of death at the individual level. In particular, Keyfitz' entropy $\bar{H}$ represents the elasticity of life expectancy to a change in mortality and it has been used as an indicator of lifespan variation. However, it is unknown how this measure changes over time and whether a threshold age exists, as it does for other lifespan variation indicators. RESULTS The time derivative of $\bar{H}$ can be decomposed into changes in life disparity $e^\dagger$ and life expectancy at birth $e_o$. Likewise, changes over time in $\bar{H}$ are a weighted average of age-specific rates of mortality improvements. These weights reflect the sensitivity of $\bar{H}$ and show how mortality improvements can increase (or decrease) the relative inequality of lifespans. Further, we prove that $\bar{H}$, as well as $e^\dagger$, in the case that mortality is reduced in every age, has a threshold age below which saving lives reduces entropy, whereas improvements above that age increase entropy. CONTRIBUTION We give a formal expression for changes over time of $\bar{H}$ and provide a formal proof of the threshold age that separates reductions and increases in lifespan inequality from age-specific mortality improvements.