Decomposing demographic change into direct vs. compositional components

We present and prove a formula for decomposing change in a population average into two components. One component captures the effect of direct change in the characteristic of interest, and the other captures the effect of compositional change. The decomposition is applied to time derivatives of averages over age and over subpopulations. Examples include decomposition of the change over time in the average age at childbearing and in the general fertility rate for China, Denmark and Mexico. A decomposition of the change over time in the crude death rate in Denmark, Germany and the Netherlands is also presented. Other examples concern global life expectancy and the growth rate of the population of the world.