The effect of changes in fertility on the age distribution of stable populations

When fertility increases in a previously stable population, the new stable population will be younger than the old one, and the two age distributions will cross at the mean age of the populations. 1. Relationship Consider a stable population with a growth rate r. Suppose that, in that population, agespecific fertility rates change to new levels and stay constant thereafter, while mortality remains constant. Eventually, a new stable population with a different growth rate will emerge. For example, if age-specific fertility rates increase, the growth rate of the new stable population will be higher than that of the old one. The effect of this fertility-induced change in r on the proportionate age distribution of the stable population can be formally expressed as follows: (1) d ln[c(a)] dr = AP − a where c(a) da is the proportion of the stable population in the age interval a to a + da, and AP is the mean age of the stable population. In Equation (1), the derivative of ln[c(a)] is positive at ages below AP and negative at ages above AP . This means that, in the case of a fertility-induced increase in r, the proportion of the population at ages below AP will increase, while the proportion at ages above AP will decrease. At age AP , the derivative is zero, so the proportion of the population at that age remains unchanged. In other words, the new stable population will be younger than the initial one, and the two age distributions will cross at the mean age of the populations. Conversely, in the case of a fertility-induced decrease in r, the new stable population will be older. Equation (1) also shows that the further away an age is from AP , the larger the relative change in the age distribution. 1 University of Wisconsin-Madison http://www.demographic-research.org 595 Guillot: The effect of changes in fertility on the age distribution of stable populations 2. Proof The age distribution of a stable population, c(a), is given by (2) c(a) = be−rap(a) where b is the crude birth rate, r is the population growth rate, and p(a) is the proportion surviving from birth to age a. The impact of an infinitesimal, fertility-induced change in r on c(a) is analyzed by developing the derivative of ln[c(a)] with respect to r: (3) d ln[c(a)] dr = d [ln b− ra + ln p(a)] dr Since b = 1/ ∫∞ 0 e−rap(a)da in a stable population, equation (3) becomes: (4) d ln[c(a)] dr = d [ln 1− ln ∫∞ 0 e−rap(a)da− ra + ln p(a)] dr . This demonstration is about the impact on c(a) of a fertility-induced change in r, so p(a) is not affected and remains constant. Therefore, equation (4) simplifies as follows: d ln[c(a)] dr = −d ln ∫∞ 0 e−rap(a)da dr − dra dr = ∫∞ 0 ae−rap(a)da ∫∞ 0 e−rap(a)da − a = ∫∞ 0 ac(a)da ∫∞ 0 c(a)da − a = AP − a (5)