The time to decline: tracing a cohort's descendants in below replacement populations

A number of contemporary populations are exhibiting sustained fertility at levels substantially below long-term replacement. Nonetheless, relatively few populations are actually diminishing in size. Here, we approach that apparent paradox by analyzing the time before the number in a birth cohort, and its descendants, falls below the initial number in the cohort. First, models are examined with constant below replacement fertility, cohort extinction at age 75 or 85, and no mortality below the highest age attained. For a net reproduction rate (NRR) of 0.75, it takes 150 years for the cohort's descendants to be fewer than the cohort's original size if persons live to age 85, and over 130 years if persons live to age 75. If the NRR is at least 0.60, it takes a century before the descendants are fewer in number than the original cohort. Second, projections are done for the USA 2012, Italy 2012, and Hong Kong 2011 assuming that fertility and mortality remain constant. The results resemble the projections. For example, in Italy, with actual mortality and an NRR of 0.70, it takes over 125 years before the descendants of a cohort are fewer in number than the initial cohort. A relatively simple equation for the long term "time to decline" is presented, showing that it depends primarily on the level of fertility, secondarily on longevity, and only modestly on the mean age of fertility.