Optimal fertility along the lifecycle

We explore the optimal fertility age-pattern in a four-period OLG economy with physical capital accumulation. For that purpose, we firstly compare the dynamics of two closed economies, Early and Late Islands, which differ only in the timing of births. On Early Island, children are born from parents in young adulthood, whereas, on Late Island, children are born from parents in older adulthood. We show that, unlike on Early Island, there exists no stable stationary equilibrium on Late Island, which exhibits cyclical dynamics. We also characterize the social optimum in each economy, and show that Samuelson's Serendipity Theorem still holds. Finally, we study the dynamics and social optimum of an economy with interior fertility rates during the reproduction period. It is shown that various fertility age-patterns are compatible with the social optimum, as long as these yield the optimal cohort growth rate. The Serendipity Theorem remains valid in that broader demographic environment.