To grow or to fluctuate – optimal demographic paths

This paper establishes a rigorous foundation for population policy by applying intertemporal optimization theory to derive the optimal trade-off between population growth (or decline) and age-structure fluctuations caused by fertility rate changes. The model is based on the McKendrick-von Foerster partial differential equation, with an objective functional representing the discounted adaptation costs associated with the net reproduction rate. To enhance analytical tractability, we assume concentrated vitality rates. Our findings suggest that, over short-time horizons, an initial under- or overshooting of the net reproduction rate is optimal, while over longer horizons, an oscillating net reproduction rate emerges as the best trajectory. Numerical simulations using a stylized population structure illustrate how variations in the net reproduction rate influence the total population dynamics and age-group distributions over time. This study highlights the critical role of optimization in designing adaptive and sustainable population policies.