To grow or to fluctuate: Optimal paths to demographic equilibria

Wrzaczek, S., Feichtinger, G., Fent, T., & Novak, A. (2024). To grow or to fluctuate: Optimal paths to demographic equilibria. IIASA Working Paper. Laxenburg, Austria: WP-24-008

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Abstract

The principal goal of this paper is to establish a firm fundament of population policy. It is shown how intertemporal optimisation theory can be used to fulfill this important task. In particular, this will be illustrated by calculating the optimal trade-off between the further growth (or shrinking) of a population and the fluctuations of its age-structure generated by the decline (or the increase) of the fertility.
While the system dynamics of the age-structured optimal control model considered in this paper is described by the McKendrick-von Foerster partial differential equation, its objective functional is given by the discounted stream of the adaptation costs of the net reproduction rate (NRR) and the aforementioned trade-off. For the sake of obtaining analytic results the model is reduced to concentrated vitality rates. Its essential results for growing and declining populations are that under-/over shooting of the NRR is optimal for a short time horizon, whereas a fluctuating NRR takes over if the time horizon is extended. Numerical simulations for a stylized population structure show how the change in the NRR carries over to the total population and age-groups along time.

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