Life span and the problem of optimal population size

We reconsider the optimal population size problem in a continuous time economy populated by homogenous cohorts with a fixed life span. This assumption is combined with a linear production function in the labor input and standard rearing costs. A general social welfare function is specified, admitting the Millian and Benthamite cases as polar parameterizations. It is shown that if the lifetime is low enough, population is asymptotically driven to extinction whatever the utility function and the level of inter-generational altruism. Moreover, population is driven to extinction at finite time whatever the values of lifetime and altruism provided the utility function is negative. When the utility function is positive, it is shown that the Millian welfare function leads to optimal extinction at finite time whatever the lifetime. In contrast, the Benthamite case is much more involved: for isoelastic positive utility functions, it gives rise to two threshold lifetime values, say T_0 < T_1: below T_0, finite time extinction is optimal; above T_1, balanced growth paths are optimal. In between, asymptotic extinction is optimal. Last, intermediate welfare functions are studied, resulting in more complex optimal consumption and fertility dynamics compared to the two polar cases but delivering similar optimal extinction features.