Constant savings rates and quasi-arithmetic population growth under exhaustible resource constraints

In the Dasgupta–Heal–Solow–Stiglitz (DHSS) model of capital accumulation and resource depletion we show the following equivalence: if an efficient path has constant (gross and net of population growth) savings rates, then population growth must be quasi-arithmetic and the path is a maximin or a classical utilitarian optimum. Conversely, if a path is optimal according to maximin or classical utilitarianism (with constant elasticity of marginal utility) under quasi-arithmetic population growth, then the (gross and net of population growth) savings rates converge asymptotically to constants.