Optimal economic growth and non-stable population

An analysis is conducted of optimal economic growth in the simple one-sector neoclassical growth model when the exogenous rate of population growth changes. The optimal growth path is characterized by a "Golden Rule," which is a very straightforward generalization of the traditional steady-state Golden Rule of Accumulation. Comparative statics analysis reveals that a rise in the long-run growth rate of population should generally be accompanied by an increase of the optimal rate of savings and should result in a decrease in consumption per capita. The results allow for a generalization of Samuelson's (1975) analysis of the lower bound for the optimal rate of population growth. It is shown that, under plausible assumptions, the optimal savings rate follows either an inverted U-shaped or U-shaped pattern during periods of demographic transition, according to whether the rate of births is falling or rising.