Population Dynamics in Optimally Controlled Economic Growth Models: Case of Linear Production Function

This paper discusses optimally controlled economic growth models with linear aggregate production function of capital and labour. It compares and contrasts real per capita income performance over time in situations where the labour (population) growth dynamics vary from purely exponential to strongly logistic. The study seeks to identify, by means of analytical and qualitative methods, as well as numerical simulations, the causal factors and parameters, particularly population related ones, which induce qualitative changes in the performance of real per capita income over time. Furthermore, the paper discusses the concept of maximum sustainable population growth for the models and the conditions for exiting the Malthusian trap. Consumption per (effective) labour is used as the control variable, and capital per (effective) labour, as state variable, whereas income per (effective) labour is considered the output variable. A time-discounted welfare functional is applied as the models’ objective functional, maximized subject to a differential equation in the control and state variables. Each system is found to be controllable and observable. The models’ simulation results are reasonably intuitive and realistic. The results also indicate that, consistently, real per capita income grows faster and generates greater time values, however marginal, as the population growth dynamics tends increasingly logistic.