Population Dynamics in Optimally Controlled Economic Growth Models: Case of Cobb-Douglas Production with Human Capital

In this paper, optimally controlled economic growth models with Cobb-Douglas aggregate production function are formulated to compare and contrast real per capita GDP performance as the population growth dynamics vary from purely exponential to strongly logistic. Using analytical and qualitative techniques, as well as numerical simulations, the population related parameters which induce qualitative changes in real per capita GDP over time are investigated. Consumption per effective labour and investments per effective labour in respect of (physical and human) capital are used as control variables, and (physical and human) capital per effective labour applied as state variables. Income per effective labour is used as the output variable. A time-discounted quadratic cost functional of the state and control vectors is used as the objective functional. It is found that, generally, under research and development (R & D) technological process, real per capita income grows faster and establishes higher time values to the extent that the population growth dynamics is purely exponential and far from strongly logistic. On the contrary, under any other case besides R & D, real per capita income grows faster and establishes higher time-values to the extent that the underpinning population growth dynamics is strongly logistic and far from being purely exponential. These results have far reaching implications in respect to the management of both underdeveloped (with exponential population growth) and developed (with logistic population growth) economies.