Population Dynamics in Optimally Controlled Economic Growth Models: Case of Cobb-Douglas Production Function

Abstract

In this paper, we discuss optimally controlled economic growth models with Cobb-Douglas aggregate production function, comparing real per capita income performance in scenarios where the labour (population) growth dynamics range from purely exponential to strongly logistic. The paper seeks to ascertain, by means of analytical and qualitative methods, as well as numerical simulations, the causal factors and parameters, especially population related ones, which induce qualitative changes in the performance of real per capita income. The models use consumption per effective labour as their control variable, and capital per effective labour as the state variable. Income per effective labour is here used as the output variable. A time-discounted welfare functional is used as the objective functional, maximized subject to a differential equation in the state and control variables. Each system is found to be stable in the neighbourhood of its non-trivial critical value. The models are both locally controllable and observable. The models’ simulation values, in control, state and output, appear plausible and consistent with reality. It is found out that under R & D technological process, economies with exponential population growth consistently out-perform those with logistic population growth. On the contrary, in all other instances, economies with exponential population growth consistently perform worse than those with logistic population growth. These findings have far reaching inferences with regard to the running of economies.