The world population development according to a dynamic extension of the Wicksellian production function

Abstract

Based on UN figures for the World Population Development combined with the Wicksellian production function, we develop expressions for how the population development depends on capital (saved income) and the maximisation of future discounted per capita consumption.

We prefer to adopt the term Wicksellian function, rather than the widely used name Cobb-Douglas function, since it was first published by Knut Wicksell (1851 - 1926) and two years after his death, the same function was published by Charles W. Cobb and Paul H. Douglas, (Wicksell 1916, Cobb and Douglas 1928, Olsson 1971).

In a recent paper the Wicksellian production function has been extended by means of the Calculus of Variations to take care of the fact that the production factor capital is an accumulation of previously saved income, which is a result of previous production activities, (Grubbström 2024). There, the population development was assumed as given. Instead, in this paper using the same method, it is assumed to be a consequence of the opportunity to consume that is offered by labour and capital according to this production function.

It is shown that if the population develops in this “natural” way, then there is no risk for any Malthusian Catastrophe, and that the living standard of the population (consumption per capita) will grow at a rate determined by the discount rate times the propensity to save, but the size of the population will decrease, once it has reached its peak (at the end of this current century).

We also show that with our approach, the propensity to consume is the weight of labour input in the Wicksellian function (apart from earlier interpretations of this weight).

Our approach is analogous to Hamilton's principle of stationary action for finding the behaviour of dynamical mechanical systems in a general configuration space.