Extended von Bertalanffy Equation in Solow Growth Modelling

The aim of this work is to model the growth of an economic system and, in particular, the evolution of capital accumulation over time, analysing the feasibility of a closed-form solution to the initial value problem that governs the capital-per-capita dynamics. The latter are related to the labour-force dynamics, which are assumed to follow a von Bertalanffy model, studied in the literature in its simplest form and for which the existence of an exact solution, in terms of hypergeometric functions, is known. Here, we consider an extended form of the von Bertalanffy equation, which we make dependent on two parameters, rather than the single-parameter model known in the literature, to better capture the features that a reliable economic growth model should possess. Furthermore, we allow one of the two parameters to vary over time, making it dependent on a periodic function to account for seasonality. We prove that the two-parameter model admits an exact solution, in terms of hypergeometric functions, when both parameters are constant. In the time-varying case, although it is not possible to obtain a closed-form solution, we are able to find two exact solutions that closely bound, from below and from above, the desired one, as well as its numerical approximation. The presented models are implemented in the Mathematica environment, where simulations, parameter sensitivity analyses and comparisons with the known single-parameter model are also performed, validating our findings.