Solow differential equations on time scales - A unified approach to continuous and discrete Solow growth model

Abstract

In this paper we reformulate the axioms of the well-known Solow macroeconomic growth model by means of the mathematical calculus on time scales. We derive a system of differential equations on a time scale T which is a generalization of the classical Solow fundamental differential equation for the continuous case as well as its discrete version. We also prove sufficient conditions for the exponential stability of equilibrium points of this system having positive coordinates. Applications of these results to the case of the Cobb-Douglas production function are given.

Katugampola Generalized Conformal Derivative Approach to Inada Conditions and Solow-Swan Economic Growth Model

“…In time, the model is represented as in Eq. (12) where Y is the national production, K and L (state variables) are the quantities of capital and labor factors used in production, both measured with the appropriate units and A represents a technological constant that is usually interpreted as the total productivity of all factors. For a period of time t, national production is the result of combining capital and labor, given a certain technological constant, represented mathematically as:…”

“…The main advantage of the proposed approach is the non-locality property of these fractional derivatives that are convenient for the modeling of real financial situations and macroeconomic systems. Models based on other types of derivatives have also been studied, for example in [12], axioms for the system of differential equations (time scales) of Solow type are formulated under the assumption that a certain function is constant, proving stability and balance results in positive coordinates. A Cobb-Douglas type production function is also considered.…”