On the Gradient-Hamiltonian Systems to the Derivation of Economic Multivariate Total Functions
John Awuah Addor *
Department of Mathematics, Statistics and Actuarial Science, Takoradi Technical University, P.O.Box 256, Takoradi, Ghana
Kwadwo Ankomah
Department of IT Business, Ghana Technology University College, Takoradi Campus, Takoradi, Ghana
Emmanuel Benson
Department of Mathematics, Statistics and Actuarial Science, Takoradi Technical University, P.O.Box 256, Takoradi, Ghana
*Author to whom correspondence should be addressed.
Abstract
This paper highlights an application of Gradient or Hamiltonian (Grad-Ham) Systems in deriving multivariate total functions. The objective is to establish a relationship between Gradient or Hamiltonian systems and economic-oriented multivariate marginal functions, and demonstrate how they can significantly be applied to the derivation of economic multivariate total functions. The multivariate marginal functions are represented by the Grad-Ham systems of differential equations whose analytical solutions are based on the partial antiderivative technique. The paper establishes that all economic multivariate marginal functions can respectively be expressed as exact differential equations. It also uncovered that functions that can be optimized are conservative along their optimal paths and that these functions become the first integrals of their respective marginal systems. Finally, it introduces two model examples- one hypothetical and the other based on the Cobb-Douglas Production function- and presents their derivations thereof.
Keywords: Analytical solution, economics, exact differential equations, gradient systems, hamiltonian systems, marginal systems, multivariate and univariate functions, total functions