A General Delta-Nabla Calculus of Variations on Time Scales with Application to Economics

TL;DR

This paper develops a unified calculus of variations framework on time scales using delta and nabla derivatives, with applications to optimizing production and investment strategies in economics.

Contribution

It introduces a general delta-nabla calculus of variations on time scales and derives Euler-Lagrange equations applicable to economic decision-making models.

Findings

Derived Euler-Lagrange delta-nabla equations for the calculus of variations on time scales.

Provided insights into discretization processes for economic models.

Applied theoretical results to optimize production and investment policies.

Abstract

We consider a general problem of the calculus of variations on time scales with a cost functional that is the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange delta-nabla differential equations are proved, which lead to important insights in the process of discretization. Application of the obtained results to a firm that wants to program its production and investment policies to reach a given production rate and to maximize its future market competitiveness is discussed.