Diamond-Phi_h dynamics on time scales with an application to economics

Abstract

Conventional dynamic models in economics are usually expressed in discrete or continuous time. A new modelling technique-time scales calculus-unifies both of these approaches into a general framework. We present and construct a dynamic optimization problem from economics in which the utility function is $phi_{h}$-concave and the value function and constraints are on different time scales. The calculus of variations and optimal control are employed, with the aid of the newly introduced diamond-$phi_{h}$ dynamic calculus by the authors [12] on time scales, to obtain a solution. The Hermite-Hadamard inequality with the diamond-$phi_{h}$ dynamic integral follows a proof of the new model. The new diamond-$phi_{h}$ time scale model unifies various related existing models involving general and more complex time domains.