Concavity and Unimodality of Expected Revenue Under Discrete Willingness to Pay Distributions

Abstract

Most pricing and revenue management models have at their core an optimization problem; one needs to determine the optimal price or quantity to maximize a profit or revenue function. To ensure tractability, conditions that assure the objective function has a unique solution are enormously helpful. So far, several technical assumptions have been proposed for the continuous case, but comparatively little attention has been given to the discrete counterpart despite its prevalence in practice. Thus, this study aims to develop new technical assumptions, built upon relevant economic concepts, to guarantee the tractability of revenue management models in discrete settings. In particular, we present two sufficient conditions for the revenue function to be concave, in terms of quantity or price and propose a condition for the revenue function to be unimodal, called discrete increasing generalized failure rate (IGFR). Our definition has an appropriate economic interpretation and offers comparable properties to those of the continuous version. Finally, we show the discrete IGFR property holds for several discrete distributions.