Minimization of communication expenditure for seasonal products

Abstract

Abstract. We consider a rm that sells seasonal goods. The rm seeks to reach a xed level of goodwill at the end of the selling period, with the minimum total expenditure in promotional activities. We con-sider the linear optimal control problem faced by the rm which can only control the communication expenditure rate; communication is performed by means of advertising and sales promotion. Goodwill and sales levels are considered as state variables and word-of-mouth eect and saturation aversion are taken into account. The optimal control problem is addressed by means of the classical Pontryagin Maximum Principle and the solution can be easily found solving, in some cases numerically, a system of two non linear equations. Moreover, a para-metric analysis is performed to understand how the total expenditure in communication should be divided between advertising and sales pro-motion.