Optimal control of non-linear advertising models with replenishable budget

Abstract

This paper deals with an optimal advertising model with two state variables. The revenue and efficiency functions are concave. The aim is to choose the rate of advertising expenditures over time in a way that maximizes the present value of net profit streams over an infinite horizon subject to a replenishable budget. It is shown that the optimal advertising expenditures start at a high level and decrease over time if the initial stock of goodwill is low. Low initial budget as well as low initial stock of goodwill can cause shortness of budget. It turns out that, in contrast to comparable linear models, the remaining budget is always positive.