Advertising and exogenous interference in a segmented market

We propose a model of a monopolist firm which advertises a product in a segmented market where a constant exogenous interference is present. Using the framework of Nerlove- Arrow advertising model, we describe the interference using a constant negative addendum in the goodwill ODE. This effect may vary over the different market segments. Hence, we admit that the firm goodwill concerning any segment may become negative and we associate 0 demand with negative goodwill values. By using the simplest demand model, i.e. a piece- wise linear function, we formulate a nonsmooth infinite horizon optimal control problem. The features of an optimal advertising policy depend on the different effects in each segment of the advertising effort and of the exogenous interference. We describe an optimal solution for a single wide-spectrum medium, which requires a constant advertising effort after some finite time. We characterize the segment sets which may be a possible target for the firm using a given advertising medium.