Advertising and production of a seasonal good for a heterogeneous market: from total segment separability to real media

Market segmentation is a fundamental topic of marketing theory and practice. We bring some market segmentation concepts into the statement of an advertising and production problem for a seasonal product with Nerlove-Arrow's linear goodwill dynamics, along the lines of some analyses concerning the introduction of a new product. We consider two kinds of situations. In the first one, we assume that the advertising process can reach selectively each segment. In the second one, we assume that one advertising medium is available and that it has a known effectiveness segment-spectrum for a non-trivial set of segments. In both cases we study the optimal control problems in which goodwill productivity of advertising is either linear or concave, and good production costs are (convex and) quadratic. We obtain the explicit optimal solutions using the Pontryagin's Maximum Principle conditions.