On continuous-time optimal advertising under S-shaped response

Continuous-time monopolistic models of advertising expenditure that rely on strict response concavity have been shown to prescribe eventual spending at a constant rate. However, analyses of discrete analogs have suggested that S-shaped response (convexity for low expenditure levels) may allow for the periodic optima encountered in actual practice. Casting the dynamic between advertising and sales in a common format (an autonomous, first-order relationship), the present paper explores extensions along three dimensions: an S-shaped response function, the value of the discount rate, and the possibility of diffusion-like response. Supplementing the treatment by Mahajan and Muller, a flexible class of S-shaped response models is formulated for which it is demonstrated that, in contrast to findings in the literature on discretized advertising models, continuous periodic optima cannot be supported. Further, a set of conditions on the advertising response function are derived, that contains and extends that suggested by Sasieni. Collectively, these results both suggest a set of baseline properties that reasonable models should possess and cast doubt on the ability of first-order models to capture effects of known managerial relevance.