Classical continuous-time models of advertising expenditure tends to fall into two categories, those that prescribe spending at a constant level and those that prescribe switching infinitely quickly between several different levels of spending. The latter practice, chattering, though impossible literally, can be interpreted in practice to imply that the faster the switching the better. Empirical evidence, however, sometimes suggests the superiority of pulsing, alternating between different spending levels with finite frequency, for example in a periodic fashion. Furthermore, the actual behavior of marketing managers, who often advertise in flights or pulses, appears to differ from the optimal policies many current models prescribe. Showing how structural properties of common advertising models rule out finite-frequency pulsing a priori, the paper develops a continuous-time model for which finite-frequency pulsing can be optimal. Relaxing these structural assumptions yields a new class of models which, for certain values of their parameters, lead to periodic pulsing optima; this is accomplished by inclusion of both S-shaped response and an exponential-smoothing filter. These theoretical results are illustrated through simulation of a variant of the Vidale-Wolfe model.
