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 (1986), 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 (1971). 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.