On Continuous-Time Optimal Advertising Under S-Shaped Response

Fred M. Feinberg ()
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Fred M. Feinberg: University of Michigan Business School, 701 Tappan St., Ann Arbor, Michigan 48109

Management Science, 2001, vol. 47, issue 11, 1476-1487

Abstract: 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.

Keywords: Advertising Models; Optimal Control; Contagion; Diffusion Models; S-Shaped Response (search for similar items in EconPapers)
Date: 2001
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Citations: View citations in EconPapers (33)

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