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On Continuous-Time Optimal Advertising Under S-Shaped Response

  • Fred M. Feinberg


    (University of Michigan Business School, 701 Tappan St., Ann Arbor, Michigan 48109)

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    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.

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    Article provided by INFORMS in its journal Management Science.

    Volume (Year): 47 (2001)
    Issue (Month): 11 (November)
    Pages: 1476-1487

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    Handle: RePEc:inm:ormnsc:v:47:y:2001:i:11:p:1476-1487
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