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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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    File URL: http://dx.doi.org/10.1287/mnsc.47.11.1476.10246
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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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    1. Ram C. Rao, 1986. "Estimating Continuous Time Advertising-Sales Models," Marketing Science, INFORMS, vol. 5(2), pages 125-142.
    2. Gustav Feichtinger & Richard F. Hartl & Suresh P. Sethi, 1994. "Dynamic Optimal Control Models in Advertising: Recent Developments," Management Science, INFORMS, vol. 40(2), pages 195-226, February.
    3. Hartl, Richard F., 1987. "A simple proof of the monotonicity of the state trajectories in autonomous control problems," Journal of Economic Theory, Elsevier, vol. 41(1), pages 211-215, February.
    4. Fred M. Feinberg, 1992. "Pulsing Policies for Aggregate Advertising Models," Marketing Science, INFORMS, vol. 11(3), pages 221-234.
    5. Vijay Mahajan & Eitan Muller, 1986. "Reply—Reflections on Advertising Pulsing Policies for Generating Awareness for New Products," Marketing Science, INFORMS, vol. 5(2), pages 110-111.
    6. Jinn-Tsair Teng & Gerald L. Thompson, 1983. "Oligopoly Models for Optimal Advertising When Production Costs Obey a Learning Curve," Management Science, INFORMS, vol. 29(9), pages 1087-1101, September.
    7. Vijay Mahajan & Eitan Muller, 1986. "Advertising Pulsing Policies for Generating Awareness for New Products," Marketing Science, INFORMS, vol. 5(2), pages 89-106.
    8. S. A. Ozga, 1960. "Imperfect Markets Through Lack of Knowledge," The Quarterly Journal of Economics, Oxford University Press, vol. 74(1), pages 29-52.
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