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On Continuous-Time Optimal Advertising Under S-Shaped Response

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  • Fred M. Feinberg

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

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.

Suggested Citation

  • Fred M. Feinberg, 2001. "On Continuous-Time Optimal Advertising Under S-Shaped Response," Management Science, INFORMS, vol. 47(11), pages 1476-1487, November.
  • Handle: RePEc:inm:ormnsc:v:47:y:2001:i:11:p:1476-1487
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    File URL: http://dx.doi.org/10.1287/mnsc.47.11.1476.10246
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Eftichios Sartzetakis & Anastasios Xepapadeas & Emmanuel Petrakis, "undated". "The role of information provision as a policy instrument to supplement environmental taxes: Empowering consumers to choose optimally," DEOS Working Papers 1012, Athens University of Economics and Business.
    2. Navdeep Sahni, 2015. "Effect of temporal spacing between advertising exposures: Evidence from online field experiments," Quantitative Marketing and Economics (QME), Springer, vol. 13(3), pages 203-247, September.
    3. Thales S. Teixeira & Michel Wedel & Rik Pieters, 2010. "Moment-to-Moment Optimal Branding in TV Commercials: Preventing Avoidance by Pulsing," Marketing Science, INFORMS, vol. 29(5), pages 783-804, 09-10.
    4. Beltran-Royo, C. & Zhang, H. & Blanco, L.A. & Almagro, J., 2013. "Multistage multiproduct advertising budgeting," European Journal of Operational Research, Elsevier, vol. 225(1), pages 179-188.
    5. Toker Doganoglu & Daniel Klapper, 2006. "Goodwill and dynamic advertising strategies," Quantitative Marketing and Economics (QME), Springer, vol. 4(1), pages 5-29, March.
    6. Marshall Freimer & Dan Horsky, 2012. "Periodic Advertising Pulsing in a Competitive Market," Marketing Science, INFORMS, vol. 31(4), pages 637-648, July.
    7. Du, Rong & Hu, Qiying & Ai, Shizhong, 2007. "Stochastic optimal budget decision for advertising considering uncertain sales responses," European Journal of Operational Research, Elsevier, vol. 183(3), pages 1042-1054, December.
    8. Nobuhiko Terui & Masataka Ban & Greg M. Allenby, 2011. "The Effect of Media Advertising on Brand Consideration and Choice," Marketing Science, INFORMS, vol. 30(1), pages 74-91, 01-02.
    9. Eftichios Sartzetakis & Anastasios Xepapadeas & Emmanuel Petrakis, 2012. "The Role of Information Provision as a Policy Instrument to Supplement Environmental Taxes," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 52(3), pages 347-368, July.
    10. Vahideh Sadat Abedi, 2017. "Allocation of advertising budget between multiple channels to support sales in multiple markets," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 68(2), pages 134-146, February.
    11. Sofia Berto Villas-Boas & J. Miguel Villas-Boas, 2008. "Learning, Forgetting, and Sales," Management Science, INFORMS, vol. 54(11), pages 1951-1960, November.
    12. Huang, Jian & Leng, Mingming & Liang, Liping, 2012. "Recent developments in dynamic advertising research," European Journal of Operational Research, Elsevier, vol. 220(3), pages 591-609.
    13. Chen, Chien-Ming & van Dalen, Jan, 2010. "Measuring dynamic efficiency: Theories and an integrated methodology," European Journal of Operational Research, Elsevier, vol. 203(3), pages 749-760, June.
    14. El Ouardighi, Fouad & Feichtinger, Gustav & Grass, Dieter & Hartl, Richard & Kort, Peter M., 2016. "Autonomous and advertising-dependent ‘word of mouth’ under costly dynamic pricing," European Journal of Operational Research, Elsevier, vol. 251(3), pages 860-872.
    15. Mesak, Hani I. & Ellis, T. Selwyn, 2009. "On the superiority of pulsing under a concave advertising market potential function," European Journal of Operational Research, Elsevier, vol. 194(2), pages 608-627, April.
    16. Demetrios Vakratsas & Fred M. Feinberg & Frank M. Bass & Gurumurthy Kalyanaram, 2004. "The Shape of Advertising Response Functions Revisited: A Model of Dynamic Probabilistic Thresholds," Marketing Science, INFORMS, vol. 23(1), pages 109-119, April.

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