On Continuous-Time Optimal Advertising Under S-Shaped Response

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 thatS-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.

[1]  G. Thompson,et al.  Optimal control theory : applications to management science , 1984 .

[2]  Gerald L. Thompson,et al.  Oligopoly Models for Optimal Advertising When Production Costs Obey a Learning Curve , 1983 .

[3]  John D. C. Little,et al.  Feature Article - Aggregate Advertising Models: The State of the Art , 1979, Oper. Res..

[4]  Maurice W. Sasieni,et al.  Optimal Advertising Strategies , 1989 .

[5]  Fred M. Feinberg,et al.  Pulsing Policies for Aggregate Advertising Models , 1992 .

[6]  Maurice W. Sasieni,et al.  Optimal Advertising Expenditure , 1971 .

[7]  Bart J. Bronnenberg.  Advertising Frequency Decisions in a Discrete Markov Process under a Budget Constraint , 1998 .

[8]  Minhi Hahn,et al.  Pulsing in a Discrete Model of Advertising Competition , 1991 .

[9]  J. Simon.  The shape of the advertising response function , 1980 .

[10]  Suresh P. Sethi,et al.  A Survey of the Maximum Principles for Optimal Control Problems with State Constraints , 1995, SIAM Rev..

[11]  S. Sethi.  Dynamic Optimal Control Models in Advertising: A Survey , 1977 .

[12]  Minhi Hahn,et al.  Advertising cost interactions and the optimality of pulsing , 1991 .

[13]  S. Ożga,et al.  Imperfect Markets Through Lack of Knowledge , 1960 .

[14]  Frank M. Bass,et al.  Testing Distributed Lag Models of Advertising Effect , 1972 .

[15]  Suresh P. Sethi,et al.  Optimal advertising policy with the contagion model , 1979 .

[16]  J. P. Gould,et al.  Diffusion Processes and Optimal Advertising Policy , 1976 .

[17]  Ambar G. Rao,et al.  Modeling Response to Advertising and Pricing Changes for “V-8” Cocktail Vegetable Juice , 1986 .

[18]  Vijay Mahajan,et al.  Advertising Pulsing Policies for Generating Awareness for New Products , 1986 .

[19]  Richard F. Hartl,et al.  A simple proof of the monotonicity of the state trajectories in autonomous control problems , 1987 .

[20]  Richard D. Edie,et al.  On the optimal control of the Vidale-Wolfe advertising model , 1997 .

[21]  Suresh P. Sethi,et al.  Optimal Control of the Vidale-Wolfe Advertising Model , 1973, Operational Research.

[22]  R. Hartl,et al.  Dynamic Optimal Control Models in Advertising: Recent Developments , 1994 .

[23]  Ram C. Rao,et al.  Estimating Continuous Time Advertising-Sales Models , 1986 .

[24]  M. L. Vidale,et al.  An Operations-Research Study of Sales Response to Advertising , 1957 .