Advertising for a new product introduction: A stochastic approach

We formulate a stochastic extension of the Nerlove and Arrow’s advertising model in order to analyze the problem of a new product introduction. The main idea is to introduce some uncertainty aspects in connection both with the advertising action and the goodwill decay, in order to represent the random consequences of the advertising messages and of the word-of-mouth publicity, respectively. The model is stated in terms of the stochastic optimal control theory and a general study is attempted using the stochastic Maximum Principle.Closed form solutions are obtained under linear quadratic assumptions for the cost and the reward functions. Such optimal policies suggest that the decision-maker considers both the above mentioned phenomena as opportunities to increase her/his final reward. After stating some general features of the optimal solutions, we analyze in detail three extreme cases, namely the deterministic model and the stochastic models with either the word-of-mouth effect only, or the lure/repulsion effect only. The optimal policies provide us with some insight on the general effects of the advertising action.

[1]  Suresh P. Sethi,et al.  Deterministic and Stochastic Optimization of a Dynamic Advertising Model , 1982 .

[2]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

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

[4]  G. Zaccour,et al.  Cooperative Advertising in a Marketing Channel , 2000 .

[5]  Ke Te Le Mei Ru He Deng Yi.  Marketing management: analysis. planning. implementation and control , 2000 .

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

[7]  Charles S. Tapiero,et al.  Optimum Advertising and Goodwill under Uncertainty , 1978, Oper. Res..

[8]  Charles S. Tapiero,et al.  A Generalization of the Nerlove-Arrow Model to Multi-Firms Advertising under Uncertainty , 1979 .

[9]  K. Arrow,et al.  OPTIMAL ADVERTISING POLICY UNDER DYNAMIC CONDITIONS , 1962 .

[10]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[11]  Xun Yu Zhou,et al.  Stochastic Linear Quadratic Regulators with Indefinite Control Weight Costs. II , 2000, SIAM J. Control. Optim..

[12]  Lars Tyge Nielsen.  Pricing and Hedging of Derivative Securities , 1999 .

[13]  A. Bensoussan.  Stochastic Control of Partially Observable Systems , 1992 .

[14]  K. Raman,et al.  Stochastically optimal advertising policies under dynamic conditions: the ratio rule , 1990 .

[15]  Bruno Viscolani,et al.  New product introduction: goodwill, time and advertising cost , 2002, Math. Methods Oper. Res..

[16]  Daniela Favaretto,et al.  Optimal communication mix to maximize brand image , 2002 .

[17]  Charles S. Tapiero,et al.  A Stochastic Model of Consumer Behavior and Optimal Advertising , 1982 .

[18]  G. Zaccour,et al.  Equilibrium Pricing and Advertising Strategies in a Marketing Channel , 1999 .

[19]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[20]  Charles S. Tapiero.  Applied stochastic models and control in management , 1988 .

[21]  Charles S. Tapiero,et al.  On-Line and Adaptive Optimum Advertising Control by a Diffusion Approximation , 1975, Oper. Res..

[22]  Charles S. Tapiero,et al.  Stochastic diffusion models with advertising and word-of-mouth effects , 1983 .

[23]  John B. Moore,et al.  Indefinite Stochastic Linear Quadratic Control and Generalized Differential Riccati Equation , 2002, SIAM J. Control. Optim..

[24]  P. Kotler.  Marketing Management: Analysis, Planning, Implementation and Control , 1972 .