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Explicit Solutions of Optimization Models and Differential Games with Nonsmooth (Asymmetric) Reference-Price Effects

School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Department of Industrial Engineering and Management, Ben-Gurion University, Beer-Sheva 84105, Israel
School of Management, Ben-Gurion University, Beer-Sheva 84105, Israel
fibich{at}math.tau.ac.il
ariehg{at}bgumail.bgu.ac.il
odedl{at}bgumail.bgu.ac.il

Models in marketing with asymmetric reference effects lead to nonsmooth optimization problems and differential games which cannot be solved using standard methods. In this study, we introduce a new method for calculating explicitly optimal strategies, open-loop equilibria, and closed-loop equilibria of such nonsmooth problems. Application of this method to the case of asymmetric reference-price effects with loss-aversive consumers leads to the following conclusions: (1) When the planning horizon is infinite, after an introductory stage the optimal price stabilizes at a steady-state price, which is slightly below the optimal price in the absence of reference-price effects. (2) The optimal strategy is the same as in the symmetric case, but with the loss parameter determined by the initial reference-price. (3) Competition does not change the qualitative behavior of the optimal strategy. (4) Adopting an appropriate constant-price strategy results in a minute decline in profits.

Subject classifications: Marketing, pricing:optimal pricing with reference effects; Games, differential:nonsmooth differential games; Optimal control, applications: nonsmooth problems.
History: Received December 2000; revision received April 2002; accepted October 2002.

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