A mathematical reformulation of the reference price
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A mathematical reformulation of the reference price Kevin D. Dayaratna & P. K. Kannan
Published online: 10 June 2012 # Springer Science+Business Media, LLC 2012
Abstract Reference prices have long been studied in applied economics and business research. One of the classic formulations of the reference price is in terms of an iterative function of past prices. There are a number of limitations of such a formulation, however. Such limitations include burdensome computational time to estimate parameters, an inability to truly account for customer heterogeneity, and an estimation procedure that implies a misspecified model. Managerial recommendations based on inferences from such a model can be quite misleading. We mathematically reformulate the reference price by developing a closed-form expansion that addresses the aforementioned issues, enabling one to elicit truly meaningful managerial advice from the model. We estimate our model on a real world data set to illustrate the efficacy of our approach. Our work is not only important from a modeling perspective, but also has valuable behavioral and managerial implications, which modelers and non-modelers alike should find useful. Keywords Reference price . Logit choice models . Logistic regression . Non-iterative estimation . Heaviside step function . Maximum likelihood estimation . Finite mixture models . Misspecified models
1 Introduction For decades, it has been well understood that people use benchmarks in making purchasing decisions. For example, if a customer at a grocery store sees several brands of a particular product, her choice about which brand to purchase will depend on a variety of factors. Such factors may include how much she has paid for these K. D. Dayaratna (*) Department of Mathematics, University of Maryland, College Park, MD, USA e-mail: [email protected] P. K. Kannan Robert H. Smith School of Business, University of Maryland, College Park, MD, USA
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Mark Lett (2012) 23:839–849
brands in the past as well as her overall experiences having tried those brands before (Hardie et al. 1993). Marketing researchers have used the concept of the “reference price” to quantify how customers incorporate price stimuli experienced during purchase occasions. Some research has modeled reference prices, referring to them as stimulus-based reference prices, based on information available at the time of the purchase (Hardie et al. 1993, Rajendran and Tellis 1994, Mazumdar and Papatla 1995). Other work has modeled the reference price as a weighted average of past prices encountered with varying carry-over weights, appropriately called memory-based reference prices (Lattin and Bucklin 1989, Kalyanaram and Little 1994; Mazumdar and Papatla 1995). There has also been some work incorporating explanatory variables in addition to past prices, such as price trends and market share (Winer 1986; Kalwani et al. 1990, Kalwani and Yim 1992, Kopalle et al. 1996). Such alternative formulations of reference prices are usually the first step in using them as exogenous variables
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