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Optimal criteria for selecting price discrimination metrics when buyers have log-normally distributed willingness-to-pay

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Abstract

This paper investigates how a monopoly seller should determine the optimal set of pricing variables (pricing metrics) for third-degree price discrimination applications in which buyers have log-normally distributed willingness-to-pay (WTP). In a setup that closely resembles linear and probit regressions, this paper shows that when the monopoly seller is restricted to using one metric and no price discrimination cost exists, the pricing metric that best reduces the residual variance of buyers’ willingness-to-pay is the one that maximizes revenue. Equivalently, the explanatory power of willingness-to-pay is the ordering criterion. This paper also shows that this criterion is not universally true when willingness-to-pay follows other distributions. When the seller incurs price discrimination costs associated with different metrics, the ordering criterion becomes the explanatory power of each pricing metric divided by its cost. This paper also discusses how to apply this model to solve real-world pricing problems with contingent valuation models or using probit regression.

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Notes

  1. In this study, WTP is defined as the largest amount of money that an individual or group will pay for a product or service without being worse off with the purchase than without it.

  2. In this example, readers may consider pricing by “weight” or “distance” as second-degree price discrimination cases. In fact, conventional categorization by second- and third-degree price discrimination is not exhaustive. Economists find different ways to categorize price discrimination (e.g., Png and Lehman 2007). This paper categorizes price discrimination into three cases: (1) buyers are not allowed to change the values of their metrics (3rd degree price discrimination); (2) buyers are allowed to change the values of their metrics, but rarely do so in practice; (3) buyers can and do change metrics values in practice (2nd degree price discrimination). Our model applies well to the first two cases and the DHL example applies to the second case. Specifically, a one-time customer who does not use DHL frequently, essentially, cannot change the distance or the weight of a package. Other examples include the following: eTrade’s users may not change the dollar value or number of their transactions because transaction fee is relatively small, and eBay’s auction sellers are charged a percentage of final auction prices, which are beyond their control. Some apparel retailers price by size, although consumers can self-select different sizes, they won’t choose other sizes because of small price differences.

  3. Montgomery (1997) assumes that demand (or the logarithm of demand) is a linear function of the RHS variables. This setup is different from the discrete choice models and is briefly discussed in Section 3.3. The metric selection problem in this setup is discussed in detail in Huang (2009).

  4. The objective of this paper is to advise marketing decision makers via a data-driven model. Also, competitive equilibrium results are not an appropriate focus because few marketers are equipped with sufficient analytical tools to make rational decisions about the metric selection problem (Shugan 2002). Relaxing the monopoly assumption also conflicts with the normality assumption, which is required for linking the present model to empirical models.

  5. A rigorous model about unfairness in price discrimination should be included as part of the WTP of each type of customer, rather than an additional cost. This is an important topic for future research, but is beyond the scope of the present paper.

  6. Residual Variance = \( \tfrac{SSE}{n - q}; \) \( SSE = (1 - {R^2}) \times n \times \)Prior Variance. n = 777 (the number of sample). q = 30 (the number of independent variables).

  7. If we estimate β 0 and β 1 by the probit model, then we cannot derive both μ and σ from Equation (8). If we try to estimate β 0, β 1 and σ at the same time using the probit model, then the first-order conditions of the maximum likelihood function will be a system of linearly dependent equations.

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Authors and Affiliations

  1. Department of Economics and Department of Information Systems, National University of Singapore, SoC1 #05-21, Lower Kent Ridge Rd., Singapore, Singapore, 117543

    Ke-Wei Huang

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  1. Ke-Wei Huang
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Correspondence to Ke-Wei Huang.

Additional information

This paper is based on my dissertation at New York University. I would like to thank my advisor, Roy Radner, for encouragements and numerous helpful suggestions. I would also like to thank the following people for their comments on the earlier drafts of this paper: Joel Steckel, William Greene, Lorin Hitt, DJ Wu, Ivan Png, Anindya Ghose, Arun Sundararajan, Ying-Ju Chen, the co-editor, the two anonymous associate editors, seminar participants at Workshop on Information Systems and Economics 2006, Stern School of Business, University of Connecticut, Georgia Tech., Wharton School of Business, and National University of Singapore. This project has been supported by Taiwan Merit Scholarship(TMS-094-1-A-043). The usual disclaimer applies

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Huang, KW. Optimal criteria for selecting price discrimination metrics when buyers have log-normally distributed willingness-to-pay. Quant Mark Econ 7, 321–341 (2009). https://doi.org/10.1007/s11129-009-9069-9

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