Abstract
We investigate direct and indirect specification of the distribution of consumer willingness-to-pay (WTP) for changes in product attributes in a choice setting. Typically, choice models identify WTP for an attribute as a ratio of the estimated attribute and price coefficients. Previous research in marketing and economics has discussed the problems with allowing for random coefficients on both attribute and price, especially when the distribution of the price coefficient has mass near zero. These problems can be avoided by combining a parameterization of the likelihood function that directly identifies WTP with a normal prior for WTP. We show that the typical likelihood parameterization in combination with what are regarded as standard heterogeneity distributions for attribute and price coefficients results in poorly behaved posterior WTP distributions, especially in small sample settings. The implied prior for WTP readily allows for substantial mass in the tails of the distribution and extreme individual-level estimates of WTP. We also demonstrate the sensitivity of profit maximizing prices to parameterization and priors for WTP.
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Notes
The reservation price is the price that induces indifference between purchase and non-purchase in the category. The equalization price (Swait et al.1993) is the price that induces indifference between two choice alternatives within the category.
We adopt a fully Bayesian approach to inference in this paper. However, the points made apply in the context of hierarchical models independent of the estimation technique.
Another motivation for directly parameterizing the model in WTP terms is that the researcher often has variables such as demographics that may be useful in the characterization of consumer heterogeneity. In such instances it seems more reasonable to build hierarchical regression structures for WTP instead of parameters with less clear interpretation. We do not explore this issue here.
Note that the median of the posterior mean of the individual-level estimates will not be the same as the posterior mean of the median of the population distribution.
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Acknowledgement
The authors would like to thank Peter Rossi, JP Dubé, Jordan Louviere, Kenneth Train, and Greg Allenby for helpful insights. We also thank seminar participants at The Ohio State University, Duke University, University of Michigan and the University of Chicago for providing useful comments.
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Sonnier, G., Ainslie, A. & Otter, T. Heterogeneity distributions of willingness-to-pay in choice models. Quant Market Econ 5, 313–331 (2007). https://doi.org/10.1007/s11129-007-9024-6
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DOI: https://doi.org/10.1007/s11129-007-9024-6