A Bayes multinomial probit model for random consumer-surplus maximization

Willingness to pay measures derived from discrete choice model not only have a clear economic interpretation, but also are a relevant input for welfare analysis, developing marketing strategies, and policy-making. Because inference on parameter ratios is associated with statistical issues, transforming the original utility maximization problem into a consumer surplus maximization model that provides direct inference on willingness to pay is thus desirable. Recent literature uses a parameter reparameterization from the original preference space to the desired willingness-to-pay space. However, we propose a slight variation to this reparameterization that is based on normalizing the marginal utility of income and that works particularly well for models with a general covariance matrix. (For logit-based models the normalization is equivalent to working with the standard reparameterization in willingness-to-pay space.) In fact, we show that Bayes implementation of the normalization of the marginal utility of income solves well-known problems of current multinomial probit samplers. The estimator that we propose effectively avoids defining proper priors on unidentified parameters as well as identification of priors for and making draws of a constrained covariance matrix, and improves the behavior of the predictive posterior of the choice probabilities. In addition, willingness-to-pay estimates from previous studies can easily be used as proper priors in the proposed Gibbs sampler.