Assuring Finite Moments for Willingness to Pay in Random Coefficient Models

Assuring Finite Moments for Willingness to Pay in Random Coefficient Models


A Daly, RAND Europe / ITS, University of Leeds, UK; S Hess, ITS, University of Leeds, UK; K Train, UC Berkeley, US



Over recent years, there has been growing interest in the use of random coefficients models in the analysis of travel behaviour, primarily in the form of the Mixed Multinomial Logit model (MMNL). While these models were initially used primarily in academic research, they are also gradually making their way into practice. Other than issues related to specification and estimation, which have been addressed in a vast number of papers, the main problem facing actual users of these models is how to generate useful results from such models.

Indeed, apart maybe from the computation of elasticities, there is relatively little interest in individual coefficients, and we are primarily interest in measures calculated on the basis of these coefficients, most notably in the form of willingness-to-pay (WTP) indicators, such as the valuation of travel time savings. This paper specifically looks in detail at ways of computing WTP measures in the presence of random taste heterogeneity. In particular, interest focuses on taste heterogeneity with respect to cost, which is of course the denominator of the WTP measure. Here, the discussion in the paper is split into three parts.

The first part of the paper is concerned with continuous mixture models, such as for example MMNL models based on Normal distributions. Here, we first of all show empirically that the common practice of computing WTP measures on the basis of simulating a ratio of random draws is inadequate. As a next step, we discuss a proof that shows that for many choices of random distributions for the cost coefficients, the distribution of the WTP is not defined. Special cases, such as the Lognormal distribution, are also looked at in detail. Finally, we discuss how the use of individual-specific coefficients, another commonly used approach, has few further benefits in this context.

The second part of the paper looks at models estimated in WTP space as opposed to preference space, thus avoiding the need to work with a ratio of random coefficients. Here, we revisit the issue of the interaction between the scale coefficient and the WTP parameter, and also discuss the situation where our dataset contains multiple cost coefficients or where multiple denominators may be of interest, making standard WTP space approaches inapplicable.

The final part of the paper moves away from the use of continuous mixture models and looks at how the situation is different when relying on discrete mixture approaches. Here, we look at various different approaches, including simple discrete mixtures, latent class models as well as models based on a grid estimation approach.


Association for European Transport