International Choice Modelling Conference, International Choice Modelling Conference 2015

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Willingness to Pay-Inference in the Absence of Rejected Propositions
Caspar Chorus

Last modified: 11 May 2015


Inference of Willingness to Pay (WtP) has a high priority in a variety of fields, among researchers and practitioners alike. Classical examples include consumers’ WtP for quality improvements of consumer goods; travelers’ WtP for travel time reductions; patients’ WtP for more effective treatments; citizens’ WtP for reductions in flood risk, to name just a few. Motivated by this broad interest in WtP, a wide variety of methods for WtP-inference have been developed over the years. In terms of data collection, it is widely acknowledged that – if available – Revealed Preference (RP) data provide the preferred empirical context for such inference, given its high level of external validity. However, a well-known drawback of RP data is that it is often more noisy and less detailed, than data collected in Stated Preference-settings such as discrete choice experiments. This paper focuses on a particular type of data-imperfection which may occur in RP-settings, and provides a new method for WtP-inference in the context of such imperfect data.Before I discuss the type of data-imperfection that is the focus of this paper, it should be noted that most, if not all, types of data that are being used for WtP-inference can be conceptualized as a series of propositions to individuals to ‘buy’ an increased level of ‘quality’ (e.g., a larger smartphone-screen, a lower travel time, a better medicine, or a higher dike) for a given price. When there is sufficient variation in propositions, decision-makers’ WtP can be inferred from observed patterns of acceptance and rejection of different propositions. Acceptance of a proposition helps to determine a lower bound of WtP, while a rejected proposition helps determine an upper bound. Jointly, observations of accepted and rejected propositions enable the analyst to pinpoint the mean WtP for a given sample.In some cases, however, only accepted propositions are observed. Take for example the situation where only choices of individuals who are willing to pay a premium for a larger smartphone screen are observed. Or the situation where only travelers on a faster but more expensive toll road are observed. Such a failure to observe rejected propositions may be due to particularities – or even flaws – of a data set or data collection process; or they may be the consequence of intrinsic factors. An example of the latter may be that the data are supplied by an entity (e.g., firm or road operator) which only provides the ‘premium product’ (i.e., smartphone with larger screen, faster toll road) and hence has no data available concerning individuals who chose not to buy the premium product. Conceptually and econometrically speaking, it appears impossible to infer WtP from these observations of accepted propositions – or: choices for the premium product – alone. At best one might expect to be able to derive lower bounds of WtP from such observations. In this paper, I develop a method that can be used to infer WtP in the absence of observations concerning rejected propositions. The method is fairly straightforward conceptually and econometrically, and is based on three assumptions of which the first two are trivial: first, WtP is assumed to vary across individuals; second, propositions (i.e., offers to pay a particular price for a particular quality increase) are assumed to vary as well. That is, not every individual receives the same proposition; third, the distribution of propositions that are presented to individuals is related to the distribution of WtP in the population. A possible relation would be that the distribution of propositions is the same as, but independent from, the distribution of WtP; that is, individuals with a higher or lower WtP are equally likely to receive a proposition with a given price. Another possible relation might be that the distribution of propositions is the same as the distribution of WtP, and that the two distributions are to some extent correlated: that is, individuals with a higher WtP are more likely to receive a higher priced proposition. Using a series of examples, I argue in the paper that this set of assumptions – of which the third one might at first seem somewhat unrealistic – are likely to be reasonably accurate in many practical applications. Given this set of assumptions, the likelihood of observing an accepted proposition can be conceived as being a function of i) the likelihood that a particular proposition is drawn from the distribution of propositions, and ii) the likelihood that an individual is drawn from the WtP-distribution, whose WtP is at least equal to the price embedded in the proposition. Using a simulated dataset, I show that this approach is able to recover the ‘true’ distribution of WtP, based on a sample of accepted propositions only. Subsequently, I use a real (RP) data set of shopping location choices to test the method empirically. In this context, WtP is framed as the amount of extra travel time someone is willing to ‘pay’ in order to reach a larger shopping center; that is, WtP is measured in seconds per square meter. I first infer WtP by means of estimating a conventional Logit-model on the full dataset of shopping location choices., and subsequently I take the subset of observations where an individual was observed to select a shopping center that is further away than a randomly selected non-chosen shopping center, but also same larger. These imperfect data thus contain only ‘accepted propositions’ to ‘buy’ additional floorspace for a given amount of extra travel time. On these imperfect data, I am able to infer a WtP distribution whose mean (median) is reasonably (very) close to the WtP inferred from the Logit-model using the full data.To conclude, I formulate a series of further research steps that need to be taken, before more definitive conclusions about the usefulness and limitations of the approach can be drawn.Wordcount: 975

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