Estimating distributions of willingness to pay for heterogeneous populations.

• Author: Das, Chhandita

1. Introduction

   Discrete choice methods have been used by many disciplines to evaluate trade-offs presented by policy alternatives. Examples of discrete choice applications can be found in the fields of environmental valuation, transportation, marketing, and public health. Published studies have used this approach to, among many other purposes, value impacts of noxious facilities (Opaluch et al. 1993) and water quality (Spencer, Swallow, and Miller 1998) and to assess consumer preferences for medical providers (Bolduc, Lacroix, and Muller 1996) and alternative transportation modes (Gonzalez-Savignat 2004).

   The discrete choice analysis proceeds by presenting the decision makers with choices between alternative outcomes. Each respondent selects the outcome with the attribute combination she most prefers, stating her preferences for the trade-offs implicit among the listed attributes of alternative outcomes. Given a large number of responses, a statistical discrete choice model, such as a logit or probit model, is used to recover the relative weight each attribute is given in the utility function of the respondent. These weights can be monetized by dividing by the marginal utility of income (Hanemann 1984), yielding a willingness to pay (WTP) or willingness to accept (WTA) for changes in an attribute level. Policy makers may use these values to understand how alternative policies affect the welfare of the sample population.

   More recent discrete choice analysis has moved away from straightforward logit or probit models to models that recognize the heterogeneity of preferences over choice alternative attributes. Such models offer greater statistical efficiency and information about the distributional effects of policies, which may be of interest to political decision makers. Several approaches have been taken by researchers to incorporate preference heterogeneity into their analysis. For example, Boxall and Adamowicz (2002) use a latent class model to value the recreational demand for wilderness parks. Breffle and Morey (2000) specify different parametric methods to model heterogeneity in the choice of sites for salmon fishing. These methods include modeling utility as a function of individual characteristics, a random parameters model, and a utility function with a heterogeneous scale parameter.

   A commonly adopted approach is random parameters logit (RPL), also known as random coefficients logit. RPL explicitly accounts for heterogeneity by allowing each individual to have a different preference pattern; instead of estimating a common set of attribute marginal utilities for all respondents, each respondent has his or her own marginal utilities, and the distributions of marginal utilities in the population are estimated. This approach has become popular for several reasons. First, RPL removes three limitations of standard logit by allowing heterogeneity in preferences, unrestricted substitution patterns in multinomial choice situations, and correlations in unobserved factors over time. Second, it is also highly flexible so that it can approximate any random utility model (Train 2003, chap. 6). Finally, compared to logit models with homogeneous parameters, RPL is a more robust specification that captures distributions of preference measures rather than a single preference measure for a representative individual.

   RPL has been used to identify welfare effects on both stated preference and revealed preference data across a range of applications and sub-disciplines. Revelt and Train (1998) estimated a RPL for households' choices of appliances with repeated choices. Layton and Brown (2000) examined the heterogeneity of preferences in mitigating climate change impacts. Morey and Rossmann (2003) used RPL to model the preservation of marble monuments in Washington, D.C. Train (1998) modeled a RPL of anglers' choices of fishing sites. Nahuelhual, Loureiro, and Loomis (2004) studied heterogeneity of preferences for protection of public open space. Brownstone, Bunch, and Train (2000) estimated a mixed logit model merging both stated preference and revealed preference data for alternative-fuel vehicles. Rouwendal and Meijer (2001) estimated a mixed logit model to analyze the preferences of workers with respect to housing, job, and commuting. Anderson, Das, and Tyrrell (2006) modeled a RPL of parking preferences among tourists. Bhat and Sardesai (2006) used RPL to model commuter's mode choice using both revealed preference and stated preference data. Hall et al. (2006) estimated a multinomial logit model with random coefficients to determine the factors that influence consumers' preferences for genetic carrier testing. Berry (1994) and Berry, Levinsohn, and Pakes (1995) applied random coefficients models to estimate market demand in markets with product differentiation. Many more applications can be found in the literature.

   In standard application, RPL yields distributions of marginal utilities, leaving analysts to produce WTP distributions by dividing the distributions of attribute coefficients by the distribution of the alternative cost coefficient. However, calculating a ratio of two distributions can be difficult, especially if the distribution of the denominator, the cost coefficient in this case, has mass close to 0 and can lead to high means and WTP distributions with very thick tails. To address this problem, the RPL model is often specified with a fixed (non-random) cost coefficient; it is straightforward to divide the attribute distribution by a constant to obtain a distribution of WTP. (1)

   An alternative way to avoid the complications stemming from potential heterogeneity in the marginal utility of income is to impose the heterogeneity structure directly on the WTP, rather than on the utility coefficients. This approach adapts Cameron's (1988) censored logistic regression (CLR) model, which estimates a random expenditure function normalizing the cost coefficient to 1 (since the marginal expenditure of a dollar spent is a dollar for everyone), and instead estimates the logit scale parameter. While this "new paradigm" presented a different way to think about welfare measure estimation, few were willing to hand-code likelihood functions to achieve direct measures, which are readily available by transforming results from canned logit routines.

   However, random parameters models significantly complicate the transformation necessary to get WTPs, warranting a reconsideration of Cameron's approach. We show that a random parameters version of CLR yields distributions of WTP for outcome attributes, which facilitate policy formulation more directly and efficiently and require no complicated transformation, as in the case of RPL. However, whether RPL or random parameters CLR predicts better is an empirical question, and that the latter provides easier interpretation may dominate economically unimportant differences in fit. The objectives of this paper are to demonstrate this alternative methodology and to point out its advantages in a sample environmental application. (2)

   Two previous studies estimated identical models for directly estimating distributions of WTP using Bayesian methods. Sonnier, Ainslie, and Otter (2003) analyzed automobile choice as a function of attributes and branding using a RPL model, with the price coefficient normalized to 1 and an estimated scale parameter. In a Monte Carlo analysis resampling from their actual survey design and computing choices based on assumed WTP distributions, they find RPL actually fits best but that direct distribution estimation yields better true WTP recovery; on their actual response data, the WTP model had lower in-sample log-likelihood but higher out-of-sample log-likelihood. Train and Weeks (2005) estimated the reciprocal of the scale parameter as a log-normally distributed price coefficient, which also multiplies the WTP distribution for each attribute, in an analysis of preferences for alternative-fuel cars. They find that RPL fits their data better than estimating distributions of WTP, but that RPL yields incredibly large WTP values; the directly estimated WTP distributions are more sensible. Our paper complements this analysis in three ways. First, it is the first to adopt a classical maximum-likelihood estimation approach in demonstrating the advantages of direct estimation of WTP distributions. Second, our model presentation is based on Cameron's CLR model. Basing our exposition in a familiar model and common estimation paradigm clarifies the similarities and differences introduced by the random parameters implementation with familiar models. Third, our empirical analysis adds to the collective body of evidence on the use and interpretation of models directly estimating distributions of WTP.

   The next section presents the theoretical framework of the RPL model and our alternative model. Section 3 describes the Rhode Island landfill siting application we use to demonstrate the model, and section 4 discusses the results and how they improve upon the homogeneous version of our paper and RPL results for policy purposes.

2. The Theoretical Framework

   We develop our model within a standard contingent choice environment, wherein an individual makes a choice among K hypothetical alternatives, based upon the alternative attributes and the cost of provision.

   Estimating Distributions of Marginal Utilities

   RPL is based on a random utility model and thus estimates distributions of attribute marginal utilities. In this model, the utility obtained by individual i from alternative j is given by

   [U.sub.ij] = [c.sub.ij][[beta].sub.ci] + [x.sub.ij[[beta].sub.i] + [u.sub.ij], (1)

   where [c.sub.ij] is the cost to i of alternative j, [[beta].sub.ci] is i's cost coefficient, [x.sub.ij] is the vector of non-cost attributes for alternative j, and [[beta].sub.i] is the vector of attribute coefficients, which varies randomly in the population. Within the random utility framework, the...