Modeling starting point bias as unobserved heterogeneity in contingent valuation surveys: an application to air pollution.

Contingent valuation (CV) studies are increasingly used to evaluate nonmarket goods and services, environmental goods or health care programs, and thus affect decision making and policy formulation. These studies raise methodological issues that have been extensively documented in Mitchell and Carson (1989), and critically assessed in Hausman (1993). In particular, attention has been paid to biases, such as systematic nonrandom sources of error between the "true" value and the stated value of the respondent.

Willingness to pay (WTP) elicitation is certainly a central issue in the CV literature on these potential discrepancies between stated and "true" WTP values. The two oldest and most widely used formats to elicit WTP are open-ended questions (Davis 1964) and dichotomous (closed-ended or referendum) questions (Randall, Ives, and Eastman 1974). Each of these has pros and cons (Loomis 1990). The National Oceanic and Atmospheric Administration (NOAA) panel recommends the dichotomous choice (DC) introduced by Bishop and Heberlein (1979): "Would you be willing to contribute (or be taxed) D dollars to cover the cost of ...", where D is a bid proposed to respondents. The main arguments in favor of the DC format are that it mimics the decision-making tasks individuals face in everyday life (Herriges and Shogren 1996) and that it reduces incentives for strategic responses. Further, a question with a yes or no answer is easier to answer than a question requiring independent quantitative estimation on goods unfamiliar to the respondent. As a consequence, closed-ended question formats have been fairly widely used in CV surveys. However, the DC method reveals little about individuals' WTP and requires large sample sizes to reach a sufficient level of precision (Hanemann and Kanninen 1999).

Therefore, Hanemann (1985) and Carson (1985) proposed adding a follow-up question to improve efficiency of DC questionnaires, known as the double-bounded model. This amounts to asking the respondent for a second bid. More recently, Milon (1989) and Green et al. (1998) propose adding an open-ended question rather than a DC follow-up question since "it provides far more information on WTP as well as information on plausibility of the response" (Green et al. 1998, p. 111).

However, the key disadvantage of starting with a DC question is the influence that the value proposed may have on the individual's answer to the subsequent question: different initial bids may lead to respondents stating different WTP. As a result, different initial bids can lead to different welfare estimates and, consequently, to different public decisions. This initial bid effect is known as the starting point bias (Tversky and Kahneman 1974) or, more generally, the anchoring bias.

This phenomenon is observed in many psychological studies dealing with beliefs about purely objective quantities: the length of the Amazon or the height of the tallest redwood (Tversky and Kahneman 1974), the number of physicians and surgeons listed in the local yellow pages (Wilson et al. 1996), the monthly number of gallons of gasoline used by a car owner, the yearly number of inches of rainfall at the wettest spot on earth (Green et al. 1998), the yearly average mileage traveled by car (O'Conor, Johannesson, and Johansson 1999), etc. The initial bid therefore plays a part in basic cognitive constructs used for decisions under uncertainty. When CV studies elicit monetary values, the initial bid may be considered by the respondent as an indication of the quality of the contingent good. The respondent may therefore think the bid gives an approximate range of the "correct" value of this good. Indeed, "confronted with a dollar figure in a situation where he is uncertain about an amenity's value, a respondent may regard the proposed amount as conveying an approximate value of the amenity's true value and anchor his WTP amount on the proposed amount" (Mitchell and Carson 1989).

To deal with this problem, Herriges and Shogren (1996) consider a double-bounded model in which the respondents combine their true WTP with the first bid amount to form a revised WTP. They show that when anchoring occurs, the estimations provided by the usual double-bounded model are largely biased. Their analysis as well as all subsequent analyses on anchoring concludes that this effect must be corrected for, in order to obtain reliable WTP estimates. However, anchoring has up to now always been considered as a process that homogeneously affects respondents: all individuals anchor in the same way.

This is a particularly strong assumption that may lead to misleading results if individuals react differently to the proposed bids, since individual heterogeneity can lead to major misspecifications. The possible existence of individual heterogeneity is mentioned by various authors such as Herriges and Shogren (1996), Lechner, Rozan, and Laisney (2003), or Green et al. (1998): "one might expect the strongest anchoring effects when primitive beliefs are weak or absent, and the weakest anchoring effects when primitive beliefs are sharply defined" (Green et al. 1998, p. 95). Although very intuitive, this affirmation is hard to test since "primitive beliefs" is not a clearly defined notion, making it difficult to identify characteristics that explain anchoring.

In this article, we examine the possibility of heterogeneous anchoring when a first DC question is used with an open-ended follow-up. Our contribution regarding this assumption is twofold. First, we show both analytically and by way of Monte Carlo simulations that using a misspecified homogeneous anchoring model when the true anchoring process is heterogeneous leads to biased parameter and mean WTP estimates. Second, in order to solve this problem, we propose an econometric model in which anchoring is heterogeneous and related to unobserved heterogeneity, which basically involves defining the starting point bias as a random variable distributed in the population.

The article is divided into three sections. A starting point bias DC model with an open-ended follow-up question and heterogeneous anchoring is developed and studied in the section "Starting point bias as an unobserved effect." The section "An application" applies this model to a CV survey dealing with air quality improvements. The section "Conclusion" concludes the article.

Starting Point Bias as an Unobserved Effect

To model the starting point bias, we adopt the general framework of Herriges and Shogren (1996), based on the literature on bidding games (Randall, Ives, and Eastman 1974; Bishop and Welsh 1985 for instance). It assumes that an uncertain respondent may consider the starting bid as providing information on the "correct" WTP value. In this case, the respondent combines his true WTP with the starting bid to form a revised WTP that he ends up stating as his WTP. Stated WTP may then be upwardly biased if the starting point is set above true WTP and downwardly biased if the starting point is set below true WTP. This approach covers extreme cases, from no anchoring (i.e., the respondent is not influenced by the initial bid) to complete anchoring (i.e., the respondent completely ignores his true WTP). It has been widely used in the literature because prior WTP and starting bid are usually combined in quite a simple way: either linear combination or weighted average.

Econometric Models

Let us call [W.sup.*.sub.i1] the unobserved respondent i's true estimate of his WTP, which is defined as follows:

(1) [W.sup.*.sub.i1] = [X.sub.i][beta] + [u.sup.i]

where [X.sub.i] is a set of explanatory variables which represents respondent i's tastes, [u.sub.i] are normally and independently distributed (NID) error terms, and [beta] and [sigma] are unknown parameters. This article focuses on a particular elicitation mechanism to gather information on this true WTP, i.e., a DC question with an open-ended follow-up.

This kind of elicitation design can be econometrically modeled as follows. When the respondent answers the bid ([b.sub.i]), a censoring rule links the observed answer [W.sub.i1] = 0, or 1 (respectively "no" or "yes") to the unobserved variable [W.sup.*.sub.i1]:

(2)

[W.sub.i1] = 1 if [W.sup.*.sub.i1] [greater than or equal to] [b.sub.i] and [W.sub.i1] = 0 otherwise.

An open-ended follow-up question that provides more information on individuals' WTP values is then proposed and [W.sub.i2] is the WTP given by respondent i.

This format, like the more widely used double-bounded DC format, may induce a starting point bias. That is, respondents may view the bid [b.sub.i] as providing information on the "correct" WTP and combine their true WTP, [W.sup.*.sub.i1], with [b.sub.i] to form a revised WTP, [W.sup.i2]. (1) Following Herriges and Shogren (1996), we model this revised WTP as a convex combination of the true WTP and the bid:

(3) [W.sub.i2] = (1 - [gamma])[W.sup.*.sub.i1] + [gamma][b.sub.i], [gamma] [member of] [0, 1]

where [gamma] is a constant parameter for the whole sample, meaning that anchoring is assumed to be homogeneous. Combining equations (1) and (3) leads to:

(4) [W.sub.i2] = (1 - [gamma])[X.sub.i][beta] + [gamma][b.sub.i] + [u.sub.i](1 - [gamma]).

[W.sub.i2] contains all information required to estimate the set of parameters. This model (referred to as Model I) is easily estimated by maximum likelihood using the following likelihood:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]