Consequences of using different question formats in contingent valuation: a Monte Carlo study.

I. INTRODUCTION

As with any survey method, the validity of contingent valuation estimates depends on the format of the questionnaire. To assess the value of nonmarket goods, researchers using the contingent valuation method (CVM) have employed various question formats to elicit an individual's willingness-to-pay (WTP) (Mitchell and Carson 1989). The direct question that asks a respondent to report the exact amount he or she is willing to pay has been criticized by CVM researchers as difficult to answer (Hanemann 1985). Two other frequently used formats are the referendum (e.g., Bishop and Heberlein 1979; Boyle 1990; Kristrom 1990; Loomis 1988; Sellar, Stoll and Chavas 1985, 1986) and payment-card (also called check-list) questions (e.g., Cocheba and Langford 1978; Loehman and De 1982; Mitchell and Carson 1989; Smith and Desvousges 1986). The referendum (dichotomous-choice, closed-ended, or take-it-or-leave-it) question starts with a specific dollar amount and asks the respondent whether he or she is willing to pay that amount for the good in question. The payment-card format offers respondents a range of values and asks them to circle the highest amount she or he would be willing to pay. With the referendum format, all that is known is whether the respondent's true WTP is more or less than the offered amount. The payment-card format allows the researcher to know the range in which the WTP lies. Thus, the payment-card format should not require as large a sample as the referendum method to estimate WTP. Cameron and Huppert (1991) recently compared WTP estimates from an actual payment-card format survey to estimates that would have been obtained had the respondents been asked a referendum question. They used Monte Carlo experiments to construct 200 samples by randomizing the offered amounts (bids) among respondents. They used the lower bounds of the payment-card intervals as the offered bids. In another set of the experiments, they used the upper bounds. They controlled for behavior bias by making responses to the bids (referendum) consistent with the actual payment-card responses. So, if the randomly assigned bid was less than what the respondent had circled in the payment-card, the assigned referendum response would be "yes." Cameron and Huppert compared the average WTP from the actual payment-card survey to the average WTP from the simulated samples. The latter is an average of averages, because for each sample (of the 200 samples), an across-respondent average WTP was obtained and then these averages were summed and divided by 200. The conclusions from the two sets of experiments did not vary significantly. Cameron and Huppert concluded from these experiments that referendum questions can easily lead to a wide range of WTP and parameter estimates. This paper redoes the type of analysis carried out by Cameron and Huppert but goes one step further. We ensure that the "true" population WTP model is known by relying on artificially generated Monte Carlo data rather than field data. Random samples were drawn from this population and both the referendum and payment-card models were estimated using data from these samples. Results from the model estimations were then compared. Hence, unlike Cameron and Huppert's (1991) study, neither the payment-card model nor the referendum model was assumed to be the true model. This way, we were able to compare estimates from both models to the true population parameters.

II. ESTIMATING WTP

The usual practice with referendum data is to fit a logit or probit model and integrate the area bounded by the curve (Bishop and Heberlein 1979; Hanemann 1984). However, Cameron (1988) proposed an alternative and simpler model. Cameron's model allows for the estimation of individual's WTP based on, for example, respondent's socioeconomic and demographic characteristics and changes in the quality of the good being valued. Cameron's approach is used in this paper and it utilizes the full information maximum likelihood method to estimate the WTP model (Cameron 1988; Cameron and James 1987).

The ith respondent's true WTP, |Y.sub.i~, is expressed as:

g(|Y.sub.i~) = |X|prime~.sub.i~|Beta~ + |e.sub.i~ |1~

where g(|center dot~) is a function of a known form, |X.sub.i~ is a vector of explanatory variables, |Beta~ is a parameter vector, and |e.sub.i~ is independently normally distributed with mean zero and variance ||Sigma~.sup.2~. The variable |S.sub.i~ takes the value one if the ith respondent says "yes" to a bid |t.sub.i~, and zero if not. The probability of a yes response, Pr||S.sub.i~ = 1~, is given by:

Pr||S.sub.i~ = 1~ = Pr||Y.sub.i~ |is greater than or equal to~ |t.sub.i~ |is equivalent to~ Pr|g(|Y.sub.i~) |is greater than or equal to~ g(|t.sub.i~)~ = Pr(|e.sub.i~/|Sigma~) |is greater than or equal to~ {g(|t.sub.i~) - |X|prime~.sub.i~|Beta~)}/|Sigma~~ = 1 - |Phi~|{g(|t.sub.i~) - |X|prime~.sub.i~|Beta~}/|Sigma~~ |2~

where |Phi~||center dot~~ is the standard normal distribution function. The log-likelihood function is given by:

ln L = |summation of~ {|S.sub.i~ ln(1 - |Phi~||z.sub.i~~) where i = 1 to n + (1 - |S.sub.i~)ln(|Phi~||z.sub.i~~)} |3~

where

|z.sub.i~ = {g(|t.sub.i~) - |X|prime~.sub.i~|Beta~}/|Sigma~ |4~

and ln stands for natural logarithm.

Maximization of |3~ allows for both |Beta~ and |Sigma~ to be estimated due to the presence of g(|t.sub.i~). This is in contrast to the conventional probit model, where |Sigma~ is normalized to one (Cameron 1988).

With the payment-card, the true WTP, |Y.sub.i~, lies between the circled value (|t.sub.il~, lower bound of WTP interval) and the next higher value (|t.sub.iu~, upper bound). The probability that |Y.sub.i~ lies between |t.sub.il~ and |t.sub.iu~, Pr||t.sub.il~ |is less than or equal to~ |Y.sub.i~ |is less than~ |t.sub.iu~~ is given by:

Pr||t.sub.il~ |is less than or equal to~ |Y.sub.i~ |is less than~ |t.sub.iu~~ |is equivalent to~ Pr|g(|t.sub.il~) |is less than or equal to~ g(|Y.sub.i~) |is less than~ g(|t.sub.iu~)~ = |Phi~|{g(|t.sub.iu~) - |X|prime~.sub.i~|Beta~}/|Sigma~~ - |Phi~|{g(|t.sub.il~) - |X|prime~.sub.i~|Beta~}/|Sigma~~. |5~

The corresponding log-likelihood function is given by

ln L = |summation of~ {ln(|Phi~||z.sub.iu~~ - |Phi~||z.sub.i~~)} where i = 1 to n |6~

where

|z.sub.iu~ = {g(|t.sub.iu~) - |X|prime~.sub.i~|Beta~}/|Sigma~

and

|z.sub.il~ = {g(|t.sub.il~) - |X|prime~.sub.i~|Beta~}/|Sigma~.

Equation |6~ is similar to an ordered-probit model (OPM) with known threshold values and can be estimated by LIMDEP's Grouped Data procedure (Greene 1990a). Unlike the OPM where |Sigma~ is normalized to one, both |Beta~ and |Sigma~ can be estimated.

The |Beta~- and |Sigma~-estimates from the referendum and payment-card models will be used to obtain the expected WTP, E||Y.sub.i~~ = E||g.sup.-1~(|Y.sub.i~)~, where …