3.3.2.4.1 Open Ended Responses
Responses to open ended questions are straight forward to analyze as each bid amount (maximum WTP) may be taken to correspond to a point on the demand curve. The mean and median of the sample are computed mathematically from the sample data. The sample is assumed to be representative of the population and the distribution of WTP values in the sample is assumed to be similar to that of the population so that the expected sample mean and median are good estimates of the population mean and median.
To obtain the aggregate population WTP, the sample mean or median is multiplied by the population size. This is equivalent to the horizontal summation of individual WTP across all individuals who constitute the population. For private goods, individual demand curves are horizontally summed to obtain the market demand curve. For public goods, theory suggests that the summation of the individual demand curves should be vertical (Bradford, 1970; Mitchell & Carson, 1989). The WTP estimates may be reported as; mean WTP per household (or per person), aggregate WTP per year, or as a
net present value8, depending on how the valuation question is framed and the researcher’s requirements.
3.3.2.4.2 Binary Choice Responses
Binary choice responses cannot be processed in the same way as responses to open- ended questions. “The mean and median WTP measures for [dichotomous choice] responses are inferred from the underlying statistical distribution of the probability that respondents say yes or no to the different bid amounts used” (Brouwer & Bateman 2005, p. 3). Pate and Loomis (1997) state that, the dichotomous structure of the dependent variable (yes/no responses) requires the use of a non-linear probability model such as the logit model which is most commonly used in CV studies. Hanemann (1984) developed the first economic-theoretic framework for the calculation of the mean and median WTP estimates from dichotomous choice response data.
Literature suggests two utility theoretic approaches for dichotomous choice models; the ‘utility difference’ and ‘tolerance approach’ (Duffield & Patterson, 1991). The utility difference theoretic model which allows for the specification of an indirect utility function was developed by Hanemman (1984). Cameron (1988) adopted the tolerance approach which involves specifying a functional form for the expenditure function.
The tolerance approach assumes that each individual has a maximum amount they are willing to pay (WTP) for a particular change in environmental quality improvement and would answer “yes” to a valuation question if presented with a bid amount less that WTP or “no” otherwise. The probability that an individual will say “yes” to the bid amount ‘x’ [Pr(yes|x)] is modelled as a function of the bid amount ‘x’. Pr(yes|x) is given by the formula (Duffield & Patterson, 1991):
Pr(yes|x) = Pr [WTP>x] = 1 – F(x) (1a)
Where F(.) is the distribution function of WTP values in the population. F(.) usually belongs to a parametric family such as the logistic CDFs (which give the logit model) or
8
The values may also be per person or household per year or some other units such as cubic meters, square meters, hectares etc.
normal CDFs (which give the probit model). Duffield & Patterson estimate mean WTP using the formula:
∫
+ − + − = max 0 ) ln (ln max (1b) 1 1 ) ( X X dX e X WTP E α βBoyle, Welsh and Bishop (1988) argue that using an equation that truncates the expected value of WTP is not correct as this under estimates the value of expected WTPMAX because truncating the integration at XMAX violates the condition that the area under the probability density function (pdf) is exactly equal to one. The area under the (pdf) over the range of integration truncated at XMAX is less than 1. They suggest that the correct range of integration should be from -∞ to ∞.
The utility difference approach starts by specifying indirect utility functions for the status quo and the environmental improvement, and assumes that the individual knows for sure which choice maximizes his/her utility (Hanemann, 1984) and will select a choice that reveals his/her true preferences as a rational agent. The probability that the respondent will say “Yes” to a given bid amount $A is given by the formula of the form (Hanemann, 1984; Lee & Han, 2002; Amirnejad et al., 2006):
Pr (Yes| $A) = Pr [V (q1, Y-A, z, ε1) ≥ V (q0, Y, z, ε0)] (1c)
Where V (q1, Y-A, z, ε1) and V (q0, Y, z, ε0) are the indirect utilities associated with the environmental improvement and the status quo respectively.
Pate and Loomis (1997) use a logistic regression model developed to analyze their data as follows:
log{prob(yes)/1 - prob(yes)} = C0(constant) – C1(log D )– C2(bid) + C3(know) - C4(substitutes)+ C5(SpRec) + C6(member) + C7(age) + C8(sex) + C9(angler) (1d)
The dependent variable is the log of odds or logit, and the Ci’s are the slope coefficients including the intercept. The variables in parentheses are those hypothesised to influence the respondents’ responses to the bid offered. Each coefficient was interpreted as the
change in the log odds (log{prob(yes)/1 - prob(yes)}) or logitassociated with a one-unit change in the independent variable. The logit coefficients were then transformed into WTP coefficients using the method of Cameron (1988) which allows the researcher to rescale the logit equation into the more familiar WTP function (Pate & Loomis’ 1997). The transformation is accomplished by dividing the constant term (intercept) and all of the slope coefficients in the model (other than the bid amount) by the absolute value of the coefficient on the bid amount variable. “This transforms the coefficients in the equation into coefficients with ordinary least squares interpretation, insofar as the estimation of the impact on WTP” (Pate & Loomis, 1997, p. 203). WTP is then estimated using the equation:
WTP = ∑{(mean Xi)*(Ci/C2))} + C1/C2 (log D) (1e)
Bishop, Heberlein, and Kealy (1983) analyzed their data with a logit model of the form:
πi = (1 + eβYi) - 1 (1f)
Where πI is the probability that the ith hunter will say yes to the offered bid; Yi is a vector of explanatory variables; and β is a vector of regression coefficients. In their model (1) the natural logarithm of the bid amount (lnbid) was used as the only explanatory variable in the model.