Stated Preference Approach

The valuation of public goods and resources, such as drinking water quality, is based on welfare economics. Markets cannot efficiently allocate public goods or resources with pervasive externalities, or for which property rights are not clearly defined (Haab and McConnell, 2003). An example of this market failure is agricultural activities for food production, where farmers do not take account of the negative effects of their farming practices on water quality, such as contamination from agrochemicals, erosion from certain land practices and deforestation for land use expansion.

An improvement in resource allocation requires the measurement of benefits and costs, and the former need to exceed the latter. Economists have devised and refined methods for measuring benefits and costs; and thus whether and to what extent resources are being allocated efficiently. Two basic approaches are used for benefit estimation: indirect or behavioural methods (i.e. revealed preferences) and direct or stated preference methods. The need for statistical inference and econometrics arises because individual actions, whether observed behaviours or

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responses to hypothetical questions, almost never reveal precisely the economic value that needs measurement. First one needs to infer a preference function such as a utility function, or behavioural relation such as a demand function, and then benefit measures such as willingness to pay are calculated (Rodriguez et al., 2009).

The process of benefit estimation begins with the measurement of the net change in income that is equivalent to or compensates for changes in the quantity or quality of public goods; and then these are expanded to the relevant population. To start, let 𝑢 (x, q) be the individual preference function, where x = 𝑥1… 𝑥𝑚 is the vector of private goods and q = 𝑞1… 𝑞𝑛 is the vector of public goods. Individuals choose their x but their q is exogenous. The individual maximises utility subject to income 𝑦 and 𝑝 represents price. The indirect utility function 𝑉 (p, q, 𝑦), is given by (equation 3.1)

𝑉 (𝑝, 𝑞, 𝑦) = 𝑚𝑎𝑥_{𝑥 {𝑢 (x, q)|p ∙ x ≤ 𝑦 }}

The minimum expenditure function 𝑚 (p, q, 𝑢) is dual to the indirect utility function (equation 3.2)

𝑚 (p, q, 𝑢) = 𝑚𝑖𝑛_𝑥 {p ∙ x | 𝑢 (x, q) ≥ 𝑢 }

The indirect utility function and the expenditure function provide the theoretical structure for welfare estimation. For stated preferences approaches, the changes in these functions are needed. For revealed preference approaches, a conceptual path from observations on behaviour to these constructs is needed. There are two equally valid ways of describing money welfare measures: compensating and

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equivalent variation, and WTP and WTA. They measure the same phenomenon, the increment in income that makes an individual indifferent to a change in some public good.

WTP is the maximum amount of income an individual will pay in exchange for an improvement in circumstances, or to avoid a decline in circumstances. WTA is the minimum amount an individual will accept for a decline in circumstances or to forego an improvement in circumstances. Compensating variation is the amount of income paid or received that leaves the individual at the initial level of wellbeing, and equivalent variation is the amount of income paid or received that leaves the individual at the final level of wellbeing. WTP and WTA relate to the right to a utility level; so for this study where individuals are required to pay to achieve a higher wellbeing, the right to that level of wellbeing lies elsewhere. Equivalent and compensating variation rely on the initial versus final wellbeing for their distinction. Recent practice adopts WTP and WTA terms chiefly because contingent valuation surveys use the terms, so we follow this focus here (Haab and McConnell, 2003; Smith, 2006).

WTP is preferred over WTA for several reasons, including the fact that WTA always exceeds WTP in empirical settings but not in behavioural methods, the general belief that WTA is not an incentive-compatible measure, and the recommendations of the NOAA that researchers should measure WTP. Thus, this study utilises the WTP approach to valuation. For an individual, WTP is the amount of income that compensates for an increase in the public good (equation 3.3):

𝑉 (p, q*_{, 𝑦 − 𝑊𝑇𝑃) = 𝑉 (p, q, 𝑦)}

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𝑊𝑇𝑃 = 𝑚 (p, q, 𝑢) − 𝑚 (p, q*_{, 𝑢) 𝑤ℎ𝑒𝑛 𝑢 = 𝑉 (p, q, 𝑦)}

The dichotomous choice approach to asking the question that leads directly to WTP has become the presumptive method of elicitation for CV practitioners. This is mainly due to this question format being incentive-compatible in theory. Since the contingent valuation responses are binary variables, yes or no, a statistical model appropriate for a discrete dependent variable is necessary. The aim is to estimate a probability distribution for the true WTP in a CV setting, using information on upper and lower bounds. The CV responses are analysed using statistical models, but the models need to make sense from the point of view of economic theory. This places significant restrictions on the statistical models that can be used (Hanemann et al.; Haab and McConnell, 2003).

Parametric models provide the most information from an economic point of view, but they can be fragile if misspecified. Non-parametric models are more robust and offer greater flexibility in the shape of the response function, but they provide less economic information (Hanemann et al., 1999). Thus, a parametric model is used to estimate the preference function that allows the calculation of WTP given the estimated parameters. The basic model for analysing dichotomous CV responses, including parameter estimation, is the random utility model (Hanemann, 1984; Hanley, 1997). This model assumes that while the individual knows their own preferences, these are not observable by the researcher. In the CV case there are two choices, so that the indirect utility for respondent 𝑗 can be written:

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Where 𝑖 = 1 is the state that prevails when the CV scenario is implemented, that is, the final state, and 𝑖 = 0 for the status quo. The factors that determine utility are 𝑦𝑗 (for income), 𝑧𝑗 (for household characteristics and attributes of the given scenario) and 𝜀𝑖𝑗 (a component of random preferences known to the individual respondent but not observed by the researcher; Hanemann, 1984). Due to these unobserved random preferences researchers can only make probability statements about yes and no. The probability of a yes response is the probability that the respondent thinks that he or she is better off in the proposed scenario, even with the required payment, so that 𝑢1 > 𝑢0. For respondent 𝑗, where 𝑡𝑗 is the bid amount offered to the jth respondent, the general probability estimation is:

Pr(𝑦𝑒𝑠𝑗) = Pr [𝑣1(𝑦𝑗− 𝑡𝑗, 𝑧𝑗) + 𝜀1𝑗 > 𝑣0(𝑦𝑗, 𝑧𝑗) + 𝜀0𝑗]

This probability statement is too general for parametric estimation, so modelling decisions are needed. In order to understand the decision to answer positively, the utility difference between the yes and no responses needs to be examined. That is, the probability of a certain response is examined as a function of the differences in the utilities at the base and final states. Given that the random term can be rewritten as 𝜀𝑗 = 𝜀1𝑗 − 𝜀0𝑗, the probability of a positive response is:

Pr(𝑦𝑒𝑠_𝑗) = 1 − 𝐹_𝜀 [− (𝑣₁(𝑦_𝑗− 𝑡_𝑗, 𝑧_𝑗) − 𝑣₀ (𝑦_𝑗, 𝑧_𝑗))]

where 𝐹𝜀 (𝑎) is the probability that the random variable 𝜀 is less than 𝑎 known as the cumulative distribution function (CDF). The above equation is the point of departure for all the random utilities with different functions. In the linear utility function specification the deterministic part of a respondent’s preferences is linear both in covariates and income:

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𝑣𝑖𝑗 = 𝛼𝑖𝑧𝑗+ 𝛽𝑖𝑦𝑗

where 𝛼𝑖 denotes an m-dimensional vector of parameters and 𝛽𝑖 is the marginal utility of income. The deterministic utility for the initial and final states are:

𝑣0𝑗(𝑦𝑗) = 𝛼0𝑧𝑗+ 𝛽0𝑦𝑗 𝑣1𝑗(𝑦𝑗) = 𝛼1𝑧𝑗+ 𝛽1 (𝑦𝑗− 𝑡𝑗)

and assuming that the marginal utility of income is constant in the quality change (i.e. 𝛽0 = 𝛽1 ), the change in deterministic utility for respondent j can be written as:

𝑣1𝑗− 𝑣0𝑗 = (𝛼1− 𝛼0)𝑧𝑗+ 𝛽1 (𝑦𝑗− 𝑡𝑗) − 𝛽0𝑦𝑗 = 𝛼𝑧𝑗− 𝛽𝑡𝑗

and the probability of a yes response becomes:

Pr(𝑦𝑒𝑠_𝑗) = Pr(𝛼𝑧_𝑗− 𝛽𝑡_𝑗+ 𝜀_𝑗 > 0)

Once the response model to the CV responses is built, a measure of welfare (i.e. people’s WTP for the change to be valued) is estimated. The expression for the expectation of WTP with respect to preference uncertainty, following Hoyos, 2010) is:

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𝐸_𝜀 (𝑊𝑇𝑃_𝑗 |𝛼, 𝛽, 𝑧_𝑗) = 𝛼𝑧_𝑗/𝛽

Logistic and probit models are both parametric models that play a key role in the analysis of discrete CV data. The aim is to estimate a probability distribution for the true WTP in the CV setting using information on upper and lower bounds (Hanemann et al., 1999). Both models focus on proportions of cases in two categories of the dependent variable and they are akin to multiple regression in that the dependent variable is predicted from a set of variables that are continuous or coded to be dichotomous. They produce an estimate of the probability that the dependent variables equal to 1 given a set of independent variables. The difference between the two models lies in the transformation applied to the proportions forming the dependent variable that, in turn, reflects assumptions about the underlying distribution of the dependent variable. In probit analysis each observed proportion is replaced by the value of the standard normal curve (𝑧 value) below which the observed proportion is found, that is, it assumes a normally distributed dependent variable. The assumption of a normal distribution makes probit analysis a bit more restrictive than logistic regression. However, the shapes and the results of the probit and logit distributions are quite similar (Tabachnick and Fidell, 2013). Thus, the commonly used probit model is used to analyse the CV data.