Random Utility Model

The Random Utility Model interprets respondents’ answers to the WTP question as the result of their comparison of the underlying utility of two circumstances, i.e. the status quo and the valuation scenario of paying the offered price for the specified ecosystem service. The utility that respondent j derives from a specified circumstance Uij is assumed to consist of a deterministic, observable component Vij and a random, unobservable error component ɛij.

𝑈𝑖𝑗 = 𝑉𝑖𝑗 + 𝜀𝑖𝑗 (3.12)

where i = 1 if the respondent answers yes to the offered price tj, and i = 0 if he/she answers no. It is reasonable to suppose that the respondent would answer yes only if U1j > U0j. So the probability of yes answer is

P r�yesj� = Pr�𝑉1𝑗 + 𝜀1𝑗 > 𝑉0𝑗+ 𝜀0𝑗� (3.13)

which can be rewritten as

P r�yesj� = Pr�𝜀1𝑗 − 𝜀0𝑗 > −(𝑉1𝑗− 𝑉0𝑗)� (3.14)

Define the difference between the two random components as ɛj = ɛ1j - ɛ0j, and denote the change in utility as ∆V = V1j – V0j. Equation 3.14 becomes

P r�yesj� = Pr(𝜀𝑗 > −∆𝑉) = 1 − Pr (𝜀𝑗 < −∆𝑉)

(3.15)

It can be noted that Pr�𝜀_𝑗 < −∆𝑉� is the cumulative distribution function of ɛj with regard to -∆V. In order to construct a Random Utility Model, assumptions are needed to 1) specify the function of the deterministic utility

component Vij so that the utility change ∆V can be specified, and 2) to specify the statistic distribution of the random component ɛj.

The most commonly used function for the deterministic utility component is the linear utility function (Hanemann 1984; Loomis et al. 2000; Haab and McConnell 2002; Moreno-Sanchez et al. 2012).

𝑉𝑖𝑗 = 𝛼𝑖𝑐𝑗+ 𝛽𝑖𝑦𝑗 (3.16)

where 𝑐_𝑗 is a vector of m explanatory variables with respect to respondent j, 𝑎𝑗 is the vector of m coefficients (i.e. 𝛼𝑖𝑐𝑗 = ∑𝑚𝑘=1𝛼𝑖𝑘𝑐𝑗𝑘 ), 𝑦𝑗 is the

respondent’s income and βi is the coefficient of income. If the respondent answers no to the WTP question, 𝑉_0𝑗 = 𝛼₀𝑐_𝑗+ 𝛽₀𝑦_𝑗; if the answers is yes, it means the respondent is willing to forgo a portion of income (the offered price 𝑡_𝑗) in exchange for the ecosystem service specified in the valuation scenario, i.e. 𝑉_1𝑗 = 𝛼₁𝑐_𝑗+ 𝛽₁(𝑦_𝑗− 𝑡_𝑗). So the utility change is presented as ∆𝑉 = 𝑉1𝑗− 𝑉0𝑗 = (𝛼1− 𝛼0)𝑐𝑗+ 𝛽1�𝑦𝑗 − 𝑡𝑗� − 𝛽0𝑦𝑗 (3.17)

It is reasonable to assume that β1 = β0 (denoted as β for simplicity), i.e. the marginal utility effect of income is constant in the status quo (no answer) and valuation scenario (yes answer) unless the offered price could cause substantial change to the respondent’s income. Define α = α1 – α0, Equation 3.17 becomes

∆𝑉 = α𝑐𝑗− 𝛽𝑡𝑗 (3.18)

Now Equation 3.15 becomes

P r�yesj� = 1 − Pr[𝜀𝑗 < −(α𝑐𝑗− 𝛽𝑡𝑗)] (3.19)

In order to estimate α and β, the statistic distribution of ɛj needs to be specified. It is usual to assume that ɛj is independently and identically distributed (IID) with the mean of 0. Accordingly, there are two widely used statistic distributions, i.e. the normal and logistic distributions, both of which are symmetric and exhibit the nature that 𝐹(𝑥) = 1 − 𝐹(−𝑥). So Equation 3.19 can be simplified as

P r�yesj� = Pr�𝜀𝑗 < 𝛼𝑐𝑗− 𝛽𝑡𝑗� (3.20)

If ɛj follows a normal distribution with mean 0 and variance σ2, this distribution of ɛj can be transformed into a standard normal distribution of 𝜀𝑗⁄ with mean 0 and variance 1. Then Equation 3.20 becomes 𝜎

P r�yesj� = Pr �𝜀_𝜎𝑗< α𝑐𝑗_𝜎−𝛽𝑡𝑗� = 𝛷 �α𝑐𝑗−𝛽𝑡_𝜎 𝑗� (3.21)

where 𝛷( )is the cumulative distribution function of standard normal distribution. This is the standard probit model in statistics/econometrics. For

a sample of T respondents, define 𝐼_𝑗 = 1 if respondent j answered yes and 𝐼𝑗 = 0 if the answer is no, the log-likelihood function for the responses of the

whole sample is

ln 𝐿 (𝑦𝑒𝑠) = ∑𝑇𝑗=1𝐼𝑗ln �𝛷 �α𝑐𝑗−𝛽𝑡_𝜎 𝑗�� + (1 − 𝐼𝑗) ln �1 − 𝛷 �α𝑐𝑗−𝛽𝑡_𝜎 𝑗��

(3.22)

The model parameters, α 𝜎⁄ and − β 𝜎⁄ , can be readily estimated by standard probit regression procedure of statistics/econometrics software packages which utilise the Maximum Likelihood Estimation algorithm to find the parameters that maximize the log-likelihood of Equation 3.22.

On the other hand, if ɛj follows a logistic distribution with mean 0 and variance 𝜋2𝜎_𝐿2⁄ , this distribution of ɛ3 j can be converted to a standard logistic distribution of 𝜀_𝑗⁄ with mean 0 and variance 𝜋𝜎_𝐿 2⁄ . Then Equation 3.20 3 becomes

P r�yesj� = Pr �_𝜎𝜀𝑗_𝐿 <α𝑐𝑗_𝜎−𝛽𝑡_𝐿 𝑗� = [1 + exp �−𝛼𝑐𝑗_𝜎−𝛽𝑡_𝐿 𝑗�]−1 (3.23)

This is the standard logit model, and the log-likelihood function of the logit model is ln 𝐿 (𝑦𝑒𝑠) = ∑ 𝐼𝑗ln ��1 + exp �−𝛼𝑐𝑗_𝜎−𝛽𝑡_𝐿 𝑗�� −1 � T j=1 + �1 − 𝐼_𝑗� ln �1 − �1 + exp �−𝛼𝑐𝑗−𝛽𝑡𝑗 𝜎𝐿 �� −1 � (3.24)

Likewise, the parameters, α 𝜎⁄ and −β 𝜎_𝐿 ⁄ , can be estimated by standard _𝐿 logit regression procedure of statistic software packages to maximize the log-likelihood in Equation 3.24.

So far, a Random Utility Model (probit or logit) based on the linear utility function in Equation 16 has been established to use respondents’ characteristics 𝑐_𝑗 and the offered price 𝑡_𝑗 to explain the probability of yes answer in a Contingent Valuation survey. The next step is to calculate respondents’ mean WTP for the ecosystem service under valuation.

As a welfare measure of the monetary value of the ecosystem service, WTP is the amount of money/income that the respondent is willing to forgo in exchange for the environment service. In other words, WTP is the amount of money that makes U0 = U1, i.e. the loss of utility caused by forgoing that amount of money is equivalent to the utility that the respondent can derive from the environment service. For the Random Utility Model defined by Equations 3.12 and 3.16, U0 = U1 means

As mentioned above, define α = α1 – α0, β = β1 = β2 and ɛj = ɛ1j - ɛ0j, Equation 3.25 yields

𝑊𝑇𝑃𝑗 =α_β𝑐𝑗+1_β𝜀𝑗 (3.26)

In probit and logit models where 𝜀_𝑗 is assumed to follow the normal and logistic distribution respectively, the mean of 𝜀_𝑗 is 0. So the expected value (mean) of respondent j’s WTP is simply

𝐸�𝑊𝑇𝑃𝑗� = 𝛼_𝛽𝑐𝑗 (3.27)

The ratio α/β can be easily calculated using the estimated parameters of the probit or logit model, thus the mean WTP of respondents in the whole sample is

𝐸(𝑊𝑇𝑃) =_{β 𝜎}α 𝜎⁄_⁄ 𝑐𝑗 (probit model) 𝑜𝑟 _{β 𝜎}α 𝜎⁄_⁄ 𝐿_𝐿𝑐𝑗 (logit model) (3.28)

where 𝑐_𝑗 is the sample mean of the explanatory variables.

Equations 3.21, 3.23 and 3.28 construct the widely used linear random utility model (probit or logit) for the estimation of respondents’ mean WTP which adopts a linear function (Equation 3.16) to specify the deterministic utility component (Vij) in a random utility model (Equation 3.12) and assumes a normal or logistic distribution of the random component (ɛj).

One drawback of the linear random utility model is that the variable of income, which may significantly influence respondents’ answers in Contingent Valuation surveys, is not included in the probability functions (Equations 3.21 and 3.23). This is because the income term is incorporated in the function of Vij in a linear form (Equation 3.16), and the marginal utility of income is assumed to be constant between the status quo and the valuation scenario (i.e. β0 = β1 in Equation 3.17). Thus when the utility difference ∆V between the two circumstances is specified, the income variable 𝑦_𝑗 is removed and only the change in the income, i.e. the offered price 𝑡_𝑗 is included in the probability functions. For relaxing the assumption of constant marginal utility of income and retaining the income variable in the probability functions, non-linear forms of income term, such as the log-linear form, can be used in specifying Vij

𝑉𝑖𝑗 = 𝛽 ln(𝑦𝑗) + 𝛼𝑖𝑐𝑗 (3.29)

where the marginal utility of income is 𝜕𝑉𝑖𝑗

𝜕𝑦𝑗 =

𝛽

𝑦𝑗, and the corresponding

probability function is

P r�yesj� = Pr �𝜀𝑗 < α𝑐𝑗+ 𝛽ln (𝑦𝑗_𝑦−𝑡𝑗

An even more complicated form of the income term is the Box-Cox Transformation 𝑉𝑖𝑗 = 𝛽 ln𝑦𝑗 𝜆₋₁ 𝜆 + 𝛼𝑖𝑐𝑗 (3.31)

where the marginal utility of income is 𝜕𝑉𝑖𝑗

𝜕𝑦𝑗 = 𝛽𝑦𝑗

𝜆−1_{, λ is the transformation}

parameter which can be flexibly chosen by researchers, and the corresponding probability function is

P r�yesj� = Pr �𝜀𝑗 < α𝑐𝑗+ 𝛽(𝑦𝑗−𝑡𝑗)

𝜆 _{− 𝑦} 𝑗𝜆

𝜆 � (3.32)

Probit and logit models can be constructed depending on the normal or logistic distribution assumption about ɛj in Equation 3.20 and 3.32 (Bateman, Willis and Arrow 2001; Haab and McConnell 2002; Carson and Hanemann 2005).

In practice, the models with complicated, non-linear income terms are rarely used in empirical Contingent Valuation studies because of the consequent complication in WTP calculation. In fact, as the offered prices in Contingent Valuation surveys usually account for a very small portion of respondents’ income, it is not necessary to suppose the marginal utility of income to vary with this small change (Haab and McConnell 2002). Moreover, since the offered price 𝑡_𝑗 in Equation 3.30 and 3.32 is no longer a separate independent variable but a component in the complicated income term, the effect of the offered price on the probability of yes answer, which is important to researchers and policy makers, can no longer be clearly revealed by the model estimation. Additionally, income information is usually prone to a great deal of measure error and often collected in the categorical form (income groups/ranges) rather than exact values in real surveys, which further weakens the rationale of using the models with complicated non-linear income terms.

However, the problem of removing the income variable from the probability function remains if the linear random utility model is preferred to the non- linear models. A more critical but rarely mentioned problem regarding the linear random utility model is that if the assumption of constant marginal utility of the income variable is plausible because the offered price would merely cause a small change to respondents’ income, why the same assumption is not made for other explanatory variables such as respondents’ age and education level given these variables are unchanged at all between the status quo and the evaluation scenario? If such

assumption is made (i.e. α0=α1 in Equation 3.17), all the explanatory variables should be excluded from the probability functions just like the income variable when the utility difference is specified, then the offered price (the change in income) would be the only explanatory variable in the probability functions. This was indeed the case in Hanemann’s original paper where the variables of respondents’ characteristics were “suppressed” and represented by a constant term in his random utility model (Hanemann 1984). However, Contingent Valuation studies following Hanemann (1984)’s approach, such as Loomis et al. (2000)’s highly cited paper, just added other explanatory variables in their models without addressing why the assumption of constant marginal utility is made only for the income variable. Nevertheless, this does not mean that Hanemann (1984)’s original univariate model is satisfactory because it cannot reveal the effects of respondents’ characteristics on their answers to the WTP question, and more problematically from the statistical aspect, the univariate model is likely to exhibit poor goodness of fit in practice.

The discussion above indicates that the random utility model exhibit a contradiction between the necessity for retaining income and other explanatory variables in the probability function and the illogicality of assuming inconstant marginal utility of these variables in order to retain them in the probability function. The Random Willingness to Pay Model, an alternative approach of constructing parametric models for Contingent Valuation studies, can avoid this contradiction and thus was adopted by this study.