However, the contingent valuation estimates of VSL, which are based on equation (5), can have significant biases from the theoretically correct estimates as defined in equation (4). First, the proposed risk reduction in a CV study cannot be too small in order to be sensible to the respondents. Therefore the approximation of equation (5) to equation (4) can be a question in reality. Secondly, it can be a difficult task for a respondent to calculate his/her WTP for a small risk reduction. A systematic error may exist in a WTP calculation with an individual. Previous empirical studies find that the VSL estimates as defined in equation (5) are sensitive to the scales of the risk reduction proposed in CV studies. WTP estimates are found not proportional to the level of risk reduction. Krupnick et al. (2002), Alberini et al. (2004) and Alberini and Chiabai (2006) studied the VSLs for USA, Canada and Italy with the CV approach for two different levels of mortality risk reduction, with one to be 5 times the other, but found that the WTP is not 5 times high, and therefore different VSL estimates can be produced for a same sample of population.<sup>6</sup> Similar findings are also reported in Muller and Reutzel (1984), Vassanadumrongdee and Matsuoka (2005), Hultkrantz et al (2006), and Corso et al. (2001), Andersson and Svensson (2008), Bhattacharya et al. (2007), where WTP increases less than proportionally with the size of risk reduction. Hammit and Graham (1999) provided a synthetic analysis for the 19 CV studies published before 1995 that have employed more than one level of risk reduction in the surveys, and found that, although many studies did find an increasing trend of WTP with risk reduction, none of the 19 studies shown the proportionality between WTP and the proposed risk reduction. Persson et al. (2001) elicited WTP information for risk reductions of 10, 30, 50 or 99 percent of mortality risk for one year and provided a graphical relationship between WTP and absolute mortality risk reduction. They find that WTP is increasing with the level of risk reduction but the increasing rate is decreasing.

No theoretical studies have been found on the relationship of WTP with risk reduction. While a theoretically-correct functional form of WTP with respect to risk reduction is believed to be dependent on the utility function and the wealth function as presented in equations (1) and (2) above, one can project that it is an increasing function and to be concave when the risk reduction level is high enough due to the budget constraint.

To overcome the shortcomings of contingent valuation studies in estimating VSL as discussed above, it is logical to require researchers to estimate functional forms of WTP with respect to risk reduction and to derive a marginal WTP estimate of risk reduction. This study represents such an effort, where marginal WTPs are estimated and simulated with different WTP functional forms, and the results are compared with the average WTPs of risk reduction.

# **5. The Health Survey**

In 2000, we conducted a household survey on public health and environment in three rural areas in China: Danyang (Jiangsu Province), Liupanshui (Guizhou Province) and Tianjin suburban. Inperson surveys, where respondents complete the questionnaires with close guidance from enumerators<sup>7</sup> , are conducted. About a half of the sample is workers who were working in local

> 6 In Alberini et al. (2004), the mean WTPs for 1/10000 and 5/10000 risk reductions are $370 and $466  in Canada, and $487 and $ 770  in the United States.

> 7 The major intention is to minimize the potential interviewer bias. The interviewers read another copy of identical questionnaire to the respondents, but can not directly work on the questionnaire, and the respondents do not need to

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