Abstract
The aim of the study reported in this paper is to test the hypothesis that individual utility of wealth functions may violate the assumption of smoothness that underpins the standard analysis of the Value of Statistical Life (VSL) and safety. In order to do so we examine the way in which the Willingness to Accept/Willingness to Pay (WTA/WTP) ratio varies as the severity of a health complaint is reduced. We find that as the severity of the health effect is reduced, the WTA/WTP ratio converges across the sample and tends to a level that does not significantly exceed unity. While we acknowledge that this does not constitute conclusive evidence of smoothness, it does suggest that the case in favour of the assumption that individual utility of wealth functions tend to display a kink at the current level of wealth is less than wholly persuasive.
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Notes
For a comprehensive survey of the WTA/WTP literature see Horowitz and McConnell (2002).
However, they do not find a similar decay in preference reversals under the same controls, so its support for standard economic theory is far from complete.
These results can be formally demonstrated using the Mean Value Theorem.
Full protocol and materials are available on request.
Resulting in an expected value which ranged, dependent upon group size, from £0.42 to £1.00.
This is the terminology used by us to represent minimum WTA, as defined in our experiment, since respondents seemed more comfortable with it than “minimum willingness to accept”, although the two are synonymous.
In addition, the term “reserve price” was not used in the WTP questions.
Studies such as Isoni et al. (2011) which show that the WTA-WTP gap persists in the case of lottery tickets is of little direct concern here—there may be many valid reasons for this and, indeed, we have reasons to expect a gap may still persist in our raw WTA-WTP health states data: as stated our main aim was to use the Plott and Zeiler mechanism as a ‘teaching’ vehicle to improve understanding of what is meant by minimum (maximum) WTA (WTP) and the “risks” of not revealing it in a trade—however, whether respondents choose to reveal it in experimental lotteries or contingent valuations is entirely up to them.
Given the small number of rounds and the fact that any data is likely to be biased by (unidentifiable) “learning effects” it cannot be analysed subsequently to unambiguously confirm that respondents understood these mechanisms. We can only infer this indirectly through analysis of the data in the stated preference study.
Although of necessity hypothetical, it does not necessarily follow that the health valuation responses are also perceived as inconsequential i.e. will have no influence whatsoever on policy/decision making and hence no effect on a respondent’s utility function, as defined by Carson and Groves (2007). While this cannot be proved either way, the plausibility and economic consistency of the resulting data suggests that answers were not randomly generated by careless respondents.
Respondent No. 96 refused to answer WTA for the stomach complaint but gave well-behaved responses for the throat complaint, despite having suffered from the former but not the latter; Respondent No. 141 was scope insensitive in the case of the stomach complaint (requiring £10,000 in compensation regardless of duration); and Respondent 148 asked for £1 m (1 week), £2 m (1 month) and £3 m (3 months) for the stomach complaint, continuing with requests of £4 m, £5 m and £6 m for the throat complaint. As s/he declined to disclose income, we are unable to judge whether these responses could in fact be plausible (e.g. if s/he happened to be a millionaire).
A t-test for equality of mean WTA responses cannot reject equality with a probability of: 0.2116 for 1 week, 0.6913 for 1 month, and 0.2164 for 3 months. For WTP a t-test for equality of means cannot reject equality with a probability of: 0.5003 for 1 week, 0.3003 for 1 month, and 0.5422 for 3 months. Additionally, a t-test of equality of the WTA mean values, for each duration period, for each complaint was rejected at the 5% level as was the same test for WTP values. As such, the aggregate data passes a conventional scope test.
At the individual level, all respondents offered strictly larger WTP amounts for longer durations of the complaints. While two respondents stated WTA amounts that were the same size for two of the durations, all other respondents stated strictly larger amounts for longer durations. The vast majority of respondents therefore can be seen to have passed a basic rationality test.
The reason for using only the type 1 responses in the regression analysis is that if the full sample had been used it would have resulted in larger standard errors for the coefficients (due to the divergence of the means at 3 months duration) and would therefore be a weaker test of our hypothesis.
Full results can be obtained from the authors.
Given that both standard theory and the “kinked” utility function hypothesis entail strict concavity of the utility function in the neighbourhood of current wealth, neither hypothesis would imply a constant strictly less than unity. It therefore seems appropriate to subject the hypothesis that the constant is unity to what is in this case the stricter one-tailed test.
As a further robustness check we also ran the OLS regression on Logarithm of duration (clustered by individual) for the entire sample, i.e. including all types. The Constant for the stomach complaint was 1.71 with a standard error of 4.24, whereas for throat the Constant was 4.14 with a standard error of 2.08. The Z scores are 0.1674 and 1.51 respectively both of which are below the T critical of 1.67.
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Chilton, S., Jones-Lee, M., McDonald, R. et al. Does the WTA/WTP ratio diminish as the severity of a health complaint is reduced? Testing for smoothness of the underlying utility of wealth function. J Risk Uncertain 45, 1–24 (2012). https://doi.org/10.1007/s11166-012-9145-5
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DOI: https://doi.org/10.1007/s11166-012-9145-5