2. Theoretical background

While the VSL is generally defined as the WTP for a marginal change in one’s risk of dying from any cause (Schelling, Reference Schelling and Chase1968), it is possible to derive two metrics that are more specific to cancer (or other serious diseases). The first is the cancer VSL, that is, the WTP for a marginal change in one’s unconditional risk of dying from cancer; the second, the VSCC, is the WTP for a marginal change in one’s risk of developing cancer—a necessary precursor to dying of cancer. Both the cancer VSL and the VSCC are useful ex ante measures of the benefits of health and safety regulations, as they can be applied to value reductions in exposure to a variety of pollutants and other toxic substances.

However, in many cases such regulations protect against either exposure to different chemicals (giving rise to so-called mixture toxicity) or different exposure modes (e.g., respiratory and dermal exposure to a toxic substance). Therefore, it is often impossible to predict in which organ cancer will develop. From a policy perspective, it is important to account for differences in incidence and survival rates of various cancers, as this will often be the key determinant of a regulation’s benefits (Cropper et al., Reference Cropper, Hammitt and Robinson2011). The question is thus whether different survival prospects should affect the rate at which people trade off income against reductions in the risk of developing cancer.

In this section, we present a simple theoretical model to examine this question. Formally, we define the cancer VSL as the rate at which one is prepared to trade off income $ y $ for reductions in the risk of dying from cancer $ m $ . To account for the fact that people may die from other causes than cancer (Eeckhoudt & Hammitt, Reference Eeckhoudt and Hammitt2001), we use $ \theta $ to denote the risk of dying from a competing risk. Individuals will develop cancer with probability $ p $ and, if they do develop cancer, they will have a conditional probability $ q $ of dying from it. The unconditional probability of dying from cancer is thus given by $ m= pq $ .Footnote ³

As in Alberini and Ščasný (Reference Alberini and Ščasný2021), we assume that the utility of income in the healthy state is $ U(y) $ , the utility of income when the individual has or has had cancer but is alive is $ V(y) $ , the utility of income when dying from cancer is $ W(y) $ , and that when dying from any other cause is $ G(y) $ . Combining these assumptions yields the following expression for expected utility:

(1) $$ \mathrm{EU}=\left(1-\unicode{x03B8} \right)\left(1-\mathrm{p}\right)\mathrm{U}\left(\mathrm{y}\right)+\left(1-\unicode{x03B8} \right)\mathrm{p}\left(1-\mathrm{q}\right)\mathrm{V}\left(\mathrm{y}\right)+\left(1-\unicode{x03B8} \right)\mathrm{p}\mathrm{qW}\left(\mathrm{y}\right)+\unicode{x03B8} \mathrm{G}\left(\mathrm{y}\right). $$

If we normalize the bequest utility, that is, the utility of income when dead, to zero (Rosen, Reference Rosen1988), we have W(y)=G(y)=0. It then becomes straightforward to derive the cancer VSL as:

(2) $$ {\mathrm{V}\mathrm{SL}}_C\equiv \frac{\mathrm{dy}}{\mathrm{dm}}={\mathrm{q}}^{-1}\frac{\mathrm{U}\left(\mathrm{y}\right)-\left(1-\mathrm{q}\right)\mathrm{V}\left(\mathrm{y}\right)}{\left(1-\mathrm{p}\right){\mathrm{U}}^{\prime}\left(\mathrm{y}\right)+\mathrm{p}\left(1-\mathrm{q}\right){\mathrm{V}}^{\prime}\left(\mathrm{y}\right)}. $$

The second term in the right-hand side of Eq. (2) is the usual utility differential between the healthy and the sick state, divided by the expected marginal utility of income.Footnote ⁴ Eq. (2) shows that this term gets scaled by the inverse of the risk of dying when sick.

On differentiating Eq. (1) with respect to the probability of developing cancer, we obtain a formal definition of the VSCC:

(3) $$ \mathrm{VSCC}\equiv \frac{\mathrm{dy}}{\mathrm{dp}}=\frac{\mathrm{U}\left(\mathrm{y}\right)-\left(1-\mathrm{q}\right)\mathrm{V}\left(\mathrm{y}\right)}{\left(1-\mathrm{p}\right){\mathrm{U}}^{\prime}\left(\mathrm{y}\right)+\mathrm{p}\left(1-\mathrm{q}\right){\mathrm{V}}^{\prime}\left(\mathrm{y}\right)}. $$

Eq. (3) clarifies the proportional relationship between the cancer VSL and the VSCC—the former is equal to the latter divided by the conditional probability of dying from cancer.

Based on these definitions, we ask whether differences in the survival prospects of cancer affect the two cancer metrics. On taking the partial derivative of the cancer VSL and VSCC with respect to the conditional probability of dying from cancer, we obtain

(4a) $$ \frac{\mathrm{\partial VS}{\mathrm{L}}_C}{\mathrm{\partial q}}=\frac{-{\mathrm{q}}^{-2}\left(\mathrm{U}\left(\mathrm{y}\right)-\mathrm{V}\left(\mathrm{y}\right)\right)\left[\left(1-\mathrm{p}\right){\mathrm{U}}^{\prime}\left(\mathrm{y}\right)+\mathrm{p}\left(1-\mathrm{q}\right){\mathrm{V}}^{\prime}\left(\mathrm{y}\right)\right]+\mathrm{p}{\mathrm{V}}^{\prime}\left(\mathrm{y}\right)\left[\left(\mathrm{U}\left(\mathrm{y}\right)-\mathrm{V}\left(\mathrm{y}\right)\right)+\mathrm{qV}\left(\mathrm{y}\right)\right]}{{\left[\left(1-\mathrm{p}\right){\mathrm{U}}^{\prime}\left(\mathrm{y}\right)+\mathrm{p}\left(1-\mathrm{q}\right){\mathrm{V}}^{\prime}\left(\mathrm{y}\right)\right]}^2}, $$

(4b) $$ \frac{\partial \mathrm{VSCC}}{\mathrm{\partial q}}=\frac{\mathrm{V}\left(\mathrm{y}\right)\left[\left(1-\mathrm{p}\right){\mathrm{U}}^{\prime}\left(\mathrm{y}\right)+\mathrm{p}\left(1-\mathrm{q}\right){\mathrm{V}}^{\prime}\left(\mathrm{y}\right)\right]+\mathrm{p}{\mathrm{V}}^{\prime}\left(\mathrm{y}\right)\left[\left(\mathrm{U}\left(\mathrm{y}\right)-\mathrm{V}\left(\mathrm{y}\right)\right)+\mathrm{qV}\left(\mathrm{y}\right)\right]}{{\left[\left(1-\mathrm{p}\right){\mathrm{U}}^{\prime}\left(\mathrm{y}\right)+\mathrm{p}\left(1-\mathrm{q}\right){\mathrm{V}}^{\prime}\left(\mathrm{y}\right)\right]}^2}. $$

From Eqs. (4 a) and (4 b) one sees immediately that $ \partial VSCC/\partial q $ is strictly positive, whereas $ \partial VS{L}_C/\partial q $ cannot be unambiguously signed. We empirically test these predictions by developing a survey instrument that elicits information about the WTP for various reductions in the risk of developing cancer. The questionnaire is administered to two independent samples of individuals. One sample was told that the chance of 5-year survival since the diagnosis is 60 %, which is the average survival chance in Western countries across all types of cancers for people aged 45–60; the other was told that it is 75 %, roughly corresponding to the 5-year survival chance for kidney cancer patients.Footnote ⁵

If our theoretical predictions hold, we should obtain a lower VSCC for the setting that resembles kidney cancer—assuming that the expected utility model is a reasonable approximation of people’s preferences and that they are capable of processing probabilities correctly. How much larger is an empirical question, which we seek to answer below. We also seek to answer the question whether the cancer VSL held by the two samples differs from each other, and, if so, by how much.

3. Study design

In our survey questionnaire, respondents were asked to indicate whether they would pay a specified amount of money to reduce the risk of developing cancer from its current level (25 in 1000 over 5 years, which is equivalent to 5 in 1000 per year). While this baseline risk was the same for all respondents, the risk reductions varied across and within respondents, as each respondent was asked a total of three valuation questions. The risk reductions were randomly selected out of an array comprised of 2, 3, and 4 in 1000 over 5 years.

The questionnaire informed the respondents about the survival rate at 5 years from diagnosis. Specifically, we told respondents in Wave 1 that the 5-year survival rate was 60 %, for an annual mortality rate of 0.097 (9.7 %). In Wave 2, which was conducted 3 months later and with different subjects, the 5-year survival rate was 75 %, which implies an annual mortality rate of 0.056 (5.6 %). Clearly, respondents in Wave 2 were asked to consider less fatal cancers, although they saw the same combinations of side effects and other cancer descriptors as the respondents in Wave 1. In both waves, the respondents were told the 5-year survival rate, but we did not present the corresponding annual mortality rate to them. The study design is summarized in Table 1.

Table 1.

Summary of the survey design.

Since the unconditional chance of dying from cancer is the product of the risk of developing cancer and the conditional chance of dying from it, a reduction in either of these risks implies a reduction in the unconditional chance of dying from cancer. Holding conditional mortality constant, a larger reduction in the risk of developing cancer therefore means a larger unconditional mortality risk reduction. To illustrate, the baseline cancer risk was 25 in 1000 over 5 years for each respondent. When multiplied by 0.097 (the annual mortality rate assigned to all respondents in Wave 1), this yields an absolute cancer mortality risk of 48.6 in 100,000 per year. If the risk of cancer were reduced by 5 in 1000 over 5 years (which is equivalent to 1 in 1000 per year), then the reduction in unconditional mortality risk would be 9.7 in 100,000 per year. Similarly, when multiplied by 0.056 (the annual mortality rate assigned to all respondents in Wave 2), the baseline cancer mortality risk would be equal to 28.0 in 100,000 per year and the mortality risk reduction following from a 1 in 1000 reduction in annual cancer risk would be 5.6 in 100,000 per year.

We did not present the above calculations to the respondents but, effectively, the subjects in the two waves combined were asked to consider a total of six hypothetical reductions in unconditional cancer mortality risk, ranging from 2.24*10^-5 to 9.71*10^-5 per year. Table 2 provides a summary of the risk variations in both waves. Moreover, we did not mention which organ would be affected by cancer, but we did describe the cancer disease in terms of its impact on quality of life during and after treatment (from perfectly normal life to being confined to bed half of the time) and pain (mild or moderate). We varied quality-of-life impacts and pain—if one gets cancer—from one valuation question to the next, but not within a valuation question. In other words, respondents could reduce their risk of developing cancer (and of experiencing the described consequences), but they could not “buy” a change in the severity of these consequences.Footnote ⁶

Table 2.

Risk variations in Wave 1 and Wave 2.

In sum, respondents received three dichotomous choice WTP questions. Each question asked whether they would choose to pay a specified sum to reduce their risk of getting some form of cancer that would be accompanied by certain impacts on quality of life and pain. The cancer risk reduction and the cancer severity (as measured by impacts on quality of life and pain) varied from one WTP question to the next, and across respondents. The 5-year survival chance was held constant within a questionnaire but varied across the two samples of respondents. Figure 1 shows a sample choice card for each of the two waves.

Figure 1.

Sample choice card from Wave 1 (top) and Wave 2 (bottom).

4. Econometric model

The responses to the choice questions can be used to estimate the WTP for either the cancer risk reduction or the mortality risk reduction implied by it and derive the two key metrics for benefit-cost analysis—the VSCC and the cancer VSL.Footnote ⁷ We assume that the responses to the WTP questions are driven by the respondent’s true WTP, which we denote as WTP*, which remains unobserved because our questionnaire does not ask respondents to report exact WTP amounts (more on this below). We assume that this unobserved WTP* for the risk-reducing alternative depends on the magnitude of the risk reduction as follows:

(5) $$ {\mathrm{WTP}}_{\mathrm{ij}}^{\ast }=\exp \left(\unicode{x03B1} \right)\ast {\left({\Delta \mathrm{R}}_{\mathrm{ij}}\right)}^{\unicode{x03B3}}\ast \exp \left({\unicode{x025B}}_{\mathrm{ij}}\right), $$

where $ \Delta R $ denotes the reduction in either the risk of developing cancer stated in that alternative (e.g., 1 in 1000 per year) or the unconditional mortality risk implied by that alternative (e.g., 9.71*10^-5), and $ \varepsilon $ is a normally distributed error term with mean zero and variance $ {\sigma}^2 $ .

On taking logs, Eq. (5) becomes:

(6) $$ \log \left({\mathrm{WTP}}_{\mathrm{ij}}^{\ast}\right)=\unicode{x03B1} +\unicode{x03B3}\ \mathrm{log}\left({\Delta \mathrm{R}}_{\mathrm{ij}}\right)+{\unicode{x025B}}_{\mathrm{ij}}. $$

The sign and magnitude of $ \gamma $ are of particular interest. At a minimum, $ \gamma $ should be positive and significant, as one would expect the WTP to be greater for a larger risk reduction. If $ \gamma $ is equal to one, this means that the WTP is perfectly proportional to the size of the risk reduction (Corso et al., Reference Corso, Hammitt and Graham2001), implying a single cancer VSL (if $ \Delta R $ stands for the unconditional mortality risk reduction) or a single VSCC (if $ \Delta R $ denotes the reduction in the risk of developing cancer) equal to exp( $ \alpha $ ).

If we pool the data obtained from both waves, Eq. (6) needs to be amended to:

(7) $$ \log \left({\mathrm{WTP}}_{\mathrm{i}\mathrm{j}}^{\ast}\right)=\unicode{x03B1} +\unicode{x03B3}\ \mathrm{log}\left({\Delta \mathrm{R}}_{\mathrm{i}\mathrm{j}}\right)+\unicode{x03B4} {\mathrm{D}}_{\mathrm{i}}+{\unicode{x025B}}_{\mathrm{i}\mathrm{j}}, $$

where $ {D}_i $ is a dummy that takes on a value of 1, if subject $ i $ is assigned the higher survival chance. Our theoretical model suggests that $ \delta $ should be negative when WTP* is the WTP to reduce the risk of getting cancer (in which case $ \Delta R $ denotes the change in the chance of getting cancer), as we expect the VSCC to be larger, the smaller the baseline survival chance is. Our model does not offer an unambiguous prediction for the sign of $ \delta $ when WTP* is the WTP to reduce unconditional mortality risk (in which case $ \Delta R $ is the reduction in the unconditional mortality risk).

As mentioned, we do not observe the respondent’s exact WTP* for a specified risk reduction. All we can infer from the responses to the WTP questions is whether the latent WTP* amount is greater than the cost of the risk-reducing alternative—if the respondent chooses that alternative—or otherwise. This results in a binary choice model that describes the probability of selecting the risk-reducing alternative as a function of the magnitude of its risk reduction and cost:

(8) $$ {\displaystyle \begin{array}{l}\Pr\;\left(\mathrm{i}\;\mathrm{chooses}\;\mathrm{j}\right)=\Pr\;\left({\mathrm{WTP}}_{\mathrm{ij}}^{\ast}\ge {\mathrm{C}}_{\mathrm{ij}}\right)=\Pr\;\left(\log \hskip0.24em \left({\mathrm{WTP}}_{\mathrm{ij}}^{\ast}\right)\ge \log\;\left({\mathrm{C}}_{\mathrm{ij}}\right)\right)\\ {}\hskip9.3em =\Phi \left(\mathrm{a}+\mathrm{b}\;\log\;\left({\Delta \mathrm{R}}_{\mathrm{ij}}\right)+\mathrm{c}\;\log\;\left({\mathrm{C}}_{\mathrm{ij}}\right)\right),\end{array}} $$

where $ \varPhi \left(\bullet \right) $ denotes the standard normal cdf, $ \mathrm{a}=\unicode{x03B1} /\unicode{x03C3} $ , $ \mathrm{b}=\unicode{x03B3} /\unicode{x03C3} $ , and $ \mathrm{c}=-1/\unicode{x03C3} $ .

This model is appropriate if one choice task is considered in isolation (or if the respondent only answers one WTP question) and results in a probit model that, for now, contains only one regressor in addition to the log of the risk reduction, namely the log of cost. The original parameter $ \alpha $ in Eq. (5) is recovered as the intercept from the probit model, divided by the negative of $ \hat{c} $ ; similarly, the standard deviation of $ \varepsilon $ is obtained as ( $ -1/\hat{c} $ ). In our survey, however, each respondent answered three WTP questions and the corresponding error terms are likely to be correlated due to unobserved characteristics of the respondent or individual perceptions that affect each choice task. If we assume that the correlation between any two pairs of responses is the same, that is, $ E\left({\varepsilon}_{i1}\;{\varepsilon}_{i2}\right)=E\left({\varepsilon}_{i1}\;{\varepsilon}_{i3}\right)=E\left({\varepsilon}_{i2}\;{\varepsilon}_{i3}\right) $ , then the model becomes a random effects probit.

If we further constrain $ \gamma $ = 1, Eq. (7), which refers to the WTP for a cancer mortality risk reduction, simplifies to:

(9) $$ \log \left(\frac{{\mathrm{WTP}}_{\mathrm{ij}}^{\ast }}{{\Delta \mathrm{R}}_{\mathrm{ij}}}\right)\equiv \log \left(\mathrm{cancer}\;{\mathrm{VSL}}_{\mathrm{ij}}^{\ast}\right)=\unicode{x03B1} +{\unicode{x025B}}_{\mathrm{ij}}, $$

and $ \log \left({\mathrm{C}}_{\mathrm{ij}}/{\Delta \mathrm{R}}_{\mathrm{ij}}\right) $ , that is, the natural logarithm of the cost per unit of risk reduced, enters in its binary choice econometric counterpart. In its simplest variant, the binary choice model includes only the log cost per unit of risk reduced. In more complex variants, $ \alpha $ is replaced by a linear combination of variables denoting respondent sociodemographics and risk perceptions.

5. Data

The survey questionnaire was administered in two waves—in May and October 2019, respectively—to two samples selected to be representative for income and education of the population of the Czech Republic aged 45–60. Attention was restricted to persons aged 45–60 for two main reasons. First, our experience is that it is extremely difficult to get younger persons to focus on adverse health outcomes—especially mortality outcomes—in surveys. Second, their cancer risk is very low, as 9 out of 10 cancers are diagnosed in people 45 and older, which makes it challenging to represent it using the conventional risk communication graphs (Ancker et al., Reference Ancker, Senathirajah, Kukafka and Starren2006).

By contrast, among older people the chance of getting cancer increases sharply with age. Theory suggests that the WTP to reduce the risk should increase with the baseline risk (Pratt & Zeckhauser, Reference Pratt and Zeckhauser1996), an effect that may be partially or completely offset by the elderly’s shorter remaining life expectancy (Krupnick et al., Reference Krupnick, Alberini, Cropper, Simon, O’Brien, Goeree and Heintzelman2002; Krupnick, Reference Krupnick2007). To avoid having to disentangle these opposing effects, we limit the sample to 45-60-year-olds and we show the respondents the average risk of getting cancer in this age group.

The questionnaire was identical across the two waves in all respects but the survival prospect if one develops cancer. This prospect was 60 % at 5 years in Wave 1 and 75 % at 5 years in Wave 2, which correspond to 40 and 25 % 5-year mortality rates, respectively.

The valuation section of the questionnaire was preceded by (i) questions about the health status of the respondents (in SF-36 type of format), (ii) a simple probability tutorial, (iii) the cancer incidence rate and average 5-year survival rate, (iv) a short description of the possible effects of cancer on quality of life (including possible impacts on family life and mental health), and (v) measures that can reduce the chance of getting cancer and/or the chance of dying from it, such as regular health screenings (mammograms, pap smears, colonoscopies, etc.), a healthy diet and lifestyle, and environmental programs.

As previously mentioned, throughout the questionnaire we always referred to a generic cancer, without identifying the organ that might develop cancer or naming specific types of cancer. We felt that doing so would have interfered with the respondent’s comprehension of the probabilities and would have limited our ability to apply the results of our study to a broad range of cancer diseases. In the valuation questions, the impact of cancer on quality of life was described in terms of limitations to everyday activities (none; unable to do heavy physical work; unable to work; bedridden half of the time) and pain level (mild or moderate).

The respondents—all members of European National Panels (formerly known as Czech National Panel)—completed the questionnaire online.Footnote ⁸ After dropping “speeders” and respondents who failed a simple probability quiz, our final sample size is comprised of 468 completed questionnaires from Wave 1 (containing 1404 WTP responses) and 550 from Wave 2 (containing 1650 WTP responses), respectively. Descriptive statistics of the two samples are displayed in Table 3. They suggest that, after dropping speeders and persons who failed the probability quiz, the final samples were faithful to the sampling quotas and similar in terms of sociodemographics, as well as familiarity with and dread of cancer diseases.Footnote ⁹

Table 3.

Sample sizes and characteristics of the respondents.

A slight majority of the respondents were male. About one-third had a university degree and some 45 % had earned their high school diploma. More than half of the respondents reported that a family member had or had had cancer, and over 70 % knew of a friend or acquaintance that had or had had cancer. The level of cancer dread was similar across the two waves, with more than half of the respondents indicating that they had “high” or “very high” levels of cancer dread.

Comparison with the sociodemographics of the Czech population aged 45–60 (shown in Table A.1 in the Appendix) shows that our two samples are very similar to the population in terms of income. They were also somewhat more highly educated, and we had a slight overrepresentation of men.