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# Valuation of Health Risks

## Summary and Keywords

Many public policies and individual actions have consequences for population health. To understand whether a (costly) policy undertaken to improve population health is a wise use of resources, analysts can use economic evaluation methods to assess the costs and benefits. To do this, it is necessary to evaluate the costs and benefits using the same metric, and for convenience, a monetary measure is commonly used. It is well established that money measures of a reduction in health risks can be theoretically derived using the willingness-to-pay concept. However, because a market price for health risks is not available, analysts have to rely on analytical techniques to estimate the willingness to pay using revealed- or stated-preference methods. Revealed-preference methods infer willingness to pay based on individuals’ actual behavior in markets related to health risks, and they include such approaches as hedonic pricing techniques. Stated-preference methods use a hypothetical market scenario in which respondents make trade-offs between wealth and health risks. Using, for example, a random utility framework, it is possible to directly estimate individuals’ willingness to pay by analyzing the trade-offs they make in the hypothetical scenario. Stated-preference methods are commonly applied using contingent valuation or discrete choice experiment techniques. Despite criticism and the shortcomings of both the revealed- and stated-preference methods, substantial progress has been made since the 1990s in using both approaches to estimate the willingness to pay for health-risk reductions.

# Introduction

Does it make economic sense to implement human papillomavirus (HPV) vaccination in national child-vaccination programs? What are the economic consequences of stricter vehicle emission standards that improve air quality and therefore reduce cardiovascular mortality and morbidity risks?

Public policy commonly addresses questions in which the trade-offs involve comparing economic costs to the health benefits associated with investments and regulations. The input from economic analyses on the merits of such regulations is typically based on cost-effectiveness analysis (CEA) or cost-benefit analysis (CBA). In CEA, the health risks are not evaluated using monetary values, and therefore these analyses cannot be (directly) used to assess the economic-welfare effects of public policies.1 The aim of a CBA is to assign monetary values to the predicted benefits and costs using a societal perspective: identifying, measuring, and valuing the consequences for individuals, firms, and the public sector. If the net present value is positive—that is, the present value of all benefits outweighs the present value of all costs, the regulation is said to increase social welfare (the Hicks-Kaldor criterion).

Cost-benefit analyses of policy proposals have increased in use and importance over time. In the United States, the use of CBA to inform policymakers about the consequences of policy proposals has been explicitly encouraged since the early 1980s. For proposals with an economic impact over 100 million U.S. dollars, federal agencies are required to carry out analyses using CBA (Obama, 2011). Several federal agencies, such as the Department of Transportation and the Environmental Protection Agency, have also published guidelines on how to implement CBA in their specific domains, which include recommended monetary values of, for example, mortality and morbidity health risks (Department of Transportation, 2016a, 2016b; Environmental Protection Agency, 2010). In Europe, many national agencies also demand and carry out CBAs of major regulations and investments (Andersson, 2018). In the United Kingdom, the government outlines the principles of CBA in the Green Book, in which it stipulates that each new policy, program, or project must be assessed asking the questions, are there better ways to achieve this objective, and are there better uses for these resources? (HM Treasury, 2003). At the level of the European Union, the European Commission (2014) has also developed a framework for evaluating new regulations and investments using CBA.

To be able to carry out CBA of policies that involve improved (or worsened) population health, it is necessary to value the predicted health changes in a monetary metric, for example, U.S. dollars, Chinese yuan, euros, and so on. Considering that health risks are not directly traded in markets, the valuation of health risks is commonly one of the main challenges in CBA. The appropriate economic value of a health change is the willingness to pay (WTP) for or the willingness to accept (WTA) the health change. The WTP is defined as the maximum price a person is willing to give up to receive a specific good or service. The WTA is defined as the minimum amount a person is willing to accept to refrain from receiving a good or service, or the compensation for an increase in a disamenity (e.g., in a health risk).

For public policy regulations and investments, the changes in average individual health risks are typically very small, and the valuation exercise thus concerns individuals’ WTP for a very small change in the health risk (e.g., the mortality risk). The total economic value of such a policy is, then, the population aggregate of each individual’s WTP. Most research efforts have been devoted to WTP estimates of changes in mortality risk, which may be due to the importance of the mortality-risk consequences of many public policies and to the fact that death is a relatively easy concept to define compared to many morbidity risks that are less objective. The economic term used for the trade-off between mortality risk and wealth is (commonly) the value of a statistical life (VSL). The term “statistical” highlights the fact that it considers unidentified, rather than identified, lives.

To illustrate VSL, assume a population of 10,000 individuals, where the baseline mortality risk (e.g., cancer risk) is 5 per 10,000; that is, each year five persons in the population will die of cancer. Now imagine that a preventive treatment (e.g., a screening program) is estimated to reduce the annual mortality risk to 4 per 10,000; that is, one less person will die each year, but it is not identified (ex ante) who that person will be, and that everyone will benefit from the same risk reduction. Now estimate the mean WTP for the screening program to be $100. This implies that the population WTP for the program and thus the VSL is$100 × 10,000 = $1 million, which we can interpret such that the population is in aggregate willing to pay$1 million to reduce deaths by one person (“1 statistical life”).

Lacking market prices for WTP estimates of mortality (and morbidity) risks, economists have developed several approaches to derive monetary values using what can be broadly classified as revealed-preference and stated-preference approaches. Revealed-preference (RP) approaches rely on observed market behavior, and the most common RP approach has been to assess the wage premium necessary to compensate for jobs with higher mortality or morbidity risks or both. Stated-preference (SP) studies are based on surveys and experiments in which individuals are asked to make hypothetical choices between different programs or investments with varying costs and health risks.

The rest of this article will explore in greater detail the theoretical foundations and empirical methods for estimating the WTP for small changes in health risks. The article is organized as follows: The section “Theory” covers the general theoretical background of the valuation of mortality risks, which can also be extended and applied to value other types of health risks. The section “Empirical Methods” describes the empirical approaches in greater detail, with a particular focus on hedonic price regressions as the main RP approach used in valuation of health risks and the two main SP approaches of discrete choice experiments (DCE) and contingent valuation method (CVM) studies. The section “Examples of Applications” presents some applications in the form of empirical studies that have used hedonic pricing, DCE, and CVM to estimate the WTP for small changes in mortality and morbidity risks. The concluding section, “Discussion,” considers some of the main challenges and criticisms regarding the use of empirical methods to derive the values of health risks. It also highlights some views about the most fundamental knowledge gaps in order to increase our understanding of individuals’ valuation of health risks.

# Theory

## Monetizing Preferences

As explained, the willingness-to-pay (WTP) concept is considered the appropriate approach to elicit monetary values. However, at the time of Thomas Schelling’s (1968) discussion of the use of a WTP approach, the human-capital (HC) approach dominated as a means to estimate the social value of reducing mortality risk. The underlying assumption of the HC approach is that the market goods and services produced by an individual during his lifetime reflect his value to society (Mishan, 1982). It is calculated as the expected future earnings, and hence does not consider individual preferences to reduce the risk of death. Under some (plausible) assumptions and restrictions, it has been shown that the HC can serve as a lower bound for the value of a statistical life (VSL; Bergstrom, 1982; Rosen, 1988), but it is in general considered a poor proxy for a preference-based—that is, WTP—measure of individual risk preferences (e.g., Freeman, Herriges, & Kling, 2014).

This section focuses on the elicitation of preferences to reduce the risk of death. The models described can be generalized to also cover morbidity risks (Andersson, Hammitt, & Sundström, 2015), but because mortality risk is a clearly defined outcome, using it avoids having to address heterogeneity issues related to morbidity risks.2 As described, the VSL is the aggregated monetary value of avoiding one statistical death in society. It is the population mean of the marginal WTP to reduce mortality risk and should not be interpreted as the value of an identified life. However, because the term “value of a statistical life” contains the word “life,” the VSL concept leads to confusion and misunderstandings, and some have suggested changing the name to something more neutral (e.g., Cameron, 2010). But this is an ongoing discussion, and for now, VSL remains the terminology used, and hence it is used it in this article.

The subsection “One-Period Model” presents the standard one-period model for VSL, together with some predictions from the model and the literature, and then describes a multiperiod model. The extension from the one- to the multiperiod model is highly relevant for examining health risks, not only because it makes it possible to examine how individuals’ WTP may vary over the life cycle, but also because many health risks are characterized by an often-substantial time interval between a change in exposure and the health effect. This is usually referred to as latency, and its effect on WTP can be examined in the multiperiod model.

## One-Period Model

The VSL refers to the population mean of the marginal rate of substitution (MRS) between mortality risk and wealth, under the assumption that the individual MRS and the personal change in risk are uncorrelated (see, e.g., Jones-Lee, 2003). The theoretical model assumes that individuals maximize their utility in a state-dependent expected utility framework (Dreze, 1962; Jones-Lee, 1974; Rosen, 1988). Let $p$ define survival probability and $us(w)$ the state-dependent utilities of wealth $(w)$, where the states are either staying alive (s = a) or dead (s = d), and individuals are assumed to maximize

(2.1)

$Display mathematics$

Following the standard assumptions in the literature, we assume that the utility functions are twice differentiable, that utility of wealth is higher if alive than dead, that marginal utility of wealth is nonnegative and higher if alive than dead, and that individuals are weakly risk averse to financial risks; that is,

(2.2)

$Display mathematics$

and

$Display mathematics$

Hence, at any wealth level, both the utility and the marginal utility are higher if alive than dead, and given these assumptions, the indifference curves over wealth and survival probability are, as illustrated in Figure 1, decreasing and strictly convex.

Using Equation (2.1) we can derive the compensating and equivalent surplus—that is, the WTP and the willingness to accept (WTA)—for a change in the mortality risk $Δp≡ε$ (Freeman et al., 2014). Let $EU0$ be defined by Equation (2.1) and $C(ε)$ denote the WTP for the risk reduction $ε$, then $C(ε)$ is given by

(2.3)

$Display mathematics$

and, similarly, if we let $P(ε)$ denote the WTA for the risk increase $ε$, then $P(ε)$ is given by

(2.4)

$Display mathematics$

As is evident from Equations (2.3) and (2.4) and illustrated in Figure 1, WTP and WTA will depend on the size of $ε$. The larger the change in the mortality risk, the larger the WTP or WTA, depending on whether the change is an increase or decrease in $ε$. However, when eliciting WTP and WTA for changes in mortality risks, the size of $ε$ will be small. We therefore expect WTP and WTA to be nearly equal and that they are near-proportional to $ε$ (Hammitt, 2000).

Click to view larger

Figure 1. The value of a statistical life.

Source: Lectures notes, Henrik Andersson, Toulouse School of Economics, inspired by lectures notes by James Hammitt, Harvard University.

The VSL measures the WTP or WTA for an infinitesimal change in risk, and it can be obtained by taking the limit of WTP or WTA when $ε≅0$. That is, as has already been explained, it is the MRS between wealth and mortality risk and is defined as follows:

(2.5)

$Display mathematics$

It is derived by totally differentiating Equation (2.1) and keeping utility constant. Hence, it is the ratio between the utility difference and expected marginal utility, and given the assumptions in Equation (2.2), VSL is always strictly positive.

Equation (2.5) can be empirically estimated, as will be described and shown in the sections “Empirical Methods” and “Examples of Applications.” However, in empirical applications, the risk reduction may be small but finite. This could be the case in surveys, where it does not make sense to ask respondents about a truly marginal risk change, or in studies that look at discrete decisions in markets. In those cases, the VSL is given by the ratio between the WTP and the change in risk, as shown in Equation (2.6),

(2.6)

$Display mathematics$

Equation (2.6) suggests that the WTP is proportional to the change in risk. But, as noted, the true relationship between WTP and the size of Δ‎p is only near-proportional, which is a necessary, but not sufficient, condition for WTP to be a valid measure of individuals’ preferences (Hammitt, 2000).

### The Wealth Effect and the Dead-Anyway Effect

For the empirical applications of the theoretical framework described in the section “One-Period Model,” predictions are important to test the construct validity of the findings. Two standard predictions are the wealth effect and the dead-anyway effect. First, examining how wealth influences WTP is central to test the validity of preference estimates, not only for health risk, but also for other nonmarket goods (Arrow et al., 1993). The wealth effect in this scenario describes how VSL increases with wealth (Weinstein, Shepard, & Pliskin, 1980). The intuition is clear—that is, wealthier individuals (everything else equal) can pay more for a good, and in this scenario, it can be broken down to (i) the numerator in Equation (2.5) increases in wealth, and (ii) the expected marginal utility in the denominator decreases in wealth. Both effects are assured by the assumptions in Equation (2.2), and the results state that wealthier individuals have more to lose and that their utility cost of spending is smaller.

Second, the dead-anyway effect (Pratt & Zeckhauser, 1996) suggests that WTP increases with the baseline risk; that is, it decreases with the survival probability (p). Since p only shows up in the denominator of Equation (2.5) the dead-anyway effect does not depend on the utility difference in the numerator. Instead, the effect is driven by the fact that the size of the denominator increases when p increases because . The intuition is that a person at a high risk has little incentive to limit his spending on increasing his survival probabilities.

### The Model and Selected Predictions

The two predictions for wealth and baseline risk, and the one provided before on WTP and WTA being sensitive to the size of the mortality risk reduction, can be considered the basic predictions of the standard one-period VSL model. The model has also been used to provide several other predictions that are useful for understanding and testing the validity of empirical findings. For instance, the survival probability presented so far has been the overall chance of survival. However, individuals face many different types of risks, and individuals’ WTP to reduce one risk may be influenced by the other risks that they face. These can be referred to as background risks, and depending on whether these background risks are independent (Eeckhoudt & Hammitt, 2001) or additive (Andersson, 2008) to the specific risk, it can be shown that VSL can either decrease or increase. Moreover, it has been shown that “although aversion to financial risk increases VSL in definable cases, under many plausible assumptions the relationship between risk aversion and VSL is ambiguous” (Eeckhoudt & Hammitt, 2004, p. 13), and that VSL increases with ambiguity aversion (Treich, 2010).

Another example is the effect on health status on the VSL. Assuming that the utility of wealth is higher in good health than in bad health and that the marginal utility of wealth is increasing in health status, then both the numerator and denominator of Equation (2.5) will increase; hence, the effect on VSL will be ambiguous (Hammitt, 2002; Strand, 2006). This article does not go into any details about these predictions and instead refers to the above provided references for readers interested in these topics.

## Multiperiod Model

The multiperiod model may be considered more realistic than the single-period model because individuals do have preferences for how survival probability and consumption opportunities are distributed over their length of their lives. Moreover, the multiperiod model also allows for an examination of how age and latency affect individual WTP to reduce mortality risk.

The theoretical foundation is the life-cycle model in which individuals are assumed to maximize their expected value of the utility of consumption (see, e.g., Johansson, 2002; Yaari, 1965). Let $τ,u(ct),i,$ and denote the point of reference, the utility of consumption at time t, the utility discount rate, and the probability at $τ$ of surviving to t. The individual’s expected utility is then given by

(2.7)

$Display mathematics$

To simplify the description of the multiperiod model, we follow Hammitt and Liu (2004) and illustrate it with a two-period model (assuming for the sake of simplicity that the marginal utility of a bequest is equal to zero):

(2.8)

$Display mathematics$

subject to the budget constraint

(2.9)

$Display mathematics$

where, as above, $p,u(c)$, and i are the survival probabilities (conditional on being alive at the beginning of each time period), utility of consumption, and the discount rate, with subscripts 1 and 2 referring to the first and second time periods. In a multiperiod framework, the VSL will depend on the optimal consumption path, and it can be shown that the optimization of consumption between periods is given by

(2.10)

$Display mathematics$

Let VSLj,k denote the marginal WTP of a survival probability that appears in k, but where the consumption is given up in j (i.e., the individual pays for the risk reduction in j). By totally differentiating Equation (2.8), we obtain

(2.11)

$Display mathematics$

which is the corresponding expression for a risk reduction that appears today and where the individual gives up other consumption (wealth) today, as in Equation (2.5). However, many health risks are characterized by a time period between exposure and the health effect, referred to as a latency period. It is therefore of interest to also estimate the WTP for latent risks, and the corresponding expression for a WTP today for a future risk reduction is given by

(2.12)

$Display mathematics$

Equations (2.10) and (2.11) assume temporary risk reductions; that is, the mortality risk is only reduced in a single time period, and we use these two equations when we discuss predictions of age, latency, and multiperiod WTP.

### Age and Latency and Multiperiod WTP

The relationship between age and preferences for health has received a lot of attention, and there is a vast theoretical and empirical literature on the subject (see, e.g., Huang, Andersson, & Zhang, 2017). Intuitively, it would make sense if WTP to reduce mortality risk were to decline with age, because in older age (everything else equal), an individual has less to gain from a reduction in the risk of dying. However, it has been shown that this expectation is not necessarily true. The relationship between the WTP and age will depend on the optimal consumption path over the individual’s life cycle, as shown by Equation (2.10), which will depend on assumptions of the model. For instance, Shepard and Zeckhauser (1984) showed that in an economy in which individuals can optimize their consumption path only by saving, but not by borrowing, their VSL will have an inverted U-shape over their lifetime; whereas in an economy in which they can also borrow against future earnings, their VSL declines monotonically with age. However, Johansson (2002) showed that the relationship is ambiguous since it depends on the assumptions of the model; that is, it can, in addition to the predictions in Shephard and Zeckhauser (1984) also be positive or independent.

The concept of latency was introduced in this article with Equation (2.12) showing the marginal WTP for a latent risk reduction. Intuition tells us that WTP for a current risk should exceed that of a latent risk, because an individual would also benefit from an early risk reduction in future time periods. However, as Hammitt and Liu (2004) pointed out, this intuition is misleading, and the WTP for a future risk reduction could also be equal to or greater than the WTP for a current risk reduction. They show this by subtracting Equation (2.12) from Equation (2.11), which results in

(2.13)

$Display mathematics$

which suggests,

(2.14)

$Display mathematics$

It is reasonable to assume that the survival probability will decline with age, and hence the right-hand side of the element to the right of Equation (2.14) will be positive. The condition on the left can be satisfied if the utility of consumption in the first period is sufficiently small compared to the one in the second period, which “seems unlikely but cannot be ruled out” (Hammitt & Liu, 2004, p. 78).

So far, this article has discussed temporary risk reductions that last for one time period. The multiperiod model can also be used to estimate the WTP for risk reductions that last several time periods or are permanent, because this WTP can be calculated as the summation of the WTP for future time periods with the risk reduction (Johannesson, Johanssonm, & Löfgren, 1997). This approach has also frequently been used in the empirical literature as a means to make the change in risk more understandable by making the risk change larger. However, Andersson, Hammitt, Lindberg, and Sundström (2013) have shown, this approach can introduce a nonnegligible bias if the time period is too long, or if the discount rate is sufficiently high.

# Empirical Methods

Because no easily available market prices exist for risk-reduction policies, researchers instead rely on what are usually referred to as nonmarket valuation techniques to estimate the WTP for such policies. As explained, these techniques can be broadly classified as either revealed- preference (RP) or stated-preference (SP) methods. The former refer to methods that use individuals’ actual decisions in markets that are related to the good of interest, and the latter are based on individuals’ responses to hypothetical choice situations. This section reviews the main approaches used in the literature and refers to additional sources for readers interested in learning more about a particular methodology.

## Revealed-Preference Methods

The RP approach most commonly used to estimate the WTP for risk reductions is the hedonic regression method (Rosen, 1974), which is based on the notion that the price of a good is a function of its attributes. For example, the price of a house will depend on characteristics such as the number and size of the rooms, as well as its location (proximity to amenities, transport links, etc.). The hedonic price function can then be written as

(3.1)

$Display mathematics$

where P is the price of the good and $Q=(q1,q2…,qk)$ is a vector of attributes. Rosen showed that in a competitive market with utility-maximizing individuals and profit-maximizing firms, the marginal willingness to pay (MWTP) for an attribute will equal its equilibrium implicit price. The implicit price of attribute $qk$ is given by the partial derivative of the hedonic price function with respect to that attribute:

(3.2)

$Display mathematics$

The hedonic wage method (Viscusi & Aldy, 2003) is a variant of the hedonic regression technique that uses data on individuals’ job choices to infer the trade-offs workers are prepared to make between wages and risk. Holding other characteristics of the individuals and the job constant, the increase in wages associated with an increase in workplace risk can be interpreted as the compensation needed to keep the workers’ utility constant. The hedonic wage regression is typically specified along the following lines (Viscusi & Aldy, 2003):

(3.3)

$Display mathematics$

where $wi$ is the wage rate of worker $i;Xi$ is a vector of individual characteristics; $Wi$ is a vector of job characteristics; $pi$ is the probability of a fatal workplace accident; $qi$ is the probability of a nonfatal workplace accident; and $εi$ is an error term. The model can be extended by including interactions between $pi$ and $Xi$, which allows the trade-off between wages and risk to vary across groups of workers, depending on their characteristics and interactions between qi and measures to compensate workers for nonfatal accidents. Based on the wage regression, the MWTP for a reduction in risk can be derived as the effect on wages of a unit increase in the probability of fatal accidents:

$Display mathematics$
(3.4)

If the model is linear, the MWTP is simply given by γ‎1, although it is common in the literature to compare estimates using different functional forms (such as semi-log or log-linear), or use a Box-Cox regression that nests several functional forms. A more comprehensive overview of hedonic regression methods is given in Freeman et al. (2014).

The hedonic wage method has been used extensively to estimate the WTP for risk reductions, especially in the United States (Environmental Protection Agency, 2016). There is also a substantial literature that utilizes data from the housing market by recognizing that housing associated with differential health risks will be capitalized in the market price. Examples include studies assessing how house prices are affected by local cancer clusters associated with increased risks of pediatric leukemia (Davis, 2004) and how hazardous water pollution affects residential land prices (Leggett & Bocksstael, 2000). An example of another variant of the hedonic regression method is given in Andersson (2005), who studies the relationship between car prices and fatality risks. The observation that consumers can reduce health risks by purchasing products associated with lower risks (such as safer cars or houses in less polluted areas) is the foundation for studies on averting behavior in consumption, which also includes studies using alternatives to the hedonic regression method (see, e.g., Blomquist, 2004).

## Stated-Preference Methods

As the name suggests, the SP approach is based on respondents’ stated choices in hypothetical market scenarios. There is a wide range of different SP methods, but the approaches most commonly used to elicit individual WTP are the contingent valuation method (CVM) and discrete choice experiments (DCE; Bateman et al., 2004). SP methods have been used to evaluate a wide range of health risks, such as from contaminated water (Adamowicz, Dupont, Krupnick, & Zhang, 2011), cancer risks (Hammitt & Haninger, 2010), road mortality risks (Andersson et al., 2013), and fire and drowning risks (Carlsson, Daruvala, & Jaldell, 2010). The advantage of the SP approach is that it offers flexibility in creating specific markets of interests and allows the analysts to control the decision alternatives. Johnston et al. (2017) provide best-practice recommendations for SP studies used to inform decision-making.

### The Contingent Valuation Method

The CVM presents a sample of survey participants with a hypothetical policy scenario that would reduce the risk of fatalities. The respondents are also presented with background information regarding the nature of the risk that the policy would reduce. The respondents are either asked to state their maximum WTP for the policy (open-ended CVM) or told how much they would have to pay if the policy were introduced (the “bid” in CVM terminology) and asked whether they are willing to pay this amount or not (closed-ended CVM).3 See Table 1 for an example of a closed-ended CVM question based on the application in Andersson, Hole, and Svensson (2016). An influential review of the CVM method carried out by the National Oceanic and Atmospheric Administration panel (Arrow et al., 1993) recommended the closed-ended approach, because it mimics a referendum in which respondents vote on whether a policy should be introduced in exchange for an increase in taxes. It is also similar to a market transaction in which consumers are presented with the price of a good and then decide whether or not to buy it.

Open-ended CVM data are straightforward to model using standard regression techniques, where the maximum WTP is specified as a function of individual characteristics. Closed-ended CVM data can be modeled using a latent variable framework, in which the latent (unobserved) WTP is specified as

(3.5)

$Display mathematics$

where $Xi$ is a vector of individual characteristics and $εi$ is an error term that is typically assumed to be normally distributed with mean 0 and variance $σ2$. The probability that respondent $i$ accepts a bid with value $ri$ is then given by

(3.6)

$Display mathematics$

where $Φ$ denotes the standard normal cumulative distribution function. The $α,β$, and $σ$ parameters can be estimated by maximum likelihood in standard software using interval regression (e.g., intreg in Stata).4

It is well established that the model specification can have a substantial impact on the estimated WTP, which indicates that analysts should conduct extensive sensitivity analyses using different specifications. It may also be advisable to estimate CVM data using distributional-free estimators such as the Turnbull estimator. See Haab and McConnel (2002) for an extensive introduction to the econometrics of CVM data.

Table 1. Example CVM Question

 Introductory text Campylobacter is a bacteria that can infect humans via food or water. Campylobacteriosis affects about 63,000 people in Sweden each year, which means that 700 people are affected annually in a medium-sized city with 100,000 inhabitants. The symptoms of the disease vary from case to case, but simply put, one can say that there are mild, intermediate, and serious versions of the disease. It is not known which version of the disease one will get before being affected by campylobacteriosis. What is known is that among those affected, 77 out of 100 get the mild version, 22 out of 100 the intermediate version, and 1 out of 100 get the serious version. In very rare cases, affected individuals die from the disease (fewer than 5 per year in Sweden). CVM question Assume that a government authority is considering introducing a stricter water sanitation policy that will reduce the occurrence of campylobacter. Would you be willing to pay 2,000 Swedish krona for a policy that would imply that 2 fewer persons will die in Sweden from campylobacteriosis?

### Discrete Choice Experiments

In DCEs, the participants are presented with two or more hypothetical policies and asked to choose their preferred policy or the status quo (not introducing either policy). As an example, we use a simple experiment with only two attributes: the number of fewer individuals who die when the policy is implemented and the cost of the policy (see Table 2). Respondents are asked to choose their preferred option between two hypothetical scenarios and the status quo (the choice set). In this simple form, the DCE method shares many similarities with the closed-ended CVM method, but the advantage of DCEs is that further attributes of the alternatives can easily be accommodated in the experiment. For example, we could include an additional attribute representing the reduction in nonfatal cases of campylobacteriosis, which would allow us to investigate how individuals trade off reductions in fatal and nonfatal cases (Andersson et al., 2016). The levels of the attributes presented to the participants in the experiment—that is, the number of fatalities prevented and the cost of the policy—are varied according to an experimental design (Carlsson & Martinsson, 2003). Typically, each respondent is presented with several hypothetical choice sets with different attribute levels.

Data from DCEs are typically analyzed using a random utility model framework. The utility that respondent n derives from choosing alternative j in choice set t is given by

(3.7)

$Display mathematics$

where $β0$, $β1$, and $β2$ are coefficients to be estimated; $sqnjt$ is a dummy variable for the status quo alternative; $dienjt$ is the number of fewer individuals who die when the policy is implemented; $costnjt$ is the cost of the policy; and $εnjt$ is a random error term that is assumed to be IID type I extreme value.

The MWTP for a reduction in risk equivalent to saving one life is given by the marginal rate of substitution between cost and lives saved:

(3.8)

$Display mathematics$

Given these assumptions, the probability that respondent n chooses alternative j in choice set t has a multinomial logit (MNL) form:

(3.9)

$Display mathematics$

The parameters in the multinomial logit (MNL) model5 can be estimated using maximum likelihood in standard software (e.g., asclogit in Stata). Lancsar, Fiebig, and Hole (2017) present a practical guide to modeling DCE data with examples that use Stata and other software packages.

Other more advanced discrete choice methods that overcome the main limitations of the MNL model are commonly used in the DCE literature. In particular, the assumption that respondents have the same preferences for changes in the attributes is usually thought to be unrealistic. Although this assumption can be relaxed by augmenting the MNL model with interactions between the attributes and respondent characteristics, the researcher does not typically observe all the characteristics that are related to heterogeneity in preferences. The mixed logit model overcomes this limitation by allowing the parameters in the model to vary randomly. The vector of parameters $β$ is specified to have a particular distribution, $f(β|θ)$, whose parameters $θ$ can be estimated. If the coefficients are normally distributed, for example, $θ$ represents the mean and covariance of the distribution. The mixed logit probability that respondent n makes a particular sequence of choices is given by

(3.10)

$Display mathematics$

where $ynjt$ is a dummy variable that is equal to one if alternative j is chosen and to zero otherwise, and $β=(β0,β1,β2)$. Besides allowing for preference heterogeneity, the mixed logit model allows for the fact that respondents make multiple choices, as the individual preferences are assumed to remain constant over the choices made by the same individual. The integral in Equation (3.10) cannot be solved analytically, and it is therefore approximated using simulation methods. See Train (2009) for a comprehensive review of the mixed logit model and other advanced discrete choice methods using simulation.

An alternative to assuming that the coefficient distribution is continuous (e.g., normal or log-normal) is to specify that it is discrete. A mixed logit model with a discrete coefficient distribution is often referred to as a latent class logit model (e.g., Greene & Hensher, 2003; Hole, 2008). The latent class logit probability that respondent n makes a particular sequence of choices is given by

(3.11)

$Display mathematics$

where the coefficients are given a class (c) subscript to indicate that preferences vary across (but not within) classes. $Hnc$ is the probability that individual n belongs to class c, which is typically specified to have a multinomial logit form:

(3.12)

$Display mathematics$

where $Zn$ is a vector of characteristics relating to individual $n$ and $γC$ is normalized to zero for identification purposes. Stata implementations of the mixed logit model and latent class logit model are described in Hole (2007) and Pacifico and Yoo (2013).

Table 2. Example of Discrete Choice Experiment Question

 Introductory textAssume that a government authority is considering introducing one of the following two stricter water-sanitation policies that will reduce the occurrence of campylobacter. Policy A Policy B Number of fewer individuals who die when the policy is implemented 1 2 Your cost 1,000 Swedish krona 2,000 Swedish krona Which option do you prefer?□ Policy A□ Policy B□ Neither of the suggested policies (today’s situation remains, and at no additional cost for you)

# Examples of Applications

## Using Hedonic Price Regressions to Value Mortality Risks

An example of a hedonic pricing study valuing mortality risks is Gentry and Viscusi (2016), in which they specified the basic hedonic regression equation as (p. 93)

(4.1)

$Display mathematics$

The equation shows that the natural logarithm of wage $(w)$ for an individual $i$ in industry $j$ and occupation $k$ was regressed on a set of individual demographic controls $(X)$, such as age, sex, race, years of education, and so forth; a set of industry- and occupation-specific controls $(W)$, the fatality $(p)$ rate, and injury $(q)$ rate. The coefficient estimate of interest to value the mortality risk is $γ1$, whereas $γ2$ shows the impact on wage of the nonfatal injury rate. Data on wages and control variables were collected on the individual level from the Current Population Survey, whereas the fatality and injury rates data were based on average risks per industry and occupation retrieved from the U.S. Bureau of Labor Statistics Census of Fatal Occupational Injuries.

The fatality rate is based on average industry-occupation risks and is calculated as

(4.3)

$Display mathematics$

where $Njk$ is the number of fatal injuries and $EHjk$ is total hours worked by everyone in the specific industry-occupation group. The rate is multiplied by 2,000 (hours worked per year per worker) and subsequently by 100,000 to express the rate as the annual risk per 100,000 people. The reported overall annual fatality rate was 6.23 fatalities per 100,000 employees, but this varied from, for example, 0.75 fatalities per 100,000 healthcare employees to 70.1 fatalities per 100,000 employees in forestry, fishing, and hunting occupations. The injury rate is calculated in a similar manner, the difference being that N is the number of work days lost due to injury per year.

The results from estimating Equation (4.1) showed that a higher fatality and injury rate were both statistically significantly related to higher hourly wages. An increase in the fatality rate by 1 in 100,000 was associated with a 0.16% higher hourly wage. Hence, the evidence suggests that workers are compensated to take on risk, and based on Equation (4.1), the VSL was estimated as

(4.3)

$Display mathematics$

where the mean hourly wage ($wage¯$) is multiplied by 2,000 to convert the hourly wage into an annual wage. The annual wage is then multiplied by 100,000 to take into account that the risk is per 100,000 workers. Based on the regression result and the mean wage for the full sample, the implicit VSL was estimated at $5.36 million.6 A large number of similar studies have been carried out using wage-risk equations to estimate the implicit VSL. Viscusi (2014) reviewed many of the wage-risk studies, and updated to 2016 (based on the GDP deflator), the mean predicted VSL ranged from$7.5 million to $10.3 million in the U.S. sample. Results reported in this literature have been important in policymaking and have influenced the recommended VSL estimates for economic evaluations, especially in the United States, by for example, the U.S. Department of Transportation and the U.S. Environmental Protection Agency (Department of Transportation, 2016a, 2016b; Environmental Protection Agency, 2010). ## Using Stated Preferences to Value Health Risks Because SP methods can be tailored to value any specific condition, they have been applied to a broader range of health risks than has the RP method. For example, SP methods have been used to estimate the WTP for primary-care cancer tests, insecticide-treated mosquito nets to reduce malaria risks, and genetic testing for inherited retinal disease (Biadgilign, Reda, & Kedir, 2015; Hollinghurst et al., 2016; Tubeuf et al., 2015). However, the single largest area of study has been to estimate the WTP to reduce mortality risks, that is, comparable to the case with hedonic price regressions. A recent meta-analysis covering only a subset of all VSL estimates (Lindhjelm, Navrud, Braathen, & Biausque, 2011) includes almost 1,000 published VSL estimates using SP methods. Here we borrow from two studies to provide examples of how SP methods can be used to estimate the WTP for reductions in health risks. ### A CVM Application Hammitt and Haninger (2017) used the CVM method to estimate the WTP for small reductions in the risk of suffering nonfatal health conditions. The survey was administered to a sample drawn randomly from a U.S. internet panel. The survey respondents were asked to choose whether they would participate in a health-protection program that would reduce the risk of developing a particular illness from exposure to environmental contaminants. The illness varied across survey versions and was described using the EQ-5D health-state descriptive system (EuroQol Group, 1990). In addition, half of the respondents were presented with the condition name (e.g., “migraine headaches”). Depending on the question, the risk reduction either related to the respondents themselves, another adult living in the household, or a child younger than 18 years. The baseline mortality risks and risk reductions used in the survey are shown in Table 3. Table 3. Mortality Risks and Risk Reductions Baseline Risk Risk if Participating in the Program Risk Reduction 3 in 10,000 3 in 10,000 4 in 10,000 4 in 10,000 2 in 10,000 1 in 10,000 3 in 10,000 2 in 10,000 1 in 10,000 2 in 10,000 1 in 10,000 2 in 10,000 Note: All figures are per year. Source: Hammitt and Haninger (2017). The baseline, reduction, and final risks were presented using a grid containing 10,000 squares, with the number of red squares corresponding to the risk after reduction (1, 2, or 3) and the number of white squares to the risk reduction (1 or 2). The total number of red and white squares represented the baseline risk (3 or 4). The risk information was also presented numerically, as in Table 3. The WTP elicitation format was double-bounded dichotomous choice, in which respondents are asked two sequential closed-ended CVM questions in each valuation task. The respondents were first asked if they would be willing to pay a cost$X to participate in the program, where X ranged from $10 to$2,000 per year. If they answered yes (no) to the initial WTP question, a second question followed in which they were asked if they would participate in the program if the cost was $X × 2 ($X × 0.5). To set the range of the initial cost (bid) vector, the authors assessed the implicit minimum and maximum value per statistical illness covered. With a range from $10 to$2,000, the implicit value ranges from < $50,000 (which results from not paying$10 for the 2/10,000 risk reduction) to > $20 million (resulting from being willing to pay$2,000 for the 1/10,000 risk reduction).

The authors estimated the following model using interval regression:

(4.3)

$Display mathematics$

where $ri$ is the reduction in the probability of illness, $qi$ is the reduction in health-related quality of life (HRQL) as a result of the illness (on a scale from 0 to 1), and $ti$ is the duration of the illness. The vector $Xi$ includes additional control variables, such as current HRQL, and $εi$ is a normally distributed error term. This model specification implies that $γ1$ is the elasticity of WTP with respect to a change in the probability of illness. The standard theory implies that for small changes in the probability, this elasticity should be close to 1, implying that WTP is near-proportional to small changes in risk.

The authors estimated the model in Equation (4.3) on four different samples: including all responses or including only responses in which the target of the risk reduction was the respondent, another adult living in the household, or a child younger than 18 years. In all four samples the coefficient on the reduction in the probability of illness was found to be close to (and insignificantly different from) 1, confirming the theoretical prediction that WTP is near-proportional to small changes in risk. The coefficients on the reduction in health-related quality of life and the duration of the illness were found to be positive, but less than 1, indicating that the WTP increases with these variables but not proportionally. Including dummy variables for the target type in the pooled model indicated that the respondents have a higher WTP for reducing the probability of illness for a child younger than 18 years (200% more) and for another adult living in the household (150% more) than for themselves. The authors carried out various sensitivity analyses, which generally did not affect the main findings.

The modeling results can be used to estimate the WTP for different illnesses characterized by their duration and reduction in health-related quality of life. The predicted WTP for a statistical illness can be calculated by predicting log (WTP) using the regression results, exponentiating the predicted value (which yields an estimate of median WTP) and dividing by the risk reduction. Some illustrative values are presented in Table 4:7

Table 4. Values Per Statistical Illness (in Dollars)

Duration(years)

Health-Related Quality of Life Loss

Household Member at Risk

Self

Child

1

0.1

678,000

2,010,000

1,670,000

1

0.8

1,380,000

4,090,000

3,400,000

5

0.1

817,000

2,420,000

2,020,000

5

0.8

1,660,000

4,930,000

4,100,000

It can be seen from the Table 4 that the predicted WTP for children in the household is about 3 times as high as the value for the respondents themselves, while the WTP for another adult is about 2.5 times as high. This corresponds to the magnitudes of the estimated target type dummy coefficients. It is also apparent that while the WTP increases in the HRQL loss the increase is not proportional (the WTP of a 0.8 decrease in HRQL is less than 8 times the WTP of a 0.1 decrease), which is a result of the HRQL coefficient being positive but less than one.

### A Discrete-Choice Experiment Application

Adamowicz et al. (2011) used the DCE method to elicit consumers’ preferences for reductions in the health risks associated with tap water. A sample drawn from a panel of Canadian internet users was invited by email to complete an online survey, which presented the respondents with information about the health risks and different programs to reduce such risks before introducing the DCE choice tasks. In the DCE, the tap water delivered to households was described in terms of the following attributes: two types of mortality risks (cancer and microbial), two types of morbidity risks (cancer and microbial), and the costs to the household of disinfection and treatment methods to reduce the health risks. The respondents were presented with four DCE choice tasks of the form presented in Table 5.8

Table 5. Example of Discrete Choice Experiment Choice Task

 This is the first scenario we want you to vote on. For every 100,000 People, the NUMBER Who Would . . . Current Situation Proposed Program A Proposed Program B Get sick from microbial illness in a 35-year period 23,000 15,000 7,500 Die from microbial illness in a 35-year period 15 10 5 Get sick from bladder cancer in a 35-year period 100 50 100 Die from bladder cancer in a 35-year period 20 10 20 Change to your water bill starting in January, 2005 No change Increase $75 per year ($6.25 per month) Increase $75 per year ($6.25 per month) If there were a referendum I would vote for . . .CHECK ONE ONLY□ Current Situation□ Proposed Program A□ Proposed Program B

Note: The choice sets used in the actual survey also included a graphical representation of the mortality and morbidity risk attributes.

The respondents’ utility function was specified to be a linear function of the attributes, that is, the number of microbial deaths and illnesses averted ($mdienjt,msicknjt$), the number of cancer deaths and illnesses averted ($cdienjt,csicknjt$) and the cost of the program ($costnjt$), as well as a status quo dummy variable ($sqnjt$) and an error term ($εnjt$):

(4.3)

$Display mathematics$

The authors estimated the parameters in the utility function using an MNL model with and without interactions between the status quo variable and respondent characteristics. Evidence of preference heterogeneity was explored using a mixed logit model with lognormally distributed mortality and morbidity risk coefficients and a fixed (nonrandom) cost coefficient, and a latent class logit model with two classes in which the class membership was modeled as a function of individual characteristics.

Based on the results from the MNL model, the authors found that the WTP for one fewer microbial death in the community was $12.6, and that the WTP for one fewer cancer death was$10.4 (including respondent characteristics in the model made little difference to the results).9 The difference between these MWTP estimates was not found to be statistically significant. The difference between the MWTP estimates for morbidity risk reductions, on the other hand, was found to be statistically significant, with cancer risk reductions being valued more highly ($2.43 vs.$0.02). The corresponding mean MWTP estimates from the mixed and latent class logit models are $15.0/$13.6 (microbial mortality risk), $10.8/$12.2 (cancer mortality risk), $0.02/$0.02 (microbial morbidity risk), and $3.05/$2.32 (cancer morbidity risk).

While the mean MWTP estimates were found to be similar across models, there is evidence of a large degree of preference heterogeneity among respondents. The results from the latent class model suggest that there are two distinct groups of respondents, one who are reluctant to move away from the status quo and show an unwillingness to make trade-offs for improved water quality. The authors argue that this suggests that a discrete representation of preference heterogeneity is appropriate in this context because it allows such behavior to be identified, whereas a continuous representation of preference heterogeneity (as in the mixed logit model) does not.

The VSL was found to be $17 million for microbial death and$14 million for cancer death, according to the MNL model. The VSL estimates are derived by multiplying the MWTP estimates by the number of households in the community (38,500) over a 35-year period.10 The estimates are at the high end of the range reported in the VSL literature, and the authors suggest that this may be because their estimates are based on the WTP to reduce public mortality risks. This means that they include both the WTP to reduce the risk of death to oneself and the WTP to reduce risk of death to others in the community, including family members. This is in contrast to the majority of studies in the VSL literature, which estimate the value of private mortality risk reductions.

# Discussion

This article has provided an overview of the valuation of health risks as carried out in economics. The willingness-to-pay (WTP) approach is well established as the appropriate approach to assign monetary values to health risks that reflect individual preferences. As described in the section “Theory,” the monetizing of health risks is based on well-developed theoretical models that provide predictions for the empirical applications of the WTP approach. Among economists, valuation of health is quite uncontroversial; individuals make daily decisions in their lives that suggest that they have a finite WTP to reduce their risk of fatality, injury, illness, and so on. Moreover, when health policies are implemented, it is not known who will benefit from the policies, only that a certain number of deaths, injuries, and so forth, will be prevented (veil of ignorance). However, among noneconomists, assigning monetary values to health outcomes is controversial, especially assigning monetary values to the “value of life.” This means that the values obtained in empirical studies are often heavily scrutinized, whether they come from revealed-preference (RP) studies or stated-preference (SP) studies, simply because they are considered to put a “price” on health outcomes. Advocates of WTP studies for health outcomes obviously do not agree with the ethical objections to valuing health, but they can agree that analysts face several difficulties in implementing health valuation studies.

Economists have traditionally been more willing to accept estimates from RP methods considering the reliance on actual behavior. A drawback of the RP approach is that markets do not always exist for the good of interest. For example, it may be that individuals have different preferences for different kinds of risks, such as the risk of dying from cancer versus the risk of a suffering a fatal workplace accident. Furthermore, the hedonic wage method depends crucially on the analyst having access to accurate measures of risk and all the relevant individual and job-related characteristics, to avoid biases arising from either measurement error or omitted variables.

A larger literature focuses on the limitations and drawbacks of SP methods (see, e.g., Hausman, 2012). The main criticism of SP methods is that the decisions are hypothetical, which means that respondents do not have an incentive to become well informed when making their decisions and that their stated responses may not reveal how they would have acted if the decision had been real. A particular issue that has received much attention in the literature is the lack of sensitivity to scope. Although most CVM studies find evidence that the WTP for risk reductions increases with the size of the reduction in risk, the increase is typically not near-proportional, which is a necessary validity criterion for SP-based WTP estimates (Corso, Hammitt, & Graham, 2001; Hammitt & Graham, 1999; Robinson & Hammitt, 2015). Although there is less evidence on scope sensitivity in discrete-choice experiments (DCEs), especially using between-subject designs in which different respondents are presented with different risk reductions, recent findings have suggested a lack of scope sensitivity in DCEs as well (Andersson et al., 2016).11 This evidence implies that SP estimates of the WTP for risk reductions must be interpreted with caution. However, despite the criticism of eliciting preferences based on hypothetical scenarios, there has been a large increase in the use of SP studies since the 1990s (Carson & Hanemann, 2005).

Despite this criticism and the shortcomings of both approaches—that is, for revealed preferences, for example, the problems of nonexisting markets and that the analysts may not be well informed about the decision alternatives individuals do face, and for stated preferences, as listed in the previous paragraph, both have a key role to play in eliciting preferences for health risk reductions. Because there are no readily available prices for health risk reductions, nonmarket valuation methods are necessary to monetize these preferences. There has also been a lot of progress in estimating WTP for health risk reductions since the 1990s, both regarding access to data for RP studies, and methodological improvements in SP studies, resulting in improved validity and reliability of the estimated values (Viscusi, 2014).

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## Notes:

(1.) There is a very large literature on CEA and the health-risk metrics used in CEA, such as quality-adjusted life years (QALYs) and disability-adjusted life years (DALYs). These methods and metrics will not be covered in this article, whose focus is on the welfare economics approach to health-risk valuation.

(2.) Such as large variation in possible health outcomes and context dependence related to survival when presenting the theoretical model.

(3.) Other versions of the CVM method, such as the use of payment cards, have also been used in practice, although the open- and closed-ended approaches are most commonly used.

(4.) If the respondents are presented with a single bid, the observations are either left-censored or right-censored in the interval regression terminology. If respondents are additionally presented with a follow-up bid, then some observations will be intervals (if the respondent accepts one bid and rejects the other), left-censored (if both bids are rejected), or right-censored (if both bids are accepted).

(5.) A multinomial logit model with alternative-specific attributes is often referred to as a conditional logit model (see, e.g., Lancsar et al., 2017).

(6.) The authors also extend the basic model to incorporate what they call morbidity risks associated with a fatal injury, which captures the number of days from the injury until death. For example, in some cases, death occurs immediately after an injury; whereas in other cases, the time from injury to death (when it occurs) may stretch up to several months.

(7.) We refer interested readers to Hammitt and Haninger (2017) for information on the values chosen for the different explanatory variables when generating the predicted values of log (WTP).

(8.) Some respondents were instead randomly chosen to complete DCE tasks with only two alternatives (one program and the current situation) and DCE tasks in which the ratio of morbidity and mortality reductions were held constant. The survey also included three double-bounded CVM choice tasks, but here the focus is on the DCE results.

(9.) The MWTP estimates are calculated as the ratios between the mortality risk and cost coefficients, as exemplified in Equation (3.8).

(10.) The calculations assume that the risk of death is equally likely over the entire 35-year period. The authors also present a sensitivity analysis in which this assumption is relaxed.

(11.) Between-subject designs are considered a more stringent approach to test for scope sensitivity, as within-subject designs may lead to a failure to reject scope sensitivity due to respondents’ desire to be “internally consistent” when completing the survey.