Age effects in monetary valuation of reduced mortality risks: the relevance of age-specific hazard rates

Abstract

This paper highlights the relevance of age-specific hazard rates in explaining the age variation in “value of statistical life” (VSL) figures. The analysis—which refers to a stated preference framework—contributes to the ongoing discussion of whether benefits resulting from reduced mortality risk should be valued differently depending on the age of the beneficiaries. By focussing on a life-threatening environmental phenomenon I show that the consideration of the individual’s age-specific hazard rate is important. If a particular risk affects all individuals regardless of their age so that their hazard rate is age-independent, VSL is rather constant for people at different age; if hazard rate varies with age, VSL estimates are sensitive to age. The results provide an explanation for the mixed outcomes in empirical studies and illustrate in which cases an adjustment to age may or may not be justified. Efficient provision of live-saving measures requires that such differences to be taken into account.

Introduction

To reduce mortality risks and hence save people’s lives, various risk reduction programmes are generally available. They differ in their effectiveness regarding the number and groups of beneficiaries. For example, reducing air pollution leads to a reduction in respiratory diseases, which saves the lives of primarily the youngest and oldest groups of the population, while only working people benefit from reduced work place risks. If budget constraints are relevant, the question emerges how restricted resources should be distributed across the available life-saving measures so that social benefits are maximised.

An economic approach to evaluate the benefits resulting from reduced health risks is the concept of “value of statistical life” (VSL). The VSL provides a monetary measure of health benefits and represents the ratio at which people are willing to exchange income for risk changes. The VSL is often inferred from information about the individual willingness-to-pay (WTP) to prevent mortality risks. It is an ongoing discussion whether the financing of life-saving measures should be adjusted to individual characteristics (for a comprehensive discussion, see Sunstein 2004). Adjustment according to age has provoked particularly intense debate. Government agencies pursue different approaches regarding the use of such a “senior discount”. For example, while the European Commission DG Environment recommends an adjustment to age (European Commission 2000), an age-dependent VSL is not applied by the US Environmental Protection Agency (Environmental Protection Agency 2003, p. 37). These different governmental approaches may result from ambiguous findings in the literature with respect to age effects on VSL.

The purpose of this paper is to investigate empirically the importance of hazard rates in explaining the ambiguity discussed in theoretical and empirical literature regarding the age-sensitivity of the VSL. In contrast to previously published preference studies, the relevance of an individuals’ probability of dying for VSL estimates is analysed by using information on a particular risk specification that allows for distinguishing explicitly between two groups exposed to the same risk type. In this paper, mortality risk is defined as the probability of dying in a snow avalanche. While one group (skiers) faces an age-dependent hazard rate, the hazard rate of the others (non-skiers) is rather insensitive to individual characteristics. The paper addresses the question of whether people benefit differently from reduced mortality risks, in particular, whether WTP and VSL is sensitive to age once the individual mortality risk (which can be either sensitive or insensitive to age) is taken into account. Age is considered by calculating the WTP and VSL for particular age classes.

I find that the individual hazard rate associated with a particular risk plays a crucial role in explaining the age dependency of VSL measures. Age effects do occur for skiers but are not present for non-skiers. Accordingly, adjustment to age may be justified only in cases where the individual probability of dying varies with age.

The paper is organised into the following sections: "Review of literature" examines the literature dealing with the estimation of VSL and its age dependency, "Survey design" describes socio-economic characteristics, survey setting and wording of risk. "Estimation procedure" explains model specification, VSL, and age effects. "Results" summarises the overall effects. "Conclusion" presents resolving remarks.

Review of related literature

The literature dealing with estimation of VSL is extensive. VSL is discussed in various contexts, such as valuation of reduced health risks (e.g. cancer, respiratory diseases, cardiovascular diseases), work place risks, traffic risks, or risks arising from natural hazards.1 In general, VSL estimates are dependent on whether they are based on one’s overall risk of dying or on a very specific risk of dying, the profile of future income, the remaining life expectancy or the individuals’ baseline risk. Among these determinants, the relevance of age in mortality risk assessment is of ongoing interest and has been discussed and analysed in several studies. 2 The theoretical literature defines two key components of how age affects the monetary valuation of reduced mortality risks: (1) the individual hazard rate (probability of dying), and (2) the level of optimal consumption.

Shepard and Zeckhauser (1984) discuss the trade-off between an increase in survival probability and the level of consumption. In their simulations, the authors demonstrate that the profile of WTP with respect to age depends on the existence of markets for borrowing and annuities, as such markets influence the individuals’ income over time and, hence, the individual consumption path. If no markets exist, the WTP for risk reductions follows an inverted-U shape, with lower values for the youngest and oldest individuals in the population and a peak of payments at the age of 40. However, this shape is dampened considerably when perfect markets for borrowing and annuities are available. A further theoretical discussion of age effects on VSL is provided in Johansson (2002) or Alberini et al. (2004). These authors also analyse the role of individual hazard rates and the utility of income and argue that theory does not support a general decline of VSL with age. They conclude that, depending on the age sensitivity of the hazard rate and the utility of income, the VSL can either remain age-independent, increase, or decrease with increasing age.

Such theoretical discussions show that age can influence the VSL quite differently. Empirical studies provide evidence for this ambiguous picture. For example, Smith et al. (2004) estimate the VSL for older workers by observing their behaviour (wage-risk trade-offs) on the labour market. Their findings show an increase of compensation requirements with increasing age, which is mirrored in higher VSL figures for older employees. Viscusi and Aldy (2007) examine how VSLs vary with age across the working population considering age-specific fatality risks. Their VSL estimates for different age classes project an inverted U-shape, indicating that VSL figures first rise and then fall with age. A considerable decrease of VSL with age is found in Alberini et al. (2005), in which the authors estimate the WTP for reduced cardiovascular and respiratory risks. According to their results, people in their 70s state values that are about three-quarters lower than the statements of 30-year-old respondents. Alberini et al. (2004) examine the influence of age and health risks on VSL using data from two contingent valuation studies conducted in Canada and the US. They analyse the individuals’ WTP for a reduction in their overall mortality risk—without pointing at a specific risk—and find weak (no) evidence for decreasing VSL with increasing age in the Canadian (US) sample. Liu et al. (2005) refer to mortality risks due to severe acute respiratory syndrome (SARS) and do not find any influence of age on WTP to eliminate the chance of becoming infected.

Based on these examples, I argue that the individual probability of dying associated with a particular risk type plays a crucial role. For instance, while workplace risks or cardiovascular risks are age-dependent and affect a particular group of individuals (job risks: working population; cardiovascular risks: elderly people), people of all ages are potentially in danger of becoming infected with some viruses. The following sections describe in detail how the assumption that the individual probability of dying is an important factor in the monetary valuation process for reduced health risks is tested in a stated preference framework. 3

Survey design

Socio-economic characteristics

This paper analyses responses from a contingent valuation survey conducted in September/October 2004 and February 2005 in Tyrol, a federal state of Austria. A randomised quota sample of 1,997 individuals was drawn from the Tyrolean population aged over 15 years. The quota applied to the subjects’ district of residence and size of domicile. Within the quota, random sampling was used. The respondents were asked in face-to-face interviews at their permanent residences about their WTP to prevent an increase in the risk of dying in a snow avalanche. 4 Table 1 compares the socio-demographic characteristics of survey participants and the population of Tyrol to give an insight into how well the sample represents the population.

Table 1.

Sample and population characteristics

Variable	Sample		Census
	ObservationsI	Mean	Mean
Sex of respondent (female = 1, 0 otherwise)	1996	0.53	0.52a
Age of respondent (in years)	1954	37.08	43.79a
Marital status of respondent (single = 1, 0 otherwise)	1958	0.39	0.35a
Number of household members	1982	2.82	2.56b,e
Children per capita	1997	0.42	0.23a
Birthplace of respondent (Austria = 1, 0 otherwise)	1997	0.89	0.88a,e
Smoking behaviour (smoker = 1, 0 otherwise)	1988	0.51	0.30c
Personal take-home income per month (in 1,000 €)	1128	1.08	1.11d
Self reported health status of respondent
Healthy	1937	0.76	0.80c
Moderate illness	1937	0.20	0.16c
Bad illness/bad disability	1937	0.04	0.04c
Education of respondent
Elementary/junior high school	1967	0.22	0.37a
Apprenticeship	1967	0.33	0.33a
Vocational school	1967	0.16	0.13a
Secondary school/post-secondary coursesII	1967	0.20	0.10a
College/university	1967	0.09	0.07a
Employment status of respondent
Employed full time (38 working hours per week)	1967	0.53	0.48c
Employed part-time	1961	0.10	0.07c
Employed short-time (monthly income not over € 323.46)	1967	0.02	0.03c
Retired	1961	0.12	0.22c
Homemaker	1961	0.03	0.10c
Student	1961	0.11	0.06c
Unemployed	1961	0.02	0.03c
Others	1961	0.06	0.02c

The survey sample refers to Tyroleans ≥15 years old interviewed in September/October 2004 and February 2005. The Census represents the whole population of Tyrol (=673,504) in 2001 (exceptions are noted). Where feasible, children <15 (=123,855) were excluded from comparisons

^IDifferences in numbers of observations due to missing data

^IIThe Austrian educational system provides a 2-year-programme (“post-secondary courses”) designed for students who did not get vocational education in their secondary school

^aPopulation in 2001. Source: Statistics Austria. Statistical Yearbook 2005, Table 2.14

^bSource: Tyrolean Provincial Government 2004. Tyrolean Population—Results of the Census 2001, Table 25

^cPopulation in 1999 >15 years. Source: Tyrolean Provincial Government 2003. Gesundheitsbericht 2002, Table 3.4.1

^dMonthly take-home income (=annual income/14; in 1,000 €) of employees in 2003. Source: Statistics Austria, Statistical Yearbook 2005, Table 9.07

^eExclusion of children was not possible

A comparison of the sample characteristics with the census attributes shows a good approximation to the population characteristics in gender, birth place, and income, 5 while the divergence in age, children per capita, and smoking behaviour is considerable. The self-reported health 6 in the sample conforms quite well with the census average. The differences between the sample and census with respect to education and employment may result from the younger age of individuals in the sample.

Survey setting and wording of risk

In order to control for the sensitivity of WTP to the dimension of risk variation (=“scope effects”) two different risk changes were incorporated in the questionnaire: participants in the winter survey were asked about their WTP to prevent a doubling or quadruplication of the baseline risk. 7 The initial risk level was inferred from official statistics. Risk prevention was assumed to come from passive control measures against avalanches, i.e. fatal avalanche accidents could be reduced by means of constructions such as tunnels or barriers. In the underlying study it was assumed that an increase in risk is avoidable by means of maintenance of pre-existing risk reduction programmes. The reduction in question was described as follows (divergence in wording for the larger risk variation in parentheses):

Several risk reduction programmes against avalanches on roads and in residential areas have been implemented in Tyrol. At present, on average 2.35 people out of 100,000 inhabitants are killed by avalanches annually. Assume that all public funds for maintaining these measures will be cut, and so servicing costs henceforth have to be paid exclusively out of private funds. If aggregate private contributions are too small, maintenance is not carried out, and the probability of a fatal avalanche doubles (quadruples). Then, on average, 4.7 (9.4) people out of 100,000 inhabitants die in avalanche-related accidents (see Fig. 1). Would you be willing to pay—given your income constraint—a monthly insurance premium of € 2.5/5/10 to maintain the effect of previous risk reduction programmes to save human lives?

Fig. 1.

Causes of deaths in Tyrol in the year 2002

Respondents were assigned randomly to one of three versions that differ in the bid vector. The initial bids of versions 1, 2 and 3 are 2.5, 5, and 10, respectively. Depending on the response to the first question, participants were asked whether they would also pay 5 (version 1)/10 (version 2)/20 (version 3) Euros in case of a “yes” answer to the initial question, or 1.3 (version 1)/2.5 (version 2)/5 (version 3) Euros if the reply to the first question was “no”. 8 If the interviewees’ answers were “no; no” or “do not know; no” they were asked whether they would pay anything at all and why they refused payment. 9 Individual responses were classified as protest answers if the interviewees stated that they generally refuse payments for protection against natural hazards or if it was argued that the protection of citizens was the responsibility of the government. Protesters (N = 329) were excluded from further analyses. 10

Avalanche protection of roads and residential areas are usually financed by national, federal and municipal funding. Analogous to the Austrian social security system, which finances publicly provided health services through compulsory insurance premiums, ‘insurance premium’ was used as a parafiscal payment vehicle that respondents are familiar with. This approach was chosen to avoid respondents’ negative attitudes towards taxes. 11

In order to reduce the difficulty of interpreting small risk variations (Kunreuther et al. 2001, Shanteau and Ngui 1989), I considered the recommendation of Corso et al. (2001) and visualised the risk change in question. In the Tyrolean study, a logarithmic scale was implemented to picture the risk variation (see Fig. 1; graph shown for the smaller change). The bottom and top indicate low and high risk, respectively. Along the risk ladder, other mortality risks, such as cancer, AIDS, or car accidents were plotted to enable a comparison and relative estimate. Additionally, the magnitude of the risks was also stated as number of affected persons in differently sized populations (e.g. small town, city).

Estimation procedure

Model specification

This section links the theoretical background of VSL and the econometric approach chosen in this paper and refers to the arguments given in Johansson (2001).

Reductions in mortality risk are commonly evaluated within a state-dependent utility framework (Jones-Lee 1974; Pratt and Zeckhauser 1996; Weinstein et al. 1980). It is assumed that individuals substitute income, y, for a risk change, Δp, so as to maximise their expected utility. Johansson (2001) looks at this maximisation problem using a life-cycle VSL approach, explicitly considering the influence of age on the VSL. This approach assumes that individuals maximise their expected remaining present value utility (ERPVU) subject to a dynamic budget constraint. Solving the maximisation problem results in the maximal ERPVU at age τ, denoted by V _τ(τ), conditional on having survived until age τ (for details, see Johansson 2001):

1

denotes the utility derived from the optimal consumption level at age t,   with t ≥ τ,  θ represents the marginal rate of time preference and is assumed to be a constant, and stands for the survival function conditional on attaining age τ with and the hazard function γ(s).12 Consider a decrease in the hazard rate and the associated change in the ERPVU:

2

with τ + T referring to the point in time when the risk reduction starts and dμ(t;τ) representing the time change in the survival function resulting from the change in the hazard rate. In the VSL context, this change in the hazard rate is outweighed by a payment CV so that the ERPVU remains unchanged. Using the first order condition, one obtains

3

The first term represents the change in utility due to the change in risk. stands for the dynamic Lagrange multiplier that results from differentiating the Hamiltonian function (Johannesson et al. 1997). The dynamic Lagrange multiplier can be interpreted as the expected present value of marginal utility of income at age τ (Johansson 2001). dCV describes the amount of money an individual at age τ is willing to pay today to achieve a decrease in the hazard rate for the rest of the individual’s lifetime. If consumption does not change over time,13 Eq. 3 can be written as

4

which can be again converted to

5

The left hand side of Eq. 5 denotes the ERPVU, which is transformed to monetary units by dividing by the marginal utility of income The numerator on the right hand side summarises the expected remaining discounted life years, and the denominator indicates the expected increase in discounted life years due to risk reducing programmes. Equation 5 defines the VSL as the trade-off between income (dCV = WTP) and reduced mortality risk (gain in life expectancy).

In the present contingent valuation survey, respondents were asked directly about their maximum WTP to avoid an increase in avalanche risks (Δp) from current levels. Individuals’ WTP statements are summarised into six age classes. In this case, the VSL in a particular age group j is approximated by

6

The payment question has a double bounded dichotomous choice (DBDC) format. It follows that the “true” WTP cannot be observed but has to be estimated from the yes/no answers of the respondents. One can thus infer whether WTP is above (below) a specific amount if the respondent answers “yes” (“no”) to the payment question. Let us assume that WTP can be described as unobserved latent variable by:

7

where X _i is a 1  ×  k matrix of variables representing characteristics of individual i as well as risk relating attitudes, β is a k  ×  1 vector of coefficients which has to be estimated and ε stands for the error term. What is known is the probability of an affirmative response to the payment question. The sequence of the yes (y) and no (n) answers is pictured by the following dummy variables:

8

where B ^H,  B ^I, and B ^L represents the higher, initial, and lower bids, respectively. WTP is estimated using a maximum likelihood procedure. In the corresponding likelihood function, each response is represented by the probability:

9

In the underlying valuation, it is assumed that an increase in risk can be avoided by maintenance of pre-existing risk reduction programmes. Thus, negative aspects of new constructions (e.g. interference with the environment, natural scenery), which may cause negative WTP values, should not impact the valuation process. Given the individuals’ and the public’s preference for safety, a reduction in mortality risks increases welfare. Hence, respondents are expected to have at least non-negative WTP values so that a distribution that allows only zero or positive values is appropriate. A naturally positive distribution is therefore used to estimate WTP.

The estimates in this paper are based on a Weibull distribution assumption of the error term. 14 The cumulative distribution function for a Weibull distribution is

10

with the bid level B ^•_i, shape parameter ρ and scale parameters ψ_i. 15 Thus, using Eqs. 8, 9 and 10, the log-likelihood function for the Weibull can be written as

11

The maximum likelihood estimates resulting from Eq. 11 are used to calculate the mean WTP, which is the appropriate welfare measure informing cost benefit analysis (CBA) of avalanche risk reduction programmes. 16 In the case of the Weibull distribution, mean WTP is calculated as

12

where Γ(•) represents the gamma function (see Cameron and Trivedi (2005), p. 584).

Value of statistical life, age effects and age-specific hazard rate

The VSL describes the rate at which individuals are willing to forgo money for a (small) risk reduction. It is calculated by dividing the annual WTP by the given risk change. Here, WTP and VSL figures are presented for six different age classes to examine variations in VSL with age.

In order to examine the influence of age on WTP for the two groups of hazard rates (skiers/non-skiers), indicator variables for age classes—defined according to Viscusi and Aldy (2007)—are included as explanatory variables in the regressions. 17 The observations of respondents aged 18 and older were used, and were classified into six age classes: 18–24, 25–34, 35–44, 45–54, 55–62, and 63+.

The theoretical arguments given in Shepard and Zeckhauser (1984), Johansson (2002), and Alberini et al. (2004) show that the impact of age on the VSL is ambiguous and depends primarily on the individual hazard rate and optimal level of consumption. Under the assumption that skiers and non-skiers differ only in their hazard rate (higher age-specific avalanche risks for skiers vs age-independent risks for non-skiers), the skier’s VSL is expected to have a larger age-gradient than the non-skier’s VSL. For non skiers, the VSL should even be quite constant over age as their hazard rate is age-independent (see the theoretical analysis above).

The empirical evidence also suggests that the age-specific mortality risk (hazard rate) may be a crucial argument. Thus, the following analysis focuses on examining the relevance of the hazard rate in individual valuations. If it is true that the type of hazard rate (age-dependent vs age-independent) influences individual utility differently, this pattern has to be considered in the evaluation of corresponding life-saving measures and in designing an optimal policy.

Due to the topographical characteristics of residential areas in Tyrol (see Footnote 1), the present survey of WTP to reducing avalanche risks allows two types of hazard rates to be distinguished: one refers to the risk for non-skiers and the other to mortality risks for skiers. While the former face a baseline risk of dying in an avalanche, which is relevant for most residents (independent of their age), the group of skiers are confronted with additional avalanche risks by pursuing their hobby. The following paragraphs describe the different hazard rates with respect to avalanche risks for skiers and non-skiers.

Non-skiers as well as skiers who live in Alpine regions are confronted with a basic avalanche risk. Exposure to this basic avalanche risk is related mainly to the location of residential areas and the traffic routes chosen by residents while carrying out their daily business (working, shopping, …). The risk of getting caught in an avalanche in their own homes or while in a car, bus or train is independent of the individual resident’s age. 18 Therefore, it is reasonable to assume that the avalanche risk for non-skiers is age-independent, as all people who live in Alpine regions are—independent of their age—exposed to this basic avalanche risk.

The difference in the hazard rate between skiers and non-skiers emerges from the skiers’ additional risk of dying in an avalanche due to their presence in snowy Alpine areas. From the official statistics of avalanche fatalities [provided by the Alpine Safety and Information Centre (ASI-Tirol) and the Kuratorium fuer Alpine Sicherheit] in the past five winters it can be inferred that all deaths listed occurred among the group of skiers. 19 Furthermore, the number of fatalities varies across age groups. On a 5-year average (winter 2004/2005—winter 2008/2009), the highest number of avalanche-related deaths, namely 4.6, occurred among 25- to 34-year-olds. The average death toll among those aged 35–44 and 45–54 years amounts to 3.8 and 4.4, respectively (see sixth column in Table 2). What is of relevance in this context is not the absolute number of skiers who died in avalanches but their probability of dying, which depends on the number of people exposed to additional avalanche risks (i.e. skiers). I combined information given in the survey with data from official statistics to obtain a rough estimate of this probability.

Table 2.

Skiing behaviour and avalanche fatalities per age class

Age (years) class	Skiers in sample		Populationa		Average deaths/yearb
	N	Sharec	N	N skiersd	N	Conditional probabilitye
18–24	261	63.20	63,651	40,225	1.4	0.004
25–34	232	61.21	95,299	58,336	4.6	0.008
35–44	162	57.65	119,248	68,748	3.8	0.006
45–54	90	45.92	93,961	43,145	4.4	0.010
55–62	43	36.13	60,971	22,032	0.6	0.003
63+	34	21.66	118,328	25,625	1.2	0.005

^aTyrolean population on a 5-year (2004–2008) average

^bFive-year average (winter 2004/2005–2008/2009) of avalanche fatalities. Source: ASI-Tirol, and Kuratorium fuer Alpine Sicherheit, Austria

^cPercentage of skiers (per age class) in the survey sample

^dNumber of skiers in Tyrol, estimated by multiplying the proportion of skiers in the survey sample by the number of residents

^eAge-dependent fatality risk (in percent). Calculation: Number of avalanche-related deaths divided by number of skiers in the population (e.g. 1.4/40,225 × 100)

The second and third column in Table 2 report the number and proportion of skiers per age class in the sample. Overall, 822 out of 1,545 respondents were skiers but the proportion of skiers decreases steadily with age. The number of skiers in the population (fifth column) was calculated by multiplying the age specific proportion of skiers in the sample (third column) by the corresponding 5-year average of population size in each age class (fourth column), which gives the number of skiers in the Tyrolean population assuming that the sample’s proportion of skiers is constant over time and that the sample is representative of the Tyrolean population. Dividing the average number of avalanche-related fatalities of (back country) skiers in the 5-years period 2004/2005–2008/2009 by the number of skiers in the population leads to the skiers’ conditional probability of dying in an avalanche (seventh column). This probability—conditional to being a skier—shows that the probability of dying in an avalanche is lowest for 55- to 62-year-olds and peaks for 45- to 54-year-olds. 20

The level and intensity of risk exposure (e.g. how often skiers go on ski trips or back country tours; the ability to adequately respond to challenging terrains) among skiers depend on their (physical) abilities and skills. As Table 2 indicates, the probability of being a skier and therefore being potentially at (higher) risk varies with age. Age-related variation is also seen in the absolute number of avalanche fatalities and, more important, for the probability (conditional on being a skier) of dying in an avalanche.

By splitting the sample into skiers and non-skiers, it is possible to examine age effects on the VSL implicitly by considering the different levels of individual hazard rates. According to Eq. 5, the VSL among the group of skiers is expected to vary across age classes and should be lower for individuals at higher avalanche risk [higher hazard rate, i.e. lower survival function μ (t;τ)]. On the contrary, the age-gradient of non-skiers should be less pronounced, and their VSL should be more constant due to their age-independent hazard rate. The relevance of age is considered by calculating the WTP and VSL for particular age classes.

Further explanatory variables

Apart from the socio-demographic characteristics of the participants (age, gender, income, …), the survey data provide information about risk-related attributes, such as individual behaviour in risky situations (e.g. using sunscreen, wearing seat belts), preferences for alternative risk reduction programmes (other than protection against avalanches), and risk perception. 21 These variables are included in the regression to control for internal validity of WTP estimates. Table 3 provides a brief description of these explanatory factors.

Table 3.

Explanatory variables

Variable	Description
Age 25–34	Dummies. 1 if respondent is 25–34, 35–44, 45–54, 55–62, and above 62 years old, respectively (reference group: age 18–24)
Age 35–44
Age 45–54
Age 54–62
Age 63+
Alevel	Dummy. 1 if respondent has a university entrance diploma; 0 otherwise
Anthropogen	Dummy. 1 if respondent always regards avalanches as a human-caused event; 0 otherwise
Famexp	Dummy. 1 if respondent has had personal experience with avalanches; 0 otherwise
Female	Dummy. 1 if respondent is female; 0 otherwise
Healthy	Dummy. 1 if respondent states that she is in good health; 0 otherwise
Housemember	Number of persons in the respondent’s household
Impalter	Dummy. 1 if the respondent prefers alternative risk reduction programmes; 0 otherwise
Jobrisk	Dummy. 1 if respondent states that she faces workplace risks; 0 otherwise
Largereduct	Dummy. 1 if the predetermined risk variation = 3/42,500; 0 otherwise
Lnincome	Logarithm of personal monthly take home income
Lowrisk	Dummy. 1 if respondent assesses her personal risk of dying in an avalanche as below average
Lowriskvol	Interaction term: low risk and volunteer
Missaversion	Dummy. 1 if missing observations of risk aversion are replaced by zero; 0 otherwise
Missincome	Dummy. 1 if missing observations of income are replaced by mean income in the individual’s residence; 0 otherwise
Natural	Dummy. 1 if respondent always regards avalanches as a natural event; 0 otherwise
Riskaversion	Respondent’s behaviour in risky situations. Ranges between 0 (risk taker) and 21 (risk averse)a
Riskpercept	Respondent’s perception of fatal avalanche risks. Ranges between 0 (no risk) and 131 (death)
Skiing	Dummy. 1 if respondent is a skier; 0 otherwise
Volunteer	Dummy. 1 if respondent volunteers; 0 otherwise
Winter	Dummy. 1 if the survey took place in February 2005; 0 otherwise

^aIn this context, risk aversion refers not only to financial risk aversion but also to aversion to life-threatening risks

Results

Maximum likelihood estimates

The intensive discussion of how age influences WTP for risk reduction programmes encourages a detailed analysis of age effects. I estimated an econometric model using all observations and also ran regressions for each sub sample (skiers and non-skiers) separately. This procedure allows one to determine whether the influence of age on the respondents’ WTP differs between the two types of hazard rates. Table 4 presents the estimated coefficients.

Table 4.

Willingness-to-pay (WTP)—regressions

Variable	Total	Skier	Non-skier
Largereduct	0.491***	0.694***	0.250
Winter	−0.163*	−0.335***	0.039
Ae 25–34	−0.117	−0.208	0.070
Age 35–44	−0.261**	−0.322**	−0.126
Age 45–54	−0.186	−0.240	−0.045
Age 55–62	−0.148	0.027	−0.156
Age 63+	−0.303*	−0.501*	−0.152
Female	−0.031	−0.029	−0.036
Lnincome	0.186**	0.103	0.291**
Missincome	−0.137	−0.099	−0.206
Alevel	−0.278***	−0.430***	−0.057
Housemember	0.027	0.037	0.013
Volunteer	0.418**	0.529***	0.112
Famexp	0.334***	0.332***	0.330*
Riskpercept	0.011***	0.013***	0.008**
Skiing	0.195**
Lowrisk	0.091	0.191	−0.088
Lowriskvol	−0.321	−0.679***	0.238
Anthropogen	0.085	0.116	0.051
Natural	−0.054	−0.001	−0.106
Riskaversion	0.019	0.034*	0.016
Missaversion	0.048	0.339	−0.119
Impalter	−0.389***	−0.116	−0.653***
Healthy	−0.017	0.014	−0.044
Jobrisk	0.181**	0.229**	0.172
Constant	−0.425	0.066	−0.999
Observations	1,481	799	682
Log-L	−1,808	−1,005	−781

Dependent variable = WTP intervals; protest answers (N = 329) and respondents younger 18 (N = 87) excluded; reference group: age 18–24 years

* P < 0.1; ** P < 0.05; *** P < 0.01

Estimates using the information in the total sample are discussed first. The coefficient of the scope dummy (largereduct) indicates that the results pass the sensitivity test; respondents who value higher risk prevention reveal a significantly higher WTP than those whose statements are based on the smaller variation. As expected, WTP increases with increasing income (lnincome) and increasing perception of avalanche risks (riskpercept). WTP is significantly higher for respondents who had personal experience with avalanches in the past (famexp), who volunteer their time for welfare services (volunteer), who are skiers (skiing), and who face workplace risks (jobrisk). 22 Negative impacts on WTP show higher education (alevel) and existing preferences for alternative risk reduction programmes (impalter). Surprisingly, respondents in the winter sample state a significantly lower financial contribution although avalanche events often occur in this period. From previous analyses it can be inferred that interviewees in winter who consider avalanches as human-caused events state a significantly lower WTP, and that the “outrage effect”, as discussed in Kahneman et al. (1993) or more recently in Bulte et al. (2005), is weaker than the “responsibility effect” (Walker et al. 1999) for respondents in winter. 23 The lower WTP statements of those who consider avalanches as human-caused events indicate that these respondents are not willing to support risk-reduction programmes in situations where people at risk voluntarily choose their level of risk exposure and, hence, are themselves responsible for their risk experience. This result also indicates that the claim for responsible behaviour outweighs the potential upset (outrage) involved with man-made adverse situations (i.e. those which would not have occurred without human intervention) and the individual’s willingness to correct man-made harm accordingly.

Once the sample is split into two groups, an interesting pattern concerning these significant variables can be observed. Skiers are in general much more sensitive to the above-mentioned characteristics than non-skiers—with two exceptions. (1) While the coefficient on income (lnincome) is positive in both groups it is significant only for non-skiers. (2) If non-skiers think that other risk reduction programmes to save human lives are more important than protection against avalanches they state a significantly lower WTP (impalter). This effect is not significant for skiers.

Which influences of age on WTP can be observed? Using the full sample, I found that respondents aged 35–44 (age35–44) and 63+ (age63+) reveal a significantly lower WTP (21 and 27%, respectively). The remaining age classes do not differ significantly from the youngest group. When the sample is divided into “Skier” and “Non-Skier”, the significantly lower WTP statements at higher age occur only among respondents facing an age-sensitive hazard rate (see column “Skier”). Furthermore, the significant effects are even more pronounced for skiers: people aged 35–44 and 63+ state significantly lower payments, which are 28% and 40% below the WTP of the reference group of 18- to 24-year-old individuals. No significant age effect is observable for the group of non-skiers who are equally affected by avalanches—independent of their age.

These results point to an age dependency of WTP for respondents whose hazard rate varies with age (skiers), while no age-related variation is observable for non-skiers whose avalanche-related fatality rate is independent of age.

WTP for preventing fatalities and VSL

Mean WTP for risk reduction programmes against avalanches is calculated by using the estimated coefficients shown in Table 4. The coefficients reported for “Skiers” and “Non-skiers” are combined with the characteristics of an average respondent in the particular group. Mean WTP is computed using Eq. 12. VSL figures can be calculated from WTP values by dividing the annual WTP by the dimension of prevented risks for the average respondent (=1/42,500). 24 Table 5 depicts WTP and VSL for each age class and type of hazard rate referring to the smaller risk change.

Table 5.

WTP/month and value os statistical life (VSL) per age class

Age class (years)	Observations	WTP (€)a	VSL (million €)b
Skiers
18–24	250	5.69 (0.977)	2.90
25–34	225	4.62 (0.881)	2.36
35–44	159	4.13 (0.755)	2.10
45–54	89	4.48 (1.003)	2.28
55–62	43	5.85 (1.572)	2.98
63+	33	3.45 (1.048)	1.76
Non-skiers
18–24	137	5.58 (1.456)	2.85
25–34	143	5.98 (1.578)	3.05
35–44	110	4.92 (1.336)	2.51
45–54	101	5.34 (1.425)	2.72
55–62	73	4.77 (1.347)	2.43
63+	118	4.79 (1.350)	2.44

^aStandard errors (based on the delta method) in parentheses

^bCalculation of VSL: annual WTP/risk variation = (monthly WTP×12)/(1/42,500)

Monthly mean WTP (and therefore VSL) across age classes is quite similar for non-skiers and ranges from €4.77 to 5.98 (from €2.4 to 3.1 million). For non-skiers whose risk of dying in an avalanche is age-independent, no significant differences occur between the age classes. However, a more pronounced variation among age classes is observable for skiers: their WTP figures range from €3.45 to 5.85 (=monthly WTP) and from €1.8 to 3.0 million (=VSL), respectively. The differences in WTP between the reference group of 18- to 24-year-old skiers and the age classes 35–44 and 63+, respectively, are statistically significant. 25

This analysis supports the hypothesis made: age variation is more pronounced for the group of skiers and is statistically significant within this group only. There is evidence that age-sensitivity matters for the valuation of fatality risks and that age-dependent hazard rates play a crucial role in explaining the variation in VSL figures: age effects occur for individuals whose mortality risk due to avalanches is age-dependent, while no age effects are observable for respondents whose risk level is independent of their age. 26 These results suggest that policies that aim at an efficient provision of live-saving measures have to take into account such variations in the beneficiaries’ hazard rates. An overall approach would miss the diversity in benefits and would lead either to an over- or under-supply of risk-reducing measures.

Assuming that the utility of consumption and life expectancy is constant, another question can be addressed: do skiers know that they belong to a higher risk group? Jones-Lee (1974) or Weinstein et al. (1980) argue that the higher the individual’s mortality risk, the higher the WTP for reducing life-threatening risks. This means that, based on the tendency shown in Table 2, the 45–54 (55–62) year-olds with the highest (lowest) conditional probability of dying in an avalanche should reveal the highest (lowest) WTP. But the estimated WTP paints a different picture: WTP of 55–62 (45–54) year-olds is the highest (third lowest) value. This admittedly rough and simplified comparison could lead to a premature conclusion that skiers do not correctly predict their risk of dying in an avalanche (i.e. they rather base their decision on subjectively perceived risk levels). However, one has to keep in mind that such a statement is based on strong assumptions. The skiers’ remaining life expectancy and their potentially time-variant utility of consumption are further determinants of the VSL (see "Estimation procedure"), which may explain these deviations.

Robustness checks

The estimation procedure described in "Estimation procedure" assumes a Weibull distribution of the error term. Both responses (initial and follow-up) to the payment questions are used, which implies that respondents show a consistent behaviour and think of a single amount when stating their WTP. It is often questioned whether it is appropriate to use both answers from a DBDC format. The resulting estimates are inconsistent when the responses to the follow-up questions do not reveal utility-maximising behaviour or rational preferences but are influenced by various factors such as yea-saying, weariness or free-riding (see for example, Herriges and Shogren 1996; Bateman et al. 2001; McFadden 1994; Park 2003). An alternative to deal with these problems would be to consider only the first response to the payment question. The disadvantage of this approach is that one forgoes a gain in statistical efficiency provided by the follow-up responses (e.g. Alberini 1995; Hanemann et al. 1991). This means that a trade off between bias and efficiency exists.

In order to check how the results vary using the single-bounded approach the same regressions discussed for the DBDC format but ignoring the responses to the second question are conducted. The coefficients estimated do not change considerably. For example, using the sample of skiers, the corresponding coefficients on the age dummies age25–34, age35–44, age45–54, age55–62, and age63+ are −0.202, −0.077, −0.205, 0.349, and −0.655. However, the age effects cannot be estimated precisely any more due to the large standard errors, which are at least twice as high as in the double-bounded case.

A second approach—the Turnbull distribution-free estimator described in Haab and McConnell (2002)—that imposes as few restrictions as possible on the estimation procedure is conducted to test for the robustness of age effects. 27 The calculation of the welfare measure is quite straightforward but this is at the cost of testing for covariate effects, 28 i.e., the Turnbull mean WTP is a descriptive unconditional estimate. Comparing the Turnbull WTP among age classes for the group of skiers, one finds that the WTP for people aged 25–34 and 63+ is significantly lower than WTP for the youngest. Further significant differences are observable between other age classes as well. The depreciation is strongest, and always highly significant, for the oldest. In the group of non-skiers, such a pronounced age pattern is not observable. There are only three pairs of age classes where WTP differs significantly from each other. 29 But WTP for non-skiers aged 63 and older does not show a statistically significant deviation from the mean values in the other age classes. As the Turnbull estimates represent a lower bound of WTP, mean WTP virtually always lies below the parametric mean values.

To summarise, among the group of skiers, strong evidence for a lower WTP for older people is found using the Turnbull estimator but this clear pattern does not occur among non-skiers. As the Turnbull estimator is a distribution free estimator that uses only the first response to the bid question, one can conclude that the difference in age sensitivity between the two groups is dependent neither on the chosen distributional assumption nor on the presumptions imposed by using the DBDC format. The single-bounded estimates also suggest that the dimensions of age effects do not differ considerably when only the first response is used, so that the double-bounded approach is favoured and the additional information from the follow-up response is used.

Conclusion

This paper discusses a relevant feature of monetary valuation of reduced mortality risks: the age-sensitivity in an individuals’ valuation of live-saving measures. I focus on comparing the WTP statements of two groups of beneficiaries who focus on the same type of risk but who face different hazard rates: for one group the hazard rate varies across age, for the other it is constant. The results indicate that the consideration of age-specific hazard rates is important and can explain the different findings regarding the age dependency of VSL estimates in the empirical literature.

The analysis is based on a CV survey carried out in Austria in two waves (autumn 2004/winter 2005). WTP for preventing fatal avalanche accidents is estimated by the method of maximum likelihood using an interval-data model and assuming a Weibull distribution of the error term. The sample is split into two groups (skiers and non-skiers) who differ in their levels of fatality risks. Age class dummies are included in the regression model to test for age effects among the two groups.

The results highlight the fact that the non-skiers’ evaluation of risk reduction is almost constant and independent of age, while the skiers’ WTP statements considerably vary over age. In other words, the estimates reveal age-sensitive VSLs for respondents to whom risk exposure is age-dependent (skiers). But no significant differences between the age classes are observable for non-skiers who face only an age-independent “baseline” risk due to the location of their residences. Average VSL for non-skiers ranges from 2.4 to 3.1 million Euro. For the group of skiers the VSL lies between 1.8 and 3.0 million Euro. Within the latter group a statistically significant depreciation of VSL for the age classes 35–44 and 63+ is observable: VSL is 28 and 40% lower than the VSL for the youngest.

What implications can be drawn from these findings for the efficient use of public funds to finance life-saving measures? The results in this paper indicate that the age pattern of individual hazards rates is a decisive determinant in explaining age variations in VSL figures and, hence, in justifying an adjustment of those measures to age. If protection measures that aim at protecting a wide population whose mortality risks do not depend on individual characteristics such as age are at stake (i.e. risk exposure is the same for all beneficiaries), an adjustment of the resulting benefits to age is not justified. In this setting, non-skiers belong to this group as they reveal similar WTP across all ages. A “senior discount” as sometimes proposed would contradict the individual statements. In the context of preventing fatal avalanche accidents, an age adjustment in VSL measures is appropriate solely for skiers. In this case, policies that simply use an overall value but do not control for such variation ignore the diversity in benefits and, hence, lead to an insufficient provision of (environmental) risk-reducing measures.

Considering the growing number and intensity of weather extremes it is of increasing importance to learn how residents evaluate protection measures against natural hazards to provide an efficient level of life-saving policies. Further research remains to be done to learn about the validity of these findings for, and the transferability to, other environmental risk or health risk valuations.