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ROBUSTNESS OF VSL VALUES FROM CONTINGENT VALUATION SURVEYS *
By
Anna Alberini
Department of Agricultural and Resource Economics
2200 Symons Hall
University of Maryland
Address for correspondence:
Anna Alberini
AREC, 2200 Symons Hall
University of Maryland
College Park, MD 20742
USA
Phone 001 301405-1267
Fax 001 301314-9091
e-mail: aalberini@arec.umd.edu
ABSTRACT. This paper examines factors that may influence the estimates of the Value
of a Statistical Life obtained from contingent valuation surveys that elicit the willingness
to pay (WTP) for mortality risk reductions. We examine the importance of distributional
assumptions, the choice of the welfare statistics of interest, the procedure for computing
them, outliers, undesirable response effects, and internal validity of the WTP responses.
We illustrate the importance of these factors using dichotomous-choice and open-ended
WTP data from four recent contingent valuation surveys.
Key words: contingent valuation, VSL, WTP, risk reductions, robustness, outliers,
endogeneily.
Subject Area Classifications: Health, Valuation, Benefit Cost Analysis
* This research was conducted under Cooperative Agreement 015-29528 with the U.S. Environmental
Protection Agency. I wish to thank Drs. Johannesson, Persson and Gerking for making their data available
to me. I also wish to thank my project officer, Nathalie Simon of the US Environmental Protection Agency,
for her guidance with this project, seminar series participants at the U.S. EPA National Center for
Environmental Economics and session attendees at the annual EAERE meeting in Budapest, Joel Huber,
Trudy Cameron, Melonie Sullivan and Kelly Maguire for their comments on this work.
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ROBUSTNESS OF VSL VALUES FROM CONTINGENT VALUATION SURVEYS
By
Anna Alberini
1. Introduction.
The Value of a Statistical Life (VSL) is the rate at which people are prepared to
trade off income for a reduction in their risk of dying. The VSL is a key input for
computing the mortality benefits of environmental and safety policies that save lives. In
recent retrospective analyses of the Clean Air Act and of the Clean Air Act Amendments,
for example, the US Environmental Protection Agency has used a VSL of $6.1 million in
its base analyses ($3.7 million in "alternate" analyses), and the resulting monetized
mortality benefits account for over 80% of total benefits of these environmental statutes.
Within the European Commission, DG Environment uses central VSL estimates of about
1.2 million euro, with adjustments for age and for the futurity of the risk.l
The VSL figures for cost-benefit analysis purposes ars typically derived using
three possible methods: (i) compensating wage studies, (ii) consumer behavior studies,
and (iii) contingent valuation surveys. Compensating wage studies use data from labor
markets to infer how much workers have to be compensated to accept riskier jobs—or the
sacrifice in income they would agree to in exchange for an improvement in their
workplace safety. Viscusi and Aldy (2003) document over 60 compensating wage studies
conducted in 10 countries, noting that in the US most of the labor market studies produce
estimates of the VSL in the range of $4-9 million.
Consumer behavior studies observe tradeoffs between time and risk, or money
and risk, to place a value on mortality risk reductions. An early such study (Blomquist,
1 See http://europa .cu.mt/contm/environment^euvecQ/Qlhers/recoinniended interim values.odt'.
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1979) observed whether or not individuals fasten their seatbelts when driving. The VSL
is calculated as the value of the time required for buckling up, divided by the reduction in
the risk of dying in a traffic accident afforded by the use of seatbelts. Atkinson and
Halvorsen (1990) obtain an estimate of the VSL from the higher price of cars with more
sophisticated safety equipment. The VSL figures from labor market or consumer studies
are often transferred to the environmental policy context. Doing so implicitly assumes
that the preferences of individuals for income and risk do not vary with the context.
In contingent valuation surveys, respondents are asked to report their willingness
to pay (WTP) for a specified—and hypothetical—risk reduction. Contingent valuation
studies have the potential to circumvent many of the shortcomings for which the other
approaches are sometimes criticized. For example, they lend themselves to valuing risk
reductions in many contexts, and are thus not limited to workplace risks. Rather than
assuming that people know the exact magnitude of the risks they face, in a well-designed
CV study respondents are educated about them, and the extent of the risk reduction is
spelled out explicitly for them.
Last but not least, CV allows the researcher to survey directly the beneficiaries of
any proposed risk-reduction measure. This is regarded as a particularly advantageous
feature of the method, because when the risk reduction measure is an environmental
program, these beneficiaries (e.g., the elderly) are likely to be very different than the
population covered in compensating wage studies, and may have different preferences for
income and risk.
Despite these advantages and the flexibility of the approach, much debate
surrounds the estimates of VSL from contingent valuation surveys. This paper focuses on
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two main difficulties associated with the V|SL figures from CV studies. The first is that
many recent high-quaJity CV surveys have elicited information about WTP using
dichotomous-choice questions. Dichotomops-choice question:; have been shown to be
incentive compatible (Hoehn and Randall, 1987), and are generally thought to be easier
to answer than open-ended questions. However, the researcher must rely on assumptions
about the distribution of the underlying WTP in order to obtain estimates of mean and
median WTP, and these in turn are typically sensitive to the upper tail of the distribution
of WTP.
Second, it is, in general, difficult to \alue risk reductions. Respondents are not
used to dealing with probabilities, especially when risks are very small, and the cognitive
burden imposed upon them in the survey—or the failure to communicate risks to them in
a meaningful way—may result in undesirable effects, including failure to distinguish
between risk reductions of different sizes (Hammitt and Graham, 1999), confusion
between absolute and relative risk reductions (Baron, 1997), protest responses,
completely random answers to the payment questions, etc. (Carson, 2000). One possible
approach for uncovering these problems is to test for the internal validity of the
responses, i.e., to check that WTP depends on certain variables in the ways predicted by
economic theory. Alternatively, one may try to fit models that explicitly seek to identify
abnormal responses.
The purpose of this paper is to discuss the importance of distributional and
modeling assumptions, and the effect of econometric misspecifications and of the
presence of abnormal response patterns in the sample on the estimates of the VSL. We
illustrate our robustness criteria and checks using the data from four recent CV surveys.
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We examine robustness with respect to four possible criteria The first three are specific
to dichotomous-choice WTP responses. First, we examine how the VSL changes with (a)
the distribution WTP is assumed to follow, (b) the welfare statistic of interest (e.g.,
median or mean), given the distribution of WTP, and (c) given the distribution and the
welfare statistic, different procedures for calculating the latter.
Our second set of analyses is similar to the first, except that it focuses on how (a),
(b) and (c) impact the estimated relationship between WTP and specific covariates. Third,
we examine outliers and abnormal response patterns in dichotomous-choice CV surveys.
Fourth, we look at internal validity checks.
The remainder of this paper is organized as follows. Section II provides a
definition of VSL and outlines the robustness criteria discussed in this paper. Section III
focuses on dichotomous choice CV responses, examining the robustness of VSL to the
assumed distribution of WTP, the welfare statistics used, and the procedure used for
computing such welfare statistics. Section IV focuses on outliers and response
mechanisms that do not comply with the economic paradigm, such as yea-saying, nay-
saying, and completely random responses. Section V examines how internal validity
tests, such as scope tests, are affected by the possible endogeneity of risks and WTP, and
section VI focuses on the relationship between WTP and income. Section VII provides
concluding remarks.
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II. Econometric Robustness of VSL Figures
A. Definition of VSL
The Value of a Statistical Life is the rate at which individuals are prepared to
trade off income for risk reductions. In an expected utility framework, let U(w) be the
(state-dependent) utility associated with incpme w if the individual is alive, and V(w) the
utility of income if the individual is dead. If the probability of dying is p, expected utility
is defined as (l-p)U(w)+pV(w). This expression can be further simplified to (l-p)U(w) if
it is assumed that V(w)=0 (i.e., the utility of income is zero when one is dead). The VSL
is the rate of substitution between income and risk that keeps expected utility unchanged,
and is in this context equal to — = - — — - .
dp (l'
The VSL is, therefore, a derivative, but in practice contingent valuation surveys
ask people to report information about their willingness to pay (WTP) for a gjecified —
and finite — reduction in their risk of dying, Ap. The VSL is estimated as WTP/Ap.
Accordingly, in this paper the robustness of the VSL estimates and the robustness of
WTP estimates are regarded as interchangeable.
B. Robustness Criteria
This paper examines robustness with respect to four criteria. Because many recent
high-quality CV studies have deployed dichotomous-choice questions to elicit
information about WTP, the first series of robustness checks refers to dichotomous-
choice CV data Specifically, we examine by how much WTP changes with (a) the
distribution WTP is assumed to follow, (b) the welfare statistic of interest (e.g., median or
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mean), given the distribution of WTP, and (c) given the distribution and the welfare
statistic, different procedures for calculating the latter.
Our second set of analyses is similar to the first, except that it focuses on how (a),
(b) and (c) impact the estimated relationship between WTP and specific covariates, such
as the age of the respondent.
Next, we turn to the issue of whether it is possible to identify abnormal responses
to dichotomous-choice WTP questions. To identify outliers, we first use a "reduced-
form" approach based on a regression equation relating the response to the payment
question to observable individual characteristics, checking if the inclusion of these
observations in the sample affects appreciably the estimates of WTP, and, if so, by how
much. In our next step, we seek to model explicitly abnormal response patterns. By
abnormal response patterns, we mean answers to the payment questions that do not
comply with the economic paradigm, such as responses motivated by "yea-saying" or
"nay-saying" behaviors, or completely random responses.
Fourth, we focus on internal validity. We first work with a dataset where
respondents subjectively assessed their baseline risk and were asked to value a given
reduction in this initial risk. Economic theory posits that WTP should increase
appreciably with the size of the risk reduction. Moreover, meaningful VSL figures can be
computed only if individuals are valuing the absolute risk reduction. We examine
whether our ability to empirically check that the WTP responses are consistent with
economic theory and with the VSL construct is affected by treating WTP as
econometrically endogenous with risk. The specific application we use to explore these
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issues elicits WTP using open-ended questions, producing observations of WTP on a
continuous scale.
Economic theory also predicts that WTP should increase with respondent's
income. We check how the estimated income elasticity of WTP changes as respondents
who would be willing to commit a large fraction of their income to the risk reduction are
excluded from the sample. This has potentially important consequences for benefit
transfer, i.e. the practice of applying the results of the study conducted at one locale to
another population or context.
III. The Data.
We illustrate out robustness checks using data from four applications, which we
summarize in table 1. In the first application (Johannesson et al., 1997), a representative
sample of Swedish adults aged 18-74 were surveyed over the telephone about their WTP
for a reduction in their risk of dying. The survey was conducted in November 1996, and
produced WTP data for a total of 2029 individuals, for a response rate of 83 percent.
The goal of the study was to study the relationship between the VSL and the age of an
individual.
Respondents were told that X out of 10000 people of their gender and age would
die during the next year.2 They were also asked to assume that a preventive and painless
treatment was available that would reduce by 2 in 10000 the risk of dying in the next
year, but have no effects thereafter. Information about WTP was elicited using single-
bounded dichotomous-choice questions, with bid values ranging from 300 to 10000
2 The baseline risk of death over the next year was 10, 30, 70, and 200 for males in the age groups 18-39,
40-49, 50-59, and 60-69, respectively. For females, the baseline risk values were 5, 20, 40, and 100. All
baseline risks are out of 10000.
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SEK.3 Johanesson et al. estimate the mean WTP to be equal to 6300 SEK (about 954 US
dollars), which corresponds to a VSL of 31.4 million SEK (4.75 million US dollars), and
detect a quadratic relationship between age and WTP that peaks at 40 years of age.
The second and third applications used in this study employed dichotomous
choice questions with follow-ups, and a virtually identical survey instrument for
Hamilton, Ontario (Krupnick et al., 2002) in Spring 1999 and a national sample of US
respondents (Alberini et al., 2004) in August 2000. Both questionnaires were self-
administered by the respondent using the computer. In the Hamilton study, respondents
were asked to go to a centralized facility to take the survey, whereas in the US studies
they received the questionnaire via Web-TV™.
By asking people to value immediate and future risk reductions (for which
payment would have to start immediately), and by recruiting individuals of various ages,
including the elderly, and health statuses, these studies explore four main research
questions. The first is the relationship between VSL and age. The second is the
relationship between WTP and the health status of the respondent. This is important for
policy purposes, as some agencies have argued in favor of using Quality Adjusted Life-
Years (QALY), a construct widely used in medical decisionmaking where values are
adjusted for quality of life, which is presumably lower for chronically ill people.
The third research question is whether the WTP for a future risk reduction is less
than the WTP for an immediate risk reduction (as is implied by discounting and by the
fact that the individual may die before he reaches the age when the future risk reduction
3 The payment question read as follows: "It is estimated that X(Y) men (women) out of 10,000 in the same
age as you will die during the next year. Assume that you could participate in a preventive and painless
treatment which would reduce the risk that you will die during the next year, but has no effects beyond that
year. The treatment reduces the risk of your dying during the next year from X(Y) to X-2 (Y-2) out of
10,000. Would you at present choose to buy this treatment if it costs SEK I?"
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would begin), and the fourth is how large is the implicit discount rate (Alberini et al.,
2004).
Hie fourth study (Persson et al., ,2001) is a mail survey eliciting WTP for
reductions in the risk of dying in a road-traffic related fatality. The survey was conducted
in Sweden in Spring 1998. Questionnaires were mailed to a representative sample of
Swedes of ages 18-74, for a total of 2884 returned questionnaires (the response rate was
51%). Two versions of the questionnaire were created. The first focused on the risk of
non-fatal injuries, while the other focused on the risk of dying in a road-related accident.
In this paper, attention is restricted on the 935 completed questionnaires about fatal risks.
In this questionnaire, risks were expressed as X in 100,000, and depicted using a
grid of squares. People were first shown, as an example, the risks of dying for various
causes (all causes, heart disease, stomach or esophageal cancer, traffic accident) for a 50-
year-old. They were then asked to assess subjectively their risk of dying for any cause,
and in a road-related traffic accident. They were also asked directly to report their WTP
for a reduction in each of these two risks. Unlike the previous studies, the payment
question used an open-ended format, resulting in observations about WTP on a
continuous scale.
In this paper, we focus on the WTP for a reduction in the risk of dying in a traffic
accident. This risk reduction is expressed as a proportion (10%, 30%, or 50%, depending
on the questionnaire version4) of the baseline risk. The risk reduction is a private
commodity (safety equipment and preventive health care) and is valid for one year. A
reminder of the respondent's budget constraint is provided.
4 Respondents were randomly assigned to one of these possible risk reductions.
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Table 1. Mortality Risk Studies examined in this paper. All monetary figures in US
dollars, unless otherwise indicated.
Study
Description and VSL
Johannesson et al.
(1997)
Telephone survey of Swedes aged 18-74. Dichotomous-choice
questions about WTP for 2 in 10,000 reduction in their risk of
dying (from all causes).
VSL = $4.5 million.
Perssonetal. (2001)
Mail survey in Sweden. Elicits WTP for X% reduction in the
risk of dying in a road-traffic accident. Subjective baseline
risks. Open-ended WTP questions.
VSL = $2.84 million (based on WTP for 2 in 100,000 risk
reduction).
Krupnick et al.
(2002)
Survey of persons aged 40-75 years in Hamilton, Ontario.
Self-administered computer questionnaire, centralized facility.
Dichotomous-choice payment questions with dichotomous -
choice follow-up question.
VSL = Can $1.2 to 2.8 million (US $ 0.96 to 2.24 million).
Alberini et al. (2004)
US national survey conducted over Web-TV. Dichotomous-
choice payment questions with dichotomous-choice follow-up
question.
VSL =$700,000 to $1.54 million (based on 5 in 1000 risk
reduction)
III. Estimation of VSL with Dichotomous Choice Data
Dichotomous-choice payment questions ask respondents whether they would be
willing to pay a specified amount of money to obtain the risk reduction stated to them in
the questionnaire. The amount of money (usually termed "the bid") is randomly assigned
to the respondent out of a list of preselected values, and is varied across respondents.
Respondents are offered two possible response categories: "yes" and "no."5 Their
5 When the risk reduction is delivered by a public program, the payment question is often phrased in terms
of vote in a referendum on a ballot. The respondent is told that the program would be implemented only if
there are a majority of votes in favor of the program, and that the cost of the program—usually, in the form
of an income tax—for his household is SX. If a majority is not reached, the program is abandoned, and no
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responses imply that WTP is greater ("yes") or less than ("no") the bid, but the exact
WTP amount is not observed.
Estimates of mean WTP and other welfare statistics (eg., median WTP) can be
obtained by estimating binary data models that rely on this mapping from the unobserved
WTP amount to the response to the payment question. For example, if latent WTP is
normal (logistic) with mean u. and scale a, a probit (logit) model is estimated where the
dependent variable is a dummy indicator that takes on a value of one if the response to
the payment question is a "yes" and zero otherwise, and the right-hand side includes the
intercept and the bid. Cameron and James (1987) show mat mean/median WTP is equal
to -a/fl, where a and B are the probit (logit) intercept and slope, respectively.6
In the remainder of this section, we examine the sensitivity of the estimates of
WTP (and hence VSL) based on dichotomous choice data to the distribution WTP is
assumed to follow, the welfare statistic one wishes to work with, and the procedure used
by the researcher in computing it To illustrate the consequences of assumptions and
procedures, we use the data in Johannesson et al. (1997), a telephone survey of Swedes
aged 18-74 about their WTP for a 2-in-10,000 reduction in their risk of dying over the
next year.
Respondents were informed about the chance of dying for a person of their age
and gender over the next year, and were queried about their WTP to reduce that risk
additional income taxes are incurred by households. The two possible response categories are "in favor"
and "against" the program.
6 Symmetric distributions like the normal and logistic imply that mean WTP is equal to median WTP. If
WTP is assumed to be a lognormal, the probit equation is amended by replacing the bid in the right-hand
side of the model with its logarithmic transformation. Median WTP is equal to exp(-o/p), and mean WTP
is equal to exp(0.5'(l/p)2 - o/P). With a Weibull distribution, a binary choice model is estimated where
Pr(yes|B)=exp(-{B/a)e)> where 9 and a are the scale and shape parameters, respectively, of the Weibull
variate. Mean WTP isaT( 1/9+1) and median WTP is
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using dichotomous choice questions. The bid values ranged between 300 and 10,000 SEK
(about $40 to $1400, implying VSL values of $200,000 to $7 million). Johannesson et al.
estimate mean WTP to be 6300 SEK, or about $900.
A. WTP Responses and WTP Distribution
We begin our examination of the data from the Johannesson et al. study by
checking whether (i) the percentage of "yes" responses decline with the bid amount, and
(ii) the bids cover a reasonably wide portion of the range of WTP values.7 As shown in
Figure 1, the percentage of "yes" responses declines from 51.36% at the lowest bid
amount, 300 SEK, to 28.83% at the higher bid amount 10,000 SEK, satisfying the first of
these two requirements. Figure 1 also implies that all bids are greater than median WTP,
failing to satisfy requirement (ii), and raising concerns about the stability of the estimates
of WTP. Median WTP is pegged between 300 and 500 SEK.
7 Cooper (1993) emphasizes the importance of using a vector of bids that covers the entire range of possible
WTP values. Kanninen (1993) and Alberini (1995) derive c-optimal and d-optimal designs for
dichotomous-choicc CV surveys that ruly on only two bid values.
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Figure 1.
Percent "yes'1 by bid amount
Johannesson etal. (1997)
n
300 500 1000 2000
Bid Amount
5000
10000
We estimate mean and median WTP under four alternative distributional
assumptions. Mean and median WTP are derived directly from the estimated parameters
of the binary-response models, as explained above. Results are reported in table 2.
The most surprising result of table 2 is that the estimates of mean and median
WTP are negative when WTP is assumed to follow the normal or the logistic distribution.
The model based on the normal distribution predicts that 54% of the respondents have
negative WTP values. Using the Weibull and lognormal distributions, which admit only
non-negative values of WTP and fit the data better,8 circumvents this problem, but results
in a large discrepancy between median and mean WTP. Mean WTP is very large. The
median WTP amounts predicted by the two distributions are relatively close to one
another (239 and 250 SEK for Weibull and lognormal, respectively), but both are less
8 The lognormal and Weibull distribution results in higher Akaike Information Criterion (AIC) values. The
AIC is computed as the log likelihood minus the number of parameters to be estimated, and is frequently
used in applied work to assess the fit of a model.
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than what would be inferred by examining the responses to the payment questions, and
less than the smallest bid value offered to the respondents in the study.
Table 2. Mean and Median WTP for various distributional assumptions
(Johannesson et al. study, 1997).
Mean WTP
Median WTP
Log L
Normal
-2096.08
-2096.08
-1349.19
Logistic
-2007.75
-2007.75
-1349.10
Weibull
2,894,292
238.39
-1344.01
Lognormal
Infinity
254.30
-1343.84
Clearly, these figures are very different from those reported by Johannesson et al.
(1997), who rely on a completely different procedure for estimating mean WTP.
Specifically, they start with fitting a logit model, which implicitly admits negative WTP
values, but compute mean WTP as the area under the survival curve for positive WTP
values:
(1)
where G(y) is the cdf of WTP. When WTP is a logistic variate with mean n and scale a,
it can be shown that (1) is equal to (-1 / j3)hi[l -f exp(a)], where a=u/
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B, Changing the Procedure for Estimating f^fean WTP
In table 3, we experiment with alternative calculations of mean WTP that are
variants on four basic procedures. The first procedure follows Cameron and James
(1987). We fit a probit or logit model of the "yes" or "no" responses to the payment
questions, and compute mean WTP as
(2) m,=-a/p.
The second is the procedure followed by Johannesson et al., who fit a logit model
but effectively disregard the portion of the distribution corresponding to negative values.
If WTP follows the logistic distribution, this yields:
(3) n*-(-l/0)]n[l + exp(a)].
Our third procedure continues to rely on the fact that mean WTP is the area under
the survival curve, i.e., [l-F(os + /3y)]s where F( ) is tie cdf of the standardized WTP
variate. In earlier applications of the CV method, researchers estimated mean WTP by
computing the area under the fitted survival curve up to the largest bid amount offered in
the survey (10,000 SEK in the Johannesson et al. study). Our third estimate of mean WTP
is thus:
(4) m3
Finally, Chen and Randall (1998) and Creel and Loomis (1997) describe
semiparametric approaches to modeling the WTP responses. Specifically, they propose to
estimate mj by improving the fit of F(«) through augmenting its argument to include terms
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such as the sine and cosine transformations of the bid and of other regressors, in the spirit
of fast Fourier transform approximations.9 The argument of F(.), therefore, becomes:
(5)
A J
a=l j=l
[i/jfc cos(/kas(x)) - vja sin( jkas(x))],
where x is a vector that includes the bid and other determinants of WTP. For a subset, or
all, of these variables (the dimension of this subset being A), we introduce a scaling
function s(x). This scaling function subtracts the minimum value of x, divides the result
by the maximum value of x (thus forcing the rescaled variables to be between zero and
1), and then multiplies it by (2rc-0.00001). For this rescaling function to be possible, there
must be at least three distinct values for x, which rules out applying this transformation to
dummy variables. The ks are vectors of indices, and the us are parameters to be
estimated. Chen and Randall (1998) and Creel and Loomis (1997) suggest that for most
dichotomous choice CV survey applications it is sufficient to consider J=l, in which case
z is simplified to:
(6)
z = x/3 + 2^ [ua cos(.y(x))- va sin(.
We apply the semiparametric approach defined by equations (5) and (6) to the
Johannesson et al. data, where F(.) is the standard normal (logistic) cdf, and z, the
argument of the standard normal (logistic) cdf, includes an intercept, the bid and its sine
and cosine transformations (after rescaling). Formally, we compute mean WTP as
[-F(a + fiy
Chen and Randall also consider polynomial terms in the variables x.
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where /= (2n -0.00001)
B.
max
, Bmin | and BmK are the smallest and largest bid
amounts used in the survey, and B is the upper limit of the integration.
Following Creel and Loomis (199t), we first set B equal to he largest bid
amount used in the study (B-B^. This defines our estimate n> of mean WTP. We
subsequently compute n* by letting B in (6) tend to infinity. 1C
The results from these alternative calculations are shown in table 3. Table 3
shows that the largest change in estimated mean WTP occui's when going from m—
which yields a negative mean WTP—to approaches rn>-nis, which restrict integration to
the positive semiaxis (or a portion of it). Using the standard normal or the standard
logistic cdf gives similar results (table 3, second and third rows). The estimate of mean
WTP is sensitive to the upper limit of integration. As shown in table 3, third and fifth
rows, for the probit model, when the upper limit of integration is B^, mean WTP is
roughly half the figure that is obtained ,by letting B tend to infinity. A similar
comparison for the logit model (table 3, second and fourth rows) confirms these findings.
The semiparametric approach results in an estimated mean WTP similar to that of
regular probit and logit models when B-B^^, but that is considerably more
conservative than the regular probit and logit equations when B tends to infinity.
Assuming that WTP is normally distributed and that a conventional probit model is fit,
10 Cooper (2002) points out that it is not clear a priori which of these two estimates—104 or ms—is greater.
This is because, unless additional restrictions are imposed, when we adop: the semiparametric approach
F(z) as defined in (7) can no longer be interpreted as the cdf of the standardized WTP variate. Also see
Crooker and Herriges (2004) for a comparison between the Chen and Randall approach and other models
of dichotomous choice CV responses.
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nk is 6434 SEK, whereas ITS based on the semiparametric probit is equal to 4338 SEK (a
33% reduction).
A comparison between the probabilities of "yes" responses predicted by the
semiparametric and conventional probit at various bid values suggests that the former
outperforms the latter. For example, the former predicts that when the bid is 300 SEK the
probability of "yes" is 0,5172, which is closer to the relative frequency (0.5136) than the
prediction from the conventional probit (0.4794). When the bid amount is 10000 SEK,
the semiparametric probit predicts that the probability of a "yes" is 0.2857 against 0.2669
from the conventional probit. For comparison, the empirical frequency is 0.2883.
Table 3. Alternative procedures for computing mean WTP.
Johannesson et al. (1997) data
Approach
mi (Cameron and James, 1987)
rife (Johannesson et al., closed-
form expression)
rrt (Numerical integration of the
survival function to infinity)
n% (Numerical integration of the
survival function up to max. bid)
rrfe (Numerical integration of the
survival function up to max. bid)
m* (Creel and Loomis (1998)
semiparametric approach.
Numerical integration up to max.
bid)
ms (Creel and Loomis (1998)
semiparametric approach.
Numerical integration with
*„=«>)
Distribution F( )
Standard logistic
Standard logistic
Standard normal
Standard logistic
Standard normal
Standard normal; probit
model with bid, sin(bid)
and cos(bid)
Standard normal; probit
model with bid, sin(bid)
and cos(bid)
Mean WTP (in SEK)
-2007
6849
6434
3522
3528
3732
4339
18
-------
C. The Effect of Regressors
Would we obtain similar results in situations where regressors are included in the
I
model, and mean WTP is calculated conditionally on specific values of the regressors?
This question is appropriate, for example, when seeking to answer the question of how
the WTP for a risk reduction varies with age. Epidemiological evidence (e.g., Pope et al.,
1995) suggests that the majority of the lives saved by air quality regulations and
environmental policies are those of the elderly, and some observers have argued that
older people should be willing to pay less for a risk reduction—and their VSL should be
lower—mirroring their fewer remaining life years. Economic theory, however, does not
offer unambiguous predictions about the effect of age on WTP (Alberini et al., 2004).''
Johannesson et al. run a logit regression that includes age and age squared, plus
gender and education dummies, income and the respondent's quality-of-life rating,12 and
report finding a quadratic relationship between age and WTP that peaks when the
individual is about 40 years old. To check the sensitivity of these results to the procedure
used in the calculation, we ran logit and probit models with their same regressors (or
subsets of them), and predicted mean WTP at different ages using approaches m, and m*.
In both cases, we let B tend to infinity.
The results of these calculations are shown in Figure 2, panels (a)-(d). Panel (a)
plots rrj against age, confirming Johannesson et al.'s finding: the relationship between
11 Because a large proportion of the lives saved appear to be those of the elderly, there has been much
recent debate about whether the VSL should be lower for the elderly to reflect their fewer remaining life
years. In the US, the Office of Management and Budget recently repudiated making such adjustment for
age, on the grounds of insufficient evidence that the VSL is lower for elderly persons (Skrzycki, 2003).
In the Johannesson et al. survey, respondents were asked to rate their quality of life on a scale from 1 to
10, where 1 represents the worst possible quality, and 10 is the best possible quality.
19
-------
age and WTP is an inverted U that peaks at age 40.13 Similar results are observed when,
as shown in panel (b), the logit model is replaced by a probit, and mean WTP is
r
computed as I [1 - <&(*,« + fiy)]cfy, where x, = [1 age age2 ], a is the vector of probit
0
coefficients on these variables, and fl=- I/a.
In Figure 2, panel (c), we compare the predictions based on the conventional logit
model with Ihose from semiparametric probit models. Probit Fourier 1 is based on a
probit regression where the right-hand side variables are the bid, age and age squared,
and trigonometric functions of these variables. In this case, the shape of the relationship
between WTP and age is no longer an inverted-U. Moreover, this approach produces
estimates of mean WTP that are consistently smaller than those from the conventional
probit model. The curve labeled Probit Fourier 2 is based on a similar model, except that
the sine and cosine transformations are applied only to the bid. This time, the relationship
between WTP and age resumes its quadratic shape, but the estimated WTP values remain
consistently lower than those of the regular probit model.
Figure 2, panel (d) displays the results based on probit models that are similar to
those used for panel (c), except that more regressors—household income, the quality-of-
life rating reported by the respondent, a gender dummy and an educational attainment
dummy—are entered in the right-hand side of the model. The curves labeled Probit
Fourier 1 and Probit Fourier 2 are different from one another in that the former includes
trigonometric functions of all of the continuous variables, while the latter includes only
the sine and cosine transformations of the bid.
13 Dummy indicators for the "yes" or "no" responses to the payment questions were regressed on an
intercept, the bid amount, age and age squared.
20
-------
The mean WTP figures plotted in pinel (d) refer to a 50-year-old male (SEX=1)
with high school education (DEDU=1), the average household income of the sample
(24,490 SEK), and the same quality-of-life rating as the avenige respondent (7.34 on a
scale from 1 to 10). As in panel (c), Probit Fourier 1 results in a non-monotonic
relationship between WTP and age, while Probit Fourier 2 implies a quadratic
relationship. Both predict lower WTP figures 1han the conventional probit model.
21
-------
Figure 2.
Legend: in panel (c), Probit Fourier 1 includes bid, age, age squared, and sine and cosine functions of these
variables; Probit Fourier 2 includes bid, sin(bid), cos(bid), age and age squared.
In panel (d), Probit Fourier 1 includes bid, age, age squared, income, quality of life rating, a gender dummy
and an education dummy, plus sine and cosine terms of ail continuous variables. Probit Fourier 2 includes
bid, sin(bid), cos(bid), age, age squared, income, quality of life rating, a gender dummy and an education
dummy.
WTP in SEI
8000 •
6000 •
4000 •
3000 '
1000
(a) Mean WTP by age: Logit Model, Johannesson et al. Data
<
\
^ ^^
^^"•V.
i
20 30 40 50 60 70
Age
(b) Mean WTP by age: Logit and Probit Model,
Johannesson etal. Data
WTP in SEK
8000
log it
1000
20
70
22
-------
(c) Mean WTP by Age: Johanneeeon et al. Data
8000
WTP in SEK
7000
20 30 40 50
twobit
60 70
6000 i
WTP in SEK
5000
Mean WTP by Age: Probit Model* with Covarfates, Jotiannesson
et al. Data
Predictions for a male with high school education, average income and quality
of life rating
Probit
Fourier 1
Probit
10 20 30 40 50 60 70 80
23
-------
In sum, panels (c) and (d) in Figure 2 suggest that there are no easily discernible
patterns, and that claims about the relationship between age and WTP, and the magnitude
of WTP at various ages, are not robust and may be an artifact of restrictive assumptions.
We further investigate this matter by switching to a lognormal distribution for
WTP and to median WTP for specific ages, which we expect to result in more
conservative estimates. The results, shown in table 4, suggest that the lognormal model
implies a quadratic, inverted-U relationship between age and WTP. It also suggests,
however, that the curvature of the relationship is much sharper than that predicted by
Johannesson et al. For example, the WTP of a 70-year-old for a reduction in risk of 2 in
10,000 is only 90 SEK, or only about 20% of the WTP predicted for a 40-year-old person
(440 SEK).
Taken together with the evidence from the semiparametric approach, these results
suggest that detecting the shape of the relationship between WTP and a regressor of
interest depends crucially on, and is very sensitive to, three factors: (i) the distribution
assumed for WTP, (ii) the welfare statistic used (mean or median WTP), and (iii) the
procedure used for computing it.
24
-------
Table 4. The relationship between age and WTP for a risk reduction:
lognorrnal WTP and median WTP v. Johannesson et al. logit and truncated mean WTP.
Age
20
30
40
50
60
70
Johannesson et al, 1997.
Mean WTP in
SEK
6100
6900
7200
6900
6000
4600
Implied VSL in
million SEK
30.3
34.6
36.1
34.3
29.8
23.3
Alternative calculation using log
normal WTP.
'Median WTP in
SEK
137.18
307.31
440.77
Implied VSL in
million SEK
0.672
1.505
2.160
404.75 1.983
237.97 1.166
89.57 0.439
IV. Treatment of Outliers
In this section, we investigate the effect of outliers on the estimates of WTP. We
begin by tackling 1he problem of identifying outliers in dichotomous-choice CV surveys,
and explore the effect of including or excluding these observations from the sample. We
use logit regressions of the WTP responses on individual characteristics to classify
observations as outliers. Because outliers can be caused, among other reasons, by a
number of undesirable response effects, we then examine whether it is possible to
estimate "structural" models of these response effects.
A. Outliers
Collett (1991) defines as outliers as "observations that are surprisingly far away
from the remaining observations in the sample," and points out that such values may
occur as a result of measurement errors, execution error (i.e., use of a faulty experimental
procedure), or be a legitimate, if extreme, manifestation of natural variability.
25
-------
Our first order of business is to define outliers when the variable of interest is
binary, as is the case with the responses to dichotomous-choice CV questions. Copas
(1988) defines an outlier as an observation for which we predict a low probability of a
one (zero), but we do observe a one (zero).
We use the Johannesson et al. (1997) data to check (i) how many observations
could be classified as outliers according to several alternative cutoff levels, and (ii) by
how much mean WTP would change if these outliers were excluded from the sample.
Specifically, we wish to see for how many observations the predicted probability of a
"yes" is less than 0.05, 0.10, etc., but the response to the payment question is a "yes." The
predicted probability is based on Johannesson et al.'s logit regression of the "yes" or "no"
response indicator on respondent age, age squared, income, an education dummy, and a
quality-of-life rating subjectively reported by the respondent in the interview:
pi = [1 + exp(-(x,d + )3 • £,)]"', where a and /3 are the estimated logit coefficients, x is
a vector of regressors, and B is the bid assigned to respondent i.
For ease of comparison, we use Ihe same procedure for estimating mean WTP as
in Johannesson et al.'s work (see section III). The resulting mean WTP figures are
reported in column (C) of table 5. We also fit a binary data model of the responses based
on an alternate distribution—the Weibull—and report estimates of mean and median
WTP based on the latter in columns (D) and (E) of table 5, respectively.
26
-------
Table 5. Outliers in the Johannesson et al. data (based on logit regression, n=1660):
All WTP figures in SEK
(A)
Definition of
outlier
No outliers
identified
Prob(yes)< 0.05
and yes observed
Prob(yes)<0.10
and yes observed
Prob(yes) <0,20
and yes observed
Prob(yes) <0.25
and yes observed
Prob(yes) <0.30
and yes observed
(B)
How
many?
None
None
None
5
26
59
(C)
Johannesson et
al. procedure
Mean WTP*
6732
673(2
6732
6141
4846
3767
(D)
Weibull:
Mean WTP*
2.894 million
2.894 million
2. 894 million
1.1 50 million
193,481
36,114
(E)
Weibull:
Median
WTP*
238
238
238
302
338
369
*.
: Welfare statistics after excluding outliers.
Table 5 shows that outliers according to the Copas' definition were found only
when the cutoff for identifying an outlier was set to 0.15 or higher. When the cutoff is set
to 0.25, for example, a total of 26 observations would be considered outliers, and
dropping them from the usable sample would reduce Johannesson et al.'s mean WTP
from 6732 to 4846 SEK—a 30% reduction. A cutoff of 0.30 results in the exclusion from
the usable sample of 33 more individuals, and in a further reduction of mean WTP to
3767 SEK—a 45% reduction.
As shown in column (D) of table 5, using a Weibull distribution generally results
in implausibly large mean WTP values. The mean WTP, however, does get smaller when
outliers are excluded from the sample. By contrast, median WTP (shown in column (E)),
which is 338 SEK for the full sample, increases slightly when outliers are omitted from
27
-------
the sample, which suggests that the skewness of the distribution of WTP has become a
little less pronounced.
B. Undesirable Response Effects: Yea-saying, Nay-saying, and Random Responses
Contingent valuation studies about mortality risk reduction rely crucially on the
respondent's comprehension of the risk and risk reductions being valued. Many recent
survey questionnaires deploy visual aids and practice questions about risks, but, despite
these efforts, it is possible that some respondents still remain confused about the
commodity being valued, and that their answers to the payment questions may be
affected by undesirable response effects.
Carson (2000) describes three types of undesirable response effects that may
occur in dichotomous-choice CV surveys. The first is yea-saying, whereby a respondent
answers "yes" to the bid question with probability 1, regardless of the bid amount. This
may be done in an effort to please the interviewer, or in the hope of terminating the
interview sooner.
The opposite phenomenon is nay-saying, which occurs when the respondent
answers "no" to the payment question with probability 1, regardless of the bid amount.
Respondents engaging in nay-saying may dislike new public programs and new taxes, or
might be afraid of committing to something they do not fully understand.
It is also possible that some people give completely random responses, answering
"yes" to the payment question with probability 0.5 (and hence, "no" with probability 0.5),
regardless of the bid amount. Completely random responses may be due to confusion
28
-------
about the scenario, failure to understand th$ commodity being valued, no interest in the
survey, poorly written survey questions or survey materials, or simply a data entry error.
In practice, not all respondents in a contingent valuation survey will be subject to
these undesirable effects. To accommodate for this possibility, we consider discrete
mixtures. For simplicity, attention is restricted to discrete mixtures with two components,
where a fraction of the sample (a-100%) is affected by one of these undesirable response
effects, while the remainder answers the payment questions in the usual fashion (i.e., by
saying "yes" if latent WTP is greater than the bid, aid "no" otherwise). The researcher's
problem is that—unless respondent or interviewer debriefs are used—it is not possible to
tell from which component of these two populations the respondent is drawn. This
requires estimating a (discrete) mixture of distributions.
Figures 3 and 4 show how mixtures alter the estimated survival curve of WTP.
Figure 3 depicts the true and observed survival curve when one of the two mixing
components is yea-saying. (By true survival curve, we mean the survival curve that refers
to the non-degenerate component of the mixture.) Yea-saying raises the observed survival
curve at every bid value, which implies that mean WTP will be overestimated. The extent
of this positive bias depends on severity of the yea-saying (i.e., on the value of a).
The converse will be true with a nay-saying component. Figure 4 shows that if the
sample is contaminated with a small number of people who answer the payment question
in a completely random fashion, the observed survival curve is tilted, crossing the true
survival function from below at the median, and remaining above it for higher bid values.
This implies that both the mean and variance of WTP will be overestimated, but that the
29
-------
median WTP should not be affected by the presence of completely random responses, as
long as the mixture contains only these two components.
C. Likelihood Functions for Two-component Mixtures
Absent any other undesirable response effects, the log likelihood function with
dichotomous choice responses is:
(8)
log/, = J>, log Pt(yi = 1 1 5.) + (1 -x)log Pi-a = 0 1 Bt)
where .y is an indicator that takes on a value of 1 if the response is a "yes" and zero if it is
a "no," B denotes the bid, and F(Bi;B) is the cdf of WTP evaluated at the bid value, 0
being the vector of parameters indexing F.
In this paper, we consider a two-component mixture where (l-a)-lOO percent of
the population answers the payment questions according to the usual assumption ("yes" if
WTP is greater than the bid, "no" otherwise). The distribution of WTP for this
component of the mixture is non-degenerate. If the remaining ct-100 percent of the
population consists of yea-sayers, who answer "yes" to the payment question with
probability I,14 the probability of observing a "yes" in equation (8), Pr^, = 1 1 B,), is
equal to:
(9)
14 In practice, one would expect the data from a contingent valuation survey to come from mixtures with
more than two components. The simple cases considered in this paper should, therefore, be interpreted as
the situations where the researcher stands his or her best chance to identify the components.
30
-------
"No" responses must come from the non-degenerate component of the mixture:
(10) Pr(>v = 0| Bf) = (!-«)• F(B.;d).
If the sample is described by a discrete mixture with nay-sayers, the probability of
a "yes" is:
(11) Pt(y,;=1| B,) = (!-«)• [1-
and the probability of observing a "no" is:
(12)
Finally, if the sample is a mixture comprising persons who answer in a completely
random fashion, the probabilities of a "yes" and "no" response are
(13)
(14)
respectively.
Pr(>>, = 0] Bt) = (1 -a) - F(B, ;fl) + a - 0.5,
31
-------
Figure 3. Effect of yea-saying.
% willing to pay •
l-F(-)
Observed
True curve
Bid amount
Figure 4. Effect of Completely Random Responses.
% willing to pay
l-F(-)
Observed
Median
WTP
Bid amount
32
-------
C. Application
We apply these two-component mixtures to the data from the CV survey
conducted in Hamilton, Ontario, by Krupnipk et al. (2002). Participants in this study took
a self-administered computer questionnaire at a centralized facility. After a probability
tutorial, the study participants were shown the risk of dying for all causes for a person of
their age and gender.
They were asked to value a total of three risk reductions by answering payment
questions in a dichotomous-choice format with follow-up.15 Each of these risk reductions
would be taking place over 10 years, and would be delivered by a hypothetical product.
The risk reduction was context-free, in that it was not associated with air pollution
reductions or another public program.
In this paper we focus on WTP for the 5 in 1000 risk reduction. We assume that
WTP is a Weibull variate, and begin with fitting the models described by equations (9)-
(14) to the responses to the initial payment questions. In a subsequent round of
estimation, we amend equations (9)-(14) to accommodate for the responses to follow-up
questions (double-bounded models). Table 6 reports mean and median WTP for the
component of the mixture that behaves following the usual economic paradigm (after
filtering out the degenerate components).16
As shown in table 6, panel (i), in our mixture based on single-bounded models we
find no evidence of yea-saying: in column (ii), the estimated a is identically equal to
zero. By contrast, column (iii) suggests that as much as 26% of the sample may be
15 The three risk reductions were 5 in 1000 and 1 in 1000 occurring over the next 10 years, and 5 in 1000
beginning at age 70,
16 We use the constrained maximum likelihood routine in GAUSS to force a to lie between 0 and 1.
33
-------
comprised of nay-sayers.17 Column (ii) displays an even more implausible result: in the
mixture with completely random responses, we estimate that over 50% of the sample
answers the payment questions in a completely random fashion.
Because we do not know whether a specific hdividual does or does not answer
the payment question at random, this implies that each observation has a probability
equal to 0.51 of being a completely random response. This finding is in sharp contrast
with the answers to debriefing questions and the gpod internal consistency and validity
shown by study participants, raising doubts about the ability of single-bounded models of
WTP to capture the correct proportion of subjects engaging in degenerate response
mechanisms.
In theory, double-bounded estimation should produce more reliable results, thanks
to the narrower intervals around the respondent's unobserved WTP amounts afforded by
the follow-up payment question. Table 6, panel (ii) reports estimation results based on
double-bounded models. While any evidence of nay-saying seems to have vanished
(column (iii)), almost two-thirds of the respondents are predicted to be answering the
payment questions in a completely random fashion (column (ii)). Clearly, this result is
not plausible.
To seek for an explanation for this finding, we turn to evidence from Monte Carlo
simulations. Alberini and Carson (2001) report that mixture models like (9)-(14) give
reliable results only when (a) F(«) is correctly specified, (b) the mixture type is correctly
specified, and (c) a is not too small. In most other cases, the estimated coefficient a
appears to pick up any divergence between the assumed F(«) and the true distribution, or
17 We view this as a somewhat alarming result, but this percentage is comparable to the probability of yea-
saying (20%) estimated by Kanninen (1995) using double-bounded data from a contingent valuation survey
about wildlife and wetland habitat protection in California's San Joaquin Valley.
34
-------
any other misspecification in the type of mixture. We conclude that it nay be difficult to
estimate mixtures using existing CV data about mortality risk reductions, and that a study
capable of identifying the components of a mixture would probably have to be designed
specifically for that purpose.
Table 6.
Discrete1 Mixture Models:
Results for Krupnick et al. (2002) (Canada study) Bootstrap standard errors in
parentheses.
1. using only the responses to the initial payment questions (single-bounded)^
a (probability of
yea-sayers, nay-
sayers, etc.)
Mean WTP ($)
Median WTP($)
(i)
No mixture
1176.94
(306.55)
445.99
(44.02)
(ii)
Mixture with
completely
random
responses
0.5108
(0.13)
594.55
(n/a)
551.04
(146.05)
(iii)
Mixture
with nay-
sayers
0.2644
(0.05)
969.57
(n/a)
859.61
(134.10)
(iv)
Mixture with
yea-sayers
1176.94
(n/a)
445.99
(44.02)
II. using the responses to the initial and follow-up payment questions (double-bounded).
a (probability of
a yea-say er, nay-
say er, etc.)
Mean WTP ($)
Median WTP ($)
(0
No mixture
826.41
(70.85)
323.83
(20.96)
Mixture with
completely
random
responses
0.6577
(0.0448)
831.24
(70.92)
323.75
(21.10)
(iii)
Mixture
with nay-
sayers
826.41
(70.85)
323.83
(20.96)
(iv)
Mixture with
yea-sayers
826.41
(70.85)
323.83
(20.96)
35
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V. Internal Validity: Scope Tests and Endogenous Risks.
A. Subjective Risks and Risk Reductions
In this section, we consider the situation where respondents are asked to assess
their subjective baseline risks (Gerking et al., 1988; Johannesson et al., 1991 ;18 Perssonet
al., 2001). If the risk reduction is expressed as a specified proportion of the baseline risk
(e.g., 20%), then the risk reduction varies across respondents, allowing one to test for
sensitivity of WTP with respect to the size of the risk reduction—the so-called "scope"
effect 19
In this section, we ask three related questions. First, should WTP and risk
reduction be treated as endogenous in such studies? Second, does this affect conclusions
about the "scope" effect? Third, does treating risk reduction as endogenous with WTP
affect our conclusions about whether subjects respond to absolute or relative risk
reductions when they announce their WTP amounts (Baron, 1997; McDaniel, 1992)?
To answer these questions, we use the data from the Persson et al. (2001) study
about transportation safety. Persson et al. conducted a mail survey of 18-74-year-old
Swedes in 1998. Respondents were asked to estimate their own risk of dying from any
cause (EIGRISK), and to value a specified reduction in this risk. The reduction was
ls Johannesson et al. (1991) ask hypertensive patients in a health care facility to assess both baseline risks
and risk reductions associated with hypertension medication. In this case, it seems reasonable to expect that
WTP and risk reductions should be endogenous.
19 Hammitt and Graham (1999) review over 25 contingent valuation studies where the size of the risk
reduction was exogenously varied to the respondents. They find that the scope effect is more often satisfied
when the size of the risk reduction is varied within a respondent (an internal test) than when it is varied
across respondents (an external test), and that there are several studies where WTP does not vary
systematically with the size of the risk reduction. They also report that in internal tests WTP is rarely found
to be strictly proportional to the size of the risk reduction. In studies that conducted external scope tests
WTP generally exhibits little responsiveness to the size of the risk reduction, and generally fails to be
strictly proportional to the risk reduction.
36
-------
expressed as a percentage of the baseline rjsk, this percentage being selected at random
among a predetermined set of values (10%, 30% or 50%).
Respondents were next asked to assess their own risk of dying in a road-traffic
accident (DEGRISK), and to value a risk reduction of 10%, 30%, 50% or 99%
(RISKMD). Both risk reductions are private goods and are valid for one year only. In this
paper, attention is restricted to the risks of dying in road accidents.
B. Hypotheses
We assume that:
(15) WTP = exp(x,0,) • ABSR1SK?1 • exp(e,)
where x is a Ixk vector of individual characteristics thought to influence risks, ABSRISK
is the absolute risk change (ABSRISK=DEGRlSKxRISKMD), and e is an error term. On
taking logs,
(16) log WTP = x, ft + j32 log ABSRISK, + et,
which can be re-written as:
(17) log WTP = x,/j, + J32 log DEGRISK. + /J3 log RISKMD, + e.,
where RISKMD is the percentage risk reduction, which is randomly assigned to the
respondent in the survey, and is therefore considered an exogenous variable. If log
DEGRISK and log WTP share common, unobservable respondent-specific factors, they
are econometrically endogenous, which makes the OLS estimate of /32 biased and
inconsistent. This problem can be addressed using instrumental-variable estimation
techniques, such as two-stages least squares (2SLS).
37
-------
In equation (17), the coefficients of log DEGRISK and log RISKMD are allowed
to be potentially different from each other in order to test hypotheses about the
determinants of WTP and about scope effects. Specifically, if /32 = /33 , then respondents
correctly valued the absolute risk reduction, which allows us to estimate the VSL. If
/32 = /J3 = 1 , then WTP is strictly proportional to the size of the risk reduction, as
economic theory suggests should be the case with small risks (Hammitt and Graham,
1999).
Should we find that /32 * /33 , and that both /32 and /33 are different from zero,
we would conclude that at least some weight has been given to the baseline risk (Baron,
1997; McDaniels, 1992). By contrast, if & # & , )32 =0 and ft, * 0 , then WTP depends
exclusively on the proportion, but not on the absolute risk reduction, and it is not possible
to compute a meaningful VSL. One would expect the outcome of these tests of
hypotheses to depend on whether baseline risks, DEGRISK, are treated as endogenous
with WTP.
C. Endogeneity of Risks and WTP
If log DEGRISK is endogenous with log WTP, we need an additional equation
explaining log DEGRISK:
(18) logDEGRISK = zj, + w.y2 -l-r],
where z is a vector of instruments that overlap with some of the regressors x in equation
(15), w is a vector of instruments excluded from the right-hand side of the WTP equation
38
-------
to ensure identification, y, and y2 are vectors of coefficients, and 7], is an error term, e
and T| are potentially correlated within a respondent.
We estimate the system of equations (17)-(18) by 2SLS using the broader of the
two samples used by Persson et at. (2001).20 The results of OLS estimation of (18) are
reported in table 7 because they are of independent interest and because they are the first
stage of the 2SLS procedure. Table 7 shows that subjective risks are related to age and
age squared, and to miles driven per year. Other individual characteristics, such as
gender, do not appreciably influence subjective risks.
As shown in table 7, we included jn this regression various dummies, such as
whether the respondent travels by bicycle, moped or motorcycle, and whether he wears a
helmet when doing so, for two reasons. First, these variables are thought to influence the
subjective chance of dying in a road traffic accident Second, because they are excluded
from the WTP equation, these variables provide identification restrictions for the 2SLS.
However, table 7 shows that they are not significant predictors of subjective risks. We
reach the same conclusion for the two education dummies (high school diploma and
college degree), which were included in this equation for the same reasons.
Table 8 reports the results of OLS and 2SLS regressions of WTP on various
regressors, including subjective risks. In addition to the risk variables, the regressors
include the logarithmic transformations of income and age, the square of log age,21 log
miles driven, a dummy indicating whether the respondent has ever been injured in an
20 Specifically, following Persson et al., we form the sample by taking all observations with positive WTP,
and replacing zero WTP values with a small positive number (i.e., WTP=2 SEK). Observations with
missing baseline risk and/or missing WTP, baseline risk smaller than 1 in 100,000, WTP less than 1 and or
WTP greater than 5% of annual income are excluded.
21 Including the logarithmic transformation of age and its square, rather than age and age squared, which
were included in the first-stage regression, should help in identifying the coefficients of the WTP equation.
39
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accident, and education and family composition dummies. As shown in table 8,
regardless of the specification, the estimation techniqus used, and the restrictions placed
on the parameters, WTP is well predicted by income, miles traveled, and the risk
reduction. The effect of age is weak at best, and education and family status are not
important.
In columns (A) and (B) of table 8, the restriction that /32 = /33 is imposed, which
means that WTP is regressed on the size of the absolute risk reduction. Column (A)
reports the results of OLS estimation that treats subjective risks, and hence the risk
reduction, as exogenous, whereas column (B) reports the results of 2SLS estimation that
treats subjective risk as endogenous with WTP and imposes the abovementioned
restriction.
Column (A) shows that WTP does increase systematically with the size of the risk
reduction, but in a less-than-proportional fashion: /32 is less than 1. The responsiveness
of WTP to the size of the risk reduction is weak: doubling the size of the risk reduction
would increase WTP by only 18%. With 2SLS (column (B)), one concludes that WTP E
more responsive to the size of the risk reduction, since the 2SLS /52 is more than twice as
large as its counterpart in column (A), implying that doubling the risk reduction increases
WTP by 41%. The null hypothesis that absolute risk is exogenous is rejected soundly.
Columns (C) and (D) use OLS and 2SLS, respectively, but relax the restriction
that J32 = /33. F tests of the null that these coefficients are equal reject the null at the
conventional levels with OLS estimation, but fail to reject it with 2SLS. Inspection of the
coefficient estimates and their t statistics, however, suggest that in both cases people were
responding to the proportions, rather than the absolute risk reductions. This suggests that
40
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I
with CV surreys about mortality risk reductions that present the risk reduction as a
specified proportion of the baseline risk one should check whether subjects were truly
responding to the absolute risk reduction, or simply to the percentage risk reduction,
without considering the baseline. In testing |these results, however, it is important to pay
attention to the possible endogeneity of baseline risk with WTP, as conclusions may be
<
sensitive to whether baseline risk is treated as endogenous with WTP.
Table 7. First-stage regression - Person et ai. data.
Dependent variable: log DEGRISK. N=S18.
Intercept
Age
Age squared
Male
Log km traveled in a car
Travels by moped or
motorcycle (dummy)
Travels by bicycle (dummy)
Wears helmet when bicycling
(dummy)
Uses seatbelt when in back
seat of car (dummy)
High school diploma (dummy)
College degree (dummy)
Coefficient
1.1095
-0.0647**
0.00068**
-0.0613
0.2816**
Q.0077
0.0735
0.1875
-0.1374
0.0489
-0.0578
T statistic
1.85
-3.44
3.16
-0.69
3.83
0.06
0.42
1.36
-1.28
0.39
-0.46
** =
significant at the 1% level.
41
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Table 8. Second-stage results - Persson et at. data
Dependent variable: log WTP. T stats in parentheses.
Intercept
Log income per
household member
Log km traveled in car
LogDEGRISK
LogRlSKMP
Injured in accident
(Dummy)
Log age
Log age squared
High school diploma
College degree
Household members
jigesO-3
Household members
ages 4- 10
Household members
ages 11-17
Household members
ages 18+
A
OLS
n=676
-1.0221
(-0.45)
0.4213**
(2.47)
0.4949**
(3.19)
0.1850**
(2.32)
0.1850**
(2.32)
0.3779
(1.58)
-0.4943A
(-1.68)
-0.0085
(-0.17)
-0.1275
(-0.47)
0.1298
(0.47)
0.1754
(0.75)
0.2401
(1-59)
-0.0445
(-0.26)
0.0543
(0.43)
B
2SLS
N=579
-2.4185
(-0.97)
0.4777**
(2.50)
0.4368**
(2.60)
0.4092**
(2.38)
0.4092**
(2.38)
0.4372A
(1.71)
-0.3308
(-1.03)
-0.0105
(-0.18)
0.0829
(0.28)
0.3539
0.16)
0.1008
(0.42)
0.3410*
(2.14)
0.0366
(0.20)
0.1545
(1.14)
C
OLS
n=676
0.0810
(0.04)
0.3747*
(2.19)
0.5502**
(3.52)
0.0671
(0.73)
0.5467*
(3-35)
0.3659
(1.53)
-0.5770*
(-1.96)
-0.0211
(-0.42)
-0.0635
(-0.23)
0.1627
(0.59)
0.1441
(0.62)
0.2396
(1.60)
-0.1594
(-0.09)
0.0372
(0.30)
D
2SLS
N=579
-0.5672
(-0.18)
0.4373*
(2.19)
0.6573**
(2.47)
-0.6332
(-0.65)
0.4420**
(2.47)
0.4262
(1.63)
-0.6910
(-1.48)
0.0350
(0.45)
0.1567
(0.50)
0.3249
(1.03)
0.0552
(0.22)
0.3088A
(1.87)
0.0269
(0.14)
0.1494
(1.07)
Test: p2=P3
F=6.44
Pval=0.011
4
F=1.18
Pval=0.277
6
Observations with missing baseline risk and missing WTP, observations with baseline
risk smaller than 1 in 100,000, observations with WTP less than 1 and with WTP greater
than 5% of annual income are excluded. Observations with WTP equal to zero are
replaced by WTP=2.
42
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VI. Internal Validity: The Relationship Between WTP and Income
Income is an important independent variable in regressions relating WTP to
individual characteristics of the respondent. There are at least two reasons why
researchers regress WTP on household (or personal) income. First, this is a common
practice for testing the internal validity of the WTP responses, as theory suggests that
WTP for mortality risk reductions should be positively associated with income. Second,
there is much interest in the income elasticity of WTP for the purpose of predicting WTP
at specified levels of income within the sample, or for benefit transfer purposes.22 23
In many contingent valuation surveys about environmental quality or other public
goods, researchers expect WTP to be a small fraction of the respondent's income. This
expectation has led them, in some cases, to exclude from the sample respondents whose
implied WTP is greater than, say, 5% of the respondent's income.
With reductions in one's own risk of dying, there is no particular reason to believe
that WTP should be a small proportion of income. However, Persson et al. do omit from
the usable sample respondents whose WTP exceeds 5% of household income, and Lanoie
et al. (1995) find that their estimate of VSL for workers in the Montreal area, which is
Can $22-27 million, drops to Can $15 million after excluding from the sample three
22 It is recognized, however, that knowing the income elasticity of WTP in a cross-sectional sample sense
does not answer the important policy question of whether VSL should change over time, as income grows
and the tradeoffs people are prepared to make between income and risk reductions change. Costa and Kahn
(2002) circum'ent this problem by estimating compensating wage studies for different years in the US.
Using Census micro-data and fatality risk figures from the Bureau of Labor Statistics for 1940, 1950, 1960,
and 1980, Costa and Kahn conclude that the quantity of safety has increased over time, and that the
compensating differential has increased. The implied elasticity of VSL witi respect to per capita GNP is
1.5 to 1.7. A meta-analysis of compensating wage studies by Viscusi and Aldy (2003) pegs the income
elasticity of VSL to be 0.5-0.6, and certainly less than one. DeBlaeij et al. (2000) conduct a meta-analysis
of the WTP to reduce transport risks, finding a considerable higher income elasticity of 1.33. Liu et al.
(1997) compare estimates of the VSL from compensating wage studies in T&iwan based on 1982-1986 data
with predictions based on VSL-income relationship from developing countries.
23 Alberini (2004) discusses difficulties associated with measuring income ir surveys.
43
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workers whose WTP is one-third of their pre-tax income.24 It seems, therefore,
appropriate to check for respondents whose announced WTP is a relatively large
proportion of income, and to examine how much the estimates of mean WTP and of the
income elasticity of WTP change when these respondents are excluded from the sample.
Respondents with very high announced WTP relative to income may, for example, have
failed to give proper consideration to their budget constraint, have intentionally
misrepresented their income, or have simply miscalculated it.
We use the data from two mortality risk surveys conducted in the US and
Sweden, respectively—Alberini et al. (2004), and Persson et al. (2001)—to investigate
these issues. We begin with the data from the Alberini et al. study. Results from
estimating mean WTP after excluding respondents with WTP greater than a given
percentage of income are shown in Table 9.25 We vary this percentage from 25 (the least
stringent criterion) to 2.5 (the most stringent criterion), showing how doing so excludes
from 68 to 133 respondents (almost one-third of the sample). As shown in table 9, mean
and median WTP do decline as we exclude more observations from the sample, but the
change is within 10-12% of the original figures.
By contrast, what does change dramatically is the income elasticity of WTP, a key
quantity when one wishes to (i) extrapolate study results to the general population, (ii)
focus on the economically disadvantaged, and (iii) attempt benefit transfers to other
countries or locales where income levels are different. As shown in table 9, the income
24 Lanoie et al. (1995) value risk reductions in the car safety and workplace safety contexts. They survey
employees of firms in the Montreal area, asking both willingness-to-pay and willingness-to-accept (WTA)
questions. The Can $22-27 million VSL figure refers to WTP for job safety (which is judged as more
reliable than WTA).
25 These estimates are based on a Weibull interval-data model that combines the responses to the initial
WTP questions and to the follow-up questions in a double-bounded fashion.
44
-------
elasticity of WTP is 0.16 when the full sample is used, 0.29 when persons whose implied
WTP amount is greater than 10% of household income are excluded, 0.52 when we
exclude persons whose WTP is greater than 5% of household income, and, finally, 0.92
when the most stringent criterion is used. Predictions for WTP as income changes would,
therefore, vary dramatically, depending on' which of these "cleaned" samples, and tie
corresponding income elasticity of WTP, one opts for.
Table 9. Outliers with respect to income.
Alberini et al. US Survey. WTP for 5 in 1000 risk reduction, wave 1, cleaned sample*
Least
stringent
Most
stringent
Exclude if. . .
(all sample)
WTP > 25% of
household
income
WTP > 10% of
household
income
WTP > 5% of
household
income
WTP > 2.5% of
household
income
N
551
483
477
458
418
Mean WTP
($)
752.84
(88.37)
755.56
(90.84)
747.53
(90.02)
719.25
(89.21)
678.39
(91.64)
Median WTP
($)
346.21
(28.45)
362.38
(31.97)
355.14
(29.24)
339.33
(30.02)
302.26
(28.67)
Income
elasticity of
WTP
0.16
0.16
0.29
0.52
0.92
* Excludes those respondents who failed the probability quiz and the probability choice.
Further investigation reveals that the 65 respondents who violated the most
stringent exclusion criteria were slightly older than the remainder of the sample, but not
significantly so (average ages were 57 and 54, t statistic of the null of no difference -
1.38), significantly less educated than the remainder of the sample (11.75 years of
45
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schooling vs. 13.3, t statistic = -6.27), and reported much lower annual household income
than the rest of the sample (sample averages: $17,942 v. 56,151, t statistic -22.18).26 27
Moreover, they were twice as likely to indicate, in the debriefing section of the survey,
that they had misunderstood the timing of the payment (27% of this group versus 13% of
the remainder of the sample, t statistic = -4.66).
Results for the Persson et al. data are displayed in tables 10 and 11. One
respondent in Persson et al.'s sample reports a WTP amount that is 83% of annual
household income. Fortunately, tiie rest of the sample is more reasonable: Ninety-nine
percent of the sample holds a WTP amount for reducing the risk of a fatal auto accident
that is equal to or less than 12.5% of household income. In their analysis, Persson et al.
discard from the usable sample observations such that WTP accounts for more than 5%
of annual household income. This loses 29 observations.
Table 10 displays mean WTP for the full sample, and when persons with
relatively high WTP/income ratios are excluded from the usable sample. This table shows
that while median WTP remains the same for the various exclusion criteria, mean WTP
jumps from 1875 to 2778 SEK when we reinstate into the sample those respondents
whose WTP was more than 5% of household income. This is a 50% increase in WTP.
26 The median annual household income is $17,500 and $55,000, respectively.
27 It is possible that these respondents miscalculated or intentionally underreported their income. We
regressed log income on age, age squared, education and the gender dummy for the full sample, and used
the results of this regression to compute predicted income. For the 65 respondents with high WTP/income
ratio, income predicted on the grounds of education, gender and age was always larger than reported
income.
46
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Table 10. Person et al. study.
Effect of excluding observations with large annual WTP/household income ratios.
All values in 1998 SEK.
Exclusion criterion
None
Respondents with zero income but
positive WTP
WTP greater than 50% of household
income (least stringent)
WTP greater than 25% of household
income
WTP greater than 20% of household
income
WTP greater than 12.5% of
household income
WTP greater than 10% of household
income
WTP greater than 5% of household
income (most stringent)
Number of
observations in
the sample
637
;637
636
633
632
631
629
618
Sample
average
wrp
2778
2778
2635
2163
2151
2143
2134
1875
Sample
Median WTP
1000
1000
1000
1000
1000
1000
1000
1000
Table 11 displays the income elasticity of WTP when observations where WTP
accounts for a relatively large share of household income are omitted from the sample.
As explained in section V, we estimate a system of simultaneous equations for log
baseline risk and log WTP. The right-hand side of the WTP equation includes the
logarithmic transformation of the absolute risk reduction, log miles traveled in a car in a
year, a dummy accounting for previous injuries sustained in a car accident, log age, log
age squared, two education dummies, and dummies for the size of the household in
various age groups. Table 11 shows that income elasticity of WTP doubles when we
move from the sample created with the least restrictive criterion to the most stringent
criterion. It remains, however, relatively low (0.28).
47
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We conclude that while there is no unambiguous criterion for considering one's
WTP "large" relative to this person's income, researchers should experiment with
checking how the estimates of WTP and other coefficients of interest are affected by
including/excluding from the usable sample those respondents whose announced WTP is
high relative to income. In the two examples presented in this section, doing so had a
completely different impact on the estimates of mean WTP and on the income elasticity
of WTP. These checks also pointed out that observations with a large WTP relative to
income may be due to the respondent's failure to understand or retain details of the risk
reduction scenario, as shown by the example based on the Albermi et al. data. They may
also result from an inaccurate calculation or a deliberate misreporting of income on the
part of the respondent.
48
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Table 11. Persson et al. study.
Effect of excluding observations with large WTP/household inc ome ratio on the income
elasticity of WTP. 2SLS estimation, dependent variable: log WTP for risk of dying in a
road traffic accident.
Exclusion criterion with respect to income •
None
Respondents with zero personal income but
positive WTP
WTP greater than 5% of household income
WTP greater than 10% of household,
income
WTP greater than 12.5% of household
income
WTP greater than 20% of household
income
WTP greater than 25% of household^
income
WTP greater than 50% of household
income
Number of
observations
514
514
501
509
510
511
512
514
Income
elasticity
ofWTP
0.1475
0.1475
0.2850
0.2264
0.1937
0.1668
0.1418
0.1475
Standard
error
0.1136
0.1136
0.1109
0.1139
0.1129
0.1126
0.1119
0.1136
Observations with missing baseline risk and missing WTP, observations with baseline risk smaller
than 1 in 100,000, observations with WTP less than 1. Other regressors in the WTP equation: log
degrisk, log riskmd, log miles traveled in a car, previously injured in a traffic accident (dummy),
log age, log age squared, two education dummies, dummies for household members.
Coefficients of log degrisk and tog riskmd are restricted to be equal.
VII. Discussion and Conclusions
This paper examines the issue of the robustness of the estimates of the VSL from
CV studies that elicit WTP for a reduction in the risk of dying. We illustrate the effects of
maintained assumptions and techniques using the data from four recent CV surveys.
When dichotomous choice questions are used, we emphasize the importance of
spreading bids nicely over a broad portion of the range of WTP (Cooper, 1993). We also
show that the estimate of mean WTP from dichotomous-choice CV data can be extremely
sensitive to the distributional assumption made by the researcher about the latent WTP
49
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variable, and to the procedure used for calculating the welfare statistics. Median WTP
tends to be less affected by these factors. We also find that claims about the shape of the
relationship between WTP and certain covariates of interest (e.g., age) may no longer be
valid when alternative distributional assumptions are made, alternative welfare statistics
are used, or alternative (e.g., semiparametric) procedures are employed.
Outliers can, in general, be defined as observations that are distant from the
remainder of the sample. How exactly "distant" is defined is, of course, a matter of
interpretation. In this paper, we examine outliers in dichotomous choice CV samples in
the sense of Copas (1988).
We also attempt to estimate discrete mixtures to identify yea-sayers, nay-sayers,
and persons who answer the payment questions in a completely random fashion—another
possible cause for outliers—but obtain implausible results that we attribute to the fact that
the distribution assumed for WTP and/or the mixture we choose to work with fits the data
poorly.
Next, we examine the internal validity of the WTP responses. Specifically, we
look at the scope effect and at the association between WTP and income. Carson (2000)
emphasizes that a CV study about mortality risk reductions must satisfy the "scope" test
for the quality of the survey and its data to be considered acceptable. Testing for scope
requires that the risk reduction be varied to the respondents by virtue of the experimental
design. WTP will pass a "weak" scope test if it increases systematically with the size of
the risk reduction. It will pass a "strong" scope test if it is proportional to the size of the
risk reduction.
50
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Testing for scope is straightforward when people are asked to value objective risk
reductions which are varied to them in the study.28 Testing for scope can be more
complex when people are subjectively assessing their own baseline risks and/or risk
reductions. This is because both of these variables may be affected by unobserved
individual factors, which results in their econometric endogeneity. The Persson et al.
survey is an example of one such study.
Using the Persson et al. data, we find that the sensitivity of WTP to the size of the
risk reduction is indeed more pronounced when risk reductions meant as a specified
proportion of a subjectively assessed baseline are treated as econometrically endogenous
with WTP. We also show that with CV surveys conceived in this fashion it is important
to test for whether people were truly valuing absolute risk reductions, or the mere
proportion, without applying the latter to their baseline risks. In the Persson et al. study,
people indeed appear to be responding to the percentage risk reduction, but not to the
baseline risk, which makes it problematic to compute the VSL.
Turning to the other internal validity issue, we tackled the question whether one
should omit observations with large WTP relative to income. We illustrate our checks
using two applications where the omissions of these variables from the sample has the
opposite effect on mean WTP and on the income elasticity of WTP, a key quantity when
one wishes to (i) extrapolate study results to the general population, (ii) focus on the
economically disadvantaged, and (iii) attempt benefit transfers to other countries or
locales where income levels are different.
28 Ideally, the scope tests should be external, i.e., performed using independent samples where respondents
are faced with risk reductions of different size.
51
-------
For the Alberini et al. application, further checks suggest that individuals with
high announced WTP relative to income may have misunderstood some key aspects of
the survey's risk reduction provision. This and the earlier checks imply that researchers
should explore and report variants of their models driven by different assumptions, and
attempt to recognize outliers and respondents who may have misunderstood some aspects
of the valuation scenario.
52
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Viscusi, W. Kip and Joseph E. Aldy (2003), "The Value of a Statistical Life: A Critical
Review of Market Estimates throughout the World," Journal of Risk and
Uncertainty, 27(1), 5-76.
U.S EPA Headquarters Library
Mai! code 3404T
1200 Pennsylvania Avenue NW
'A^lilfjI'Vl. DC ?f'<60
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