NCEE 0
NATIONAL CENTER FOR
ENVIRONMENTAL ECONOMICS
Valuation of Human Health: An Integrated Model of
Willingness to Pay for Mortality and Morbidity Risk
Reductions
Shelby Gerking, Mark Dickie, and Marcella Veronesi
Working Paper Series
Working Paper# 12-07
October, 2012
sta^ U.S. Environmental Protection Agency
-0 *
* A rc National Center for Environmental Economics
33
1200 Pennsylvania Avenue, NW (MC 1809)
W
V c<-° http://www.epa.gov/economics
"U proI*-
* Washington, DC 20460
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Valuation of Human Health: An Integrated Model
of Willingness to Pay for Mortality and Morbidity Risk
Reductions
Shelby Gerking, Mark Dickie, and Marcella Veronesi
NCEE Working Paper Series
Working Paper # 12-07
October, 2012
DISCLAIMER
The views expressed in this paper are those of the author(s) and do not necessarily represent those
of the U.S. Environmental Protection Agency. In addition, although the research described in this
paper may have been funded entirely or in part by the U.S. Environmental Protection Agency, it
has not been subjected to the Agency's required peer and policy review. No official Agency
endorsement should be inferred.
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Valuation of Human Health:
An Integrated Model of Willingness to Pay for
Mortality and Morbidity Risk Reductions
Shelby Gerkinga
Mark Dickieb
Marcella Veronesi0
January 24, 2012
Key words: willingness to pay, children, environmental hazards, health, integrated analysis,
morbidity, mortality, value of a statistical life, cancer, stated preference
Subject matter categories: 62, 63
JEL code: Q51
a Corresponding author, affiliations: Department of Economics, University of Central Florida, P.O. 1400,
Orlando, FL 32816-1400, United States and Department of Economics and Tilburg Sustainability Center,
Tilburg University, P.O. Box 90153, 5000 LE Tilburg, the Netherlands, email: 56f 100pu@gmail.com
phone: 407-349-9093 fax: 407-823-3669
b Department of Economics, University of Central Florida, P.O. 1400, Orlando, FL 32816-1400, United
States email: mdickie@bus.ucf.edu.
c ETH Zurich, Universitaetstrasse 22, 8092 Zurich, Switzerland email: marcella. veronesi@env.ethz.ch
*The US Environmental Protection Agency (USEPA) partially funded the research described here under
R-82871701-0 and RD-3159201-0. The research has not been subjected to USEPA review, may or may
not reflect the views of the Agency, and no official endorsement should be inferred. Gerking
acknowledges the hospitality of the Department of Spatial Economics, Free University of Amsterdam,
where the research was initiated. We thank Anna Alberini, Kathryn Anderson, Michael Caputo, Diane
DuPont, Reyer Gerlagh, and seminar participants at the 18th Annual Conference of the European
Association of Environmental and Resource Economists (EAERE) at University of Tor Vergata in Rome,
at the Tinbergen Institute and the Canadian Resource and Environmental Economics Study Group for
numerous helpful suggestions on previous versions of this paper.
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Valuation of Human Health:
An Integrated Model of Willingness to Pay for
Mortality and Morbidity Risk Reductions
ABSTRACT
This paper develops and applies an integrated model of human mortality and morbidity valuation
that is consistent with principles of welfare economics. The standard expected utility model of
one person facing two health states (alive and dead) is extended to a setting in which two family
members (a parent and a child) face three health states (healthy, sick, and dead). A key finding
is that total health benefits of public programs equate to the sum of willingness to pay for
reduced mortality risk plus a fraction of the willingness to pay for reduced morbidity risk.
Implications of the integrated model are tested using two field data sets from the U.S. on skin
cancer and leukemia risk reductions. Results obtained show how the integrated model can be
used to increase the accuracy of health benefit estimation for benefit-cost analyses as well as for
the design of public hazard reduction programs.
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Valuation of Human Health:
An Integrated Model of Willingness to Pay for
Mortality and Morbidity Risk Reductions
1. Introduction
Valuation of improvements in human health is a central feature in benefit-cost analyses
of public programs to reduce environmental hazards such as air pollution (e.g., U.S.
Environmental Protection Agency 2011). Benefit-cost studies generally divide human health
improvements into two categories, reduced mortality and reduced morbidity. Valuation of
reduced mortality is achieved using an estimate of the value of a statistical life (VSL) that is
assumed to be applicable across many types of mortality risk.1 Total health benefits then are
obtained by adding the value of reduced mortality to independently estimated, disease-specific
values of reduced morbidity. A recent white paper released by the U.S. Environmental
Protection Agency's National Center for Environmental Economics (2010, p. 8) concludes that
this procedure is deficient because mortality risks and morbidity risks are intertwined; thus
benefits of reduction in each would ideally be based on an integrated model. Developing such a
model, however, was considered to be beyond the scope of the white paper and near the frontier
of the empirical valuation literature.
This paper: (1) develops an integrated model of human mortality and morbidity valuation
that is consistent with principles of welfare economics and (2) demonstrates how this model can
be empirically implemented. In particular, the standard one-person/two-health state (alive and
dead) expected utility model used to support estimation of the VSL is extended in two directions.
1 Cameron (2010) discusses alternative ways to scale reductions in mortality risk to avoid misconceptions associated
with the term "value of a statistical life." The units in which mortality risk is measured, however, do not materially
affect the results presented here. In any case, this paper retains the more traditional terminology used in benefit-cost
studies of environmental hazards that affect human health.
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First, to develop a theoretical link between mortality and morbidity valuation, the model
proposed envisions three possible health states (healthy, sick, and dead). Inclusion of the
"sick" health state allows an illness to leave a person alive, but with a lower level of utility than
would be enjoyed in the healthy state. Second, to allow for the possibility that parents value
risks to their children's health differently than they value risks to their own health, the model is
"3
developed in a setting with two family members, a decision-making parent and a child. This
framework then is applied to obtain morbidity and mortality values both for parents and their
children using two data sets on cancer risk reduction in the U.S.
The theoretical model yields two main insights. The first turns on the possibility that the
risk of getting sick (S), referred to as "illness likelihood risk", and the conditional risk of death
given sickness (D), referred to as "conditional mortality risk", are perfect substitutes. As the
term is used in this paper, perfect substitution means indifference between alternative
combinations of S and D that leave unconditional mortality risk (r = S ¦ !)) unchanged. In the
special case in which illness likelihood risk and conditional mortality risk are perfect substitutes,
total health benefits are entirely captured by the willingness to pay to reduce unconditional
mortality risk obtained from the standard two-health state model (i.e, the monetized difference
between utility if alive and utility if dead).
In the more general case in which illness likelihood risk and conditional mortality risk are
not perfect substitutes, total health benefits are equal to the willingness to pay for reduced
2 In policy oriented studies, the value of reduced morbidity often is established from an ex post perspective using
medical costs and lost work time incurred by those suffering from particular diseases. Alberini and Krupnick (2002,
p. 234) point out that this approach is not consistent with principles of welfare economics. This paper, in contrast,
takes an ex ante view in which the value of reduced morbidity rests on the monetized difference between utility if
healthy and utility if sick.
3 Recent empirical studies (e.g., Dickie and Messman (2004); Hammitt and Haininger (2010)) have estimated that
parents are willing to pay more to reduce hazards to their children than they are to reduce the same hazards to
themselves.
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unconditional mortality risk (r) plus a fraction of the willingness to pay for reduced morbidity
risk(g), where morbidity risk is defined as the risk of getting sick given that death does not
occur.4 An important additional nuance in this calculation is that willingness to pay to reduce
unconditional mortality risk accounts for the risk of getting sick and willingness to pay to reduce
morbidity risk accounts for the risk of death. Thus, in the model developed, willingness to pay to
reduce these two types of risk differs from the corresponding values that would be obtained by
making independent estimates of each.
A second result from the model is that a parent's willingness to pay to reduce a given
health risk to herself will not generally equal her willingness to pay to reduce this same risk to
her child. She may be willing to pay a higher or lower amount to protect herself as compared to
the amount that she would pay to protect her child depending on the utility levels that she would
experience in various health states, together with the relative likelihood of these possible
occurrences. This result highlights the difficulties inherent in the practice sometimes followed
by policy-makers either in implicitly or explicitly transferring to children prior estimates of the
VSL obtained for adults.
The theoretical model also is used to: (1) motivate empirical estimation of parents'
willingness to pay to reduce both illness likelihood risk and conditional mortality risk, (2)
convert these estimates into values of willingness to pay to reduce unconditional mortality risk
and morbidity risk, and (3) test whether illness likelihood risk and conditional mortality risk are
perfect substitutes. In the extensive literature on the VSL, the implicit assumption of perfect
substitution has never been tested.
4 In line with recent empirical studies (Cameron, DeShazo, and Stifler 2010; Evans and Shaur 2010), the approach
taken here allows for heterogeneity in the willingness to pay for reduced mortality that depends on illness
characteristics.
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Empirical estimates are based on stated preference data obtained in two field studies of
skin cancer and leukemia risks to parents and their children. Use of stated preference data is
controversial partly because respondents are often thought to overstate their purchase intentions
when they do not actually have to pay. Econometric methods applied here, however, confine any
systematic tendency to misstate willingness to pay in a regression constant term that plays no
role in the analysis.
Estimates obtained suggest that for skin cancer, parents view illness likelihood risk and
conditional mortality risk as perfect substitutes both for themselves and for their children. This
outcome means that for this disease: (1) parents are willing to pay the same amount of money for
X% reductions in illness likelihood risk, conditional mortality risk, and unconditional mortality
risk and (2) parents' willingness to pay to reduce morbidity risk is equal to zero. Estimates
obtained for leukemia also, and perhaps surprisingly, suggest that parents view illness likelihood
risk and conditional mortality risk as perfect substitutes for their children, but for themselves,
they view utility if sick as the same as utility if dead. In this case, total willingness to pay to
reduce unconditional mortality risk and morbidity risk turns out to be a multiple of the
willingness to pay to reduce illness likelihood risk.
2. Model
A. Overview
The model extends the standard von Neumann-Morgenstern expected utility model to
consider a "family" consisting of one parent and one child. The family behaves as if maximizing
a single expected utility function subject to a single budget constraint and household choices can
be described as if made by a single individual (a parent). This perspective is adopted for two
reasons. First, the field data focus on health risk tradeoffs facing one parent and her child.
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Second, the model developed can be closely linked to the expected utility framework
traditionally used to support estimation of the VSL.5
The model envisions that both the parent (assumed to be the mother) and the child are
initially healthy and that each face three prospective health states, healthy (H), sick (S), and
dead (D). The parent's expected utility (EU) is the probability weighted sum of utility levels
that she would experience in each of nine possible health states. These probabilities and thus
expected utility can be expressed in two equivalent ways. Both formulations of the model are
described because each is useful in the analysis later on.
One way to write the model would be to let S denote the parent's perception of her own
illness likelihood risk (i.e., the probability that she will become sick), Sk denote the parent's
perception of her child's illness likelihood risk, Dp denote the parent's perception of her own
conditional mortality risk (i.e., the probability that the parent will die if she becomes sick), and
Dk denote the parent's perception of her child's conditional mortality risk. In this case, EU can
be expressed as
EU = ( I- ^)(1 - Sp)UHH + (1 - Sk)Sp (1 - Dp)UHS + (1 - Sk)SpDpUHD
+ Sk(l-Dk)(l-Sp)USH+Sk(l-Dk)Sp(l-Dp)Uss+Sk(l-Dk)SpDpUSD (1)
+ SkDk (1 - Sp )UDH + SkDkSp (1 - Dp )UDS + SkDkSpDpUDD
where, Ut refers to the parent's utility in health state // where i refers to the child's health state
(/ = H, S,D), and j refers to the parent's health state(j = H,S,D).6 For example, denotes
5 Alternative family decision making frameworks recently have been proposed that may also serve as a useful basis
in environmental benefits assessment (e.g., Blundell, Chiappori and Meghir 2005).
6 This framework has broad similarities to the model applied by Smith and Desvousges (1987) who split the
unconditional risk of death from exposure to hazardous wastes into the probability of exposure and the conditional
probability of premature death given exposure.
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the parent's utility if the child is healthy and the parent is sick. Equation (1) can be written more
compactly as
EU = (1 -Sk)EUHn + Sk(l-Dk)EUsn + DkSkEUDn. (2)
Here EUHa, EUsa, and I'AIiy respectively denote the parent's expected utility given that the
child is healthy, sick, or dead (EUm = (1 -Sp)UiH + Sp (1 - Dp)UiS + DpSpUiD, i = H,S,D).
Equation (2) focuses explicitly on risks faced by the child, but could be alternatively expressed to
n
focus on risks to the parent.
A second and equivalent way of expressing the model is to denote the unconditional
mortality risk to either the parent or child as r2 = S2 -Dz, z = p,k and define morbidity risk (i.e.,
the conditional risk of illness given that death does not occur) as q2 = (S2 -r2)/(\-r2). In this
case, and again focusing on risks to the child, the parent's expected utility is
EU = (1 - rk )(i - qk )EUh d + qk (1 - rk )EUsa + rkEUDQ (3)
where EUm = (1 -rp )(1 - qp )UlH +qp( 1 - rp )UlS + rpUlD ,i = H,S,D.*
Regardless of whether expected utility is expressed as equation (2) or equation (3), six
assumptions are adopted to simplify the analysis. First, Utj = I/,,(/); where / denotes the parent's
income when she and her child are healthy. Of course, the parent may not have the same amount
of income in other health states. If she is sick or her child is sick, for example, her capacity to
7 To focus on risks to the parent EU can be written as EU = (1 - Sp )EUQH +Sp(l- Dp )EUQS + DpSpEUQD where
EUaj = (1 -Sk )UHj +Sk( 1 - Dk )USj + DkSkUDj, j = H, S, D .
8 To focus on risks faced by the parent, EU can be written as EU - {\- rp){\- qp)EUUH + qp{\-rp)EUus + rpEUnD
Where EUDj. = (1 -rk )(1 - qk )UHj +qk( I- rk )U Sj + %U Dj ,j = H,S,D.
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earn income may be lower than if both are healthy. Nonetheless, income differences between
health states are subsumed into the various health state utility functions.
Second, the model assumes that decision making occurs in a one-period, rather than in a
multi-period, framework. This approach rules out the possibility that a person might be sick for
a period of time and then die later on. Because exposure to environmental hazards may result in
a long illness leading to death, extension to a multi-period framework may be a useful direction
in future research. Nonetheless, factors important in the one-period analysis presented here may
also emerge in a multi-period setting.
Third, the expected utility model abstracts from treatment of illness. For instance, if the
parent becomes sick, she either can remain sick or die. She cannot return to the healthy state or
take action to increase her utility if she becomes ill. Thus, utility in the sick health state in this
model is interpreted as net of any treatment received and can be thought of as a situation in
which the parent has lower income than in the healthy state or has a reduced capacity to enjoy a
given amount of income available in both health states.
Fourth, the possibility that the parent might purchase insurance to transfer income
between health states is not considered. Fifth, the expected utility that the parent derives from
bequests may depend on the health state of the child. Sixth, the parent's marginal expected
utility of income generally obeys EU'Ha > EU'sa > EU'm > 0 and EJJ'W > EU^ > EU'm > 0, where
the prime notation denotes the derivative of expected utility with respect to income. An
exception could occur in the case where an illness is considered a "fate worse than death" in
which case utility if dead exceeds utility if sick and it is possible that EU'm > EU'sa and
EU'm > . The condition EU'Ha > EU'Da > 0 reflects a common assumption in related
literature.
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B. Willingness to pay to reduce mortality and morbidity risks (r and qj
In this sub-section, the parent's willingness to pay to reduce unconditional mortality risk
rk and morbidity risk qk is treated explicitly for the case of the child. Her willingness to pay to
reduce the corresponding risks to herself takes a similar form and is therefore not treated in
detail. In computing the expressions for willingness to pay, the (small) reduction in rz(qz)
considered is just large enough to save the life of one person (prevent one person from becoming
sick given that death does not occur) in a large population of identical persons.
Using equation (3), the parent's ex ante willingness to pay to reduce unconditional
mortality risk to the child is
& (EUHa-EUDa)-qk(EUHa-EUsa)
drk (EU'Hn-rk(EU'Hn-EU'Dn)-qk(\-rk)(EU'Hn-EU'sn)
In equation (4), the numerator denotes the difference in the parent's expected utility when the
child is alive rather than dead, recognizing that the child will be healthy with probability (1 -qk)
and sick with probability qk . The denominator is interpreted as the marginal expected utility of
income (assumed to be positive) hereinafter denoted MEUj. The value of dl / drk differs from
that derived from the two-health state (alive and dead) model in which EUm and EUare
implicitly assumed to be equal. In equation (4), if qk> 0, departures of Elifrom Eli,, make
both the numerator and denominator smaller. Thus, the resulting value of dl / drk: (1) either may
be larger or smaller than in the two-health state case and (2) will vary with the illness considered
(i.e., with the magnitude of (EUHa -EUsa)).
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Equation (3) also can be used to find the willingness to pay for a reduction in morbidity
risk qk.
dl (1 -rk)(EUHa-EUsa)
8qk meu\
Thus, the parent's willingness to pay for a reduction in qk is the monetized difference in
expected utility between the state in which the child is healthy and the state in which the child is
sick, multiplied by the probability that death does not occur (1 - rk). Notice that dl / dqk accounts
for utility in the death state through the marginal expected utility of income (see equation (4)). If
the marginal expected utility of income if sick declines with declines in expected utility if sick,
then larger values of (EUHa -EUsa) indicate smaller values ofMEU7 and therefore larger
values for willingness to pay to reduce morbidity risk by a given amount. In any event, the value
of dl / dqk will vary according to the illness considered.
Benefit-cost studies of changes in environmental policy sometimes calculate a value for
total health benefits by adding an estimate of the value of a statistical life that is assumed to be
applicable to many types of unconditional mortality risks, to disease-specific values of reduced
morbidity risk. In the context of the model developed here (neglecting the scale factor for
population size), the correct calculation would be obtained by adding equation (4) to equation
(5).
dl | dl (EUHa-EUDa) + (l-rk-qkXEUHa-EUsa)
drk 8qk MEU\
Equation (6) shows that a parent's total willingness to pay for reductions in unconditional
mortality risk and in morbidity risk equates to the sum of willingness to pay to avoid mortality
((EUhq - EUDa) / MEUj) plus a fraction (1 -rk-qk) of the willingness to pay to avoid an illness
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given that it is not fatal (EUHn - EUsa) / MEUj 9 An advantage of the integrated model
described here is that both utility differences are monetized using a common value for the
marginal expected utility of income. In any case, failure to account for the fraction (1 -rk-qk)
when adding the illness avoidance value to the death avoidance value will lead to an
overestimate of total health benefits.
C. Willingness to pay to reduce children '.s mortality risk (rk)
To illustrate how a parent's willingness to pay to reduce her child's unconditional
mortality risk dl / 5rk might depend on the nature of the illness examined, consider two situations
that arise when the parent's expected utility given that the child is sick is: (1) less than her
expected utility when the child is healthy and (2) equal to her expected utility when the child is
healthy.
Illness reduces expected utility. Values of willingness to pay for reductions in rk,qk ,Sk,
and Dk can be compared for alternative values of (EUsa -EUm) and (EUHa -EUsa) using
equations (4) and (5) together with equation (2). It is convenient to begin by considering
willingness to pay for percentage reductions in Sk and Dk. Relationships developed turn out to
be useful in interpreting empirical results presented in Section 4.
Willingness to pay for percentage reductions in Dk and Sk are
dl rk(EUsn-EUDn)
ci)k / i)k Mia i,
9 Notice that it is possible that rk + qk could exceed unity, nonetheless if mortality and morbidity risks are "small"
then {\~rk -qk)>0.
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dl _Sk(EUHQ-EUsa) , dl
dSk/Sk MEUj dDk / Dk
+ (8)
If (EUsa-EUDa) ^ 0, indicating that sickness is preferred to death, then as shown in equations
(7) and (8), willingness to pay for an X% reduction in Sk will exceed willingness to pay for an
X% reduction in /), , In this situation, the parent has more to gain from a given percentage
reduction in one component of rk (illness likelihood risk) than a reduction in the other component
(conditional mortality risk) by that same percentage.
If instead (EUsa-EUDa) = 0, indicating that the illness is as bad as death, then
dl / dDk = 0 and using equations (4) and (5), dl / drk = (1 - qk)(dl / dSk) and
dl/ dqk = (1 -rk)(c)I/ dSk). Total health benefits in this case, obtained by summing
dl / drk +dl / dqk as shown in equation (6), can be expressed as a multiple (2-rk-qk) of
dI/dSk. If (EUsa-EUDa) < 0, then the illness is viewed as a fate worse than death. In this
situation, because the individual would prefer death to illness, dl / dDk < 0. Nonetheless,
dl/dqk> 0, dI/8Sk>0,md dl/drk> O.10
Finally, using equation (4) it is straightforward to show that d2I / crkcrk > 0. Thus,
willingness to pay to save the lives of two children cannot be greater than twice the willingness
to pay to save the life of just one.
Illness has no effect on expected utility. Next consider the special case in which the
parent's expected utility is the same whether the child is healthy or sick (EUHQ = EU,, ). In this
situation, perhaps the child's illness is so minor that the parent does not suffer a utility reduction,
10 These results assume that MEUf > o even if eudo > EUsa
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or perhaps the parent simply does not care about the child's health. As shown in equation (5),
the value of reduced morbidity risk is zero. Thus, total health benefits are fully captured by the
willingness to pay for a reduction in unconditional mortality risk. This willingness to pay value
is equal to
where the denominator of equation (9) is just MEUj when (EUHn = EUsn) at all income levels.
Additional calculations using equations (7) and (8) show that if (EUHa = EUsa) Sk and
Dk are perfect substitutes in the sense that the parent is willing to pay the same amount of money
for an X% reduction in Sk as for an X% reduction in I\ (i.e., the marginal rate of substitution
between equal percentage changes inSk and I\ equates to -1). Thus, in this case, the parent will
sacrifice the same amount of money for an equal percentage reduction in rk as in either ,VA or/), .
D. Marginal rate of substitution between children '.s and parents' mortality risk
Is the parent willing to pay more to protect her child from unconditional mortality risk
than she is to protect herself? The parent's willingness to pay to protect her child from
unconditional mortality risk is given in equation (4) and her willingness to pay to protect herself
is given below in equation (10).
Equations (4) and (10) can be more easily compared using the marginal rate of substitution
between rp and rk.
dl (EUHa-EUDa)
drk (1 -rkXEU'Ha) + rk(EU'Da)
(9)
dl (-EUm-EUm)-qp(EUm -EUW)
(10)
MEUj
drp (EUhd - EUdd) - qk (.EUHn - EUsn)
drk (EUm-EUaD)-qp(EUm-EUas)
(11)
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Equation (11) demonstrates that the parent's willingness to pay to reduce her child's
unconditional mortality risk dl / crk either may be larger or smaller than the parent's willingness
to pay to reduce unconditional mortality risk to herself dl / drp .11 Even in the perfect
substitution case in which (EUHQ = ) and (EUm = EAJ ..) the marginal rate of substitution
between rp and rt will equate to -1 only in the unlikely event that the parent's expected utility loss
from her child's death is equal to her expected utility loss from her own death
((EUHa - EUDa) = (EUm - El J D)). This outcome suggests that adult VSL values may be a
questionable guide to corresponding VSL values for children.
3. Data
Section 4 presents empirical estimates of parents' willingness to pay to reduce mortality
and morbidity risks to their children and to themselves (i.e., dl / cr and dl / dqt) for two diseases
that have been associated with exposure to environmental hazards, skin cancer and leukemia.
Two similarly structured instruments, one for each disease, were used to collect the data. The
skin cancer data set, collected in Hattiesburg MS in 2002, is fully described in Dickie and
Gerking (2007). In consequence, discussion in this section presents only a few details about the
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skin cancer data and focuses mainly on the leukemia data.
A. Overview of leukemia data
Data on leukemia risk perceptions were collected in a field study conducted in Orlando,
FL between December 2008 and February 2009. Parent respondents were drawn from a list of
more than 20,000 residents of the Orlando area maintained by the market research firm Insight
Orlando, Inc. Parents were contacted by dialing randomly selected telephone numbers from this
11 A similar result easily can be developed for the marginal rate of substitution between parent morbidity risk and
child morbidity risk.
12 Further details about the leukemia data set may be found in Dickie and Gerking (2009).
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list and were asked whether they had children living at home between the ages of 1 and 16 years.
If so, they were asked to participate in a federally-funded, computer-assisted survey dealing with
health risks to parents and their children. Prospective participants were told that they would
receive $40 for completing the survey instrument. Prior to administering the instrument, a
preliminary version was tested in two focus groups consisting of a total of 24 parents in May
2008. The instrument was revised in light of focus group input and pretested with 68 subjects in
early December 2008.
13
The sample analyzed below consisted of 815 parents. Questions focused on the parent
and one child aged 1-16 years. Among the parents, 68.5% were white, 14.2% were African-
American, 15% were Hispanic, and 21% were under the age of 40. Most of the parents were
female (77.9%), employed full-time (56%), and mean household income was $76,000. Most
parents indicated that they were aware of leukemia; 90% said that they had heard of the disease,
43%) knew someone personally who had had it, 25% had thought about the possibility that they
themselves might get it, and 28% had thought about the possibility that one of their children
might get it. For the 68% of parents with two or more children living at home, one child was
randomly selected and designated as the sample child. Roughly half (52.8%) of the sample
children were male and the average age of sample children was 10 years.
B. Leukemia questionnaire design and data description
The leukemia questionnaire was divided into four segments that elicited: (1) initial
perceptions of the likelihood of getting leukemia, (2) revised perceptions of the likelihood of
getting leukemia, (3) perceptions of conditional mortality risk from leukemia defined as the
13 A total of 839 parents participated in the survey. Responses from 24 parents were disregarded: 3 parents did not
answer the question about the number of children in their family, 10 parents were ineligible because they responded
that no children lived with them, and 11 parents failed to answer key questions about their perceptions of leukemia
risks.
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conditional probability of dying within five years given a diagnosis of this disease, and (4)
willingness to pay to reduce leukemia likelihood risk and conditional mortality risk.
Initial perceptions of leukemia likelihood risk (Sk and S ). Parents made an initial
estimate of the lifetime risk of contracting this disease (illness likelihood risk) using an
interactive scale similar to that used by Krupnick el al. (2002) and Corso, Hammitt, and Graham
(2001). The scale depicted 1000 squares in 25 rows and 40 columns. Squares were numbered
from 1 to 1000 and all initially were colored blue. Parents re-colored squares from blue to red to
represent amounts of risk. For example, a parent could use the computer mouse to indicate a risk
level of 200 in 1000 by selecting the square numbered 200 in the scale, causing all the squares
from 1 to 200 to turn red. Beneath the scale, the level of risk was indicated by displaying both
the number and the percentage of the 1000 squares that were colored red. Parents could make as
many changes to the scale as desired before pressing the "Continue" button to record the final
answer.
Parents practiced using the risk scale before making estimates of the risk of getting
leukemia. First, they were shown four examples of scales representing risk levels of 25%, 50%,
75%) and 100% and were told the relationship between these percentages and "chances in 1000."
Second, parents were asked to identify which of two hypothetical people had the smallest chance
of getting into an auto accident; Ms. B, a relatively safe driver who had a 1% chance of an
accident, or Mr. A, a relatively careless driver who had a 33.3% chance of an accident. After
viewing risk scales depicting these two individuals' accident risks, 11% of respondents wrongly
answered that Mr. A had the lowest risk. All of these respondents got the correct answer (Ms. B)
when given a second chance at the question. After completing the practice problems, parents
used the risk scale to estimate lifetime chances of getting leukemia for themselves and then for
16
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their sample child. These estimates are interpreted as subjective assessments of illness likelihood
risk.
The first three rows of Table 1 characterize the frequency distributions of these leukemia
likelihood risk estimates. Three features of parents' initial risk perceptions are worth noting.
First, there is considerable variation in risk estimates. Second, parents on average estimated that
their own lifetime risk of getting leukemia was about the same as that for their sample child. The
null hypothesis that mean perceived leukemia risks are equal for parents and children is not
rejected at the 1% level in a matched-samples test. This outcome is broadly consistent with data
on actual leukemia risks faced by adults and children. According to the National Cancer Institute
(Ries el al. 2006), lifetime risk of contracting leukemia falls by less than 1 in 1000 between ages
10 and 50 years. Third, parents appear to have overestimated leukemia risk both for themselves
and for their children. On average, parents estimated their own risk of getting leukemia at about
9.6% and estimated their children's chances at about 9.7%. These estimates are about 6-7 times
higher than the actual risk of approximately 1.3% that can be estimated from National Cancer
Institute data (Ries et al. 2006). Median risk estimates are 3-4 times higher than actual risk.
Overestimation of relatively small risks is a well-known phenomenon (see, for example,
Lichtenstein et al. 1978).
Revised perceptions of leukemia likelihood risk (Sk and S ). Parents were given an
opportunity to revise their estimates of the chances of getting leukemia after considering
information about this disease. After making their initial risk estimates, they were presented
with the National Cancer Institute estimate of 1.3% and told that an individual's risk may differ
from this average because of many factors including cigarette smoking, exposure to pesticides,
exposure to benzene as might occur if the parent lived in an area with high automobile traffic, as
17
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well as genetic factors. After answering questions about their exposure to these risk factors,
parents were shown their initial leukemia risk estimates (both for themselves and for their sample
child) as previously marked on the risk scales and were given a chance to revise their answers.
As shown in Table 1, about 57% of parents revised their own and their children's lifetime
risk estimates. Downward revisions predominated. Revised leukemia likelihood risk estimates
for parents averaged about 5.6% and revised perceived risk estimates for children averaged about
5.0%. Even though the downward revisions are substantial, mean perceived risk still
overestimates actual risk by a factor of about four. Median revised risk estimates are equal to
actual risk (1.3%).
Perception of conditional mortality risk from leukemia (Dk and Dp). After revising
initial risk estimates, parents were told to imagine that they had received a diagnosis of leukemia
from a doctor and were asked to estimate the chances in 1000 of dying within five years of the
diagnosis. Parents were unaware that they would be asked about the chance of dying from
leukemia when they answered the previously described questions about the risk of getting this
disease. Estimates of conditional mortality risk were obtained for each parent and her child
using the previously described risk scale.
As shown in Table 1, the average parent believed that the conditional risk of death from
leukemia is about 29.9% for herself and about 25.8% for her child. The difference in mean
conditional death risks between parents and children is significant at the 1% level in a matched-
samples test. Over three-quarters of parents saw the conditional risk of dying from leukemia as
higher than the risk of getting leukemia. The null hypothesis that mean subjective assessments
of risk of illness and conditional mortality from illness are equal is rejected at 1% for both
18
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parents and children, whether risk of illness is measured by initial or by revised assessments.
Thus, parents appear to have distinguished between the two risks.
Although these risk estimates suggest that parents were aware that leukemia can be fatal,
parents appear to have overestimated the chance of dying conditional on a diagnosis of leukemia
for their children and underestimated this risk for themselves. As reported in Ries el al. (2006),
the five-year survival probability for leukemia falls with age, from 85% for children younger
than 5 years to 49% for those between ages 15 and 19 years. The overall five-year survival
probability for all adults is 49% and falls with age beyond age 45 years.
Willingness to pay for reduced illness likelihood risk and conditional mortality risk.
Parents valued leukemia risk reductions by expressing purchase intentions for each of
two hypothetical vaccines. One of the vaccines permanently reduced the risk of getting leukemia
for both parent and child, while the other permanently reduced conditional mortality risk from
leukemia for both persons. Each type of vaccine was described as similar to newly developed
vaccines against cervical cancer. As the vaccine was described, its effectiveness was varied
randomly across parents and children. In all, there were eight descriptions of the effectiveness of
the vaccine. Four vaccine types reduced risk of getting leukemia (by 10% for both parent and
child, by 90% for both persons, or by 10% for one and 90% for the other) from the revised
assessments of likelihood risk for the parent and the child. The other four vaccine types reduced
conditional mortality risk from leukemia by 10% or 90% from the previously assessed
conditional mortality risk for the parent and the child (see Table 2).
Each parent was randomly assigned two of the eight vaccine types. One of the assigned
vaccines offered reduced risk of getting leukemia and the other offered reduced conditional
mortality risk from leukemia. The two vaccines were presented one at a time in randomized
19
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order. The parent was asked to read the description of the vaccine and then was shown the
previously marked risk scales for herself and for her child, which now indicated the risk
reduction that the vaccine would offer and the amount of risk remaining if it was purchased.
For the first of the two vaccines, the parent was asked, "Now please think about whether
you would buy the new vaccine for yourself and your child. Please do not consider buying it for
anyone else. Suppose that buying the vaccine would cost $X. Of course, if you did buy it, you
would have less money for all of the other things that your family needs. Would you be willing
to pay $X to vaccinate you and your child?" The cost ($X) was randomly selected from among
five values ($150, $300, $600, $1200, $2400).14 To introduce the second type of vaccine, the
parent was told, "Suppose that instead of the previous vaccine, we showed you the following
one." Information then was presented and purchase intentions were elicited as before.
Responses to leukemia willingness to pay questions are summarized in Table 2. The
right-most column of Table 2 shows how the proportion of parents expressing a willingness to
purchase the vaccine varies with the risk change, after aggregating over all costs. Considering
all risk changes and costs, 49% of parents said that they would purchase a vaccine offering a
reduction in the risk of getting leukemia and 67% of parents said that they would purchase a
vaccine offering a reduction in conditional mortality risk of this disease. Parents were more
likely to say they would purchase the vaccine to reduce risk of getting leukemia both for
themselves and for their child when it offered larger as compared to smaller risk reductions.
These differences in proportions achieve statistical significance at the 5% level using a two-tail
14 In the focus groups mentioned previously, particular attention was devoted to selecting prices at which some
respondents would agree to purchase the vaccine while others would decline. The idea here was to avoid a situation
where all or nearly all respondents presented with the lowest price would offer to purchase the vaccine, and/or all or
nearly all respondents presented with the highest price would refuse to purchase it.
20
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independent samples test.15 Parents also were more willing to purchase the vaccine at lower
costs than at higher costs.
For the vaccine to reduce conditional mortality risk, increases in risk protection for the
child significantly increased the propensity to purchase at 5% using the same type of differences
in proportions test as before. Increases in risk protection for the parent, however, lowered the
purchase propensity, but these differences in proportions are too small to achieve statistical
significance at conventional levels. This outcome appears to be driven mainly by respondents'
reluctance to buy at the two highest cost values (either $1200 or $2400). The bottom row of
Table 2 shows how the proportion of parents who would buy a vaccine varies with the cost, after
aggregating over all four magnitudes of risk change. These rows show a general tendency for
the propensity to purchase to decline as cost increases.
A parallel design was used in the skin cancer survey. Each parent expressed purchase
intentions for two hypothetical sun lotions. Whereas a single vaccination permanently reduced
leukemia risks, parents were told that sun lotions would have to be applied to exposed areas
when outdoors over a lifetime in order to be effective. One sun lotion reduced the risk of getting
skin cancer for the parent and child, and the other reduced conditional mortality risk of skin
cancer for the parent and child. The sun lotions offered either 10% or 50% risk reductions at
annual costs ranging from $20 to $125.
4. Econometric methods and results
Estimates of parents' willingness to pay to reduce illness likelihood risk, conditional
mortality risk, unconditional mortality risk, and morbidity risk are presented in this section
together with tests for perfect substitution between illness likelihood risk and conditional
15 At given prices, differences in proportions measuring increased risk protection for both parents and children are
too small to be statistically significant at conventional levels.
21
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mortality risk. Econometric methods are developed in Subsection A and estimates based on the
skin cancer and leukemia data are presented in Subsections B and C.
A. Development of econometric methods
Recall that in the field studies: (1) the parent uses the same good (sun lotion in the case of
skin cancer or a vaccine in the case of leukemia) to reduce illness likelihood risk for herself (S )
and for her child (Sk) and the same good to reduce conditional mortality risk for herself (l)p) and
for her child (Dk) and (2) reductions in the two types of risk were presented to respondents in the
form of percentage changes. Additionally, assume that the risk reducing goods are valuable to
the parent only insofar as they reduce her own risk and the risk faced by her child.
Consequences of possible violations of this assumption are considered momentarily.
In this case, ex ante willingness to pay for the good to reduce illness likelihood risk
(WTPS) equals the change in income that holds expected utility constant in the face of
simultaneous proportionate changes in Sk and S . Using the model of Section 2, this amount
equals
WTPS =¦
dl
{dsp^
dl
f dSk]
dsp/sp
u,
' ssk/sk
U J
(12)
Similarly, willingness to pay for the good to reduce conditional mortality risk is
WTP = ¦
dl
dl
dD>)
dDp/Dp
lD,J
8DJD,
V Dk ,
(13)
Equations (12) and (13) are re-expressed in more compact notation as
t = S,D.
WTP1 = / a; +
k5
(14)
22
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Equation (14) pertains to the Mi sample parent although the observation index has been
suppressed, WTP* denotes this parent's willingness to pay for the good (sun lotion in the case of
skin cancer or a vaccine in case of leukemia) that reduces risk t (= S,D), and A*p and Aj, are
variables representing the randomly assigned proportionate reductions in the fth component of
risk for the parent and child, respectively. Also, y'k and y'p denote parameters to be estimated
that are interpreted as the parent's willingness to pay to eliminate risk t (i.e., for a 100% risk
reduction) to the child and to herself. Thus, equation (14) treats willingness to pay to reduce a
given risk as proportionate to the size of the risk reduction (Hammitt and Graham 1999).
Stated willingness to pay (W*) is allowed to differ from true willingness to pay {WTP*).
A discrepancy between stated and true willingness to pay may arise, for example, because the
choice of whether to buy the risk reducing good was presented as a hypothetical question and/or
this question may not have been adequately considered in light of preferences and financial
constraints. The difference W* - WTP* is modeled as sum of systematic (a') and random (v')
factors
W* -WTP' = a* + v', V ~ #(0,(C*,
where C* denotes the randomly assigned cost of the risk reducing good, or if
(a' / ct) + (ytp / 0 (17)
23
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Thus, equations are estimated for whether the risk reducing goods are purchased with
covariates for the randomly assigned percentage risk changes and costs. A supplementary
covariate also was included to indicate the order in which the good was presented. Estimation is
by bivariate probit to allow for the possibility that E(vsv°) ^ 0 .
An advantage of this estimation procedure is that the randomly assigned risk reductions
and costs are independent of each other and also independent of parent and child characteristics
that are included in the error term. For instance, parent and child demographic characteristics
and the parent's discount rate vary across observations and may influence the decision to
purchase risk reducing goods however, these variables are distributed independently of the
randomly assigned risk reductions and costs. This means that if the functional form of equation
(16) is correct: (1) endogeneity problems in estimating y'k and y'p are avoided, (2) estimates of
parameters are unaffected by whether observed parent characteristics are included as additional
covariates, and (3) any systematic tendency for respondents to misstate their purchase intentions
affects only the constant term which plays no role in testing the hypotheses of interest.16 The
coefficient of cost is an estimate of -1 / a' that can be used to recover un-normalized coefficients
17
of risk reductions from the normalized estimates (see Cameron and James 1987).
16 This specification does not remedy all potential problems with hypothetical valuation questions. For example,
Carson, Groves and Machina (2007) examine incentive compatibility of binary discrete-choice valuation questions
for new private goods. In their analysis, if subjects believe that survey responses will influence whether the new
good will be provided, then overstatement of purchase intentions and insensitivity to risk changes and costs would
result. On the other hand, if subjects believe that the new good will be provided but that survey responses will
influence the price of the good, then excess sensitivity to costs would result. The specification in equation (15) does
not address these problems but instead treats misstatement of willingness to pay as an individual-specific error that
is unrelated to risk changes or costs.
17 If the risk reducing goods are direct sources of utility, a possibility suggested earlier, the correct price to use in
equation (17) would be the randomly assigned price of the risk reducing goods net of monetized utility/disutility.
Suppressing subscripts and superscripts, suppose that the price of the risk reducing good is C* = C + u, where C
denotes the value of price used in the survey and u denotes monetized utility/disutility. Then substitute C + u into
the equation (17) to replace the true value C*. The term -(11 a )u then can be treated as an additional component
of the error already present in the equation. This term will affect the estimate of the constant term if it has a non-
24
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B. Results
Tables 3 and 4 present maximum likelihood bivariate probit estimates of equations for
whether parents stated that they would buy the goods to reduce illness likelihood risk and
conditional mortality risk for skin cancer and leukemia together with sample means of all
covariates. Binomial probit coefficients (not presented) were used as starting values. The tables
report estimates of normalized coefficients. Estimates of un-normalized coefficients are
discussed momentarily. Table 3 shows that in the skin cancer estimates, the error correlation
between the illness likelihood risk and conditional mortality risk equations is 0.767. The
corresponding error correlation in the leukemia estimates is 0.608 (see Table 4). Both estimates
significantly differ from zero at 1% indicating that bivariate probit offers an efficiency gain over
binomial probit applied separately to the two equations.
For skin cancer, estimates of normalized coefficients of both parent and child risk change
variables are positive and differ significantly from zero at 1% in both equations. The coefficients
of cost are negative and are significantly different from zero at 1%. Order of presentation of
18
risk reducing goods does not matter at the 1% level.
For leukemia, estimates of normalized coefficients of both parent and child risk change
variables are positive and differ significantly from zero at 1% in the likelihood risk equation. In
the conditional mortality risk equation, the normalized coefficient estimate of child risk
reduction is positive and differs from zero at 1%; however, the normalized coefficient estimate
of parent risk change is negative and does not differ from zero at conventional significance
levels. Order of presentation of the risk reducing goods again does not matter at the 1% level.
zero mean, but will not affect the point estimates of the coefficients of price and parent/child risk reduction because
these variables were randomly assigned.
18 Whether presentation order of valuation tasks influences respondents' intended purchase decisions has long been a
concern in repeat-response stated preference studies (see Day et al. (2012) for further discussion).
25
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Further perspective on these estimates can be obtained by testing whether illness
likelihood risk and conditional mortality risk are perfect substitutes. Wald tests for perfect
substitution, reported in Table 5, are based on the null hypotheses: (1 )H0: Yk-Yk =0and (2)
Ha: ysp ~/p = 0. These tests make use of the un-normalized coefficient estimates of the risk
change variables; thus, they involve comparisons of parents' willingness to pay for 100% risk
reductions. Point estimates of the un-normalized coefficients of the risk change variables are
computed by multiplying the normalized coefficients reported in Tables 3 and 4 by estimates of
-------
For leukemia, the hypothesis H0\ fl ~Y^ = 0 is not rejected at conventional significance
levels, but p-values indicate that the hypothesis Ha: - y't' = 0 would be rejected at levels of
significance greater than or equal to 1%. A Wald test jointly rejects the two null hypotheses at
significance levels above 2.3%. Also, supplementary tests (not shown in Table 5) for
Ho: ysp = 0 and Ho: yDp = 0 indicate that the first of these hypotheses is rejected at conventional
significance levels, while the second hypothesis is not rejected at conventional significance
20
levels (p=0.333). This outcome suggests that parents' willingness to pay to reduce their own
conditional mortality risk from leukemia is equal to zero.
Test results for leukemia are more difficult to interpret than those for skin cancer. The
implication of perfect substitution between illness likelihood risk and conditional mortality risk
for children is unexpected. Leukemia symptoms are debilitating and the suffering of a child
would be expected to result in a utility loss for parents. Also, while the five-year survival rate
for childhood leukemia is 85%, remission is the goal in treating this disease, rather than a once-
and-for-all cure (American Cancer Society 2010). Once a child is in remission, parents still may
be concerned about a possible relapse or a future diagnosis of another type of cancer. Among
adult leukemia patients, activity limitations, and treatment goals are similar to those for children,
although five-year survival rates are considerably lower and decline with age at onset. In any
case, a finding of a zero willingness to pay to reduce conditional mortality risk is plausible if
adults view the leukemia illness state as a fate equivalent to death.
An additional question of interest centers on whether parents are willing to pay the same
amount to eliminate a given risk for their children as for themselves. Table 5 also shows results
20 In the skin cancer estimates, both of these hypotheses are rejected at 1%. Parallel hypotheses for children are
rejected at 1% based on both skin cancer and leukemia estimates.
27
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of Wald tests of the null hypotheses: (3) Ho: ysk -ysp = Oand (4) Ho:y'k' - y'J = 0, In the skin
cancer case, p-values indicate that both hypotheses are not rejected individually and jointly at
conventional levels of significance. Using the leukemia data set, the hypothesis for illness
likelihood risk is not rejected (p-value = 0.911), but the hypothesis for conditional mortality risk
is rejected at significance levels above 1.7%. Additionally, the two hypotheses are jointly
rejected at significance levels above 1.6%. In summary, parents appear to be willing to pay an
equal amount for an equal percentage reduction in a given risk, whether the risk affects the
parent or the child, in 3 of 4 cases considered.
C. Willingness to pay estimates to reduce mortality and morbidity risks
Estimates of willingness to pay to reduce unconditional mortality risk to parents and
children from skin cancer and leukemia can be found by re-estimating the equations reported in
Tables 3 and 4 with cross-equation constraints to impose perfect substitution in the three
situations where this null hypothesis was not rejected. These constraints (yk = yk and ysp = yDp
for skin cancer and yk = yk for leukemia) are motivated by discussion in Section 2 which
indicates that perfect substitution leads to common values of willingness to pay to eliminate
21
illness likelihood risk, conditional mortality risk, and (therefore) unconditional mortality risk.
Willingness to pay values obtained from the constrained estimates are presented in Table
6. These estimates show that parents are willing to pay $70.06 annually for a 100% reduction in
unconditional mortality risk from skin cancer for themselves and are willing to pay $105.44
annually for a 100% reduction in unconditional mortality risk from skin cancer for their children.
21 Likelihood ratio tests were used to compare restricted estimates to unrestricted estimates presented in Tables 4 and
5. Using the skin cancer data, the two restrictions imposed are not rejected, p=0.914. Using the leukemia data, the
one restriction is not rejected. /?=().491.
28
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For leukemia, constrained estimates suggest that parents are willing to pay $811.29 to
eliminate their children's unconditional mortality risk. This estimate exceeds the corresponding
estimates for skin cancer possibly because in the leukemia study, parents were asked for an
immediate, one-time payment for vaccines that would be effective for life; whereas in the skin
cancer study, parents were asked how much they would pay in the first year for sun lotions when
a stream of annual purchases would be required to maintain effectiveness. Table 6 also indicates
that parents are willing to pay $760.81 to eliminate leukemia likelihood risk for themselves and
-$295.02 (estimate not significantly different from zero at conventional levels) to eliminate
conditional mortality risk for this disease for themselves.
Table 7 reports estimates of parents' willingness to pay to reduce illness likelihood risk,
conditional mortality risk, unconditional mortality risk, and morbidity risk by an absolute amount
(1 in 10,000), rather than a by proportional amount. Estimates presented make use of the
constrained estimates reported in Table 6 together with sample means of the risk variables (see
Table 7, footnotes a and b). Estimates of mean unconditional mortality risk for both diseases and
for parents and children were computed by multiplying revised illness likelihood risk by
conditional mortality risk for each parent and then averaging across all sample members.
An "average" parent's willingness to pay to reduce her children's and her own
unconditional mortality risk of skin cancer by 1 in 10,000 are $0.48 and $0.21, respectively.
These point estimates suggest that parents are willing to pay more than twice as much to reduce
unconditional mortality risk for their children than they are willing to pay to reduce their own
22
risk. In light of the finding of perfect substitution between illness likelihood risk and
22 Results in Table 6 and 7 may overstate the child health premium for skin cancer. As discussed previously, the
hypotheses that parents are willing to pay an equal amount to eliminate a given risk, whether the risk affects the
parent or the child, were not rejected for illness likelihood risk and for conditional mortality risk. If equality of
willingness to pay to eliminate risk is imposed along with perfect substitution when re-estimating equations reported
29
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conditional mortality risk, the value of reduced morbidity from skin cancer is zero. The
"average" parent's willingness to pay to reduce unconditional mortality risk from leukemia by 1
in 10,000 to their child is $4.28. In light of perfect substitution, the value of reduced morbidity
risk for children implied by the estimates is equal to zero.
Parents' willingness to pay to reduce their own unconditional mortality risk and
morbidity risk by 1 in 10,000 can be obtained from the corresponding estimate of willingness to
pay to reduce leukemia likelihood risk shown in Table 7. If (EUsa-EUDa) = 0 (recall that
Ho'.yDp = 0 was not rejected), then dl / drk = (1 -qk)(dl / dSk) and dl / dqk = (1 -rk)(dl / dSk) (see
Section 2). Willingness to pay to reduce the "average" parent's own unconditional mortality risk
of leukemia by 1 in 10,000 can be computed therefore by multiplying the value of willingness to
pay to reduce leukemia likelihood risk by 1 in 10,000 ($1.37) by an estimate of
(1 -qp) = [1 -((Sp -rp)/(l-rp))]=[1.0- ((0.06 - 0.02)/(l.0 - 0.02)) ]= 0.96. 23 This calculation
yields $1.32. Correspondingly, the "average" parent's willingness to pay to reduce morbidity
risk by 1 in 10,000 can be found by multiplying $1.37 by (1.0-0.02) to obtain $1.34. Thus, total
health benefits sum to $2.66, which is about 62% of the corresponding value for children.
Although the adjustment factors of (1 - q ) and (1 - rp) applied to the estimate of dl / dSk are
relatively close to unity in this illustrative calculation for leukemia, they may well be smaller and
therefore more important to consider for a different illness with larger values of r and q .
VSL estimates can be obtained by multiplying the estimates of willingness to pay to
reduce unconditional mortality risk shown in Table 7 by 10,000. The skin cancer estimates,
in Table 3, results indicate that parents are willing to pay $87 to eliminate illness likelihood risk or conditional
mortality risk to either the parent or the child. The implied values of 1/10,000 reductions in unconditional mortality
risk are $0.26 for the parent and $0.40 for the child. The 50% child health premium is similar to that obtained by
Hammitt and Haininger (2010).
23 The mean value of S reported in Table 1 is 0.06 and the mean value of rp reported in Table 7 is 0.02.
30
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which pertain to one-year risk reductions, suggest VSL values of $4,800 for children and $2,100
for adults. The corresponding VSL values for leukemia, which pertain to permanent risk
reductions, are $42,800 for children and $13,100 for adults.
Two factors may help to explain why these VSL values are much lower than those
generally obtained in labor market studies (Viscusi and Aldy 2003). First, labor market
estimates of VSL are interpreted as the marginal value of saving one life; whereas estimates
presented here are interpreted as the average willingness to pay to save one life in the case where
24
unconditional mortality risk is eliminated. Marginal willingness to pay for a unit of risk
reduction would be expected to decline as successive increments in risk reduction are considered
(see Section 2), thus the VSL obtained here is expected to be lower than the estimates of VSL
obtained in other studies. Second, parents tended to overestimate illness likelihood risk and
conditional mortality risk in both the skin cancer and leukemia data sets. A given value for
willingness to pay to eliminate reduced unconditional mortality risk may therefore underestimate
willingness to pay to reduce a unit of risk.
A possible concern about the results just presented is that analyzing the skin cancer and
leukemia samples as a whole may mask differences in willingness to pay among various
subsamples. This concern is motivated by three observations. First, the coefficients in
equation (14) in general depend on individual parents' incomes and the risks faced by parents
and children (see equations (7) and (8)). This motivates testing for differences in parents'
willingness-to-pay by income and by the levels of risk faced. Second, findings in previous
studies of parental health investments in children suggest testing for differences by child gender
24
For instance, the mean of unconditional mortality risk for children's leukemia (see Table 7, footnote b) suggests
that eliminating this risk would save the lives of 189 children in 10,000. An interpretation of estimates presented
here would be that willingness to pay to save all of these lives would be $8,089,200, so that average willingness to
pay per life saved is $42,800.
31
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and age. In developing countries Haddad et al. (1996) and Liu el al. (2000) find evidence that
parents may favor sons over daughters and studies in developed countries, particularly the United
States and Europe, have investigated whether child gender prompts parents to make different
intra-family resource allocation decisions (e.g., Behrman, Pollack, and Taubman 1986; Lundberg
and Rose 2004; Alberini and Scasny 2011). Recent evidence also suggests that when making
health investments, younger children are favored over older children (Dickie and Messman 2004;
Blomquist, Dickie, and O'Conor 2010). Third, few studies have examined whether mothers and
fathers make different health investment decisions for their children, but this possibility still is
worth investigating.
Additional bivariate probit regressions (available from the authors on request) were
estimated to determine the extent to which differences in parents' willingness to pay to reduce
illness likelihood risk and conditional mortality risk were associated with selected demographic
factors. For instance, to test for differences in willingness to pay by child gender, each sample
was stratified by girls/boys. The regressions reported in Tables 3 and 4 were estimated
separately for each of the subsamples. A restricted model, constructed by augmenting the Table
3 and 4 regressions with a dummy variable indicating child gender was estimated for the full
sample. A test of whether child gender "matters" then was carried out by testing whether slope
coefficients of willingness to pay functions are equal between the two groups, allowing different
intercepts, using a likelihood-ratio test at the 5% significance level. A similar approach was used
to test for differences in willingness-to-pay functions by whether parent income was above or
below its sample median, age of the child was above or below its median, and by parent gender.
Tests performed revealed no effect of the demographic factors considered on parents'
32
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willingness to pay to eliminate illness likelihood risk or conditional mortality risk either for skinc
cancer or leukemia.
5. Summary and conclusions
This paper develops and applies an integrated model of human mortality and morbidity
valuation consistent with principles of welfare economics. The model extends the standard one-
person/two health-state VSL model in two directions. First, it allows for a third health state
(sick) with a utility level that is intermediate to utility in the healthy state and utility in the death
state. Second, it adopts a family instead of one-person perspective by envisioning a parent who
makes choices about risk exposure both for herself and for her child.
Three key findings are obtained from the model. (1) Total health benefits equate to the
sum of willingness to pay for reduced unconditional mortality risk plus a fraction of the
willingness to pay for reduced morbidity risk. (2) Willingness to pay to reduce unconditional
mortality risk accounts for the risk of getting sick and willingness to pay to reduce morbidity risk
accounts for the risk of death. Thus, willingness to pay to reduce these two types of risk differs
from the corresponding values that would be obtained by making independent estimates of each.
(3) A parent's willingness to pay to reduce a given health risk to herself may be larger or smaller
than her willingness to pay to reduce this same risk to her child. This outcome suggests that
adult VSL values may be a questionable guide to corresponding VSL values for children.
The model is applied to demonstrate how to calculate estimates of willingness to pay to
reduce mortality and morbidity risks using two similarly structured field studies dealing with
skin cancer and leukemia in the U.S. The empirical analysis rests on an examination of stated
preference values for hypothetical goods used by both parents and children. A novel
econometric approach is used to control for effects of possible misstated purchase intentions for
33
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hypothetical goods so that they do not contaminate the willingness to pay estimates. Evidence
for both diseases suggest that parents are willing to pay more to reduce unconditional mortality
risk for their children than for themselves. Estimates of willingness to pay to reduce
unconditional mortality risk also are quite low in comparison to related estimates from the labor
market. In this study, however, VSL estimates are obtained from parents' willingness to pay to
eliminate unconditional mortality risk of an illness; whereas in most prior studies VSL estimates
are based on small reductions in unconditional mortality risk from a job-related accident.
A future extension might adapt the theoretical and econometric framework developed in
this study to investigate whether alternative sources of mortality risk (e.g., an auto accident and
cancer) are perfect substitutes. Such an extension would be of possible interest for three reasons.
First, if all sources of risk are perfect substitutes, then a value of mortality risk reduction
obtained in one setting (e.g., the labor market) would be fully transferrable to other settings such
as those arising from environmental health hazards. Second, whether people are willing to pay
more to reduce one cause of death than another has attracted attention in recent empirical papers
(e.g., Hammitt and Haininger 2010; Alberini and Scasny 2011). Third, government policy-
makers have considered whether a "cancer-premium" should be applied in computing monetized
benefits of public programs to reduce mortality risk (U.S. Envrionmental Protection Agency,
National Center for Environmental Economics 2010; Postle et al. 2003 on behalf of the
Directorate General-Environment of the European Commission; U.K Department of Food, Rural,
and Environmental Affairs 2003).
34
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Table 1. Parents' Perceived Leukemia Likelihood Risk and Conditional Mortality Risk.
Chances per 1000 (N=815).
Risk Sample Statistics
Component Mean Std. Dev. Min 25%-tile Median 75%-tile Max
Illness
Parent (Sr)
96.27
140.72
0
10
50
100
1000
Likelihood
Child (Sk)
97.20
134.96
0
10
37
145
985
Risk
(Initial)
Difference
-0.93
7.00
-450
-25
0
5
850
Illness Parent (Sp) 55.60 108.68 0 10 13 50 1000
Likelihood child ^ 5Q 24 97 26 0 10 13 50 975
Risk
(Revised) Difference 5.35 73.86 -490 -3 0 5 950
Illness
Likelihood parent -40.67 109.54 -897 -65 0 0 570
Rlsk: , Child -46.96 108.33 -860 -70 0 0 249
Revised
minus
Initial
Parent (Dp) 298.51 261.03 0 50 250 500 1000
A' ¦f* 1
MortaHt^ Child ^ 258 34 255.33 0 40 200 500 1000
Risk Y Difference 40.17 143.44 -700 0 0 100 700
39
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Table 2. Responses to Questions to Assess Willingness to Pay to Reduce Leukemia Likelihood
Risk and Condit
tional Mortality
Risk.
Risk
Proportion of Parents that
Reduction
would Buy the Leukemia Vaccine
Risk
(%)
(Number of Observations)
Component
Parent Child
$150
$300
$600
$1200
$2400
Total
10
10
0.37
0.56
0.36
0.38
0.29
0.39
(27)
(39)
(45)
(42)
(45)
(198)
Illness
Likelihood
10
90
0.81
(31)
0.74
(43)
0.45
(47)
0.30
(47)
0.34
(38)
0.51
(206)
Risk
90
10
0.78
0.58
0.51
0.39
0.21
0.48
(41)
(31)
(47)
(51)
(47)
(217)
90
90
0.75
(37)
0.71
(45)
0.60
(35)
0.45
(44)
0.42
(33)
0.59
(194)
Total
0.70
0.66
0.47
0.38
0.31
0.49
(136)
(158)
(174)
(184)
(163)
(815)
10
10
0.73
0.76
0.61
0.53
0.56
0.64
(44)
(37)
(41)
(40)
(39)
(201)
Conditional
10
90
0.91
0.82
0.54
0.59
0.67
0.73
Mortality
(57)
(33)
(39)
(29)
(37)
(195)
Risk
90
10
0.77
0.67
0.63
0.44
0.48
0.60
(43)
(43)
(43)
(39)
(40)
(208)
90
90
0.92
(48)
0.86
(37)
0.68
(47)
0.46
(37)
0.52
(42)
0.70
(211)
Total
0.84
0.77
0.62
0.50
0.56
0.67
(192)
(150)
(170)
(145)
(158)
(815)
40
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Table 3. Sun Lotion Purchase Intentions: Bivariate Probit Estimates (N=610).
Sample Mean or Proportion
Coefficient Estimate
(standard error)
Covariate
(Parameter Notation)
Illness
Likelihood
Risk
Conditional
Mortality
Risk
Illness
Likelihood
Risk
Conditional
Mortality
Risk
Parent's percentage risk
reduction (= 0.1 or 0.5)
0.299
0.299
0.624*
(0.238)
0.778*
(0.236)
Child's percentage risk
reduction (= 0.1 or 0.5)
0.300
0.296
1.000*
(0.241)
1.115
(0.236)
Cost of sun lotion ($/year)
62.560
63.840
-0.010*
(0.001)
-0.010*
(0.001)
Order (=1 if risk change in
column presented last, 0 if
first)
0.497
(0.500)
0.503
(0.500)
-0.127
(0.108)
-0.188
(0.109)
Constant
0.665*
(0.157)
0.559*
(0.154)
Correlation of errors
between equations
0.767*
(0.041)
Log-likelihood
-649.330
"Denotes significantly different from zero at 1% statistical level.
41
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Table 4. Leukemia Vaccine Purchase Intentions: Bivariate Probit Estimates (N=815).
Sample Mean or Proportion
Coefficient Estimate
(standard error)
Covariate
Likelihood
Risk
Conditional
Mortality Risk
Likelihood
Risk
Conditional
Mortality
Risk
Parent's percentage risk
reduction (=0.1 or 0.9)
0.503
0.511
0.349*
(0.105)
-0.106
(0.108)
Child's percentage risk
reduction (=0.1 or 0.9)
0.492
0.499
0.332*
(0.105)
0.340*
(0.109)
Cost of vaccine ($)
962.210
894.480
-0.0005*
(0.00005)
-0.0003*
(0.00005)
Order (=1 if risk change in
column presented last, 0 if
first)
0.497
0.497
-0.159
(0.091)
0.031
(0.093)
Constant
0.161
(0.111)
0.634*
(0.110)
Correlation of errors between
equations
0.608*
(0.048)
Log-likelihood
-968.130
*Denotes significantly different from zero at 1% statistical level.
42
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Table 5. p-values for Tests of Differences between Un-Normalized Coefficients of Risk
Reduction.
Skin Cancer
Leukemia
(1)
Child illness likelihood risk - Child
conditional mortality risk
h0 -- rl-r? = o
0.823
0.508
(2)
Parent illness likelihood risk - Parent
conditional mortality risk
H : ys-yD =0
O / p / p
0.708
0.009
Wald test, both differences, (1) - (2) = 0
0.925
0.023
(3)
Child illness likelihood risk - Parent illness
likelihood risk
H0.ysk-ysp= o
0.257
0.911
(4)
Child conditional mortality risk - Parent
conditional mortality risk
ho-.y°-ydp= o
0.305
0.017
Wald test, both differences, (3) - (4) = 0
0.323
0.016
43
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Table 6: Parents' Willingness to Pay to Eliminate Illness Likelihood Risk and Conditional
Mortality Risk: Constrained Estimates
Skin Cancer
Parents' WTP to eliminate
illness likelihood risk
to themselves (ys )
$70.06
(17.27)
Parents' WTP to eliminate
conditional mortality risk
to themselves {y®)
$70.06
(17.27)
Parents' WTP to eliminate
illness likelihood risk
to their child (ysk)
$105.54
(18.99)
Parents' WTP to eliminate
conditional mortality risk
to their child (ff )
$105.54
(18.99)
Leukemia
Parents' WTP to eliminate
illness likelihood risk
to themselves (ys )
$760.81
(245.00)
Parents' WTP to eliminate
conditional mortality risk
to themselves {y®)
-$295.02
(308.65)
Parents' WTP to eliminate
illness likelihood risk
to their child (ysk)
$811.29
(199.69)
Parents' WTP to eliminate
conditional mortality risk
to their child (ff )
$811.29
(199.69)
44
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Table 7: Parents' Willingness to Pay to Reduce Illness Likelihood Risk, Conditional
Mortality Risk, Unconditional Mortality Risk, and Morbidity Risk by 1 in 10,000: Based on
Constrained Estimates
Skin Cancer"
Illness
likelihood risk
to parent
$0.03
Conditional
mortality risk
to parent
$0.06
Unconditional
mortality risk
to parent
$0.21
Morbidity
risk to
parent
$0
Illness
likelihood risk
to child
$0.06
Conditional
mortality risk
to child
$0.11
Unconditional
mortality risk
to child
$0.48
Morbidity
risk to child
$0
Leukemiab
Illness
likelihood risk
to parent
$1.37
Conditional
mortality risk
to parent
$0
Unconditional
mortality risk
to parent
$1.32
Morbidity
risk to
parent
$1.34
Illness
likelihood risk
to child
$1.62
Conditional
mortality risk
to child
$0.31
Unconditional
mortality risk
to child
$4.28
Morbidity
risk to child
$0
a Estimates are based on sample means of: (1) 0.2391 for parent revised illness likelihood risk, (2) 0.1897 for
child revised illness likelihood risk, (3) 0.1224 for parent conditional mortality risk, (4) 0.0943 for child
conditional mortality risk, (5) 0.0340 for parent unconditional mortality risk, and (6) 0.0218 for child
unconditional mortality risk. Mean unconditional risk estimates computed by multiplying revised illness
likelihood risk times conditional mortality risk for each observation and averaging across the sample.
b Estimates are based on sample means of revised illness likelihood risk and conditional mortality risk reported in
Table 1. Mean unconditional risk estimates computed by multiplying revised illness likelihood risk times
conditional mortality risk for each observation and averaging across the sample. These means are: (1) 0.0205 for
parent unconditional mortality risk and (2) 0.0189 for child unconditional mortality risk.
45
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