Comparing Risk Preferences over Financial and Environmental Lotteries
Mary Riddel*
May 2011
Abstract: Monetary equivalency, a basic assumption underlying willingness to pay for
environmental risk reductions, requires that the marginal utility of an additional dollar’s
worth of an environmental good should be the same as the marginal utility of an
additional dollar. One upshot of this is that risk aversion, defined as diminishing
marginal utility of wealth, must be the same whether measured in dollars or dollardenominated environmental goods. Nevertheless, studies demonstrate that people display
preferences that differ with the outcome domain.
Using data from two field experiments and a student control group, I show that
preference functions for environmental risks differ significantly from those of financial
risks, with most of the difference arising from the differences in the way that people
emphasize low probability outcomes rather than differences in the standard measure of
risk aversion. The results generally support monetary equivalency. However, the finding
that probability weighting is an important component of environmental preferences
indicates that the standard assumption that environmental preferences can be modeled
using expected utility or subjective expected utility is rejected in favor of richer models
that consider probability weighting.
*Professor, Economics Dept., University of Nevada, Las Vegas. Box 6005. 4505
Maryland PKWY. Las Vegas, NV. 89154. [email protected].
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I. INTRODUCTION
Expected utility (EU) theory requires that for a given individual, risk aversion is
constant over outcome domains such as wealth and consumption of market or nonmarket
goods. This is because risk aversion is synonymous with diminishing marginal utility of
wealth and the utility of consumption of market and nonmarket goods can both be stated
in terms of wealth. This assumption, sometimes called monetary equivalency, requires
that the marginal utility of an additional dollar’s worth of an environmental good should
be the same as the marginal utility of an additional dollar. Intuitively, this means that
people that have distaste for financial risks will be similarly averse to environmental
risks.
Monetary equivalency is a basic assumption underlying willingness to pay for
environmental risk reductions. Willingness to pay for a given level of risk mitigation is
the maximum amount of money that a subject would be willing to forgo to reduce the
probability of an environmental hazard. As such, willingness to pay assumes that
environmental goods can be valued in monetary terms and that the utility of the foregone
wealth is equal to the utility gained from the risk reduction.
However, psychologists and economists alike note that people appear to display
preferences that differ with the outcome domain. For example, Weber, Blais and Betz
(2002) find variation in risk preferences across different activities such as recreation and
food consumption. Soane and Chmiel (2005) also find that preferences over health and
financial risks deviate for a significant portion of their experimental subjects. Einav et al.
(2010) find that observed preferences for financial investments and different forms of
health insurance suggest variation in risk aversion between the health and financial
domains for some subjects.
In EU, a subject’s degree of risk aversion is determined solely by the utility
function component of their preference function. Alternatives to EU theory, such as
cumulative prospect theory (CPT), add probability weights to the preference function that
transform objective probabilities into weighted subjective probabilities. Probability
weighting allow preferences to vary with the outcome domain while maintaining the
assumption of monetary equivalency. Studies examining preferences over financial and
health risks have found that curvature and the crossing point (where the weighted
2
probability is equal to the native probability) of the probability-weighting function
depend on the outcome domain (Bleichrodt and Pinto 2000). For example, the
probability-weighting function the financial domain takes an inverse-S shape that is
concave for low probabilities and convex for higher probabilities. This shape represents
“probabilistic risk aversion” as people place more weight on low-probability extreme
outcomes (Abdellaoui 2000; Diecidue and Wakker 2001, Starmer 2000, Wu and
Gonzalez 1996, Tversky and Kahneman 1992, Camerer and Ho 1994, Bleichrodt and
Pinto 2000, Prelec 1999). The probability-weighting function for life duration takes the
same general shape but, perhaps not surprisingly, displays greater probabilistic risk
aversion, i.e. more extreme curvature, than that for financial risks (Wakker and Deneffe
1996).
The shape of the probability-weighting function for environmental risks is an
unexplored empirical question. Notably, I cannot find any work within the
environmental-risk valuation literature that explicitly accounts for probability weighting.
This paper examines if, how, and why environmental and financial preferences differ
using data from two field experiments aimed at eliciting risk attitudes together with
responses to similar questions from a control-group experiment conducted in the
laboratory. In an effort to include a broad swath of risk preferences, the field
experiments involved subjects who appear to exhibit more risk-loving behavior than the
average person. Subjects in the first field experiment were members of the Porsche Club
of America (PCA) who were interviewed at a club track event. The second field
experiment involved rock climbers contacted at a large annual meeting that serves as both
a training and social event.
In each of the three experiments, I elicit preferences over financial and
environmental gambles using a new multiple-price list (MPL) approach described in
Tanaka, Camerer, and Nguyen (2010). Financial gambles are defined in dollar terms,
whereas environmental gambles involve the success of different clean-up strategies
following a deep-ocean oil spill. Assuming constant relative risk aversion (CRRA) utility
and a one-parameter probability weighting function, I elicit coefficients of risk aversion
for environmental and financial risks, σ e and σ f , respectively as well as environmental
and financial probability weighting coefficients α e and α f , respectively.
3
My goals in these experiments are two-fold. First, I explore the extent and
determinants of environmental and financial probability weighting. More to the point, I
examine how behavioral, attitudinal, and demographic characteristics of each subject
affect their probability weighting functions first for financial risks, then for oil spill risks.
Further, I test whether the probability weighting function for environmental risks is
statistically distinguishable from that applied to financial risks. If they are statistically
distinct, then this alone could explain why people’s preferences over environmental risks
seem to be at odds with their preferences over financial risks.
Second, I explore how behavioral, attitudinal, and demographic attributes of the
subject affect the coefficients of risk aversion for environmental risks and financial, σ e
and σ f . I also compare estimates of the risk aversion parameter for financial gambles to
that of risk aversion parameter for an oil spill using first a simple two-sample t-test. I
then test for differences in the conditional means of σ e and σ f , controlling for attitudinal,
behavioral, and demographic characteristics of each subject. If the means or conditional
means differ, this is evidence that variation in risk aversion across the outcome domains
is at least one source of the difference in financial and environmental preferences. It also
would tend to undermine the standard assumption of monetary equivalency underlying
willingness to pay models.
As a quick preview of the findings, the statistical models indicate that attitudes
and behavior are good predictors of both the degree of probability weighting and risk
aversion. The results indicate that subjects are more likely to overemphasize low
probability environmental outcomes than low probability financial outcomes when
making decisions over environmental and financial gambles, leading subjects to offer
more support for mitigating environmental gambles than financial gambles with the same
odds. There is only weak support for differences in financial and environmental risk
aversion coefficients. A two-sample t-test fails to reject the null hypothesis of σ e = σ f .
However, regression results suggest that the conditional means of σ e and σ f may differ
for some subjects. Taken together, this evidence suggests that preference functions for
environmental risks differ significantly from those of financial risks, with most of the
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difference arising from the differences in the way that people emphasize low probability
outcomes rather than differences in the standard measure of risk aversion.
The finding that probability weighting plays an important role in environmental
preferences has important implications for those seeking to value environmental risk
reductions. In the past, researchers have assumed that the risk reduction enters the
preference function in s standard linear fashion as in an EU or subjective EU model.
Ignoring probability weighting in general, and the heterogeneity in probability weighting
in particular, may lead to biased estimates of willingness to pay for an environmental risk
reduction.
II. LITERATURE REVIEW
2.1
Cumulative Prospect Theory
Violations of the assumptions of EU theory are frequently observed in the
experimental setting. Many of the observed violations entail deviations from the
independence axiom which requires that the outcomes of a lottery are independent of
their corresponding probability and consequently, the expected marginal utility of a
change in probability is constant (Starmer 2000). Under the independence axiom, moving
from a situation of certainty to a lottery with a risk of 1 offers the same change in
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expected utility as moving from 1 to 3 .1 Contrary to the independence axiom,
2
5
behavioral experiments have consistently found that people tend to place more mental
weight on extreme outcomes in their decision process, particularly when small
probabilities are considered (Abdellaoui 2000; Diecidue and Wakker 2001, Starmer 2000,
Bleichrodt and Pinto 2000, Prelec 1999, Rabin 1998, Wu and Gonzalez 1996, Tversky
and Kahneman 1992). A common example of this type of thinking is “Murphy’s Law:”
the worst thing that can happen, probably will. Similarly, within the realm of
environmental policy, the precautionary principle is based on the idea that extremely
unfavorable environmental outcomes should be given extra consideration when designing
public programs.
1
The independence axiom holds that if for preference functions F,G, and H, if F is
preferred to G, then for any 0 ≤ a ≤ 1 , aF + (1 − a) H is preferred to aG + (1 − a) H .
5
Non-expected utility models, including cumulative prospect theory of Tversky
and Kahneman (1992), allow for violations of the independence axiom by transforming
objective probabilities into decision weights using a probability-weighting function.
Under CPT, three coefficients define preferences over lotteries: 1) a risk aversion
coefficient defines the curvature of the utility function 2) a coefficient that defines the
curvature of a probability weighting function and 3) a coefficient of loss aversion. For the
purposes of this study, I focus on the first two coefficients, and query subjects only about
financial and environmental gains. I hope to extend this work to include loss aversion in
the future.
2.2. Multiple Price List Auction for CPT Preferences
Many risk elicitation procedures performed in the laboratory are discussed at
length by Harrison and Rutström (2008) and Chetan et al. (2008). In the multiple price
list (MPL) approach popularized by Holt and Laury (2002) subjects are presented with a
table of lottery pairs and asked to choose one lottery for each pair. Assuming a
functional form for risk aversion, such as CRRA, their choices define a range for a
coefficient of risk aversion.
The standard MPL approach assumes that expected utility (EU) applies, so that
the preference function is a probability-weighted utility function where the probability
are exogenous and determined by the researcher. In a recent paper, Tanaka, Camerer and
Nguyen (2010) extend the MPL approach to allow for estimation of the parameters of
non-expected utility functions such as CPT. For CPT without loss aversion addressed
here, this method entails presenting subjects with two series of lottery pairings for each
risk domain. The subjects’ choices imply ranking over six lotteries. One can solve for
ranges of the risk aversion and probability-weighting coefficients that together are
consistent with the offered rankings. The current study uses this approach to elicit
coefficients of risk aversion and probability-weighting coefficients for first financial, then
environmental gambles.
2.3. Heterogeneity in Risk Preferences
The extant research suggests that there is significant heterogeneity in risk
preferences. For example, more educated and older subjects have been shown to be more
risk averse (Harrison, Lau, and Rutstrom 2007, Tanaka, Camerer, and Nguyen 2010).
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Gender has been shown to affect risk preferences, but this result does not appear to be
very robust. Whereas early studies of the influence of gender on risk preferences tended
to indicate that women tend to be more risk averse than men (Jianakoplos and Bernasek
1998, Eckel and Grossman 2008), more recent studies have not observed any variation in
risk preferences across gender (Harrison, Lau, and Rutstrom 2007, Tanaka, Camerer, and
Nguyen 2010).
Others have studied whether income affects preferences over financial risks. In a
study of Vietnamese farmers, Tanaka, Camerer, and Nguyen (2010) find that mean
village income significantly affects risk preferences, with residents of wealthier villages
exhibiting more risk tolerance, all else equal. Nevertheless, this result is not robust to
model specification. They do not find any relationship between risk preferences and
household income. In a study of risk preferences for Danish citizens, Harrison, Lau, and
Rutstrom (2007) find that household income does not affect risk preferences.
III. THE EXPERIMENTS
3.1 Eliciting the CPT Coefficients
The theoretical model is based on cumulative prospect theory (Tversky and
Kahneman 1992) with preference function:
(1) U ( x, p; y, q ) = w( p + q)v( x) + w(q)(v( y ) − v( x))
U ( x, p; y; q) gives the expected prospect function for outcome x with probability p and
outcome y with probability q . The value of prospects x and y are denoted by v( x) and
v( y ) , respectively. The weighting function, w( p ), assigns weights to the objectively
given probabilities p and q allowing for differential weighting of outcomes.
The probability-weighting function is:
(2)
w( p ) = exp[−(− ln p)α ]
The parameter σ determines the curvature of the value function. Subjects are risk averse
for 0 < σ < 1 subjects, risk neutral for σ = 1 , and risk loving for σ > 1 . The probability
weighting parameter α defines the curvature of the probability weighting function.
When α < 1 the probability weighting function takes on an inverse-S shape, indicating
that subjects tend to over-emphasize low-probability events and under-emphasize high-
7
probability events when making decisions under risk. When α = 1 , probability weighting
is absent and the model is equivalent to the EU model.
The experiments are based on a variation of the MPL approach described by
Tanaka, Camerer, and Nyguen (2010). In their experiments, subjects are faced with two
series of financial lotteries (See Table 1 for these series.) Subjects are asked to state their
choice between lotteries A and B for the sequence of lotteries in series 1. The sequence
is defined so that E[A]-E[B] decreases and eventually becomes negative. The analyst
notes where the subject switches from preferring lottery A to preferring lottery B. Later
switch points indicate higher levels of risk aversion. Adding the series 2 choice sequence
gives a second switch point, allowing for the estimation of α simultaneously with σ .
Because the switch points imply rankings, the resulting equations are based on
inequalities and therefore gives ranges for α and σ .
Given the inequalities, one can solve for a set of pairs (α , σ ) that are consistent
with the inequalities. Because the statements are inequalities, the solution implies a
range (rather than a unique value) for each parameter (α , σ ) . As others often do, I use
the mid-point of the ranges for the subsequent analysis (Tanaka, Camerer, and Nguyen,
2010).
I contacted members of the PCA club of Las Vegas at one of the club’s quarterly
track events held at Spring Mountain Motorsports Ranch in December, 2010. Subjects
were offered a free lunch ticket worth $10 for completing the experiment. The track event
has several facets. Novice drivers are given classroom and track instruction and allowed
to finally go solo on the track when deemed ready by an instructor. There are also openlapping events for intermediate and advanced drivers where cars are driven at speeds in
excess of 120 mph, but passing zones are limited to avoid accidents. These track events
are intended for training and no formal racing takes place. However, many of those
present are also involved in formal wheel-to-wheel open-passing races at other times. Of
the 62 people at the event, 40 completed the experiment.
Rock climbers were contacted at the Red Rock Rendezvous in Red Rock Canyon
near Las Vegas, NV in March, 2011. The Red Rock Rendezvous is a meeting of over
300 climbers from all over the US, Canada, and Mexico. Attendees enroll in seminars to
advance their skills and several social events are also scheduled. Attendees range from
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novice climbers to some of the world’s most accomplished rock climbers and
mountaineers. Climbers who completed the experiment were entered into a drawing for
two climbing ropes, worth about $210 a piece, and two belay devices worth about $70 a
piece. Of the roughly 300 people at the event, 85 subjects completed the experiment.
The subjects who form the control group were graduate and undergraduate
students at the University of Nevada, Las Vegas. They were offered extra credit for
completing the experiment during the Fall 2010 and Spring 2011 semesters.
Seventy-seven students completed the experiment, bringing the total sample size
for the field and laboratory experiments to 202. Four subjects did not give complete
demographic information, and are excluded from the analysis. Thus the total sample size
for estimation is 198.
For both the field and laboratory subjects, the experiment involved completing a
questionnaire booklet. The questionnaire booklet has four sections for the control group
and five sections for the field experiments. Section 1 demonstrates the MPL technique
using two examples based on financial lotteries. Section 2 elicits switch points for two
choice sequences of financial gambles. These gambles are given in Table 1. Section 3
outlines a hypothetical oil-spill scenario. Lotteries are stated in terms of two different
mitigation technologies that have different levels of effectiveness. For example, gamble
A may be a 30% chance of 20 sq. miles mitigated and 70% chance of 5 sq. miles
mitigated whereas gamble B has a 10% chance of 34.5 sq. miles mitigated and 90%
chance of 2.5 sq. miles mitigated. I elicit switch point for the two sequences given in
Table 2. Note that these gambles are framed in terms of clean-up, so we are evaluating a
“good” rather than a “bad.” Further note that the gambles in the Table 2 sequences are a
monotonic transform of the gambles in Table 1 such that the outcomes in Table 2 are
those in Table 1 divided by 20. Since they are monotonic transformations, if the
observed switch points in Table 1 match those for Table 2, the subject displays the same
preference function for financial and environmental gambles.
Section 4 elicits demographic characteristics and whether the subject participates
in a set of risky behaviors such as smoking or not regularly wearing a seatbelt. The
climbers and PCA members have a fifth section that assesses their level of participation
in their respective sports. For the climbers, I query subjects about their abilities, asking
9
them the highest grades they have climbed in different subsets of climbing such as freeclimbing and bouldering. For the PCA members, I establish their level of experience and
whether they participate in wheel-to-wheel race events, time trials, or structured open
lapping events.
3.2 Model Variables.
Using the switch points elicited in sections 2 and 3 of the experiment booklet, I
calculate values of the parameters of the CPT function α fi , σ fi , α ei and σ ei where the
subscript i indicates that corresponding coefficient for experimental subject i. These are
the dependent variables for the models. Past research has shown that demographic,
attitudinal, and behavioral variables affect risk preferences. As such, I model the
parameters of the CPT function using three sets of variables: group indicators for the field
experiments, behavioral variables, and demographic variables.
I control for attitudinal differences across subjects using the three experiment
groups: PCA membership, attendance at the rock climbers’ meet (the Red Rock
Rendezvous) and the control group. The variable CONTROL=1 if the subject is a
member of the student control group and zero otherwise; the variable CLIMBER=1 if the
subject was contacted at the Climber’s Red Rock Rendezvous and zero otherwise and
PCA=1 if the subject is a member of the Porsche club who was contacted at the track
event and zero otherwise. Club membership or more informal group relationships such
as those established by rock climbers at the Red Rock Rendezvous might well reveal
distinct attitudes about risk. The Porsche brand implies luxury, but more importantly
speed and daring. Porsche advertizing typically depicts a car on a racetrack at speed. But
there are both health and environmental consequences associated with high horsepower
vehicles. High insurance premiums on all Porsche models reflect the increased
likelihood of an accident over less sporty automobiles. Further, high horsepower means
low gas mileage, and may suggest less concern about environmental risks on the part of
the driver than for a more gas mileage-conscious driver. Rock climbing is also associated
with an increased risk of accident or death which is reflected in higher life-insurance
premiums for climbers. However, most climbing takes place outdoors in highly scenic
areas and climbers may well show a preference for mitigating risk to the environment in
general as well as the specific mountainous environments in which they recreate.
10
Behavioral variables include the indicator variable SMOKE=1 if the subject
admits to being a regular smoker within the past year and zero otherwise, RACER=1 if
the subject is a member of the PCA race group and zero otherwise and, finally
CLIMBELITE=1 if the subject is an elite climber and zero otherwise. Racers make up a
distinct group within the PCA. For one, they must qualify to enter PCA-sponsored races
where speeds are significantly higher than in time trials or open-lapping events. Also,
passing is permitted anywhere on the track, rather than limited to straight-aways as is the
case for non-race events. High speeds and open passing make for a much more
challenging and potentially dangerous track experience. Thus, racers tolerance for risk
may well be different than other PCA drivers who prefer to drive in a slower (but still
high speed) and more controlled environment. An elite climber is classified as a climber
who has free-climbed 5.12 or harder and/or bouldered V7 or harder.2 Like the racers in
the PCA, these elite climbers likely represent a group that has significantly different risk
preferences than the average climber. Nevertheless, one can argue that racers and elite
climbers are engaging in less hazardous activities than their novice counterparts because
experience likely offsets some of the inherent hazards of the sport. Still, experienced
racers and climbers are likely more aware of hazards than novices, and are therefore
showing a tolerance for well-defined risks. In the end, the models reported below allow
me to test for differences in risk preferences between the elite classes within the sports to
determine whether a taste for risk is part of the motivation for engaging in the activity.
2
Free-climbing involves climbing with ropes and other safety gear without relying on the
gear to physically ascend the climb. Free-climbing grades range from 5.1 to 5.15.
Bouldering involves climbing without ropes or safety gear, however climbs are typically
less than 20 feet high. Bouldering grades range from V1 to V16. After a discussion with
several experienced, professional climbers, I chose to rank those free-climbing 5.12 and
bouldering V7 and higher as elite climbers.
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IV. MODEL RESULTS
4.1 Probability weighting functions
Table 3 reports two censored logistic models for the dependent variables
α e and α f as a function of group indicators, demographic, and behavioral variables.3
Censoring occurs outside the range 0.05 < α f < 1.45 and 0.05 < α m < 1.45 .4 The models
include the group indicator variables CONTROL, PCA and CLIMBER; the set of
behavioral variables RACER, CLIMBELITE, and SMOKE; as well as the demographic
variables FEMALE, AGE and INCOME.
The basic model is linear in the coefficients:
α j = γ j ' Group + λ j ' X + ϕ j ' D + ε
where j=financial or environmental, Group represents the vector of the group indicator
variables (CONTROL, PCA, and CLIMBER); X represents the vector of behavioral
variables (CLIMBELITE, RACER, AND SMOKE); D represents a vector of
demographic characteristics; γ j , λ j , and ϕ j are vectors of model coefficients; and ε j is
normally-distributed independent random error term.
Recall that values of 0 < α < 1 indicate an S-inverse probability-weighting
function where the subject tends to overweight small probabilities and underweight
higher probabilities. When α = 1 , probability weighting is absent and EU preferences
apply. When α > 1 , the subject tends to underweight low probabilities and overweight
high probabilities in their preference function.
Columns I and II report the results for probability weighting in the financial
domain using α f as the dependent variable. According to the model, all of the group
indicators are positive and statistically significantly different from zero. The probability
weighting function for the PCA group, at 0.944, is almost a straight line. A Wald test of
the null hypothesis of γ PCA
= 1 fails to reject the null, indicating that the PCA group tends
f
3
Each model is estimated separately. I hope to conduct joint estimation of the models
soon so that I can take advantage of the correlation between the two dependent variables
to improve the coefficient standard errors.
4
The censored model is used since border solutions are inequalities.
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to weight probabilities quite accurately, consistent with the EU model. The model
indicates that climbers and the student control group exhibit significant probability
= 1 and γ Climber
= 1 reject the null
weighting. Wald test of the null hypotheses that γ Control
f
f
in both instances (p-values = 0.01 and 0.02, respectively). Climbers exhibit the most
probability weighting. The coefficient of 0.785 indicates that they tend to overweight
low probability events and underweight high probability events in their financial decision
making. For example, a probability of 0.10 is transformed into a weight of 0.15 whereas
a probability of 0.9 is transformed into 0.85. As a result, a low probability financial gain
carries more weight in the subject’s preference function than it would under subjective
expected utility. Sometimes this is referred to as probabilistic risk aversion since subjects
over-emphasize small probability outcomes relative to high probability outcomes.
Of the behavioral variables, SMOKE and CLIMBELITE are significantly
different from zero. The coefficient of RACER is not significantly different from zero,
indicating that racers do not display different financial probability-weighting functions
than the non-racing PCA group. Still, the PCA group showed very little probability
weighting and the racer group is consistent with a group that deviates only modestly from
expected utility preferences, if at all. However, elite climbers are distinctly different
from their climbing peers. The estimated probability weighting coefficient for elite
climbers is 1.09. A Wald test of the null hypothesis that γ Climber
+ λ Climbelite
= 1 fails to
f
f
reject the null, indicating that the EU model adequately characterizes this group’s
preferences for financial risks. Finally, the coefficient of smoker is negative and
significant, indicating that smokers, independent of group membership, tend to have more
s-inverse type curvature in the probability weighting function than non-smokers.
Columns III and IV of Table 3 give the coefficients and standard errors,
respectively, for the dependent variable α e (the coefficient of environmental probability
weighting). The coefficients of the group variables are all statistically significantly
different from zero. For each group, the null hypothesis that the coefficient is greater
than 1 is rejected, indicating that the respective group probability weighting functions
take on the familiar s-inverse shape. Like the models of financial probability weighting,
the PCA group exhibits the least probability weighting. However, unlike the results for
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the financial probability weighting function, the control group shows the most probability
weighting with the most pronounced s-inverse shape.
Interestingly, none of the behavioral or demographic variables are statistically
significant. Thus, I infer that attitudinal characteristics reflected in group membership
capture most of the heterogeneity in the probability weighting function for environmental
risk.
4.2 Curvature in the Utility Function
Table 4 gives the results of two models aimed at explaining heterogeneity in the
coefficient of risk aversion for financial and environmental risks. Both models are
estimated using a tobit specification with censoring at 0.05 and 1.5. Again, the basic
model is linear in the variables:
σ j = β j ' Group + θ j ' X + ϑ j ' D +ν j
where β j , θ j , and ϑ j are vectors of model coefficients, and ν j is an independent
normally-distributed random error term.
Columns I and II of Table 4 give the coefficients and standard errors,
respectively, for the dependent variable σ f (the coefficient of financial risk aversion).
All of the groups are financially risk averse with a Wald test indicating that
β Control
< 1, β fPCA , and β Climber
. Climbers are the least risk averse, followed by the control
f
f
group then the PCA group.
The racing group appears to be more risk loving than the control group and the
non-racing PCA group. A Wald test of the null hypothesis that β Control
= β fPCA + β fRacer
f
and rejects the null (p-value=0.04), indicating that the RACER group is significantly
more risk loving than the control group. Also, a Wald test of the null hypothesis that
β fPCA = β fPCA + β fRacer rejects the null (p-value=0.02), indicating that the RACER group is
significantly more risk loving than the non-racing PCA group. However, A Wald test of
the null hypothesis that β Control
= β fPCA fails to reject the null at the 10% level, indicating
f
that the non-racing PCA subjects do not display significantly increased taste for risk over
the control group.
14
Climbers do not display significantly different financial risk aversion than the
control group or the non-racing PCA group. Wald tests of the null
= β Climb
+ β Climbelite
, β Climb
= β Climb
+ β Climbelite
, and β Control
= β Climb
all
hypotheses β Control
f
f
f
f
f
f
f
f
fail to reject the null hypothesis at any standard level of significance. Finally, smokers
and non-smokers are equally financially risk averse.
The model results indicate that there is some demographically based
heterogeneity in financial risk aversion. For each ten years of age, the coefficient of
financial risk aversion decreases by 0.04, indicating that older subjects are more
financially risk averse than their younger counterparts. Income and gender do not
significantly affect financial risk aversion.
Columns III and IV in Table 4 report results of the tobit models for the coefficient
of environmental risk aversion, σ e . Although all groups are environmentally risk averse,
climbers are the least environmentally risk averse, followed by the control group and the
PCA group. Racers and elite climbers do not display different environmental risk
aversion than their overall group average, as indicated by the insignificant coefficients of
RACER and CLIMBELITE. Interestingly, smokers are more environmentally risk loving
than non-smokers, all else equal. Income, age and gender do not influence environmental
risk aversion.
V. COMPARING ENVIRONMENTAL AND FINANCIAL RISK AVERSION AND
PROBABILITY WEIGHTING
4.1 Simple tests of Equality of Means
The sample average coefficients of risk aversion for environmental and financial
risks are σ e = 0.59 and σ f = 0.58 , respectively. A two-sample t-test of the null
hypothesis σ e = σ f versus the alternative σ e ≠ σ f accepts the null (p-value=0.76). The
results indicate that the curvatures of the utility functions are not statistically
distinguishable from a simple comparison of means perspective. In other words, the
standard assumption that the marginal utility of a dollar’s worth of an environmental
good is equal to that of a dollar’s worth of any other consumption good is supported.
The sample average probability weighting parameters for environmental and
financial risks are α e = 0.69 and α f = 0.77 , respectively. Both estimated means are less
15
than one, indicating that the probability weighting functions take on the inverse-s shape
frequently found in other studies. A two-sample t-test of the null hypothesis that
α e ≥ α f versus the alternative α e < α f rejects the null (p-value<0.01). I infer that the
probability weighting parameter for environmental risk is less than that for financial risk.
This implies that subjects are more probabilistically risk averse for environmental risks
than they are for financial risks i.e. the probability-weighting function for environmental
risks has a more pronounced curvature. In other words, subjects are more likely to place
additional weight on low probability environmental outcomes in decisions related to
environmental risks than decisions related to investment and other financial risks.
Taken together, these results appear to support the hypothesis that the difference
in preference functions between environmental and financial preferences is driven by
differences in probability weighting. This means that standard risk aversion, defined as
diminishing marginal utility of money, is invariant to the outcome domain. The
underlying reason that environmental and financial preference function differ is that
probability weighting is sensitive to the outcome domain. In the sample considered here,
subjects tend to over-emphasize low-probability environmental outcomes relative to
financial outcomes.
Though risk aversion appears to be stable across the financial and environmental
risk domains with a simple equality of means test, this may not hold when we look at
conditional means. It could be that on average, risk aversion is stable across the domains,
but for some groups or individuals, it varies. Further, it is interesting to see whether
demographic, attitudinal, or behavioral characteristics of subjects explain the differences
noted between the financial and environmental probability weighting functions. To
explore these questions, I estimate two models aimed at determining what causes the two
coefficient sets to differ, if at all.
Columns I and II of Table 5 report the parameter estimates and standard errors,
respectively, of the results of an ordinary least square regressions based on the dependent
variable α fi − α ei . Coefficient standard errors are corrected for heteroskedasticity using
the White correction. The independent variables explain less than 4% of the variation
in α fi − α ei . The negative and statistically significant estimated coefficient of SMOKER
16
indicates that for smokers, the deviation between the probability weights is less than for
nonsmokers, so that the probability weighting function varies less in the outcome domain
than for nonsmokers. None of the other model variables are statistically significant.
Suspecting possible multicollinearity issues, I tried several specifications that excluded
sets of independent variables, but still found that the group and demographic
characteristics and remaining behavioral characteristics did not significantly influence the
difference in probability weighting coefficients. I surmise that this difference is highly
idiosyncratic and generally unpredictable based on the simple behavioral and
demographic variables I include here.
Columns III and IV of Table 5 report the parameter estimates and standard errors,
respectively, of the results of an ordinary least square regressions based on the dependent
variable σ fi − σ ei . Again, standard errors are adjusted for heteroskedasticity. With an R2
value of 0.09 and 4 of the 9 coefficients of the explanatory variables significantly
different from zero, this model offers better predictive power than the difference in
probability weighting model. The positive and significant coefficient of the CONTROL
group variable indicates that the risk aversion coefficient differs with the outcome
domain so that the control subjects are more environmentally risk averse than they are
financially risk averse. The PCA group shows even more divergence between
environmental and financial risk aversion than the CONTROL group, whereas risk
aversion is invariant to the outcome domain for the CLIMBER group. The coefficient of
age is negative and statistically significant, indicating that older subjects show less
variation with the outcome domain than do younger subjects. Finally, smokers’
coefficients of risk aversion are less likely to diverge than those of nonsmokers.
The finding that conditional means of environmental and financial risk aversion
coefficients diverge has interesting implications. For one, this indicates that for some
groups of subjects, that the marginal utility of a dollar’s worth of environmental goods is
less than a dollars worth of other consumption goods.
VI.
CONCLUSIONS
This study demonstrates that preferences for financial risks are distinct from
preferences for environmental risk for the average subject. Probably the most important
finding is that the difference is primarily driven by differences in the probability-
17
weighting function rather than the utility function. In fact, the assumption of monetary
equivalency for environmental goods, a standard of EU theory, is shown to hold in most
instances. Nevertheless, for a subset of the subjects, there does appear to be some
differences in the risk aversion coefficient across domains, but that effect diminished for
older subjects. Still, these findings suggest that future research further exploring the
plausibility of environmental-monetary equivalency would be helpful, given the
importance of this assumption for valuing environmental goods.
Probability weighting in the environmental risk domain has important
implications for modeling environmental risk reductions. Current risk valuation models
base willingness to pay estimates on some stated risk reduction offered by researchers
(such as in Alberini et al. 2007) or elicit a subjective estimate of a risk reduction related
to some policy (such as Riddel and Shaw 2006, Cameron 2005). In either case, the
probabilities enter the preference function as linear weights corresponding to the standard
EU model in the former case and the subjective EU model in the latter case. The finding
that probability weighting is an important component of the experimental subjects’
environmental preference functions indicates that the standard assumption that
environmental preferences can be modeled using EU or subjective EU is rejected in favor
of richer models that consider probability weighting.
Another problem with current risk valuation models is that they assume that risk
aversion is constant across subjects. Models typically assume either risk neutrality or a
function based on the log of income or wealth.5 The results reported in this paper indicate
that there is significant heterogeneity in financial risk aversion. Moreover, the log
income model implies far more risk aversion than found in this study. The findings
reported here suggest that a more thorough consideration of the effects of risk aversion on
environmental valuation models may be in order.
5
Log income is equivalent to CRRA preferences with a risk aversion coefficient
approaching 0.
18
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20
Table 1. Choice tasks for financial outcome domain
Series I
Lottery A
probability
probability
($400)
Payoff
($100)
0.3
$400
0.7
0.3
$400
0.7
0.3
$400
0.7
0.3
$400
0.7
0.3
$400
0.7
0.3
$400
0.7
0.3
$400
0.7
0.3
$400
0.7
0.3
$400
0.7
0.3
$400
0.7
0.3
$400
0.7
0.3
$400
0.7
0.3
$400
0.7
0.3
$400
0.7
Lottery B
Payoff
$100
$100
$100
$100
$100
$100
$100
$100
$100
$100
$100
$100
$100
$100
probability
probability
($?)
Payoff
($50)
Payoff EV(A)-EV(B)
0.1
$680
0.9
$50
$77
0.1
$750
0.9
$50
$70
0.1
$830
0.9
$50
$62
0.1
$930
0.9
$50
$52
0.1
$1,060
0.9
$50
$39
0.1
$1,250
0.9
$50
$20
0.1
$1,500
0.9
$50
-$5
0.1
$1,850
0.9
$50
-$40
0.1
$2,200
0.9
$50
-$75
0.1
$3,000
0.9
$50
-$155
0.1
$4,000
0.9
$50
-$255
0.1
$6,000
0.9
$50
-$455
0.1
$10,000
0.9
$50
-$855
0.1
$17,000
0.9
$50
-$1,555
Series II
Lottery A
probability
probability
($400)
Payoff
($300)
0.9
$400
0.1
0.9
$400
0.1
0.9
$400
0.1
0.9
$400
0.1
0.9
$400
0.1
0.9
$400
0.1
0.9
$400
0.1
0.9
$400
0.1
0.9
$400
0.1
0.9
$400
0.1
0.9
$400
0.1
0.9
$400
0.1
0.9
$400
0.1
0.9
$400
0.1
Lottery B
Payoff
$300
$300
$300
$300
$300
$300
$300
$300
$300
$300
$300
$300
$300
$300
probability
($?)
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
Payoff
$540
$560
$580
$600
$620
$650
$680
$720
$770
$830
$900
$1,000
$1,100
$1,300
probability
($50)
Payoff EV(A)-EV(B)
0.3
$50
-$3
0.3
$50
-$17
0.3
$50
-$31
0.3
$50
-$45
0.3
$50
-$59
0.3
$50
-$80
0.3
$50
-$101
0.3
$50
-$129
0.3
$50
-$164
0.3
$50
-$206
0.3
$50
-$255
0.3
$50
-$325
0.3
$50
-$395
0.3
$50
-$535
21
Table 2. Choice tasks for environmental outcome domain, stated in terms of number of
acres of oil spill that is cleaned up
Cleanup Option A
probability
(20 sq
miles)
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
Square
probability Square
Miles
(5 sq
Miles
Cleaned
miles)
Cleaned
20 sq miles
0.7
5 sq miles
20 sq miles
0.7
5 sq miles
20 sq miles
0.7
5 sq miles
20 sq miles
0.7
5 sq miles
20 sq miles
0.7
5 sq miles
20 sq miles
0.7
5 sq miles
20 sq miles
0.7
5 sq miles
20 sq miles
0.7
5 sq miles
20 sq miles
0.7
5 sq miles
20 sq miles
0.7
5 sq miles
20 sq miles
0.7
5 sq miles
20 sq miles
0.7
5 sq miles
20 sq miles
0.7
5 sq miles
20 sq miles
0.7
5 sq miles
Cleanup Option A
probability
(20 sq
miles)
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
Series I
Cleanup Option B
probability
(? sq
miles)
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
Square Miles
Cleaned
34 sq miles
37.5 sq miles
41.5 sq miles
46.5 sq miles
53 sq miles
62.5 sq miles
75 sq miles
92.5 sq miles
110 sq miles
150 sq miles
200 sq miles
300 sq miles
500 sq miles
850 sq miles
probability
(2.5 sq
miles)
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
Square
Miles
Cleaned EV(A)-EV(B)
2.5 sq miles
3.85
2.5 sq miles
3.5
2.5 sq miles
3.1
2.5 sq miles
2.61
2.5 sq miles
1.95
2.5 sq miles
1
2.5 sq miles
-0.25
2.5 sq miles
-2
2.5 sq miles
-3.75
2.5 sq miles
-7.75
2.5 sq miles
-12.75
2.5 sq miles
-22.75
2.5 sq miles
-42.75
2.5 sq miles
-77.75
Series II
Cleanup Option B
Square
probability Square
probability
Square
Miles
(15 sq
Miles
Square Miles (2.5 sq
Miles
Cleaned
miles)
Cleaned
probability
Cleaned
miles)
Cleaned EV(A)-EV(B)
20 sq miles
0.1
15 sq miles
0.7
27 sq miles
0.3
2.5 sq miles
-0.15
20 sq miles
0.1
15 sq miles
0.7
28 sq miles
0.3
2.5 sq miles
-0.85
20 sq miles
0.1
15 sq miles
0.7
29 sq miles
0.3
2.5 sq miles
-1.55
20 sq miles
0.1
15 sq miles
0.7
30 sq miles
0.3
2.5 sq miles
-2.25
15 sq miles
0.7
31 sq miles
0.3
2.5 sq miles
-2.95
20 sq miles
0.1
20 sq miles
0.1
15 sq miles
0.7
32.5 sq miles
0.3
2.5 sq miles
-4
20 sq miles
0.1
15 sq miles
0.7
34 sq miles
0.3
2.5 sq miles
-5.05
20 sq miles
0.1
15 sq miles
0.7
36 sq miles
0.3
2.5 sq miles
-6.45
20 sq miles
0.1
15 sq miles
0.7
38.5 sq miles
0.3
2.5 sq miles
-8.2
20 sq miles
0.1
15 sq miles
0.7
41.5 sq miles
0.3
2.5 sq miles
-10.3
20 sq miles
0.1
15 sq miles
0.7
45 sq miles
0.3
2.5 sq miles
-12.75
15 sq miles
0.7
50 sq miles
0.3
2.5 sq miles
-16.25
20 sq miles
0.1
20 sq miles
0.1
15 sq miles
0.7
55 sq miles
0.3
2.5 sq miles
-19.75
20 sq miles
0.1
15 sq miles
0.7
65 sq miles
0.3
2.5 sq miles
-26.75
22
Table 3. Censored Tobit models for Probability Weights: Dependent Variables are the
Probability Weighting Coefficients for the Financial and Environmental Risks
Dependent Variable
αf
αe
standard
standard
Variable
coefficient error coefficient error
CONTROL
0.825*** 0.070 0.785*** 0.067
PCA
0.944*** 0.124 0.824*** 0.120
CLIMBER
0.785*** 0.095 0.814*** 0.092
RACER
0.064
0.094
0.100
0.095
CLIMBELITE
0.224** 0.098
0.016
0.097
AGE
-0.002
0.002
-0.002
0.002
FEMALE
0.014
0.048
-0.015
0.047
INCOME (thous $) -0.278
0.461
-0.435
0.462
SMOKER
-0.117** 0.059
-0.010
0.061
SCALE
0.173*** 0.011
0.170
0.010
Akaike info criterion
0.672
0.559
Schwarz criterion
0.838
0.724
Log likelihood
-56.523
-45.585
*,**,*** represent coefficients that are statistically significantly from the zero at the
0.10, 0.05, and 0.01 level, respectively.
Table 4. Censored Tobit Models for Risk Aversion: Dependent Variables are the Risk
Aversion Coefficients for the Financial and Environmental Risks
Dependent Variable
σf
σe
standard
standard
Variable
coefficient error coefficient error
CONTROL
0.673*** 0.078 0.400*** 0.086
PCA
0.646*** 0.141
0.271*
0.156
CLIMBER
0.731*** 0.102 0.553*** 0.113
RACER
0.265*** 0.109
0.051
0.120
CLIMBELITE
0.086
0.095
-0.098
0.106
AGE
-0.004*
0.002
0.003
0.003
FEMALE
-0.031
0.052
-0.002
0.057
INCOME (thous $)
0.272
0.529
0.257
0.584
SMOKER
-0.001
0.065 0.159** 0.072
SCALE
0.327*** 0.017 0.159*** 0.072
Akaike info criterion
0.754
0.981
Schwarz criterion
0.920
1.147
Log likelihood
-64.653
-87.620
*,**,*** represent coefficients that are statistically significantly from the zero at the 0.10,
0.05, and 0.01 level, respectively.
23
Table 5. Ordinary Least Squares Models of the Differences between Probability Weights
and Risk Aversion Coefficients: Dependent Variables are α f − α e and σ f − σ e
Dependent Variable
α f − αe
σ f −σe
standard
standard
Variable
coefficient error coefficient error
CONTROL
0.037
0.083 0.275*** 0.107
PCA
0.118
0.139 0.366** 0.187
CLIMBER
0.004
0.134
0.175
0.125
RACER
-0.024
0.101
0.198
0.141
CLIMBELITE
0.179
0.141
0.179
0.143
AGE
0.000
0.003 -0.007** 0.003
FEMALE
0.042
0.060
-0.035
0.071
INCOME (thous $)
0.239
0.547
-0.045
0.608
SMOKER
-0.101** 0.052 -0.155* 0.083
R-squared
0.036
0.090
Akaike info criterion
0.961
1.194
Schwarz criterion
1.110
1.343
Log likelihood
-86.099
-109.202
24