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Deciphering the Noise: The Welfare Costs of Noisy Behavior

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Theoretical work on stochastic choice mainly focuses on the sources of choice randomness, and less on its economic consequences. We attempt to close this gap by developing a method of extracting information about the monetary costs of noise from structural estimates of preferences and choice randomness. Our method is based on allowing a degree of noise in choices in order to rationalize them by a given structural model. To illustrate the approach, we consider risky binary choices made by a sample of the general Danish population in an artefactual field experiment. The estimated welfare costs are small in terms of everyday economic activity, but they are considerable in terms of the actual stakes of the choice environment. Higher welfare costs are associated with higher age, lower education, and lower income.
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Deciphering the Noise:
The Welfare Costs of Noisy Behavior
Aleksandr Alekseev, Glenn W. Harrison, Morten Lau and Don Ross
August 2019
Abstract
Theoretical work on stochastic choice mainly focuses on the sources of choice ran-
domness, and less on its economic consequences. We attempt to close this gap by
developing a method of extracting information about the monetary costs of noise from
structural estimates of preferences and choice randomness. Our method is based on
allowing a degree of noise in choices in order to rationalize them by a given structural
model. To illustrate the approach, we consider risky binary choices made by a sample
of the general Danish population in an artefactual field experiment. The estimated
welfare costs are small in terms of everyday economic activity, but they are consider-
able in terms of the actual stakes of the choice environment. Higher welfare costs are
associated with higher age, lower education, and certain employment status.
Keywords: stochastic choice, choice under risk, welfare costs, behavioral welfare eco-
nomics
JEL codes: D61, D81, C93
Economic Science Institute, Chapman University, USA (Alekseev); Department of Risk Management &
Insurance and Center for the Economic Analysis of Risk, Robinson College of Business, Georgia State Univer-
sity, USA (Harrison); Copenhagen Business School, Denmark (Lau); School of Sociology, Philosophy, Crimi-
nology, Politics and Government, University College Cork, Ireland; School of Economics, University of Cape
Town, South Africa; and Center for Economic Analysis of Risk, Robinson College of Business, Georgia State
University, USA (Ross). Harrison is also affiliated with the School of Economics, University of Cape Town.
E-mail contacts: alekseev@chapman.edu,gharrison@gsu.edu,mla.eco@cbs.dk and don.ross@uct.ac.za. We
are grateful to the Danish Social Science Research Council (Project #12-130950) for financial support.
1 Introduction
Stochastic choice has become an active area of research in recent years, motivated primar-
ily by two considerations. First, a large body of empirical evidence shows that stochastic
choice is a robust empirical phenomenon,1and much work has been devoted to explaining
this behavior.2Second, models of stochastic choice provide researchers with econometric
tools to estimate structural models in a broad range of applications. The primary inter-
est in applying a model of stochastic choice is to recover the structural parameters of the
deterministic part of a model, such as risk or time preferences. Little attention has been
given, however, to the systematic economic interpretation of the parameter estimates of the
stochastic part, which determine the magnitude of choice randomness. The interpretation of
these parameters is important for understanding the economic value of choice randomness,
which has implications for the quality of decision making, and also for a better understanding
of the underlying “source” models of stochastic choice. We study the economic consequences
of stochastic choice by developing an intuitive method of translating the estimates of the
stochastic part into economically tractable terms.
Consider a generic structural model of discrete choice that uses a standard multinomial
logit model of stochastic choice,3which assigns each discrete alternative a choice likelihood
Paccording to
P(a|β, µ) = exp(U(a|β))
Pa0Aexp(U(a0|β)).(1)
In this expression, aand a0are alternatives, such as lotteries or dated outcomes, from a
set of all alternatives A. The deterministic part of this structural model is parametrized
by a vector of behavioral parameters β, which could represent, for instance, an agent’s
risk or time preferences. For example, in the case of risk preferences, βcould be a risk
1Nogee and Mosteller (1951) provide the earliest evidence of stochastic choice, followed by Tversky
(1969), Starmer and Sugden (1989), Camerer (1989), and Ballinger and Wilcox (1997).
2Wilcox (2008) provides an excellent overview of many popular stochastic models of choice under risk.
Recent examples include Swait and Marley (2013), Wallin, Swait, and Marley (2018), Matˆejka and McKay
(2015) and Agranov and Ortoleva (2017).
3Also known in the literature as the strong utility model or the Fechnerian model.
1
aversion parameter and Ucould be the expected utility of a risky alternative; in the case
of time preferences, such as the quasi-hyperbolic discounting model, βwould comprise the
exponential and hyperbolic discounting parameters and Uwould be the discounted utility
of an income stream. The behavioral parameters determine the aggregate utility function U
assigned to each alternative.
The stochastic part of the model is parametrized by µ, often called the noise parameter.4
The noise parameter determines how sensitive choice likelihoods are to the maximization of
utility Uaccording to a given structural model. As noise tends to zero, an agent will almost
surely choose the alternative with the highest utility. When noise goes to infinity, the agent
will assign equal likelihoods to choosing each alternative regardless of their utilities. Higher
values of µthus imply a higher magnitude of choice randomness in this popular specification.
Three issues arise with the interpretation of the estimates of the noise parameter. First,
while the effect of µon choice likelihoods is clear, one cannot readily interpret a particular
estimate of noise in economic terms.5A monetary value assigned to a noise estimate, on
the other hand, would provide clear information about the economic consequences of choice
randomness. Second, since the noise parameter is unbounded from above, it is difficult to
judge whether the randomness of an agent’s choices is high or low. A value defined on the
unit interval would solve this problem.6Third, the raw estimates of µare not well suited for
interpersonal comparisons, since behavioral parameters βalso change across people. Having
choice randomness expressed in common units, such as money, and taking into account the
interpersonal differences in βwould help to overcome this issue. Aspects of these three issues
4In the game theory literature on Quantal Response Equilibrium due to McKelvey and Palfrey (1995),
which applies stochastic choice to strategic settings, it is common to use an alternative parametrization
λ1.
5In the existing literature (von Gaudecker et al.,2011;Bland,2018), an estimate of noise is sometimes
interpreted as the likelihood of choosing the best alternative (among the two available) for a given difference
in utilities (or certainty equivalents) between them. While this number is informative of the economic
consequences of choice randomness, it does not provide a monetary measure of the welfare costs associated
with stochastic choice.
6The parameter of the tremble model of stochastic choice (Harless and Camerer,1994) has this property
and thus allows one to evaluate the relative magnitude of choice randomness. However, an estimate of
the tremble parameter would still require an economic interpretation. See Carbone and Hey (2000) for a
comparison between the tremble model and the Fechnerian model.
2
arise not only in the standard multinomial logit model but also in its modifications, such
as the contextual utility model of Wilcox (2011) or specifications that substitute certainty
equivalents of alternatives for their expected utilities, such as von Gaudecker et al. (2011).
We address these issues by converting an estimate of µinto two intuitive measures.7
The first measure, absolute welfare cost (AWC), puts a dollar value on choice randomness.
It shows how much money, in certainty equivalent terms, an agent would be allowed to
“waste” compatibly with rationalization of her choices by an underlying structural model.8
The second measure, relative welfare cost (RWC), scales the absolute welfare cost by the
monetary value at stake in a choice context. The relative welfare cost is thus defined on the
unit interval. It shows what proportion of the total monetary value at stake an agent would
be allowed to waste compatibly with rationalization of her choice by the model.9
Our approach rests on a careful interpretation of the concepts of “noise” and “waste.”
We follow the descriptive, structural literature on risk preferences by assuming a specific
model of the manner in which choice randomness is rationalized. In the language of Infante,
Lecouteux, and Sugden (2016, p. 21), this is
...not an inference about the hypothetical choices of the client’s inner rational
agent, but rather a way of regularising the available data about the client’s
preferences so that it is compatible with the particular model of decision-making
that the professional wants to use. Regularisation in this sense is almost always
needed when a theoretical model comes into contact with real data.
In our case the subject being evaluated is the “client,” and we are the “professional.” Thus
we consistently use the expression “noise,” or some synonym, rather than “error.” When
it comes to us using this regularised model of the agent, we may then adapt the “inten-
7While the discussion below focuses on the multinomial logit model and its modifications, a similar logic
can be applied to other models of stochastic choices, such as the trembles model (Harless and Camerer,1994)
or the random preferences model (Loomes and Sugden,1995;Gul and Pesendorfer,2006).
8While our discussion focuses on individual decision-making, our method can also be used to study
stochastic choice in group decision-making (Bone, Hey, and Suckling,1999).
9Other ways to measure the welfare costs of stochastic choice might exist, however we find that using
monetary measures based on certainty equivalents to be intuitive and transparent. It might be the case that,
depending on a particular research question, one might be more interested in an absolute measure than a
relative measure, or vice versa. Our goal here is to provide the general tools, which can then be adapted to
a particular research question.
3
tional stance” towards the evaluation of an agent’s behavior, using a philosophical approach
developed by Dennett (1987), theoretically interpreted for use in economics by Ross (2014,
ch. 4), and explicitly applied to behavioral welfare economics by Harrison and Ross (2018,
§5). This perspective, which has become the dominant one in the philosophy of psychology,
emphasizes that preferences and beliefs are not fixed internal states of people, but are ra-
tionalizations of choice behaviors that people rely on to interpret one another. This applies
mutatis mutandis to self-interpretation. Preference and belief attributions pick out “real
patterns” in choice behaviors (Dennett,1991), and these patterns, which typically involve
some noise, are the basis for assessing people’s goals, and hence, for economics, their welfare.
Only then can we use the expression “waste.” Similarly, when we characterize behavior as
being “imperfectly rational” below, that also reflects our intentional stance, rather than a
claim that the agent has made an error in cognitive processing or problem representation.
Our measures of the welfare costs of noisy behavior are consistent with the model-based
approach advocated by Manzini and Mariotti (2014). This means that in order to calculate
our welfare cost measures, we assume specific deterministic and stochastic models of the
decision-making process. These assumptions allow us to derive precise (in the sense of being
point estimates) and efficient (in the sense of efficiently using available data, explained in
Section 2.3) values of welfare costs. We recognize the potential sensitivity of our results to
these assumptions, and address them in Appendix A.
Our absolute and relative welfare cost measures allow one to conveniently evaluate the
economic significance of choice randomness, its relative magnitude, and to compare the mag-
nitude of choice randomness across people. The implications of these measures for an agent’s
behavior, however, will ultimately depend on the underlying model of the source of choice
randomness adopted by a researcher. This is an important point since different “source”
models of stochastic choice often lead to the same choice likelihoods, such as the likelihoods
generated by the multinomial logit model presented above. For instance, the Random Utility
model due to Marschak (1960) assumes that when an agent makes an optimal choice, the
4
choice randomness is due to the perturbations in her utility function that are unobservable
to a researcher. The noise parameter in the Random Utility model is then proportional to
the variance of the unobserved component of utility. High estimated welfare costs would
imply that the stochastic part of the structural model dominates the deterministic part, i.e.,
the structural model cannot explain the agent’s choices well. The welfare costs can then be
viewed as measures of a model’s fit.10
Recent studies offer an alternative view on choice randomness as an optimal response to
costly frictions in the decision-making process. For example, these frictions may be caused
by the need to collect the relevant information to make a choice, as in Rational Inattention
models of Caplin and Dean (2015) and Matˆejka and McKay (2015). The noise parameter
in a Rational Inattention model represents marginal information costs. The estimates of
welfare costs in this type of models can then be interpreted as aggregate information costs,
or losses that an agent incurs relative to an ideal case of no information costs. Another
example of frictions is the pursuit of multiple goals that cannot be obtained simultaneously
(Swait and Marley,2013;Wallin et al.,2018). An agent is assumed to balance the goal of
choosing the best available alternative with the goal of having diversity in choices. Noise
parameters in this model represent the relative weight of the second goal. The estimates of
welfare costs in this type of models can be interpreted as the economic value that an agent
places on the goal of having diversity or, alternatively, as the loss an agent incurs relative to
a case of having a single goal of choosing the best alternative.
We apply our method to the data from an artefactual field experiment in Denmark. The
subjects came from a sample of the general Danish population and were asked to make a series
of choices between two risky alternatives. Each subject answered a detailed demographic
survey, which we use to characterize the effects of demographic characteristics on the observed
heterogeneity in the AWC and RWC. We find that the average AWC are around 67 Danish
10 Recent work by Halevy et al. (2018) provides a promising example of how welfare costs can be used as
a measure of fit.
5
kroner ($10)11 and thus negligible for the subjects’ natural economic environment. However,
the RWC are quite significant, at 0.87 on average. There is also considerable variation
among the subjects in terms of their AWC and RWC. Regression analysis shows that certain
demographic characteristics are associated with higher costs. In particular, subjects who
are older, less educated, and have a particular employment status, have larger welfare costs.
Females have higher AWC than males, but do not differ in RWC.
Section 2describes the method of converting an estimate of noise into welfare costs
measured in monetary terms and provides an explicit algorithm for computation in a binary
choice case. Section 3applies the method to data from an artefactual field experiment in
Denmark involving choice under risk and studies the properties of the welfare costs, as well
as their demographic correlates. Section 4discusses connections with previous literature.
Section 5concludes.
2 Method
We first look at a general case when the set of alternatives is continuous. This case allows us
to clearly demonstrate the logic behind our method of extracting the welfare cost information
from a noise estimate. Then we turn to a more common discrete case with two alternatives
and explicitly describe the algorithm to implement our method.
11 Throughout the text, we use an exchange rate of 1 Danish krone = $0.15 that was prevalent at the time
of the experiment.
6
2.1 General Case
Consider an agent choosing from a set of alternatives indexed by real numbers on a compact
interval A= [ al, ah]. Each alternative generates a lottery12
l(a) = {x1(a), . . . , xk(a); q1(a), . . . , qk(a)},
aA, xiR, qiR+,i= 1, . . . , k,
k
X
i=1
qi= 1,
where xiare monetary outcomes and qiare respective probabilities of obtaining those out-
comes.
This setting could represent allocating resources between two state-contingent accounts,
as in Choi et al. (2007). Each allocation in this example is an alternative with k= 2
outcomes, x1(a) and x2(a), and equal probabilities of each outcome. The minimum (al= 0)
and maximum (ah>0) amounts an agent can allocate to account 1 will define the interval
of alternatives A. Then x1(a) = aand x2(a) = b(aha), where b < 0 is the slope of the
budget line and q1(a) = q2(a),aA.13
The risk elicitation task of Gneezy and Potters (1997) is another example of such a
setting. In this example, the minimum (al= 0) and maximum (ah>0) amounts a subjects
can allocate to a risky asset define the set of alternatives A, where ahis the initial endowment.
A subject’s choice of how much of the endowment to allocate to a risky asset, a, generates
lotteries with two outcomes given by x1(a) = aha(the asset yields no return) and x2(a) =
ah+a(k1) (the asset yields a positive return k1). The probabilities of outcomes are
given exogenously and do not depend on a.
Each alternative ahas an aggregate utility U(a)U(l(a)) defined by an assumed struc-
tural model of choice under risk. Monetary outcomes are transformed using u:R7→ R, the
von Neumann-Morgenstern utility function. Each value of U(a) can be translated into a cer-
12 The lottery itself does not need to be discrete. An alternative can generate a continuous probability
density.
13 In an actual experiment, the set of alternatives is, of course, discrete. This choice set, however, comes
close to being continuous.
7
tainty equivalent m(a), defined by u(m(a)) = U(a). The ordering of alternatives is preserved
for the certainty equivalent transformation: U(a)>U(b)m(a)>m(b),a, b A.
Assume that Uis concave and reaches its unique maximum (minimum) at a(a), as does
the certainty equivalent function. Define the maximum certainty equivalent as mm(a),
and the minimum certainty equivalent as mm(a). If the agent always chooses the best
alternative a, we call this behavior perfectly rationalizable (by an assumed model of choice
under risk). On the other extreme, if the likelihood of choosing ais the same as for any other
alternative, we call such a behavior non-rationalizable. We are concerned with the behavior
in between, which is neither perfectly rationalizable nor non-rationalizable, a behavior that
we call imperfectly rationalizable.
The degree of this imperfection14 is characterized by a number ε, 0 6ε6m, with
mmm. Choices that lead to certainty equivalents within εdistance from the
maximum certainty equivalent can be viewed, from the perspective of a model, as imperfectly
rationalizable.15 These choices form an optimal region Adefined by
A(ε) = aA|m(a)>mε.(2)
The degree of imperfection εshows how much monetary welfare an agent would be allowed to
waste to make her choices rationalizable by the model, and effectively includes these choices
in the optimal region. In other words, εrepresents the welfare costs measured in monetary
units. Our goal is to link these costs to noise.
The allowed degree of imperfection co-varies with the width of the optimal region. If εis
set to 0, the optimal region will consist only of the best alternative a. If εis high enough,
the optimal region will coincide with the whole set of alternatives A. Figure 1illustrates
how the optimal region varies with the degree of imperfection. Geometrically, the optimal
14 This term should be understood as an imperfection of a given model to regularise data, rather than a
statement about an agent making decision errors.
15 The idea of allowing an agent some degree of imperfection in choices is not new. For example, Harrison
(1994) introduces a similar quantity based on an agent’s subjective cost of choosing one alternative versus
the other to explain many EUT violations.
8
region is the line segment a
l, a
h.
A*
m*
m*− ε
al*a*ah*
Alternatives
Certainty equivalent
(a) Low Degree of Imperfection
A*
m*
m*− ε
al*a*ah*
Alternatives
Certainty equivalent
(b) High Degree of Imperfection
Figure 1: Optimal Region and Degree of Imperfection
The optimal region and the degree of imperfection are the first two components that
we need to interpret an estimate of noise. The third component comes from a stochastic
model p:A7→ R+, which generates choice likelihoods over the set of alternatives. Some
alternatives fall into the optimal region, by definition. By integrating the density p(a) over
this region we get the proportion of choices that are counted, from the perspective of a
model, as imperfectly rationalizable for a given degree of imperfection. We call this measure
adegree of rationalizability (DoR):
ρ(µ, ε) = ZA(ε)
p(a) da. (3)
The DoR has several intuitive properties, two of which turn out to be crucial for our
analysis, and can be represented graphically. Figure 2shows that as the degree of imper-
fection increases, the optimal region expands and the DoR, represented by the gray shaded
area, increases. Figure 3shows that as the noise goes up, the density flattens out and the
9
probability mass shifts from the optimal region to the outside area, reducing the DoR.
A*
ρ(µ, ε)
Alternatives
Density
(a) Low Degree of Imperfection
A*
ρ(µ, ε)
Alternatives
Density
(b) High Degree of Imperfection
Figure 2: Degree of Rationalizability and Imperfection
The DoR for certain values of noise and imperfection has attractive interpretations. The
quantity ρ(, ε) tells us what proportion of choices are counted as rationalizable for a given
imperfection εwhen they are, in fact, non-rationalizable. It represents a Type II error in
a test to detect rationalizability, and the quantity 1 ρ(, ε) is the power of this test.
This power will decrease as the allowed degree of imperfection increases or as the set of
alternatives shrinks. The value of DoR at ρµ, 0) measures the proportion of rationalizable
choices for an estimated level of noise ˆµand no imperfection. We refer to it as the default
degree of rationalizability or DDoR.
We now have all the tools to decipher the noise. We do this by linking an estimate of µ,
the value of which is hard to interpret, to the degree of imperfection, a monetary measure that
has an intuitive economic interpretation as the welfare cost, or monetary welfare required
to rationalize the agent’s choices by a model. In order to link them, we need to reverse the
steps we followed so far. Currently, we introduced a degree of imperfection εthat defines an
optimal region A. The optimal region combined with a stochastic model, parameterized by
10
Alternatives
Density
(a) Low Noise
Alternatives
Density
(b) High Noise
Figure 3: Degree of Rationalizability and Noise
µ, yields a value of DoR. Now suppose that instead we start with a DoR measure and fix it
at some target level α. Let an estimated value of the noise be ˆµ. The question is how much
imperfection should be allowed for 100 ×α% of the choices to be rationalized for a given
noise. In other words, we need to find εthat satisfies
ρ(ˆµ, ε) = α. (4)
This equation establishes an implicit function, ε(ˆµ;α). For the purpose of our analysis, the
following property of this function is important.
Proposition 1. For a given α, the degree of imperfection as a function of noise, ε(µ;α), is
monotonically increasing:16
>0.
16 We note that in the case when alternatives are discrete rather than continuous, as discussed below, the
DoR as a function of imperfection will not be continuous and thus it will not be possible to match the target
DoR αperfectly. We address this issue by using a discrete grid for εand an interpolated version of ρµ, ε).
The interpolated DoR function on a discrete grid is continuous and thus Proposition 1applies.
11
Proof. See Appendix B.
This property implies that noise and imperfection are in a direct and monotonic relation.17
This property is important since more noise should imply higher welfare costs, which in our
case are measured by imperfection. If imperfection and noise were not in a direct and
monotonic relation, such an interpretation would be impossible. The relation between εand
µcomes from the fact that the DoR is decreasing in noise and increasing in imperfection.
From these properties it also follows that higher values of αimply higher values of ε. The
more choices we wish to rationalize, for a given value of a noise, the more imperfection we
should allow. The choice of the target αis left to the discretion of a researcher. In our
empirical analysis we use the values of 0.9,0.95, and 0.99, which appear to be reasonable
targets.
So far we have focused on a single choice context, but in practice we observe agents make
choices over a series of rounds of a choice task. Suppose we observe an agent’s choices over n
rounds indexed by j= 1, . . . , n, and in each round the mapping of alternatives ainto lotteries
lj(a) is different. In the context of an allocation task, the variation is introduced by changing
the slope of a budget line bjand a maximum amount aj
hthat can be allocated to one of the
accounts: Aj= [ 0, aj
h], xj
1(a) = a, xj
2=bj(aj
ha). We can repeat all the previous steps in
deriving the DoR, but now it will differ by the round: ρj(µ, ε). What remains common across
rounds, however, is the degree of allowed imperfection ε. We assume that µand preferences
remain fixed for the duration of a choice task. We can then aggregate the DoR from all the
choices by averaging across the DoR for single choices:
ρ(µ, ε) = 1
n
n
X
j=1
ρj(µ, ε).(5)
Naturally, the average follows all the properties of the DoR for a single choice. In particular,
it increases in εand decreases in µ. We can then use the aggregate DoR in (5) to calculate
17 This property holds for a given agent, or rather given risk preferences. This property will not hold
perfectly across agents whose preferences are different.
12
the imperfection needed to reach a target αin equation (4).
After calculating the value of εthat satisfies equation (4), εµ, α), it makes sense to
adjust this value to take into account the fact that the degree of imperfection should not
exceed the difference between the maximum and minimum certainty equivalents for a given
choice. Since a common εis applied to all the rounds of a choice task, for some rounds it can
actually exceed ∆m. Increasing imperfection beyond this difference does not have any effect
on the DoR and would imply that we allow an agent to waste more monetary welfare than
there actually is. This issue can be addressed by bounding εby ∆m, and then averaging
across all the rounds:
¯ε(ˆµ, α) = 1
n
n
X
j=1
min ε(ˆµ, α),mj.(6)
We call the resulting measure of imperfection Absolute Welfare Costs (generated by noise
ˆµ, with 100 ×α% of choices rationalized), or AWC. It represents the monetary welfare that
the agent would be allowed to give up for exactly 100 ×α% of her choices to be rationalized
by the model, given noise ˆµ. For any estimated value of noise and any desired proportion of
choices we would like to rationalize we can, therefore, always find a precise dollar value of
the welfare costs.
We can go further and translate the welfare costs into relative terms, to compare these
costs with the actual stakes of a choice context. For example, an AWC of $1 may not look
like much, but if ∆mjare close to $1 in all the rounds, almost all the welfare would have to
be sacrificed to rationalize an agent’s choices. We divide the degree of imperfection by the
difference between the maximum and minimum certainty equivalents for every round, and
average across all the choices:18
˜ε(ˆµ, α) = 1
n
n
X
j=1
min (ε(ˆµ, α)
mj
,1).(7)
The resulting degree of imperfection represents Relative Welfare Costs (generated by
18 Since the resulting quantity has to be a fraction, we bound this ratio by 1.
13
noise ˆµ, with 100 ×α% of choices rationalized), or RWC. Another benefit of this measure
is that it allows one to appreciate the relative magnitude of noise, since RWC are bound
between 0 and 1, while a raw estimate of noise is unbounded from above. If rationalizing
100 ×α% of the choices requires on average almost all the difference between the maximum
and the minimum certainty equivalents, in which case RWC are close to 1, that clearly
indicates that the choices are close to being non-rationalizable, from the perspective of the
model. On the other hand, if it requires only a small fraction of this difference, in which case
RWC are near 0, then choices are close to being perfectly rationalizable, from the perspective
of the model.
2.2 Binary Choice
An important special case arises when an agent has only two alternatives to choose from.
This is one of the most common experimental designs in risk elicitation tasks.19 In this case
the set of alternatives in each round is A={a1, a2}. Without loss of generality, assume that
alternative a2always gives the highest utility, so that Uj(a2)> Uj(a1), j = 1, . . . , n, i.e.,
a
j=a2, using the notational convention Uj(a)Ulj(a). The maximum and the minimum
certainty equivalents in each round jare m
jmj(a2) and mjmj(a1), respectively. The
optimal region and the DoR can then take only two values:
A
j(ε) =
a2, ε < mj,
A, ε >mj,
ρj(µ, ε) =
pj(a2), ε < mj,
1, ε >mj,
(8)
where pj(a2) is the likelihood of choosing alternative a2in round j.
Suppose we observe a series of binary choices made by a subject and estimate a structural
model of risk preferences in which ˆγis a vector of estimated risk parameters and ˆµis an
estimate of noise. The ˆγvector in the Expected Utility Theory (EUT) case is typically just
19 For example, the risk elicitation tasks developed and popularized by Hey and Orme (1994) and Holt
and Laury (2002) apply to the binary choice case.
14
the relative risk aversion. In the case of Cumulative Prospect Theory (CPT) ˆγincludes
the risk aversion parameter(s), the probability weighting parameter(s), and the loss aversion
parameter.20 The computation of AWC and RWC (rationalizing 100 ×α% of the choices)
from these data can be performed using the following algorithm.
1. For each round, compute the aggregate utilities of both alternatives, Uj(a1; ˆγ), Uj(a2; ˆγ),
j= 1, . . . , n.
2. Compute the certainty equivalents of both alternatives, mj(a1), mj(a2), using the in-
verse transformation, mj(a) = u1(Uj(a; ˆγ); ˆγ), a A, and the difference between
them, ∆mj.
3. Compute the likelihoods of each alternative using the stochastic model, pj(a; ˆγ, ˆµ),
aA.
4. Start with ε= 0. Compute the DoR in each round ρj(ˆµ, ε) using (8). Compute the
aggregate DoR ρ(ˆµ, ε) using (5).
5. If ρ( ˆµ, ε)< α, increase εby a small number ∆ε > 0.
6. Repeat Step 5 until the aggregate DoR reaches the target level of α.21
7. Compute the AWC ¯ε( ˆµ, α) using (6). Compute the RWC ˜ε(ˆµ, α) using (7).
2.3 Alternative Measures
Note that the proposed computation of welfare costs does not involve actual choices. After
estimating risk parameters and noise, we ignore whether the actual choices corresponded
to the maximum certainty equivalent or not. A question then arises: what choices do we
20 The parametrization will also depend on the utility and probability weighting functions used. For
example, if an expo-power utility function is used, it will have two parameters rather than one.
21 In practice, due to the discreteness of ρ(ˆµ, ε) it will often be impossible to match the target αexactly.
We use linear interpolation to handle this issue.
15
rationalize, if not actual choices? This question also suggests an equivalent computation
based on actual choices rather than on likelihoods.
Consider the following alternative algorithm. Start by computing the implied (by the
model) decisions based on certainty equivalents. Compare actual and implied decisions
by looking at the proportion of times when implied and actual decisions coincide. This
proportion gives the actual default DoR. Next, calculate the vector of the differences in
the certainty equivalents (CE differences) for the cases when implied and actual decisions
disagree. These are the “mistakes,” from the perspective of the model, we need to “correct,”
or regularise by adding a structural model of behavioral noise. Start with ε= 0 and increase
it by a small positive amount. When εis lower than the CE difference, the DoR in that
round is zero; otherwise it equals one, meaning that implied and actual decisions become
equivalent. After that, compute the relative proportion of times when rationalized decisions
coincide with the actual ones. Increase εuntil this proportion reaches the target level.
Compute the average of the bounded (by CE difference) εfor the absolute actual welfare
costs, and the average of their ratios to CE differences for the relative actual welfare costs.
Although this alternative algorithm is almost identical to the previous algorithm, there
is a subtle difference. This difference makes us choose in favor of the method described in
§2.2, which involves rationalizing potential choices, as opposed to actual ones. Consequently,
we obtain the estimates of the potential welfare costs, while the alternative method would
give us the actual welfare costs. The key difference between the two methods lies in the fact
that the likelihoods of choices represent what could have been chosen if the same options
were presented many times. We view the approach of using potential choices as extracting
more information from the same data points. The informational gain is obtained through
the introduction of a particular structure that describes the choice likelihoods.
Of course, if the two methods gave completely different estimates, one would need a
stronger argument in favor of one method against the other. Comparing the potential and
actual welfare costs, however, shows that the measures are tightly associated in practice (not
16
reported here). In principle, one could easily substitute one method for another.
Another alternative method of computing the absolute welfare costs would arise if we
reconsidered equation (6), which involves bounding the value of imperfection by the differ-
ence in certainty equivalents. This is not required and we could, as well, have computed
the unbounded absolute welfare costs.22 One might expect that we would obtain higher esti-
mates of the AWC in that case. Indeed, our calculations show (not reported here) that the
unbounded AWC are on average twice as large as the bounded AWC, and both measures
are tightly associated. We prefer to use the bounded measure, however, since it represents
only the welfare costs that can be potentially incurred, while the unbounded measure allows
wasting more monetary welfare than there actually is.
3 Empirical Analysis
3.1 Data
We present the results for 218 adult Danes, a subsample of a larger field study by Harrison,
Jessen, Lau, and Ross (2018). The subjects for the original study were recruited from two
internet-based panels with 165,000 active members combined. The sample frame consisted of
65,592 adult Danes between ages 18 and 75. The sample was stratified by sex and age across
three regions of Denmark: greater Copenhagen, Jutland, and Funen and Zealand.23 The
completed sample consisted of 8,405 respondents, or 12.8% of the sample frame. Invitations
were sent out by email, and the subjects could participate in a survey using internet-browsers
on their computers or mobile devices. The experiment was implemented as an artefactual
field experiment (Harrison and List,2004).
Table 1provides a summary of the socio-demographic characteristics of our subsample
22 The computation of the RWC must involve bounding, since they represent a fraction that must lie in
the unit interval.
23Greater Copenhagen area was assigned a weight of 50%, and the other two regions were assigned equal
weights of 25%.
17
who were invited to participate in an experiment after completing the online survey. Slightly
less than half of the sample were females and the average age was just less than 50 years. The
majority of the sample had college education, and the distribution of income across different
income brackets was roughly equal. Most of the participants were either employed as public
servants or retired. More than 75% of our sample comes from the Greater Copenhagen area.
The subjects made binary choices across 60 pairs of lotteries and answered a set of
demographic questions. Once all the lottery choices were made, one of the choices was
selected randomly for payoff. Table C.1 in the Appendix contains the battery of lotteries
that were given to the subjects. This battery is based on designs by Loomes and Sugden
(1998), Wakker, Erev, and Weber (1994) and Cox and Sadiraj (2008). Together they provide
a powerful test of EUT and RDU.
3.2 Estimation Procedure
Computation of the welfare costs relies on structural estimates of risk preferences γand
noise µ. We implement the estimation in the standard fashion by maximizing the Bernoulli
log-likelihood function at the level of a subject:24
γ, ˆµ) = arg max
γ,µ
n
X
j=1 yjln pj(a2;γ, µ) + (1 yj) ln pj(a1;γ , µ),
where yjI(a=a2)jis an indicator variable that takes a value of 1 whenever an alternative
a2is chosen in round j. The alternative a2is taken to be the one on the right side of the
screen without loss of generality, and we no longer assume that it gives the highest aggregate
utility in all the rounds.
We assume that the choice probability pj(a2;γ, µ) is given by the strong utility model in
24 We find that the estimation procedure successfully converges for 183 subjects (84% of the sample). For
the rest of our subjects, the estimation procedure terminates after a number of iterations and yields the
best parameter values at the time of termination. The results in subsequent sections are reported for the
full sample of subjects. Using only the subset of sub jects with successful convergence yields quantitatively
similar results, e.g., compare Figure 4that uses the full sample and Figure D.7 (in Appendix) that uses the
subset of subjects.
18
Table 1: Socio-Demographic Characteristics of the Sample
Characteristic Mean
Female 0.46
Age 48.06
Education
Vocational training 0.19
Low level of formal education 0.21
College, less than 3 years 0.09
College, 3 to 4 years 0.27
College, 5 or more years 0.24
Annual household income, before tax
Less than 300k DKK 0.23
300k–500k DKK 0.23
500k–800k DKK 0.23
More than 800k DKK 0.17
Not reported 0.14
Occupation
Public servant 0.42
Student 0.12
Unemployed 0.04
Retired 0.23
Skilled worker 0.03
Unskilled worker 0.06
Self-employed 0.06
Other 0.04
Family
Has children 0.25
Lives with a partner 0.54
Geographic area
Copenhagen 0.78
Central Denmark 0.07
Zealand 0.09
Southern Denmark 0.06
19
the logit form25
pj(a2;γ, µ) = exp Uj(a2;γ)
exp Uj(a2;γ)+ exp Uj(a1;γ)= Λ Uj(a2;γ)Uj(a1;γ)
µ,
where Λ(·) denotes the logistic cumulative density function, and pj(a1;γ , µ) = 1pj(a2;γ, µ).
We also assume that the lotteries are compared according to their expected utilities
(dropping an index for the round)
U(a;γ) =
k
X
i=1
qi(a)u(xi(a); γ),
and the ufunction takes the constant relative risk aversion form:
u(x;γ) = x1γ
1γ.
Nothing in our approach relies on assuming an EUT model or a strong utility model in the
logit form. In fact, we could have proceeded in a way suggested by Harrison and Ng (2016)
and estimated different models for different subjects, classifying our subjects as EUT or
RDU, for example. Alternatively, as suggested by Monroe (2017), we could have assumed an
RDU model for all the subjects, since correct classification has significant data requirements.
Appendix Ademonstrates this important generalization by regenerating all results assuming
an RDU model of risk preferences, as well as assuming a different utility function, the expo-
power utility function, and a different model of stochastic choice, the contextual utility model
of Wilcox (2011).
25 One could alternatively use certainty equivalent functions minstead of the aggregate utility functions
Uin the specification of the stochastic model, as in Bruhin et al. (2010) or von Gaudecker et al. (2011).
Using this alternative specification only changes the scale of the estimates of the noise parameter, and does
not change the estimates of risk parameters or the estimated likelihoods of choosing each alternative. The
algorithm for computing welfare costs and their magnitude would remain the same.
20
3.3 Welfare Costs
Figure 4(Panel A) shows the distribution of the individual-level estimates of AWC (in Danish
kroner, DKK) and Table 2(Panel A) presents their summary statistics.26 The distribution
of the AWC is composed of two clusters: a major cluster on the left end and a minor cluster
on the right end of the support, so that overall the distribution is right-skewed. The major
cluster is bell-shaped and fairly symmetric. The minor cluster appears to be bell-shaped but
the number of observation in it is small. As the target DoR increases, the distribution of
the AWC flattens out and slides to the right end of the support.
The AWC are, on average, quite modest in size. For α= 0.95, the mean AWC are only
66.96 DKK (10.04 USD) and the median AWC are even smaller, 58.56 DKK (8.78 USD).
For 50% of the subjects, the AWC lie within 38.37 DKK (5.76 USD) and 80.76 DKK (12.11
USD) at this level of DoR. As the target level of DoR increases, the mean AWC also increase,
as expected. For α= 0.99, the mean AWC reach 88.66 DKK (13.3 USD), and the median
AWC reach 79.44 DKK (11.92 USD).
There is substantial variation among subjects in their AWC. At α= 0.95 the smallest
AWC are just 1.24 DKK (0.19 USD) while the maximum AWC are 224.23 DKK (33.63 USD),
which is roughly 3 times as large as the mean AWC. The standard deviation of AWC at this
level of DoR is 42.2 DKK (6.33 USD). The variation in AWC increases as the target DoR
goes up, which is reflected in higher standard deviations and higher ranges. At α= 0.99 the
minimum AWC are still tiny, just 4.56 DKK (0.68 USD), while the maximum AWC become
271.3 DKK (40.69 USD), which is again roughly 3 times as large as the mean AWC at this
level of DoR. The standard deviation reaches 48.05 DKK (7.21 USD) at this level of DoR.
While the variation in the AWC between subjects is substantial, there is also consider-
able uncertainty at the individual level. Figure 5(Panel A) shows the point estimates of
the AWC (for α= 0.95) for each subject along with the 95% confidence interval around
26 As discussed in §2.3, we rationalize the potential choices, which allows us to use a fine grid for the target
DoR. If we were rationalizing the actual choices instead, we would have to deal with the target DoR’s that
are fractions of 60 (the number of choice pairs in the experiment).
21
Figure 4: Distributions of AWC and RWC in the Sample
α= 0.99
α= 0.95
α= 0.9
0 50 100 150 200 250
0.000
0.005
0.010
0.015
0.020
0.000
0.005
0.010
0.015
0.020
0.000
0.005
0.010
0.015
0.020
AWC (DKK)
Density
Panel A.
α= 0.99
α= 0.95
α= 0.9
0.4 0.6 0.8 1.0
0
5
10
15
20
25
0
5
10
15
20
25
0
5
10
15
20
25
RWC
Density
Panel B.
Note: The graph shows the distributions of the individual-level estimates of AWC (Panel A) and
RWC (Panel B) for three target levels of α: 0.9, 0.95, and 0.99. The bars are the histograms
and the smooth lines are the kernel density estimates. The dashed lines show the medians of the
distributions. The AWC numbers are in DKK. For the RWC, we truncate the support at 0.4 to
improve readability of the graph. This results in dropping 11 observations for which the RWC are
below 0.4.
22
Table 2: Summary Statistics for AWC and RWC
αMean SD Min Q1 Median Q3 Max
Panel A. AWC (DKK)
0.9 50.10 36.60 0 27.60 41.00 58.90 200.00
0.95 67.00 42.20 1.24 38.40 58.60 80.80 224.00
0.99 88.70 48.00 4.56 56.70 79.40 110.00 271.00
Panel B. RWC
0.9 0.77 0.15 0 0.73 0.81 0.89 0.93
0.95 0.87 0.13 0.18 0.83 0.91 0.94 0.98
0.99 0.95 0.09 0.41 0.95 0.98 0.99 1.00
Notes: The table reports the summary statistics for the three
samples of the individual-level estimates of AWC and RWC com-
puted at different target levels of DoR: 0.9, 0.95, and 0.99. The
AWC numbers are in DKK. RWC are measured as proportions.
those estimates. The confidence intervals are computed using bootstrap methods. We rank
subjects based on their point estimates of AWC. The vertical axis represents the percentile
rank of each subject. The uncertainty in the individual-level estimates of AWC stems from
the combined uncertainty in the estimates of risk aversion and noise and tends to increase
with the percentile rank.
Figure 4(Panel B) shows the distribution of the individual-level estimates of RWC and
Table 2(Panel B) presents their summary statistics. At α= 0.9 the distribution is very
flat and has a long left tail resulting in a negative skew. As the target DoR increases, the
distribution of the RWC shifts to right and becomes more concentrated while preserving a
long left tail. The distribution features some observations on the left tail with unusually low
RWC. For some of these outcomes, RWC are less than a half, even for the highest level of α.
In contrast to the AWC, the RWC are extremely high, which implies that while the
AWC are modest in size, these costs represent a substantial portion of the monetary welfare
available in the choice environment. For α= 0.95, the mean RWC are 0.87 and the median
RWC are 0.91, so that around 90% of the relative welfare has to be sacrificed in order to
23
Figure 5: Uncertainty in the Individual-Level Estimates of AWC and RWC
0.00
0.25
0.50
0.75
1.00
0 50 100 150 200 250 300 350 400
AWC (DKK)
Percentile Rank
Panel A.
0.00
0.25
0.50
0.75
1.00
0.00 0.25 0.50 0.75 1.00
RWC
Percentile Rank
Panel B.
Note: The graph shows the point estimates and the confidence intervals for AWC (Panel A) and
RWC (Panel B) computed at α= 0.95. Each point on the graph represents an individual-level
estimate for a given subject. The estimates are ranked from lowest to highest, and the percentile
rank of each subject is shown on the vertical axis. The horizontal error bars show the bootstrapped
95% confidence intervals around the point estimates.
24
rationalize this proportion of choices. For 50% of the subjects, the RWC lie within 0.83 and
0.94 at this level of DoR. As the target level of DoR increases, the mean RWC increase even
further. For α= 0.99, the mean RWC reach 0.95, and the median RWC reach 0.98: almost
all the welfare must be sacrificed in this case.27
The RWC numbers also show significant variation across subjects. At α= 0.95 the
smallest amount of RWC is 0.18, while the maximum amount is 0.98, which is roughly 1.13
times as large as the mean amount. The standard deviation at this level of DoR is 0.13.
At α= 0.99 the minimum RWC is slightly below a half, 0.41, while the maximum amount
becomes 1, and the standard deviation is 0.09.
There is considerable uncertainty in the RWC at the individual level, just as we found
for the AWC. Figure 5(Panel B) shows the point estimates of RWC (at α= 0.95) for each
subject along with the 95% confidence interval around those estimates. Contrary to the
AWC, however, the uncertainty in the RWC is highest at the lower percentile ranks. This
uncertainty tends to decrease with the percentile rank.
The preceding analysis allow us to formulate the following result.
Result 1. The welfare costs are low in terms of everyday economic activity, but are sub-
stantial for the choice environment in which they occurred.
Comparing our results to Choi et al. (2014), we find that the subjects in our sample
require a larger fraction of the total monetary welfare to rationalize their choices. This
difference can be explained by the differences in methods. The GARP-based measure used
by Choi et al. is well-known for its relatively mild requirements on choice consistency (Beatty
and Crawford,2011).28 For example, their primary measure does not even require choices
to satisfy first-order stochastic dominance.
27 Rationalizing all the choices would definitionally require RWC of 1 for every subject with a non-zero
noise, however small, which is the reason to use 0.99 as the highest level of α, and not 1.
28 Another potential explanation could be that there are systematic differences in the samples used.
The Choi et al. (2014) experiment was conducted in Netherlands, while our experiment was conducted in
Denmark. We believe this explanation to be unlikely a priori. The results in Blow et al. (2008) support our
claim, employing a revealed preference approach and showing that Danes are generally consistent in other
choice domains.
25
3.4 Marginal Welfare Costs
So far we have looked at the distributions of AWC and RWC for only three levels of the target
DoR. This analysis does not tell us how quickly welfare costs grow as the rationalizability
requirements become tighter, and in general what the shape of the costs as functions of α
is. Figure 6provides an answer to these questions by showing the median welfare costs as
functions of αacross all the subjects in the sample, with the dashed lines corresponding to
the 5% and 95% empirical quantiles. The lowest possible target DoR in our context is 0.5,
since the choice is binary. However, the welfare costs stay at 0 for the median subject until
αcrosses the 0.62 mark.
Figure 6: Welfare Costs as Functions of the Target DoR
0
50
100
150
200
0.5 0.6 0.7 0.8 0.9 1.0
α
AWC (DKK)
Panel A.
0.00
0.25
0.50
0.75
1.00
0.5 0.6 0.7 0.8 0.9 1.0
α
RWC
Panel B.
Note: The graph shows the AWC (Panel A) and the RWC (Panel B) as functions of the target
DoR (α). The black solid lines are the median welfare costs for each level of α. The dashed lines
below (above) the solid line represent the 5% (95%) quantiles of the welfare costs.
Panel A on Figure 6shows the graphs of the AWC in relation to α. The median AWC
tend to be a convex function of α: at first increasing the DoR requires relatively little AWC,
but as the target becomes higher, each additional percentage point of DoR costs more and
more in terms of AWC. The graph for the RWC on Panel B of Figure 6is in a sense the mirror
image of the AWC. The RWC tends to be a concave function of α. For small values of DoR
extra percentage points of change require high welfare costs, but as the target increases these
extra points become less costly in relative terms. These observations allow us to formulate
26
the next result.
Result 2. The marginal absolute (relative) welfare costs are increasing (decreasing) with the
increase in the target DoR.
This result can be explained by the way our measures of welfare costs are computed.
Starting from a given default DoR, ρµ, 0), we gradually increase εuntil the DoR reaches
the target, ρ(ˆµ, ε) = α. The low marginal AWC at low αtargets imply there are many choices
that can be easily rationalized by small ε, since the difference in the certainty equivalents be-
tween the alternatives must be low. At high target DoR more choices have to be rationalized,
but no “easily rationalizable” choices are left. Increasing DoR requires tapping into choice
pairs with higher differences in certainty equivalents, and hence higher marginal AWC. The
implications for the RWC graphs are the converse. At low target DoR the marginal RWC are
high, since rationalizing many choices with small differences in certainty equivalents requires
the whole difference. At high targets fewer such choice pairs remain and the marginal RWC
decrease.
3.5 Relationship Between the Measures
We now turn to the relationship between the two welfare costs measures and the default
DoR (DDoR). We ask whether people with lower AWC also have lower RWC, and formally
test the previous observation that people with lower DDoR tend to have higher costs. The
motivation behind these questions is that it is intuitive to expect the positive relation between
AWC and RWC. It does not follow, however, directly from the method of their computation.
Only if preferences are held constant must higher AWC imply higher RWC, but there is
no such prediction when preferences are not constant, as is typically the case when making
comparisons across subjects. Likewise, even though it is natural to expect that people with
higher DDoR have lower costs we cannot formally expect this observation to hold a priori.
Figure 7(Panel A) shows a scatterplot of the RWC (y-axis) against the AWC (x-axis)
27
Figure 7: Relationship Between AWC, RWC, and DDoR
0.25
0.50
0.75
1.00
0 50 100 150 200
AWC (DKK)
RWC
Panel A.
0
50
100
150
200
0.5 0.6 0.7 0.8 0.9 1.0
DDoR
AWC (DKK)
Panel B.
0.25
0.50
0.75
1.00
0.5 0.6 0.7 0.8 0.9 1.0
DDoR
RWC
Panel C.
Note: The graph shows the scatterplots between three pairs of measures: AWC, RWC, and DDoR.
The welfare costs are evaluated at α= 0.95. The dots represent individual subjects. The dashed
lines are the smooth fitted lines estimated using local polynomial regressions, and the shaded regions
are the estimated 95% confidence intervals.
computed at α= 0.95.29 A clear positive association between the two measures can be
observed. The Kendall rank correlation between the two measures is 0.45 (the ranking of
subjects by the two measures is the same 73% of the time) and is highly significant, with
p-value <0.001. The relation between RWC and AWC is non-linear, and has a concave
shape.
Figure 7(Panel B) shows the scatterplot of the AWC (y-axis) against the DDoR (x-axis),
and confirms our earlier observation from the analysis of marginal costs. There is a moderate
negative association between the DDoR and the AWC. The Kendall rank correlation between
the two measures is 0.5 (the ranking of subjects by the two measures is the opposite 75%
of the time) and is highly significant, with p-value <0.001. The relation between them is
again non-linear, and has a convex shape.
Figure 7(Panel C) shows the scatterplot of the RWC (y-axis) against the DDoR (x-axis).
We can immediately see a very tight negative association between the two measures. The
Kendall rank correlation between them is 0.85 (the ranking of subjects by the two measures
is the opposite 93% of the time) and is highly significant, with p-value <0.001. The relation
is again slightly non-linear and has a concave shape.
29 The results in this section remain quantitatively similar if we use α= 0.9 or 0.99.
28
These observations allow us to formulate the following result.
Result 3. People with higher absolute welfare costs tend to have higher relative welfare costs.
People with higher default degree of rationalizability tend to have lower absolute welfare costs
and relative welfare costs.
This result implies that there is a certain degree of consistency between the measures we
introduce. Moreover, this consistency works in the way we expect. This is a nice property, but
it could not have been deduced from the method by which these measures are constructed.
If risk preferences were the same across subjects, higher AWC must have implied higher
RWC, but we cannot say much about the case when preferences and noise are different
across subjects. It is possible, that a subject with high AWC has preferences such that the
differences in the certainty equivalents are even higher, and the RWC are actually low. We
do, in fact, observe such cases. But the general tendency is for the subjects to have the same
ordering, whether it is measured according to the absolute or relative measure of welfare
costs.
There is also a negative relationship between the DDoR and the welfare two costs mea-
sures. This implies that people who make more consistent choices, measured by the DDoR,
also require less welfare costs to rationalize their choices. This is an intuitive property, but
it is hard to see a priori why it should hold even though the data indicate that it does, with
the relation between the default DoR and the RWC being particularly strong. The relative
strength of this relationship, compared to the relationship with the AWC can be partially
attributed to the fact that both the default DoR and the RWC are relative measures defined
on the unit interval. Nonetheless, such a strong relationship is remarkable, given that the
two measures address two very different questions.
3.6 Welfare Costs and Noise
Our approach is in part motivated by the desire to attach an economic meaning to the noise
parameter. It is, therefore, of interest to look at the relationship between the two welfare
29
cost measures we introduced and noise, as well as the relationship between the DDoR and
noise. Higher noise does translate into higher welfare costs if preferences are kept constant,
but no prediction is available for comparisons between subjects, whose preferences are not
kept constant. It is natural to expect, however, that this property should also hold between
subjects. Given the negative association between the DDoR and the costs, it is also natural
to expect that higher noise translates into lower default DoR, but whether it does is an
empirical question.
Figure 8: Relationship Between Welfare Costs and Noise
0
50
100
150
200
-5 0 5 10 15
Log Noise
AWC (DKK)
Panel A.
0.3
0.6
0.9
-5 0 5 10 15
Log Noise
RWC
Panel B.
0.5
0.6
0.7
0.8
0.9
-5 0 5 10 15
Log Noise
DDoR
Panel C.
Note: The graph shows the scatterplots between three measures (AWC, RWC, DDoR) and (the
logarithm of) noise. The welfare costs are evaluated at α= 0.95. The dots represent individual
subjects. The dashed lines are the smooth fitted lines estimated using local polynomial regressions,
and the shaded regions are the estimated 95% confidence intervals. The graphs drop one subject
with log noise higher than 15.
Figure 8shows the scatterplots of (from left to right) the AWC and RWC and the DDoR
(on the y-axis) against the logarithm of noise (x-axis).30 The three panels confirm our
hypotheses. We do see that higher noise is associated with higher AWC and RWC and lower
DDoR, although the strength of this association differs across the measures. It is small,
though statistically significant, for the AWC. The Kendall rank correlation between the two
measures is 0.13 with p-value = 0.003 (the ranking of subjects by the two measures is the
same 57% of the time). The weakness of the association can be seen by the substantial
variation in the AWC at the high values of noise, which means that there are many subjects
30 We truncate the logarithm noise at 15, in order to make the graph more readable. This excludes one
subject.
30
with high estimates of noise but low AWC. The association with the RWC is much stronger.
The Kendall rank correlation between the two measures is 0.48 with p-value <0.001 (the
ranking of subjects by the two measures is the same 74% of the time). Finally, the association
with the DDoR (in absolute terms) is slightly weaker than the association with the RWC, but
much stronger than the association with the AWC. The Kendall rank correlation between
the two measures is 0.41 with p-value <0.001 (the ranking of subjects by the two measures
is the opposite 70% of the time).
A notable feature in these results, most pronounced in the relationship between noise
and the DDoR and the RWC, is that there is an outer boundary that constrains the values.
On Panel B of Figure 8this boundary constrains the values of the RWC from above, and
on Panel C of Figure 8this boundary constrains the values of the DDoR from below. This
pattern suggests that for given noise the RWC (DDoR) cannot be higher (lower) than a
certain value, defined by this boundary.
These findings lead us to the next result.
Result 4. People with higher noise tend to have higher absolute and relative welfare costs and
lower default degree of rationalizability. For any given value of noise there appears to exist a
maximum (minimum) amount of absolute and welfare costs (degree of rationalizability) that
one can have.
The first part of this result confirms our intuitive guesses. We do see some association
between noise and welfare costs, which implies that noise contains some information about
welfare costs and choice consistency, but this information is imprecise. Despite big differences
in noise estimates between some subjects, their RWC need not be that different. Similarly,
some subjects might appear to have high welfare costs based on the noise measure, while in
fact their AWC are not nearly as large.
The second part of the result is unexpected and remarkable. It says that there is a
regularity in the relation between noise, welfare costs and default DoR. This regularity is
in the form of a boundary that constraints the possible values. The existence of such a
31
boundary is likely to be related to the estimation and computation procedures, however it
is not clear why it exists and what determines its shape. We leave this question for further
research.
3.7 Socio-Demographic Covariates of Welfare Costs
We have seen that the estimates of welfare costs vary substantially between subjects in our
sample. Here we attempt to attribute this variability to the observable socio-demographic
characteristics of the subjects. We focus on sex, age, education, work, income, housing,
family, and health characteristics. The demographic covariates are defined as indicator
variables, relative to a base category. The base category is male, age 18–29, vocational
training, employed as a student, household income less than 300,000 DKK, living in an
apartment, owning apartment/house, living alone, no children, has not experienced death,
has not been hospitalized, and not smoking.
Figure 9provides descriptive regression results by plotting the estimates of regression
coefficients along with 95% confidence intervals (using robust standard errors). The model
on Panel A uses the logarithm of AWC, computed at α= 0.95, as the dependent variable,31
ln(AW C)i=constant +β Demographic controlsi+i,
and is estimated using OLS. The model on Panel B uses the RWC, computed at α= 0.95,
as the dependent variable. Since the RWC are defined only on the unit interval, we use a
fractional regression model due to Papke and Wooldridge (1996) to estimate the coefficients.
Several patterns emerge from Figure 9. Females tend to have higher AWC than males.
The RWC, however, are not significantly different between males and females. Welfare costs
tend to be higher for older subjects. The AWC tend to increase monotonically with the
age group, but the effect is not precisely estimated. The RWC are higher for subjects older
31 Using alternative target DoR, 0.9 or 0.99, produces quantitatively similar results.
32
Figure 9: Regression Results
Smoker: Yes
Hospitalized: Yes
Experienced death: Yes
Children: Yes
123Civil status: Lives w/partner
Ownership: Rented
Ownership: Cooperative
Housetype: House
Income: Not reported
Income: over 800k DKK
Income: 500k–800k DKK
Income: 300k–500k DKK
Employment: Retired
Employment: Public servant
Employment: Worker
Education: College (5 yrs)
Education: College (3–4 yrs)
Education: College (<3 yrs)
Education: Low formal
Age: over 50
Age: 40–49
Age: 30–39
Female
-0.4 0.0 0.4
Coefficient
Panel A. Log AWC
-1.0 -0.5 0.0 0.5 1.0
Coefficient
Panel B. RWC
Note: The graph shows descriptive regression results for the AWC (Panel A, OLS) and RWC
(Panel B, fractional regression). Bars correspond to coefficient estimates and error bars show 95%
confidence intervals based on robust standard errors. Number of observations: 217.
33
than 30 years than for younger subjects, but there is no statistically significant difference
between the three age groups above 30 years. College education has a beneficial impact on
the welfare costs relative to vocational training. The effect is most pronounced for the RWC
and subjects with 5 or more years of college. Interestingly, subjects who are employed as
public servants have significantly lower AWC and RWC than subjects employed as students
or workers. Retired subjects tend to have lower AWC and RWC, on average, but the effect
is not statistically different from zero. The effect of income is mixed and not precisely
estimated. Subjects with medium and medium-high levels of income tend to have higher
welfare costs, while subjects with very high levels of income tend to have lower welfare costs.
The type of housing a subject occupies and the type of ownership does not appear to have
a meaningful impact on welfare costs. Similarly, the effect of parenting status is small and
not statistically significant, except for the effect having children on the RWC. Subjects show
some systematic variation by their health status with the effects most pronounced for RWC.
For instance, smokers tend to have higher RWC than non-smokers.
These observations lead us to the following result.
Result 5. Having higher welfare costs is associated with higher age, lower education, and
particular employment status. The RWC are not significantly different for males and females,
although the AWC for females tend to be higher.
Overall, even the rich set of socio-demographic characteristics that we use does little to
explain the observed variance in welfare costs. The regression for the AWC, for instance,
is able to explain only 12% of the observed variation. After correcting for the number
of covariates included, the R2actually becomes negative. Such low explanatory power of
socio-demographic characteristics for elicited economic variables is typical in the literature
(l’Haridon et al.,2018;Noussair et al.,2014;Choi et al.,2014;von Gaudecker et al.,2011).
One explanation for low predictive power of socio-demographic characteristics is the sampling
error in the estimates on the left-hand side.32 On the other hand, part of the heterogeneity
32 Using weighted OLS in the AWC regression in which weights are proportional to the inverses of the
34
in the estimates of welfare costs that we observe might be truly idiosyncratic, which in our
view is not necessarily an undesirable property as suggested by some, such as l’Haridon et al.
(2018). If an elicited economic quantity (such as welfare costs, in our case) could be perfectly
decomposed into a linear combination of socio-demographic characteristics, this quantity
would have nothing to contribute to explaining variation in other behavioral outcomes.
4 Related literature
Our approach connects to a large theoretical literature on stochastic choice, which we briefly
summarize. The early work on stochastic choice dates back to Fechner (1860) and Thurstone
(1927). It was subsequently developed into the Random Utility Model (RUM) by Marschak
(1960) and summarized by McFadden (2001). Luce (1959) introduced and axiomatized the
strong utility (or multinomial logit) model, as well as other models of stochastic choice.
McFadden (1976) established necessary and sufficient conditions under which a RUM is
equivalent to the multinomial logit model.
Wilcox (2011) extends the standard multinomial logit model by allowing for the noise
heterogeneity that is caused by the range of monetary stakes in a choice context. This exten-
sion allows one to preserve the deterministic notion of being more risk averse in a stochastic
setting. The stronger utility model developed by Blavatskyy (2014) also allows for noise
heterogeneity, but focuses on preserving the first-order stochastic dominance relation in a
stochastic choice setting. Gul, Natenzon, and Pesendorfer (2014) modify the multinomial
logit model by considering the attributes of choice alternatives rather than alternatives them-
selves, to address some of the criticism of the original formulation. Apesteguia, Ballester,
and Lu (2017) characterize the RUM that satisfies a single-crossing property.
Conceptually, our measures are similar to the Critical Cost Efficiency Index (CCEI) of
Afriat (1972), which is used to evaluate the degree of consistency with the Generalized
Axiom of Revealed Preference (GARP). Just like our relative cost measure, CCEI is defined
squared standard errors substantially improves fit.
35
on the unit interval, and its complement shows what proportion of monetary value an agent
should be allowed to waste in order to rationalize her choices by some utility function. While
GARP provides qualitative statements, we put more structure on it in a flexible manner to
complement it and provide quantitative evidence.
Viewing our approach as a structural extension of GARP allows us to position our ap-
proach again in a broader methodological setting. Ross (2014, ch. 4) carefully lays out
the full case for interpreting economic experimentation as an application of the intentional
stance of Dennett (1987), noted earlier. This is the methodology that Ross (2014) calls
“neo-Samuelsonian,” a label that tries to nudge economists toward seeing that the inten-
tional stance is what they have always been doing when they applied Revealed Preference
Theory to actual, finite, choice data. In other words: our approach is not novel, exotic
economic methodology. Instead we view it as just a sophisticated, structural interpretation
of the good old-time religion for economists.
The intuition behind the computation of our measures also links it to a literature on payoff
dominance in experiments (Harrison,1994,1992;Harrison and Morgan,1990;Harrison,
1989). This literature shows that allowing for small deviations from optimal behavior, just
as we do, allows one to rationalize supposedly anomalous effects observed in experimental
studies.
Harrison and Ng (2016,2018) use an approach similar to ours in order to evaluate the loss
of consumer surplus resulting from suboptimal insurance choices. Harrison and Ross (2018)
apply the same approach to evaluate suboptimal portfolio investments. Their measure of
lost consumer surplus is similar to our AWC measure, with both being based on computing
certainty equivalents. One key difference, however, is that these studies use two experimental
tasks: one for preference estimation and the other for welfare evaluation, while we rely on a
single task to estimate welfare costs resulting from stochastic choice. The approach that we
take in this study does not rely on an independent risk metric, as is the case in Harrison and
Ng (2016,2018) and Harrison and Ross (2018), but rather relies on a specific noise structure
36
to “bootstrap” a measure of welfare costs.
Our approach is related to studies that estimate structural models of choice under risk
and over time. Holt and Laury (2002) study subjects’ choices under risk in a laboratory
experiment. Subjects make choices between a “safe” and a “risky” lottery across different
pairs of lotteries, in which the probabilities of lottery outcomes vary from one pair to the
next. HL estimate the Expected Utility model with a flexible Expo-Power utility function
using the strict utility model of stochastic choice.33 Andersen, Harrison, Lau, and Rutstr¨om
(2008) also use the strict utility model to structurally estimate risk and time preferences
of a representative sample of the Danish population. They note that noise estimates are
higher in the risk task than in the discounting task. von Gaudecker et al. (2011) uses a
representative sample of the Dutch population to estimate subjects’ risk preferences using
a model of stochastic choice that is a hybrid between the multinomial logit and tremble
models, and features two measures of choice randomness due to noise: a Fechner parameter
that is common to all subjects, and a “tremble” parameter that is specific to each subject.34
While these studies typically focus on estimates of risk and time preferences, and do not
interpret the estimates of the stochastic part, we explicitly focus on the estimates of the
stochastic part and provide a systematic approach to economically interpret the estimates
of choice randomness. Finally, Bland (2018) considers mixture specification over pooled
choices, contrasting one “rational” model as one of the data generating processes (DGP)
with a “behavioral” model as the other DGP. He then calculates CE of choices using the
deterministic core of the “rational” model DGP, thereby evaluating potential welfare losses
from using a “behavioral” DGP as well as the existence of noise for both DGP. We reject
the simplistic identification of one model as “rational” and the explicit assumption that the
“behavioral” model is therefore “irrational.” But the general logic of allowing the estimated
33The strict utility model of Luce (1959) differs from the multinomial logit model in the way the noise
parameter enters choice likelihoods.
34 Their model also allows for random coefficients on core parameters for risk preferences. Although this
specification allows for stochastic variation in those parameters, it is conventionally interpreted as reflecting
unobserved heterogeneity in these parameters across subjects, “unobservable” only in the sense that the
researcher cannot account for the variation with observable characteristics of the subject or task.
37
structural model of noise to provide a basis for welfare evaluation is consistent with our
approach.
We provide economic measures of choice randomness (or consistency), which link to
studies on the quality of decision-making. Choi et al. (2007) study decision-making under
risk in a laboratory experiment in which they present subjects with convex budget sets
for two Arrow securities. This design allows them to gauge the subjects’ decision-making
quality using a measure of GARP-consistency, a standard technique in the revealed preference
approach to consumer demand. They find that subjects’ behavior is highly consistent with
GARP. Choi et al. (2014) expand the analysis by using a representative panel of the Dutch
population. They also find a high degree of GARP-consistency in risky choices, which
varies, however, with education, sex, and age. Beatty and Crawford (2011) show that while
behavior in a wide range of situations is highly GARP-consistent, this might be a result of
a misspecified measure of consistency. They propose an alternative to the traditional CCEI
measure, which is based on predictive success, and show that the CCEI measures of GARP-
consistency are overinflated, and hence that the actual consistency of choices is much lower.
Hey (2001) studies decision-making quality in a laboratory experiment on choice under risk
and asks whether choice consistency improves with experience. He finds mixed evidence of
a positive effect of experience on choice consistency. We rely on a parametric measure of
choice consistency and find a lower degree of consistency than in the studies that use the
non-parametric revealed preference approach.
Finally, our approach is also related to recent literature on rational inattention. Matˆejka
and McKay (2015) show that when an agent faces information costs, optimal behavior is
stochastic choice, and that under certain conditions choice likelihoods are represented by
the multinomial logit specification. Cheremukhin, Popova, and Tutino (2015) apply a model
of rational inattention to risky choices and estimate the shape of the cost-of-information
function in a laboratory experiment with student subjects. Caplin and Dean (2015) develop
a revealed preference test of rational inattention theories with general cost-of-information
38
functions. Since the noise parameter in the rational inattention models has the interpretation
of marginal information costs, our method allows one to convert these costs into monetary
or percentage terms.
5 Conclusion
Stochastic choice has become an active area of both theoretical and empirical research. While
the existing literature mainly focuses on the sources of choice randomness, its economic
consequences are less well understood. We develop tools to assess the economic significance
of noise and apply them to a sample from the general Danish population in an artefactual
field experiment.
We introduce three interconnected concepts: rationalizing imperfection, optimal region,
and degree of rationalizability. Fixing the degree of rationalizability at a certain target
level, we vary the amount of imperfection, which in turn affects the optimal region, to
make the proportion of subjects’ choices falling in the optimal region equal the target level.
This amount of imperfection represents the welfare costs, or monetary welfare allowed to be
wasted, that is required to rationalize by a model a given proportion of choices. The resulting
welfare costs can be expressed both in absolute (dollar) and in relative (to the actual stakes
of the choice environment) terms.
We compute the absolute welfare costs and relative welfare costs at the individual level
in an experiment with binary-choice lotteries. Several patterns emerge from our analysis,
some of which coincide with previous findings, and some of which are new. We find that
the AWC are not economically significant in our sample, while the RWC are economically
significant. In other words, the welfare costs are tiny if viewed from a broad perspective of
economic activity, but they are substantial if viewed from the perspective of this particular
choice experiment. As compared to Choi et al. (2014), who employ a relative measure based
on the consistency with GARP, our estimates of RWC are much larger. We attribute the
39
difference in results to the difference in the methods, with our method imposing stricter
requirements.
Since our welfare costs measures depend on the target level of rationalizability α, we study
the shape of the relation between αand these welfare costs. We find that the AWC increase in
αat an increasing speed, while the RWC increase in αat a decreasing speed. The difference in
these two relations is explained by the way our method of computation works. Subjects with
higher AWC tend to have higher RWC. Also, a lower DDoR is associated with higher AWC
and RWC: subjects who start out with low default degree of rationalizability require a higher
cost to reach a given degree of rationalizability. Looking at the relationship between our
cost measures and raw estimates of noise reveals that they are positively associated, though
our measures do not have such a wide range, which allows for sensible comparisons across
subjects and allows us to make judgments about the magnitudes of choice inconsistencies.
The analysis of observable heterogeneity and its role in predicting welfare costs suggests
patterns similar to those reported by Choi et al. (2014). We find that welfare costs increase
with age, decline with education, and are lower for certain occupations.
Finally, we take seriously the need for consistent methodological and philosophical po-
sitions when it comes to undertaking behavioral welfare economics. The reason is simple:
one cannot question the consistency of observed choices by agents on the one hand and then
turn around and effortlessly infer the preferences of those agents on the other hand. This
isolates the deep normative challenge raised by the core descriptive insight of behavioral
economics, as stressed by Ross (2014, ch. 4), Infante, Lecouteux, and Sugden (2016), and
Harrison and Ross (2018,§5). Dennett (1987)’s intentional stance, as applied to economics
by Ross (2014)’s “neo-Samuelsonian” methodology, provides a general and consistent ap-
proach to address this challenge, and permits concrete applications illustrated by Harrison
and Ng (2016,2018), Harrison and Ross (2018) and the present study.
40
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44
Appendices
A Robustness Checks
Here we present additional results derived from alternative assumptions about risk prefer-
ences and stochastic choice.
First, we consider an alternative to the EUT, the Rank-Dependent Utility (RDU) model
due to Quiggin (1982), which allows for probability weighting. The RDU model has been
used extensively in applied and theoretical work. Under this alternative assumption the
aggregate utilities of the lotteries are computed as
U(a;γu, γq) =
=
k
X
i=1 ωq(1)(a) + . . . +q(i)(a); γqωq(1)(a) + . . . +q(i1)(a); γq×
×ux(i)(a); γu,
where ω:[0,1] 7→ [0,1] is the probability-weighting function, and outcomes are ranked
from highest x(1) to lowest x(k), with corresponding probabilities. We assume that ωis the
two-parameter (Prelec,1998) probability weighting function,35
ω(q;γ1
q, γ2
q) = exp(γ2
q(ln q)γ1
q).
Figure A.1 shows the calculated absolute and relative welfare costs under the assumption
of the RDU model for each individual. Figure A.1 shows that the distributions look very
similar to those under EUT, Figure 4.
Taking a closer look at the differences between the EUT and RDU-based calculations,
we can see from Figure A.2a that the AWC calculated using the EUT model are lower. For
35We do not restrict the shape parameter γ1
qto the unit interval, and thus do not impose an inverse-S
shape on the probability weighting function.
A.1
0.000
0.002
0.004
0.006
0 200 400
Absolute welfare costs
density
α
0.9
0.95
0.99
(a)
0
5
10
0.00 0.25 0.50 0.75 1.00
Relative welfare costs
density
α
0.9
0.95
0.99
(b)
Figure A.1: Absolute and Relative Welfare Costs for Three Levels of α, RDU.
α= 0.9 the difference in the medians between the AWC calculated using EUT vs. RDU is
59.04 (Wilcoxon signed rank test, p-value <0.001). The mean of the differences is 84.89
DKK (approximately 13 USD): RDU-based AWC are almost 3 times higher on average.
The RWC, however, are slightly higher under EUT, as shown in Figure A.2b. The
difference in the medians between the RWC calculated using EUT vs. RDU is 0.02 (Wilcoxon
signed rank test, p-value = 0.02). The mean of the differences is 0.03. The difference in the
RWC for RDU and EUT disappears at higher values α, while the difference in the AWC
persists. All the other qualitative results on marginal welfare costs, relations between the
measures, and observable heterogeneity hold under the RDU assumption.
0.000
0.005
0.010
0.015
0.020
0 200 400
Absolute welfare costs
density
model
EUT
RDU
(a)
0
1
2
3
4
0.00 0.25 0.50 0.75 1.00
Relative welfare costs
density
model
EUT
RDU
(b)
Figure A.2: Absolute and Relative Welfare Costs for EUT vs. RDU, α= 0.9.
A.2
Second, we consider a different specification for the utility function under EUT, an expo-
power (EP) utility, which generalizes the CRRA and CARA utility functions
u(x;γa, γr) = 1exp(γax1γr)
γa
,
where γaand γrare the two parameters to be estimated. This specification does not do so
well in modeling subjects’ risk preferences in our data. For a large (40%) fraction of subjects
the estimation procedure yields unreasonably high parameter values, which impedes the
calculation of certainty equivalents and welfare costs. We use CRRA specification for these
subjects when presenting the results on Figure A.3. They look very similar to the baseline
specification with the CRRA utility function.
0.000
0.005
0.010
0 100 200 300
Absolute welfare costs
density
α
0.9
0.95
0.99
(a)
0
5
10
15
20
25
0.00 0.25 0.50 0.75 1.00
Relative welfare costs
density
α
0.9
0.95
0.99
(b)
Figure A.3: Absolute and Relative Welfare Costs for 3 Levels of α, EP.
Looking at the differences between the AWC calculated under the two utility specifica-
tions, our baseline specification again provides lower values (see Figure A.4a). The difference
in the medians between the AWC calculated using CRRA vs. EP is 16.17 (Wilcoxon signed
rank test, p-value <0.001), for α= 0.9. The mean of the differences is 24.7 DKK (approx-
imately 4 USD). The AWC under the EP utility function are roughly 60% higher than in
the baseline, which is even higher than in the case of the RDU model as an alternative.
At the same time there are no significant differences in the RWC between the two utility
specifications (Wilcoxon signed rank test, p-value = 0.52). The same pattern of results hold
A.3
for other values of α. Under the EP-utility assumption the marginal welfare costs have a
similar shape, but the association between the measures becomes weaker, as do the effects
of observable heterogeneity.
0.000
0.005
0.010
0.015
0.020
0 100 200
Absolute welfare costs
density
model
CRRA
EP
(a)
0
1
2
3
4
5
0.00 0.25 0.50 0.75
Relative welfare costs
density
model
CRRA
EP
(b)
Figure A.4: Absolute and Relative Welfare Costs for CRRA vs. EP, α= 0.9.
Finally, we look at an alternative stochastic choice specification, the contextual utility
model due to Wilcox (2011), which allows for a heterogeneous noise term and preserves the
“more risk averse” relation in the stochastic domain. This specification of noise has been
shown by (Wilcox,2015) to have good out-of-sample predictive power. Under the assumption
of contextual utility the choice probabilities become
p(a2;γ, µ) = Λ U(a2;γ)U(a1;γ)
µu(x(1);γ)u(x(k);γ)!,
where we drop the index for the decision round, and p(a1;γ, µ) = 1 p(a2;γ, µ). As before,
x(1) and x(k)denote the highest and lowest outcomes, but this time they are defined only
among the outcomes that occur with positive probabilities, and outcomes are ranked across
both lotteries in the choice.
Figure A.5 shows the calculated AWC and RWC under the assumption of contextual
utility. These graphs, again, look very similar to those under EUT and no contextual utility
(Figure 4), except that the right tails in the distributions of the AWC become thicker.
A.4
0.000
0.005
0.010
0.015
0.020
0 100 200 300 400 500
Absolute welfare costs
density
α
0.9
0.95
0.99
(a)
0
10
20
30
40
0.00 0.25 0.50 0.75 1.00
Relative welfare costs
density
α
0.9
0.95
0.99
(b)
Figure A.5: Absolute and Relative Welfare Costs for Three Levels of α, Contextual Utility.
Figure A.6 contrasts the AWC and RWC for the baseline and alternative specifications
of noise. The densities of the AWC are very much alike, except for a thicker right tail in
the case of contextual utility, which leads to higher welfare costs. The difference in the
medians between the AWC calculated using non-contextual vs. contextual models is 1.87
(Wilcoxon signed rank test, p-value <0.001). The mean of the differences is 16.33 DKK
(approximately 2 USD). This result is comparable to the non-contextual noise specification
with RDU as an alternative. Again, there is no significant difference between the RWC for
these two models (Wilcoxon signed rank test, p-value 0.47). All the results reported for
the baseline model hold in the case of contextual utility model as well.
0.000
0.005
0.010
0.015
0.020
0 100 200 300 400 500
Absolute welfare costs
density
model
Contextual
Non−Contextual
(a)
0
1
2
3
4
0.00 0.25 0.50 0.75 1.00
Relative welfare costs
density
model
Contextual
Non−Contextual
(b)
Figure A.6: Absolute and Relative Welfare Costs for Non-contextual vs. Contextual Utility,
α= 0.9.
A.5
B Proofs
Consider an implicit function ρ(µ, ε) = α. From the implicit function theorem, it follows
that
=∂ρ/∂µ
∂ρ/∂ ε .
The denominator of this expression is
∂ρ
∂ε =
∂ε Za
h(ε)
a
l(ε)
p(a)da =p(a
h(ε))a0
h(ε)p(a
l(ε))a0
l(ε)>0,
since a0
h(ε)>0, and a0
l(ε)60.
In order to show the sign of the numerator, we restrict our attention to the binary choice
case, since it is the setting of our primary interest. Recall that
p(a2;γ, µ) = Λ U(a2;γ)U(a1;γ)
µ.
Then
∂p(a2;γ , µ)
∂µ = Λ0U(a2;γ)U(a1;γ)
µ(U(a2;γ)U(a1;γ))(µ2)<0,
since alternative a2gives the highest certainty equivalent by our assumption. Therefore,
∂ρ
∂µ =
∂p(a2;γ )
∂µ , ε < m,
0, ε >m,
so that ∂ρ/∂µ 60. Together the two results imply that dε/dµ >0.
A.6
C Additional Tables
Table C.1: The Battery of Lotteries
ID La1 Lp1 La2 Lp2 La3 Lp3 La4 Lp4 Ra1 Rp1 Ra2 Rp2 Ra3 Rp3 Ra4 Rp4
1 450 0.50 1,350 0 2,250 0.50 0 0 450 0.10 1,350 0.80 2,250 0.10 0 0
2 450 0.50 1,350 0 2,250 0.50 0 0 450 0 1,350 1 2,250 0 0 0
3 450 0.10 1,350 0.80 2,250 0.10 0 0 450 0 1,350 1 2,250 0 0 0
4 450 0.70 1,350 0 2,250 0.30 0 0 450 0.50 1,350 0.40 2,250 0.10 0 0
5 450 0.70 1,350 0 2,250 0.30 0 0 450 0.40 1,350 0.60 2,250 0 0 0
6 450 0.50 1,350 0.40 2,250 0.10 0 0 450 0.40 1,350 0.60 2,250 0 0 0
7 450 0.40 1,350 0 2,250 0.60 0 0 450 0.10 1,350 0.75 2,250 0.15 0 0
8 450 0.40 1,350 0 2,250 0.60 0 0 450 0 1,350 1 2,250 0 0 0
9 450 0.30 1,350 0 2,250 0.70 0 0 450 0.15 1,350 0.25 2,250 0.60 0 0
10 450 0.10 1,350 0.75 2,250 0.15 0 0 450 0 1,350 1 2,250 0 0 0
11 450 0.70 1,350 0 2,250 0.30 0 0 450 0.60 1,350 0.25 2,250 0.15 0 0
12 450 0.70 1,350 0 2,250 0.30 0 0 450 0.50 1,350 0.50 2,250 0 0 0
13 450 0.60 1,350 0.25 2,250 0.15 0 0 450 0.50 1,350 0.50 2,250 0 0 0
14 450 0.40 1,350 0 2,250 0.60 0 0 450 0.20 1,350 0.60 2,250 0.20 0 0
15 450 0.40 1,350 0 2,250 0.60 0 0 450 0.10 1,350 0.90 2,250 0 0 0
16 450 0.20 1,350 0.60 2,250 0.20 0 0 450 0.10 1,350 0.90 2,250 0 0 0
17 450 0.60 1,350 0 2,250 0.40 0 0 450 0.50 1,350 0.30 2,250 0.20 0 0
18 450 0.30 1,350 0 2,250 0.70 0 0 450 0 1,350 0.50 2,250 0.50 0 0
19 450 0.60 1,350 0 2,250 0.40 0 0 450 0.40 1,350 0.60 2,250 0 0 0
20 450 0.50 1,350 0.30 2,250 0.20 0 0 450 0.40 1,350 0.60 2,250 0 0 0
21 450 0.25 1,350 0 2,250 0.75 0 0 450 0.10 1,350 0.60 2,250 0.30 0 0
22 450 0.25 1,350 0 2,250 0.75 0 0 450 0 1,350 1 2,250 0 0 0
23 450 0.10 1,350 0.60 2,250 0.30 0 0 450 0 1,350 1 2,250 0 0 0
24 450 0.50 1,350 0.20 2,250 0.30 0 0 450 0.40 1,350 0.60 2,250 0 0 0
25 450 0.55 1,350 0 2,250 0.45 0 0 450 0.40 1,350 0.60 2,250 0 0 0
26 450 0.55 1,350 0 2,250 0.45 0 0 450 0.50 1,350 0.20 2,250 0.30 0 0
27 450 0.15 1,350 0.25 2,250 0.60 0 0 450 0 1,350 0.50 2,250 0.50 0 0
28 450 0.15 1,350 0.75 2,250 0.10 0 0 450 0 1,350 1 2,250 0 0 0
29 450 0.60 1,350 0 2,250 0.40 0 0 450 0 1,350 1 2,250 0 0 0
30 450 0.60 1,350 0 2,250 0.40 0 0 450 0.15 1,350 0.75 2,250 0.10 0 0
31 135 0.55 1,620 0.25 1,890 0.20 0 0 135 0.55 1,215 0.25 2,430 0.20 0 0
32 810 0.40 675 0.40 1,620 0.20 0 0 810 0.40 405 0.40 2,025 0.20 0 0
33 1,485 0.40 675 0.40 1,620 0.20 0 0 1,485 0.40 405 0.40 2,025 0.20 0 0
34 2,160 0.40 675 0.40 1,620 0.20 0 0 2,160 0.40 405 0.40 2,025 0.20 0 0
35 675 0.70 1,485 0.10 2,835 0.20 0 0 675 0.70 945 0.10 3,375 0.20 0 0
36 1,620 0.70 1,485 0.10 2,835 0.20 0 0 1,620 0.70 945 0.10 3,375 0.20 0 0
37 2,565 0.70 1,485 0.10 2,835 0.20 0 0 2,565 0.70 945 0.10 3,375 0.20 0 0
38 3,510 0.70 1,485 0.10 2,835 0.20 0 0 3,510 0.70 945 0.10 3,375 0.20 0 0
39 0 0.50 540 0.10 540 0.40 0 0 0 0.50 0 0.10 810 0.40 0 0
40 540 0.50 540 0.10 540 0.40 0 0 540 0.50 0 0.10 810 0.40 0 0
41 1,080 0.50 540 0.10 540 0.40 0 0 1,080 0.50 0 0.10 810 0.40 0 0
42 945 0.55 1,620 0.25 1,890 0.20 0 0 945 0.55 1,215 0.25 2,430 0.20 0 0
43 1,620 0.50 540 0.10 540 0.40 0 0 1,620 0.50 0 0.10 810 0.40 0 0
44 540 0.50 1,080 0.10 1,080 0.40 0 0 540 0.50 540 0.10 1,350 0.40 0 0
45 1,080 0.50 1,080 0.10 1,080 0.40 0 0 1,080 0.50 540 0.10 1,350 0.40 0 0
46 1,620 0.50 1,080 0.10 1,080 0.40 0 0 1,620 0.50 540 0.10 1,350 0.40 0 0
47 2,160 0.50 1,080 0.10 1,080 0.40 0 0 2,160 0.50 540 0.10 1,350 0.40 0 0
48 1,755 0.55 1,620 0.25 1,890 0.20 0 0 1,755 0.55 1,215 0.25 2,430 0.20 0 0
49 2,565 0.55 1,620 0.25 1,890 0.20 0 0 2,565 0.55 1,215 0.25 2,430 0.20 0 0
50 135 0.65 945 0.20 1,485 0.15 0 0 135 0.65 810 0.20 1,620 0.15 0 0
51 675 0.65 945 0.20 1,485 0.15 0 0 675 0.65 810 0.20 1,620 0.15 0 0
52 1,215 0.65 945 0.20 1,485 0.15 0 0 1,215 0.65 810 0.20 1,620 0.15 0 0
53 1,755 0.65 945 0.20 1,485 0.15 0 0 1,755 0.65 810 0.20 1,620 0.15 0 0
54 135 0.40 675 0.40 1,620 0.20 0 0 135 0.40 405 0.40 2,025 0.20 0 0
55 0 0 0 0 0 0 1,200 1 0 0 0 0 975 0.50 1,440 0.50
56 0 0 0 0 0 0 1,275 1 0 0 0 0 1,155 0.50 1,410 0.50
57 0 0 0 0 0 0 450 1 0 0 0 0 225 0.50 690 0.50
58 0 0 0 0 0 0 1,950 1 0 0 0 0 1,725 0.50 2,190 0.50
59 0 0 0 0 0 0 2,025 1 0 0 0 0 1,905 0.50 2,160 0.50
60 0 0 0 0 0 0 225 1 0 0 0 0 105 0.50 360 0.50
Notes. The columns are coded as follows: “L” and “R” denote left and right lottery, “a” denotes amounts (in DKK) and “p”
denotes probabilities. The amounts in the table are baseline amounts. In addition to these amounts, 1.5x and 2x amounts
were used. The sub jects were randomized across the baseline, 1.5x and 2x amounts.
A.7
D Additional Graphs
Figure D.7: Distributions of AWC and RWC in the Subset of Subjects
α= 0.99
α= 0.95
α= 0.9
0 50 100 150 200 250
0.000
0.005
0.010
0.015
0.020
0.000
0.005
0.010
0.015
0.020
0.000
0.005
0.010
0.015
0.020
AWC (DKK)
Density
Panel A.
α= 0.99
α= 0.95
α= 0.9
0.4 0.6 0.8 1.0
0
10
20
30
0
10
20
30
0
10
20
30
RWC
Density
Panel B.
Note: The graph shows the distributions of the individual-level estimates of AWC (Panel A) and
RWC (Panel B) for three target levels of α: 0.9, 0.95, and 0.99. The sample is restricted to
include only the subjects for whom the estimation procedure successfully converged. The bars
are the histograms and the smooth lines are the kernel density estimates. The dashed lines show
the medians of the distributions. The AWC numbers are in DKK. For the RWC, we truncate the
support at 0.4 to improve readability of the graph. This results in dropping 8 observations for
which the RWC are below 0.4.
A.8
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