Estimating Risk Preferences in the Field - Francesca Molinari
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Journal of Economic Literature 2018, 56(2), 1–64
https://doi.org/10.1257/jel.20161148
Estimating Risk Preferences
in the Field†
Levon Barseghyan, Francesca Molinari, Ted O’Donoghue,
and Joshua C. Teitelbaum*
We survey the literature on estimating risk preferences using field data. We
concentrate our attention on studies in which risk preferences are the focal object and
estimating their structure is the core enterprise. We review a number of models of risk
preferences—including both expected utility (EU) theory and non-EU models—that
have been estimated using field data, and we highlight issues related to identification
and estimation of such models using field data. We then survey the literature, giving
separate treatment to research that uses individual-level data (e.g., property-insurance
data) and research that uses aggregate data (e.g., betting-market data). We conclude
by discussing directions for future research. ( JEL C51, D11, D81, D82, D83, G22,
I13)
1. Introduction and public
economics, particularly in the
study of incentives and social insurance
R isk preferences are integral to mod-
ern economics. They are the primary
focus of the literature on decision making
programs. And risk preferences are a major
driver in models of consumption, invest-
ment, and asset pricing in macroeconomics.
under uncertainty. They play a central role While much of the literature is theoretical
in insurance and financial economics. The in nature, deriving qualitative predictions
topics of risk sharing and insurance are in different environments, there is also a
prominent in development, health, labor, large empirical literature that estimates risk
preferences, both their magnitude and their
nature.
* Barseghyan: Cornell University; Molinari: Cornell Most of the literature uses expected util-
University; O’Donoghue: Cornell University; Teitelbaum:
Georgetown University. We thank Steven Durlauf, four ity (EU) theory to model risk preferences.
anonymous referees, Pierre-André Chiappori, Liran Einav, Under EU theory, there are two potential
and Bernard Salanié for comments, and Heidi Verheggen sources of variation in attitudes toward risk:
and Lin Xu for excellent research assistance. Financial
support from National Science Foundation grant SES- people might differ in (i) their degree of
1031136 is gratefully acknowledged. In addition, Molinari diminishing marginal utility for wealth (their
acknowledges financial support from National Science utility curvature), or (ii) their subjective
Foundation grant SES-0922330.
†
Go to https://doi.org/10.1257/jel.20161148 to visit the beliefs. Over the years, however, economists
article page and view author disclosure statement(s). have come to recognize additional sources of
1
03_Barseghyan_562.indd 1 5/11/18 9:53 AM2 Journal of Economic Literature, Vol. LVI (June 2018)
variation in attitudes toward risk, and have We begin in section 2 with a motivating
integrated these into “non-EU” models. The example designed to address the question
most prominent of these additional sources of why economists should care about the
are (iii) probability distortions (such as in structure of risk preferences. More and
rank-dependent EU) and (iv) reference-de- more, economists are engaging in analyses
pendent utility (as in loss aversion). that investigate the quantitative impact of a
Early empirical studies on risk preferences change in the underlying environment (e.g.,
focus on the EU model and rely on data a legal reform). In such analyses, risk pref-
from laboratory experiments (e.g., Preston erences are often a required input, even if
and Baratta 1948; Yaari 1965); for reviews, only as part of a broader model. Our exam-
see Camerer (1995) and Starmer (2000). ple highlights two reasons that the specifica-
Laboratory experiments generated many tion of risk preferences matters. First, many
insights about risk preferences, and most quantitative analyses attempt to make out-
notably demonstrated both substantial het- of-sample predictions for behavior based on
erogeneity in risk preferences and substan- the broader model. We demonstrate in our
tial deviations from EU theory. However, example how different assumptions about
the limitations commonly associated with risk preferences can lead to different out-
the laboratory setting—e.g., concerns about of-sample predictions for behavior. Second,
ecological and external validity—motivated many quantitative analyses attempt to reach
economists to look for suitable data from welfare conclusions. We discuss how differ-
field settings—i.e., environments in which ent assumptions about risk preferences can
people’s real-world economic behavior is lead to different welfare conclusions.
observable. In section 3, we provide a detailed review
As a result, there is a relatively small but of several models of risk preferences.
growing literature that takes on the difficult Section 3 does not contain an exhaustive list
task of estimating risk preferences using of all models of risk preferences, but rather
field data. Our goal in this review is to sur- focuses on those that have been estimated or
vey and assess this literature, with a partic- otherwise studied using field data. We begin
ular emphasis on clarifying the differences with EU theory, and proceed to describe
among potential sources of variation in risk several non-EU models that were origi-
attitudes and highlighting how one might nally motivated by experimental evidence,
tease them apart. We concentrate our atten- but which subsequently have been studied
tion on studies in which risk preferences are using field data, including rank-dependent
the focal object and estimating their struc- expected utility (RDEU) theory and cumula-
ture is the core enterprise. In particular, tive prospect theory (CPT). In section 4, we
we generally exclude papers that estimate a then provide a discussion of identification,
structural model of risk preferences, but do and in particular describe what types of data
not treat the risk preference parameters as are needed to estimate and distinguish the
the parameters of main interest. Although various models.
there are many excellent papers in this cat- In section 5, we discuss research that
egory that make important contributions estimates risk preferences, and sometimes
to numerous fields of economics, they are heterogeneity in risk preferences, using indi-
beyond the scope of this review.1 vidual-level data. We begin with an overview
1 As we explain below, however, we discuss a handful valuable contributions to the methodology of estimating
of papers that, although they fall into this category, make risk preferences using field data.
03_Barseghyan_562.indd 2 5/11/18 9:53 AMBarseghyan et al.: Estimating Risk Preferences in the Field 3
of the general approach used throughout the risk preferences—and also to gain insight
literature. Next, we describe in detail research on the question of whether experimental
that estimates risk preferences using data on results can be directly applied to make field
property-insurance choices. We then briefly predictions. Next, we describe the recent lit-
discuss studies that use data from television erature on using surveys to measure risk per-
game shows. Lastly, we review a handful of ceptions, and we discuss the extent to which
recent papers that analyze data on health-in- survey data might be usefully combined with
surance choices. Although health insurance field data to identify and estimate risk prefer-
is an important field context, we limit the ences under weaker assumptions. Finally, we
depth of our coverage because the papers discuss the importance of “mental account-
that use health-insurance data do not focus ing,” by which we mean assumptions about
on estimating risk preferences. We believe how people translate a complex field context
this is because estimating risk preferences into a set of concrete lotteries to be evalu-
using health-insurance data is especially ated. We encourage future research to pay
challenging. Nevertheless, we highlight a more careful attention to such assumptions.
few recent papers that address some of these
challenges and whose contributions could
2. Motivating Example
facilitate future work that focuses on esti-
mating risk preferences. In this section, we present a stylized
In section 6, we turn to research that example designed to motivate why econo-
estimates risk preferences, and sometimes mists should care about the structure of risk
heterogeneity in risk preferences, using preferences. The setting of our example is a
market-level, or aggregate, data. Once again, hypothetical insurance market. We make a
we begin with an overview of the general number of strong assumptions—about the
approach of the literature, highlighting how setting and the data—that make identifica-
the use of aggregate data naturally requires tion and estimation more straightforward.
a stronger set of assumptions in order to In later sections, we highlight some of the
identify risk preferences. Next, we describe identification and estimation challenges
in detail research that estimates risk prefer- that economists face in more realistic field
ences using data on betting markets, specif- settings.
ically data on betting in pari-mutuel horse Imagine that there is a continuum of house-
races. We then discuss a select assortment holds of measure one who each face the pos-
of papers that use macroeconomic data to sibility of a loss L
that occurs with probability
estimate risk preferences, including data on μ. Both Land μare the same across house-
consumption and investment (asset returns) holds, and their values are fixed and known.2
and on labor supply. To fix ideas, let L = 10,000and μ = 0.05.
Finally, in section 7 we discuss a number of There is insurance available to the house-
directions for future research. An under-re- holds—full insurance at a price p. Moreover,
searched issue is the extent to which risk pref- there is sufficient exogenous price variation
erences are stable across contexts. We review (e.g., over time or across various identical sub-
the few studies that use field data to inves- sets of the households; see Einav, Finkelstein,
tigate this issue, and we highlight the ques- and Cullen 2010 for an example) to non-
tions left open by these studies. Relatedly, parametrically identify the m arket-demand
we also discuss the possibility of combining
data from laboratory and field settings in 2 Note that by assuming μis fixed, we are abstracting
order to paint a more complete picture of from moral hazard.
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function for full insurance, Q F (p), which Equation (1) defines r F (z)—the coefficient
returns the fraction of households willing to of absolute risk aversion of a household
purchase full insurance at price p. Panel A of with willingness to pay zfor full insurance.
figure 1 depicts one such demand function, A household purchases full insurance when
namely Q F (p) = 2 − 0.001p. It is a typical its r > r F (p), and hence the demand for full
demand function—as the price of insurance insurance satisfies Q F (p) = 1 − FEU ( r F (p)).
F
decreases, the fraction of households willing It follows that, given Q (p), we can recover
to purchase it increases. It also reflects aver- F EU. Panel B of figure 1 displays the F EU
sion to risk—households demand insurance that corresponds to the Q F (p)depicted in
at actuarially unfair prices. panel A.
Given FE U, it is straightforward to con-
2.1 Out-of-Sample Predictions
struct the demand for deductible insurance
Understanding the underlying structure Q D (p). A household’s willingness to pay for
of risk preferences matters for making out- deductible insurance is the zsuch that
of-sample predictions. Consider a regulatory
proposal to require all insurance policies to μ exp (r(z + d)) + (1 − μ) exp (rz)
(2)
carry a deductible d < L. In order to assess
this proposal, we need to know how the = μ exp (rL) + (1 − μ) .
demand for insurance would respond to the
introduction of the deductible d. The demand Equation (2) defines r D (z)—the coefficient
function for full insurance Q F (p)—which we of absolute risk aversion of a household with
observe—provides, by itself, limited infor- willingness to pay zfor deductible insur-
mation about the market-demand function ance. A household purchases deductible
for deductible insurance, Q D (p). However, insurance when its r > r D (p) , and hence
if we know the underlying model that gen- the demand for deductible insurance is
erates Q F (p)
, we can use that model to QE DU (p) = 1 − FE U ( r D (p)). Panel C of fig-
construct Q D (p). ure 1 depicts the Q EDU (p)that corresponds
Assume for the moment that the under-
to the Q F (p)depicted in panel A, assuming
lying model is EU. In addition, assume
d = 2,500 . Because deductible insurance
that (i) the utility function exhibits con-
provides less coverage than full insurance,
stant absolute risk aversion (CARA), specif-
ically u(y) = − exp (−ry)/r, where ris the naturally QE DU (p) Barseghyan et al.: Estimating Risk Preferences in the Field 5
Panel A: Demand for full insurance
2,000
1,500
p
1,000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
F
Q (p)
Panel B: Implied distribution of the risk-aversion coefficient
1
F(r)
0.5 FEU
FΩ
0
0 0.5 1 1.5 2 2.5 × 10−4
r
Panel C: Demand for deductible insurance
D
1,800 QEU (p)
1,600 QΩD (p)
1,400
p
1,200
1,000
800
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Q(p)
Figure 1. Demand for Insurance and Underlying Risk Preferences
(and maintaining the additional assumptions Given Ω̅ , we can proceed as before to use
specified above), a household’s willingness to the known demand for full insurance Q F (p)
pay for full insurance is the zsuch that to construct the counterfactual demand for
deductible insurance Q ΩD (p).
(3) exp (rz) = Ω(μ) exp (rL) + (1 − Ω(μ)) , Let FΩ denote the distribution of r
given the probability distortion model
where Ω(μ)is the weight on the loss out- with loss weight Ω̅ . We can recover F Ω
come. Suppose that Ω (μ) = Ω̅ > μis the from the demand for full insurance,
same across households, and that Ω ̅ is known. Q F (p) = 1 − FΩ ( r ΩF (p)), where r ΩF (z) is
defined by equation (3) with Ω (μ) = Ω̅ .
models of choice under risk. See section 4.4 for further Panel B of figure 1 displays the F Ω that cor-
discussion. responds to the Q F (p)depicted in panel A,
03_Barseghyan_562.indd 5 5/11/18 9:53 AM6 Journal of Economic Literature, Vol. LVI (June 2018)
assuming Ω ̅ = 0.10. Given FΩ , we can 2.2 Welfare Analysis
construct the demand for deductible insur-
D (p) = 1 − FΩ
ance, QΩ ( r ΩD (p)), where r ΩD (z) Understanding the underlying structure of
is defined by risk preferences is also important for welfare
analysis. There are two key issues here.
First, when one uses a structural model
Ω ̅ exp (r(z + d)) + (1 − Ω̅ ) exp (rz)
of preferences to conduct welfare analysis,
a misspecified model can yield misleading
= Ω̅ exp (rL) + (1 − Ω̅ ),
conclusions. In simple terms, if a misspeci-
fied model leads to incorrect out-of-sample
the equation that implicitly defines a house- predictions for the behavioral impact of a
hold’s willingness to pay zfor deductible policy change (as in the prior subsection),
insurance. Panel C of figure 1 depicts then of course welfare conclusions will be
the QΩ D (p)that corresponds to the Q F(p) misleading. However, even if the misspec-
depicted in panel A, assuming d = 2,500 and ified model leads to correct predictions for
Ω̅ = 0.10. Observe that Q ΩD (p) μ,6 and For instance, Einav, Finkelstein, and Cullen
thus the implied demand for deductible (2010) propose an approach to empirical wel-
insurance is greater under the EU model. fare analysis in insurance markets that relies
only on estimating the demand function.
However, this type of welfare analysis is valid
5 Take our example: although the deductible insurance
only if people’s revealed willingness to pay
provides 75 percent of the coverage of full insurance, is indeed a sufficient statistic for consumer
under both modes a household’s willingness to pay for the welfare. The behavioral economics literature
deductible insurance is greater than 75 percent of the will- has suggested a variety of reasons people’s
ingness to pay for full insurance (see figure 1).
6 Intuitively, this is because under the EU model a
household’s aversion to risk (which generates its insur-
ance demand) is driven solely by the concavity of its utility 7 Such analyses are confined to within-sample wel-
function, whereas under the probability distortion model fare analysis, because without an underlying model of
a household’s aversion to risk is driven also by the over- preferences, one cannot make out-of-sample behavioral
weighting of its distortion function. predictions.
03_Barseghyan_562.indd 6 5/11/18 9:53 AMBarseghyan et al.: Estimating Risk Preferences in the Field 7
behavior might deviate from what maximizes DEFINITION 1: Let X ≡ (x 1, μ 1; x 2, μ 2; … ;
their welfare. Indeed, Baicker, Mullainathan, x N, μ N)denote a lottery that yields outcome
and Schwartzstein (2015) describe how the x n with probability μ n, where ∑
N n = 1.
μ
n=1
standard revealed-preference approach to
welfare might fail in the context of health Models of risk preferences describe how a
insurance. person chooses among lotteries of this form,
The question of whether and, if so, when where we often use X to denote a choice set.
we should drop the revealed-preference Throughout, we express lottery outcomes
assumption in welfare analysis has been hotly in terms of increments added to (or sub-
debated—see, in particular, Kőszegi and tracted from) the person’s prior wealth w.
Rabin (2008), Bernheim (2009), and Chetty In other words, if outcome x n is realized,
(2015). Estimating the underlying structure then the person will have final wealth
of preferences can help frame this debate w + x n. The probabilities should be taken to
because the more one understands the be a person’s subjective beliefs. In particular,
forces that drive behavior, the better one can the models below describe how a person’s
assess whether those forces should be given subjective beliefs impact his or her choices.
normative weight. To illustrate in the context The models are silent on the source of those
of risk preferences, suppose we estimate that subjective beliefs—we return to this issue in
a probability distortion model (as described section 5.1.
in section 2.1) best explains behavior, and
3.1 Expected Utility
suppose we are able to further establish that
probability distortions primarily reflect risk According to EU theory, given a choice set
misperceptions (i.e., incorrect subjective ∈ X that
X, a person will choose the option X
beliefs). We have then reframed the debate maximizes
into one about whether we should evaluate
N
EU(X) ≡ ∑ μ n u(w + x n),
welfare using a person’s (incorrect) subjec-
tive beliefs or more objective probabilities. n=1
where uis a utility function that maps final
3. Models of Risk Preferences wealth onto the real line.
Under EU theory, a person’s attitude
In this section we describe in detail sev- toward risk is fully captured by her util-
eral models of risk preferences. We begin ity function u(and her prior wealth w). In
by reviewing the standard EU model. We broad terms, a person will be risk averse if u
then proceed to introduce several alternative is concave, risk loving if uis convex, and risk
models. Our goal is not to provide an exhaus- neutral if uis linear. More narrowly, one can
tive list, but rather to focus on models of risk derive a local measure of absolute or relative
preferences that have been prominent in the risk aversion (or risk lovingness) that char-
literature that uses field data to estimate risk acterizes how a person will react locally to
preferences.8 choices between lotteries.
We start by introducing notation that we Hence, when one estimates an EU model,
use throughout this section. the main object to estimate is the utility
function u . As we shall see, occasionally
8 In the online appendix, we provide further details
researchers have taken a nonparametric
about these models and illustrate their differences by approach to estimating u , but most often
describing their predictions in three examples. they assume a specific parametric functional
03_Barseghyan_562.indd 7 5/11/18 9:53 AM8 Journal of Economic Literature, Vol. LVI (June 2018)
TABLE 1
Functional Forms Used in this Review
Panel A. Utility functions
{y
CARA − __1r exp (−ry) for any r ≠ 0
u(y) =
for r = 0
CRRA 1 y 1−ρ for any ρ ≠ 1
___
{ln y
1 − ρ
u(y) =
for ρ = 1
HARA ⎧___ γ y 1−γ
⎪ (η + __γ ) for any γ ≠ 1
u(y) = ⎨
1 − γ
⎪
⎩γ ln(η + γ )
y
__ for γ = 1
u(w + Δ)
NTD ũ (Δ) ≡ ______
− ____ ≅ Δ − __r Δ 2
u(w)
u′ (w) ′ (w)
u 2
Panel B. Probability weighting functions
Karmarkar (1978) μ
γ
π(μ) = ________
+ (1 − μ)
μ
γ γ
Tversky and Kahneman (1992) μ γ
π(μ) = ___________
[μ + (1 − μ) γ]
γ 1/γ
Lattimore, Baker, and Witte (1992) δ μ γ
π(μ) = _________
δ μ + (1 − μ)
γ γ
Prelec (1998) π(μ) = exp (−(−ln μ) α)
Panel C. Value function
for y ≥ 0, α ∈ (0, 1)
{− λ (− y) for y < 0, β ∈ (0, 1), λ > 1
Tversky and Kahneman (1992) y α
v(y) =
β
form for u. Perhaps the most common func- implies a person’s prior wealth wis irrelevant
tional forms are the constant absolute risk to her choices. This is advantageous from
aversion (CARA), the constant relative risk the econometrician’s viewpoint, because w
aversion (CRRA), and the hyperbolic abso- frequently is unobserved. At the same time,
lute risk aversion (HARA) families, reported however, this is disadvantageous from the
in panel A of table 1. economic theorist’s viewpoint, because econ-
When one uses the CARA family, one omists typically believe that people exhibit
estimates the parameter r , which is the coef- decreasing absolute risk aversion—i.e., as a
ficient of absolute risk aversion (higher r person becomes wealthier, she becomes less
means more risk averse). The CARA family averse to risk.
03_Barseghyan_562.indd 8 5/11/18 9:53 AMBarseghyan et al.: Estimating Risk Preferences in the Field 9
When one uses the CRRA family, one relevant in a particular application. As such,
estimates the parameter ρ, which is the we label this approach the negligible third
coefficient of relative risk aversion (higher ρ derivative (NTD) approach.
means more risk averse). The CRRA family The NTD family is convenient to work
has the advantage of implying decreasing with because it does not require prior
absolute risk aversion (among those who are wealth as an input. However, one must be
risk averse). However, the CRRA family has careful to assess whether the approximation
the major drawback that it requires prior method is appropriate for the particular
wealth was an input. Hence, when research- application under consideration. This will
ers use the CRRA family and do not observe depend on the magnitude of the increments
prior wealth, they typically either posit some to wealth relative to the estimated degree of
reasonable value for prior wealth (and check risk aversion.10
robustness for other values) or proxy for
3.1.1 Utility Curvature and the Rabin
wealth using some aspect of the data (e.g.,
Critique
home value).
Finally, when one uses the HARA fam- As it is usually applied—and as it is
ily, one estimates the parameters η and γ, described above—EU theory is defined for
which together determine the degree of static choices wherein a person faces a sin-
absolute risk aversion r(y) = (η + y / γ) −1. gle decision problem that involves choosing
The HARA family has the property that it between lotteries that add to or subtract
nests the CARA and CRRA families as spe- from her initial wealth.11 For such choices,
cial cases, with γ
→ +∞yielding CARA and Rabin (2000) demonstrates that if one
η = 0yielding CRRA.9 assumes that a person uses the same util-
A third technique is to use an approxima- ity function in every choice—an assump-
tion approach; see Cohen and Einav (2007), tion one would want to make if the theory
Barseghyan, Prince, and Teitelbaum (2011), is to have any predictive power—then EU
and Barseghyan et al. (2013b). Specifically, theory is problematic. In particular, Rabin
if one takes a second-order Taylor approxi- demonstrates that if a person exhibits any
mation of the utility function around prior noticeable risk aversion over small stakes
wealth wand then normalizes by mar- for a range of initial wealth, then she must
ginal utility evaluated at prior wealth w, exhibit extremely large risk aversion over
one gets moderate stakes for that same range of ini-
tial wealth, so large as to be clearly coun-
u(w + Δ) terfactual. For instance, if a person rejects
− ____ ≅ Δ − __r Δ 2 ,
(4) ũ (Δ) ≡ _______
u(w)
u′(w) u′(w) 2 a 50–50 gamble to lose $10 and win $10.10
for any initial wealth, then she must also
where r ≡ −u″(w)/u′(w)is local absolute reject a 50–50 gamble to lose $1,000 and
risk aversion. This approximation is accurate win any positive sum, no matter how large.
when the third- and higher-order derivatives Because people arguably do exhibit notice-
of the utility function uare negligible, at least able risk aversion over small stakes, but
relative to the increments to wealth that are
10 One obvious concern is that utility must be increas-
ing, which for risk averse individuals (with r > 0) holds
9 Some researchers—e.g., Cicchetti and Dubin (1994) only for Δ < 1 / r.
and Jullien and Salanié (2000)—assume a simpler HARA 11 Under EU theory, one can equivalently convert this
specification u (y) = (η + y) γ. This simplification necessi- problem into a static choice between lotteries defined over
tates restricting γto lie in the interval ( 0, 1 ]. final wealth states.
03_Barseghyan_562.indd 9 5/11/18 9:53 AM10 Journal of Economic Literature, Vol. LVI (June 2018)
also reasonable risk aversion over moderate In other words, we replace the EU equation
stakes, Rabin concludes that EU theory with
cannot be a good explanation for b ehavior.
N
V(X) ≡ ∑ ω n u(w + x n),
This argument is known as the “Rabin
critique.” n=1
Motivated by the Rabin critique, parts of
the empirical literature have focused on cali- where ω nis a decision weight associated
brational “rejections” of EU theory, by which with outcome x nand may not be equal to a
they mean a finding of too much utility cur- person’s belief μ n. The original idea was pro-
vature over small or moderate stakes. In our posed by Edwards (1955, 1962) and popu-
review of the literature, we describe some larized in Kahneman and Tversky’s (1979)
examples of such calibrational rejections, prospect theory, which assumes ω n = π( μ n).
where authors conclude that the estimated That is, there is an increasing function π —
degree of utility curvature is “too large.” We often labeled a probability weighting func-
also attempt to clearly distinguish when EU tion—that transforms each probability into
theory is being rejected for calibrational rea- a decision weight (still normalizing π(0) = 0
sons and when it is being rejected because an and π(1) = 1). With this formulation, how-
alternative model statistically better explains ever, for any π (μ) ≠ μ, it is possible to con-
the data. struct examples in which the theory predicts
One possible response to the Rabin cri- violations of stochastic dominance—i.e., that
tique is that the static EU framework is people would choose a lottery over another
merely a simplification, as people are in fact that stochastically dominates it. The source
solving dynamic life-cycle problems with of such predictions is that, unlike under EU
many decisions taking place over time. If theory, when evaluating lotteries, the weights
we think of the static EU framework as an need not sum to one.12
“as-if” way of analyzing one of these many Quiggin (1982) proposed a rank-
decisions, it becomes less clear that we dependent model to solve this problem.
should be applying the same utility function Under the rank-dependent approach, when
to every decision that the person makes. evaluating a lottery X ≡ (x 1, μ 1; x 2, μ 2; … ,
For instance, for some decisions uncer- x N , μ N), a person first ranks the outcomes
tainty resolves quickly (such as horse race from best to worst. Specifically, if the out-
bets or laboratory gambles), while for other comes are ordered such that x 1 0
theory emerged from a tradition in psychol- ̅
such that the model predicts a person would choose the
ogy of relaxing the feature of EU theory that lottery (x, 1 /3; x − y, 1 / 3; x − 2y, 1 / 3)over the lottery
outcomes are weighted by their p robabilities. (x, 1)for all y ∈ (0, y ̅).
03_Barseghyan_562.indd 10 5/11/18 9:53 AMBarseghyan et al.: Estimating Risk Preferences in the Field 11
where πis a probability weighting function. the endpoints, reflecting a notion that as the
With this approach, when evaluating a lot- probability of an event gets small enough,
tery, the weights sum to one by construction, people ignore that possible event. The sub-
and there are no violations of stochastic sequent literature seems to have introduced
dominance.13 the excess steepness near μ = 0and μ = 1
The implications of RDEU theory, of to eliminate this discontinuity. However, it
course, depend on the specific probability is unclear how much evidence there is for
weighting function that is used. The litera- this excess steepness. As we shall see, in
ture—in large part based on experimental field applications, it is important to assess
results—has emphasized an inverse-S- whether and how low-probability events
shaped probability weighting function: for are incorporated into a person’s decision
small μ , π(μ)is concave and has π(μ) > μ, calculations.
while for large μ, π(μ)is convex and has
3.3 Cumulative Prospect Theory
π(μ) < μ.
Beyond the general inverse-S shape, a Kahneman and Tversky’s (1979) pros-
number of parameterized functional forms pect theory has two key features: probabil-
have been proposed in the literature on ity weighting and loss aversion. As discussed
probability weighting. Some prominent func- above, probability weighting derived from
tional forms are reported in panel B of table an older tradition in psychology, and is fully
1,14 and depicted in figure 2.15 Note two fea- incorporated into RDEU theory. Loss aver-
tures of these functions. First, except for the sion represents a second departure from
Karmarkar function, they are not symmetric the EU model: instead of a utility function
around μ = 1 / 2, but rather they typically udefined over final wealth, there is a value
cross the forty-five-degree line at μ < 1/ 2. function v defined over gains and losses rela-
Second, the functions exhibit excess steep- tive to some reference point.
ness near μ = 0and μ = 1—in the sense of Tversky and Kahneman (1992) propose
π′(μ) >> 1. In fact, in their original discus- an improved version of their theory, labeled
sion of probability weighting, Kahneman “cumulative prospect theory” (CPT). CPT
and Tversky (1979) instead suggested that requires as an input a reference outcome s,
probability weighting is discontinuous at and each outcome is coded as a gain or
loss relative to this reference outcome.16
Consider a lottery X ≡ (x 1, μ 1; … ; x N, μ N)
13 While some view rank dependence as merely a tech- and a reference point s, and suppose
nicial solution, others attempt to offer intuitive arguments
for rank dependence (e.g., Diecidue and Wakker 2001).
x 1 < ⋯12 Journal of Economic Literature, Vol. LVI (June 2018)
1
0.9
0.8
0.7
0.6
π(μ)
0.5
0.4
0.3
0.2
Karmarkar (1978)
Kahneman and Tversky (1979)
0.1 Lattimore, Baker, and Witte (1992)
Prelec (1998)
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
μ
Figure 2. Probability Weighting Functions
where the weight on outcome x n is and losses.17 Of course, this differential
weighting creates the potential for violations
⎧π ( μ 1)
−
for n = 1
of stochastic dominance.
⎪π (∑ j=1μ j)− π (∑ j=1 μ j) for n ∈ {2, ..., n ̅− 1}
− −
The value function vis assumed to have
three key properties: (i) v(0) = 0and it
n n−1
ω n = ⎨
+ N .
⎪π (∑
j=nμ j)− π +(∑
Nj=n+1μ j) for n ∈ { n ̅,..., N − 1} assigns positive value to gains and negative
⎩π +( μ N) for n = N
value to losses; (ii) it is concave over gains
and convex over losses (often labeled “dimin-
ishing sensitivity”); and (iii) it is steeper in
In this formulation, π − and π + are proba-
bility weighting functions applied to the loss
and gain events, respectively. Thus, the the- 17 If π + (μ) = 1 − π − (1 − μ)
, then the distinction
ory permits differential weighting for gains between π − and π + becomes irrelevant.
03_Barseghyan_562.indd 12 5/11/18 9:53 AMBarseghyan et al.: Estimating Risk Preferences in the Field 13
the loss domain than in the gain domain Kőszegi and Rabin (2006, 2007) propose a
(often labeled “loss aversion”). model of loss aversion in which the reference
To estimate a CPT model, one often needs point is taken to be one’s expectations about
functional form assumptions (although occa- outcomes. Moreover, because such expecta-
sionally researchers have attempted more tions could involve uncertainty about future
nonparametric approaches). In terms of outcomes, they extend the model of loss
the probability weighting functions π − and aversion to use a reference lottery instead of
π , the CPT literature has used the same
+
a reference outcome.
functional forms as the RDEU literature— Specifically, under Kőszegi–Rabin (KR)
indeed, the Tversky and Kahneman function loss aversion, the utility from choosing lot-
reported in panel B of table 1 was suggested tery X ≡ ( x n , μ n) N given a reference lottery
X̃ ≡ ( x̃ m , μ̃ m) M
n=1
as part of CPT. The value function pro-
m=1 is
posed by Tversky and Kahneman (1992) is
N M
V(X|X̃ ) ≡ ∑ ∑ μ n μ̃ m
reported in panel C of table 1. In that spec-
ification, α ∈ (0, 1)and β ∈ (0, 1) generate n=1 m=1
diminishing sensitivity in the gain and loss
domains, respectively. The parameter λ > 1 × [u(w + x n) + v(w + x n |w + x̃ m)].
reflects loss aversion, as it implies the nega-
tive value generated by a loss is greater than The function urepresents standard “intrin-
the positive value generated by an equally sic” utility defined over final wealth, just as
sized gain. Based on their experimental data, in EU. The function vrepresents “gain–loss”
Tversky and Kahneman (1992) suggest that utility that results from experiencing gains or
λ = 2.25, α = β = 0.88, and for their prob- losses relative to the reference lottery. Gain–
ability weighting function, γ − = 0.69 and loss utility depends on how a realized out-
γ = 0.61.
+
come x nis compared to all possible outcomes
When applying CPT, researchers must that could have occurred in the reference
specify a reference point, and typically this lottery. For the value function, KR use
is done using some external intuitive argu-
{ηλ[u(y) − u(ỹ )] if u(y) ≤ u(ỹ )
ment. For instance, in experiments it is typi- η[u(y) − u(ỹ )] if u(y) > u(ỹ )
v(y|ỹ ) =
.
cally argued that the reference point should
be zero or experimentally endowed wealth.
In field settings, researchers often argue for In this formulation, the magnitude of gain–
a natural reference point given the setting loss utility is determined by the intrinsic
(e.g., in his recent analysis of tax evasion, utility gain or loss relative to consuming the
Rees-Jones 2018 argues that a zero bal- reference point. Moreover, gain–loss utility
ance due is a natural reference point). This takes a two-part linear form, where η ≥ 0
extra “degree of freedom” in CPT is often captures the importance of gain–loss utility
seen as a limitation and it has led to various relative to intrinsic utility and λ ≥ 1 captures
ideas about how to tie down the reference loss aversion. The model reduces to EU
point. when η = 0or λ = 1.
KR propose that the reference lottery
3.4 Expectations-Based Models
equals recent expectations about out-
A class of “expectations-based” models comes—i.e., if a person expects to face
advances the idea that expectations about lottery X̃ , then her reference lottery becomes
outcomes set reference points and influence X̃ . However, because situations vary in terms
choices. of when a person deliberates about and then
03_Barseghyan_562.indd 13 5/11/18 9:53 AM14 Journal of Economic Literature, Vol. LVI (June 2018)
commits to her choices, KR offer multiple largest utility conditional on that lottery being
solution concepts for the determination of the reference lottery. Two field contexts in
the reference lottery. Here, we focus on two which a person commits to a choice well in
solution concepts that are perhaps most rele- advance of the resolution of uncertainty are
vant for field data. property insurance and health insurance.
When estimating the KR model, one
DEFINITION 2 (KR–PPE): Given a choice needs to estimate the parameters ηand λ
set X, a lottery X ∈ Xis a personal equilib- along with the utility function u (y). Because
rium if for all X′ ∈ X, V(X | X) ≥ V(X′|X), and the latter is meant to be standard utility over
it is a preferred personal equilibrium if there final wealth, as in EU, any of the functional
does not exist another X′ ∈ Xsuch that X′is a forms for u(y)in table 1 might be used.
personal equilibrium and V (X′|X′) > V(X | X). Models of “disappointment aversion” also
assume that choices are influenced by expec-
DEFINITION 3 (KR–CPE): Given a tations. The concept of disappointment aver-
choice set X, a lottery X ∈ Xis a choice- sion was proposed by Bell (1985) and further
acclimating personal equilibrium if for all developed by Loomes and Sugden (1986) and
X′ ∈ X, V(X | X) ≥ V(X′|X′). Gul (1991). The basic idea is that one is dis-
appointed (or elated) if the realized outcome
KR suggest that PPE is appropriate of a lottery is worse (or better) than expected.
when, faced with a choice set X, a person Bell (1985) proposes a variant of disap-
thinks about the choice situation, decides pointment aversion in which disappointment
on a planned choice X ∈ X, and then makes is determined from a comparison of one’s
that choice shortly before the uncertainty is realized utility to one’s EU, and the person
resolved. An option X is a personal equilib- accounts for expected disappointment when
rium if, when a person plans on that option making a choice. Formally, a lottery X ≡
and thus that option determines her refer- ( x n , μ n) N
n=1is evaluated as
ence lottery, it is indeed optimal to make that N
choice. Among the set of personal equilib- V(X) = ∑ μ nu(w + x n)
ria, the PPE is the personal equilibrium that n=1
yields the highest “utility.” In terms of field N
contexts, then, PPE is an appropriate solu- − β ∑ μ n[I(u(w + x n) < U̅ )(U̅ − u(w + x n))],
n=1
tion concept when a person is able to think
about a choice situation for some duration where Iis an indicator function and
and then make a choice shortly before the U̅ ≡ ∑ N
n=1 μ n u(w + x n). The first term is the
uncertainty is resolved. Among those that we standard EU of lottery X. The second term
discuss in sections 5 and 6, the field context reflects the expected disutility from disap-
that perhaps best fits this scenario is betting pointment that arises when the realized util-
on horse races. ity from an outcome is less than the standard
The idea behind CPE is that, when faced EU of the lottery. The parameter β captures
with a choice set X, a person commits to a the magnitude of disappointment aversion,
choice well in advance of the resolution of where the model reduces to EUfor β = 0.18
uncertainty. By the time the uncertainty is
resolved, the person will have become accus-
18 Bell (1985) further assumes that (i) u(x) = x, and (ii)
tomed to her choice and hence expect the
a person might also experience utility from elation when
lottery induced by her choice. Hence, the the realized outcome is larger than the expected util-
person chooses the lottery that yields the ity. Even with the latter, however, his model reduces to
03_Barseghyan_562.indd 14 5/11/18 9:53 AMBarseghyan et al.: Estimating Risk Preferences in the Field 15
Gul (1991) proposes another variant of characterizing the model, one has that the
disappointment aversion in which disap- model (when applied with each of these
pointment is determined from a comparison parameter vectors) yields a different pre-
of one’s realized outcome to one’s certainty dicted distribution for the observable data.19
equivalent for the lottery. Formally, a lottery _ The subsections below discuss, for each of
X ≡ ( x n , μ n) N
n=1 is evaluated as
V (X) =
V the models presented in the prior section,
such that the conditions under which the model’s
parameters are point identified.
_ N
= ∑ μ nu(w + x n)
To facilitate our discussion, we focus
V
n=1 throughout this section on the exam-
ple of households purchasing insurance.
N _ _
− β ∑ μ n[I(u(w + x n) < V)(V
− u(w + x n))]. Specifically, we consider the situation in
n=1 which a household incurs a loss L with prob-
_ ability μ, but also has the option to purchase
The zthat solves u(w + z) = Vis one’s cer- insurance against this loss with a deductible
tainty equivalent for lottery X in this model. d ≥ 0. The willingness to pay zfor such an
When Bell disappointment aversion is insurance policy must satisfy the indiffer-
applied to binary lotteries, the model is ence condition
equivalent to the KR–CPE model. Gul dis-
appointment aversion yields a slightly differ- (w − z, 1 − μ; w − z − d, μ)
ent model, though the structure is still quite
similar. (For equations in the binary insur- ∼ (w, 1 − μ; w − L, μ) .
ance case, see section 4.4.) For lotteries with
more than two outcomes, the three models Table 2 reports, for each of the models pre-
are more distinct. For details, see the online sented in the previous section, the equation
appendix. implied by this indifference condition—i.e.,
When estimating models of disappoint- the equation one would solve to obtain a
ment aversion, one needs to estimate the value for z.
parameter β along with the utility function In the empirical applications discussed in
u(y). Because the latter is standard utility this review, typically the observable data are
over final wealth, as in EU, any of the func- comprised of (i) a discrete choice set (e.g., a
tional forms for u(y)in table 1 might be used. set of insurance products); (ii) the character-
istics of that choice set (e.g., the premiums
associated with each insurance product);
4. Model Predictions and Identification
and (iii) the option selected from that choice
Our goal in this section is to develop intu- set.20 The willingness to pay z for an insur-
ition for the types of data that may yield ance product is a useful tool in generating,
point identification of a model’s parameters. for such discrete choice sets, the model-im-
Point identification obtains when, given any plied joint distribution of premiums and opti-
two distinct values for the parameter vector mal choices. Consider, for instance, when
the choice set is composed of two options,
the model in the text where β represents the difference
between the marginal disutility from disappointment and 19 See Lewbel (2017) for a thorough discussion of iden-
the marginal utility from elation. Loomes and Sugden tification in econometrics.
(1986) also use this formulation, except they study nonlin- 20 In some cases, the data also contain some characteris-
ear disappointment. tics of the household making the choice
03_Barseghyan_562.indd 15 5/11/18 9:53 AM16 Journal of Economic Literature, Vol. LVI (June 2018)
TABLE 2
Willingness to Pay (z) for Insurance with Deductible d,
against the Possibility of Losing L with Probability μ
Model WTP
EU μ u(w − d − z) + (1 − μ) u(w − z) = μ u(w − L) + (1 − μ) u(w)
RDEU π(μ) u(w − d − z) + (1 − π(μ)) u(w − z) = π(μ) u(w − L) + (1 − π(μ)) u(w)
CPT π − (μ) v(−d − z) + (1 − π − (μ)) v(−z) = π − (μ) v(−L)
KR-CPE { μ[1 + Λ(1 − μ)] u(w − d − z) = { μ[1 + Λ(1 − μ)] u(w − L)
+ [ 1 − μ[1 + Λ(1 − μ)]] u(w − z)} + [1 − μ[1 + Λ(1 − μ)]] u(w)}
Bell–DA {μ[1 + β(1 − μ)] u(w − d − z) = μ[1 + β(1 − μ)] u(w − L)
{
+ [ 1 − μ[1 + β(1 − μ)]] u(w − z)} + [1 − μ[1 + β(1 − μ)]] u(w)}
=
{ {_____
Gul–DA (1 + β) μ
_____ u(w − d − z) (1 + β) μ
u(w − L)
1 + βμ 1 + βμ
+ (
1 − _____ ) u(w − z)
} + (1 − _____ ) u(w)
}
(1 + β) μ (1 + β) μ
1 + βμ 1 + βμ
Note: In KR–CPE, Λ ≡ η(λ − 1).
the option to purchase a particular insurance pay also depends on w, L, and μ. Of course,
product and the option to remain uninsured. in practice, there is heterogeneity in w, L,
If z is the willingness to pay for the insurance and μ(and other household characteristics).
product, then the model-implied joint dis- If these variables were observable, then all
tribution of premiums and optimal choices identification arguments below would hold
involves choosing the insurance for all pre- conditional on these observables—indeed,
miums less than zand choosing no insurance as we’ll see the literature often views μ as
for all premiums greater than z. an observed variable. If these variables are
To simplify the exposition (and notation), unobserved, then identification can become
our discussion of identification assumes hav- somewhat more complicated—in this and
ing data on a population who share the same subsequent sections, we discuss ways to
wealth w , potential loss L, and loss proba- deal with various forms of unobserved
bility μ. Insurance products are defined by heterogeneity.
their deductible d — i.e., a choice set will be
4.1 Expected Utility
a set of available deductibles (and possibly
also the option not to insure). The data will We first consider point identification under
include, for each household, a premium asso- EU. From table 2, under EU z(d) satisfies
ciated with each available deductible, along
u(w − z(d) − d) − u(w − L) 1−μ
with the household’s choice. For such data, = ___
________________
μ .
u(w) − u(w − z(d))
it is natural to use z(d)to denote the will-
ingness to pay for insurance with deductible When estimating risk preferences, much
d, supressing the fact that this willingness to of the literature assumes a parametric
03_Barseghyan_562.indd 16 5/11/18 9:53 AMBarseghyan et al.: Estimating Risk Preferences in the Field 17
functional form for u —e.g., CARA, CRRA, averse the household is) the larger is z( d0 )
or NTD—with a single parameter capturing (the more the household is willing to pay
the magnitude of risk aversion. These func- for insurance). It follows that a data set in
tional forms all fall in a class of utility func- which all households make a choice between
tions that satisfy assumption 1 below. the same two options—insurance with
Denote by u (y; ϕ)the parametric utility deductible d0 versus no insurance—can be
function, where yis a final wealth state and ϕ sufficient for point identification of ϕ . In
is a taste parameter. Assume that uis contin- particular, because each ϕimplies a unique
uous in both y > 0and ϕ ∈ ℝ, and that ϕ = 0 z( d0 ), each ϕalso implies a unique joint distri-
if and only if u(y; ϕ) = y. In addition, main- bution of premiums and optimal choices, and
tain the following. thus there is point identification as long as the
data contain sufficient variation in premiums.
ASSUMPTION 1: (i) u(y; ϕ)is increas- The literature most often assumes a para-
ing in y , and for any y 0 > y 1 > y 2, the metric functional form for u, not only when
ratio R ≡ [u(y 1; ϕ) − u(y 2; ϕ)]/[u(y 0; ϕ) − estimating EU but also when estimating the
u( y 1; ϕ)]is strictly increasing in ϕ . (ii) alternative models we discuss below. As we
lim ϕ
→∞
R = ∞ and l
im
ϕ→−∞ R = 0 . 21
have seen, this assumption dramatically sim-
plifies identification, but it is a strong restric-
Assumption 1 naturally associates ϕ tion. It would be desirable to be able to trace
with the magnitude of an individual’s risk out the utility function nonparametrically
aversion.22 In particular, assumption 1 over the relevant support. Doing so can be
holds if and only if for any y 0 > y 1 > y 2 straightforward if one is willing to assume
and μ ∈ (0, 1) , there exists a ϕ ̅ such that homogeneous preferences (and has access to
( y 0 , 1 − μ; y 2 , μ) ≻ ( y 1 , 1) for ϕ ∈ [0, ϕ̅ ), data exhibiting the type of variation described
( y 0 , 1 − μ; y 2 , μ) ∼ ( y 1 , 1) for ϕ = ϕ̅ , and above). In practice, however, there is hetero-
( y 0 , 1 − μ; y 2 , μ) ≺ ( y 1 , 1) for ϕ > ϕ̅ . In geneity across households, and allowing for
words, whenever a person compares a binary this heterogeneity dramatically complicates
risky lottery to a certain amount in the inter- relaxing the parametric assumption on u. We
val ( y 0 , y 2), the person chooses the riskier return to this issue in section 6.2.1.
lottery if her risk aversion ϕ is small enough,
and she chooses the certain amount if her 4.2 Rank-Dependent Expected Utility
risk aversion ϕ is high enough. We next consider point identification
Consider the willingness to pay z ( d0 ) for under RDEU. From table 2, under RDEU
one specific deductible d 0. Under EU, for z(d) satisfies
any u (y; ⋅ )satisfying assumption 1, each
preference parameter ϕimplies a unique u(w − z(d) − d) − u(w − L)
1 −
= ___
________________
π
z( d0 ), where the larger is ϕ(the more risk π ,
u(w) − u(w − z(d))
where we use πin place of π(μ)given our
21 The limit assumption is made merely to guarantee maintained assumption that all households
interior solutions in any formal results below. In practice, have the same μ.23
this assumption is unlikely to be important. NTD does not
satisfy this assumption, but result 1, below, holds for NTD
as well.
22 Assumption 1 is equivalent to condition (e) in Pratt 23 In other words, the relationship between π and μ
(1964 theorem 1). As shown there, it is equivalent to plays no role in the discussion in this section. Hence, the
assuming that an increase in ϕcorresponds to an increase identification results in this section also hold in an EU
in the coefficient of absolute risk aversion. model when one attempts to estimate both risk aversion
03_Barseghyan_562.indd 17 5/11/18 9:53 AM18 Journal of Economic Literature, Vol. LVI (June 2018)
In this case, model predictions depend on deductible d1 > d0 . The willingness to pay
both the utility function u and the decision z( d1 )is consistent with another set of (ϕ, π)
weight π , which complicates identification pairs represented by the curve π ̅ (ϕ | d1 , z( d1 )),
even when the utility function is parametri- again as depicted in figure 3. As we establish
cally specified as u(y; ϕ). As above, consider in result 1 below, these two curves cross at
the willingness to pay z( d0 ) for one specific only one point, yielding a unique (ϕ, π) pair
deductible d0 . Unlike above, it is not the case consistent with both z ( d0 ) and z( d1 ).
that each vector of preference parameters
(ϕ, π)implies a unique z( d0 ). Rather, there RESULT 1: If u(y; ϕ)satisfies assump-
is a set of (ϕ, π)pairs consistent with z ( d0 ). tion 1, then for any 0 z( d1 ) while z( d0 ) + d0 <
__________________________________
[ u(w; ϕ) − u(w − z ; ϕ) + u(w − z − d; ϕ) − u(w − L; ϕ)
̅ ] [ ̅ ]
z( d1 ) + d1 (otherwise the household would
Given this function, any preference–param- violate dominance). Define A (ϕ) ≡ u(w; ϕ) −
eter pair ( ϕ, π ̅ (ϕ | d0 , z( d0 ))) is consistent with u(w − z(d0 ); ϕ), B(ϕ) ≡ u(w − z(d0 ) − d0 ; ϕ)
z( d0 ). For any uthat satisfies assumption 1, − u(w − L; ϕ), A′ (ϕ) ≡ u(w; ϕ) − u(w −
π ̅ (ϕ | d0 , z( d0 ))is decreasing in ϕ, as depicted z(d1 ); ϕ), and B′ (ϕ) ≡ u(w − z(d1 ) − d1 ; ϕ) −
in figure 3. Intuitively, both an increased risk u(w − L; ϕ) , in which case π ̅ (ϕ| d0 , z( d0 ))
aversion and an increased decision weight = A(ϕ)/[A(ϕ) + B(ϕ)] and π ̅ (ϕ| d1 , z( d1 ))
on the loss state imply an increased willing- = A′ (ϕ)/[A′ (ϕ) + B′ (ϕ)]. Hence
ness to pay for insurance. Hence, for a fixed
willingness to pay, as risk aversion increases, π̅ (ϕ| d0 , z( d0 )) ≷ π̅ (ϕ| d1 , z( d1 ))
the decision weight on the loss state must
decline in order to keep the willingness to A′ (ϕ)
≷ ________
A(ϕ)
pay unchanged. ⇔ ________
A(ϕ) + B(ϕ) A′ (ϕ) + B′ (ϕ)
Hence, unlike for EU, under RDEU one
B′ (ϕ) A′ (ϕ)
cannot point identify the vector of prefer- ⇔ ____
≷ ____
.
B(ϕ) A(ϕ)
ence parameters (ϕ, π)using a data set in
After algebraic manipulations, assumption 1
which all households make a choice between
yields that A′ (ϕ)/A(ϕ)is a strictly decreasing
the same two options. However, it can suf- A′ (ϕ)
fice to observe households choosing between function of ϕ , where lim ϕ →∞
____ = 0 and
A(ϕ)
A′ (ϕ)
three options. For example, consider house-
lim
ϕ→−∞
____ = 1. Analogously, assumption 1
holds choosing between no insurance, insur- A(ϕ)
ance with deductible d 0, and insurance with yields that B′ (ϕ)/B(ϕ)is a strictly increasing
B′ (ϕ)
function of ϕ , where lim ϕ→∞ ____ = 1 and
B(ϕ)
′ (ϕ)
____ = 0. The result follows. ∎
B
ϕand unobserved subjective beliefs (or unobserved risk
lim
ϕ→−∞
B(ϕ)
types), where π would be those subjective beliefs (or risk
types). If instead one observes μ and wants to identify the
function π ( ⋅ )over some range of values for μ
, one needs
The key intuition behind result 1 is that
data as described in the text for all those values of μ . probability distortions in isolation yield, for
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