AN EXPERIMENT ON REFERENCE POINTS AND EXPECTATIONS∗
Changcheng Song†
March 13, 2015
Abstract
I conducted a controlled laboratory experiment to test to what extent expectations and the
status quo determine the reference point. In the experiment, I explicitly manipulated expectations
by exogenously varying the fluctuations of lagged beliefs and tested whether expectations affect
risk attitudes. I also exogenously varied the time of receiving new information and tested whether
and how fast individuals adjust their reference points to new information. Moreover, I derived
and tested a new theoretical prediction to distinguish two expectation-based reference dependent
models: fixed reference points and stochastic reference points. I find that both expectations and the
status quo influence the reference point significantly. The results also suggest that subjects adjust
their reference points quickly, which further confirms the role of expectation as the reference point.
Moreover, both reduced form and structural estimation support the model of the stochastic reference
point reflecting full distribution of expected outcomes rather than that with a fixed reference point.
Keywords: Stochastic Reference Points, Reference Point Adjustment, Disappointment
Aversion, Expectations, Status Quo, Loss Aversion, Probability Weighting
JEL Classification Numbers: C91, D01, D81, D84
∗
Song: Department of Economics, National University of Singapore, 1 Arts Link, AS2 #05-37, Singapore,
129789, Singapore https://sites.google.com/site/songchch02/. Email: ecssccnus.edu.sg. I am extremely
grateful to Stefano DellaVigna for the encouragement, guidance, and helpful suggestions. I thank Teck Ho,
Lorenz Goette, Botond Köszegi, Shachar Kariv, David I. Levine, John Morgan, Juanjuan Meng, Matthew
Rabin, and seminar participants in National Univeristy of Singapore, PE non-lunch at UC Berkeley, Stanford
Institute for Theoretical Economics Summer workshop for helpful comments and suggestions. This research
was supported financially by the UC Berkeley Experimental Social Sciences Laboratory (Xlab), the UC
Berkeley Program in Psychology and Economics (PIPE) and the startup grant from National University of
Singapore.
†
Department of Economics, National University of Singapore, 1 Arts Link, AS2 05-37, Singapore,117570,
Singapore (email: ecssccnus.edu.sg)
1
1
Introduction
Kahneman and Tversky (1979) ’s prospect theory is well documented in the economic and
psychology literature. In this theory, the evaluation of an outcome is influenced by how
it compares to a reference point, the degree of diminishing sensitivity, loss aversion and
nonlinear probability weighting. What determines a reference point is an important question
for discussion. The status quo is one candidate for the reference point, which implies that
individuals are reluctant to give up things they currently possess. Alternatively, expectations
are taken to be reference points (Kőszegi and Rabin (2006) , Kőszegi and Rabin (2007)),
meaning that individuals are reluctant to fall short of their beliefs. These theories about
reference point determination have different implications due to loss aversion below the
reference point.
In the theories of expectation-based reference points, expectations are modeled in two
ways. In Disappointment Aversion (DA) model, expectations are modeled as the expected
utility certainty equivalent of a gamble (Bell (1985); Loomes and Sugden (1986); Gul (1991)).
The outcome is evaluated by comparing it to a fixed number which equals the expected
utility certainty equivalent. In Kőszegi and Rabin (2006) (KR) Model, expectations and the
reference points are the full distribution of expected outcomes. The outcome is evaluated
by comparing it with each expected outcome and then integrating over the distribution of
expected outcome. These two expectation-based reference-dependent models are difficult to
distinguish since they generally have similar predictions.
This paper uses several controlled laboratory experiments to accomplish three objectives.
First, I design the Payoff treatment to test to what extent expectations and the status
quo determine the reference point based on different theoretical implications. Second, I
test whether individuals assimilated new information into their reference points quickly.
Finally, I derive new theoretical predictions from DA model and KR model, and then design
the Probability treatment to distinguish these two expectations-based reference-dependent
models.
To test to what extent expectations and the status quo determine the reference point,
I design the Payoff treatment in a controlled laboratory experiment in which I explicitly
manipulated expectations by exogegously varying the fluctuations of lagged beliefs. I first
randomly split the sample into the control group and the treatment group. Then I sent
information to these groups in an email 24 hours before the experiment. For the control
group, the email said that they would receive a fixed payment for the experiment ($10, $15,
or $20). For the treatment groups, the email said that they would receive payment through
a lottery (1/3 chance to receive $10, 1/3 chance to receive $15, and 1/3 chance to receive
2
$20). When the subjects were in the lab, the lottery resolved and then those in the treatment
groups ascertain whether they would receive $10, $15, or $20. Then both the control group
and the treatment group would answer 60 (incentivized) risk-attitude questions to elicit their
risk attitudes following the Holt and Laury (2002) procedure. The difference between the
two groups will help to identify the role of expectations and the status quo as reference
point. I varied the payoffs and probabilities in the questions measuring risk attitudes so that
I could use maximum likelihood estimation (MLE) to jointly estimate the reference points
and the preferences based on the reference points.
Expectation-based reference-dependent models predict that those who are treated to
expect higher payoffs are in loss domain, and thus are less risk averse. Indeed, I find that
those who are treated to expect the lottery payment but receive $10 from the lottery are
significantly less risk averse than those who are treated to expect $10 payment (about 0.5
standard deviation in the control group). In contrast, those who are treated to expect the
lottery payment but receive $20 from the lottery are more risk averse than those who are
treated to expect $20 payment (about 0.6 standard deviation in the control group). Moreover,
the manipulation of expectation is effective: both the self-reported expectations of payoff
from post surveys and the estimated reference points from MLE are higher in the group
with higher exogenous expectations of payoffs. These results suggest that expectations play
a role to determine the reference point. I also find moderate diminishing sensitivity (α =
0.88), significant loss aversion (λ = 2.06), and significant nonlinear probability weighting
(γ = 0.51).
To investigate the relative importance of expectations and the status quo, I nested the
two models in the structural estimation and estimated the weight on each model. I find that
the weight on expectation is 0.5, which suggests that both expectations and the status quo
influence the reference point significantly.
To study the adjustment of reference points, I explore the second source of variation:
I exogenously varied the time of receiving new information and tested whether individuals
assimilated new information into their reference points, and if so, at what rate. I randomly
split the overall treatment group into two groups: the “no-waiting” treatment group and
the “waiting” treatment group. The “no-waiting” group answered the questions immediately
after they discovered the realization of the lottery. The “waiting” group filled out a survey
about their social economic background after they knew the realization of the lottery, and
then - after a few minutes - answered the questions. The key difference was that the risk
attitudes of the waiting group were elicited about ten minutes later than the no-waiting
group. I further study the decaying treatment effects to test adjustment of reference points.
Since I use 60 (incentivized) risk-attitude questions to elicit their risk attitudes, when they
3
finish the early questions, they had “waited” as if they were in the waiting group and thus
might already have adjusted their reference points to the new information.
I find that those who are treated to expect the lottery payment but receive $10 in the
waiting group are more risk averse than those in the no-waiting group, and the difference
is significant at the 5% level. Individuals in the waiting group have similar risk attitudes
to those who are treated to expect $10. Since there are only 10 minutes more waiting time
in the waiting group, the results suggest that individuals assimilated new information into
their reference points quickly. Moreover, I show the decaying treatment effects: although
subjects in the no-waiting group are less risk averse, when they are making decisions a few
minutes after receiving the new information, the treatment effects become smaller. The
evidence further supports that individuals assimilated new information into their reference
points quickly. In the structural estimation, I nested the model of full adjustment and that
of no adjustment, and estimated the weight on each model. The weight on the model of full
adjustment is 0.50, which suggests that subjects adjust reference points quickly.
Finally, I derive different theoretical predictions from DA model and KR model, and
then design the Probability treatment to distinguish these two expectations-based referencedependent models. The Probability Treatment incudes three probability of expectations:
the 10% $20 group, the 50% $20 group, and the 90% $20 group. The timeline is the same
as in the “no-waiting” treatment group except the content of the email interventions. In the
q $20 group, the email said, “During the experiment, you will finish a short survey. After
the survey, you have q probability to receive $20, and 1-q probability to receive $10.” I
show that when the probability q increases from 50% to 90%, DA model and KR model
have different predictions for those who receive $10 in the Probability treatment groups. DA
model predicts that increase in q from 50% to 100% does not change the risk attitudes for
those who receive $10. In contrast, KR model predicts that the increase in q from 50% to
90% makes those who receive $10 less risk averse. I distinguish DA model and KR model
by comparing the risk attitudes between those who receive $10 from the 50% $20 group and
those who receive $10 from the 90% $20 group.
I find that those who receive $10 from the 90% $20 group are less risk averse than those
who receive $10 from the 50% $20 group. The difference is significant at the 5% level.
The results are not consistent with DA model but support KR model. In the structural
estimation, I nested the DA model and KR model, and estimated the weight on each model.
The result suggests that the model of the stochastic reference point fits my data better than
that of the certainty-equivalent reference point.
My work contributes to the literature in the following ways. First, it provides evidence on
the question “What determines the reference point?” I test to what extent expectations and
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the status quo play a role. Many previous empirical studies assumed the status quo, lagged
status quo as the reference point or treated reference points as latent variables in different
contexts.1 In some recent research, reference points are treated as expectations in the context
of taxi drivers labor supply (Doran (2014); Crawford and Meng (2011)), large stake risky
choices (Post et al. (2008)), insurance choices (Barseghyan et al. (2013)), professional golf
(Pope and Schweitzer (2011)) and competition in a real effort sequential-move tournament
(Gill and Prowse (2012)). This paper differs from the above in that I exogenously manipulate
expectations and expectations are induced to be stochastic.
Recent studies use laboratory experiments to explicitly manipulate subjects’ expectations, and then check whether this manipulation influences their effort provision (Abeler et
al. (2011); Gneezy et al. (2013)), valuation for some products (Smith (2008); Ericson and
Fuster (2011)), or willingness to trade (Ericson and Fuster (2011); Heffetz and List (2014);
Goette et al. (2014)). The results are mixed. Some studies find consistent results with
reference-dependent models and support the notion that reference points are expectations
(Abeler et al. (2011); Smith (2008); Ericson and Fuster (2011)), others find no support for
expectation-based reference-dependent models when they vary the probability of expectations (Gneezy et al. (2013); Heffetz and List (2014); Goette et al. (2014)). Most of above
studies create a lottery expectation during the experiment and elicit individual behavior
bef ore the resolution of lottery (Abeler et al. (2011); Ericson and Fuster (2011); Gneezy et
al. (2013); Heffetz and List (2014); Goette et al. (2014)). These manipulations create expectations without lagged belief. In this paper, I apply a new design to vary the fluctuations of
lagged beliefs: I use a lottery to manipulate the stochastic expectations before the experiment, and then elicit the risk attitudes af ter the resolution of lottery during the experiment.
The manipulation holds the outcome constant and varies the fluctuations of lagged beliefs,
which is similar to Smith (2008). While this paper is not designed to compare lagged beliefs
and other manipulation of expectations, my results suggest that lagged beliefs might have a
stronger impact on forming expectations. This paper is also different from above paper since
I study whether the manipulation of expectations influences risk attitudes, not effort provision, valuation or willingness to trade. Moreover, I exogenously varied the time of receiving
new information, which further identifies the role of expectation as reference point.
As far as I know, this paper is among the first to jointly estimate the reference points and
the shape of the utility function. In the previous research estimating structural parameters
1
For example, Tversky and Kahneman (1992), and Tanaka et al. (2010) assume the status quo as reference
point in their lab experiments. Reference points are assumed to be lagged status quo (purchase price) for
small investors (Odean (1998)) and for homeowners (Genesove and Mayer (2001)). In the literature of
negative elasticity of labor supply and income targeting, most research treated reference points as latent
variables (Camerer et al. (1997); Farber (2005a); Fehr and Goette (2007); Farber (2005b)).
5
in reference-dependent models, reference points are either assumed to be the status quo
(Tversky and Kahneman (1992); Tanaka et al. (2010)) or in a Preferred Personal Equilibrium
(Sprenger (forthcoming)) or in a Choice-acclimating Personal Equilibrium (Barseghyan et
al. (2013)).2 Since it is difficult to fully control expectations, and subjects might adjust new
information, they might not be in personal equilibrium. The advantage of joint estimations
is to allow the freedom that subjects are not in personal equilibrium. Rabin and Weizsacker
(2009) jointly estimate the reference points and other preference parameters in referencedependent models. This paper is similar with regard to joint estimation, but differs in that
I manipulated the stochastic expectations and varied the time of receiving new information.
The second contribution of this paper is to provide evidence on the speed of adjustment
of the reference point by exogenously varying the time of receiving new information. Slow
adjustment can generate lower risk aversion after losses and after gains. Post et al. (2008)
specified a lagged function for adjustment of the reference point, and estimated the influence
of initial expectations and recent outcomes. Their results suggest that reference points
tend to stick to earlier values; this effect is stronger for recent outcomes than for initial
expectations. Gill and Prowse (2012) estimate the adjustment of reference points in a real
effort sequential-move tournament. They find that reference points of the second mover
adjust to their own effort choice quickly, which is consistent with Choice-acclimating Personal
Equilibrium. This paper differs in that I not only estimated the speed of adjustment of
reference points, but also exogenously varied the time of receiving new information. This
second source of variation further identified the role of expectation as reference point and
adds more evidence on the adjustment of reference points. To the best of my knowledge, this
experiment is the first to exogenously vary the time of receiving new information to study
whether subjects adjust reference points quickly.
Third, I derive new theoretical predictions from DA model and KR model, and design
a Probability treatment to distinguish these two expectations-based reference-dependent
models. Sprenger (forthcoming) uses the differences of utility elicitation between probability
and certainty equivalent methodology to distinguish the DA model from the KR model.
The result provides support for the KR model. In my experiment, I explicitly manipulated
expectations to be stochastic and elicited risk attitudes after the stochastic expectations had
been realized, either shattered by unfavorable outcomes, or surpassed by favorable outcomes.
2
Kőszegi and Rabin (2006),Kőszegi and Rabin (2007) present these rational expectations equilibrium
concepts. The Unacclimating Personal Equilibrium (UPE) is the personal equilibrium in which individuals’
choices correspond to expectations. The Preferred Personal Equilibrium (PPE) is the UPE with the highest
ex-ante expected utility. The Choice-acclimating Personal Equilibrium (CPE) is the personal equilibrium in
which individuals’ choices correspond to expectations and the choices are committed well in advance of the
resolution of uncertainty.
6
Moreover, the detailed choice data helped me to estimate the certainty-equivalent reference
points and the weights on the stochastic reference points as well as the preferences based
on the reference points. My results further provide support for KR model and deepen our
understanding about different expectations-based models.
The rest of the paper proceeds as follows. Section 2 describes the experimental design.
Section 3 presents the theoretical framework and derives different predictions under different
assumptions of reference points. The main empirical results are discussed in Section 4.
Section 5 presents the structural estimation, including the certainty-equivalent reference
points and the weights on the stochastic reference points as well as the preferences based on
the reference points. Section 6 concludes.
2
Experiment Design
The timeline of the experiment is described in Figure 1:
[Insert Figure 1]
In the Payoff Treatment, I first randomly split the sample into a control group, “nowaiting” treatment group, and “waiting” treatment group. Then I sent information to these
groups in an email 24 hours before the experiment. The control group is randomly split into
three subgroups: the $10 control group, the $15 control group and the $20 control group. For
the $10 control group, the email says, “During the experiment, you will finish a short survey.
After the survey, you will receive $10.” For the $15 and $20 control groups, the email says
similar information except the amount they are going to receive. For the treatment groups,
the email said, “During the experiment, you will finish a short survey. After the survey, you
have 1/3 chance to receive $10, 1/3 chance to receive $15, and 1/3 chance to receive $20.”
When the subjects were in the lab, the experimenter read the email again to the subjects.
Then, they would fill out a short survey about their social economic background for around
ten minutes on average. After the survey, the treatment groups would play a lottery and
then ascertain whether they would receive $10, $15, or $20. Then both the control group
and the two treatment groups would answer 60 risk-attitude questions to elicit their risk
attitudes following Holt and Laury (2002) procedure (discussed below). One out of these
risk-attitude questions will be randomly chosen ex post by drawing one card out of three
and rolling a 20-sided dice individually. If the subject draw a number 2 and roll a dice for
17, the question for her payment is question 37.
In order to test whether subjects would adjust new information into their reference point
and how quickly, the overall treatment group was split into two groups: “no-waiting” and
“waiting.” As described in Figure 1, the “no-waiting” group answers the questions immedi7
ately after they discover whether they will receive $10, $15, or $20. The “waiting” group
fills out the survey about their social economic background after they know whether they
will get $10, $15, or $20, and then - after a few minutes - answers the questions. The key
difference is that the risk attitudes of the waiting group are elicited about ten minutes later
than that of the no-waiting group.
[Insert Table 1]
My design allows me to split all the subjects into 15 groups and undertake 5 comparisons
about the subjects’ risk attitudes in Table 1. For example, in comparison 1, I compared the
risk attitudes among people who receive $10 from Control 1; people who receive $10 from
the lottery in the no-waiting treatment; and people who receive $10 from the lottery in the
waiting treatment. Since the only differences among the groups were expectations at the
time of choice and how long ago (24 hours vs 10 minutes) they were formed, I was able to
test whether and how expectations matter.
For those who receive $10 (comparison 1 and 4), I use Tables A1, A2, and A3 to elicit
risk attitudes. For those who receive $15 (comparison 2) and $20 (comparisons 3 and 5),
the tables are similar, but payoffs increase by $5 and $10. For example, for those who
receive $20, they choose between option (1) $20 and option (2) 10% probability to get $25
and 90% to get $15 in the first row in Table A1. In the experiment, subjects choose from
option 1 and option 2 for each question. For table A1, when the probability of a high
payoff increases (moving down the table), a subject should switch from option 1 (riskless
option) to option 2 (risky option). The more riskless options the subject takes, the more
risk averse the subject is. I use the number of riskless options taken as a measurement of
risk aversion. The measurement from Table A1 is “measure 1”. After Table A1, subjects
answer one summary question in Table A4: “Now you have a choice between (1) Keep the
$10 (2) Take the following bet: p% probability to get $15 and (100-p) % probability to get
$5. What is the minimum probability p% that you will choose choice option 2?” 3 Subjects
enter the minimum probability that they will choose option 2 and the number is transformed
to “measure 2”. For example, if p=52, then measure 2 is 9 because the subject would take 9
riskless options if he/she answers the questions in Table 1.
The subject then answers the questions in Table A2 and Table A3. The measurement from
Table A2 is “measure 3”. After Table A2 and Table A3, subjects answer similar summary
questions in Table A4. “Measure 4” and “measure 6” are calculated from these summary
3
Plott and Zeiler (2007) shows that languages might influence subjects’ behavior. In this setting, the
word “keep” might influence subjects and thus they are more likely to keep $10 and become more risk averse.
Although it might cause the level of risk measurement inaccurate, the effects are likely to be similar to
different groups and the difference between groups (my treatment effect) should not be influenced by the
languages.
8
questions.
These tables differ in the following way. In Table A1, I fix the payoff but change the
probability in the risky options. In Table A2, I fix the probability but change the payoff
in the risky options. In Table A3, I fix the risky options but change the riskless options.
There are summary questions after Tables A1, A2 and A3. The purpose is to have several
measures of subjects’ risk attitudes so that I can check the robustness of the results.
Expectations-based reference-dependent models also predict that risk attitudes respond
to the change of probability distribution of stochastic expectation. I derive different theoretical predictions from DA model and KR model, and then design the Probability treatment
to distinguish these two expectations-based reference-dependent models. The Probability
Treatments include three groups: the 10% $20 group, the 50% $20 group, and the 90% $20
group. The timeline is the same as in the “no-waiting” treatment group except the content
of the email interventions. In the q $20 group, the email said, “During the experiment, you
will finish a short survey. After the survey, you have q probability to receive $20, and 1-q
probability to receive $10.” This approach allowed me to undertake comparison 4 and 5
about risk attitudes among people in different groups: those who receive $10 ($20) from
the 10% $20 group; people who receive $10 ($20) from the 50% $20 group; and people who
receive $10 ($20) from the 90% $20 group.
3
Theoretical Framework and Predictions
This section analyzes the predictions of the interventions if expectations determine reference
points. In Cumulative Prospect Theory, I employ the specification from Post et al. (2008):
(
u(x|RP ) =
(x − r)α if x ≥ r
−λ(r − x)α if x < r
(1)
λ > 0 is the loss-aversion parameter, r is the reference point that separates losses from gains,
and α > 0 measures the curvature of the value function, i.e. diminishing sensitivity.4
I also consider the one-parameter form of Prelec (1998) axiomatically derived weighting
function: φ(p) = exp(−(lnp)γ ). The probability weighting function is linear if γ = 1, as it
4
Kőszegi and Rabin (2006),Kőszegi and Rabin (2007) assume that overall utility has two components:
consumption utility and gain-loss utility. They also assume η > 0 to be the weight the consumer attaches
to gain-loss utility and λKR > 1 to be coefficient of loss aversion in gain-loss utility. Since η and λKR are
KR η
not jointly identifiable, I just use gain-loss utility in my specification and the estimated λ is 1+λ
under
1+η
KR’s assumptions and the reference point r is zero. The original formulation of prospect theory allows for
different curvature parameters for the domain of losses and the domain of gains. To reduce the number of
free parameters, I assume here that the curvature is equal for both domains.
9
is in EU. If γ < 1, the weighting function is inverted S-shaped, i.e., individuals overweight
small probabilities and underweight large probabilities, as shown by Tversky and Kahneman
(1992). If γ > 1, then the weighting function is S-shaped, i.e., individuals underweight small
probabilities and overweight large probabilities.
There are three properties in the utility function: diminishing sensitivity, loss aversion
and nonlinear probability weighting. Given the specification (1), these properties have the
following implications on risk attitudes elicited from Table A1, A2 and A3. More diminishing
sensitivity ( α increases) implies more risk aversion in the gain domain and more risk seeking
in the loss domain. More loss aversion ( λ increases) implies that people are more averse
to risk around the reference point. The inverted S-shaped weighting function implies the
fourfold pattern of risk attitudes: risk-seeking for small-p gains and larger-p losses, and a
risk-aversion for small-p losses and larger-p gains (Tversky and Kahneman (1992)). In my
risk elicitation, either the subjects’ switch points are close to 50% (in Table A1) or 50% bets
are used in the risky options (in Table A2 and A3). Since 50% is greater than the fixed point
0.37 in weighting function (Prelec (1998)). More nonlinear probability weighting ( γ < 1
and γ decreases) implies more risk aversion in the gain domain (larger-p gains) and more
risk seeking in the loss domain (larger-p losses).
All three properties imply more risk seeking in the loss domain than around the kink.
Diminishing sensitivity and nonlinear probability weighting imply more risk aversion in the
gain domain than around the kink but loss aversion implies less risk aversion in the gain
domain than around the kink. The comparison of risk attitudes between the gain domain
and the kink is ambiguous. It depends on which effects of the three properties dominate others. For example, if the effects of diminishing sensitivity and nonlinear probability weighting
dominate the effects of loss aversion, we should observe more risk aversion in the gain domain than around the kink. Therefore, status quo and expectation as reference points have
different predictions.
Prediction 1 (Payoff treatment) : If reference points are status quo, those who
expect higher payoffs than $10 have similar risk attitudes to those who expect $10 payoffs.
In contrast, expectation-based reference-dependent models predict that those who expect
higher payoffs than $10 are less risk averse than those who expect $10 payoffs.
These predictions can be tested in comparison 1 since those who receive $10 in the
treatment group are in the loss domain and those in the $10 control group around the kink.
Expectation-based reference-dependent models predict that those who receive $10 in the
treatment group are less risk averse. The proof is provided in Appendix A1.
I do not have a clear prediction in comparison 3 when those who receive $20 in the
treatment group are in the gain domain and those in the $20 control group around the kink.
10
Those in the gain domain might be less risk averse, more risk averse, or equally risk averse.
If the effects of diminishing sensitivity and nonlinear probability weighting on risk attitudes
are greater than that of loss aversion, those in the treatment group are more risk averse.
Otherwise, those in the treatment group are less risk averse.
Prediction 2 (Probability treatment): DA model predicts that increase in q from
50% to 100% does not change the risk attitudes for those who receive $10. In contrast, KR
model predicts that the increase in q from 50% to 90% makes those who receive $10 less risk
averse.
[Insert Figure 2]
These predictions can be tested in comparison 4. I distinguish DA model and KR model
by comparing the risk attitudes between those who receive $10 from the 50% $20 group and
those who receive $10 from the 90% $20 group. Figure 2 illustrates these predictions under
the piecewise linear gain-loss utility. Panel A shows the utility function of DA model as q
increases from 10% to 90%. For those who receive $10 in the 10% $20 group, when they
answer the risk attitude questions from Table A1, they are making a choice of keeping $10
or playing a lottery to receive $15 with some probability. Since they have a fixed reference
point and play a lottery around the kink, loss aversion predicts that they are risk averse.
When q increases from 10% to 50%, the maximum payoff they can receive is $15, which
equals the fixed reference point. In this case, they play the lottery in the loss domain and
thus become risk neutral. When q increases from 50% to 90%, the maximum payoff ($15)
they can receive is always less than the fixed reference point. In this case, they are still risk
neutral and their risk attitudes will not change. Therefore, DA model predicts that those
who receive $10 from the 50% $20 group and those who receive $10 from the 90% $20 group
have similar risk attitudes. The key intuition is that the increase in q increases the certainty
equivalent reference point, but does not change the gain-loss situation even if the individual
wins the lottery in the 60 risk attitude questions.
Panel B shows the utility function of KR model as q increases from 10% to 90%. For
those who receive $10 in the 10% $20 group, the utility is the weighted average of the utility
with $10 as reference point and that with $20 as reference point. When they answer the risk
attitude questions, they play the lottery around the kink with low reference point ($10) and
in the loss domain with high reference point ($20). When q increases from 10% to 90%, it
increases the weight on the utility with high reference point ($20) and reduces the weight
on the utility with low reference point ($10). Thus it increases the weight on the utility in
the loss domain. Due to loss aversion, individual would like to take more risk to reduce the
utility in the loss domain as the weight on the loss domain increases. Therefore, KR model
predicts that those who receive $10 from the 90% $20 group are less risk aversion those who
11
receive $10 from the 50% $20 group. The proof with a piecewise linear gain-loss utility is
provided in Appendix A2.5
Panel C summarizes the predictions of DA model and KR model. The horizontal line is
the manipulated probability of expectations, q. The vertical line is the predicted level of risk
aversion, measured by the minimum winning probability (p∗ ) they would like to take the bet
in Table A1. The higher p∗ is, the more risk averse. Based on the proof in Appendix A2,
DA model predicts that individual becomes less risk averse when q increases from 10% to
50% but the risk attitude does not change when q increases from 50% to 90%. In contrast,
KR model predicts that individuals become less risk averse when q increases from 10% to
90%. DA model and KR model generate different predictions when q increases from 50% to
90%.
[Insert Table 2]
I further generalize the utility function to nonlinear case. The summary of predictions is
described in Table 2. If the value function satisfies constant absolute risk aversion (CARA)
or decreasing absolute risk aversion (DARA), DA model and KR model generate different
predictions when q increases from 50% to 90%. If the value function satisfies increasing
absolute risk aversion (IARA), DA model and KR model have similar predictions when q
increases from 50% to 90%. The intuition is similar to the piecewise linear case. The proof
is presented in Appendix A3.
Similarly, I do not have a clear prediction in comparison 5 since those who receive $20 in
the 10% $20 group are mostly in the gain domain and those who receive $20 in the 90% $20
group are close to the kink. If the effects of diminishing sensitivity and nonlinear probability
weighting on risk attitudes are greater than that of loss aversion, those in the gain domain
are more risk averse. Otherwise, those in the gain domain are less risk averse.
4
Empirical Results
The experiment was conducted at the Experimental Social Science Laboratory (Xlab) at the
University of California, Berkeley and Center for Behavioral Economics Lab at the National
University of Singapore. The subjects in the experiment were recruited from undergraduate
students. Each experimental session lasted about half an hour. Payoffs were calculated in
US dollars at the University of California, Berkeley and Singapore dollars at the National
University of Singapore. The earnings were paid in private at the end of the experimental
session.
5
Kőszegi and Rabin (2006), Kőszegi and Rabin (2007) consider that the utility function for small stakes
decisions are approximately linear, and a piecewise linear gain-loss utility function is adopted.
12
In the payoff treatment, a total of 856 subjects signed up for the experiments and received
emails, and 721 of them actually showed up in 44 sections. In the probability treatment, a
total of 1158 subjects signed up for the experiments and received emails, and 1032 of them
actually showed up in 72 sections. Table 2 presents the summary statistics of the experiment.
[Insert Table 3]
The overall non-show up rate was 13.0%. The non-show up rates for the $10 control
group, the $15 control group, the $20 control group and the treatment group were 17.1%,
9.8%, 14.4% and 16.3%, respectively. The Wald test shows that I cannot reject the equality
of non-show up rates across different groups (p=0.50). The non-show up rates for the 10%
$20 group, the 50% $20 group, and the 90% $20 group were 10.4%, 10.3%, and 12.1%,
respectively. The Wald test shows that I cannot reject the equality of non-show up rates
across different groups (p=0.70).
Post-treatment surveys show that 85% of subjects expect to get money in the range of my
manipulations. In the $10 control group, 68% of subjects expected to get $10, 23% expected
to get money between $10 and $15. In the $15 control group, 83% of subjects expected to
get $15. In the $20 control group, 61% of subjects expected to get $20, 18% expected to
get money between $15 and $20. In the payoff treatment groups, 93% of subjects expected
to get money between $10 and $20. In the probability treatment groups, 97% of subjects
expected to get money between $10 and $20. Therefore, subjects expected to get slightly
more than my manipulation in the $10 control group and less than my manipulation in the
$20 control group. Thus, the effects of expectation from my estimation are likely to be lower
bounds.
Statistical analysis further shows that the manipulation of expectation changes subjects’
expectations significantly. Wilcoxon-Mann-Whitney test rejects the hypothesis that the distribution of expected payoff in the $10 control group is equal to that in the payoff treatment
group (p=0.0000). Wilcoxon-Mann-Whitney test also rejects the hypothesis that the distribution of expected payoff in the $20 control group is equal to that in the payoff treatment
group (p=0.0000). In the probability treatments, the expectations are different among three
groups and they are all significant at the 1% level. These results provide the evidence that
the manipulation of expectation is effective.
4.1
The Effects of Expectations on Risk Attitudes: Payoff Treatments
The main results of payoff treatments are described in the following figures:
[Insert Figure 3]
13
Figure 3 shows the risk attitudes for those in the payoff treatment. The vertical axis
stands for the average number of riskless options taken in Table A1 to A3 and it measures
risk averse. The horizontal axis stands for those who receive $10, $15, $20 in the control
group, the no-waiting treatment groups, and the waiting treatment group.
The left bars shows the risk attitudes for those who receive $10. A risk neutral subject
should take 9 riskless options in Table A1, 11 riskless options in Table A2 and 9 riskless
options in Table A3. The average is 9.67. In the $10 control group, the average number of
riskless options taken is 10.60. Thus subjects are slightly risk averse in the control group.
For those who receive $10 in the no-waiting group, the average of riskless options taken is
9.27. This suggests that subjects are slightly risk seeking or risk neutral in the no-waitng
group.6 The results show that losers in the no-waiting groups are more risk seeking than
those in the control group. Since the only differences between the control group and the
no-waiting group are the expectations of payoffs, expectations affect risk attitudes. This is
consistent with expectations as reference points in Prediction 1. For those who receive $10
in the waiting group, the average of riskless options taken is 9.91. Subjects are slightly risk
averse in the waitng group, and become similar to those in the control group. These results
suggest that subjects get acclimated to the new reference points quickly.
The middle bars in Figure 3 shows the risk attitudes for those who received $15 in the
three groups. The pattern suggests that those who receive $15 in the no-waiting groups are
slightly more risk averse than those in the control group. Those in the waiting group are
similar to those in the control group. The right bars in Figure 3 shows the risk attitudes
for those who received $20 in the three groups. The pattern suggests that winners in the
no-waiting groups are more risk averse than those in the control group. Similarly, those
in the waiting group are similar to those in the control group. These results suggest that
subjects get acclimated to the new reference points quickly. Appendix Figure A1 shows the
robustnesss check for all six measurements of risk attitudes, shows a clear and consistent
pattern as in Figure 3.
[Insert Figure 4]
Figure 4, Panel A shows the cumulative distributions of risk attitudes for those who
receive $10 in the three different groups. The horizontal axis stands for the average number of
riskless options taken in Table A1 to A3. The figure shows that losers in the no-waiting groups
are more risk seeking than those in the control group. Kolmogorov-Smirnov test rejects the
hypothesis that the distribution of risk attitudes in the control group is equal to that in
6
Note here the terms risk neutral, risk averse, risk seeking are based on expected utility. Risk neutral
means subjects are indifferent between zero and zero-sum bet. Risk averse means subjects will reject zero-sum
bet. Risk seeking means subjects will accept zero-sum bet.
14
the no-waiting group and it is significant at the 5% level. This is also consistent with the
theory of reference-dependent utility with prospect theory value function and expectations
as reference points.
Panel B shows the cumulative distributions of risk attitudes for those who receive $15 in
the three different groups. The figure shows that those who receive $15 in the no-waiting
groups are more risk averse than those in the control group. Kolmogorov-Smirnov test cannot
reject the hypothesis that the distribution of risk attitudes in the no-waiting group is equal
to that in the control group (p=0.206).
Panel C shows the cumulative distributions of risk attitudes for those who receive $20
in the three different groups. The figure shows that winners in the no-waiting groups are
more risk averse than those in the control group. Kolmogorov-Smirnov test cannot reject
the hypothesis that the distribution of risk attitudes in the no-waiting group is equal to that
in the control group (p=0.730).
In sum, those who receive $10 in the no-waiting groups are more risk seeking than those
in the control group. The relationship reverses for those who receive $15 or $20. Those
in the waiting group are similar to those in the control group. These reulsts support that
expectation affect risk attitudes and subjects get acclimated to the new reference points
quickly.
In order to take into account other controls, I estimate the treatment effect of expectation
on risk attitudes through OLS regression. For comparisons 1 and 3, I use the following
specification:
yi = α + β1 Tnowaiti + β2 Twaiti + φXi + εi
(2)
where yi is the number of riskless options taken by subject i. Tnowaiti is an indicator for
no-waiting treatment and Twaiti is an indicator for waiting treatment. Xi is other control
variables such as age and gender. β1 captures the treatment effect of the expectations in the
no-waiting group. β2 captures the treatment effect of the expectations in the waiting group.
β1 − β2 captures the effect of waiting on risk attitudes after the realization is revealed. εi
is assumed to be i.i.d. error term. I focus my analysis on three subsamples: those who
receive $10 from the control group, the no-waiting group, or the waiting group (comparison
1 in Table 1), those who receive $15 from the control group, the no-waiting group, or the
waiting group (comparison 2 in Table 1), and those who receive $20 from the control group,
the no-waiting group, or the waiting group (comparison 3 in Table 1). The results for six
measures are presented in six columns in Tables 4.
[Insert Table 4]
Table 4, Panel A presents the results of comparison 1 in Table 1. The coefficient of Tnowaiti
for measure 1 is -1.94 and is significant at the 5% level. So the losers in the no-waiting group
15
choose about 1.94 fewer riskless options in Table A1 than those who receive $10 from the
control group. It is the same pattern for other measures. The coefficients of Tn owaiti are
negative for all six measures. And they are significant at the 10% level for measure 1 to
4. These results show a clear and consistent pattern that the losers from the no-waiting
group are less risk averse (more risk seeking) than those who receive $10 from the control
group, and thus are consistent with Prediction 1. The results are strong and consistent in
the loss domain. The reason might be that all three properties - diminishing sensitivity, loss
aversion and nonlinear probability weighting - imply more risk seeking in the loss domain
than around the kink, as discussed in Section 3.
The coefficient of Twaiti for measure 1 is -0.36 and is not significant. The magnitude of
β2 (0.36) is smaller than β1 (1.94). The possible explanation is that the losers adjust their
reference points to the realized payoffs ($10) and thus become more risk averse. The p-value
of the Wald test β1 = β2 is 0.035 and it is significant at the 5% level. Thus, losers in the
waiting group are more risk averse than those in the no-waiting group with measure 1. These
results suggest that subjects get acclimated to the new reference points quickly. In measures
2 to 6, β2 is similar to β1 . Thus, the risk attitudes of losers in the waiting group are similar
to those in the no-waiting group. The reason might be that the subjects had to first finish
the questions about measure 1, and then answer the questions about other measures. So
there is a time lag between the realization of their lottery and the answers to the questions
after measure 1. They had “waited” as if they were in the waiting group. We analyze the
decaying treatment effects in Table 5.
Table 4 Panel B presents the results of comparison 2 in Table 1. The coefficients of
Tnowaiti are generally positive but not significant. The results indicate that those who receive
$15 in the treatment groups have similar risk attitudes to those who receive $15 in the control
group.
Table 4 Panel C presents the results of comparison 3 in Table 1. The coefficient of Tnowaiti
for measure 1 is 1.94 and it is significant at the 5% level. The pattern is similar for measures
1 to 4. The results indicate that the winners from the no-waiting group are more risk
averse than those who receive $20 from the control group. The results are less consistent
in the gain domain compared to the loss domain. The reason might be that diminishing
sensitivity and nonlinear probability weighting imply more risk aversion in the gain domain
than around the kink but loss aversion implies less risk aversion in the gain domain than
around the kink. The results suggest that the effects of diminishing sensitivity and nonlinear
probability weighting on risk attitudes are greater than that of loss aversion. Section 5 will
estimate these preference parameters.
I further investigate the time variation of risk attitudes within each treatment. The
16
results in Table 4 show the decaying treatment effects: measures from first table (measure
1 and 2) are generally significant but measures from later tables (measure 3-6) have more
noise in the no waiting group. Since I use Tables A1, A2, and A3 to elicit risk attitudes in
order, it is possible that when they finish the questions in Table A1, they had “waited” as if
they were in the waiting group and thus might already adjusted their reference points to the
new information. To test this hypothesis, I construct a panel data with each individual and
their risk attitudes measured in Table A1-A3 in three different times. I use the following
specification:
yit = α + β1 Tnowaiti + β2 Idelayi t + β3 Tnowaiti Idelayi t + φXi + εit
where Idelay_it is an indicator for delayed decisions that equals 1 if the risk attitudes are
measured in Table A2 and A3. t indicates the order of risk attitudes measurement. t =1,2
or 3 indicates the measurement from Table A1, A2 or A3. β1 captures the treatment effect
on measure 1 (Table A1) between the no-waiting group and the combination of the waiting
group and the control group. β2 captures the effects of delay on the measurement of risk
attitudes in the combination of the waiting group and the control group. β3 captures the
differences in treatment effects between measure 1 (Table A1) and the delayed measurement
(Table A2 and Table A3). Since the number of riskless options is not comparable between
Table A1, A2 and A3, I transform the number of riskless options to the forgone expected
payoff. For example, if a subject takes 10 riskless options in Table A1, the expected payoff
difference is between -0.5 and -1. I take the midpoint of range and change the sign and get
the forgone expected payoff 0.75. It means that the subject would like to forgo 0.75 dollars
to remain riskless. The greater the forgone expected payoff, the more risk averse. Since the
forgone expected payoffs are in the unit of dollars, I can compare the measures from Table
A1, A2 and A3.
[Insert Table 5]
Table 5 presents the results of time variation of risk attitudes. The dependent variable
is the number of riskless options in Column (1), (3) and (5), while it is the forgone expected
payoff in Column (2), (4) and (6). For those who receive $10, the coefficient of Tnowaiti in
Column (1) is -1.91 and is significant at the 1% level. The coefficient of Tnowaiti in Column
(2) is -0.90 and is significant at the 1% level. These results are consistent with Table 4. β3 in
Column (1) is 1.34 and it is significant at the 5% level. In column (2), the coefficient is 0.73
and it is significant at the 1% level. These results show that although subjects in the nowaiting group are less risk averse in measure 1, when they are making decisions a few minutes
after receiving the new information, the treatment effects become smaller. For those who
17
receive $20, β3 in Column (5) is -0.87 and it is not significant. In column (6), the coefficient
is -0.65 and it is significant at the 5% level. These results show that although subjects in the
no-waiting group are more risk averse in measure 1, when they are making decisions a few
minutes after receiving the new information, the treatment effects become smaller. These
results suggest that subjects get acclimated to the new reference points quickly.
4.2
The Effects of Expectations on Risk Attitudes: Probability
Treatments
In section 4.1, I study the impact of stochastic expectations in which I both randomize the expected payoff and the corresponding probabilities. Expectations-based reference-dependent
models also predict that risk attitudes respond to the change of probability distribution of
stochastic expectations. In this section, I study the impact of stochastic expectations in
which I only randomize the probabilities of expected payoffs.
[Insert Figure 5]
I have three treatment groups: the 10% $20 group, the 50% $20 group, and the 90%
$20 group. Figure 5 shows the risk attitudes for those in the probability treatment. The
vertical axis stands for the average number of riskless options taken in Table A1 to A3 and
it measures risk averse. The horizontal axis stands for those who receive $10 or $20 in the
10% $20 group, the 50% $20 group, and the 90% $20 group.
The left bars shows the risk attitudes for those who receive $10. The results show that
when expectations about $20 increase from 10% to 90%, those who receive $10 become less
risk averse. This result supports the KR model as discussed in Prediction 2. The right bars
in Figure 5 shows the risk attitudes for those who received $20. The results show that when
expectations about $20 increase from 10% to 50%, and to 90%, those who receive $20 become
less risk averse. Appendix Figure A2 shows the robustnesss check for all six measurements
of risk attitudes, shows a clear and consistent pattern as in Figure 5.
I then estimate the treatment effect of expectation on risk attitudes through OLS regression with the following specification:
yi = α + β90 Tp90i + β50 Tp50i + φXi + εi
where Tp90i is an indicator for the 90% $20 group, Tp50i is an indicator for the 50% $20
group, and the omitted group is the 10% $20 group. β90 captures the treatment effect of
the expectations in the 90% $20 group compared to the 10% $20 group. β50 captures the
treatment effect of the expectations in 50% $20 group compared to the 10% $20 group. I
also conduct the Ward tests, β90 = β50 , to test whether the risk attitudes are similar in the
18
50% $20 group and the 90% $20 group.
[Insert Table 6]
Table 6 shows the results of the regression. Panel A shows the results for those that
received $10 from the lottery. The coefficient of Tp90i for measure 1 is -2.15 and is significant
at the 5% level. So the losers from the 90% $20 group choose about 2.15 fewer riskless
options in Table A1 than those losers from the 10% $20 group. It is the same pattern for
other measures. The coefficients of Tp50i are negative for all six measures. And they are
significant at the 10% level for measure 1 and measure 3. These results show a clear and
consistent pattern that the losers from the 90% $20 group are less risk averse (more risk
seeking) than those losers from the 10% $20 group. The coefficients for Tp50i are negative for
all 6 measures but they are not significant. The coefficient of Tp90i is greater in magnitude
than that of Tp50i for measure 1, and the difference is significant at the 5% level. Hence,
those who receive $10 from the 90% $20 group are less risk aversion those who receive $10
from the 50% $20 group. According to Prediction 2, DA model predicts that increase in q
from 50% to 100% does not change the risk attitudes. In contrast, KR model predicts that
the increase in q from 50% to 90% makes individuals less risk averse. Therefore, these results
are not consistent with DA model but support KR model.
Table 6, Panel B shows the results for those that received $20 from the lottery. Similar
to Table 4, Panel C, the results are less consistent in the gain domain (Panel B) compared to
the loss domain (Panel A). The reason might be that diminishing sensitivity and nonlinear
probability weighting imply more risk aversion in the gain domain than around the kink but
loss aversion implies less risk aversion in the gain domain than around the kink. The results
suggest that the effects of diminishing sensitivity and nonlinear probability weighting on risk
attitudes are greater than that of loss aversion.
4.3
Alternative Explanations
In previous sections, the results from both the payoff treatment and the probability treatment
support that expectations influence reference points and the formation of expectation is
consistent with the KR model. In this section, I discuss several alternavtive explanations.
One alternative explanation is the selection into treatment groups. The randomized
interventions were conducted 24 hours before the experiment, and overall 13.0% of subjects
did not show up after they receive emails. In the payoff treatment, if the treatment group
attracts more risk-seeking subjects to show up, it can explain the difference in comparison
1 without the theory of expectations as reference points. I provide three pieces of evidence
to show this is unlikely to be the case. First, as discussed in Section 4, the Wald test shows
19
that I cannot reject the equality of non-show up rates across different groups (p=0.50) in
the payoff treatment. Second, if the treatment group attracts more risk-seeking subjects
and expectations do not matter, the subjects who randomly assigned to receive $20 in the
treatment group should be more risk-seeking than those in $20 control group. However,
my results show the opposite in comparison 3. Second, I conducted a pilot experiment
for comparison 1 with 46 students in the classroom before the main experiment. Since
the expectations were delivered in the beginning of one-hour class, and risk attitudes are
elicited in the end of class, there is no selection into treatment. I find similar results in the
pilot: losers in the treatment groups are more risk seeking than those in the control group.
Therefore, the selection into treatment group is unlikely to explain my treatment effects.
Another alternative explanation about the result is emotion. In psychology literature,
there are two distinct theories about the relationship between mood and risk attitudes. One
is Affect Illusion Model (AIM), which suggests that a positive mood is expected to increase
risk taking tendencies whereas a negative mood is more likely to reduce the tendency to
take risks (Forgas (1995)). The model asserts that individuals in a positive mood are more
likely to access thoughts prone to positive aspects of risky situations than those who are in a
negative mood. This model predicts the opposite direction of my results so it cannot be an
alternative explanation. The other theory is Mood-Maintenance Hypothesis (MMH), which
predicts that individuals in a negative mood make more risky judgments than individuals in
a positive mood for the purpose of improving mood (Isen and Patrick (1983)). Since losers in
the treatment group might have more negative moods, MMH is consistent with my results.
My design could not distinguish whether the results are due to mood or that expectations
change reference points in KR model. However, these two explanations have a very similar
component. Both models have “reference” points: it is good mood in MMH and expected
payoff in KR model. The difference is that MMH is in the emotion domain while KR is in
the monetary domain. It is likely that emotion is just one source of KR model rather than
the confounding of KR model.
Another alternative explanation is that the survey might serve as priming or distraction
rather than time during which reference points adjust to new information. As explained in
Figure 1, all the subjects should fill a survey before the answered risk attitude questions
so they are primed by the same questions. Thus, it is unlikely that different priming can
explain the results.
20
5
Structural Estimation
I have so far shown that expectations influence risk attitudes. This is consistent with
expectations-based reference-dependent models. In this section, I use the detailed choice
data to estimate reference points and the preference based on the reference points. There
are two ways to model expectation-based reference points: the reference point could be modeled as a fixed number, which is the expected utility certainty-equivalent. Then outcome
is evaluated by comparing it to a fixed number which equals the expected utility certainty
equivalent. The reference point could also be modeled as the full distribution of expected
outcomes (KR model). Then outcome is evaluated by comparing it with each expected
outcome and then integrating over the distribution of expected outcome. I estimate these
two models to deepen our understanding about different expectations-based models. In particular, I use the detailed choice data to estimate the certainty-equivalent reference points
and the weights on the stochastic reference points as well as the preferences based on the
reference points.
I can provide insights into the identification of these parameters. The lottery choice
task identifies the utility function parameters. The subjects’ choices made in Table A3 are
used to estimate the curvature of utility function, since I fix the risky options but change
the riskless options for all above exogenous reference points. The loss-aversion parameter
is estimated using Table 2, since I fix the probabilities to 50%/50% but change the payoffs
in risky options. The probability weighting parameter is estimated using Table A1, since I
fix the payoffs but change the probabilities in risky options in that table. The experimental
manipulation of expectations identifies the reference points in different groups.
To estimate the parameters in the utility function, I use a random-utility model (McFadden (1974)) with a nonlinear component:
1
ũ(x) = u(x) + ε =
σ
(
1
(x − r)α + ε
σ
− σ1 λ(r − x)α +
if x ≥ r
ε if x < r
(3)
where ε is assumed to be i.i.d. error term and modeled as type I extreme value. The utility
is scaled by 1/σ and the parameter σ is the scale parameter, because it scales the utility to
reflect the variance of the unobserved portion of utility.
Suppose the subject is asked to choose between (1) x2 and (2) take the following bet:
p probability to get x3 and (1 − p) probability to get x1 . Let U (a)denote the utility as a
function of their choices of bets. a = 1 if the subject chooses riskless options (option 1) and
a = 2 if the subject chooses risky options (option 2). The probability to choose risky options
21
can be presented by the usual logit formula:
P (a = 2) =
exp(U (a = 2))
exp(U (a = 1)) + exp(U (a = 2))
(4)
With this formula and the data about subjects’ choices, I could use maximum-likelihood
estimation to estimate the parameters in the structural model. Given that the underlying
logistic model becomes highly nonlinear in the parameters, I code my own estimator in Stata
to estimate the parameters and account for potential correlations within clusters. Below,
I consider two expectations-based models: the model of the certainty-equivalent reference
points and the model of the stochastic reference points.
5.1
The Model of the Certainty-Equivalent Reference Points
According to equation (3), I jointly estimate α,λ,γ,σ and certainty-equivalent reference points
in the payoff treatment. The log-likelihood is calculated through equation (4), where U (a =
1) = u(x2 ) and U (a = 2) = pu(x3 ) + (1 − p)u(x1 ). I constrained 0 ≤ α ≤ 1 and 0 ≤ γ ≤ 1.
I also allow the coefficients of certainty-equivalent reference points in the following seven
groups to be different from each other: those who receive $10 in the control group (rp1),
those in the no-waiting treatment (rp2), those who receive $10 in the waiting treatment
(rp3), those who receive $15 in the waiting treatment (rp4), those who receive $20 in the
waiting treatment (rp5), those who receive $20 in the control group (rp6), and those who
receive $15 in the control group (rp7). In each specification, I apply 32 initials values in
MLE and present the maximum of log-likelihood in the 32 estimations.
[Insert Table 7]
Column 1 estimates the level of fixed reference points in the seven groups. The point
estimate of α is 0.79 and it is significantly less than one at the 1% level. This is consistent
with diminishing sensitivity. The point estimate of λ is 0.87 and it is not consistent with loss
aversion. The point estimate of γ is 0.63 and it is significantly less than one at the 1% level.
The value of γ is lower than estimates from other contexts, which is close to 0.7 (Tanaka et
al. (2010); Barseghyan et al. (2013)). The estimated fixed reference points have the following
pattern: rp1 < rp2 < rp6. The differences between rp1,rp2, and rp6 are significant at the
1% level. The results suggest that my interventions give different expectations to different
groups, which further confirms that the manipulation of expectation affect reference points.
In column 2, I analyze how quickly the certainty-equivalent reference points adjust. I
assume subjects will put some weight on the utility from new reference points and the rest on
the utility from old reference points. The subjects do not adjust reference points if the weight
is zero and they fully adjust reference points to new ones if the weight is one. I estimate
22
the weight on the utility from new reference points.The point estimate of aa is 0.675, and it
is significantly greater than zero at the 1% level. The positive weight in the waiting group
is consistent with the reduced form results in Table 4 and supports that subjects adjust
reference points quickly.
In column 3, I compare the model of certainty-equivalent expectations as reference points
with that of the status quo as reference points. I construct two utility functions: In the first
function, the reference points are fixed expectations, i.e., rp1 = 10, rp2 = rp3 = rp4 =
rp5 = rp7 = 15, and rp6 = 20; in the second one, the reference points are the status quo, i.e.
all reference points are zero. Then I estimate the weight on the model of expectations. The
estimated weight is 0.47. This result suggests that expectations and the status quo influence
reference points similarly under the assumption of certainty-equivalent reference points.
I also jointly estimate α,λ,γ,σ and certainty-equivalent reference points in the probability
treatment. I allow the coefficients of certainty-equivalent reference points in the following
three groups to be different from each other: those in the 10% $20 group (rpa), those in the
50% $20 group (rpb), those in the 90% $20 group (rpc).
[Insert Table 8]
I present the results in column 1 in Table 8. The pattern is similar to colunm 1 in
Table 7. There is no significant loss aversion but strong probability weighting (γ = 0.64).
The estimated reference points suggests that my interventions give different expectations to
different groups, which further confirms that the exogenous variations of expectation affect
reference points.
5.2
The Model of the Stochastic Reference Points
In the KR model, the reference point is the full distribution of expected outcomes. In
the payoff treatment, subjects expected to receive $10 with 1/3 probability, $15 with 1/3
probability and $20 with 1/3 probability in the no-waiting group. Therefore, the reference
points should be stochastic reference points with weights equal to the probability: $10 with
1/3 probability, $15 with 1/3 probability and $20 with 1/3 probability. In the KR model,
I will jointly estimate α,λ,γ,σ and certainty-equivalent reference points. I constrained 0 ≤
α ≤ 1 and 0 ≤ γ ≤ 1. The log-likelihood is calculated through equation (4), where
U (a = 1) = wi1 · u(x2 |10) + wi2 · u(x2 |15) + wi3 · u(x2 |20)
U (a = 2) = π(p) · {wi1 · u(x3 |10) + wi2 · u(x3 |15) + wi3 · u(x3 |20)}
+π(1 − p){wi1 · u(x1 |10) + wi2 · u(x1 |15) + wi3 · u(x1 |20)}
23
Similarly, I allow the coefficients of the weights on stochastic reference points in the following
seven groups (i = 1, ..., 7) to be different from each other: those who receive $10 in the control
group (w11, w12 and w13), those in the no-waiting treatment (w21, w22 and w23), those
who receive $10 in the waiting treatment (w31, w32 and w33), those who receive $15 in the
waiting treatment (w41, w42 and w43), those who receive $20 in the waiting treatment (w51,
w52 and w53), those who receive $20 in the control group (w61, w62 and w63), and those
who receive $15 in the control group (w71, w72 and w73). Since wi1 + wi2 + wi3 = 1, there
are 18 parameters to estimate ( α,λ,γ,σ and 14 weights in seven groups) in the KR model
and it is not identified. Therefore, I constraint wi2 = wi3 and estimate only 11 parameters.
Column 4 in Table 7 estimates the weight on stochastic reference points in the seven
groups in the payoff treatment. The point estimate of α is 0.88 and it is significantly less
than one at the 5% level. The point estimate of λ is 2.06 and it is significantly greater than
one at the 1% level. This value of loss aversion is consistent with loss aversion estimates from
other contexts (Tversky and Kahneman (1992); Gill and Prowse (2012); Pope and Schweitzer
(2011)). The estimated weights on stochastic reference points have the following pattern:
w11 > w21 > w61 . The difference between w11 and w21 is not significant. The results are
consistent with my interventions that give different expectations to different groups, which
further confirms that the exogenous variations of expectation affect reference points.
In column 5 in Table 7, I analyze how quickly the stochastic weights on reference points
adjust in the similar way to column 2 in Table 7. The point estimate of aa is 0.57 in the
KR model, and it is significantly greater than zero at the 1% level. The positive weight in
the waiting group is consistent with the reduced form results in Table 4 and supports that
subjects adjust reference points quickly.
In column 6 in Table 7, I compare the model of stochastic expectations as reference
points with that of the status quo as reference points in the similar way to column 3 in
Table 7. I construct two utility functions: In the first function, the reference points are fixed
expectations, i.e.,w11 = 1, w21 = w31 = w41 = w51 = w71 = 13 , w61 = 0; in the second one, the
reference points are the status quo, i.e. all reference points are zero. The estimated weight
on expectations is 0.50. This result suggests that expectations and the status quo influence
reference points similarly under the assumption of the KR model.
In column 7 in Table 7, I compare the model of the stochastic reference point with that
of the fixed reference point. I construct two utility functions: In the first function, I use
stochastic expectations as reference points, i.e., w11 = 1, w21 = w31 = w41 = w51 = w71 = 31 ,
w61 = 0; In the second function, I use fixed expectations as reference points, i.e. rp1 = 10,
rp2 = rp3 = rp4 = rp5 = rp7 = 15, and rp6 = 20. Then I estimate the weight on the first
model. The estimated weight is 0.7. Thus, the model of the stochastic reference point fits
24
my data better than that of the certainty-equivalent reference point.
I also estimate estimate α,λ,γ,σ and weights on stochastic reference points in the probability treatment. The log-likelihood is calculated through equation (4), where
U (a = 1) = wi1 · u(x2 |10) + (1 − wi1 ) · u(x2 |20)
U (a = 2) = π(p)·{wi1 ·u(x3 |10)+(1−wi1 )·u(x3 |20)}+π(1−p){wi1 ·u(x1 |10)+(1−wi1 )·u(x1 |20)}
Similarly, I allow the coefficients of the weights on stochastic reference points in the following
seven groups (i = a, b, c) to be different from each other: those in the 10% $20 group (wa1),
those in the 50% $20 group (wb1), those in the 90% $20 group (wc1).
Column 2 in Table 8 estimate the weight on stochastic reference point in the three
groups in the probability treatment. There is no significant diminishing sensitivity but
strong loss aversion (λ = 2.53) and probability weighting (γ = 0.64). The estimated weights
on stochastic reference points have the following pattern: wa1 > wb1 > wc1. The difference
between wa1 and wb1 is close to marginally significant (p=0.107). The difference between
wb1 and wc1 is significant at the 5% level. The difference between wa1 and wc1 is significant
at the 1% level. The estimated reference points suggests that my interventions give different
expectations to different groups, which further confirms that the exogenous variations of
expectation affect reference points.
In column 3 in Table 8, I compare the model of the stochastic reference point with
that of the fixed reference point. I construct two utility functions: In the first function, I
use stochastic expectations as reference points, i.e., wa1 = 0.9, wa1 = 0.5, wa1 = 0.1; In
the second function, I use fixed expectations as reference points, i.e., rpa = 11,rpb = 15,
rpc = 19. Then I estimate the weight on the first model. The estimated weight is 1.00. Thus,
the model of the stochastic reference point fits my data better than that of the certaintyequivalent reference point. The results are consistent with the reduced form estimation in
Table 6 that support the KR model.
6
Conclusion and Discussion
What determines a reference point is an important question. This paper provides evidence
whether expectations and the status quo determine the reference point. I explicitly manipulated expectations and exogenously varied expectations in different groups. Then I tested
whether expectations change subjects’ risk attitudes. I find that both expectations and the
status quo determine the reference point but expectations play a more important role.
I also exogenously varied the time of receiving new information and tested whether in25
dividuals assimilated new information into their reference points, and the speed of the adjustment. I find that subjects can incorporate much new information into reference points
in a few minutes. I also find the decaying treatment effects of the no waiting group in both
comparison 1 and 3. These results provide suggestive evidence that subjects adjust reference
points quickly.
Finally, I derive different theoretical predictions from DA model and KR model, and
then design the Probability treatment to distinguish these two expectations-based referencedependent models. I find that those who receive $10 from the 90% $20 group are less risk
averse than those who receive $10 from the 50% $20 group. The results are not consistent
with DA model but support KR model. Moreover, the structural estimation suggests that
the model of the stochastic reference point fits the data better than that of the certaintyequivalent reference point.
Recent studies use laboratory experiments to explicitly manipulate subjects’ expectations
and test expectation-based reference-dependent models. The results are mixed. Some studies find consistent results with the notion that reference points are expectations (Abeler et
al. (2011); Smith (2008); Ericson and Fuster (2011)), others find no support for expectationbased reference-dependent models when they vary the probability of expectations (Gneezy
et al. (2013); Heffetz and List (2014); Goette et al. (2014)). My results are consistent
with expectation-based reference-dependent models when varying payoff of expectations and
probability of expectations. One possible reason is that I use a lottery to manipulate the
stochastic expectations before the experiment, and then elicit the risk attitudes af ter the
resolution of the lottery during the experiment. The manipulation holds the outcome constant and varies the fluctuations of lagged belief, which is similar to Smith (2008). However,
most of the above studies create a lottery expectation during the experiment and elicit individual behavior bef ore the resolution of the lottery (Abeler et al. (2011); Ericson and
Fuster (2011); Gneezy et al. (2013); Heffetz and List (2014); Goette et al. (2014)). These
manipulations create expectations without lagged belief. While this paper is not designed
to compare lagged beliefs and other manipulation of expectations, my results suggest that
lagged beliefs might have a stronger impact on forming expectations. Future research should
study under what conditions the manipulation of expectations works and why.
Prior work and this paper suggest that expectations, the status quo, the time of holding
previous beliefs and the time of adjusting to new information contribute to determine reference points together. This paper shows that expectations play a more important role than
the status quo, and reference points are sensitive to time. Future work should investigate
more on the time of holding previous beliefs and the time of adjusting to new information,
with both field and laboratory evidence.
26
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29
Table 1. Summary of Lottery-Choice Treatments
Panel A: Payoff Treatment
Control group
Comparison 1
People who receive $10
from Control 1
Comparison 2
People who receive $15
from Control 2
Comparison 3
People who receive $20
from Control 3
Panel B: Probabilty Treatment
10% $20 group
Comparison 4
People who receive $10
from lottery (loser)
Comparison 5
People who receive $20
from lottery (winner)
“No waiting” treatment
People who receive $10
from lottery (loser)
People who receive $15
from lottery
People who receive $20
from lottery (winner)
“Waiting” treatment
People who receive $10
from lottery (loser)
People who receive $15
from lottery
People who receive $20
from lottery (winner)
50% $20 group
90% $20 group
People who receive $10 People who receive $10
from lottery (loser)
from lottery (loser)
People who receive $20 People who receive $20
from lottery (winner)
from lottery (winner)
Note: Panel A shows the payoff treatment for those in the control group and the treatment groups. Panel B
shows the probability treatment for three treatment groups: (1) those who receive $10 with 10% probability
and $20 with 90% probability, (2) those who receive $50 with 50% probability and (3) those who receive
$20 with 10% probability and $10 with 90% probability.
30
Table 2. Predictions of DA model and KR model
(1)
q is below 50%
q is below 50%
q is below 50%
q is abve 50%
q is abve 50%
q is abve 50%
(2)
CARA
DARA
IARA
CARA
DARA
IARA
sign of derivative in DA
model
(3)
?
0
+
-
sign of derivative in KR
model
(4)
-
Note: According to the table and the sign of derivatives in different models and cases, the following conclusions can be generated:
31
Table 3. Summary statistics
Number of
sessions
Panel A: Payoff Treatment
$10 Control
6
$15 Control
3
$20 Control
6
"No-waiting"
15
treatment group
"Waiting" treatment
14
group
Total
44
Panel B: Probabilty Treatment
90% $20 group
23
50% $20 group
24
10% $20 group
25
Total
72
Number of
subjects
Average earnings
(Dollars)
Standard
deviation
87
46
89
10.92
16.91
20.97
3.93
3.88
3.04
251
16.08
5.67
248
15.96
5.69
20.85
16.41
13.17
5.05
6.64
5.02
721
313
390
329
1032
Note: Panel A shows the summary statistics three control groups, the no-waiting group and the waiting
group in the payoff treatment. Panel B shows the summary statistics for three treatment groups in the
probability treatment: (1) those who receive $10 with 10% probability and $20 with 90% probability, (2)
those who receive $50 with 50% probability and (3) those who receive $20 with 10% probability and $10
with 90% probability.
32
Table 4. The effect of expectation on risk attitudes: payoff treatment
Specification:
OLS Regression
Dep. Var.:
Number of riskless options taken
Measure 1 Measure 2 Measure 3 Measure 4 Measure 5 Measure 6
1
2
3
4
5
6
Panel A: Those who receive $10 from control, no waiting group or waiting group
No waiting group with stochastic
expectations
Waiting group with stochastic
expectations
No waiting = waiting (p-value)
Omitted group
Omitted group mean
Obs.
Controls & Months fixed effect
R-square
-1.94
-1.48
-1.40
-1.94
-0.50
-0.25
(0.75)**
(0.69)**
(0.83)*
(0.79)**
(0.74)
(0.76)
-0.36
-1.46
-0.78
-1.09
-1.34
-0.42
(0.74)
0.035
9.67
273
Y
0.0400
(0.71)**
(0.85)
(0.85)
(0.68)*
0.984
0.471
0.303
0.256
Those who receive $10 from the control group
10.93
12.21
13.53
9.12
273
273
273
182
Y
Y
Y
Y
0.0452
0.0233
0.0432
0.0455
(0.76)
0.834
8.26
182
Y
0.0433
Panel B: Those who receive $15 from control, no waiting group or waiting group
No waiting group with stochastic
expectations
Waiting group with stochastic
expectations
No waiting = waiting (p-value)
Omitted group
Omitted group mean
Obs.
Controls & Months fixed effect
R-square
1.64
1.89
0.95
1.16
0.57
0.41
(1.00)
(0.94)**
(1.06)
(1.01)
(0.83)
(0.94)
0.83
1.35
0.19
0.69
-0.13
-0.58
(1.07)
0.332
10.67
195
Y
0.0594
(0.99)
(1.15)
(1.09)
(0.90)
0.428
0.425
0.553
0.375
Those who receive $15 from the control group
11.24
13.50
13.78
9.67
195
195
195
159
Y
Y
Y
Y
0.0634
0.0787
0.1218
0.0551
(0.96)
0.241
9.17
159
Y
0.0285
Panel C: Those who receive $20 from control, no waiting group or waiting group
No waiting group with stochastic
expectations
1.94
2.02
(0.79)** (0.76)***
Waiting group with stochastic
expectations
No waiting = waiting (p-value)
Omitted group
Omitted group mean
Obs.
Controls & Months fixed effect
R-square
0.48
(0.85)
0.063
11.37
253
Y
0.0599
0.67
0.86
1.32
-0.80
-1.52
(0.68)
(0.79)*
(0.80)
(0.85)*
-0.17
0.78
-0.85
-0.16
(0.82)
(0.76)
(0.74)
(0.61)
0.086
0.143
0.434
0.943
Those who receive $20 from the control group
11.26
14.61
14.60
9.92
253
253
253
214
Y
Y
Y
Y
0.0809
0.0570
0.1162
0.0663
(0.68)
0.071
9.42
214
Y
0.0534 Note: Dependent variable is the number of riskless options taken measured by different tables; Robust
standard errors are in the parentheses. Panel A(B, C) presents the results for those who receive $10
($15,$20) in the control group, the no-waiting group and the waiting group. Columns 1 to 6 report the
results of measures 1 to 6, respectively.
33
Specification:
Sample:
Dep. Var.:
No waiting group
with stochastic
expectations
Indicator for
delayed decisions
(measure from
Table A2 and A3)
No waiting
*Indicator for
delayed decisions
Table 5. The effect of expectation on risk attitudes
OLS
Those who receive $10
Those who receive $15
Those who receive $20
Number of Expected
Number of Expected
Number of Expected
riskless
payoff
riskless
payoff
riskless
payoff
options taken difference options taken difference options taken difference
1
2
3
4
5
6
-1.91
-0.90
0.55
0.47
1.35
0.73
(0.64)***
(0.33)***
(0.75)
(0.38)
(0.64)**
(0.33)**
1.09
-0.16
1.09
-0.33
0.75
-0.63
(0.29)***
(0.16)
(0.37)***
(0.20)*
(0.29)**
(0.17)***
1.34
0.73
0.44
-0.18
-0.87
-0.65
(0.52)**
Omitted group
Obs.
Control & Month
fixed effect
R-square
728
(0.28)***
(0.63)
(0.31)
(0.53)
(0.29)**
Measure 1 for those in the control group and waiting group
728
549
549
720
720
Y
Y
Y
Y
Y
Y
0.0419
0.0229
0.0704
0.0560
0.0517
0.0782
Note: Dependent variable is the number of riskless options
taken or the expected payoff difference; Robust
standard errors are in the parentheses. Column 1-2 present the results for those who receive $10 in the
control group, the no-waiting group and the waiting group.
Column 3-4 presents the results
for those who
receive $15 in the control group, the no-waiting group and the waiting group. Column 5-6 presents the
results for those who receive $20 in the control group, the no-waiting group and the waiting group. ***
significant on 1% level; ** significant on 5% level, * significant on 10% level.
34
Table 6. The effect of expectation on risk attitudes: probability treatment
OLS Regression
Specification:
Number of riskless options taken
Dep. Var.:
Measure 1 Measure 2 Measure 3 Measure 4 Measure 5 Measure 6
1
2
3
4
5
6
Panel A: Those who receive $10
Group with 90% chance of
receiving $20 (beta90)
-2.15
-0.21
-2.01
-0.30
-0.07
-0.11
(0.87)**
(0.83)
(1.07)*
(0.96)
(0.88)
(0.97)
-0.31
-0.11
-0.61
-0.23
-0.18
-0.17
(0.47)
(0.44)
(0.52)
(0.48)
(0.45)
(0.43)
Group with 50% chance of
receiving $20 (beta50)
beta90 = beta50 (p-value)
Omitted group
0.03
0.90
0.19
0.94
0.90
Those with a 10% chance of receiving $20
0.94
Mean of Omitted Group
Obs.
Controls & Months fixed effect
R-square
10.35
554
Y
0.0252
11.03
554
Y
0.0451
13.53
554
Y
0.0146
13.66
554
Y
0.0112
8.67
554
Y
0.0187
7.77
554
Y
0.0126
-1.34
0.14
-0.34
1.33
0.12
-1.38
(1.01)
(1.09)
(0.92)
(1.11)
(0.87)
(1.02)
-0.33
0.67
0.04
1.49
0.08
-1.34
(0.98)
(1.10)
(0.90)
(1.13)
(0.88)
(1.02)
Panel B: Those who receive $20
Group with 90% chance of
receiving $20 (beta90)
Group with 50% chance of
receiving $20 (beta50)
beta90 = beta50 (p-value)
Omitted group
0.06
Mean of Omitted Group
Obs.
Controls & Months fixed effect
R-square
13.16
478
Y
0.0277
0.30
0.47
0.76
0.94
Those with a 10% chance of receiving $20
12.16
478
Y
0.0226
15.41
478
Y
0.0404
13.72
478
Y
0.0285
9.56
478
Y
0.0106
0.92
9.97
478
Y
0.0352 Note: Dependent variable is the number of riskless options taken measured by different tables; Robust
standard errors are in the parentheses. Panel A (B) presents the results for those who receive $10 ($20)
in the 10% $20 group, the 50% $20 group and the 90% $20 group. Columns 1 to 6 report the results of
measures 1 to 6, respectively.
35
Table 7. Maximum likelihood estimation for payoff treatment
Model:
DA model with the fixed reference point
Payoff Treatment
Sample
α
λ
σ
γ
KR model with stochastic reference
DA vs KR
points
1
0.789
(0.030)
0.871
(0.104)
0.726
(0.059)
0.629
(0.034)
Weight on the new
outcome for waiting
group (aa)
2
0.978
(0.028)
1.613
(0.056)
1.402
(0.088)
0.560
(0.030)
3
0.977
(0.048)
1.495
(0.123)
1.291
(0.167)
0.646
(0.041)
4
0.884
(0.055)
2.058
(0.246)
1.161
(0.198)
0.514
(0.030)
5
1.000
(0.000)
1.717
(0.076)
1.502
(0.067)
0.543
(0.031)
6
1.000
(0.000)
1.618
(0.156)
1.410
(0.062)
0.630
(0.090)
0.675
(0.089)
Weight on expectationsbased model (aa)
0.568
(0.098)
0.470
0.504
(0.083)
(0.090)
Weight on KR model (aa)
Reference point for $10
control group (rp1/w11)
Reference point for nowaiting group (rp2/w21)
Reference point for
those who receive $10 in
waiting group(rp3/w31)
Reference point for
those who receive $15 in
waiting group(rp4/w41)
Reference point for
those who receive $20 in
waiting group(rp5/w51)
Reference point for $20
control group (rp6/w61)
Reference point for $15
control group (rp7/w71)
7
1.000
(0.000)
1.774
(0.090)
1.610
(0.076)
0.601
(0.035)
0.699
(0.127)
8.974
0.617
(0.406)
(0.063)
9.747
0.516
(0.240)
(0.083)
9.175
0.582
(0.312)
(0.056)
11.754
0.838
(0.927)
(0.133)
16.041
0.000
(0.125)
(0.000)
14.588
0.000
(0.163)
(0.000)
9.048
0.947
λ=1 (p-value)
(0.431)
0.213
rp1=rp2 (p-value)
No of individuals
No of observation
Log likelihood
0.049
721
37943
-18031.389
0.000
721
37943
-18246.487
0.000
721
37943
-18069.899
(0.161)
0.000
0.000
0.000
0.000
0.159
721
721
721
721
37943
37943
37943
37943
-17989.4 -18254.986 -18084.268 -18434.681 Note: In the payoff treatment, I jointly estimate α, λ, γ and σ and reference points in utility function in
Equation (5) through MLE. In each specification, I apply 32 initials values in MLE and present the maximum
of log-likelihood in the 32 estimations. Standard errors are clustered at the individual level. Columns 1 to 3
present the estimiation with fixed reference points. Columns 4 to 6 present the estimiation with stochastic
reference points. Column 1 estimates the level of fixed reference points, and Column 4 estimate the weight
on stochastic reference point. In columns 2 and 5, I estimate the weight on the utility from new reference
points.In columns 3 and 6, I compare the model of expectations as reference points with that of the status
quo as reference points, and estimate the weight on the first model. In column 7, I compare the model of the
stochastic reference point with that of the fixed reference point, and estimate the weight on the first model.
36
Table 8. Maximum likelihood estimation of utility function
Model:
DA model with the fixed KR model with stochastic
reference point
reference points
Probability Treatment
Sample
α
λ
σ
γ
1
0.923
(0.087)
1.340
(0.243)
1.239
(0.302)
0.640
(0.073)
2
1.000
(0.000)
2.527
(0.147)
2.116
(0.146)
0.618
(0.047)
8.483
0.708
(2.568)
(0.035)
9.219
0.602
(1.026)
(0.072)
10.576
0.533
(0.798)
0.162
0.71
1032
58824
-30044.444
(0.068)
0.00
0.11
1032
58824
-29344.646
Weight on KR model (aa)
Reference point for the
10% $20 group (rpa/wa)
Reference point for the
50% $20 group (rpb/wb)
Reference point for the
90% $20 group (rpc/wc)
λ=1 (p-value)
rpa=rpb (p-value)
No of individuals
No of observation
Log likelihood
DA vs KR
3
0.959
(0.029)
2.036
(0.080)
2.109
(0.155)
1.000
(0.000)
1.000
(0.000)
0.000
1032
58824
-29885.597
Note: In the probability treatment, I jointly estimate α, λ, γ and σ and reference points in utility function
in Equation (5) through MLE. In each specification, I apply 32 initials values in MLE and present the
maximum of log-likelihood in the 32 estimations. Standard
errors are clustered at the individual level.
Column 1 estimates the level of fixed reference points, and Column 2 estimate the weight on stochastic
reference point. In column 3, I compare the model of the stochastic reference point with that of the fixed
reference point, and estimate the weight on the first model.
37
Panel A: Payoff Treatment
Panel B: Probability Treatment
Figure 1. Treatments
Note: Panel A shows the payoff treatment. Panel B shows the probability treatment.
38
Panel A Panel B Panel C Figure 2 Figure 2. Utility functions
Note: The vertical line is the predicted level of risk aversion and Pˆ* is the minimum winning probability (p)
people would like to take the bet.
39
8
Number of Riskless Options Taken
10
12
14
Risk Attitudes for Payoff Treatment
Those who receive $10
Those who receive $15
CONTROL
Those who receive $20
No-WAITING
Figure 3. Risk attitudes
for
Figure 3 the Payoff Treatment
WAITING
Note: This figure shows the risk attitudes for the payoff treatment. The vertical axis stands for the average
number of riskless options taken in Table A1 to A3 and it measures risk averse. The horizontal axis stands
for those who receive $10, $15, $20 in the control group, the no-waiting treatment groups, and the waiting
treatment group.
40
Panel A Cumulative Distributions of Risk Attitudes
0
.2
.4
CDF
.6
.8
1
For Those Who Receive $10
0
5
10
15
The Number of Riskless Options Taken
CONTROL
NO-WAITING
20
WAITING
Panel B Cumulative Distributions of Risk Attitudes
0
.2
.4
CDF
.6
.8
1
For Those Who Receive $15
0
5
10
15
The Number of Riskless Options Taken
CONTROL
NO-WAITING
20
WAITING
Panel C Cumulative Distributions of Risk attitudes
0
.2
.4
CDF
.6
.8
1
For Those Who Receive $20
0
5
10
15
The Number of Riskless Options Taken
CONTROL
NO-WAITING
Figure 4. Cumulative Distributions of Risk Attitudes 20
WAITING
Figure 4. Cumulative distributions of risk attitudes for the payoff treatment
Note: Panel A shows the cumulative distributions of risk attitudes for those who receive $10 in the three
different groups. Panel B shows the cumulative distributions of risk attitudes for those who receive $15 in
the three different groups. Panel C shows the cumulative distributions of risk attitudes for those who receive
$20 in the three different groups. The horizontal axis stands for the average number of riskless options taken
in Table A1 to A3.
41
8
Number of Riskless Options Taken
10
12
14
Risk Attitudes for Probability Treatment
Those who receive $10
10% $20
Those who receive $20
50% $20
90% $20
Figure 5. Risk attitudesFigure 5 for the Probability Treatment
Note: This figure shows the risk attitudes for the probability
treatment. The vertical axis stands for the
average number of riskless options taken in Table A1 to
A3 and it measures risk averse. The horizontal axis
stands for those who receive $10, $20 in the 10% $20 group, the 50% $20 group, the 90% $20 group.
42 Panel A Cumulative Distributions of Risk Attitudes
0
.2
.4
CDF
.6
.8
1
For Those Who Receive $10
0
5
10
15
The Number of Riskless Options Taken
10% Group
50% Group
20
90% Group
Panel B Cumulative Distributions of Risk Attitudes
0
.2
.4
CDF
.6
.8
1
For Those Who Receive $20
0
5
10
15
The Number of Riskless Options Taken
10% Group
Figure 6.
50% Group
20
90% Group
Figure 6. Cumulative Distributions of Risk Attitudes Cumulative distributions of risk attitudes for the probability
treatment
Note: Panel A in the figure shows the cumulative distributions
of risk attitudes for those who receive $10
in the three different groups. Panel B in the figure shows the cumulative distributions of risk attitudes for
those who receive $20 in the three different groups. The horizontal axis stands for the average number of
riskless options taken in Table A1 to A3.
43
Table A.1 The Paired Lottery-Choice Decisions with Probability Changing
Option 1
Payoff
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
Probability
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
Option 2
Payoff
Probability
15
90%
15
85%
15
80%
15
75%
15
70%
15
65%
15
60%
15
55%
15
50%
15
45%
15
40%
15
35%
15
30%
15
25%
15
20%
15
15%
15
10%
15
5%
15
0%
Payoff
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
Expected payoff
difference
4
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
Note: In Table A1 I fix the payoffs but change the probabilities in risky options. For each question in a row, subjects are asked to
choose between option 1 and option 2. Subjects cannot see the expected payoff difference.
44
Table A.2 The Paired Lottery-Choice Decisions with Payoff Changing
Option 1
Payoff
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
Probability
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
Option 2
Payoff
Probability
10
50%
10.5
50%
11
50%
11.5
50%
12
50%
12.5
50%
13
50%
13.5
50%
14
50%
14.5
50%
15
50%
15.5
50%
16
50%
16.5
50%
17
50%
17.5
50%
18
50%
18.5
50%
19
50%
Payoff
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
Expected payoff
difference
2.5
2.25
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-1
-1.25
-1.5
-1.75
-2
Note: In Table A2 I fix the probabilities but change the payoffs in risky options. For each question in a row, subjects are asked to
choose between option 1 and option 2. Subjects cannot see the expected payoff difference.
45
Table A.3 The Paired Lottery-Choice Decisions with Payoff Changing
Option 1
Payoff
10
10.5
11
11.5
12
12.5
13
13.5
14
14.5
15
15.5
16
16.5
17
17.5
18
18.5
19
Probability
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
50%
Option 2
Payoff
Probability
20
50%
20
50%
20
50%
20
50%
20
50%
20
50%
20
50%
20
50%
20
50%
20
50%
20
50%
20
50%
20
50%
20
50%
20
50%
20
50%
20
50%
20
50%
20
50%
Payoff
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
Expected payoff
difference
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Note: In Table A3 I change the payoffs in riskless options. For each question in a row, subjects are asked to choose between
option 1 and option 2. Subjects cannot see the expected payoff difference.
46
Table A.4 Summary Questions for Each Table
Summary Questions
Now you have a choice between (1) Keep the $10 (2) Take the following bet:
After Table A1 p% probability to get $15 and (100-p) % probability to get $5. What is the
minimum probability p% that you will choose choice option 2?
Now you have a choice between (1) Keep the $10 (2) Take the following bet:
After Table A2 50% probability to get $X and 50% probability to get $5. What is the
minimum X that you will choose choice 2?
Now you have a choice between (1) Keep $X (2) Take the following bet: 50%
After Table A3 probability to get $10 and 50% probability to get $20. What is the minimum X
that you will choose choice 1?
47
Panel A 6
Number of Riskless Options Taken
8
10
12
14
Risk Attitudes for Those Who Receive $10
measure 1
measure 2
measure 3 measure 4
Different Measurement
CONTROL(87)
measure 5
measure 6
TREATMENT(186)
Panel B 6
Number of Riskless Options Taken
8
10
12
14
Risk Attitudes for Those Who Receive $15
measure 1
measure 2
measure 3 measure 4
Different Measurement
CONTROL(46)
measure 5
measure 6
TREATMENT(149)
Panel C 6
Number of Riskless Options Taken
8
10
12
14
Risk Attitudes for Those Who Receive $20
measure 1
measure 2
measure 3 measure 4
Different Measurement
CONTROL(89)
measure 5
measure 6
TREATMENT(164)
Figure A1. Risk Attitudes by Amount Received Figure A1. Risk attitudes for the payoff treatment
Note: Panel A, B, C shows the risk attitudes for those who receive $10, $15, $20 in the payoff treatment,
respectively. The vertical axis stands for the number of riskless options taken in Holt and Laury table and
it measures risk averse. The six bars stand for different measures from the tables. “Measure 1” is derived
from Table A1 that fixes payoffs but changes probability in risky options. “Measure 2” is calculated from
the question (for comparison 1) “Now you have a choice between (1) Keep the $10 (2) Take the following
bet: p% probability to get $15 and (100-p) % probability to get $5. What is the minimum probability p%
that you will choose choice 2?” For example, if p=52, then measure 2 is 9 because the subject would take 9
riskless options if he/she answer the questions in Table 1. “Measure 3” is derived from Table A2 that fixes
the probability to 50%/50% but change payoffs in risky options. “Measure 4” is similar to measure 2 but
calculated from the question (for comparison 1) “Now you have a choice between (1) Keep the $10 (2) Take
the following bet: 50% probability to get $X and 50% probability to get $5. What is the minimum X that
you will choose choice 2?”. “Measure 5” is derived from Table A3 that fixes the risky options but change the
riskless options. “Measure 6” is similar to measure 2 and 4, but calculated from the question (for comparison
1) “Now you have a choice between (1) Keep $X (2) 48
Take the following bet: 50% probability to get $10 and
50% probability to get $20. What is the minimum X that you will choose choice 1?”.
Panel A – Those who received $10
14
12
10
8
6
Number of Riskless Options Taken
Risk Attitudes for Different Probability Treatments
Measure 1
Measure 2
10% Group(297)
Measure 3 Measure 4
Different Measurement
50% Group(216)
Measure 5
Measure 6
90% Group(41)
Panel B – Those who received $20
14
12
10
8
6
Number of Riskless Options Taken
Risk Attitudes for Different Probability Treatments
Measure 1
Measure 2
10% Group(32)
Measure 3 Measure 4
Different Measurement
50% Group(174)
Measure 5
Measure 6
90% Group(272)
Figure A2. Risk attitudes for the probability treatments
Note: Panel A, B shows the risk attitudes for those who receive $10, $20 in the probability treatment,respectively. The vertical axis stands for the number of riskless options taken in Holt and Laury
table and it measures risk averse. The six bars stand for different measures from the tables. “Measure 1” is
derived from Table A1 that fixes payoffs but changes probability in risky options. “Measure 2” is calculated
from the question (for comparison 1) “Now you have a choice between (1) Keep the $10 (2) Take the following
bet: p% probability to get $15 and (100-p) % probability to get $5. What is the minimum probability p%
that you will choose choice 2?” For example, if p=52, then measure 2 is 9 because the subject would take 9
riskless options if he/she answer the questions in Table 1. “Measure 3” is derived from Table A2 that fixes
the probability to 50%/50% but change payoffs in risky options. “Measure 4” is similar to measure 2 but
calculated from the question (for comparison 1) “Now you have a choice between (1) Keep the $10 (2) Take
the following bet: 50% probability to get $X and 50% probability to get $5. What is the minimum X that
you will choose choice 2?”. “Measure 5” is derived from Table A3 that fixes the risky options but change the
riskless options. “Measure 6” is similar to measure 2 and 4, but calculated from the question (for comparison
1) “Now you have a choice between (1) Keep $X (2) Take the following bet: 50% probability to get $10 and
50% probability to get $20. What is the minimum X49
that you will choose choice 1?”.
For Online Publication
Appendices
A
A.1
Proofs
Payoff Treatment
Following Koszegi and Rabin (2006, 2007) small stakes decisions are considered such that
consumption utility can plausibly be taken as approximately linear, and a piecewise-linear
gain-loss utility function is adopted,
(
u(x|r) =
x − r if x − r ≥ 0
λ(x − r) if x − r < 0
)
where the utility parameter λ represents the degree of loss aversion, λ > 1.
Subjects choose between option (1) a fixed payoff of x2 or (2) get payoff x1 with probability
(1-p) and x3 with probability p , 0 ≤ p ≤ 1 in Table A1-A3. Define u1 as the utility if subjects
choose option (1) and u2 as the utility if subjects choose option (2). In the control group,
the reminder email says that the participants will receive payoff r1 with probability of 1.
Based on these expectations, the utilities to choose option (1) and (2) are:
u1 = x2 − r1 = 0
u2 = p(x3 − r1 ) + (1 − p)λ(x1 − r1 )
Let p∗ to be the minimum winning probability people would like to choose option (2) and
take the bet. The higher p∗ is, the more risk averse. Let u1 = u2 (p∗1 ),
p∗1 =
λ
(x2 − r1 ) + λ(r1 − x1 )
=
(x3 − r1 ) + λ(r1 − x1 )
λ+2
In the treatment group, the reminder email says that the participants will receive payoff
r1 with probability of 1/3, payoff r2 with probability of 1/3 and payoff r3 with probability
50
of 1/3. Based on these expectations, the utilities to choose option (1) and (2) are:
1
1
1
(x2 − r1 ) + λ(x2 − r2 ) + λ(x2 − r3 )
3
3
3
1
1
=
[(1 − p)λ(x1 − r1 ) + p(x3 − r1 )] + [(1 − p)λ(x1 − r2 ) + p(x3 − r2 )]
3
3
1
+
[(1 − p)λ(x1 − r3 ) + pλ(x3 − r3 )]
3
u1 =
u2
Similarly, let u1 = u2 (p∗1 ),
3
p∗1 =
3
2λ
(x2 − r1 ) + λ(2x2 + r1 − 3x1 )
=
(2x3 − r1 − r2 ) + λ(r1 + r2 + x3 − 3x1 )
3λ + 3
Hence, when λ > 1,
p∗1 > p∗1
3
Those who receive $10 in the treatment group will be less risk averse than those in the $10
control group.
A.2
Probability Treatment: Linear Case
I consider two expectation-based referece dependent models: fixed reference points in models
of Disappointment Aversion (DA) and stochastic reference points in models of Kőszegi and
Rabin (KR). I use the same ultility function and notation as in payoff treatment. In the probability treatment, the reminder emails says that participants will receive r2 with probability
q and r1 with probability (1-q), 0 ≤ q ≤ 1 and r2 > r1 = x2 . Let r = qr2 + (1 − q)r1 ≥ x2 ,
i.e., x1 ≤ x2 = r1 ≤ x3 ≤ r2 . In our treatment, r1 = 10, r2 = 20, and the values of q are
10%, 50% and 90% in the three groups.
A.2.1
DA model
Reference point is a fixed point, r. We consider two cases according to different r values:
Case 1: r ≥ x3 . The utilities to choose option (1) and (2) are:
u1 = λ(x2 − r)
u2 = (1 − p)λ(x1 − r) + pλ(x3 − r)
Let u1 = u2 (p∗ ),
p∗ =
x2 − x1
x3 − x1
51
∂p∗
=0
∂r
p∗ reminds constant as r increases.
Hence, in Case 1, p∗ remains constant as q increases.
Case 2: x2 ≤ r < x3 . The utilities to choose option (1) and (2) are:
u1 = λ(x2 − r)
u2 = (1 − p)λ(x1 − r) + p(x3 − r)
Let u1 = u2 (p∗ ),
p∗ =
λ(x2 − x1 )
(x3 − λx1 ) + (λ − 1)r
λ(x2 − x1 )(λ − 1)
∂p∗
=−
<0
∂r
[(x3 − λx1 ) + (λ − 1)r]2
∂r
p∗ decreases as r increases. Since ∂q
= r2 − r1 > 0, r increases as q increases.
Hence, in Case 2, p decreases as q increases.
A.2.2
KR model
In the KR model, reference point is full distribution of r1 and r2 with respect to q. The
utilities to choose option (1) and (2) are:
u1 = qλ(x2 − r2 ) + (1 − q)(x2 − r1 )
u2 = q[(1 − p)λ(x1 − r2 ) + pλ(x3 − r2 )] + (1 − q)[(1 − p)λ(x1 − r1 ) + p(x3 − r1 )]
Let u1 = u2 (p∗ ),
p∗ =
λ(r1 − x1 ) − (x3 − r1 )
q(λ − 1)(x3 − r1 ) − λ(x1 − r1 ) + (x3 − r1 )
∂p∗
λ(x2 − x1 )(λ − 1)(x3 − x2 )
<0
=−
∂q
[q(λ − 1)(x3 − x2 ) + (x3 − x2 ) + λ(x2 − x1 )]2
Hence, p∗ decreases as q increases. The relationship between p∗ and q is summrized in Figure
2, Panel C.
52
A.3
Probability Treatment: General Case
In this subsection, I use a general gain-loss utility function:
(
u(y|r) =
f (x − r) if x − r ≥ 0
λ · f (x − r) if x − r < 0
)
where the utility parameter λ represents the degree of loss aversion, λ > 1. We assume
f (x) = −f (x). f 0 > 0, f 00 ≤ 0, when x > 0; f 0 > 0, f 00 ≥ 0, when x < 0; f 0 is not defined
when x = 0.
A.3.1
DA model
Reference point is a fixed point, r. We consider two cases according to different r values:
Case 1: r ≥ x3 . The utilities to choose option (1) and (2) are:
u1 = λf (x2 − r)
u2 = (1 − p)λf (x1 − r) + pλf (x3 − r)
Let u1 = u2 (p∗ ),
p∗ =
f (x2 − r) − f (x1 − r)
f (x3 − r) − f (x1 − r)
1
∂p∗
=
·Q
∂q
[f (x3 − r) − f (x1 − r)]2
where Q = f (x2 − r)[f 0 (x3 − r) − f 0 (x1 − r)] + f (x1 − r)[f 0 (x2 − r) − f 0 (x3 − r)] + f (x3 −
r)[f 0 (x1 − r) − f 0 (x2 − r)]}
Q = f (x2 − r)[f 0 (x3 − r) − f 0 (x2 − r) + f 0 (x2 − r) − f 0 (x1 − r)]
+ f (x1 − r)[f 0 (x2 − r) − f 0 (x3 − r)] + f (x3 − r)[f 0 (x1 − r) − f 0 (x2 − r)]
f 0 (x2 − r) − f 0 (x3 − r)
= [f (x2 − r) − f (x3 − r)][f (x1 − r) − f (x2 − r)]
f (x2 − r) − f (x3 − r)
f 0 (x1 − r) − f 0 (x2 − r)
− [f (x1 − r) − f (x2 − r)][f (x2 − r) − f (x3 − r)]
f (x1 − r) − f (x2 − r)
= [f (x1 − r) − f (x2 − r)][f (x2 − r) − f (x3 − r)](R23 − R12 )
f 0 (x −r)−f 0 (x −r)
where Rij = f (xii −r)−f (xjj−r) .
Then I discuss the different functional forms of f (x) based on the coefficients of absolute
risk aversion. Define coefficients of absolute risk aversion Ax in the loss domain:
53
Ax =
f 00 (x)
f 0 (x + ∆x) − f 0 (x)
=
lim
∆x→0 f ( x + ∆x) − f (x)
f 0 (x)
1. f (x) satisfies constant absolute risk aversion (CARA) in the loss domain:
Intuitively, CARA has the property that change in wealth does not change risk attitudes. In
this setting, when all payoffs are in loss domain, change in reference points will not change
risk attitudes. Hence, p∗ remains constant as q increseases.
Formally, by the defination of CARA, A(x2 −r) = A(x2 −r+∆x) = ... = A(x3 −r)
where
f 0 (x2 − r + ∆x) − f 0 (x2 − r)
A(x2 −r) = lim (
∆x→0 f x2 − r + ∆x) − f (x2 − r)
f 0 (x2 − r + 2∆x) − f 0 (x2 − r + ∆x)
∆x→0 f ( x2 − r + 2∆x) − f (x2 − r + ∆x)
A(x2 −r+∆x) = lim
f 0 (x3 − r) − f 0 (x3 − r − ∆x)
∆x→0 f ( x3 − r) − f (x3 − r − ∆x)
A(x3 −r) = lim
R23 =
f 0 (x3 − r) − f 0 (x2 − r)
f (x3 − r) − f (x2 − r)
f 0 (x3 −r)−f 0 (x3 −r−∆x)+f 0 (x3 −r−∆x)−...−f 0 (x2 −r+∆x)+f 0 (x2 −r+∆x)−f 0 (x2 −r)
f (x3 −r)−f (x3 −r−∆x)+f (x3 −r−∆x)−...−f ( x2 −r+∆x)+f ( x2 −r+∆x)−f (x2 −r)
0
f (x2 − r + ∆x) − f 0 (x2 − r)
lim
= A(x2 −r)
∆x→0 f ( x2 − r + ∆x) − f (x2 − r)
= lim∆x→0
=
0
0
2 −r)−f (x1 −r)
Similarly, R12 = ff (x
= A(x2 −r) = R23
(x2 −r)−f (x1 −r)
∂p∗
Since R23 − R12 = 0, we have ∂q = 0.
Hence, p∗ remains constant as q increseases.
2. f (x) satisfies decreasing absolute risk aversion (DARA) in the loss domain:
By the defination of DARA, in the loss domain,
A(x1 −r) < A(x1 −r+∆x) < ... < A(x2 −r) < A(x2 −r−∆x) < ... < A(x3 −r)
R23 =
f 0 (x3 − r) − f 0 (x2 − r)
f (x3 − r) − f (x2 − r)
= lim∆x→0
R12
f 0 (x3 −r)−f 0 (x3 −r−∆x)+f 0 (x3 −r−∆x)−...−f 0 (x2 −r+∆x)+f 0 (x2 −r+∆x)−f 0 (x2 −r)
f (x3 −r)−f (x3 −r−∆x)+f (x3 −r−∆x)−...−f ( x2 −r+∆x)+f ( x2 −r+∆x)−f (x2 −r)
r) − f 0 (x1 − r)
f 0 (x2 −
=
f (x2 − r) − f (x1 − r)
= lim∆x→0
f 0 (x2 −r)−f 0 (x2 −r−∆x)+f 0 (x2 −r−∆x)−...−f 0 (x1 −r+∆x)+f 0 (x1 −r+∆x)−f 0 (x1 −r)
f (x2 −r)−f (x2 −r−∆x)+f (x2 −r−∆x)−...−f ( x1 −r+∆x)+f ( x1 −r+∆x)−f (x1 −r)
According to Lemma in Appendix A4, we have R23 > R12 .
Since f (x1 − r) − f (x2 − r) < 0 , f (x2 − r) − f (x3 − r) < 0, and R23 − R12 > 0,
54
∗
we have ∂p
> 0. Hence, p∗ increseases as q increases.
∂q
3. f (x) satisfies increasing absolute risk aversion (IARA) in the loss domain:
By the defination of IARA in the loss domain,
A(x1 −r) > A(x1 −r+∆x) > ... > A(x2 −r) > A(x2 −r−∆x) > ... > A(x3 −r) Similarly, according to
Lemma in Appendix A4, we have R23 < R12 .
Since f (x1 − r) − f (x2 − r) < 0 , f (x2 − r) − f (x3 − r) < 0, and R23 − R12 > 0,
∗
we have ∂p
< 0.
∂q
Hence, p∗ decreases as q increases.
Case 2: x2 ≤ r < x3 . The utilities to choose option (1) and (2) are:
u1 = λf (x2 − r)
u2 = (1 − p)λf (x1 − r) + pf (x3 − r)
Let u1 = u2 (p∗ ),
p∗ =
λf (x2 − r) − λf (x1 − r)
f (x3 − r) − λf (x1 − r)
∂p∗
1
=
·λ·M
∂r
[f (x3 − r) − λf (x1 − r)]2
where M = f 0 (x1 − r)[f (x3 − r) − f (x2 − r)] + f 0 (x2 − r)[λf (x1 − r) − f (x3 − r)]
+ f 0 (x3 − r)[f (x2 − r) − f (x1 − r)]
M = f (x2 − r)[f 0 (x3 − r) − f 0 (x1 − r)] + f (x1 − r)[λf 0 (x2 − r) − f 0 (x3 − r)]
+ f (x3 − r)[f 0 (x1 − r) − f 0 (x2 − r)]
f 0 (x3 − r) − f 0 (x2 − r)
f (x3 − r) − f (x2 − r)
f 0 (x2 − r) − f 0 (x1 − r)
− [f (x3 − r) − f (x2 − r)][f (x2 − r) − f (x1 − r)]
f (x2 − r) − f (x1 − r)
0
+ f (x1 − r)(λ − 1)f (x2 − r)
= [f (x2 − r) − f (x1 − r)][f (x3 − r) − f (x2 − r)]
= [f (x2 − r) − f (x1 − r)][f (x3 − r) − f (x2 − r)](R23 − R12 ) + f (x1 − r)(λ − 1)f 0 (x2 − r)
f 0 (x −r)−f 0 (x −r)
where Rij = f (xii −r)−f (xjj−r) .
1. f (x) satisfies constant absolute risk aversion (CARA) in the loss domain:
Similar to Case 1, we can show R23 = R12
∗
< 0.
Since R23 − R12 = 0 and f (x1 − r)(λ − 1)f 0 (x2 − r) < 0, we have ∂p
∂q
Hence, p∗ decreases as q increases.
55
2. f (x) satisfies decreasing absolute risk aversion (DARA) in the loss domain:
Similar to Case 1, according to Lemma in Appendix A4, we have R23 > R12 .
Since f (x2 − r) − f (x1 − r) > 0 , f (x3 − r) − f (x2 − r) > 0, R23 − R12 > 0,
∗
< 0 is ambiguous.
and f (x1 − r)(λ − 1)f 0 (x2 − r) < 0, the sign of ∂p
∂q
3. f (x) satisfies increasing absolute risk aversion (IARA) in the loss domain:
Similar to Case 1, according to Lemma in Appendix A4, we have R23 < R12 .
Since f (x2 − r) − f (x1 − r) > 0 , f (x3 − r) − f (x2 − r) > 0, R23 − R12 < 0,
∗
and f (x1 − r)(λ − 1)f 0 (x2 − r) < 0, we have ∂p
< 0.
∂q
∗
Hence, p decreases as q increases.
A.3.2
KR model
In the KR model, reference point is full distribution of r1 and r2 with respect to q. The
utilities to choose option (1) and (2) are:
u1 = qλf (x2 − r2 ) + (1 − q)f (x2 − r1 ) = qλf (x2 − r2 )
u2 = q[(1 − p)λf (x1 − r2 ) + pλf (x3 − r2 )] + (1 − q)[(1 − p)λf (x1 − r1 ) + pf (x3 − r1 )]
Let u1 = u2 (p∗ ),
p∗ =
λqf (x2 − r2 ) − λ(1 − q)f (x1 − r1 )
(1 − q)f (x3 − r1 ) + λqf (x3 − r2 ) − λ(1 − q)f (x1 − r1 )
The derivative is the following:
λf (x2 − r2 )f (x3 − r1 ) + λ2 f (x1 − r1 )[f (x3 − r2 ) − f (x2 − r2 )]
∂p∗
=
∂q
[(1 − q)f (x3 − r1 ) + λqf (x3 − r2 ) − λ(1 − q)f (x1 − r1 )]2
Since f (x2 − r2 ) < 0, f (x3 − r1 ) > 0,f (x1 − r1 ) < 0, and f (x3 − r2 ) − f (x2 − r2 ) > 0,
∗
for all λ > 0, we have ∂p
<0
∂q
∗
Hence, p decreases as q increases.
Summary
—————————————————————————————————
In DA model:
−r1
−r1
Case 1: x2 ≤ r < x3 , i.e., xr22−r
≤ q < xr23−r
,
1
1
∗
for CARA case, p decreases as q increases.
for IARA case, p∗ decreases as q increases.
∗
for DARA case, the sign of ∂p
< 0 is ambiguous.
∂q
56
−r1
Case 2: r ≥ x3 , i.e., q ≥ xr23−r
, for CARA case, p∗ remains constant as q increases.
1
for IARA case, p∗ decreases as q increases.
for DARA case, p∗ increases as q increases.
In KR model:
For all cases, p∗ decreases as q increases.
—————————————————————————————————
A.4
Lemma
Suppose
then
Proof:
a2
a2n
a1
>
> ... >
, ai , bj > 0, i, j = 1, 2, ..., n,
b1
b2
b2n
a1 + a2 + ... + an
an+1 + an+2 + ... + a2n
>
b1 + b2 + ... + bn
bn+1 + bn+2 + ... + b2n
a1 + a2 + ... + an
an+1 + an+2 + ... + a2n
>
b1 + b2 + ... + bn
bn+1 + bn+2 + ... + b2n
⇐⇒
(a1 + a2 + ... + an )(bn+1 + bn+2 + ... + b2n ) > (b1 + b2 + ... + bn )(an+1 + an+2 + ... + a2n )
⇐⇒
a1 (b1 +b2 +...+bn )+...+an (bn+1 +bn+2 +...+b2n ) > b1 (an+1 +an+2 +...+a2n )+...+bn (an+1 +an+2 +...+a2n )
To prove the above inequality, we just need to prove
a1 ((bn+1 + bn+2 + ... + b2n ) > b1 (an+1 + ... + a2n )
···
an (bn+1 + bn+2 + ... + b2n ) > bn (an+1 + ... + a2n )
To prove these series of inequalities, we just need to prove
a1
an+1
a1
a2n
an
an+1
an
a2n
>
, ...,
>
, ...,
>
, ...,
>
.
b1
bn+1
b1
b2n
bn
bn+1
bn
b2n
Done.
57
B
Experiment Instructions
Before the experiment:
I randomly split the sample into a control group, "no-waiting" treatment group, and
"waiting" treatment group. Then I sent emails to thes groups 24 hours before the experiment.
Control group is randomly split into three subgroups: the $10 control group, the $15 control
group and the $20 control group. For the $10 control group, the email is below:
"Subject line: Xlab experiment XXXX (please read to participate)
Content:
Thank you for participating experiment XXXX.
During the experiment, you will finish a short survey. After the survey, you will receive
$10.
Please come to the lab at (time and location). We look forward to seeing you soon."
For the "no-waiting" treatment group, and "waiting" treatment group, the email is below:
"Subject line: Xlab experiment XXXX (please read to participate)
Content:
Thank you for participating experiment XXXX.
During the experiment, you will finish a short survey. After the survey, you have 1/3
chance to receive $10, 1/3 chance to receive $15 and 1/3 chance to receive $20.
Please come to the lab at (time and location). We look forward to seeing you soon."
In the lab:
1. For the $10 control group:
"Welcome and thank you for participating in this study. Please don’t talk with other
students during the experiment. If you have a question, please raise your hand. "
"The study will last around 30 minutes, during which we ask you to fill out a survey.
After the survey, you will receive $10. Please fill the survey in the next page."
The survey contains the basic social economic information, monthly spending budget, past
experiences about lab experiments, voluteer acticities, language speaking at home and on
campus, living enviroment on campus.
After the survey:
"Now you have finished the survey. Next, you will make choices between option 1 and 2
in the following questions.
58
One question will be randomly chosen after you make all your choices. Your payoff from
the experiment is determined by your choice in that question.
For example, question 19 is asking you to choose between (1)$10 and (2) $15. If your
choice is 2 and question 19 is picked after all your choice. Your payoff from the experiment
is $15.
Please think carefully and state your true preference one by one. It is important to keep
in mind that there is no right or wrong answer here. Which choice you make is a matter of
personal preference."
Then subjects would answer 60 risk-attitude questions to elicit their risk attitudes following
Holt and Laury (2002) procedure. The questions are in Table A1-A4. When subjects finished all the questions, they filled the short post-treatment survey including their expected
payoff when they come to the experiment, past experiment about lotteries and stocks, and
comments about the experiment. One out of 60 risk-attitude questions will be randomly
chosen by drawing one card out of three and rolling a 20-sided dice individually. If the
subject draw a number 2 and roll a dice for 17, the question for her payment is question 37.
2. For the no-waiting treatment group
"Welcome and thank you for participating in this study. Please don’t talk with other
students during the experiment. If you have a question, please raise your hand. "
"The study will last around 30 minutes, during which we ask you to fill out a survey.
After the survey, you have 1/3 chance to receive $10, 1/3 chance to receive $15 and 1/3
chance to receive $20. Please fill the survey in the next page."
The survey contains the same questions as in the control group
After the survey,
"Now you have finished the survey. Please wait and the computer will take a random
draw and tell you whether you get $10, $15 or $20.
The computer took a random draw and you receive X dollars.
Next, you will make choices between option 1 and 2 in the following questions.
One question will be randomly chosen after you make all your choices. Your payoff from
the experiment is determined by your choice in that question.
For example, question 19 is asking you to choose between (1)$10 and (2) $15. If your
choice is 2 and question 19 is picked after all your choice. Your payoff from the experiment
is $15.
59
Please think carefully and state your true preference one by one. It is important to keep
in mind that there is no right or wrong answer here. Which choice you make is a matter of
personal preference."
Then subjects would answer 60 risk-attitude questions to elicit their risk attitudes following
Holt and Laury (2002) procedure. The questions are in Table A1-A4. When subjects finished
all the questions, they filled the same post-treatment survey and randomly draw a question
for their payment as in the control group.
3. For the waiting treatment group
"Welcome and thank you for participating in this study. Please don’t talk with other
students during the experiment. If you have a question, please raise your hand. "
"The study will last around 30 minutes, during which we ask you to fill out a survey.
After the survey, you have 1/3 chance to receive $10, 1/3 chance to receive $15 and 1/3
chance to receive $20. Please wait and the computer will take a random draw and tell you
whether you get $10, $15 or $20.
The computer took a random draw and you receive X dollars.
Please fill the survey in the next page."
The survey contains the same questions as in the control group
After the survey,
"Next, you will make choices between option 1 and 2 in the following questions.
One question will be randomly chosen after you make all your choices. Your payoff from
the experiment is determined by your choice in that question.
For example, question 19 is asking you to choose between (1)$10 and (2) $15. If your
choice is 2 and question 19 is picked after all your choice. Your payoff from the experiment
is $15.
Please think carefully and state your true preference one by one. It is important to keep
in mind that there is no right or wrong answer here. Which choice you make is a matter of
personal preference."
Then subjects would answer 60 risk-attitude questions to elicit their risk attitudes following
Holt and Laury (2002) procedure. The questions are in Table A1-A4. When subjects finished
all the questions, they filled the same post-treatment survey and randomly draw a question
for their payment as in the control group.
60