Are Individuals Prudent?
An Experimental Approach Using Lottery Choices
Marcela Tarazona-Gomez
LERNA - EHESS
10 Rue des Amidonniers
31000 Toulouse France
E-Mail: [email protected]
Tel: +33.561.23.99.73
September 24, 2004
Abstract
An agent is prudent when facing a mean preserving increase in risk (MPIR) prefers
to see it attached to the best outcomes of a lottery rather than to the worst ones.
We run an experiment to analyze if agents are prudent and the relationship between
risk aversion and prudence. Players are asked to compare two lotteries (with MPIRs
attached respectively to their best or worst outcomes) to several certainty equivalents
in order to …nd out their preferences between lotteries and estimations of their degree of
prudence and risk aversion. We …nd no correlation between prudence and risk aversion.
In addition, between one half and two thirds of the players exhibit a prudent behavior
and a similar proportion is risk averse.
Key words: prudence estimation, risk aversion estimation, experiments, downside
risk.
JEL: C91, D81
1
There are two economic de…nitions for both risk aversion and prudence under the expected
utility framework. Risk aversion is de…ned either by the second derivative of the utility
function being negative or by the rejection of any mean preserving increase in risk. Similarly,
for prudence, we have the classical de…nition under which “an agent is prudent if adding an
insurable zero-mean risk to his future wealth raises his optimal saving”(Kimball, 1990), and
a second de…nition given by a statistical transformation of a density function: “a risk averter
who is forced to undergo a mean preserving increase in risk (which he dislikes) will prefer
to see it attached to the best outcomes of a lottery than to the worst ones” (Eeckhoudt et
al., 1995). Both de…nitions of prudence are equivalent to the third derivative of the utility
function being positive.
The importance of risk aversion when analyzing behavior under risk has been broadly
investigated in economics. In contrast, prudence has been less studied. This seems rather
surprising, giving that prudence is also a key concept when analyzing behavior under risk.
Prudence is a necessary condition for decreasing absolute risk aversion (DARA) which is a
widely accepted assumption.1 If we consider the expected-utility framework, risk aversion
explains the preferences of an individual, while prudence a¤ects directly his choices. A
prudent agent can be thought either as one who increases his savings when uncertainty
a¤ects his future income, or even simpler, as someone who prefers to face a risk attached
to a good state (the best outcome of a lottery), rather than to a bad one (to the worst
outcome).
Up to now there have been several attempts to study individual’s risk aversion behavior in
2
the experimental framework. Laury and Holt (2002) mention three types of experiments for
doing so: experiments using lottery data from …eld experiments, experiments inferring risk
aversion from bidding and pricing tasks, and experiments inferring it from elicited buying and
/ or selling prices from simple lotteries. But so far, no experiments have studied prudence.
Some of the possible reasons for this matter are mentioned in the following paragraph and
discussed in the …fth section of this article.
Eckel el al (2002) talk about three empirical tools to measure preferences and behaviors: outcome-based measures, attitudinal survey questions and experiments. Examples of
outcome-based measures (which use indirect information to infer preferences and behaviors)
for studying prudence are due to Carroll (1994), Merrigan and Normandin (1996) and Dynan
(1993). The …rst two studies conclude the existence of prudence, while Dynan …nds that the
precautionary savings motive (prudence) cannot explain a signi…cant fraction of savings in
face of uncertainty. These measures, although widely used in economic literature are noisy
and imprecise because of the use of data not necessarily produced to study behaviors or
preferences of individuals. There have also been attempts to study prudence with attitudinal survey questions. This is a more ‡exible method based on asking hypothetical questions
and self-reported measures. The problem with this measure is that truth may be misrepresented and answers may be biased given that subjects are faced to hypothetical situations.
Guiso et al (1992) test the presence of prudence using a self-reported measure of earnings
under uncertainty. Finally, the increase of experimental methods to study preferences and
behaviors is evident in economic literature. Experiments are known for their advantage of
3
allowing the experimenters control for situational variation when placing subjects in identical
settings and facing them with real decisions. Further, no misrepresentation problem arises
because decisions involve real payo¤s and therefore subjects are expected to reveal their real
preferences and behaviors.
This paper presents an experiment which design is based on the economic de…nitions of
risk aversion and prudence and studies individual’s behavior related to these two variables.
The experiment allows to use the same data for studying risk aversion, prudence and the
relationship between the two variables. We …nd that between one half and two thirds of
the players behave in a prudent way. Similarly, between one half and 70% are risk averse.
Of these risk averse players, more than 60% are also prudent. Nevertheless, we …nd no
correlation between risk aversion and prudence. We develop a methodology to estimate
coe¢ cients for measuring risk aversion and prudence. The estimated coe¢ cients for the
majority of the players are very close to risk and prudence neutrality. This means that even
if we …nd evidence of a majority of prudent and risk averse cases, these behaviors are subtle.
Agents appear to be prudent or (and) risk averse but in a delicate way.
In the second section of this article, we present the experimental design followed by the
description of the way in which data are analyzed. This section includes the analysis of
experimental results. Then, in the third section we introduce a theoretical framework used
to make some estimations for prudence and risk aversion coe¢ cients using our experimental
data. In the next section, we provide an alternative analysis of the data, under the assumption of individuals behaving according to a constant relative risk aversion utility function.
4
In the …fth section we include a synthesis of the discussion about prudence, the link between
both de…nitions for it and some empirical measures that have been done. Section seven
presents our conclusions.
1
Risk aversion and prudence
Our experimental design is mainly based on the second de…nition of prudence.2 Given that
this de…nition is equivalent to the notion of downside risk aversion, we start by introducing
this concept in the present section. In such a way we pretend to make clear the intuition of
prudence and of risk aversion, and the way in which the experiment approaches both of these
concepts. Then, we present the experiment, the way in which the results show if players are
risk averse and/or prudent, and the results.
1.1
Prudence and downside risk aversion
Downside risk is associated to the placement of risk in a distribution. One way to measure
it is with skewness, the third moment of a distribution around the mean: a distribution is
said to have more downside risk than another if it is more skewed to the left.3 Menezes et al
(1980) provide a characterization of increasing downside risk which involves the unambiguous
transfer of risk from the right to the left of a distribution. Such a transfer does not change
the overall “riskiness”of the distribution, but simply alters the placement of the risk.4 The
authors show that a distribution more skewed to the right is preferred to another more
skewed to the left by all downside risk averse individuals if and only if they are characterized
5
by a von Newmann-Morgenstern utility function with a positive third derivative. This is,
downside risk aversion and prudence are equivalent.
Eeckhoudt et al (1995) con…rm this equivalence as can be seen in the Appendix 1 of this
paper. As quoted in Corollary 1, “upwards shifts of any increase in risk are bene…cial to
expected utility if and only if the individual is prudent, i.e. u000 (z) is uniformly positive”.
This means that translating a mean-preserving increase in risk (MPIR) to the right a¤ects
the welfare of the agent, increasing his expected utility, if and only if he is prudent. Even
though, this translation does not change the expected value or the variance of the initial
density.
Eeckhoudt and Schlesinger (2003) give an intuition for the link between the concepts of
downside risk aversion and prudence. Following the authors, prudence (as well as downside
risk aversion) is a preference for the location of a risk: if an individual is told he must accept
a zero-mean random variable, the prudent individual will always prefer to attach the risk to
the better outcome of a lottery rather than to the worst.5 We de…ne two lotteries (~
y ; z~) that
have the same mean and the same variance, but z~ has a mean preserving increase in risk
attached to the high payo¤ while y~ has the (same) mean preserving increase in risk attached
to the low payo¤. Then, we say that an agent is prudent if he prefers z~ to y~, this is, if
the expected utility of the individual is higher for the lottery that has the mean preserving
increase in risk attached to the high payo¤ than for the other lottery.
6
1.2
The experiment
Following the design of experiments done by Laury and Holt (2002), we propose an experiment to study simultaneously individual’s behavior regarding risk aversion and prudence.
The purpose of our experiment is to …nd certainty equivalents for some lotteries that are
designed according to a theoretical framework. We make comparisons and some estimations
based on these certainty equivalents in order to study the behavior of the players. Specifically, we investigate whether answers imply risk aversion and prudence. Besides this, we
study the relationship between risk aversion and prudence.
We estimate the certainty equivalents by presenting to the players the choice between
playing a given a lottery and gaining a sure quantity. Each lottery is compared with a list
of possible sure quantities. This list presents values around the expected value of the lottery
(higher and lower) that are ordered starting with the lowest and ending with the highest.
The players are asked to compare the same lottery with each one of the certain values and
answer each time, if they prefer playing the lottery or gaining the sure quantity. The answer
that we are looking for is the “turning point” from choosing to gain the sure quantity to
choosing to play the lottery. This answer gives a range of the certainty equivalent that each
player has for each lottery and therefore, about his preferences towards and between them.6
Two pairs of lotteries are presented to the players. All lotteries have the same mean
but lotteries in the …rst pair have a lower variance than the lotteries in the second pair.
Comparison between results for the low and the high variance cases allows us to study the
robustness of our results.
7
We describe next the questionnaire that was given to the players. Each player received
two pages, one page for each pair of lotteries that are studied in the experiment. Each page
contains two tables (one for each lottery) with three columns. The …rst column of the table
has the lottery, the second column the list of possible certainty equivalents, and the third,
the space for the players to write down their answers. All tables are included in Annex 1.
Tables 1a, 1b, 2a and 2b report the list of decisions that the players were asked to take in
the game.
The …rst pair of lotteries that was presented to the players is given by y~ = (3750; 0:25; 8750; 0:75)
and z~ = (6250; 0:75; 11250; 0:25):7 These two lotteries have the same mean (7500) and the
same standard deviation (2165,06). Lotteries y~ and z~ are the result of a mean preserving increase in risk given by x~ = (2500; 0:50; 2500; 0:50) attached to an initial lottery
m
~ = (6250; 0:50; 8750; 0:50) either to the low payo¤ of a lottery (6250 for lottery y~) or to
the high payo¤ of the lottery (8750 for lottery z~). The chosen turning points of these two
lotteries give the certainty equivalent ranges for lottery y~ (ECye) and for lottery z~ (ECze).8
As in the prior case, both lotteries in the second pair have the same mean (7500) and
the same standard deviation (3061,86). They have the same mean as the …rst pair of
lotteries, but a di¤erent (higher) variance. Lotteries in this second pair result from the
same mean preserving increase in risk as for the two pairs of lotteries (~
x), this time attached to a lottery n = (5000; 0:50; 10000; 0:50). Resulting lotteries are given by y~0 =
(2500; 0:25; 7500; 0:25; 10000; 0:50) (when attaching the MPIR to the low payo¤) and z~0 =
(5000; 0:50; 7500; 0:25; 12500; 0:25) when attaching the MPIR to the high payo¤). The turn-
8
ing points for these lotteries give the range for the certainty equivalents of lottery y~0 (ECye0 )
and for lottery z~0 (ECze0 ).
The questionnaires are divided in four groups between the players. Each group has the
lotteries presented in a di¤erent order for studying order e¤ects later in the article. This is,
if we assume four lotteries y; z; y;0 z;0 the four groups are (i) y 0 z 0 yz, (ii) yzy 0 z 0 , (iii) z 0 y 0 zy,
and (iv) zyz 0 y 0 .
1.3
Analyzing the data: When is an individual risk averse and/or
prudent?
We introduce now the link between the certainty equivalent and risk aversion and prudence.
Four lotteries (that we call y~; z~; y~0 and z~0 ) are presented to the players in pairs (~
y ; z~ and y~0 z~0 ):
Following the theoretical framework, both lotteries in each pair have the same mean and the
same variance. But one of them has a mean preserving increase in risk attached to the high
payo¤ (lotteries z~ and z~0) while the other has the (same) mean preserving increase in risk
attached to the low payo¤ (lotteries y~ and y~0). We use certainty equivalents obtained with
the experiment to study risk and prudence behaviors of the players.
An individual is said to be risk averse if he prefers the expected value of a lottery to a
lottery itself. In our experiment, this means that he has a certainty equivalent to the lottery
which is lower than the mean of the lottery. Players having a certainty equivalent equal to
the mean of the lottery (this is, being indi¤erent between the expected value of the lottery
and the lottery) are called risk neutral. Players having a certainty equivalent higher than
9
the expected value of the lottery (preferring to play the lottery over gaining its expected
value) are said to be risk lovers.
Correspondingly, we have said that the notion of prudence is related to a location preference. The idea is that a prudent agent prefers to face a pure risk when it is linked to a high
payo¤ than to face the risk when it is linked to a low payo¤ and this is independent of the
particular attitude towards risks. For our experiment, this means that an agent is prudent if
he prefers the lottery z~ to the lottery y~, which is, if the expected utility of the individual is
higher for the lottery that has the mean preserving increase in risk attached to the high payo¤ (~
z ) than for the other lottery (~
y ). Comparisons between the certainty equivalents for the
two lotteries in each pair allow us to infer player’s preferences between the two lotteries, and
therefore, if they follow a prudent behavior. This is, if ECy~ < ECze (and if ECy~0 < ECze0 )
the agent is prudent. If a player presents the same certainty equivalent for both lotteries in
a pair, he will be said to be “neutral to prudence” And if he presents a behavior opposite
to prudence (ECy~ > ECze or ECy~0 > ECze0 ), he will be called ”imprudent”. As shown
latter, the di¤erence between the two certainty equivalents (ECze
proportional to the measure of prudence.9
1.4
ECy~ or ECze0
ECy~0 ) is
Results
We concentrate our analysis on one pair of lotteries, the ones characterized by having a lower
variance (~
y and z~) but it can be extended to lotteries y~0 and z~0 . This is due to the fact that
both results of the high and low variance cases lead to similar conclusions, as will be seen
10
next.
Experiment took place in …ve sessions that gathered a total of 96 students at the University of Los Andes in Bogota. Players received monetary incentives for their choices. They
were asked to make a total of 30 decisions knowing that one of these decisions would be
randomly selected and played for determining their payo¤s. After making their decisions,
players were asked to answer some demographic questions that are used for analyzing the
results.10
Individuals who switched back and forward from the option of playing a lottery to the
option of gaining the sure quantity are excluded from the sample. We do this because their
behavior appears to be irrational and impedes the estimation of the certainty equivalent
range. There were 9 students behaving in such a way. Answers of the remaining 87 students
are used for making estimations and for analyzing the results of the experiment. For the
analysis, we assemble the four types of questionnaires in one group.
We have said that an individual is strictly risk averse if he strictly prefers the expected
value of a random variable to the random variable itself. This means that an individual is
strictly risk averse if in the experiment, he chooses as a turning point a certainty equivalent
lower than the expected value of the lottery. All lotteries included in the experiment have an
expected value of “7500”. Therefore, students choosing values lower than “7500”as turning
points are described as strictly risk averse. Students choosing “7500 ”as the turning point
are called risk neutral and those choosing values higher than “7500 ” are risk lover (or
risk seeking). As we have already mentioned, agents are prudent if their answers imply
11
ECy~ < ECze (or ECy~0 < ECze0 ). We also look for agents being simultaneously risk averse and
prudent.
For analyzing the behavior of players in the experiment we study their answers for each
pair of lotteries. We say that a player is risk averse if his answers imply risk aversion for
both lotteries in the pair, or if they imply risk aversion for one lottery and risk neutrality for
the other. Players are risk neutral if their answers imply risk neutrality for both lotteries in
the pair. And they are considered as risk lovers when answers imply that they are risk lover
for both lotteries in the pair, or when they imply risk neutrality in one case and risk seeking
in the other.11 Given these criteria, we estimate the percentage of students exhibiting each
one of these three behaviors towards risk.
Statistics for lotteries y~ and z~ are included in Table 3a (and correspondingly on Table 3b
for lotteries y~0 and z~0 ). Of the 87 players, 59 are risk averse, 9 risk neutral and 11 risk seeking.
We can see that slightly more than two thirds of the players present a risk averse behavior
(67,82%). Regarding prudence, 55 players are prudent (63,22%), 24 neutral to prudence and
8 “imprudent” (this is, presenting a behavior contrary to prudence). Once again, slightly
more than two thirds of the players are prudent. But not all of the prudent agents are risk
averse or vice versa: there are 37 individuals being risk averse and prudent (42,53% of the
total sample and 62,71% of the risk averse cases).
>From the results for lotteries y~0 and z~0 , we can say that risk aversion and prudence
are characteristics of almost 55% of the students. Besides this, more than 60% of these risk
averse players exhibit a prudent behavior. Even if the percentages are lower than for lotteries
12
y~ and z~, we still have evidence of a majority of players being prudent and risk averse. As we
have mentioned, for both prudence and risk averseness, between one half and two thirds of
the individuals present these behaviors. This is a common result for both pairs of lotteries.
Figures 1a, 1b, 2a and 2b show these results by presenting the distribution of risk aversion
and prudence cases.
For studying the relationship between prudence and risk aversion we develop some correlation tests. We …nd that the correlation between being risk averse and prudent is very
low for either of the two pairs of lotteries. The Spearman Correlation Index is -0.11 (for y~0
and z~0) and 0.05 (for y~ and z~).12 This result suggests that even if the proportion of players
being risk averse and prudent is very similar, prudence and risk aversion are independent
behaviors.
The four types of questionnaires (di¤ering just by the order in which the lotteries were
presented to players) were used for checking the presence of order e¤ects. No evidence was
found according to the tests performed. This means that in general, answers didn’t change
signi…cantly due to the ordering in which tables were presented.13
The fact that answers to all type of lotteries and questionnaires follow a similar pattern
gives robustness to our results.
2
Empirical speci…cation and parameter estimation
Estimations for risk aversion coe¢ cients have been made multiple times. On the other hand,
estimations for prudence coe¢ cients are almost inexistent in economic literature, at least
13
from experimental data. Given the importance of these coe¢ cients for simulation purposes
among others, we consider an important issue to start exploring this matter.
We made some estimations as prior steps for the design of the experiment, the main
results are presented in this section. We already mentioned that experimental data give the
ranges for the certainty equivalents of the lotteries. We start by introducing the theoretical
de…nition of the certainty equivalent and of the risk premium and their relationship with
risk aversion and prudence. Following Gollier (2001) we study the characteristics of the
risk premium for small risks and …nd an approximation of the risk premium that is used to
compute a risk aversion coe¢ cient and a prudence coe¢ cient expressed in terms of the data
that we obtain in the experiment.
For the description of the empirical speci…cation of the certainty equivalent, we talk
about a lottery y~, but a similar analysis can be applied to all lotteries that are part of the
experiment. We start by recalling some classical de…nitions and relationships. Call w0 the
individual’s initial wealth,
the mean of lottery ye and x
e a pure risk such that ye =
+x
e
and Ee
x = 0. Then, following Gollier (2001), we de…ne the certainty equivalent of lottery y~
as ECy~ (w0 ; u; + x
e) such that
Eu (w0 + ye) = u (w0 + ECye (w0 ; u; + x
e)) ;
and the risk premium associated to the pure risk x~ as
Eu (w + x
e) = u (w
14
(1)
(w; u; x
e) such that
(w; u; x
e)) :
(2)
Therefore, comparing the two conditions we have that the left hand sides are equal if
w = w0 + , which gives
u (w0 + ECye (w0 ; u; + x
e)) = u (w0 +
(w0 + ; u; x
e)) ;
or
ECye (w0 ; u; + x
e) =
(w0 + ; u; x
e) :
(3)
So the risk premium measures the di¤erence between the expected value of the lottery
and the certainty equivalent.
Based on these equations we obtain from experimental data a value for the risk premium
(w0 + ; u; x
e) of each lottery.
2.1
The risk premium in terms of risk aversion and of prudence
We examine next the characteristics of the risk premium for small risks. The purpose of this
analysis is to …nd measures of risk aversion and prudence for the risk premia. We consider a
pure risk k~
x and denote as g(k) the associated risk premium (w; u; k~
x) which is, according
to (2)
Eu(w + k~
x) = u(w
g(k));
(4)
where w represents total wealth (excepting the pure risk). We make a Taylor approximation of third degree of g around k = 0, since we are interested in the third derivative of
the utility function and its relation with the risk premium. Complete calculations, provided
15
in the Appendix 2, imply that the risk premium is approximately given by
g(k)=
~
1 2 u00 (w) 2
k
E x~
2 u0 (w)
1 3 u000 (w) 3
k
E x~ =
~ (w; u; k~
x):
6 u0 (w)
Our experimental data give the value of
(w; u; ke
x) for w = w0 +
=
(making w0 = 0)
and k = 1. These are, therefore, the speci…cations we use to perform the estimations. The
risk premium for lottery y~ is given by
e) =
y ( ; u; x
1 u00 ( ) 2
E x~y
2 u0 ( )
1 u000 ( ) 3
E x~y :
6 u0 ( )
(5)
For rede…ning this expressions, we recall the Arrow Pratt coe¢ cient which is given by
A( ) =
u00 ( )
:
u0 ( )
And prudence coe¢ cient, which is de…ned by Kimball (1990) and Gollier (2001) as
P( )=
u000 ( )
:
u00 ( )
Using these de…nitions, we can write that14
u000 ( )
=
u0 ( )
u000 ( )
u00 ( )
u00 ( )
u0 ( )
16
= P ( )A( ):
(6)
Then equation (5) can be rewritten as
y(
1
; u; x
e) = A( )E x~2y
2
1
P ( )A( )E x~3y :
6
(7)
We use these expressions in the next section for presenting estimations of a risk aversion
and a prudence coe¢ cient in terms of experimental data.
2.2
Estimations of risk aversion and prudence coe¢ cients
The experiment gives the certainty equivalents for lotteries lotteries y~ and z~. With these
answers, we estimate the corresponding risk premiums according to (3). Taking the de…nition
for the risk premium for y~, given by (7) and de…ning a similar coe¢ cient for lottery z~ we
obtain
1
A( )E x~2y
2
1
e) =
A( )E x~2z
z ( ; u; x
2
y(
; u; x
e) =
1
P ( )A( )E x~3y
6
1
P ( )A( )E x~3z :
6
For the lotteries used in the experiment, we know that E x~2y = E x~2z and that
Then, the di¤erence between the two risks premiums is given by
y
z
=
1
P ( )A( )E x~3y ;
3
17
(8)
E x~3y = E x~3z .
which implies that
P ( )A( ) =
3
(
z)
y
E x~3y
:
Replacing this result in (8) we obtain
A( )=
~
(
+ z)
;
E x~2y
y
(9)
which means that
P( ) =
u000 ( )
=
~
u00 ( )
3
(
(
y
y
+
~2y
z) Ex
:
~3y
z) Ex
(10)
Equations (9) and (10) are expressions for risk aversion and prudence coe¢ cients which
can be estimated using experimental data.
We have estimated these coe¢ cients for the possible certainty equivalents that are given
to the players in the experiment. Results are presented in Tables 4a and 4b. First two
columns in both tables include the certainty equivalents for lotteries z~ and y~ (or z~0 and y~0 ),15
while the corresponding risk premiums are given in the third and fourth columns. The …fth
and sixth columns contain the risk aversion and prudence coe¢ cients for players choosing the
preceding certainty equivalents. Possible answers imply an absolute risk aversion coe¢ cient
between (-0,0064 and 0,064) for lotteries y~ and z~, and between (-0,00010667 and 0,0002133)
for lotteries y~0 and z~0.
Seventh (and eighth) column includes the total (and the percentage) cases for each possible answer. For lotteries y~ and z~;the behavior of most of the players implies the following
coe¢ cients: for risk averse players, 25 have an Arrow-Pratt coe¢ cient A = 0; 00001067 and
18
23 of them, A = 0; 00002133. This represents correspondingly 28% and 26% of the answers,
for a total of 54% of the total sample. This means, that 81,35% of the risk averse players (as
we have said, 59 for lotteries y~ and z~) are included in these cases. This leads us to conclude
that a big majority of risk averse players ’behavior implies these two absolute risk aversion
coe¢ cients. For risk seeking players, most of the cases are concentrated on A =
0; 0001067
(11% of the players). According to results presented in Section 2.2, 10% of the players are
risk neutral, and therefore A = 0.
For lotteries y~0 and z~0 most of the risk averse cases are concentrated on three ArrowPratt coe¢ cients: for 19 players (22%) A = 0; 00002133, for 15 (17%) A = 0; 00004267,
and for 10 (11%), A = 0; 000064. These 44 individuals represent more than 90% of risk
averse players. Most of the risk seeking players (9 of them) have a risk aversion coe¢ cient
of A =
0; 00002133, this is, 60% of them. For 13 players A = 0.
According to results for lotteries y~ and z~, for most of the prudent agents P = 0; 0012 (34
players, 61,82%) and for 8 (14,54%) of them P = 0; 0006. This is, more than 75% of the
prudent players are included in these cases. 24 agents are neutral to prudence, and therefore
P = 0. For 5 of the 8 imprudent players, P =
0; 0012. Answers for lotteries y~0 and z~0 imply
that for 24 risk averse players P = 0; 0012 and for 8 P = 0; 0004. 25 players are neutral to
prudence (P = 0) and of the 13 imprudent, 7 have a P =
0; 0012. We can see a common
pattern for both pairs of lotteries: answers imply for most of the cases that P = 0; 0012 if
agents are prudent, and P =
0; 0012 if players are imprudent.
We see that even if most of the players appear to be risk averse and prudent, coe¢ cients
19
implied but their answers are small. This evidence suggests that although risk aversion
and prudence appear to be common behaviors, they appear to be subtle, at least for the
lotteries played. General characterization of the distributions for prudence and risk aversion
coe¢ cients for both pairs of lotteries are included in Table 5 and Figures 3a, 3b, 4a and 4b.
Some tests similar to the ones presented on Section 2.4 were made in order to study
correlation between prudence and risk aversion coe¢ cients. The simple correlation coe¢ cient
between the coe¢ cient of risk aversion and the coe¢ cient of prudence is very low; they are
-0.16 and 0.01 for the respective lotteries.
For studying the relationship between prudence, risk aversion and some demographic
variables, linear regressions are run. Table 6 presents four regressions run with the prudence
coe¢ cient depending on the risk aversion coe¢ cient and some demographic variables. The
risk aversion coe¢ cient is not signi…cant for either case (either pair of lotteries), whether we
include demographic variables or not. This result is con…rming the independence between
risk aversion and prudence. As can be seen on the table, no demographic variable is signi…cant. Table 7 includes results of the regressions run for the risk aversion coe¢ cient on
demographic variables and for the prudence coe¢ cient on demographic variables. Again, we
…nd no signi…cant variable. Similar results are found for an ordered probit model included
in Table 8. This result may be explained by the fact that the respondents were quite homogeneous. Therefore, the question of the impact of age, personal income and civil status
(among others) remains open for future studies.
20
3
Alternative analysis of the data: the CRRA case
There are some common hypotheses made when studying the economics of risk and uncertainty. In general, it is assumed that individuals are risk averse (increasing and concave
utility functions), that absolute risk aversion is decreasing in wealth (DARA), and that
an increase in wealth does not decrease relative risk aversion (this is, relative risk aversion remains constant or increases). Utility functions characterized by constant relative risk
aversion (CRRA) are used frequently because they ful…l these requirements. CRRA utility
functions correspond to the case where relative risk aversion is not dependent of wealth (and
therefore, “constant”). Rephrasing Gollier (2001), this implies that given a multiplicative
risk, the risk premium that one is ready to pay is independent of wealth. Besides this, CRRA
functions are DARA. The principal advantage of this type of utility functions is that they are
easy to manipulate and therefore, analytical results can be derived.16 For this reason, CRRA
utility functions are strongly used in theoretical and empirical economic studies. We make
use of this type of utility function for making an alternative and complementary analysis of
our data set. We present some estimations for relative risk aversion and relative prudence
coe¢ cients for agents behaving according to a CRRA utility function.
There are some possible answers in the experiment that may imply that individuals
behavior is compatible with a CRRA utility function. For studying this case, we start by
recalling the de…nition for the certainty equivalent in equation (1). Applying this de…nition
21
to lottery y~ we have17
u (w0 + ECy~) = [0:5u (w0 + 10000) + 0:25u (w0 + 7500) + 0:25u (w0 + 2500)] ;
and for lottery z~
u (w0 + ECz~) = [0:5u (w0 + 5000) + 0:25u (w0 + 7500) + 0:25u (w0 + 12500)] :
We assume a CRRA function given by
u (W ) =
where W is the total income and
u
1
W1
1
;
is the relative risk aversion coe¢ cient. Therefore
(W ) = (1
1
)W 1 :
Replacing this function in the de…nition for the certainty equivalent and simplifying for
lottery y~ we obtain
w0 + ECy~ = 0:5 (w0 + 10000)1
+ 0:25 (w0 + 7500)1
22
+ 0:25 (w0 + 2500)1
1
1
;
and similarly for z~
w0 + EC z~ = 0:5 (w0 + 5000)1
+ 0:25 (w0 + 7500)1
+ 0:25 (w0 + 12500)1
1
1
:
Estimation for these equations gives the ranges for the relative risk aversion coe¢ cients
corresponding to each possible certainty equivalent found in the tables presented to the
players. For the analysis of CRRA cases we concentrate on agents being risk averse and
also prudent. In Tables 9 and 9b, we present the implied ranges of each possible certainty
equivalent, and their implied relative risk aversion coe¢ cient ranges for individuals behaving
according to a CRRA utility function under w0 = 0.18 This is, if an individual chooses as a
turning point from playing lottery y~ to gaining the sure quantity the certain value “7400”,
the implied certainty equivalent that he has for the lottery is higher than 7300 (the precedent
option) but lower than or equal to 7400. This ranges are presented in the third column (for
lotteries y~ and y~0 ) and seventh column (for lottery z~ and z~0 ) of the tables.
The relative risk aversion coe¢ cients implied by the certainty equivalents that are presented as choices to the players include numbers higher than 0 and lower than 1 (see second
and sixth columns of the tables). Certain values that are presented as possible choices to
the players in the experiment are chosen for the case of a CRRA agent having a relative risk
aversion coe¢ cient belonging to this interval. This is consistent with estimations found in
other experimental studies.19
As we have mentioned, some answers may imply that individuals’behavior is consistent
with a CRRA utility function. For determining the individuals who follow this behavior,
23
we take into account both choices made regarding lotteries y~ and z~ (or y~0 and z~0 ). Let us
study the case of lotteries y~ and z~, but a similar analysis can be done for lotteries y~0 and
z~0 . If a player chooses the certainty equivalent “7300”as the turning point for lottery y~ and
“7400” for lottery z~, we can see that there is a common range for the relative risk aversion
coe¢ cients of these two choices given by [0:51; 0:74]. Similarly, if the player chooses “7300”
for lottery y~0 and “7100” for lottery z~0 , the implied common range is [0:48; 0:72].20 It is
important to remark that we are not claiming that all players making these choices actually
behave according to a CRRA utility function. But they do not contradict this hypothesis
either. Implied common ranges for the relative risk aversion coe¢ cient (third column) as well
as common ranges for the implied relative prudence coe¢ cient (fourth column) are presented
in Tables 10a and 10b for the CRRA case. The …rst two columns of these tables include the
certainty equivalents for lotteries y~ and z~ (or y~0 and z~0 ) that may imply a CRRA behavior.
Results for agents behaving according to a CRRA utility function are included in the
…fth and sixth columns of the tables. For lotteries y~ and z~ we …nd that 23 students belong to
this group (one fourth of the players, and almost 40% of the risk averse players). Of them,
17 (73,91%) have a relative risk aversion
2 [0; 26; 0; 36] and a relative prudence coe¢ cient
RP 2 [1; 26; 1; 36].21 For lotteries y~0 and z~0 we found 22 players behaving according to a
CRRA function. Percentages are similar than for the y~ and z~ case: one fourth of the total
amount of players and more that 40% of the risk averse players behave according to CRRA.
50% of the CRRA players have a
2 [0; 25; 0; 35] and a RP 2 [1; 25; 1; 35]. It is clear that
result for the two pairs of lotteries are very similar, giving robustness to our results.
24
4
Prudence: downside risk aversion and precautionary
savings motive
We have said that there are two equivalent de…nitions for prudence: one linked to the concept of downside risk, which has been clearly developed through the article, and another
linked to the concept of precautionary savings. In fact, the question of extra savings generated by uncertainty on future income has been discussed for a long time. Leland (1968),
Sandmo (1970) and Dreze and Modigliani (1972) develop the theoretical conditions about
preferences under which an increase in an uninsurable risk leads to higher savings. They
show that precautionary saving in response to risk is associated with convexity of the marginal utility function, or a positive third derivative of a von Neumann-Morgenstern utility
function. However, Kimball (1990) is the …rst who introduces the concept of prudence. Prudence is “the sensitivity of the optimal choice of a decision variable to risk”. This term is
“meant to suggest the propensity to prepare and forearm oneself in the face of uncertainty,
in contrast to “risk aversion”, which is how much one dislikes uncertainty and would turn
away from uncertainty if possible”.
The intuition for the equivalence of these two de…nitions is explained next. Under the
downside risk de…nition, the idea for prudence is that you prefer to face a pure risk when it
is linked to a high payo¤ (when you are richer) than to face the risk when it is linked to a low
payo¤ (when you are poorer). Why is this similar to “the classical”de…nition of prudence?
Well, under the savings de…nition, you increase your savings in the present in order to be
25
richer in the moment when you will face your risk (in the future). This means, that you
prefer to be richer in the moment when you will face the risk. In Eeckhoudt et al (2003)
words, “we are more willing to accept an extra risk when wealth is higher, rather than when
wealth is lower. Indeed, this logic helps to explain why someone opts for a higher savings
when second period income is risky in a two-period model. The resulting higher wealth in
the second period helps one to cope with the additional risk, exactly as in Kimball (1990),
who uses prudence as equivalent to a precautionary demand for savings.”
Up to now, some estimations of prudence have been made using the …rst two tools. Some
of these studies conclude that the precautionary saving motive is strong, while some others
lead to the opposite …ndings. Examples of studies using outcome-based measures are the
following: Carroll (1994) tests if current consumption depends on expected lifetime income
by constructing estimates of future income for a sample of households whose consumption
is directly observed and …nds a positive coe¢ cient for prudence. Merrigan and Normandin
(1996) examine the strength of precautionary savings motives by estimating the coe¢ cient for
prudence from the UK Family Expenditure Survey data set. Instrumental variable estimates
of the prudence coe¢ cient reveal that greater uncertainty systematically leads to larger
current saving.
On the other hand, Guiso et al. (1992) show that uncertainty has unimportant e¤ects on
the consumer’s behavior. The authors test the presence of precautionary saving using a selfreported measure of earnings uncertainty drawn from the 1989 Italian Survey of Household
Income and Wealth . The e¤ect of uncertainty on wealth accumulation is consistent with
26
the theory of precautionary saving, but explains only a small fraction of saving. Besides
earnings uncertainty, other risks, such as health and mortality, may be important determinants of wealth accumulation. Dynan (1993) …nds that precautionary motives cannot
explain a signi…cant fraction of saving when uncertainty is measured by the variance of the
consumption growth.
These articles evidence the fact that prudence and precautionary savings have been empirically studied in a parallel way. And this can be the reason why no experimental studies
have been made up to now for examining prudence. Estimation of savings through experiments is commonly restricted to the use of hypothetical questions about decisions that would
be taken in the future. Answers to this type of questions may be biased and may not re‡ect
real behaviors but the way in which individuals think today about possible future scenarios.
Besides this, creating incentives for individuals to give real answers for long-term decisions is
not straightforward. Another matter is that for studying savings we make use of the wealth
level of the individuals. The issue of which wealth level to use and which variables to include
when estimating this variable has been strongly debated. These problems may reduce the
robustness of experimental results. We take advantage of the fact that the downside risk
aversion de…nition for prudence is not attached to the concept of savings and of future decision making and therefore, studying prudence (and at the same time risk aversion) with an
experiment turns out to be less problematic.
Anyhow, it is important to mention that some recent …eld experiments study long-term
decision making. Such is the case of Eckel et al (2002) who study saving decisions in Canada
27
and the case of Harrison et al (2002) who make estimations for discount rates in Denmark.
These experiments give light to a possible future development of …eld experiments for precautionary savings and prudence.
5
Conclusions
The importance of risk aversion and prudence for analyzing individual behavior under risk
has been clearly stressed through the article. Under the expected utility framework, the
marginal expected utility a¤ects the choices of an agent (through prudence), while the total expected utility explains his behaviors (through risk aversion). This makes clear why
studying both variables is crucial for theoretical and applied economics.
We have also pointed out the existence of two equivalent de…nitions for prudence and for
risk aversion. Empirical literature makes evident the di¢ culties for evaluating the precautionary savings motive, and therefore, prudence under the savings de…nition. No experiment
has been done up to now for studying individuals behavior regarding this variable.
We take advantage of the downside risk aversion de…nition (based on the transformation
of a density function) that avoids some problematic for building up an experiment and design
an experiment for studying behavior of agents in face of risky choices. Individuals are asked
to take decisions about some lotteries and their answers have implications for behaviors
related to risk aversion and prudence. We de…ne a risk aversion and a prudence coe¢ cient in
terms of our experimental data. We also study relative risk aversion and relative prudence
coe¢ cients under the CRRA assumption.
28
Our experiment allows for studying risk aversion and prudence at the same time. Results
are interesting: between one half and two thirds of the players are prudent and also between
one half and slightly more than two thirds are risk averse. Of the risk averse players, more
than 60% are also prudent. Even though, we …nd that there is no correlation between the
two variables. This result is robust across lotteries. Our results highlight the existence of
a prudent behavior (as well as a risk averse behavior) among the majority of players. This
implies that individuals prefer to face risk when they can link it to a higher payo¤. Given the
equivalence between the two de…nitions for prudence, our results may imply that an agent
will prefer to be “prepared” for facing a risk, this is, he will prefer to be in a more wealthy
state. This result is consistent with the common belief that uncertainty a¤ecting (future)
incomes will increase savings.
Our study is a …rst step for analyzing prudence under the experimental framework, but
several questions remain opened. We stress the importance of studying which variables might
incentive prudent and risk averse behaviors and how.
29
Appendix 1
Following Eeckhoudt et al (1995), we de…ne z~ as a continuous random variable denoting
wealth. A Rothschild-Stiglitz increase in risk will be described by the change of the distribution of z~ from cumulative distribution function (CDF) F to CDF G0 . We denote a
mean-preserving spread (MPS) with the function H = G0
F which is assumed to satisfy
the following conditions:
T (x) =
Z
x
H (z) dz
1
0; 8x 2 R
(11)
T (+1) = 0
H ( 1) = H (+1) = 0:
Equation (11) means that the change in risk does not a¤ect the expected value of z~.
From Rothschild and Stiglitz (1970) we know that an increase in risk can be obtained by a
sequence of mean preserving spreads (MPS). A shift of an increase in risk (SIR) is obtained
by translating this sequence of MPSs. Gt , the upwards SIR is obtained from F by adding
the same sequence of MPSs characterized by H, but applying a translation of MPSs by a
distance t to the right. Therefore,
Gt (z) = F (z) + H (z
t) :
(12)
In simpler words, we can say that an upwards SIR “induces more risk in wealthier states
30
and less risk elsewhere”. For analyzing the e¤ects of a positive SIR, we assume a utility
function u with u0
0. The expected utility of …nal wealth z~ (distributed following the
cumulative distribution function Gt ) will be given by
U (t; H) =
Z
u (z) dGt (z) =
Z
u (z) dF (z) +
Z
u (z) dH (z
t) :
(13)
We see that the expected utility depends upon the initial distribution F only through
the constant term
R
u (z) dF (z). In other words, the change in the expected utility due to a
SIR is independent of F . Integrating (13) by parts twice and using (11) and (12) we obtain
U (t; H) =
Z
u (z) dF (z) +
Z
u00 (z + t) T (z) dz:
Let U (n) (t; H) and u(n) (z) denote the nth derivative with respect to t of these functions.
Then, from (??) we know that for n = 1; 2; :::
U
(n)
(t; H) =
Z
u(n+2) (z + t) T (z) dz:
Proposition 1 (Eeckhoudt et al, 1995) Let n = 1; 2; :::; U (n) (t; H)
H satisfying (11) if and only if u(n+2) (z)
0 for any t 2 R and
0 for all z. Similarly, U (n) (t; H)
t 2 R and H satisfying (11) if and only if u(n+2) (z)
0 for any
0 for all z.
Corollary 1 (Eeckhoudt et al, 1995) Upwards shifts of any increase in risk are bene…cial to
expected utility if and only if the individual is prudent, i.e. u000 (z) is uniformly positive”.
31
Appendix 2
First and second derivatives of (4) are presented next:
E x~u0 (w + k~
x) =
g 0 (k)u0 (w
g(k))
E x~2 u00 (w + k~
x) =
g 00 (k)u0 (w
g(k)) + [g 0 (k)]2 u00 (w
(14)
g(k)):
We know that g(0) = 0 and (14) implies that g 0 (0) = 0, since E x~ = 0. Then, we can
simplify the last expression for k = 0, which gives
g 00 (0) =
E x~2
u00 (w)
:
u0 (w)
The third derivative is given by:
E x~3 u000 (w + k~
x) =
g 000 (k)u0 (w
+2g 0 (k)u00 (w
g(k)) + g 0 (k)g 00 (k)u00 (w
g(k))
[g 0 (k)]3 u000 (w
g(k))
g(k)):
For k = 0 we have that
E x~3 u000 (w) =
g 000 (0)u0 (w):
And therefore, the Taylor approximation of third degree around k = 0 gives the following
result:
g 000 (0) =
E x~3
32
u000 (w)
:
u0 (w)
This means that the risk premium is approximately given by
1
1
g(k)=g(0)
~
+ kg 0 (0) + k 2 g 00 (0) + k 3 g 000 (0)
2
6
g(k)=
~
1 2 u00 (w) 2
k
E x~
2 u0 (w)
1 3 u000 (w) 3
k
E x~ =
~ (w; u; k~
x):
6 u0 (w)
33
Appendix 3
Complete questionnaire - Instructions
You have just received from the instructor a “decisions page” including some tables and questions. You will make 30 decisions, but just one of them will be used to determine the payment that
you will receive at the end of the game. We will explain you next the way in which you should
answer and the way in which your payment will be determined. The questionnaire is divided in
two parts, in the …rst one you will be asked to take some decisions presented in tables and in the
second you will be asked to answer some questions:
Tables: In the …rst column of each table you will …nd a lottery (Option A), in the second a
value (Option B) and in the third a white space. Compare individually each value of the second
column (Option B) with the lottery in the column Option A. Select which one of these two options
is more interesting for you in each case and write down your answer in the third column. In each
table you will compare the same lottery with di¤erent sure values. On the contrary, the lotteries
presented in the diverse tables are di¤erent.
Final questions: At the end of the questionnaire you will …nd some questions that will be used
for analyzing your answers. Please answer them.
When all players …nish answering the questionnaire the instructor will collect the answers.
Then, one of the decisions taken will be randomly chosen and all players will be paid according to
the answer that they took for that decision.
Selection of the decision that will be paid: The instructor will choose randomly a player that will
throw a ten face die twice. With the resulting numbers a number from 1 to 30 will be determined.
34
This number will give the decision that will be paid.
Payo¤s: If the player chose to gain the sure quantity in the decision that was chosen, he will
receive this quantity. If he chose to play the lottery he will receive his payo¤ according to the
procedure that will be explained next. A number from 0 to 99 will be determined by throwing
twice a die of ten faces. This number will determine the percentage that will determine the …nal
payo¤s.
If you have any questions, raise your hand. Now you can start to make your decisions. Please,
do not talk with no one while you are doing so
(Here we include the tables presented in the Annex 1 ).
Please answer the following questions:
1. Gender:
Male_____
2. What is your year of birth?
Female_____
_____
3. Who in your household is principally charged of the expenses and budget decisions?
a. You,
b. Husband or wife,
c. Parent (s),
d. Other (specify),
e. Do not
know
4. Which is your civil status?
a. Married,
b. Single,
c. Divorced,
d. Widowed,
e. Other
5. Which is your current employment situation?
a. Part-time employed out of the university,
university as research assistant,
b. Student only,
c. Work at the
d. Other (specify)
6. Please select the best range for your household income in 2002 (after taxes).
35
a. $10.000.000 or less,
$30.000.001 and $50.000.000,
$70.000.001 and $90.000.000,
b. Between $10.000.001 and $30.000.000,
d.
c. Between
Between $50.000.001 and $70.000.000,
e.
Between
f. Higher than $90.000.001.
7. How many people conform your household?
8. Please select the best range for your personal income in 2002 (after taxes).
a. $10.000.000 or less,
$30.000.001 and $50.000.000,
b. Between $10.000.001 and $30.000.000,
d.
$70.000.001 and $90.000.000,
c. Between
Between $50.000.001 and $70.000.000,
e.
Between
f. Higher than $90.000.001
9. How do you receive your income?
a. Fixed salary,
e. Parents,
b. Hourly rate,
c. Hourly rate plus tips,
d. Scholarships,
f. Other
10. Which is your student status?
a. Full time student,
b. Part time student,
c. No student
11. In which year of your (graduate) studies are you?
a. Undergraduate …rst year,
third year,
g. No student,
b. Undergraduate second year,
d. Undergraduate fourth year,
c. Undergraduate
e. Undergraduate …fth year,
f. Master,
h. Specialization
12. Who is the primarily responsible for your tuition and living expenses while you are studying
at the University of Los Andes?
a. Self,
b. Parent(s),
c. Parent(s) and yourself,
f. Combination / Others (specify)
36
d. Scholarship,
e. Loans,
13. In which country have you lived mostly in your life?
37
Notes
1
Expected utility remains the most dominant theory in economics of decision making under uncertainty,
in spite of the empirical evidence of its failure. An extensive example of empirical and theoretical articles
published every year proofs it.
2
Through the paper, we call “second de…nitions” of risk aversion and prudence the de…nitions based on
statistical transformations, given that they appeared latter in the economic literature.
3
See Menezes et al (1980) for a complete discussion on measuring downside risk.
4
Two examples of this type of distributions are presented in the following section when we introduce the
two pairs of lotteries used in our experiment.
5
See Section 5 for a more detailed discussion on the intuition for explaining the equivalence between the
two de…nitions.
6
Answers won’t give a single value for the certainty equivalent of each lottery but a range. This means,
that the certainy equivalent of the lottery is a value lower than or equal to the sure quantity of the turning
point but higher than the preceding (lower) sure quantity of the list.
7
Quantities are given in Colombian pesos (national currency) and US$1 = 2600 Colombian pesos.
8
Latter in the article (see Section 4 ) we show the ranges implied by each choice and also the criteria
followed to choose the certainty equivalents that are used as options.
9
This relation will be used in Section 3 for making the estimation of the prudence coe¢ tient.
10
The complete questionnaire is included in Appendix 3.
11
When answers imply risk neutrality in one case and risk aversion (risk seeking) in the other, the player is
38
considered risk averse (risk seeking). This is because if he was “really”risk neutral, he would have chosen the
expected value in both cases. Given that possible answers don’t include all potential certainty equivalents
for the lottery, we assume that the player was “forced” to choose the expected value as a turning point
because his “real” certainty equivalent was not present as an option in the experiment, but was a quantity
between the expected value of the lottery and the next lower (higher) option. In the case of a player being
risk averse in one lottery and risk seeking in the other, the observarion is not included in the analysis.
12
We have included four categories for risk aversion (risk averse, risk neutral, risk lover, ambiguous) and
three for prudence (prudent, neutral to prudence and imprudent). A Kendall rank coe¢ cient gives the same
results.
13
The mean answer for each one of the four groups was compared with the mean answer of the other three
groups using a t-test. In all cases and for both prudence and risk aversion, results implied no presence of
order e¤ects.
14
The Arrow Pratt coe¢ cient is positive for risk aversion and negative for risk seeking. A risk averse agent
being prudent (imprudent) will have a positive (negative) prudence coe¢ cient. The sign of eq (6) depends
on both risk aversion and prudence.
15
The range for the optional certainty equivalents may appear small, but prior pilot experiments showed
that most players chose as a turning point values that are very close to the expected value of the lotteries.
In order to have a more precise estimate of the real certainty equivalent of the lottery, the …nal version of
the experiment presents certainty equivalents with several options that are close to their expected value.
16
This is a common advantage of harmonic absolute risk aversion utility functions (HARA), characterized
by having the inverse of the absolute risk aversion (called absolute tolerance) linear in wealth. CRRA utility
functions belong to this class.
17
Similar analysis was done for lotteries y~0 and z~0 .
39
18
For the CRRA case we assume that the reference level W
W = 0, 0 <
0 and
> 0. Moreover, we know that if
< 1: The assumption of W = 0 is very common in experimental literature.
19
For a complete revision of this litterature, see Holt and Laury (2002).
20
An individual choosing 7200 for y~ and 7400 for z~ is not consistent with CRRA because there is no
common range for her implied relative risk aversion coe¢ cient.
21
It is easy to show that RP = 1 +
for a CRRA utility function.
40
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41
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[15] Menezes, C., Geiss, C., and Tressler, J. (1980),
”Increasing Downside Risk”,
American Economic Review, Vol.70, No.5, 921-932.
[16] Merrigan, P. and Normandin, M. (1996), “Precautionary Saving Motives: An Assessment from UK Time Series of Cross-Sections”, The Economic Journal, Vol.106, 11931208.
[17] Rothschild, M. and Stiglitz, J. (1970), “Increasing Risk:
Journal of Economic Theory, Vol.2, 225-243.
42
I. A De…nition”,
[18] Sandmo,
A.
(1970),“The
E¤ect
of
Review of Economic Studies, Vol.37, 353-360.
43
Uncertainty
on
Saving
Decisions”,
ANNEX 1: Tables and Figures
Options Presented to the Players
Table 1a
Table 1b
Option A
Play lottery y
You win
3750 with 25% of probability
8750 with 75% of probability
Your
Option B Choice
7200
7300
7400
7500
7600
7700
7800
Table 2a
Option A
Play lottery z
You win
6250 with 75% of probability
11250 with 25% of probability
Your
Option B Choice
7200
7300
7400
7500
7600
7700
7800
Table 2b
Option A
Play lottery y'
You win
2500 with 25% of probability
7500 with 25% of probability
10000 with 50% of probability
Your
Option B Choice
6500
6850
7100
7300
7500
7700
7850
8000
44
Option A
Play lottery z'
You win
5000 with 50% of probability
7500 with 25% of probability
12500 with 25% of probability
Your
Option B Choice
6500
6850
7100
7300
7500
7700
7850
8000
Results
Table 3a: Characteristics of the distribution of the answers
Lotteries y and z
Number of players
Total
87
%a
100,00
%b
RISK AVERSION
Risk Averse (RA)
Risk Neutral
Risk Seeking
CRRA (and RA)
59
9
11
23
67,82
10,34
12,64
26,44 38,98
PRUDENCE
Prudent (P)
Neutral to Prudence
Imprudent
55
24
8
63,22
27,59
9,20
RA AND P
RA and P
37
42,53 62,71
RA and NP
15
17,24 25,42
RA and IP
7
8,05 11,86
a. % with respect to all cases
b. % with respect to risk averse cases
Table 3b: Characteristics of the distribution of the answers
Lotteries y'and z'
Number of players
Total
87
%a
100,00
%b
RISK AVERSION
Risk Averse (RA)
Risk Neutral
Risk Seeking
CRRA (and RA)
47
13
15
22
54,02
14,94
17,24
25,29 46,81
PRUDENCE
Prudent (P)
Neutral to Prudence
Imprudent
49
25
13
56,32
28,74
14,94
RA AND P
RA and P
30
34,48 63,83
RA and NP
10
11,49 21,28
RA and IP
7
8,05 14,89
a. % with respect to all cases
b. % with respect to risk averse cases
45
Possible Absolute Risk Aversion and Prudence Coefficients
and Amount of Cases
Table 4a: Lotteries y and z
ECz
ECy
a
πz
7200
7200
300
7200
7300
300
7200
7400
300
7200
7500
300
7200
7600
300
7200
7700
300
7200
7800
300
7300
7200
200
7300
7300
200
7300
7400
200
7300
7500
200
7300
7600
200
7300
7700
200
7300
7800
200
7400
7200
100
7400
7300
100
7400
7400
100
7400
7500
100
7400
7600
100
7400
7700
100
7400
7800
100
7500
7200
0
7500
7300
0
7500
7400
0
7500
7500
0
7500
7600
0
7500
7700
0
7500
7800
0
7600
7200
-100
7600
7300
-100
7600
7400
-100
7600
7500
-100
7600
7600
-100
7600
7700
-100
7600
7800
-100
7700
7200
-200
7700
7300
-200
7700
7400
-200
7700
7500
-200
7700
7600
-200
7700
7700
-200
7700
7800
-200
7800
7200
-300
7800
7300
-300
7800
7400
-300
7800
7500
-300
7800
7600
-300
7800
7700
-300
7800
7800
-300
a. Risk premium for lotteries z and y
πya
A
P
Total
Players
%
300
200
100
0
-100
-200
-300
300
200
100
0
-100
-200
-300
300
200
100
0
-100
-200
-300
300
200
100
0
-100
-200
-300
300
200
100
0
-100
-200
-300
300
200
100
0
-100
-200
-300
300
200
100
0
-100
-200
-300
0,00006400
0,00005333
0,00004267
0,00003200
0,00002133
0,00001067
0,00005333
0,00004267
0,00003200
0,00002133
0,00001067
-0,00001067
0,00004267
0,00003200
0,00002133
0,00001067
-0,00001067
-0,00002133
0,00003200
0,00002133
0,00001067
0,00000000
-0,00001067
-0,00002133
-0,00003200
0,00002133
0,00001067
-0,00001067
-0,00002133
-0,00003200
-0,00004267
0,00001067
-0,00001067
-0,00002133
-0,00003200
-0,00004267
-0,00005333
-0,00001067
-0,00002133
-0,00003200
-0,00004267
-0,00005333
-0,00006400
0,00000
-0,00024
-0,00060
-0,00120
-0,00240
-0,00600
0,00024
0,00000
-0,00040
-0,00120
-0,00360
-0,00600
0,00060
0,00040
0,00000
-0,00120
-0,00360
-0,00240
0,00120
0,00120
0,00120
0,00000
-0,00120
-0,00120
-0,00120
0,00240
0,00360
0,00120
0,00000
-0,00040
-0,00060
0,00600
0,00360
0,00120
0,00040
0,00000
-0,00024
0,00600
0,00240
0,00120
0,00060
0,00024
0,00000
0
0
0
0
0
0
0
1
0
0
1
0
0
0
7
5
15
4
3
0
0
0
7
17
9
0
0
0
0
3
3
9
0
0
0
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0,01
0
0
0,01
0
0
0
0,08
0,06
0,17
0,05
0,03
0
0
0
0,08
0,20
0,10
0
0
0
0
0,03
0,03
0,10
0
0
0
0,01
0
0,01
0,01
0
0
0
0
0
0
0
0
0
0
46
Table 4b: Lotteries y' and z'
a
πz
ECz
ECy
6500
6500
1000
6500
6850
1000
6500
7100
1000
6500
7300
1000
6500
7500
1000
6500
7700
1000
6500
7850
1000
6500
8000
1000
6850
6500
650
6850
6850
650
6850
7100
650
6850
7300
650
6850
7500
650
6850
7700
650
6850
7850
650
6850
8000
650
7100
6500
400
7100
6850
400
7100
7100
400
7100
7300
400
7100
7500
400
7100
7700
400
7100
7850
400
7100
8000
400
7300
6500
200
7300
6850
200
7300
7100
200
7300
7300
200
7300
7500
200
7300
7700
200
7300
7850
200
7300
8000
200
7500
6500
0
7500
6850
0
7500
7100
0
7500
7300
0
7500
7500
0
7500
7700
0
7500
7850
0
7500
8000
0
7700
6500
-200
7700
6850
-200
7700
7100
-200
7700
7300
-200
7700
7500
-200
7700
7700
-200
7700
7850
-200
7700
8000
-200
a. Risk premium for lotteries z and y
a
πy
1000
650
400
200
0
-200
-350
-500
1000
650
400
200
0
-200
-350
-500
1000
650
400
200
0
-200
-350
-500
1000
650
400
200
0
-200
-350
-500
1000
650
400
200
0
-200
-350
-500
1000
650
400
200
0
-200
-350
-500
47
A
0,00021333
0,00017600
0,00014933
0,00012800
0,00010667
0,00008533
0,00006933
0,00005333
0,00017600
0,00013867
0,00011200
0,00009067
0,00006933
0,00004800
0,00003200
0,00001600
0,00014933
0,00011200
0,00008533
0,00006400
0,00004267
0,00002133
0,00000533
-0,00001067
0,00012800
0,00009067
0,00006400
0,00004267
0,00002133
-0,00001600
-0,00003200
0,00010667
0,00006933
0,00004267
0,00002133
0,00000000
-0,00002133
-0,00003733
-0,00005333
0,00008533
0,00004800
0,00002133
-0,00002133
-0,00004267
-0,00005867
-0,00007467
P
0,00000
-0,00025
-0,00051
-0,00080
-0,00120
-0,00180
-0,00249
-0,00360
0,00025
0,00000
-0,00029
-0,00064
-0,00120
-0,00227
-0,00400
-0,00920
0,00051
0,00029
0,00000
-0,00040
-0,00120
-0,00360
-0,01800
-0,01080
0,00080
0,00064
0,00040
0,00000
-0,00120
-0,00440
-0,00280
0,00120
0,00120
0,00120
0,00120
0,00000
-0,00120
-0,00120
-0,00120
0,00180
0,00227
0,00360
0,00120
0,00000
-0,00033
-0,00051
Total
Players
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
2
0
2
0
0
0
0
0
1
8
10
5
2
2
0
0
1
5
11
13
2
0
0
0
0
3
4
7
2
0
0
%
0
0
0
0
0
0
0
0
0,01
0
0
0
0
0
0
0
0,01
0,02
0
0,02
0
0
0
0
0
0,01
0,09
0,11
0,06
0,02
0,02
0
0
0,01
0,06
0,13
0,15
0,02
0
0
0
0
0,03
0,05
0,08
0,02
0
0
Table 4b (continuation)
a
πz
ECz
ECy
7850
6500
-350
7850
6850
-350
7850
7100
-350
7850
7300
-350
7850
7500
-350
7850
7700
-350
7850
7850
-350
7850
8000
-350
8000
6500
-500
8000
6850
-500
8000
7100
-500
8000
7300
-500
8000
7500
-500
8000
7700
-500
8000
7850
-500
8000
8000
-500
a. Risk premium for lotteries z and y
πya
1000
650
400
200
0
-200
-350
-500
1000
650
400
200
0
-200
-350
-500
48
A
0,00006933
0,00003200
0,00000533
-0,00001600
-0,00003733
-0,00005867
-0,00007467
-0,00009067
0,00005333
0,00001600
-0,00001067
-0,00003200
-0,00005333
-0,00007467
-0,00009067
-0,00010667
P
0,00249
0,00400
0,01800
0,00440
0,00120
0,00033
0,00000
-0,00021
0,00360
0,00920
0,01080
0,00280
0,00120
0,00051
0,00021
0,00000
Total
Players
0
0
0
1
0
1
0
0
0
0
0
0
0
2
1
0
%
0
0
0
0,01
0
0,01
0
0
0
0
0
0
0
0,02
0,01
0
Table 5: Characteristics of the distributions
Racoef1
Racoef2
Pcoef1
Pcoef2
25%
0
0
0
0
50%
0.0000213
0.0000107
0.00036
0.0006
75%
0.0000427
0.0000213
0.0012
0.0012
mean
0.0000202
0.0000134
0.0004061
0.0005402
Sd(mean)
0.0000441
0.0000154
0.0012821
0.0012398
Table 6: Regressing Prudence Coefficient on Risk Aversion Coefficient
racoef1
(1)
pcoef1
1.30781
(3.17508)
(2)
pcoef1
2.08790
(3.56604)
racoef2
age
female
Married
Ingrhog123
ingrepe1
Colombia
Scholarship only source
Parent is only source
Nonstudent
Only parents pay tuition
Constant
0.00038
(0.00016)*
81
0.00
0.00006
(0.00004)
0.00050
(0.00034)
0.00020
(0.00051)
0.00021
(0.00045)
-0.00045
(0.00039)
0.00039
(0.00043)
0.00070
(0.00053)
0.00030
(0.00062)
0.00075
(0.00047)
0.00052
(0.00044)
-0.00292
(0.00137)*
79
0.14
Observations
R-squared
Standard errors in parentheses
+ significant at 10%; * significant at 5%; ** significant at 1%
49
(3)
pcoef2
(4)
pcoef2
-5.28402
(8.90117)
-2.19984
(10.57766)
0.00002
(0.00004)
0.00043
(0.00032)
0.00036
(0.00052)
0.00016
(0.00043)
0.00004
(0.00038)
0.00027
(0.00042)
0.00003
(0.00051)
0.00015
(0.00060)
0.00024
(0.00048)
-0.00021
(0.00044)
-0.00072
(0.00137)
81
0.05
0.00061
(0.00018)**
83
0.00
Table 7: Regressing Prudence coefficient and Risk Aversion Coefficient on
demographic variables
age
female
married
ingrhog123
ingreper1
colombia
scholar
Parent
nonstudent
Only parents paying tuition
Constant
(1)
racoef1
0.00000
(0.00000)
-0.00001
(0.00001)
0.00001
(0.00002)
0.00001
(0.00001)
0.00001
(0.00001)
-0.00002
(0.00001)
0.00001
(0.00002)
-0.00001
(0.00002)
-0.00001
(0.00002)
-0.00000
(0.00002)
0.00001
(0.00004)
85
0.14
(2)
racoef2
0.00000
(0.00000)
-0.00000
(0.00000)
0.00001
(0.00001)
-0.00000
(0.00000)
0.00000
(0.00000)
0.00000
(0.00000)
-0.00000
(0.00001)
-0.00001
(0.00001)+
-0.00001
(0.00001)
-0.00000
(0.00001)
0.00001
(0.00001)
85
0.19
Observations
R-squared
Standard errors in parentheses
+ significant at 10%; * significant at 5%; ** significant at 1%
(3)
pcoef1
0.00006
(0.00004)
0.00052
(0.00032)
0.00037
(0.00049)
0.00047
(0.00045)
-0.00041
(0.00034)
0.00031
(0.00039)
0.00001
(0.00071)
-0.00016
(0.00059)
-0.00204
(0.00062)**
0.00052
(0.00069)
-0.00194
(0.00129)
79
0.20
(4)
pcoef2
0.00002
(0.00004)
0.00046
(0.00032)
0.00039
(0.00051)
0.00020
(0.00045)
0.00007
(0.00035)
0.00032
(0.00041)
-0.00007
(0.00066)
0.00008
(0.00053)
-0.00041
(0.00071)
0.00028
(0.00066)
-0.00067
(0.00133)
81
0.05
Table 8: Ordered Probit Regressions
(1)
(2)
(3)
ra1
ra2
prudent1
age
-0.11697
-0.04362
-0.07384
(0.05150)*
(0.05060)
(0.04912)
female
0.09250
0.03850
-0.47566
(0.33904)
(0.34328)
(0.32008)
Married
0.91000
-0.00450
-0.25724
(0.59769)
(0.64652)
(0.51812)
Ingrhog123
0.19365
0.50553
-0.38332
(0.46828)
(0.47205)
(0.45764)
ingrepe1
-0.50661
0.13872
0.16851
(0.36847)
(0.37224)
(0.34066)
Colombia
0.90478
0.54352
0.05524
(0.44326)*
(0.42963)
(0.39015)
Only scholarship
-0.04455
-0.23484
-0.52466
(0.68534)
(0.72925)
(0.64817)
Only parent
0.46834
0.37254
0.09862
(0.56612)
(0.54539)
(0.55827)
Nonstudent
0.19333
0.21366
0.96385
(0.65775)
(0.64734)
(0.64786)
Parents only pay tuition
-0.53370
0.06442
-0.38452
(0.66553)
(0.69043)
(0.64854)
Observations
73
73
73
Standard errors in parentheses
+ significant at 10%; * significant at 5%; ** significant at 1%
50
(4)
prudent2
-0.04048
(0.05176)
-0.36508
(0.32842)
-0.56155
(0.55699)
1.02945
(0.51228)*
0.28189
(0.35988)
-0.12321
(0.41189)
0.47597
(0.73144)
-0.70332
(0.70197)
0.69290
(0.68920)
-0.31306
(0.69348)
73
Ranges for Relative Risk Aversion Coefficients: CRRA Case
Table 9a: Lotteries y and z
Lottery y
Lottery z
ECy
γ
Range ECy
Range γ
ECz
γ
Range ECz
7200
0,99
<=7200
0,99
7200
0,74
<=7200
(7200;7300]a
7300
0,75
[0,75;1)
7300
0,510000 (7200;7300]
7400
0,36
(7300;7400] [0,36;0,75)
7400
0,260000 (7300;7400]
7500
0,01
(7400;7500]
[0;0,36)
7500
0,01
(7400;7500]
a. () mean that the limit is not included while [] mean that they are.
Range γ
[0,74;1)
[0,51;0,74)
[0,26;0,51)
(0;0,26)
Table 9b: Lotteries y' and z'
Lottery y'
Lottery z'
ECy'
γ
Range ECy' Range γ
ECz'
γ
Range ECz'
6500
0,99
<=6500
0,99
6500
0,99
<=6500
(6500,6850]a
0,99b
6850
0,99
6850
0,74
(6500,6850]
7100
0,72
(6850,7100]
[0,72;1)
7100
0,48
(6850,7100]
7300
0,35
(7100,7300] [0,35;0,72)
7300
0,25
(7100,7300]
7500
0,00
(7300,7500]
(0;0,35)
7500
0,01
(7300,7500]
a. () mean that the limit is not included while [] mean that they are.
b. No range appears in this case because both 6500 and 6850 imply γ = 0,99.
Range γ
0,99
[0,74;1)
[0,48;0,74)
[0,25;0,48)
(0;0,25)
Common Ranges for Relative Risk Aversion and Relative Prudence
Coefficients: CRRA Case
Table 10a: Lotteries y and z
Range γ Range RP
ECz
ECy
Total
7300
7200
[0,75;1)
[1,75;2)
1
7400
7300
[0,51;0,74] [1,51;1,74]
5
7500
7400
[0,26;0,36] [1,26;1,36]
17
() mean that the limit is not included while [] mean that they are.
%
4,35
21,74
73,91
Table 10b: Lotteries y' and z'
Range γ Range RP
ECz'
ECy'
Total
6850
6500
0,99
1,99
1
7100
6850
[0,74;1)
[1,74;2)
2
7300
7100
[0,48;0,72] [1,48;1,72]
8
7500
7300
[0,25;0,35] [1,25;1,35]
11
() mean that the limit is not included while [] mean that they are.
51
%
4,55
9,09
36,36
50,00
Distribution of risk averseness and prudence cases
Figure 1b
Risk aversion for lotteries y’ and z’
0
0
20
20
Percent
40
Percent
40
60
80
60
Figure 1a
Risk aversion for lotteries y and z
1
2
3
ra2
1
4
2
ra1
3
4
1: Risk averse, 2: Risk neutral, 3: Risk lover, 4: Ambiguous
Figure 2a
Prudence for lotteries y and z
Figure 2b
Prudence for lotteries y’ and z’
0
0
20
20
Percent
Percent
40
40
60
60
1: Risk averse, 2: Risk neutral, 3: Risk lover, 4: Ambiguous
1
1.5
2
prudent2
2.5
3
1: Prudent, 2: Prudent neutral, 3: Imprudent
1
1.5
2
prudent1
2.5
3
1: Prudent, 2: Prudent neutral, 3: Imprudent
Distribution of risk aversion and prudence coefficient
Figure 3b
Risk aversion coefficient for y’ and z’
0
0
10
10
Percent
20
Percent
20
30
30
40
Figure 3a
Risk aversion coefficient for y and z
-.00002
0
.00002
racoef2
.00004
.00006
-.00005
0
.00005
racoef1
.0001
.00015
Figure 4b
Prudence for lotteries y’ and z’
0
0
10
10
Percent
20
30
Percent
20
30
40
40
50
50
Figure 4a
Prudence for lotteries y and z
-.0001
-.004
-.002
0
pcoef2
.002
.004
.006
-.004
-.002
0
pcoef1
.002
.004