Experimental Economics (2022) 25:1202–1233
Present bias for monetary and dietary rewards
Stephen L. Cheung1 · Agnieszka Tymula1 · Xueting Wang2
Received: 2 March 2021 / Revised: 21 February 2022 / Accepted: 22 February 2022 /
Published online: 18 March 2022
© Crown 2022
Economists model self-control problems through time-inconsistent preferences.
Empirical tests of these preferences largely rely on experimental elicitation using
monetary rewards, with several recent studies failing to fnd present bias for money.
In this paper, we compare estimates of present bias for money with estimates for
healthy and unhealthy foods. In a within-subjects longitudinal experiment with
697 low-income Chinese high school students, we fnd strong present bias for both
money and food, and that individual measures of present bias are moderately correlated across reward types. Our experimental measures of time preferences over both
money and foods predict feld behaviors including alcohol consumption and academic performance.
Keywords Self-control · Quasi-hyperbolic discounting · Present bias · Adolescents ·
JEL Classifcation C91 · D12 · D80 · D91
Self-control is viewed in economics and other disciplines as a key individual characteristic responsible for efective self-regulation and personal goal attainment (Moftt
et al., 2011). Lack of self-control is thought to explain suboptimal choices and outcomes in many life domains, including fnancial decision making, health, and education. Given the importance of self-control, this individual trait is widely studied
theoretically and empirically in many diferent felds (Duckworth et al., 2018).
In the economics literature, researchers usually model problems of self-control
through time-inconsistent preferences that predict choices such as planning to go
* Xueting Wang
1 School of Economics, The University of Sydney, Sydney, Australia
2 School of Economics, Finance, and Marketing, RMIT University, Melbourne, Australia
Present bias for monetary and dietary rewards
on a diet starting next week but not going on the diet when next week arrives. Two
well-known models that can capture such behaviours are the hyperbolic (Loewenstein & Prelec, 1992) and quasi-hyperbolic (Laibson, 1997) discount models. The
latter model has attractive analytical features that have contributed to its popularity in economics (Frederick et al., 2002), and for this reason we focus on it in our
paper. The underlying assumption of the model is that agents have a “present bias”
toward current consumption, as the values of all future rewards are downweighed
relative to rewards in the present (in addition to the standard exponential discounting
of delayed rewards). Economists have applied quasi-hyperbolic discounting theoretically and empirically to explain problematic behaviours across a wide variety of
domains such as fnancial decision making (Laibson et al., 1998), health behaviours
(DellaVigna & Malmendier, 2006; Gruber & Kőszegi, 2001; Schilbach, 2019), and
work efort (Augenblick et al., 2015; Kaur et al., 2015).
In stark contrast to these diverse domains of application, most experimental
research aimed at quantifying present bias has focused on a single specifc reward
type, namely money, and on samples from developed countries, in particular students at research universities. Further, most studies have used a cross-sectional
design, which is not a true test of time inconsistency (Halevy, 2015; Read et al.,
2012).1 Only a longitudinal design permits a test of inconsistent planning, the key
prediction of the quasi-hyperbolic model (O’Donoghue & Rabin, 1999). In this
paper, we address each of these shortcomings that have characterised much of the
existing literature. We next expand upon each of these points in turn.
In this paper, we estimate and compare time preferences for money, healthy
foods, and unhealthy foods. We thus contribute to the literature by identifying the
shape of time preferences for food rewards. While the quasi-hyperbolic model has
been applied to explain behaviour across a variety of domains, several recent experimental studies fnd no present bias for money (Andersen et al., 2014; Andreoni &
Sprenger, 2012a; Andreoni et al., 2015; Augenblick et al., 2015). An infuential
interpretation of these fndings (Cohen et al., 2020) holds that experiments using
money will fail to detect present bias if subjects engage in arbitrage: if subjects
integrate experimental earnings with borrowing and savings opportunities outside
the experiment, they will simply switch from sooner to later payment at the market
interest rate, revealing linear utility and no present bias.
Clearly, to shed light on this issue it is necessary to compare present bias for
monetary and non-monetary rewards. Augenblick et al. (2015) compare present
bias for money and real efort, fnding present bias for efort but not money. In their
experiment, choices over efort (which is aversive) may be interpreted as revealing
preferences toward leisure (a reward) if it is assumed that time not spent working
for the experimenters is instead devoted to the consumption of leisure. However,
since it is only the efort choice that is elicited and controlled for in the experiment,
it is possible that this efort instead displaces another aversive use of time—such as
domestic work, study, or an outside form of market employment—as opposed to leisure. This, in efect, is the real efort analogue to the potential confound of arbitrage
1 A cross-sectional design compares, at a single point in time, preferences between two or more pairs of
temporal prospects, separated by a common interval but preceded by diferent front-end delays.
1204 S. L. Cheung et al.
in experiments using money. While previous studies (discussed next) have compared
impatience and risk preference for monetary and direct consumption rewards, ours
is the frst to do so for present bias.2 For now, we point out that if present bias is real
but confounded by arbitrage in experiments using money, then we would expect to
fnd present bias for food but not money. We return to the issue of arbitrage in the
With regard to other economic preferences, it has been found that people tend to
be less patient (as distinct from present biased) for primary rewards than for money
(Estle et al., 2007; Odum & Rainaud, 2003; Reuben et al., 2010; Tsukayama &
Duckworth, 2010; Ubfal, 2016) but that risk preferences estimated for money and
food rewards are essentially the same (Levy & Glimcher, 2012).3 These contrasting
results highlight the importance of studying the consistency of preferences across
domains separately for each economic preference. Moreover, within the domain
of foods, unhealthy foods may be more tempting, triggering more present bias. It
is thus also important to compare time preferences between healthy and unhealthy
A key feature of this paper is our sample of 697 relatively poor adolescents in
China. Most previous studies have focused on the so called WEIRD subject pool
(Henrich et al., 2010). WEIRD refers to samples drawn from populations that are
Western, Educated, Industrialised, Rich and Democratic. One reason why carefully
designed studies do not fnd present bias for money may simply be that the participants, having been admitted into top universities, did not have serious self-control
problems to begin with.4 Indeed, several recent studies provide evidence that participants from developing countries show present bias for money (Balakrishnan et al.,
2020; Banerji et al., 2018; Clot & Stanton, 2014; Giné et al., 2018; Janssens et al.,
2017). Aycinena et al., (2020) fnd impatience for money and a preference to smooth
payments over time in a sample of low-income Guatemalans, but do not fnd present bias. There is also evidence that certain clinical populations (e.g. prescription
drug program enrolees and diabetes patients) show present bias for money (Abaluck
et al., 2018; Mørkbak et al., 2017). Our paper contributes to the still relatively limited evidence on the time preferences of non-WEIRD samples.
2 Other papers have identifed self-control for consumption rewards without quantifying present bias.
Read and Van Leeuwen (1998) asked participants to make choices between healthy and unhealthy snacks
that they would receive in one week. When the appointed time came, participants were given an opportunity to change their choice. Carbone (2008) asked participants to decide which investment goods and
activities (e.g. salad, textbook reading) and temptation goods and activities (e.g. ice cream, video games)
to consume in two treatments in which consumption occurred either immediately or after a delay of four
hours. Sadof et al. (2020) used the demand for commitment to understand time-inconsistent behaviour
for food choice. Each of these studies identify inconsistencies in the types of reward chosen for immediate delivery, as opposed to the quantities.
3 At the neural level, evidence suggests the existence of a “common neural valuation system” (Montague
& Berns, 2002). In two meta-analyses, Bartra et al. (2013) and Clithero and Rangel (2013) fnd that brain
regions that respond to both primary and secondary incentives overlap.
4 In a meta-analysis, Imai et al. (2021) suggest that university students tend to show stronger present bias
than the general population. However, this fnding is confounded by collinearity between the location
of the study and the subject pool: laboratory experiments tend also to have student subjects, while feld
studies are more likely to recruit from the general population.
Present bias for monetary and dietary rewards
Our subjects difer not only on each of the dimensions of the WEIRD samples,
but also in their age. Self-control established early in life is critical to personal
development, yet few studies to date have estimated time preferences in children and
adolescents.5 Research in psychology has shown that poor self-control in childhood
is associated with a range of damaging behaviours, for example cigarette smoking.
Moreover, children with greater self-control are signifcantly more likely to be from
socioeconomically advantaged families (Moftt et al., 2013).
To identify present bias we conduct a longitudinal experiment in schools. Halevy
(2015) distinguishes three properties of standard preferences over temporal payments relative to a dated collection of such preferences. Stationarity implies that the
ranking of two temporal payments at time t depends only on the diference between
the two payments and their relative delay. The standard cross-sectional design is
a test of this property. Time invariance implies that preferences are not a function
of calendar time. Time consistency requires that the ranking of temporal payments
does not change as the evaluation perspective changes from t to t’. Only a true longitudinal design can test for this property. Halevy (2015) fnds that people can be
time inconsistent and have stationary preferences at the same time, implying that the
results of a cross-sectional design may be misleading.
Finally, conducting our experiment in school allows us to avoid selection into the
study as well as attrition from it. Further, with access to administrative data from
schools, we test the ability of our experimental measures to predict feld outcomes
such as academic performance.
697 Chinese high-school students participated in a fve-week, incentivised longitudinal experiment using a modifed version of the Convex Time Budget design
(Andreoni & Sprenger, 2012a) to elicit individual preferences for three reward
types: money, healthy food and unhealthy food. Subjects faced the same set of decisions, featuring the same reward amounts delivered on the same dates, at two points
in time. In the frst session, all choices involved rewards to be received at two dates
in the future, while in the second session the sooner rewards were available today.
Our design also incorporates a test of rationality in the form of violations of the
Generalised Axiom of Revealed Preference (GARP). We conducted our experiment
during regular class time and all 697 subjects completed both sessions, resulting in
We highlight several key fndings. First, we provide the frst estimates of present
bias for consumption rewards. At the median, averaging over all trials, our subjects choose to receive 2% more food on the sooner payment date when the decision is made on that day than when it is made in advance. Our structural estimate
of 훽 for a representative agent is 0.69 for healthy food and for unhealthy food it is
0.71 (both are signifcantly less than one, but not signifcantly diferent from one
another). Food consumption is highly consequential for people’s health. Focusing
5 The seminal study investigating the lifelong impact of self-control is Walter Mischel’s “marshmallow”
test (Mischel & Ebbesen, 1970; Mischel et al., 1989), but see McGuire and Kable (2013) and Cohen
et al. (2020) for discussion of confounds in the interpretation of this task as a measure of time preference.
Sutter et al. (2013) investigate the link between children’s and adolescents’ time preference for money
and feld behaviours, however they fnd little evidence of present bias in their sample. Alan and Ertac
(2018) and List et al. (2021) provide evidence on discounting (but not present bias) for adolescents.
1206 S. L. Cheung et al.
on the amount consumed in the moment, a representative agent who has a healthy
BMI = 21 and who participates in our experiment every week would become overweight in 4 years.
In contrast to some recent literature, we also fnd strong present bias for money.
At the median, subjects choose to receive 4% more money on the sooner payment
date when the decision is made on that day than when it is made in advance. Our
structural estimate of 훽 for a representative agent is 0.65 for money (also signifcantly less than one, as well as signifcantly diferent from our estimates for food).
Next, in contrast to previous fndings in the domain of risk, we fnd diferences
in the curvature of utility between monetary and primary rewards. For money, we
confrm recent fndings in the time preference literature that instantaneous utility is
at best only mildly concave (Abdellaoui et al., 2013; Andreoni & Sprenger, 2012a;
Cheung, 2020). However, for both healthy and unhealthy foods we fnd strong evidence of concave utility (implying a preference to spread rewards evenly over time),
more in line with conventional fndings in the domain of risk.
At an individual level, we fnd signifcantly positive and moderate correlations between individual measures of present bias for all reward type pairs
[휌 ∈ (0.47, 0.60)], as well as between individual measures of impatience
[휌 ∈ (0.59, 0.66)]. We fnd even stronger correlations for a measure of the preference
to smooth consumption over time [휌 ∈ (0.81, 0.85)].6 Together, these fndings imply
that conventional choices over money are moderately predictive of choices for food.
Finally, we fnd that our experimental measures of time preferences for both monetary and dietary rewards are predictive of subjects’ feld behaviours. Adolescents
who make less patient choices for any reward type are more likely to drink alcohol
and have lower grades. Moreover, those who are more present biased for money and
healthy food are more likely to drink alcohol and have lower grades.
The paper proceeds as follows: Sect. 2 describes our experimental design, Sect. 3
explains our empirical approach, Sect. 4 presents the results, and Sect. 5 provides a
discussion of our fndings.
2 Experimental design
2.1 Subject pool
We collected data from 697 adolescents (331 girls; average age 16.1 years, standard
deviation 0.15 years) from four public high schools in Guiyang City, China in February and March 2019. We randomly selected 16 classes in tenth and eleventh grades
to participate in the study. The University of Sydney Human Research Ethics Committee and principals of each collaborating high school approved the study. Teachers
of the participating classes permitted the experiments to be conducted in class during regular school hours. No students opted out, and all participating students and
6 For comparison, Levy and Glimcher (2011) fnd that the correlation between risk preferences for
money and food was 0.65 (Spearman’s rank test; n = 65; p < 0.0001).
Present bias for monetary and dietary rewards
their parents gave informed consent. The experiment was conducted in Mandarin
(see Online Appendix 1 for an English translation of the instructions).
Our experimental task is an extension of the convex time budget (CTB) design of
Andreoni and Sprenger (2012a), which allows us to estimate subjects’ utility and
discounting parameters using data from a single task. To simplify this task, we
implement a discrete version of the CTB based upon Andreoni et al. (2015).
Following the CTB framework, we provide options that allocate amounts of a
reward between two payment dates subject to a future-value budget constraint:
t denotes the amount of reward received at the sooner payment date t, ct+k
denotes the amount of reward received at the later payment date t + k, and r denotes
the simple interest rate between the two dates. Between trials, we systematically
vary the interest rate r keeping the future value of the endowment fxed at 70. The
back-end delay k was always equal to three weeks.
Figure 1A shows a sample budget with an interest rate of 0%. In that case, regardless of which bundle a subject chooses, the amounts received on the two dates
always sum to 70. To discretise this choice, we ofer six evenly spaced options
(shown as dots in Fig. 1A) along the budget line that a subject can choose from.
There were always six options in every trial to keep choice difculty constant. We
exclude corner bundles [i.e. (ct, 0) and (0, ct+k)] from the choice set, as previous
studies fnd that subjects who consistently choose corner bundles generate issues for
structural estimation (Harrison et al., 2013). Another advantage of this procedure is
that by forcing subjects to receive payments on both dates, we equalise transaction
costs without the use of a show-up fee.
Figure 1B shows the corresponding decision screen for the 0% interest rate trial.
As well as stating the amounts of a reward that are available on each payment date,
we also visualise these quantities to facilitate comparison of the alternatives. The
order of presentation of the six options on the screen was randomised for each subject, and the subject chose their most preferred bundle by clicking on it.
The other simple interest rates we use are- 9%, 11%, 25%, 43%, 67% and
100% (see Fig. 2A for these seven budget sets). As the interest rate varies, a subject’s choices trace out a price expansion path in terms of sooner and later rewards,
with the optimal choices depending upon both utility curvature and discounting
We further enrich this framework by adding an additional seven decisions to
allow for a test of the consistency of subjects’ choices with the Generalised Axiom of
Revealed Preference (Varian, 1982), as recommended by Chakraborty et al. (2017).
We derive these additional choice sets from a present-value budget constraint:
(1 + r) × ct + ct+k = 70,
1208 S. L. Cheung et al.
and in these trials we vary the interest rate while holding the present value of the
endowment fxed at 56. The interest rates r for these additional trials are -13%,
0%, 13%, 25%, 38%, 50%, and 63%. Figure 2B shows the complete set of budgets
used in our design. The two sets of budget lines intersect one another, allowing us to
count the number of times a subject’s choices violate GARP. The maximum number
of GARP violations in this task is 91, while a random chooser would be expected to
commit 12 violations. Note also that the trial with a 25% interest rate is presented
twice (with other trials interleaved in between), allowing us to check for the consistency of subjects’ choices when making the same decision twice.7
Figure 3 shows the timeline of our fve-week longitudinal experiment. In the frst
session in week one, subjects were presented with decisions where the sooner payment is in one week’s time (hence in week two) and the later payment is in four
weeks’ time (hence in week fve). In the second session in week two, the same subjects made the same sets of decisions over bundles of rewards received in weeks two
and fve, where the sooner payment is now available today.8 This longitudinal design
identifes dynamic inconsistency by comparing initial allocations in week one (when
all rewards are in the future) with subsequent allocations in week two (when the
sooner reward is in the present). In each school, all sessions were conducted at the
same time of day and on the same day of the week to keep other variables such as
hunger constant; for logistical reasons, the timing of the sessions difered slightly
between schools. Before making their decisions in week one, subjects were told that
they would be making decisions again in week two, and that one out of all their decisions would be randomly selected at the end of session two to be realised for payment. In the third session which took place in week fve, subjects did not make any
decisions and only received rewards. The experiment dates were between 25 February and 29 March 2019. Over this period, there were no public holidays, school
vacations or examinations.
After completing their decisions, subjects flled out a questionnaire which
included demographic characteristics (in the frst session) as well as current hunger
and fatigue level,9 and appetite ratings (in both sessions); see Online Appendix 2 for
an English translation of these questionnaires.
1 + r
t+k = 56,
7 Our design also includes two choice sets with a 0% interest rate (but diferent sized budgets), allowing
for an examination of the income efect.
8 Online Appendix 3 shows sample choice screens of the same trial as faced by a subject in week one
and week two, respectively; everything is the same except the delays until the reward dates.
9 In each session, we asked subjects to report their hunger level on a scale from 1 (not hungry at all) to 7
(very hungry). The average score is 3 and it is not signifcantly diferent between the two sessions.
Present bias for monetary and dietary rewards
2.4 Reward types
To compare time preferences for monetary and food rewards, we use a within-subjects design. Each subject faced the same sets of choices for three diferent reward
types: money, healthy food, and unhealthy food. Before making any choices in week
Fig. 1 Experimental design. A: Budget constraint with 0% interest rate. The six dots on the budget line
indicate bundles available to the chooser. B: Decision screen for the 0% interest rate trial. Each row represents one bundle. On the left is the amount received on the sooner date and on the right is the later
date. Dots represent the quantity of a reward to be received on that date. The six bundles are presented in
random order for each participant
A Seven standard budget constraints B Complete set of budget constraints
Fig. 2 Budget constraints
1210 S. L. Cheung et al.
one, we asked each subject to choose their preferred healthy food reward and preferred unhealthy reward from three alternatives in each category. We did this to cater
for diferent tastes and hence ensure that all subjects made decisions for foods that
they liked. For healthy food, the available options were pecans, raisins, and almonds.
For unhealthy food, the options were Skittles, M&M’s, and Lays. We chose these
food rewards based on a pre-experiment survey of students’ favourite snacks.
A single food item—one Skittle, one chip, one raisin, etc.—counted as one unit
of the good. For example, in a 0% interest trial, subjects may choose between 40
Skittles in one week and 30 Skittles in four weeks, 20 Skittles in one week and 50
Skittles in four weeks, and so on. For money, the budget was halved such that one
unit of money equated to RMB 0.5 to equalise the value of diferent reward types.
To summarise, in a given session each subject made 14 decisions for each of three
reward types, with all 42 decisions repeated in two separate sessions. The order of
rewards was either healthy-money-unhealthy or unhealthy-money-healthy. This
order was randomly selected for each subject in the frst session, and then held constant for the second session. Thus, choices over the two food rewards were always
separated by choices over money. The experimental interface was programmed
At the end of the second session, one decision of each subject (from either the frst
or second session) was randomly selected as the one that would count for payment.
If this was a money trial, the payments were made in cash. If it was a food trial,
the subject received the amounts of food they had chosen. Sooner payments (both
money and food) were delivered one hour after the second session. In week fve,
research assistants returned to the schools at the same time as in week two to deliver
the later payments. To protect privacy, regardless of reward type, we used nontransparent zip-lock bags to pack subjects’ payments. Therefore, monetary and food
rewards were delivered to subjects in the same way.
Fig. 3 Timeline of the experiment
Present bias for monetary and dietary rewards
Since we conducted the experiment during regular class hours in schools, the transaction costs to participate and receive payments are equalised throughout the study.
Moreover, since subjects need to come to school anyway, we did not pay any additional
show-up fee, and their compensation from the study was solely based on the choices
that they made. Participants indicated a high level of trust in the experimental procedures, on average 5 on a scale from 1 (don’t trust at all) to 7 (no doubt at all).
3 Empirical approach
We next outline two approaches we adopt to measure subjects’ time preferences and
utility curvature. Our frst approach is to use descriptive measures of time preference
and preference for smoothness that are based on simple proportions of rewards allocated to sooner versus later payment dates. These descriptive measures provide evidence on the behaviours we are interested in without needing to commit to specifc
structural assumptions. However, since descriptive measures cannot always cleanly
distinguish between parameters, our second approach is to impose a quasi-hyperbolic discounted utility model (Laibson, 1997) and jointly estimate three parameters: the discount factor 훿, present bias 훽, and utility curvature 훼. We fnd that these
two approaches yield broadly consistent results.
3.1 Descriptive measures
To investigate subjects’ impatience, without confounding it with present bias, we
consider decisions made in the frst session (week one) which result in bundles of
rewards received in weeks two and fve. Since all rewards are received in the future,
present bias does not play any role. Subjects who select a bundle with a larger proportion of rewards allocated to the sooner payment date (week two) relative to the
later date (week fve) can be classifed as more impatient (equivalently less patient).
i,j be the amount of a reward that a subject would receive in week i based
on a decision made in week j. We defne impatience for each of the 14 week one
decisions (Impatiencek, k ∈ [1, 14]) for a given reward type as the proportion of the
reward allocated to week two relative to the total amount of rewards in the chosen
bundle, when the choice is made in week one:
Then, for each reward type separately, to measure an individual’s impatience we
take the average of Impatiencek for that reward type over all 14 decisions10:
c2,1 + c5,1
10 We acknowledge that impatience defned in this manner may be confounded with utility curvature. We
address this issue in our structural estimation.
1212 S. L. Cheung et al.
By construction, this measure is bounded between zero (most patient) and one
(most impatient), although in practice because we removed corner bundles from the
choice sets the measure cannot go all the way to these limits in our design.
3.1.2 Present bias
Present bias occurs when an individual allocates a larger proportion of a reward to
the sooner date when the sooner payment is immediate relative to when it is delayed,
other things equal. To construct a descriptive measure of present bias, we frst compare an individual decision made in week two when the sooner payment is today to
the same decision made in week one when the sooner payment is delayed. We thus
defne present bias for a given decision scenario (Present biask, k ∈ [1, 14]) as the
diference in the proportion of the reward allocated to week two when making a
choice in week two compared to when making the same choice in week one:
Then, for each reward type separately, to measure an individual’s present bias we
take the average of Present biask for that reward type over all 14 decision scenarios:
By construction, this measure is bounded between negative one (most future
biased) and one (most present biased). Again, because we removed corner bundles
from our choice sets, the measure does not go all the way to these limits in our
3.1.3 Preference for smoothness
In addition to their time preferences, a subject’s choices in the experiment depend on
the strength of their preference to smooth payofs over time, as captured by the curvature of the utility function in a discounted utility model. A subject who has highly
concave utility for a reward will have a strong preference for more mixed (temporally balanced) bundles, while one who has near-linear utility will tend to choose
more extreme bundles near the corners of the budget set. To construct a descriptive
measure of preference for smoothness, for a given decision trial (k ∈ [1, 28]), we
calculate the diference between the sum of the amounts of a reward allocated to
both dates and the absolute diference in those amounts, normalised by the sum of
Impatience = 1
c2,2 + c5,2
c2,1 + c5,1
Present bias = 1
Present bias for monetary and dietary rewards
1 represents the amount of a reward allocated to the sooner date and c2 represents the amount of a reward allocated to the later date.
In the limiting case of a corner solution (where one of the cs is zero), the numerator collapses to zero and so Smoothk goes to zero. At the opposite extreme of perfect
smoothing (such that c1 = c2), it is the absolute diference term that collapses to zero
and so Smoothk goes to one.
Then, for each reward type separately, to measure an individual’s preference for
smoothness we take the average of Smoothk for that reward type over all 28 decision
By construction, this measure is bounded between zero (no preference for
smoothing) and one (maximum preference for smoothing), although in practice it
does not go to these limits because we removed the corner bundles in our design.
3.2 Structural model
To conduct a parametric estimation of the discount factor, present bias, and utility
curvature we assume a CRRA utility function and quasi-hyperbolic discount function (Laibson, 1997; O’Donoghue & Rabin, 1999). The instantaneous utility from
experimental payments, c, is:
The parameter 훼 is CRRA utility curvature, where 훼 = 0 indicates linear utility,