How long is a minute ?
∗
Juan D. Carrillo
University of Southern California
and CEPR
Isabelle Brocas
University of Southern California
and CEPR
Jorge Tarrasó
University of Southern California
February 2015
[Preliminary]
Abstract
Psychophysics studies on time perception argue that the way we represent time intervals is not linear in true time intervals. Economics research on time discounting
reveals that future rewards are valued less the longer the delay involved. Here we combine both strands and present experimental data in which subjects perform two tasks.
First, they produce short intervals of time (seconds, minutes). Second, they choose
between different rewards separated by long delays (weeks, months). We estimate the
time perception functions and the time discounting functions at the individual level.
We find that subjects for whom one unit of time feels longer than true time discount
the future at a higher rate than subjects for whom one unit of time feels shorter than
true time. The result suggests that our capacity to delay consumption is related to
our mental representation of time delays between now and the future. They also indicate the existence of an underlying internal clock that governs time representations
irrespective of the unit of time.
Keywords: laboratory experiments, time perception, time discounting, time estimation.
JEL Classification: C91, D03, D91.
∗
We are grateful to members of the Los Angeles Behavioral Economics Laboratory (LABEL) for their
insights and comments in the various phases of the project. We also acknowledge the financial support
of the National Science Foundation grant SES-1425062. Address for correspondence: Isabelle Brocas,
Department of Economics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA
90089, USA. E-mail: <brocas@usc.edu>.
Time is too slow for those who wait, too swift for those who fear, too long for those
who grieve, too short for those who rejoice, but for those who love, time is not.
Henry Van Dyke (Music and Other Poems, 1904)
1
Introduction
Time is subjective. It flies when you enjoy and virtually stops when you suffer. Tomorrow
is “in a very long time” for kids and “practically now” for seniors. The objective of this
paper is to explore the relationship between time perception and time discounting. Our
conjecture is simple: if one person perceives one week to be longer than another person,
it seems natural that he will be less willing to delay a reward by that (objective) amount
of time. If this hypothesis is correct, it can help understand the paradoxical tendency of
older adults to save more than younger adults (Banks et al., 1998) despite their shorter life
expectancy. It may also explain why an individual will choose whether to postpone gratification or not depending on the prospective experience during that time interval. More
generally, it suggests that eliciting discount rates is a valuable but incomplete measure
to understand intertemporal tradeoffs. Indeed, an individual who succumbs to immediate
gratification may in fact be someone with a highly distorted perception of time duration,
and that his preference for the present can be dramatically affected just by correcting such
perception.
To address this question we ask our subjects to perform two tasks in a controlled
laboratory environment. First, we elicit their time discount rates using the method proposed by Andreoni and Sprenger (2012a) (hereafter, [AS]), where subjects allocate a fixed
amount of tokens between two dates. We use their convex time budget (CTB) method
due to its robustness and stability, and structurally estimate a quasi-hyperbolic discount
function and curvature of utility.1 Second, we elicit their time perception estimates by
asking subjects to reproduce intervals of lengths ranging between 20 seconds and 4 minutes. Formally, we ask them to click the start box to begin a time interval and click again
when a predetermined amount of time (e.g., 2 minutes and 31 seconds) has passed. This
task is performed in conjunction with a distractor task that prevents them from counting
seconds. We estimate for each individual a time perception power function that maps
true time intervals into perceived time intervals. Finally, we correlate the impatience or
preference for the present derived from the time discounting task (T D) with the subjective
evaluation of time obtained from the time perception task (T P).
1
An advantage of CTB over the traditional methods is that it controls for diminishing marginal utility,
although it has also received some criticisms recently. Our paper does not attempt to innovate on the
method to elicit time preferences. We realize that different methods have different advantages and use one
which has proved simple and reliable.
1
Our results can be summarized as follows. We find substantial dispersion in the time
discounting of our subjects. The estimated parameters in the T D task are in line with
those found in [AS], with low levels of impatience, little evidence of present bias and some
small (but positive) concavity in the utility function. Perception estimates in the T P task
are also heterogeneous. Although a majority of individuals underestimate time, we still
observe the opposite tendency in some subjects. More generally, we find evidence of both
concave and convex time perception functions.
The main novelty of the study, however, is to analyze the relationship between time
perception and time discounting. To this purpose, we perform an upward extrapolation
of time perception estimates to intervals of time as high as 7 days and a downward extrapolation of time discounting estimates to intervals of time as low as 1 hour. We then
correlate the individual estimates in both tasks. For all intervals of time between 1 hour
and 7 days we find a strong negative correlation between the level of impatience estimated
in the T D task and the subjective perception of time estimated in the T P task (Pearson
coeff. between -0.24 and -0.26 and p-value between 0.024 and 0.035). In other words and
consistent with our hypothesis, subjects who perceive 1 day as a longer time interval than
it really is are less likely to delay consumption by that amount of time than subjects who
perceive 1 day as a shorter time interval than it really is.
We then perform a cluster analysis and endogenously identify four groups of people.
One large group of subjects is close to what we would expect of rational economic agents:
patient, time-consistent, and with a perception of time reasonably close to the true time.
A second group is composed of subjects with a concave time perception function. They
underestimate time and are more willing to delay consumption than subjects in the first
group. The remaining two (small) groups of subjects exhibit the opposite tendency: convex
time perception functions, significant overestimation of time and lower discount parameters
than the rest of the population.
Before proceeding with the analysis, we provide a brief review of the existing literature. Time discounting has received much attention in economics. Researchers have
proposed different parametric formulations of the time discounting function as well as
different experimental designs to elicit them, and the empirical and experimental estimates vary significantly across studies (see Frederick et al. (2002) for a review). There
are two main challenges for the elicitation of time discounting. First, subjects may not
be time-consistent and place a greater value on immediate gratification compared to any
future consumption. This has motivated hyperbolic specifications of time discounting as
opposed to the standard exponential formulation.2 Second, time is inherently uncertain
2
The quasi hyperbolic formulation was first developed by Phelps and Pollak (1968) and later reintroduced by Laibson (1997). The most general hyperbolic specification is due to Lowenstein and Prelec
2
and deciding to postpone consumption amounts to choosing a lottery. It is therefore important to be able to disentangle risk preferences from time preferences. Moreover, when
choosing between consumption now (for sure) and consumption in the future, a subject
may choose the former because of his intrinsic willingness to not delay immediate gratification, or because uncertainty about the future makes the safer option more desirable.
Said differently, the two challenges are inherently related. The recent literature proposes
methods to jointly estimate time and risk preferences (Harrison et al. (2005); Andersen et
al. (2008); Andreoni and Sprenger (2012a and 2012b)) and reports less or no evidence of
a present bias.3 Our analysis relies on this last line of research which allows us to better
isolate time discounting.
Time perception has also been extensively studied in the psychophysics literature. It
is centered on prospective time evaluations, where subjects are informed beforehand that
they have to make a time related judgment. These studies mostly focus on extremely
short intervals (milliseconds, seconds) and use methods in which subjects have either to
verbally assess a duration, reproduce or produce a time interval, or compare the duration
of intervals presented successively (Lorraine (1979); Grondin (2010)). There are two major
results in this literature. First, prospective time evaluation is often consistent with WeberFechner’s law of human perception, implying that subjective time is not linear in true time
but rather proportional to its logarithm (Grondin, 2001). Second, individual evaluations
are qualitatively similar for short and long intervals of time (Lewis and Miall, 2009),
suggesting the existence of a single ‘internal clock’ mechanism that governs prospective
timing. Our study draws on the concept and methods developed in this literature and
also focuses on prospective timing. However, it departs in several respects. First, we focus
on significantly longer time intervals than the majority of the literature (several minutes).
Second, we introduce a new and incentivized elicitation method together with a distractor
task that prevents subjects from counting. Third, we provide a structural estimation
of a two-parameter power time perception function instead of imposing a logarithmic
functional form. Finally, we estimate the perception function at the individual not the
aggregate level. This allows us to study heterogeneity in perception across subjects.4
(1992). Laibson (1998) discusses its relevance to life cycle consumption. Kirby et al. (1999) report that
time discounting is hyperbolic while Benhabib et al. (2010) find support for a present bias but their data
is best fitted with yet an alternative specification.
3
See also Augenblick et al. (2013) for a comparison between time-dated monetary rewards and real
effort. The authors find evidence of a present bias only in the case of effort.
4
Some studies investigate instead retrospective time evaluation, where subjects are not informed beforehand that they will have to make a time related judgment (Block and Zakay, 1997). By definition,
under this approach only one measure can be obtained per individual. The studies find that retrospective
time evaluations are usually shorter than prospective time evaluations (Fraisse, 1984) and they draw on
different (memory) processes. We performed a one-shot retrospective time evaluation task and also found
3
Finally, there is a small literature relating time perception to time discounting. Ray
and Bossaerts (2011) propose a novel theory where they show that if individuals discount
the future exponentially with respect to biological time but the internal representation
of time is stochastic and autocorrelated, then choices will be present-biased with respect
to calendar time. Cui (2011) demonstrates that the scalar property of time perception
(the idea that the incremental amount of change required for a stimulus to be noticeable
is proportional to the current stimulus magnitude) also implies hyperbolic discounting.
Zauberman et al. (2009) and Han and Takahashi (2012) study time discounting and time
perception in the laboratory. They elicit time discount rates through hypothetical questionnaires and time perception estimates by asking subjects to place a mark on a line that
represents their perception of different horizons. Their aggregate analysis reveals hyperbolic discounting with respect to objective time but exponential discounting with respect
to subjective time. Besides the methodological differences, our study departs from these
experimental papers in that our goal is not to “explain” hyperbolic discounting at the
aggregate level based on subjective time perception (in fact, there is little evidence in our
data in favor of hyperbolic discounting). Instead, we are interested in the fundamental
relationship between time discounting and time perception at the individual level, and the
idea that subjects who perceive objective time as subjectively longer should be less prone
to delay consumption.
2
2.1
The experiment
Design and procedures
The experiment was conducted in the Los Angeles Behavioral Economics Laboratory (LABEL) at the University of Southern California.5 A total of 124 subjects participated in
the study in 14 groups of 6 to 10 participants each. In order to participate in the experiment, subjects were required to be enrolled USC students with a USC Discretionary Card
Account. Students frequently use their USC Card to pay in businesses on campus and
the surrounding area. By special arrangement with the USC Card Department, we were
able to deposit money into their accounts at our convenience. Sessions lasted for about
1h30min and started either at 10am or 12pm. They consisted in two main tasks, always
administered in the same order (time discounting task followed by time perception task).
Instructions were read out loud at the beginning of each task.
Time perception task. Participants were asked to produce 9 time intervals τ of 24,
more underestimation than under prospective time evaluation (see Appendix A2).
5
For information about the lab, please visit http://dornsife.usc.edu/label.
4
31, 41, 53, 69, 89, 116, 151 and 196 seconds respectively, without knowing in advance the
number of intervals to produce.6 We designed a Matlab-based program to implement the
elicitation of the participants’ time perception. The program presented the instructions
on the screen and guided subjects to estimate time intervals. Subjects were prompted the
length of the interval τ to be estimated. Then, subjects marked the beginning and end of
the interval by clicking on a button on the top right corner of the screen. The order for
the 9 intervals was randomly selected but it was the same for all subjects.7
To make sure that subjects did not count time, we asked them to solve filler (distracting) tasks while estimating time intervals. In these tasks, subjects were presented a
4 × 6 table where each row and column had a name and they were instructed to click on a
specific cell. In the example of Figure 1, subjects were asked “Please click the cell where
the column to the right of the column labeled athena intersects the row above the biology
row”.
Figure 1: Example of a filler task
The names of the rows and columns as well as the phrasing of which cell to click on
changed from task to task to make sure subjects would pay attention. There was a random
and unspecified time limit to complete each task (between 10 and 15 seconds) and failure
to complete it counted as an incorrect answer.8
6
This is called a prospective time estimate in a production paradigm. Prospective (as opposed to
retrospective) refers to a case where subjects know in advance that they will be requested to estimate the
elapsed time. Production occurs when subject are informed about the length of the interval they must
produce (Nichelli, 1996). This is different from reproduction, where subjects experience a time interval
(without knowing its real length) and are then asked to reproduce a second interval of the same size.
7
Before coming to the laboratory, subjects were asked to put away any time-keeping devices such as
watches, music players and cell phones. An experimenter made sure that subjects placed these items in
their bag and monitored that they did not use any such device.
8
The task required sufficient effort to prevent subjects from counting but was easy enough to make sure
all subjects could complete it if they put attention. Participants were informed that if they reported the
5
The amount earned depended on the proportion of correct answers in the filler tasks
and the distance between time estimates and true time intervals. For each subject, one
time interval was randomly selected for payment. For this time interval, the subject earned
money only if at least 80% of the filler tasks were correctly answered. The subject would
then earn $25 if the estimate was within ±5% of the real length of the interval, $15 if it
was within ±10% and $5 if it was within ±20%. If less than 80% of the filler tasks were
correctly answered, the subject did not earn anything no matter how good the estimation
of the time interval was.9 The entire procedure was announced beforehand. We chose this
method because it is easier to explain and more intuitive than a quadratic scoring rule.
Time discounting task. Since the goal of the paper was not to provide an innovative way
to elicit time discounting, we followed closely the CTB design and allocation procedure
in [AS]. We provided subjects with a budget of experimental tokens to allocate either to
a sooner time t or a later time t + k, at different token exchange rates. The relative rate
at which tokens translated into money determined the gross interest rate, (1 + r). The
amounts allocated at dates t and t + k were denoted by ct and ct+k respectively. We
implemented a 3 × 3 design with three sooner payment dates starting from the experiment
date (t ∈ {0, 7, 21}) and three delay lengths (k ∈ {21, 42, 63}). For each of the 9 pairs of
(t, k), there were 5 different budgets and exchange rates for a total of 45 sooner-later token
allocation tasks. Dates were selected to avoid holidays, vacations and examination dates.
To avoid differential weekday effects, t and k were both multiples of 7 so that payments
were scheduled to arrive on the same day of the week.
Subjects were given 10 tokens for each of the 45 allocation tasks. Tokens assigned to
sooner and later payments had values vt and vt+k , respectively. Since vt+k /vt = (1 + r)
is the gross interest rate over k days, (1 + r)1/k is the daily interest rate. Values were
never multiples of $0.05 to avoid gravity point effects. To formally implement choices, we
provided paper booklets. Subjects had to circle their preferred token allocation among
the eleven possible combinations of tokens sooner vs. token later in each of the 45 tasks.
Appendix A1 shows the token rates, standardized daily interest rates and corresponding
annual interest rates for all 45 budget set. It also shows the presentation of the first 5
tasks of the paper booklet, corresponding to t = 0 and k = 21 (Figure 10).
To avoid in-lab vs out-lab payments at different dates, all sooner and later payments,
including those for t = 0, were deposited into the subjects’ USC Card Accounts by 4pm on
end of an interval during a task, that task would not count as correct or incorrect. Subjects also estimated
a 10th time interval of 219 seconds that was not used for analysis. The objective was to make sure that
the 9 relevant intervals were estimated while all subjects could hear all other subjects clicking cells and
therefore could not use the absence of clicking as a cue that the others had finished their time estimations.
9
In 84% of the trials, subjects answered correctly at least 80% of the filler tasks and therefore were
eligible for payment.
6
the specified date.10 Subjects were described the payment method and the arrangement
made with the USC Card Department. They were told that they would receive a $4.64
thank-you payment for participating in two payments, $2.32 at the sooner and $2.32 at
the later date regardless of their choices, and that all experimental earnings were added to
these two payments. Subjects knew in advance that one of the 45 choices was selected for
payment by drawing a numbered ball from a bingo cage. They were given Professor Juan
Carrillo’s business card and they were told to contact him if payments did not reflect in
their account, in which case a payment would be hand-delivered immediately. Subjects
were asked if they trusted the payment method at the end of the experiment and 95% of
respondents replied yes.11
Other tasks. We conducted three peripheral tasks: a one-shot retrospective time estimate
task, a cognitive ability test, and a survey to collect demographic information. Details of
the procedures and results obtained in these tasks can be found in Appendix A2.
2.2
Limitations
An experimental study of time perception vs. time discounting is subject to two limitations.
First, the temporal horizons are different. We can realistically elicit multiple prospective
time estimates that are on the range of minutes whereas meaningful monetary tradeoffs
must involve temporal delays that are on the range of weeks. We will therefore extrapolate
our estimates upwards for time perception and downwards for time discounting to make
them comparable.
Second and related, we are interested in comparing perception and discounting across
individuals. If time perception functions are not linear in true time and/or time discounting functions are not exponential, rankings may depend on the horizon (for example, a
hyperbolic discounting subject may be more impatient in the short run and less impatient
in the long run than another hyperbolic discounting subject). In the analysis, we will
put special emphasis in determining the time range for which the estimated rankings are
preserved.
10
This removes the salience of immediate payment. It is likely to result in the later option being chosen
more frequently but it also makes the uncertainty and potential anxiety over payment similar whether it
is today or in the future (for a discussion, see Andreoni and Sprenger (2012a, 2012b)).
11
The full list of differences relative to [AS] are (first item refers to our design, second item to theirs): (i)
payment to USC card vs. payment by check; (ii) thank you payments of $2.32 vs. $5; (iii) slight differences
in (r, t, k, m) but calibrated to equalize daily gross interest rates (see Figure 10); (iv) 11 vs. 101 choices
per budget; (v) pen and paper vs. computerized implementation; and (vi) 124 vs. 97 subjects.
7
Time perception (T P)
3
3.1
Modeling time perception
In this section we present the theoretical framework and experimental results of our time
perception task. Time perception refers to the fact that an objective length of time may
be inaccurately perceived, leading to under- or over-estimation of true delays. We consider
a simple model of time perception in which subject i formulates a subjective duration θi
of a true time interval of length τ according to the function:
θi (τ ) = ai τ bi
(1)
where ai is the reaction to time and bi the sensitivity to time (notice that ai = bi = 1
corresponds to a correct time perception). This representation is borrowed from Steven’s
law, which proposes a relationship between the magnitude of a physical stimulus (brightness of an image, level of a sound, sugar component of a meal, etc.) and its perceived
strength (Stevens, 1957). It has been applied to a variety of such problems, including time
perception (Stevens (1975); Luce (2002)).
We fitted this model to the data obtained from the time perception task, hereafter
referred to as T P. For each individual i, we estimated by non linear least squares (NLS)
the parameters ai and bi of the following regression:
yis = ai τsbi + is
(2)
where, for trial s ∈ {1, ..., 9}, the reported perception of individual i is yis , the true
length in seconds is τs ∈ {24, 31, 41, 53, 69, 89, 116, 151, 196}, and the noise in the process
is is ∼ N (0, σi2 ). The software did not record the choices of 4 subjects and 4 others were
outliers. Figure 2 graphically depicts the estimated parameters (âi , b̂i ) of the remaining 116
subjects.12 For illustrative purposes, Figure 3 presents the choices of three representative
subjects (with b̂i > 1, b̂i ' 1 and b̂i < 1, respectively).
We obtained three main findings. First, the power model explains remarkably well the
data: the average R2 is 0.97 and 105 out of 116 individuals have an R2 greater than 0.95.13
In other words, subjects typically reported estimates that were very close to the best time
perception power fit. Second, there is substantial heterogeneity across individuals in our
sample. Indeed, 77 and 39 subjects had an estimated parameter âi greater and smaller
12
Four subjects clearly double clicked in one trial, so we estimated their time perception parameters
based on one fewer observation.
13 2
R is expected to be high since we fit two parameters with 9 observations. Still, since we impose
θi (0) = 0, our regression has 7 degrees of freedom. The subjects in Figure 3 are representative of the fit.
We also tried a linear model but the fit dropped substantially.
8
6
a
4
2
0
0.50
0.75
1.00
1.25
b
Figure 2: Estimations of individual time perception parameters (âi , b̂i ).
than 1, respectively. Similarly, 81 and 35 subjects had an estimated parameter b̂i smaller
and greater than 1, respectively. Third and perhaps most significantly, time perception
parameters ai and bi are not independent across individuals. More precisely, we found a
strong hyperbolic relation between the two parameters. We ran the regression:
1
âi = q + ηi
b̂i
where ηi ∼ N (0, σ 2 ) is an error term, and obtained an estimate q̂ = 1.52 (p-value =
0.000). This means that both parameters cannot be studied in isolation: an individual
with a concave perception of time (b̂i < 1) is extremely likely to exhibit a steep reaction
to time (âi > 1) and vice versa.
3.2
Ordering of subjects’ time perceptions
Remember that our main interest in this section is to compare the degree of under- or overestimation of perceived time across individuals. The non-linearity of the time perception
function is potentially problematic for this comparative analysis (see section 2.2): a subject
with b̂i < 1 may over-estimate short intervals and under-estimate long intervals whereas
another subject with b̂j > 1 may exhibit the opposite bias. However, the link between
the variables âi and b̂i is extremely helpful. Indeed, consider the following one-parameter
time perception function that exploits the hyperbolic relationship âi = q/b̂i :
q
θ̃(τ ; b̂i ) = τ b̂i
b̂i
It is immediate that ∂ θ̃/∂ b̂i > 0 if τ > exp1/b̂i . Since min{b̂i } = 0.5 in our sample, it
means that if the hyperbolic relationship were exact, then rankings would be stable for
9
!
250
200
200
200
150
100
50
0
Perceived time
250
250
Perceived time
Perceived time
Id 105: a=3.91, b=0.65
Id 67: a=1.03, b=1.00
Id 52: a=0.38, b=1.23
150
100
50
0
50
100
150
Real time
200
0
150
100
50
0
50
100
Real time
150
200
0
0
50
100
150
200
Real time
Figure 3: Three examples of choices in time perception task
all τ > exp2 ' 7.4 seconds. Stated differently, if b̂i > b̂j then θ̃(τ ; b̂i ) > θ̃(τ ; b̂j ) for all
τ > 7.4s. This monotonicity property is extremely useful as it may allow us to rank the
subjects’ degree of under- or over-estimation at all the relevant time intervals on the basis
of just one parameter.
In our data, the relationship between âi and b̂i is not exact so a perception ranking
independent of the time interval is not possible. We can, however, determine whether
the hyperbolic relationship is accurate enough that the ranking of the time perception
functions estimated for intervals between 24s and 196s is preserved for longer time intervals,
such as 588s (three times the highest measured interval), 1 hour, 1 day, 7 days and 14
days.
To address this question, we performed the following exercise. For each subject i and
given their estimated parameters (âi , b̂i ), we evaluated his perception of an interval of
length τx (that is, θ̂i (τx ) = âi τxb̂i ) and then ordered the subjects from maxi {θ̂i (τx )} to
mini {θ̂i (τx )}. We repeated the same exercise for an interval of length τx0 . Finally, we
determined how much the time perception ranking of subjects changed between τx and
τx0 . We found that the ranking of 28% of our subjects changed by more than 5 positions
between 588s and 1 hour. This percentage dropped to 20% between 1 hour and 1 day, to
1% between 1 day and 7 days and to 0% thereafter. Overall, subjects in our sample can
be ranked very steadily regarding their perception of time for intervals above 1 hour.
10
!
4
4.1
Time discounting (T D)
Modeling time discounting
In this section we present the theoretical framework and experimental results of our time
discounting task. We refer to [AS] for more details of the theory and estimation. Subject i
chooses at time 0 to allocate a budget m between consumption at t, ci,t , and consumption
at t + k, ci,t+k , continuously along a convex budget set. Denoting (1 + r) the gross interest
rate, the budget constraint can be written as:
(1 + r)ci,t + ci,t+k = m
(3)
We assume a time separable time discounting function Φi (t) of time t from the perspective
of time 0, and a CRRA utility of money:
U0 (ci,t , ci,t+k ) = Φi (t)
1
1
(ci,t )αi + Φi (t + k) (ci,t+k )αi
αi
αi
(4)
where αi > 0 is the curvature parameter. To estimate the inter-temporal utility function of
each individual, we arbitrarily restrict attention to quasi-hyperbolic discounting functions,
that is, functions of the form:
βi δit t > 0
Φi (t) =
1
t=0
where δi ∈ (0, 1) is the one period discount and βi > 0 the time inconsistency parameter.
Note that βi = 1 corresponds to the standard exponential discounting model. A subject
is time inconsistent when βi 6= 1, exhibiting a present-bias when βi < 1 and a future-bias
when βi > 1. The subject chooses ci,t and ci,t+k by maximizing (4) subject to (3). The
optimal amounts are:
m
m
and
c∗i,t =
(5)
c∗i,0 =
1
1
1−αi
1−αi
k
k
(1 + r) + (1 + r)βi δi
(1 + r) + (1 + r)δi
We fitted the model to the data obtained from the time discounting task, hereafter
referred to as T D. For each individual, we estimated by NLS and MLE the parameters
αi , δi and βi of the following regressions:
m
m
ci,0 =
1 +εi,0 and ci,t =
1 +εi,t (6)
1−αi
1−αi
k
k
(1 + r) + (1 + r)βi δi
(1 + r) + (1 + r)δi
where εi,0 ∼ N (0, σi2 ) and εi,t ∼ N (0, σi2 ). Notice that variations in delay lengths k and
interest rates (1 + r) allow for the identification of αi and δi . Variations in starting times
t allow for the identification of βi .
11
From the 124 subjects who participated in the study, we removed 21 subjects who
had very little to no variance in behavior.14 The algorithm did not converge for 5 more
subjects and 16 other were outliers, with estimates that did not make sense (e.g., α̂i < 0
or δ̂i >> 1).15 Figure 4 presents the distributions of the (β̂i , δ̂i , α̂i ) estimated parameters
for the 82 remaining subjects using MLE.
12
24
10
10
20
8
6
6
4
4
2
2
0
0
0.8
1.0
1.2
beta
!
count
16
count
count
8
1.4
12
8
4
0
0.980
0.985
0.990
delta
0.995
1.000
0.25
0.50
0.75
alpha
Figure 4: Distribution of parameters in the time discounting task (β̂i , δ̂i , α̂i )
We obtained reasonable estimates. The estimates are also similar (and generally consistent) with those in [AS] (see their Figure 3 in p. 3351).16 In particular, the vast majority
of our β̂i estimates are close to 1, implying no evidence of present-biased behavior. If
anything, just like in [AS] we observe a future bias, although this is likely due to small
measurement errors. As expected, the overwhelming majority of the δ̂i estimates are
between 0.99 and 1.0 and most of the α̂i estimates are above 0.85 (but still below 1).
Notice also that the one-period discounting and time inconsistency parameters are
correlated in our T D data. Figure 5 reports the (β̂i , δ̂i ) estimated pairs for each individual
and the best linear fit between the two.
A simple linear regression suggests a statistically significant negative relationship between the two parameters (δ̂i = 1.017 − 0.02β̂i , p-value = 0.000). We conjecture that this
relationship must be present in most datasets on time discounting estimates. It may not
surprise some readers, although we have not seen reported in any paper. On the other
hand and as expected, there is no statistically significant relationship between δ̂i and α̂i or
14
More precisely, those who chose the later option more than 95% of the time. We could have estimated
the parameters of those individuals but we would have obtained only range estimates and they would have
not been very reliable, so we preferred to omit them.
15
Having non-reliable estimates for some subjects is not unusual in this type of exercise. For example,
among the 97 subjects in [AS], 2 did not converge, 2 made automatic choices, 22 chose the later option
more than 95% of the time and 7 were outliers according to our definition.
16
As expected, MLE and NLS gave extremely close estimates. The fit of the model was good. The
average R2 in our NLS estimation was 0.80.
12
1.00
1.000
delta
0.995
0.990
0.985
0.980
0.9
1.0
1.1
1.2
1.3
beta
Figure 5: Estimations of individual time discounting parameters (β̂i , δ̂i )
between β̂i and α̂i . As developed in the next section, this empirical relationship is useful
when we attempt to rank individuals by their degree of impatience.
4.2
Ordering of subjects’ time discounting
As it is well-known, as soon as we move away from exponential discounting, the impatience
ranking of individuals may depend on the time horizon considered. However, just like the
hyperbolic relationship between the variables in the time perception task helped rank
subjects by their time perception, the negative and (roughly) linear empirical relationship
between the discounting variables also helps rank subjects by their impatience. Indeed, if
the relationship between the variables was exactly δ̂i = 1.017 − 0.02 β̂i , one could rewrite
the discount function of subject i with only one parameter:
t
Φ̃(t; β̂i ) = β̂i 1.017 − 0.02 β̂i
This would imply that ∂ Φ̃/∂ β̂i > 0 if t <
1.017
0.02β̂i
− 1. Since max{β̂i } = 1.37, the ranking
would be unambiguously determined by β̂i for all t < 36 days.
Naturally, the relationship between β̂i and δ̂i is not exact in our data, so the ranking
of impatience across individuals may not be fully preserved within that horizon. We
therefore conducted an exercise similar to that in section 3.2, where we use the estimated
(β̂i , δ̂i ) parameters to determine what is the value at date 0 for subject i of one unit of
consumption at date tx , where tx is 1 hour, 2 hours, 6 hours, 12 hours, 18 hours, 1 day, 7
days and 14 days. Again, the objective is not to find meaningful discount rates for such
short intervals but rather to evaluate if the ranking of impatience across individuals is
13
preserved in that range of time. We found that 27% of the subjects changed ranks by
more than 5 positions between 1 day and 7 days and none between any of the shorter
intervals. Overall, we are confident that subjects are ranked steadily in terms of their
time discounting for all time horizons below 1 day.
5
From time perception to time discounting
The previous sections suggest that the time perception and time discounting of each individual i are well summarized by (ai , bi ) and (βi , δi , αi ), respectively. Furthermore and to
address the challenges described in section 2.2, we have shown that upward extrapolation
of time perception to intervals as high as 1 day and downward extrapolation of time discounting to intervals as low as 1 hour are reasonable in our setting, in the sense that they
preserve the rankings across individuals. In this section, we address the main question of
the paper, namely the relationship between time discounting and time perception.
We hypothesize that objective delays between consumption dates are evaluated in a
subjective manner, and the subjective estimates are used to choose between consumption
options over perceived delays. We first investigate this conjecture at the aggregate level.
More precisely, we hypothesize that objective time intervals are perceived subjectively and
differently across individuals resulting in different evaluations of future consumption. This
in turn generates heterogeneity in fitted discount functions.
5.1
Hypotheses
Consistent with our conjecture, we hypothesize that there exists a time-weighting function
f (·). The discounting of an objective time interval for individual i, Φi (·), corresponds to
the time-weighting f (·) of the perceived length of that interval, θi (·). Assuming for the
time being that f (·) is identical for all individuals, we formally have:
Φi (t) = f (θi (zt)) ∀i
(7)
where z is the conversion rate between units of time in the discounting and perception
tasks.17 According to (7), differences in discount functions across individuals are due to
differences in their subjective perception of time. We impose the natural assumption that
rewards at more distant perceived dates are valued less (f 0 < 0) but, for now, we do not
posit any specific functional form on f (·). This alone immediately implies:
θi (zt) ≷ θj (zt) ⇔ Φi (t) ≶ Φj (t)
17
(8)
In our case, given we formulated the time perception task in seconds (τ = 1 second) and the time
discounting task in days (t = 1 day), we have z = 60 × 60 × 24 = 86, 400.
14
The relationship in (8) captures the intuitive idea that if an objective length of time zt is
subjectively perceived as a longer interval by subject i than by subject j (θi (zt) > θj (zt)),
then subject i is less willing than subject j to postpone a reward by that amount of
time (Φi (t) < Φj (t)). Stated differently, our first testable hypothesis is that subjects who
underestimate time should be patient while subjects who overestimate time should be
impatient.
Hypothesis 1 The individual time discounting estimate Φ̂i (t) = β̂i δ̂it obtained from T D is
negatively correlated with the individual time perception estimate θ̂i (zt) = âi (zt)b̂i obtained
from T P.
We have also highlighted that the close relationship between âi and b̂i and between β̂i
and δ̂i in our data implies that both the time perception and time discounting rankings
across individuals are stable (at least in the range of 1 hour to 1 day). It also implies that
focusing on one parameter (âi or b̂i for time perception and β̂i or δ̂i for time discounting)
may sometimes be sufficient to capture the main properties of the functions. Therefore,
if Hypothesis 1 is satisfied, we should also observe that the estimated time discounting
obtained from T D is negatively correlated with b̂i (and, by the negative relationship
between the variables, positively correlated with âi ). Similarly, we should also observe
that the estimated time perception obtained from T P is negatively correlated with β̂i
(and, by the negative relationship between the variables, positively correlated with δ̂i ).
Hypothesis 2 For all time intervals between 1 hour and 1 day:
(i) The time discounting estimate Φ̂i (t) = β̂i δ̂it obtained from T D is decreasing in the
parameter b̂i and increasing in the parameter âi obtained from T P;
(ii) The time perception estimate θ̂i (zt) = âi (zt)b̂i obtained from T P is decreasing in
the parameter β̂i and increasing in the parameter δ̂i obtained from T D.
We will test Hypotheses 1 and 2 in the remainder of the section. As already discussed,
time perception estimates have been obtained for relatively small time intervals (minutes)
while time discounting estimates have been obtained for relatively large time intervals
(weeks). It implies that the point estimates are not expected to predict accurately data
in the extrapolated intermediate time intervals. For instance, a small error in estimating
bi produces a larger mistake when converted in the perception of hours, making this prediction quantitatively inaccurate. We will look for qualitative properties of our estimates
rather than quantitative ones.
15
5.2
Aggregate analysis
Given the exclusion criteria considered earlier for our estimations, we kept for this analysis
only the 77 subjects for whom we obtained reliable estimates of ai , bi , δi , βi and αi . We
removed one more subject whose time perception estimate seemed reasonable but became
an outlier when extrapolated: his perception of 1 day was 9.6 days (keeping this subject
did not alter our main results). This left us with a total of 76 subjects.
To test our hypotheses, we constructed a hypothetical experiment in which subjects
would be asked to produce intervals of time T (for the perception task) and to determine
the discount applied to one unit of consumption at that same T (for the discounting task).
Since we feel confident with upward extrapolation of time perception as high as 1 day and
downward extrapolation of time discounting as low as 1 hour, we considered the following
time intervals: 1 hour, 2 hours, 6 hours, 12 hours, 18 hours and 1 day (to which we added
7 days and 14 days for the purpose of comparison).
For each time interval T and for each subject i, we predicted both time perception
and time discounting based on the estimates obtained from the T D and T P datasets.
Therefore, we explicitly assumed that the same time evaluation process governs time perception in seconds, minutes and hours and that the same time valuation process governs
discounting in weeks, days and hours.18 We computed Pearson’s correlation coefficients
(PCC) between time perception and time discounting for each interval T . Table 1 summarizes the findings. Figure 6 represents the scatterplot of predicted time perception and
predicted time discounting when T = 1 day.
At
At
At
At
At
At
1 hour
2 hours
6 hours
12 hours
18 hours
1 day
At 7 days
At 14 days
PCC
p-value
-0.25
-0.25
-0.26
-0.26
-0.26
-0.26
0.032
0.027
0.024
0.024
0.024
0.024
-0.24
-0.21
0.035
0.073
Table 1: Correlation between time perception estimate (θ̂i (zt) = âi (zt)b̂i ) and time
discounting estimate (Φ̂i (t) = β̂i δ̂it )
18
Obviously, we were careful to control for the different units of times used to obtain our estimates in
the previous sections, and we applied the relevant conversion rate.
16
1.3
1.1
0.9
Discount (1 day)
The result provides support for Hypothesis 1. Impatience is associated with the perception that time passes slowly, as predicted by the model. More precisely, subjects who
are predicted to produce a high time interval in the time perception task are also predicted
to consume early in the time discounting task.
0
1
2
3
4
5
Time perception (1 day)
Figure 6: Time perception and time discounting estimates for T = 1 day
From Figure 6 it appears that the negative relationship is convex and that there is
substantial heterogeneity across individuals in the valuation of future rewards and the
subjective evaluation of delays. Notice also that the discounting is above 1 for the majority of subjects. Taken literally, it would mean that individuals strictly prefer 1 unit of
consumption tomorrow to 1 unit today. This is obviously not sensible. The reason is that
many β̂i -estimates are above 1, so the extrapolation of the parameters to one day results
in unreasonably high patience.19 In Appendix A3, we present the same scatterplot when
T = 14 days. For that interval, estimates of time discounting become more reasonable,
although there is still a fraction of subjects with discounting above 1.20 On the other hand,
the correlation between time discounting and time perception when T = 14 days is weaker
and statistically less significant due to the previously mentioned problem of extrapolating
time perception to intervals above 1 day.
Following Hypothesis 2, our next step in the investigation of the relationship between
discounting and perception consists in a correlation analysis of the relevant parameters
of the two datasets. The first two columns of Table 2 present the correlation coefficients
between the predicted time discounting function β̂i δ̂it from dataset T D and the parameters
19
Remember, however, that the main objective of our analysis is not to estimate levels of time discounting
and time perception but to be able to perform comparisons across individuals.
20
We need to set T = 35 days (5 weeks) in order to have a discounting below 1 for more than 90% of
our subjects.
17
âi and b̂i of the time perception function. The last two columns present the correlation
coefficients between the predicted time perception function âi (zt)b̂i from dataset T P and
the parameters β̂i and δ̂i of the time discounting function. These correlations are performed
for the same time intervals T as previously.
Φ̂i (t) ≡ β̂i δ̂it
âi
At
At
At
At
At
At
1 hour
2 hours
6 hours
12 hours
18 hours
1 day
0.21
0.21
0.21
0.21
0.21
0.21
(0.064)
(0.064)
(0.065)
(0.066)
(0.066)
(0.067)
θ̂i (zt) ≡ âi (zt)b̂i
β̂i
δ̂i
b̂i
-0.25
-0.25
-0.25
-0.24
-0.24
-0.24
(0.033)
(0.033)
(0.033)
(0.033)
(0.033)
(0.034)
-0.25
-0.25
-0.26
-0.26
-0.26
-0.26
(0.032)
(0.027)
(0.024)
(0.023)
(0.023)
(0.023)
0.21
0.21
0.21
0.21
0.21
0.21
(0.075)
(0.067)
(0.064)
(0.065)
(0.067)
(0.068)
(p-values in parenthesis)
Table 2: Correlation between parameters across datasets
The results are very much in line with those in Table 1. The correlations (Φ̂i (t), b̂i ) and
(θ̂i (zt), β̂i ) are remarkably similar to the correlation (Φ̂i (t), θ̂i (zt)) studied previously. This
is not surprising since we know that the estimates b̂i and β̂i capture the main elements of
perception and discounting, respectively. The correlations (Φ̂i (t), âi ) and (θ̂i (zt), δ̂i ) go in
the right direction but they are statistically significant only at the 10% level. It suggests
that if we have to focus on only one parameter for discounting and perception, we should
prioritize the degree of time-inconsistency and the sensitivity to time perception.
It is worth emphasizing that the estimates (âi , b̂i ) on one hand and (β̂i , δ̂i ) on the other
are obtained from independent and unrelated datasets, T P and T D. There is a priori no
exogenous reason why the measures constructed from these two dataset should correlate.
And yet, there is striking evidence that the mechanisms underlying time perception and
time discounting are closely linked. The sign of the correlation coefficients obtained in
Tables 1 and 2 remain the same for larger time intervals. However, and due at least in
part to the inherently noisy nature of our data, correlations are no longer significant after a
certain interval length. Overall, the data provides support for the link between subjective
perception of time and impatience, and the existence of a time-weighting function f (·)
that transforms perceived time into discount rates.
18
5.3
Cluster analysis
The aggregate analysis shows that differences in time perception are associated with differences in impatience but it also suggests substantial heterogeneity in behavior. The
objective of this section is to investigate in more detail the differences across subjects.
To study heterogeneity, we use the time perception and discount estimates to group
individuals with the objective of finding common patterns of behavior. We focus on the
time interval T = 1 day.21 We consider a model-based clustering method to identify
the clusters present in our population. A wide array of heuristic clustering methods
are commonly used but they typically require the number of clusters and the clustering
criterion to be set ex-ante rather than endogenously optimized. Mixture models, on the
other hand, treat each cluster as a component probability distribution. Thus, the choice
between numbers of clusters and models can be made using Bayesian statistical methods
(Fraley and Raftery, 2002). We implement our model-based clustering analysis with the
Mclust package in R (Fraley and Raftery, 2006). We consider ten different models with a
maximum of nine clusters each, and determine the combination that yields the minimum
Bayesian Information Criterion (BIC).22 For our data, the diagonal model with equal
volume and equal shape that endogenously yields four clusters minimizes the BIC. Table
3 presents the average statistics of the time perception and time discounting parameters
for subjects within each cluster. Figure 7 provides the same scatterplot as Figure 6 with
subjects coded by cluster (the ellipses superimposed on the classification plot correspond
to the within-cluster covariances and the mean of each cluster is marked with a ∗ sign).
Cluster 1 is close to what we would expect of rational economic agents. Their subjective perception of time is linear and reasonably close to the true time, with a slight
underestimation. They are also very patient and time-consistent. Cluster 2 is a group
of subjects exhibiting a concave time perception function. They tend to underestimate
time and, as a consequence, they are more willing to delay consumption than subjects in
cluster 1. This is reflected by an extremely high patience and a future bias (as discussed
previously, the fact that the estimated discount is above 1 is likely due to a measurement
error of extrapolation). Clusters 3 is a small group of subjects exhibiting moderately
convex time perception functions and a significant overestimation of time. Their discount
parameter is lower than the previous subjects. Finally, cluster 4 is an extreme version of
21
We conducted the same analysis with the other time intervals reported in Tables 1 and 2 and obtained
consistent results. Only a few subjects shifted from one group to another as we shifted the time interval,
and none of the differences were significant.
22
Hierarchical agglomeration first maximizes the classification likelihood and finds the classification for up
to nine clusters for each model. This classification then initializes the Expectation-Maximization algorithm
which does maximum likelihood estimation for all combinations of models and number of clusters. Finally,
the BIC is calculated for all combinations with the Expectation-Maximization generated parameters.
19
Perception (1 day)
a
b
Discounting (1 day)
β
δ
α
Cluster 1
Cluster 2
Cluster 3
Cluster 4
0.68
1.49
0.92
1.03
1.03
0.996
0.93
0.47
2.04
0.85
1.26
1.28
0.992
0.90
2.06
0.70
1.10
1.00
1.00
0.997
0.95
4.84
0.42
1.22
0.94
0.95
0.998
0.94
# subjects
(0.05)
(0.12)
(0.01)
(0.01)
(0.01)
(0.00)
(0.01)
56
(0.10)
(0.46)
(0.05)
(0.02)
(0.02)
(0.00)
(0.02)
10
(0.15)
(0.11)
(0.02)
(0.02)
(0.02)
(0.00)
(0.01)
7
(0.33)
(0.07)
(0.01)
(0.03)
(0.03)
(0.00)
(0.00)
3
Standard errors in parenthesis
Table 3: Summary statistics by cluster for T = 1 day
cluster 3, with strongly convex time perception, extreme overestimation of time and the
lowest discount. They are the only subjects to exhibit a present bias.
1.1
1.3
Cluster
Cluster
Cluster
111
Cluster
Cluster
Cluster
222
Cluster
Cluster
Cluster
333
Cluster
Cluster
Cluster
444
0.9
Discount (1 day)
Classification
0
1
2
3
4
5
Time perception (1 day)
Figure 7: Time perception and time discounting by cluster
To investigate the significance of differences across clusters, we ran a series of t-tests.
We found that the 1-day time discounting is significantly different between the most patient
cluster 2 and any of the other clusters but not between clusters 1, 3 and 4. By contrast, the
1-day time perception is significantly different between all pairs of clusters except between
1 and 2.
20
5.4
Discounting of perceived time
In section 5.2 we have emphasized a relationship between time perception and time discounting. In section 5.3, we have highlighted the substantial heterogeneity in choices
across individuals. In this section, we study how much of this heterogeneity is driven by
the differences in the subjective perception of time.
To study this question, we propose the following exercise. Let xi = θi (t) denote
individual i’s subjective time perception of t objective units of time, where θi is increasing
for all i. Inverting the function, we can determine ti = θi−1 (x). This function represents
individual i’s real time interval that corresponds to x perceived units of time. So, for
example, if i overestimates time (θi (t) > t) and j underestimates time (θj (t) < t), then it
requires a shorter objective time interval to subject i than to subject j in order to fill the
same perceived amount of time x (θi−1 (x) < θj−1 (x)). Next, we can compute how much
individual i discounts x units of perceived time:
θ−1 (x)
Φi (x) ≡ βi δi i
.
0.8
0.4
0.0
Discount
1.2
Finally, we can use the discount estimates (β̂i , δ̂i ) determined in section 4.1 for the 76
individuals in our sample to compute the discounting of each individual i for x units of
perceived time. Figure 8 presents a boxplot of these estimates for different perceived
intervals of time x (1 hour, 1 day, 7 days, 14 days and 21 days).
1 hour
1 day
7 days
14 days 21 days
Perceived times
Figure 8: Discounting of perceived time intervals
The line in the middle of the box is the median. The top and bottom lines are the
1st and 3rd quartile (Q1 and Q3) of the distribution, the notches are the 95% confidence
21
interval for the median and the whiskers’ edges indicate the smallest and highest nonoutlier observations. Outliers are defined as observations below Q1 − 1.5 [Q3 − Q1] or
above Q3 + 1.5 [Q3 − Q1] and are plotted separately with a hollow circle.
According to Figure 8, the median of discounting per unit of perceived time is in a
reasonable range (between 1.0 and 0.8). Discount estimates are, with some exceptions,
concentrated for intervals up to 7 perceived days, as shown by the relatively small height
of the boxes. This means that heterogeneity in discounting is attenuated when we use
perceived rather than true time as the temporal unit of measure, at least for short intervals.
Table 4 presents summary statistics of the average discounting (including standard
error, standard deviation and coefficient of variation) for different units of perceived time
x using the full sample of 76 individuals. It also contains the same information after
excluding the 16 outliers identified in Figure 8.
Full sample [76]
Perceived
interval (x)
1 hour
1 day
7 days
14 days
21 days
Avg. discount
1.055
1.018
0.905
0.833
0.778
(0.01)
(0.02)
(0.03)
(0.03)
(0.04)
Excluding outliers [60]
st. dev.
C.V.
0.10
0.16
0.26
0.29
0.31
9.9
15.6
29.0
35.1
39.4
Avg. discount
1.021
1.015
0.973
0.924
0.879
(0.01)
(0.01)
(0.01)
(0.01)
(0.02)
st. dev.
C.V.
0.06
0.06
0.07
0.10
0.14
6.1
6.0
7.1
11.1
16.0
Standard errors in parenthesis, number of individuals in brackets
Table 4: Average discount as a function of perceived time intervals.
The results in Table 4 confirm those in Figure 8. As before, average discounting is
above 1 for very short perceived intervals (1 hour, 1 day) due to the β̂i -estimates but it
becomes less than 1 for intervals above 7 days and, as one would expect, it monotonically decreases as the interval increases. Discounting is concentrated for short perceived
intervals. It is more dispersed for long perceived intervals but this is due to a few number of outliers. Once these individuals are removed, standard deviations (and therefore
coefficients of variation) decrease substantially. Overall, this section reaffirms the conclusion that considering perceived units of time rather than objective units of time helps
understanding the decision of individuals to postpone rewards.
22
6
Individual time-weighting function
From the T P dataset, we showed that individual i’s time perception is well summarized
by θi (zt) = ai (zt)bi . From the T D dataset, we found that individual i’s discount function
can be approximated by Φi (t) = βi δit . Our aggregate analysis revealed that the discount
function can be interpreted as the time weighting of perceived time. Even though a given
perceived time interval is reached for different true time intervals for different individuals,
its weighting is similar across individuals. Overall, we have shown that the data can
be summarized reasonably well by a common weighting function f (·), that transforms
perceived times into discount rates Φi (t) = f (θi (zt)). Still, we noted some individual
differences. The purpose of this section is to investigate this heterogeneity in more detail.
To do so, we posit that the weighting function that transforms perceived time into
discount has the same structure for every participant but it is parametrized individually.
More precisely, we assume:
a (zt)bi
Φi (θi (zt)) = β̃i d˜i i
This quasi-hyperbolic formulation is the same as the one we used to estimate discounts,
except that now participants are assumed to discount payoffs with respect to their perception of time rather than the true time. It also uses one second as the unit interval of
b
time, so d˜i can be interpreted as the discount per second. Notice that if we set δ̃i ≡ d˜iz i ,
then we can rewrite the previous function using one day as the (standard) unit of time:
ai (t)bi
Φi (θi (t)) = β̃i δ̃i
Our objective is to revisit the T D data and to propose a new discounting model driven
by perceived time rather than true time. Following the very same optimization procedure
as in section 4.1, the optimal consumption of individual i at date t is:
c∗∗
i,0 =
m
ai (k)bi
(1 + r) + (1 + r)β̃i δ̃i
1
1−α̃i
and c∗∗
i,t =
m
ai (t+k)bi −ai (t)bi
(1 + r) + (1 + r)δ̃i
1
1−α̃i
We imported the time perception parameters âi and b̂i estimated from the dataset T P and
we estimated by MLE the remaining three parameters (β̃i , δ̃i , α̃i ) in dataset T D exactly as
we did in section 4.1. We excluded one outlier with an estimated δ̃i = 1.809 (the results
are virtually unchanged if we keep that subject). Figure 9 presents the distributions of
the (β̃i , δ̃i , α̃i ) estimated parameters for the 75 remaining subjects.
A comparison between Figure 4 and Figure 9 suggests that the distribution of estimated
parameters are remarkably similar when we consider perceived time rather than true time.
We find that β̂i and β̃i are positively correlated (PCC = 0.54, p-value = 0.000) and so are
23
16
14
14
12
18
16
10
8
6
4
14
count
10
count
count
12
8
12
10
6
8
4
6
4
2
2
0
0
0.75
20
1.00
1.25
1.50
1.75
2.00
2
0
0.980
0.985
beta
0.990
0.995
1.000
0.25
delta
0.50
0.75
alpha
Figure 9: Distribution of parameters (β̃i , δ̃i , α̃i )
!
α̂i and α̃i (PCC = 0.82, p-value = 0.000). Said differently, participants have very similar
time inconsistency and curvature estimates in both models.
Interestingly, even though the distributions of δ̂i and δ̃i are very similar, the parameters
are not significantly correlated (PCC = 0.22, p-value = 0.0919). This is not surprising
because δ̃i is now applied to perceived time and individuals are highly heterogeneous
in their time perception. The new parameter is therefore adjusting for the individual
b
perception biases. Formally, the counterpart of δ̂it is now δ̃iat so, unlike for parameters β
and α, correlations of δ across models cannot be studied independently of t (for that very
same reason, δ̃i cannot be interpreted as the daily discount factor). Overall, the model
introduced here is a reinterpretation of the standard quasi-hyperbolic discounting model
in terms of perceived times and, according to AIC, both models perform equally well.
Last, we present in Table 5 the average statistics of the time discounting parameters
using the clusters obtained in section 5.3.
Perceived
Discounting (1 true day)
β̃
δ̃
α̃
# subjects
Cluster 1
Cluster 2
Cluster 3
Cluster 4
1.05
1.06
0.995
0.93
1.33
1.36
0.990
0.89
0.99
0.99
0.997
0.95
0.93
0.93
0.998
0.93
(0.02)
(0.02)
(0.01)
(0.02)
56
(0.04)
(0.05)
(0.00)
(0.02)
10
(0.02)
(0.02)
(0.00)
(0.02)
7
(0.03)
(0.03)
(0.00)
(0.01)
3
Standard errors in parenthesis
Table 5: Summary statistics by cluster for T = 1 day under perceived time
Again, a comparison between Table 3 and Table 5 reveals that the results obtained
earlier are qualitatively unchanged. Consistent with our previous discussion, the discount
24
1.00
parameter δ̃ is adjusting for the individual perception biases. When t = 1, the average
discount adjusted by the average perceived time in each cluster is very close to the average
discount from Table 3 (formally, δ̂ ' δ̃ a ), meaning that the differences across clusters are
preserved when we consider perceived rather than true time.
7
Concluding remarks
This paper provides experimental evidence on the relationship between time perception
and time discounting. Our data reveals a negative correlation between the two: subjects
for whom one unit of time feels longer than true time also discount rewards at a higher
rate than subjects for whom it feels shorter than true time. This result suggests that our
ability to delay consumption is related to our mental representation of time delays between
now and the future. Our result also indicates the existence of an underlying internal clock
that governs time representations irrespective of the unit of time.23
Prospective timing has been associated with working memory, a function performed
by the dorsolateral prefrontal cortex (dlPFC) (Grondin (2010); Lewis and Miall (2006)).
Interestingly, recent evidence in neuroscience supports the idea that this region is also
implicated in time discounting (Hare et al., 2014). This provides a rationale for why time
perception and time discounting should be related, as indicated by our data.
Our results are also in line with findings obtained in the time discounting and time
perception literature over the life cycle. It has been shown that the subjective perception
of the passing time tends to speed up with age. People increasingly underestimate time as
they age. Ordinary days appear longer for children and shorter for older adults (Block et
al. (1999); Coelho et al. (2004)). In parallel, children succumb to immediate gratification
while older adults are typically willing to wait for rewards (Green et al. (1999); Lockenhoff
et al. (2011)). In other words and consistent with the results presented here, children are
impatient and overestimate time whereas older adults are patient and underestimate time.
Interestingly, the dlPFC, which we conjecture is at the core of time related judgments,
is late to develop in children (Casey et al., 2005) and early to age (Raz et al., 2005).
These points taken together suggest that the relationship between time perception and
time discounting and the changes over the life cycle are no coincidence.
23
This conclusion is strengthened by the fact that, according to our retrospective task, there is also
a relationship between retrospective and prospective time evaluation (see Appendix A2). Overall, we
conjecture that a single mechanism is involved in all time related evaluations.
25
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29
Appendix
Appendix A1. Time discounting tasks
Sheet2
Set
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
!
t
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
k
21
21
21
21
21
42
42
42
42
42
63
63
63
63
63
v(t)
$1.484
$1.438
$1.339
$1.236
$1.532
$1.484
$1.438
$1.339
$1.236
$1.532
$1.482
$1.326
$1.159
$0.981
$1.517
v(t+k)
$1.532
$1.532
$1.532
$1.532
$1.752
$1.532
$1.532
$1.532
$1.532
$1.752
$1.532
$1.532
$1.532
$1.532
$1.752
(1+r) Daily (1+r)%
1.032
0.152
1.065
0.302
1.144
0.643
1.239
1.028
1.144
0.641
1.032
0.076
1.065
0.151
1.144
0.321
1.239
0.512
1.144
0.320
1.034
0.053
1.155
0.229
1.322
0.444
1.562
0.710
1.155
0.229
Set
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
t
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
k
21
21
21
21
21
42
42
42
42
42
63
63
63
63
63
v(t)
$1.484
$1.438
$1.339
$1.236
$1.532
$1.532
$1.484
$1.438
$1.339
$1.236
$1.482
$1.326
$1.159
$0.981
$1.517
v(t+k)
$1.532
$1.532
$1.532
$1.532
$1.752
$1.532
$1.532
$1.532
$1.532
$1.532
$1.532
$1.532
$1.532
$1.532
$1.752
(1+r) Daily (1+r)%
1.032
0.152
1.065
0.302
1.144
0.643
1.239
1.028
1.144
0.641
1.000
0.000
1.032
0.076
1.065
0.151
1.144
0.321
1.239
0.512
1.034
0.053
1.155
0.229
1.322
0.444
1.562
0.710
1.155
0.229
Set
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
t
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
k
21
21
21
21
21
42
42
42
42
42
63
63
63
63
63
v(t)
$1.484
$1.438
$1.339
$1.236
$1.532
$1.484
$1.438
$1.339
$1.236
$1.532
$1.482
$1.326
$1.159
$0.981
$1.517
v(t+k)
$1.532
$1.532
$1.532
$1.532
$1.752
$1.532
$1.532
$1.532
$1.532
$1.752
$1.532
$1.532
$1.532
$1.532
$1.752
(1+r) Daily (1+r)%
1.032
0.152
1.065
0.302
1.144
0.643
1.239
1.028
1.144
0.641
1.032
0.076
1.065
0.151
1.144
0.321
1.239
0.512
1.144
0.320
1.034
0.053
1.155
0.229
1.322
0.444
1.562
0.710
1.155
0.229
!
Budget!sets!in!the!time!discounting!task!
!
Page 1
!
!
First!5!time!discounting!tasks:!(t,!k)!=!(0,!21)!
Figure 10: Parameters and presentation of the time discounting task
30
Appendix A2. Other tasks in the experiment
1. Description of tasks.
The timing was the same in all sessions. We started with the sound of the first bell.
We then proceeded to the time discounting task. At the end of the time discounting task,
we rang the second bell and asked subjects to estimate the time that passed between
the two bells (the one-shot retrospective time estimate task). We the moved to the time
perception task. We ended the experiment with the cognitive ability test and the survey.
Details of the tasks are provided below.
a. Retrospective time estimate task. This task began with the sound of a bell. Subjects were told “the bell you just heard marked the beginning of the experiment. From
now on we would like to have your undistracted attention.” A second bell sound rang later
during the experiment. Subjects were then told to estimate the interval of time between
the two bells and that they would earn $5.00 paid in cash at the end of the experiment if
their estimate was within the real length of the interval ±10%.
b. Cognitive ability test. At the end of the experiment we conducted an incentivized
cognitive ability test. We used a simplified version of Raven’s IQ test, namely Set I
of the Raven’s Advanced Progressive Matrices (APM) with a five minute time control,
as developed by Raven et al. (1998). This set consists of 12 non-verbal multiple choice
questions that become progressively more difficult. Each test item consists of a pattern
with a missing element. From the eight choices below the pattern, the subject is to identify
the piece that will complete the pattern. Instructions for the test were read directly from
the script provided. Subjects were made familiar with the format of the test and method
of thought required through two practice problems preceding the test. During this time,
they were allowed to ask questions from the experimenters. We incentivized subjects by
paying $5 in cash to the top 2 scorers in the test.
c. Demographic survey. We also administered a survey to collect demographic information such as gender, GPA and primary language spoken.
2. Results of the retrospective time evaluation task
The retrospective time estimate task is interesting in that it gives a measure of time
perception for intervals longer than a few minutes. However, the data is extremely volatile
as it contains a single observation. In our experiment, the time interval between the two
bells, µ, varied between 23min 35s and 41min 36s depending on the sessions.
There is a fundamental difference between the prospective time production task (the
one studied in the main paper) and the retrospective time evaluation task. In the prospective task, participants are informed that they have to make a time related judgment and
31
they base such judgment on their experienced duration. Prospective timing problems have
been demonstrated to involve attention (Block and Zakay (1997); Macar (1996); Brown
(2008)). By contrast, in the retrospective task, participants receive no prior warning that
they will have to make a time related judgment and their report relies on remembered
duration, a piece of information retrieved from memory (Block, 1990). Retrospective timing problems are therefore associated with memory processes and they do not involve
attention (Zakay and Fallach (1984); Block (2003)).
Research on time perception has mostly focused on prospective timing and in particular
on the properties of the ‘internal clock’ –a central mechanism responsible for estimating
time– as well as the relationship between time perception and attention (see Grondin
(2010) for a review). Even though retrospective and prospective timing are likely to
involve different processes, they should also share some. Indeed, estimating a length of
time retrospectively or keeping track of a starting time to produce a length is likely to
involve similar abilities to “travel in time.” We investigate this possibility by looking at
the relationship between the results obtained in both tasks.
For each subject i, we computed the percentage difference between the reported interval
between the two bells, ri (µ), and the real interval µ in the corresponding session. This
gives a measure of the perception bias in the retrospective task: P B = ri (µ)−µ
. Figure 11
µ
presents a histogram with the perception bias of all subjects, after removing four outliers
whose bias was below -0.70.
count
9
6
3
0
−0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Perception bias
Figure 11: Distribution of “perception bias”
We notice from Figure 11 that the majority of subjects underestimate time. The
average P B is -0.196 (st. error = 0.025). Next, we computed the percentage difference
between the reported interval between the two bells, ri (µ), and the interval we predicted
the individual would report based on the estimates (âi , b̂i ) in the prospective time production task. This gives a measure of the excess bias, above and beyond our subjective
32
b̂i
i (µ)
time estimate: EB = ri (µ)−â
. Again, reports are typically lower than the predicted
µ
estimates. The average EB is -0.095 (st. error = 0.055). It is therefore lower than the
average P B but also more dispersed. The results on P B and EB are consistent with
the existing literature, which suggests that retrospective time is subjectively perceived as
shorter than both prospective time and real time (Zakay and Block (2004); El Haj et al.
(2013)).
Interestingly, we also found that the excess bias (EB) was strongly correlated with
the sensitivity parameter bi (PCC = -0.78, p-value = 0.000). Figure 12 depicts this
relationship.
Excess bias
1
0
−1
−2
0.50
0.75
1.00
1.25
b
Figure 12: Correlation between bi and “excess bias”
Subjects with an approximately linear time perception in the prospective task (b ' 1)
were reasonably well predicted in the retrospective task. By contrast, subjects with a
convex (respectively concave) evaluation of time, that is b > 1.05 (respectively b < 0.95),
had a lower (respectively higher) estimate of the time delay between the two bell sounds
compared to what we predicted. These findings suggest that time evaluation in both
tasks are related, indicating that retrospective and prospective time evaluation rely on a
common subset of processes. They also point to systematic differences between the two
and a specific pattern: subjects with a convex time evaluation perceive time as longer than
it truly is and remember past events as passing relatively faster whereas subjects with a
concave time evaluation perceive time as shorter than it truly is and remember past events
as passing relatively slower.24
24
The relationship between prospective and retrospective time evaluation is consistent with the findings
in El Haj et al. (2013). The authors found systematic differences for both time evaluations between
Alzheimer’s disease patients and healthy controls, indicating the existence of a correlation between the
processes underlying both. They also administered a mental time travel task and found that the results
obtained in that task were strongly correlated with both retrospective and prospective time evaluation.
33
3. Results of the IQ test and survey
We found almost no associations between our results and the answers to the questionnaire. In particular, we did not find any gender effect. At the aggregate level, we found
that GPA scores and performance in the IQ test were positively correlated (PCC = 0.28,
p-value = 0.01). However, none of them were found to have an effect on time perception
or discounting, and their distributions were similar across clusters.
We also investigated the effect of language. A recent study (Chen, 2013) suggests that
the way we represent time and we allocate consumption over time might be associated with
factors such as culture or language. We found only a small difference between subjects who
reported to use English (N = 49) and Chinese (N = 21) as their primary spoken language
(6 subjects reported “other” as their primary language). Chinese speakers had higher
time perception estimates than English speakers, driven mostly by a higher sensitivity
(bi ) parameter: b̄ = 1.01 vs. b̄ = 0.92, p-value = 0.025. However, they were not allocated
differently across clusters 1 and 2.25
1.1
0.9
Discount (14 days)
Appendix A3. Time discounting and time perception at 14 days
0
20
40
60
80
100
120
140
Time perception (14 days)
Figure 13: Time perception and time discounting estimates for T = 14 days
This indicates that some processes are involved in both mental time travel and time evaluation.
25
Notice that our subjects are either domestic students or international students living in the US, so
cultural differences are likely to be less pronounced than if we compare populations living in different
countries (as in Chen (2013) for example).
34