Psychology
2012. Vol.3, No.8, 555-557
Published Online August 2012 in SciRes (http://www.SciRP.org/journal/psych) http://dx.doi.org/10.4236/psych.2012.38082
Tempospect Theory of Intertemporal Choice
Taiki Takahashi, Ruokang Han
Department of Behavioral Science, Center for Experimental Research in Social Sciences,
Hokkaido University, Hokkaido, Japan
Received May 3rd, 2012; revised June 5th, 2012; accepted July 6th, 2012
Anomalies in intertemporal choice (e.g. hyperbolic discounting, subadditive discounting, a sign effect, a
magnitude effect, and a delay-speedup asymmetry) have been investigated in neuroeconomics and be-
havioral neuroeconomics. In this study we propose a “tempospect” theory of intertemporal choice which
can account for these anomalies in intertemporal choice. The key features of the present theory are: 1) de-
cision over time is made with psychological time; and 2) psychological time is determined by a change in
delay until receipt (i.e., positive or negative time-interval between options); 3) psychological time is less
sensitive to a decrease in delay in comparison to an increase in delay; and 4) psychological time is influ-
enced by the sign and magnitude of the delayed outcomes. Implications of the present theory for neu-
roeconomics are discussed.
Keywords: Time Discounting; Psychological Time; Psychophysics; Behavioral Economics;
Neuroeconomics
Introduction
People tend to devalue the subjective value of outcomes as
delay until its receipt increases, which is referred to as temporal
discounting (intertemporal choice). Studies in behavioral eco-
nomics (Loewenstein & Prelec, 1992; Frederick et al., 2002;
Read & Roelofsma, 2003; Scholten & Read, 2010) and neu-
roeconomics (McClure et al., 2004; Kable & Glimcher, 2007;
Takahashi, 2009) demonstrated several anomalies in intertem-
poral choice (temporal discounting). Neuropsychopharmacolo-
gical studies indicate that these anomalies are related to addic-
tion (Bickel & Marsch, 2001). Since its introduction by
Samuelson (1937), an exponential discounting model for in-
tertemporal choice has dominated in economic theory. In ex-
ponential discounting, people are assumed to be time-consistent,
because time-discount rate is constant over time. However, later
empirical evidence suggests that several anomalies exist in
human and animal intertemporal choice behavior (Thaler, 1981;
Loewenstein & Prelec, 1992; Frederick et al., 2002). The im-
portant anomalies are 1) hyperbolic discounting; 2) subadditive
discounting; 3) a sign effect; 4) a magnitude effect; and 5) a
delay-speedup asymmetry (see Loewenstein & Prelec, 1992;
Frederick et al., 2002; Scholten & Read, 2010, for a review).
We briefly explain these anomalies below.
Hyperbolic Discounting
Time preferences between two delayed outcomes often
switch when both delays are increased by a given constant
time-interval. For instance, a person might prefer one apple
today to two apples tomorrow, but at the same time, prefer two
apples in 101 days to one apple in 100 day. This behavioral
tendency gives rise to time-inconsistent behavior (Strotz, 1956).
In other words, people tend to make patient plans in the distant
future, but make impulsive (impatient) actions in the near future.
It is to be noted that in exponential discounting, this type of
time-inconsistency does not exist because the time-discount
rate is constant over time. Recently, McClure et al. (2004) and
Kable and Glimcher (2007) examined the neural correlates of
hyperbolic discounting.
Subadditi ve D iscounting
Similar to human probability judgment, temporal discounting
is subadditive. Consider someone judging the present value of
an outcome to be received in one month. He or she can sepa-
rately discount for each of the four weeks in the month, or dis-
count once for the unbroken one month. Subadditive discount-
ing (Read, 2001) means that the total discounting is greater
when the month is divided into weeks. This anomaly is a sharp
evidence against Samuelson’s discounted utility theory (Schol-
ten & Read, 2010).
Sign Effect
Empirical studies reported that loss is less steeply time-dis-
counted than gain (Thaler, 1981). In Thaler (1981)’s study,
time-discount rates for gains were three to ten times greater
than those for losses. A more recent study also found the sign
effect in temporal discounting (Estle et al., 2006).
Magnitude Effect
Behavioral economic studies revealed that larger gains are
less steeply time-discounted than smaller ones (Thaler, 1981;
Estle et al., 2006). Similar to the sign effect in intertemporal
choice, this anomaly challenges the assumptions in Samuelson’s
discounted utility model (Loewenstein & Prelec, 1992; Freder-
ick et al., 2002).
Delay-Speedup Asymmetry
Loewenstein (1988) demonstrated that time-discount rates
can be dramatically affected by whether the change in delivery

T. TAKAHASHI, R. HAN
time of an outcome is framed as an acceleration (“speedup”) or
a “delay” from a certain temporal reference point. For instance,
people who didn’t expect to receive a product for another year
would pay an average of $54 to receive it immediately, but
those who thought they would receive it immediately demanded
an average of $126 to delay its receipt by a year. In other words,
time-discount rate is larger for “delay” than for “speedup” of
receiving the delayed reward. This anomaly also contradicts
with the assumptions in discounted utility theory (Loewenstein
& Prelec, 1992; Frederick et al., 2002).
Taken together, there has been accumulating evidence
against the standard exponentially discounted utility model. To
date, however, no theory is capable of accounting for these
important anomalies, although some behavioral economists
proposed descriptive models of intertemporal choice (Frederick
et al., 2010; Scholten & Read, 2010). In this study, we propose
a model which can explain all of the anomalies in intertemporal
choice by incorporating nonlinear and delay-speedup asymmet-
rical subjective time in decision over time.
Tempospect Theory for Intertemporal
Decision-Making
Temporal Cognition in Intertemporal Choice
In order to account for the anomalies in intertemporal choice,
various theoretical models have been proposed (Loewenstein &
Prelec, 1992; Takahashi, 2005; Takahashi, 2006; Takahashi,
2009; Scholten & Read, 2006; Scholten & Read, 2010). Gener-
ally speaking, Loewenstein and Prelec’s theory is a value-based
account of the anomalies in intertemporal choice, in contrast,
Scholten and Read’s group and Takahashi’s theories are time-
based accounts. In Takahashi’s theory, both hyperbolic (Taka-
hashi, 2005) and subadditive discounting is due to nonlinearity
in psychological perception (weighting) of delay or time-in-
terval between options. Specifically, time-interval in the distant
future is perceived as a shorter psychological time in compari-
son to that in the near future, which can explain why people are
patient regarding the distant future but impatient regarding the
near future (hyperbolic discounting). Zauberman et al. (2009)
experimentally confirmed the predictions from Takahashi’s
nonlinear time-perception theory of hyperbolic and subadditive
discounting. However, neither Loewenstein-Prelec theory,
Scholten-Read theory, nor Takahashi’s theory totally explains
all the anomalies mentioned earlier. In order to explain all the
anomalies mentioned, the present study generalizes the time-
based accounts proposed by Scholten-Read and Takahashi. In
generalizing the time-based accounts, we specify the functional
forms of psychological time and temporal discounting, in order
to future parametric applications in behavioral economics and
neuroeconomics, although the main conclusions of the present
study do not depend on the precise functional forms.
Time Discounting with Psychological Time
Takahashi (2005) proposed that subjects utilize the following
psychological time in their intertemporal choice:
  
ln 1.DD (1)
where τ(D) is subjective time (or time-weighting) of delay D
when the delayed outcome is obtained. If we further assume
that subject discount the delayed outcome x exponentially with
the psychological time, we obtain








,,0expVxDVxk D. (2)
where V(x, D) is the time-discounted value of the delayed out-
come x obtained at delay D. Equation (2) is hyperbolic in
physical delay D (Takahashi, 2005). It is to be noted that Equa-
tion (2) is equivalent to Loewenstein and Prelec’s generalized
hyperbola (Loewenstein & Prelec, 1992) and the “q-exponential”
time-discount model developed in econophysics (Cajueiro,
2006; Takahashi et al., 2007; Takahashi, 2007; Takahashi et al.,
2008; Destefano & Martinez, 2011). Takahashi (2005) claimed
that Equation (2) can account for decreasing impatience (hy-
perbolic discounting) over physical time, although the discount
rate with respect to psychological time (k > 0) is independent of
psychological time, and Zauberman et al. (2009) confirmed this
claim in their behavioral economic experiment.
Tempospect Theory of Intertemporal Choice
Scholten and Read (2010) and Takahashi (2006) stated that
intertemporal choice occurs with respect to time-intervals rather
than delay (time-point) per se at which the delayed outcome is
obtained. Therefore, it is natural to generalize Equation (1) to
the following equations:




ln 1
dd
DD

  (for ΔD > 0). (3)






ln 1
ss
DD  (for ΔD < 0). (4)
where ∆τ(∆D) is subjective time-interval (what we call “tem-
pospect”) between options which are obtained at the physical
time-interval of ∆D (∆D > 0 and ∆D < 0 correspond to “delay”
and “speedup” of the delayed option, respectively). Here we
make a natural assumption, as is the case with the gain-loss
asymmetrical value function in Kahneman-Tversky’s prospect
thery (Kahneman & Tversky, 1979; Tversky & Kahneman,
1992) in which an increase in outcomes less dramatically im-
pacts subjective valuation than a decrease in outcomes (referred
to as “loss aversion”), that ∆τ is larger for “delay” (an increase
in delay until receipt, which may be perceived as “loss”) than
“speedup” (a decrease in delay until receipt, which may be
perceived as “gain”), when the outcome is positive (gain). Op-
positely, ∆τ may be smaller for “dalay” than for “speedup”
when the outcome is negative (loss), because an increase in
delay in waiting for loss may be perceived as gain. Under these
assumptions, the time-interval discount function may be:








,,expVxtD VxtkD  (for ∆D > 0). (5)








,,expVxtD VxtkD  (for ∆D < 0). (6)
where V(x, t + ∆D) is the subjective value of the delayed out-
come obtained at delay t + ∆D. As can be seen from Equation
(5), when ∆τ is large, the subjective value of the outcome is
small. Considering the assumption that ∆τ may be larger for
“delay” than “speedup” in waiting for delayed gain, we can
suppose that subjective value of the delayed reward changes
more dramatically when the option is delayed than sped up.
Concerning the characteristics of the outcome x, when x is
negative (loss), ∆D may have smaller impact on decision over
time, than when x is positive (gain), because subjective valua-
tion of the outcome, rather than ∆D, may have stronger impact
when x is negative, due to loss aversion in the value function in
prospect theory (Kahneman & Tversky, 1979; Tversky & Kah-
neman, 1992). This can explain the sign effect in intertemporal
556

T. TAKAHASHI, R. HAN
choice. Also, when the outcome is large, ∆D may have smaller
impact on decision over time, again the subjective value of the
outcome, rather than ∆D, may have stronger impact. This may
account for the magnitude effect in intertemporal choice. Taken
together, our present “tempospect” theory can account for the
important anomalies in intertemporal choice.
Implications for Behavioral Economics and
Neuroeconomics
Since Loewenstein and Prelec (1992)’s proposal, studies in
intertemporal choice have mainly focused on the value-based
account of the anomalies in intertemporal choice. Subsequent
studies examined the roles of the shape of value functions
which can explain the anomalies in intertemporal choice (Al-
Nowaihi & Dhami, 2006; Al-Nowaihi & Dhami, 2009). Our
present study emphasizes, in line with perspectives proposed by
Scholten and Read’s group (Scholten & Read, 2006, 2010), the
roles of psychological time regarding time-intervals between
options. By estimating the functional forms of the psychologi-
cal time for time-intervals, we will be able to establish more
precise functional forms of temporal discounting. Also, inves-
tigations into neural processing underlying psychological time
in decision over time may help establish more effective medical
treatments for addiction and other impulsive and problematic
behaviors observed in psychiatric illnesses (Takahashi, 2009,
for a review).
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