for by a conventional model of temporal discounting in microeconomic theory
(”exponential discounting”; see Frederick et al., 2002). As a consequence, both
impulsivity (strong discounting) and inconsistency in temporal discounting (i.e.,
hyperbolic and subadditive discounting) have extensively been investigated in
neuroeconomic studies by employing neuroimaging techniques (Boettiger et al.,
2007; Hariri et al., 2006; Kable & Glimcher, 2007; McClure et al., 2004; Mc-
Clure et al., 2007; Monterosso et al., 2007; Wittmann, Leland, & Paulus, 2007),
stimulating thereby much further research and debate.
Recently, behavioral neuroeconomic and econophysical studies established
discount models in order to better describe neural and behavioral correlates of
impulsivity and inconsistency in intertemporal choice. In order to analyze hu-
man and animal intertemporal choice behavior in a manner that would allow
for a dissociation between impulsivity and inconsistency, recent econophysical
studies (Cajueiro 2006; Takahashi, Oono, & Radford, 2007) proposed and ex-
amined the following q-exponential discount function for subjective value V (D)
of a delayed reward:
V(D)=
A
exPq (kqD)
A
[1 + (1 - q)kqD]1/(1-q)
(1)
where expq (x) := [1 + (1 - q)x]1/(1-q) is a q-exponential function, D is a delay
until receipt of a reward, A is the value of a reward at D = 0, and kq is a param-
eter of impulsivity at delay D = 0 (q-exponential discount rate). We can easily
see that this generalized q-exponential function approaches the usual exponen-
tial function in the limit of q → 1. The q-exponential function has extensively
been utilized in econophysics, where the application of Tsallis’ non-extensive
thermostatistics (Tsallis et al., 2003) may possibly explain income distributions
following power functions (Michael & Johnson, 2003). It needs to be noted
here that when q = 0, the equation (1) becomes the same as the ”hyperbolic”
discount function (i.e., V (D) = A/(1 + kq D)), while in the limit of q → 1, it
reduces to the ”exponential” discount function (i.e., V (D) = A exp(-kq D)). In
exponential discounting (when q → 1 in equation (1)), intertemporal choice is
consistent, because the discount rate := -(dV /dD)/V = kq is time-independent
when q → 1. The q-exponential discount function is capable of continuously
quantifying human sub jects’ inconsistency in intertemporal choice (Takahashi
et al., 2007). Namely, human agents with smaller q values are more inconsistent
in intertemporal choice. If q is less than 0, the intertemporal choice behavior is
more inconsistent than hyperbolic discounting. Thus, 1 - q can be utilized as
an inconsistency parameter. Moreover, it is possible to examine neuropsycho-
logical modulation of kq (impulsivity in temporal discounting) and q (dynamic
consistency) in the q-exponential discount model. It is now important to note
that in any continuous time-discounting functions, a discount rate (preference
for sooner rewards over later ones) is defined as -(dV (D)/dD)/V (D), inde-
pendently of functional forms of discount models, with larger discount rates
indicating more impulsive intertemporal choice. In the q-exponential discount
model, the q-exponential discount rate qEDR (”impulsivity”) is then defined as:
qEDR =
________kq________
[1 + kq (1 - q)D]
(2)
We can see that when q = 1, the discount rate is independent of delay D, corre-
sponding to the exponential discount model (consistent intertemporal choice);