In one of the major contributions to behavioral economics, Loewenstein and Prelec (1992) set the foundations for the behavioral approach to decision making over time. We correct a number of errors in Loewenstein and Prelec (1992). Furthermore, we provide a correct, more direct and simpler derivation of their generalized hyperbolic discounting formula that has formed the basis of much recent work on temporal choice.Anomalies of the DU model; Intertemporal choice; Generalized hyperbolic discounting

Ali al-Nowaihi

Sanjit Dhami

English

Research Papers in Economics

 
 
 
 
 
DEPARTMENT OF ECONOMICS 
   
 
 
 
A NOTE ON THE LOEWENSTEIN-PRELEC 
THEORY OF INTERTEMPORAL  
CHOICE 
 
 
 
 
 
 
 
 
Ali al-Nowaihi, University of Leicester, UK 
Sanjit Dhami, University of Leicester, UK 
 
 
 
 
 
 
 
 
Working Paper No. 05/18 
July 2005 
 
 A Note On The Loewenstein-Prelec Theory Of
Intertemporal Choice
Ali al-Nowaihi∗ Sanjit Dhami†
July 2005
Abstract
In one of the major contributions to behavioral economics, Loewenstein and Pr-
elec (1992) set the foundations for the behavioral approach to decision making over
time. We correct a number of errors in Loewenstein and Prelec (1992). Furthermore,
we provide a correct, more direct and simpler derivation of their generalized hyper-
bolic discounting formula that has formed the basis of much recent work on temporal
choice.
Keywords: Anomalies of the DU model, Intertemporal choice, Generalized hyper-
bolic discounting.
JEL Classiﬁcation Codes: C60(General: Mathematical methods and program-
ming); D91(Intertemporal consumer choice).
∗Department of Economics, University of Leicester, University Road, Leicester. LE1 7RH, UK. Phone:
+44-116-2522898. Fax: +44-116-2522908. E-mail: aa10@le.ac.uk.
†Department of Economics, University of Leicester, University Road, Leicester. LE1 7RH, UK. Phone:
+44-116-2522086. Fax: +44-116-2522908. E-mail: Sanjit.Dhami@le.ac.uk.1. Introduction
It is well known that the discounted utility model of intertemporal choice (henceforth,
DU) is contradicted by a relatively large body of empirical and experimental evidence; see
for instance Thaler (1981). Furthermore, it appears that these anomalies are not simply
mistakes; see for instance, Frederick, Loewenstein and O’Donoghue (2002). If we wish
to develop models that better explain economic behavior, then we have no choice but
to take account of these anomalies. Furthermore, certain types of behavior, and several
institutional features, can be explained by decision makers attempting to deal with time-
inconsistency problems that arise from non-exponential discounting1.
Loewenstein and Prelec (1992) (henceforth LP) give a formal statement of the known
anomalies of the DU model. Of the anomalies mentioned in LP, the subsequent literature
has focussed largely on the evidence for and implications of declining discount rates (DU in
contrast assumes constant discount rates). The importance of LP’s contribution2 is that it
remains, as far as we know, the only theoretical contribution that provides an explanation
of the other anomalies of the DU model, in particular, the magnitude eﬀect and the gain-
loss asymmetry;w eg i v ed e ﬁnitions below. Furthermore, LP give the ﬁrst statement and
axiomatic derivation of the generalized hyperbolic discounting formula which has been the
main, but not the only, alternative to the exponential discounting model3.
The purpose of our note is two-fold. First, we correct a number of errors in LP (1992)
which have potentially serious implications4. Second, we provide a correct, more direct
and simpler derivation of their generalized hyperbolic discounting formula.
2. Loewenstein-Prelec theory of intertemporal choice
Consider a decision maker who, at time t0, formulates a plan to choose ci at time ti,
i =1 ,2,...,n,w h e r et0 <t 1 < ... < tn. LP assume that the utility to the decision maker,
at time t0,i sg i v e n( L P(9), p579) by :
U ((c1,t 1),(c2,t 2),...,(cn,t n)) = Σ
n
i=1v(ci)ϕ(ti) (2.1)
1Time inconsistency problems can lead individuals to make suboptimal decisions about, for instance,
savings, pensions, retirement etc. The existence of madatory pension plans, retirement age, compulsory
insurance of several sorts etc. are possible institutional responses to these time inconsistency problems;
see, for instance, Frederick, Loewenstein and O’Donoghue (2002).
2It is fair to characterize LP’s work as seminal in the behavioral economics literature. This is reﬂected
in the citations count. Our check of the social sciences citations database revealed 134 citations of their
paper.
3The simpler quasi-hyperbolic formulation, due to Phelps and Pollack (1968) and later popularized by
Laibson (1997) often tends to be used in applied theoretical work on account of its tractability. However,
LP’s formulation is the most general form of the hyperbolic discounting function.
4We have checked a more recent reprint of the LP article in Kahneman and Tversky (2000) but the
same errors remain.
1We get the standard DU model for the special case of exponential discounting:
ϕ(ti)=e
−βti,β>0 (2.2)
Aside from its tractability, the main attraction of DU is that it leads to time-consistent
choices (at least, in none game-theoretic situations). If the plan (c1,t 1),(c2,t 2),...,(cn,t n)
is optimal at time t0,t h e na tt i m etk the plan (ck+1,t k+1),(ck+2,t k+2),...,(cn,t n) is also
optimal. But this may no longer be true for more general speciﬁcations of the discount
factor ϕ.
LP adopt the utility function (2.1) taking v to be the value function introduced by
Kahneman and Tversky (1979). Thus v satisﬁes (among other properties)
v :( −∞,∞) −→ (−∞,∞) is continuous, strictly increasing,
v(0) = 0 and is diﬀerentiable except at 0 (2.3)
They deﬁne the elasticity of v (LP (16), p583) by:
 v (c)=
c
v
dv
dc
,c6=0 (2.4)
LP introduce ﬁve assumptions, all with good experimental basis (LP, II pp574-578).
The ﬁrst four of these are:
A0 (impatience). ϕ :[ 0 ,∞) −→ (0,∞) is strictly decreasing (in an arbitrarily small
interval).
A1 (gain-loss asymmetry). If 0 <x<yand v(x)=v (y)ϕ(t),t h e nv(−x) >
v(−y)ϕ(t).
A2 (the magnitude eﬀect). If 0 <x<y , v (x)=v(y)ϕ(t) and a>1,t h e nv(ax) <
v(ay)ϕ(t).
A3 (common diﬀerence eﬀect). If 0 <x<y , v(x)=v (y)ϕ(t) and s>0,t h e n
v(x)ϕ(s) <v(y)ϕ(s + t).
A0 is only implicit in LP, however, it is essential for Theorem 3 (below). In this note,
we make no use of their ﬁfth assumption: delay-speedup asymmetry (LP, p578).
To derive the LP formula for generalized hyperbolic discounting (LP (15), p580), a
specialized form of A3 is needed. We adopt:
A3a (common diﬀerence eﬀect with quadratic delay). If 0 <x<y , v(x)=v(y)ϕ(t) and
s>0,t h e nv(x)ϕ(s)=v (y)ϕ(s + t + αst), α>0.
2Note that A3a =⇒ A3 and that α =0gives exponential discounting.
Three theorems follow.
Theorem 1 : A1 implies that the value function is more elastic for losses than for gains:
x>0= ⇒  v (−x) >  v (x).
Theorem 2 : A2 implies that the value function is less elastic for outcomes of larger
absolute magnitude:
(0 <x<yor y<x<0) −→  v (x) >  v (y).
Theorem 3 : A0 and A3a imply that the discount factor is a generalized hyperbola:
ϕ(t)=( 1+αt)
−
β
α, β>0, t ≥ 0 (α is as in A3a).
Corollary 1 : A0 and A3a imply that −
•
ϕ
ϕ =
β
1+αt. Hence, the discount rate is positive
and declining.
We discuss the errors in LP more fully in the Appendix. For the moment we indicate
the main errors. Theorem 2 is stated incorrectly in LP and their proof contains an error
(LP pp583-584, V3 p584). The generalized hyperbola given in Theorem 3 does not follow
from the assumptions made in LP. Our A3a is a corrected version of the one that appears
in LP (LP (11), p579) and enables us to derive the required generalized hyperbola. There
are further errors in an example of intertemporal consumption-savings choice. We correct
and sharpen the LP results. We also give a more direct, simpler and correct5 proof of
Theorem 3. In Section 3 we give the correct proofs.
3. Proofs
P r o o fo fT h e o r e m1 :See LP, V2 p583.
P r o o fo fT h e o r e m2 :Let
0 <x<y (3.1)
Assume a consumer is indiﬀerent between receiving the increment x now and receiving
the increment y, t-periods from now. Then, letting v be the value function and ϕ the
discount factor, we get
v(x)=v(y)ϕ(t) (3.2)
Let
a>1 (3.3)
5In the Appedix we show that LP’s generalised hyperbolic discounting formula contradicts their spe-
cialized form of A3. Hence, it cannot follow from it.
3then the magnitude eﬀect, A2, predicts that
v(ax) <v(ay)ϕ(t) (3.4)
(3.2) gives
v (x)
v (y)
= ϕ(t) (3.5)
Since y,a are positive, it follows that ay, and hence, v(ay) are also positive. Hence, (3.4)
gives
v(ax)
v(ay)
<ϕ(t) (3.6)
(3.5) and (3.6) give
v (x)
v(y)
>
v (ax)
v (ay)
, 0 <x<y ; a>1 (3.7)
It follows from (3.7) that the value function, v,i ssubproportional6.I t a l s o f o l l o w s t h a t
ln(v) is a concave function of ln(x) (for x>0)a n dt h a tt h ed e r i v a t i v eo fln(v) with
respect to ln(x) is decreasing. It then follows that the value function is less elastic for
outcomes that are larger in absolute magnitude:
(0 <x<yor y<x<0) −→  v (x) >  v (y)
P r o o fo fT h e o r e m3 :Let
0 <x<y (3.8)
Assume a consumer is indiﬀerent between receiving the increment x now and receiving the
increment y, t-periods from now. Then, letting v be the value function and ϕ the discount
factor, we get
v(x)=v(y)ϕ(t) (3.9)
Multiply (3.9) by ϕ(s),w h e r es>0,t og e t
v(x)ϕ(s)=v(y)ϕ(s)ϕ(t) (3.10)
A3a, (3.8)and (3.9) give
v(x)ϕ(s)=v(y)ϕ(s + t + αst),α>0 (3.11)
(3.10) and (3.11) give
ϕ(s + t + αst)=ϕ(s)ϕ(t) (3.12)
6See Kahnman and Tversky (1979, p282) for the deﬁnition of subproportionality. Note that our, and
LP’s, a>1 corresponds to their 0 <r<1.
4Let
X =1+αs,Y =1+αt (3.13)
Hence
s =
X − 1
α
,t =
Y − 1
α
,s + t + αst =
XY − 1
α
(3.14)
Deﬁne the function G :[ 1 ,∞) −→ (0,∞) by
G(X)=ϕ
µ
X − 1
α
¶
(3.15)
Hence,
G(Y )=ϕ
µ
Y − 1
α
¶
,G(XY)=ϕ
µ
XY − 1
α
¶
(3.16)
From (3.12),(3.14),(3.15),(3.16)
G(XY)=ϕ
µ
XY − 1
α
¶
= ϕ(s + t + αst)=ϕ(s)ϕ(t)
= ϕ
µ
X − 1
α
¶
ϕ
µ
Y − 1
α
¶
= G(X)G(Y ) (3.17)
Deﬁne the function h :[ 0 ,∞) → (0,∞) by
h(x)=G(e
x),x≥ 0 (3.18)
Hence, and in the light of A0, h satisﬁes:
h :[ 0 ,∞) → (0,∞) is strictly decreasing (in an arbitrarily small interval)
and h(x + y)=h(x)h(y) (3.19)
As is well known, see for example Corollary 1.4.11 in Eichhorn (1978), the solution of
(3.19) is the exponential function
h(x)=e
cx,x ≥ 0,c < 0 (3.20)
(3.13), (3.15), (3.18), (3.20) give
ϕ(t)=( 1+αt)
c (3.21)
Let
β = −αc (3.22)
(3.21), (3.22) give
ϕ(t)=( 1+αt)
−
β
α ,α,β>0,t≥ 0 (3.23)
where β>0 because α>0 and c<0.
54. Optimal consumption plans
LP provide a rich set of applications. However, one of them needs correction (their appli-
cation 5 pp591-593). The corrected version, given in this section, has sharper conclusions.
Consider a consumer with an exogenously given stream of real income whose present
value at time 0 is I. Let the real interest rate, r, be positive and constant. Let the con-
sumer’s reference real consumption, c, be a non-negative constant. At time 0 the consumer
chooses her consumption plan c(t)=c∗ (t) so as to maximize
R T
t=0 v(c(t) − c)ϕ(t)dt sub-
ject to
R T
t=0 c(t)e−rtdt ≤ I. Suppose that the consumer is able to commit to this (generally
time-inconsistent) plan. Following Kahneman and Tversky (1979), assume that the con-
sumer’s value function, v, is strictly concave for gains but strictly convex for losses, i.e.,
for x>0, v00 (x) < 0,v00 (−x) > 0.
4.1. The consumer is always in the domain of gains: c(t) > c
Here the value function is strictly concave. The problem can be transformed into a standard
control problem as follows. Let y (t) be the exogenously given stream of real income and
s(t) the stock of saving at time t.T h e nc∗ (t) is the solution to the following problem
Maximize
Z T
t=0
v(c(t) − c)ϕ(t)dt
subject to
•
s = y (t)+rs− c; s(0) = s(T)=0
where the dot represents the time derivative. The Hamiltonian is
H (c,s,λ,t)=v(c − c)ϕ(t)+λ(y (t)+rs− c)
where c is the control variable, s is the state variable and λ is the costate variable. vϕ(t)
is strictly concave in c (hence jointly in c and s,s i n c es does not occur in vϕ(t))a n d
•
s is linear in s and c. Hence, Hamilton’s equations are both necessary and suﬃcient
for a global optimum (Kamien and Schwartz, 1981, II.3). Solving Hamilton’s equations:
∂H
∂c =0 ,
•
λ = −∂H
∂s,
•
s = y (t)+rs− c,g i v e s
•
c
∗
=
"
r −
Ã
−
•
ϕ
ϕ
!#
v0
(−v00)
(4.1)
(4.1) gives:
(i) If r(0) ≥
³
−
•
ϕ
ϕ
´
t=0
then, since −
•
ϕ
ϕ is declining (by Corollary 1), we get r>−
•
ϕ
ϕ for
all t>0. Hence,
•
c
∗
> 0 for all t>0.
6(ii) Suppose r(0) <
³
−
•
ϕ
ϕ
´
t=0
.H e r ew eh a v et w os u b - c a s e s :
(iia) r<−
•
ϕ
ϕ for all t<T.T h e n
•
c
∗
< 0 for all t<T .
(iib) r(t0)=
³
−
•
ϕ
ϕ
´
t=t0
for some t0,0 <t 0 <T .T h e n
•
c
∗
< 0 for t<t 0 and
•
c
∗
> 0 for
t>t 0.
4.2. The consumer is always in the domain of losses: c(t) < c
Here the value function is strictly convex. Hence, it is optimal for the consumer to consume
all her income at one point in time. This point has to be at the upper boundary, t = T.
There are two reasons for this. First, the value of real life-time income at time t is Iert,
which is increasing with time. Hence, postponing consumption as long as possible pushes
consumption closer to reference consumption and, hence, reduces loss. Second, because of
discounting, postponing consumption reduces this loss even further.
Here is a formal derivation. If the consumer consumes all her income at time t,i t s
present value in utility terms will be u(t)=v(Iert − c)ϕ(t). The consumer will postpone
consumption as long as
•
u ≡
•
ϕv +ϕrIertv0 > 0. But this quantity is always positive, since
•
ϕ<0,v<0,ϕ>0,r>0,I>0,e rt > 0,v0 > 0. It follows that the consumer will consume
all her income at the last possible moment, i.e., at t = T.
5. Conclusions
Lowenstein and Prelec (1992) is a foundational paper in economics. To the best of our
knowledge, it provides the ﬁrst and only available theoretical framework to explain several
important anomalies to the DU model. Furthermore, it provides an axiomatic derivation
of the generalized hyperbolic discounting formula that forms the basis of much recent
research in temporal choice.
We correct several errors in the paper, some with potentially serious implications. We
also provide direct and simple proofs of some their most important results and sharpen
others.
6. Appendix : Errors in Loewenstein and Prelec (1992)
Error 1. LP state Theorem 2 incorrectly. What they state (LP, V3 p584) is equivalent
to:
Theorem 2: A2 implies that the value function is more elastic for outcomes of larger
absolute magnitude:
7(0 <x<yor y<x<0) −→  v (x) <  v (y).(more should be less and the last <
should be >).
In the course of their proof, LP derive, incorrectly, the formula (LP (18) p583) :
v(x)
v(y)
<
v (ax)
v (ay)
,0 <x<y ;a>1 (the ﬁrst < should be > ) (6.1)
They then conclude that the value function is subproportional. This is correct, but it follows
from (3.7), not (6.1). They follow this with the two incorrect statements that ln(v) is a
convex function of ln(x) (for x>0) and that the derivative of ln(v) with respect to ln(x)
is increasing. Of course, what they should have said is that ln(v) is a concave function of
ln(x) (for x>0) and that the derivative of ln(v) with respect to ln(x) is decreasing.
Error 2. To derive the formula for generalized hyperbolic discounting, a specialized form
of A3 is needed. What LP adopt (LP, (11) p579) is equivalent to:
A3b (common diﬀerence eﬀect with linear delay). If 0 <x<y , v (x)=v(y)ϕ(t) and
s>0,t h e nv (x)ϕ(s)=v(y)ϕ(ks+ t),w h e r ek = k (x,y) is a function of x and y but
not of s or t. (Note that A3b =⇒ A3).
We will now show that generalized hyperbolic discounting implies that k =1+αt.
Hence, k depends on t but is independent of x and y.
Let
0 <x<y (6.2)
Assume that a consumer is indiﬀerent between receiving the increment x now and
receiving the increment y, t-periods from now. Then, letting v be the value function and
ϕ the discount factor, we get equation (10) of LP :
v(x)=v(y)ϕ(t) (6.3)
The common diﬀerence eﬀect, A3, then implies that the consumer, now, strictly prefers y,
(s + t)-periods ahead to x, s-period ahead. LP say (p579 just above (11))“ W en o wd e r i v e
a ... general functional form, by postulating that the delay that compensates for the larger
outcome [y] is a linear function [t + ks] of the time [s] to the smaller, earlier outcome [x]
(holding ﬁxed the two outcomes x and y)”. Hence, their equation (11) :
v(x)=v (y)ϕ(t) ⇒ v(x)ϕ(s)=v(y)ϕ(t + ks) (6.4)
LP continue: “for some constant k, which, of course depends on x and y”.
From the above equation they prove that the discount factor ϕ must be given by their
equation (15), reproduced immediately below.
ϕ(t)=( 1+αt)
−
β
α ,α,β>0 (6.5)
8We shall show that if the discount factor, ϕ, is given by their equation (15) ((6.5), above),
then,
k =1+αt (6.6)
(From (6.6) it is clear that k is a function of t but not of x or y.)
Multiplying (6.3) by ϕ(s) gives
v(x)ϕ(s)=v(y)ϕ(s)ϕ(t) (6.7)
Comparing (6.4) and (6.7), we see that
ϕ(t + ks)=ϕ(s)ϕ(t) (6.8)
From (6.5) and (6.8), we get
(1 + α(t + ks))
−
β
α =( 1+αs)
−
β
α (1 + αt)
−
β
α (6.9)
Successive simpliﬁcations of (6.9) give
(1 + αt + αks)
−
β
α =[ ( 1+αs)(1+αt)]
−
β
α (6.10)
(1 + αt + αks)
−
β
α =
£
1+αs + αt + α
2st
¤−
β
α (6.11)
1+αt + αks =1+αs + αt + α
2st (6.12)
αks = αs + α
2st (6.13)
k =1+αt (6.14)
It follows that, to get their equation (15) ((6.5) above) k must be given by (6.14). Hence,
their equation (11) ((6.4) above) has to be written as
v(x)=v (y)ϕ(t) ⇒ v(x)ϕ(s)=v(y)ϕ(s + t + αst) (6.15)
Error 3. (The optimal consumption plan when the consumer is always in the domain of
gains) LP’s equation (23) p592 is incorrect. What they state is equivalent to:
•
c
∗
= r −
Ã
−
•
ϕ
ϕ
!
v0
(−v00)
(6.16)
The correct equation is given by (4.1) above. However, their conclusions are correct but
follow from (4.1), not (6.16).
Error 4. LP derive the incorrect equation (LP (24) p592):
The consumer will postpone consumption as long as r<
−
•
ϕ/ϕ
 v (Iert)
(6.17)
9Furthermore they claim, incorrectly, that  v (Iert) is increasing.I nf a c t , v (Iert) is decreas-
ing (see Theorem 2). They merely conclude that, in the domain of losses, consumption
will be concentrated at a single point in time, which could be anywhere in the interval
[0,T]. The correct form of (6.17) is:
The consumer will postpone consumption as long as r>
−
•
ϕ/ϕ
Iert
Iert−c v (Iert − c)
(6.18)
Since the right hand side of (6.18) is negative, the inequality in (6.18) always holds.
Hence, we get the sharper result that the consumer will consume all her income at the last
possible moment, i.e., at t = T. However, we gave a simpler proof of this result in section
4.2, above.
References
[1] Eichhorn W. (1978). Functional Equations in Economics. Addison-Wesley.
[2] Frederick, S., Loewenstein, G., & O’Donoghue, T. (2002) “Time Discounting and Time
Preference: A Critical Review.” Journal of Economic Literature, 40, 351-401.
[3] Kahneman D. and Tversky A. (1979) “Prospect Theory : An Analysis of Decision
Under Risk” Econometrica, 47, 263-291.
[4] Kahneman D. and Tversky A. (2000) Choices, Values, and Frames (Cambridge Uni-
versity Press: Cambridge).
[5] Laibson, D. (1997) “Golden Eggs and Hyperbolic Discounting” Quarterly Journal of
Economics,1 1 2 ,4 4 3 - 4 7 7 .
[6] Loewenstein G. and Prelec D. (1992) “Anomalies in Intertemporal Choice : Evidence
and an Interpretation.” The Quarterly Journal of Economics, 107, 573-597.
[7] Phelps, E.S. and Robert Pollak. (1968) “On Second-Best National Saving and Game-
Equilibrium Growth.” Review of Economic Studies, 35, 185-199.
[8] Thaler (1981) “Some Empirical Evidence on Dynamic Consistency” Economic Letters,
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York).
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A Note On The Loewenstein-Prelec Theory Of Intertemporal Choice

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