A Model of Utility Smoothing
Katsutoshi Wakai
May 2007
Abstract
Experimental studies have found that a decision maker prefers spreading good and bad outcomes evenly over time. We propose, in an axiomatic framework, a new model of discount factors that captures this
preference for spread. The model provides a re…nement of the discounted utility model while maintaining dynamic consistency. The derived discount factors incorporate gain/loss asymmetry recursively: the
di¤erence between average future utility and current utility de…nes a
gain or a loss, and gains are discounted more than losses. This notion
of utility smoothing explains a preference for spread: if bad outcomes
are concentrated on future periods, moving one of the bad outcomes to
today would be bene…cial because such an operation eliminates a large
loss and replaces it with a small gain.
Keywords: Discount factor, gain/loss asymmetry, recursive utility,
reference point, utility smoothing
JEL Classi…cation Numbers: D90, D91
Department
Bu¤alo,
427
of
Economics,
Fronczak
Hall,
The
State
Bu¤alo,
http://pluto.fss.bu¤alo.edu/classes/eco/kwakai/
1
University
NY
14260;
of
New
York
at
kwakai@bu¤alo.edu;
Acknowledgement
This paper is a revised version of the …rst half of Chapter II of my dissertation at Yale University (Wakai (2002)). I would like to express my appreciation
to my advisor, Stephen Morris, and dissertation committee members, John
Geanakoplos and Benjamin Polak, for their invaluable advice and support. I
would also like to thank Larry Epstein and Itzhak Gilboa for valuable discussions and suggestions. I have bene…ted from comments by Richard Braun,
Atsushi Kajii, Takao Kobayashi, Giuseppe Moscarini, Robert Shiller, Leeat
Yariv, and seminar participants at the State University of New York at Buffalo, the University of Iowa, the University of Rochester, the University of
Texas Austin, the University of Tokyo, Washington University, the RUD 2004
conference, and the 2006 Spring Midwest Theory conference. I am also grateful to the Editor, Eddie Dekel, and anonymous referees, whose comments and
suggestions greatly improved this paper.
2
1
Introduction
Intertemporal choices involve an allocation of resources over time. To analyze
this choice problem, Samuelson (1937) proposes a special form of the weighted
summation decision rule, called the discounted utility model. For example, at
each time t, the conditional utility of a consumption sequence c = (c0 ; :::; cT )
is expressed in an additively separable form
Vt (c)
T
X
t
(1)
U (c );
=t
where 0 <
< 1 is called a single-period discount factor and U is called an
instantaneous utility function with diminishing marginal utility. Since it is
analytically tractable as well as axiomatically driven (Koopmans (1960)), the
discounted utility model has become the standard framework for analyzing
intertemporal choices. One of the de…ning characteristics of this model is
utility independence: each utility sequence (U (ct ); :::; U (cT )) is discounted by
the same series of discount factors (1; :::;
T t
), regardless of the distribution
of utility over time.
Although utility independence is an analytically attractive assumption,
many experimental studies have reported results that seem to contradict utility
independence. For example, Loewenstein (1987) asked subjects to choose a
plan from alternatives, each of which involved a sequence of consumption and
changed the locations of good outcomes. In a symbolic term, his …nding is
summarized by
(G, B, B)
(B, G, B) but (G, B, G)
(B, G, G),
where G stands for “good” consumption and B stands for “bad” consumption. This result violates utility independence because changing common consumption B, located in the last period, to G altered the preference ordering.
3
Moreover, the result cannot be attributed to the diminishing marginal utility
of the instantaneous utility function. Instead, the result suggests a preference
for spread, that is, a decision maker (DM) prefers spreading good and bad
outcomes evenly over time.
Motivated by the above study, we propose a new model of discount factors
that captures this preference for spread. We provide a simple re…nement of (1)
in an axiomatic approach: by relaxing the assumption of utility independence,
we introduce an appropriate notion of utility smoothing while maintaining a
key feature of (1), dynamic consistency. Formally, at each time t, the conditional utility of a consumption sequence c is expressed in the recursive form,
Vt (c)
min
[(1
t+1 )U (ct )
+
t+1 Vt+1 (c)];
(2)
t+1 2[ t+1 ; t+1 ]
where
t+1
and
t+1
describe the upper and lower bounds of single-period
discount factors, respectively, and satisfy 0 <
t+1
t+1
< 1. Our axiom-
atization adapts a method developed in a di¤erent context by Gilboa and
Schmeidler (1989) and Epstein and Schneider (2003).
The representation (2) introduces a key feature called recursive gain/loss
asymmetry: the di¤erence between future utility Vt+1 (c) and current utility
U (ct ) de…nes a gain or a loss, and gains are discounted more than losses.
Intuitively, from the reference point U (ct ), the cost of reducing the future
utility Vt+1 (c) outweighs the bene…t of increasing it. Thus, (2) is a class
of reference-based preferences, where discount factors incorporate referencedependence. Since Vt+1 (c) is the average utility of future periods, recursive
gain/loss asymmetry generates an aversion to utility changes that can explain
the motive behind a preference for spread: If good outcomes are concentrated
on future periods, the DM may …nd it bene…cial to move one of the good
outcomes to today because such an operation eliminates a gain but signi…cantly
4
increases current utility U (ct ) without decreasing future utility Vt+1 (c) much.
On the other hand, if bad outcomes are concentrated on future periods, the
DM may …nd it bene…cial to move one of the bad outcomes to today because
such an operation eliminates a large loss and replaces it with a small gain.
Preference for spread is observed in short and …nite horizon experiments. In
contrast, much empirical evidence supports gain/loss asymmetry, the existence
of which seems independent of the time horizon.1 To stress the application
of gain/loss asymmetry, we also provide an in…nite-horizon extension of (2),
where we assume stationarity.
The remainder of the paper proceeds as follows: Section 2 formally introduces the representation (2). Section 3 compares our model with other
intertemporal utility functions. All proofs are collected in the appendices.
2
Representation
We adapt the Anscombe-Aumann (1963) framework with a temporal interpretation: preferences are de…ned on the set of sequences whose outcome at any
period is a lottery de…ned over a consumption set. This domain is rich enough
to model a preference for utility smoothing independently of a functional form
of instantaneous utility. The domain of our interest, the set of consumption
sequences, is identi…ed with a subset of this larger domain, where each element
of the subset is a sequence of degenerate lotteries.
We adopt notation that is compatible with both …nite and in…nite horizon
settings. Time is discrete and varies over T = f0; 1; :::; T g, where T > 0; T is
either …nite or countably in…nite, and for the latter case, we set T = 1. For
1
See Frederic, Loewenstein, and O’Donoghue (2002) for their recent survey.
5
each t 2 T with t < 1, let Tt
ft; t + 1; :::; T g. Denote by Bt a collection
of all bounded and real-valued functions on Tt , where Bt is endowed with the
sup norm. We use the weak topology on Bt , i.e., the dual space of Bt .
Let X be a non-empty consumption set; X can be a convex set in R or
a …nite set of objects. Let M be the set of all probability distributions over
X with …nite support. An act is a function l : T ! M , where l = (l0 ; l1 ; :::).
Denote by L = M T the collection of all acts. A constant act p is a function
l : T ! M such that lt = p 2 M for all t 2 T ; p is also identi…ed with p 2 M .
Let C be a collection of all constant acts. Given a probability mixture operation
+ on M , de…ne a mixture operation + on L by ( l +(1
for
)l0 )t = lt +(1
)lt0
2 [0; 1] and l; l0 2 L.
The primitive of the model is the collection of complete and transitive
preference orderings f
tg
conditional ordering
t
f
t
j t 2 T g, whose element is de…ned on L. Each
is assume to be Archimedean, non-degenerate, and
independent of the payo¤ history (l0 ; l1 ; :::; lt 1 ). In addition, each conditional
ordering on C induces a conditional ordering on M , also denoted by
t,
on
which we assume the following:
Assumption 1:
represents
t
(i) There exists an a¢ ne function Ut : M ! R that
on M . (ii) Ut is independent of time t.
Later we show that this assumption is implied by the axioms we impose below.
The …rst axiom relates the evaluation of lotteries to the evaluation of acts.
A1 (Strict Monotonicity-SMT): For each t 2 T and for all l; l0 2 L,
if l
t
then l
l0 for all
t
2 T , then l
t
l0 . In addition, if for some
t, l
t
l0 ,
l0 .
A1 assumes that a lottery is evaluated independently at each time. Thus,
6
given Assumption 1, a preference ordering on L induces a preference ordering
on utility sequences, also denoted by
t.
Moreover, the mixture operation +
on L induces the addition between the corresponding utility sequences.
The discounted utility model fails to explain a preference for spread because
of utility independence, which is characterized by the following property:
A20 (Independence-ID): For each t 2 T , for all l; l0 ; l00 2 L, and for all
2 (0; 1), l
t
l0 if and only if
)l00
l + (1
t
l0 + (1
)l00 .
We replace A20 with the following weaker axiom:
A2 (Constant-Independence-CI): For each t 2 T , for all l; l0 2 L,
p 2 C, and for all
2 (0; 1), l
t
l0 if and only if
l +(1
)p
t
l0 +(1
)p.
Since p does not involve utility changes, the pattern of utility changes
implied by l is preserved in the pattern of utility changes in
l + (1
)p.
Hence, A2 allows for the possibility that the same discounting is applied only
to a subset of utility sequences sharing a similar pattern of utility changes. A2
also implies the existence of an a¢ ne function Ut : M ! R that represents
t
on M (i.e., Assumption 1(i)).
Next, to capture a preference for spread, we introduce a notion that expresses the aversion to utility changes:
A3 (Time-Variability Aversion-TVA): For each t 2 T , for all l; l0 2
L, and for all
2 (0; 1), l 't l0 implies
)l0
l + (1
t
Given A1 and A2, A3 assumes
u 't u0 implies u + (1
7
)u0
t
u;
l.
where u and u0 are utility sequences implied by l and l0 , respectively. Thus,
A3 implies utility smoothing, which is independent of the functional form of
Ut .
Finally, we must specify the relationship between conditional orderings.
We maintain the following assumption of the discounted utility model:
A4 (Dynamic Consistency-DC): For each t 2 T with t < T and for
all l; l0 2 L, if l = l0 for all
t and if l
l0 , then l
t+1
t
l0 ; the latter
ranking is strict if the former is strict.
Dynamic consistency has a normative appeal; without this assumption, we
must model how the con‡ict between di¤erent selves is resolved. Also, A1 and
A4 imply that Ut : M ! R is time homogeneous (i.e., Assumption 1(ii)).
The following proposition is the main result of this paper, which is the
representation in a …nite-horizon setting (for the proof, see Appendix A):
Proposition 1: For 0 < T < 1, the following statements are equivalent:
(i) f
tg
satisfy A1 to A4.
(ii) There exist an a¢ ne function U : M ! R and a collection of sets of
discount factors f[ t ; t ]g1
t T
such that: for each t 2 T ,
t
satisfying 0 <
t
t
< 1 for all t 2 f1; :::; T g
is represented by Vt (:); where fVt (l)g0
t T
are
recursively de…ned by
Vt (l)
min
[(1
t+1 )U (lt )
+
t+1 Vt+1 (l)]
(3)
t+1 2[ t+1 ; t+1 ]
and VT (l)
U (lT ).2 Moreover, [ t ; t ] is uniquely de…ned for all t 2 f1; :::; T g,
and U is unique up to a positive a¢ ne transformation.
2
The scale of future utility is compatible with the scale of current utility because U (p) =
Vt+1 (p), where p is a constant act that pays p at each time.
8
The proof proceeds as follows: First, A1 to A3 imply that each conditional
ordering is represented by a version of Gilboa-Schmeidler’s (1989) multiplepriors utility.3 Formally, the conditional utility of l at time t is expressed
by
Vt (l) = min
t
b 2Dt
where Dt
T
X
bt Ut (l );
(4)
=t
Bt is a non-empty, closed, and convex set of strictly positive
P
weighting functions bt : Tt ! R++ such that T=t bt = 1. We interpret this
weighting function bt as a discount function de…ned on Tt .
Second, dynamic consistency forces the minimization on a set of discount
factors [
t+1 ; t+1 ]
to be applied recursively. Hence, the set of discount func-
tions Dt is recursively constructed. Indeed, Epstein and Schneider (2003)
derive a similar condition on Dt where each bt is regarded as a probability
de…ned on the -algebra in Tt . Our proof exploits this similarity. On the other
hand, the realization of a state is di¤erent from the evolution of time. Thus,
their setting is not compatible with our setting, so their proof does not directly
apply (see Appendix C).
We now consider an in…nite-horizon setting. Under the obvious identi…cation, let B
B0 = Bt for all t 2 T . To ensure that the utility sequence
implied by act l is in B , we assume the existence of best and worst lotteries:
A5 (Best–Worst-BW): Assume T = 1. For each t 2 T , there exist l
and l 2 L such that l
t
l
t
l for all l 2 L.4
We also maintain another key attribute of the discounted utility model.
3
In Gilboa and Schmeidler (1989), A2 is called certainty-independence and A3 is called
uncertainty aversion.
4
This axiom is adapted from Epstein and Schneider (2003).
9
A6 (Stationarity-ST): Assume T = 1. For each t 2 T and for all
l; l0 ; e
l; e
l0 2 L, if ls = e
ls and ls0 0 = e
ls0 0 for all s < t and s0 < t + 1 and if
0
0
for all s 2 T , then l
and e
lt+s = e
lt+1+s
lt+s = lt+1+s
l0
t+1
t
e
l0 .
e
l if and only if
A6 states that the passage of time does not alter a preference ordering.
Under this assumption, in (4),
D0 = Dt for any t 2 T .
D
Then, the following proposition is the stationary and in…nite-horizon version
of (3) (for the proof, see Appendix B):
Proposition 2: For T = 1, the following statements are equivalent:
(i) f
tg
satisfy A1 to A6.
(ii) There exist an a¢ ne function U : M ! R, a set [ ; ]
< 1, and a non-empty, closed, and convex set D
0<
R satisfying
B , each element
of which, b 2 D, is a strictly positive discount function b : T ! R++ , satisfying
P1
t is represented by Vt (:); where
=0 b = 1 such that: for each t 2 T ,
Vt (l)
minf
b2D
1
X
b
t U (l
)g =
min [(1
t+1 2[
=t
t+1 )U (lt )
+
t+1 Vt+1 (l)]:
(5)
; ]
Moreover, ; ; and D are unique; U is unique up to a positive a¢ ne transformation; and maxp2M U (p) and minp2M U (p) exist. Furthermore, D is recursively constructed from [ ; ], as shown in Appendix B.
At t = 0, our model (5) can be written as follows:
V0 (c)
min
B2B
1
X
t=0
10
Bt F (ct ;
t+1 );
(6)
where F (ct ;
t+1 )
(1
t+1 )U (ct )
processes B de…ned by B0
and B is a collection of discount-factor
1 and by Bt+1
t+1 Bt
with
t+1
2 [ ; ].
b de…ned by
Analogously, let Bb be a collection of discount-factor processes B
Rt
b0
bt = exp(
r d ), where the discount rate rt satis…es
B
1 and by B
=0
0<r
rt
r. Then a continuous time analogue of (6) can be written as
follows:
V0 (c)
min
B2Bb
Z
1
(7)
Bt F (ct ; rt )dt;
0
where a felicity function F satis…es appropriate conditions. Indeed, (7) is a
special form of variational utility as proposed by Geo¤ard (1996); (7) is also
shown to be equivalent to the recursive utility of Epstein (1987). Hence, in a
discrete time setting, this paper provides axiomatic foundations for a particular
form of those models.
The representations (3) and (5) also permit a binary relation of “more
time-variability averse”for pairs of preference relations that embody the same
ranking of a single-period consumption lottery. For example, in (3), Vt is more
time-variability averse than Vt0 if and only if U = U 0 , and
[ ;
where [ ;
] and [ ;
]
[ ;
]0 for all
2 ft + 1; :::; T g;
]0 represent sets of discount factors for Vt and Vt0 ,
respectively. A similar condition de…nes the relation for (5).
In addition, the discounted utility model is a special case of (3) or (5). This
relationship is most clearly shown by (5): if
=
= , (5) is equivalent to the
discounted utility model with the discount factor 0 <
Vt (l)
(1
1
X
)
t
U (l ):
< 1, i.e.,
(8)
=t
Therefore, (8) is the least time-variability averse in the class of (5), where the
DM is time-variability neutral.
11
3
Related Literature
Consider the following pair of questions from Loewenstein (1987), answered
by the same set of subjects, who were asked …rst Q1 then Q2:
[insert Table 1 about here]
We maintain the assumption of time homogeneous instantaneous utility. Without loss of generality, let U (Fancy French)
1 and U (Eat at home)
0. We
also restrict our attention to the subjects whose preferences show U (Fancy
lobster) > 0:5 because if U (Fancy lobster) is small, Q2 is similar to Q1.
As mentioned in the introduction, the answers to Q1 and Q2 suggest a
preference for spread. This preference may be attributed to an attitude toward
the utility variations between adjacent periods. To capture such an attitude,
Gilboa (1989) derives the following representation for the ex-ante preference:
V0 (c)
0 U (c0 )
+
T
X
t=1
where
t
j tj + j
t+1 j
with
0
f t U (ct ) +
0 and
T +1
t jU (ct )
U (ct 1 )jg ;
(9)
0. Motivated by Kahneman
and Tversky’s (1979) loss aversion (similar to gain/loss asymmetry in our context), Shalev (1997) extends this model and derives the representation under
which a positive increment is weighted di¤erently from a negative increment.
Loss aversion used in Shalev (1997) can explain A
C
B, but it also predicts
D, which is opposite to the pattern actually observed. On the other hand,
variation liking (
t
> 0) permitted in Gilboa (1989) can explain preference
patterns shown in Table 1. However, Loewenstein and Prelec (1993) conducted
a …ve-period version of Loewenstein’s (1987) experiment, in which a large
percentage of subjects answered as follows:
(F; H; H; H; H)
(H; H; F; H; H) but (F; H; H; H; L)
12
(H; H; F; H; L);
where F stands for “Fancy French”, H stands for “Eat at home”, and L stands
for “Fancy lobster”. Neither Gilboa (1989) nor Shalev (1997) explains this result because the utility change between the last two periods is identical in each
option in the questions. Thus, a preference for spread cannot be attributed to
an attitude toward the utility variations between adjacent periods.
On the contrary, Loewenstein and Prelec (1993) propose the following
heuristic model for the ex-ante preference that depends on the utility changes
in cumulative utility sequences:
T
X
V0 (c)
U (ct ) +
t=0
where dT
+
max fdt ; 0g +
min fdt ; 0g ;
(10)
0 and
dt
(T
t)
(
1
T
t
T
X
U (c )
=t+1
)
T
1 X
U (c ) if t < T .
T + 1 =0
The variable dt is de…ned by the departure of the average utility of future periods from a reference point, that is, the average utility of the whole interval.5
Thus, if good and bad outcomes are spread evenly over the whole interval, dt
‡uctuates less, and vice versa. In particular, if
+
<0<
, the DM strongly
dislikes the utility changes and prefers uniform utility sequences. These parameter values can support a preference for spread. For example, if
=
and
+
=
, (10) is consistent with A
0:8
+
B and C
+
< 0 and
D. Similarly, if
+
<0
, (10) is consistent with Loewenstein and Prelec’s (1993)
experiment (see Appendix D).
For comparison, we rewrite (3) as follows: VT (c)
Vt (c)
5
U (ct ) +
t
max fVt+1 (c)
U (ct ); 0g +
t
In Loewenstein and Prelec (1993), dt is de…ned by dt
which is identical to the quantity we de…ned in the text.
13
U (cT ) and if t < T ,
min fVt+1 (c)
t+1
T +1
T
=0 U (c
U (ct ); 0g .
)
t
=0 U (c
),
A key feature of (10) is still present: the future utility Vt+1 (c), which is a
preference-adjusted weighted average, is compared with the reference point.
In particular, observe that
if
t
=
Thus, if
t
t
T
=
T
<
t+1
for each t > 0, then Vt (c) =
t+2
T
T t+1
T t+2
<
t
X
1
f
U (c )g:
t + 1 =t
T
for each t > 0, the DM strongly dislikes the utility
changes and prefers uniform utility sequences. These parameter values can
support a preference for spread. For example, if [ 1 ;
[ 2;
2]
= [0:45; 0:56], (3) is consistent with A
1]
B and C
= [0:58; 0:75] and
D. Similarly, under
a certain set of discount factors, our model is consistent with Loewenstein and
Prelec’s (1993) experiment (see Appendix D).6
The main di¤erences between Loewenstein and Prelec (1993) and our model
are as follows: First, our model provides axiomatic foundations for a preference
for spread, whereas Loewenstein and Prelec (1993) is nonaxiomatic. Second,
the decisions based on our model are dynamically consistent, whereas those
based on Loewenstein and Prelec (1993) may be dynamically inconsistent.
More precisely, let T > 1 and 0 < t < T . As in Section 2, assume that each
conditional preference is independent of a consumption history. Then, the exante preference represented by (10) is inconsistent with conditional preference
at t if we assume
+
<0<
.7 This di¤erence is re‡ected in the de…nition of
reference utility: in our model, history independence and dynamic consistency
force the reference point to be U (ct ), whereas in Loewenstein and Prelec (1993),
the reference point is the average utility of the whole interval.
6
[ 4;
7
For example, [ 1 ;
4]
1]
= [0:63; 0:91]; [ 2 ;
2]
= [0:60; 0:81]; [ 3 ;
3]
= [0:58; 0:75]; and
= [0:45; 0:56].
Under the same assumption, the ex-ante preference represented by (9) is inconsistent
with conditional preference at t if
t
6= 0. A similar result holds for Shalev (1997).
14
Finally, gain/loss asymmetry is often interpreted in terms of reference dependence in the instantaneous utility function while maintaining utility independence. These are models of reference-dependent utility, where a gain or a
loss is de…ned by a departure of current consumption from reference consumption. As shown in Loewenstein and Prelec (1992), reference-dependent utility
is designed particularly to explain gain/loss asymmetry related to a status quo
e¤ect. On the contrary, our model introduces gain/loss asymmetry in discount
factors, where a gain or a loss is de…ned by a di¤erence in utility. Since the
reference sequence implied by our model is endogenous and speci…c to each
consumption sequence, gain/loss asymmetry does not derive from a status quo
e¤ect.
Appendix A: Proof of Proposition 1 (Finite Horizon)
Necessity of the axioms is routine. The proof of su¢ ciency is based on Lemmas
A.1-A.4. We adopt notation that is compatible with both …nite and in…nite
horizon settings. First, for an a¢ ne function Ut : M ! R, de…ne a function
Ut (:)t : L ! Bt by Ut (l)t = Ut (l ) for each
sequence de…ned on Tt induced by act l. Let
2 Tt ; Ut (l)t denotes a utility
t
be the -algebra that consists
of all subsets of Tt . By construction, each element in Bt is
A
be an indicator function of A 2
t,
t -measurable.
Let
t
and let 1 2 Bt be a constant function
of one. Assume that T < 1.
Lemma A.1: Assume A1-A4. Then for each t 2 T , there exist an a¢ ne
function Ut : M ! R and a non-empty, closed, and convex set Dt
Bt , each
element of which, bt 2 Dt , is a …nitely additive weighting function bt on
R
R t
2 Tt such that:
satisfying Tt 1 dbt = 1 and Tt f g dbt > 0 for all
15
t,
t
is
represented by Vt (:); where
Vt (l)
min
bt 2Dt
Z
Ut (l)t dbt :
(A.1)
Tt
Moreover, Dt is unique, and Ut is unique up to a positive a¢ ne transformation.
Proof. This follows from Theorem 1 of Gilboa and Schmeidler.
Lemma A.2: Assume A1-A4. Then for each t, there exist real numbers
and
t
satisfying
t
> 0 such that for all p 2 M , U (p)
U0 (p) =
t Ut (p) +
t
t.
Proof. For a given t < T , consider l; l0 2 L such that l = p 2 M and
t + 1 and l = l0 for all
l0 = q 2 M for all
and SMT, l
t+1
if and only if l
l0 if and only if p
t
t+1
< t + 1. By completeness
q. By completeness and DC, l
l. By completeness and SMT, l
t
t+1
l0 if and only if p
t
l0
q.
Given Lemma A.1, the conclusion follows by induction on t.
For the remaining lemmas, we use U in place of Ut . By non-degeneracy,
assume that [ 1; 1]
U (M ). In particular, let p[0] 2 M satisfy U (p[0] ) = 0.
For each t < T , let Ut
[ 1; 1]
f(U (lt ); Vt+1 (l))j l 2 Lg. By Lemmas A.1-A.2,
Ut . De…ne the following function It : Ut ! R by
[ 1; 1]
It (U (lt ); Vt+1 (l))
(A.2)
Vt (l):
By SMT and DC, (A.2) is well de…ned and increasing on Ut . Also, It (1; 1) = 1.
For Lemmas A.3-A.4, we adapt the arguments in Epstein and Schneider.
Lemma A.3:
satisfying 0 <
t+1
For a given t < T , there exists a unique [
t+1
It (U (lt ); Vt+1 (l)) =
t+1 ; t+1 ]
R
< 1 such that for all l 2 L,
min
[(1
t+1 2[ t+1 ; t+1 ]
16
t+1 )U (lt )
+
t+1 Vt+1 (l)]:
(A.3)
Proof. We adapt the arguments in Lemma 3.3 of Gilboa and Schmeidler.
(i) It is homogeneous on Ut : for any x; x0 2 Ut with x0 = x for 0 <
)p[0] and (U (lt ); Vt+1 (l)) = x.
It (x0 ) = It (x). Let l; l0 2 L satisfy l0 = l+(1
Since U is a¢ ne, U (lt0 ) =
1,
U (lt ). By (A.1), Vt+1 (l0 ) =
Vt+1 (l). By (A.2),
It (x) = Vt (l) and
It (x0 ) = It ( U (lt ); Vt+1 (l)) = It (U (lt0 ); Vt+1 (l0 )) = Vt (l0 ):
By (A.1), Vt (l0 ) = Vt (l) so that It (x0 ) = It (x).
(ii) Extend It by homogeneity to R
(iii) It is monotone on R
R.
R: for any x; x0 2 Ut , x
x0 implies It (x)
It (x0 ). This follows by construction.
(iv) It satis…es constant additivity on R
R: for any x; x0 2 R
R with
x0 = (x01 ; x02 ) and x01 = x02 , It (x + x0 ) = It (x) + It (x0 ). By homogeneity, we
may assume that 2x; 2x0 2 Ut . Let l; p 2 L satisfy (U (lt ); Vt+1 (l)) = 2x and
1
1
1
1
(U (p); Vt+1 (p)) = 2x0 . Since U is a¢ ne, U (lt ) + U (p) = U ( lt + p). By
2
2
2
2
1
1
1
1
(A.1), Vt+1 (l) + Vt+1 (p) = Vt+1 ( l + p). Hence, by (A.1), (A.2), and
2
2
2
2
homogeneity,
1
1
1
1
It (x + x0 ) = Vt ( l + p) = Vt (l) + Vt (p) = It (x) + It (x0 ):
2
2
2
2
(v) It is superadditive on R
R: for any x; x0 2 R
R, It (x + x0 )
It (x) + It (x0 ). By homogeneity, we may assume that 2x; 2x0 2 Ut . Let l; l0 2 L
satisfy (U (lt ); Vt+1 (l)) = 2x and (U (lt0 ); Vt+1 (l0 )) = 2x0 , where l0 = l0 0 for
1
1
1
1
all ; 0 > t. Since U is a¢ ne, U (lt ) + U (lt0 ) = U ( lt + lt0 ). By (A.1),
2
2
2
2
1
1
1 1 0
0
Vt+1 (l)+ Vt+1 (l ) = Vt+1 ( l+ l ). Hence, by (A.1), (A.2), and homogeneity,
2
2
2 2
1
1
It (x + x0 ) = Vt ( l + l0 )
2
2
1
1
Vt (l) + Vt (l0 ) = It (x) + It (x0 ):
2
2
17
The conclusion follows from Lemma 3.5 and Theorem 1 of Gilboa and
Schmeidler.
For given t+1 2 [ t+1 ;
Z
t t
at dbt (1
b (a ) =
Tt
and bt+1 2 Dt+1 , construct bt 2 Bt by
Z
t
at+1 dbt+1 for each at 2 Bt , (A.4)
t+1 )at + t+1
t+1 ]
Tt+1
where at = (att ; at+1 ) with att 2 R and at+1 2 Bt+1 . De…ne a nonempty set
D0t
D0t
Bt by
fbt 2 Bt jbt satis…es (A.4) for some
t+1
2[
t+1 ; t+1 ]
and bt+1 2 Dt+1 g:
R t
Each bt 2 D0t is a …nitely additive weighting function on t , where Tt 1 dbt = 1
R
and Tt f g dbt > 0 for all 2 Tt . Since [ t+1 ; t+1 ] and Dt+1 are closed and
convex in the respective space, so is D0t in Bt ; D0t is also compact in Bt .
Lemma A.4: For a given t < T , D0t = Dt .
Proof. For all l 2 L,
Z
U (l)t dbt
Vt (l) = min
t
b 2Dt
=
Tt
min
t+1 2[ t+1 ; t+1 ]
=
min
Z
= min
U
0
t
t+1 2[ t+1 ; t+1 ]
b 2Dt
f(1
t+1 )U (lt )
f t+1min
b
2Dt+1
f(1
(by (A.3))
Z
U (l)t+1 dbt+1 gg
t+1 )U (lt ) + t+1
+
t+1 Vt+1 (l)g
Tt+1
(l)t dbt :
Tt
Hence, D0t with U represents
t.
The conclusion follows from the uniqueness
of Dt , as implied by non-degeneracy.
Finally, (ii) of Proposition 1 follows by recursively applying Lemmas A.3
to A.4 from time T
1 to 0.
18
Appendix B: Proof of Proposition 2 (In…nite Horizon)
Necessity of the axioms is routine. For su¢ ciency, by BW, Lemmas A.1-A.4
follows from Proposition 4.1 of Gilboa and Schmeidler, where maxp2M U (p)
and minp2M U (p) exists. We also use B ,
, and b in place of Bt ,
t,
and bt ,
respectively. Then the following Lemmas A.5-A.7 prove (ii) of Proposition 2.
There exist unique D
Lemma A.5:
Dt = D in (A.1) and [
t+1 ; t+1 ]
B and [ ; ]
(0; 1) such that
= [ ; ] in (A.3) for all t 2 T .
Proof. This follows from ST and the uniqueness of Dt .
Given Lemma A.5, let f t g1
1 be a sequence of single-period discount factors,
where
t
1
1. From f t g1
1 , de…ne a sequence f t g0 by
2 [ ; ] for each t
1 and
0
t
t t 1
for t > 0:
1
Construct b 2 B from f t g1
1 and f t g0 as follows: for each t 2 T ,
t
b(a )
1
X
b
ta
t
=t
for each at 2 Bt , where bs
De…ne a nonempty set
s
s+1
for s
0:
(A.5)
B by
1
fb 2 B jb satis…es (A.5) for some admissible f t g1
1 and f t g0 g:
Each element of
is a strictly positive discount function b : T ! R++ such
P1
P1
that
+1 ) = 0 = 1; b is also a discrete and countably
=0 (
=0 b =
additive weighting function on . In addition,
Lemma A.6:
Proof. For any
is closed and compact in B .
is convex.
2 (0; 1) and any b; b0 2
eb is based on fet g1 such that e1 =
1
et =
t
+
(1
t
, let eb
+ (1
)
0
0
t 1( t
t)
1
)
1 + (1
19
)
0
t 1
0
1
b + (1
)b0 . Then,
2 [ ; ] and for t > 1,
2 [ ; ];
1
0 1
0 1
0
e
where f t g1
1 and f t g0 de…ne b and f t g1 and f t g0 de…ne b . Hence, b 2
so that for all l 2 L, Vt (l) = minf
Lemma A.7: D =
b2
Proof. Given (A.5), it su¢ ces to show D = . For D
P1
=t
b
t U (l
.
)g.
, let bb 2 D. For
b
a given t, denote by f g1
t+1 a sequence of single-period discount factors of b
de…ned by (A.4). Let
k
t+k k 1
0, there exists b(n) 2
for each integer n
all
1 and let
0
for k > 0. Then, by (A.5),
such that b
(n)
t
=
+1 t
t
for
2 ft; :::; t + ng. By recursively applying (A.4), for each at 2 Bt ,
Z
t+n
X
t
bb(at ) =
( t
at+n+1 dbbt+n+1 , and
+1 t )a + n+1
Tt+n+1
=t
b(n) (at ) =
t+n
X
(
t
t
t )a +
+1
n+1
=t
Therefore,
jbb(at )
b
(n)
t
(a )j =
<
1
X
b
=t+n+1
n+1 j
Z
a
t+n+1
Tt+n+1
n+1 (3 sup ja
2Tt
The bound is independent of n, and
t
j):
n+1
dbbt+n+1
(n)
t
at+n+1 :
n+1
1
X
b
=t+n+1
(n)
t
n+1
at+n+1 j
converges to zero. Hence,
bb(at ) = lim b(n) (at ):
(A.6)
n!1
If bb = b(n) for some n, bb 2
is a limit point of
. Otherwise, since (A.6) holds for any at 2 Bt , bb
. Hence, bb 2
because
is closed in B .
Given that D is closed in B , a similar argument proves D
.
Appendix C: Comparison with Epstein and Schneider (2003)
Consider the following setting: Let
be a state space. Assume a …xed in-
formation structure de…ned by a …ltration fFt g, where each Ft is based on a
20
…nite partition and Ft (!) is the partition element containing ! 2 . An act is
a function h = (ht ), where each ht :
! M is measurable with respect to Ft .
At each t and !, the preference ordering
t;!
is de…ned on a set of all acts.
By adapting a version of Gilboa-Schmeidler’s (1989) multiple-priors axioms, Epstein and Schneider (2003) derive the representation for each
t;! .
In
particular, to derive a representation similar to (3), we assume the following
information structure that resembles the passage of time:
= T ; Ft (!) = f!g if ! < t, and Ft (!) = Tt if !
Then
t.
(A.7)
is represented by
8
9
+
< (1
=
t;t )[U (ht (t)) + Vt+1;t (h)]
;
Vt;t (h)
min +
+
+
: + + [U (h (t + 1)) + V
;
t;t 2[ t;t ; t;t ]
t
t+1;t+1 (h)]
t;t
(A.8)
t;t
where 0 <
< 1 is a discount factor and [
+
+
t;t ; t;t ]
(0; 1) is a set of one-step-
ahead priors.
The crucial di¤erence between (A.8) and our model is most clearly shown
by the key axiom of dynamic consistency used in Epstein and Schneider (2003):
If h (:) = h0 (:) for all
t and if h
t+1;! 0
h0 for all ! 0 ; then h
The above assumes that the preference ordering
!. However, in our setting,
t+1;!
t+1;!
t;!
h0 :
is well de…ned for all
is unde…ned if ! 6= t + 1 because it is not
possible to go back to t from t + 1 or to go to t + 2 if we are still at t + 1.
Hence, the crucial utility, Vt+1;t (h), is unde…ned. This implies that even if we
simplify acts, adapting (A.8) under (A.7) is inconsistent with our setting.
21
Appendix D: Parameter Values for (3) and (10)
For (10), A
<
B implies
U (L) + 1
U (L) 2
>
+
. To obtain C
B and C
< 0 and
< 0 if U (L)
2.
D if
+
0:5
D,
+
if 0:5 < U (L) < 2; and
Hence, for any U (L) > 0:5, A
+
+
0:5
+
<
.
A similar computation shows that to explain Loewenstein and Prelec’s (1993)
experiment for any U (L) > 0:5,
+
For (3), (i) A
3
7
< 0 and
B implies
+
1 (1
<
2)
+
0:818
> (1
1 ).
.
To obtain C
D:
(ii) 0:5 < U (L) < 1:
1 2
(iii) U (L)
1 2
> 0 and (
1 2
1 and V1 (C) < 1: (
(iv) U (L) > 1 and V1 (C)
It is easy to see that [ 1 ;
1 2 )U (L)
1]
1 (1
1 2 )U (L)
1 2
1: (1
>
1)
>
1 (1
= [0:58; 0:75] and [ 2 ;
2)
>
1 (1
(1
1 ):
2)
(1
1 ).
2 ).
2]
= [0:45; 0:56] satisfy
(i)-(iv) for any U (L) > 0:5. A similar computation shows that under sets of
discount factors shown in footnote 6, (3) is consistent with Loewenstein and
Prelec’s (1993) experiment for any U (L) > 0:5.
The above computations indicate that for both (3) and (10), it becomes
harder to support a preference for spread if U (L) < 1. In this case, A
B implies a preference for improving sequences, whereas C
D implies a
preference for declining sequences; a preference for spread does not frequently
dominate these two con‡icting motives. It also becomes harder to support a
preference for spread in Loewenstein and Prelec’s (1993) experiment because
sequence C positions good outcomes too far apart.
22
4
References
1. Anscombe, F., and R. Aumann (1963): “A De…nition of Subjective Probability,”Annals of Mathematical Statistics, 34(1), 199-205.
2. Epstein, L. (1987): “The Global Stability of E¢ cient Intertemporal Allocations,”Econometrica, 55(2), 329-355.
3. Epstein, L., and M. Schneider (2003): “Recursive Multiple-Priors,”Journal of Economic Theory, 113(1), 1-31.
4. Frederic, S., G. Loewenstein, and T. O’Donoghue (2002): “Time Discounting and Time Preference: A Critical Review,”Journal of Economic
Literature, 40(2), 351-401.
5. Geo¤ard, P. (1996): “Discounting and Optimizing: Capital Accumulation Problems as Variational Minmax Problems,” Journal of Economic
Theory, 69(1), 53-70.
6. Gilboa, I. (1989): “Expectation and Variation in Multi-Period Decisions,”Econometrica, 57(5), 1153-1169.
7. Gilboa, I., and D. Schmeidler (1989): “Maxmin Expected Utility with
Non-Unique Prior,”Journal of Mathematical Economics, 18(2), 141-153.
8. Kahneman, D., and A. Tversky (1979): “Prospect Theory: An Analysis
of Decision under Risk,”Econometrica, 47(2), 263-292.
9. Koopmans, T. (1960): “Stationary Ordinal Utility and Impatience,”
Econometrica, 28(2), 287-309.
10. Loewenstein, G. (1987): “Anticipation and the Valuation of Delayed
Consumption,”Economic Journal, 97(387), 666-684.
23
11. Loewenstein, G., and D. Prelec (1992): “Anomalies in Intertemporal
Choice: Evidence and an Interpretation,” Quarterly Journal of Economics, 107(2), 573–597.
12. Loewenstein, G., and D. Prelec (1993): “Preferences for Sequences of
Outcomes,”Psychological Review, 100(1), 91-108.
13. Samuelson, P. (1937): “A Note on Measurement of Utility,” Review of
Economic Studies, 4(2), 155-161.
14. Shalev, J. (1997): “Loss Aversion in a Multiple-Period Model,” Mathematical Social Sciences, 33(3), 203-226.
15. Wakai, K. (2002): “Linking Behavioral Economics, Axiomatic Decision
Theory and General Equilibrium Theory,”Ph.D. Dissertation, Yale University.
Option Weekend 1
8
Table 18
Weekend 2
Weekend 3
Choices
Eat at home
16%
Fancy French Eat at home
84%
Q1 A
Fancy French Eat at home
Q1 B
Eat at home
Q2 C
Fancy French Eat at home
Q2 D
Eat at home
Fancy lobster 54%
Fancy French Fancy lobster 46%
Percentages shown in this table are from Example 3 of Loewenstein and Prelec (1993).
In Loewenstein (1987), the choice for C is reported as 57%.
24