Discounted Utility and Present
Value— A Close Relation
Han Bleichrodta , Umut Keskina , Kirsten I.M. Rohdea ,
Vitalie Spinua , & Peter P. Wakkera *
a
: Erasmus School of Economics, Erasmus University Rotterdam,
P.O. Box 1738, Rotterdam, 3000 DR, the Netherlands;
+31-(0)10 - 408.91.62 (F); [email protected];
*: corresponding author
May, 2013
Abstract
We introduce a new type of preference conditions for intertemporal choice, requiring independence of present values from various other
variables. The new conditions are more concise and more transparent than traditional ones. They are natural, and directly related to
applications, because present values are widely used in intertemporal choice. Our conditions give more general behavioral foundations,
which facilitates normative debates and empirical tests of time inconsistencies and related phenomena. Like other preference conditions,
our conditions can be tested qualitatively. Unlike other preference
conditions, however, our conditions can also be directly tested quan-
1
titatively, e.g. to verify the required independence of present values
from predictors in regressions. We show how similar types of preference conditions, imposing independence conditions between directly
observable quantities, can be developed for other decision contexts
than intertemporal choice, and can simplify behavioral foundations
there.
JEL-classi…cation: D90, D03, C60
Keywords: present value, discounted utility, time inconsistency, preference
foundation
2
1
Introduction
Debates about appropriate models of intertemporal choice, both for making
rational decisions and for describing and predicting human decisions, usually focus on the critical preference conditions that distinguish between those
models. The two most discussed conditions are time consistency, distinguishing between constant and hyperbolic discounting, and intertemporal separability, pertaining to habit formation, addiction, and sequencing e¤ects.1 Both
these conditions are assumed in the classical models but are usually violated
empirically. Their normative status has also been questioned.2
We introduce a new kind of preference conditions, stated directly in terms
of present values (PVs). PVs are simple and tractable, and have been widely
used in intertemporal choice throughout history (de Wit 1671; Fisher 1930).
Hence preference conditions in terms of PVs are simple and natural too. In
general, PVs can depend on many variables, such as the times of the receipt
of future payo¤s, the initial wealth levels at those times, and the wealth levels
1
See Attema (2012), Crawford (2010), Dolan & Kahneman (2008 p. 228), Epper, Fehr-
Duda, & Bruhin (2011), Frederick, Loewenstein, & O’Donoghue (2002), Gilboa (1989),
Herrnstein (1961), Gul & Pesendorfer (2007), Loewenstein & Prelec (1993), Tsuchiya &
Dolan (2005), Wakai (2008), and others.
2
See Broome (1991), Gold et al. (1996 p. 100), Hammond (1976), Par…t (1984 Ch. 14),
Strotz (1956 p. 178), and others.
3
at other times. Our preference conditions will impose independence of PVs
from part of those other variables, and we show that these independencies
characterize various models of intertemporal choice. Like all preference conditions, our conditions can be tested qualitatively. Unlike other preference
conditions, our conditions can also be directly tested quantitatively, amounting to independence of PVs from particular predictors in regressions. The
latter tests are more widely known than qualitative tests and can be used in
econometric techniques, using better established error theories.
To illustrate our new PV conditions clearly, we will apply them to wellknown models. A concise presentation of these models and their representations is in Table 1. Details of the table will be explained in following sections.
The table illustrates the organization of the …rst four sections.
We provide the most concise preference foundations presently available in
the literature for: (a) constant discounted value as commonly used in …nancial economics (Hull 2013); (b) constant discounted utility (Samuelson 1937);
(c) general discounted utility, which includes hyperbolic discounting; (d) nobook and no-arbitrage for uncertainty; (e) Gorman (1968)-Debreu (1960)
additive separability for general aggregations. All our preference conditions
are given in the PV condition column in Table 1. Their statements take little space in this column, but yet give complete de…nitions as explained later.
Extensions to decision under uncertainty, and to other decision contexts, are
4
a topic for future research, as are extensions to more general intertemporal
models (Section 5). The follow-up paper Keskin (2013) provides extensions
for some popular hyperbolic discount models.
2
Preferences and subjective PVs
We consider preferences < over outcome sequences x = (x0 ; : : : ; xT ) 2 RT +1 ,
yielding outcome xt in period t; t = 0 denotes the present. To avoid trivialities, we assume T
2; T is …xed. Besides indexed Roman letters xt that
specify the period t of receipt, Greek letters
; ; : : : are used to designate
outcomes (real numbers) when no period of receipt needs to be speci…ed. By
tx
we denote x with xt replaced by .
V represents < if V : RT +1 ! R and x < y , V (x)
V (y), for
all x; y 2 RT +1 . If a representing V exists, then < is a weak order; i.e.,
< is complete (x < y or y < x for all x; y) and transitive. We assume
throughout that < is a weak order. Strict preference
, indi¤erence
,
reversed preference 4, and strict reversed preference ( ) are as usual. We
also assume monotonicity (strictly improving an outcome strictly improves
the outcome sequence) and continuity of < throughout.
The following condition considers sums of outcomes xt = ' + et . Here we
call et an initial endowment. We sometimes use the term payo¤ instead of
5
6
t=0
PT
t=0
PT
t=0
PT
t=0
PT
t=0
PT
t=0
PT
xt
U (xt )
Ut (xt )
t
U (xt )
t xt
t
xt
Functional
Separable
Stationary
Separable +
Symmetric
Separable +
Additive
Stationary
Additive +
Symmetric
Additive +
('; t; x0 ; xt )
0
1
(U (et + ')
( t U (et + ')
U (et ) + U (e0 ))
1
Ut (et ) + U0 (e0 ))
U0 1 (Ut (et + ')
t
U
U (et ) + U (e0 ))
U
e0
e0
e0
is a
= 1; U is a strictly increasing continuous utility function; Ut is a period-
t
“PV formula”gives the formula of PV under each theory.
discounting) used in our preference foundations. Note that these cells contain complete de…nitions. The column
condition” gives the PV condition (how ('; t; e) can be rewritten, with an extra equality imposed for constant
traditionally used in preference foundations of the functional forms, and de…ned in the Appendix. The column “PV
dependent strictly increasing continuous utility function. The column “Pref. conditions”gives preference conditions
period-dependent discount factor with
'
t'
t
'
PV formula
is a discount factor; 0 <
('; t + 1; e01 = e0 ; e0t+1 = et )
('; t; e0 ; et ) =
('; x0 ; xt )
('; t)
('; t) = ('; t + 1)
(')
Pref. conditions PV condition
Table 1: The column “Functional” gives the functional forms. Here: 0 <
discounted utility
Time-dependent
discounted utility
Constant
discounted utility
Non-
discounted value
Time-dependent
discounted value
Constant
discounted value
Non-
Theory
outcome for '. Our terminology only serves to facilitate interpretations, and
there is no formal role for reference dependence.
DEFINITION 1
is the present value (PV ) of outcome ' received in period
t with (prior) endowment e = (e0 ; : : : ; eT ) if (e0 + )0 e
(e0 + ; e1 ; : : : ; et ; : : : ; eT )
(et + ')t e, i.e.
(1)
(e0 ; : : : ; et + '; : : : ; eT ):
Eq. 1 means that, with e the current endowment, receiving an additional
future payo¤ ' in period t is exactly o¤set by receiving an additional present
outcome
. That is, the PV of a future payo¤ ' in period t is
. For
simplicity, we assume throughout that a PV always exists. By monotonicity,
it is unique. The de…nition of PV
can be naturally extended to an outcome
sequence x with endowment e: (e0 + )0 e
e + x. However, we will not need
this concept in this paper.
The PV can in general depend on '; t, and e, and is a function ('; t; e).
For t = 0, it trivially follows that ('; 0; e) = '. Note that
is subjective,
depending on < and, thus, re‡ecting the tastes and attitudes of the decision maker. The preference conditions presented in the following sections
all amount to independence of PV from some of the variables ('; t; e). We
denote this independence by writing only the arguments that
7
depends on.
For example, if ('; t; e) depends only on ' (i.e. is independent of t and e)
then we write
= ('):
Similarly, if
(2)
depends on e only through e0 and et , then we write
= ('; t; e0 ; et ):
(3)
These conditions, like all other PV conditions in this paper, are indeed preference conditions. Our conditioins impose restrictions on indi¤erences. Preference conditions should be directly veri…able from preferences, the observable
primitives in the revealed preference paradigm, without invoking theoretical
constructs such as utilities. That our PV conditions satisfy these requirements is further illustrated by a rewriting, directly in terms of preferences,
in Eq. 11 in the appendix. Our conditions can thus be tested in the same
manner as all qualitative preference conditions can be.
Because our preference conditions are stated as usual independence conditions between variables, they can also be directly tested using analyses
of variance and regression techniques. We can take PV as the dependent
variable, and then Eq. 3 predicts that '; t; e0 , and e1 may be signi…cant predictors, but the ej ’s with j 6= 0; t will not be. This way we can readily use
the probabilistic error theories underlying econometric regressions, which are
easier to use than the more recently developed error theories for preferences
8
(Wilcox 2008).
3
Linear Utility
This section considers models with linear utility, as commonly used in …nancial markets. They can serve as approximations for subjective individual
choice if the stakes are small (Epper, Fehr-Duda, & Bruhin 2011; Luce 2000
p. 86; Pigou 1920 p. 785). The …rst model in Table 1 maximizes the sum of
outcomes:
Non–Discounted Value:
T
X
(4)
xt :
t=0
This model does not involve subjective parameters and, hence, is directly
observable. It serves well as a …rst illustration of the nature of PV conditions.
THEOREM 2 The following two statements are equivalent:
(i) Non-discounted value holds.
(ii)
= (').
Throughout this paper, Condition n.(ii) refers to the condition in Statement (ii) of Theorem n. Theorem 2 shows that if
depends only on '
(Condition 2.(ii)), then in fact it must be the identity function ( (') = ').
This is also displayed in the last two columns of the corresponding row of
Table 1. It may also be surprising that Condition 2.(ii) implies the addition
9
operation of non-discounted value. An informal proof in the appendix will
clarify these points.
Our next model involves a subjective parameter, the discount weight :
Constant Discounted Value:
T
X
t
xt f or
(5)
> 0:
t=0
Under the usual assumption that the decision maker is impatient, we have
1. In PV calculations of cash ‡ows, …nancial economists commonly
use constant discounted value. Then
= 1=(1 + r) with r the interest
or discount rate. The discount factor
then, however, is not a subjective
parameter re‡ecting the attitude of a decision maker, as in our approach,
but it is a given constant, publicly known and determined by the market.
The latter approach can be related to our approach where
is subjective
if we interpret the market as a decision maker, setting rational PVs. Then
re‡ects the market attitude. Condition 2.(ii) rationalizes this common
evaluation system. If the parameter
re‡ects the attitude of an individual
decision maker, then it will probably be in‡uenced by the market discount
factor but need not be identical to it.
As a preparation for the preference foundation, we de…ne tomorrow’s
value as an analogue to PV:
DEFINITION 3
is tomorrow’s value of payo¤ ' received in period t (t
10
1)
with endowment e, denoted
= ('; t; e), if (e1 + )1 e
(e0 ; e1 + ; e2 ; : : : ; et ; : : : ; eT )
Now
(et + ')t e, i.e.,
(e0 ; e1 ; : : : ; et + '; : : : ; eT ):
(6)
is the extra payo¤ in period 1 that exactly o¤sets the extra payo¤
' in period t. It is tomorrow’s PV.
THEOREM 4 The following two statements are equivalent:
(i) Constant discounted value holds.
(ii)
3
= ('; t) = ('; t + 1) = .
Uniqueness of the parameter , and all other uniqueness results of this
paper, are given in the proofs in the Appendix. Condition 4.(ii) entails that
and
are independent of the endowments, and that tomorrow’s perception of
future income is the same as today’s. The condition implies that PV depends
only on stopwatch time (time di¤erences) and not on calendar time (absolute
time).
3
The notation here is short for:
= ('; t; e) = ('; t) = ('; t + 1) = ('; t + 1; e0 ).
The two endowments e and e0 are immaterial, and are allowed to be di¤erent. Because of
the shift by one period, the condition is imposed only for all t < T . Existence of
the equality in Statement 4.(ii) imply that
also exists.
11
and
Many studies have shown that constant discounting is violated empirically. Hence the following model is of interest.
Time-Dependent-Discounted Value:
T
X
t xt (with
0
= 1):
(7)
t=0
This model allows for general discount weights that depend on time in any
manner possible. Many special cases of such discount weights have been
studied in the literature, including hyperbolic weighting (Doyle 2013). The
representation in Eq. 7 is not a¤ected if all
positive factor. The scaling
0
ts
are multiplied by the same
= 1 re‡ects the most common convention in
the literature.
THEOREM 5 The following two statements are equivalent:
(i) Time-dependent discounted value holds.
(ii)
= ('; t).
An implication, displayed in the last two columns of Table 1, is that if
is any function of ' and t, then it must be the function
4
t '.
Nonlinear Utility
The models presented in the preceding section take a sum, weighted or unweighted, of the outcomes. Thus, these models assume constant marginal
12
utility in the sense that an extra euro received in a particular period gives
the same utility increment regardless of the endowment of that period. In
individual choice this condition is often empirically violated and not normatively imperative. More realistic and more popular models allow for nonlinear
utility. If marginal utility does depend on the endowment, then the models
of the preceding section become:
Non-Discounted Utility:
Constant Discounted Utility:
T
X
t=0
T
X
t
U (xt );
(8)
U (xt );
(9)
Ut (xt ):
(10)
t=0
Time-Dependent Discounted Utility:
T
X
t=0
Eq. 9 is Samuelson’s (1937) discounted utility, the most popular model for
intertemporal choice. Each utility-model reduces to the corresponding valuemodel if utility is taken linear. In particular, in time-dependent discounted
utility, if the Ut ( ) are linear then they can be written as
can renormalize such that
0
t
, where we
= 1. Then time-dependent discounted value
results.
The mathematics underlying the preference foundations for the utility
models in Eqs. 8-10 is more advanced than it was for Theorems 2 4, and
5. Those theorems in fact solved linear equalities. We now have to deal
13
with general equalities, with nonlinear utilities intervening. Fortunately, the
increased complexity of the underlying mathematics does not show up in
the preference conditions and, consequently, in the empirical tests of the
models. The relevant PV preference conditions are obtained directly from
those de…ned in §3 by adding dependence on the endowment levels e0 ; et .
This way we readily obtain Theorems 6, 7, and 8 from Theorems 2, 4, and
5, respectively. The transparancy of our preference conditions clari…es the
logical relations between the di¤erent results.
THEOREM 6 The following two statements are equivalent:
(i) Non-discounted utility holds.
(ii)
= ('; e0 ; et ).
4
An implication, displayed in the last two columns of Table 1, is that if
is any function of ', x0 , and xt , then it must be of the form displayed there.
Studies providing preference foundations for non-discounted utility (sums
of utilities) include Fishburn (1992), Kopylov (2010), Krantz et al. 1971), Pivato (2013), and Wakker (1989). Blackorby, Davidson, & Donaldson (1977)
4
Here t refers to the period where ' is added and et speci…es the endowment in that
period. Note that there is no explicit dependence of
suppressing period t from the arguments of (: : :).
14
on t, fact that we denote by
provided results for lotteries with T +1 equally likely lotteries, taking the representation as von Neumann-Morgenstern expected utility. Several authors
have argued that any discounting, even if consistent over time, is irrational,
and have thus recommended non-discounted utility for intertemporal choice
(Jevons 1871 pp. 72-73; Ramsey 1928; Rawls 1971; Sidgwick 1907). Condition 6.(ii) characterizes this proposal. We now turn to discounting.
THEOREM 7 The following two statements are equivalent:
(i) Constant discounted utility holds.
(ii)
= ('; t; e0 ; et ) = ('; t + 1; e01 = e0 ; e0t+1 = et ).5
The …rst preference foundation of constant discounted utility was in Koopmans (1960, 1972), with generalizations in Harvey (1995), Bleichrodt et al.
(2008), and Kopylov (2010). Condition 7.(ii) requires that tradeo¤s are made
5
The condition implies that the endowment levels ej ; j 6= 0; t of the PV endowment e,
and similarly the endowment levels e0j ; j 6= 1; t + 1 of the tomorrow-value endowment e0 ,
are immaterial. Existence of
holds for all 0
t
T
and the equality imply that
also exists. The condition
1, and whenever e01 = e0 and e0t+1 = et . We use the convenient
argument matching notation popular in programming languages (R: see R Core Team 2013,
Python: see Python Software Foundation 2013, Scala: see École Polytechnique Fédérale
de Lausanne 2013, among others) to express the latter two restrictions. Here the formal
argument of a function (e01 or e0t+1 ) is assigned a value (e0 or et ).
15
the same way tomorrow as they are today. Such requirements have sometimes
been taken as rationality requirements (see introduction).
THEOREM 8 The following two statements are equivalent:
(i) Time-dependent discounted utility holds.
(ii)
= ('; t; e0 ; et ).
Statement 8.(ii) expresses that the tradeo¤s between periods 0 and t are
independent of what happens in the other periods, re‡ecting a kind of separability. Time-dependent discounted utility is a general additive representation,
which has been axiomatized several times before.6 It implies intertemporal
separability, which is arguably the most questionable assumption of most
intertemporal choice models (see introduction).
5
Applications to contexts other than intertemporal choice
The mathematical theorems provided in previous sections, and the type of
preference conditions that we used, can be used in contexts other than intertemporal choice. For instance, Theorem 5 is of special interest for decision
6
Debreu (1960), Gorman (1968), Krantz et al. (1971); Wakker 1989).
16
under uncertainty, capturing nonarbitrage in …nance. To see this point, we
reinterpret the periods t as states of nature. Exactly one state obtains, but
it is uncertain which one (Savage 1954). Now x = (x0 ; : : : ; xT ) refers to an
uncertain prospect yielding outcomes (payo¤s) xt if state of nature t obtains.
For simplicity, we focus on a single time of payo¤ here, so that discounting
plays no role. If we divide the discount weights
laxing the requirement of
0
t
by their sum
PT
t=0
t
(re-
= 1), then they sum to 1, and the representation
becomes subjective expected value.
Subjective expected value was …rst axiomatized by de Finetti (1937) using
a no-book argument, called no-arbitrage in …nance. Then the representation
is as-if risk neutral, and the decision maker is the market which sets rational
prices for state-contingent assets. For state j, a state-contingent asset x =
(0; : : : ; 0; 1; 0; : : : ; 0) yields outcome 1 if j happens and nothing otherwise. In
this interpretation,
j
becomes the market price of this state-contingent asset,
and PV is the o¤setting quantity of state-0-contingent assets. Condition
5.(ii) provides the most concise formulation of the no-book and no-arbitrage
argument presently available in the literature.
For decision under uncertainty, certainty equivalents are more natural
quantities than state-contingent prices. Characterizing models using independence conditions for certainty equivalents, or other natural quantities in
decision under uncertainty, is a topic for future research. Analogs of time con17
sistency for uncertainty are central in derivations of expected utility (Karni
& Safra 1989).
Eq. 8 can be interpreted as von Neumann-Morgenstern expected utility for
equal-probability lotteries, which essentially covers all lotteries with rational
probabilities (writing every probability i=j as i probabilities 1=j). Eq. 8
can also be interpreted as ambiguity under complete absence of information
(Gravel, Marchant, & Sen 2012). Eq. 10 concerns Debreu’s (1960) additively
separable utility. Here again, our Statement 8.(ii) provides the most concise
preference axiomatization presently available in the literature.
6
Conclusion
We have introduced a new kind of preference conditions for intertemporal
choice, requiring that (quantitative) PVs are independent of particular other
variables. The resulting conditions can be used both in qualitative and in
quantitative tests, and are more concise and transparent than preceding conditions in the literature. The technique of obtaining preference conditions by
taking a quantity with direct empirical meaning and then imposing independence conditions can be extended to other decision contexts such as decision
under uncertainty. Then new, more concise characterizations result in those
domains, which can easily be tested empirically.
18
Appendix. Proofs, uniqueness results,
and clari…cation of the empirical status
of PV conditions
We present the proofs, with added the uniqueness results, of our theorems
in reversed order, from most to least general, because this approach is most
clarifying and also most e¢ cient. We present the proofs in such a way that
they clarify how our PV conditions are related to well known preference
conditions. Thus we further clarify that PV conditions indeed are preference
conditions. In each proof, we start from our PV condition, which is always
weaker than the conditions that are derived and that are commonly used in
the literature. We thus show that our PV conditions give stronger results.
Because each Statement (ii) is immediately implied by substitution of the
functional, we throughout assume Statement (ii) and derive Statement (i)
and the uniqueness results.
Proof of Theorem 8, and uniqueness of its parameters
The following uniqueness result holds for Statement 8.(i), which uses Eq.
10: A time-dependent real constant
t
can be added to every Ut , and all
Ut can be multiplied by a joint positive constant . This follows from well-
19
known uniqueness results in the literature (Krantz et al. 1971 Theorem 6.13;
Wakker 1989 Observation III.6.6’). We next derive Statement 8.(i).
The equality ('; t; e) = ('; t; e0 ; et ) in Condition 8.(ii) means that
is
independent of ej ; j 6= 0; t. This holds if and only if
(e0 ; e1 ; : : : ; et 1 ; et + '; et+1 ; : : : eT ) )
(e0 + ; e1 ; : : : ; et 1 ; et ; et+1 ; : : : eT )
(e0 + ; e01 ; : : : ; e0t 1 ; et ; e0t+1 ; : : : e0T )
(e0 ; e01 ; : : : ; e0t 1 ; et + '; e0t+1 ; : : : e0T ):(11)
This holds if and only if the implication holds with twice preference < instead
of indi¤erence
.7 Eq. 11 with preference instead of indi¤erence is known
as separability of f0; tg (Gorman 1968). By repeated application of Gorman
(1968), separability of every set f0; tg holds if and only if < is separable; i.e.,
every subset of f0; : : : ; T g is separable (preferences are independent of the
levels where outcomes outside this subset are kept …xed, as with separability
of f0; tg). As is well known, this holds if and only if an additively decomposable representation holds8 , which we call time-dependent discounted utility
in the main text.
7
To wit, if the upper indi¤erence is changed into a strict preference, then we …nd
to give indi¤erence. The lower indi¤erence follows with
replacing
8
0
by
0
0
<
instead of . By monotonicity,
leads to the lower strict preference.
See Debreu (1960), Gorman (1968), Krantz et al. (1971 Theorem 6.13), Wakker (1989
Theorem III.6.6).
20
Proof of Theorem 7, and uniqueness of its parameters
The following uniqueness result holds for Statement 7.(i), which uses Eq.
9: A real-valued time-independent constant
can be multiplied by a positive constant .
can be added to U , and U
is uniquely determined. These
uniqueness results follow from those in Theorem 8, where, in terms of Eq.
10,
is the proportion Ut+1 =Ut for any t. It is useful to note that the sum
of weights,
P
t
, is the same for each outcome sequence evaluated, implying
that there is no special role for utility value 0. We next derive Statement
7.(i).
The equality ('; t; e0 ; et ) = ('; t + 1; e01 = e0 ; e0t+1 = et ) in Condition
7.(ii) means that
is independent of ej ; j 6= 0; t; T (as in Theorem 8, but
now only for t < T ), but also of whether it is measured in period 0 or period
for e0 = e01 and
1. This holds if and only if, writing
( + ; e1 ; : : : ; et 1 ; ; et+1 ; : : : eT )
(e00 ;
for et = e0t+1 ,
( ; e1 ; : : : ; et 1 ; + '; et+1 ; : : : eT ) ,
+ ; : : : ; e0t 1 ; e0t ; ; : : : e0T )
(e00 ; ; : : : ; e0t 1 ; e0t ; + '; : : : e0T ):(12)
It implies separability of f0; tg for all t < T , as in Theorem 8, but now,
instead of separability of f0; T g we have separability of all f1; t + 1g for all
t < T . The latter separability can for instance be seen by replacing all primes
in Eq. 12 by double primes, which should not a¤ect the second indi¤erence
because of the maintained equivalence with the …rst indi¤erence. By repeated
21
application of Gorman (1968), we still get separability of <. Hence the above
condition holds if and only if: time-dependent discounted utility holds with,
further, U0 ; Ut additively representing the same preference relation over R2
as U1 ; Ut+1 do. We can set Ut (0) = 0 for all t. Then, by standard uniqueness results (Wakker 1989 Observation III.6.6’), U1 =U0 = Ut+1 =Ut =
for a
positive constant . This proves the equivalence in Theorem 7. Because Condition 7.(ii) implies constant discounted utility, it implies stationarity, used
by Koopmans (1960, 1972) to axiomatize the model. The latter condition is
de…ned as follows:
(x0 ; x1 ; : : : ; xT
1 ; cT )
< (y0 ; y1 ; : : : ; yT
1 ; cT )
, (c0 ; x0 ; : : : ; xT
1)
< (c0 ; y0 ; : : : ; yT
(13)
Our condition is weaker by considering tradeo¤s between two periods, keeping
the outcomes in all other periods …xed.
Proof of Theorem 6, and uniqueness of its parameters
The following uniqueness result holds for Statement 6.(i), which uses Eq.
8: A real-valued time-independent constant
can be added to U , and U can
be multiplied by a positive constant . These uniqueness results follow from
those in Theorem 7. We next derive Statement 6.(i).
The equality ('; t; e) = ('; e0 ; et ) in Condition 6.(ii) means that
22
is
1 ):
not only independent of ej ; j 6= 0; t, as in Theorem 8, but also of t. This
holds if and only if9
(e0 + ; e1 ; : : : ; et 1 ; et ; et+1 ; : : : eT )
(e0 + ; e01 ; : : : ; e0t0 1 ; (et )t0 ; e0t0 +1 ; : : : e0T )
(e0 ; e1 ; : : : ; et 1 ; et + '; et+1 ; : : : eT ) )
(e0 ; e01 ; : : : ; e0t0 1 ; (et )t0 + '; e0t0 +1 ; : : : e0T ):(14)
It readily follows that the above condition holds if and only if we have all
the conditions of Theorem 8 and its representation, with the extra condition
Ut (et + ')
Ut (et ) = Ut0 (et + ')
Ut0 (et ), implying that we can take all
functions Ut the same, independent of t. It implies symmetry, the condition commonly used in the literature to axiomatize non-discounted utility.
Symmetry requires invariance of preference under every permutation of the
outcomes. Symmetry immediately implies that
is independent of t, which
is what Condition 6.(ii) adds to Condition 8.(ii). This shows once again that
the PV conditions are weak relative to conditions commonly used in the literature.
Proof of Theorems 2, 4, and 5, and uniqueness of their parameters
9
In the following equation we write (et )t0 for e0t0 to indicate that the t0 level of e0 is the
same as et , the tth level of e.
23
The discount parameters in Theorems 5 and 4, based on Eqs. 7 and 4, are
uniquely determined. This follows from the uniqueness results of Theorems
8 and 7. We next derive Statements (i).
The proofs of Theorems 5 [4, 2] readily follow from Theorem 8, [7, 6], as
follows. The theorems to be proved are the linear counterparts of the theorems from which they follow. The preference conditions are always the same
except that dependence of the endowment levels e0 ; et has been dropped. In
the notation of Theorem 8 this means that Ut (et + ')
Ut (et ) is indepen-
dent of et , which implies linearity of Ut . Similarly, the utility functions in
Theorems 7 and 6 are linear. Then Theorems 5, 4, and 2 follow.
For completeness, we show how the conditions of Theorems 2 - 5 can be
restated directly in terms of preferences:
The equality
= ('; t) in Condition 5.(ii) holds if and only if
(e0 ; e1 ; : : : ; et 1 ; et + '; et+1 ; : : : eT ) )
(e0 + ; e1 ; : : : ; et 1 ; et ; et+1 ; : : : eT )
(e00 + ; e01 ; : : : ; e0t 1 ; e0t ; e0t+1 ; : : : e0T )
The equality
=
(e00 ; e01 ; : : : ; e0t 1 ; e0t + '; e0t+1 ; : : : e0T ):(15)
('; t) = ('; t + 1) in Condition 4.(ii) holds if and
only if
(e0 ; e1 ; : : : ; et 1 ; et + '; et+1 ; : : : eT ) ,
(e0 + ; e1 ; : : : ; et 1 ; et ; et+1 ; : : : eT )
(e00 ; e01 + ; : : : ; e0t 1 ; e0t ; e0t+1 ; : : : e0T )
24
(e00 ; e01 ; : : : ; e0t 1 ; e0t ; e0t+1 + '; : : : e0T ):(16)
The equality
= (') in Condition 2.(ii) holds if and only if
(e0 + ; e1 ; : : : ; et 1 ; et ; et+1 ; : : : eT )
(e00 + ; e01 ; : : : ; e0t0 1 ; et0 ; e0t0 +1 ; : : : e0T )
(e0 ; e1 ; : : : ; et 1 ; et + '; et+1 ; : : : eT ) )
(e00 ; e01 ; : : : ; e0t0 1 ; et0 + '; e0t0 +1 ; : : : e0T ):(17)
We …nally compare the PV conditions in Theorems 2 - 5 with other conditions used in the literature to axiomatize the models in question. Condition
5.(ii) implies that the extra value of an extra payo¤ ' is independent of the
level et to which it is added. This is implied by the well known additivity
condition, requiring that a preference x < y not be a¤ected by adding the
same constant
to xt and yt . By adding the same constant to et and et + ',
we can change them into e0t and e0t + ', implying our preference condition.
Additivity is necessary and su¢ cient for time-dependent discounted utility
(Wakker 2010 Theorem 1.6.1). It is more restrictive than our condition because we only consider tradeo¤s between two periods, keeping the outcomes
in all other periods …xed. Similar observations apply to the elementary Theorem 4 and the trivial Theorem 2.
Informal proof of Theorem 2
The extra value of any extra future outcome is always the same and,
hence, it may as well be added to today’s wealth. Then all that matters is
the sum of all future outcomes, which may as well be received immediately
25
today.
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