A Theory of Intergenerational Altruism
Simone Galperti

Bruno Strulovici∗

UC, San Diego

Northwestern University
January 25, 2017
Abstract

Modeling intergenerational altruism is crucial to evaluate the long-term consequences of current decisions, and requires a set of principles guiding such altruism.
We axiomatically develop a theory of pure, direct altruism: Altruism is pure if
it concerns the total utility (rather than the mere consumption utility) of future
generations, and direct if it directly incorporates the utility of all future generations. Our axioms deliver a new class of altruistic, forward-looking preferences,whose weight put on the consumption of a future generation generally depends
on the consumption of other generations. The only preferences lacking this dependence correspond to the quasi-hyperbolic discounting model, which our theory
characterizes. Our approach provides a framework to analyze welfare in the presence of altruistic preferences and addresses technical challenges stemming from theinterdependent nature of such preferences.
Keywords: pure altruism, non-paternalistic altruism, time inconsistency, present
bias, intergenerational welfare, quasi-hyperbolic discounting, welfare criterion.
JEL Classification: D01, D60, D90∗

The authors are grateful to Nageeb S. Ali, Nabil Al-Najjar, James Andreoni, Geir B. Asheim, Doug
Bernheim, Eddie Dekel, Jeff Ely, Drew Fudenberg, David Laibson, Bart Lipman, Ehud Kalai, Emir
Kamenica, Peter Klibanoff, Mark Machina, Paul Milgrom, David G. Pearce, Todd Sarver, Marciano
Siniscalchi, Joel Sobel, and Asher Wolinsky, as well as seminar participants at UC San Diego, Stanford,l’Université de Montréal, RUD 2014, University of Chicago, Yale, NASMES 2014, SITE 2014, UCL, WU
St. Louis, UC Berkeley, QSPS Utah, and Pennsylvania State University for useful comments. An earlyversion of this project was developed by the second author and presented at Northwestern University
and the 2011 SAET conference in Faro, Portugal under the title “Forward-Looking Behavior, Well-Being
Representation, and Time Inconsistency.” A subset of the results appeared in a working paper by thesame authors, titled “From Anticipations to Present Bias: A Theory of Forward-Looking Preferences.”
Strulovici is grateful for financial support from the National Science Foundation (No.1151410) and a
Sloan Research Fellowship, as well as for the hospitality of Microsoft Research, New England.

11Introduction

Intergenerational concerns lie at the heart of many economic decisions: They determine the bequests from parents to their descendants (Bernheim et al. (1985)), which act
as a fundamental transmission mechanism in macroeconomic analysis (Barro (1974));
they affect our perception of social mobility and economic inequality (Becker and Tomes(1979)); they play a central role in weighing the environmental consequences of current
economic activities, as prominently illustrated by the Stern Review (Stern (2007)) andthe ensuing debate on the appropriate level of discounting. Intergenerational altruism
also underlies models of capital accumulation (Phelps and Pollak (1968)) and affects thestability of political regimes as already suggested by Tocqueville (1835).1 By shaping thepreferences of households, policymakers, and other economic agents, intergenerational
altruism influences how society evaluates and chooses between feasible courses of action.
Although many models of intergenerational altruism have been proposed, an investigation of the fundamental properties of preferences that exhibit altruism towards future
generations seems to be missing.2 Understanding these properties is essential to assess
the merits and flaws of the different models available as well as to propose new ones.
This paper addresses this issue, exploiting the conceptual and analytical apparatus ofmodern decision theory to describe and represent altruistic preferences.
We propose an axiomatic theory of pure, direct altruism. A generation exhibits pure
altruism if it derives utility—besides from its own consumption—from its descendants’
overall utility, or “well-being.” Pure altruism is sometimes called nonpaternalistic (Ray(1987), Pearce (2008)) as it depends on future generations’ overall well-being, not on thespecific consumption bundles and other considerations which determine it; in particular,
it entails no value judgment about the nature of these components. When future generations are also altruistic, the current generation’s pure altruism ascribes to each of thema well-being that includes their own altruism. Pure altruism is commonly assumed inmodels of intergenerational altruism, such as Barro (1974), Kimball (1987), and Loury(1981).3 Altruism is said to be direct if it directly depends on the well-being of all future
1

On Tocqueville’s hypothesis, see also Lipset (1960).
An exception is the axiomatization in Koopmans (1960), which may be interpreted as a theory ofpure indirect altruism, as explained in this paper. Like Koopmans (1960), Phelps and Pollak (1968), anda large body of the ensuing literature, this paper focuses on the case of forward-looking preferences inthe sense that they do not depend on past generations’ consumption. The inclusion of backward-looking
preferences raises interesting challenges discussed in Section 6. A large literature on normative social
choice, discussed in Section 2, analyzes the implications of axioms such as equity and anonymity to rank
consumption streams, which may be compared to the altruistic preferences studied here.
3
Pure altruism also plays a key role in Bergstrom’s (1989) analysis of reciprocal altruism.
2

2

generations. For example, environmental policies chosen with a regard towards all future generations reflect direct altruism towards these generations. This property is also
common in models of intergenerational altruism (Fels and Zeckhauser (2008), Saez-Marti
and Weibull (2005)).4
Our analysis takes as primitive the preference  of the present generation (hereafter,
generation 0) over consumption allocations to itself and future generations.5 We begin
by characterizing the class of generation 0’s preferences that have the following representation: For every stream (c0 , c1 , . . .), where ct is the consumption of generation t ≥ 0,
generation 0’s utility from it can be expressed as
U (c0 , c1 , . . .) = V (c0 , U1 , U2 , . . .)(1)

where Ut = U (ct , ct+1 , . . .) for t ≥ 1. We interpret Ut as the well-being that generation 0
assigns to generation t based on the stream (ct, ct+1 , . . .). In particular, these well-beings
are all computed with the same function U , which means that generation 0 “projects” its
own preference onto future generations to evaluate continuation streams. A motivation
for this assumption, beyond tractability, is that generation 0 may not know the preferences of generations who will be born many years into the future and may thus use its
own preferences as the best guess. Alternatively, generation 0 may view its preferenceas embodying some normatively desirable properties which future generations will also
adopt. The interpretation of Ut also serves to motivate our axioms and provide insights
into the resulting class of models and their properties. If  has the utility representation (1), then it is as if the current generation exhibited pure altruism. A special caseof (1) corresponds to Koopmans’ (1960) model of indirect pure altruism, in which V
depends only on c0 and U1 . By contrast, direct pure altruism requires that V depend on
Ut for all t ≥ 1. Koopmans’ model obtains as a limit of direct altruism, as discussed in
Section 3.3.
Representation (1) allows for arbitrary relations in generation 0’s treatment of distinct
future generations. On normative and practical grounds, it is valuable to impose morestructure on these relations. We introduce two substantive axioms which, together with4

See also Abel (1985) and Pearce (2008). Ray and Wang (2001) analyze a model of direct paternalistic
altruism.
5
The domain of this preference relation consists of infinite consumption streams, as in Koopmans(1960) and Diamond (1965). In this decision-theoretic framework, eliciting a preference relation andtesting whether it satisfies some axiom requires a standard observability condition: One must know
which of any two streams the current generation would choose if it could implement either of them. In
this respect, our approach is identical to these earlier works and a large subsequent literature.

3

ancillary conditions, narrow the representation down to the additive form∞X
U (c0 , c1 , . . .) = u(c0 ) +αt G(U (ct , ct+1 , . . .)),

(2)

t=1

where 0 < α < 1 and the function G is interpreted as the utility that generation 0
derives from future generations’ well-being. The axioms imply that G is bounded, strictlyincreasing, and Lipschitz continuous with modulus (1 − α)/α, a property which turns out
to have important consequences, in particular concerning the continuity and uniquenessof representation (2).
Our first axiom, intergenerational separability, adapts to pure altruism a principle which
often arises in the literature on impure (or paternalistic) altruism. The axiom rulesout any complementarity or substitutability in how the well-being of future generations
affects the well-being of generation 0. It thus expresses a form of “neutrality,” which theliterature has often used in relation to future consumption rather than to future wellbeing (Koopmans (1960), Asheim (2010)). In a theory of pure altruism, this neutrality
naturally concerns future generations’ well-being rather than their consumption, since
their consumption does not matter per se to generation 0. Importantly, however, theaxiom—as all axioms of our theory—is formulated in terms of consumption streams,
since consumption streams are the primitive objects over which generation 0’s preference
is defined.
Our second axiom is altruism stationarity. It focuses on the altruistic components ofgeneration 0’s preference by considering only changes in the consumption of future generations. To illustrate, suppose that c = (c0 , c1 , . . .) and c0 = (c00 , c01 , . . .) give the same
consumption to a grandmother (c0 = c00 ) who thinks that her son is indifferent between(c1 , c2 , . . .) and (c01 , c02 , . . .) after accounting for his consumption and the well-being of
his daughter, grandson, and so on. Suppose that the future generations’ well-beings
induce the grandmother, who cares directly about them, to prefer c to c0 . Then, intuitively, altruism stationarity says that if her son were removed from the genealogicalsequence—bringing all remaining generations one step closer to the grandmother in thesequence—his mother would continue to prefer (c2 , c3 , . . .) to (c02 , c03 , . . .) for her granddaughter and so on. From a normative standpoint, the axiom may be viewed as imposing coherence on how generation 0 directly cares about future generations. Altruism
stationarity is the axiom, in our theory, which distinguishes direct from indirect pure
altruism—hence our model from Koopmans’ (1960) model and in particular from theexponentially-discounted-utility model (hereafter, EDU). In Koopmans’ (1960) model, if
the son is indifferent between (c1 , c2 , . . .) and (c01 , c02 , . . .), then his mother must also be4

indifferent, given her initial consumption. Once her son is removed from the genealogicalsequence, however, the grandmother can strictly prefer (c2 , c3 , . . .) to (c02 , c03 , . . .) for her
granddaughter and so on. As direct altruism becomes arbitrarily negligible, the EDU
model arises as a limit of our theory.
Representation (2) has distinctive implications concerning how generation 0 resolves
intergenerational trade-offs. First, its well-being can be expressed as a function of theconsumption utility of all generations (u(c0 ), u(c1 ), and so on), which allows us to examine how generation 0 trades off its own consumption utility against that of any future
generation t. We find that, even though generation 0 cares directly about the well-beingof all future generations, it exhibits positive discounting towards future consumption,in the following sense: If generation 0 can swap consumption utility between itself andsome future generation, it always chooses the stream starting with the higher of the two
consumption utilities.
Perhaps more surprisingly, generation 0 exhibits present bias: Roughly speaking, it ismore reluctant to transfer consumption from itself to the next generation, than from any
future generation to a later one. To see why, consider a grandmother who cares about
her son’s well-being (understanding that it depends on his daughter’s well-being), butalso directly about her granddaughter’s well-being. The grandmother can disagree withher son on whether he should transfer more resources to his daughter (say, as a bequest),
because they internalize differently the effects of changing her consumption. The sontakes into account the daughter’s utility through his well-being, while in addition to her
son’s well-being, the grandmother also cares directly about her granddaughter’s wellbeing. As a result, the grandmother thinks that her son should transfer more resources
to his daughter than he wants to. At the same time, however, if she were in her son’s
position, she would agree on keeping more resources for herself. The grandmother thus
appears more willing to delay consumption when it affects future generations than when
it directly affects herself. Remarkably, therefore, present-generation bias is not an assumption of our theory, but a logical consequence of direct pure altruism. We will showthat this consequences emerges already from representation (1), which does not use theseparability and stationarity axioms.
Under representation (2), how generation 0 discounts the consumption utility of somefuture generation t generally depends on the entire consumption stream it is facing. If thefunction G is concave, for instance, the consumption utilities of successive generations
are substitutes: Increasing the consumption utility of some generation t raises the wellbeing of earlier generations, thereby reducing the marginal value of later generations’5consumption utility. This reduces the effective discount factor applied to these later
generations, while the opposite holds if G is convex.
These observations suggest that consumption interdependence disappears if G is linear.
This is indeed the case, and with linear G our model takes on a well-known form: quasihyperbolic (or β-δ) discounting (Phelps and Pollak (1968)), where β and δ are simple
functions of α and the constant slope of G. Linearity of G corresponds to an additional
axiom, consumption independence, which has two parts: (i) how a grandmother trades off
her consumption with that of her son does not depend on the consumption—and hencewell-being—of his descendants, and (ii) how she trades off her consumption with that ofher son’s descendants does not depend on his consumption. A byproduct of our general
analysis is thus to provide an axiomatization of Phelps and Pollack’s (1968) model ofimperfect intergenerational altruism as a model of direct pure altruism. While Phelps
and Pollak (1968) viewed the EDU model as capturing perfect intergenerational altruism(p. 186), this paper’s perspective suggests that EDU exhibits only indirect pure altruism,
offering an opposite view on which of these discounting models—the exponential and thequasi-hyperbolic one—captures a deeper degree of intergenerational altruism.
Direct pure altruism alone has significant implications if we now imagine a sequenceof generations, each having the same preference relation—a property commonly known
as time invariance.6 If each preference has representation (1) with V strictly increasing in all U ’s, then the resulting sequence of preferences cannot be time consistent:7 A
course of action for generation t and its descendants that is optimal from the viewpointof generation t need not be optimal from the viewpoint of generation s < t. This conclusion identifies direct pure altruism as a possible source of time inconsistency acrossgenerations. Importantly, the model predicts a specific form of inconsistency: By thepresent-generation bias, each generation tends to disproportionately favor its own satisfaction over that of any future generation.
Time-inconsistent preferences create well-known challenges for welfare analysis as dis6

Time invariance is not needed to characterize the preference of generation 0: The former property
compares preferences across multiple generations, while the latter characterization is entirely focusedon the preference of a single generation. However, time invariance is a useful additional assumption toassess the time consistency of successive generations whose preferences we have characterized. Besides
its analytical convenience, time invariance has additional justifications in our setting: First, under anormative interpretation of our axioms, time invariance captures the idea that all generations are subject
to the same normative considerations. Second, it describes a coherence between the interpretation thatthe current generation projects its own preference onto future generations and the future generations’actual preferences. While the first generation’s projection may turn out to be false, assuming it is correct
is a useful and natural benchmark.
7
For this conclusion to hold, it is enough that each generation cares about the well-being of its
immediate descendant as well as some other future generation.

6

cussed, among others, by Rubinstein (2003) and Bernheim and Rangel (2007, 2009).
This paper makes several contributions to this discussion. As noted earlier, time inconsistency is a necessary consequence of direct pure altruism combined with time invariance.
This observation weakens the case for paternalistic interventions, which usually relies onviewing time inconsistency as a form of irrationality. Another implication of our analysis concerns the use of a “libertarian” criterion which measures social welfare with thewell-being of generation 0 (that is, U (c0 , c1 , . . .))—this is what is usually done in thecase of Koopmans’ (1960) model. Such a criterion seems more appropriate for directpure altruism—hence time inconsistency—than for indirect pure altruism—hence time
consistency. This assessment is based on the other properties of purely and directly altruistic preferences, such as neutrality and sensitivity towards future generations, whose
importance relative to time consistency seems a priori not obvious.8
One may argue that, despite generation 0’s altruism, a social planner should aggregate
the preferences of all generations, assigning an appropriate positive weight to each generation’s well-being. We show that pure altruism—whether direct or indirect—makes it
difficult to find an aggregator that also renders the planner time consistent.9 One remarkable exception is representation (2) with linear G, the quasi-hyperbolic model. We showthat the standard welfare criterion for β-δ preferences obtained by setting β = 1 corresponds to an aggregator that weighs the well-being of each generation t (including t = 0)by αt , where α comes from representation (2) and satisfies α = δ(1 − β).102

Related Literature

The relevance of intergenerational altruism is highlighted by its key role in various economic applications, including optimal national savings (Ramsey (1928), Phelps and Pollak (1968)), economic growth (Bernheim (1989)), charitable giving (Andreoni (1989)),
family economics (Bergstrom (1997)), public finance (Barro (1974)), and environmental
economics (Weitzman (1999), Dasgupta (2008), Schneider et al. (2012)).
Despite its importance, intergenerational altruism lacks a clear conceptual framework,
as emphasized by Ray (1987):
8

Similar properties have been proposed in the normative social-choice literature as desirable featuresof a welfare criterion (see, for example, Asheim (2010) and the references therein).
9
In his study of hedonistic altruism and welfare, Ray (2014) examines welfare criteria that aggregate
well-being of altruistic, time-consistent, generations and that are formally similar to those in Section 5(see also Bernheim (1989)).
10
For other discussions on the “right” welfare criterion to use for the quasi-hyperbolic discounting
model see, for example, O’Donoghue and Rabin (1999), Rubinstein (2003), Bernheim and Rangel (2009).

7

“the representation of non-paternalistic functions [in terms of total utilities,
U ] in paternalistic form [in terms of consumption utilities, u] has been thesubject of limited attention. A systematic analysis of the relationship between
these two frameworks [...] appears to be quite a challenge, especially formodels with an infinite horizon (pp. 113–114).
Saez-Marti and Weibull (2005) and Fels and Zeckhauser (2008) derive the mathematicalequivalence between the β-δ formula and expression (2) with linear G. Bergstrom (1999)
studies systems of utility functions that include altruism towards others, focusing on
the infinite regress that they may generate. None of these papers, however, provides anaxiomatic foundation of either representation.
Several axiomatizations of the intertemporal preference of a decision maker have been
proposed. Koopmans (1960) derives a general model which contains as a special case
Samuelson’s (1937) EDU model. The difference between Koopmans’ analysis and ours is
explained when we formally state our axioms. The EDU model obtains as a limit of ourtheory as direct altruism becomes arbitrarily negligible. Other models that include quasihyperbolic discounting have been characterized by Hayashi (2003), Olea and Strzalecki(2014), and Echenique et al. (2016). We compare their approaches to ours in Section 3.3.
Time inconsistency of preferences has been extensively studied since Strotz’s (1955)
seminal work, and has been given several explanations. In Akerlof (1991), present bias
is based on a principle of cognitive psychology which says that decision-makers unduly
overweigh relatively more salient or tangible events, such as present consumption relative
to future one. In Gul and Pesendorfer (2001), time inconsistency can arise from a generalchange in the decision-maker’s preference over time. Halevy (2008), Saito (2011), andChakraborty and Halevy (2016) link present bias to the certainty effect. In a settingwith uncertainty about the length of the horizon, this literature establishes an equivalence between the decision-maker being disproportionately sensitive to certainty, as inAllais (1953) and Kahneman and Tversky (1979), and exhibiting present bias. In Dasgupta and Maskin (2005) hyperbolic discounting arises from an evolutionary responseto the uncertain time of payoff realizations, whereas in Farmer and Geanakoplos (2009)it stems from uncertainty about future discount rates. In Kőszegi and Szeidl (2012),
time inconsistency and present bias arise because, when facing a present decision, theagent focuses too much on its immediate consequences, but when considering that samedecision ex ante, he is able to focus more on its overall consequences over time. Our
theory provides a novel explanation for present bias, as part of a broader conceptual
framework of how future consequences of current decision enter into the decision-maker’s
8intertemporal preference.
This paper is also related to the normative social-choice literature studying how to rankintergenerational consumption streams.11 Its typical approach is to introduce properties
that those rankings should satisfy and derive their implications and functional representations. Often suggested properties include sensitivity to and equal treatment of all future
generations’ well-being. The literature highlights, however, that these properties collidewith the standard goal of having a numerical representation of preferences (for example,
Basu and Mitra (2003)); the present work considers weaker properties consistent witha numerical representation (see also Asheim et al. (2012)). The social-choice literature
also commonly excludes sentiments like altruism from the generations’ well-being. From
this perspective, one of our contributions is to add this dimension to intergenerational
preferences and explore the properties that it implies.
From a methodological perspective, our derivation of representation (2) builds on known
results in Debreu (1960) and Koopmans (1960) applied to the sequences of generation 0’s
consumption and future generations’ well-beings induced by the consumption streams.
A complication arises, however, because the space of such sequences does not have a
Cartesian-product structure: Since a generation’s well-being depends on its descendants’
well-beings, one cannot modify those well-being independently of one another. We propose a way to overcome this difficulty which may be useful in other settings where players’
preferences are interrelated. Segal (1992, 1994) provides a framework and general conditions encompassing non-Cartesian spaces to derive representation results. While theinterdependence structure arising here makes it challenging to directly verify these conditions, the present analysis suggests an approach to usefully connect such a structurewith Segal’s framework.

3

Preference Representations

3.1

Preliminaries

Consider an infinite sequence of generations, indexed by t ∈ N = {0, 1, 2, . . .}. Generation
t’s consumption is denoted by ct and belongs to a connected, separable, metric space X;
for example, ct can be a finite vector representing the lifetime consumption of generation t.
The streams of consumption starting from generation 0 are denoted by c = (c0 , c1 , . . .) and11

Asheim (2010) provides a detailed review of this literature. Intergenerational welfare has also been
extensively discussed in the moral-philosophy literature—for example, to apply to the set of all generations the principles laid out by Rawls (1971).

9belong to the set C = X N , which we endow with the sup-norm: kc − c0 kC = supt d(ct , c0t ),where d is a bounded metric on X.12 For t ≥ 1, the set of streams starting at t isN
t C = X , whose elements will be denoted by t c = (ct , ct+1 , . . .).
The purpose of this paper is to study the single preference of generation 0 over streams
of consumption for itself and all future generations, i.e., over C. This preference isdenoted by , with the symbols % and ∼ having the usual meaning, and represents the
choices that generation 0 would make for itself and all its descendants. A basic premise isthat  has a utility representation (as in Koopmans’ (1960) Postulate 1). This is ensured
by the following axioms.13
Axiom 1 (Weak Order). % is a complete and transitive binary relation.
Axiom 2 (Continuity). For all c ∈ C, {c0 ∈ C : c0 - c} and {c0 ∈ C : c0 % c} are closed.
Axiom 3 (Future Constant-Flow Dominance). For all c ∈ C, there exist x, y ∈ X suchthat (c0 , x, x, . . .) - c - (c0 , y, y, . . .).
Axioms 1 and 2 are standard. Axiom 3 captures the following idea: For every stream c,from generation 0’s viewpoint there is a sufficiently bad consumption x and a sufficiently
good one y such that forcing each future generation to consume x (resp. y) is worse (resp.
better) than forcing them to consume according to 1 c. These axioms lead to the following
standard result.14
Theorem 1 (Utility Representation). Under axioms 1-3, there exists a continuous function U : C → R such that c  c0 if and only if U (c) > U (c0 ).
Hereafter, U (c) will represent the total utility of generation 0 from stream c. To avoid
confusion later with the utility that generation 0 derives from its own consumption, we
will use the term well-being to refer to U (c).
Since we are interested in studying altruism towards future generations, by assumption
the well-being of generation 0 depends on the consumption of later generations. It is also
natural that generation 0 cares about its own consumption.
Axiom 4 (Non-triviality). There exist x, x0 , x̂ ∈ X and c, c0 , ĉ ∈ C such that (x, ĉ) (x0 , ĉ) and (x̂, c)  (x̂, c0 ).
ˆ ·) =
Assuming that d(·, ·) is bounded is without loss of generality, as we can always replace it by d(·,d(·, ·)/(1 + d(·, ·)), which is another metric respecting the initial distances.
13
This paper assumes, as in Koopmans’ (1960, 1964) classic analysis and a large subsequent literature, that generation 0’s preference does not depend on past generations’ consumption. Relaxing thisassumption raises interesting challenges, discussed in Section 6.
14
The proofs of the main results are in the Appendix A. The remaining proofs are in the Online
Appendix.
12

10

3.2

Pure-Altruism Representation

Our objective is to investigate which properties of  correspond to pure altruism towards
future generations. Intuitively, generation 0 exhibits pure altruism if it derives utility
from the total utilities—or well-being, in our terminology—of future generations (see,for example, Ray (1987) and Pearce (2008)).15 Since  corresponds to the subjective
attitude of generation 0 towards future generations,  can only reveal how this generation
perceives the well-being that each future generation t derives from the stream t c. However,generation 0 may not know future generations’ preferences—for instance, for those born
many years into the future; or it may have no reason to believe that their preference
will systematically differ from its own—for instance, because it views its preference asbased on some generally appealing and sensible principles. In these cases, generation 0
may simply “project” its preference onto future generations and use it to assess theirwell-being. This is the approach that we take in this paper. Other axiomatic models of
altruism assume that the decision-maker projects its preference onto its peers (see, forexample, Saito (2015)).
This leads to the following general class of representations of purely altruistic preferences. Fixing a representation U of , let U be the range of U and define
F = {(f1 (c), f2 (c), . . .) : ft (c) = U (t c) for c ∈ C and t ≥ 1}.

(3)

Note that F ⊂ U N , but F need not be a Cartesian product because the well-being
U (t c) ascribed to generation t depends on the well-being ascribed to its descendants. For
example, generation 1’s maximal well-being—when it is finite—may only be achieved bygiving all future generations their highest well-being.
Definition 1 (Pure-Altruism Representation). Preference  has a pure-altruism representation if and only if it can be represented by
U (c) = V (c0 , U (1 c), U (2 c), . . .)(4)for some function V : X × F → R that is nonconstant in c0 and U (t c) for some t ≥ 1.
If generation 0 is purely altruistic, its ranking of any two streams c and c0 depends only
on its own consumption, c0 and c00 , and on how it perceives—through the lens of its
current preference—that each future generation t evaluates the continuation streams, t c
and t c0 . Hereafter, we will call U (t c) the well-being of generation t, keeping in mind that15

By contrast, generation 0 is impurely (or paternalistically) altruistic if it cares only about theutility that other generations derive from what they actually consume. To the extent that generation 0
thinks that future generations are also altruistic, this utility from actual consumption differs from eachgeneration’s total utility.

11this well-being is evaluated from generation 0’s perspective.
Axiom 5 below is the key to obtaining a pure-altruism representation. It captures aminimal property of purely altruistic preferences: Given its own consumption, if generation 0 thinks that two consumption streams will render all future generations indifferent,
it should also be indifferent.
Axiom 5 (Pure Altruism). If t c ∼ t c0 for all t ≥ 1, then (c0 , 1 c) ∼ (c0 , 1 c0 ).
The premise t c ∼ t c0 captures the thought experiment of generation 0 facing the same
continuation stream as generation t will face and using its current preference to compare0t c and t c . Axiom 5 rules out the possibility, for instance, that generation 0 prefers c
to c0 because, despite generating the same stream of current consumption and future
well-being, they allocate future consumption differently across generations.
Theorem 2. Axioms 1–5 hold if and only if  has a pure-altruism representation.
Proof. Define F0 = X × F and let f0 (c) = c0 and f (c) = (f0 , f1 , f2 , . . .).
(⇒) First, we define equivalence classes on C as follows: c is equivalent to c0 if ft (c) =ft (c0 ) for all t ≥ 0.16 Let C ∗ be the set of equivalence classes of C, and let the function U ∗be defined by U on C ∗ . Then, the function f ∗ : C ∗ → F, defined by f ∗ (c∗ ) = f (c) for cin the equivalence class c∗ , is by construction one-to-one and onto. Let (f ∗ )−1 denote its
inverse and, for any f ∈ F0 , define
V (f ) = U ∗ ((f ∗ )−1 (f )).
By Axiom 5, V is a well-defined function, and V (f (c)) = U (c) for every c. By Axiom 4,
V is nonconstant in f0 and ft for some t ≥ 1.
(⇐) See Appendix A.
Axiom 5 requires that generation 0 be indifferent between two streams only if it perceives that all future generations—not just generation 1—will be indifferent between therelevant continuations of these streams. Axiom 5 clearly holds if generation 0 cares only,
besides its own consumption, about the well-being of generation 1—as in Koopmans’(1960) model. By allowing generation 0’s preference to depend on future generations’
well-being in a richer way, Axiom 5 is a steppingstone in our approach to modelingintergenerational preferences differently from Koopmans.
16

In general, there may be several consumption streams in an equivalence class. For a simple illustration, take U (c) = c0 + c1 + c2 + c3 , c = (1, 1, −1, −1, 1, 1, −1, −1, . . .) and c0 =(1, −1, −1, 1, 1, −1, −1, 1, 1, . . .).

12

To distinguish our approach from the standard one, we introduce the following terminology. As noted, Koopmans’ (1960) model satisfies
U (c) = V (c0 , U (1 c));in particular, for EDU we have U (c) = u(c0 ) + δU (1 c). In this case, generation 0 is purely
altruistic only towards generation 1, and by expecting that all future generations will be
altruistic in a similar way, it ends up caring about how c affects all of them but only indirectly through the well-being of generation 1. We will thus say that generation 0 exhibits
indirect pure altruism.17 We will say that generation 0 exhibits direct pure altruism if its
well-being U (c) depends directly on the well-being of all future generations (that is, Vin (4) depends on U (t c) for all t ≥ 1).18 Concretely, a grandmother usually cares about
her son’s well-being—which depends on his daughter’s well-being—and directly about
her granddaughter’s well-being. By contrast, in Koopmans’ (1960) model, it is as if the
grandmother cares about her granddaughter’s well-being only indirectly to the extent
that it affects her son’s well-being.

3.3

Additive Pure-Altruism Representation

To obtain sharper results, we refine the general representation V in (4) by considering
preferences that satisfy some form of intergenerational separability and stationarity.
The first axiom captures the idea that  is separable across generations—that is, separable between generation 0’s consumption and future generations’ well-being, as well as
across their well-being. Intuitively, this means that the well-being of future generations
does not affect how generation 0 enjoys its own consumption. Moreover, the well-beingof any group of future generations does not influence how generation 0 trades off thewell-being of the remaining generations. As such, this notion of separability captures aform of neutrality of generation 0’s altruism towards future generations. To formalize
these properties, recall that t c ∼ t c0 means that U (t c) = U (t c0 ) and let Π consist of allunions of subsets of {{1}, {2}, {3, 4, . . .}}.
Axiom 6 (Intergenerational Separability). Fix any π ∈ Π. If c, ĉ, c0 , ĉ0 ∈ C satisfy(i) t c ∼ t ĉ and t c0 ∼ t ĉ0 for all t ∈ π,(ii) t c ∼ t c0 and t ĉ ∼ t ĉ0 for all t ∈ N \ π,17

An alternative interpretation of the EDU model is that generation 0 does care directly about all
future generations, but ignores the fact future generations are altruistic, reducing the well-being ofgeneration t to its consumption utility u(ct ).
18This classification is non-exhaustive: Partially direct altruism would entail caring about a subset of
future generations, beyond the immediate descendant. This paper focuses on the two extreme cases.

13

(iii) either c0 = c00 and ĉ0 = ĉ00 , or c0 = ĉ0 and c00 = ĉ00 ,then c  c0 if and only if ĉ  ĉ0 .
To illustrate, set π = {1, 2} and consider streams c and c0 which give the same consumption to generation 0 and yield the same well-being to all future generations, exceptthat generation 1 is better off under c while generation 2 is better off under c0 . Suppose,
moreover, that generation 0 prefers c to c0 . Now, replace c by ĉ and c0 by ĉ0 , with the
new streams chosen so that going from c to ĉ and from c0 to ĉ0 leaves the well-being ofgenerations 1 and 2 unchanged, while generation 0’s consumption and the well-being of
all other generations change by the same amount. According to the axiom, generation 0
should prefer ĉ to ĉ0 , thereby continuing to trade off the well-beings of generations 1
and 2 in favor of the former. Axiom 6 is inspired by Debreu’s (1960) and Koopmans’(1960) separability axioms, but it differs substantively by requiring that certain continuation streams render generation 0 indifferent (that is, U (t c) = U (t c0 )), rather that certain
physical consumptions be equal (that is, ct = c0t ). While expressed purely in terms of
consumption streams, as with earlier theories, the axiom thus captures separability infuture generations’ well-being rather than in their consumption.
The next axiom imposes a weak monotonicity property of altruistic preferences. First,
all else equal, generation 0 is better off if it thinks that generation 1 will be better off.
Second, suppose that, for any horizon T and any common continuation stream c00 after T ,generation 0 prefers the consumption of the first T generations implied by stream c tothat implied by c0 ; then, generation 0 also prefers the whole c to c0 . Intuitively, this rulesout the possibility that the well-being of a generation in the infinite future could overturnhow generation 0 ranks c and c0 , even though, having fixed that generation’s well-being,
all intermediate generations are better off with c than with c0 .
Axiom 7 (Monotonicity). Let c and c0 be any streams in C.
(i) If c0 = c00 , 1 c  1 c0 , and t c ∼ t c0 for all t ≥ 2, then c  c0 .
(ii) If for every T and continuation stream c00 we have (c0 , c1 , ..., cT , c00 ) % (c00 , c01 , ..., c0T , c00 ),then c % c0 .
The axioms introduced so far are all satisfied by Koopmans’ (1960) model U (c) =V̂ (u(c0 ), U (1 c)), where V̂ strictly increasing in each argument, hence in particular by the
EDU model. They thus define a class of preferences which can exhibit either direct orindirect altruism. It turns out that it is possible to distinguish these kinds of altruism bymeans of a single additional property, stationarity, which marks the watershed between
Koopmans’ model and ours. Adding Koopmans’ (1960) notion of stationarity to Axioms 1-7 completes the characterization of indirectly altruistic preferences, as explained14

at the end of this section. The notion of stationarity introduced next takes our theory
on a different path which generates direct altruism.19
Altruism Stationarity and Direct Pure Altruism
Our notion of stationarity focuses on the altruistic component of generation 0’s preference by fixing its own consumption and considering only changes in consumption streams
that involve future generations. Intuitively, if generation 0 cares directly about the wellbeing of generations beyond its immediate descendant in a coherent way, it should bepossible to “remove” generation 1 and preserve how generation 0 ranks the consumption
streams starting from generation 2 onward. Clearly, for this comparison to be meaningful, we must start from a situation in which how generation 0 ranks two streams in thepresence of generation 1 depends only on the well-being of subsequent generations but
not on that of generation 1.
Axiom 8 (Altruism Stationarity). If c, c0 ∈ C satisfy c0 = c00 and 1 c ∼ 1 c0 , then c % c0
if and only if (c0 , 2 c) % (c00 , 2 c0 ).
To gain intuition, suppose that c and c0 give the same consumption to a grandmotherand that she thinks that her son is indifferent between 1 c and 1 c0 after he takes into
account his consumption as well as the well-being of his daughter and later descendants.
Also, suppose that later descendants’ well-beings induce the grandmother, who cares
directly about them, to prefer c to c0 . The axiom says that if the son were to die—whichmoves all remaining generations one step closer in the genealogical sequence—his mothershould continue to prefer 2 c to 2 c0 for her granddaughter and later descendants.
These axioms lead to the representation in Theorem 3 below. Intuitively, by the theorem
it is as if generation 0 derives a utility u from its consumption, as usual, and an altruism
utility G from the well-being of each future generation, which it discounts exponentiallyas generations move farther away in the lineage.20 Moreover, G is bounded, so future
generations’ well-being can have only a limited impact on generation 0’s well-being. Intuitively, this generation cannot become infinitely happy or unhappy just from its altruism
towards its descendants. The consumption utility u, however, can be unbounded.
19

As noted, indirect altruism arises at the limit, as direct altruism becomes negligible. This point isparticularly clear for the quasi-hyperbolic discounting case discussed in Section 4.3.
20
Rogers (1994) suggests an evolutionary justification based on genetic relationship for why generation 0 may progressively care less about its children, grandchildren, and so on: Every step in the lineage
reduces the share of genes that generation 0 can expect to have in common with that generation.

15

Theorem 3 (Additive Pure-Altruism Representation). Axioms 1-8 hold if and only if
the function U may be chosen so that∞X
U (c) = u(c0 ) +αt G(U (t c))(5)
t=1

where α ∈ (0, 1), u : X → R and G : U → R are continuous, nonconstant functions,and G is strictly increasing and bounded. Moreover, if (u, α, G) and (û, α̂, Ĝ) both represent , then α̂ = α and there exist a > 0 and b ∈ R such that û(x) = au(x) + b
and Ĝ(Û ) = aG( Û a−b ) for all x and Û .
The proof relies on known results in Debreu (1960) and Koopmans (1960). However,a complication arises in our setting. As noted, the set F of streams of future generations’ well-being induced by consumption streams in C may not be a Cartesian product(see (3)). Roughly speaking, to deal with this issue, the key is to show that (i) if wetake any stream f in F, there is an open neighborhood of f which belongs to F and has
the structure of a Cartesian product, and (ii) it is possible to “cover” F with countably
many of such neighborhoods which intersect with each other. Given (i) and (ii), we canobtain a preliminary additive representation on each neighborhood. Relying on these
representations’ uniqueness up to positive affine transformations, we can then “glue” allof them into a single representation over the entire set F.21
To work with this class of models, it is natural to ask whether expression (5) is welldefined for any function G. Proposition 1, below, provides necessary and sufficient restrictions for (5) to be well defined. It also proves that the representation U is continuous
at infinity in the following sense: The effect on generation 0’s well-being of future consumption changes becomes arbitrarily small when these changes take place arbitrarily
far in the future.
Definition 2. A function U : C → R is H-continuous22 if, for every ε > 0, there exists
T ∈ N such that |U (c) − U (c̃)| < ε whenever ct = c̃t for t ≤ T .
Proposition 1.
(i) In representation (5), U is H-continuous and for ν 0 , ν ∈ U
1−α 0|ν − ν| .
|G(ν 0 ) − G(ν)| <α
21This approach may be useful more generally to obtain separable models of a decision-maker who
exhibits pure altruism towards other individuals in society and takes into account that they are also
purely altruistic towards each other.
22
This notion is similar to the concept of “continuity at infinity” of payoff functions in infinite-horizon
games (see, for example, Fudenberg and Tirole (1991)).

16

(ii) Suppose that G is strictly increasing, bounded, and Lipschitz continuous with constantK < 1−α. Then, there exists a unique H-continuous function U : C → R that solves (5).
αThis result helps to choose G appropriately in applications, and implies several propertiesof directly altruistic preferences, to be described shortly.
Koopmans’ Stationarity and Indirect Pure Altruism
Koopmans’ (1960) model is based on the general representation U (c) = V̂ (u(c0 ), U (1 c))with V̂ strictly increasing in each argument. To obtain such a representation in theframework of the present paper, it is enough to add Koopmans’ stationarity axiom to
Axioms 1-7.23
Axiom 9 (Koopmans’ Stationarity). For some c0 and all 1 c, 1 c0 ,(c0 , 1 c) % (c0 , 1 c0 ) if and only if 1 c % 1 c0 .
As Koopmans (1960) showed, if we want to specialize V̂ to obtain the EDU model, we
need stronger separability assumptions (cf. Section 4.3).
To see why Axiom 9 rules out direct pure altruism, note that Axiom 6 with π ={1, 2, . . .} implies that, if the condition of Axiom 9 holds from some c0 , then it holds for allc0 ∈ X. Axiom 9 thus implies that, given her consumption c0 , how a grandmother ranks
the consequences of streams 1 c and 1 c0 for her son, granddaughter, great granddaughter,and so on is always pinned down by how she thinks that her son ranks these consequences
for himself and his descendants. This property seems unduly restrictive and unrealistic:
Why, when assessing the consequences of its actions on future generations, should ageneration rely exclusively on its immediate descendant’s preferences? This exclusive
reliance arises as a limit of our model, when direct pure altruism becomes negligible.

4

Intergenerational Trade-offs

This section studies the consequences of direct pure altruism on how a generation trades
off consumption between generations. A special case of representation (5) corresponds tothe β-δ discounting model introduced by Phelps and Pollak (1968) to capture “imperfect”
intergenerational altruism. Our theory provides a very different perspective, suggestingthat the altruism captured by the β-δ discounting model is pure and direct.
23

This set of axioms imply postulates 1-4 in Koopmans (1960).

17

4.1

Positive Discounting and Present-generation Bias

Pure direct altruism implies that a generation derives utility from the well-being of future
generations and recognizes that the well-being of each future generation depends on its
descendants’ well-being. Can this induce a generation to sacrifice its own consumption
utility in favor of that of a future generation? In the case of representation (5), the answer is negative: Generation 0 always discounts positively its descendants’ consumption
utility, relative to its own. To formalize this, we introduce a precise definition of positive
discounting.
Definition 3 (Positive Discounting).  is said to exhibit positive discounting if thefollowing holds: For any x, y ∈ X such that (x, c)  (y, c) for all c ∈ C, and any t ≥ 1,
we have cx  cy whenever cx0 = x, cxt = y, cy0 = y, cyt = x, and cxs = cys for all s ∈ N\{0, t}.
The first part of the condition says that x is unambiguously preferred to y. The second
part says that, given any two streams which differ only at times 0 and t, with the streams
swapping x and y at times 0 and t, the stream starting with x is always preferred to thestream starting with y. The necessary restrictions derived in Proposition 1, above, yield
the following key result.
Corollary 1. If axioms 1-8 hold, then  exhibits positive discounting.24
Positive discounting refers to how generation 0 trades off its own consumption utility
with future generations’, but how does it trade off satisfaction between future generations? Does it do so in the same way in which it trades off its own satisfaction againstthat of any of those generations? As it turns out, pure direct altruism alone—without imposing stationarity or separability—introduces a systematic difference between trade-offs
which directly involve generation 0 and trade-offs which involve only future generations.
Perhaps more surprisingly, direct pure altruism leads generation 0 to be biased in favorof its present satisfaction in the following sense.
Definition 4 (Present-generation bias).  exhibits present-generation bias if the following holds: For any x, y such that (x, c)  (y, c) for all c, and streams such that(z0 , . . . , zt−1 , x, x̂, c0 ) ∼ (z0 , . . . , zt−1 , y, ŷ, c0 ) for some t ≥ 1, we have (x, x̂, c0 )  (y, ŷ, c0 ).
24

In a finite-horizon setting, a representation of the form in (5) is consistent with negative discounting:Generation 0 may be willing to sacrifice its satisfaction for the benefit of future generations that are
sufficiently close to it in the lineage. Unlike the infinite-horizon case, (5) yields well-defined preferenceseven if the function G is “steeper” than 1−α
α . See the earlier version of this paper (Galperti and Strulovici(2014)).

18

Intuitively, suppose that x is unambiguously preferred to y and that generation 0 is
indifferent if generation t is forced to give up x in exchange for y, provided that the change
is “compensated” by giving ŷ to generation t + 1 instead of x̂. Then, if generation 0 itself
faced the choice between x and y in the present, it would strictly prefer not to give up x
for y even if generation 1 got ŷ in exchange.
As the next result shows, this form of present bias is already implied by pure direct
altruism, without requiring the separability and stationarity axioms used to characterize
representation (5).
Proposition 2. Suppose that U (c) = V (c0 , U (1 c), U (2 c), . . .) represent  and that V isstrictly increasing in U (t c) for all t ≥ 1. Then,  exhibits present-generation bias.
The proof of the proposition establishes a stronger form of present bias: The conclusionof Definition 4 holds when x̂ and ŷ concern any period s ≥ t + 1, instead of just t + 1.
The logic of Proposition 2 may be explained as follows. Set t = 1 in Definition 4 andimagine that x and y correspond to higher and lower levels of consumption. From theviewpoint of a grandmother (generation 0), a reduction in her son’s consumption from x toy can be compensated by an appropriate increase in her granddaughter’s consumption, ŷ.
However, because the grandmother takes directly into account her granddaughter’s wellbeing, the level ŷ that renders her indifferent may be insufficient to render her son
indifferent. This implies that if the grandmother were in the same situation of her son,she would also strictly prefer not to reduce her own consumption from x to y even if herchild received ŷ. Thus, the grandmother appears to be more willing to shift consumption
to a later generation when only future generations are affected than when she is directly
affected.
Proposition 2 generalizes and offers an interpretation to the results obtained by SaezMarti and Weibull (2005) and Fels and Zeckhauser (2008). These papers assume thatgeneration 0’s preferences are represented by an aggregator V which is additive and linear
in future generations’s well-beings, U . For this specific case, they show that U (c) canbe rewritten as the discounted sum of future generations’ consumption utilities—given
by some function u(ct )—where discounting takes the quasi-hyperbolic form as in Phelps
and Pollak (1968) (see Corollary 3 below). This, of course, implies that the preference
corresponding to such a U (c) exhibits present-generation bias. Proposition 2 shows that
this conclusion holds more generally as a consequence of direct pure altruism.

19

4.2

Intergenerational Rate of Utility Substitution

Representation (5) exhibits separability in future generations’ well-being but not necessarily in their consumption. In fact, how generation 0 trades off its consumption and thatof generation t generally depends on the consumption of all other generations, because
generation t’s consumption affects the well-being of all preceding generations, and in a
way that depends on the consumption of all subsequent generations.
To examine this, we consider generation 0’s discount function between 0 and t. Clearly,
intergenerational trade-offs involving consumption also depend on the curvature of theconsumption utility u. To bypass this dependence, we start with a convenient result,which follows from Theorem 3.
Corollary 2 (u-Representation). Given representation (5), there exists a nonconstantfunction Û : IuN → R (where Iu is u’s range) such that, for all c ∈ C,
U (c) = Û (u(c0 ), u(c1 ), . . .).
This implies that, given a stream c, what matters for generation 0’s preferences is thestreams of consumption utility us = u(cs ), rather than consumption per se. This simplification is useful because utility levels are real numbers, whereas consumption bundles
take values in a potentially much more complex space. The discount function can now
be defined as∂ Û (u0 , u1 , . . .)/∂ut.
(6)d(t, c) =∂ Û (u0 , u1 , . . .)/∂u0
That is, d(t, c) is the marginal rate at which generation 0 substitutes consumption utility
between 0 and t. In the EDU model, d(t, c) = δ t . For d(t, c) to be well defined, thederivatives in (6) must exist. This is always the case when G is differentiable.25
Proposition 3. Suppose that G is differentiable. Then, d(1, c) = αG0 (U (1 c)) and, fort ≥ 2,
"
#
t−1
τY
−1
X
d(t, c) = αt G0 (U (t c)) 1 +G0 (U (t−τ c)) (1 + G0 (U (t−s c))) ,
τ =1

where

s=1

Qτ −1

0
s=1 (1 + G (U (t−s c))) ≡ 1 if τ = 1.

To understand this formula, suppose that u(ct ) rises by a small amount. This has
two effects: (i) generation t’s well-being rises, which explains the term G0 (U (t c)); consequently, (ii) the well-being of each generation τ ∈ [1, t] also rises, which explains the25

Differentiability is a mild additional assumption, since G’s monotonicity already implies that it isdifferentiable almost everywhere.

20

summation. Moreover, the rise in U (t c) affects U (t−τ c) through the well-being of all
generations between t − τ and t, which explains the product.
For an arbitrary altruism function G, the discount function d(t, c) thus depends not
only on generation t’s consumption, but on the entire stream c. Suppose, to illustrate,that G is concave. A drop in the consumption of some future generation s reduces thewell-being of earlier generations, which increases the G0 terms and, hence, the discountfactor d(t, c): Generation 0 becomes more willing sacrifice its current satisfaction u(c0 )in favor of future generations’. The only case in which d(t, c) is independent of c is when
G is linear, in which case, the discount factor takes a well-known form.26). Then, for all t ≥ 1,
Corollary 3. Suppose that G(U ) = γU with γ ∈ (0, 1−ααd(t, c) = βδ t ,γ
where β = 1+γ< 1 and δ = (1 + γ)α < 1.

Proof. By Proposition 3, the result is immediate for t = 1, 2. For t ≥ 3,
"
#
t−1
X
d(t, c) = αt γ 1 + γ(1 + γ)τ −1 = αt γ(1 + γ)t−1 .
τ =1Thus the class of preferences that Phelps and Pollak (1968) invented to model “imperfectly” altruistic generations is actually equivalent to a specific version of preferencesthat exhibit direct pure altruism. By contrast, the EDU model—which Phelps and Pollak (1968) viewed as modeling “perfectly” altruistic generations—is actually a specificversion of preferences that exhibit only indirect pure altruism. The present paper reverses the common view concerning which of the exponential and β-δ discounting models
captures a deeper degree of intergenerational altruism.
The EDU model obtains at the limit of ours as direct altruism towards generationst ≥ 2 becomes negligible. This limit is obtained by taking α to zero while keeping the
product αγ equal to some constant δ̂—which necessitates that γ becomes arbitrarily
large. The weight that generation 0 puts on generation t is γαt , which converges to zero
except for t = 1. In this case, Corollary 3 implies that the preference has a beta-delta
representation, where β = γ/(1+γ) converges to 1 and δ = (1+ γ)α converges to δ̂ = γα,which corresponds to the EDU model.
26

A formally equivalent result appears in Saez-Marti and Weibull (2005) and Fels and Zeckhauser(2008).

21

4.3

Quasi-hyperbolic Discounting of Consumption Utilities

Since the linearity of G implies quasi-hyperbolic discounting, we wish to determine which
properties of the preference imply this linearity. When G is nonlinear, we observed earlier
that generation 0’s trade-off between its consumption and that of generation t typicallydepends on the well-being of other generations. Building on this observation, Axiom 10
rules out this type of dependence, and yields the desired conclusion.
Axiom 10 (Consumption Independence).
(i) (c0 , c1 , 2 c)  (c00 , c01 , 2 c) if and only if (c0 , c1 , 2 c0 )  (c00 , c01 , 2 c0 );(ii) (c0 , c1 , 2 c)  (c00 , c1 , 2 c0 ) if and only if (c0 , c01 , 2 c)  (c00 , c01 , 2 c0 ).
Intuitively, condition (i) says that a grandmother trades off her consumption with thatof her son in a way that does not depend on the consumption and, hence, the well-beingof her son’s descendants. Condition (ii) says that she trades off her consumption withthat of her son’s descendants in a way that does not depend on her son’s consumption.
Theorem 4 (Linear Pure-Altruism Representation). Axiom 1-8 and 10 hold if and only
if the function U may be chosen so that∞X
U (c) = u(c0 ) +αt γU (t c),
t=11−αwhere α ∈ (0, 1), γ ∈ (0, α ) and u : X → R is a continuous nonconstant function.

Corollary 4 (Quasi-hyperbolic Discounting). Axiom 1-8 and 10 hold if and only if there
exist β, δ ∈ (0, 1) and a continuous nonconstant function u : X → R such that∞X
δ t u(ct ).
U (c) = u(c0 ) + β
t=1

Proof. By Theorem 4, U (c) is a strictly increasing, linear function of the well-beings
U (t c) and of u(c0 ). By induction , U (c) is a strictly increasing, linear function of theconsumption utilities u(ct ) for t ≥ 0. There exists then a function κ(t) : N \ {0} → R++such that∞X
U (c) = u(c0 ) +κ(t)u(ct ).
t=1

Clearly, for all t ≥ 1, κ(t) = d(t, c) defined in (6). Corollary 3 then shows the claim.
This result allows us to understand β-δ discounting of consumption utilities in terms of
simple properties of generation 0’s directly altruistic preference. This preference depends
on two, conceptually different, entities: generation 0’s physical consumption and the22overall well-being of all future generations. This sheds light on why generation 0 treats
its consumption utility differently from the consumption utilities of all future generations,in a way that is, however, coherent across them as captured by altruism stationarity (Axiom 8). Moreover, by expecting all future generations to also be altruistic, generation 0
develops the view that future generations should be more willing to sacrifice their consumption for the good of their descendants than how much it itself is willing to sacrifice
its own consumption for them (present-generation bias). These properties contributeto explaining why the different treatment of future consumption utilities takes a starkform in Phelps and Pollack’s (1968) model, where those utilities are all scaled uniformlyby β < 1. Finally, generation 0 evaluates its consumption and future generations’ wellbeing in a separable way, and trades off its consumption against that of any future generation independently of other generations’ well-being. This delivers additivity in terms of
consumption utilities. Note that Axiom 10 coincides with Koopmans’ (1960) strong separability axiom (Postulate 30 ), which in his model delivers EDU. Therefore, according to
the present paper, one single axiom—the stationarity one—distinguishes exponential andquasi-hyperbolic discounting. Put differently, in terms of the consumption-utility representation (4), Koopmans’ stationarity corresponds to setting β = 1, whereas altruism
stationarity corresponds to setting β < 1.
Corollary 4 provides a tight link between the degree of present bias, β, and the marginalutility from altruism, γ. In this linear representation, it is possible to interpret γ as thedegree to which generation 0 finds future generations’ well-being “imaginable” or “vivid.”
Note that this is distinct from how much generation 0 cares about its descendants as theintergenerational gap increases, which is captured by αt . Thus, we identify two possible
determinants of how much generation 0 is altruistic towards generation t. Corollary 4
implies that β is directly proportional to γ, so generation 0 becomes less present biased
when it anticipates more vividly the well-being of future generations.27
Other Axiomatizations of β-δ Discounting
Other papers axiomatize preferences over consumption streams that correspond to theβ-δ model (Hayashi (2003); Olea and Strzalecki (2014)). These papers take the perspective of a single individual instead of different generations.28 To attempt a comparison
with the present paper, note that generation 0—the decision-maker in our framework—
27

Vividness of the well-being of future generations implied by today’s decisions may be influenced withspecific information campaigns. For example, consider the dramatic images that media and environmental organizations report on the catastrophic consequences of manmade climate change.
28
While Phelps and Pollak (1968) focused on an intergenerational interpretation of quasi-hyperbolic
discounting, the single-individual interpretation is equally widespread (following Laibson (1997)).

23

essentially treats its descendants as future copies of itself. Thus, from a single-individual
perspective, one may view our analysis as describing an agent who perceives himself asa collection of selves, one for each period t, and exhibits direct pure altruism towards
his future selves, expecting that they will do the same. Therefore, self 0 directly cares
about the well-being (that is, U (t c)) of all his future selves through what we may callintrapersonal altruism,29 an attitude that is revealed by his current preference over consumption streams.
The property that self 0 directly cares about all his future selves’ well-being is thefirst, key conceptual difference from the previous axiomatizations of β-δ discounting.
Both Hayashi (2003) and Olea and Strzalecki (2014) adopt the common view that thedecision-maker cares only about his per-period consumption utilities.30 Within thisframework, Koopmans’ (1960) stationarity (Axiom 9) is replaced by a weaker notion,
quasi-stationarity, which means stationarity from period 1 onward. However, if thedecision-maker evaluates streams based only on consumption utilities and views consumption in the same way in all periods, it is unclear why stationarity should holdbetween period 1 and later periods, but not between period 0 and period 1. This issue
does not arise with altruism stationarity (Axiom 8), as self 1’s well-being is equivalent
to the well-being of all future selves, but is different from self 0’s physical consumption.
Second, to obtain the β-δ representation, Olea and Strzalecki (2014) need to ensurethat current and future per-period utilities are cardinally equivalent. Their axioms leadto novel, practical experiments to identify and measure β and δ, but may be difficult tointerpret. That paper imposes an additional, explicit present-bias axiom to obtain β ≤ 1.
By contrast, when self-0 directly cares about the well-being of all his future, similarly
altruistic selves, present bias follows as a general, logical consequence (Proposition 2). In
our theory, β-δ discounting is tightly linked with an intuitive consumption-independence
condition (Axiom 10).
Echenique et al. (2016) propose a different method to characterize several models of
29

The idea that an individual over time consists of a collection of selves is not new in economics (see,for example, Strotz (1955) who thought that “the individual over time is an infinity of individuals” (p.
179) and Frederick (2003)); it also has a long tradition in philosophy (see, for example, Parfit (1971,
1976, 1982)). In this case, direct pure altruism simply means that the individual’s present self cares
about his future selves and anticipates that they will continue to do the same.
30
The second paper, in particular, is based on the idea of annuity compensation: To avoid relying
on assumptions of the form of u to elicit β and δ separately, the idea is to consider fix compensationlevels—and hence fixed u’s—and vary the time horizon at which they occur, so as to find exact points
of indifference for the decision-maker and hence infer the parameters of the model. The decision-makerhas different subjective views of the time distance between period 0 and 1 and between any two future
periods. If he cares only about the u’s that he gets in each period, then it is possible to objectively spaceout these u’s in an appropriate way so as to identify β and δ.

24

intertemporal preferences, including the EDU and β-δ model. Their starting point isa dataset consisting of a decision-maker’s choices of finite consumption streams fromstandard intertemporal budget sets and the dated prices that define them. They discussversions of the Generalized Axiom of Revealed Preference that satisfy various separability
requirements adapted to intertemporal settings. Using real data from the experimentin Andreoni and Sprenger (2012), they apply their axioms to classify subjects acrossmodels: Roughly, only one third of the 97 subjects is consistent with either EDU or β-δ
discounting, and about half violates time separability in consumption.

4.4

Time Inconsistency

The analysis thus far has concerned the preference of a single generation. This section
considers the implications of direct pure altruism on how different generations evaluate
a common consumption stream.
It is well known that if each generation has the same preference and this preference canbe represented by Koopmans’ (1960) recursive model, then as a whole these preferences
form a sequence which satisfies time consistency. That is, if a course of action startingat time t is preferable according to generation t’s preference, then it remains preferable
for time t onward according to generation t − 1’s preference. To formalize this, considert
31
the family {t }∞t=0 , where  is the preference relation of generation t.
Definition 5 (Time Consistency). {t }∞t=0 exhibits time consistency if, for every t ≥ 0,s ≥ 1, and c, c0 , ĉ ∈ C, the following condition holds:(ĉt , . . . , ĉt+s−1 , c) %t (ĉt , . . . ĉt+s−1 , c0 ) if and only if c %t+s c0 .
Despite its similarity with Axiom 9 when s = 1, time consistency is conceptuallydifferent because it involves multiple, possibly different, preference relations.32 Indeed,
Koopmans et al. (1964) write,“[Stationarity] does not imply that, after one period has elapsed, the ordering
then applicable to the ‘then’ future will be the same as that now applicable to the‘present’ future. All postulates are concerned with only one ordering, namely that
guiding decisions to be taken in the present. Any question of change or consistency
31

Definition 5 expresses in formal terms the idea of time consistency informally introduced by Strotz(1955) in his seminal paper (p. 171).
32
This point has also been recently emphasized in the experimental analysis of Halevy (2015). It is
straightforward to construct examples of preferences {t }∞t=0 that satisfy Axiom 9 but are not time
consistent, and vice versa.

25

of preferences as the time of choice changes is therefore extraneous to the present
study.” (Koopmans et al. (1964), p. 85, emphasis in the original)

Nonetheless, under the assumption that {t }∞t=0 is time invariant—that is, there exists
ˆ such that t = ˆ for all t ≥ 0—time consistency is
a common preference relation 
equivalent to Koopmans’ (1960) stationarity. As noted, this notion of stationarity implies
that each generation exhibits indirect altruism. The converse of this is that direct altruism
is incompatible with time consistency, a statement proved in the Supplemental Material.
t0
Proposition 4. Consider a family of preferences {t }∞t=0 and suppose that  = forall t ≥ 1 and that 0 has a pure-altruism representation. Then, {t }∞t=0 satisfies time
consistency if and only if

V (c0 , U (1 c), U (2 c), . . .) = V (c0 , U (1 c))
for all c ∈ C, and V is strictly increasing in its second argument.
Proposition 4 identifies and highlights a possible source of time inconsistency across generations. This source corresponds to the intuitive property that each generation directly
cares about the well-being of future generations beyond its immediate descendant—as inour running example of a grandmother, her son, and her granddaughter. It is common
to view time consistency of intergenerational preferences as the norm and time inconsistency as an exception. Proposition 4 reverses this view. If we deem plausible thateach altruistic generation cares about future generations’ well-being beyond its immediate descendant and expects them to do the same, then we have to conclude that time
inconsistency should be the norm, not the exception.33
In general, time inconsistency can take many forms. For instance, if we allow thepreferences to violate time invariance and differ in arbitrary ways across generations, we
could have that generation t prefers higher consumption (say, of fossil fuels) for itself andall future generations, whereas generation t + 1 prefers lower consumption for itself andall subsequent generations. This kind of inconsistency is clearly possible, but differs fromthe one arising here. In particular, directly altruistic preferences are time inconsistent
To see the intuition behind Proposition 4, suppose that 1 c and 1 c0 represent two courses of actions
that start in period 1 and involve different consumptions only for generations 1 and 2 (that is, c1 6= c01 ,c2 6= c02 , but 3 c = 3 c0 ). In this case, generation 0 (the grandmother) and generation 1 (her son) may
disagree on the ranking of 1 c and 1 c0 for the following reason. The son trades off how his well-being varies
because his current consumption changes from c1 to c01 and because his daughter’s well-being is affectedby the change from c2 to c02 . In addition to these two effects on her son’s well-being, the grandmotheralso takes directly into account how her granddaughter’s well-being varies between c2 and c02 . Therefore,
the grandmother and her son internalize the effects of changing c2 in different ways, which can inducethem to disagree on which course of action is better.
33

26

even if all generations have the same preference relation. Moreover, by Proposition 2
this inconsistency takes a very specific form: Each generation tends to disproportionatelyfavor its own satisfaction over that of any future generation.

4.5

Intergenerational Trade-offs with Backward-looking Altruism

While our theory’s focus on forward-looking agents is standard, incorporating backwardlooking altruism to the theory would be instructive and realistic. An axiomatic treatmentof two-sided altruism raises conceptual and technical challenges beyond the scope of thepresent paper—some of which is discussed in Section 6 along with the related literature.
As a first step in this direction, this section explores how the intergenerational trade-offs
discussed earlier would be affected by the addition of backward-looking altruism. As itturns out, backward-looking altruism has rather counter-intuitive consequences.
We illustrate these consequences in settings with two and three generations. Suppose,first, that there are only two generations, the second of which may be affected by the
first generation’s well-being. Specifically, consider the linear specification
U0 = u0 + αU1

U1 = u1 + α̂U0where ui is generation i’s utility from consumption and α and α̂ are nonnegative factors
used to discount future and past generations’ well-being. This system has a uniquesolution as long as the product αα̂ is strictly less than 1, yielding
1
1(u0 + αu1 )U1 =(u1 + α̂u0 ).
U0 =1 − αα̂
1 − αα̂Ignoring the scaling factor 1/(1 − αα̂), notice that the weight put by generation 0 on
generation 1’s consumption utility, relative to its own, is equal to α and thus independentof generation 1’s altruism coefficient α̂. Remarkably, therefore, adding backward-looking
altruism does not affect how generation 0 trades off the consumption utilities of bothgenerations. This result has the following intuition. With backward-looking altruism,
increasing 0’s consumption utility now raises 0’s well-being by increasing 1’s well-being,
which 0 finds valuable. But increasing 1’s consumption utility increases 1’s well-being
directly, and both increases, whether coming from 1’s backward-looking altruism or froma direct increase of u1 , get injected into 0’s well-being in the same way through 0’s
forward-looking altruism. Therefore, consumption utilities u0 and u1 have the same
relative impact on 0’s well-being with this new channel as they did over the original
forward-looking channel.

27

The effect of backward-looking altruism gets even more surprising—in fact, counterintuitive—if one adds a third generation. Suppose that
U0 = u0 + αU1 + α2 U2U1 = u1 + αU2 + α̂U0U2 = u2 + α̂U1 + α̂2 U0 .
This system has a unique solution provided αα̂ < 31 , yielding
1[(1 − αα̂)u0 + α(1 + αα̂)u1 + 2α2 u2 ].
U0 =(1 + αα̂)(1 − 3αα̂)Now the weights of u1 and u2 have increased, relative u0 ’s weight, as a result of generation 1’s and 2’s altruism towards older generations (α̂ > 0).34 This is counterintuitive
as one would expect that generation 0’s consumption matters more now that it affects
generation 1’s and 2’s well-being. To gain intuition, it is useful to compare this situation with the two-generation case, where the relative weights on 0’s and 1’s consumption
utility on 0’s well-being were independent of α̂. Consider the increase of u2 ’s weight
relative to u0 ’s weight. Backward-looking altruism causes generation 0’s well-being toaffect the well-being of generations 1 and 2. By forward-looking altruism, u2 also affects
the well-being of generation 1 and 2 by affecting generation 0’s well-being. Therefore,
if these were the only effects of u2 , its weight relative to u0 would not depend on theamount of backward-looking altruism, just as in the two-generation case. In addition,
however, now there is an intermediate generation between 0 and 2. Therefore, u2 also
enters the backward-looking channel that links 1’s well-being to 2’s well-being; and thisadditional effect—from which u0 is excluded—increases the impact of u2 relative to u0 .
The increase in u1 ’s relative weight has a similar intuition.

5

Welfare Analysis with Intergenerational Altruism

Models featuring time-inconsistent preferences raise serious conceptual problems whendefining welfare criteria and discuss policy questions (see, for example, Rubinstein (2003);
Bernheim and Rangel (2007, 2009)). Concerning β-δ discounting, Rubinstein (2003)
notes that
“Policy questions were freely discussed in these models even though welfare assessment is particularly tricky in the presence of time inconsistency. The literature
34

Note that the weight of u1 relative to u0 , α(1 + αα̂)/(1 − αα̂), as well as the weight of u2 relative
to u0 , 2α2 /(1 − αα̂), are increasing in α̂.

28often assumed, though with some hesitation, that the welfare criterion is the utility
function with stationary discounting rate δ (which is independent of β).” (p. 1208)

A more fundamental question is whether time inconsistency across generations’ preferences justifies some form of paternalistic social planner. An immediate consequence ofthe present paper is to weaken the case for paternalistic interventions. If intergenerational time inconsistency were the result of some form of bounded rationality or lack ofconsideration for future generations, one may be tempted to argue that society benefits
from delegating its choices to a paternalistic planner. This argument, however, is mootif time inconsistency is the logical consequence of direct pure altruism which every generation fully takes into account. In this case, the current generation 0 is already taking
into account the preferences that it expects all future generations to have. Thus, unlessthe social planner believes that generation 0 anticipates these preferences in a systematically biased way, one may argue that she should simply adopt a “libertarian” stanceand measure the welfare of any stream c using W L (c) = U (c).
This libertarian criterion coincides with the criterion that is usually applied for standard
time-consistent models, such as EDU, but has significantly different implications in thepresent model. For standard models, W L corresponds to evaluating consumption streams
for the entire society based only on the consumption of generation 0 and how it thinksthat generation 1 will rank the remaining entire streams. In our model, by contrast, W L
evaluates streams based on the consumption of generation 0 and how it thinks that all
future generations will rank the streams they will face for their remaining future. For
example, consider two policies, A and B, inducing streams cA and cB such that 1 cA ∼ 1 cB
but t cA  t cB for all t ≥ 2. Then, the criterion W L based on the standard model implies
that policy A is as desirable as B. By contrast, the criterion W L based on our model
implies that A is strictly more desirable. Thus, the second criterion is more sensitive to apolicy long-run consequences for all future generations’ well-being. From this perspective,
it is far from obvious that such a criterion should be dismissed simply because it violates
time consistency.
Despite generation 0’s direct pure altruism, one may still argue that a libertarian criterion does not sufficiently take into account the preferences of future generations andattempt to include these preferences above and beyond how they affect generation 0. This
leads to a paternalistic welfare criterion aggregating the preferences (that is, well-being)across all generations (Bernheim (1989), Phelan (2006), Farhi and Werning (2007)). Crucially, one may ask whether there exists an aggregator which, despite the preferences’

29

time inconsistency, renders the social planner time consistent. For the additive criterion∞XP
W (c) =
w(t)U (t c),(7)
t=0

where w : N → R++ , we obtain the following result.
Proposition 1. Suppose that all generations have the same preference representations,
given by Theorem 4 and Corollary 4 with parameters (u, α, γ) and (u, β, δ), respectively.
Then,
∞
X
P
W (c) =
δ t u(ct )t=0t

if and only if w(t) = α .
This proposition provides a novel justification for the welfare criterion, consisting indropping the β factor, cited by Rubinstein. We focus here on proving the “if” direction,
which has an intuitive justification, relegating the proof of the “only if” direction to
P∞ t
Appendix A. Let W (c) =t=0 α U (t c). Using the linear representation of U (c) in
Theorem 4, we haveX
W (c) = u(c0 ) + (1 + γ)ααt−1 U (t c) = u(c0 ) + (1 + γ)αW (1 c).
t≥1Comparing the extremes of this equation already shows why the planner must be time
consistent. It also shows that W (c) must equal the discounted sum of consumption
utilities with discount factor δ̂ = (1 + γ)α. We know from Corollaries 3 and 4 that δ̂ = δin the β-δ version of U (c), which proves the “if” part. There is a more intuitive argument,
however. Note that the linear representation satisfies
γα X tU (c) = u(c0 ) + γαW (1 c) = u(c0 ) +δ̂ u(ct ).
δ̂ t≥1
It follows that U (c) must have a quasi-hyperbolic representation in terms of consumption
utilities, where the long-run discount factor coincides with the planner’s factor.
Proposition 1 implies, for the widespread Phelps–Pollak model of intergenerational
altruism, the existence of a paternalistic welfare criterion that aggregates the timeinconsistent preferences of all generations, yet renders the planner time consistent. This
criterion is completely determined by generation 0’s revealed preference, summarized byu, α, and γ and is easily interpreted by viewing α as a survival probability used by boththe social planner and each successive generation to assess the relevance of each successive period. However, the conclusion of Proposition 1 does not hold for more general
directly altruistic preferences.35
P∞
For a general function G, the criterionPW P (c) = t=0 αt UP(t c) need not satisfy time
P∞consistency. It is∞∞
possible to construct examples in which t=1 αt G(U (t c)) > t=1 αt G(U (t c0 )) and t=1 αt [G(U (t c)) +35

30

By contrast, aggregating time-consistent preferences by a similar process leads to a timePtinconsistent welfare criterion. In particular, using the EDU model U (c) = ∞t=0 δ u(ct ),we may rely on δ to aggregate well-being across generations using the criterion Ŵ (c) =P∞ t
t=0 δ U (t c) (cf. Ray (2014)). Doing so makes the planner time inconsistent, since it canPtbe shown that Ŵ (c) = u(c0 )+ ∞t=1 δ (1+t)u(ct ). Except for Phelps and Pollack’s (1968)
quasi-hyperbolic model, pure altruism across generations—whether direct or indirect—
thus makes it difficult to construct a welfare criterion that includes the preferences of all
generations and renders a planner time consistent.

6

Concluding Remarks

We have examined how pure direct altruism towards future generations shapes the preference of a generation over courses of action that have long-lasting consequences. Eachgeneration exhibits positive discounting and a present-generation bias. The dependenceof each generation’s well-being on that of its descendants gives rise to rich discounting
patterns, the simplest of which turns out to be quasi-hyperbolic discounting. Our analysis
characterizes directly altruistic preferences within a classic decision-theoretic frameworkand in terms of axioms that are easy to interpret. Given the importance of uncertainty
for long-run problems—such as poverty or climate change—future work should extend
the theory to allow for stochastic consumption streams.
A natural question is whether the theory in this paper is descriptive or normative. Weview these interpretations as non-exclusive and both valid and provide rigorous conceptual tools which may be used for both. On the one hand, our axioms are meant tocapture general properties that one might expect to describe how each generation ranks
consumption streams for itself and its descendants. Our results then identify which
utility functions satisfy these properties and help understand their key behavioral implications. On the other hand, one can view our axioms as crystallizing some ethicalprinciples—whose origin lies outside the model—that all generations adopt and share as
guides for intergenerational altruism. Such principles might be that one should be sensitive to all of one’s descendants’ preferences (to the extent that they may be anticipated)and should try to treat those preferences in a neutral way. For instance, the ongoing
public debate on the control of greenhouse-gas emissions often stresses the importanceof taking into account the consequences of these decisions for many generations far intoP∞P∞P∞
U (t c)] > t=1 αt [G(U (t c0 )) + U (t c0 )], but t=1 αt U (t c) < t=1 αt U (t c0 ). In this case, W P (c0 , 1 c) >W P (c0 , 1 c0 ), but W P (1 c) < W P (1 c0 ).

31the future, but usually does not consider complementarities or complicated dependence
across these generations. These principles may provide a normative foundation for our
main axioms—intergenerational separability and altruism stationarity—and hence forour main representation theorem. More generally, the paper points out that requiring
preferences to be time consistent on the basis of some normative consideration comes atthe cost of violating other properties which may also have a normative appeal.
Our welfare analysis also lends itself to positive and normative interpretations. On
the one hand, it connects generations’ altruistic preferences with the welfare criterion
used by a social planner. This is important if, for instance, the planner coincides witha democratically elected government which responds only to the preference of the current generation: Its altruism will be the only channel through which the government
will take into account future generations. In this case, the paper emphasizes that the
government will inherit the current generation’s desire for long-run sustainable policies(such as adopting renewable sources of energy) but also its present bias, which may render commitments to those policies desirable. On the other hand, the aggregating rulesthat we considered may again be based on some external ethical principles—for example,
that the planner should directly weigh the well-being of each generation in a way that
includes its own altruism. These principles, combined with direct pure altruism, may
give a normative meaning to the time-consistent criterion that we derive in our welfare
analysis. Our results highlight that, at least for a well-known and broadly-used model ofintertemporal preferences, it is possible to reconcile time inconsistency at the generations’level with time consistency at the planner’s level.
This paper proposes a theory of forward-looking altruism, which is consistent with alarge literature on intergenerational preferences and instrumental in comparing our approach with other theories of forward-looking agents. The addition of backward-looking
altruism can affect intergenerational trade-offs in surprising and even counter-intuitive
ways, as we saw in Section 4.5. Beyond this, incorporating backward-looking altruism
into an axiomatic approach raises a number of conceptual challenges.36 One is an observability problem: how can we determine how a generation’s preference depends on the
consumption of its ancestors, when this consumption is past and thus unchangeable? A
potential resolution is to extract from a generation’s preference information about how
this generation perceives that its descendants will care about itself, so as to back out
the backward-looking component of altruism. Another challenge concerns the infinite
36

Backward-looking altruism is considered by Peleg and Yaari (1973), Pearce (2008), Kimball (1987),
Bergstrom (1989), and Bergstrom (1999), who assume specific utility functions and derive conditions
leading to well-defined preferences and equilibrium existence.

32regress stemming from reciprocal altruism, and the technical difficulties that it raises
for an axiomatic formulation. Addressing the reciprocity challenge would pave the way
to an axiomatic theory of altruism across contemporaneous members of society, extending the analysis beyond backward-looking altruism. Incorporating backward-looking andcontemporaneous altruism into an axiomatic approach is an exciting avenue for future
research; the framework and techniques developed in this paper should prove helpful tothis end.37

37

In particular, a well-known problem when writing a reciprocal altruism model is that the utility
function may “blow up.” However, if one starts with preferences as primitives, then the class of model
emerging from axioms capturing reciprocal altruism will come with specific parametric restrictions—also
implied by the axioms—guaranteeing that the models of the class are well specified and do not blow up.
Even in our theory of forward-looking altruism, the altruism utility G must be (1 − α)/α-Lipschitz continuous, by Proposition 1, which guarantees that the discounting effect dominates the forward-lookingaltruism effect so that utility is always well-defined and finite in the additive representation (5).

33

A

Appendix: Proofs of the Main Results

A.1

Proof of the “only if ” part of Theorem 2

Suppose that there exists a continuous function U : C → R which represents  and that U (c) = V (f (c))
where V : F0 → R is a function which is nonconstant in f0 and ft for some t ≥ 1. This last property
immediately implies that  satisfies Axiom 4. Since U is a continuous representation of %, standard
arguments imply that this relation satisfies Axioms 1-3. Finally, consider any streams c, c0 ∈ C thatsatisfy c0 = c00 and t c ∼ t c0 for all t ≥ 1. Then, since U represent , we have U (t c) = U (t c0 ) for all t ≥ 1;that is, ft (c) = ft (c0 ) for all t ≥ 1. Since by assumption f0 (c) = c0 = c00 = f0 (c0 ) and V is a function,
we must have V (f (c)) = V (f (c0 )) or, in other words, U (c) = U (c0 ), which in turn implies that c ∼ c0 as
required by Axiom 5.

A.2

Proof of Theorem 3

By the definition of F in (3), note that F need not be a Cartesian product, as ft depends on fs fors > t. Letting t f = (ft , ft+1 , . . .), we can denote elements in F by (f1 , f2 , . . . , ft−1 , t f ). On F0 = X × F(where f0 (c) = c0 ), the primitive  induces a ∗ with representation V : F0 → R; by Theorem 2, ∗is well defined. If F0 were a Cartesian product, we could mimic the steps in Debreu (1960) (Theorem3) and Koopmans (1972) on the domain F0 to prove our theorem. However, this is not possible. We
will then proceed as follows. In step 1, we show that ∗ satisfies the essentiality and strong separability
properties at the heart of Debreu’s (1960) Theorem 3. In step 2, we show that the ranking of streams(f0 , f1 , f2 , 3 f ) ∈ F0 depends only on a function of 3 f ; so we can restrict attention to a four dimensional
space. In step 3, we show that this space is a Cartesian product ‘locally;’ so we can apply Debreu’s result
to obtain an additive representation ‘locally.’ Since additive representations are unique up to positive,affine transformations, we can extend uniquely the additive representation to the entire F0 . In step 4,
we show that this representation takes the form in our Theorem 3.
Step 1. Lemma 1 says that, if (f0 , f1 , f2 , 3 f ) ∗ (f00 , f10 , f20 , 3 f 0 ), then changing the common components of (f0 , f1 , f2 , 3 f ) and (f00 , f10 , f20 , 3 f 0 ) in the same way leaves the ranking under ∗ unchanged.

Lemma 1. Fix any nonempty subset π of {0, 1, 2, 3}. Then(f0 , f1 , f2 , 3 f ) ∗ (f00 , f10 , f20 , 3 f 0 ) ⇔ (fˆ0 , fˆ1 , fˆ2 , 3 fˆ) ∗ (fˆ00 , fˆ10 , fˆ20 , 3 fˆ0 ),where ft = fˆt , ft0 = fˆt0 , 3 f = 3 fˆ, and 3 f 0 = 3 fˆ0 if t or 3 are in π, and ft = ft0 , fˆt = fˆt0 , 3 f = 3 f 0 , andˆˆ0
3 f = 3 f if t or 3 are not in π.

Proof. Recall that t c ∼ t c0 is equivalent to ft = ft0 . Then, by Axiom 6, for any π
V (f0 , f1 , f2 , 3 f ) > V (f00 , f10 , f20 , 3 f 0 ) ⇔ V (fˆ0 , fˆ1 , fˆ2 , 3 fˆ) > V (fˆ00 , fˆ10 , fˆ20 , 3 fˆ0 ).

Using Lemma 1 with π = {0} and π = {1, 2, 3}, we obtain the following.

34

Lemma 2. The function V : F0 → R can be written in the form
V (f ) = W (u(f0 ), d(1 f )),(8)

where u : X → I u ⊂ R and d : F → D ⊂ R. W is jointly continuous in its two arguments and strictlyincreasing in each of them, u is continuous, and Iu and D are non-degenerate intervals.

Proof. Consider ∗ on X × F and Lemma 1 with π = {0} and π = {1, 2, 3}. By an argument similar
to that in Section 5 of Koopmans’ (1960), for any f we can write V (f ) = W (u(f0 ), d(1 f )), whereu(f0 ) = V (f0 , 1 fˆ) for some 1 fˆ ∈ F and d(1 f ) = V (f00 , 1 f ) for some f00 ∈ X. Recall that V (f (c)) = U (c)for all c ∈ C. Hence, the continuity property of U implies continuity of u. By Axiom 4, neither u
nor d can be constant. Since X is connected, u takes all values in a connected interval Iu ⊂ R. Sinced(1 f (c)) = U (f00 , 1 c), U is continuous, and X is connected, d takes all values in a connected interval
D ⊂ R. By definition of u and Lemma 1 with π = {0}, W must be strictly increasing in its first argument
on Iu . Similarly, by definition of d and Lemma 1 with π = {1, 2, 3}, W must be strictly increasing in
its second argument on D. Given ĉ, U (·, 1 ĉ) takes values in an interval. Then the strictly increasing
W (·, d(1 f (ĉ))) also takes values in an interval and hence must be continuous in its first argument on Iu .
By a similar argument, W must be continuous in its second argument, and hence jointly continuous on
Iu × D.

Hereafter, let u = sup Iu and u = inf Iu . Also note that the function d in Lemma 2 defines a ranking
on F.

Lemma 3. There exist x, y, z, x0 , y0 , z 0 ∈ X and c ∈ C such that (i) (z, c)  (z 0 , c), (ii) (y, z, c) ∼(y 0 z 0 , c), and (iii) (x, y, z, c) ∼ (x0 , y 0 , z 0 , c).

Proof. By Axiom 4, there exist z, z 0 ∈ X and c ∈ C such that (z, c)  (z 0 , c). Using representation(8), we have u(z) > u(z 0 ). Now, consider (y 0 , z, c) and (y, z 0 , c) where y = z and y 0 = z 0 . By Axiom 7(i),(y 0 , z, c)  (y 0 , z 0 , c).
Case 1: (y, z 0 , c) % (y 0 , z, c). Since Iu is connected, we can modify y to y 00 ∈ X so that u(y 00 ) takes any
value in [u(y 0 ), u(y)]. By Axiom 2, there exists y 00 such that (y 00 , z 0 , c) ∼ (y 0 , z, c); moreover, we musthave u(y 00 ) > u(y 0 ). Now consider (x, y 00 , z 0 , c) and (x0 , y 0 , z, c) where x = z and x0 = z 0 . By Axiom 7(i),(x0 , z, c)  (x0 , z 0 , c); so, by Axiom 8, (x0 , y 0 , z, c)  (x0 , y 00 , z 0 , c).
Case 1.1: (x, y 00 , z 0 , c) % (x0 , y 0 , z, c). Since we can modify x to x00 ∈ X so that u(x00 ) takes any value
in [u(x0 ), u(x)], by Axiom 2, there exists x00 such that (x00 , y 00 , z 0 , c) ∼ (x0 , y 0 , z, c).
Case 1.2: (x, y 00 , z 0 , c) ≺ (x0 , y 0 , z, c). We can modify z and y 00 to ỹ, z̃ ∈ X so that u(z̃) and u(ỹ) takeany value in [u(z 0 ), u(z)] and [u(y 0 ), u(y 00 )]. Moreover, we can do so maintaining (ỹ, z 0 , c) ∼ (y 0 , z̃, c)
by Axiom 2. Since (x, y 0 , z 0 , c)  (x0 , y 0 , z 0 , c), by Axiom 2, there exist ỹ and z̃ such that (x, ỹ, z 0 , c) ∼(x0 , y 0 , z̃, c). Finally, we must have u(z̃) > u(z 0 ), so (z̃, c)  (z 0 , c).
Case 2: (y, z 0 , c) ≺ (y 0 , z, c). We can modify z to ẑ ∈ X so that u(ẑ) takes any value in [u(z 0 ), u(z)].
Since (y, z 0 , c)  (y 0 , z 0 , c), by Axiom 2 there exists ẑ such that (y, z 0 , c) ∼ (y 0 , ẑ, c), and by Axiom 7(i)we must have (ẑ, c)  (z 0 , c) and hence u(ẑ) > u(z 0 ). Now consider (x, y, z 0 , c) and (x0 , y 0 , ẑ, c) wherex = z and x0 = z 0 .

35

Case 2.1: (x, y, z 0 , c) % (x0 , y 0 , ẑ, c). We can modify x to x̂ so that u(x̂) takes any value in [u(x0 ), u(x)].
By Axiom 7(i), (x0 , z 0 , c) ≺ (x0 , ẑ, c); so, by Axiom 8, (x0 , y, z 0 , c) ≺ (x0 , y 0 , ẑ, c). Then, by Axiom 2 there
exists x̂ such that (x̂, y, z 0 , c) ∼ (x0 , y 0 , ẑ, c).
Case 2.2: (x, y, z 0 , c) ≺ (x0 , y 0 , ẑ, c). We can modify y and ẑ to ŷ and ẑ 0 so that u(ŷ) and u(ẑ 0 ) takeany value in [u(y 0 ), u(y)] and [u(z 0 ), u(ẑ)]. Moreover, we can do so maintaining (ŷ, z 0 , c) ∼ (y 0 , ẑ 0 , c) by
Axiom 2. Since (x, y 0 , z 0 , c)  (x0 , y 0 , z 0 , c), by Axiom 2 there exists ŷ and ẑ 0 such that (x, ŷ, z 0 , c) ∼(x0 , y 0 , ẑ 0 , c).

Hereafter, for t ∈ {0, 1, 2, 3}, we will refer to the factor t of F0 as the component of position t + 1
in the representation (f0 , f1 , f2 , 3 f ) of every f ∈ F0 (e.g., the factor 2 is the third component of every(f0 , f1 , f2 , 3 f ) ∈ F 0 ).

Definition 6 (Debreu (1960)). For t ∈ {0, 1, 2}, if f ∗ f 0 for some f, f 0 ∈ F0 with fs = fs0 for
all s 6= t, then the factor t of F0 is called essential for ∗ . If (f0 , f1 , f2 , 3 f ) ∗ (f00 , f10 , f20 , 3 f 0 ) for somef, f 0 ∈ F0 with fs = fs0 for s = 0, 1, 2, then the factor 3 is called essential for ∗ .

Lemma 4. For all t ∈ {0, 1, 2, 3}, the factor t of F0 is essential.
Proof. By Axiom 4, the factor 0 is essential. Using the streams in Lemma 3, let 1 c = (x, y, z, c) and
0
0 0 0
0
0
0
1 c = (x , y , z , c) and consider the corresponding f and f in F0 with any f0 = f0 . We have f1 = f1 ,f2 = f20 , f3 > f30 , and ft = ft0 for all t ≥ 4. By Axiom 7(i), (f0 , f3 , f4 , 5 f ) ∗ (f00 , f30 , f40 , 5 f 0 ), hence thefactor 1 is essential. By Axiom 8, (f0 , f2 , f3 , 4 f ) ∗ (f00 , f20 , f30 , 4 f 0 ) and (f0 , f1 , f2 , 3 f ) ∗ (f00 , f10 , f20 , 3 f 0 ).

So the factors 2 and 3 are essential.

Step 2. By Lemma 1 with π = {2, 3}, ∗ also satisfies the following property:(f0 , f1 , 2 f ) ∗ (f0 , f1 , 2 f 0 ) ⇔ (fˆ0 , fˆ1 , 2 f ) ∗ (fˆ0 , fˆ1 , 2 f 0 ).
Define Q̃ = {(f1 (c), d(2 f (c))) : c ∈ C}. Note that Q̃ ⊂ U × D, but it need not be a Cartesian productbecause the value of d affects that of f1 .

Lemma 5. There exists a continuous function Ṽ : X × Q̃ → R such that, for all f ∈ F0 ,V (f ) = Ṽ (f0 , f1 , d(2 f )),(9)where d is the function defined in Lemma 2. For any f1 , f10 , d0 , and d00 we have the following:(5.i) if (f1 , d00 ) and (f10 , d00 ) are in Q̃, Ṽ (f0 , f1 , d00 ) > Ṽ (f0 , f10 , d00 ) iff38 f1 > f10 ;(5.ii) if (f1 , d0 ) and (f1 , d00 ) are in Q̃, Ṽ (f0 , f1 , d0 ) > Ṽ (f0 , f1 , d00 ) iff d0 > d00 .

Proof. Recall that d(·) defines a ranking on F and that 2 f ∈ F. For any (f0 , f1 , 2 f ) and (f00 , f10 , 2 f 0 )such that both (f0 , f1 , 2 f 0 ) and (f00 , f10 , 2 f ) are in F0 , by Lemma 1 with π = {2, 3}, V (f0 , f1 , 2 f ) ≥38

Hereafter, “iff” stands for “if and only if.”36

V (f0 , f1 , 2 f 0 ) iff V (f00 , f10 , 2 f ) ≥ V (f00 , f10 , 2 f 0 ). Moreover, for any (f1 , 2 f ) and (f1 , 2 f 0 ) in F, by Axiom
8, V (f0 , f1 , 2 f ) ≥ V (f0 , f1 , 2 f 0 ) iff W (u(f0 ), d(2 f )) ≥ W (u(f0 ), d(2 f 0 )), and therefore iff d(2 f ) ≥ d(2 f 0 ).
So, the ranking of (f0 , f1 , 2 f ) and (f0 , f1 , 2 f 0 ) depends only on the value of d(·). Now, for any f ∈ F0 ,setṼ (f0 , f1 , d(2 f )) = V (f0 , f1 , 2 f ).
The previous argument implies property (5.i).
Ṽ is well defined for the following reasons. First, if (f0 , f1 , 2 f ) and (f00 , f10 , 2 f 0 ) are such that ft = ft0 for
t = 0, 1 and d(2 f ) = d(2 f 0 ), then V (f0 , f1 , 2 f ) = V (f00 , f10 , 2 f 0 ) again by Axiom 8. Second, if (f0 , f1 , 2 f )and (f00 , f10 , 2 f 0 ) are such that either (f0 , f1 , 2 f 0 ) ∈/ F0 or (f00 , f10 , 2 f ) ∈/ F0 , then (f0 , f1 ) 6= (f00 , f10 ). So,000even if d(2 f ) = d(2 f ), Ṽ (f0 , f1 , d(2 f )) can be different from Ṽ (f0 , f1 , d(2 f 0 )).
Consider now (f1 , d00 ), (f10 , d00 ) ∈ Q̃. There exist c, c0 ∈ C, such that ft = ft (c) and ft0 = ft (c0 ) fort = 0, 1, and d(2 f (c)) = d(2 f (c0 )) = d00 . By Lemma 2, without loss, we can assume that 2 c = 2 c0 sothat 2 f (c) = 2 f (c0 ) = 2 f 00 . By Axiom 7(i), then V (f0 , f1 , 2 f 00 ) > V (f0 , f10 , 2 f 00 ) iff f1 > f10 , and property(5.ii) follows from (9).
Finally, Ṽ is continuous for the following reasons. For any (f1 , d) ∈ Q̃, Ṽ (·, f1 , d) = U (·, 1 c) for any csuch that f1 = f1 (c) and d = d(2 f (c)). Hence, the continuity property of U implies that Ṽ is continuous
in its first argument. Given any f0 and d ∈ D, Ṽ (f0 , ·, d) = U (f0 , ·, 2 c) for some c such that f0 (c) = f0
and d = d(2 f (c)). Hence, Ṽ must take value in a connected interval and, being strictly increasing, it
must be continuous in its second argument given f0 and d. By a similar argument, for any (f0 , f1 ),Ṽ (f0 , f1 , ·) must take values in a connected interval and, being strictly increasing, it must be continuous
in its last argument. It follows that Ṽ must be continuous on the connected set X × Q̃.

Now define Q = {(f1 (c), f2 (c), d(3 f (c))) : c ∈ C}. By an argument similar to that in the proof of
Lemma 5, using Lemma 1 with π = {3}, we obtain the following.

Lemma 6. There exists a continuous function V : X × Q → R such that, for all f ∈ F0 ,V (f ) = V (f0 , f1 , f2 , d(3 f )),(10)

where d is the function defined in Lemma 2. Moreover, for any f1 , f10 , f2 , f20 , d0 , and d00 we have thefollowing:(6.i) if (f1 , f2 , d0 ), (f10 , f2 , d0 ) ∈ Q, then V (f0 , f1 , f2 , d0 ) > V (f0 , f10 , f2 , d0 ) iff f1 > f10 ;(6.ii) if (f1 , f2 , d0 ), (f1 , f20 , d0 ) ∈ Q, then V (f0 , f1 , f2 , d0 ) > V (f0 , f1 , f20 , d0 ) iff f2 > f20 ;(6.iii) if (f1 , f2 , d0 ), (f1 , f2 , d00 ) ∈ Q, then V (f0 , f1 , f2 , d0 ) > V (f0 , f1 , f2 , d00 ) iff d0 > d00 .
Hereafter, for any c ∈ C, let d3 (c) = d(3 f (c)). Also, we say that c ∈ C induces (f0 , f1 , f2 , d3 ) ∈ X × Q¯ on X × Q;
if ft (c) = ft for t = 0, 1, 2 and d3 (c) = d3 . Note that the function V defines a preference 0moreover, by definition, for c, c ∈ C¯ 0 (c0 ), f1 (c0 ), f2 (c0 ), d3 (c0 )) ⇔ f (c) ∗ f (c0 ).
(f0 (c), f1 (c), f2 (c), d3 (c))(f¯ satisfies the following property (see Definition 4 in Debreu (1960)). Fix
Lemma 7. The preference 37

any nonempty subset π of {0, 1, 2, 3}. Then
¯ 00 , f10 , f20 , d03 ) ⇔ (fˆ0 , fˆ1 , fˆ2 , dˆ3 )(¯ fˆ00 , fˆ10 , fˆ20 , dˆ03 )(f0 , f1 , f2 , d3 )(fwhere ft = fˆt , ft0 = fˆt0 , d3 = dˆ3 , and d03 = dˆ03 if t or 3 are in π, and ft = ft0 , fˆt = fˆt0 , d3 = d03 , anddˆ3 = dˆ03 if t or 3 are not in π.

Proof. Given π, let πc be its complement. If 3 ∈ πc , then there exist c, c0 , ĉ, ĉ0 ∈ C such that, fort = 0, 1, 2, ft = ft (c), ft0 = ft (c0 ), fˆt0 = ft (ĉ0 ), fˆt = ft (ĉ), d(3 f (c)) = d(3 f (c0 )), and d(3 f (ĉ)) = d(3 f (ĉ0 )).
Then, by Lemma 2, f2 (c) = f2 (c0 , c1 , c2 , 3 c0 ) and f2 (ĉ) = f2 (ĉ0 , ĉ1 , ĉ2 , 3 ĉ0 ). Similarly, by Lemma 5,
f1 (c) = f1 (c0 , c1 , c2 , 3 c0 ) and f1 (ĉ) = f1 (ĉ0 , ĉ1 , ĉ2 , 3 ĉ0 ). Therefore, we can take 3 c = 3 c0 and 3 ĉ = 3 ĉ0 , sothat 3 f = 3 f 0 and 3 fˆ = 3 fˆ0 .39 It follows from Lemma 1, thatV (f0 , f1 , f2 , 3 f ) > V (f00 , f10 , f20 , 3 f 0 ) ⇔ V (fˆ0 , fˆ1 , fˆ2 , 3 fˆ) > V (fˆ00 , fˆ10 , fˆ20 , 3 fˆ0 ).
Hence, by (10), the result follows.
Suppose 3 ∈ π. Again, there exist c, c0 , ĉ, ĉ0 ∈ C, each inducing the respective element of X ×Q—in particular, d(3 f (c)) = d(3 f (ĉ)) and d(3 f (c0 )) = d(3 f (ĉ0 )). Then, by Lemma 2, f2 (c) = f2 (c0 , c1 , c2 , 3 ĉ) andf2 (c0 ) = f2 (c00 , c01 , c02 , 3 ĉ0 ). Similarly, by Lemma 5, f1 (c) = f1 (c0 , c1 , c2 , 3 ĉ) and f1 (c0 ) = f1 (c00 , c01 , c02 , 3 ĉ0 ).
Therefore, we can take 3 c = 3 ĉ and 3 c0 = 3 ĉ0 , so that 3 f = 3 fˆ and 3 f 0 = 3 fˆ0 . It follows again from
Lemma 1, thatV (f0 , f1 , f2 , 3 f ) > V (f00 , f10 , f20 , 3 f 0 ) ⇔ V (fˆ0 , fˆ1 , fˆ2 , 3 fˆ) > V (fˆ00 , fˆ10 , fˆ20 , 3 fˆ0 ).
Hence, by (10), the result follows.

Step 3: Let O be the set of vectors (f1 (c), f2 (c), d(3 f (c))) for c ∈ C, such that u < u(ct ) < u fort = 1, 2 and d(3 f (c)) ∈ intD. It is straightforward to check that O is nonempty and that Q is includedin the closure of O.40

Lemma 8. For any (f1 , f2 , d3 ) ∈ O, there exists η > 0 such that the rectangle
R(f1 , f2 , d3 ; η) = (f1 − η, f1 + η) × (f2 − η, f2 + η) × (d3 − η, d3 + η)lies in O.
Recall that by Lemma 6, if (f0 , f1 , f2 , 3 f˜) and (f0 , f1 , f2 , 3 f˜0 ) are in F0 and d(3 f˜) = d(3 f˜0 ), thenV (f0 , f1 , f2 , 3 f˜) = V (f0 , f1 , f2 , 3 f˜0 ).
40
To see that O =6 ∅, consider any constant c0 ∈ C such that u < u(c00 ) < u. By changing c03 so thatu(c3 ) varies continuously in an open interval around u(c03 ), by continuity of U we can continuously spanan open interval around f3 (c0 ). By Axiom 7(i), this variation in c3 leads to variations in f2 (c), whichmust span an open interval around f2 (c0 ), again by continuity of U . Since we are not changing c02 , byLemma 2, d3 (c) must change in an open interval around d3 (c0 ). Finally, by Lemma 5, f1 (c) must also
vary continuously in an open interval around f1 (c0 ). To see that Q ⊂ clO, notice that any point of Q
induced by some c ∈ C can be approximated, by slightly modifying c, by a c0 such that d3 (c0 ) ∈ intDand u(c0t ) ∈ intIu for t = 1, 2, i.e., a point in O.
39

38

Proof. Fix (f1 , f2 , d3 ) ∈ O and, for the inducing c, let ut = u(ct ) for t = 1, 2. Since d3 ∈ intD,there is an interval (d3 , d3 ) ⊂ D containing d3 . Since u < u2 < u, given d3 , there is an interval(f 2 (d3 ), f 2 (d3 )) ⊂ U, containing f2 and spanned by u2 ∈ intIu . Let η 0 > 0 be such that [d3 −η 0 , d3 +η 0 ] ⊂(d3 , d3 ). By the properties of W in Lemma 2, there exists η 0 > 0 such that f 2 (d3 ) < f 2 (d3 + η 0 ) < f2
and f 2 (d3 ) > f 2 (d3 − η 0 ) > f2 . Hence, for all d03 ∈ [d3 − η 0 , d3 + η 0 ], all f20 ∈ [f2 − ε(η 0 ), f2 + ε(η 0 )]are achievable by changing only u2 , where ε(η 0 ) = min{f2 − f 2 (d3 + η 0 ), f 2 (d3 − η 0 ) − f2 }. Sinceu < u1 < u, given f2 and d3 , there is an interval (f 1 (f2 , d3 ), f 1 (f2 , d3 )) ⊂ U, containing f1 andspanned by u1 ∈ intIu . By the properties of Ṽ in Lemma 5, there exist η 00 > 0 and ε00 > 0 such that[d3 −η 00 , d3 +η 00 ] ⊂ (d3 , d3 ), [f2 −ε00 , f2 +ε00 ] ⊂ (f 2 (d3 ), f 2 (d3 )), and f 1 (f2 , d3 ) < f 1 (f2 +ε00 , d3 +η 00 ) < f1
and f 1 (f2 , d3 ) > f 1 (f2 − ε00 , d3 − η 00 ) > f1 . Hence, for all (f200 , d003 ) ∈ [f2 − ε00 , f2 + ε00 ]× [d3 − η 00 , d3 + η 00 ],all f100 ∈ [f1 − δ(ε00 , η 00 ), f1 + δ(ε00 , η 00 )] are achievable by changing only u1 , where δ(ε00 , η 00 ) = min{f1 −f 1 (f2 + ε00 , d3 + η 00 ), f 1 (f2 − ε00 , d3 − η 00 ) − f1 }. Let η̂ = min{η 0 , η 00 }, ε = min{ε(η̂), ε00 }, and δ = δ(ε, η̂).
Noting that ε(η̂) ≥ ε(η 0 ) and letting η = min{η̂, ε, δ}, we have that all (f10 , f20 , d03 ) in R(f1 , f2 , d3 ; η) are
induced by some c ∈ C and belong to O.

Lemma 9. O is connected.
Proof. We will show that O is path connected and hence connected. Take any (f10 , f20 , d03 ), (f100 , f200 , d003 ) ∈O with inducing streams c0 , c00 ∈ C. By definition, u(c0t ), u(c00t ) ∈ intIu for t = 1, 2 and d03 , d003 ∈ intD.
Since D is an interval, we can vary consumption from t = 3 onward, creating a path from 3 c0 to 3 c00 so
as to cover the interval between d03 and d003 . Along this path d3 remains in intD; moreover, by Lemma 2,
f2 varies covering an interval between f20 and f2 (c00 , c01 , c02 , 3 c00 ), and by Lemma 5, f1 varies covering aninterval between f10 and f1 (c00 , c01 , c02 , 3 c00 ). Since c01 and c02 are unchanged, all (f1 , f2 , d3 ) along the path
are in O. Now fix 3 c = 3 c00 and vary c2 to create a path from c02 to c002 so as to cover the interval betweenu(c02 ) and u(c002 ). Along this path u(c2 ) remains in intIu ; moreover, by Lemma 2, f2 varies covering theinterval between f2 (c00 , c01 , c02 , 3 c00 ) and f2 (c00 , c01 , c002 , 3 c00 ), and by Lemma 5, f1 varies covering an interval
between f1 (c00 , c01 , c02 , 3 c00 ) and f1 (c00 , c01 , c002 , 3 c00 ). Since c01 is unchanged, again all (f1 , f2 , d3 ) along thissecond path are in O. Finally, fix 2 c = 2 c00 and vary c1 to create a path from c01 to c001 so as to cover
the interval between u(c01 ) and u(c001 ). Along this path u(c1 ) remains in intIu ; moreover, by Lemma 2,
f1 varies covering the interval between f1 (c00 , c01 , c002 , 3 c00 ) and f1 (c00 , c001 , c002 , 3 c00 ). Since c002 is unchanged,
again all (f1 , f2 , d3 ) along this third path are in O. The three paths together form a connected pathfrom (f10 , f20 , d03 ) to (f100 , f200 , d003 ) which never leaves O.

¯ relying on Debreu (1960).
We are now ready to obtain an additive representation of ,¯ over X × Q can be represented by an additive function
Lemma 10. The preference 
V 0 (f0 , f1 , f2 , d3 ) = û(f0 ) + a(f1 ) + b(f2 ) + ζ(d3 ),where û, a, b, and ζ are continuous, and a, b, ζ are strictly increasing on U.
¯ has an additive representation over X × O. By continuity, we then extend
Proof. We first show that 39¯ over X × Q immediately implies the desired
this representation to X × Q. The representation of ∗
representation of  on F0 .
The set O may be expressed as a countable union of open rectangles {Ri }i∈N of the form in Lemma
8, and such that for any j there is an i < j such that Ri ∩ Rj 6= ∅. To construct {Ri }i∈N , proceed asn
3follows. Let {B }∞n=1 be the sequence of closed balls of radius n centered at the origin in R . Then, letnKn = {o ∈ O | o ∈ B , B 1/n (o) ⊂ O},
where B 1/n (o) is the open ball of radius 1/n centered at point o. For each n, Kn is compact41 and thenincreasing sequence {Kn }∞n=1 converges to O. So, each K can be covered by finitely many rectangles ofthe form in Lemma 8. Since Kn ⊂ Kn+1 , when moving from Kn to Kn+1 , one can cover Kn+1 by simply
adding rectangles to those used to cover Kn . Without loss, any added rectangle contains a point withrational coordinates not contained in other rectangles, so that the list of rectangles needed to cover O,
denoted by {Ri }i∈N , is countable. Finally, since O is connected, each Rj must intersect at least anotherRi . For simplicity, we can relabel the rectangles so that, for each j, we have Rj ∩ Ri 6= ∅ for some i < j.
For any Ri , Lemmas 4 and 7 guarantee that the hypotheses of Debreu’s (1960) Theorem 3 are satisfied¯ may be expressed over each X × Ri as
on X × Ri . Therefore, V i (f0 , f1 , f2 , d3 ) = ûi (f0 ) + ai (f1 ) + bi (f2 ) + ζ i (d3 ),for functions ûi , ai , bi , and ζ i that are continuous and, except for ûi , strictly increasing by the properties¯ 42
of V̄ which induces .
By construction, R0 and R1 have a nonempty open intersection. Over X × (R0 ∩ R1 ) representations
V 0 and V 1 must be positive affine transformations of each other (Debreu’s (1960) Theorem 3). So there
exist constants ρ > 0 and χ ∈ R such that, on X × (R0 ∩ R1 ),û0 (f0 ) = ρû1 (f0 ) + χ,

a0 (f1 ) = ρa1 (f1 ),b0 (f2 ) = ρb1 (f2 ),ζ 0 (d3 ) = ρζ 1 (d3 ).

Using these conditions, we can extend û0 , a0 , b0 , and ζ 0 to the set X × (R0 ∪ R1 ). Indeed, each function
ai , bi , and ζ i is defined on Rik which denotes the projection of Ri on the k th dimension of Q. Consider
a0 . By extending a0 over R11 \ R01 using a1 , the resulting function a0 is well defined and continuous on
R01 ∪ R11 . By a similar reasoning for b0 and ζ 0 , the function V 0 can be extended to X × (R01 ∪ R11 )×(R02 ∪ R12 )× (R03 ∪ R13 ). Since this product includes X × R0 ∪ R1 , the function V 0 is, in particular, welldefined and continuous on it.
Finally, since for each j > 0 we have Rj ∩ Ri 6= ∅ for some i < j, we can extend by induction
representation V 0 from X × R0 to X × (∪i∈N Ri ) = X × O, in countably many steps. Notice that thefunctions a, b, and ζ (we henceforth omit the superscript ‘0’) entering the formula of V 0 are defined,
through the induction, over the respective projections of O.
n

Kn is clearly bounded. Consider any sequence {om } ⊂ Kn converging to o0 . Since B is closed,n0o ∈ B . There remains to show that B 1/n (o0 ) ⊂ O. Let o00 be any point such that ||o0 − o00 || = r < 1/n.
Then ||o00 − om || ≤ r + ||o0 − om ||. So, for m large enough, o00 ∈ B 1/n (om ) and hence o00 ∈ O.
42
While Debreu’s theorem requires that the preference domain be a Cartesian product, it does notassume compactness of the sets forming the product.
41

40

Since any point of X × O is contained in X × Ri for some i ∈ N, V 0 and its components û, a, b,¯ on X × O. To see this, we need toand ζ are continuous over X × O. Moreover, V 0 represents 
0
0
0
0
00
00
00 00
check that for any (f0 , f1 , f2 , d3 ) and (f0 , f1 , f2 , d3 ) in X × O, V 0 (f00 , f10 , f20 , d03 ) > V 0 (f000 , f100 , f200 , d003 ) iff¯ (f000 , f100 , f200 , d003 ). Note that (f10 , f20 , d03 ) and (f100 , f200 , d003 ) must both belong to some Kn in(f00 , f10 , f20 , d03 )
¯ on X × Kn , it ranks (f00 , f10 , f20 , d03 ) and (f000 , f100 , f200 , d003 )the previous construction. Since V 0 represents correctly, which proves the claim.
It remains to show that V 0 can be extended to the entire domain X × Q, additively, and that it¯ over this domain. We first show that V 0 can be extended to a continuous function overrepresents ¯ over X × Q—and hence over X × O. So, there
X × Q. Recall that V is continuous and represents exists a strictly increasing map φ : Y → Y 0 such that V 0 = φ ◦ V , where Y 0 and Y are the ranges of V 0
and V on X × O. Y 0 and Y are intervals of R because X × O is connected and V 0 and V are continuous
over this domain. Since φ is strictly increasing, it must be continuous on its domain, otherwise it would
not cover Y 0 . Let Y be the range of V over X × Q. Since X × Q ⊂ cl(X × O) and V is continuous, Y
contains at most two more points than Y (its boundaries), and this may occur only when the relevantboundaries are finite. One can extend φ to these points, whenever applicable, by taking the limit of φ:/ Y , one may define φ(ȳ) as limy↑ȳ φ(y).43 Finally,for example, if ȳ denotes the upper bound of Y and y ∈we can extend V 0 to X × Q by letting V 0 = φ ◦ V over this domain. By construction, V 0 is continuous
as the composition of continuous functions.
Next, we show that this extension of V 0 to X × Q still obeys the additive representation obtained on
X ×O in terms of û, a, b and ζ. We first show that a, b, and ζ can be extended on the relevant projectionsof Q (not just of O). Since O is connected and Q ⊂ clO, the extension is only needed (possibly) at thetwo boundaries of D for ζ, and at the boundaries of U for a and b; these extensions are necessary only
if these boundaries are achieved by some (f1 , f2 , d3 ) ∈ Q.
To extend ζ, suppose that there is an (f1 , f2 , d3 ) ∈ Q such that d3 is the upper bound of D—theother case follows similarly. Without loss, we can choose f1 , f2 ∈ intU.44 By perturbing c3 , we canthen construct a sequence {(f1n , f2n , dn3 )} such that f1n and f2n are in some compact K ⊂ intU anddn3 ∈ intD for all n, and dn3 → d3 . By construction, each (f1n , f2n , dn3 ) ∈ O. Fixing some f0 , the sumû(f0 ) + a(f1n ) + b(f2n ) + ζ(dn3 ) is well defined and equal to V 0 (f0 , f1n , f2n , dn3 ) for each n. Moreover,possibly moving to subsequences, f1n → fˆ1 and f2n → fˆ2 for some fˆ1 , fˆ2 ∈ K. Since a and b are43

One can show that for ȳ ∈ Y \ Y , limy↑ȳ φ(y) must be finite. Suppose not: First, there exist i)¯ ∈ X × Q such that V̄ (s̄) = ȳ, which means that the agent prefers s̄ to any other stream;s̄ = (f¯0 , f¯1 , f¯2 , d)and ii) a sequence sn = (f0n , f1n , f2n , dn ) ∈ X × O that converges to s̄, and such that V 0 (sn ) diverges to+∞. Because V 0 is additive, this means that there must be at least one sequence, among û(f0n ), a(f1n ),b(f2n ), and ζ(dn ), with a subsequence diverging to +∞. For example, suppose that dn is such that ζ(dn )¯ we have V̄ (f0 (c), f1 (c), f2 (c), d(c))¯
diverges to +∞. Then, for any stream c such that d(3 f (c)) = d,= ȳ.
Indeed, fix any c0 and ct ’s such that u < u(ct ) < ū for t ∈ 1, 2. Choosing the sequence of continuation¯ V 0 evaluated at those streams
streams (c3 , . . .) corresponding to the sequence of dn converging to d,(and the fixed c0 , c1 , c2 ) must diverge to +∞. This implies that V̄ converges to ȳ for that sequence. By¯ wecontinuity of V̄ over its entire domain, this implies that when choosing (c3 , . . .) such that d(3 f (c)) = d,have V̄ (f0 (c), f1 (c), f2 (c), d(3 f (c))) = ȳ, regardless of the values of c0 , c1 , c2 . This, however, violates thefact that preferences are strictly increasing in u(c0 ) (Lemma 2), a contradiction. A similar contradiction
can be derived if instead û(f0n ), or a(f1n ), or b(f2n ) has a subsequence diverging to +∞. This shows thatnecessarily Y0 is bounded above whenever ȳ ∈ Ȳ \ Y . By a similar argument for the lower bound, weconclude that φ is bounded at any boundary for which it needs to be extended.
44
This can be achieved by changing c1 and c2 of the stream inducing (f1 , f2 , d3 ), without affecting d3 .

41

continuous over K, a(f1n ) and b(f2n ) converge on these subsequences. Therefore, ζ(d3 ) is well definedas the difference V 0 (f0 , fˆ1 , fˆ2 , d3 ) − û(f0 ) − a(fˆ1 ) − b(fˆ2 ), because V 0 has already been extended to(f0 , fˆ1 , fˆ2 , d3 ). Moreover, since V 0 was extended continuously over X × Q, ζ must also be continuous atd3 .
We can similarly extend b to the boundary of U, whenever needed. To see this, take any (f1 , f2 , d3 ) ∈ Q
such that f2 lies at a boundary of U, say f2 = ν̄—again, the other case follows similarly. Moreover, we
can choose c1 in the inducing stream c ∈ C so that f1 ∈ intU. By perturbing c2 , we can build a sequence{(f1n , f2n , d3 )} such that f1n is in a compact K ⊂ intU and f2n ∈ intU for all n, and f2n → ν. Possibly
taking a subsequence such that f1n → fˆ1 for some fˆ1 ∈ K, we obtain a well define limit for V 0 , a, andζ, from which we can obtain the value of b(ν̄). The argument for a is identical.
In conclusion, the function û(·) + a(·) + b(·) + ζ(·) is equal to V 0 over the entire set X × Q, and¯ over this domain.
represents Step 4: By Lemma 1 with π = {1, 2, 3}, for any f0 ∈ X, the induced preference ∗−0 on F isindependent of f0 . By Lemma 10, we can conclude that ∗−0 has a representation∗
V−0(f1 , f2 , 3 f ) = a(f1 ) + b(f2 ) + ζ(d(3 f )).

(11)

Note that Axiom 8 holds for any f0 . So if f1 = f10 , (f1 , f2 , 3 f ) %∗−0 (f10 , f20 , 3 f 0 ) iff (f2 , f3 , 4 f ) %∗−0
(f20 , f30 , 4 f 0 ).

Lemma 11. There exist α > 0, ξ ∈ R, and G : U → R continuous and strictly increasing such that,for any finite T ≥ 2 any f ∈ F,∗
V−0(f ) =TX

˜ T +1 f ) + ξ
αt G(ft ) + αT d(

t=1

T
−2
X

αt .

(12)

t=0

Proof. Consider again R0 in the proof of Lemma 10. By definition of a rectangle, if (f1 , f2 , 3 f ) and(f10 , f20 , 3 f 0 ) are such that (f1 , f2 , d(3 f )), (f10 , f20 , d(3 f 0 )) ∈ R0 , then all fˆ1 ∈ R01 are feasible with both
(f2 , 3 f ) and (f20 , 3 f 0 ). By the stationarity property of ∗−0 , we havea(fˆ1 ) + b(f2 ) + ζ(d(3 f )) ≥ a(fˆ1 Let) + b(f20 ) + ζ(d(3 f 0 ))if and only ifa(f2 ) + b(f3 ) + ζ(d(4 f )) ≥ a(f20 ) + b(f30 ) + ζ(d(4 f 0 )).
Hence, since additive representations are unique up to positive affine transformations, for all (f2 , 3 f )such that (f1 , f2 , d(3 f )) ∈ R0 ,α(a(f2 ) + b(f3 ) + ζ(d(4 f ))) + ξ = b(f2 ) + ζ(d(3 f ))(13)

for some α > 0 and ξ ∈ R.
The argument used for R0 can be equivalently applied to any Ri in the covering {Ri }i∈N of O.
Moreover, since for each j > 0 we have Rj ∩ Ri 6= ∅ for some i < j, it is clear that the α in (13) must42

be the same for all f ∈ F such that (f1 , f2 , d(3 f )) ∈ O. That (13) must hold for all f ∈ F is impliedby the following two observations. First, if c ∈ C induces (f1 , f2 , d3 ) ∈ O, it imposes no restriction ond(4 f (c)), which can take any value in D—hence 4 f can take any value in F. To see this, recall that forf ∈ F we defined d(f ) = V (fˆ0 , f ) for some fˆ0 ∈ X, and V (fˆ0 , f ) = Ṽ (fˆ0 , f1 , d(2 f )) by Lemma 5. So,
since Ṽ is strictly increasing in its second and third argument, the condition d3 (c) ∈ intD only implies
f3 (c) ∈ intU, but d(4 f (c)) can be at the boundary of D. Therefore, (13) already holds for any value ofn
4 f ∈ F. Second, suppose that f is such that (f1 , f2 , d(3 f )) is at boundary of Q. Take a sequence {f }nnnsuch that (f1 , f2 , d(3 f )) ∈ O for all n and converges to (f1 , f2 , d(3 f )). The sequence can be chosen
so that 4 f is fixed: perturbing only c1 , c2 , and c3 is enough to guarantee that we are in O. Now recall
that the functions a, b, and ζ are continuous by Lemma 10. Then, the right-hand side of (13) converges,
as do the first two terms of the left-hand side. The last term is constant and equal to ζ(d(4 f )), so it
converges trivially. Therefore (13) holds everywhere.
We conclude that, for all f ∈ F,∗∗
V−0
(f1 , f2 , 3 f ) = a(f1 ) + ξ + αV−0(f2 , f3 , 4 f ).

Therefore, using this condition recursively and (11), for any f ∈ F and finite T > 2, we have∗
V−0
(f1 , f2 , 3 f ) =T−1X

αt a(ft+1 ) + αT −1 (b(fT +1 ) + ζ(d(T +2 f ))) + ξ

T
−2
X

αt .

t=0

t=0

˜ = α−1 (b(·) + ζ(d(·))).
The result then follows by defining G = α−1 a and d(·)

By Lemma 11, for any finite T ≥ 2, we can represent  for streams c as
U (c) = û(c0 ) +TX˜ (T +1 c), U (T +2 c), . . .) + ξ
αt G(U (t c)) + αT d(U

t=1

T
−2
X

αt .

(14)

t=0

The next two technical lemmas will be useful to complete the proof of our theorem.

Lemma 12. For any constant streams c, c0 ∈ C, c  c0 iff û(c0 ) > û(c00 ).
Proof. Suppose û(x) > û(y) and consider c = (x, x, . . .) and ĉ = (x, y, . . .). For any t ≥ 0 and c00 ∈ C,
define ct = (c0 , . . . , ct , c00 ) and ĉt = (ĉ0 , . . . , ĉt , c00 ). For t = 0, we have U (ct ) = U (ĉt ). For any t ≥ 1,
using (14), we first have U (t ct ) > U (t ĉt ). Then, using again (14) backward recursively and monotonicityof G, we conclude that U (ct ) ≥ U (ĉt ). Since this is true for any t ≥ 0 and c00 ∈ C, Axiom 7(ii) implies
c % ĉ. Now note that, again by (14), ĉ  (y, y, . . .). Hence, by Axiom 1, c  (y, y, . . .).
Now suppose û(x) = û(y) and consider c = (x, x, . . .) and ĉ = (y, y, . . .). For any t and c00 ∈ C, define
ct and ĉt as before. Using again (14) backward recursively and the fact that G is a function, we conclude
that U (ct ) = U (ĉt ). Since this is true for any t and c00 ∈ C, Axiom 7(ii) implies c ∼ c0 .

Lemma 13. For any c ∈ C, there exists x ∈ X such that c ∼ (x, x, . . .).
43

Proof. By Lemma 19 in the Supplemental Material, for any c ∈ C, there exists y ∈ X such thatc ∼ (c0 , y, y, . . .). Suppose (c0 , y, y, . . .) 6∼ (y, y, . . .). If (c0 , y, y, . . .)  (y, y, . . .), then û(c0 ) > û(y). Letĉ = (c0 , c0 , . . .) and c̃ = (c0 , y, y, . . .). For any t ≥ 0 and any c00 ∈ C, consider ĉt = (ĉ0 , . . . , ĉt , c00 ) andc̃t = (c̃0 , . . . , c̃t , c00 ). We have ĉt % c̃t . Indeed, for t = 0, ĉt = c̃t . For t ≥ 1, we can proceed using (14).
Since û(c0 ) > û(y), U (t ĉt ) > U (t c̃t ). For s < t, since û(ĉts ) ≥ û(c̃ts ) and G is strictly increasing, we have
U (s ĉt ) ≥ U (s c̃t ). By Axiom 7(ii), we then have ĉ % c̃ and hence (c0 , c0 , . . .) % c  (y, y, . . .). Since X isconnected, by Axiom 2, there exists x ∈ X such that (x, x, . . .) ∼ c. The case (c0 , y, y, . . .) ≺ (y, y, . . .)follows similarly.

We can now prove that α < 1.

Lemma 14. α < 1.
Proof. Consider consumption streams that are constant from t = 3 onward. Then ft is constant fort ≥ 3. So we can write d(3 f ) = d(4 f ) = e(f3 ) in (13) and thus obtain(1 − α)e(f3 ) = αb(f3 ) + αa(f2 ) − b(f2 ) + ξ.
First, note that f3 > f30 implies e(f3 ) > e(f30 ). By Lemma 12, f3 > f30 implies u(c3 ) > u(c03 ). Definec = (c3 , c3 , . . .) and c0 = (c3 , c03 , c03 , . . . ). Replicating the argument in the proof of Lemma 13, we havec % (c3 , c3 , c03 , c03 , . . . ). Moreover, by Axiom 7(i), (c3 , c3 , c03 , c03 , . . . )  c0 . Then, by Axiom 1 and Lemma
2, W (u(c3 ), d(f3 , f3 , . . .)) > W (u(c3 ), d(f30 , f30 , . . .)), which holds iff d(f3 , f3 , . . .) > d(f30 , f30 , . . .).
Second, we can find ĉ, c̃ ∈ C, constant from t = 3 onward, such that f2 (ĉ) = f2 (c̃) and f3 (ĉ) > f3 (c̃).
Consider x, y ∈ X with u(x) > u(y) and the streams (x, y, y, . . .) and (y, x, x, . . .). By the previous
argument based on Axiom 7(ii), (x, x, x, . . .)  (x, y, y, . . .). If (x, y, y, . . .) % (y, x, x, . . .), then by(14) and continuity of û there exists z ∈ X such that (x, y, y, . . .) ∼ (z, x, x, . . .). In this case, letĉ = (c0 , c1 , z, x, x, . . .). If (x, y, y, . . .) ≺ (y, x, x, . . .), then by Axiom 2 there exists w ∈ X such that(x, y, y, . . .) ∼ (y, w, w, . . .). Moreover, u(w) > u(y). Otherwise, since (y, y, y, . . .) % (y, w, w, . . .) foru(y) ≥ u(w) (again by the same argument as before), we would have (x, y, y, . . .)  (y, w, w, . . .) by (14)and Axiom 1. In this case, let ĉ = (c0 , c1 , y, w, w, . . .). Finally, let c̃ = (c0 , c1 , x, y, y, . . .).
To conclude the proof, note that for c ∈ {ĉ, c̃}, (1 − α)e(f3 (c)) = αb(f3 (c)) + ξ 0 for some constant ξ 0 .
Since b and e are strictly increasing, we must have α < 1.

Note that, if c is constant from any T ≥ 3 onward, by Lemma 14 and (13)˜ (T c), U (T +1 c), . . .) =d(UαξG(U (T c)) +.
1−αα(1 − α)So, for eventually constant streams, we can write
U (c) = û(c0 ) +

T
X
t=1

αt G(U (t c)) +αT +1
1 + α(1 − αT −1 )G(U (T +1 c)) +ξ.
1−αα(1 − α)44

(15)

Lemma 15. G is bounded on U.
∗
Proof. By Axiom 3, V−0
is finite for all c ∈ C. Suppose that G is unbounded above—the othercase follows similarly. Then, for each r ∈ R++ , there must be a stream cr with utility U r such that00Gr ≡ G(U r ) ≥ r. Moreover, for r > r0 , we can choose cr and cr so that Gr > Gr , relying on continuityof G and connectedness of U. By Lemma 13, for each r we can also let cr be constant. As a preliminary
0observation, note the following: given r0 > r, a stream c that equals cr for the first k periods and cr
forever after must satisfy G(U (c)) ≥ r. This is because, by definition, U (t c) > U (t cr ) for t ≥ k; then,by monotonicity of G and using (15) backward recursively, we have U (t c) > U (t cr ) for 0 ≤ t < k.
Now construct stream ĉ as follows. For some M > 1 and each t ≥ 1, consider the constant streamtcwith the property αt G(M/α) ≥ M t . Then, let ĉ0 be such that u < û(ĉ0 ) < u and, for each t ≥ 2,
t
T
(M/α)
let ĉt = ct
. Now, for any T > 0, let cT be equal to ĉ up to T and to c(M/α) thereafter. Using (15),
we have(M/α)tU (cT ) =û(ĉ0 ) +T−1X

αt G(U (t cT )) +t=1≥û(ĉ0 ) +

T
−1
X

Mt +

t=1

T
1 + α(1 − αT −2 )αT
G(M/α) +ξ
1−αα(1 − α)1
1 + α(1 − αT −2 )
MT +
ξ,
1−α
α(1 − α)where the inequality follows by recursively applying our preliminary observation. Note that the lower
bound on U (cT ) goes to +∞ as T → ∞.
Now fix any T and cT . To simplify notation, let c̃ = cT . Using Axiom 7(ii), we have U (ĉ) ≥ U (c̃). To
see this, consider any t ≥ 0 and c00 ∈ C, and let ĉt = (ĉ0 , . . . , ĉt , c00 ) and c̃t = (c̃0 , . . . , c̃t , c00 ). For t ≤ T ,we have ĉt ∼ c̃t because the two streams are identical. For t ≥ T + 1, we first have that u(ĉs ) > u(c̃s )for T < s ≤ t by Lemma 12. Hence, U (ĉt , c00 ) > U (c̃t , c00 ). Second, using again monotonicity of G and(12) recursively, we conclude U (ĉt ) ≥ U (c̃t ). By Axiom 7(ii), we then have the claimed property.
∗
It follows that, for any T , U (ĉ) ≥ U (cT ) and hence, since û(ĉ0 ) is bounded by assumption, V−0(f (ĉ))must be infinite, violating Axiom 3.

Lemma 16. For any c ∈ C, U (c) = û(c0 ) +P∞

t
t=1 α G(U (t c)).

∗is finite for all c ∈ C. Using (12) for any finite T and observing that
Proof. Again by Axiom 3, V−0˜

T f can take any value in F, we conclude that the function d must be finite because G is bounded. The

result then follows by letting T → ∞, relying on α < 1 and ignoring the additive constant.

To conclude, both functions U and U represent  over C. So, they are strictly increasing transformations of one another. Letting G denote the function of U such that G(U (c)) = G(U (c)) for all c,
we obtain representation (5). For uniqueness, note that the additive form of U is unique up to positiveaffine transformations, i.e., Ũ = ρU + χ for ρ > 0 and χ ∈ R. So,Ũ (c) = ρû(c0 ) + χ +∞Xtα ρG(U (t c)) = ρû(c0 ) + χ +

t=1

∞
X
t=1

45

t

α ρG

Ũ (t c) − χ
ρ

!
.

A.3

Proof of Proposition 1

Part (i). Take ν 0 , ν ∈ U. By definition, there exist c0 , c ∈ C such that U (c0 ) = ν 0 and U (c) = ν. By
Lemma 13, we can take c0 = (x, x, . . .) and c = (y, y, . . .) for some x, y ∈ X. Suppose u(x) > u(y). Then,by Lemma 12, U (x, . . .) > U (y, . . . ). By representation (5),U (x) −αα
G(U (x)) > U (y) −G(U (y)).
1−α1−αRearranging, we get that for any ν 0 > ν in U
G(ν 0 ) − G(ν) <1−α 0(ν − ν).
α1−α0Lemma 17. For any ε > 0, there exists a constant K ∈ ( 1−α2α , α ) such that, for all ν > ν in U,

G(ν 0 ) − G(ν) ≤ max{K(ν 0 − ν), ε}Proof. See the Supplemental Material.

To show that U is H-continuous, consider any c, c̃ ∈ C and define cT = (c0 , c1 , . . . , cT , c) and c̃T =(c0 , c1 , . . . , cT , c̃). Using Lemma 17, we will show that for any ε > 0, there exists T such that
|U (cT ) − U (c̃T )| <

2αε
.
1−α

α
To do so, let M = 1−α2 supν∈U |G(U )| and δ = (1 + K)α). Since K < (1 − α)/α, we have δ < 1. Let
T denote the first time such that KM δ T < ε. Note that for all t < T , we have max{KM δ t , ε} = KM δ t .

We first show that for all t < T , we have |U (ct ) − U (c̃t )| ≤ M δ t . The proof works by induction. For
t = 0, we have ct0 = c̃t0 , so
|U (c0 ) − U (c̃0 )| =∞X

αs |G(U (s c0 )) − G(U (s c̃0 ))| ≤ M

s=1

Suppose the claim holds for t < T − 1, we will show it holds for t + 1. We have|U (ct+1 ) − U (c̃t+1 )|≤ α|G(U (1 ct+1 )) − G(U (1 c̃t+1 ))|∞X+ααs |G(U (s+1 ct+1 )) − G(U (s+1 c̃t+1 ))|.

(16)

s=1

By the induction hypothesis, the sum in (16) is bounded above by M δ t . And because t < T − 1, we
have KM δ t ≥ ε. Therefore,|U (c) − U (c0 )| ≤ αKM δ t + αM δ t ≤ M δ t+1 ,which shows the claim.

46

Finally, for t = T , (16) still applies, but this time the first term is bounded by αε, because KM δ T < ε.
This implies that
|U (c) − U (c0 )| ≤ αε + αM δ T ≤ αε + αε/K = δε/K.
Since δ < 1 and K > (1 − α)/2α, (A.3) follows.
Part (ii). Let C(M ) be the set of consumption streams such that |u(ct )| ≤ M for all t, and B(M ) be
the space of bounded real-valued functions with domain C(M ). Endowed with the sup norm kU k∞ =supc∈C(M ) |U (c)|, B(M ) is a complete metric space. Let J be the operator on B(M ) defined by
J (U )(c) = u(c0 ) +∞X

αt G(U (t c)).

t=1

By construction, J (U ) is bounded over C(M ), as u is bounded by M and U is bounded over C(M ).
Moreover, since G is K-Lipschitz continuous with K < (1 − α)/α, J must be a contraction, as is easily
checked. So, J has a unique fixed point; call it UM . As M increases, the domain of UM increases.
However, for any M, N , uniqueness of the fixed point guarantees that UM and UN coincide on theintersection of their domains. Thus, we obtain a unique solution U ∗ to (5) over C(B) = ∪M C(M ).
Let H be the set of H-continuous functions. To verify that U ∗ ∈ H, it suffices to show that (a) J maps
H onto itself, and (b) H is closed under the sup norm. Indeed, this will guarantee that J ’s fixed-point
belongs to H. To show (a), take any U ∈ H and ε > 0. Since α < 1 and G is bounded, there isT
2Ḡ
T > 0 such that α1−α< ε/2, where Ḡ = supν∈U |G(ν)|. Moreover, since U ∈ H, there exists N such
|U (c) − U (c̃)| < ε/2 whenever ct = c̃t for all t ≤ N . For any c and c̃,|J (U )(c) − J (U )(c̃)|≤∞
X

αt [G(U (t c)) − G(U (t c̃))]

t=1

≤

K

T
−1
X
t=1

αt |U (t c) − U (t c̃)| + αT

2Ḡ
.
1−α

Kα
where K is the Lipschitz constant of G. The first term is less than (1−α)
maxt≤T −1 |U (t c) − U (t c̃)|.
0
Now suppose that ct = c̃t for all t ≤ N = N + T . This implies that (t c)t0 = (t c̃)t0 for all t ≤ T
and t0 ≤ N , because t c is truncating at most T elements of c, and c and c̃ were identical up to time
T + N , by construction. By definition of N , we have |U (t c) − U (t c̃)| < ε/2 for all t ≤ T and, hence,|J (U )(c) − J (U )(c̃)| < ε. Setting T (ε) = N 0 shows that J (U ) satisfies H-continuity. To prove (b),consider a sequence {U m } in H that converges to some limit U in the sup norm. Now fix ε > 0. There ism such that kU m − U k∞ < ε/3. Since U m ∈ H, there is N such that |U m (c) − U m (c̃)| < ε/3 wheneverct = c̃t for all t ≤ N . Thus, for such c, c̃,|U (c) − U (c̃)| ≤ |U (c) − U m (c)| + |U m (c) − U m (c̃)| + |U m (c̃) − U (c̃)| < ε,which shows that U ∈ H.
To extend the definition of U ∗ from C(B) to C, for any c ∈ C \ C(B), consider any sequence {cn } inC(B) such that cnt = ct for all t ≤ n, and let U ∗ (c) = limn→+∞ U ∗ (cn ). This limit is well-defined andindependent of the chosen sequence. To see this, note that, for any such sequence {cn } and any ε > 0,
H-continuity of U ∗ implies that there is T such that |U ∗ (c) − U ∗ (c̃)| < ε whenever ct = c̃t for all t ≤ T .

47

Hence, |U ∗ (cn ) − U ∗ (cm )| < ε for all n, m ≥ T , since the consumption levels of cn and cm coincide up
to min{n, m}. So, {U ∗ (cn )} forms a Cauchy sequence in R and thus converges. Moreover, the limit isindependent of the chosen sequence, as for any ε > 0, |U ∗ (cn ) − U ∗ (c̃n )| < ε for n large enough andsequences {cn } and {c̃n } of the type constructed above.
The limit U thus defined satisfies representation (5). Since U ∗ is a fixed point of J on C(B) and cn
belongs to C(B), for each n∞X
U ∗ (cn ) = u(cn0 ) +αt G(U ∗ (t cn ))t=1∗

The left-hand side converges to U (c). Moreover, for each t, U ∗ (t cn ) converges to U ∗ (t c) (which issimilarly well defined). Since G is continuous, G(U ∗ (t cn )) converges to G(U ∗ (t c)) for each t. Since
α < 1 and G is bounded, by the dominated convergence theorem, the right-hand side converges toP∞u(c0 ) + t=1 αt G(U ∗ (t c)), which proves that (5) holds for all c ∈ C.
Finally, there is a unique H-continuous extension of U ∗ from C(B) to C that solves (5). To see this,let U be any H-continuous solution to (5). Since U is a fixed point of J and the fixed point is uniqueon C(B), U must coincide with U ∗ on C(B). Take any c ∈ C \ C(B) and ε > 0. By H-continuity of
U and U ∗ , both |U (c) − U (c̃)| and |U ∗ (c) − U ∗ (c̃)| are less than ε/2 for some c̃ ∈ C(B) equal to c for
all t up to a large N . Since U and U ∗ must be equal at c̃, |U (c) − U ∗ (c)| < ε. Since ε was arbitrary,U (c) = U ∗ (c) for all c, establishing uniqueness.

A.4

Proof of Proposition 2

Let U (c) = V (c0 , U (1 c), U (2 c), . . .) where V is strictly increasing in U (t c) for all t ≥ 1. By definition,(x, c)  (y, c) means that U (x, c) > U (y, c). Hence, for all 0 ≤ s ≤ t,
U (s zt , x, c) > U (s zt , y, c),where, for s < t, s zt = (zs , . . . , zt ) and t zt = zt . This follows by induction. For s = t,
U (t zt , x, c) = V (t zt , U (x, c), U (c), . . .) > V (t zt , U (y, c), U (c), . . .) = U (t zt , y, c).
Now suppose that the claim holds for r + 1 ≤ s ≤ t, with 0 ≤ r < t. ThenU (r zt , x, c) = V (zr , U (r+1 zt , x, c), . . . , U (t zt , x, c), U (x, c), . . .)> V (zr , U (r+1 zt , y, c), . . . , U (t zt , y, c), U (y, c), . . .) = U (r zt , y, c).
By definition, (0 zt , x, x̂, c0 ) ∼ (0 zt , y, ŷ, c0 ) means thatV (z0 , U (1 zt , x, x̂, c0 ), . . . , U (t zt , x, x̂, c0 ), U (x, x̂, c0 ), U (x̂, c0 ), . . .)= V (z0 , U (1 zt , y, ŷ, c0 ), . . . , U (t zt , y, ŷ, c0 ), U (y, ŷ, c0 ), U (ŷ, c0 ), . . .).
Since U (0 zt , x, c) > U (0 zt , y, c) for all c,
V (z0 , U (1 zt , y, x̂, c0 ), . . . , U (t zt , y, x̂, c0 ), U (y, x̂, c0 ), U (x̂, c0 ), . . .)48

< V (z0 , U (1 zt , y, ŷ, c0 ), . . . , U (zt , y, ŷ, c0 ), U (y, ŷ, c0 ), U (ŷ, c0 ), . . .).
This implies that U (ŷ, c0 ) > U (x̂, c0 ). Otherwise, U (y, ŷ, c0 ) ≤ U (y, x̂, c0 ) and, by induction, U (s zt , y, ŷ, c0 ) ≤U (s zt , y, x̂, c0 ) for all 0 ≤ s ≤ t, which is a contradiction.
Finally, we must have U (x, x̂, c0 ) > U (y, ŷ, c0 ). Otherwise, again by induction, for all 0 ≤ s ≤ t
U (s zt , y, ŷ, c0 ) > U (s zt , x, x̂, c0 ),which contradicts (0 zt , x, x̂, c0 ) ∼ (0 zt , y, ŷ, c0 ).
Suppose that we replace condition (0 zt , x, x̂, c0 ) ∼ (0 zt , y, ŷ, c0 ) with (0 zt , x, t+2 zs , x̂, c0 ) ∼ (0 zt , y, t+2 zs , ŷ, c0 )where s ≥ t + 2. By the same argument as before, (0 zt , y, t+2 zs , x̂, c0 ) ≺ (0 zt , y, t+2 zs , ŷ, c0 ) and so(ŷ, c0 )  (x̂, c0 ). If not, by induction (τ zs , x̂, c0 ) % (τ zs , ŷ, c0 ) for all 0 ≤ τ ≤ s (where zt+1 = y). Then,we must have (x, t+2 zs , x̂, c0 )  (y, t+2 zs , ŷ, c0 ). If not, since (τ zs , ŷ, c0 )  (τ zs , x̂, c0 ) for all t + 2 ≤ τ ≤ s,
we would have (0 zt , y, t+2 zs , ŷ, c0 )  (0 zt , x, t+2 zs , x̂, c0 ).

A.5

Proof of Theorem 4

Using Axiom 10 and Theorem 3, we also have(c0 , c1 , 2 c)  (c0 , c01 , 2 c) ⇔ (ĉ0 , c1 , 2 c0 )  (ĉ0 , c01 , 2 c0 )(17)

(c0 , c1 , 2 c)  (c0 , c1 , 2 c0 ) ⇔ (ĉ0 , c01 , 2 c)  (ĉ0 , c01 , 2 c0 )(18)

(c0 , c1 , 2 c)  (c00 , c1 , 2 c) ⇔ (c0 , c01 , 2 c0 )  (c00 , c01 , 2 c0 )(19)

(c0 , c1 , 2 c)  (c0 , c01 , 2 c0 ) ⇔ (ĉ0 , c1 , 2 c)  (ĉ0 , c01 , 2 c0 )(20)

By Debreu’s (1960) Theorem 3, conditions (17)-(20) and (i)-(ii) in Axiom 10 imply that  can be
represented by
w0 (c0 ) + w1 (c1 ) + w2 (2 c),for some continuous and nonconstant functions w0 , w1 , and w2 . By Theorem 3,  is also represented byu(c0 ) + αG(u(c1 ) + g(2 c)) + αg(2 c),
where g(2 c) =

P∞

t=2 α

t−1

G(U (t c)). It follows thatu(c0 ) + αG(u(c1 ) + g(2 c)) + αg(2 c) = ξ [w0 (c0 ) + w1 (c1 ) + w2 (2 c)] + χ,
for some ξ > 0 and χ ∈ R. This implies thatαG(u(c1 ) + g(2 c)) + αg(2 c) = ξ [w1 (c1 ) + w2 (2 c)] ,and therefore G must be affine. Since G must be increasing, without loss of generality let G(U ) = γUwith γ > 0. Finally, by Proposition 1, γ < 1−α
α .

49

A.6

Proof of Proposition 5

By assumption, for all t,∞
X

U (t c) = u(ct ) +βδ τ −t u(cτ ),(21)

τ =t+1γ
where 0 < β = 1+γ< 1, 0 < δ = (1 + γ)α < 1, 0 < α < 1.

For the “if part” see the main text. For the “only if” part, using (21), we get∞
X

w(t)U (t c) = w(0)u(c0 ) +

t=0

∞
X

"
u(ct ) w(t) + βδ

t=1

t

t−1
X
w(τ )
τ =0

δτ

!#
.

P∞
P∞
By assumption, t=0 w(t)U (t c) = t=0 δ t u(ct ). So the coefficients of u(ct ) must match for all t. For
t = 0, w(0) = 1. Then, for t = 1, w(1) = (1 − β)δ = α. Now suppose w(t) = αt for all t = 0, . . . , τ .
Then,
ατ +1
τ +1
τ +1 1 − δ τ +1
w(τ + 1) = δ− βδ= ατ +1 .
1 − αδ
Hence, by induction, w(t) = αt for all t.

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