Page 1

ARTICLE IN PRESS

Journal of Economic Theory

()–

www.elsevier.com/locate/jet

Ordering infinite utility streams?

Walter Bosserta,Yves Sprumontb, Kotaro Suzumurac,∗

aDépartement de Sciences Economiques and CIREQ, Université de Montréal, C.P. 6128, succursale Centre-ville,

Montréal QC, Canada H3C 3J7

bDépartement de Sciences Economiques and CIREQ, Université de Montréal, C.P. 6128, succursale Centre-ville,

Montréal QC, Canada H3C 3J7

cInstitute of Economic Research, Hitotsubashi University, Kunitachi, Tokyo 186-8603, Japan

Received 8 March 2005; final version received 16 March 2006

Abstract

We reconsider the problem of ordering infinite utility streams. As has been established in earlier contri-

butions, if no representability condition is imposed, there exist strongly Paretian and finitely anonymous

orderingsofintertemporalutilitystreams.Weexaminethepossibilityofaddingsuitablyformulatedversions

of classical equity conditions. First, we provide a characterization of all ordering extensions of the gener-

alized Lorenz criterion as the only strongly Paretian and finitely anonymous rankings satisfying the strict

transfer principle. Second, we offer a characterization of an infinite-horizon extension of leximin obtained

by adding an equity-preference axiom to strong Pareto and finite anonymity.

© 2006 Elsevier Inc.All rights reserved.

JEL classification: D63; D71

Keywords: Intergenerational justice; Multi-period social choice; Leximin

1. Introduction

Treating generations equally is one of the basic principles in the utilitarian tradition of moral

philosophy. As Sidgwick [19, p. 414] observes, “the time at which a man exists cannot affect

the value of his happiness from a universal point of view; and […] the interests of posterity

?The paper was presented at the University of Maastricht and the University of Oregon.

∗Corresponding author. Fax: +81425808353.

E-mail addresses: walter.bossert@umontreal.ca (W. Bossert), yves.sprumont@umontreal.ca (Y. Sprumont),

suzumura@ier.hit-u.ac.jp (K. Suzumura).

0022-0531/$-see front matter © 2006 Elsevier Inc.All rights reserved.

doi:10.1016/j.jet.2006.03.005

Page 2

2

W. Bossert et al. / Journal of Economic Theory()–

ARTICLE IN PRESS

must concern a utilitarian as much as those of his contemporaries”. This view, which is formally

expressed by the anonymity condition, is also strongly endorsed by Ramsey [16].

Following Koopmans [14], Diamond [9] establishes that anonymity is incompatible with

the strong Pareto principle when ordering infinite utility streams. Moreover, he shows that if

anonymity is weakened to finite anonymity—which restricts the application of the standard

anonymity requirement to situations where utility streams differ in at most a finite number

of components—and a continuity requirement is added, an impossibility results again. Hara

et al. [12] adapt the well-known strict transfer principle due to Pigou [15] and Dalton [7] to

the infinite-horizon context. They show that this principle is incompatible with strong Pareto

and continuity even if the social preference is merely required to be acyclical. Basu and Mitra

[5] show that strong Pareto, finite anonymity and representability by a real-valued function are

incompatible.

Faced with these impossibilities, it seems to us that the most natural assumption to drop is

that of continuity or representability.We view the strong Pareto principle and finite anonymity as

being on much more solid ground than axioms such as continuity or representability, especially

in the context of the ranking of infinite utility streams where these conditions may be consid-

ered to be overly demanding. Svensson [21] proves that strong Pareto and finite anonymity are

compatible by showing that any ordering extension of an infinite-horizon variant of Suppes’[20]

grading principle satisfies the required axioms. The Suppes grading principle is a quasi-ordering

that combines the Pareto quasi-ordering and finite anonymity. GivenArrow’s [1] version of Szpil-

rajn’s [22] extension theorem, this establishes the compatibility result. As noted by Asheim et

al. [2], Svensson’s possibility result is easily converted into a characterization: ordering exten-

sions of the Suppes grading principles are the only orderings satisfying strong Pareto and finite

anonymity.

Once the possibility of satisfying these two fundamental axioms is established, another natural

questiontoaskiswhatorderingssatisfyadditionaldesirableproperties.AsheimandTungodden[3]

provide a characterization of an infinite-horizon version of the leximin principle by adding an

equity-preference condition (the infinite-horizon equivalent of Hammond equity; see [10]) and a

preference-continuity property to strong Pareto and finite anonymity.An infinite-horizon version

of utilitarianism is characterized by Basu and Mitra [6] by adding an information-invariance con-

ditiontothetwofundamentalaxioms.Furthermore,theynarrowdowntheclassofinfinite-horizon

utilitarian orderings to those resulting from the overtaking criterion [23]. This is accomplished

by using a consistency condition in addition to the three axioms characterizing their utilitarian

orderings.

In this paper, we focus on equity properties. One of the most fundamental equity properties (if

not the most fundamental) is the Pigou–Dalton transfer principle, adapted to the infinite-horizon

frameworkbyHaraetal.[12].OurfirstresultcharacterizesallorderingsthatsatisfystrongPareto,

anonymity and the strict transfer principle.They are extensions of an infinite-horizon formulation

of the well-known generalized Lorenz quasi-ordering [18].

In the presence of strong Pareto, the axiom of equity preference (the infinite-horizon version

of Hammond equity) is a strengthening of the strict transfer principle. We use it to identify a

subclass of the class of orderings satisfying the three axioms just mentioned. These orderings are

extensionsofaparticularinfinite-horizonincompleteversionofleximin.Thissecondresultleaves

a larger class of orderings than that identified byAsheim andTungodden [3] because they employ

an additional axiom. The relationship between our leximin characterization and that of Asheim

and Tungodden is analogous to the relationship between Basu and Mitra’s [6] characterizations

of infinite-horizon utilitarianism and of the overtaking criterion.

Page 3

ARTICLE IN PRESS

W. Bossert et al. / Journal of Economic Theory()–

3

2. Basic definitions

ThesetofinfiniteutilitystreamsisX = RN,whereRdenotesthesetofallrealnumbersandN

denotesthesetofallnaturalnumbers.AtypicalelementofXisaninfinite-dimensionalvectorx =

(x1,x2,...,xn,...) and, for n ∈ N, we write x−n= (x1,...,xn) and x+n= (xn+1,xn+2,...).

The standard interpretation of x ∈ X is that of a countably infinite utility stream where xnis the

utility experienced in period n ∈ N. Of course, other interpretations are possible—for example,

xncould be the utility of an individual in a countably infinite population.

Our notation for vector inequalities on X is as follows. For all x,y ∈ X, (i) x?y if xn?yn

for all n ∈ N; (ii) x > y if x?y and x ?= y; (iii) x?y if xn> ynfor all n ∈ N. For n ∈ N

and x ∈ X, (x−n

being broken arbitrarily.

R ⊆ X × X is a weak preference relation on X with strict preference P(R) and indifference

relation I(R).A quasi-ordering is a reflexive and transitive relation, and an ordering is a complete

quasi-ordering.Analogously, a partial order is an asymmetric and transitive relation, and a linear

order is a complete partial order. Let R and R?be relations on X. R?is an extension of R if R ⊆ R?

and P(R) ⊆ P(R?). If an extension R?of R is an ordering, we call it an ordering extension of R,

and if R?is an extension of R that is a linear order, we refer to it as a linear order extension of R.

A finite permutation of N is a bijection ?:N → N such that there exists m ∈ N with ?(n) = n

for all n ∈ N \ {1,...,m}. x?= (x?(1),x?(2),...,x?(n),...) is the finite permutation of x ∈ X

that results from relabelling the components of x in accordance with the finite permutation ?.

Two of the most fundamental axioms in this area are the strong Pareto principle and finite

anonymity, defined as follows.

Strong Pareto: For all x,y ∈ X, if x > y, then (x,y) ∈ P(R).

Finite anonymity: For all x ∈ X and for all finite permutations ? of N,

(x?,x) ∈ I(R).

Szpilrajn’s [22] fundamental result establishes that every partial order has a linear order exten-

sion. Arrow [1, p. 64] presents a variant of Szpilrajn’s theorem stating that every quasi-ordering

has an ordering extension; see also [11]. This implies that the sets of orderings characterized in

the theorems of the following sections are non-empty.

(1),...,x−n

(n)) is a rank-ordered permutation of x−nsuch that x−n

(1)? ··· ?x−n

(n), ties

3. Transfer-sensitive infinite-horizon orderings

Now we examine the consequences of adding the strict transfer principle to strong Pareto and

finiteanonymity.APigou–Daltontransferisatransferofapositiveamountofutilityfromabetter-

off agent to a worse-off agent so that the relative ranking of the two agents in the post-transfer

utility stream is the same as their relative ranking in the pre-transfer stream. The strict transfer

principle requires that any Pigou–Dalton transfer leads to a utility stream that is strictly preferred

to the pre-transfer stream.

Stricttransferprinciple:Forallx,y ∈ X andforallm,n ∈ N,ifxk= ykforallk ∈ N\{m,n},

ym> xm?xn> ynand xn+ xm= yn+ ym, then (x,y) ∈ P(R).

The strict transfer principle is the natural analogue of the corresponding condition for finite

streams; see also [12]. An alternative (equivalent) formulation of the strict transfer principle

involves the explicit expression of the amount transferred from m to n when moving from y to

Page 4

4

W. Bossert et al. / Journal of Economic Theory()–

ARTICLE IN PRESS

x (this amount is ? = ym− xm = xn− ynand is readily obtained from our statement of the

axiom). Although this alternative may be more standard in the literature, we use the version

introduced above because it is parallel in structure to the equity-preference axioms to be defined

in the following section.

To define the class of orderings satisfying the three axioms introduced thus far, we begin with

a statement of Shorrocks’ [18] generalized Lorenz quasi-ordering Rn

of n ∈ N individuals. This quasi-ordering generalizes the standard Lorenz quasi-ordering by

extending the relevant dominance criterion to comparisons involving different average (or total)

utilities. For all x,y ∈ X,

k

?

The relation Rn

G⊆ X × X is defined by letting, for all x,y ∈ X,

(x,y) ∈ Rn

Clearly, Rn

Gis a quasi-ordering for all n ∈ N. The infinite-horizon extension of the generalized

Lorenz quasi-ordering that is of interest in this paper is defined by RG=?

in the following theorem.

gfor a society consisting

(x−n,y−n) ∈ Rn

g

⇔

i=1

x−n

(i)?

k

?

i=1

y−n

(i)

for all k ∈ {1,...,n}.

G⇔ (x−n,y−n) ∈ Rn

g

and

x+n?y+n.

n∈NRn

G. The relation

RGcan be shown to be a quasi-ordering and we characterize the class of its ordering extensions

Theorem 1. An ordering R on X satisfies strong Pareto, finite anonymity and the strict transfer

principle if and only if R is an ordering extension of RG.

Proof. ‘If.’Step 1: We show that the relations Rn

P(Rn

G) are nested, that is, for all n ∈ N,

Rn

G

Gand their associated strict preference relations

G⊆ Rn+1

(1)

and

P(Rn

G) ⊆ P(Rn+1

G

).

(2)

To prove (1), suppose that (x,y) ∈ Rn

thus,

G. By definition, (x−n,y−n) ∈ Rn

gand x+n?y+nand,

k

?

xn+1?yn+1

i=1

x−n

(i)?

k

?

i=1

y−n

(i)

for all k ∈ {1,...,n},

(3)

(4)

and

x+(n+1)?y+(n+1).

(5)

Because of (5), it is sufficient to prove that

k

?

i=1

x−(n+1)

(i)

?

k

?

i=1

y−(n+1)

(i)

for all k ∈ {1,...,n + 1}.

(6)

Page 5

ARTICLE IN PRESS

W. Bossert et al. / Journal of Economic Theory()–

5

If k = n + 1, we have

n+1

?

and

i=1

x−(n+1)

(i)

=

n

?

i=1

x−n

(i)+ xn+1

n+1

?

i=1

y−(n+1)

(i)

=

n

?

i=1

y−n

(i)+ yn+1.

Adding (3) for k = n and (4), we obtain (6) for k = n + 1.

Now let k ∈ {1,...,n}. We distinguish the following four cases which cover all possibilities.

Case 1: xn+1?x−n

i ∈ {1,...,k}, and (6) for this k follows immediately from (3).

Case 2: xn+1?x−n

(k)and yn+1?y−n

(k). This implies x−(n+1)

(i)

= x−n

(i)and y−(n+1)

(i)

= y−n

(i)for all

(k)and yn+1?y−n

(k). This implies

k

?

i=1

x−(n+1)

(i)

=

k−1

?

i=1

x−n

(i)+ xn+1

and

k

?

i=1

y−(n+1)

(i)

=

k−1

?

i=1

y−n

(i)+ yn+1.

Adding (3) and (4), we obtain (6) for this k.

Case 3: xn+1< x−n

(k)and yn+1> y−n

(k). This implies

k

?

i=1

x−(n+1)

(i)

=

k−1

?

i=1

x−n

(i)+ xn+1

and

k

?

i=1

y−(n+1)

(i)

=

k−1

?

i=1

y−n

(i)+ y−n

(k).

Combining (4) and the inequality yn+1> y−n

it follows that xn+1?y−n

Case 4: xn+1> x−n

(k)(which is valid by definition of the present case),

(k).Adding this inequality and (3), we obtain (6) for this k.

(k)and yn+1< y−n

(k). This implies

k

?

i=1

x−(n+1)

(i)

=

k−1

?

i=1

x−n

(i)+ x−n

(k)

and

k

?

i=1

y−(n+1)

(i)

=

k−1

?

i=1

y−n

(i)+ yn+1.