This note provides an axiomatic analysis of a social welfare ordering over infinite utility streams. We offer two characterizations of an infinite-horizon version of the Nash criterion.Infinite generations, intergenerational equity, the Nash criterion

Susumu Cato

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 3 
  
Characterizing the Nash social welfare relation for infinite utility streams: a 
note 
  
 
 
Susumu Cato  
Institute of Social Science, The University of Tokyo 
Abstract 
This note provides an axiomatic analysis of a social welfare ordering over infinite utility streams. We offer two 
characterizations of an infinite-horizon version of the Nash criterion.
I am extremely grateful to an associate editor and an anonymous referee of the journal for valuable comments. Needless to say, I am 
responsible for any remaining errors. Further, I gratefully acknowledge the Japan Society for the Promotion of Science (JSPS) Research 
Fellowships for Young Scientists and Grant-in-Aids for JSPS Fellows of the Ministry of Education, Culture, Sports, Science and Technology, 
Government of Japan.  
Citation: Susumu Cato, (2009) ''Characterizing the Nash social welfare relation for infinite utility streams: a note'', Economics Bulletin, Vol. 
29 no.3 pp. 2372-2379. 
Submitted: Mar 26 2008.   Published: September 18, 2009. 
 
     For all x,y 2 X, x + y = (x1 + y1,x2 + y2,...). For all x,y 2 X, x ¢ y = (x1y1,x2y2,...). A
constant sequence satisﬁes xi = a for all i 2 N for some a 2 R+, and it written as (a)con
For x,y 2 X, x ¸ y if xi ¸ yi for all i 2 N. For x,y 2 X, x > y if x ¸ y and x 6= y.
A social welfare relation is a binary relation º on X, which is reﬂexive and transitive. The
symmetric and asymmetric part of º is deﬁned as usual sense. Hence, x » y if and only if x º y
and y º x; and, x Â y if and only if x º y and :(y º x). A social welfare ordering is a binary
relation º on X, which is reﬂexive, complete and transitive. Let ºS and ºT be social welfare
relations. If ºS)ºT and ÂS)ÂT, we call ºT is an extension of ºS. If an extension ºT of ºS
is an ordering, we call ºT an ordering extension of ºS.
A ﬁnite permutation π is a permutation, such that there exists m 2 N with π(i) = i for all
i ¸ m. We write π(x) for the vector (xπ(1),xπ(2),...,xπ(i),...).
3 The Nash Social Welfare Relation
We propose an inﬁnite-horizon version of the Nash criterion. Let us deﬁne a social welfare relation
ºN on X by
x ºN y if and only if 9n 2 N such that (
n ∏
i=1
xi,x
+n) ¸ (
n ∏
i=1
yi,y
+n).
Note that a binary relation ºN is reﬂexive and transitive, but it is not necessarily complete. This
deﬁnition is a simple extension of the standard deﬁnition of the Nash criterion. For the argument
for the ﬁnite version of the Nash criterion, see Kaneko and Nakamura (1979) and Roberts (1980).
A social welfare relation ºN has the following properties:
9n 2 N such that (
n ∏
i=1
xi,x
+n) ¸ (
n ∏
i=1
yi,y
+n) ) 8m > n,(
m ∏
i=1
xi,x
+m) ¸ (
m ∏
i=1
yi,y
+m),
9n 2 N such that (
n ∏
i=1
xi,x
+n) > (
n ∏
i=1
yi,y
+n) ) 8m > n,(
m ∏
i=1
xi,x
+m) > (
m ∏
i=1
yi,y
+m).
By the contributions of Arrow (1951) and Szpilrajn (1930), we know that every binary relation
that is reﬂexive and transitive has an ordering extension. Therefore, there exists a social welfare
ordering º that is an ordering extension of ºN.
4 The Results
We introduce four axioms on º. The following axiom is well-known and therefore requires no
explanation.
Pareto: For all x,y 2 X, x > y ) x Â y.
Next, we propose ratio-incremental equity.
Ratio-incremental equity: For all x,y 2 X, for all s,t 2 N, and for all   2 R++, if (i) [ys = xs 
^ yt = xt/ ], and (ii) xk = yk for all k 2 N n fs,tg, then x » y.
2This is an equity axiom that requires an impartial treatment of a utility ratio change.3 This
axiom has a role similar to the incremental equity axiom proposed by Blackorby et al. (2002).4
The following axiom requires the equal treatment of generations.
Anonymity: For all x 2 X and all ﬁnite permutations π of N, x » π(x).
Note that our deﬁnition of anonymity does not allow an inﬁnite permutation. Lauwers (1997b,
1997c) and Mitra and Basu (2007) discuss classes of permutations that include inﬁnite permuta-
tions.5
The following axiom is an adaptation of the invariance transformation condition used in clas-
sical social choice theory.6 This axiom is an appropriate counterpart of partial-unit comparison,
introduced by Basu and Mitra (2007).
Partial ratio-scale invariance: For all x,y,a 2 X, and for all n 2 N, if x+n = y+n and x º y,
then x ¢ a º y ¢ a.
We present our results. Our ﬁrst result characterizes all ordering extensions of ºN by ratio-
incremental equity.
Theorem 1. A social welfare ordering º on X satisﬁes Pareto and ratio-incremental equity if
and only if º is an ordering extension of ºN.
Note that in this characterization, we do not impose anonymity.
Our second result characterizes all ordering extensions of ºN by partial ratio-scale invariance.
Theorem 2. A social welfare ordering º on X satisﬁes Pareto, anonymity, and partial ratio-scale
invariance if and only if º is an ordering extension of ºN.
5 Concluding Remarks
In this note, we characterize the Nash social welfare relation for inﬁnite utility streams in two
ways. In our ﬁrst characterization, the key axiom is ratio-incremental equity. This axiom is in the
spirit of impartiality assumptions emphasized by many authors. Bossert et al. (2007) investigate
two classical equity axioms, which have an ethical motivation. They characterize the inﬁnite
version of the generalized Lorenz criterion and of leximin by the Pigou-Dalton equity principle
and the Hammond equity principle, respectively. In our second characterization, the key axiom
is partial ratio-scale invariance. This axiom speciﬁes the informational structure of individual’s
utilities.
3An anonymous referee points out the problem of the ethical appeal of ratio-incremental equity as the impar-
tiality of utility ratio changes. Consider x = (a)con 2 X for some positive constant a 2 R++. Let   be a very
large positive value. Let y 2 X be such that [ys = xs  ^ yt = xt/ ] and xk = yk for all k 2 N n fs,tg. The
ratio-incremental equity requires that x » y for any  . Note that ys ! 1 and yt ! 0 as   ! 1. These facts imply
that ratio-incremental equity allows inequality among individuals.
4Kamaga and Kojima (2008a) characterize an inﬁnite version of utilitarian social welfare relation by the incre-
mental equity axiom proposed by Blackorby et al. (2002). Their characterization is closely related to Theorem 1
above. They also investigate the extended anonymity axiom, which is proposed and studied in Mitra and Basu
(2007), and characterize the extended versions of the generalized Lorenz and leximin criteria.
5See also Banerjee (2006), Kamaga and Kojima (2008a), and Kamaga and Kojima (2008b).
6For example, see Roberts (1980).
3Appendix
Proof of Theorem 1. Suﬃciency: Suppose that º is an ordering extension of ºN. We check
that Pareto is satisﬁed. Suppose that x > y. Obviously, we have that there exists n 2 N such
that (
∏n
i=1 xi,x+n) > (
∏n
i=1 yi,y+n). Then, x ÂN y. Since º is an ordering extension of ºN, we
have x Â y. Now, we show that ratio-incremental equity is satisﬁed. We consider two sequences
x,y 2 X such that (i) ys = xs  and yt = xt/ , and (ii) xk = yk for all k 2 N n fs,tg. Obviously,
we have that for n = maxfs,tg 2 N, (
∏n
i=1 xi,x+n) = (
∏n
i=1 yi,y+n). Then, by deﬁnition of ºN,
x »N y. Since º is an ordering extension of ºN, we have x » y.
Necessity: Suppose that a social welfare ordering º satisﬁes Pareto and ratio-incremental
equity. To prove º is an ordering extension of ºN, we have to show that ºN)º and ÂN)Â.
We take x,y 2 X such that x ÂN y. By deﬁnition of ºN, there exists n 2 N such that
n ∏
i=1
xi >
n ∏
i=1
yi and x
+n ¸ y
+n.
Now we prove that x Â y. Ratio-incremental equity implies the following results:
(x
−n,x
+n) »
(
(
n ∏
i=1
xi)
1/n,x2,...,
x1xn
(
∏n
i=1 xi)1/n,x
+n
)
»
(
(
n ∏
i=1
xi)
1/n,(
n ∏
i=1
xi)
1/n,...,
x1x2xn
(
∏n
i=1 xi)2/n,x
+n
)
. . .
»
(
(
n ∏
i=1
xi)
1/n,(
n ∏
i=1
xi)
1/n,...,(
n ∏
i=1
xi)
1/n,x
+n
)
.
Therefore,
(x
−n,x
+n) » (ˆ x
−n,x
+n) (1)
where ˆ x = ((
∏n
i=1 xi)1/n)con. By the same argument, we obtain
(y
−n,y
+n) » (ˆ y
−n,y
+n) (2)
where ˆ y = ((
∏n
i=1 yi)1/n)con. Note that in this case, we have (
∏n
i=1 xi)1/n > (
∏n
i=1 yi)1/n Hence,
Pareto implies that (ˆ x−n,x+n) Â (ˆ y−n,y+n). Therefore, in combination with (1) and (2), the
transitivity of º implies that x Â y.
Next, we take x,y 2 X such that x ºN y. By deﬁnition of ºN, there exists n 2 N such that
(
n ∏
i=1
xi,x
+n) ¸ (
n ∏
i=1
yi,y
+n).
If (
∏n
i=1 xi,x+n) > (
∏n
i=1 yi,y+n), then x Â y by the above argument. Hence, we have to consider
the case where (
∏n
i=1 xi,x+n) = (
∏n
i=1 yi,y+n). In this case, (ˆ x−n,x+n) = (ˆ y−n,y+n). Hence, by
(1) and (2), the transitivity of º implies that x » y. Therefore, º is an ordering extension of
ºN. ¥
4Before proving Theorem 2, we refer to the result of Basu and Mitra (2007). They propose the
following criterion. Let us deﬁne a social welfare relation ºU on RN by
x ºU y if and only if 9n 2 N such that (
n ∑
i=1
xi,x
+n) ¸ (
n ∑
i=1
yi,y
+n).
Basu and Mitra (2007) introduce the following axiom.
Partial-unit comparability: For all x,y,b 2 RN, and for all n 2 N, if x+n = y+n and x º y,
then x + b º y + b.
They show that a social welfare ordering º on RN satisﬁes Pareto, anonymity, and partial-unit
comparability if and only if it is an ordering extension of ºU (Basu and Mitra (2007), Theorem
1).7
Proof of Theorem 2. Suﬃciency: Suppose that º is an ordering extension of ºN. In the proof
of Theorem 1, we have already checked that Pareto is satisﬁed. Now, we show that anonymity
is satisﬁed. Let x 2 X and π be a ﬁnite permutation of N. There exists m 2 N such that
xi = π(xi) for all i ¸ m. Obviously, we have that (
∏m
i=1 xi,x+m) = (
∏m
i=1 yi,y+m). Then,
x »N y. Since º is an ordering extension of ºN, we have x » y. Finally, we show that ratio-scale
invariance is satisﬁed. We take x,y 2 X such that x+n = y+n for some n 2 N, and x º y. Cleary, ∏n
i=1 xi ¸
∏n
i=1 yi. This implies that for all a 2 X,
∏n
i=1 xiai ¸
∏n
i=1 yiai and x+n¢a+n = y+n¢a+n.
Since º is an ordering extension of ºN, we have x ¢ a º y ¢ a.
Necessity: Suppose that a social welfare ordering º∗ on RN satisﬁes Pareto, anonymity,
and partial-unit comparison. By Theorem 1 of Basu and Mitra (2007), if º∗ satisﬁes Pareto,
anonymity, and partial-unit comparability, then º∗ is an ordering extension of ºU. Now we
deﬁne an ordering º on X as follows: for all x,y 2 X,
(e
x1,e
x2,...) º (e
y1,e
y2,...) , x º
∗ y.
It is straightforward to show that º satisﬁes Pareto and anonymity. Furthermore, by taking
ai = ebi, we can check that º also satisﬁes ratio-scale invariance. By deﬁnition of º, x º y holds
if and only if (logx1,logx2,...) º∗ (logy1,logy2,...). Since º∗ is an ordering extension of ºU, if
there exists n 2 N such that (
∑n
i=1 logxi,logxn+1,...) ¸ (
∑n
i=1 logyi,logyn+1,...), then x º y.
Note that
∑n
i=1 logxi = log
∏n
i=1 xi. This implies that x ºN y ) x º y. Similarly, we can show
that x ÂN y ) x Â y. Therefore, º is an ordering extension of ºN. ¥
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7

(2008a) “Q-anonymous social welfare relations on inﬁnite utility streams” Social Choice and Welfare (forthcoming).

(2008b) “On the leximin and utilitarian overtaking criteria with extended anonymity” 21COE-GLOPE Working Paper Series,

Aggregating inﬁnite utility streams with intergenerational equity: The impossibility of being Paretian” Econometrica 71, 1557–1563. 7Basu and Mitra

An axiomatic approach to intergenerational equity”

Continuity and egalitarianism in the evaluation of inﬁnite utility streams”

Continuity and equity with inﬁnite horizons”

Equitable intergenerational preferences on restricted domains”

Equity, continuity, and myopia: A generalization of Diamond’s impossibility theorem”

Impossibility theorems and inﬁnite horizon planning”

Inﬁnite utility: Insisting on strong monotonicity”

Interpersonal comparability and social choice theory”

Intertemporal equity and the extension of the Ramsey criterion”

On the Existence of Paretian Social Welfare Relations for Inﬁnite Utility Streams with Extended Anonymity”

On the extension of the utilitarian and Suppes-Sen social welfare relations to inﬁnite utility streams”

Ordering inﬁnite utility streams”

Rawlsian equity and generalized utilitarianism with an inﬁnite population”

Social Choice and Individual Values

Sur l’extension de l’ordre partiel”

The evaluation of inﬁnite utility streams”

The Nash social welfare function”

Transfer principles and inequality aversion, with an application to optimal growth”

Utilitarianism and the Theory of Justice”

Characterizing the Nash social welfare relation for infinite utility streams: a note

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Characterizing the Nash social welfare relation for infinite utility streams: a note

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