We reexamine the analysis of Chambers (Social Choice and Welfare, 2009), that produces a characterization of a family of social welfare functions in the context of intergenerational equity: namely, those that coincide with either the sup, inf, lim sup, or lim inf rule. Reinforcement, ordinal covariance, and monotonicity jointly identify such class of rules. We show that the addition of a suitable axiom to this three properties permits to characterize each particular rule. A discussion of the respective distinctive properties is provided.Social welfare function; Intergenerational equity; Lim sup ; Lim inf

Alcantud, José Carlos R.

García-Sanz, María D.

English

Research Papers in Economics

A comment on “Intergenerational equity: sup,
inf, lim sup, and lim inf”
Jos´ e Carlos R. Alcantud 1, Mar´ ıa D. Garc´ ıa-Sanz∗,1
Facultad de Econom´ ıa y Empresa, Universidad de Salamanca, E 37008
Salamanca, Spain
Abstract
We reexamine the analysis of Chambers (2009), that produces a characterization
of a family of social welfare functions in the context of intergenerational equity:
namely, those that coincide with either the sup, inf, lim sup, or lim inf rule. Rein-
forcement, ordinal covariance, and monotonicity jointly identify such class of rules.
We show that the addition of a suitable axiom to this three properties permits to
characterize each particular rule. A discussion of the respective distinctive properties
is provided.
Key words: Social welfare function, Intergenerational equity, Lim sup, Lim inf
∗ Corresponding author
Email addresses: jcr@usal.es (Jos´ e Carlos R. Alcantud), dgarcia@usal.es
(Mar´ ıa D. Garc´ ıa-Sanz).
URL: http://web.usal.es/jcr (Jos´ e Carlos R. Alcantud).
1 Both authors acknowledge ﬁnancial support by the Spanish Ministerio de Ciencia
e Innovaci´ on, and by Junta de Castilla y Le´ on under Project SA024A08.
Preprint submitted to Elsevier April 21, 20091 Introduction
The resolution of distributional conﬂicts among an inﬁnite and countable num-
ber of generations is subject to intense debate. When it comes to evaluating
inﬁnite utility streams diﬀerent rules may be employed. Providing sets of prop-
erties that distinguish them is crucial as to the assessment of their normative
appeal. After Koopman’s (1960) axiomatic characterization of the discount-
ing utilitarian rule (see also Lauwers, 1997) many authors have contributed to
this aim. The Rawlsian inﬁmum rule (also referred to as inf) is axiomatized in
Lauwers (1997). Diﬀerent versions of leximin and utilitarianism are character-
ized in Asheim and Tungodden (2004), and Basu and Mitra (2007) reobtain
characterizations of the overtaking and catching up criteria that Asheim and
Tungodden had axiomatized in terms of “preference continuity”. Related ref-
erences are d’Aspremont (2008), Asheim and Banerjee (2009), Bossert et al.
(2007) among others.
Chambers (2009) has given a set of three conditions that identiﬁes a family
of criteria formed by the sup, inf, lim sup, and lim inf rules. Because those
rules can not be diﬀerentiated according to Chambers’ axiomatics, we build
on Chambers (2009) in order to characterize the sup (resp. inf, lim sup, lim
inf) rule in terms of suitable axioms.
We introduce our setting and properties in Section 2. Section 3 contains the
characterizations as Corollaries to Chambers (2009), Theorem 1. Our conclu-
sions are summarized in Section 4.
2 Notation and deﬁnitions
2.1 Chambers’ characterization: the framework
Let X denote a subset of RN, that represents a domain of utility sequences or
inﬁnite-horizon utility streams. We restrict ourselves to study bounded real-
valued sequences, that is, X = l∞. The usual notation for utility streams
applies: x = (x1,...,xn,.......) ∈ X. The constant sequence (y,y,....) is abbre-
viated as (ycon), and (x,(y)con) holds for (x,y,y,y,....). We write x > y if
xi > yi for each i = 1,2,..., and x ≫ y if xi > yi for each i = 1,2,.... Also,
x > y means x > y and x  = y.
A social welfare function (SWF) is a function W : X −→ R. Next we deﬁne
particular SWFs with relevance in the literature.
2Deﬁnition 1 The Rawlsian inﬁmum (or inf) rule is deﬁned by:
for each x = (x1,...,xn,.......) ∈ X, WR(x) = inf{x1,...,xn,...}
Deﬁnition 2 The supremum (or sup) rule is deﬁned by:
for each x = (x1,...,xn,.......) ∈ X, Ws(x) = sup{x1,...,xn,...}
Deﬁnition 3 The lim inf rule is deﬁned by:
for each x = (x1,...,xn,.......) ∈ X, Wli(x) = liminf{x1,...,xn,...}
Deﬁnition 4 The lim sup rule is deﬁned by:
for each x = (x1,...,xn,.......) ∈ X, Wls(x) = limsup{x1,...,xn,...}
The following axiom captures eﬃciency displayed by each of these rules.
Monotonicity, also M. If x,y ∈ X and x > y then W(x) > W(y).
Together with Monotonicity, Chambers (2009) uses the following two proper-
ties to axiomatize the family of rules formed by Deﬁnitions 1-4:
Reinforcement, also RI. For K ∈ N and a bijection σ : N → N × {1,...,K},
we write for each i ∈ N,σ(i) = (σ1(i),σ2(i)), where σ1 : N → N and σ2 : N →
{1,...,K}.
Let {xj}K
j=1 ⊆ X, where K < ∞. Suppose that W(xj) = W(xk) for all
j,k ∈ {1,...,K}. Let σ : N → N × {1,...,K} be a bijection. Deﬁne xσ(i) ≡
xσ2(i)(σ1(i)). Then W(xσ) = W(x1).
Ordinal Covariance, also OC. Let x ∈ X, and let ϕ : R → R be continuous
and strictly increasing. Then W(ϕ(x)) = ϕ(W(x)).
2.2 Additional properties
In order to complement Chambers’ argument we make use of further axioms
on a social welfare function W.
Restricted Sensitivity, also RS. There are y,x ∈ R such that y > x and
W(y,(x)con) > W(xcon).
3Along Deﬁnitions 1-4, only the sup rule agrees with RS. In the presence of
OC, RS is equivalent to the following stronger property:
Sensitivity, also S. Whenever y > x, W(y,(x)con) > W(xcon).
S is weaker than Koopmans’ sensitivity in the ﬁrst coordinate, which requests
W(y,x1,x2,....) > W(x,x1,x2,....) whenever y > x irrespective of x1,x2,....
Restricted Lower Sensitivity, also RLS. There are y,x ∈ R such that y > x
and W(ycon) > W(x,(y)con).
Along Deﬁnitions 1-4, only the inf rule agrees with RLS. In the presence of OC,
RLS is equivalent to the following stronger property (that is called Restricted
Dominance (RD) in Asheim et al., 2008).
Lower Sensitivity, also LS. Whenever y > x, W(ycon) > W(x,(y)con).
Weak Sensitivity, also WS. There are x,y ∈ R and z = (z1,...,zn,...) ∈ X
such that W(x,z1,...,zn...) > W(y,z1,...,zn...).
WS is weaker than RS, RLS, and LS. Both the inf and the sup rules agree
with WS, but neither the lim inf nor the lim sup rule do.
All the axioms above in this Subsection are speciﬁcations of the Weak Domi-
nance axiom, which reads as follows: if x,y ∈ X and there is j ∈ N such that
xj > yj, with xi = yi for all i  = j, then W(x) > W(y). None of the rules
under inspection displays this kind of eﬃciency.
Independent Future, also IF. For each x = (x1,...,xn,...), y = (y1,...,yn,...),
and x ∈ R: W(x,x1,...,xn,...) > W(x,y1,...,yn,...) if and only if W(x1,...,xn,...) >
W(y1,...,yn,...).
Weak Non-Substitution, also WNS. For each z,t, for each y > x: W(z,(y)con) >
W(t,(x)con).
Both the lim inf and the lim sup rules agree with IF and WNS, but neither
the inf nor the sup rule fulﬁl any of these two properties. WNS appears in
Asheim et al. (2008) as a weaker version of Lauwers’ (1998) Non-Substitution
condition. It is weaker than Hammond Equity for the Future.
In order to distinguish the lim inf and the lim sup rules we introduce two rein-
forcements of Monotonicity that are weaker than the standard Pareto axiom.
Although neither of the rules in Deﬁnitions 1 to 4 are Paretian –in the sense
x > y implies W(x) > W(y)– they do exhibit some Paretian eﬃciency that
we intend to capture.
4For any threshold t ∈ R, with each x ∈ X we associate the streams x∗(t) and
x∗(t) whose i-th component are deﬁned by:
x
∗(t)i =

 
 
xi + 1 if xi > t
xi otherwise
x∗(t)i =

 
 
xi − 1 if xi 6 t
xi otherwise
(1)
Above-t Monotonicity, also AtM. The SWF satisﬁes M and W(x∗(t)) > W(x)
for each t ∈ R and x ∈ X such that xi > t for an inﬁnite number of
generations.
Below-t Monotonicity, also BtM. The SWF satisﬁes M and W(x) > W(x∗(t))
for each t ∈ R and x ∈ X such that xi 6 t for an inﬁnite number of
generations.
The SWFs deﬁned by lim sup and supremum satisfy the AtM property, which
is not satisﬁed by the lim inf and the inﬁmum rules. Opposite, BtM is satisﬁed
by the lim inf and the inﬁmum rules, but not by the lim sup and the supremum.
Figure 1 shows the relationships among the eﬃciency axioms in use.
Weak Dominance
Lower Sensitivity/Restricted Dominance Sensitivity
Restricted Lower Sensitivity Restricted Sensitivity
Weak Sensitivity
Pareto
AtM BtM
Monotonicity
Fig. 1. The eﬃciency axioms under study.
3 Results
In this Section we provide Corollaries to the following result.
5Theorem 1 (Chambers, 2009) A SWF deﬁned on X = l∞ agrees with
either the supremum, the inﬁmum, the limit superior, or the limit inferior
rule, if and only if it satisﬁes M, OC, and RI.
From Theorem 1 the next Corollaries follow:
Corollary 1 A SWF deﬁned on X = l∞ is the inﬁmum rule if and only if
it satisﬁes M, OC, RI, and RLS (resp., LS).
Proof: The necessity part has been already stated. As for suﬃciency, apply
Theorem 1 and discard the supremum, lim inf, and lim sup because they do
not satisfy RLS, which is equivalent to LS under M. ￿
Corollary 1 adds to Lauwers’ (1997) characterization of the inﬁmum rule in
terms of related axioms (v. Chambers, 2009, Section 4 for a discussion on this
topic).
By using analogous direct arguments, further Corollaries identify the supre-
mum / lim sup / lim inf rules.
Corollary 2 A SWF deﬁned on X = l∞ is the supremum rule if and only
if it satisﬁes M, OC, RI, and RS (resp., S).
Corollary 3 A SWF deﬁned on X = l∞ is the lim sup rule if and only if it
satisﬁes AtM, OC, RI and WNS (or IF).
Corollary 4 A SWF deﬁned on X = l∞ is the lim inf rule if and only if it
satisﬁes BtM, OC, RI and WNS (or IF).
4 Summary of results and conclusions
Chambers (2009) has provided three axioms that, together with suitable com-
parisons of streams, permit to identify four rules for assessing inﬁnite utility
streams. For example, the inﬁmum rule is characterized by M, OC, RI, and
WR(0,(1)con) = WR((0)con). The supremum rule is characterized by M, OC,
RI, and Ws(1,(0)con) > Ws((0)con). The lim inf rule is characterized by M,
OC, RI, Wli(1,0,1,0,1,...) = Wli((0)con), and Wli(0,(1)con) > Wli((0)con).
Finally, the lim sup rule is characterized by M, OC, RI, Wls(1,0,1,0,1,...) >
Wls((0)con), and Wls(1,(0)con) = Wls((0)con). The distinction of cases in
Chambers’ proof already hints these consequences.
We have built on Chambers’ theorem to produce characterizations in terms
of normative axioms for the four rules under inspection. Table 1 collects the
6properties that have served us to state diﬀerent Corollaries to that focal result.
Axioms
Criteria RLS/LS RS/S AtM BtM WS IF WNS
inf + — — + + — —
sup — + + — + — —
lim inf — — — + — + +
lim sup — — + — — + +
Table 1. Properties of the criteria under inspection.
References
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8

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