The resource allocation problems withinterpersonal comparisons of welfare:
An axiomatization of the impartial Walras rule
Ryo-Ichi Nagahisa
Department of Economics, Kansai University
3-3-35 Yamatecho Suita 564-8680 Japan
July 2019
Abstract
The extended sympathy approach, which has been studied so far in
the abstract framework of social choice, is applied to the resource allocation problem of exchange economies with a …nite number of agents anda …nite number of private goods. The central issue in this paper is theaxiomatic analysis of what we call the impartial Walras rule: It associates
each (subjective) preference pro…le with the set of impartial Walras allocations, which are de…ned as Walrasian allocations operated from permuted
initial endowments. We show that the impartial Walras rule is the uniquerule that satis…es Suppes nondiscrimination, impartial rationality, Suppes
equity, and extended local independence.

1

Introduction

The notion of extended preference makes it possible to compare the welfare
of di¤erent individuals in social choice. It has been developed mainly in theliterature of Arrow’s impossibility theorem, in the abstract framework of social
choice1 . The purpose of this paper is applying this approach to the resource
allocation problem of exchange economies with a …nite number of agents andwith a …nite number of private goods.
1 d’Aspremont (1985), d’Aspremont and Gevers (2002), Sen (1970,1977,1986), and Suzumura (1983) surveyed this area. Blackorby et al. (1984) give a diagrammatic introduction.
The most recent contribution is Yamamura (2017).
The exceptions are Sen (1974a,b) and Deschamps and Gevers (1978), which consider theincome distribution problem in a single commodity economy.

The central issue of the paper is an axiomatic analysis of what we call theimpartial Walras rule, which associates with each (subjective) preference pro…lethe set of impartial Walrasian allocations, which are de…ned as Walrasian allocations operated from permuted initial endowments. Theorem 1 is the main result,which says that the impartial Walras rule is the unique rule satisfying Suppes nondiscrimination (SN), impartial rationality (IMR), Suppes equity (SE),and extended local independence (ELI). SN is a generalization of anonymity or
nondiscrimination and says that any allocations which are supposed to be identical through permuting agents should be dealt with equally in social choice.
IMR is a generalization of individual rationality and gives each agent a utility
level that is better than the utility attained through a permutation of agents.
SE is an interpersonal extension of Pareto optimality. ELI, a generalization of
local independence due to Nagahisa (1991), is a requirement of informational
economization for interpersonal comparisons of welfare. It works over extended
preferences in a similar way as local independence does over subjective preferences. Theorem 1 is the counterpart of the main result of Nagahisa (1991), an
axiomatization of the Walras rule2 : If interpersonal comparisons of welfare areallowed, the similar axioms as those in Nagahisa (1991) lead us to the impartial
Walras rule, not to the Walras rule.
We describe the impartial Walras rule as the rule chosen in a …ctional social
choice arena. An example of the arena is an original position assumed by Rawls(1971) that is covered with the veil of ignorance. A state assumed by Hare (1981)
and Harsanyi (1955) where each agent replaces the other’s position equally isalso another example. We also discuss that if each agent has the identicalendowment, Theorem 1 gives an alternative axiomatization of the equal income
Walras rule studied by Thomson (1988), Nagahisa and Suh (1995), Maniquet
2 The study of the axiomatic analysis of the Warlas rule with a …nite number of agents is
initiated by Gevers (1986) and Hurwicz (1979). Hammond (2002) is a comprehensive surveyof the …eld.

2

(1996), and Toda (2004).
The paper is organized as follows. Section 2 gives notation and de…nitions,where extended preferences and Suppes criterion are introduced. The main
result and the proof are given in Sections 3 and 4 respectively. Section 5 is theconclusion.

2

Notation and De…nitions

2.1

Exchange Economies

We consider exchange economies with a …nite number of agents and a …nite
number of private goods. Let N = f1; 2; :::; ng and L = f1; 2; :::; lg be the set ofthe agents and the set of the private goods respectively. All the agents have thenlll
and z = (z1 ; :::; zn ) 2 R+
. Let zi = (zi1 ; :::; zil ) 2 R+
same consumption set R+lbe agent i’s consumption and an allocation respectively. Let ! i 2 R++
be agent

i’s initial endowment and …xed throughout the paper. Let ! = (! 1 ; :::; ! n ). An
X
X
allocation z is feasible ifzi =! i . Let Z be the set of feasible allocations.
i2N

i2N

l
Let <i be agent i’s preference on R+. A pro…le <= (<i )i2N is a list of allagents’preferences. We consider two types of the set of pro…les. Let l := fp 2
l
l
P
P
l
l
R+
:
pi = 1g and int: l := fp 2 R++:
pi = 1g. Let L be the set of pro…les
i=1

i=1

such that <= (<i )i2N 2 L if and only if there exists some p 2 int:
for each <i , x <i y () px

lsuch thatlpy for all x; y 2 R+
. That is, a pro…le < is in L if

and only if every agent has the same preference represented by a linear utility
function. We use the notation <p = (<pi )i2N if we need to specify p. Let Q belthe set of preferences satisfying (i)-(iii). (i) <i is continuous and convex on R+land continuously di¤erentiable on R++. (ii) xlin addition x 2 R++, this implies xlfy 2 R+: y <i xg

y&x 6= y implies x <i y, and ifli y (monotonicity). (iii) for any x 2 R++ ,
n

z
l
R++
(boundary condition). Let Qn = Q}|{
Q be thesecond type of the set of pro…les. Let D = L [ Qn be called the domain. The
3

domain is the same as that of Nagahisa (1991)3 .
Let a pro…le <2 D be given. The Pareto optimal, individually rational, andWalrasian allocations for the pro…le are de…ned as usual: (i) z 2 Z is Pareto
optimal if and only if there is no feasible allocation z 0 such that zi0 <i zi for
all i 2 N and zi0

i zi for some i 2 N :(ii) z is individually rational if andonly if zi <i ! i for all i 2 N :(iii) z is a Walrasian allocation if and only ifthere is a price vector p 2 int:lxi 2 R+such that pxi

lsuch that for all i 2 N , zi <i xi for all

p! i . Let P O(<), IR(<), and W (<) stand for theset of Pareto optimal, individually rational, and Walrasian allocations for <respectively. We occasionally use the notation W (<; !) and IR(<; !) insteadof W (<) and IR(<).

2.2

Extended Preferences

The notion of extended preferences is based on the principle of the extended
sympathy, mentioned by Arrow (1963) and initiated by Suppes (1966) and Sen(1970). The basic idea is that a hypothetically existing ethical observer compares the welfare of di¤erent persons from a social point of view while respecting(or sympathizing with) their subjective preferences. Given a pro…le <2 D, an
extended preference <E generated from < is a complete and transitive binarylrelation on R+

N . We read (x; i) <E (y; j) as "being agent i with con-sumption x is at least as well o¤ as being agent j with consumption y".4 We
lassume that for any i 2 N , the restriction of <E to R+

fig is identical to <i ;lx <i y () (x; i) <E (y; i) for any x; y 2 R+
and any i 2 N . This assump-tion has been referred to in the literature as the axiom of identity. It requiresthat individual preferences be respected in extended preferences. This axiom
3 The domain Nagahisa and Suh (1995) employed is more natural. But we prefer mathematical tractability here.
4 Note that we admit interpersonal welfare comparisons, but still deny cardinality.

4

constitutes the core idea of the sympathy of the ethical observer.5
The examples below illustrate two extended preferences, both of which play
important roles in the subsequent sections.
Example 1 Let <2 D and p 2 int:lbe given. The extended preference <E(p)is de…ned by(x; i) <E(p) (y; j) () minfpq : q

i xg

minfpq : q

6
i yg .

Let p be prices. In <E(p) , the price p is used as a common indicator ofinterpersonal comparisons of welfare: Being agent i with consumption x is atleast as well o¤ as being agent j with consumption y if and only if the minimum
amount of expense that agent i needs to spend to achieve the level of utility at
x is not smaller than that at agent j’s y. If <p 2 L, then <pE(p) reduces to asimple form as follows.
Example 2 (x; i) <pE(p) (y; j) () pxpy.

We can identify subjective preference itself with the extended preference. To
simplify the notation, we denote this extended preference by <pE hereafter.

2.3

Suppes criterion

Given an allocation z and a permutation

on N , we let z be an allocationsuch that zi = z (i) for every i 2 N . Let

be the set of permutations. We

introduce the Suppes criterion (Suppes 1966), occasionally called the grading
principle and interpreted as an interpersonal extension of the Pareto criterion.
Let <E and allocations z; z 0 be given: According to the criterion, (i) z is atleast as just as z 0 for <E if and only if there is some2

such that (zi ; i)<E (z 0 (i) ; (i)) for all i 2 N : (ii) z is more just than z 0 for <E if and only
if (i) holds with strict preference relation (zi ; i)0
E (z (i) ;(i)) for at least one

5 This is implicitly assumed in almost every literature related to extended preferences. Refer
to Sen (1970) and d’Aspremont (1985) for more details.
6 Note that minfpq : q
l
= R++
.
i xg = 0 if <i 2 Q and x 25member i : (iii) z is equally as just as z 0 for <E if and only if there is some2

such that (zi ; i)0
E (z (i) ;(i)) for all i 2 N . It is obvious that all of

the three relations are well de…ned and transitive, and that "equally as just as"
relation is symmetric whereas "more just than" relation is asymmetric. The
Suppes criterion is used in the axiomatization of the utilitarian rule and theleximin rule in the literature of Arrovian social choice7 .
Let <E be given. A feasible allocation z is Suppes equitable for <E if andonly if there are no feasible allocations z 0 that are more just than z for <E . Let
SE(<E ) be the set of Suppes equitable allocations. Let IM R(<E ) be the setof feasible allocations that are at least as just as !, called the set of impartial
rational allocations. The axiom of identity implies that SE(<E )IR(<)

P O(<) andIM R(<E ) for all <E . Note that SE(<E ) is nonempty if there exist

utility functions ui (i 2 N ) such that (x; i) <E (y; j) () ui (x)uj (y) for allXl: Feasible allocations z = (zi )i2N maximizingui (zi )i; j 2 N and x; y 2 R+
i2N

on Z are Suppes equitable. This assures that SE(<E(p) ) is nonempty.

2.4

Rules

Let f : D ! Z be a social choice rule, simply a rule hereafter, which associateswith each pro…le a nonempty subset of feasible allocations. A rule f decides
f (<) through interpersonal comparisons of welfare. In other words, it uses
extended preferences generated from < to decide f (<). But not all the extended
preferences are used.
Given a pro…le <2 D, let D(<) be a nonempty subset of <E . A rule f
decides f (<) based on extended preferences in D(<), and no extended preferences excluded from D(<) are used in this decision. Let us call [ D(<) the<2D

extended domain. We consider the following three assumptions on the extended
7 The contributions are Sen (1976), d’Aspremont and Gevers (1977), Deschamps and Gevers(1978), Gevers (1979), Hammond (1976,79), and Strasnick (1976).

6

domain.
D.1. For any <p 2 L, then D(<p ) = f<pE g.
D.2. For any <2 D and any <E 2 D(<), (x; i) <E (0; j) for all i; j 2 N andlall x 2 R+.

D.3. For any <2 D and any p 2 int:l

, <E(p) 2 D(<).

There exist many extended domains satisfying the three conditions. The
smallest one is D(<) = f<E(p) : p 2 int:lg for any <2 D. The largest one

is de…ned by comparison of utilities among agents subject to ui (0) being equal
across all i.
There is no compelling reason to dismiss D.1: If everyone has the same linear
preference, that preference needs to be the extended preference and no otherextended preferences are considered possible. It is very hard to reject D.2, whichre‡ects our intuition that as long as other conditions are equal, people without
wealth are the most miserable in the world. D.3 can be justi…ed as follows. Let p
be the supporting price at z. If we look only around z, we can think of <E(p) as
being identical to <p : They are locally identical around z. To create extended
preferences, let us request that only the local information about preferences inthis sense be used. Then, as long as <p is allowed as an extended preference,
so is <E(p) . This is the legitimacy of D.3.
The Walras rule W associates with each pro…le the set of Walrasian allocations for the pro…le. The impartial Walras rule is a generalization of the Walras
rule, which is de…ned as follows. Let <2 D and2

be given. Let W (<; ! )be a set such that z 2 W (<; ! ) if and only if z is a Walrasian allocation
when all agents’ endowments are permuted throughp 2 int:l: Given a price vector, every agent i chooses zi as his best choice when his endowment is(i)’s one. Let [ W (<; ! ) be called the set of impartial Walrasian allocations2

for <. The impartial Walras rule IW : D ! Z is a rule that associates witheach <2 D the set of impartial Walrasian allocations for the pro…le. The equal
7income Walras rule is the Walras rule when every agent has the same initialendowment. The impartial Walras rule reduces to the equal income Walras rulein the case.

2.5

Axioms

A rule f satis…es Suppes Nondiscrimination (SN) if for any <2 D, and forany z; z 0 2 Z, z is equally as just as z 0 for any <E 2 D(<), then z 2 f (<) ()Sz 0 2 f (<). A rule f satis…es Suppes Equity (SE) if f (<)
SE(<E )
<E 2D(<)

for any <2 D. A rule f satis…es Impartial Rationality (IMR) if f (<)
S
IM R(<E ) for any <2 D.
<E 2D(<)

SE is well de…ned: Take z 2 P O(<) arbitrarily. Let p 2 int:lbe thesupporting price at z. It is easy to see z 2 SE(<E(p) ), and hence z 2 SE(<E(p)
D:3

)

#

S

<E 2D(<)

SE(<E ) 6= ;.

Suppes Equity and Impartial Rationality correspond to Pareto Optimality
and Individual Rationality of the framework of subjective preferences respectively. Just as Pareto Optimality is replaced by Suppes Equity, Individual Rationality must also be replaced by Impartial Rationality. Note that the strengthof each axiom depends on the size of D(<). The richer the D(<), the weaker
the axiom.
l
Let <2 D and zi 2 R++
be given. We say that p 2 int:lis the supporting

price of < at zi if agent i’s indi¤erence curve passing through zi is tangent tonl
the line fx : px = pzi g at zi . Let z 2 R++
be given. We say that p 2 int:

the supporting price of < at z if for any i, p 2 int:

l

l

is

is the supporting price of< at zi .
nlLet <= (<i )i2N 2 D and z = (zi )i2N 2 R++
be given. Let p(<i ; zi ) be thesupporting price of <i at zi .
A rule f satis…es Local Independence (LI) if for any <; <0 2 D and anynlz 2 Z \ R++, if p(<i ; zi ) = p(<0i ; zi ) for all i, then z 2 f (<) () z 2 f (<0 ).

8

LI was used in the axiomatization of the Walras rule by Nagahisa (1991).8
We can show that LI together with the other axioms characterizes the impartial
Walras rule. However, LI is a requirement de…ned in subjective preferences, andhence we must seek a more sophisticated version de…ned in extended preferences.
The idea behind LI is that only the local information around z is used for social
choice. Just as LI applies this idea to individual preferences, the extended local
independence, a generalization of LI, applies it to extended preferences. Roughly
speaking, it says that if two extended preferences with totally di¤erent forms
as a whole are considered as identical around z, they have the same implication
for social choice. The formal de…nition is as follows.
nl
Let <; <0 2 D and z 2 R+
be given. Let <E and <0E be extended preferences

generated from < and <0 respectively. We say that <E is essentially identical tol<0E around z if there exist functions ui , "i : R+
! R such that for all i; j 2 Nl,and all x; y 2 R+

(x; i) <E (y; j) () ui (x)uj (y);(x; i) <0E (y; j) () ui (x) + "i (x)where "i (zi ) = 0 and jjx"i (x)zi jj

uj (y) + "j (y);! 0 as xi ! zi , and the same property holds

for "j .
We regard ui and "i as a utility function representing <i and a higher ordererror term respectively. We regard uj and "j as well.
A rule f satis…es Extended Local Independence (ELI) if for any <2 Dnl
and any z 2 Z \ R++, if there exists some <E 2 D(<) and some p 2 int:

l

such

that <E is essentially identical to <pE around z, then z 2 f (<) () z 2 f (<p ).
Suppose that p 2 int:lnl
is the supporting price of <2 D at z 2 Z \ R++. If

extended preferences that are used to decide whether z 2 f (<) or not should be
8 Refer to Nagahisa (1991) for more details on LI. Generalizations and related axioms of LI
are found in Yoshihara (1998), Fleurbaey, et.al (2005), and Miyagishima (2015). An ordering
version of LI is given by Sakai (2009). Urai and Murakami (2015) uses LI for economies withmoney.

9created from the information of individual preferences only around z, it mustbe essentially identical to <pE around z. Such the extended preferences areused for deciding z 2 f (<) or not9 , not all extended preferences in D(<) areused: Extended preferences that are not essentially identical to <pE around z
are not considered. As D.1 requires that <pE be the only extended preferencesin D(<p ), this means that deciding whether z 2 f (<) or not is equivalent todeciding whether z 2 f (<p ) or not. This is the essence of ELI.
We show that ELI is equivalent to LI if PO is satis…ed (Lemma 5). ELI
says nothing in the case of no supporting price. However, this is enough for theaxiomatization of the impartial Walras rule. No further discussion is necessary.

3

Results

Theorem 1 Assume D.1, D.2, and D.3. The impartial Walras rule is theunique rule satisfying SN, IMR, SE, and ELI.
The proof of Theorem 1 follows along the similar line as that of the axiomatization of the Walras rule due to Nagahisa (1991). As Suppes nondiscrimination, Suppes equity, and extended local independence are essentially equivalent
to nondiscrimination, Pareto optimality, and local independence in Nagahisa
respectively, we conclude that impartial rationality is the most responsible forthe axiomatization of the Impartial Walras rule.
It is believed that the impartial Walras rule is chosen in a …ctional social
choice. Two interpretations of "…ctional" are open to us. One interpretation
is to think of it as a social choice problem where no one knows who has which
initial endowment.10 This interpretation reminds us of Rawls’s veil of ignorance
9 The existence of such extended preferences is assured by Lemma 3 in Section 4, whichshows that <E(p) is one of the extended preferences.
1 0 Speaking more accurately, we consider a problem where everyone knows the set of initialendowments, but no one knows who owns which of them.

10(Rawls 1971). The similar logic as Rawls works with this interpretation. The
impartial Walras rule probably may not be supported if the veil of ignorance
was removed, i.e., if everyone knew their endowments perfectly. In that state,
the Walras rule will be more favored. But otherwise, the impartial Walras rule
could be justi…ed.
A social choice problem in which everyone expects and/or imagines to own
other agent’s initial endowment with equal probability can also be the other interpretation. This idea, putting oneself in another’s shoes, is reminiscent of the
suppositions of Hare (1981) and Harsanyi (1955). Moreover, it is semanticallythe same as the situation where everyone has the same initial endowment, i.e.,Peveryone owns n1! i initially. In this case, the impartial Walras rule is equivi2N

alent to the equal income Walras rule studied by Thomson (1988), Nagahisa andSuh (1995), Maniquet (1996), and Toda (2004), and hence Theorem 1 can be
regarded as an axiomatization of the equal income Walras rule in the case.
The four examples below illustrate that the axioms are independent and that
Theorem 1 does not hold if one of the axioms lacks.

Example 3 (The impartial Walras rule operated from di¤erent endowments)
Let $ be a new initial endowment such that $i = !2i (i = 1; :::; n 1) andn
X1[! i . The rule f is given by f (<) =$n = ! n + 12
IW (<; $ ) for any
i=12<2 D. This rule satis…es all the axioms except for IMR.

Example 4 (The Walras rule) The Walras rule satis…es all the axioms except for SN.
[
Example 5 (The impartial Walras plus rule) Let (<) =z 2 Z : zi
28=<n
< IW (<) [ (<) if <2 Qn and <1 =f (<) =
for any <2 D:IW (<)
otherwise11

i ! (i) 8i.

This rule satis…es all the axioms except for SE. We can easily understandthat f satis…es IMR if we notice z 2 IM R(<E(p) ) for any z 2 (<).11 Refer tothe next section to see that f satis…es SN and ELI. Note that ! 2(<) is notalways Pareto optimal. Thus SE is violated.
Example 6 (The impartial core rule) Let Core(<) be the set of core allocations for <12 . Let f be a rule such that z 2 f (<) if and only if there exists
some z 0 2 Core(<) that is equally as just as z for any <E 2 D(<). This rule, ageneralization of the core rule, is well de…ned because of Core(<)

f (<). This

rule satis…es all the axiom except for ELI.

4

ProofsThe proof follows along the same line as Nagahisa (1991). The four axioms, impartial rationality, Suppes nondiscrimination, Suppes equity, and extended local
independence play the same role as in individual rationality, nondiscrimination,
Pareto optimality, and local independence there respectively.

Lemma 1 The Impartial Walras rule satis…es SE, SN, and IMR.
Proof. SE: Let z 2 IW (<) and p 2 int:lbe the price associated with z. Take

<E(p) 2 D(<). We show z 2 SE(<E(p) ) that completes the proof. Suppose
not. There exists some z 0 2 Z such that z 0 is more just than z for <E(p) . Thisimplies that there exists some2

such that (zi0 ; i) <E(p) (z (i) ; (i)) for all(i)) for some i. By de…nition of <E(p) , this further

i and (zi0 ; i)E(p) (z (i) ;implies pzi0

pz (i) for all i and pzi0 > pz (i) for some i. This contradicts thefeasibility of z and z 0 .

q

1 1 For any z 2(<), minfpq : q

i zi gz2 (<)#<i =< (i)#=minfpq : q i ! (i) g=
minfpq :(i)).
1 2 Core allocations are de…ned as usual by using weak and strict preference relations.
(i) ! (i) g. Thus (zi ; i) <E(p) (! (i) ;12

SN: Let z 2 IW (<) and p 2 int:lbe the price associated with z. Letz 0 2 Z be such that z is equally as just as z 0 for any <E 2 D(<). By de…nitionof IW , there is some2

such that for all i 2 N ,l(1) zi <i xi for all xi 2 R+
such that pxi

p! (i) .

By noting that <E(p) 2 D(<) and that z is equally as just as z 0 for <E(p) ,there is some2(2) (z (i) ; (i))

such that0
E(p) (zi ; i) for all i 2 N .

Thus we have(3) pz (i) = minfpq : q(i) z (i) g = minfpq : q0
i zi g.

The …rst equation of (3) follows from that p is the supporting price of z,
and the second from the de…nition of <E(p) and (2). Obviously (3) implies
pz (i)

pzi0 for all i 2 N , which, due to the feasibility of z and z 0 , implies(4) pz (i) = pzi0 for all i 2 N .
Substituting (4) for (3), we know pzi0 = minfpq : q0
i zi g and hence(5) p is the supporting price at zi0 .
On the other hand, (1) implies pzi = p! (i) for all i 2 N and hence(6) pz (i) = p! ( (i)) for all i 2 N .
(4) and (6) imply(7) pzi0 = p! ( (i)) for all i 2 N ,which together with (5) assures that z 0 is a Walrasian allocation when allagents’endowments are permuted through, which completes the proof of

SN.
IMR: Let z 2 IW (<). By de…nition of IW , there are some p 2 int:
and somethat pxi

2llsuch that for all i 2 N , zi <i xi for all xi 2 R+suchlp! (i) . Let x (i) be the best on the set fx (i) 2 R+
: px (i)p! (i) g with respect to < (i) . Noting that <E(p) 2 D(<) and p is the supporting price at zi and x (i) , we have (zi ; i)E(p)(x (i) ; (i)). By de…nition ofx (i) , we have x (i) < (i) ! (i) , which together with axiom of identity implies13

(x (i) ; (i)) <E(p) (! (i) ; (i)). Thus we conclude (zi ; i) <E(p) (! (i) ; (i)).
This is true for all i 2 N , and so z 2 IM R(<E(p) ; !)[<E 2D(<)

which completes the proof of IMR.
SLet Z p =fz 2 Z : pzi = p! (i) 8ig.

IM R(<E ; !),2

Lemma 2 Let f a rule satisfying IMR and SN. Then we have f (<p ) = Z p forlany p 2 int:.

Proof. Let z 2 f (<p ). IMR and D.1 mean that there is somepzi

2such thatp! (i) for all i 2 N . Due to the feasibility of z, we have pzi = p! (i) for

all i 2 N , which means z 2 Z p . Thus we have f (<p )Z p.

Next, take z 2 Z p arbitrarily. The previous argument assures the existence ofz 0 2 f (<p )Z p . By de…nition of Z p , there exist

andand pzi0 = p! (i) for all i. Thus pz 0 1 (i) = p! i = pzsuch that pzi = p! (i)1 (i)for all i. By noting

D.1, SN and z 0 2 f (<p ) imply z 2 f (<p ). Thus we also have f (<p )

Z p.

nl
Lemma 3 For any <2 D and any z 2 Z \ R++, the followings are equivalent:(i) There exists the supporting price p 2 int:lof < at z.

(ii) <E(p) is essentially identical to <pE around z.
Proof. It is easy to see that (ii) implies (i), just by noting that u0i = ui + "irepresents <0i . We show the converse. Let ui represent <i such that ui (x) =minfpw : w

i xg. The proof goes on with several steps.

Step 1. For each h 2 L, the marginal utility of h at zi is equal to ph ;i)i.e., @u@xi (z= ph .
ihh

Proof of Step 1: Take " 6= 0 arbitrarily. Let "h = (0; :::; 0; "; 0; :::; 0) 2 Rl . Asui (zi ) = pzi and ui (zi +"h ) = minfpq : qhhave ui (zi +" ") ui (zi )pzi +ph " pzi= ph .
"

i zi +"hgp(zi +"h ) = pzi +ph ", wei)Letting " ! 0, we have @u@xi (zihNext, imagine the budget constraint line with the initial endowment of zi

14

ph .
"h

and the price of p.13

Let zi" be i’s optimal consumption under the budget"h )constraint. Obviously, we have ui (zihhhence ui (zi ) u"i (zi " ) = pzi ui"(zi " )i)have @u@xi (zihui (zi" ) = pzi" = pzi

pzi (pzi ph ")= ph .
"

ph ", andLetting " ! 0, we

ph , which completes the proof of Step 1.

Step 2. ui (x) = pxi +"i (xi ), where jjx"ii(xzi i)jj

! 0 as xi ! zi and "i (zi ) = 0.

Proof of Step 2: Because of the di¤erentiability of ui , we havelP@ui (zi )ui (xi ) = ui (zi ) +zih ) + "i (xi ), where jjx"ii(xzi i)jj
@xih (xihh=1! 0 asxi ! zi and "i (zi ) = 0.

i)= ph (Step 1), we haveAs @u@xi (zihlPui (xi ) = ui (zi ) +ph (xih zih ) + "i (xi ) = ui (zi ) + p(xizi ) + "i (xi ).

h=1

As ui (zi ) = pzi , we haveui (xi ) = pzi + p(xizi ) + "i (xi ) = pxi + "i (xi ), which completes the proofof Step 2.
Step 3. We complete the proof.
Step 2 shows(x; i) <E(p) (y; j) () ui (x)uj (y) () px + "i (x)py + "j (y)This means that <E(p) is essentially identical around z to <pE .
We say that a rule f satis…es Pareto Optimality (PO) if f (<)

P O(<) for

any <2 D.
Lemma 4 For any rule f satisfying PO, f satis…es ELI if and only if it satis…es
LI.
Proof. ELI implies LI: Suppose that p(<i ; zi ) = p(<0i ; zi ) for any i, wherenl<; <0 2 D and any z 2 Z \ R++. Let z 2 f (<). As f satis…es PO, this implies

that for some p 2 int:l

, p(<i ; zi ) = p(<0i ; zi ) = p for any i. Lemma 3 implies

that <E(p) and <0E(p) are essentially identical to <pE around z. Thus we have
1 3 We make " small enough to satisfy z

i

"h

15

0.

ELI
#

ELI
#

z 2 f (<) =) z 2 f (< ) =) z 2 f (<0 ), which completes z 2 f (<) =) z 2 f (<0p). We can prove z 2 f (<) (= z 2 f (<0 ) similarly.
LI implies ELI: Suppose that there exists some <E 2 D(<) and some p 2
int:

l

nl
such that <E is essentially identical to <pE around z 2 Z \ R++. Thisimplies that p is the supporting price of < at z. Thus LI implies z 2 f (<) ()z 2 f (<p ), the desired result.
Proof of Theorem 1. By noting that SE implies PO, Lemma 4 implies that
ELI is equivalent to LI. It is easy to see that the impartial Walras rule satis…es
LI. Let f be a rule satisfying SN, IMR, SE, and LI. The only remaining thing
to prove is f = IW .
IW

f : Take z 2 IW (<) arbitrarily. By de…nition of IW , i weakly preferslnl
for some i. If <2 Qn , the. This implies zi 2
= R++
zi to ! (i) . Suppose z 2= R++boundary condition requires ! (i)i zi , which is a contradiction. So we assume<2 L. Let p be such that ui (x) = px represent <i . As i weakly prefers zi to! (i) , we have pzi

p! (i) . As this holds for all i, we have pzi = p! (i) for alli, and hence z 2 Z p . Lemma 2 implies z 2 f (<), the desired result.
nl
Next, consider the case of z 2 R++. Let p be an equilibrium price associatedwith z. We have pzi = p! (i) for all i, and hence z 2 Z p . Lemma 2 showsnl
z 2 f (<p ). As we assumed z 2 R++, LI implies z 2 f (<), the desired result.

f

IW : Take z 2 f (<) arbitrarily. Suppose <2 L. Lemma 2 implies

that there exists some2

such that pzi = p! (i) for all i, where p 2 int:land <i is represented by ui (x) = px. Thus z 2 IW (<), the desired result.
nllNext, consider the case of <2 Qn . Assume z 2= R++. This implies zi 2
= R++

for some i. IMR implies that there exist some <E 2 D(<) and some2

such that (zi ; i) <E (! (i) ; (i)). On the other hand, the boundary condition
together with the continuity and monotonicity of <i implies 0
axiom of identity implies (zi ; i)E

i zi .

Thus the(0; i), which together with D.2 and the16axiom of identity requires (! (i) ; (i))

E

(

! (i)
2 ;

(i)) <E (zi ; i), which is anlcontradiction. Now we can assume z 2 R++. Then SE implies z 2 P O(<),
l

and hence there exists some p 2 int:

such that p is the supporting price at z.

LI implies z 2 f (<p ), which together with Lemma 2 implies z 2 IW (<p ) andhence z 2 IW (<), the desired result.
Proof of Example 5.
z2

Let us show SN. For this, it su¢ ces to show that if(<) and z 0 2 Z is equally as just as z for all <E 2 D(<), then z 0 2

Take p 2 int:
and minfpq : qlarbitrarily. There exist someizi g = minfpq : q

same preference, this implies z 0 (i)such that zi

0(i) z (i) g for all i.

i zi g = minfpq : q

z 0 (i) g = minfpq : q

and

i

(<).

i ! (i)Thus minfpq : q(i)! (i) g. As i and (i) have the0(i) ! (i) , i.e., zi

i !

1(

(i)) , which meansz 0 2 (<).
Next, we show ELI. Let <E 2 D(<) and p 2 int:lbe such that <E isnl. It su¢ ces to show z 2
essentially identical to <pE around z 2 Z \R++z2(<p ). If z 2(<), then there exists some2(<) ()

such that zi

i ! (i) 8i.

As <E is essentially identical to <pE around z, Lemma 3 shows pzi

p! (i) 8i.

(<p ). If z 2Thus pzi = p! (i) 8i and hence z 2(<p ), then for some, pzi = p! (i) 8i. As <E is essentially identical to <pE around z, Lemma 3
shows that z is a Walrasian allocation for < with the endowments of ! . Thusz 2 (<).

5

Conclusion

In this paper, we combined two studies in social choice, the studies of interpersonal comparisons of welfare and that of axiomatic analysis of resource allocation problems, both of which had advanced independently of each other.14 We
conclude with one more remark below.
1 4 Chamber and Hayashi (2017) proposed an alternative axiomatic approach of the Walras
rule that can cope with income distribution problems.

17

It seems to have not very much been investigated so far to what extent it
is possible to create extended preferences from a given set of preference pro…lesand what they are.15 In most of the literature on interpersonal comparisons of
welfare, extended preferences are given a priori, and not asked for the ground.16
In contrast, we have stated that there must be a convincing basis for the extended preferences created from a given preference pro…le. D.1-D.3 is the basis.
We hope that this paper will provide new insights into research in this direction.

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