agreement, separability, and other axioms for quasi-linear social choice problems

Agreement, separability, and other axioms for quasi-linear social choice problemsAuthor(s): Youngsub ChunSource: Social Choice and Welfare, Vol. 17, No. 3 (2000), pp. 507-521Published by: SpringerStable URL: http://www.jstor.org/stable/41106372 .

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Soc Choice Welfare (2000) 1 7: 507-521 -

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Social . Choice «* Welfare

© Springer-Verlag 2000

Agreement, separability, and other axioms for quasi-linear social choice problems Youngsub Chun

Division of Economics, Seoul National University, Seoul 151-742, Korea (e-mail: [email protected])

Received: 18 May 1998/Accepted: 1 July 1999

Abstract. A quasi-linear social choice problem is concerned with choosing one among a finite set of public projects and determining side payments among agents to cover the cost of the project, assuming each agent has quasi-linear preferences. We first investigate the logical relations between various axioms in this context. They are: agreement, separability, population solidarity, consistency, converse consistency, and population-and-cost solidarity. Also, on the basis of these axioms, we present alternative characterizations of egalitarian solutions; each solution assigns to each agent an equal share of the surplus derived from the public project over some reference utility level, but uses a different method to compute the reference utility level.

1 Introduction

We consider the following class of quasi-linear social choice problems (Moulin 1985b). A society must choose one among a finite set of public projects; money is available to perform side payments. Each agent has quasi-linear preferences (separably additive with respect to the public projects and money, and linear with respect to money). We are interested in determining which public project should be chosen and what side payments should be performed.

This is a revised version of my paper entitled "Egalitarian Solutions for Quasi-Linear Social Choice Problems." This paper was written during my visit to the University of Rochester. I thank its Department of Economics for its hospitality. I am also grateful to Lionel McKenzie, William Thomson, a referee, and an associate editor for their comments, and the LG Yonam Foundation for its financial support. Without William Thomson, this paper would not exist. I retain, however, the responsibility for any shortcomings.



508 Y. Chun

In this context, we investigate the implications of various axioms. They are: (i) agreement, which requires that changes in the preferences of some members of the society, whether or not they are accompanied by changes in the cost vector, should affect the agents whose preferences have not changed in the same direction; all gain or all lose; (ii) separability, which requires that if for two problems, a subgroup of agents have the same preferences and the total amounts awarded to them are the same, then the amount awarded to each agent in the subgroup should be the same; (iii) population solidarity, which requires that the arrival of additional agents unaccompanied by changes in the cost vector should affect all of the original agents in the same direction; (iv) consistency, which requires that if the solution chooses a certain alternative for some problem, then for the "reduced" problem obtained by imagining the departure of some agents with their awards and reassessing the situation from the viewpoint of the remaining agents, the solution should assign to the remaining agents the same awards as before; (v) converse consistency, which requires that if an alternative has the property that for all two- person subgroups, the solution chooses the restriction of the alternative to the subgroup for the associated reduced problem this subgroup faces, then the alternative should be the solution outcome for the problem; and (vi) population-and-cost solidarity, which is a strengthening of population solidarity, requiring that all of the original agents be affected in the same direction even when the cost vector changes.

Of these axioms, only consistency and population solidarity have been studied in the context of quasi-linear social choice. Moulin (1985b) charac- terizes on the basis of consistency1 various families of egalitarian solutions: each solution assigns to each agent an equal share of the surplus derived from the public project over some reference utility level, but uses a different method to compute the reference utility level. On the other hand, Chun (1986) presents alternative characterizations of egalitarian solutions on the basis of population solidarity.2 Related results can be found in Moulin (1985a, 1987a) and Chun (1989, 1998).

In this paper, we first examine the logical relations between the axioms. In general, they are not directly related. However, it turns out that they are related on the class of quasi-linear social choice. Then, we provide alternative characterizations of the egalitarian solutions studied by Moulin (1985b) and Chun (1986) on the basis of agreement, separablity, and population-and-cost solidarity.

2 Preliminaries

Let / = { 1 , 2, . . . } be an (infinite) universe of "potential" agents. Agent i in / is indexed by the subscript i. Let Jf be the collection of nonempty and finite

1 Under the name of separability. 2 Under the name of solidarity.



Quasi-linear social choice 509

subsets of /, with generic element denoted by N. Let A be a set of public projects. Each project a e A has a cost c(a). In addition, a private good, "money," can be used for compensation. The preferences of agents i g N, defined over the product ̂xR, admit a quasi-linear representation by a utility vector Ui = (ui(a))aeA in R^: given the project aeA and agent f s holdings of money m¡ e R, /'s utility is iii(a) + m¡. Let c = (c(a))aeA in RA be the cost vector and u = (ui)i€N in [RA]N be the utility profile.

Given a society AT g Jf, a quasi-linear social choice problem or simply a problem is a pair (m,c) g [Ra]n x R'4 made up of a utility profile and a cost vector. Let 5^ be the class of quasi-linear social choice problems for N9 and let 1 = v£lN. We denote by 1 g Ra the vector whose coordinates are all equal to 1.

A solution is a function S : Ü -» uRN, that associates to any N e Jf and any (u,c) eâN, a vector S(u,c) = (Sfac))ieN of utility levels. The vector S(u, c) is the solution outcome of{u, c).

We will impose the following axioms. Efficiency requires that a society should choose the project that maximizes the difference between the sum of individual utilities and the cost. Anonymity requires that the naming of each agent should not matter in the decision process. Independence of the zero of the individual utility vector or utility independence requires that the choice of zero for the individual utility vector should not have any effect on the other agents. Finally, independence of the zero of the cost vector or cost independence requires that the choice of zero for the cost vector should affect all agents by the same amount. These are minor requirements satisfied by most well-known solutions.

Let S be a solution.

Efficiency. For all N e JÍ and all (u, e) e âN ,

V Sfa c) = max aeA

J ̂ u^a) - c(a) I . i^N aeA

[i¿N J

Anonymity. For all N e jV, all permutations p of N, and all (w, c), (w', cr) g 2N9 if u1 = (up{i))i€N and c' = c, then Sfa c) = Sp{i) (uf, c) for all i e N.

Utility independence. For all N e JÍ, all i g N, all (w, c), (w', e') e £N, and all a g R, if u' - Ui + al, uj = Uj for all j ^ i, and c' = c, then Sz(mx, cf) =

Sfa c) + a and S/(i/', c') = s/(u, c) for all j # i.

Cost independence. For all NeJT, all (k,c), (w',c;) g 5^, and all a g R, if

u' = u and c' = c + al, then ̂(w', c;) = Sfa c) - -^- for all i e N.

Notation: For all x e RA, we define xmax = maxaeA x(a) and all r £ TV, we define w^ = (ui)ieT, St(u) = (^(w))^^, and so on. Vector inequalities are:

Now we are ready to introduce our main axioms, which are concerned with certain changes in the parameter of the problem, but assuming the pop-



510 Y. Chun

ulation to be fixed. The first axiom, agreement, was introduced by Moulin (1987b) for surplus sharing problems. It requires that changes in the preferences of some members of the society, whether or not they are accompanied by changes in the cost vector, should affect the agents whose preferences have not changed in the same direction: all gain or all lose.

Agreement. For all M, N e jV such that M cN, and all (w, e), (w', e') e £N, if u'M = uM, then either SM(u', c1) ̂ SM(u, c) or SM(u', c') g SM(u, c). This axiom and the next one, separability, have been investigated by Moulin (1987b) in the context of surplus sharing and by Chun (1999) in the context of bankruptcy. A weaker version of agreement, which requires that the agents whose preferences have not changed should be affected in the same direction only when the cost vector is unaffected, has been studied by Moulin (1987a) under the same name in the context of binary choice, and by Thomson (1993, 1997) under the name oí replacement principle in various contexts (for a survey, see Thomson 1999).

Our second axiom, separability, was introduced by Moulin (1987b) for surplus-sharing problems. It requires that if for two problems, a subgroup of agents have the same preferences and the total amounts awarded to them are the same, then the amount awarded to each agent in the subgroup should be the same.

Separability. For all M, N e JÍ such that M cN, and all («, c), («', e') e âN , iîu'M = um and Y.ieM *(*'.*') = ZieM *(«,c), then SM{u',c') = SM{u,c).

Our next four axioms are concerned with certain changes in the population. The first axiom, population solidarity, requires that the arrival of additional agents unaccompanied by changes in the cost vector should affect all of the original agents in the same direction: all gain or all lose.

Population solidarity. For all M, N e JT such that M c]'í, all (w, e) e £M , and all (w', e') e 2N ' if u'M = u and c1 = c, then either SM{u', cf) ̂ S(«, c) or SM{u'c')^S[u,c). An axiom in this spirit was introduced and studied by Thomson (1983a,b) for bargaining problems under the name of population monotonicity. Since then, it has been studied in various contexts.3 In the context of quasi-linear social choice, its implications have been analyzed by Chun (1986).

Our second axiom is consistency. It requires that if the solution chooses a certain alternative for some problem, then for the "reduced" problem obtained by imagining the departure of some agents with their awards and reassessing the situation from the viewpoint of the remaining agents, the solution should assign to the remaining agents the same awards as before. In the current context, the reduced problem is defined by (i) the utility vectors of the

3 See Sprumont (1990), Ching and Thomson (1993), Thomson (1995a), Chun (1998), and for a survey, Thomson (1995b).



Quasi-linear social choice 5 1 1

remaining agents, and (ii) the modified cost vector to honor the awards given to the departing agents.

Consistency. For all M, N e Jf such that M c AT, and all (w, e) e £N ' if c1 = c - EjeN'M UJ + EjeN'M SAU, c)^ then S((ui)ieMi c') = Sm(", c).

This axiom has been studied for a great variety of models (for a survey, see Thomson 1998). In particular, it has been analyzed in the context of quasi- linear social choice by Moulin (1985b).

Our third axiom, converse consistency, can be regarded as a dual requirement to consistency. It requires that if an alternative has the property that for all two-person subgroups, the solution chooses the restriction of the alternative to the subgroup for the associated reduced problem this subgroup faces, then the alternative should be the solution outcome for the problem. Although the axiom has been studied in a great variety of contexts (for a survey, see Thomson 1998), it has not been investigated in the context of quasi-linear social choice. To state it formally, we use the following notation. Given a problem (w,c) and a solution 5, let c.con(u,c,S) = {xeR^ | J2ìgn xi ; =

(EieN Ui - C)™'XP = S((Ui)i€Pi C - ZieN'P *i + ZieN'P *¿) for a11 P C N such that 'P' = 2}.

Converse consistency. For all TV e jV, and all (w, e) e 1N, if x e c.con(u, c' 5), then x = S(u,c).

To relate these axioms, we introduce our final axiom, population-and-cost solidarity, which is a strengthening of population solidarity. It requires that all of the original agents be affected in the same direction by the arrival of additional agents even if the cost vector changes. It has been analyzed in the context of bankruptcy by Chun (1999).

Population-and-cost solidarity. For all M, N e Jf such that M ci N, all (w, e) e £M, and all (i/, e') e âN, iíu'M = w, then either SM(uf, c') ^ S(w, c) or SM(u'c')£S(u,c). Remark 1. Our six axioms are vacuously satisfied if | JV| = 1 or 'N' = 2. Con- sequently, in Sect. 3, we assume that |AT| > 3.

3 Logical relations between the axioms

In this section, we investigate the logical relations between our axioms. For completeness, we begin with population-and-cost solidarity and population solidarity.

Proposition 1. Population-and-cost solidarity implies population solidarity.

Proof. This follows from the fact that the hypotheses of population-and-cost solidarity is weaker than the hypotheses oí population solidarity and the con- clusions are the same. ■



512 Y. Chun

Now we investigate the logical relation between population-and-cost solidarity and consistency. In the context of quasi-linear social choice, efficiency and population-and-cost solidarity together imply consistency, but not vice versa.

Proposition 2. Efficiency and population-and-cost solidarity together imply consistency.

Proof. Let 5 be a solution satisfying efficiency and population-and-cost solidarity. Let N eJf, (w, e) e £N ' and x e RN be such that x = S(w, c). Let M c N, and («', c') g j2m be such that u' = {u¡)ieJá and <r' = c - J2íen'm uí +

^2i€N'M Si{u,c)'. By population-and-cost solidarity, either (i) Si(u,c)> Si(u' c') for all i e M or (ii) £(«, c) < ^(w', c') for all i e M.

If (i) holds, by summing up these 'M' inequalities, we obtain

leAf ieM

On the other hand, by efficiency } max

(' ^«,-c) - £ S,-(«,c) /eAT y ieN'M

and

X)*(«/,c') = x;s'f(i"W'c- E "'+ E 5'(«^)iN) ieM ieM ' ieN'M ieN'M J

('max Y,ui-c+ Y, «-- E 5'("'c)M

max

(' 2«,-c) - x: Si(u,c)- ieN J ieN'M

Therefore, the inequality should hold as an equality and thus consistency holds.

A similar argument can be established for the case (ii). ■

Remark 2. The converse statement that consistency implies population- and-cost solidarity is not true in general. The utilitarian solution, studied by Moulin (1985b), satisfies consistency, but noi population solidarity. Of course, it does not satisfy population-and-cost solidarity.

Next, we investigate the relation between consistency and converse consistency. Although these two axioms are not related in general,4 efficiency, cost

4 For bargaining problems, the Nash solution satisfies consistency, but not converse consistency. On the other hand, the egalitarian solution does not satisfy consistency, but satisfies converse consistency. See, for detail, Thomson (1998).



Quasi-linear social choice 5 1 3

independence, and consistency together imply converse consistency in the context of quasi-linear social choice.5

Proposition 3. Efficiency, cost independence, and consistency together imply converse consistency.

Proof. Let S be a solution satisfying efficiency, cost independence, and consistency. To show that S satisfies converse consistency, we need to show that for all N e JÍ and all («, e) e âN , if x e c.con(u, c' 5), then x = S(u, c).

Let x = S(u,c). Suppose, by way of contradiction, that there exist N = {ly...,n} c/T, (u,c)e£N and yeRN such that y ̂ x and ye c.con(u, c' S). Since y e c.con(u, c; S), then for all P c= N such that 'P' = 2,

By cost independence, for all / e P,

' jtp ) Ljtp

or equivalently

Si((uj)jep>c-Yiun =y^iY,yy (*)

On the other hand, by consistency, for all / e P with |P| = 2,

^ = S,(«, c) = Si ( (uj)jeP, c - J2 uj + E 5;(M' c)* ) '

and by cost independence,

' jiP ) LjiP

Substituting the equation (*) to the above equation, we obtain

5 Converse consistency can be strengthened by imposing an additional requirement that for all iV e Jf and all (w, e) e 1N, c.con{u, c; S) # 0. In fact, such a strengthened version of converse consistency implies consistency.



514 Y. Chun

By setting i = 1 and changing P from {1, 2} to {1, n},

^7*1,2 7^1,2

Summing up these (n - 1) equations yields

Note that E/6JV J'y = (52jeN uj ~ c)™*- Also, by efficiency, ^jeN Sj(u,c) =

E/eJv "y ~ c)maX- Altogether, we conclude that

xi =yx.

Since the choice of agent 1 was arbitrary, for all i e N,

Xi = y¡,

which is a contradiction. Consequently, if x e cxon(u, c' 5), then x = S(u, c), and thus converse consistency holds. ■

We now investigate the relation between population-and-cost solidarity and agreement. As it turns out, efficiency and population-and-cost solidarity implies agreement, but not vice versa.6

Proposition 4. Efficiency and population-and-cost solidarity together imply agreement.

Proof. We use Proposition 2 which states that efficiency and population- and-cost solidarity together imply consistency.

Let S be a solution satisfying efficiency and population-and-cost solidarity. Let TV g Jf, and (w, c), (V, e') e âN be two problems satisfying the hypotheses of agreement, i.e., for some M cz N,u'M = um> Let x - 5(w, c) and c" = c- EjeN'M UJ + EjeN'M Sj(u,c)l. First, consider ((ui)ieM,c») e 1M.

By consistency, S((ui)ieMic") = xM- Next, add the other |JV'Af| agents to obtain (w',c'). By population-and-cost solidarity, either Sm(u',c') ^xm or Sm(u', c1) g xm- Therefore, agreement holds. ■

6 For bankruptcy problems, by additionally imposing the axiom of dummy, these two axioms are equivalent (Chun 1999).




Finally, we relate agreement and separability. Moulin (1987b) shows with the help of minor additional requirements the equivalence of two axioms in the context of surplus sharing, and so does Chun (1999) in the context of bankruptcy. As it turns out, these axioms are also equivalent in the context of quasi-linear social choice when cost independence is additionally required.

Proposition 5. Agreement implies separability.

Proof. Let S be a solution satisfying agreement. Let M, N e jV be such that M c N, Let (w,c), (u',c') e £N be such that uM = u'M and Y,ieMSì(uic) =

E/etfW.O- By agreement, either SM(u,c) ^ SM{u',c') or SM(u,c)£ SM{u'c'). Since E/eM^(w^) = E/eM^c/)5 we have SM(u,c) =

SM{ur, cf), and thus separability holds. ■

Proposition 6. Cost independence and separability together imply agreement.

Proof. Let S be a solution satisfying cost independence and separability. Let M, N e Jf be such that Af .c N. Let (k,c), (w',cr) g âN be such that wM =

u'M. For all leM, let ol¡ = Si(u,c) - S¡(u',cf). Note that EieAf ̂ ("»^ ~

Let « = |JV|, m = |M|, and y = - 2/gm a¿- By awf independence, m

53s,(«,c)-X;s'(«'»c'-yi)

= S *(«, c) -S *(«',«') -^y

a,- ¿-J y (Xi Yl fri ¿-J

ieM Yl nmieM fri

= 0.

By separability, for all / e M,

Si(uìc) = Si(u'ìcf-yl) = Si(u'ìc') + l-yì

which implies that either SM(u' ',c') = Sm(u,c) or Sm{u',c') ^ Sa/Íw,^). Therefore, agreement holds. ■

The logical relations obtained so far are summarized in Fig. 1 .

4 Characterization results

For quasi-linear social choice problems, various egalitarian solutions consti- tute important classes, and consequently characterized in a variety of ways. This solution assigns each agent an equal share of the surplus derived from the project over some reference utility level, but uses a different method to compute the reference utility level.



516 Y. Chun

population-and-cost solidarity

Proposition 2*/ ' '£*

consistency agreement

population solidarity

3*t 5 6t

converse consistency separability

Fig. 1 The logical relations between axioms. * indicates imposing efficiency additionally; f indicates imposing cost independence additionally

Definition. Let g : RA - * R be a function such that

(1) g(x + al) = g(x) + a for all xeRA and all a e R, (2)flf(0)=0.

Then, for all N e Jf, all i e N, and all (w, e) e £N, the egalitarian solution with the reference function g (in short, g-egalitarian solution) is defined by

«M-*^{(£.»-.)~-£i<.»>}. For the ̂-egalitarian solutions, the reference level is determined by the function g. However, depending on the function g, each solution assigns a different utility level to each agent.7

Remark 3. It is easy to check that the ̂ -egalitarian solutions satisfy the six axioms studied in Sect. 3.

Now we identify all solutions satisfyng efficiency, anonymity, utility independence, cost independence, as well as agreement. In fact, these axioms char- acterize the ̂-egalitarian solutions for |iV| > 3, assuming the population to be fixed.

7 Various subfamilies of the g-egalitarian solutions are studied in Moulin (1985b) and Chun (1986).




Theorem 1. Let 'N' > 3 be given. A solution satisfies efficiency, anonymity, utility independence, cost independence, and agreement if and only if it is a g- egalitarian solution.

Proof. It is easy to check that the ̂ -egalitarian solutions satisfy the five axioms. We now prove the converse statement that if a solution satisfies the five axioms, it is a ̂ -egalitarian solution. Let S be a solution satisfying the five axioms. Let M, N e Jf be such that M c N and N = { 1 ,...,«} where n > 3. Let (w, c), («', e') e âN be such that uM = u'M. For all i e M, let a, = Sz(w, c) -

Si{u'9c'). Now suppose that for some /, j e M, a,- ̂ ocj. Without loss of generality, we

may assume that az > a,. By agreement, we have either (i) a,- > a7 > 0 or

(ii) 0 > a, r> oLj. First, suppose that (i) holds. Let y = x(a* + a/). By cost

independence, we have:

S,(k, c) - SK«', c' - yl) = Sfa c) - SK«', c;) - 7- n

n ' n , x = a,- - - - (af

, + ay) x

>0,

whereas

Sy(«, c) - Sj(u', c' - yl) = Sj(u, c) - Sj(u', c') - y-

» y 1 n

1 n. . = ay~ñ2(a/ + a^

.

<0.

These two inequalities contradict agreement. A similar argument can be established for the case (ii).

Therefore, for all (w, c), («', e') e 1N such that uM = u'M and all i, j e M,

St(u, c) - Stiu1, c1) = Sy(«, c) - 5,(«', c'),

which implies

S,(», c) - Sj(u, c) = Siiu', c') - Sj(u', c').



518 Y. Chun

This is true if and only if, for all i, j e N, there exist a function ftj : [RA]2 -> R such that

Sf(u, c) - Sj{u, c) = fy(uh Uj).

By anonymity, there is a function / such that fy=f for all i, jeN. Altogether, for all (w, e) e 1N and all i, j e N,

St(u,c)-Sj(u,c)=f(uhuj). Note that 'N' > 3. Moreover, for all i,j, keN, and all «,-, wy, wa:, /(«/, wy) +

f(uj,Uk)+f(uk,Ui) =0. This is true if and only if there is a function8 h : RA - ► R such that for all i, y e N, and all w/5 Uj, f(ui, Uj) = A(wz) - h(uj). Define gf : RA -> R by ̂ (wz) = A(wz) - A(0).

Now we have

Si (w, c) - 52(w, c) = ff (i/i ) - g(u2

Si (w, c) - S3 (m, c) = g(ui ) - ö^(w3

Si(u,c)-Sn(u,c) = g(u') - g{un).

Summing up these (n - 1) equations, we obtain

(n-l)Si(u,c)-

By efficiency, T,jeN sj(u>c) = (EjeN uj ~ C)™X> so that this last equatio: becomes

jeN J / jeN JeN J ) J / jeN JeN J )

Dividing both sides by n, s / ' max x

Since the choice of agent 1 is arbitrary, for all / g N,

lC)=(l(Ui)+-f( n['-N J }Tn J

Finally, we investigate the properties of the reference function g. By utility independence, for all aeR, g{ut + al) = g{ui) 4- a. Also, from the definition of g, g(0) = 0. ■

For a detail, see Footnote 6 in Chun (1986).




From Propositions 5 and 6, agreement implies separability, and cost independence and separability together imply agreement. Consequently, in Theo- rem 1, we can replace agreement by separability.

Corollary. Let 'N' > 3 be given. A solution satisfies efficiency, anonymity, utility independence, cost independence, and separability if and only if it is a g- egalitarian solution.

Remark 4. Although the ̂-egalitarian solutions are characterized in Theorem 1 and its Corollary, these results are somewhat different from the results of Moulin (1985b) and Chun (1986) since the population is assumed to be fixed. Even though the function g changes across societies of different cardinalities, the resulting solution could satisfy agreement and separability. However, it would not satisfy population solidarity, consistency, converse consistency, or population-and-cost solidarity.

Remark 5. In this remark, we discuss what additional solutions would be made possible by removing one axiom at a time from the list appearing in Theorem 1 and its Corollary.

(i) Efficiency. Pick any a* e A. Then, for all N eJi, all i g TV, and all (w, c) g âN, let g : RA - ► R be a function as in the definition of

the ̂ -egalitarian solution and S}(u,c) = g(ui) +-{^2jGN uj(a*) - c(a*) -

^2jeNg{uj)}. It can easily be checked that Sl satisfes the remaining four axioms.

(ii) Anonymity. Without anonymity, all agents are not required to have the same g function. Therefore, for all i g N, let g¡ : RA - ► R be a function such that (1) flf/(* + al)=0,-(jc) + a for all xgR^ and all ocgR, and that (2) 0,(0) = 0. Now, for all N g JT, all i g N, and all (w, c) g âN , let Sf(u, c) =

gi{ui)^wMY,jeN UJ - c)maX - ¿2jeN9j(uj)}. It can easily be checked that

'2 satisfes the remaining four axioms.

(Hi) Utility independence. Dropping utility independence allows some flexibility on the choice of the function g. For example, for all i g N, let g : RA -» R be defined by g(u¿) = ^2aeA Ui(a). Then, the resulting solution satisfies the remaining four axioms.

(iv) Cost independence. For all N g Jf, all / g N, and all (w, e) e âN , let ui = minui(a), üt ■ = maxima), u = (wz)/e^, and ü = {ü¡)ieN. If u # w, then let S3(u, c) be the outcome on the line passing through u and ü satisfying Eye^5/(w^) = (EjeN UJ ' CT™> otherwise, let S?(w,c) = w,+

AŒtjeN UJ ~ c)maX ~ EjeNUj}- lt can easily ̂ checked that S3 satisfies

the remaining four axioms.



520 Y. Chun

(v) Agreement or separability. Since the remaining four axioms are minor, most solutions satisfy the four axioms, including the utilitarian and the equal allocation of non-separable cost solutions characterized in Moulin (1985b).

Chun (1986) in his Theorem 3 shows that a solution satisfies efficiency, anonymity, utility independence, cost independence, population solidarity, and consistency if and only if it is a ^-egalitarian solution. In Propositions 1 and 2, we show that population-and-cost solidarity implies population solidarity, and efficiency and population-and-cost solidarity together imply consistency. Moreover, it is easy to check that the g-egalitarian solutions satisfy population-and-cost solidarity. Altogether, we obtain the following result.

Proposition 7. A solution satisfies efficiency, anonymity, utility independence, cost independence, and population-and-cost solidarity if and only if it is a g- egalitarian solution.

5 Concluding remarks

As pointed out by Moulin (1987a) and Thomson (1999), it is an interesting open question whether a characterization can be established by imposing the replacement principle, which requires that changes in the preferences of some members of the society should affect the agents whose preferences have not changed in the same direction only when the cost vector is unaffected.

In fact, Moulin (1987a) establishes a related characterization of the egalitarian solution in the context of binary social choice: a society must choose one out of two costless public projects; money is available to perform side payments; each agent has quasi-linear preferences. His characterization is based on the following axioms: efficiency, continuity,9 fair ranking,10 and replacement principle. However, this result can not be applied to quasi-linear social choice problems (with more than two public projects) since fair ranking cannot be generalized.

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Chun Y (1989) Monotonicity and independence axioms for quasi-linear social choice problems. Seoul J Econ 2: 225-244

9 A small change in the problem should not cause a large change in the solution outcome. 10 An agent with a larger incremental utility from the project should receive a greater share of the surplus.




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agreement, separability, and other axioms for quasi-linear social choice problems

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