Bargaining with Claims in Economic Environments [original]

Revista Española de Economía, Vol. 15, nº 1, 1998

3-13

Bargaining with Claims in Economic

Environments

Carmen Herrero

Universitat d’Alacant. Departament de Fonaments de l’Anàlisi Econòmica.

Recibido:

enero de 1997

Aceptado:

septiembre de 1997

Abstract

In this paper a reconstruction of the theory of bargaining with claims in economic environ-

ments is addressed. The spirit of that reconstruction is similar to that of the standard bar-

gaining theory made by Roemer.

Key words:

economic environments; bargaining with claims.

Resumen.

Negociación con derechos en entornos económicos

En este trabajo se aborda el problema de la reconstrucción de la teoría de negociación con

derechos en entornos económicos. El espíritu de la reconstrucción y las técnicas son seme-

jantes a las de Roemer para el problema clásico de negociación.

Palabras clave:

entornos económicos, negociación, derechos.

1. Introduction

Chun & Thomson (1992) enriched the traditional model of axiomatic bargaining

(Nash (1950)), by adding to the disagreement point and the feasible set a third

element, the

claims point

, an unfeasible point which has to be taken into account

when proposing a solution to the conflict.

A problem of bargaining with claims is understood as the mathematical

abstraction of a physical bargaining problem in which the claims point plays the

role of

founded expectations

of the agents. Examples of such a type of situation

may be either bankruptcy-like problems, or arbitration problems in which agents'

(*) Thanks are due to Antonio Villar and María del Carmen Marco for helpful comments. My special

gratitude to John Roemer who supported wonderfully this work during my visit to the University

of California, Davis, in the winter of 1994. Financial support from the IVIE and the DGICYT,

project PB92-0342, are gratefully acknowledged.

Vol. 15, nº 1, 1998 Carmen Herrero

expectations correspond to previous agreements, environmental variables, etc.

Finding solutions to this type of problems entails providing with satisfactory

answers to the conflict at hand.

As in any arbitration scheme, two main ingredients are always present when

looking for satisfactory answers: efficiency and fairness. The traditional axiomatic

approach is particularly useful, since by means of a suitable combination of a small

group of

nice

criteria, a

unique

outcome is selected as the solution of each problem.

Despite the formal beauty of the axiomatic approach, it is not immune from

criticism. While the motivation for the bargaining with claims model is economic

(agents bargain over some kind of pie, with founded expectations on the result),

all the relevant economic information is sterilized out of the model, since it con-

centrates only on utility data. As a consequence, problems which have nothing to

do with each other in terms of their economic information (different utility func-

tions, different goods, different physical claims) collapse into the same data in

utility terms, and thus become indistinguishable. This fundamental difficulty is

particularly apparent if we take into account that individuals' perceptions of what

they consider as being

just

vary, depending upon the particular nature of the prob-

lem at hand (see Yaari & Bar-Hillel (1984)).

As in the case of bargaining theory, working on a domain of economic envi-

ronments is entirely different from using the domain of triples {S,d,c}, where

a convex, closed, comprehensive set (in

ℜ

is a point in

, and

is a point in

ℜ

outside

. Notice that some

natural

economic mechanisms are not mecha-

nisms in classical bargaining with claims theory. This is the case, for instance, for

the classical

proportional to the claims division

mechanism. It is interesting to

observe that, nevertheless, properties (or axioms) in the classical bargaining with

claims framework, are justified in terms of some underlying economic environ-

ment situation, in order to motivate their rational or their fairness content. In con-

trast, those properties are much more demanding than intuition suggests.

The said difficulty is shared by both the traditional axiomatic bargaining and

the bargaining with claims approaches. In the case of classical bargaining, Roem-

er (1988) reconstructed the traditional axiomatic bargaining theory focusing on

economic environments.

This paper explores the reconstruction of bargaining with claims theory tak-

ing economic environments as the domain of the allocation mechanism. Two

main questions arise: First, why it is considered to be an interesting exercise, and

second, whether or not the technical aspects of such an exercise are different from

that in Roemer (1988) for the classical bargaining theory.

In answer to the first question, the interest of the exercise derives from clari-

fying the limits of concentrating only on utility information in this particular case.

If the claims point is understood as indicating property rights, and the disagree-

ment point indicates that unless an agreement is reached the available resources

are thrown out, it may be that the classical bargaining with claims theory is clos-

est to its reconstruction from an economic environments perspective than is the

case for the classical bargaining theory.

Bargaining with Claims in Economic Environments Vol. 15, nº 1, 1998

As for the second question, some remarks are in order:

1. The literature on bargaining problems with claims was initiated by Chun &

Thomson (1992), meaning that it is still a novel topic. Somehow, we may

think of a bargaining with claims problem as an extension of both the classical

bargaining theory and the bankruptcy theory. This explains why there are two

different types of mechanisms analyzed so far: those adapted from the classi-

cal bargaining theory and those adapted from the bankruptcy literature.

2. The reconstruction of those mechanisms inspired in classical bargaining

mechanisms, such as the

proportional solution

(Chun & Thomson (1992)),

and the

extended-claims egalitarian solution

(Bossert (1993), Marco (1994),

(1995)) can be done by suitably adapting Roemer's techniques. Notice that

two main ingredients of these techniques are Billera & Bixby (1973) and

Howe (1987). As a consequence, we also need the commodity space to be of

unbounded dimension.

3. This is not the case for those mechanisms inspired in the bankruptcy model,

especially when

minimal concession points

are involved, as they are in Herre-

ro (1994), (1998). These solutions are out of the scope of this paper.

In Section 2 we present the set-up, as well as the main axioms used in the

aforementioned reconstruction of the theory. Section 3 presents characteriza-

tion results for a fixed number of agents. Section 4 is devoted to the correspond-

ing characterizations for a variable population. The proofs are relegated to an

Appendix.

2. The set-up

Let

be the set of all real valued monotone concave continuous functions

defined on . An

economic environment

is a tuple

= <

where

≥

1 are integer numbers, , ,

= (

,…,

), such that

for all

= 1,…,

, and

= (

,…,

), where for all

= 1,…,

The interpretation is that we deal with a problem involving

agents, each one

characterized by her utility function

, having claims on

commodities

, but

the available amounts of the commodities to be divided among them is

. If for

some

= 1,…,

, , then a conflict arises, which shall be called a conflict

of bargaining with claims. Agents may agree upon any feasible distribution of the

available commodities, otherwise they will end up with no goods at all. The ques-

tion is how to allocate w among the

individuals in the previous problem, name-

ly, to suggest an agreement.

We start by dealing with a family of problems in which

(the number of

agents) is fixed, but

(the number of goods to be distributed) varies. Let us call

ℜ+

wℜ+

∈cℜ+

∈

ciℜ+

∈uiUh

∈

cij

j1=

∑wj

Vol. 15, nº 1, 1998 Carmen Herrero

∑

the class of all admissible environments for

agents. In this section, when

we specify an economic environment , no explicit mention to the number

of agents will be made, that is, = <

If , an allocation is a vector

= (

,...,

), where . Let us

denote

(

) = [

(

),…,

(

)]. Let

= { } be the set of feasible

allocations, and the utility pos-

sibility set of .

is a closed, comprehensive, convex set in , containing

the point

(0). Convexity comes from concavity of the utility functions; com-

prehensiveness and closedness come from free disposability of the goods and

continuity of the utility functions, respectively.

allocation mechanism

is a correspondence which associates to each

economic environment , a set of feasible allocations. It will be assumed

that

induces a function in utility space, namely, if

, , then

(

) =

(

Call the induced utility function , where is the set of

bundles which

assigns to every agent. Furthermore, it is assumed that

chooses all the feasible allocations associated with a given point in the utility

space. Roemer (1988) call this type of correspondence a «full» correspondence.

Associated to any

economic environment

, we can define

a classical

bargaining with claims problem

in a natural way: If = <

u,h,w,c>, take

as the feasible set, u(0) as the disagreement point and u(c) as the claims

point. Thus, we may look at as a classical bargaining problem with claims.

Two main ingredients are necessary in order to reconstruct the classical bar-

gaining with claims theory as a theory on economic environments: (1) To

ensure that the information contained in the utility possibilities set, the utilities

of the origin and the utilities of the claims point are all that are needed, (2) To

ensure that any triple (S,d,c), where S is a convex comprehensive subset of

and comes from a suitable economic environment

Point (2) is answered affirmatively by Billera & Bixby (1973), whenever

the commodities space is of unbounded dimension. In order to fulfill point (1),

we need the following axiom:

Axiom of Welfarism (W).— Let = <u,h,w,c>, and = <u',h',w',c'> be two

economic environments in such that = , u(0) = u'(0) and

u(c) = u'(c'). Then, .

Axiom W says that an allocation mechanism F must treat identically (in terms

of utility) any two economic environments which have the same utility possibili-

ties set and in which the utility levels of the origin and of the claims point are

identical for all individuals. W is extremely strong, since it has to hold even in cir-

cumstances in which the dimension of the goods space is different. The welfarist

axiom is thus named because it requires the allocation mechanism to ignore all

ξΣ

∈

ξΣ

∈x

iℜ

∈

ξ()

∑

≤

Aξ() u

1…u

,,()ℜ

∃

ξ()∈∈ u

()

≥,{}

ξξ() ℜ

ξΣ

∈yFξ()∈

Fξ() uFξ()()=Fξ()

ξΣ

∈ξ

Aξ() ξ

ℜ

ndS∈, cℜ∈n

cS∉,

ξΣ

∈

ξξ'

Aξ()

Aξ'()

Fξ() µ

Fξ'()=

Bargaining with Claims in Economic Environments Vol. 15, nº 1, 1998

information about an environment which is not summarized both in the utility

possibilities set and in the utilities of the disagreement and of the claims point.

In order to present the next axiom we need the concept of

personal good

Let be an economic environment, = <

∈

{1,2,…,

We shall say that good jth is a

personal good

for agent ith if for any

∈

≠

for any , and for any , if stands for a vector in having all

components equal to zero but the jth component, equal to

, we get that

(

) = .

Axiom of Consistency (CON)

. Let , = <

,h+h',(w,w'),(c,c')> be an envi-

ronment such that , where each of the y-goods (k = h+1,…h+h') is

a personal good for at most one of the agents. Define v by vi(xi) = ui.

Consider now the environment = <v,h,w,c>. If , and ,

v(0) = u(0,0), v(c) = u(c,c'), then .

CON asks for a particular type of consistency in the allocation proce-

dure. First, the mechanism allocates those commodities which are only liked

by a particular agent (personal goods). Next, the remaining commodities are

allocated. The mechanism satisfies this property whenever the allocation is

immune to this two-step procedure. The interest of the axiom CON is that it

will imply the axiom of welfarism, even though it is an axiom relying on the

economic data of the problem.

The main result of this section is the next lemma, which is a straightforward

adaptation of Lemma 5 in Roemer (1988):

Lemma 1. CON and fullness of F imply W.

3. Bargaining with Claims and economic information with a fixed

number of agents

For any , = <u,h,w,c>, consider the sets of efficient and weakly effi-

cient allocations: PO = , WPO =

= .

Now, consider the following axioms:

Axiom of Efficiency (PO). .

Axiom of Weak Efficiency (WPO). .

Axiom of Symmetry (SY). Let , = <u,h,w,c> be such that ∀i,j ∈ N, ui = uj, ,

ci = cj,. Then, (w/n,…,w/n) ∈.

PO and WPO ask the solution allocations for different degrees of effi-

ciency. SY establishes that whenever we face identical individuals, then the

equal division of the available resources is a solution allocation.

Let Λn be the class of transformations λ:

ℜ

n →

ℜ

n such that for any i ∈ N,

there exist a ai ∈

ℜ

++, bi ∈

ℜ

, such that for all x ∈

ℜ

n , λi(x) = aixi +bi. Let Tn be

the subclass of Λn with bi = 0 for all i ∈ N.

ξΣ

∈ξ

xℜ

∈α0≥α

jℜ

xα

+()

ξΣ

∈ξ

xy,()Fξ()∈ x

,()

ξ'

ξ'Σ

∈

Aξ() Aξ'()=

xFξ'()∈

ξΣ

∈ξ ξ() xZξ()∈

()

()>

ξ()∉⇒{}ξ()

xZξ()∈

()

ξ()∉⇒{}

ξ∀Σ

∈

ξ(), PO

ξ()⊂

ξ∀Σ

∈

ξ(), WPO

ξ()⊂

ξΣ

∈ξ

Fξ()

Vol. 15, nº 1, 1998 Carmen Herrero

Consider now the following axioms:

Axiom of Cardinality and Non Comparability (CNC).

Let , .

= <

u,h,w,c

>, = <

u',h,w,c

>, with

. Then, = .

Axiom of Ordinal Level Comparability (OLC).

Let , , = <

u,h,w,c

= <

u',h,w,c

>, with

' =

∈

. Then, = .

CNC and OLC deal with two problems in which all the physical data are

identical and the only difference relies in the utilities. If these utilities are

related by allowed transformations, then the solutions must be identical. In

CNC the allowed transformations belong to

, and therefore we are confined

to a universe in which utilities are cardinal and no interpersonal comparability

can be performed. Under OLC, the transformations belong to

, and in con-

sequence, interpersonal comparability is possible, and utilities are ordinal.

These axioms play the role of Scale and Translation Invariance, respectively.

Axiom of Boundedness (BDD)

. For all , = <

u,h,w,c

(0)

≤

(

The previous axiom has the same meaning as in traditional bargaining

problems with claims. BDD requires that at any solution allocation no agent

is either better off than he is at the claims point or worse off than he is at the

disagreement point.

The axioms of consistency we present below are two different ways of

strengthening CON.

Axiom of Strong Consistency (CON*).

Let , = <

u,h+h',

(w,w'),(c,c')> be

an environment such that , where each of the

-goods

(

+1,…,

') is a personal good for at most one of the agents. Define

(

) =

(

, ). Consider now the environment = <

u,h,w,c

>. If ,

and

(0) =

(0,0),

(

) =

(

'), then .

Axiom of Rational Strong Consistency (RCON*).

Let ,

= <

w,w'),(c,c')> be an environment such that , where

each of the y-goods (k=h+1,…,h+h') is a personal good for at most one of the

agents. Define v by vi (x

) = ui(xi, ). Consider now the environment

= <v,h,w,c>. If , and v(0) ≥ u(0,0), v(c) = u(c,c'), u ≥ v(0), then

The axiom below is the version of the monotonicity axiom in economic envi-

ronments:

Axiom of Resource Monotonicity (R.MON). Let , , = <u,h,w,c>,

= <u,h,w',c>, with w' ≥ w. Then ≥ .

We can now define the Proportional Mechanism P in economic environ-

ments: For any , = <u,h,w,c>, P = { u(x) = λu(c)+(1–

λ)u(0), 0 ≤ λ ≤ 1, x ∈ WPO }.

ξξ'Σ

∈

ξξ'

λΛ

∈

Fξ()

Fξ'()

ξξ'Σ

∈ξ

ξ'Fξ() Fξ'()

ξΣ

∈ξ

ξ()

ξ'Σ

∈ξ

xy,()Fξ()∈

iξ'ξ'Σ

∈

xFξ'()∈

ξΣ

∈

ξxy,()Fξ()∈

ξ'ξ'Σ

∈xy,()

xFξ'()∈

ξξ'Σ

∈ξ

ξ'µ

ξ'() µ

Fξ()

ξΣ

∈ξ ξ()

xZξ()∈

ξ()

Bargaining with Claims in Economic Environments Vol. 15, nº 1, 1998 9

Similarly, we may consider the Claim-Egalitarian Mechanism E, as follows:

For any , = <u,h,w,c>, E = { ui(ci)–ui(xi) = uj(cj)–uj(xj),

i,j = 1,…,n, x ∈WPO}. Now, the Extended Claim-Egalitarian Mechanism EE, is

defined in the following way: For any , = <u,h,w,c>,

EE = { if for some i,j ∈N, ui(ci)–ui(xi) > uj(cj)–uj(xj), thus

uj(xj) = 0, x ∈WPO}.

By means of previous axioms the following characterization results are

obtained [see Appendix]:

Theorem 1. The proportional mechanism is the unique full mechanism in

∑

n satisfying Weak Efficiency, Consistency, Resource Monotonicity, Symme-

try and Cardinality and Non Comparability.

Theorem 2. The extended claim-egalitarian mechanism is the unique full

mechanism in

∑

n satisfying Weak Efficiency, Ordinal Level Comparability,

Strong Consistency, Resource Monotonicity, Symmetry and Boundedness.

4. Bargaining with Claims in economic environments with variable

population

Let us consider now an infinite set of potential agents I = {1,2,…}, such that only

a finite group of them are present in every concrete problem. Let M be the class of

finite subsets of I. For a given M ∈M, denotes the cartesian product of |M|

copies of indexed by the elements of M. Let be the class of economic envi-

ronments for |M| agents indexed by the elements of M. Let From

now on, an economic environment is a tuple = <M,u,h,w,c>, where

M∈M, and the environment <u,h,w,c> ∈ , where m = |M|.

An allocation mechanism F is a full correspondence defined on associating

to any economic environment a set of feasible allocations.

Consider now the following axiom:

Axiom of Anonymity (AN). For all M, M', with |M|=|M'|, for any ∈, =

<u,h,w,c> and for any one to one mapping γ:M →M', if ∈, is such that

= <γ(u),h,w,γ(c)>, then iff γ(x) ∈.

Axiom of Population Monotonicity (POP.MON). For any M,N ∈M, for any

=<M,u,h,w,c> ∈, =<N,u',h',w,c'> ∈, if M⊂N, uM

'=u, cM

'=c,

then ≥.

Axiom of Continuity (CONT). For all N, for all sequence { }, ∈,

= <ut,h,wt,ct>, = <u,h,w,c>∈, if ut →u in the pointwise convergence

topology, wt→w, ct→c in the Hausdorff topology, then → in

the pointwise convergence topology.

ξΣ

∈ξ ξ() xZξ()∈

ξΣ

∈ξ

ξ() xZξ()∈

ℜ

ℜΣ

Σ=Σ

MM∈

∪

ξΣ∈

ξΣ

ξ'xFξ()∈

Fξ'()

ξΣ

ξ'Σ

ξ'() µ

Fξ()

tξΣ

()

ξ()

10 Vol. 15, nº 1, 1998 Carmen Herrero

Now, the following characterization results are obtained [see Appendix]:

Theorem 3. The proportional mechanism is the unique full mechanism in

∑

satisfying Weak Efficiency, Boundedness, Continuity, Anonymity, Cardi-

nality and non Comparability, Strong Consistency and Population Monoto-

nicity.

Theorem 4. The extended claim-egalitarian mechanism is the unique full

mechanism in

∑

satisfying Weak Efficiency, Boundedness, Continuity, Sym-

metry, Ordinal Level Comparability, Rational Strong Consistency and Pop-

ulation Monotonicity.

Appendix: Proofs of the Theorems

We shall prove that the right combination of axioms on economic environments

imply the corresponding counterparts in utility terms. Then, by using the charac-

terizations of the proportional and the extended claim-egalitarian solutions in the

classical setting, we obtain our characterization results.

For the sake of completeness, we include the definitions of the axioms in util-

ity terms as well as the characterization theorems in the classical setting. For an

environment = <u,h,w,c>, and a permutation π of the set of agents, call

π = <π (u),h,w,π(c)>.

U-Symmetry (U-SY). Let ∈ such that for any permutation π:N →N,

A( ) = A[π( )]. Then, [ ]i = [ ]j , ∀ i,j ∈N.

U-Anonymity (U-AN). For all M, M', with |M| = |M'|, for any ∈, ∈

=<u,h,w,c>, =<u',h',w',u'> and for any one to one mapping γ:M →M',

if A( ) = γ[A( )], u'(0) = γ[u(0)], u'(c') = γ[u(c)], then = γ[].

Scale Invariance (SC.INV). Let , ∈. If there exist such that

z ∈ Z iff , then, = λ[].

Translation Invariance (T.INV). Let , ∈. If there exist such that

z ∈ Z iff , then, = λ[].

Strong Monotonicity (ST.MON). If = <u,h,w,c>, = <u',h',w',c'> ∈ are

such that u(0) = u'(0), u(c) = u'(c'), A() ⊂ A( ), then ≥ .

For = <u,h,w,c>, call K( ) = {z ∈WPO( )| u(0)≤ u(z)≤ u(c)}.

Independence of Pareto irrelevant, non-individually rational and unclaimed alter-

natives (IND). If , ∈ are such that A[K( )] = A[K( )], then

= .

U-Continuity (U-CONT). For all N, for all sequence { }, ∈, =<ut,h,wt,ct>,

= <u,h,w,c>∈, if A() → A( ), \ ut(0)→u(0) and u(ct)→u(c), then

→ where the convergences are taken in the Hausdorff topology.

For = <u,h,w,c>∈, call IR( ) ={z∈Z()| u(z) ≥ u(0)}.

ξ()

ξΣ

ξξ µ

ξ() µ

Fz()

ξΣ

ξ'

ξξ'

ξ'ξµ

ξ'()

Fξ()

ξξ'Σ

nλΛ

∈

ξ() λzZξ'()∈ µ

Fξ'()

Fξ()

ξξ'Σ

nλT

∈

ξ() λzZξ'()∈ µ

Fξ'()

Fξ()

ξξ'Σ

ξ'

ξ'()

Fξ()

ξξξ

ξξ'Σ

nξξ'

ξ() µ

Fξ'()

tξ

tΣ

nξ

ξΣ

() µ

ξ()

ξΣ

ξξ

Bargaining with Claims in Economic Environments Vol. 15, nº 1, 1998 11

Rational Contraction Independence other than Claims point (RCIC). For any ,

∈, if A(IR( )) ⊂ A(IR( )), u(c) = u'(c'), ∈A(IR( ), then =

Fixed Claims Contraction Independence (FCCI). For any , ∈, if

u(0) = u'(0), u(c) = u'(c'), A() ⊂ A( ), ∈A( ), then = .

Let us consider a set of agents, N, and a subset of it, M. If x ∈ℜN, let xM

be the projection of x onto ℜM, and given (S,d) such that S ⊂ ℜN, d ∈S, let

(S) = { x ∈ℜM | (x,dN/M ) ∈S}.

U-Population Monotonicity (U-POP.MON). For all M,N, such that M ⊂ N, for all

=<n,u,h,w,c> ∈, = <m,u',h',w',c'> ∈, if uM(0) =u'(0), uM(c) =

u'(c'), and [A()]= A( ), then, for all i ∈M, ui[F( )] ≤ u'i[F( )].

Chun & Thomson (1992), theorem 1, characterize the proportional solution in

utility terms by means of WPO, U-SY, SC.INV. and ST.MON.

Lemma 1. CON and fullness of F imply W.

Proof: Let , ∈, = <u,h,w,c>, =<u',h',w',c'> such that

A( ) = A( ), u(0) = u'(0), u(c) = u'(c'). It must be proven that = .

As in Roemer (1988, lemma 5), construct * = ⊗, * =

<v,h+h',(w,w'),(c,c')>, where vi(xi,yi) = min {ui(xi), ui'(yi)}. By Billera &

Bixby (1973), A( *) = A( ) = A( ); v(0) = u(0) = u'(0). Furthermore,

v(c,c') = u(c) = u'(c').

It has to be proven that = and = . The previous

result is obtained as in Roemer (1988, lemma 5), by means of a series of interme-

diate problems, suitably modified by adding the appropriate claims point to each

one, in order to maintain not only the equality of the utilities possibilities sets, but

also the utilities of the claims points.■

Lemma 2. Fullness, R.MON and CON imply ST.MON.

Proof: Let = <u,h,w,c>, =<u',h',w',c'> such that u(0) = u'(0), u(c) = u'(c'),

and A() ⊇ A( ). We have to show that ≥ . Act as in Roemer

(1988, lemma 6), constructing the auxiliary problems, suitably modified by add-

ing the claims point in such a way that the utility of the claims points is always

preserved.■

In an immediate way we obtain:

Lemma 3. Fullness, W and SY imply U-SY.

Lemma 4. Fullness, CNC and W imply SC.INV.

Theorem 1 follows from Lemmas 1, 2, 3, 4, and theorem 1 in Chun & Thomson

(1992).

Bossert (1993), theorem 1, characterizes the extended claim-egalitarian solu-

tion by means of WPO, U-SY, T.INV., ST.MON, BDD and IND.

ξ'Σnξξ'

ξ'()

ξµ

ξ()

Fξ'()

ξξ'Σ

ξξ'

ξ'()

ξµ

ξ()

Fξ'()

ξΣ

ξ'Σ

u'0() ξξ'ξξ'

ξξ'Σ

nξξ'

ξξ'µ

ξ()

Fξ'()

ξξξ'ξ

ξξξ'

() µ

ξ() µ

Fξ

() µ

ξ'()

ξξ'

ξ()

ξ'()

12 Vol. 15, nº 1, 1998 Carmen Herrero

Lemma 5. CON* and fullness imply IND and FCCI.

Proof: Let us prove that IND follows. The other implication goes similarly.

1. Let = <u,h,w,c>, = <u',h',w',c'> such that u(c) = u'(c'), and

K() ⊆ K( ). Let x' ∈ F( ) such that u'(x') ∈ K( ). To prove:

= .

2. Construct * = ⊗ . * = <v,h+h',(w,w'),(c,c')>, where vi(xi,yi) =

min {ui(xi), ui'(yi)}. K( *) = K()∩K( ) = K( ), and v(c,c') = u(c) = u'(c').

3. Consider now = < ,h+h',(w,w'),(c,0)>, where (xi,yi) = ui(xi), i = 1,...,n.

Obviously, K( ) = K( ), u(c) = (c,0).

4. Notice that (xi,yi) ≥vi(xi,yi) for all i = 1,…,n. Now by Howe (1987,

Prop. 3), it is possible to find functions Ui: →, such that

Ui(xi,yi,1) = (xi,yi), Ui(xi,yi,0) = vi(xi,yi). Now, construct the functions

Vi: →, Vi(x,y,z) = Ui(x,y,zi), for z ∈, i = 1,…,n.

5. Consider now the environment: ζ = <V,h+h'+n,(w,w',1,…,1),(c,0,f)>,

where fi = ei, the ith vector of the canonical basis in . Thus, K(ζ) = K() =

K( ). Moreover, Vi (ci ,0,ei) = Ui(ci ,0,1) = (ci ,0) = ci(ci,ci').

6. Construct the environment * = <V,h+h'+n,(w,w',0);(c,c',0)>. Thus, K=

K( ). Moreover, V(ci ,ci',0) = Ui(ci, ci',0) = vi (ci ,ci').

7. From previous constructions, A( ) = A(ζ). Consider an allocation in F(ζ),

zi = (xi,yi,ei). Then, there must be an allocation ti = (x'i,y'i,0) in Z( ), induc-

ing the same utility point, i.e., V(zi) = V(ti). Thus, since ti is feasible in ζ, and

because of fullness of F, t ∈F(ζ).

Now, by CON*, (x'i,y'i)i=1,…,n∈F( *), obtaining that ( *)= (ζ). And

therefore ( *) = ( ).■

Lemma 6.- OLC, W and fullness imply T.INV.

Theorem 2 follows from lemmas 1, 2, 3, 5, 6 and theorem 1 in Bossert (1993).

Chun & Thomson (1992), theorem 6, characterize the proportional solution

for a variable population by means of WPO, BDD, U-CONT, U-AN, SC.INV,

FCCI and U-POP.MON. Marco (1995), theorem 1 characterizes the extended

claim-egalitarian solution for a variable population by means of WPO, U-SY,

T.INV., U-CONT., BDD, RCIC and U-POP.MON.

In a similar way to previous lemmas we obtain now:

Lemma 7. Fullness of F, W and POP.MON imply U-POP.MON.

Lemma 8. Fullness of F, W and RCON* imply RCIC.

Lemma 9. Fullness of F, W and CONT. imply U-CONT.

ξξ'

ξξ'ξ'ξ

ξ() µ

Fξ'()

ξξξ'ξ

u u

ξξ u

ℜ

hh'1++

ℜ

hh'n++ ℜℜ

ℜ

ξu

ξξ

()

ξµ

Bargaining with Claims in Economic Environments Vol. 15, nº 1, 1998 13

Lemma 10. Fullness of F, AN and W imply U-AN.

Theorem 3 follows from lemmas 1, 4, 5, 7, 9, 10 and theorem 6 in Chun & Thom-

son (1992). Theorem 4 follows from lemmas 1, 3, 6, 7, 8, 9 and theorem 1 in

Marco (1995).

References

BILLERA, L.; BIXBY, R. (1973). «A Characterization of Pareto Surfaces». Proc. Amer.

Math. Soc., 41: 261-267.

BOSSERT, W. (1993). «An Alternative Solution to Bargaining Problem with Claims». Math-

ematical Social Sciences, 25: 205-220.

CHUN, Y.; THOMSON, W. (1992). «Bargaining Problems with Claims». Mathematical So-

cial Sciences, 24: 19-33.

HERRERO, C. (1994). «Bargaining with Reference Points-Bargaining with Claims: Egalitar-

ian Solutions Reexamined». A Discusión, WPAD 94-15.

— (1998). «Endogenous Reference Points and the Adjusted Proportional Solution in Bar-

gaining Problems with Claims». Social Choice and Welfare, 15: 113-119.

HOWE, R. (1987). «Sections and Extensions of Concave Functions». Journal of Mathemat-

ical Economics, 16: 53-64.

MARCO, M.C. (1994). «A Characterization of the Extended Claim Egalitarian Solution».

Economics Letters, 45: 41-46.

— (1995). «Efficient Solutions for Bargaining problems with Claims». Math. Soc. Sciences,

30: 57-69.

NASH, J.F. (1950). «The Bargaining Problem». Econometrica, 18: 155-162.

ROEMER, J.E. (1988). «Axiomatic Bargaining Theory on Economic Environments». Jour-

nal of Economic Theory, 45: 1-30.

YAARI, M.E.; BAR-HILLEL, M. (1984). «On Dividing Justly». Social Choice and Welfare,

1: 1-24.