Surplus-sharing methods

Mathematical

Social Sciences

21 (1991) 287-301

287

North-Holland

SURPLUS-SHARING Andreas

METHODS

PFINGSTEN

Fachbereich 5, VWL, Universittit Gesamthochschule Siegen, Htilderlinstrasse 3, W-5900 Siegen, Germany Communicated Received

by H. Moulin

10 November

Revised

27 February

In a recent were jointly

1990 1991

contribution characterized

which is a value judgement The present others,

study

sharing

equal

axioms.

of a class of sharing sharing

sharing;

absolute

sharing

One essential of dropping methods

as polar

proportional

and proportional

requirement

of real (as opposed

the consequences

and proportional

Key words: Equal absolute

(1987),

as soon as a notion

examines

two characterizations

absolute

by Moulin by appealing

to nominal)

investments

the homogeneity

are provided.

axiom.

This class contains

(and in fact as the only homogeneous) sharing;

homogeneity;

sharing

was homogeneity, exists. Among equal cases.

surplus.

1. Motivation Quite frequently in real world economics, situations of the following type are found. A number of agents have invested in a common project, and since it was successful they now must distribute the surplus among themselves. The question arises how this should be done. An analysis of such a problem can follow very different tracks. One is in terms of incentives, i.e. the focus is on situations when all agents decide to take part in the project as opposed to conditions such that some agents are not willing to engage. In this regard there exist some connections to game theory, in particular to bargaining models. (For references to work on the behavioural aspects of the topic see, for example, Giith, 1987.) Our approach takes the decision to invest in the project, as well as its surplus, as given and asks for a method to distribute the gains. We suggest properties, axioms, that a reasonable mechanism should have, and then derive all methods with certain (combinations of) properties. The main starting point of our paper is an article by Moulin (1987) to whom we owe a great deal of inspiration. Some of the definitions and results could be traced back to other work by Moulin and other authors, e.g. Young (1988). We will mainly refer to Moulin’s contribution, however, since this is probably the most accessible collection, and it also provides suitable references to the related literature. 0165-4896/91/$03.50

0

1991-Elsevier

Science

Publishers

B.V. All rights

reserved

In his paper, Moulin has jointly characterized equal absolute sharing and proportional sharing. He has also derived several parametric (in the sense of Young, 1988) classes of sharing methods that contain these mechanisms as polar cases. Throughout his analysis he requires the homogeneity axiom, i.e. if all investments and the surplus change by the same factor, then the individual dividends must also change by this factor. We shall deviate from Moulin’s approach by dropping the homogeneity axiom: Moulin’s interpretation, ‘whether we count in dollars or cents should make no difference’ (Moulin, 1987, p. 165), captures only the case when investments are in nominal terms. But if there exists a notion of investments in real terms, then this property is a value judgement and other views are, of course, possible. (A nominal investment of US$lOOOO represents very different real values in the United States and in a developing country, respectively.) There do indeed exist real surplus-sharing methods violating the homogeneity axiom. One brief example may suffice to indicate a case where this is so. Consider the salary distribution in German soccer teams. There are usually three main components: a basic monthly salary depending on the player’s ‘market value’, per capita premiums for victories, and an amount per ticket sold. When a team becomes more successful the opportunity costs of the players (their investments) increase. Suppose all investments increase proportionally. Yet there is no guarantee that total salaries increase proportionally as well. For instance, commonly observed changes in the basic monthly salary, keeping the other components equal, modify the share of surplus distributed according to opportunity cost versus the share of the surplus distributed per capita. The methods we are going to characterize form a parametric class and have equal absolute sharing and proportional sharing as polar cases. Except for these ‘extremes’, none of the mechanisms is homogeneous. Instead, they have the interesting property that the share of the surplus that is distributed proportionally with respect to investments increases with the size (total investments) of the project. After an outline of Moulin’s model (Section 2) we present our main result (Section 3). Most of the technical parts of the proofs are relegated to an appendix which follows the brief summarizing Section 4.

2. Moulin’s

model

There is a fixed number n (n 2 3) of agents, and the set of agents is N= { 1, . . . , n}. Agent i invests an amount ui (u;> 0) and receives (u, + d,), where dj? 0. The surplus to be divided in total is s (~20). Notice that investments and surplus are treated as being given. This assumption is perfectly alright in the ex post analysis performed here, but could (and probably should) be modified in game-theoretic settings where investments are not given but decided upon.

A. Pfingsten / Surplus-sharing methods

289

Denoting the non-negative (positive) reals by R, (R, +) and using d;(u, . . . , u,, s) for agent i’s dividend, we can rewrite Moulin’s basic definition of a (homogeneous) sharing

method

(henceforth

a HSM).

Definition. A homogeneous sharing method (HSM) is a mapping d from iR: + x R, into R: satisfying the following five properties (TOT) through (MOS) for all i E N.

(TOT)

Total

distribution

of the surplus:

Edj(U1,...,u,,s)=s

j=l

for all UE R:+ and all SE R,. (HOM)

(2.1)

Homogeneity: d;(~.u,~.s)=~.d,(u,s) for all UER:+,

(ANO)

and all AEIR++.

(2.2)

Anonymity: For all UER:+, u;=u~

(MOI)

all SER,,

Monotonicity

implies w.r.t.

all SEE,,

and alljEN:

d;(u,s)=dj(u,s). investments:

d;(u ],...,u;+~,...,u,,s)~~~(u,,...,u~,...,u,,s) for all u~lR:+, (MOS)

Monotonicity

w.r.t.

for all u~fR:+,

all SER,,

and all AER++.

(2.3)

and all IER++.

(2.4)

surplus:

all SER,,

Most of these properties are quite straightforward: (TOT) is simply a type of budget balance constraint; (ANO) requires equal treatment of equals; (MOI) is important from an incentive point of view (without (MOI) mechanisms might be admissible that decrease somebody’s dividend although his investment has increased); and (MOS) finally requires that nobody is worse off if the total surplus increases. The homogeneity property (HOM), which also could be called scale invariance, is not quite that indisputable (see above). Apart from the five properties required in the above definition, Moulin also suggests four further axioms:

290

(SEP)

A. Pfingsten / Surplus-sharing methods

Separability: For every proper S,S’E IR,:

T, 0# TCNf

coalition

ui = U: for all i E T and

c

T, all U, U’E IR: +, and all

di(U, S) = c d,(u’, s’) implies

i6T

ic T

for all in T.

di(u,s)=d,(u’,s’)

The property (SEP) requires that dividends depend only on investments within a coalition and the share of the surplus rewarded to this coalition. (This is some subgroup consistency property.) (NAR)

No advantageous

reallocation:

For every proper SE IR,:

T, 0# TC N# T, all U, U’E k?:+,

coalition

and all

iFT ui = iFT u,! and Uj = u,! for all j E N \ T implies

The axiom (NAR) requires that transfers of investments within a coalition do not change the coalition’s share of the surplus, i.e. the distribution of investments within a coalition does not matter for the coalition’s surplus share. The property is important since it excludes the possibility that a coalition increases its surplus share by misreporting the individual contributions to the coalition’s total investments. Notice that, by assuming a fixed number of agents, we cannot examine the role of ‘dummy’ agents. (ADD)

Additivity: For all ~EN, all UE!?:,,

and all .s,s’EIR+:

d,(u,s+s’)=di(u,S)+dj(U,S’). Additivity requires that two (parallel) projects vestments are the basis dividends are calculated ly suited when dealing (PID)

Path

(2.5)

it does not matter whether or not one project is split into such that, on the one hand, in both projects the same infor distributing the surplus, but, on the other hand, the separately. This convenient property seems to be particularwith joint production.

independence: For all iEN,

all u~lR:+,

and all s,s’EIR+:

di(u,s+s’)=di(u,s)+di(u+d(u,s),S’).

(2.6)

291

A. Pfingsten / Surplus-sharing methods

The axiom

(PID)

requires

that it does not matter

whether

one project

is split into

two (subsequent) projects with the dividends for the second project calculated on the basis of initial investments plus dividends on the first project. This property of dynamic consistency seems to be useful, for example, when analyzing the potential withdrawal of profits (with other agents joining in to fill the capital gap). For some of Moulin’s results an additional quires that rates of return are nonincreasing. (PRO)

condition

is used. This condition

re-

Progressivity: For all i,j~N, Uil

Uj

implies

all UE L?:,,

and all SE IR,:

di(U,S)/Ui2dj(U,S)/Uj.

This is also an interesting property to link surplus sharing, inequality, and income taxation - a line of research not pursued in this paper. Moulin’s main result is that a homogeneous sharing method satisfies any three of the properties (SEP), (NAR), (ADD), and (PID) if and only if it is either proportional sharing, i.e.

@(u,s):=s~ ui

I

i

uj

j=l

for all UE Ry+, all SE IR,, and all ieN, or equal

absolute

sharing,

(2.7)

i.e.

d,~(u, s) := s/n for all u~lRy+,

3. Characterizations

all SEIR,,

of nonhomogeneous

and all iEN.

(2.8)

sharing methods

As stated earlier, homogeneity (HOM) is a value judgement as soon as investments as well as the surplus are measured in real terms. Having in mind the problem of appropriately defining investment and reward standards (see Guth, 1987), some might feel particularly unhappy: with homogeneity, nominal investments provide basically the same basis for distributing the surplus as real investments, no matter whether people think this is fair or not. In inequality and tax progressivity measurement the importance of distinguishing between real incomes and nominal incomes has gained some recognition in Pfingsten (1986) and the subsequent literature. There the concepts of relative and absolute inequality have been generalized to intermediate inequality (see, for example, Bossert and Pfingsten, 1990). Similarly, we are going to demonstrate here

292

A. Pfingsten / Surplus-sharing methods

that, by dropping (HOM), Moulin’s homogeneous sharing methods can be further generalized to allow for methods other than equal or proportional sharing even if all of the properties (SEP), (NAR), (ADD), (PID), and (PRO) are required. As indicated, we deviate from Moulin’s approach by not assuming homogeneity and require from a sharing method (SM) only the four properties (TOT), (ANO), (MOI), and (MOS) for all ieN. We can now state our main result. It is the characterization of a parametric class of sharing

methods

by two (equivalent)

sets of axioms:

Theorem. The following three statements are equivalent: (a) The function d: IR: + x R, --+R: satisfies (TOT), (ANO), (ADD), and (PID). (b) The function d: IR: + x R, + R: satisfies (TOT), (ANO),

(MOS),

(NAR),

(MOS),

(NAR),

(ADD), and (SEP). (c) Either

d;(u, s) = d,!‘(u,s) for all i E N, for all u E I?: +, and all s E R,, or d;(u,s)=dr(u,s)

for all ieN, for all UE:R:+, and all .sER+,

or there exists a constant A E II?,+ such that d;(u,s)=d;(u,s):=s+/(A+jy)

+s* [ 14, for all ieN,

uj/(A+j;,u#

for all UER:+,

(3.1)

and all se&?+.

This is all proved in the appendix. The methods in equation (3.1) are quite intuitive: part of the surplus is distributed on an investment and part on a per capita basis. This is evident from the following expression:

(3.2)

For 13-+ 0 and L + 03 the methods (3.2) approach proportional sharing and equal absolute sharing, respectively. From equation (3.2) two appealing observations are obvious. (1) Given a fixed positive value of the parameter A, an increase in the size of the project (sum of all investments uj) increases the share of the surplus that is distributed proportionally.

A. Pfingsten / Surplus-sharing methods

293

(2) Given a fixed size of the project (sum of all investments uj), the share of the surplus that is distributed proportionally is decreasing in A. These comments further enlighten the nice intuitive interpretation of 1: it is just that size of the project for which half of the surplus is distributed per capita and half of it proportional to investments. In any case, a positive 1 says that there is a reward for participation as such. The mechanisms (3.1) are not homogeneous. That is to say, changing all investments and the surplus by the same factor does not in general change all dividends by this same common factor. In other words, the methods are not scaleinvariant, i.e. they are not currency-independent. (Notice that if during inflation the parameter A is changed by the same factor that applies to all investments and the surplus, then invariance in real terms is obtained.) We have, once again, a conflict between two objectives: a choice has to be made between more variety in parametric value judgements on the one hand, and more simplicity in application on the other. It is also straightforward to show (see the appendix) that we obtain some of the properties discussed earlier for free. In particular it can be seen that the property (MOI) is dispensable in some of Moulin’s results. Stated as another observation: (3) The properties (TOT), (ANO), (MOS), (NAR), (ADD), and (PID) imply that (a) di(U,s) is partially differentiable w.r.t. every argument; (b) d also satisfies (MOI) and hence is a sharing method (SM); (c) d also satisfies (SEP); and (d) d also satisfies (PRO). Obviously, (PID) and (SEP) could be interchanged in this observation. A referee has pointed out that a result related to our theorem was derived in Wakker (1987). In a somewhat different framework, Wakker’s requirements include strong monotonicity, a variant of (NAR), and a condition on the relative changes of surplus shares of agents due to investment changes. Adding anonymity to the conditions of his Theorem 3.4 (p. 169) yields another characterization of our class of sharing methods. 4. Summary Moulin has jointly characterized equal and proportional sharing. He was able to provide four different characterizations by showing that adding any three of the four axioms (SEP), (NAR), (ADD), and (PID) to his basic definition of a homogeneous sharing method (HSM) is necessary and sufficient, By dropping the assumption of homogeneity, we have characterized a parametric class of sharing methods, where only the polar cases, equal and proportional sharing, are homogeneous. For this class we have provided two different characterizations, picking two different three-out-of-four combinations of the four axioms mentioned. Trying to characterize the same class with the remaining two combinations would be an interesting (and maybe difficult) task, which is left to the reader as a matter of further research.

294

A. Pfingsten / Surplus-sharing methods

Appendix Proof of the theorem. That (c) implies both (a) and (b) is a simple matter of algebra and will not be shown. To show that (a) implies (c), we start out by assuming only that no advantageous reallocations of investments are possible within coalitions. This precondition implies that agent i’s dividend only depends on his own investment, on total investments, on the number of agents, and on the surplus (this observation is in essence contained in Moulin, 1987, p. 167). By choosing T= N\ {i} and equating investments within T we obtain from (NAR): c;=, uk-u; , u, Ci=, uk-“i c;=, u/C-u; s d; (u, S) = dj n-l ” n-l ‘...’ > n-l ‘...’ From this expression it is immediate that d;(u,s) for given n may only depend on ui, ci= 1 ukr and s. That is to say, there must exist functions fi: IR: + x R, + R, (i= 1, . ..) n) such that, among others, for all u E IR: + and all s E II?,

d;@,s) =.A U;, t

c

k=l

uk,s

.

Adding anonymity, it follows that these individual dividend functions f; must be identical, i.e. there must exist a function f: Rf + x R, -+ R, such that for all in N, for all UEIR:+, and all sEIR+

From now on, U:= Ci= 1 uk will be used wherever this simplification of the notation seems to be useful. The constraint on total dividends (they must be equal to the surplus) turns out to impose a severe restriction on the function f: only affine functions of the individual investments are admissible. Formally, we can state the following general result:

Lemma.

A function

k$,f

(ukv

f: I?: + x IR, + R, satisfies the condition u,s)

=s 64.1)

n

for all UEIRT+ and all sER+, where I/=

c

uk,

k=l

if and only if there exists a function g : I?+ + x IR, -+ IR such that for all u E IR: + and all SE fR+

A. Pfingsten / Surplus-sharing

= g( U, s) . ( uk

-

f(u,,

Us)

+m,,

UsI

for all u~[Ry+, By letting

E=(u~-

u,)/2,

(A4

U].

u3, . . . , u, constant

Keeping

295

U/n) + s/n

ands/U1g(U,s)2--s/[(n-1). Proof of the Lemma. (A. 1) requires

methods

=f(u,

all SELF,,

+ E, u

and varying

4 +.I-@42 - E, u

u1 and u2, condition

9

and E such that (u, + E),(u~ - E) E R, +.

we obtain:

u,+&=U~-&=(U,+z42)/2>0. i.e. for all ul, u2 E R, + the condition f(u*,

U,.9+.m2,

u,.Q=2*f((u,

+ u2m

UT4

must hold. This is a Jensen equation in the first argument of the functionf,.and hence necessity of f being an affine function of its first argument follows straightforward from Aczel (1966, p. 44), i.e. f(Q, Because

Q s) = g(u, s) * Uk + au, G.

of the adding-up

constraint

on f we must have

g(U,s)=s/n-U.g(U,s)/n. Non-negativity hence

of f requires,

first,

non-negativity

for arbitrarily

small

uk, and

g(U,s)
non-negativity

g(U,s)L Thus, necessity of the lemma.

for uk almost

-s/[(n-

is shown. 0

equal to U, i.e.

1). U].

Since sufficiency

is obvious,

we have completed

the proof

Proof of the theorem (continued). So far we have shown that there must exist a function g : R, + x R, --t R such that for all u E iR: + and all s E R, d;(U,S)=g(U,S).(Ui-

and s/Uzg(U,s)z

U/n)+s/n -s/[(n-1).

(A.3) U].

If g is identical to zero, we have equal absolute sharing. Negative values of g mean that those who have contributed more than the average receive less than equal absolute shares. The upper bound for g is seen to imply proportional sharing; lower values benefit those who have contributed less than the average. It is remarkable how much structure the three properties (TOT), (ANO), and

296

.A. Pfingsren

/ Surplus-sharing

merhods

(NAR) imply. Not only are the informational requirements rather weak (only the number of agents, total investments, the surplus, and agent i’s investment are needed to calculate i’s dividend), but also there are just a few ways how dividends can be calculated, namely as affine functions of individual investments. Tight connections to Lemma 2 of Moulin (1987, p. 174) should be noted: by (HOM), (MOI), and (MOS), the functionfmust be a certain type of function depending only on the ratio of surplus to total investments. Let us now briefly abandon the three properties required so far in order to derive implications of additivity. As an easy application of the theory of functional equations we obtain (see also equation (22) in Moulin, 1987, p. 182) that (MOS) and (ADD) require the existence of functions x : R:+ -+ I?+ such that

d,(u,.s)=f;(u)Ys (A.4) for all UE E?:+ and all SEE?+. (Theorem 1 in Aczkl, 1966, p. 215, applies since condition (2.5) is a Cauchy-type functional equation; (MOS) excludes irregular solutions; and non-negativity of the A is required by monotonicity in s.) This result means that individual dividends must be calculated as shares of total surplus, where the shares must not depend on the surplus. Applying equations (A.3) and (A.4) simultaneously, we are able to restrict the class of admissible functions d further. Since the affine functions of individual investments must at the same time be non-negative valued and linear in the surplus, we obtain: d;(u,s)=h(U)*s* (U;- U/n)+s/n. (A.5) where

h : R,,

---)R must

-I/[@--1). Rewriting

equation

satisfy U]sh(U)
(A-6)

(A.5) as

d;(u,s)=[U.h(l/)].s.u;/U+(l-[(I.h(l/)]).s/n.

64.7)

striking similarities to Theorem 3 of Moulin (1987, p. 170) appear. In his theorem, instead of the expression in square brackets only a constant number /?, O
A. Pfingsten / Surplus-sharing methods

297

=h(U).s.Ui+S/n-U.h(U).s/n + h( U-t

s)

-(U+s). Rearranging

. s’ - [u; + h(U) . s. ui + s/n - U. h(U) . s/n] + s’/n h(U+s).s’/n.

terms yields: h(U)+s’.u,-U.h(U).s’/n =h(U+s).s’.u;+h(U+s).s’.h(U).s.u;-h(U+s).s’.U.h(U).s/n -U.

h(U+s).s’/n.

or, equivalently, u;. [h(U)-h(U+s)-h(U+s). =U.

h(L7.sl.s

[h(U)-h(U+s)-h(U+s).h(U).~]~s’/n.

This equation must hold for all u E iR=+ and all s, S’E IR,. Varying U; while keeping U constant (by variation of some uj, j+ i) it is immediate that this can only be the case if the expression in square brackets is equal to zero for all U and all S, i.e. h(U)-h(U+s)-h(U+s).

h(U).s=O

(A.@

for all U E R, + and all s E iR+ equation for h that for all U E I?+ + . But this is not the If h is not identical to zero, then zero for all UE R,,. Suppose not,

is the functional

h(U*)=O Then,

by condition h(U)=0

If there

would

is to be solved. One obvious solution is h(U) = 0 only one. it can be shown that h must be different from i.e.

for some U*>O. (A.8), for all U> U*.

exist o<

U* such that

h(O)#O, then we could choose s = U* - C?to show a contradiction

to equation

(A.8), namely

h(O)-h(U*)-h(U*).h(O).(U*-0)=h(0)+0. Consequently,

we must have that h(U) = 0

for all U< U*

as well, i.e. if h is equal to zero for some U>O, then it is identical to zero and, the other way round, if h is not identical to zero, then it must be different from zero for all U>O.

298

A. Pfingsten / Surplus-sharing methods

It can also be shown that h cannot vestments below average are rewarded cluded. Suppose the contrary, h(U)=c
s = -l/c

take negative values, i.e. the cases that inmore than an equal (absolute) share are ex-

for some U>O. (s> 0) such that c = - 1/s. Then

we have

h(U)-h(U+s)-h(U+s).h(Cr).s

contradicting condition (A.8). In other words, if h(U) has a negative value for some (A.8) cannot be satisfied for all U E R, + and all SE R, . UE R++, then condition Thus, apart from the case of h being identical to zero, h must be positive for all (A.8) as UE R++, and we can rewrite condition h(U+.s) =

h(U) l+h(U).s

Condition

(A.9) is a functional

for all U E fR++ and all s E R, . equation

of the type

W + Y) = F[h(x), ~1, where the function

No,

F is not arbitrary

(A.9)

(A. 10) but given as

w)=&.

(A.ll)

The (nonconstant) solutions are derived by a method of Acztl (1966, pp. 17). Let x = E for some small E > 0 (see below for remarks on this choice) and define h(c) =: c. Then the solution must be of the form (A. 12)

h(t + E) = F(c, t).

where,

in particular, (A.13)

F(c, 0) = c

(with c> 0 because of the positivity of h) must hold. Substituting tion (A.12) into the functional equation (A.lO) yields: F(c,x+y-c)=F[F(c,x-c),y]

the (possible)

solu-

(A. 14)

(which would be called a translation equation (Aczel, 1966, p. 245) if F were arbitrary). This condition must hold for XZE and ~10. From AczCl(l966, p. 18) it is known that the function (A. 12) is a solution of equation (A. 10) if and only if the given function F satisfies equations (A. 13) and (A. 14) for the constant c appearing in (A.12), and in that case it is also the most general solution. (Note that the functional equation (A.lO) only has a solution if there exists some c such that condition (A.14) is satisfied.)

A. Pfingsten / Surplus-sharing methods

Thus,

let us first check for which constants

c this last condition

C

l+c.(x+y-E)=I+

C

(A.14)

h(‘+E)=& are all solutions

=

h(t) = &

l+c.

.Y

is satisfied

(x-&)+CeY

l+c*(x-E)

for all c>O.

Hence

the functions (A. 15)

cc>01

that can be rearranged

From

l+c*(x-E)

l+c.(x-E) it is seen that condition

is satisfied.

C

l+c.(x-E)

c

299

as (A. 16)

(F=(l-C.&)/C>-&).

By choosing x= E and noting that the function F is only defined on R, + x R,, it is obvious that we have only derived the solutions of h(U) for ULE. Since we are looking for a solution for all u > 0, however, we could have chosen E > 0 arbitrarily small. Letting E approach zero we hence must have 2~0. This completes the proof that (a) implies (c). To show that (b) also implies (c), we start from equation (A.5). (SEP) requires (defining U’:= C;f.=, u;) h(U).s.u;+[l-U.h(U)].s/n (A. 17)

=h(U’)~s’*u;+[1-U’~h(U’)]~s’/n

for all ie T and all ui less than U and U’ (ui= u,! for all i E T), given that s, s’, u and u’ (and hence U and U’) are such that

c d,(u,s)=;~Td,(u:s’).

ie

Condition

(A.18)

T

(A. 17) then directly

requires

h(U)*s=h(U’).s’

(A. 19)

[1-u*h(U)]~s/n=[1-U’~h(U’)]*s’/n.

(A.20)

and

An obvious solution of condition (A. 19) is h(U) = 0 for all UE L?, +. In this case condition (A. 18), together with the form (A.5) for d,, implies s = s’, and in this case condition (A.20) is satisfied as well: equal sharing emerges. If h is not identical to zero, then there exists some U’E R,, such that h(U’)#O, and we may rewrite equation (A.19) as s’=s. Inserting condition

h(U)/h(U’).

this expression into condition (h(U) #O for s’# 0)

(A.21) (A.20),

upon

rearranging

terms

yields the

300

A. Pfingsten / Surplus-sharing methods

1-

1- CT’. h(U’)

h(U)

h(U)

=

(A.22)

”

MU’)

This can only hold for all CJE R, + if there exists some c such that 1- U. h(U)

=c

(A.23)

for all UER++.

h(U) It is obvious from equation (A.23) that c cannot be negative, condition was violated for U= -c. Now it is simply a matter for non-negative c

since otherwise this of algebra to derive

h(U)=+u. This completes the proof that (b) implies (c). Since (a) and (b) are both equivalent to (c), they are also equivalent other. 0

to each

Proof of observation (3). For equal sharing all results are obvious. For the remaining mechanisms part (a) is obvious from expression (3.1), and part (b) simply follows by differentiating di(U,S) w.r.t. ui. To prove part (c) note that

and ui = u,! for all i E T imply

that

and hence this claim is immediate.

Part

for A E R, and U;E R, + is nonincreasing

(d) is obvious

in Ui.

since the expression

Cl

Acknowledgements The author is indebted to Herve Moulin and two referees for their extremely useful comments. An earlier version of this paper was presented at the 1988 European Meeting of the Econometric Society at Bologna, Italy.

301

A. Pfingsten / Surplus-sharing methods

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