Tesis doctoral de la Universidad de Alicante. Tesi...

Financing public goods. Miguel Gines Vilar.

Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant.1996

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D. FRANCISCO MARHUENDA HUKIADO, TITULAR DE

UNIVERSIDAD DEL DEPAHTAMENTO DE FUNDAMENTOS

DEL ANÁLISIS ECONÓMICO DE LA UNWERSIDAD DE

ALICANTE,

CERjIIFICA: Que la presente memoria "Financiación de Bienes

Públicos", ha sido realizada bajo su dirección,

en el Departamento de Fundamentos del Análisis

Económico de la Universidad de Alicante, por el

Licenciado en Matemáticas D. Miguel Ginés Vilar,

y que constituye su tesis para optar al grado de

Doctor en Economía.Y para que así conste,, en cumplimiento de ia

legislación vigente, presento ante la Universidad

de Alicante la referida Tesis Doctoral, fi-rmando el

presente certificado en

Alicante a 10 de Mayo de 1995.

Fdo. Francisco Marhuenda Hurtado.

f-



A Yolanday Victoria



All good work is done for the people, but to carry out this work it is

always necessary to count on the help of others. Therefore, I would first of

all like to express my thanks to the supervisor of my dissertation D' Francisco

Marhuenda for having confidence in me, for his constant support, his never-

ending patience and his total dedication to this work. He has been the main

contribuitor to my been able to finish this project'

I would also like to thank all my colleagues of the Deparment of Funda-

mentos del Análisis Económico who have encoraged me to persevere, have

provided me with a friendly working atmosphered and furthermore have con-

tributed greatly with their helpful and interesting suggestions and comments.

I would especially like to mention the great support of Professor Luis c.

corchón in this work. Apart from introducing me to the theory of Public

Goods , he has also followed the stages in the development of this project.

It shouid not be forgotten that this work was begun at the University of

Bielefeld with Professor walter Trockel, Jorg Naeve and Lisa steinweg. I am

grateful to all of them.

The special thanks must be given, firstly to my wife, Yolanda, who has

supported and encouraged me greatly and without whom this work never

have neen possible. The second special thanks should be given to my little



daughter, who must forgive me for all the time that I have not been able to

dedicate to her and for the few opportunities we have had to plav together'.

The third is for Him who I trust and who guides my steps whenevel I go' He

is the main pillar of this work and therefore I dedicate it speciall¡' to Him.

Finantial support from the Ministry of Education ancl from Instituto Va-

lenciano de Investigaciones Económicas must also be acknowloclgecl'



CONTENTS

INTRODUCTION

Chapter 1: ON THE CORE OF AN ECONOMY WITH

INFINITELY MANY PUBLIC GOODS

Chapter 2: CORE SELBCTIONS IN ECONOMIES WITH

INCREASING RETURNS AND PUBLIC GOODS ..

Chapter 3: EFFICIENCY, MONOTONICITY AI{D

RATIONALITY I}{ PUBLIC GOOD ECONOMIES

Chapter 4: EFFICIENCY AND TECHI\OLOGICALTMPROVEMENTS

Chapter 5: EQUITY AS A CO}{SEQUENCB OF SOLiDARITY



II\TRODUCTIOI\



There are several definitions of public goods, but the most common one

identifies public goods as those that ail the agents of the society may enjoy

in the same quantity which is equal to the total quantity of public goods

available. These goods could display free disposal (if agents can decide to

enjoy a quantity lower than the available quantity, for example T'v' or forest)

or pute public goods (the agents are forced to consume the total amount as

in the ca.se of the army, the police or laws). This study focuses mainly on

the second tYPe.

One of the main issues in Public Economies is the optimal provision of

public goods, that is, how to decide which quantity of the public goods to

produce and how to finance their cost. An ailocation is a bundle of pubiic

goods and a financing plan that covers its cost'

The literature has mostly concentrated on two questions in this area:

First, to design mechanisms that reveal the pleferences of the agents on the

public goods and attain an optimal allocation (e'g. Groves and Leyards

mechanism). The second question is how to decentralize an optimal alloca-

tion by means oi personalized prices, once the preferences of the agents are

knorvn.

The present study takes the preferences of the agents as known and seeks



a solutions that provid.es an optimal level of public goods and proposes a

share of the costs among the agents. We study properties of "justice" and

stability that would be desirable for the solution'

A property which reflects stabiiity is the core of the economy. This con-

cept was introduced by D. Foley (1970) to economies with public goods' If

ali the agents have access to the same technology, it would be desirable that

no coalition of agents could improve the welfare of its members operating the

technology on its own. There are two approaches for proving the existence

of allocations in the core: The first is to define a cooperative game with a

non-empty core, which, in turn, implies that the core of the economy is also

non-empty. In general, when the core is non-empty, it contains solutions that

could be considered as unfair'. The second consists of proving the existence

of particular solutions that are in the core'

Following the first line P. Champsaur (1975) proved that in economies

with one private and one public good-, the core is non-empty under very

mild conditions. Later, w. sharkey (r9s9) proved that in a special case of

increasing returns it is possible to obtain a non-empty core' on the other

hand, following ideas of E. Lindaht, D. Foley (1970) proposed a generalization

of the Lindahl equilibria, analogous to the competitive equilibrium for the



private good case. However Lindahl equilibria is in the core only under

constant returns to scale. M. Kaneko (1977) introduced the concept of Ratio

equilibrium, which is always in the core, to solve part of these difficulties'

D. Diamantaras and s. wilkie have generalized this equiiibrium to multiple

private goods. Another formalization of Lindahl's equilibrium notion is the

cost share equilibrium of A. Mas-Colell and. J. silvestre. s. weber and H'

Wiesmeth (1991) proved the equivalence between the cost share equilibrium

and the core. Most of these papers consider one private good. only the

work of Foley with the Lindahl equilibrium (under constant returns to scale)

and Diamantaras and Wilkie with the Generaiized Ratio equilibrium (with

separable production) allow for more than one private good'

The first chapter of this study ploves) by means of a cooperative game, the

non-emptyness oi the core when the spaces of private and public goods may

have an infinite dimension. For the case of a finite number of goods it also

shows the non-emptiness of the core under joint production and decreasing

returns to scale.

The second chapter, follows closely the generalization of Lindahl equilib-

rium made by J. M. Bonnisseau (1991) for the case of increasing returns' It

looks for conditions under which this equilibrium is in the core. W. Sharkey



(1gs9) proved, defining a cooperative game, that when there is only one dis-

tributive production set and some inputs of the production do not enter in

the utility function of the agents, the core is non-empty. This author, jus-

tifies the result because in the presence of increasing returns the flrms can

not maximize their profits so there is no Lindahl equilibrium' Following a

similar model, this chapter shows that, if we restrict the quantity of inputs

that a firm could use to produce the pubiic goods,, there exists a lindahlian

equilibrium in which agents maximize preferences subject to their budget

contraints and firms maximize profrts subjet to the inputs available in the

economy. Moreover, this equilibrium is in the core'

It is commomly assumed that the technoiogy to produce public goods is

jointly owned by all the agents. Then, it is reasonable to plopose that tech-

nological ad.vances increase the welfare of all the rrrembers of the society' or

at least do not hurt anybody. This condition is known as cost monotonicity'

It was introduced by J. Roemer (1986) in the economies with private goods

and analyzed by H. Moulin and J. Roemer (1986) among others' using this

property in the context of one public good, H. Moulin (1987) characterized

the egalitarian equivalent solution, originally proposed by E. A. Pazner and

D. Schmeidler (19?S), by means of the cost monotonicity and the core pr"op-



erties. In fact, he obtained a characterization with a milder set of axioms,

namely Pareto optimaiity, individual rationality and cost monotonicity'

The third chapter studies Moulin's results on cost monotonicity in the

context of several public goods. It shows that the properties of Pareto ef-

ficiency, individual rationality and cost monotonicity are compatible if and

only if all the agents order the bundles of public goods in the same way

whenever they are consumed for free. This property, that we call equal or-

d.ering, is trivially satisfi.ed for the case of one public good. However, it is

very restrictive for the case of several public goods' In the generai case differ-

ent agents might differ in their opinions about which public goods should be

given priority. clearly, under the equal ordering property, these discrepancies

in priority do not arise. It is also shown that whenever this property holds

the egalitarian equivalent mechanism is unambiguous (in the sense that every

agent is indifferent among al, the egalitarian equivalent allocations) and is

the only cost monotonic selection from the set of Pareto efrcient and individ-

ually rational allocations. On the other hand, the equal ordering property is

no longer necessary in the case of economies with quasi-lineal and additively

separable utility functions and additively separable cost functions' In this

framework, there is essentially only one Pareto efñcient and individuallv ra-



tional allocation which is cost monotonic, and it is an egalitarian equivalent

allocation. This result appears as the natural generalization from the case of

one public good.

Another monotonic property which also appears in the literature is popu-

lation monotonicity. when one agent is added to the society nobody should

suffer any loss of welfare. This property was introduced by Thomson (1983)

in the axiomatic bargaining theory and has also been used in others contexts

as problems of resource allocations or problems of fair division. In the con-

text of one public good, Moulin (1937) shows that the egalitarian equivalent

solution satisfi.es population monotonicity. When this property is studied in

the context of several public goods, one obtains the same impossibility result,

that is, the properties of Pareto efficiency, cost monotonicity and population

monotonicity are compatible if and only if the agents satisfy the equal or-

dering property. Again in separable economies there is only one solution

characterized by these three properties.

The fourth chapter also deals with the ploperty of cost monotonicity but

in the special case of Leontief production functions (production in a fixed

ratio), in the context of one public good and several non-transferable private

goods. It proves that if these types of prodr,rction functions are allowed then



the properties of Pareto efñciency and cost monotonicity are incompatible'

Although in this context the egalitarian equivalent solution may exist and it

is cost monotonic, it may not be Pareto efficient'

Ali these impossibility results are analogous to those of w' Thomson

(1987) and H. Moulin and w. Thomson (19SS) which show the incompa'tibil-

ity of the properties of Pareto efficiencS individual rationality and resource

monotonicit5' or input monotonicity'

The last chapter shows an existence result. It looks for "fair" or "equi-

tabie" solutions, taking Pareto efificiency, symmetry, that is, equal agents are

treated equally, and a solidarity axiom as desirable properties. The soiidarity

property requires that if one agent improves his ability to exploit the public

goods then no agent should. suffer any loss in welfare. In situations where

cooperation to produce the public goods is necessary and where there is no

competition among agents this axiom make sense (firms that use the public

good as input to produce non-substitutable products and there is no compe-

tition among them). These three properties, Pareto efñciency' symmetry and

the solidaritl, axiom characterize, in the context of public goods, the welfare

egalitarian solution. cooperation is the key of this result' In the context of

private goocls , where coordination is not necessaly to produce goods, there



might be solutions other than the welfare egalitarian, satisfying these three

properties.



CHAPTER 1

OI\ THE CORE OF AI\ ECOI\OMY WITH

II\FII{ITELY MANY PUBLIC GOODS

Abstract

Weprov idesuf rc ien tcond i t ionsunder rvh ich theCoreofanecon-omy wiih a finite number of agents and infinitely many private and

poúü. goods is non-empty. We give two examples of economies to

which the result aPPlies.



Introduction

In this chapter we consider the problem of finding the appropiate supply of

public goods. We analyze the problem in a context of a finite number of

agents but where either the space of public goods, or the space of private

goods, used as inputs in the production of public goods, may have an infinite

dimension.

We look for solutions which satisfy some equity and stability properties.

One of the most accepted requirements on stability which appears in the

literature is the Core property. A possible allocation will be objected to by

a group of agents whenever using the technology to its own benefit, ii could

improve the welfare of its members, regardless of the actions of the rest.

But, in the case of a finite number of pubiic goods, there exist examples for

which the Core of the economy is empty. Hence, it seems reasonable to fi.nd

sufficient conditions under which there exists a solution satisfying the Core

property.

The conditions we impose to obtain the result are essentially, convexity,

monotonicity, and some continuity in preferences and in production.

We obtain an existence theorem which guarantees the non-emptyness

l 0



of the Core. We establish the result for general topological vector spaces

which include several economic enviroments. As an illustration we give two

examples to which the result applies:

Example 1-.1 Suppose that a group of agents wants to undertake a research

project that lasts for a finite number of years and which will benefit all of

them . Each agent has to provide some resources throughout these years

to develop and mantain the project. The agents have to decide upon the

characteristics of the project and how much each of them has to pay. We

consider continuous time to aliow payments by agents at any point in time

and the number of potential characteristics of the research project is infinite.

These two conditions suggest that the problem be modeled in an infinite

dimensional seiting.

The space of private goods is the set of possible fundings for the program,

and we choose E : M(10,7]), the Borel measules on the interval [0,2],

where ? is the number of years. Each element z € M(10,7]) denotes a way

of financing the program, and z(fa, ó]) the quantity of "money" paid in the

per iod [a ,ó] c [0 ,7] .

The space of public goods is G : M(K), Borel measures on a compact

11



metric space /(. We assume that the research project is a public good for all

agents and 1( repesents the space of possible characteristics (maybe infinite)

that describe the research project. An eiement y e G : M(K) is a complete

description of a research project.

The technology is described by a function which assigns a specifi.c research

project to each possiblefinancing. For each z e M(l},Tl), F(z) € M(h) is

the complete desciption of the project to execute with the funding z.

With respect to the agents, each member has a bound, u¿, of money to-

wards financing research projects, and his preferences on the research project

and money are modeled by means of utility functions which are non decreas-

ing in the research projects and money.

Example 1.2 The next example is inspirated in the natural resources that

are public goods for the whole society as for example public forests. Suppose

we have one private and one public good across infinite periods of time. In

each period the public good needs to be rebuilt or at least some private good

is needed to maintain the public good. We assume there aÍe n infinite-lived

agents, and each of them has to contribute with some of its private good to

maintain or increase the quantity of the public good.

T2



This suggests that we use E : I* as the space for the private good and

G : l* as the space for the public good.

We choose /*, the space of bounded sequences because we consider con-

sumptions of goods across infinite periods of time and the endowments of

agents in each period are finite. For each t > !, r¿ lepleseots the amount of

good in period t, and we take o(l*,,/1) as the natural topology on -E and G.

The amount of public good produced in period n depends on the amount

of public good in period f - 1 and the private good that agents invest in

period t. Then, the technology is given as fol lows W: f (A*1,2¿) fot t > 2

and y1 : f ("r), where z¿ is lhe total amount that agents invest in produc-

ing and preserving the public good y in period f . But for each period I we

have that Ut : gt(2t,. . . , zt-t) where 9t(21,. . . , zt-t) is defined recursively by

gt(2y, . . . ,2 t - t ) : f (gr - r ( " r , " ' ,z t -z) ,z t - t ) ' Then, we get that the technol -

ogy J7 : E ' -+ Gby y : F ( z ) ,w i t ' h F¿ (z ) : : 9 t ( 2 t , " ' , z t - t ) '

The agents have preferences over goods across periods. These preferences

are modeled by means of "impatient" utility functions to express the idea

that gains or losses in consumptions in the distant future are negligible.

On the other hand, each agent has a bound ai on the resources dedicated

to the production of the public good.

I Ó



Model

In our economy there are two types of commodities, private and public goods,

and a f inite number of agents, A: {Ir. . .rn}, whose preferences are lepre-

sented by utility fuctions. There is a technology to produce public goods

using private goods as inputs. And all possible coalitions have access to the

same technology.

we consider two topological vector spaces with a partial order (8,2s)

and (G, >"). Ot E, the space of private goods, and G, the space of public

goods, we take the weak topologies, o(8, E.) and o(G, G*), and the Mackey

topologies r(8, E*), r(G,G.). Convergence in these topologies is denoted by

ro L r for the weak topology and ro 3 r for the Mackey. We denote

by ,n € E the initial endowments of private goods of agent i, and we define

[O , ru ] : { r € E :01n t 1s a ; } .

We make the following assumptions:

A.1 For each i € A, [0,c..'¿] is weakly compact on -8.

A.2 The utility function for each i e A, u; : E x G ------+ IR, is non-decreasing

in public and private goods, quasi-concave and Mackey continuous.

t4



A.3 The technology is described by a production function F : E > G,

where -F (.) is weakly continuous, non-decreasing and concave.

Assumption A.1 is the weakest assumption on compacity we may make

because compacity in another consistent topologyl implies compacity in the

weak technology. In A.2 we choose the Mackey continuity rather than weak

because it aliows for a larger set of continuous preferences with respect to

it. Assumption A.3 forces the production set to be convex; in particular,

increasing returns are ruled out.

Definit ion 2.t An econonly e : (A, (u),ro, (ro),ro, F) i 's def 'ned by a group

of agents wi,th preferences, endowments and a technology to produce Ttublic

goods. An a l locat ion i ,s a aector ( t t , . . . , rn ,U) e E" x G. Gi 'aen S C A,

we so,y that an al locqtion ("r,. . . ,rn,U) is feasi,ble for S if for each ' i e S,

0 1 r¿ 1 u¿ and y : F(D¡es(a¡ - rj)) . We sirnply say feasi,ble if it is

feasible for A.

Definit ion 2.2 A coali t ion S of consurners' is a non-empty subset of A. We

say that a coali , t i ,on S i,mproaes upon an al locati 'on (r i , . ' . ,r l ,y*) wheneuer

there eri,sts a feasi 'ble al locati 'on for S ("t, . . . ,rn,y) such that for al l i , e

1o' is consistent i f o C dt C r

15



S, u¿(r¿,y) > u¿(ri ,y-), A Core al locati 'on i ,s an al location that cannot be

i,mproaed, upon by any coali,tion. The set of all core allocations of an economy

e i,s called, the Core of the econotny and i,s d,enoted by Core(e).

we reca.ll some concepts of games without side payments. Fix a finite set

of a players A and let lú be the set of all coalitions of A, i.e., N : {S g

A : S + 0\. An n-person game is defined by a non-empty correspondence

V : N --.2R". As usuai, a coalition S can improve upon a payoff vector

z eV(A) whenever there exists a payoff vector z. eV(S) such that zi ) z¿

holds for all i e S. We say that z € core(v) if" z € v(A) and given a

coalition s g /ú and z* € y(s), there is i e ,9 such lhal zi I z¿.

A (non-empty) fa.mity B of l{ is said to be balanced whenever there exist

non-negative weights {), : S € B} satisfying D1r.",n.r1)" : 1 for each

i e A. An n-person game I/ is said to be balanced whenever e\¡ely balanced

family B of coalitions satisfies l-l"eB V(S) g V(A)'

The following Theorem is due to Scarf (i967)'

Tlreorem 2.3 A balanced n-person game v ltas a non-empty core i,f for

euery non-empty coali t ion S c A:



&) V(S) i,s closed,

b) V(S) is cornprehens ' iae, ' i , .e ' , a 1y andA e V(S) impl íes r e V(S)

c ) c€R" ,9€y (S ) and " r¿ -y¿ fo reach ie S i 'mp l i es r€V (S ) and

d,) eachv(s) is bound,ed, from aboue inFt" i,.e., for each coali,tion s there

er is ts some M,>0 sat i ,s fy i ,ngr¿1M, for a l l r € y(s) and a l l i e s .

We also use the following property of topological vector spaces which

appears in the book of Aliprantis, Brown and Burkinshaw (1989).

Propositi on 2,4 Let X be a topologi,cal uector spo'ce) let C be a non-ernpty,

conaeÍ and weakly closed subset of X and let u : C '-+ 8,, be a quasi-concaae,

Mackey-cont,inuous functi,on. Then no o , r i,n C i,mpli,es Limsupou(r") I

u \ n ) .

With these preliminaries we state the following Theorem'

Theorem 2.5 If ün economy satisf,es assumpti,ons A.1, A.2 and A.3 then

the Core of thi,s economA ds non-empty'

Proof

consider the following n-person game without side payments:

Fo reachScAwede f i ne

T7



y (S ) : { ( " r , . . . , , 2n ) € lR ' : f ( r r , . . . s rn , ¡ y ) f eas ib l e fo r ' 9

s . t .V i e S u¿ ( r¿ , y )22 ; \

We claim that the n-personal game V satisfies the conditions of Scarf's

Theorem. Properties (b) and (c) are easy to prove, and (d) follows from

the fact that each utility function u¿ is monotone and hence bounded on

[0 , r¿ ] x F ( [0 , r ] ) bV u¿(a¿ ,F ( r ) ) , where a :LT - - ta i .

The first condition (a), closedness of y(^9), is established as follows: Take

a ne t (€ í , . . . , € i l € I / ( . 9 ) such tha t ( { f l , . . . , €n - (€ t , . . . , €n ) ' Fo r each

a, p ick a feas ib le a l locat ion for S, ( *?, . . ' , r f t ,A") , such that u i ( r | ,u" ) >

# Vi e S. Moreover, for each i e S and for all a, 0 (a rf 3n rn¿ a'r7d,

since [0, c,;¿] is weakly compact, we assume rf -\ r¿ 19 e¿ Vi g S. Clearly,

( r r , . . . , rn ,A) wi th y : F(D¿es@n - r¿)) is feas ib le for ,9 because f rom the

continuity o{ F we have that y' -"+ y. On the other hand, for each i e S,

@7,a\ 3 (rn,g) and, since each utiiity function is Mackey continuous,

it fol lows f i-om Proposit ion 2.4 lhat l imsupo u¿(r1,y") < u¿(r¿,y). Hence,

€¿ : l imsup , € i " ( l imsupo u¿( r f ,A ' ) < u¿ ( r¿ ,U) , \ . e . (€ t , . . . , { ' ) e y (S )

Finally, we prove that v is balanced: Let B be a balanced family of

18



coaii t ions with weights {.\ , : s € B} and let ("r '" ' ,zn) € l lr ."V(S)' We

have to show that (rr, . - . , zn\ € V(A)'

De f i ne B¿ : {S € B : I e S } . Fo r ,9 e B , s ince ( ' , , " ' ' ' , * ) € V (S) the re

is a feasible ai location for S, (" i , . ' . ,r l ,a'), such that u¡(r i ,a") ) z¡ Vi €

S. Now, for each i e . A, le l r¿: L"eB, \ "ü l and y : F(DLt@n - *n)) '

since, r¿ \s a convex combination of rf and each rf is less than or equal to

ar¿, then r¿ is less than or equal to a¿' So, for each h € A, we have that

n n- ¡ \ a , - - . ) ) - F l \ -

U - f l ) , \ u ¿ - ' r ; i ) l : t \ / - ¿ I ,r(,0 - "f ))

: r(t .ls f (cr; - *f )).SC R . SCB ¿€S

(Since F is nondecreasing)

(Since F is concave)

i= !

-' \---\ \-(r, - ri))2-c !'\ )_, At u,SeBn ;e S

)n f r"r( f ( r , - r i ) )LJ

- '/-J

SeBn i 'eS

= \- .\"us'useBn

And then, for each i € A,

u¿(n¿,a) = ,,( I \srf ,v) > u¿(I )"r i , I r"Y")S€B¿ S€B¿ S€B;

wh ich p roves tha t (21 , . . . , 2n ) €V IA) '

ln[ow, scarf,s Theorem guarantees that the n-person game has a non-

empty Core. P ick (21, . . . ,2n) € Core(V) and le t ( '1 , " ' , r i " 'U") be afeasib le

19



allocation satisfying u¿(r[,t/*) 2 z¿ Vi e A.

we shall establish that (r{, . . . ,r l ,y.) belongs to core(e ); the proof is by

contradiction.

Suppose there is a coali t ion,g and an al location (21, " ' trn,y), feasible

for ,5, satisfying u¿(r¿,a) > u¿(ri ,y*) Vi e S. Then u¿(r¿,y) > '¿ Vi e 5'

We see that coali t ion .S improves upon the Core vector (4r... ,2n), which is

a contradict ion. Hence, (" i , . . . , t i ,?)*) e Core(e ) ' D

RtrFERENCtrS

Aliprantis, C. D.; Brown, D. J. and Burkinshaw, O' (1989):

Existence and optimality of competitive Equilibria. springer-verlag.

champsaur, P. (1975): "How to share the cost of a public good,"

International Journal of Game Theory, 4, tl3-t29'

Jones, L. E. (1983): "Existence of Equilibria

sumers and Infinitely Many Commodities"'

12 ,119 -139 .

with Infinitely Many Con-

Journal of Mathematical Economics,

20



Mas-colell, A. and zame, w. R. (1991): "Equilibrium Theory in

Dimensional Spaces," Handbook of Mathematical Economy'

Infinite

IV, ch.

and Murray Hill (1931): "Convex Games Without Side

International Journal of Game Theory, 10' 101-106'

34.

scarf, H. E. (1967): "The core of an n-Person Game," Econometrica, 35,

50-69.

Shapley, L.S. (1971): "Cores of ConvexGames," International Journal of Game Theory,

L) rL-26.

Sharkey, W. W.

Payments,"

2 l



CHAPTER 2

CORE SELECTIO}IS I}T ECO}{OMIES WITH

INCREASI}{G RETURI\S Ai\D PUBLIC GOODS

Abstract

we consider economies with increasing returns and in which flrms

follow loss-free pricing rules. In the case of only one firm with an input

distributive production set, we obtain an equilibrium which belongs

to the core of the economy and hence, is a Pareto optimal allocation.

22



Introduction

One of the first and most studied equilibrium notions in the public goods

literature is the Lindahl equilibrium, (investigated by Foley (1970)), its main

characteristic being the existence, fór each agent, of a personalized price'

These individ,ual prices provide unanimity in the demand of public goods

and optimality in the convex case. One of the difficulties of the Lindahl

equilibrium is that it does not exist in the presence of increasing returns to

scale. Many authors have introduced new notions in order to improve the

idea of Lindahl equilibrium, but as is weil-known, except in special cases

such as Mas-Colell and silvestre (1989) or Diamantaras and wilkie (1994)'

firms,behavior as profit maximizers is incompatible with increasing returns

to scale in production.

One way to deal with increasing returns to scale is to allow fi.rms to behave

differently from profit maximization. In this context, it is useful to model

the behavior of firms by means of pricing rules'

. There are two different criteria to approach the problem of increasing

returns by means of pricing rules, the positive and the normative criteria.

With respect to the latter, in Vohra and Khan (1987) the existence of

23



an equilibrium, called the Lindahl-Hotelling equilibrium, where firms follow

ma.rginal pricing rules is proved.

Bonnisseau (1991) generalizes this concept and shows the existence of

Lindahl equilibrium when firms follow loss-bounded pricing rules. In general,

the equilibrium obtained in the presence of pricing rules is not a Pareto

optimum.

We study the existence of equilibrium when firms follow loss-free pricing

rules, which only allow for non-negative profits'

We look for conditions under which this equilibrium satisfies the core

property and hence, is a Pareto optimal allocation. To obtain a selection of

the core in the presence of increasing returns, we focus on the special case in

which there is only one firm with an input distributive set, and it behaves in

accordance with profit maximization subject to input constraints.

The result we obtain constitutes the public goods version of Scarf's result

in the case of private goods (Scarf (1986)). Our result completes Sharkey's

(1g8g) on non-emptyness of the Core in the sense that we obtain the existence

of a Lindahl equilibrium in the Core which maximizes the utility of consumers

subject to the budget constraint and rvhich maximizes the profi-ts of the firm

subject to inPut constraints.

24



The methods presented here also provide an alternative proof of the exis-

bence of Lindahl equilibria in the non cofrvex case without making use of an

auxiliary economy with an enlarged space of goods as in Bonnisseau (1991)'

we borrow the tbrmulation of the problem from Khan and vorha (19s5) and

introduce pricing rules. In this setting it is possible to adapt the proof of

Bonnisseau and. cornet (1983). we provide two examples of pricing rules to

which the existence result applies'

2 The Model

First, we introduce some notation. For any X C ]Rk+¿, rve let Xg and X7

denote its projections on the first k and last / coordinates whereas for any

r € IRfr+/, ¿¡<¿ will refer to the ith coordinate ('i = I,' ' ' , ft) and r7¿ to the

(k + i)th coord.inate. we will denote the (k + t - 1) dimensional simplex by

A , and de f i ne e t : { a e Rk+ t f ye :0 } whe re e : ( 1 , " ' , 1 ) € ¡ 14+ / ' We

identify er with the (k + t - 1) dimensional Euclidean space and identify

T: eL X... X er with the Euclidean space n1?x(rr+¿-l). Given r,U Q IR", we

define

( r ¡ y ) : ( r t A t r r 2 ! 2 t . . . , r " 9 " )

we consider an economy with two types of commodities' / public and k

25



private goods. The set of consumers is M - {1,.. . ,-}. Each consumer

i e M is characterized by a triple {xu,[Jo,u,'i], where X' is the consumption

set, U¿ : Xi - IR is his utility function and c¿¿ his initial resources of private

goods. There are - lü : {1,.. . ,n} f irms. We denote by Yi C IRe+/ the

p roduc t i on se t o f f r rm j € l { and y : (A1 , . . . ,A " ) €Y :Y I x " ' x Y " i s a

vector of productions.

A pricing ruie is a correspondence which assigns a set of prices to each

element of the boundary of the production set. The firms behave followins

pricing rules, denoting by Ó¡:\Yi -, R!+¿ the rule that firm j foliows.

We define $ :ll i=tTYi -- A" by ó@) : I|;='(Ó¡lail n A).

Consider now the following assumptions:

A.1) For each i e M

L) Xi : lRf', where k is the number of private goods and /

number of public goods. c"r¿ ) 0

2 \ u¿ is a continuous function, strictly increasing in at least one private

good and non-decreasing in the rest of private goods , stlictly

increasing in public goods and satisfies that given r, y e IR!+'

such that Lr¿(r) > U¡@) then t/¿(lr + (1 - t)a) > t/,(E) for all

the

26



0<¿<1 .

3) The wealth of each agent is described with a function r; : IRÁ+r x

V -' IR such that ,o(p,A) : pKa¿ + ILt T¿ipAi , where p¡¡ and p7

are the price vectors of private and public goods respectively, and

fo r i : 1 , . . . ,m , i : 1 , . . . , n , 0¿ ¡ ) -0 i s t hesha reo fagen t i i n t he

profits of firm j.

A .2 ) Fo reachT eN ,Y i i sac losedse t , 0 eY i andy j - IR , |+¿ ay : ( f r ee

disposal).

A.3) The attainabie set

A-- l@' , . . . ,nn ;Y) € f f X i xY 0 ! ' : " ' : rT : r r '

T n n

and (l r'r ',:rr) < t ui * (co,0)).uF- ,z

is bounded, where @ : L1j-{wi > 0 is the total endowment.

A.4) 1) / is an upper hemicontinuous collespondence, with non-empty,

convex and comPact values'

2) For every a e IIT=.TYj and" q : (qi)¡eN e Ó(v), qivi Z0 for each

j€N(Loss - f r eeRu les ) .

27



3) Le t a € l l ;= rOYj and p : (0 , pL) e A . i f (p , . . . ,p ) e { (y ) , then

there is j e lf such that p;yr, > 0.

n : ( (xn,(J¿,r i ) , (Yi) , (0n¡) , ( / ¡ ) ) whereA public economy is a tuple

i : L r . . . r n i i : 1 r . . . r f f i .

We define

D:{6:(ór,...,ó,,).rt*ln1

\- rt.. - lL

- o L for each j :

where ó;¡ is the share of agent i in the price of the jth public good.

Def in i t ion 2.1 Gtuen an econorr l ,U E : ( (Xn,Un,ro) , (Y i ) , (00¡) , (ó¡ ) ) , a Gen-

e ra l i zed , L indah l equ i l i b r i um i s an e lemen t ( ( r r , . . . , r * ) , y ,p ,5 ) e f l i rX i x

I IT= rY t xAx D such tha t

a) r' mardmi,zes U¿(r) subject to

r € X ' and pxrr t (p7r6¿)r r 1r¿(p,A)

Fo ra l l j e N , y i e 7Y t andpe ó ¡@t ) .

Tlt ere is some r¿ € IRi such that for all i e M , rir : z L at"¿d

b)

c)

(DL, rT; , r r ) : ( r ,0) + Di=, y i ,

28



Note that, the vector (p¿tró¿) is the personalized price of agent i.

In the following theorem, we state the existence of a Generalized Lindahl

equilibrium. It is less general than the theorem of Bonnisseau (1991) but

on the other hand we present a direct proof which does not use an enlarged

economy.

Theorem 2.2 The publ ic econorny E: ( (X ' ,U¿,a ' ) , (Yt ) , (0n¡) , ($¡ ) ) has a

Generali,zed Li,ndahl equili,bri'um i,f A1-Al hold.

The proof of Theorem 2.2 is a variant of the proof of Theorem 3.2 below.

For completeness, in the appendix we provide the reader with the required

modifications.

We now apply Theorem 2.2 lo some pricing rules considered previously

in the literature.

First consider the classical case in which firms maximize profits on a

convex set. The next corollary shows the existence of Lindahl Equilibrium,

in the context where the production set is convex, there is more than one

private and public good and in which there are neither endowments of public

goods nor constant returns to scale in the production.



Corollary 2.3 Let E be an econonly which sati,sf'es A.1, A.2 and' such that

(a)For each j € N, Yi is conuet and"Yi n R11' : {0} '

(b) For each j e N, ó¡@i) : {p € R11 ' : pv i>pv Vv eYi }nA;

Profit matirnization.

(c) There is j e N and, ai eYi such that atL> 0'

Then, E has a Li'ndahl equt'libri,urn.

From theorem 2.2 we also derive the following two corollaries

Corollary 2.4 Let E : ((Xn,(J¿,t,¿),(Yi),(0n¡),@i)) b" o,n econonly whi 'ch

sati,sf,es assumpti,ons A.1-A.3, and cons,ists of two types of f,rms N : I u J

such that

a) Euery frr* j e I follows the Auerage Cost Prictng rule:

ó ¡ (a i ) : { p e A : pa i : 0 } .

b) J is tzon-ernpty, and for eaery frr* j € J t'he producti'on

conuer, sati,sf,es y¡ n IRfl : {0}, and follows the rule defined by

set Yr i,s

ó¡(v j ) - -

€ J a n d{p e IRf¿ : pyi >- py VE e Y;} n A. Moreouer, there i's h

yh e Yh such that y!, > 0.

Then E lt,as a Generalized Li,nd,ahl equilibrium'

30



Corollary 2.5 If assumpti,ons A.1-A.3 hold and there are two types of f ' rms

l/ = 1¿ J such that

a) Euery fir* j €. I has an output distri'butiae production setr which

satisf,es yi n IRl+' : {0} and, follows:

ó ¡@i ) : { pe IR f " pa2py ' Va '€Y w i t hA '3y i+ }oL (Vo lun ta r y

tradi,ng) where Ai+ : max{yi,0}.

b) J is non-ernpty, and, for eaery frr* j e J the production set Yj is

conaer, satisfies yi n IRI+/ : {0}, and its behauior is modeled ba ó¡(yi) :

{ pe A : py i >py Yy e Y i } . Moreouer , the re i she J and ,yh €Yh suc l t

tha t y !> 0 .

Then, there is a Generali,zed Lindahl equi,libmurn in this econonxy.

3 CORE SELECTION

We focus on the case described in Sharkey (1989).

There are two types of private goods: capital goods and ordinary com-

modities. Suppose consumers are indifferent in the consumption of some

commodities, the capital goods. Then, these commodities do not affect the

1A production set is output distribuúive ifany (nonnegative) weighted sum offeasibleproduction plans is feasible if it involves more outputs than any of the original plans(Dehez

and Dréze (1987))

J T



marginal calculations of any consumer so the agents fully utilize the set of

these commodities in their possession. Let r e IR! and let r" denote its

projection onto the space of capital goods, ro its projection onto the space

of ordinary commodities. If A e Y then y : (a",ao, b) where oc are capitai

inputs, ao ordinary inputs, with (4., e.) S 0 and ó the output vector'

Suppose also, that there is only one frrm with an input distributive sel

Y. Recall that Y is an input distributive set if any non-negative weighted

sum y : (e"raorb) of- feasible production plans Ao : (o'" 'o'" 'bn) is feasible'

whenever a'" < a'"for all i,i.e.,when y uses at least as many inputs of every

capital input as everY Y'.

Hence,weaddthefo l lowingtotheaboveassumpt ions:

A.5) If r,r ' €lR!+i ar" such that üo: stoand r¿ : f i tL' then t/, ,(r) :LI¿(" ')

f o ra l l i eM

A.6) There only exists one firm, whose production set Y is an input dis-

tributive set which satisfies that if I : (0, ao,,b) € Y then (4,, b) < 0.

And the f i rm fo l l ows thep r i c i ng r r r l e$ :0Y -+Ade f i nedby

ófu): {p e lRl* ' : pA:0 and py 2 py' Vy' € Y with a '") a ' } )L '

A.7) There is y : (a",ao,b) eY such that y* ( ' ,0) e lRf+¿ and ó ) 0 '

32



Assumption A.7 expresses that there is a feasible production of all public

goods.

The following lemma is based in a similar one in viiiar (1994)'

Lemma 3.L LetY be an i,nput d,i,sti,buti'ae set wl'¿i,ch sati,sf,es assumpti'on A'2

and , i f 9 : ( 0 , ao ,b ) €Y t hen (4 " , b ) ( 0 '

Def,ne now, the followi'ng pri'ci'ng rule $ 1 flf --+ L by

ó@) : {p e IRl* ' : pA :0 and, py Z py' Vy' €Y with üL2 a"} n L.

Then, $ i,s an upper hemi,conti,nuous correspondence wi"th non-empty, con-

aer and, compact ualues. Moreouer, i,t is a loss-free pri,ci,ng rule.

Proof

The pricing rule / is loss-free because 0 g Y. Moreovet, since Y is an

input distributive set, it follows from results by scarf (1986) that / is non-

empty. Clearly, it is a correspondence with convex values. We derive from

the upper-hemicontinuity and the fact that is defined on the simplex A that

it has compact values. we only need to prove that it is upper hemicontinuos

or in this case that it has a closed graph. Let y € Y and consider a sequence

(p",A') converging to (p,y) such thal y" e AY and po e Ó(y") for all a'

ó.)



SupposeF(ó@then the reex i s t y , €Yw i thd , " }d " such tha tpy ,>Fy .

For a sufficiently large we obtain t'hal p"y' ) p"Yo'

I f ,a" :0 then a ' . :0 and (a ' , ,ó) < 0 ' Therefore,0 ) p*(0, a 'o 'b) : poa ' )

poyo :0. Which is a contradict ion'

Now, define

a'l¡ : a'", * ¿ if o'"¡ < 0 and o'li :0 otherwise'

o'lj: o'oj - 5 and b'j : b'j - 6'

Taking e ) 0 and 6 > 0 such that V" : (o'! ,o' | ,b") € Y and py" >Ft (this

is possible because Y is comprehensive and close). B:ul a'!) a'")- n"' Now,

for a suff iciently large, (a7,a7,,b") is close to (d",oo,6) such tb'at a'! )- ai

with p"y" ) poA" which contradicts that p" e Ó(a")'

The next theorem establishes the existence of a particula'r core selection'

Theorem 3.2 An economA E : ( (X i , {J¡ , ' i ) ,Y, (0u) ,$) has a Genera l i 'zed

Li,nd,aht equr,lr,bri,um if A.1-A.3, A.5-A.7 holds. Moreouer, thi's eqttili 'brium

belongs to the Core of the economY'

Proof

34



The proof consists of three steps. In the first step we construct a non-empty,

convex and compact set. In the second, we define a collespondence to which

we apply Kakutani's theorem and the third where we check that the fi-xed

point obtained is a Gener alized Lindahl equilibrium by means of five claims.

STEP 1:

Let Q € IR+, we denoteby C: { r € IRf" r " = 0} n [0,8e] theequal

truncated consumption set for each agent. From lemma 5.1 of Bonnisseau

and Cornet (19SS) we define an homeomorphism between er and 0Y and

from it we have that y(s) : s - '\(s)e where s € er and '\(s) is defined by

the homeomorfismo. Also, we define, A. : {p € lR¿ : D'n=rpn : !,Pn Z -e

w i t h á - 1 , . . . , k+ l \ '

using lemma 5.1 of Bonnisseau and cornet (19s8) and assumption A'3,

we may choose e > 0 small enough, Q large enough and a closed cube B

centered at 0 in the Euclidean space T satisfying i) and ii) below:

i) I f z e Y +(r,0) then z KQe, where y i t th" projection, on the space of

production, of the attainable set. As a consequence of this, if r e *i

then (0, f lo,,rr) 4. Qe, where *n i t the projection, on the space of

consumption goods, of the attainable set'

35



i i ) {s € T : y (s) * (o ,0) e

Let G: f17tC x B x

compact set in the Product

R1*'] c int(B).

A. x A x D, which is a non-empty, convex and

topology.

STEP 2:

First, we define the demand correspondence' Let (p, ó, ¿) € A' x D x IR'

B¿(P,6,t) : {r € C : PP4¡ * @7rt6¿)rn S t},

t¿(p, ,5, t ) : { r e B¿(p,ó,ú) : Vr¿ e B¿(p,6, t ) } U¿lt) > Ut( 'n) '

u¿(p,6, t ) : {n e C : pvrv t (p7t5¿)r¡ : inf(p¡< C x t 1 'p"a6;)C r)} '

And

r;(p, 6,,) : { 7,((o',u;,t,\ i,ni,i,llt

(prc t< * @7n5¡)c 1)

Now we are in a position to d.efine the correspondence to which we apply

Kakutani's theorem. Let ,F : lll=, F, : G --+ G be a corlespondence such

that:

Fr(r,s,p,pt,ó) : l l l , f¿lp,6,psai lg;PU(s)) is the demand correspondence'

F"( r ,s ,p,p l ,ó) : {o e B, (p - p t ) ( " - o ' ) >0 Va' € B} ensures that in

the fixed point all frrms agree on the price of the goods'

Fu(* ,s ,p,pr ,ó) : {q e A. , (q - q ' ) (D7r@i,0, (6onrV)) - v( " ) - ( ' ,0) ) >

0 Vg, € A.) guaranties the feasibility of the equilibrium in the fi'xed point'

36



Fn(r,s,p,p!,ó) : d(y(t)) prices have to be acceptable for al l f irms' And

F r ( * , , s , p ,F t , ó ) : { óe D : 6 ¡ €a rgmax f [ , 6¿ i r¿ r i VJ :1 , " ' , ¿ ]

obtains the share of the price of public goods which equalizes the demand of

public goods by the agents.

The correspondence Fr is the classical demand and is upper-hemicontinuous,

with non-empty, convex and compact values by assumption A.1, Iemma 5'1 of

Bonnisseau and Cornet (19SS), lemma 1 in Debreu (1962) and the maximum

theorem. Fz is the maximum argument of a linear function on a non empty,

convex and compact set. Thus, Fz has non-empty) convex and compact val-

ues. It is upper-hemicontinuous by the maximum theorem. Fe satisfres these

properties by lemma 5.1 of Bonnisseau and Cornet (1988) and the maximum

theorem, and so does F¿ by lemma 3.1. To finish, F5 is again an upper-

hemicontinuous function, with non-empty' convex and compact values by

the maximum theorem. It follows that we can apply Kakutani's Theorem to

the correspond.ence,F to f ind a f ixed point ("*,s*,P*,P*1,ó*) ' That is, the

following fi.ve equations hold.

r*i € f¿(p*,6*,r i) for each i e M (3 .1 )

,l'1



(p. - p"t)r* \ / x x l r u - Da \p -p ' ) s vsc .o

p*1 )_ p'l Vp € A.

(3 .2 )

(3 3 )

(3 .4 )

(3 5)

p.t e ó(v.)

m

r i I f {a ; r r t ' ) fora l l i€Mh.=7

where rl : r¿(P* rA*), A* : Y(s.) and

r : ( f i , f (óX¡rÍ)) - (r , ,0) - y. .; - l ; - 1

STEP 3:

From these equations we wil l derive that ((r. ') i€M,U*,P*,6") is a Gener-

alized Lindahl equilibrium fot E, by proving the following fi've claims.

Cla imI : r€-R!*u{0} .

First of all, we prove that p.y. 2 0. Taking s : 0 in inequality 3.2' it

is obtained that (p. - p*l)s* ) 0, hence p*s* ) p*t"*. Moreover, since gr* :

s* - )(s*)e, appiying lemma 5.1 of Bonnisseau and Cornet (1988) and taking

38



into account that p*e: yf7e, we obtain that p*y* ) p*'y*. Consequently

p*U* ) 0 by lemma 3.1 and equation 3'4.

Since p*y* ) 0 and 0 e C¿ we obtain lhat |¿p*y* > 0 > inf((pinói)Cr)'

Moreover, ri : pi¡w¿ * g¿p*y* 2 inf (plCr) + int11pLD6;)CL): inf (pirC¡ *

@in6)C) for all i € M, because ui e CN and ?¿p*v* > inf((p¿n6i)Cü'

Taking the definition of fl1 into account we see that pirri! ¡@ia6f)rt' <

r[ for all á € M. So,

*+r** i ,p*'t : L,@it** + (einói) ri - ri) < o

and byinequality 3.3pr sp..t ( 0 for al lp € a,. This impliesthat 1€ a?,

the polar cone of A., so z e -R/++ U {0}.

Claim II: p*7 : p* and p- € Ófu.).

Since, by claim I, 7 € -R!* U {0} and C c IR!+/, we see that y* * (o, 0) e

A,l*,. From the choice of B, s* e int(B). Consequently, by inequalíty 3.2,

the gradient at s* of the linear function -* * (p* -p*t)t is equal to 0, hence

p* : p*7, and the claim is proved. In particular, p* ) 0.

Claim III: r*i e €¿(p*, ó*,, r¿(p*, (y.'):e ¡¡)).

claim II shows that p* € a. since 0 e c we have that inf(pfuc7r

@2=5i)Cr) :0 for all i e M. Suppose, for contradiction, that fo..ri

39



pk.,, + p*y* -- 0. Since, p*A* 2 0 and w : D¿ewui > 0 we obtain that

p*A* : 0 and pir : 0. In the proof of claim I, we have obtained that

r*i ) inf.(pfuC.¡ + kf¡ói)Cr):0, for aII i e M. As a consequence, we

d.erive that r*¿ : 0, for all i e M. This implies that p*1 : 0. But p* € A

and 7 e _R!* u t0). Hence, 1 : 0. Since, the demand of capital goods

of ali agents is 0 and ? : 0 we have that y* = (u",aI,b.)' To sum up'

p* : (0 ,p; ) € d(y . ) , p*y* :0 and p*y* ) p*y for a l l g - (&" ,eo 'ó) e Y such

that 4," 2 ol : ol". But this contradicts assumption A'7'

Thus, there exists h e M such that "[

> 0. since Y + (r,0) c c, we

have that y* * (u,0) < Qt. BY claim I,

(I "i., f taio";)) I (.,0) -F v",i =7 i = \

so, DLr rft I w t yk K QeN' Hence, rfr < Qex' We must have that

pk > 0, because t*h €. €n(p*,,6",ri) and [/¿ is strictly increasing in at least

one private good. From wi > 0 for all i g M and taking into account thai

p*a* ) 0, it follows that for each i e M ri > inf(p|¡cx + @LJ':)CL) and

r*¿ € t¿(p*,6*,r i ) .

claim rv: For some ri e IR¿' rT: nL' for every i' € M' and

40



, : (i,ru,i$:n,Í)) - (',0) - É v*i : o: - 1 ; - 1

Since í '+ @,0) c C, we have that y* *(*,0) < Q"' From equation

3.b, for each i e M,, ri < Dl1(óinrf). From claim II it follows that

f[r(óilrfl s aL.\Aie conclude that ri < Qe¡. Since, the preferences are

strictly increasing in public goods, we see from claim IIi that (p¿¡ót) > 0

for all i e M and p € A. Furthermore, ó¿ ) 0' Since, f[t(óf lri]) :

suprn¡,,{rf } we have that there is ri e IR¿* such that ri: zl, for all i e AI '

If. plorfr + (einó[) r{ < ri for some á e M, then, becatse tl K Qet

and the preferences are strictly increasing in public goods, there is r € C,

such that pkrN + @LJ6i)r7 1ri und {/n(r) > un(**h), contradicting claim

III.

Thus, we must have that Pi;üfr + @ia6i)rT : 'i for all i e M; so that

p*1 :0 . S ince ,p . ) 0 andT e -R ' * *U {0 } , weconc ludetha t 7 :0 '

Claim V: For each i € M, Uo(**¿) 2 U¿(r) for every r € X' such that

pkrx + (pinói)rt" I ri .

Suppose there are i € M and. r' € Xi such that Un(t.n) < U¿(*'). From

assumption A.5, we have that r : (0,r 'o,r 'r) í r 'satisf ies u¿@): u¿(r ') >

4T



Uo(r*i).Take the convex combination ¡tr * (1- p)**o with ¡-¿ € [0,1] ' Since

r*i e *¿ by i) r. i K Qe and for a ¡;* small enough p*r+(r- P*)"" ' € C' But,

by assumption A.7 we have that tJ¿(p.r * (1 - p.)r.n) > (J¿(r*;).As a conse-

quence of claim III, ¡t*r + (1 - lt*)**o does not satisfy the budget constraint

of agent 'i. After some computation we obtain that pirl¡¡ + (eilói) r'7 > r!,

and the claim is proved'

Finally, we prove that this Generalized Lindahi equilibrium obtained is

in the Core. Suppose there are S C l{ and ((rt,z);es,y) such that

(l*n,, 4 : (U,o¿, o) * y and¿€s ¿€5

[Jn(*¿,2)2U;(r f t , r i ) for a l l i e 5

with at least one strict inequality.

By claim III,

pkrd * ('pLü6;)z ) ri for all z e S

with at ieast one strict inequality..Hence,

nkl*n+\(r in6)z t I '1.¿€s i€s ¿€s

(3 .6 )

(3 .7 )

/ 1 ,



Since, 0 ( D¿.s ó; ( 1 we have that f,."bl¡6i) I pltrl : pi and

pLt*n+P7"tI"; '¿eS i€S

l\ow, by feasibility, equity 3-6, we derive that

* , \ - , r * , r \ . - ? ^ \ . \ - \ - . - *

p-(Lr ' . z ) : e - t (2_u" 'UJ * A ) > l rn .ies ¿€s ie.S

!

Finai ly, s ince p*y* :0 then Do.r t i : p.(D¿es@',0) '

Furthermore, p*((D¿ e.sai,,O) + v) > p.(D¿eso',0)' Thus, p*Y ) 0'

Since p*A* :0 and by feasibil i ty y : (&.,&o,ó) satisfies a" ) -a"' t 'he

fact that p*y ) 0 contradicts p* e Ó(y.)'

Although, it was proved by sharkey (1989), by means of the balanced

condition, that in this context the Core of an economy with public goods is

non-empty, with this theorem we find a particular selection of it in which

consumers maximize their uiility subject to the budget constraints and the

firm maximizes its profits subject to the availability of inputs. The existence

of the social equilibrium with personalized prices that holds in the core of

the economy is shown.

1 q+.)



4 APPENDIX

Proof of Theorem 2.2

we only provide the changes with respect to the proof of theorem 3'2

Pick Q € IR+ and let c: [0,Qe]. We may choose e ] 0, Q and B again

such that the following hold

i ) i + ( r , 0 ) CCand * i cC .

i i ) {s eTlDl ,=ry i ( ' ) * ( r ,0) € ]R,1* ' } c in t (B) .

The correspondence F is now defi'ned on G : I\LtC x B xA. xfl!, A x

D, where F : lll=, F, and

Fr(r,s, p, (d )¡e¡¿, á) : I\L, f o(p, 6, p x ui + D? =t 0 ¿¡ pv' (t))

Fr (* ,s ,p , ( f ) ¡ew,6) : { (or ) ¡e¡¿ e B ' DT=t(p- f ) (o¡ -o} ) 20 v(o} ) ¡eru €

D l-DJ

Fu(*,s,p, (1)¡e¡¿, ó) : {q e S. , (q-q')(D!t(r¿*,(6¿zrt))- I t t a 'G)-

( ,u ,0) ) )0 vg 'es . )

Fn(r, s, p, (f ) ¡ e¡,t, ó) : d((y' (t))¡.n¡)

Fu( r , s ,p , (1 ) ¡e ¡¿ ,ó ) : {6 e D : 6 ¡ € a rgmax f [ r 6¿ i r¿ r i V i : I , " ' , 1 ]

From Kakutani's theorem we find a frxed point (r*, s* ,p*,(p*i)ir¡r,6*)'

44



Letting

we have

ÍrL n1 n

^u : l \ - r ;1. \ - r ¡ l t r ; , ) ) - (w,o) - f u| - \ , L * ^ ) / . \ " x , L r: - 1 ; - 1 ; - 1

r"' € f¿(p*, ó*, "i) f o reach ieM 1 4 R )

n n

l lr- - p*j\s: t ) l(p. - p*j)t¡ v(sr)re¡¡ e B, L \ r r J - J -

/ _ J \ f , t / r \

; - 1 a - ]. J - r

p.l 2 pr Vp € ,5,

(P- ' ) ¡ . ¡ , € d( (Y. ' ) ¡ . t )

(4.e)

(4 .12)

/ / . 10 )

(4 .11)

n1

xf < !{fiarf) for all i e M

The proof of claims (I) and (II) is analogous to theorem 3.2. It is oniy

necessary to take into account that there are n firms. In the ploof claim (III)

i t is obtained that p*y*i :0 for al l j e N and pi¡ :0 which contradicts

assumption A.4.3. The rest is now analogous.

, t E't.)



Finally, claims (IV) and (V) foitow the same argument as previously but

with n firms.

REFERENCES

Bonnisseau, J. M. and cornet, B. (1988): "Existence of Equilibria when

Firms Follow Bounded Losses Pricing Rules,"

Journal of Mathematical Economics, L7, 119-148'

Bonnisseau, J. M. (1991): "Existence of Lindahl Equilibria in Economies

with Non-convex Production Sets," Journal of Economic Theory , 54,

409-416.

Debleu, G. (1959): Theory o! l/u]g1 l\ew York, Wilev.

Debreu, G. (1962): "New concepts and techniques for equilibrium analysis,"

Int. Economic Review, 3,257-273.

Dehez, P. and Dréze, J. (1988): "Distributive Production sets and Equi-

libria with Increasing Returns," Journal of Mathematical Economics,

t7 ,23 t -248 .



Diamantaras, D. and Wilkie, S. (1994): "A Generalization of

Ratio Equilibrium for Economies with Private and Public

Journal of Economic Theory, 62,499-512.

Kaneko's

Goods,"

Foley, D. (1970): "Lindahl's solution and the core of an economy with public

goods," Econometrica, 38, 66-72.

Guesnerie, R. and Neuefeind, W. (1985):

Firms Follow Special Pricing Rules,"

"General Equilibrium when Some

Econometrica. 53. 1369-1394.

Khan, A. and Vohra, R. (1985): "on the Existence of Lindahl Equilibria

in Economies with a Measure Space of Non Transitive Consumets,"

Journal of Economic Theory, 36, 319-332.

(1987): "On the Existence of Lindahl-Hotelling Equilibria,"

Journal of Public Economics, 34,143-158.

Mas Colel l , A. and Silvestre, J. (1989): "Cost Share Equil ibria,"

Journal of Economic Theory, 47,239-256.

Scarf, H. E. (1936): "Notes on the Core of a Production Economy,"

Contributions to Mathematical Economics (in Honor of Gerard Debreu).

ch . 21 .

47



sharkey, w. w. (1989): "Game Theoretic Modeling of Increasing Returns

to Scale," Games and Economic Behavior, 1-, 370-431'

Villar, A. (199a): "Equilibrium with Nonconvex Technologies,"

Economic Theory, 4, 629-638.

48



CHAPTER 3

EFFICIE}{CY, MONOTO}{ICITY ANTD RATIOI\ALITYI}{ PUBLIC GOOD ECO}{OMIES

Abstract

In economies with public goods, we provide a necessary and suf-ficient condition for the existence of cost monotonic selections fromthe set of Pareto optimal and individually rational allocations. Suchselections exist if and only if the preferences of the agents satisfy whatwe call the equal ordering property. This requirement is very restric-

tive in the context of more than one public good. Howevet, whenever

it holds, any such mechanism must choose an egalitarian equivalent

allocation. The equal ordering property is no longer necessary in thecase of economies with quasi-linear and separable utility functions andseparable costs. In this framework, there is essentially only one Paretoefrcient and individually rational allocation which is cost monotonic,

and it is an egalitarian equivalent allocation.

49



Introduction

Consider the question of finding the optimal allocation of a bundle of public

goods, and the way in which its cost shouid be shared by the agents who

consume it. Most of the literature has concentrated on two aspects of this

problem. Firstly, there is the controversy of designing mechanisms which

induce the agents to reveal their utilities; one would expect that, in most

cases, the agents have strong incentives to hide their true utility regarding

the public goods. Secondly, as in the present work, there is the issue of

selecting an optimal bundle of public goods and distributing the cost involved

in financing the production plan amóng the members in the Bconomy.

To address this problem we adopt the normative approach: The solution

is determined by considering some "equitable" properties which are agreed

upon by the agents and which express their sense of fairness. Once the

relevant "ethical guidelines" have been acknowledged, one tries to pinpoint

a solution complying with them. If one is found, then it is applied to the

ploblem at hand.

A universally accepted property is Pareto optimality. Allocations for

which it is possible to improve the welfare of some agents without makinE

50



the rest worse off, should not considered. However, Pareto efficiency by itself

has one major drawback; it does not determine a unique aliocation. What

is worse, it contains proposals such as "one agent absorbs all the surplus"

which are objectionable on grounds of non equitability.

Individuai rationality is another of the most widely accepted requirements

for a solution to have. Since the technology is jointly owned by all members

of the society, it seems reasonable to require that the optimal production

plan and its financing should possess a certain degree of unanimity. In this

framework, this corresponds to individual rationality; a possible allocation

will be objected to by some member who operating the technology on his

own, could improve his utility.

The same objection to the one considered above for Pareto optimality

also applies to the latter solution concept: quite often, the set of individuall.'

rational allocations, even with the added requirement of Pareto optimality,

turns out to be a very large set. And there is no obvious way of picking an

appropriate selection from it because there seems to be no single universal

solution which would satisfy everyone's sense of fairness. This naturally leads

to the question of finding relevant situations in which there is a suitable one-

point selection process'

/ 1C I



Another property considered in the literature as being desirable is "cost

(or technological) monotonicit¡" i.e., if the publicly owned technology gets

better, then no agent should be worse off. Technological monotonicity was

introduced by J. Roemer (1936) and has been used subsequently to study

some solution concepts (see, for example, Moulin (1987b), Moulin and Roe-

mer (1989), Roemer and Silvestre(1987)).

In the case of just one public good, H. Moulin (1987a) has characterized

the egalitarian-equivalent solution, proposed originally by E. A. Pazner and

D. Schmeidler (1978), as the only selection from the set of Pareto efficient

allocations which satisfies cost rnonotonicity and the Core property. Due

to the interest of this result, it seems very natural to ask whether it can be

extended to wider contexts.

In the present work, we characterize when such an extension is possible.

we find that in a setting very similar to the one in Moulin (1987a), but

with several public goods, the axioms we have just discussed are not aiways

compatible when taken together. As we prove in Section 4, under some mild

assumptions, there is a mechanism satisfying the three properties above if

and only if the preferences of the agents satisfy the equal ordering property.

This latter condition, which is very natural in the context of one public and

X O¿ L



one private good (as in Moulin (1987a)), is severely restrictive in general' For

example, with quasilinear, strictly increasing preferences in public goods, the

equal ordering property is equivalent to the assertion that all the agents have

the same ordinal (but not necessarily cardinal) preferences on public goods.

Thus, with several public goods, one needs to impose additional restric-

tions on the preferences of the consumers in order to find cost monotonic and

individuaily rational selections from the set of Pareto efficient allocations.

on the other hand, when such a mechanism exists, then: (1) It must pick

an egalitarian equivalent allocation; (2) the latter form a subset of the core;

and (3) all cost monotonic mechanisms are equivalent, i.e,, they provide the

same utility profile to the agents. Thus, the results in Mouiin (1987a) cover

essentially all the cases for which a cost monotonic selection mechanism from

the set of Pareto optimal and individually rational allocations is possible.

The difference between just one and several public goods is that, in the

first case, there is no conflict of interests: everybody likes more of the public

good. Nevertheless, with more than one public good to choose from, different

agents might differ in their opinions about which should be given priority

over thr -

ers creating, thus, a possible soulce of conflict. Clearl¡', under

the equal ordering property, these discrepancies in priority do not arise.

53



One possibility of avoiding the above impossibility result, is to restrict

the domain of the mechanisms considered. Indeed, in Section 5 we show that

there is an egalitarian mechanism characterized by cost monotonicity, Pareto

efficiency and individual rationality provided one looks only in the reduced

class of ttseparable economiest'.

The model

We consider economies with one private good and, possibly, more than one

public good. The space of public goods is Y : IRi, with m ) I. These

are produced at a cost which is financed by the members of the society.

The technoiogy available to produce the public goods is described by a cost

function c:Y -----+ IR... Given A €Y, the cost, in terms of the private good'

needed to produce the bundle y of public goods is c(y). In addition, the

technology is jointly owned by all the agents and only one bundle of public

goods is eventually produced.

We will assume that whatever technology we consider-, it exhibits some

bounded returns to scale when producing very large bundles of public goods.

Of course, this does not preclude having arbitrarily large increasing returns

to scale for public goods within some compact set.

o+



Assumption 2.1 The technology c : Y -----+ IR-¡- is cont'inuous, nondecreas-

zng, sattsf i ,es c(0) :0 and'

limsup l]qlL . +o.l lg l l*+o" c\9 J

We use the Euclidean norm llyll : "rffi^fy For the purposes of com-

puting the lim sup we adopt the following convention: Consider the extended

real line IR* : IR U {-l-oo, -oo}. We also extend the usual ordeling on IR

to IR* by defining -oo < r < *oo for any real number r € IR and we let

l ly l l l "@): *oo whenever c(y) :0 .

As usual, the ordering in Euclidean space is defined as follows. Given two

vectors r,z € R?, r ) z (resp. n )) z) means that r¿ ) z¿ (resp. r i> z¿)

for every i - - I , . . . ,mi the notat ion r ) z ind icates that r ) z and r I z .

We let N : {1, . . . ,n} denote the set of agents. The init ial endowment

of private good for agent i e N is (,¿ € IR* (rn : *oo is allowecl). Each

agent i e N has preferences over public and private goods represented by a

utility function u¡ : Y x X¿ + IR, where X, c m' is his consumption set of

private good. By abuse of notation and for simplicity, we will wlite tt4(y,t)

r -a the r t han u¿(y , r¿ - t ) . I n o the r words , f o r each i : I , . . . , f r , u ¡ ( y , t ) i s t he

utility obtained by agent i when the bundle y of public goods is implementecl



and he has to contribute the amount f from his private endowment towards

its financing. The payment f could be negative, meaning that agent i e N

receives some compensation from other agents for accepting the bundle g

instead of another one he might have preferred to y.

We use the notation r(S) : D¿e s r¿ fot a non empty subset S C 1/ and

a vector r € IRa, with S g I q ¡ú.

Assumption 2.2 For each agent i e N, the followi,ng assurnptions hold:

(ü Xn : [ - cu ( / / ) *w¿ , , t ¡ ] .

(i,i,) The utitity functi,on u¿ : Y x X¿ ------+ R" i,s conti,nuous non-decrea.si,ng in

the first argument (public goods) and, strictly decreasi,ng in the second

(tl-re priuate good).

(iii) Giaen a giaen bundle of publi,c good,s a € Y , there is one and only one

p¿(y) e X¿ such that u¿(y,p¿(y)) : u¿(0,0) : 0 ' The mapping p¿(y)

uerifi,es

lfiTHffi:oBy (ii) arrd (iii) there are no public bads and the amount of private good

which agents are willing to provide for the consumption of a fixed bundle of



public goods is limited. Note that uo(U,t) is deoeasing in the private good

to ind.icate that I denotes a payment. For convenience, we have normalized

u¿ (0 ,0 ) : 0 f o r each i : L . , . . . , 77 .

Part (i) says that no agent can contribute more than his own endowment

to fund the construction of the public good; thus restricting total investment

towards the construction of the pubiic good to ,(¡i). In the co,s€ cr.r¿ : *co,

for some i g /{ then we take u.'(1ú) - Lüi : *oo. As mentioned above, agents

are allowed to transfer part or all of their endowments to other agents to

encourage them to accept a particular bundle of public goods. The results

below will siill hold whenever transfers of private good are not allowed, as

long as the utility functions are (strictly) monotone in public goods. One

needs only to modify the argument in Remark 6.1 in an easy way.

It fol lows from (i i) and (i i i ) that, for i : L,. .-, f t , the mappings p¿ are

non decreasing and satisfy gr(0) : 9.

We let X : IIT=.X¿ and extend the utility functions of the agents to

Y xX ,by u¿ (y ; t ) : u¿ (y ,ú¿ ) , w i t h ( y ; t ) : ( a ; t t , " ' , t n ) €Y xX and i e j v '

The utility profile of the agents is the mapping u : Y x X ----+ IR' given

by u ( y ; t ) : ( u1 (y , t t ) , . . . , u . ( y , t . ) ) . L i kew ise , we w i l l a l so w r i t e p ( y ) :

kr@), . . . ,v"@)) .

É..7d f



From now on, we fix the set X and a profile of utilities satisfying assump-

tions 2.2. An economy is a pair (u, c) consisting of a utility profile and a

technology. Since the utilities of the agents are fixed throughout the paper,

we will use the notation c instead of (u, c) to denote an economy. An alloca-

t ion is a pair (z;t) eY x X. The al location (z;f) is feasible in the economy

cfor anon empty coali t ion Sc // i f c(z) <f(,9) with I € f l¿.sX¿. We wil l

simpiy say that (z;ú) is feasible whenever it is feasible for the grand coalition

A/.

Given a technology c, an allocation (z;t) is said to be Pareto optimal

in the economy c if it is feasible and u(z;t) : u(y;s) for any other feasible

allocation (y;s) such that u(z;t) 3 u(y;s). The set of Pareto optimal al lo-

cations is denoted bV P(c). A nonempty coalition ,9 C lú can improve upon

an allocation (z;f) if there is another allocation (y;"), feasible for S, such

that u¿(y;s) ) u¿(z;f) for each i e ,9 rvith at least some strict inequality.

An allocation (z;ú) is individually rational (resp. in Core(c)) if no agent can

irnprove upon it (resp. if no coalition can improve it)'

có



3 Egalitarian equivalent allocations

One of the principles we will be considering to determine the allocation which

is "optimal" for the society is cost monotonicity. As we will see, if a solution

satisfying this requisite exists, then it has to select an egalitarian equiva-

lent allocation. In the present section, we review this notion. Consider a

technoiogy c:Y -+ IR.. satisfying assumption 2.1.

Definition 3.1 The set of egalitarian equivalenf allocations i,s def,ned to be

EE(c ) : { ( r ; t ) e e ( c ) : t he re i , s zeY w i t hu ( r ; t ) : u ( r ;O ) } .

The bundle of public goods z appearing in the definiiion of EE(c) is

the reference bundle. The egalitarian equivalent solution was proposed by

E. A. Pazner and D. Schmeidler (1978) and has been characterized in Dutta

and Vohra (1993) and Moulin (1987a). An alternative procedure to describe

the set EE(c) (more appropriate for the present set up) is given in the fol-

lowing construction.

Let S--1 : {z € IR- : l l r l l :1} denote lhem - 1 sphere and consider

its posit ive orthant

S? -1 : S * -1 aYT

and

59



Given ae Si-r define

F"(") : { ) e IR¡ : there is (y; l ) € Y x X with

c(y ) < t ( /ú ) and u(y ;ú ) : u ( )a ;0 ) ]

and the application Ho : Fo(c) ---+ R^ by H"()) : u()a;0). Note that

F"(") f 0,since 0 e FL(c). The function Ho is nondecreasing because u is

nondecreasing as well.

The set of egalitarian levels along the ray o e Sf -1

in the economy c is

defined as

EL"(c) : {)a : ) € argmaxfl"}

and the set of egalitarian leveJs is

ñ T / \ | I / \t -L(c) : U I rL"(c) .

aeSf-1

The feasible allocations supporting the egalitarian levels are the sets

At rL" (c ) : { ( " ; t )eYxX : c (a )<¿( ¡ú ) and there is

z €EL*(c ) w i th u ( r ; l ) : " (z ;0 ) ] .

The relationship between the two concepts presented above is given by

the following observation.

60



Remark 3.2 Suppose assumpti,ons 2.2 hold and let c : Y --+ R"¡ satisfy

assumption 2.1. Then

U AEL*(c) cEE(c) c U AEL"(c).oesf;l aeSi-L

The second inclusion might be strict whenever preferences are constant

on the boundary of the positive orthant (e.g. Cobb-Douglas preferences).

In such cases, the boundary of the positive orthant is contained in the set

of egalitarian levels and typically will not be supported by Pareto optimal

allocations. Remark 3.2 combined with the next straightforward result shows

that under rather weak assumptions, egalitarian equivalent allocations exist.

The proof of it is in the Appendix.

Proposit ion 3.3 Suppose assumptions 2.2 hold,. Then, EL"(c) f 0, for any

c:Y ---+ R ¡ satósfyi,ng 2.1 and any a € ST;t.

As the following example shows, when the number of public goods is

greater than one, there are two problems associated with the egalitarian

equivalent allocations: firstly, in many cases there is a continuum of egaii-

tarian equivalent ailocations which are not individually rational. Secondly,

the ones which are individually rational, form a continuum of allocations;

61



furthermore, there does not seem to be a natural procedure for selecting any

one of them, since they yield different utilities to the agents.

Example 3.4 The economy c consists of two public goods (so Y : IR1) and

two consumers with quasi-linear preferences in money given by the utility

functions

ut(y ; t ) - - 2{y1 - f2r /y , - t , uz(y ; t ) : 2 \ /y1 - t

where U : (Uyyz) e Y. The cost of producing the bundle A e Y of public

goods is

c (Y ) :9 t *Yz '

It is easy to compute ( for example in Moulin (1987a) or Mouiin (1988)) that

the set of egalitarian levels is

EL(c) = {a eY : 4 t /y1 iz t /y , : 5¡

Only a strict subset of the egalitarian equivalent allocations are individu-

ally rational. The set of utilities given by individually rational egalitarian

equivalent allocations is

I J : { ( u t , u 2 ) : q I u z : 5 , u 1 ) n 2 , D 1 } 2 , u 2 } L } .

62



Hence, not all the egalitarian levels provide individually rational allocations

and, furthermore, there are several distributions of utilities in the set U.

4 Cost monotonic mechanisms

One way to overcome the difficulties mentioned at the end of the last section

is to consider alternative properties to those of optimality and technological

monotonicity in order to narrow down the solution to the cost allocation

problem. In this section, we consider whether any obstructions exist to con-

sider individual rationality as a third axiom compatible with the other two

just mentioned.

A mechanism R will be defined below to be a mapping assigning a feasible

allocation to each economy c. In order to give a precise definition we need

to specify the domain of R. As a first step, we consider the following set for

the admissible technoloqies.

Y ------+ IR*/. satisfies assumption 2.1 ]

However, there are cost functions c € Eo for which the economy c does not

have any allocation which is individually rational for the agents. In order

to avoid such economies, we will consider only technologies satisfying the

63



condition

E1 : { c € Eo : c ( r v y ) < " ( r )

+ " ( y ) } .

where , g i ven any two vec to rs * : ( r r r . . . , r ^ ) , U : (U t , . . . ,A * ) i n ]R - we

denote rV y : (max{r1, Ut}, . . . ,max{r*,U*}). The requilement c(rV y) <

c(r) * c(g) guarantees that agents can benefit from cooperating.

Remark 4.1 We observe that for any c € Er the set of individually rational

and feasible allocations of c is non-empty. Indeed, iet c € E1 and suppose

that for each agent i :Lr . . . , , f r , thevector ( * ' , t¿) € Y x X¿ is a so lut ion to

the problem

rnax {u¿ (z , r ) : ( z , r ) e Y v X ; , c ( z ) : r ¡

Assumptions 2.1 and2.2 guarantee that there is a solution to the maximiza-

t ion problem. Let r : trr V 12 V . . .Y rn. Then

n

c(r) < I r ( ' ' ) : ¿(1ú)i="1

Hence, (r;f) is feasible for the grand coalition in the economy c and every

agent i e ,¡/ wil l be at least as well off with (e;ú) as with (ct, ú¿).

The domain Er still includes some technologies which might be considered

unreasonable like cost functions which are constant on arbitralily iarge sets.

64



We will eventually show the non existence of mechanisms satisfying certain

normative axioms. Of course, showing the non existence of such mechanisms

for the smaller domain, immediately implies the same result for the larger

one. Thus no loss of generality is entailed in further restricting the domain

o fR to

E : {c € Et : "(")

> c(y) whenever r } y}

Definit ion 4.2 A mechani 'sm'is a mappi,ng

R: E ------+Y, flX,?€¡ l

assi,gni,ng to euery technology c € E an allocati'on, R(c), feasible in the econ-

orny c.

A mechanism -R is Pareto efficient (resp. egalitarian equivalent) if E(c) e

P(c) (r 'esp. R(") e EE(c)) for every c € E.It is individually r-ationai i f f t(c)

is an indiviclually rational allocation in the economy c. The mechanism .R is

said to be cost monotonic if, given two cost functions c1,c2 e E such that

ct(y) 3 c2(y) for every U €Y, i t assigns al locations R("¡), fol j : 1,2, such

that u(R(c1)) > u(R(cr) ) .

In the context of one public and one private good, H. Moulin (1987a)

has proved the existence of cost monotonic, individually rational and Pareto



efficient mechanisms. He also shows that such a mechanism must select an

egalitarian equivalent allocation. It is easy to check that c1V c2 € B whenever

c1,c2 e E, where (c1v c2)(y) : -u*{"t(y),"r(y)}. Using this remark, one

can verify that the same proof that is used in Moulin (1987a), also applies

here to obtain the followinq result.

Lemma 4.3 Let R be a Pareto optirnal and cost monoton'ic mechan'ism.

Then for each c1, cz € E we haue that eit l '¿er u(ft(c1)) > u(R(cr)) or u(f i(ci)) <

u(R(cr))

We now address the main issue: given a fixed set of agents, are the axioms

of cost monotonicity, Pareto efficiency and individual rationality compatible?

The key to answering this question lies in the equal ordering property.

Definition 4.4 We say that the agents order the bundle of publi,c goods

equally (or that the profile of utili,ti.es u satisf'es the equal ordering prop-

er ty) wheneaer for each bundle of publ ic goods A,z €Y i f u¿(y,0) > u¿(2,0)

for some agent i e l / then u¡(y ,O)> u¡(2,0) for euerA other agent j e IV.

In other wolds, the equal oldering property is fulfilled whenever given

U,z e Y, e i ther u(A,0) ) u(2,0) or e lse u(2,0) ) u(A,0) . So, i f a consumer

oo



prefers to take the bundle of public goods z for free, rather than choosing

bundle y, then so do all the other agents. This property eliminates the

possible sources of disagreement among players in ranking the bundles of

public goods. It cleariy holds in the case of one public good'

Example 4.5 We illustrate this notion in the case of quasilineal utility func-

tions. The utility of each agent i e lú is given by

u¿ (y , t ) : b¿ (y ) - t

where b¿ : Y -+ IR is the utility obtained by agent i e lú whenever he enjoys

the bundle of public goods A € Y for free. The equal ordering property

is equivaient to the following statement: for each pair of bundles of public

goods U, z € Y, either b¿(y) >- b¿(z) for every agent i e I/ or else b¿(z) 2 b;(a)

for every agent i' e N.

We can now answer the question posed above concerning the compatibility

of cost monotonicity, Pareto efficiency and individual rationality. The result

stated below makes precise the conditions under which thele is a solution

satisfying these three properties.



Theorem 4.6 Let /ú : {1, . . . ,n} be a set of agents whose profile of uti,li,ties,

u : (ur,.. . ,un) : Y x X --+ 8y satisf ies assumpti,ons 2.2. Then, there

i,s a cost monoton'ic, Pareto effici,ent and indiuidually rational mechan'isrn

R: E --+ Y x X if and only if u ueri,fies the equal orderi,ng property.

Furthermore, if such a mechanism R: E -+ Y x X erists, then for eDerA

technology c €. E,

(t) R(c) e EE(c).

( i i ,) The rnap "(.,0)

i ,s constant onEL(c), In fact, u(o,0) : u(R(c)) for

any a e EL(c) .

( i t ; ,) EE(c) c Core(c).

As a consequence, such a mechanism exists oniy if the agents ha,ve exactly

the same ordinal preferences when the bundles of public goods are free. This

condition, which clearly holds for economies with one public good, is very

restrictive in the case of several types of public goods. Thus, the fir'st part

of Theorem 4.6 limits severely the existence of individually rational and cost

monotonic selections from the set of Pareto optimal allocations.

The second part makes explicit that, whenever the equal propelty holds,

we are back in the setting of Moulin(f987a). Nameiy, part (i) implies that

68



Pareto optimal, individually rational and cost monotonic mechanisms co-

incide with the egalitarian equivalent correspondence and by part (ii) all of

them are equivaient, since the agents are indifferent among egalitarian equiv-

alent allocations. Finally, it follows from (iii) that any such mechanism is

also a selection from the Core of the economy and that the egalitarian al-

locations are individually rational and do not allow private transfers amonÉ'

the agents.

There is a related literature, in the context of monotonicity with respect

to changes in resources (Thomson (1989), Moulín and Thomson (1992)). The

conclusion therein is that Pareto optimality and resource monotonicity are

incompatible with other normative properties such as individual rationality

from equal division or envy-free. The egalitarian equivalent soiution has also

been chara cterized. by Pareto efficiency, monotonicity and a certain notion of

fairness with respect to some commodity (Dutta and Vohra (1993)). These

authors also show that the equity axiom cannot be imposed on more than one

commodity; thus their results show the strength of the monotonicity axiom

in another setting.

We finish this section with two remarks. Firstly, we could extend the do-

main of the mechanism to allow for changes in the number of agents. With



this modifi.cation we could also consider the axiom of population monotonic-

ity: roughly speaking, when the number of players increases, the cost of

fi.nancing the optimal bundle of goods is shared among more agents' Thus,

population monotonicity requires that by increasing the number of players

everybody should be no worse off than before. It is easy to argue that the

axioms of population monotonicity and Pareto optimality imply the Core

property and, hence, individual rationality. Therefore, Theorem 4.6 also

holds when we replace "individual rationaiity" with "population monotonic-

ity."

Secondly, if assumptions 2.2 hold, then there are plenty of cost monotonic,

pareto efficient (but of course) not individually rational) mechanisms defined

on Eo. It is straightforward to check the following result: Fix a € sT;t.

Then, the map,R which assigns any al location r?(c) : (A;t) e EE(c) to

each c ¿ Eo, such that "(y;t)

: u(),a;0), with ) e IR satisfying ' \a €

EL(c), is cost monotonic. The problem, as remarked above, is that for some

technologies c € E, the allocation .R(c) : (g;t) will not be individuaily

rational. This example also shows that the axiom of cost monotonicity alone

cannot discriminate among the different egalitarian equivalent allocations.



Mechanisms under non equal ordering

In this section, we exhibit a particular class of economies for which there

exists an egalitarian equivalent mechanism that provides a Core selection

which is cost monotonic. In particular, a Pareto efficient and individuaily

rational mechanism which is cost monotonic. As before, we use IRi and IR,

respectively to denote the spaces of public and private goods.

A utility function u : IRf x IR -----+ IR is said to be quasilinear and

separable if there are m nondecreasing functions b¡ : IR ------+ IR, for j :

1 . . . , r nsuch tha t

u (a , t ) :É b ¡ ( y ¡ ) - tj = 1

where A: (yt,. . . ,U^) € RT. Similarly, a cost function, c: IRf ------+ IR is

separable if there arerr¿ increasingfunctions c¡ : lR --+ IR where i : 1,.. .,?ft

such that

c(v):i",{a).j = 1

Assumption 5.L For each agent i € N, the followi,ng assurnptions hold:

(i,) x¿ - IR.

(ii,) The uti,li,ty functi,on u¿ : IRf x X¿ ------+ R" is quasi,-li,near and separa-

7T



ble, cbnti,nuo'rrs) non-decreasing i,n the fi,rst argument (public goods) and

stri,ctly decreasi,ng i,n the second (the pri,uate good).

(i,i,i,) For each j - 1,. . . )n1 ure haae that

l imsuP b¿¡(Y) : a'

g++co y

Assumption 5.2 The technology c : Y ------+ IRl is separable, continuous,

non-d,ecreasáng and sattsf"es for each j :1,. . . , , f f i that c¡(0) : 0 and

l imsup -f . +*.Y+¡a c¡\!J )

In the present context, we restrict our attention to the class of quasilinear

and separable economies

E' : {e : ( f f , {u¡ } ¡¿¡¡ ,c) : ( rn)n: t , . . . ,1¡u¡ sat is f ies 5.1,

c satisf ies 5.2).

Thus, each consumer i € l/ is characterized by a quasi-linear and sepa-

rable utility function

u¿(y , t ¡ ) : bo(y) - t i : ib , , (y , ) - r , (5 .1)J=T

whele U : (yt,. . . ,U^) € R?, ú¿ € IR and b¿¡: IR¡ ------+ IR is non-decreasing'

72



From the set of egalitarian-equivalent mechanisms we pick a mechanism,

we call 1{, on the class of economies .8".

Def in i t ion 5.3 F i r an economy e in 8" . For each publ ic good j : L , . . . ,m,

Iet

uf e arg Vf..a I b¡¡(r¡) - "¡(r¡)- r _

" ¡2o 7t

and choose y; € IR such that

\ - , , A / r , r ) : \ - ó r r l u i ) .)_-o¿¡rYj ) - c¡\a¡ / /_¿ "r \¿.r '

Now let yN : (a{ ,. . . ,yX) the quanti, ty of public goods produced, y* :

(Ai,. . . ,A:) an egali , tarian leuel and the amount of priuate good consurner

i e N has to pay to fi,nance the publi,c goods yN is dictated to be t¿ :

bn(y*) - b¿(a-) 2 0. Then, the mechani.sm K is defined by

K( " ) - - ( y * ; t r , . . . , t n ) .

There may be more than one yj but ali of them give the same utility to

each consumer since the functions b¿¡ are non decreasing and gN ) y*.

Clearly, K(e) is Pareto efficient since it maximizes the sum of the utilities.

Note that for each j :7 , . . . , f f i , the a l locat ion

@f ;br¡(vf¡ - b ' i@il , . . . ,b,¡(vf ,) - b"¡(vl))

73



is the usual egalitarian equivalent allocation for one public good as in Moulin

(19S7a). Thus, it follows from the same property of the egalitarian equivalent

selection in the one public good case that 1l is cost monotonic.

The next theorem proves that /{ is the only Pareto efficient, individual

rational and cost monotonic mechanism. In particular, this shows that al-

though all Egalitarian Equivalent mechanisms are cost monotonic not ali are

individually rational.

Theorem 5.4 The mechanism K i,s cost monoton'ic and for each e € E',

K(e) is z'n Core(e), and hence Pareto efficient and i,ndi,aidually rational. Fur-

thermore, i,f T is another cost monoton'ic, Pareto effici,ent and indi,ui,d,ually

rational mechanism, then u¿(K(e)): "¿(T(")) for each economy e in E' and

each agent z g l/.

6 Proofs

We fi.rst make a simple but useful observation.

Remark 6.1 Let ^9 c lú be a non empty coalition, let (y; t) e Y x Xs be an

allocation feasible for ,S and let u € IRs such that a¿ ) u¿(0,0) for each i e ,5.

Suppose that for every i e S we have u¿(y,t¡) ) u¿, with some inequality

74



being strict. Then, r € Xs exists such that the allocation (y;r) is feasible

for ,9 and, for every i e S, u¿(y,r¿) ) u¿.

Indeed, assume without loss of generality that u1(y,tt) > ,t ) u1(0, 0) :

ut(y,Vl(A\). Then, h < W(A) I crr, so we can find e > 0 small enough

so t ha t 11 : h * , < V r@) ,u t (U , " t ) t u1 and fo r i e S \ {1 } , r¿ :

t ; - e l (S l - 1) > - r ( ¡ / ) . Therefore, r € Xs, ¿( .9) : r (5) and u¿(y, r¿) > u¿

for every i € S. tr

The foliowing Lemma shows that in the definition of fL(.) one can replace

u(y; t ) : u( \a ;O) wi th u(y ; t ) < u()o;0) . I t w i l l be used in the subsequent

oroofs.

Lemma 6.2 Let c € Eo, u be a uector i ,n R" such that u Z u(0; 0) and

(y;r) eY x X be a feasible al location such that

u ( y ; r ) ) u ) u (0 ;0 )

Def,ne tr: lR ----- X by

I ()) : { t e X:c( )y )<¿(¡ / ) and u ( \ y ;¿ ) 2 , )

l r )



and

any

let,

se

)* : sup{.\ e IR : I()) + 0}. Then, )* < *oo, ¿(}.) I A and for

¿( ) . ) we haae tha t u ( \ *y is ) : u .

Proof

First, note that L(1) + A. Let {}r}É, be an decreasing sequence converging

to )*. For each lc : I , . . . ,6, let s¡ € ¿()e) such that ()ry;sft) is feasible

and

u(o; o) : u( \nu iPQIY)) 1 u 1 u( \ ¡ ,Y; sk)

In particular, we have that VQ¡A) > sk for every k : I,. . . , ñ. We claim

that ). is finite. Otherwise, the sequence {)¡}p, diverges to infinity and,

by assumplion 2.2, for each i : Ir. . . )n

Therefore,

ri3s;n# <rigs;nffi:o

o < limr,ro ,Í(]*'], < tim.,-ro Pitt lf : or*o"^ l l ̂ *Y l l

- f t-co l l ¡rg l l

but this contradicts assumption 2.1.

We conclude that {)¡}* is a bounded sequence and .\. is finite. Since sa (

p(\ny) < p(\-y) for every k : I ,2. . . , we a,lso have that the sequence {"*}L,

is bounded. Taking an appropiate subsequence we may assume that (,\¡g;sfr)

76



converges to, say ().y; ".).

Since X is a closed set and c is continuous, then

().y;".) € Y x X is also a feasibie al location. In addit ion, u(\*Ais*) ) u,

because u is continuous. Therefore, s. e LQ,.) I A.

Finally, Iet s € ¿().) and suppose that ,().y;s) > u. Say, u1().9isr) >

u1 ) u1(0,0). Then, st 1 gt().y), so we can f ind rt € Y with s1 ( 11 (

p r ( ) . y ) and s t i l l u t ( \ *U i " t ) > u1 . Take t t : r t and ü¿ :3 i , f o r i : 2 , . . . ) n .

Then, t € X and f(N) > "(N)

> "().y).

By continuity and Assumption2.I,

wecanf ind ) ) ) * suchthat c( )y) < ¿(¡ f ) .Then, ( )y ;¿) € YxX is feas ib le

and u(),y;¿) 2 , contradicting the definition of )*. Therefore, u(\"Ui") : ,.

!

Proof of Remark 3.2: The second inclusion is clear. To plove the first

one, let a e Sill and let (z;r) e AEL,(c) be a feasible allocation supporting

the egal i tar ian level )6a € EL"(c) wi th )o € IR+; that is u(z ; r ) : u( )0a,0) .

we only need to show that (z;r) e P(c). suppose not. Then, there is another

feasible allocation (y;t) e Y x X such that

u (a ; t ) > u ( )64 ;0 ) 2 z (0 ;0 )

77



By Remark 6.1 above, we can find r € X such that c(y) : r(/ú) and

u (A ; r ) >> u ( ) so ;0 )

Let ) : sup{) € IR : u(y;r) >> u(, \o;0)} . Since, r( lú) : " (y) 2 0, there

is some io e /ü such that r¿o ) 0. Hence, u¿o(\a,0) ) u¿o(A,rn), whenever

)a ) y. Therefore, )s < ) < oo.

It cannot be the case that u(y;r) >> z()a;0) because if i t were we may

find p > ) such lhat u(y;r) >> u(pa;O). Hence, we must have that

u ( y ; r )> u ( ) , a ;O ) > u ( )6a ;0 )

Applying now lemma 6.2 with 11 - u(\a;0) we find ). € IR and s* € X such

that ().y;s*) is feasible and u().y;"*) : u()a;O). But this contradicts that

)6a € EL"(c) .

We are now in a position to prove the

allocations.

Proof o f Proposi t ion 3.3:

existence of egalitarian equivalent

78



Let c € -E and a e Sii1. Consider a non-decreasing sequence {)*}Ér c

F"(") such that

JI* )* : u"p(F"("))'

For each k e N there is an allocation (ye; tk) eY x X feasible in c such that

u( \ ¡a ;O) : u(ak; tk) .

Assume that the sequence {go}, is unbounded. We may fi.nd an increasing

unbounded subsequence which, for simplicity, we also denote by {yu} r. Using

that u is non-decreasing in public goods, we have that

u ( yk ; t k ) : u (A ta ;0 ) 2 u (0 ; 0 ) : u ( y r ;V@\ ) .

Hence, tr < V(Ar) for every k e IN and assumption 2.2 implies that for any

agen t i : 1 . , . . . ,Q ,

rig;n#=t,ffJoffi:o

by adding these equations we get

o < l imr"o:Í ' ,1) : l imr,ro EEt / f : gA-oo l l y^ l l k *co l lY" l l

(6 .2 )

(6 3)

which contradicts assumpti on 2'I

Hence, {ar}r must be bounded. It follows now from Equation 6.2 that for'

each i € /ú, the sequence {tf}r is bounded above anu, thus from Equation 6.3,

t v



there is also an infinite subsequence of {úf }¡ bounded belorv.

By taking appropriate subsequences we may assum. {Ak}r converges to

a limit, say g and {fk}¡ converges to, say t*. By continuity, the allocation

(y;t-) is feasible.

Suppose first that sup(F'(c)) : *oo.

unbounded. Fix ko,), € IR such that .\¡oa

large enough so that )¡ ) .\, we have

Then the sequence {)¡}pt

) y and ) ) )60. Given ,k €

1 S

]R

u(a ;0 ) < u ( . \ a ; 0 ) < u ( ) ¡a ; 0 ) : u (Yk ; t k )

By taking limits as k tends to inflnity, we obtain the inequality "(y;0)

<

u ( \a ;O) I u (y ; t - ) . Le t ú t : *u * {ú1 , . . . , t L } . S ince 0 l c ( v ) ( f * ( l r r ) , we

must have ú; > 0. On the other hand, since preferences are decreasing in

the amount paid and u¿o(y;0) I u¿o(y;l [), we must also have that q" 50.

Therefore, f[ : 0 and t* : 0.

It fol lows lhat u(y;O) : r()o;0) whenever ) ) )¡0. Hence, the mapping

//, is continuous and constant for ) 2 )ro, so it must attain its maximum.

If sup(F"(c)) : lim¡*oo ),k : ) a finite number, then

u( )a ;0 ) : I * u ( ) ¡a ;0 ) : u (y r ; t r ) : u (y ; t * ) ,lim

t^ ^^

80



so .\ € F"(r).

Next, we start the preliminaries to prove Theorem

Lemma 6.3 Let i

decreasi,ng function

N and y € Y . Then, tl'¿ere i,s a, cont'inuo't!"s, non-

: IR1 ------+ B"¡ such tlzat

€

d,¿

(i,) If ) ( 1, then, d¿(\) =0.

f t i , ) A) > 1, then, u¿() ,y,d¿())) : u¡(y,0).

Proof

Fix ) ) 1 and ie t á : IR+ - IRbe def ined by h(s) :u¿( \U,g;Qy)¡ . Then,

á()) : u¿(Áv,p¿(^a)) : u¡(o,o) < z¿(v, o)

and

á(0) : u¿( \y ,e r (0 ) ) : u¿( \U,0) 2 u¿(y , 0 )

Since á is continuous, there is s()) € [0, )] such that n(s())) : u¿(\U,f;(s())y¡¡ :

o¡ (A,0) .We def ine d i (^) : f¿(s( ) )y) . Note that d¿( . \ ) is un iquelv def ined

because u is strictly increasing in the second argument. Aiso, 0 : d¿(1) <

d¿()) I v¿()y),, since g¿ is increasing.

81



In addition, d¿ is continuous and non-decreasing, since it is defined im-

plicitely by the equation

u¿(Y ,0 ) : u¿ ( \A ,d¿( ) ) )

Thus, we can extend d¿()) continuously to

in that interval.

[0,1] by requiring that it vanishes

Lemma 6.4 Let y € R?+ be a bundle of publi,c goods in the i,nteri,or of

Y. Then, a technologA cs € E7 erists such that u(z;t) : u(y;0) for any

(z;t) e Y x X whi,ch i,s an indiui,dually rational and Pareto effici,ent allocation

i,n the econorny c,s.

Proof

Let G¡ : d([O,)y]) be the boundary of the set {r e Y : 0

(The boundary is taken as a topological subset of Y.) For each i

d; as in Lemma 6.¿1 and let

d : d , t Y " ' V d n

1 r

€/v

< \y j .

choose

¿2



Defi.ne now

co(r ) :2nmax{d() ) , ' \ - 1}

where ) is the unique point in IR..,. such that r € Gx. Note that cr(r) : g 11

r < y so, in par t icu lar , (y ;0, . . . ,0) is feas ib le \n cr . In addi t ion, c(c) > 0 i f

r €Gsw i th )>1 .

Fol each r €G¡ we have l l t l l f f l l y l l and cr(r) > () - 1) . So,

limsup lE-ll < tim,,,p +ll-g+ ( *ool l " l l *+o" cg \ r / ) r *co A - L

Let r a G.l, , z € G¡". Assume, without loss of generality, that )1 ) Az'

Then r 1 ) , 1y and z 1 \ zU 1 \ú .Hence , rY z < \ ú so ¿Vz € G . r , and

c(rY z) : c( r ) < " ( t )

* c(z) . Therefore, c , e E1 '

Let now (z;t 'S : (";tr, . . . , tn) be an al iocation which is individually lat io-

nal and Pareto ef f ic ient in c , wi th z e Gx. Denote by t¿o: max{ f r : . . . , t . } .

83



Suppose l ha t z ( y and f¿o :0 . Then , t :A and u ( z ; t ) 3u (y ;0 ) ' Hence '

by Pareto optimality, u(z;t) : u(A;O) and the proposit ion is proved. If

z 3 y and t¿o ) 0) then u;o(z,t) 1 u¿o(A,O) with g feasible for is ' But, this

contradicts that (z;ú) is individually rational

Otherwise, z #10,9] , so ) > 1. Again, by feas ib i l i ty ,0 ( " r ( ' )

< ¿(N) '

Therefore,

c , , ( z \t ; )_Y>2d(^)

n

It fol lows that, ú¿o > 2d(^) > zd¿r(^). Assume that d¿o()) > 0, then

t¿o ) d,¿o(.\) and

u¿o(z ) t¿o ) ( u¿o (z ,d¿ , ( ) ) ) 1u¿o( \ y ,do . ( ) ) ) - u¿o (y ,0 )

so (z;ú) cannot be individually rational because agent i6 would be strictly

better off by deviating to the allocation (y;0), which is feasible for him.

Hence, we must have that do. ( I ) :0 . But then u¿o()g,0) : udo(g,0) , so

u ; o ( A , 0 ) : r o o ( ) y , 0 ) ) u ¿ o ( 2 , 0 ) 2 u ; o ( r , t , o )

By ind iv idua l ra t iona l i t y , u¿o(y ,0 ) : u ¡o(z , , t io ) : u¿o(2 ,0) ' Hence, Ú;o :0

and it follows that co(z) : 0. But this contradicts that z € Gx with '\ > 1'

n

84



We say that sequence {ce}p t C Eo converges to c e -E6, whenever for

every z €Y, we have l im¡-- ck(z) -- c(z).

Lemma 6.5 Let {"*}Ét C Eo be a sequence conuerging to c e Eo and let

{(rr;tr)}L, CY x X be a sequence of al locati,ons such that for eaery k e IN,

u(zk;tk) 2 ,(0,0) and (rr;tr) i ,s i ,ndi,aidually rational and' Pareto optimal

in the econorna ck. Suppose that {(tr; tr)}f=r conl)erges to (z;t)- Then, the

allocation (z;t) i.s indi,ai,d,ually rati,onal and Pareto optimal i,n the econonlA c.

Proof

The allocalion (z;Ú) is feasible because for each k e IN we have that ck(zk) <

úfr(N) with úfr € x. since the latter set is closed, by taking limits we have

that c(z) < ¿(¡/) and ú e X.

Note a lso that , by cont inu i ty , u(z ; t ) > z(0;0) . Suppose (z ;Ú) is not

Pareto optimal. Then, there is another allocation (r, r) e Y x X which

satisf ies c(z) : r(1ú) and u(r;r) > u(r;ú). By remark 6.1 we may assume

there is another feasible al location (r; s) e Y xX such that u(n; s) >> u(z;t).

By increasing s slightly, we may also assume that -s(/{) > ,(*).

óD



Since the sequence {(zk;¿*)}É, converges to (z;ú), there is Aio e N such

that whenever k ) Afo we have

u( r ; s ) >> u (zk ; t k )

The sequence cft(r) converges to c(z) < "(N)

so, by a continuity algu-

ment, we can take l/r € IN, /út > 1ú0 such that for every k > N1 r've have

"k(*) < s(N) and u(r ;s) >) u( rk ; tk) . But th is contradic ts that (zA; tA) is

Pareto optimal in ck.

A simiiar, but simpler argument shows

in the economy c.

that (z;l) is individually rational

Remark 6.6 Let c € Es and consider the set

¡

A(c ) : { (y ; t )€Yx X :c (y ) : I ro , u (y ;ú ) >0}. - 1

From assumptions 2.1 and 2.2 we have that for any technology

anyagen t i eN ,

(6 .4 )

c€Eoand

Iimsup *0r",,' : ol ls l l -o" c\Y )

ó0



Thus, if c € Eo, (A;ú) is feasible and llyll i. large enough, there must be

some agent i0 € /ú such that t¿o >- c(A)/" ) ?¿o(y). Therefore, u¿o(Y,too) <

u¿o(A ,Va@D : u¿o (0 , 0 ) : 0

It follows that there is M e IR such that llyll ( i'14 whenever (y; t) e A(c).

I n add i t i on , s ince c (y ) :DT= t | ) 0 and t¿ < V ¡@) fo r each i : ! , . . . , r¿ ) we

conclude that the set A(c) is bounded and has compact closure' Note that

the set of feasible and individually rational allocations is a subset of A(c).

Now we can prove the "only if" part of Theorem 4.6.

Proposition 6.7 Let R be a cost monoton'ic, Pareto efficient and indi'ui'du-

ally rational mechani,sm. Then,

(i) For any functi"on c € E, R(c) e EE(c)'

( i i ,) For eaery technology c and o € EL(c) we hare thatu(a,O) : "(R(")),

t , .e . the map u( ' ,0) is constant on EL(c) '

( i i i ) The uti , l i , ty l trof i le u, -- (u1,... ,u,) sati ,sf,es the equal ordering property,

Proof

(i) Fix a technology c € E and suppose -R(c) : I i;Ú). Let z e IRi* be

a bundle of public goods which is strictly positive. We will prove that

87



u(n(c)) : u(Aozi0) for some )s € IR+.

Given ) e IR1 we may apply Lemma 6.4 with A : \z to construct

c¡" € 81. Clearly, for each e e (0,1], the technologies

c"A, : ec * (1 - s)c¡"

belong to E.

The set A("r), defined by equation 6.4, with c¡(9) : min{c(d),c¡,(d)} is

a bounded subset of" Y x X. The set of feasible and individually rational

allocations of all the economies c!, with e € [0,1], being a subset of A(c6),

is also bounded. By a compactness argument, there is a sequence {e¡}p.

contained in (0,1), converging to 0 and such that the sequence {ft(cii)}p,

converges to a feasible allocation, say (z;t), in the econorny c)2.

Since -R is Pareto efficient and individually rational, so is (z;I). By Lemma

6.4 we have that

JIg u(R(ci l)) : u(z;I) : u()z;o).

For each k € N, we may now apply Lemma 4.3 to obtain that eiiher

u(ft(c)) > u(R(c!")) or else u(R(c)) ! u(f i(cl!)). By a l imit ing argument

we conclude that for each ,\ e IR+

ei ther u( f t (c) ) >- u( )z ;O) or u( / i (c) ) < u()z ;0) . (6 .5 )

88



Let ) be large enough so that \z ) r. Since 0 < c(r) < tlt ú¿, there

must be some agent, say io € N, such that ú¿o ) 0. Therefore, z¿o(.\2,0) )

u¿o(r, l ;o) and hence u(),2;0) 2 u(ft(c)). Observe also that u(0;0) i u(ft(c)).

Let

)o : i n f { ) : u ¡ (R (c ) ) l u¿ ( \ 2 , , 0 ) f o r a l l i : L , . . . , n }

By cont inu i iy , u¿(R(c)) I " r ( loz,0) , for each i : I , . . . t tu . Suppose some

inequality is strict, say

u1 ( l? (c ) ) < u1 ( )62 ,0 ) .

Again, by continuity, there is ,\' < )6 close enough to ,\s such that we still

have u1(.R(")) < u1(\'2,0). On the other hand, recalling the definition of

infimum, there must be some index, say i : 2, such that

u2 (R(c ) ) > u2 ( \ ' 2 ,0 ) .

The last two equations contradict equation 6.5. Therefore, z¿(ft(c)) : u¿()02,0)

f o r a l l ' i : 1 , . . . ) n .

(ii) Let now z e EL(c) and ) € R+; by taking U : \z in Lemma 6.4, we

may construct cs" as in part (i) above and we may find ) g IR+ such that

u(n(c ) ) : u ( \z ;g )

89



Since z € EL(c) and r?(c) is feasible, then u(z;0) ) u(B(c)). But ft(c) is

Pareto optimal, so u(zi0) : "(R(c)).

This proves (ii) for bundles of public

goods in EL(c). A simple continuity argument can be used to extend the

result to all bundles of public goods in EL(c).

(iii) Let i e /ú be an agent, and let A,z € IRf* be two bundles of public

goods. Suppose that u¿(y,0) > u¿(2,0). Construct, as above, c"ok, c""k srch

that l im**,*u(R(cik)) : u(y;0) and lim¡*- u(R(cik)) : "(t;0).

By Lemma 4.3, for each k e IN, either u(R(cih )) 2 u(n(c?* )) "r u(R(ci,)) S

u(R(cik)). gV assumption,

/1X z;(l?(ci-)) : u¿(v,0) ) u¡(2,,0) : JIl u¿(R(cir))-

Thus, for large enough k, u(R(cik)) 2 z(ft(clr)) and, taking limits we obtain

u (y ;0 ) > u ( z ;0 ) .

Hence, the equal ordering property holds U,z € RT*. A simple con-

Y : ]F "T .T

tinuity argument extends this propeúy to y,z

To finish we prove the converse of Theorem 4.6. The equal ordering prop-

erty is a sufficient condition for the existence of cost monotonic mechanisms.

for

, C

90



This property by itself also guarantees that the egalitarian equivalent allo-

cations are in the Core of the economy.

Proposition 6.8 Suppose the equal ordering property holds. Then,

(i) Any egali,tarian equi,ualent mechan'ism is cost monotonic.

( t i ) EE(c) CCore(c ) .

Proof

(i) Let -R be an egalitarian equivalent

technologies in E. For each i : I,2,

u(z i ;o) : u(Yi ; r t ¡ fo t some z i e Y '

mechanism and let c1 1 c2 be

let f i(c¿) : (yi;r¿) e EE(c¿)

two

with

By the equal ordering property, either u(n(c1)) : u(zL;0) ) u(ft("r)):

u(22;0) or e lse u( f t (c1)) I u( f t (cr ) ) . But , " t (a ' )

< " r (y2)

: r2(N) , so the

allocation R("r) is feasible in the economy ci. Since E(.t) is Pareto optimal

in c1, we cannot have that u(f i(ci)) < u(R(c2)). Therefore, z(€(c1)) >

u(R(cr) ) .

( i i ) Let r e EL(c) and (y ; t ) e EE(c) so that u(a; t ) : u( r ;0) . Suppose

that thereis a nonempty coali t ion,g C ly' and an ¿ 'cation (r;r ') e Y xXs

such that c(z): r '(S) and u¿(z,r ')) u¿(n,0) for i e S with some inequali ty

91



being str ict. Consider the al location (z; r) eY x X, where ri : 0 i f z e N\5

and r¿ : rt¿ if. i e S. Then, c(z) :"(¡/), so (z; r) is feasible for l/.

By the equal ordering property, either uQ;A) > u(*;0) or else u(z;0) <

u(r;0). However,, u(z;0) 2 u(r;O) is not possible, because we would have

thai u(z;r) > u(r;O) contradicting that the egalitarian equivalent allocations

are Pareto optimal.

Hence, we must have that u(z;0) < u(r;0). Then, for each t, e S we have

the inequali t ies u¿(z; 0) 1u¿(r,o) < u¿(z,r¿). Therefore, r¿ ( 0 for every

i e S. But 0 < "(r)

: r(S), so ?'¿ : 0 for every i e S. Hence, c(z) : g,

z : 0 and u¿ (0 ,0 ) : u¿ ( r , r¿ ) ) u¿ ( r , 0 ) 2 u ; ( 0 ,0 ) f o r i e S w i t h some s t r i c t

inequality, which is a contradiction. tr

Proof Theorem 5.4

1l is cost monotonic because it inherits this property from the egalitarian

equivalent solution with one public good.

First, we prove that 1( provides a selection of the core and hence, Pareto

efñcient and individually rational. Fix e € 8", and suppose .tí(e) : (yN; h,. . . ,tn)

is not in the core(e). Then, there exists a coalition ̂ 9 and a feasible allocation

92



(yt,, t t);es such that

c(u ' \ : \ - ¿{- \ ¿ / . / - - J " z

bo(y') - t ' ;

(6 6 )

(6 7)

where y* is the egalitarian vector defined by mechanism .I(. l'trote, first, that

y' ) y* is not possible. Otherwise, bn(y') > bn(A-) for every k e ,¡/. In

part icular, for every k ( S . But this contradicts that (yN; tt , . . . ,1,) is a

Pareto efficient allocation. Similarly, if y' 1y*, then b¿(y,) 1b¿(A.) for every

i e S, which contradicts inequality 6.7.

We conclude that y' and y* are not comparable. Then, the sets M : U €

{1 , . . . ,m } : y i > y i } and .L : { j e { 1 , . . . ,m } : A ' ¡ < y i } a re non -emp ty .

Adding in inequality 6.7 for ¿ e S, we obtain,

" , / \ n z

t ( t bn¡(y,¡) - c¡(y,¡) ) > t D,butu;)., = r \ ; e s / l = _ ; t e s

We split this sum into two parts

(Y,u,,r',) - ",{r))(6 .8 )

(>,u,,r,1 -,,rr))- Et

j € M

\-/-/

j €L i es

< bo¡(yi) for every i e N. Hence,

/ _ \

I f a'i(ul) - b¿¡tuil ) t I "¡(v'¡).\ icM / ieM

J € M ¿ C S

If j € tr, then bn¡(y'¡)

93



Thus, we can find (r¿)¿6s such that

ri : D"¡@')j eM

:t¿jeM

In addition, from the definition of M,

Y,ur¡(a) > tbr¡(y i ) forevery k#5.jeM i€M

Consider the allocation ((yj)¡ r*,'(rt¿)rcN), where r'i -- t¿ if i € S and

r '¿ :0 i t i #S-Then ,

Drn : D"¡@')i€s jeu

j eM i€M

Duot(o')

Construct a new economy as follows: Let IRe, with p : lMl, be the new

space of public goods. The utilities of the agents are

t / \ \ - r / \o¿lu) : ) .o¿¡ \u¡ )j eM

fol i : 1,. . . , n and the technology is

c(a) : D "t@t)'jett't

94

ti eS



Denote by yv and yf6 the projections of gN and y* onto the space IRp and

Iet

qi : L(b,¡(yf) - bn¡(yiD for i :

jetw

The al localion (yy;Qtt.. . ,g,,) is Pareto eff icient in this new economy

since it maximizes the sum of utilities of consumers. The utility obtained

flom this allocation by each consumer rsb¿(y¡a)-q¡ -* b¿(Aír,,t). But inequalities

6.9 show that, in the restricted economy, all consumers are no worse off with

the allocation ((yj);r*,(r'¿);eN) and the ones who belong to S are strictly

better off. This contradicts that (yu,(qr,.. . ,q")) is Pareto eff icient.

Now we prove that 1{ is the only Pareto efficient, individually rational

and cost monotonic mechanism. Let "

: (N, (rr);.r, c) be an economy in

E'. Let y* be the egalitarian vector of the allocation K(r).For each ,9 c /ú

and j -1 , . . . ,m Ie t

b'¡(v): t b¿¡(v)i€s

and define

" t ¡ (y) - max{b¡s1r¡ - b is(u}) ,0}

and

c j (u) : m¿¡{" ¡s(y) } .

95



Note that

a' 4@',)and

0 S 4@) 3 "¡(y)

for each i : 1,. . . ,ffi, because if for some 7 and

c¡(y')then there exists S C I/ such that bs¡(y') -bs¡@i) > c¡(y')

b s¡(v') - "¡(v') > b s¡(vi) (6 .10)

Since c] is nondecreasing y' > yi and because ó¿¡ are nondecreasing for each

ieN . i 4Swehave

bo¡(a') > bn¡(v.)

Adding inequalities 6.10 and 6.11 we obtain that

(6 .11)

bN ¡(a') - "¡(y') > b*¡(yI),,

which contradicts the definition of yl.

Now we define cL 1 c by

1 t , a \ 1 tc ' \ y ) : Lc ; \a ) ,i ,= I

Consider the economy et : ( lú,(r , ) , . r ,c1). I t is easy to see that fol each

i e I / we have that u¿(/ l ( . t )) : u¿(K(c)) : b¿(y-). Since (y. ,0) is a feasible

allocation with c1 fol each agent and, ? is individually rational then

u¿(1{(c1)) : u¿(K(c)) : bn(a*) < u¿(T(cr) )

96



for each ¿ e 1/. But 1((c1) is a Pareto efficient allocation, then u¿(?("t)) :

b¿(a\ and by cost monotonicity u¿QQ)) < u{T(cr)) : u¿(K(")). Also by

eflñciency u¿(T(c)) : u¿(I{(c)) for each i e lú.

REFERENCES

Champsaur, P. (1975): "How to share the cost of a public good,"

Int. Journal of Game Theory, 4,II3-I29.

Dutta, B. and Vohra, R. (1993): "A characterization of egalitarian equiva-

Ience," Economic Theory, 3, 465-479.

Mas Colell, "Remarks on game-theoretic analysisof a simple distribu-

of Game Theory, 9,tion of surplus problem," International Journal

125-140.

Moulin, H. (1987a): "Egalitarian-equivalent cost sharing of a public good,"

Econometrica, 55, 963-976.

(1987b): "A core selection for regulating a single-output monopoly,"

RAND Journal of Economics, 18, 397-407.

n

o 7



society

(1938): Axioms of "o@.

Econometric

Monographs, Cambridge University Press.

Moulin H. and Roemer J. (1989): "Public ownership of the external world

and private ownership of self," Journal of Political E"qglqy, 97 , 347 -

ñ F -

J O f .

Moulin, H. and Thomson, W. (1992): "Can everyone benefit from growth?

Two difficulties," Journal of Mathematical Economics, L7,339-345.

Pazner E. A. and Schmeidler, D. (1978): "Egalitarian equivalent allocations:

A new concept of economic equit¡" Quarterly Journal of Economics,

92,677-687.

Roemer, J. (1936): "The mismarriage of bargaining theory and distributive

justice," Ethics, 97, 88-110.

Roemer, J. and Silvestre, J. (1987): "What is public ownership," working

Paper no. 294, Univ. of California at Davis, Dept. of Economics.

Thomson, W. (1939): "Monotonic allocation rr-rles in economies with public

goods," Mimeo, University of Rochester, NY.

98



CHAPTER 4

trFFICIEI\CY AI{DTECHNOLOGICAL IMPROVEMtr}{TS

Abstract

In economies with public goods in which the endowments of theagents are not transferable, and the technology to produce the publicgoods is publicly owned, there is no efficient solution which is costmonotonic , i.e., no agent suffers any loss when the technology im-proves. We also find a necessary and sufficient condition to the exis-tence of Pareto efficient which satisfres a property weaker than costmonotonicity.

99



Introduction

Modern states quite often fund research projects. The reasoning behind these

subsidies is that technological improvements will increase the welfare of the

whole society. On the other hand, if an improvement in the technology will

benefit all the members of the society then, nobody will be against research.

We investigate this idea by studying an economy with public goods which

are produced by means of a collective technology. We allow the technology

to vary and we assume that given two technologies, one cheaper than the

other, all the agents should be better off with the cheaper technology than

with the more expensive one. This property is called cost monotonicity.

The motivation to study the property of cost monotonicity is twofold.

On one hand we pose the question whether, a society intelested only in

efñcient results will promote technological research. On the othel hand, when

some agents have to work together to produce public goods using a coilective

technology (each of them only owns one of the inputs) so that cooperation is

necessaryr it seems reasonable to require that improvements in the collective

technology should benefit all the agents.

Taking into account these ideas, we consider economies u'ith one public

100



good and several private goods, but where each agent only owns one private

good. There is a technoiogy, publicly owned by all the agents, which produces

one public good using the private goods as inputs. We have in mind situations

in which one agent owns the capital and the other one the labour, or contexts

in which the agents have specialized jobs and have to work together in a

production line.

We consider economies where technology is described by Leontief produc-

tion functions, that is, there is a fixed proportion of inputs that is needed to

produce the public good and study the existence of Pareto efñcient and cost

monotonic allocation rules.

We find that the properties of Pareto efficiency and cost monotonicity

are incompatible whenever the domain of technologies includes all Leontief

production functions. To prove this, we introduce a weak version of cost

monotonicity. Given a fixed proportion of inputs, we oniy apply the mono-

tonicity to those functions which depend on a fixed proportion of inputs. We

find a necessary and sufficient condition for the existence of a rule satisfy-

ing Pareto optimality and this weaker version of co. -ronotonicity. These

properties are compatible if and only if the agents have the same preference

relation on the set of relevant allocations, i.e., on those allocations that ale

101



not Pareto dominated by any activity at all. The fact that Leontief produc-

tion functions are included is the key of the result.

For the related literature we refer the reader to the previous chapter, the

third, which is common to both chapters.

2 Model

We consider an economy with two agents with prefelences on one plivate

good and one public good (the extension to n agents and n private goods is

straightforward). There is a technology F which converts the private good

of the agents into a public good. We assume that each agent only owns one

of the private goods, which for the sake of concreteness, we will call labour.

Assumptions

¡ A.1. - For each i : L ,2

- a) u¿: IR". x IR+ - IR is continuous, strictly increasing in the fir'st

argument (the public good), and strictly decreasing in the second

(the labour).

102



- b) For eachy € R+, there \s z;(y) € IR+ such that u¿(A,"n(y))

u¿(0,0) : 0. Furthermore, l imsupr-a *"0(y)la :0

L.2.- The technology is a mapping F : IR] -* IR+, which is continuous,

non decreasing and satisfies:

- a )F (0 ,0 ) : 0 .

- b)There exists "

e IR] such that for all t > T and ) ) 1 we

have that F(,\t) < .\¡'(¿). (Decreasing returns to scale for large

bundles of public goods).

An allocation is a vector (g;t) e IR+ x IR2* such that g is the quantity of

public good produced, and f1, t2 are the quantities of inputs that each agent

provides. It is feasible for the production function F if y : F(tbt2). we

extend the uti l i ty function of the agents to IR-., ' "

R?* by uofu;t): u¡(a,t¿),

with (y;ú) an allocation and i : 1,2. The utility profi.le of the agents is given

by u ( y ;ú ) : ( 21 (y ; t ) , u2@; t ) ) .

We fix a profile of utilities satisfying assumption A.1, and define D the set

of economies whose technology satisfies assumption A.2. Since the utilities

of the aqents are fixed we identify an economy with its technology.



AS

Given an economy F € D, we define the set of Pareto efficient allocations

PE(F) : { (y ; l ) feas ib le for F: i f (y ' ;ú ' ) is feas ib le for -F and

u(a ' ; t ' ) > u (y ; t ) , t hen u (y ' ; t ' ) : " ( y ; t ) j

Assumptions A.1) and A.2) guarantee thal PE(F) is non-empty.

We say ihat F > G if F(¿) > G(l) for all ú e IR1.

Definition 2.L A mechan'ism i,s a mappi'ng R whi,ch assi,gns a fea.si,ble

Iocation to each economy F e D. We say that R is Pareto efficient

ft(¡ ') e PE(F) for each F e D. And, R is cost monoton' ic i ' f giaen F,G e

F > G then u(R(F)) > u(,B(G)).

We study a weak version of cost monotonicity. Fix a proportion of inputs

to produce the public good, say f e IRI*. fire admissible production func-

tions are 9 : IR+ -- IR+ such that there is I e D rvith g()) : F()¿]. We

denote the set of all of them by

D({) : {s :1F € D wi th s( } ) : r ( ) t ) } .

al-

,f

D,

r04



We only take into account those allocations which belong to the ray gen-

erated by f

AA(71 : {(g()); \D' g € r(t), ) e lR+}.

From this set we define the set of non-relevant allocations with respect to

l, NRA(|, as those allocations that are Pareto dominated by the allocation

(o; o):

N RA(ü : { (y ; t ) € AA(ü : u (y ; t ) < u (0 ;0 )

with strict inequality for at least one agent]

An allocation is relevant with respect to I if ii is feasible for some pro-

duction function in D(f], and is not in ^f RA(T).

RA(D: { (y; Ál) e AA(D ' (y; t ) # NRA(T)} .

We say that a mechanism is cost monotonic

f I s e D(l) then u(.R(/)) < "(n(g))

respect to I glven

Now, we look for Pareto efficient mechanisms which satisfy cost mono-

tonicity with respect to I

The following lemma, that was already used in Moulin (1987a), also ap-

plies to this context and says that if there exists a mechanism satisfying these

105



properties then the mechanism is also path monotone. The proof is the same

as Moulin (1987a).

Lemma 2.2 If R : D(l) * IR+ "

Rl i,s a Pareto ffici,ent mechanism, cost

monoton'ic with respect to l. Then, for each ft, f' € D(ü we haue that ei,ther

, ( f i ( / ' ) ) 2 r (R( f ' ) ) or u(R( f ' ) ) l , (n( / ' ) ) '

Before presenting the first result we need two lemmas.

Lemma 2.3 Git:en (y;).¿] e RA(I), there is f e n1q such that i f R is a

Pareto efficient mechanism then u(R(f)) : u(y; .\.f).

Proof

Let (y;).0 e RA(r. I f . u(y; ).ú] : u(0;0), then /()) : 0 for al l ), t l ivial ly

satisfles the statement of the lemma.

Otherwise, choose e e (0, ).) and construct the production function, f, as

follows.

We leti f 0<) ( t r * -ei f ) * -e <)<)*i f . \> ) -

continuous and non-decreasing mapping

y. We may choose e and g()) such that

f ( ))

where g : l). - €,1*l

such that 9(). - e) :

"(g()) ; . \0 < u(y; \ "0.

lo: { e())IY

-+ lR isa

0,9( ) . ) :

106



Then f e n(ü and fi(/) : (y; ).ú] and the lemma follows. ¡

Lemma 2.a If the agents haue the same preference relat'ion on the set RA(q '

then for euery f € D(ü all the agents are i,ndi,fferent among all Pareto effi-

cient al locati,ons with f .

Proof

Let f € D(t) and (yi;)rD, @';\rü two Pareto efficient allocations. We

claim that both are in RA([). Both are feasible with / and since they are

Pareto efficient and the allocation (0;0) is also feasible with / then neither

are Pareto dominated by the allocation (0;0).

Since they have the same preference relation on -RA(l], either u(yr; \tü <

u(az; \zD or u(y1; Ir¿] Z u(Az; \zü' But both are Pareto eff icient, so u(y1; )11)

u(Yz; \zD.

Lemma 2.4 establishes that when all ihe agents have the same ordinal

preferences over the set of relevant allocations, the set of Pareto effi.cient

allocations is unambiguous because the agents are indifferent among them.

107



Although it seems a very restrictive context, to have the same preferences

relation on the set ftA(¿] is necessary and sufficient to the existence of a

Pareto efficient mechanism which is cost monotonic with respect io I as the

next result shows. Note that even in the case where both agents have the

same pref'erences, they could have a different preference relation on the set

RA(ü.

Theorem 2.5 Fir a uti,li,ty profile sati'sfyi'ng assumption 4.1. There is a

Pareto efficient mechanism whi,ch r,s cost monotontc with respect to I i'f and

onty i,f agents haue the same preference relation on the set RA(ü.

Proof

Suppose there is a Pareto efficient mechanism -R which is cost monotonic

rvith respect to f. Let (y1;AtD,(y';)21] be two al locations in RA(ü. By

lemma 2.3 there are /1 ,f" € D(l] such that u(€(/t)) : u(y1;)1f and

u(R(f2)) : u(vz; )2/). l ' trow, applying lemma 2'2 either

, (y ' ; \ rü -- " (R(f ' ) ) f " (R(f ' ) )

: u(y ' ; \ rü

, ( y t ; ) ' ¿ ) : " ( f i ( / ' ) )

2 "(R(/ ' ) ) : u(a2; ÁzD.

108



We conclude that the agents have the same preference relation on the set

RA([).

Now, if agents have the same pleference relation on the set ftA(f), by

lemma 2.4, for each / € D([) all Pareto efficient allocations provicle the

same utility to the agents. Define the mechanism T : D(t) -. IR+ x R"'z* by

letting 7(/) be any Pareto efficient allocation in the economy /, that is, for

each / € D(l),, f (f) e PE(f). This mechanism is unambiguous because all

Pareto efficient allocations provide the same utility to the agents. We claím

that 7 is also cost monotonic with respect to f. Given / < g e D(l), then

T(f),7(g) e RA(q Since all agents have the same preference relation on

RA([) and 7(/) is also feasible with the production function g we conclude

that u(T(/)) I "(T(g)). Hence T is cost monotonic with respect to f. n

Only in the case that given a fixed proportion of inputs the agents or-

dei- the relevant allocations in the same way it is possible to obtain a cost

monotonic mechanism which is Pareto efficient.

As a consequence, in the larger domain D (or any subdomain containing

the Leontief production functions) the properties of Pareto Efficiency and

109



cost monotonicity are not compatible.

Theorem 2.6 If we f,r the uti,li,ty of agents (un)n=t,, sati,sfying assumption

A.1, there is no Paret,o effici,ent and cost rnonot,onic mechanism de.fined on

any subdomain of D contai,ni,ng the Leonti,ef prod,ucti,on functions.

Proof

The result is due to the non-existence of preferences satisfying assumption

A.1 and sr.lch that, for each f € IRl+, represent the same preferences relation

on ftA(t). We may choose U\,U2 € IR++ and f : (fr, lr), , \ , < )2 such that

(s ' ; tr t0, @';\rü e RA(f) and

u t ( y t , ) t | < u { A r , \ z ü

uz(At,)r i ) > uz(az, ÁzD (2.r)

If there was a cost monotonic and Pareto efficient mechanism over D,

then it would also be a Pareto efficient and cost monotonic with respect to ú

over- D(f). But then the inequalities in 2.1 contradict Theorem 2.5. tr

REFERENCES

110



Gines, M. and Marhuenda, F. (1994): "Efficiency' monotonicity and ratio-

nality in public goods economies," Mimeo, Universidad de Alicante.

Milleron, J. C. (1972): "Theory of Vaiue with Public Goods: A Survey

Article," Journal of Economic Theory, 5, 419'477.

Moulin, H. (1987a): "Egalitarian-Equivalent Cost sharing of a Public Good,"

Econometrica, 55, 963-976.

(19S7b): Axioms of Cooperative Decision Making. Econo-

metric society Monographs, Cambridge University Press.

Moulin H. and Roemer J. (1989): "Public ownership of the external world

and private ownership of self," Journal of Political Economy, 97,347-

ñ^ -J O l .

lVlouiin, H. and W. Thomson (1992): "Can Everyone Benefrt from Growth?

Two Diff icult ies," Journal of Mathematical Economics, L7,339-345.

Thomson, W.(1989): "Monotonic Allocation Rules in Economies with Pub-

lic Goods," Mimeo, IJniversity of Rochester, l{Y.

i 11



CHAPTtrR 5

EQUALITY AS A CONSEQUE}TCE OF SOLIDARITY

Abstract

In economies with public goods, and agents with quasi-Linear pref-erences? we give a characterization ofthe welfare egalitarian correspon-dence in terms of three axioms: Pareto optimality, symmetry and sol-idarity. This last property requires that an improvement in the abilityto exploit the pubiic goods of some of the agents should not decreasethe welfare of anv of them.



Introduction

We address the issue of how the benefits derived from a publicly owned tech-

nology should be allocated. Consider a set of agents who possess a common

technology used in the production of some public goods. This technology

uses the private goods of the agentsas an input. If these agree on an optimal

bundle of public goods and a financing plan for it, then the whole project is

carried out. Otherwise, the agents remain in the status quo.

We adopt the point of view of axiomatic theory in that we seek a solution

determined by some equitable properties. The minimum rationality require-

ment which one may ask for is Pareto efficiency. It seems a waste of resources

not to pick some aliocation in the Pareto frontier. The latter is however a

very large set and there is no natural social ordering of the allocations in

it. This renders the problem posed above interesting and gives rise to one

of the central problems in social choice theory. The normative approach rec-

ommends the use of some considerations of fairness to narrow down the set

of admissible allocations. The interpretation of the word "fairness" in the

last sentence is at the core of the controversy and the-r-e is a vast literature

suggesting different axioms to implement a proposal.



For example one of the properties of fairness, which we adopt here, is the

requirement that "equal agents should be treated equally". This is usually

referred to as symmetry and in our setting it corresponds to axiom 2.3 (ii).

There are plenty of other properties which, together with the two we have

just mentioned, will determine a unique solution. In this work we considel

an axiom of solidarity which is akin to the axiom of monotonicity in social

choice theory. The axiom of soiidarity in our setting (see axiom 2.3 (iii)

below) requires that whenever the ability of some agent to use the public

goods increases, it should translate into an improvement in the welfare of

the whole society or at least should not hurt any of the othei- agents.

The principle of solidarity is not entirely new in the literature. It has

been used before in Sprumont and Zhou (1995), Sprumont (1995) and Splu-

mont(1996) in a slightly different form. The idea herein is that whenevel

thele is a change in the preferences of some agents, the ones lvhose pr-ef-

elences remain the same should be affected in a similar manner. A closelv

related necessary and sufficient condition for the existence of solidarity is pro-

vided in Keiding and Moulin (1991) where solida"rity now means that when

the exogenous parameter changes the welfare of every agent is affected in the

same direction. The concept proposed here is also related to the notions of

174



population solidarity (Chun (1986) and Thomson (1983)) and ski l l sol idarity

(Fleurbaey and Maniquet (199a)).

The purpose of this work is to show that on the set of economies whose

agents have preferences represented by quasi-linear utility functions, these

three axioms characterize a unique solution. It is the welfare egalitarian

correspondence which splits the surplus equally among the agents.

The welfare egalitarian soiution has been studied in the axiomatic bar-

gaining literature by E. Kalai (1977) where the central point is a resource

monotonicity axiom. R. B. Myerson (1977) uses a condition on decomposabil-

ity with respect to sequences of bargaining problems and enough invariance

under ordinal utility transformations to determine a solution which equalizes

the gains of the agents in some ordinal utility space. Also in the context of

ordinal preferences Y. Sprumont (1996) has axiomatized the welfare egalitar-

ian solution by means of solidarity with respect to changes in the feasibilitv

constraints and the oreferences.

H. Moulin and J. Roemer (1989) study the justification for the existence

of inequalities in a model with a publicly owned technology and two iden-

tical agents whose skills are privately owned. In addition to efficiency, they

propose three othel properties which reflect the pubiic and private prop-

115



erty rights of the agents. They also find that the produced goods must be

distributed in a way that equalizes the welfare of the agents.

Our model can also be regarded along similar lines. The role played by

the "private ownership of skill" in Moulin and Roemer (1939) is played in

our model by the private ability to exploit the public goods. In this way, our

result can also be interpreted as saying that the differences in talents of the

agents do not necessarily lead to inequalities.

The cla,ssical approa,ch taken by social choice theory is based solely on

using information derived from the utility possibilities set. In particular in-

formation about the economic environment behind is ignored. J. E. Roemer

(19SS) has criticized the axioms of axiomatic bargaining theory because they

cannot always be justified on economic grounds. The author proposes al-

ternative axioms which make use of the explicit economic information to

characterize the standard bargaining soiutions.

We also depart flom the line of research followed by classical bargaining

theory and address the issue of collective decision making but within an

economic environment. Our assumptions and modelling are on the basics of

the economy. We postpone the discussion of the differences between these

two frameworks to section 4 after we have presented the model. For the

116



moment, we only mention that, for example, it is not necessary to assume

convexity of the utility possibility set to guarantee that the solution is Pareto

optimal. It follows from natural assumptions on the economic data of the

model, nameiy, the technology and the preferences of the agents.

Even though the axioms we consider reflect ethical concerns similar to

those in axiomatic bargaining theory and are interpreted in the same way)

they are logically different and háve a meaning within the economic scenario

considered. On the other hand, we continue with the normative point of

view. Our axioms are more related to the concerns of welfare economics

than to those of the mechanism design literature.

2 Notation and the model

Given two vectors r,z in some Euclidean space R!, the notation r 2 z (resp.

ü >> z) means that r¿ 2 z¿ ( resp. *¿ > "¿)

for every i : I , . . . ,p . We wr i te

r ) z to indicate that r ) z and r I z. Fina1ly, r y' z means Lhat r¿ 1 z¿

fo r some i : I r . . . r p .

Thespaceo f pub l i cgoods i sY : RT : { y e R - : y > 0 } . The

technology to produce them is jointly owned and described by a cost function

c : Y -+ R. Throughout this paper we will consider a fixed cost function c

LI7



satisfying the following.

Assumption 2.1 The mappi,ng c is lower semicontinuous, .stri,ctly increas-

i,ng and satisfies c(O) : g.

Recall that a mapping c : Y --+ R is lower semicontinuous if for each

z € Y we have f (r) S liminfr-, f (y). A lower semicontinuous function

is bounded on every compact set and it attains its minimum value. As-

sumption 2.1 allows for technologies with jumps. In particular, initial fixed

cos ts a re i nc luded . An a l l oca t i on ( z ; t ) : ( z ; t t , . . . , t . ) €Y x X i s f eas ib le

whenever c(z) SDT=rto.

We let lú : {1,2. . . ,n} denote the set of agents and for each agent

i e 1/ we let X¿ - R. be his consumption set of private good and de-

f ine X - Xr x " ' x Xn. The preference re ia t ion of agent i : L , . . . ,n is

r-epr-esented by a quasi-linear utility function u,o(y;t) : t¿(y) - Ú;, where

(y ; t ) : ( y ; t r , . . . , t n ) e Y x X and r¿ :Y - - lR .

The mapping r¿ represents the private technology available to agent i :

1, . . . , n for using the public goods. The present set up might be applied to a

scenalio in which some companies, who do not compete among themselves,

decide to collaborate in some project which will be profitable for all of them.

118



Each of the mappings 7ri represents the ability (which is different among the

companies) of firm i e N to exploit the common project. Thus n;(y) - t is

the net benefit firm i € 1/ obtains when it has to contribute I units of its

private good for implementing the bundle y of public goods.

We identify each agent with its private technology. We will often write

vectors of such functions n : (nr,. . . ,Tn) and we will use either one of the

fo l l ow ing no ta t i on u " (y ; t ) : ¡ r (A ) - t - ( u " , (a ; t ) , . . . , un . (a i f ) ) : ( " t ( y ) -

t t , . . . ,r^(A) - t"). The vector of ut i l i ty iechnologies result ing from n' :

(nr,. . .,Tn) € -E when r.; is replaced by the new utility function z¿ is denoted

by (* - , , v i ) : ( t r , . . . , ' t r i - r ¡u i r l r i+r r . . . ,Tn) . Given two vectors of pr ivate

technologies, r- and z defined on Y we say that ¡r ) v (or un ) u',) whenever

"(y) > u(y) for every y €Y.

The foilowing assumption is made on the preferences of the agents'

Assumpt ion 2.2 For each i : 7 , . . . , f r , l i i

decreasing mapping satisfying r'¿(0) : 0 and

R is o continuous, tzon-

r . - ^ . . * t t n ( r )r r r r r D uy / il l v l l * - c \Y )

Since the technology is fixed, an economy is identified with a vectol of

private technologies n' : (nt,. .. , n,,) satisfying 2.2. We let E denote the set

i 19



of such economies

A mechani,sm is a mapping R : E -+ Y x X which assigns a. feasible

allocation R(r) to every economy n € E. We denote bV P(") (resp. P.("))

the set of Pareto optimal (resp. weakly Pareto optimal) consisting of those

feasible al locations (y;¿) e Y x X for which if :w,(y;t) < ut*(z; r) (resp.

. ' .u , (y ;¿) << u, (z i r ) ) for someother a l locat ion (z ; r ) eY x X, then (z ; r ) is

not feasible.

The problem which occupies us here is to fi.nd "the optimal" bundle of

pubiic goods and a fair share of its cost. According to the normative approach

a mechanism which is "acceptable" should satisfy certain equity require-

ments. The normative principles concerning us in this work are described by

the following three properties.

Axionrs 2.3 For euery r € E,

( t , ) R(n) e P(r ) .

( i i ) I f l r t : " ' - ' t r , * t hen , un r (n ( " ) ) : " ' : u " " (R ( r r ) ) .

(iii) Suppos€ 7r¿o 2- u¿o for some io e N. Consi,der the uector.s o.f mappings

T : ( ¡ r r , . . . , T , ) and u : (T -¿o ,u¿o ) . Then u , (R ( " " ) ) 2 u , (R (u , ) ) .

r2a



Properties (i) and (ii) are standard in the literature and need no further

comment. The novelty here lies in axiom 2.3 (iii). It is interpreted as saying

that any progress in the private technology of one firm, which is reflected in

an increase in its net profits, should,not affect negatively the other firms. It

is in this sense that 2.3 (iii) is called a solidarity axiom: an improvement in

the skill of one fi.rm translates into a benefit for the whole society or, at least,

does not hurt the other members. This assumption could be justified on the

basis that the technology to produce the public goods is jointly owned by all

the agents. Hence, they are forced to cooperate since they need to agree on

both a unique bundle of public goods and a financing plan for it.

We remark that an equivalent but more concise folmulation of axiom

2.3 (iii) is the following.

Axiom 2.3 ( i i i ) : I f r 2 u, rhen u"(R(u") ) 2 u, (R(u") ) .

Motivation: Let us consider an economy consisting of two agents i : L,2

with quasi-linear utility functions ri(A) - ú¿. The egalitalian equivaient allo-

cations were introduced by E. A. Pazner D. Schmeidler (1978) in the setting

of plivate goods and studied in the context of public goods by H. Moulin.

For our purposes we will consider them for the case of one public good and

72I



y ' yFigure 1: ,

two agents having preferences represented by quasi-linear

'

"n (y ) - l¿ w i th i : L ,2 . :

functions

Suppose the function lr1 I rz - c attains a maximurn at z € Y. The

set of egalitarian equivalent allocations consists of those feasible allocations

(";tr,ú2) for which there is y € Y such that r1(z) * n2(z) - ,(r) : nr(y) *

"r@). The bundle of public goods y is the reference bundle. The utility

obtained by each of the agents ' i : I ,2 is r¿(z) - t ; : ¡r¿(y).

Imagine now that we perturb slightly the preference relation of agent 2

so that it is represented by a new quasi-linear utilityfunction "L@)

-ú¿ such

that near y we have "L@)

> nr(y) (See figure 2) but stlII r'r(z) : rz(z).

Since "z(y)

> nz(A),, the new reference bundle, say a', is strictly below

y. Hence, in the new situation agent t has suffered a loss in utility even

though the total surplus is still the same and there is no apparent reason

why replacing agent zr2 with agent r'l should affect agent 1 negatively. (Total

r22



surplus has not changed and the strategic interplay between the agents seems

to be the same)

Next we define the welfare egalitari,an corlespondence IM : E --++ Y x X.

For each tr € E,the set W(n)consists of lhose feasible al locations (y;t) e

Y x X sa t i s f y ing un , (A ; t ) : . . . - ' t t r rn (y ; f ) and wh ich a re max ima l w i tn

respect to this property; that is, i f . unr(z;r).: " ' : l l rnQ;r) for some other'

feas ib le a l locat ion ( " ; r ) <Y x X, then u"(y ; t ) ) u" (z ; r ) . Consider the set

of attainable utilities

n

A(n) : { " (s ) - t , (y ; t ) e XxY, c (a) i> I tu}1,= I

Then P(n.) is the boundary, }AQr), of A(r) and W(n) is obtained in the

space of utilities by intersecting the diagonal, A C )Rl, with 0A(r). Since

l imsup¡ ¡ r ¡ ¡ * * t¿ (y ) l c ( y ) : 0 f o r eve ry i : I , . . . , f r . ¡ t he re i s (a t l eas t ) a so lu -

bion, say y, to the maximization problem max{fT=rn¿(y) - r@): y e Y}.

Then

which clearly intersects A. We see that A ¡Vr¡r¡ c P(r') for ever-y tr e E.

1 . )2L A ¿

(2.r)



Remark that in the classical framework of the social choice literature, welfare

egalitarian allocations are, in general, only weakly Pareto optimal, but not

optimal in the strong sense. However, in the present context, the welfare

egalitarian mechanism is (strongly) Pareto. efñcient.

Even thou gh W(r) might be multiple valued, the agents are indifferent

among the allocations in W(").Thus, we think of it as being single valued

by choosing an arbitrary selection of it. Clearly trZ satisfies axioms 2.3.

The contents of the next result is that it is essentially the only mechanism

satisfying those properties.

Theorem 2.4 A rnechani,sm R sati,sf,es arioms 2.3 i,f and only if u"(R(r)) :

u"(W (tr)) for eaery r € E .

Theorem 2.4 also admits a negative interpretation. One may ask whether

th.ere is an economic context in which an improvement in the pr-ivate tech-

nology of one agent to exploit the public goods would mostly benefit that

agent but without hurting the others. Theorem 2.4 shows that any differen-

tial advantage some agents might have from their private abilit¡, to exploit

the public goocls must vanish.

We now address the proof of Theorem 2.4. We only have to shorv that

724



every mechanism satisfying 2.3 has to coincide u'ith the weltare egalitarian

correspondence.

Proof of Theor ern 2.4: Let ft be a mechanism satisfvine 2.3 and let

R(n) : (g;r) e Y x X. Since f i(n') is Pareto optimal, then y is a solu-

t ion to maxf!, "n(y)

- c(y) and "(!)

- DT-rrn. The proof proceeds in

three steps.

Step 1: Choose another profi le of ut i l i t ies u,(y;t): r(y) - ú such that

u 1 r and u¿(z) : r¿(y) for every z ) y. Then u-(RQr)) : u,(R(r)) and

u,(R(u)) < u"(R( t r ) ) : " , ( f i (zr ) ) .

Hence, u"(R(r ) ) : u , (R(r ) ) because

R(u ) e P ( , ) .

Step 2: Fix y* >> g and let e > 0 be a real numbel such that (1 -

e)y* >> y. We choose a uti l i ty profi le u.(y;t) : , -f such that the fol lowing

holds.

( i ) , . : . . . : c ü n .

( i i) c.. ' (z) :0 for every , y' (I - e)y*.

( i i i ) u, '(z) : w(a.) for every z 2- a*.

( iv) The solution to max DT=rr¿(r) - c(r) is attained at the point y*.

125



, ( ' ) : , (y - )

Figure 2:

(v) u(z) > ,(") for every z ) y*.

(" i)

fu",(n(,)) : É u.,(R(w))

We indicate how it is possible to construct the mappings Lor ¡ . . ., co,. First,

we remark that to obtain (iv) we only need to make the functiols üJ1 :

. . . : c,J," "steep" enough in the region between (I - ,)y- and y*. Then by

taking y* large enough and i*.,1 : . . . : c.,ln applopriately defined at the point

y* €Y, we can also get (v) and (vi). I\ow, (ii) and (iii) define the mappings

uJr : " ' : u)n On the whole s'ace.

It fol lows fr"om (iv) lhat u,(R(r)) : u,(!t";r*) fol some r* e X. By (i)

we a lso have r f - - ' ' - r i .

126



To

Let

Step 3: Take the vector o f technologies B(z) : (0{z) , . . . ,0 .G)) :

( rnax{a(z) , r r ( r ) } , . . . , max{2, (z) ,a*(z) } ) and the prof i le of u t i l i t ies uB(z; t ) :

gQ) -1. We also choose e > 0 small enough so that ' the solution to

Max D!=r gn(y) - .(y) Is . t . y> ( I -e )y . J

is attained at the point y* (recall that B(z) - w(z) if z is close enough to g*).

: u,(R(u) : u,(R(a)) .

u p(R( p)) > z.(ft(c-. ') ).

(a) Assume first that y ) y*. Then, by (v) we have that 0(ü) : r(A) >

z(e) and ",(R(13))

: uB(R(13)) > u,(/?(r)). Since ft(B) is also feasible in

the economy z. and ft(c-r) e P(r), then uB(R(p)) : u,(R(a)).

(b) Suppose now that y / (1 - e)y.. then from (i i) we see thal B@):

, (a) :> , (ü) : and u, ( f i , f ) : uB( f i , f ) > u, (R(u)) : u , (v , , t ) . But , i f

some inequaiity is strict, then rB(z) would not be Pareto optimal. Hence,

"p (R(p ) ) : u , ( .R ( r ) ) .

(c) Otherwise, y 2 (I-r)y- and y y' y* and g* is in region II of f igure 2.

The Theorem follows if we can prove that uB@@))

see this, note first thaL up(R(B)) > u,(R(u) and

(Ú;f) : R(P). We consider three cases'

127



T h r r s

Lu., (R(r)) : I ro(y-) - c(y-) : I Bn@-) - c(y")

" - 1 ; - 1 t - 1 ' .

n 1 7

?'" '" '

Since, uB(ü, i ) 2 " . (n(c . . ' ) ) , then

"p(ü, , i ) : u , (R(r ) ) , that is , z tB(R(B)) :

u,(E(c,. ')).

In either of the three cases, the claim follows from

\ a / ñ / \ \ \ . . . - / ñ / \ \

L u q \ I t \ a ) ) : 4 u u i \ 1 1 " \ u

) )1 . = L X = L

and B¿(y) : max{¿¿¿(ü) , rn(ü)} for a i l i : r , . . - , f r . Thus, u"(R(r ) ) :

u,(R(w)) and this concludes the proof of Theorem 2.4. x

3 Private goods

We argued in section 2 that axiom 2.3 (iii) can be justified in a cooperative

setting anci is interpreted as a solidarity requirement among the agents. In

this section we show that, even from the point of view of the strict mathe-

matical argument, 2.3 (iii) only has bite whenever-some degree of cooperation

among the agents is necessary. We will see that Theorem2.4 no longer holcts

728



in economic environments in which the agents do not have strong incentives

to coordinate their decisions.

We abandon the setting of public goods and let Y = RT be the space

of (produced) private goods. That is, we assume that the sets X1, .. . ,Xn,

represent. as in section 2, the spaces of some private good which can be used

by means of a public technology c, to produce a bundle of goods U € LT=rY¡.

This vector y is no longer a bundle of public goods, but it has to be divided

y : y r + . . . +y ' , w i t h y i €Y¿ among t he agen ts i : I , . . . , , t 1 .

The difference from the approach in section 2 is that, in the present

context , a feas ib le a l locat ion consis ts of a vector (y t , . . . ,y" ; t ) e I , i x . . . x

h x X such t ha t " ( y t

+ . - - 1y " ) : h * . . . + tn .The res t o f t he mode l and

assumptions are translated readily into this new scenario. The question we

now pose is whether the equivalent of Theorem 2.4 holds in the settins of

private goods.

Suppose the techno logy i s l i nea r , so tha t , ( y t + . . .+y " ) : c ( l l \ + . . .+

c ( y " ) .F i x a u t i l i t y p ro f i l e , : ( u r , . . . , un ) and f b r each i - 1 , . . . , n l e t

(yn(u),t¿(";)) €V x X¿ be the solution to agent i 's maximization pr-oblem

max ,n . (yo, tn) \s.t . c(y') : t¿ J

r29



Then the mechanism,S : E - -+ Yr x . . .xY" x X ass igning the a l locat ion

^9(u) : (At@r) , . . . ,y" (u*) ; t {u) , . . . , , t , (u , ) ) to every ut i l i ty prof i le u :

(ur , , . . . ,un) ver i f ies proper t ies 2.3 but is not wel fare egal i tar ian.

The difference between public versus private goods is that, in the first. '

case, agents are forced to come up with some common identical bundl"

consumed by all of them. Whereas, in the case of private goods, the linear

technology allows each of them to behave individually; in such a way that

the different solutions proposed by each of the agents are compatible.

This suggests that, if there are sufficiently increasing returns to scale

in the technology, such that it pays off to cooperate, then a lesult similar

to Theorem 2.4 might hold again. Indeed the proof presented there can be

easily adapted to the private good context if one assumes that the technology

c has a horizontal asymptote. This is, of course, the most extreme case of

incleasing returns to scale in which it is possible to obtain arbitrarily large

bundles of goods by using only a limited amount of resources.

Since, that is clearly a very unrealistic situation we do not pursue this

point any further. We only mention it here to stless the idea that axiom 2.3

(iii) has bite only in situations in which it is advantageous for the agents to

coopelate.

130



C oncluding corrlrrrent s

We review some of the differences between the axioms as presented here and

the way they appear in social choice thgory. Since we are in the quasi-

linear context, for each r € E the Pareto frontier, which coincides with the

boundary of the set A(ri) defined in equation 2.1, is the l ine or *. . .*Dn: &,

where o is the maximum surplus that the agents can (collectively) obtain.

We make the following remarks.

(i) Axiomatic bargaining theory makes use of utility possibility sets which

look like rectangles. Assumption 2.2 excludes these domains in our setting.

As a consequence, the proof used in the classical setting (Kalai(1977)) does

not go over here. As a bonus of our approach we obtain that weifare egalitar-

ian allocations are Pareto optimal in the strong sense, not only weak Pareto

optimal.

(ii) Axiom of monotonicity as compared with solidarity: Given two utility

profiles n' and z it is always the case that either A(tr) c A(u) or A(u) C

A("). Hence, the axiom of monotonicity applies to any pair of economic

environments n and z, whereas solidarity does not (at least directly).

(iii) The axiom of symmetry is also different in our context from the



corlesponding one in bargaining. In the latter, symrrietry applies whenever

the utility possibility set is symmetric with respect to the diagonal. But

this is always the case in our context. Hence, thelaxiom of symmetry as

used in bargaining theory cannot distinguish amor1g the agents ancl would

recommend treating them equally regardless of them being very different.

This shows clearly the strong demands of bargaining theory when applied

to particular economic environments. Just the axiom of symmetry by itself

would determine, in a trivial way, the egaliiarian solution.

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