The theory of implementation of socially optimal decisions in economics

Lindahl's equilibrium is an extension of the Walrasian idea of librium to economies with public goods... Proposition 1: If there are constant returns to scale, the utility of every consu

The Theory of Implementation of Socially Optimal Decisions in

Economics Luis C Corchón

OPTIMAL DECISIONS IN ECONOMICS

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1.6 Mas-Colell and Silvestre's cost-share equilibrium 151.7 A criticism of the notions of equilibrium with an

2.2 An example of resource allocation: the case

3.3 The impossibility of truthful implementation in

3.4 The impossibility of truthful implementation in

3.5 The manipulation of the initial endowments 56

4 Implementation in Nash Equilibrium (I):

4.2 Characterization of social choice correspondences

4.3· Implementation in Nash equilibrium in economic

4.4 Implementation when the feasible set is unknown

5 Implementation in Nash Equilibrium (II):

5.2 Implementing the Lindahl and the Walras

correspondences by means of abstract mechanisms 905.3 Doubly implementing the ratio and the Walras

correspondences by means of market mechanisms 965.4 Implementation of solutions to the problem of fair

6.3 Implementation in undominated Nash equilibrium 118

Appendix II: double implementation in Nash and strict

7 Bayesian Implementation 137

7.4 Necessary and sufficient conditions for Bayesian

A major purpose of social choice theory is to study the tradeoffs tween different desiderata This book focuses its attention on' a well-defined subset of social choice theory, namely the implementation ofsocially optimal decisions in economics

be-Implementation problems arise when the social planner (sometimes

a real person, sometimes a surrogate of society) cannot distinguishbetween things that are indeed different This may be due to the factthat certain characteristics are unobservable to the planner or that even

if the planner has this information she cannot use it because of legalrequirements For example, public monopolies are sometimes bound

by laws not to discriminate among consumers

The quintessential implementation problem is that of the 'free rider',that is, the agent who hopes to improve her luck by not telling thetruth about her own unobservable characteristic (Imagine that you areasked how much you would pay for the construction of a park nearyour home and that in the case where the park is actually built thisamount is the one you would pay Would you give a truthful answer?)

A closely related question was spotted by the ancient Romans andsummarized by the question 'who controls the controllers?' In the words

of Roger Myerson: 'An organization must give its members the rect incentives to share information and act appropriately An indi-vidual cannot be relied upon to testify against himself or to exert effortsfor which he will not be rewarded' (in L Hurwicz, D Schmeidlerand

cor-H Sonnenschein (eds) 1985, Social Goals and Social Organization,

(Cambridge University Press), chapter 8) But if these incentives must

be the right ones they bind the choice of the planner as much as thescarcity of resources does: 'The basic insight of mechanism theory isthat incentive constraintsshould be considered coequally withresource constraints in the formulation of the economic problem In situationswhere individuals' private information and actions are difficult to monitor,the need to give people an incentive to share information and exertefforts may impose constraints on the economic system just as much

as the limited availability of raw materials The theory of mechanismdesign is the fundamental mathematical methodology for analysing theseconstraints' (Roger Myerson (1989), entry on 'Mechanism Design' inthe J.Eatwell, M Milgate andP Newman (eds), New Palgrave(London:

xi

Macmillan) Chapter 1 of this book presents the 'classical' theory ofresource allocation in which incentive problems are entirely disregarded.Chapter 2 presents the main ingredients of the approach that will befollowed in the rest of the book where incentives of agents are care-fully modeled.

Some writers have argued that the kind of opportunistic behaviorimplied by the 'free rider' problem accounts only for a part of thestory (see, for instance, D North, Structure and Change in Economic History, New York, Norton, 1981, chapter 5) Indeed one of the most

fundamental contributions of the Theory of Implementation has been

to show that the 'free rider' problem mayor may not occur, ing on the kind of game that agents play and on the (game-theoretical)solution concept In fact the story of implementation theory is that of

depend-a liberdepend-ation from constrdepend-aints The first mdepend-ajor development wdepend-as in thework of Gibbard-Hurwicz-Satterthwaite in the early 1970s: when in-formation is private, and thus the appropriate equilibrium concept isdominant strategies, incentives bite a lot These incentives adopt theform of incentive compatibility constraints where for each agent totell the truth about her characteristic, must be a dominant strategy.This and related topics are discussed in Chapter 3 The second majordevelopment case from Maskin in the late 1970s: when the informa-tion about the characteristics of the agents is shared by them (but not

by the planner), and thus the relevant equilibrium concept is Nash librium, incentive compatibility does not matter Thus, for instance, byputting agents in a circle, if each agent is able to monitor the charac-teristics of her neighbors the free rider problem dissolves What it biteshere is a, generally much weaker, monotonicity condition that can beexplained as follows Suppose that an allocation, say a, is optimal forsome preferences Now preferences change in such a way that a goes

equi-up in all individual rankings about allocations Then a must also besocially optimal for the new preferences Chapter 4 is devoted to ex-plaining the theory of Nash implementation The third big push came

in the mid-1980s from Moore and Repullo.' They and their followersexploited the knowledge gathered on Nash equilibrium refinements inthelate 1970s and early 1980s that followed the lead of Reinhard Selten

By discarding some Nash equilibria (because they are not subgameperfect or they are weakly dominated, etc), they were able to showthat neither incentive compatibility, nor monotonicity bite (seeChapter 6 below) Thus, in this approach, incentives do not have anycutting power

We should mention here two more highlights of implementation theory

One is the theory of implementation in Bayesian equilibrium (a cept developed by John Harsanyi) where agents act under incompleteinformation Constraints implied by Bayesian implementation are both

con-a form of incentive compcon-atibility con-and con-a certcon-ain monotonicity tion Another important topic is the construction of 'nice' mechanisms(including those inspired in the market) implementing specific socialgoals These issues are discussed in chapters 7 and 5 respectively ofthis book My guess about the direction of future research is that itwill move towards the consideration of dynamic models and a lessmechanical formalization of the rules of the game (on this matter thepaper by Hurwicz in the first issue of Economic Design is warmly

restric-recommended)

This book concentrates on the study of two questions First of all, wewin study which kind of social decisions can be implemented by non-cooperative games by means of various types of equilibrium (domi-nant strategies, Nash or some of its refinements, Bayesian equilibria,etc.) Secondly, we will study some concrete mechanisms which imple-ment 'good' social decision rules The emphasis of the applicationswill lie on the implementation of the Lindahl correspondence in economieswith public goods Pure exchange economies are also considered Eachchapter includes a collection of problems These problems are meant

to supplement the exposition of the main text and to test the standing of the reader about certain questions They also provide ad-ditional references of important work

under-This book is based on my lecture notes for a course in the PhDprogram of the University of Alicante It is a pleasure to acknowledgethe challenging intellectual atmosphere of my department led by peoplelike Carmen Herrero, Fernando Vega-Redondo, Ignacio Ortufio-Ortin,Paco Marhuenda, Subir Chattopadhyay and Antonio Villar Thanks to

a Fulbright scholarship I could visit the Department of Economics ofthe University of Rochester There I learnt Implementation Theory fromWilliam Thomson My debt to William is immense not only because

of his guidance and care but also because my contact with his studentsproved to be very fruitful for my future research Among them, SimonWilkie, Baskhar Chakravorty and Tomas Sjostrom became co-authors.Parts of our joint research have been used freely in this book Thegroup also included J.P Conley, D Diamantaras, T Shinotsuka and T.Yamato among others I also owe to Ignacio Ortufio-Ortfn countlessilluminating discussions on the role of Implementation Theory A part

of them came out as a joint paper reviewed in Chapter 7 Another

joint paper with my former student Carmen Bevia forms the basis for

a section in Chapter 3 John P Conley corrected some serious derstandings of mine in Chapters 1 and 2 Chapter 3 benefited fromthe insightful comments of Salvador Barbera I am also indebted to

misun-my fellows Jose Alcalde, Inigo Iturbe-Ormaetxe, 'Diego Moreno, JavierLopez-Cufiat and to several generations of students of the PhD pro-gram of the University of Alicante, especially to Pablo Amoros, BernardoMoreno and Socorro Puy for correcting my mistakes J Alcalde and

B Moreno are co-authors of a paper that became the backbone ofAppendix I to Chapter 6 and S Baliga is co-author of a paper re-viewed in Chapter 5 I am specially indebted to Jorg Naeve for hiscareful reading He amended several inadequacies in propositions 1(Chapter 3), 2 (Chapter 4) and 2 and 3 in Chapter 7 The proof ofthese results presented in this book are virtually his My secretary,Mercedes Mateo, did many of the drawings and with her efficiencycontributed to my dedication to the book The latter also apply to VeraEmmen It is only fair that I thank warmly all these persons for thevarious kind of help that they have given me This book could nothave been written without them, but I alone am responsible for anyerror

LUIS C CORCHON

1 Economies with Public

Goods

A public good is one for which there is non-rivalry in consumption,that is, if the good is consumed by individual i, this does not precludeindividual j from consuming it When there is neither exclusion norfree disposal a public good becomes a collective decision whose con-sequences affect the whole of society Pure public goods are those whosequantity consumed by each member of the society is identical It should

be noted that a public good is a special sort of externality

Examples of public goods include: (i) goods which are generallyoffered by central governments, such as the armed forces, the policeforce and, to some extent, roads, railways, social security and justice;(ii) Goods of a more local nature offered by autonomous governments

or town councils, such as lighting, sewer systems, the collection ofrefuse, bridges and parks; (iii) goods offered by the public or privatesector such 'as TV, radio, inventions and pollution Therefore, it must

be noted that(a) a public good is not necessarily desirable (even thoughunder free disposal its consumption might be avoided) and(b) it is notnecessarily offered by the public sector

In this chapter the following problem will be studied: Suppose wehave a society where there are both private and public goods Does anoptimal way of organizing this society exist? In order to examine thisquestion, we will first of all formally describe an economy with publicand private goods

1.2 EFFICIENCY AND PUBLIC GOODS

We will suppose that there are n consumers, l pure private goods and

m pure public goods Each consumer has a vector of initial resources

of private goods Wi. A consumption bundle for each individual i is an

l + m dimensional vector (Xi' y) E Xi ~ lRl+ m

, where the first (resp.the second) component of the vector denotes the consumption of private

(resp public) goods and Xirepresents the consumption set of the ;th

agent Note that the consumption of public goods is not denoted by asubscript as it is the same for all agents Each consumer has prefer-ences regarding the consumption bundles, which can be represented

by a utility function u i: Xi ~fR. The technology of the economy will

be described by a set Yc fRl+mor sometimes by the function Ttz ; y) =

owherezis an I-dimensional vector which represents the input tive) output (negative) vector of private goods and y is the vector of

(posi-outputs of public goods.1If Y = {O} we have an exchange economy

If (z ,y) E Y~y = 0 we have an economy where only private goods

with (z, y) E Y and i'Jd:JXi + Z :5 t:JWi (social feasibility)

Definition 1: An allocation (x, y) is Pareto efficient if it is feasible and no other feasible allocation (x, y/ ) exists, such that u/x(, y/) 2

uJ Xi' y) for all i, and with strict inequality for at least one sumer.

con-It is possible to prove that if the utility functions are continuous, satiated and concave and the set of production possibilities is closedand convex an allocation is Pareto efficient if and only if it maximizes

non-i (XiUi(X p y) on the set of feasible allocations, for some vector(al' ,an) that belongs to the simplex of dimension n - 1 That is

Let - ~(resp. ~, j =1, ,1) be the Lagrange multiplier associated

with the technological restriction T(z, y) = 0 (resp social feasibilityfor private goods !I (w;j - Xi) = z). As it was first noticed by

Samuelson, if the utility functions and the function T( ) are continuouslydifferentiable, the necessary conditions for an interior maximum are:

It is useful to have a graphical illustration of the set of feasibleallocations in an economy of public goods, equivalent to Edgeworth'sbox in economies with only private goods This construction is calledKolm's triangle (after its inventor, Serge-Christophe Kolm) and it as-sumes n = 2, 1= m = 1 and constant returns to scale A detailedexplanation can be found in the book of J.J Laffont quoted in thereferences in section 1.8 below Figure 1.1 shows Kolm's triangle wherethe set of efficient allocations is indicated.

Initial endowments

The notion of core is the application of the idea of Pareto efficiency to

a situation where any coalition (i.e a non-empty subset of the set ofagents) can be formed We assume that each coalition has free access

to the technology, in other words, that the production set only includestechnical details that are common knowledge in the society Consequently,all the specialized technical knowledge is included in the commodityspace The concept of core we will use is termed as Foley's core

Definition 2: A feasible allocation (x , y) is in Foley's core if there

is no coalition C and an allocation for the members of C, (x;, y')iEC

such that u/x(, y') ~ u ix; y) for all i E C and with strict ity for at least one j E C with (z', y') E Y and L(W j - x/) ~ z',

inequal-iEC

some z'.

In other words, an allocation is in Foley's core if there is no coalitionwhich, producing the public good with its own resources, is able toincrease the utility of some of its members without diminishing that ofany member This notion tries to capture the idea of some type of

social stability in the sense that any allocation which is not in the core

cannot be stable if there exists a possibility for binding agreementsbetween members of the society

Notice that the coalitions that 'deviate' cannot hope for anythingfrom the complementary coalition This can be interpreted as sayingthat the deviating coalition separates itself from the society and willhave to provide its own public goods In this case separation from thesociety implies an exclusion from consumption of the public good.However, members of the society cannot be excluded from this con-sumption (think about the case of the police or the armed forces) Inthis sense, society behaves like a private club, excluding non-members

In other words, we must distinguish between non-rivalry in tion (which is the essential characteristic of a public good) and non-exclusion Other notions of the core can be constructed to cope withsituations where different concepts of blocking are available

consump-It can be easily observed that if an allocation is in Foley's core it isPareto efficient but not necessarily vice versa Finally notice that, inthe case of exchange economies, individuals can obtain in any coreallocation at least the utility corresponding to the consumption of theirinitial endowments (any other allocation is improved upon by a one-person coalition) This property is called individual rationality How-ever, in our case, individuals can guarantee in any core allocation aneven greater utility since they have access to the production set Wewill call this property strong individual rationality In formal terms, anallocation(x", y") is individually rational if V i = 1, , n, u;(x;': y")

~ uj(w i' 0) This corresponds to the area inside WCAEA' in Figure

1.2 An allocation (x', y') is strongly individually rational if V i =

1, ,n, ui(x;', y') ~ Vi' where Vi = max u;(x;, y) s.t, {(Xi' y)/ Xi + Z

:5 Wi and (z, y) E fl. This corresponds to the area inside DBD' B' in

Figure 1.2 In this figure the core of the economy has been marked

with a thick line BB' The curve AA' designates the individually rational

and Pareto efficient allocations

1.4 LINDAHL'S EQUILIBRIUM

In this and the following sections we will consider a family of notions

of equilibrium based on the following characteristics: (i) the decision

, r. + -T T-~ Individually

rational set

w

Initial endowments

t

regarding the quantity of public goods is unanimous and efficient; (ii)

This decision is decentralized by some parameterized schedules thatconsumers take as given; (iii) There is an auctioneer, who announcesthe parameters of the schedules

This family of concepts of equilibrium includes: (a) Lindahl's librium; (b) ratio equilibrium (Kaneko); (c) the cost-share equilib-rium (Mas-Colell and Silvestre); and (d) Valuation equilibrium(Mas-Colell), The fundamental difference between these concepts lies

equi-in the type of parameters or functions announcedby the auctioneer

We win now concentrate on the first concept in which the auctioneerannounces (linear) prices

Lindahl's equilibrium is an extension of the Walrasian idea of librium to economies with public goods An intuition of how it workscan be obtained from a simple economy (n = 2,m =l = 1) illustrated

equi-in Figure 1.3 Suppose that (x;, y') i = 1, 2 is an efficient allocation

For both consumers we draw the supporting hyperplane for the set ofpreferred points at (x;, y/) that if the utility function is differentiable,

it will coincide with the marginal rate of substitution evaluated at

(x;, y ') Note that the implicit prices in the two consumers' budgetrestrictions are different, in other words, prices are personalized Wealso draw the production possibility set and the supporting hyperplane

at (z /,Y '). These prices, multiplied by the quantity of the public good,can be interpreted as the taxes paid by each one of the consumers inorder to finance the production of the public good Finally, the firmreceives an income equal to consumers' contributions Therefore,Lindahl's equilibrium is based on the creation of personalized marketsfor public goods, where only one purchaser exists This corresponds to

a much more general idea of the first best allocation of resources in aworld with externalities being obtained by creating a market for eachexternality and assuming a price-taking behavior on the part of theagents The problem that arises here, as we will see at the end of thischapter, is that in markets created in this manner, the price-taking be-havior is far from rational

We denote by d,the share of the consumeriin the profits generated

by the use of technology, d,~ 0 and :t d, = 1

1=1

(a) Vi = 1, , n (xf, yL) maximizes ulx i , y) over

A public goods economy can be reduced to an economy where thereare only private goods in the following way We consider that each publicgood consumed by each consumer is a private good consumedonly by

her Therefore, for each public good we have n 'private' goods jointlyproduced in fixed proportions We now define a Walrasian equilibrium inthis new economy in the usual way Then, the allocation and the price

vector of Walrasian equilibrium are a Lindahl equilibrium This suggeststhat the proof for the existence of the Walrasian equilibrium can be ex-tended to prove the existence of the Lindahl's equilibrium However, inthis case, the continuity of the budget constraint is not assured unless wesuppose that the individuals possess strictly positive initial resources ofpublic goods 1, M. The existence of a Lindahl equilibrium will beobtained as a corollary of Proposition 4 in the next section.

We will now study the relationship that exists between the core andthe Lindahl equilibrium

Proposition 1: If there are constant returns to scale, the utility of every consumer is strictly increasing for some good and never de- creasing for public goods, the Lindahl equilibrium allocation is in Foley's core.

Proof: We suppose that this is not so Therefore :IC, (x!, y' )iEC' such

and such that Y: (Wi - x/) ? z' and T(z', y') = o. Therefore if

IEC

(x], y') was not chosen at the prices of Lindahl's equilibrium it must

be true that pL X / + qfy' ~ pLxf + qfyL = t/w, 'ijEC (where the fact that the utility is increasing for some good has been made use of) with strict inequality for some JEC Summing on C we obtain

firm maximizes its profits and there are constant returns to scale,

o = 17qfyL - pLzL~ z q fy' - pL Z' > O Contradiction.

;=1 iEC

It is clear that the inverse relation does not necessarily hold in finiteeconomies (see Figure 1.4) Even in infinite economies the core can

be larger than the competitive equilibrium, in contrast with economies

of private goods where they coincide Intuitively, the reason for this isthat on replicating an economy with public goods, it is as if we werereplicating the quantity of private goods in the artificial joint productioneconomy mentioned before Therefore, the standard approach (wherethe number of goods is given) cannot be applied

Finally, we will prove that Lindahl's equilibrium generates tions that are Pareto efficient.

alloca-Proposition 2: If the utility of any consumer is strictly increasing for some good, the allocation of Lindahl 'sequilibrium is Pareto effi- cient.

Proof: It is identical to that of Proposition 1.

Note that in this case it is not necessary either that the consumers'utility be non-decreasing for public goods or that there be constantreturns to scale If both assumptions were postulated, Proposition 2would be a Corollary of Proposition 1 and the fact that any allocation

in the core is Pareto efficient

1.5 KANEKO'S RATIO EQUILIBRIUM

Lindahl's equilibrium presents a series of difficulties which it is necessary

to point out Firstly, as the personalized markets have only one pur- ,chaser, it is not natural to suppose that she accepts market prices Sec-ondly, if there are decreasing returns to scale for some vector of profitshares, Lindahl's equilibrium is not necessarily within the core Thirdly,Lindahl's equilibrium does not exist under increasing returns to scale.Finally, the vector of shares in the profits of the firm is assumed to begiven exogenously The ratio equilibrium proposed by Kaneko claims

to remedy some of these defects Specifically, this solution is always

in thecore and exists if increasing returns are small or if all the sumers are identical Furthermore, it yields an endogenous vector ofshares that is in accordance with the principle that individual pay-ments should be related to the consumers' interest in the public goods.The limitations of this equilibrium notion are that a rather specifictype of technology must be assumed and we cannot postulate the ex-istence of more than one private good

con-Paradoxically, the basic idea of ratio equilibrium is very close toLindahl's original idea One example is sufficient to illustrate this Let

us suppose that there are two agents, a public good and a private good.Let us denote with (Xi the proportion of the cost of the public good

n

paid by i = 1, 2 Of course, 1;(Xi = 1 In Figure 1.5, the desiredquantity of the public good as afunction of a, for agents 1 (Yl(<X1))

and 2 (y2(1 - (Xl)) is pictured Intuition suggests thaty j is decreasing

in al and thatY2 is increasing in (J.,j< If the proportion paid by 1 wereex! (and that paid by 2, 1 - (Xj), y* would be the unanimous choicemade by both agents concerning the quantity of the public good that is

to be produced A ratio equilibrium is a feasible allocation and a list

of ratios such that if the individuals accept them as given, they wouldcome to a unanimous decision concerning the quantity of the publicgood that is to be produced We have therefore changed the assump-tion of price-taking consumers for that of ratio-taking consumers

We now proceed to a more formal definition of the ratio rium As indicated previously, it is supposed that there exists a uniqueprivate good (which will be the numeraire) and that the technology is

y

Definition 4: A feasible allocation (x', yr) is a ratio equilibrium if

3 r i E fRm i = 1, , n such that I r," = 1and Vi = 1, ,n

'=1

m

(x~, yr) maximizes ulxi , y) subject to X i+!dl'UCj(Y} 5 "Vi

Note that the profit-maximizing firm has disappeared Kaneko speaks

of agencies that offer the public goods at a certain cost The cost tion is assumed to be given (that is, we do not investigate the greater

func-or lesser efficiency of the agencies that produce the public goods) andknown by the consumers However, an alternative interpretation is thatthere exist m firms, each one producing a PMblic good, maximizingtheir profits which would always be zero (as t:lrijcj(Yj) - ci(Yj) == 0j

= 1, m). In this model, the proportions of the total cost play therole fulfilled by the personalized prices in Lindahl's equilibrium Infact, the ratio equilibrium can be interpreted as a situation where con-sumers take non-linear prices parametrically The non-linear price (that

is, a price that depends on the quantity consumed) of public goodj,for consumer i is rijcj(y}. This explains, as we will see, how ratioequilibrium may exist evenifthere are non convexities in the production

It is easy to prove that under assumptions about preferences, equivalent

to the ones in Propositions 1 and 2, the ratio equilibrium is in the coreand is Pareto efficient We do not carry out these proofs here as theywill be special cases in the propositions proven in the section corre-sponding to the cost-share equilibrium We will concentrate on two

questions: the relationship between the Lindahl and ratio equilibriumand the existence of the latter In order to study the first question, wewill assume that there are constant returns to scale Let us denote with

cj > 0 the average and marginal cost of producing public goodj, i.e

cj(Yj) = cj Yj tIj = 1, , m.

Proposition 3: Under constant returns to scale, if(xL, yL) is a Lindahl equilibrium, it is also a ratio equilibrium and vice versa.

Proof: (XL, yL) is feasible On the other hand \Ii = 1, , n,

(xt yL) maximizes u[x, y) subject to Xi +~l'tYj S Wj Defining riP

= pflc p it is clear that (xt y L

) is a ratio equilibrium for r;j = pflc t

(x', y') is feasible If rij are the proportions of a ratio equilibrium, takin; Pij = cjrij' it is clear that (x; yT) maximizes u,{x j' y) over

Xi +ld!ijYj :5 wi' and (z', y') E 1': We now suppose that (z', v') does not ~laximize the profits of the firm at prices Pu' Then, 3(Z " v')such that:

Assumption 2: V'i = 1, , n, X j contains (IWi 0) is compact,

in-Assumptions 1 and 2 imply that the set of socially feasible allocations

is compact Assumption 2 and the first part of assumption 3 are s~an­

dard The final part of Assumption 3 will be fulfilled if Ui(Wi - ~ r ij

c(Yj)' y) were strictly quasi-concave in y. This would be true ifJ-theutility function were quasi-concave and increasing returns were small

Decreasing returns to scale

y

Increasing returns to scale

Figure 1.6

(or non-existent) in relation to the curvature of the indifference lines(see Figure 1.6) Consequently, our assumptions allow certain non-convexities in the production Then, we have:

Proposition 4: Under Assumptions 1, 2 and3 there exists a ratio equilibrium.

Wi E the relative interior of Xi' 0 E Xi and cj(O) =0, Bir.) is empty It is also closed (as Xi is closed and the c j ( ) are continuous) and bounded, as is X Therefore, Weierstrass's Theorem guarantees that M;( ) is well-defined as it results from the maximization of a con- tinuous function on a compact non-empty set Furthermore, M, ()

non-is upper semi-continuous (see Kaneko (1977), p 128 or apply Berge's maximum theorem) and, as it is single valued, it is continuous.

We denote with Ylr) the components ofM/ ) that refer to public goods It is clear that Y/ ) is a continuous function As a result of the assumption of the monotonicity ofpreferences, the ri can be assumed

to be non-negative Therefore, Y =.XYi is such that Y : S ~ [R:~n

where S is the Cartesian product} of m simplices of dimension

n - 1 Therefore a theorem of variational inequalities (see Herrero and Villar, 1991) assures us that there exists a Y*such that Yij<Yj

implies rt =0 and Yo :5y~ Vi =1, , n, Vj =1, , m The assumption of strict monotonicity preferences forbids the first case and therefore Yij = Y1, \7'j =1, ,m Consequently, all consumers are willing to consume the sanle quantity of public goods 1, , m.

On the other hand, budget constraints can be written with equality and yield the consumption of Xi i =1, n in a ratio equilibrium.

Note that the structure of the proof is very similar to that of the ence of a Walrasian equilibrium In the case where there are constantreturns to scale, Proposition 4 gives us as a corollary the existence ofLindahl's equilibrium In the following section VJe will prove (as acorollary) the existence of a ratio equilibrium under different assumptions

exist-1.6 MAS-COLELL AND SILVESTRE'S COST-SHARE

EQUILIBRIUM

It is easy to see that the main idea of ratio equilibrium (that the sumers face an schedule of non-linear prices) can be generalized Allthat is required is a system of non-linear prices through which produc-tion costs of producing public goods are shared and that the consumerstake this schedule as given This idea gives rise to the cost-share equi-librium proposed by Mas-Colell and Silvestre

con-As previously, we will assume that there exists a unique privategood that is used as an input in order to produce public goods Tech-nology will be represented by the function z - c(y) = 0, where c(y)

is the cost function of the vector of public goods y in terms of the

private good Notice that we do not assume that the cost function isadditively separable Acost-sharensystem is a family of nfunctions gi;

fR":- ~IR such that gi(O)=0and ~ gi(Y) =c(y) 'v'YEIR~.In Kaneko's

t>!

Definition 5: A cost-share equilibrium is a feasible allocation

In other words a cost-share equilibrium is a feasible allocation suchthat each consumer maximizes her utility over her budget constraintstaking the cost-share system as given Note that as in a ratio equilib-rium, a profit-maximizing firm could be introduced since by definitionits profits are ~gi(Y) - c(y) == O.The following proposition is proven

in an identical"banner to Proposition 1

Proposition 5: If the utility of any consumer is strictly increasing

= 1, , n, the allocation of cost-share equilibrium is in the core.

Proof Assuming that it is not Then 3C, (x;, y' )iEC' such that

some good has been used) with strict inequality for some jEe Adding

obtain that

In the same way, we can prove that

Proposition 6: If the utility of any consumer is strictly increasing for some good, the cost-share allocation is Pareto efficient.

Proof It is identical to that of Proposition 5.

Note that as ratio equilibrium is a special case of cost-share

equilib-rium, Propositions 5 and 6 give us as corollary that this is in the coreand is Pareto efficient.

Finally, to end this section we will prove the existence of a share equilibrium In order to do that we assume the followingAssumption 4: All of the consumers are equal, that is, they have identical preferences that can be represented by a continuous utility function, and they have the same initial resources and consumption

represen-Consequently, Proposition 7 gives us as a corollary, the fact that underAssumption 4 there exists a ratio equilibrium Note that in comparisonwith Proposition 4, the assumptions concerning the production side ofthe economy have been relaxed considerably In particular, we have'not imposed any condition on the nature of the returns (as we did inthe last part 'of Assumption 3) On the other hand, the assumptionsregarding the consumers are substantially stronger because we requireidentical consumers Finally, we point out that the so-called valuationequilibrium proposed by Mas-Colell (1980) is simply an extension ofthe cost-share equilibrium to commodity spaces without a linear structure

So, to summarize, in the previous three sections we have studiedthree concepts of equilibrium which only differ from each other inwhat the consumers take as given: the linear prices (Lindahl), the pro-portions (Kaneko) and the cost-share system (Mas-Colell-Silvestre)

1.7 A CRITICISM OF THE NOTIONS OF EQUILIBRIUM WITHAN' AUCTIONEER

In the previous three sections we have examined a family of concepts

of equilibrium for the allocation of public goods where an auctioneerannounces the (possibly non-linear) prices If the consumers take these

as given and maximize their utility over their budget sets, it will lead

Initial endowments

Figure 1.7

to a unanimous individually rational, and efficient decision on the level

of public goods that the society must be provided with Consequently,these concepts of equilibrium have a certain normative appeal, whichincreases due to the fact that we can study economies with non-convexproduction sets, even at the expense of complicating the price function.There still remains, however, one dark point: these models have anexcessively simplified vision of how markets function Firstly, it isassumed that there is an auctioneer who announces the price schedulefor each consumer Secondly, the consumers take this schedule as given.The question is, therefore, how the auctioneer discovers all the character-istics of the consumers and of the firm that are necessary in order tocalculate the equilibrium prices The simplest answer is that the auctioneermay ask the agents what their preferences are and act in response tothis The problem is that the agents are rational and will foresee thatthe information they offer will be used to determine the final alloca-tion Consequently, they will distort it This is the so-called 'free rider'problem which was noted by Samuelson (1954) in relation to Lindahl'sequilibrium: 'each person will be interested in giving false signals.'This intuition can be formalized by using Kolm's triangle In Figure 1.7

Initial endowments

Figure 1.8

an economy with a unique Lindahl equilibrium (L) corresponding to thetrue preferences (u 1, u 2)is illustrated However, if the first agent is able toconvince the auctioneer that his real preferences are u:,the Lindahl equi-librium corresponding to (u~, u 2) (denoted byL') is such that consumer1

obtains an allocation that she prefers toLin accordance with her true erences Therefore, this consumer has no incentive to tell the truth It iseasy to see that this conclusion is general: similar examples can be car-ried out using ratio or cost-share equilibrium as the starting point This type

pref-of reasoning leads many economists to feel extremely pessimistic ing the possibility of finding efficient allocations of a decentralized form

regard-in public good economies

However, as Hurwicz noticed, there is nothing in the previous examplethat depends specifically on the economy producing public goods: anidentical conclusion would be reached in an exchange economy wherethere are only private goods Thus, Figure 1.8 illustrates Edgeworth'sbox and the Walrasian equilibrium(W) corresponding to the true pref-erences(u 1, u 2) However, if consumer 1 announces preferences u~, theWalrasian equilibrium corresponding to the preferences (u~, "2) is W' and consumer 1 is better off in W' than in W in accordance with histrue preferences Identical results would be obtained if the auctioneeradjusted prices by, say, ataronnement procedure Therefore, both casesseem to suggest that there is a conflict between the decentralization ofinformation and the efficiency of the results The following chapterswill be dedicated to a deeper study of this topic froma more formaland exhaustive point of view As we will see, the pessimism that exists

regarding the efficient allocation of public goods when the planner doesnot have perfect knowledge is not completely justified.

1.8 EXERCISES

1.1 Show by means of an example that in economies with public goods theweak and strong definitions of a Pareto efficient allocation do not coincide,even if the consumers' preferences are monotone and continuous (see Tian

1.2 Construct an example where the price equals to marginal cost rule

im-plies productive inefficiency (see Beato and Mas-Colell, Journal of Economic

econ-omies with public goods?

1.3 Study the Pareto' efficient allocations in KoIm's triangle if the ences are homothetic

prefer-1.4 Study the Pareto efficient allocations in KoIrn's triangle and Edgeworth'sbox, when the preferences are quasi-linear (that is, that the utility function islinear in the private good)

1.5 If the preferences are quasi-linear, prove that if an allocation maximizes

a weighted sum of utilities over the set of feasible allocations and gives allthe consumers strictly positive quantities of the private good, then this alloca-tion must maximize the sum of utilities over the set of feasible allocations(see Moulin(1988), pp 170-1)

1.6 Prove that the two definitions of Pareto efficient allocations given on

p.2 and coincide

1.7 Define Lindahl's equilibrium and the Pareto efficient allocations in a modelwhere the public goods can be inputs (see Milleron (1972), p 427).1.8 Give examples where:

(a) Lindahl's equilibrium is not in the core

(b) The core does not exist

1.9 Prove by means of an example that the core of an economy with publicgoods does not necessarily tend towards the competitive equilibrium whenthe number of agents tends to infinity (see Milleron (1972), pp 460-3).1.10 Prove Proposition 2

1.11 Assume that each coalition has access to a different production set fine Lindahl's equilibrium and the core Prove that the first is contained in thesecond.

De-1.12 Within the framework of the previous question, demonstrate that, given

a price vector, a production plan maximizes the total profits if and only if itmaximizes the profits of each firm Use this to prove that in economies withseveral firms, Lindahl's equilibrium is Pareto efficient

1.13 Assume that the consumers have a relation of preferences not arily transitive or complete Define Lindahl's equilibrium and the core, andprove that the first is contained in the second

necess-1.14 A group of n farmers possess identical utility functions V i = Y - ri

where Xi = work done by i, and Y = the quantity of the public good (abridge, etc.) which is produced by means of constant returns to scale.(a) Calculate the value of y in the Nash equilibrium where the strategies

are the quantities (This is called a subscription equilibrium.)

(b) Calculate those Xi and ythat maximize a utilitarian welfare function.(c) Comment and explain the differences between (a) and (b) especially forlarge n.

(d) Calculate Lindahl's equilibrium

(e) What would occur in cases (a), (b) and (c) if the utility functions were

U 1 =Y / nfJ - x~, a >O? InterpretQ.

1.15 Let an economy be formed by n identical agents with utility functions

u i = ay" + Xi' 0< a < 1, in which the public good is produced under stant returns to scale

con-(a) Calculate the subscription equilibrium

(b) Calculate Lindahl's equilibrium

1.16 Interpret Cournot's equilibrium as a subscription equilibrium Prove thatthis equilibrium yields, in general inefficient allocations If linear subsidiesare introduced, prove that it is generally possible to obtain efficiency Give anexample (with just one firm) in which this is not always possible (see Guesnerieand Laffont, Journal of Economic Theory, 1978, pp. 443-8)

1.17 Give an example in which ratio equilibrium does not exist if the costfunction is not additi vely separable

1.18 Prove Proposition 6

1,19 Assume that there exists a public good and a private good and that theutility function of the consumer is quasi-linear Using the differential calcu-lus, find a sufficient condition for the final part of Assumption 3 to hold Willthis condition be fulfilled if u =yfJ. + Xi and c = y~,o, ~ >O?

1.20 Assume that there is a private good and a public good and that theutility function of the consumer is Cobb-Douglas If the cost function is C =

yP~ > 0, find a condition with which the final part of assumption 3 is filled Calculate the ratio equilibrium if all agents are identical

ful-1.21 Prove that a Generalized Lindahl equilibrium is locally Pareto efficient(see Vega-Redondo, 1987, op cit.).

1.22 Prove that an allocation is a valuation equilibrium with zero profits, ifand only if it belongs to the core (see Mas-Colell (1980), op. cit.

pp 628.; 31) Show by means of an example that when there are more thanone private good, the core may be strictly larger than the set of cost-shareequilibria (see Diamantaras and Gilles, 'The Pure Theory of Public Goods:Efficiency, Decentralization and the Core', Working Paper E94-01, Depart-ment of Economics, Virginia Polytechnic Institute and State University,Blacksburg, Virginia)

1.23 Study the optimality of the system in which the quantity of the public good

is decided by a majority referendum (see Moulin (1988), op. cit., pp 263-4).

1.24 Give an example in which the ratio equilibrium does not exist (seeMoulin (1988), op cit., p 192).

1.25 Prove that the introduction of certain types of taxes, whose rates can becalculated without the necessity of knowing the preferences, causes the sub-scription equilibrium to be Pareto efficient (see Boadway, Pestieau and Wildasin

(1989), Public Finance, no 1 pp 1-7).

1.26 Argue that the consideration of externalities introduces non-convexities

in production (see Starret, Journal of Economic Theory, pp 180-99, 1972

and Boyd and Conley, 'A Note on Fundamental Nonconvexity and Local ParetoSatiation', Working Paper, University of Rochester)

1.27 Show by means of an example that the Pareto efficient set in publicgood economies may be neither closed nor connected even if preferences arestrictly monotone and continuous and there are constant returns to scale (seeDiamantaras and Wilkie, 'On the Set of Pareto Efficient Allocations in Econ-omies with Public Goods', Economic Theory (forthcoming).

1.28 Using the Kolm triangle show that when considering how the Lindahlequilibrium changes in response to variations in the initial endowments thefollowing possibilities may arise:

(a) An agent may lose when her initial endowment increases

(b) An agent may lose when some other agent's initial endowment increases.(c) An agent may gain by transferring part of her initial endowment toanother agent and the recipient may lose

(d) An agent may lose when aggregate endowment increases

Study the same questions but by using a general equilibrium approach

which generalizes the notion of a Lindahl equilibrium (see Thomson, 'Monotonic Allocation Mechanisms in Economies with Public Goods',

op cit.).

1.9 REFERENCES

Good introductions to the topics dealt with in this chapter are: J.J Laffont,

(1988) Fundamentals of Public Economic, Introduction and Chapter 2 (MIT

Press) and W Thomson, 'Lecture on Public Goods', mimeo, University de Rochester, Sections 1-6

The articles in which the topics dealt with here were developed for the first time in a modern way are: P.A Samuelson (1954), 'The Pure Theory of Pub- lic Expenditure',Review of Economics and Statistics, 36,pp 387-9; andD.K.Foley (1970), 'Lindahl's Solution and the Core of an Economy with Public Goods', Econometrica, 38, no 1 pp. 66-72

The classical surveys on public goods and Lindahl's equilibrium are: J.C.Milleron (1972), 'Theory of Value with Public Goods: A Survey Article',Journal

of Economic Theory, 15, pp 419-77; and D.J Roberts (1974), 'The Lindahl

Solution for Economies with Public Goods', Journal of Public Economics, 3,

pp 23-42.

Both papers present proofs for the existence of Lindahl's equilibrium where

it is required that all consumers hold strictly positively endowments of public goods The existence of Lindahl's equilibrium without this assumption has been established by C Herrero and A Villar (1991), 'Vector Mappings with Diagonal Images', Mathematical Social Sciences, 122, pp 57-67.

The generalization of the Lindahl-Bowen-Samuelson condition to allow boundary allocations is analyzed in D.E Campbell and M Truchon (1988), 'Boundary Optima and the Theory of Public Goods Supply', Journal of Pub- lic Economics, 35, pp 241-9; and J.P Conley and D Diamantaras, 'General-

ized Samuelson Conditions and Welfare Theorems for Nonsrnooth Economies', Working Paper, University of Illinois.

The second welfare theorem in economies with public goods is proved in M.A Khan and R Vohra (1987), 'An Extension of the Second Welfare The- orem to Economies with Nonconvexities and Public Goods', Quarterly Jour- nal of Economics, pp 223-41.

The non-equivalence between the core and Lindahl's equilibrium was first demonstrated by T Muench (1972), 'The Core and the Lindahl Equilibrium

of an Economy with Public Goods',Journal of Economic Theory, 4, pp 241-55.

Sufficient conditions for the convergence of the core to the Lindahl tion in economies with pure public goods are studied in J.P Conley (1994), 'Convergence Theorems on the Core of a Public Goods Economy: Sufficient Conditions',Journal of Economic Theory, 62, no 1, pp 161-85; and M Wooders

alloca-(1991), 'On Large Games and Competitive Markets 1 Theo.ry 2 tions' University of Bonn, 303, DPB-195-6.

Applica- The ratio equilibria were first proposed in M Kaneko (1977), 'T Equilibrium and a Voting Game in a Public Good Economy' Journal omic Theory, vol. 16,pp. 123-36.

The assumptions on the technology used above have been genera

D Diamantaras and S Wilkie (1994), 'A Generalization of Kanekc Equilibrium for Economies with Private and Public Goods',Journal, omic Theory, vol 62 no. 2, pp. 499-512

Whereas the cost-share equilibria were proposed in A Mas-Cole Silvestre (1989), 'Cost-Share Equilibria: A Lindahlian Approach', Jo Economic Theory, vol. 47 no. 2, pp. 239-56

The relationship between the core and cost-share equilibria (unde: sumption of non-increasing returns to scale) is studied in S Webei Wiesmeth (1991), 'The Equivalence of Core and Cost-Share in an I with a Public Good', Journal of Economic Theory, vol. 54, pp. 190-Other authors have proposed different solution concepts.Alistwhi

no means exhaustive includes: A Mas-Colell (1980), 'Efficiency anctralization in the Pure Theory of Public Goods', Quarterly Journal j

omics, vol XCIV, no 4, pp. 625-641: H Moulin(1992), 'All Sorry to I

AGeneral Principle for the Provision of Nonrival Goods' ,Scandinavian

of Economics, vol. 94 no. 1pp. 37-51; F.Vega-Redondo (1987), 'Eand Non-Linear Pricing in Non-Convex Environments with Externa Generalizationofthe Lindahl Equilibrium Concept',Journal of Economic

Thomas More) In general, the advocates of such societies were notexcessively worried about the human behavior being selfish and maximiz-ing (in tune with Hobbes's sentence 'man is a wolf for man') andconsequently they did not deal with the problem of the agents havingincentives to follow the rules of such societies The adjective 'utopic'therefore came to have a pejorative connotation, indicating that suchforms of social organization were destined to fail, at least as long asmen behaved in accordance with the assumptions of the 'homoeconomicus'

In this chapter we will approach the problem of resource allocationfrom a general viewpoint in such a way that no extra agent is needed

in the resource allocation process and the selfish interests of the peopleare reflected as restrictions in the possible design of alternative societ-ies Furthermore, we will be much more careful when specifying theinformation that the agents need and how this will be transmitted inthe economy To this end, we will first develop a specific example.Subsequently we will describe the problem of resource allocation in ageneral way by introducing some concepts of equilibrium taken fromgame theory and we will define the general problem of implementa-tion We will see how the theory of implementation tries to seek the

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