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Economic Equilibrium: Optimality and PriceDecentralization C. D. ALIPRANTIS∗Department of Economics, Purdue University, West Lafayette, IN 47907–1310,USA.
E-mail : [email protected] B. CORNETCERMSEM, Maison des Sciences Economiques, Universit´ e Paris I, 106–112 Boulevard de l'Hopital, 75645 Paris Cedex 13, France.
E-mail : [email protected] R. TOURKY†Department of Economics, University of Melbourne, Melbourne, VIC 3010,Australia.
E-mail : [email protected] (Received 10 August 2001; accepted 15 December 2001) Abstract. Mathematical economics has a long history and covers many interdis-ciplinary areas between mathematics and economics. At its center lies the theoryof market equilibrium. The purpose of this expository article is to introduce math-ematicians to price decentralization in general equilibrium theory. In particular, itconcentrates on the role of positivity in the theory of convex economic analysis andthe role of normal cones in the theory of non-convex economies.
Keywords: equilibrium, Pareto optimum, supporting price, properness, marginalcost pricing, vector lattice, ordered vector space, Riesz–Kantorovich formula, normalcone, separation theorem AMS Classification: 91, 46, 47 1. A Historical Survey General equilibrium theory describes the equilibrium and disequilib-rium arising from the interaction of all economic agents in all markets.
The basic abstractions in the model of general competitive equilibriumare the notions of commodities and prices. Commodities define theuniverse of discourse within which the constraints, motivations, andchoices of consumers and producers are set. Consumers and produc-ers act independently and respond to prices. At equilibrium, a linearprice system summarizes the information concerning relative scarcities ∗ Research supported by NSF grants EIA–0075506 and SES–0128039.
† Research funded by the Australian Research Council Grant A00103450.
2002 Kluwer Academic Publishers. Printed in the Netherlands.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.1 C. D. ALIPRANTIS B. CORNET AND R. TOURKY and locally approximates the possibly non-linear primitive data of theeconomy.
Advances in the theory of general equilibrium have gone hand-in- hand with the study of the existence of at least one equilibrium pricesystem. This is not surprising since the existence problem was far moreinvolved than what many economists had anticipated in the past. Withcomplexity came the need for rigor and rigor lead to a better under-standing of not only the existence problem but also the model as awhole.
The purpose of this section is to informally and summarily trace the evolution of the general equilibrium model from L´ eon Walras' system of production and exchange equations to the ‘state of the art' modelwith infinitely many commodities and a finite number of consumersand producers.
1.1. Finite Number of Commodities eon Walras: The classical reference to the theory of general equilib- eon Walras' Elements of Pure Economics, which was published in four successive editions between 1874 and 1900 and in a further‘definitive' edition, published sixteen years after Walras' death.1 Walrasin Elements (§162–163) concedes to Gossen the priority of discoveringthe principle of ‘maximization of utility' and to Jevons the priorityof discovering the ‘equations of exchange.' Walras, however, is the onlyone out of the three to deal with the case of more than two commoditiesand two consumers.2 His own contribution, he remarks, is "the generalcase in which any number of individuals enter into mutual exchangerelations. . on the supposition that any number of commodities arebeing exchanged for one another." [114, §163, p. 206].
The first problem that Walras considers by means of "marginal utility" (first order derivatives), or to use Walras' term raret´ problem of barter. Walras derives in Lessons 11–15 of Elements a systemof equations, which he calls the ‘equations of exchange.' A solution tothis system of equations is an equilibrium for an exchange economy,even in our modern understanding of the term. A solution to Walras'‘production equations,' in Lessons 17–22, on the other hand, is very dif-ferent from the modern notion of production equilibrium. Walras doesnot include a fixed number of profit maximizing producers. Consumerstrade production inputs and transform them into consumable goods,which they either trade or consume themselves. In this way productiondecisions are primarily motivated by utility maximization. One can 1 We cite William Jaff´e's English translation of the ‘Definitive Edition' [114].
2 See "Walras on Gossen" in [113] and Schumpeter [103, pp. 103–126].
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.2 ECONOMIC EQUILIBRIUM think of Walras' production model as one in which each consumer holdsa one hundred percent share in every available technology.
Walras goes through a great deal of effort to show that each system of equations reduces to a system with an equal number of variables asequations. This appears to have been the usual practice in the physicalsciences of the late nineteenth century. The following is a quote froma review of Pareto's Manuel d' ´ Economie Politique from the June 1912 issue of the Bulletin of the American Mathematical Society: ". . it should be remembered that not so very long ago the method ofcounting constants was widely used in pure mathematics. . Moreover,in a physical science the question of rigor is very different from that inmathematics; to be ultra rigorous mathematically may be infra-rigorousphysically. To throw out Gibb's phase because its proof, being essentially acount of constants, is no proof at all, would be equally good mathematicsand equally bad physics." [114, p. 511].
Walras however fully recognizes that even his two commodity exchangemodel may have no equilibrium (§64) or multiple equilibria (§65). Wal-ras also recognizes that there can be a stable equilibrium (§66), anunstable equilibrium (§67), and multiple equilibria some of which arestable and some that are not (§68). However, in §156 he makes theinteresting (but probably wrong) assertion that when "the numberof commodities is very large" multiple equilibria "are, in general, notpossible." Abraham Wald: The first rigorous results on the existence of a so-lution to Walras' ‘production equations' and ‘exchange equations' isdue to a series of papers, appearing between 1933 and 1936, by Abra-ham Wald.3 Wald recognizes that a "theorem on the solubility of theequations under consideration can only be proven to follow from theassumptions by means of difficult mathematical analysis." [112, p. 403].
Most historical accounts concentrate on the unsatisfactory aspects of Wald's production theorem, while ignoring the tremendous break-through on the exchange front. Wald's assumptions for the consumptioncase are remarkably weak. He assumes the following: 1. Each consumer holds a non-negative amount of each commodity.
2. Each consumer holds a strictly positive amount of some commodity.
3. Each commodity is held in a strictly positive amount by some 3 See Wald [112] for a report on his work.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.3 C. D. ALIPRANTIS B. CORNET AND R. TOURKY 4. For each consumer the marginal utility of a good is independent of the amount held of other goods, and is strictly monotonicallydecreasing with the amount held of the good.
Condition (4) of Wald means that the utility functions are separableand is identical to Walras' assumptions on the raret´ e. Thus, in so far as the Walrasian ‘exchange equations' are concerned, Wald solved theexistence problem.
Concerning Wald's theorem for the ‘production equations' it suffices to say that he describes the consumption sector by inverted demandfunctions, where prices are functions of quantity. Wald's system ofequations is in fact the Cassel system of equations. Though Walrasexpressed demand as a function of prices, Cassel [41] uses the inversesof the demand functions and assumes that these inverses are themselvesfunctions. He assumes, among other things, that these inverted demandfunctions are continuous and satisfy a condition almost identical to thewell known Samuelson's weak axiom of revealed preferences; see [83].
The Arrow–Debreu–McKenzie Model: The problem of existenceof equilibrium stayed uninvestigated for about two decades after thepublication of Wald's work. In the early fifties, stimulated by the ad-vances in linear programming, activity analysis, and game theory, "itwas perceived independently by a number of scholars that existencetheorems of greater simplicity and generality than Wald's were nowpossible." [18, p.11]. By 1956 numerous equilibrium existence resultswere independently obtained. Some of these are in McKenzie [84], Ar-row and Debreu [17], Gale [62], and Nikaido [93]. One of the mostgeneral finite dimensional existence result is that of Debreu [54].
Two characteristics of these results are worth mentioning. First, the problem of existence was no longer perceived as that of solving asystem of equations but was reformulated as a problem of showing thatthe simultaneous maximization of individual goals under independentconstraints can be carried out. These saw a departure from the calculusbased marginal utilitarian analysis to functional analytical techniques;and according to Debreu [53, p. x] new influences "freed mathematicaleconomics from its tradition of differential calculus and compromiseswith logic." Second, all these results were obtained by applying a fixedpoint argument; some variant of Brouwer's fixed point theorem. It isoften stated that L´ eon Walras could not have proven the existence of equilibrium since he did not have available Brouwer's fixed pointtheorem—proven in 1912. We note, however, that L´ eon Walras was in correspondence with the famous mathematician Henri Poincar´ 1883–1884 announced the following result (see [36, p.51]): alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.4 ECONOMIC EQUILIBRIUM e) Let ξ1, ξ2, . . , ξn be n continuous functions of n vari- ables x1, x2, . . , xn. The variable xi is subject to vary betweenlimits +ai and −ai. Let us suppose that for xi = +ai, ξi is con-stantly positive, and for xi = −ai, ξi is constantly negative,; I saythat there exists a system of values of x for which all ξ's vanish.
e's theorem is now known as Miranda's fixed point theorem and is equivalent to Brouwer's fixed point theorem (see [88]) and canbe used to prove the existence of equilibrium.
Apart from this technical revolution a new notion of commodi- ties had emerged. Commodities were now contracts promising the de-livery of a good or service g, in a specific location l, on a specificdate t, and contingent on a series of uncertain events occurring up todate t (see [53, Chapter 7]). Endowed with this notion of commodi-ties, the Walrasian general equilibrium model became known as theArrow–Debreu–McKenzie general equilibrium model.
The archetype general equilibrium model with a finite number of commodities is that of Arrow and Debreu [17]. The standard refer-ence to this model is Debreu's classic monograph Theory of Value,an Axiomatic Study of Economic Equilibrium, which was published in1959. A central theme of Debreu is the axiomatic method in whichthe mathematical theory "is logically entirely disconnected from itsinterpretation." [53, p. viii].
The contribution that such a dichotomy made to economics can- not be overestimated. The axiomatic method provided for the rapidadvancement of economics: Problems that were of unimaginable com-plexity could now be clearly formulated and solved. With the axiomaticmethod, economists could now focus on the most minute features ofa suitably formulated theory. The Arrow–Debreu–McKenzie ‘privateownership economy' is formulated as follows: Commodities and prices. There is a finite number of commodi- ties. Both commodity bundles and price systems are points in The value of a commodity bundle x ∈ R relative to a price system R is p · x, the scalar product in R .
Producers. There is a nonempty finite set J of producers. Each pro- ducer j ∈ J is described by a production set Yj, a subset of thecommodity space which describes the technological possibil- ities of the producer. Points in Yj are called production plans.
The positive coordinates of a production plan are the producer'soutputs while the negative coordinates are the producer's inputs. Aproducer's profit relative to a production plan and a price system is alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.5 C. D. ALIPRANTIS B. CORNET AND R. TOURKY the value of the production plan relative to the price system. Givena price system producers choose a production plan that maximizesprofit over their production sets.
There is a nonempty finite set I of consumers. Each consumer i ∈ I is described by a consumption set Xi (a subset ofthe commodity space R ), a preference relation on Xi, a commod- ity bundle ωi ∈ Xi (called the initial endowment), and a shareθij ∈ [0, 1] of the jth producer's profits. Points in Xi are calledconsumption bundles. The positive coordinates of a consumptionbundle are the consumer's inputs while the negative coordinatesare the consumer's outputs. Given a price system and a produc-tion plan for each producer, a consumer's budget set is the set ofall consumption bundles whose values are not greater than thevalue of the initial endowment plus his share of total profits.
Given a price system and a production plan for each producer,a consumer chooses a consumption bundle that maximizes herpreference relation relative to her budget set.
Equilibrium. An attainable (or a feasible) allocation is a set consisting of a production plan for each producer and a consumption bundlefor each consumer such that the sum of all consumption bundlesis equal to the sum of all production plans plus the sum of theinitial endowments. An equilibrium price system is a price systemp such that consumer and producer choices relative to p is a feasibleallocation.
Extensions and generalizations: The literature on equilibrium witha finite number of commodities is enormous. The assumptions usedhave been considerably weakened. For instance, McKenzie [86] showedthat the assumption of irreversibility of total production is superfluous;McKenzie [85] and Shafer and Sonnenschein [106] allowed for interde-pendent preferences; McKenzie [85], Bergstrom [23] and many othersweakened the assumption of free disposability; McKenzie [87] adaptedthe excess demand approach to the case of intransitive and incompletepreferences.
In their remarkable papers Debreu and Scarf [56] and Aumann [19] characterize market equilibrium with many traders in terms of a coop-erative notion of equilibrium. Aumann considers the case of infinitelymany consumers and Debreu and Scarf consider infinite replications offinite economies.
Another approach for proving the existence of equilibrium with a finite number of commodities is that of Negishi [92], which was ex-tended by Takayama and El-Hodiri [108] and Arrow and Hahn [18].
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.6 ECONOMIC EQUILIBRIUM This approach involves finding efficient feasible allocations and pricesystems that support them and then applying a fixed point argumenton the space of utilities.
One of the most important generalizations of Debreu's assumptions is the work on intransitive and incomplete preferences. Empirical evi-dence suggests that consumers sometimes display cyclical preferences(e.g., Sonnenschein [107]). Also, a bounded rationality approach toconsumer behavior suggests that preferences may be incomplete; theremay exist two commodity bundles that cannot be compared in terms ofpreferences. The first satisfactory solution for the existence of equilib-rium with intransitive preferences was provided by Mas-Colell in [80].
Mas-Colell showed that both the assumption of transitivity and theassumption of completeness are superfluous. Mas-Colell's proof howeverwas long and very difficult. A shorter more appealing proof was laterprovided in Gale and Mas-Colell [63], which consists of three steps.
First, it defines an abstract game using the data of the economy. Second,it shows that this abstract game has an equilibrium. Third, it provesthat an equilibrium for the abstract game is also an equilibrium for theeconomy. Of course, this method of proof was not new. For example,Arrow and Debreu [17] use a similar approach in their proof of theexistence of equilibrium. However, what was new was the specific formof the abstract game. Further generalizations of the Gale–Mas-Colellresult and new proofs were provided in many papers (e.g., Shafer andSonnenschein [104, 105, 106]).
The problem of uniqueness of equilibrium has been studied using techniques from global analysis in a list of papers following Debreu'sseminal contribution [55]. The standard reference on the approxima-tion of equilibria is Scarf [102]. Recently, there have been importantadvances in the theory of general equilibrium with incomplete markets;see for instance Geanakoplos [64]. Two surprising things have emergedfrom this research. First, that Brouwer's fixed point theorem is notpowerful enough to prove the existence of equilibrium with incompleteassets markets. Second, that an equilibrium for such a market need notbe constraint efficient.
1.2. Infinite number of commodities The definition of a commodity in the Arrow–Debreu–McKenzie modellead inevitably to the need for considering a model with infinitely manycommodities. Such a situation arises if one wants to consider economiesextending over an infinite horizon, or where time or location are takenas continuous variables. It is also needed in the case of uncertainty withinfinitely many states of nature.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.7 C. D. ALIPRANTIS B. CORNET AND R. TOURKY However, the need for considering infinitely many commodities was perceived even before the Arrow–Debreu–McKenzie model was devel-oped. For example, Schumpeter [103, p. 900] in his discussion of FrankKnight's criticism of the theory of factors of production remarks that"it would hardly be easy to eliminate entirely the idea of factors. For. .
Professor Knight. . admits an indefinite variety of factors within whichthere is no economically significant difference," and that "[in] strictlogic, the number of these factors would be infinite, for conceptuallythey form a continuum." Among the first articles to consider the case of infinitely many com- modities within a formal general equilibrium framework was Debreu's1954 paper [51] Valuation Equilibrium and Pareto Optimum. Debreu'smodel is a strict mathematical generalization of the model with a finitenumber of commodities. He substitutes for the commodity space an infinite dimensional topological vector space, while generalizing theconcept of a price system to that of a continuous linear functional onthe commodity space. The same approach to the theory of value withinfinitely many commodities was taken by Hurwicz [68], who studied,among other things, linear programming in models with infinitely manycommodities.
Debreu shows that a Pareto optimal allocation for a production economy can be supported by a price system. He makes, however, astrong assumption on the production set: he assumes that the produc-tion set has a non-empty interior. He then notes that this non-interiorityassumption is satisfied if the commodity space is properly chosen and iffree disposal of commodities is assumed. By ‘properly chosen' Debreumeans the infinite dimensional topological vector space L∞, which isthe space of essentially bounded measurable real valued functions on ameasure space. The importance of L∞ lies in the fact that its canonicalpositive cone has a non-empty interior.
In a paper published in 1967, Radner [97] pointed out that Debreu's notion of a price system is too general from an interpretive point ofview. This is because some continuous linear functionals (the singularones) do not maintain the traditional interpretation of a price systemas a list of prices. He therefore proposes that for the commodity spaceL∞ an appropriate price system would be an integrable real valuedfunction, i.e., a function in L1. Such functions, he argued, both retainthe desired economic interpretation as well as being continuous linearfunctionals.
Vector lattice commodity spaces: The first result on the existenceof equilibrium in economies with commodity space L∞ and the pricespace is L1 was presented by Bewley [24]. However, it became quickly alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.8 ECONOMIC EQUILIBRIUM apparent that Bewley's result is not easily extendable to the case ofordered vector paces without order unit; or even to an ordered Banachspace whose positive cone contains no interior points.
In 1983 Aliprantis and Brown [2] proposed that the appropriate setting for infinite dimensional equilibrium theory is that of a vectorlattice dual system. Subsequently, a general existence of equilibriumresult was obtained by Mas-Colell [81] in 1986. His solution replacesthe requirement of the existence of an order unit with two assumptions.
First, that the commodity space be a locally solid vector lattice andthat the price space be its topological dual. Second, that preferencessatisfy a cone condition that he termed uniform properness; this con-dition is closely related to the condition of Klee [74] for the existenceof supporting hyperplanes. For more on such cone conditions see [11].
During the last two decades order theoretic properties of vector spaces have become central to equilibrium analysis in economics, seefor example [3, 4, 5, 8, 9, 13, 15, 20, 57, 76, 82, 95, 98, 99, 109, 110, 116].
It has become apparent that the existence of equilibrium prices in general order vector spaces can be obtained by using the famous Riesz–Kantorovich formula, where for each pair p and q of linear functionalsand each x ≥ 0 their Riesz–Kantorovich formula is defined by Rp,q(x) = sup{p(y) + q(z): y, z ≥ 0 and y + z = x} .
The above formula defines automatically a linear price if the underlyingordered vector space is a vector lattice. However, in the general case ofordered vector spaces the formula still defines non-linear functions thatcan be used to develop a new theory of economic equilibrium, see [10,13]. Remarkably, if one can find two positive linear functionals on anordered topological vector space for which the supremum exists butdoes not satisfy the Riesz–Kantorovich formula, then one can constructan economy that satisfies all the standard assumptions but fails tohave an equilibrium supported by linear prices [89]. These economicresults are closely related to the following open problem regarding linearoperators between vector lattices: If L and M are vector lattices with M not Dedekind completeand the supremum (least upper bound) S ∨ T of two operatorsS, T ∈ L∼(L, M ) exists in L∼(L, M ), does it then satisfy theRiesz–Kantorovich formula? Economic insights have provided partial solutions to this problem in the case of M = R; for details see [12, 10].
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.9 C. D. ALIPRANTIS B. CORNET AND R. TOURKY 1.3. Non-Convex Economies All the literature that we have talked about so far deals with convexeconomies, i.e., convex production sets, and both convex consumptionsets and convex preferences. We shall focus hereafter to the study of"non-convexities" in the production side. In the consumption side "non-convexities" can be treated by allowing the set of consumers to beinfinite (say [0, 1] as in Aumann [19], or more generally, a positivemeasure space, as in Hildenbrand [66]), so that every agent is "negli-gible," hence giving an idealistic formalization of perfect competition.
This is typically what cannot be assumed on the production side, since"non-convexities" are generally present in "big" firms.
The presence of "non-convexities" in the production sector is widely recognized in the economic literature and the failure of the competitivemechanism in such an environment has been known since Marshall [79].
Indivisibilities, increasing returns and fixed costs are the most com-mon forms of non-convexities in production. The study of indivisibili-ties needs different types of mathematical techniques (mainly discretemathematics). The two other forms of non-convexities arise in the caseof public monopolies, such as railways or electricity production, forwhich the state intervention is often suggested. In these cases profitmaximization is typically replaced by marginal cost pricing followingthe works of Allais [14], Hotelling [?, 67], Lange [75], Lerner [77],Pigou [94], for example. Non-convexities are part of a long traditionwhich dates back to Dupuit [61] and his work in 1844 on public utilitypricing.
The treatment of these different questions has benefited from the de- velopment over the past twenty years of new mathematical techniques,known as "Non-smooth Analysis," which are discussed in the books byClarke [43] and Rockafellar [?].
2. The Economic Model 2.1. The Commodity-Price Duality One of the basic economic characteristics associated with any economicmodel is the commodity-price duality. G. Debreu [51] expressed this interms of a dual pair hL, L0i.4 The vector space L is the commodityspace and its vectors are called commodity bundles and L0 is the pricespace and its vectors are called prices. The real number hx, x0i is the 4 That is, L and L0 are vector spaces that are related via a non-trivial bilinear function h·, ·i: L × L0 → R called the valuation of the dual pair.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.10 ECONOMIC EQUILIBRIUM value of the bundle x at price x0. It is customary in economic theory todesignate the price vectors by p, q, etc., instead of using primes. Also,the valuation hx, pi is denoted p · x, i.e., p · x = hx, pi.
We shall now consider on the commodity space L a partially (or- dered) structures and we need first to recall some definitions and re-sults.
An ordered vector space L is a real vector space L equipped with an order relation ≥ (i.e., with a transitive, reflexive, and antisymmetricrelation ≥) compatible with the algebraic structure of E in the sensethat it satisfies the following two properties: (i) if x, y ∈ L satisfy x ≥ y, then x + z ≥ y + z holds for all z ∈ L, (ii) if x, y ∈ L satisfy x ≥ y, then λx ≥ λy for each λ ∈ R with λ ≥ 0.
The notation y ≤ x will be used interchangeably with x ≥ y. The relation x > y (resp. y < x) will mean x ≥ y (resp. y ≤ x) and x 6= y.
The elements x of E with x ≥ 0 are called positive elements or positivevectors. The set L+ = {x ∈ L: x ≥ 0} is called the positive cone (orsimply the cone) of E and it satisfies the following three properties: (a) L+ + L+ ⊆ L+, where L+ + L+ = {x + y: x, y ∈ L+}; (b) λL+ ⊆ L+ for each 0 ≤ λ ∈ R, where λL+ = {λx: x ∈ L+}; (c) L+ ∩ (−L+) = {0}, where −L+ = {−x: x ∈ L+}.
Any subset C of a real vector space L satisfying properties (a), (b), and (c) is called a pointed convex cone of L, or simply a cone hereafter.
If C is a cone of L, note that the relation x ≥ y whenever x − y ∈ Cmakes L an ordered vector space whose positive cone is precisely C.
We shall later introduce an order (lattice) structure on the com- modity and price spaces. Throughout this paper we assume that L is aHausdorff locally convex space and L0 is its topological dual. Thus bothL and L0 are equipped with both a linear structure and a topologicalstructure.
There is a finite set I of consumers indexed by i. Each consumer i ∈ Ican consume vectors in the agent's consumption set Xi ⊆ L. Unlessotherwise stated, we assume that Xi = L+ for each consumer.
Each consumer i has a preference relation i on Xi. Preferences and utility functions are among the fundamental concepts of microeconomictheory. They provide the mathematical framework for modeling the alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.11 C. D. ALIPRANTIS B. CORNET AND R. TOURKY "tastes" of the consumers. We present their mathematical backgroundin this section. Hereafter we shall omit the index i of the consumer.
A preference  on a set X is a binary relation on X that is: 1. reflexive: x  x for all x ∈ X, 2. transitive: x1  x2 and x2  x3 imply x1  x3, and 3. complete: for each x1, x2 ∈ X either x1  x2 or x2  x1.
When x1  x2 holds, we say that x1 is preferred or is indifferent to x2. When both x1  x2 and x2  x1 hold, then x1 is indifferent to x2,written x1 ∼ x2. We also associate to  the strict preference relation which is defined by x1  x2 if both x1  x2 and x2 6 x1.
A preference relation  on a topological space X is: upper semicontinuous, if for each x ∈ X the strictly worse-than-x set {x0 ∈ X : x  x0} is open in X (for the relative topology), lower semicontinuous, if for each x ∈ X the strictly better-than-x set {x0 ∈ X : x0  x} is open in X, continuous, if  is both upper and lower semicontinuous, locally non-satiated at x, if x belongs to the closure of thestrictly better-than-x set {x0 ∈ X: x0  x}.
When X is a convex subset of a vector space, we shall say that: a preference relation  on X is convex, if for each x ∈ X the set{x0 ∈ X: x0  x} is convex, a strict preference relation  on X is convex, if for each x ∈ Xthe set {x0 ∈ X: x0  x} is convex.
In case the set X is partially ordered by ≥,5 then we say that a preference relation  on X is: monotone, if x1 ≥ x2 implies x1  x2, strictly monotone, if x1 > x2 implies x1  x2.
5 To any partial order ≥ we define (as usual) the relation > by letting x2 > x1 if 2 ≥ x1 and x1 6= x2. When X = R , we shall consider the partial order defined by the positive orthant R+ = {x = (xh) ∈ R : xh ≥ 0 for every h}.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.12 ECONOMIC EQUILIBRIUM A utility function for a preference relation  on X is a function u: X → R such that for every x1 and x2 we have u(x1) ≥ u(x2) ifand only if x1  x2. In this case we say u represents  on X. Everyreal-valued function u: X → R gives rise automatically to a preferencerelation  defined by x1  x2 if u(x1) ≥ u(x2). Note that a utilityfunction is quasi-concave if and only it represents a convex preferencerelation. For the existence of a utility function representing a givenpreference relation, we refer to the classical article by Debreu [52] andfor more recent generalizations to [35].
When preferences are not complete or transitive we usually describe the tastes of an agent by a correspondence P : X→ →X. The set P (x) is interpreted as the strictly better-than-x set. We assume in this case thatthe correspondences are irreflexive in the sence that x / ∈ P (x) for all x in X. Each preference relation  on X defines its own correspondencesP : X→ P (x) = {x0 ∈ X: x0  x} .
An element a ∈ X is maximal for a preference correspondence →X if P (a) = 6. When a preference  is complete and reflexive an element a is maximal if and only if it is a greatest element, i.e.,a  x for all x ∈ X. For economic applications and several results onmaximal elements of preference correspondences see [1, 115].
There is a finite set J of producers indexed by j. A producer (also calleda firm or a technology of production or even a sector of production) is anabstract "entity" to which some knowledge (technological possibility)is available that allows him to produce some outputs from one or moreinputs.
The knowledge of the producer j ∈ J is represented by a subset Yj of the commodity space L, which gathers all its technologicallypossible production plans. The set Yj is called the production set ofj. A production plan for the jth producer is a vector y ∈ Yj.
When L is a vector lattice and y = y+ − y− is a production plan, y+ is the output of the producer and y− comprises the input. Moreover, ifp ∈ L0 is a price, then the cost of production is p · y− and the revenue isp · y+. The profit of the producer for the plan y is p · y = p · y+ − p · y−,that is, revenue minus cost.
Producers seek to maximize profit. That is, given a price p ∈ L0 the producer j wants to choose some y ∈ Yj so that: p · y ≥ p · y0 for all y0 ∈ Yj .
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.13 C. D. ALIPRANTIS B. CORNET AND R. TOURKY As we shall see later, this maximizing behavior basically requires thateach production set (or the total production set Y = Pj∈J Yj) is con-vex. In the sequel, we shall consider production sets that are not convexand therefore they introduce alternative objectives for the producers.
Assumption (P) Every production set Yj is closed and satisfies thefollowing properties: 0 ∈ Yj (possibility of inaction) Yj − L+ ⊆ Yj (free disposal ) The free disposal assumption has the following interpretation in the vector lattice setting: if y ∈ Yj is a production plan, then every planz ∈ L with higher input than y (i.e., z− ≥ y−) and output lower thanthe output of y (i.e., z+ ≤ y+) is a production plan (i.e., z ∈ Yj).
When the positive cone L+ has a nonempty interior, we point out that Assumption (P) implies that the boundary ∂Yj of the productionset Yj coincides with the set of (weakly) efficient production plans, thatis, ∂Yj = {yj ∈ Yj: 6 ∃ zj ∈ Yj such that zj  yj} , where as usual x  y means that x − y ∈ int L+.
The economy has an initial endowment ω ∈ L+, and the order interval[0, ω] = {x ∈ L : 0 ≤ x ≤ ω} is known as the Edgeworth box of theeconomy.
An economy is a list E = (I, J, hL, L0i, (Xi, i)i∈I, (Yj)j∈J , ω) , The finite sets I and J are, respectively, the sets of consumers andproducers .
hL, L0i is the commodity-price duality. (As mentioned earlier, L isalso equipped with a Hausdorff locally convex topology so that L0is its topological dual.) Each consumer i ∈ I has a consumption set Xi ⊆ L, an initialendowment ωi ∈ Xi, and a preference relation i on Xi.
Each producer j ∈ J has a production set Yj ⊆ L.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.14 ECONOMIC EQUILIBRIUM ω ∈ L is the total initial endowment of the economy.
An exchange economy is an economy without production. That is, it is an economy where there are no producers; or where all producersare inactive, i.e., Yj = {0} for every producer j.
An allocation for the economy E is a list ((xi)i∈I , (yj)j∈J ) of vectors such that xi ∈ Xi for each i and yj ∈ Yj for each j. The allocation issaid to be attainable (or admissible or even feasible) if supply equalsdemand. That is, The collection of all attainable allocations is denoted A, i.e., = ((xi), (yj)) ∈ Y Xi × Y Yj: An economy is said to be convex if all consumption sets, all prefer- ences, and all productions sets are convex. An economy is said to befinite if the sets I, J of consumers and producers are finite and if thereare finitely many commodities , or equivalently if the commodity spaceis finite dimensional, say L = R ordered by the standard ordering.
A private ownership economy is an economy E in which it is further Each consumer i ∈ I has an initial endowment ωi and Pi∈I ωi = ω.
Each consumer i ∈ I has a share θij ≥ 0 in the profit of theproducer j, where for each j we have Pi∈I θij = 1.
2.5. Equilibrium and optimality We start with the fundamental economic notion of price decentraliza-tion which will be formulated as a geometric notion of supporting anallocation by prices.
DEFINITION 1. An attainable allocation ((xi), (yj)) is said to be: 1. a weak valuation equilibrium, if there exists a (non-zero) price p that supports the allocation in the sense that it satisfies thefollowing properties.
a) For each consumer i: x i xi implies p · x ≥ p · xi.
b) For each producer j: y ∈ Yj implies p · y ≤ p · yj.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.15 C. D. ALIPRANTIS B. CORNET AND R. TOURKY 2. a valuation equilibrium, if there exists a (non-zero) price p that in addition to satisfying properties (a) and (b) above, it also satis-fies the property: c) p · ω 6= 0.
REMARK 1. Property (c) above is important for the infinite dimen-sional case. If the commodity space does have strictly positive vectors(for instance let L = ca[0, 1], the Riesz space of all regular Borel mea-sures on [0, 1]), then every attainable allocation is a weak valuationequilibrium.
The preceding definition will be generalized later in two ways. First we shall allow for prices valuation function that need not be linear. Thatis, we assumed above that the valuation function p: L → R is linear andwe shall later consider a case where the valuation function, denotedRp: L → R, may not be linear. The above definitions of supportingprices and valuation equilibria, remain the same, replacing p by Rp.
Second, in the absence of convexity assumptions on the productionsets Yj and the preferred sets, we shall weaken the conditions on thesupporting price by only assuming that the "necessary condition" forprofit maximization and expenditure minimization hold (in a precisemathematical sense to be defined later).
Let us move to notions of economic optimality whose definitions do not depend on market prices. An attainable allocation ((xi), (yj)) is: Pareto optimal, if there is no attainable allocation ((x0 ), (y0 )) i xi for all i and x0i i xi for some i.
weakly Pareto optimal, if there does not exist an attainableallocation ((x0 ), (y0 )) with x0  i xi for all i.
Two of the classical propositions in economics relate the notions optimality and supporting prices. They are known as the first andsecond welfare theorems. These theorems were first proven in reason-able generality by Arrow [16] and Debreu [51]. Subject to appropriateconditions, the two theorems can be stated as follows: 1st Welfare Theorem: Every valuation equilibrium is Paretooptimal.
2nd Welfare Theorem: Every Pareto optimal allocation is avaluation equilibrium.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.16 ECONOMIC EQUILIBRIUM The importance of the welfare theorems is that they characterize the supportability by market prices in terms of normative notions.
THEOREM 1 (First Welfare Theorem). If in a production economy, (a) each consumption set is convex and contains zero, (b) each production set contains zero, and (c) preferences are lower semicontinuous , then every valuation equilibrium having a supporting price satisfyingp · ω > 0 6 is weakly Pareto optimal.
Proof. Let ((xi), (yj)) be a valuation equilibrium. This means that it is an attainable allocation supported by a non-zero price p that satisfiesp · ω > 0. Suppose by way of contradiction that ((xi), (yj)) is notweakly Pareto optimal. Then, there exists some attainable allocation((x0 ), (y0 )) such that x0  x i for each i.
The supporting property implies that p · x0 ≥ p · x −p · y0 ≥ −p · y j for each j. Therefore, p · yj = p · ω .
i for each i and p · y0j j for each j. From the condition 0 ∈ Yj, we see that p · yj ≥ 0 for each j. Consequently, wehave Pi∈I p · xi = Pj∈J p · yj + p · ω > 0, hence there exists some i suchthat p · x0 = p · x Using that the preference i is lower semicontinuous and that zero belongs to Xi, we conclude that there exists some δ ∈ (0, 1) such thatδx0  i xi. This implies δp · xi = δp · x0i = p · (δx0i) ≥ p · xi > 0 , which is not possible.
6 We first notice that p · ω ≥ 0 (and p ≥ 0) when ω ∈ L+ and the preferences are strictly monotonic. Thus the assumption p · ω > 0 will be met under one of theadditional assumption that will be considered later in this paper (i) ω ∈ int L+ andp 6= 0, and (ii) p · ω 6= 0.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.17 C. D. ALIPRANTIS B. CORNET AND R. TOURKY 3. Convex exchange economies 3.1. The positive cone L+ has a nonempty interior We begin by presenting a version of the second welfare theorem whenthe positive cone L+ has a nonempty interior, a result due to De-breu [51] for infinite dimensional commodity spaces.
THEOREM 2 (Second Welfare Theorem). If in an exchange economythe positive cone L+ has a nonempty interior, every strict preferenceis monotone and convex, then every weakly Pareto optimal (and henceevery Pareto optimal) allocation ((xi), (yj)) is a weak valuation equilib-rium.
Moreover, if the total endowment ω is an interior point of L+ and p is the supporting price, we can assume that p · ω > 0 (in which caseevery weakly Pareto optimal allocation is a valuation equilibrium).
Proof. Let ((xi), (yj)) be a weakly Pareto optimal allocation, we first Pi(xi) + Int L+ where Pi(xi) = {y ∈ L+ : y i xi}.
Indeed, if it were not true, then we could write ω = Pi∈I[x0 + u], for some x0 ∈ P i(xi) (i ∈ I ) and some u ∈ Int L+. But then for each consumer i the strict monotonicity of Pi implies x0 + u ∈ P contradicting the weak Pareto optimality of the allocation ((xi), (yj)).
Clearly the set Pi∈I Pi(xi) + Int L+ is convex (since each Pi(xi) is convex by assumption and Int L+ is convex, hence the sum is alsoconvex), open, and nonempty (since each Pi(xi) is nonempty from themonotonicity assumption, and Int L+ is also nonempty). Consequently,by the Hahn–Banach separation theorem, there exists a non-zero func-tional p ∈ L0 such that for any choices of yi ∈ Pi(xi) for each i ∈ I andall u ∈ Int L+ we have From the continuity of the functional p ∈ L0, we deduce that (?) is stillvalid for every yi ∈ cl Pi(xi) for each i ∈ I and every u ∈ L+. Noticingthat the monotonicity assumption implies xi ∈ cl Pi(xi) for each i ∈ Iand 0 ∈ L+, we infer that p · yi ≥ p · xi for every yi ∈ Pi(xi) for eachi ∈ I, and p · u ≥ 0 for every u ∈ L+. Hence p > 0 and consequently((xi), (yj)) is a valuation equilibrium.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.18 ECONOMIC EQUILIBRIUM If, in addition we assume that ω ∈ Int L+, then we can immediately deduce that p · ω > 0.
When ω is not an interior point of L+ the preceding theorem is false even in the finite dimensional case with one consumer.
EXAMPLE 1 (Arrow [16]). Consider an exchange convex economy withone consumer and two commodities. The consumer has the utility function u(x, y) = y and the initial endowment ω = (0, 1).
Clearly, ω is the only Pareto optimal allocation. A straightforwardverification shows that ω is a weak valuation equilibrium but not avaluation equilibrium, that is, there is no supporting price p such thatp · ω > 0.
This problem can be solved by requiring that ω is an interior point of the positive orthant of R . However, the standard infinite dimensional setting is one where this problem cannot be readily assumed away.
When ω is not an interior point we need to add extra conditions onpreferences and use the lattice structures of the commodity and pricespaces. The main definitions and properties that are needed hereafterare recalled in the next section.
3.2. Commodity and price spaces as vector lattices This section considers the case where the positive cone L+ may havean empty interior. We shall need to introduce first a lattice structureon both the commodity and the price spaces and second a conditionon the consumers known as "properness." A Riesz space (or a vector lattice) is an ordered vector space L with the additional property that the supremum (least upper bound) ofevery nonempty finite subset of L exists. Following the classical latticenotation, we denote the supremum of the set {x, y} by x ∨ y. If now{xi : i ∈ I} is a finite collection of elements in a Riesz space, then weshall denote their supremum by Wi∈I xi, that is, _ xi = sup {xi: i ∈ I} .
In modern equilibrium theory quite often the commodity-price du- ality is represented by a dual pair hL, L0i, where L is a Riesz spaceand L0 is also a Riesz space. For this we need to introduce some moreterminology. In an ordered vector space L an interval is any subsetof L of the form [x, y] = {z ∈ L: x ≤ z ≤ y}. A set is said to beorder bounded if it is included in an order interval. A linear functional alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.19 C. D. ALIPRANTIS B. CORNET AND R. TOURKY p: L → R on an ordered vector space is said to order bounded if itcarries order bounded sets to bounded subsets of R.
We define the order dual of L, denoted by L∼, as the collection of all order bounded linear functionals on L. The order dual L∼ is orderedby the cone of positive linear functionals and is itself an ordered vectorspace, provided that the cone L+ is generating.7 Recall that a linearfunctional p: L → R is said to be positive if x ≥ 0 implies p · x ≥ 0.
The following theorem which is the basis of our analysis (see [6] THEOREM 3 (Riesz–Kantorovich). If L is a Riesz space, then L∼ isalso a Riesz space. In particular, if {pi: i ∈ I} is a finite collectionin L∼, then for each x ∈ L+ we have the following famous Riesz–Kantorovich formula: pi · xi: xi ∈ L+ and We need now to apply the above Riesz–Kantorovich formulas to prices in the topological dual L0. For this we shall assume that (L, τ ) isan ordered topological space and that L0 is a vector sublattice (Rieszsubspace) of L∼, that is if p, q are in L0, then the supremum p ∨ q(which exist in L∼) also belongs to L0. We point out that if L locallysolid,8 then L0 is automatically a vector sublattice of L∼. For details ofthe lattice structure of the topological and order duals of a topologicalRiesz space see again [6] and [7].
We now present the "properness" condition (see Mas-Colell [81] for the standard reference) and we shall use here the one taken fromTourky [109].
DEFINITION 2 (Tourky). A preference correspondence P : L+→ is said to be ω-proper if there exists some correspondence b (which is assumed to be convex-valued if P is convex-valued) such thatfor each x ∈ L+: 1. the vector x + ω is an interior point of b 7 If L is an ordered vector space with cone L+, then L+ is said to be generating if the vector space generated by L+ is all of L, or equivalently if L = L+ − L+. IfL is also equipped with a Hausdorff locally convex topology, then the dual "cone"L0+ = {p ∈ L0: p · x ≥ 0 ∀ x ∈ L+} is indeed a cone that is also weakly closed cone.
The cone of any Riesz space is generating.
8 A subset A in a Riesz space is said to be solid if x ≤ y and y ∈ A imply x ∈ A. A linear topology is called locally solid if it has a base at zero consisting ofsolid sets.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.20 ECONOMIC EQUILIBRIUM P (x) ∩ L+ = P (x).
We are now ready to state a version of the second welfare theorem when both the commodity space and the price space are vector lattices.
THEOREM 4 (Mas-Colell–Richard [82]). Assume that L is a vectorlattice and that L0 is a vector sublattice of the order dual L∼ of L. Ifin an exchange economy preferences are ω-proper, have convex strictupper sections, and are strictly monotone, then every weakly Paretooptimal allocation is a valuation equilibrium.
Proof. Let (xi)i∈I be a weakly Pareto optimal allocation. For each i, Pi(xi) be the convex extension of the set Pi(xi) = {y ∈ L+: y i xi} given by the ω-properness assumption. Let A be the convex non-emptyset of all attainable allocations, i.e., Since (xi)i∈I is weakly Pareto optimal, it follows that Pi(xi) = 6 .
Since, according to the ω-properness assumption, (xi + ω)i∈I is an inte-rior point of Qi∈I b Pi(xi), it follows that we can separate the two convex sets A and Qi∈I b Pi(xi) of the locally convex space L. That is, by the separation theorem, there exists a non-zero functional (pi)i∈I ∈ (L0)Isuch that for every (zi)i∈I ∈ Qi∈I b pi · yi: (yi) ∈ A ≤ X pi · zi .
The preceding inequality in conjunction with the continuity of the functionals pi ∈ L0 shows that (†) is also valid for every (zi)i∈I inQ Pi(xi). Also, from the strict monotonicity of the preferences we obtain xi ∈ cl b Pi(xi) for each consumer i. Consequently, pi · yi: (yi) ∈ A = pi · y ≥ pi · xi .
Next, we define the function p: L → R by alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.21 C. D. ALIPRANTIS B. CORNET AND R. TOURKY pi · yi: yi ∈ L+ and Clearly, from the lattice assumptions made on L and L0, we deducethat p = Wi∈I pi ∈ L0; see Theorem 3. Therefore, pi · xi = p(ω) = Furthermore, for every i ∈ I we have p(xi) ≥ pi · xi, and consequentlyp(xi) = pi · xi for each i ∈ I. Therefore, from († † †) we get: p(y) ≥ pi · y ≥ pi · xi = p(xi) , which shows that the price p supports the allocation (xi)i∈I .
We end the proof by showing that the price p is non-zero, and in fact that p(ω) > 0. Indeed, we know that (xi + ω)i∈I is an interior pointof Qi∈I b Pi(xi). So by (†) and (††), Pi∈I pi · xi < Pi∈I pi · (xi + ωi).
Hence, Pi∈I pi · ω > 0. Again we have p(ω) ≥ pi · ω for every i ∈ I.
Consequently, p(ω) ≥ 1 P i∈I pi · ω > 0.
3.3. Finite dimensional commodity spaces ordered by non-lattice cones A recurrent theme in the literature concerns the sharp contrast betweenthe central role of vector lattices in infinite dimensional theory and theapparent irrelevance of lattice theoretic properties in finite dimensionalanalysis. This contrast was highlighted by Mas-Colell [81, p. 506] in thefollowing quote: "A major surprise of this paper is precisely this relevance of lattice the-oretic properties to the existence of equilibrium problem. One would nothave been led to expect it from the finite dimensional theory. In the latterit is possible to formalize and solve the existence problem using onlythe topological and convexity structures of the space (cf. Debreu [54]).
Informally, a source for the difference seems to be the following: in generalvector lattices, order intervals (i.e., sets of the form {x: a ≤ x ≤ b}) are, asconvex sets, much more tractable and well behaved than general bounded,convex sets." Mas-Colell's quote has posed a long standing open question concerningthe application of the "lattice" approach to finite dimensional models: Does Theorem 4 hold true when the commodity space is finitedimensional and ordered by a non-lattice cone? alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.22 ECONOMIC EQUILIBRIUM The next example, due to Monteiro and Tourky [89], shows that the Mas-Colell–Richard theorem does not hold for finite dimensionalcommodity spaces when ordered by a non-lattice cone.
EXAMPLE 2 (Monteiro–Tourky). This is an example of an exchangeeconomy with two consumers and three commodities, i.e., L = The commodity space L is ordered with the order induced by the "icecream" cone: = {λ(x, y, 1): λ ≥ 0 and x2 + y2 ≤ 1}.
Of course, the "ice cream" cone does not induce a lattice structure onL. But, nevertheless, L+ is a closed and generating cone—and so theorder intervals are compact sets.
The initial endowment of consumer 1 is ω1 = (0, α, α) ∈ L+, where 0 < α < 1 is a fixed number. The initial endowment of consumer 2is ω2 = (0, 1 − α, 1 − α) ∈ L+. Therefore, the total endowment of theexchange economy is ω = (0, 1, 1). Since ω defines an extremal ray, itfollows that [0, ω] = {βω: 0 ≤ β ≤ 1} = {(0, β, β): 0 ≤ β ≤ 1}.
The utility functions of the consumers are defined on L by u2(x, y, z) = x + 2z .
It is easy to check that these utility functions are linear, strictly mono-tone, and obviously ω-uniformly proper.
Let us say that an allocation x = (x1, x2) is non-trivial if xi 6= 0 for i = 1, 2. We conclude the example by establishing the followingsurprising result.
• No non-trivial attainable allocation is a valuation equilibrium. In particular, this economy does not have any competitive eqilibria.9 To establish this claim, let x = (x1, x2) be an arbitrary non-trivial attainable allocation. Therefore, there exists some 0 < α < 1 suchthat x1 = (0, α, α) and x2 = (0, 1 − α, 1 − α). Now assume by way ofcontradiction that there exists some p ∈ L0 with p · ω 6= 0 that supportsthe above allocation. From the strict monotonicity of preferences, itmust be the case that p ∈ L0+. We can therefore assume without loss ofgenerality that p·ω = 2. This implies that p·x1 = 2α and p·x2 = 2−2α.
We claim that p ≥ (1, 0, 2) and that p ≥ (0, 0, 2) in the order induced by L+. That is, for every bundle a = (x, y, z) ∈ L+ we have p · a ≥ max{x + 2z, 2z} .
9 Recall that an allocation is a competitive (or a Walrasian) equilibrium if x i xi for any consumer i implies p · x > p · ωi.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.23 C. D. ALIPRANTIS B. CORNET AND R. TOURKY To see this, let a = (x, y, z) be an arbitrary point in L+ and noticethat max{x + 2z, 2z} ≥ 0. If max{x + 2z, 2z} = 0 is the case, thenit follows that x = y = z = 0, and so p · a = 0 = max{x + 2z, 2z},as required. Now consider the case max{x + 2z, 2z} > 0. This impliesz > 0. From u1( α a) = 2α = u 1(x1), the strict monoticity of u1 and the supporting property of p, we get p · ( α a) ≥ p · x 1 = 2α and so p · a ≥ 2z.
If x + 2z ≤ 0, then p · a ≥ 2z = max{x + 2z, 2z}. Finally, we considerthe case x + 2z > 0. In this case, we have u2( 2−2α a) = 2 − 2α = u This implies p · 2−2α a ≥ p · x Now let q = p − (0, 0, 2) ∈ L+ and note that the vector (x0, y0, z0) = q − (1, 0, 0) = p − (1, 0, 2) ∈ L+ satisfies the property This implies that (1+x0, −z0, z0) = (x0, y0, z0)+(1, 0, 0) = q ∈ L+, andthus 1 + x0 = 0. Consequently, x0 = −1 and from this it follows that(x0, y0, z0) = (−1, −z0, z0) / ∈ L+, a contradiction. This contradiction shows that the allocation (x1, x2) cannot be supported by any price.
3.4. Optimality with nonlinear prices With the preceding example in mind, Aliprantis, Tourky, and Yan-nelis [13] extended the theory of general equilibrium beyond vectorlattices by introducing the notion of a personalized price system thatinduces non-linear value functions. We briefly illustrate their optimalityresults below.
We call an arbitrary linear functional p = (p1, p2, . . , pm) on Lm a list of personalized prices (or simply a list of prices). For each com-modity bundle x ∈ L+, we let Ax denote the set of all allocations whenthe total endowment is x, i.e., DEFINITION 3. The generalized price (or the Riesz–Kantorovichformula) of an arbitrary list of personalized prices p = (p1, p2, . . , pm)is the function Rp: L+ → [0, ∞] defined by alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.24 ECONOMIC EQUILIBRIUM Clearly, if p ∈ (L∼)m, then Rp is a real-valued function. The value Rp(x) is the maximum value that one can obtain by decomposing thebundle x into consumable allocations, where each consumer i pays theprice pi · xi for her assigned bundle xi.
The basic properties of the generalized prices are included in the next result that can be found in [13].
LEMMA 1. If p = (p1, p2, . . , pm) ∈ (L∼)m is a list of order boundedpersonalized prices, then its generalized price Rp: L+ → [0, ∞) is anon-negative real-valued function such that: 1. Rp is monotone, i.e., x, y ∈ L+ with x ≤ y implies Rp(x) ≤ Rp(y).
2. Rp is super-additive, that is, Rp(x) + Rp(y) ≤ Rp(x + y) for all x, y ∈ L+.
3. Rp is positively homogeneous, that is, Rp(αx) = α(Rp(x)) for all α ≥ 0 and x ∈ L+.
5. If x ∈ L+, then pi · x ≤ Rp(x).
6. If L is a vector lattice (or more generally, if L has the Riesz De- composition property10 and its cone is generating), then for eachx ∈ L+ we have Rp(x) = π(x), where π = Wm The next definition generalizes the notion of a supportability.
DEFINITION 4. An allocation (x1, x2, . . , xm) is said to be a person-alized valuation equilibrium if there exists some list of personalizedprices p = (p1, p2, . . , pm) ∈ (L∼)m such that: 1. Rp(ω) > 0, 2. y ∈ Pi(xi) =⇒ Rp(y) ≥ Rp(xi), and 3. the following arbitrage–free condition holds It turns out that the properties of generalized prices that are listed in Lemma 1 can be easily used to extend the proofs of Theorems 8 andTheorem 9 to the case of ordered vector spaces with non-linear prices.
We conclude this section by stating an extension of the first and secondwelfare theorems to ordered vector spaces.
That is, [0, x] + [0, y] = [0, x + y] for all x, y ∈ L+ alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.25 C. D. ALIPRANTIS B. CORNET AND R. TOURKY THEOREM 5. In an exchange economy every valuation equilibriumis a personalized valuation equilibrium. Furthermore, for an orderedtopological vector space L with topologically bounded order intervals wehave the following.
1. If preferences are ω-proper, have convex strict upper sections, and are strictly monotone, then every weakly Pareto optimal allocationis a personalized valuation equilibrium.
2. If preferences have open upper sections, then every personalized valuation equilibrium is weakly Pareto optimal.
3. If L is a vector lattice and L0 is a vector sublattice of the order dual L∼ of L, then a personalized valuation equilibrium is a valuationequilibrium.
For a proof of this result and more on personalized prices, we refer the reader to [13].
4. Non-convex production economies In this section, we no longer assume that the production sets Yj areconvex and the profit maximization behavior of the firms is replacedby the so-called marginal pricing rule. We consider a finite dimensionalsetting, that is, the economy has finitely many commodities , or,equivalently, the commodity space L = The firm j is said to follow the marginal pricing rule if at equilibrium the production yj satisfies the first order necessary condition for profitmaximization (for the fixed price p) on the production set Yj, in aprecise mathematical sense formalized below. As we shall see, if theproduction set Yj is additionally assumed to be convex, then the abovenecessary condition will also be sufficient. In other words, in the convexcase, the marginal pricing rule coincides with profit maximization.
4.1. Marginal Pricing and Normal Cones We now give a precise mathematical definition of the marginal pricingrule, with the help of the following normal cones. Let C be a nonemptysubset of R , we call perpendicular vector (or proximal normal vector) to C at x ∈ cl C, every vector p in the set alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.26 ECONOMIC EQUILIBRIUM and limiting normal vector to C at x, every vector p in the set lim pν : ∃ {xν } ⊆ cl C, xν → x and ∀ ν, pν ∈⊥C (xν) .
It is worth pointing out that we have not assumed above that the set C is closed. The closedness of C is quite common when C = Yjis a production set, but it is no longer the case when we consider thepreferred set C = Pi(xi) (see below). From the above definitions, oneeasily sees that: ⊥cl C(x) =⊥C(x) ⊆ b Ncl C(x) for every x ∈ cl C.
DEFINITION 5.
The producer j, with production set Y said to follow the marginal pricing rule if its equilibrium condition isformalized by the condition The term "marginal pricing rule" is justified by the fact that the stronger condition p ∈⊥Y (y j ) [and thus also the weaker condition j )] is a first order necessary condition for profit maximization.
Indeed, if, for a given price p, the production yj maximizes the profitp·y0 over the production set Y j , then, for every ρ ≥ 0, it also maximizes the function p · y0 − ρky0 − y j k2 over the set Yj ; hence, the condition j ) is satisfied. We shall see below that, if Yj is convex, we can take ρ = 0 in the cone ⊥Y (y j ), that is, the necessary condition j ) for profit maximization is also sufficient.
The condition p ∈⊥Y (y j ) has a clear economic and geometric interpretation. It means that, given the price system p, theplan yj maximizes the quadratic function p · yj − ρky0 − y the production set Yj, i.e., it maximizes the profit p · yj up to the"perturbation" −ρky0 − y j k2. It can also be interpreted in terms of "non-linear prices" by noting that it is equivalent to saying that yjmaximizes the quadratic function πj(y0 ) = [p − ρ(y0 − y over the production set Yj.
Moreover, the condition p ∈⊥Y (y j ) formalizes in a natural way the notion of "orthogonality" to a set. Indeed, it is easy to show that it isequivalent to the following geometric condition: int[B(yj + εp, εkpk)] ∩ cl Yj = 6 .
The perpendicularity condition, however, is too restrictive in many situations, i.e., it is easy to find examples for which y1 ∈ ∂Y1 and⊥Y (y 1) = {0}. This is essentially the reason why, in the following, we need to formalize the marginal pricing rule with the the weaker notionof limiting normal cone.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.27 C. D. ALIPRANTIS B. CORNET AND R. TOURKY 4.2. Properties of the Normal Cones The following propositions summarize the main properties of the nor-mal cones. A general reference on this subject is Clarke [43].
PROPOSITION 6. Let C be a nonempty closed subset of 1. For every x ∈ C, the sets ⊥C(x), and b NC(x) are closed cones with vertex 0, and ⊥C(x) ⊆ b 2. Let U be an open subset of R and let x ∈ C ∩ U , then one has ⊥C∩U(x) =⊥C(x), and b 3. If x ∈ ∂C, then b NC(x) 6= {0}, but one may have ⊥C (x) = {0}.
NC(x) 6= {0}, then x ∈ ∂C.
k be a finite family of closed subsets of R be an open set containing C := Πk∈KCk and let f : U → R be adifferentiable function at a point x = (xk)k∈K in C (resp. twicecontinuously differentiable on a neighborhood of x). If x is a mini-mum of the function f over the set C, then the following necessarycondition is satisfied: −∇x f (x) ∈ b k ) [resp. −∇xk k ) ] for all k ∈ K.
REMARK 3. It is essentially because of the above Property 2, that thecones b NY (·) is well adapted to price decentralization by the marginal pricing rule. In other words, if Yj is closed, to every element yj ∈ ∂Yj[which coincides with the set of weakly efficient production plans, underAssumption (P)], one can associate a non-zero price p that supportsthe plan yj, in the sense that p ∈ b NC(x). However, this property does not hold, in general, for the cone of perpendicular vectors.
The next proposition states that the two definitions of normal cones coincide with the classical notion when C is convex or when C has asmooth boundary.
PROPOSITION 7. For a subset C of R the following holds.
1. If C is closed and convex, then, for each x ∈ C, we have NC(x) = {p ∈ R : p · x ≥ p · x0 ∀ x0 ∈ C} .
2. If C = {x0 ∈ R : g(x0) ≤ 0}, where the function g: R twice continuously differentiable and satisfies the nondegeneracyassumption [g(x) = 0 =⇒ ∇g(x) 6= 0], then NC(x) = {λ∇g(x): λ ≥ 0} .
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.28 ECONOMIC EQUILIBRIUM 3. If C satisfies Free Disposal, that is, C − C, then, for every NC(x) ⊆ R+ .
4.3. The case of convex strict preferences Consider, as in Section 2.4, the production economy R , (Xi, ωi i)i∈I , (Yj )j∈J ) , and define for each consumer i and each x = (xi) ∈ Qi∈I Xi, the strictlypreferred sets We now consider the case of convex strict preferences for which we can state the following results.
THEOREM 8. Let ((xi), (yj)) be a weakly Pareto optimal allocationsuch that, for each consumer i, the set Pi(xi) is convex, xi ∈ cl Pi(xi)[which holds under the Nonsatiation Assumption], and, for every pro-ducer j, Yj is closed. Then there exists a non-zero price p such that: • For each consumer i, x  xi implies p · x ≥ p · xi .
• For each producer j, p ∈ b The proof of Theorem 8 is given in the next section. The interpre- tation of this result is the same as for the second welfare theorem ofconvex economies, the only difference being that here the producersstrive to follow the marginal pricing rule rather than maximizing theirprofits as in the convex case.
REMARK 4. The statement of Theorem 8 does not treat symmetri-cally the consumption and the production sectors. Indeed, the presenceof non-convexities in these two sectors is of a different nature. For exam-ple, by considering a continuum of consumers all infinitely small (i.e. ameasure space without atoms), the convexity assumption on consumers'preferences can be dropped (see Aumann [19], Hildenbrand [66]). Ananalogous treatment of the production sector, however, is not veryrealistic on economic grounds, since the presence of increasing returns(and other types of non-convexities) usually characterizes large firms.
We now state the standard version of the second theorem of welfare economics in the convex case (compare with the statement given inSection 3).
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.29 C. D. ALIPRANTIS B. CORNET AND R. TOURKY COROLLARY 9. Let ((xi), (yj)) be a weakly Pareto optimal allocationsuch that for each consumer i, the set Pi(xi) is convex, xi ∈ cl Pi(xi)[which holds under the Nonsatiation Assumption], and, for each pro-ducer j, Yj is closed and convex. Then, there exists a non-zero pricep such that: • For each consumer i, x  xi implies p · x ≥ p · xi .
• For each producer j, and each yj ∈ Yj we have p · yj ≥ p · yj.
REMARK 5. Theorem 8 and Corollary 9 do not hold, in general, ifthe cone b NY (·) of limiting normal vectors are replaced by the cone of perpendicular vectors ⊥Y (·).
4.4. Proof of the second welfare theorem The proof of Theorem 8 relies on the following result, which was usedin a similar context by Bonnisseau [28].
LEMMA 2 (Cornet–Rockafellar [49]). Let Ck (k ∈ K) be a finite fam-ily of subsets of R , let ¯ ck ∈ cl Ck (k ∈ K) and let ε > 0. Then ck, ε) =⇒ NC (¯ Proof. Step 1. We first prove the lemma under the additional as- sumption that the sets Ck are closed and that the following strongercondition holds: Pk ¯ck ∈ ∂[Pk Ck]. From this condition we deduce theexistence of a sequence (eν ) ⊆ R converging to zero such that, for allν we have: We consider the following minimization problem xk ∈ Ck (k ∈ K), (k ∈ K) are the variables of the problem (P ν ) and eν ck (k ∈ K) are fixed parameters.
Claim 1. For every ν, the problem (P ν ) admits a solution (not nec- essarily unique), denoted by (xν ) k k∈K , and the sequence (xν ck for every k. Indeed, first the existence of the solution is a clear alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.30 ECONOMIC EQUILIBRIUM consequence of the fact that the criterium of the problem is coerciveand the sets Ck (k ∈ K) are closed. We now denote by vν the value ofthe problem (P ν ), that is, Consequently, for every ν and every k one has kxν − ¯ k k2 ≤ vν . More- over, letting xk = ¯ ck for all k in the problem (P ν), one gets vν ≤ keνk, which we recall converges to zero . Thus (xν ) converges to ¯ the proof of the claim.
For all ν, the solution (xν ) k k∈K of (P ν ) satisfies the first order nec- essary conditions of the problem (P ν ), which can be written as follows[see Proposition 6.2] −pν − 2(xν − k ) for all k and ν, where pν = eν + P k )/keν + Pk∈K (xν k )k is well defined, Without any loss of generality, we can suppose that the sequence (pν ) in the unit sphere S, converges to some element ¯ non-zero). Taking the limit in (∗), when ν → ∞, for all k, one gets−¯ p = − limν pν ∈ ˆ (from Claim 1). This ends the proof of Step 1.
Step 2. We now present the proof of the lemma in the general case. Letting Dk = cl Ck ∩ B(x, ε). From Step 1, we deduce thatT k ) 6= {0}. But, from Proposition 6.2, one gets NDk∩B(¯ck,ε)(¯ck) = ˆ Ncl C∩B(¯ck,ε)(¯ck) = ˆ hence the desired conclusion follows.
We come back to the proof of Theorem 8, which is a consequence of Lemma 2 and the two following claims.
CLAIM 1. There exist e ∈ R and ε > 0 such that, for every t ∈ (0, ε), cl Pi(xi) ∩ B(xi, ε) − X cl Yj ∩ B(yj, ε) ⊆ X Pi(xi) − X Yj.
Hence, there is a sequence eν → 0 such that, for all ν we have cl Pi(xi) ∩ B(xi, ε) − X cl Yj ∩ B(yj, ε) ⊆ X Pi(xi) − X Yj.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.31 C. D. ALIPRANTIS B. CORNET AND R. TOURKY Proof of Claim 1. Since Yj is closed, it is sufficient to show that, for every i there exist ei and εi > 0 such that t ∈ (0, εi) implies: tei + cl Pi(xi) ∩ B(xi, ε) ⊆ Pi(xi).
Then e = Pi∈I ei and ε = mini εi will satisfy the desired conclusion.
We now show that the set C := Pi(xi), which is convex by assump- tion, satisfies the above inclusion (∗∗). For this, we chose ¯ relative interior of the convex set C, and we let ei := ¯ c−xi and ε ∈ (0, 1) be such that A ∩ ¯ c, ε) ⊆ ri C, (where A is the affine space spanned by C) and we let c ∈ cl C ∩ ¯ B(xi, ε). Then tei + c = t(¯ c, ε) ⊆ ri C, and c ∈ cl C. Consequently, tei + c belongs to ri C ⊆ C; see, for example, Rockafellar [100].
CLAIM 2. ω = Pi∈I xi − Pj∈J yj and cl Pi(xi) ∩ B(xi, ε) − X cl Yj ∩ B(yj, ε) .
Proof of Claim 2. Let (eν ) be the sequence converging to 0 given by Claim1. It suffices to show that for every ν one has ω − eν 6∈ X cl Pi(xi) ∩ B(xi, ε) − X cl Yj ∩ B(yj, ε).
Suppose it is not true. Then, by Claim 1, there exists some ν such cl Pi(xi) ∩ B(xi, ε) − X cl Yj ∩ B(yj, ε) ⊆ X Pi(xi) − X Yj, which contradicts the fact that ((xi), (yj)) is a weakly Pareto optimalallocation.
The above Theorem 8 was proved in Cornet [44] by showing that the following maximization problem t ∈ (−ε, +ε), cl Pi(xi) ∩ B(xi, ε), (i ∈ I) cl Yj ∩ B(yj, ε), alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.32 ECONOMIC EQUILIBRIUM admits for solution the point (0, (xi), (yj, )). Then, it was shown thatthe Lagrange multiplier p ∈ R associated to the equality constraint (using the Lagrangian multiplier rule of Clarke [43, 6.1.1, and 6.1.2(iv)]), satisfies the conclusion of Theorem 8, when the marginal pricingrule is defined by Clarke's normal cone, that is, the closed convex hullof the cone of limiting normal vectors. The convexity of Clarke's normalcone was shown to be fundamental in the existence problem of marginalcost pricing equilibria [30, 31, 32, 33, 46].
The same maximization problem is considered by Khan [70], a paper that has been circulating since 1987, to prove that, in Theorem 8,Clarke's normal cone can be replaced by the smaller cone of limitingnormal vectors, using the Lagrangian multiplier rule by Mordukhovicz[90]. The earlier work by Guesnerie [65] was using another cone, namelythe Dubovickii–Miljutin's one, with a dual approach via the tangentcone (see also, in the infinite dimensional setting, Bonnisseau–Cornet[29]). Generalizations of the above Lemma 2 to the infinite dimensionalsetting are considered by [34, 91].
4.5. The general case The proof of the Second Welfare Theorem that we gave in the previoussection works for a much larger class of economies. This leads us tointroduce the following Constraint Qualification Assumption that wasintroduced in [44]. As before we consider the economy: R , (Xi, ωi i)i∈I , (Yj )j∈J ) .
DEFINITION 6. The allocation ((xi), (yj)) of the economy E is said tobe qualified if, for every i ∈ I, xi ∈ cl Pi(xi) (which holds in particularunder Local Nonsatiation), and there exist ε > 0 and a sequence (eν ) ⊆ R converging to zero such that for every ν we have cl Pi(xi) ∩ B(xi, ε) − X cl Yj ∩ B(yj, ε) ⊆ X Pi(xi) − X Yj.
We now state a general version of the Second Welfare Theorem, in which no convexity, no differentiability, and no interiority assump-tion are made, neither on the production sets, nor on the consumers'characteristics.
THEOREM 10. Let ((xi), (yj)) be a weakly Pareto optimal allocationof the economy E , and assume that it is qualified. Then, there exists anon-zero price p in For each consumer i, −p ∈ b For each producer j, p ∈ b alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.33 C. D. ALIPRANTIS B. CORNET AND R. TOURKY The proof of Theorem 10 is exactly the same as the one given in the previous section, with the only difference that we don't have to proveClaim 1, which is exactly the Qualification Assumption of the economy.
We refer to Cornet [44] and Khan [71] for a more detailed discussionon the relationship with other results on this subject. We note againthat Theorem 10 does not hold, in general, if the cones b NY (·) of limiting normal vectors are replaced by the smaller cones of perpendicular vectors ⊥P We end this section by giving several cases of economic interest under which an allocation is qualified.
PROPOSITION 11. (a) The allocation ((xi), (yj)) of the economy Eis qualified if, for every i ∈ I, the couple (C, c) = (Pi(xi), xi) and, forevery j ∈ J the couple (C, c) = (Yj, yj) satisfies one of the followingconditions: (i) there exist ε > 0 and a sequence (eν ) converging to 0, such that eν + cl C ∩ B(c, ε) ⊆ C.
(ii) [Convexity] C is convex; (iii) cl C + Q ⊆ C, for some closed cone Q ⊆ R (which is satisfied, in particular, when C is closed, with Q = {0}) .
(b) The allocation ((xi), (yj)) of the economy E is qualified if, for some i ∈ I, the couple (C, c) = (Pi(xi), xi) or, for some j ∈ J thecouple (C, c) = (Yj, yj) satisfies one of the following conditions:(i0) there exist ε > 0 and a sequence (eν ) converging to 0, such that eν + cl C ∩ B(c, ε) ⊆ int C.
(ii0) C is convex, with a nonempty interior; (iii0) [Free Disposal] C + Q ⊆ C, for some closed cone Q ⊆ nonempty interior.
The proof of the proposition is a consequence of the two following CLAIM 3. Condition (i) (resp. (i0)) is satisfied by C ⊆ if one of the conditions (ii) or (iii) (resp. (ii0) or (iii0)) holds.
Proof. (ii) =⇒ (i). It is shown in the proof of Claim 1.
(iii) =⇒ (i). Take e ∈ Q.
(ii0) =⇒ (i0). The set int C is convex, and cl C = cl int C (see, forexample [100]). Hence, the implication [(ii) =⇒ (i)] (applied to int C),gives us the conclusion.
alipr-cornet-tourky-7-01-02.tex; 18/12/2002; 14:34; p.34 ECONOMIC EQUILIBRIUM (iii0) =⇒ (i0). We prove that every e ∈ int Q satisfies the strongercondition that te + cl C ⊆ C for every t > 0 . Indeed, if c ∈ cl C, thereexists a sequence (cn) ⊆ C converging to c. Hence, for every t > 0,te + c = t[e + (c − cn)/t] + cn. But, for n large enough, the pointen := e + (c − cn)/t ∈ int Q and from the Free Disposal Assumption,we get c + te = cn + ten ∈ C + int Q ⊆ C.
(k ∈ K) be a finite family, let ck ∈ cl Ck, and assume that the couple (Ck, ck) satisfies Condition (i) for every k(resp. Condition (i0) for some k), then there exist ε > 0 and a sequence(eν ), converging to 0, such that cl Ck ∩ B(ck, ε) ⊆ X Ck.
Proof of Claim 4. Let us first assume that the couple (Ck, ck) satisfies Condition (i) for every k. Then, there exist εk > 0 (k ∈ K) and asequence (eν ) (k ∈ K), converging to 0, such that eνk + cl Ck ∩ B(ck, εk) ⊆ Ck.
We define the sequence eν := Pk∈K eν and ε := min k εk . Taking the sum of the above inclusions, we get cl Ck ∩ B(ck, ε) ⊆ X Ck.
We now consider the case where the couple (Ck, ck) satisfies Condi- tion (i0) for some k, say k = 1. Then there exist (ε1) and a sequence(eν1), converging to 0, such that eν1 + cl C1 ∩ B(c1, ε1) ⊆ int C1.
Consequently, for every ν cl Ck ∩ B(ck, ε1) ⊆ int C1 + and one checks that int C1 + Pk6=1 cl Ck ⊆ Pk∈K Ck.
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