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When macroeconomic tools fail to respond to wealth inequality optimally, regulators can still seek to mitigate inequality within individual markets. A social planner with distributional preferences might distort allocative efficiency to achieve a more desirable split of surplus, for example, by setting higher prices when sellers are poor--effectively, using the market as a redistributive tool. In this paper, we seek to understand how to design goods markets optimally in the presence of inequality. Using a mechanism design approach, we uncover the constrained Pareto frontier by identifying the optimal trade-off between allocative efficiency and redistribution in a setting where the second welfare theorem fails because of private information and participation constraints. We find that competitive equilibrium allocation is not always optimal. Instead, when there is substantial inequality across sides of the market, the optimal design uses a tax-like mechanism, introducing a wedge between the buyer and seller prices, and redistributing the resulting surplus to the poorer side of the market via lump-sum payments. When there is significant within-side inequality, meanwhile, it may be optimal to impose price controls even though doing so induces rationing.

A mixed manna contains goods (that everyone likes) and bads (that everyone dislikes), as well as items that are goods to some agents, but bads or satiated to others. If all items are goods and utility functions are homogeneous of degree 1 and concave (and monotone), the competitive division maximizes the Nash product of utilities (Gale–Eisenberg): hence it is welfarist (determined by the set of feasible utility profiles), unique, continuous, and easy to compute. We show that the competitive division of a mixed manna is still welfarist. If the zero utility profile is Pareto dominated, the competitive profile is strictly positive and still uniquely maximizes the product of utilities. If the zero profile is unfeasible (for instance, if all items are bads), the competitive profiles are strictly negative and are the critical points of the product of disutilities on the efficiency frontier. The latter allows for multiple competitive utility profiles, from which no single-valued selection can be continuous or resource monotonic. Thus the implementation of competitive fairness under linear preferences in interactive platforms like SPLIDDIT will be more difficult when the manna contains bads that overwhelm the goods.

This paper studies competitive equilibria of economies where assets are heterogeneous and traders have heterogeneous information about them. Markets are defined by a price and a procedure for clearing trades, and any asset can, in principle, be traded in any market. Buyers can use their information to impose acceptance rules which specify which assets they are willing to trade in each market. The set of markets where trade takes place is derived endogenously. The model can be applied to find conditions under which these economies feature fire sales, contagion, and flights to quality.

© 2015 The Econometric Society. There is a widely held view within the general public that large corporations should act in the interests of a broader group of agents than just their shareholders (the stakeholder view). This paper presents a framework where this idea can be justified. The point of departure is the observation that a large firm typically faces endogenous risks that may have a significant impact on the workers it employs and the consumers it serves. These risks generate externalities on these stakeholders which are not internalized by shareholders. As a result, in the competitive equilibrium, there is under-investment in the prevention of these risks. We suggest that this under-investment problem can be alleviated if firms are instructed to maximize the total welfare of their stakeholders rather than shareholder value alone (stakeholder equilibrium). The stakeholder equilibrium can be implemented by introducing new property rights (employee rights and consumer rights) and instructing managers to maximize the total value of the firm (the value of these rights plus shareholder value). If there is only one firm, the stakeholder equilibrium is Pareto optimal. However, this is not true with more than one firm and/or heterogeneous agents, which illustrates some of the limits of the stakeholder model.

We investigate the role of search frictions in markets with price competition and how it leads to sorting of heterogeneous agents. There are two aspects of value creation: the match value when two agents actually trade and the probability of trading governed by the search technology. We show that positive assortative matching obtains when complementarities in the former outweigh complementarities in the latter. This happens if and only if the match-value function is root-supermodular, that is, its nth root is supermodular, where n reflects the elasticity of substitution of the search technology. This condition is weaker than the condition required for positive assortative matching in markets with random search. The paper initially circulated under the title “The Sorting Effect of Price Competition.” We would like to thank numerous colleagues and seminar participants for insightful discussions and comments. We greatly benefited from comments by Ken Burdett, John Kennan, Stephan Lauer- mann, Benny Moldovanu, Michael Peters, Andrew Postlewaite, Shouyong Shi, Robert Shimer, and Randy Wright. Kircher gratefully acknowledges support from the National Science Foundation, Grant SES-0752076, and Eeckhout acknowledges support from the ERC, Grant 208068.

We report recent advances concerning the package allocation problem, in which traders seek to buy or sell combinations of goods. The problems are most difficult when some goods are not substitutes. In that case, competitive equilibrium typically fail to exist but the core is non-empty and comprises the competitive solutions. Also in that case, the Vickrey auction fails to select core allocations and yield revenues that are less than competitive. The Ausubel-Milgrom auction generally selects core allocations and, when goods are substitutes, prescribes the Vickrey allocation. We also evaluate the problems and promise of mechanisms for the package exchange problem.

Extensive-form market games typically have a large number of noncompetitive equilibria. In this paper, we argue that the complexity of noncompetitive behavior provides a justification for competitive equilibrium in the sense that if rational agents have an aversion to complexity (at the margin), then maximizing behavior will result in simple behavioral rules and hence in a competitive outcome. For this purpose, we use a class of extensive-form dynamic matching and bargaining games with a finite number of agents. In particular, we consider markets with heterogeneous buyers and sellers and deterministic, exogenous, sequential matching rules, although the results can be extended to other matching processes. If the complexity costs of implementing strategies enter players' preferences lexicographically with the standard payoff, then every equilibrium strategy profile induces a competitive outcome.

We consider an exchange economy in which there are infinitely many consumers and some commodities are bads, that is, cause disutility to consumers. We give an example of such an economy for which there is no competitive equilibrium or its variants (quasi- or pseudo-equilibrium), and an example of the failure of the so-called uniform integrability condition of equilibrium allocations of increasingly populous finite economies.

We extend the standard model of general equilibrium with incomplete markets to allow for default and punishment by thinking of assets as pools. The equilibrating variables include expected delivery rates, along with the usual prices of assets and commodities. By reinterpreting the variables, our model encompasses a broad range of adverse selection and signalling phenomena in a perfectly competitive, general equilibrium framework. Perfect competition eliminates the need for lenders to compute how the size of their loan or the price they quote might affect default rates. It also makes for a simple equilibrium refinement, which we propose in order to rule out irrational pessimism about deliveries of untraded assets. We show that refined equilibrium always exists in our model, and that default, in conjunction with refinement, opens the door to a theory of endogenous assets. The market chooses the promises, default penalties, and quantity constraints of actively traded assets.

The paper studies bilateral contracting between one principal and N agents when each agent’s utility depends on the principal’s unobservable contracts with other agents. We show that allowing deviations to menu contracts from which the principal chooses bounds equilibrium outcomes in a wide class of bilateral contracting games without imposing ad hoc restrictions on the agents’ beliefs. This bound yields, for example, competitive convergence as N →� in environments in which an appropriately-defined notion of competitive equilibrium exists. We also examine the additional restrictions arising in two common bilateral contracting games: the “offer game” in which the principal makes simultaneous offers to the agents, and the “bidding game” in which the agents make simultaneous offers to the principal.

THE CENTRAL PURPOSE OF THIS PAPER is to develop a model of insider trading (i.e., trading based on private information) in the context of an imperfectly competitive multi-security market with risk-neutral agents. Imperfect competition allows us to consider strategic behavior, and a multi-security market lets us study the effect of a correlated environment on equilibrium. We employ the informational assumption that market makers can observe all order flows, and so portfolio diversification arises in this model for strategic reasons. Given correlated fundamentals, market makers can potentially learn about every security from each order flow. This causes even a risk neutral trader who does not face short-selling restrictions to refrain from determining the demand for each security independently. This contrasts with traditional multi-asset models, which focus on the incentive to reduce portfolio variance, or the effect of short-selling restrictions or budget constraints. Under imperfect competition, correlation has two effects. One, ceteris paribus, it allows the uninformed to learn from additional variables since each order flow could potentially have information about all payoffs. On the other hand, it creates an incentive for informed traders to restrict what others can learn from public information. Thus, our analysis can be viewed as an application to the multi-security, heterogeneous-information model in Admati (1985) of the imperfectly competitive equilibrium concept which Kyle (1985) first applied to the single-security, homogeneous-information model of Grossman and Stiglitz (1980). Our principal results include an explicit characterization of a linear equilibrium as a function of three general covariance matrices associated with payoffs, noise trading, and errors in private signals. Under general covariance structures, we show that there always exists an equilibrium in which the relationship between the vector of prices and the vector of order flows is governed by a symmetric positive definite matrix. The plan of the paper is as follows. In Section 2, we introduce our model. We derive the equilibrium in Section 3, and Section 4 comments on the properties of the equilibrium. The proofs of the results are in the Appendix.

The existence of competitive equilibrium is established for economies with a produc tion sector and an infinite dimensional space of commodities. The cru cial assumptions which are required (beyond those required in the fin ite dimensional setting) are bounds on marginal rates of substitution and marginal rates of transformation. Copyright 1987 by The Econometric Society.

The paper introduces the concept of a conditionally dispersed endowment distribution (given the consumption set) in the study of regularizing effect of aggregation and existence of equilibria in an exchange economy with an atomless measure space of consumers. It is shown that there is a regularizing effect of aggregation whenever the endowment distribution of an economy is conditionally dispersed. On the basis of this regularizing effect, the existence of a competitive equilibrium is proved in a very general setting where some of the commodities are indivisible and where different consumers may have different consumption sets. GENERAL EQUILIBRIUM THEORY saw, in recent years, explicit formulation of models with indivisible commodities in which existence of competitive equilibria was discussed. (References are Henry [12], E. Dierker [10], Broome [7], and Mas-Colell [19].) A pathbreaking paper by Mas-Colell [19] clearly reconfirmed the power of a continuum of traders model introduced by Aumann [1], and it emphasized, for the first time in the literature of the equilibrium existence problem, the importance of a condition relating to the form of distribution of agents' characteristics. More specifically, a competitive equilibrium was shown to exist in an economy with indivisible commodities where a single perfectly divisible commodity exists and the distribution of its initial holdings among agents is "dispersed" in the sense that it is absolutely continuous with respect to the Lebesgue measure. (See [19, Theorem 1, p. 446].) This development in the existence literature is of much interest to equilibrium theory and coincides with the way in which the analysis of a certain aggregation problem is being pursued in equilibrium theory. (See, e.g., Sonderman [33], Mas-Colell [20], and Hildenbrand [18].) In both of these problems, elaboration of the concept of "dispersedness" in the distribution of agents' characteristics is required.3 The work of Mas-Colell [19] was followed by [35] in which the concept of a dispersed endowment distribution was introduced in the study of existence of equilibria in an exchange economy with an atomless measure space of consumers where no convexity assumptions on preferences and consumption sets were

Let F be a private ownership economy. Call an allocation w "voluntary for a price vector p" if no agent can benefit by trading less at p. We prove that, under differentiability and interiority, if w is voluntary and Pareto efficient it is a competitive equilibrium for W. Perhaps surprisingly, production economies where some agents receive profit income require a stronger voluntariness condition than the one sufficient for exchange economies. The stronger condition singles out a commodity as a medium of exchange.

Reformulating marginal productivity theory by replacing productivity with respect to commodities with productivity with respect to persons and then defining perfectly competitive equilibrium as an allocation at which each person receives the marginal product of his/her contribution called a no-surplus allocation there emerges a competitive theory of price determination. Characterizations of no-surplus allocations are given in models with a nonatomic continuum of agents and an infinite-dimensional commodity space. Comparisons between the no-surplus and Walrasian equilibrium definitions of competitive equilibrium are made and some sufficient conditions are obtained for the existence of a no-surplus allocation.

Generalizing the Sharpe-Lintner capital asset pricing model, Dieffenbach [4] presents a model of securities markets in a private enterprise economy in a multiperiod competitive equilibrium with uncertainty. Risk premiums on securities depend on the covariances of holding period returns with the return on the market portfolio and with a multiperiod cost-of-living index. This paper develops a quantitative theory of that relationship suitable for empirical estimation and testing. Whether the Arrow-Pratt relative risk aversion of a representative investor is greater or less than one is important in the theory; the empirical results for the United States suggest that this value exceeds one. A theoretical and empirical application of the theory to the term structure of United States Treasury securities concludes the paper. Mean observed returns are consistent with theoretical predictions for medium and long term securities, but the differences of mean observed returns among bills of different maturities exceed the theoretical predictions.

It is known that the price mechanism whereby the rate of change of a price is proportional to the excess demand of the corresponding commodity need not converge to a competitive equilibrium for a pure exchange economy with more than two commodities. On the other hand, there exist convergent price mechanisms, similar to the Newton iterative process, where the rate of change of the prices is determined by the excess demand and the marginal excess demands of all the commodities. This is a considerable informational requirement. It is shown that this requirement cannot be substantially reduced for any convergent price mechanisms, that is for price mechanisms expressed in terms of a difference or differential equation where the solutions converge to a competitive equilibrium.

It is shown that a market with a continuum of traders possesses a competitive equilibrium even when the preferences are not complete. This generalizes further a result of Aumann, ('Econometrica'; 32: 39-50 (1964); 34: 1-17(1966)) who showed that the convexity assumption may be dispensed within the presence of a continuum of traders. The proof is inspired by the Arrow-Debreu ('Econometrica; 22: 265-290(1954)) proof for the finite case. (Author)

THE MAJOR INTEREST in the recent literature on infinite economies has lain in the results about large but finite economies that have been derived from the results proved for infinite economies. It is thus important to find simple yet general proofs for infinite economies. In this article we wish to provide a simple yet general proof of the existence of a competitive equilibrium in an infinite, nonstandard economy with production. The simplicity of our proof comes from the fact that nonstandard analysis can deal with "infinitely large" and "infinitely small" quantities very much as ordinary analysis deals with finite quantities.2 As a result, it is possible to follow very closely the proof of existence for an economy with a finite number of traders, such as that in G. Debreu's classic Theory of Value [6]. In fact, if one is willing to believe that nonstandard analysis permits us to manipulate infinite quantities as claimed above, then no further knowledge of nonstandard analysis is required in order to follow the proof. As examples of the simplicity of nonstandard analysis, it may be pointed out that no analogue of the Fatou-Schmeidler lemma [7, p. 69], a fairly difficult mathematical theorem, is required; nor is it necessary to prove separately that integration preserves upper-semicontinuity, a proof that Aumann [1] has recently simplified, because "integration" in the nonstandard model consists of an "infinite" summation, hence an appeal to 1.9.4 of Debreu [6] suffices to establish this point. As our main objective is to obtain results about large but finite economies, it is a welcome bonus to find out that no further effort is needed to obtain these desired theorems. This arises because of the following property of nonstandard analysis. Consider a sequence of real numbers {an} which tends to zero. If we could extend this sequence to the infinite integers, it would surely be a necessary property of the values of {an} at the infinite integers that they are all infinitely close to zero. What makes nonstandard analysis powerful is that the above line of reasoning can be reversed, so to speak. Suppose we have a sequence which

This paper reports both theoretical results and also computational experience with a method for approximating a competitive equilibrium in a piecewise linear economy. The algorithm consists of solving a sequence of linear programs, alternating between: (a) a "master" problem which ensures a balancing bundle of choices and generates a price vector; and (b) a "sub" problem which indicates the maximum level of utility attainable by each household--given the initial resource endowments-and also given the prices generated at the current iteration of the master problem. Each subproblem provides a "price-consistent" utility vector. The master problem determines a convex combination of the utility vectors generated at previous iterations. This convex combination is chosen so as to minimize the distance between the quantityconsistent and the price-consistent set. For the sequence of sub and master problems to approach a competitive equilibrium, this distance must approach zero. Thus far, the algorithm has failed whenever all equilibria are "unstable," and it has converged rapidly when there are "stable" equilibria. It will be shown that the algorithm does not cycle. It will also be shown that if the sequence of solutions (obtained from the algorithm) converges, then it converges to a Walrasian equilibrium.

This paper considers a market where pairs of agents who are interested in carrying out a transaction are brought together by a stochastic process and, upon meeting, initiate a bargaining process over the terms of the transaction. The basic bargaining problem is treated with the strategic approach. The paper derives the steady state equilibrium agreements; analyzes their dependence on market conditions such as the relative numbers of agents of different types; and discusses their relations with the competitive equilibrium outcome and other results in the search equilibrium literature. THIS PAPER CONSIDERS a market that operates in the following way. Pairs of agents who have mutual interest in carrying out a transaction are brought together by a stochastic process. When two agents meet, they initiate a bargaining process over the terms of the transaction. If two agents reach an agreement a transaction takes place and they leave the market. Of course, the bargaining positions and hence the agreement reached in any particular meeting will be affected by the conditions prevailing in the market. These will include the chances that each of the negotiating parties have of meeting other partners in the event that the agreement in the current negotiations is delayed; also the expected length of time required to achieve an alternative transaction, and the expected behavior of alternative partners. The study of such a market mechanism is of interest for two reasons. Firstly, it captures some realistic aspects of the trade in certain specific markets (e.g., asset markets such as housing and some labor markets). Secondly, it contributes to an understanding of the micro-mechanisms of price formation and their role in shaping market outcomes. In both cases, the features studied are largely neglected in the traditional market equilibrium analyses. The related literature includes the articles by Diamond [4, 5], Diamond and Maskin [6], Mortensen [8, 9], and Zusman and Bell [13]. These articles consider markets of the type described above, in which transactions are concluded at pairwise meetings of agents. The major difference between our work and these articles is in the approach to the basic bargaining problem. In the cited articles it is assumed that a meeting is concluded with an instantaneous agreement which divides the associated surplus in an arbitrary predetermined way (when the surplus is assumed to be divided equally, the division rule is, in fact, Nash's axiomatic bargaining solution). In contrast, the present paper treats the basic bargaining problem with the strategic approach (see Rubinstein [19]) which constitutes an attempt to look into the bargaining black-box. This approach explicitly models the time dimension of the bargaining process, describes in-detail the bargaining procedure, and justifies the agreement as a perfect equilibrium in

THE EXISTENCE of increasing returns to scale in production poses a number of difficult problems. In general equilibrium theory the problems are obvious. Standard results on the competitive equilibrium guarantee that when an equilibrium exists it is contained in the core of an economy. When the competitive equilibrium fails to exist, as is likely under increasing returns, the core may provide a useful equilibrium concept of its own. However, very little is known about the existence of a core except under conditions of constant returns to scale. One of the objectives of the present paper is to demonstrate the existence of the core in a general economy in which certain types of increasing returns are present. The second objective of this paper concerns partial equilibrium analysis. When it is asserted that a particular industry is an increasing returns industry it is not at all obvious what is being said, particularly when the industry produces more than one output. One aspect of increasing returns is the property of scale economies, which exist if by multiplying all inputs by a factor A > 1 it is possible to increase all outputs by at least a factor of A. A more primitive concept of increasing returns behavior is the concept of subadditivity of a cost function (or superadditivity of the production possibility set). Presumably the properties of subadditivity and economies of scale are both assumed of a given industry when it is said to have increasing returns. One particular problem involving increasing returns which has generated much recent research is the question of sustainability or supportability of natural monopoly.2 For this problem the existence of a core is of great interest, but at present, a set of necessary and sufficient conditions for a core to exist in a completely general model is not known. This paper is an attempt to advance the theory of natural monopoly, and of increasing returns industries in general, by considering a more general model of equilibrium than has previously been used. Thus, the paper is both a very general model of an essentially partial equilibrium problem and a rather specialized model of general economic equilibrium. In the next section, the basic model will be developed and its interpretation for both general and partial equilibrium will be discussed in greater detail.

The problem of the existence of a competitive equilibrium in models with hidden knowledge and self-knowledge has been discussed previously by M. Rothschild and J. E. Stiglitz_(1976), C. A. Wilson_(1977), and J. G. Riley_(1979). Recent analyses of such models by I. Cho and D. Kreps_(1986) and Riley argue for a particular outcome - the Pareto-dominant separating, zero-profit one. The authors prove the existence of such an outcome under very general conditions and, generalizing the reactive equilibrium concept introduced by Riley, they prove this outcome is the unique reactive equilibrium. Copyright 1987 by The Econometric Society.

Noncooperative strategic behaviors are studied in the Malinvaud-Dreze-de la Vallee Poussin decentralized planning procedure. We depart from the assumption of myopic behavior by assuming that every agent takes into account the effect over a given period of time [0, T] of his answers to the Center. One shows that, for T large, every Nash equilibrium of the ensuing game in intertemporal strategies approaches: (i) a competitive equilibrium in an exchange economy, and (ii) a Lindahl equilibrium in an economy with public goods. Thus, the Center loses any significant influence on the income distribution.

It is the purpose of this paper to show first that in a competitive system satisfying a generalized Wairas' law all principal minors of order less than n of the Jacobian determinant of the system of excess supply functions will ordinarily change sign somewhere on the domain of all positive prices whether demand is bounded or not. Conditions for uniqueness of equilibrium are proposed for the case of bounded demand which allow minors of the Jacobian to change signs. Unbounded demand was treated in Part I. Existence of equilibrium is proved under a generalized Walras law.

A number of recent works have addressed the problem of describing the allocation of resources in an economy where prices are fixed at a value that does not achieve equilibrium of supply and demand in the classical sense. In the context of a model developed by Dreze, the purpose of this paper is to describe more precisely the different states of the markets (excess supply or excess demand) that may occur near a competitive equilibrium. A general analytical picture is obtained, which associates with each state of the markets the region of the price domain where it prevails. This allows us to point out an important difficulty which arises in the local comparative statics of this class of model: local unicity is not warrantied, that is, there may exist price systems very close to some competitive price, for which all the fixed price allocations are far from the competitive allocation. Examples are shown in the macroeconomic model (two aggregate agents: households and firms, and three goods: money, output, and labor); necessary and sufficient conditions for local unicity are given in this context. They require that all commodities be Hicks-substitute for the consumer, which is another way of saying that both the marginal propensities for consumption and leisure are positive and smaller than one. On the other hand, assuming local unicity, one can look for the implications of the foregoing results for the long-run determination of prices. It is easy to show that, if the economy always reaches a fixed price allocation, an increase of the price of one commodity near the competitive equilibrium is always to the advantage of the sellers and to the disadvantage of the buyers. This suggests that the determination of prices should take the form of a struggle between buyers and sellers in each market.

This paper surveys some recent studies of economies where trading takes place sequentially over time, and where each agent makes decisions at every date in the light of his expectations about his future environment, which are functions of his information on the present and past states of the economy. The paper reviews particularly the issues raised by arbitrage in capital markets, by the consideration of money and banking activities, and by the introduction of production in temporary competitive equilibrium models. A thorough investigation of the logic of temporary equilibrium models with quantity rationing is also offered, as well as a quick review of the study of stochastic processes of temporary equilibria.

A general equilibrium model of markets for commodities and assets is considered. The author develops the notion of a price system that admits no arbitrage opportunity and demonstrates the fundamenta l role of this concept for the theory of the existence of a competiti ve equilibrium. A price system admits no arbitrage opportunity for a consumer if every bundle of desired commodities has a positive market value. The author proves that the existence of a price system that a dmits no arbitrage opportunity for all consumers is sufficient for th e existence of an equilibrium. Copyright 1987 by The Econometric Society.

This paper investigates the effect of costs of adjustment on the dynamic characteristics of intra-urban resource allocation. The analysis is concentrated on the development pattern of an open city within a system of many cities both in steady-state and in variable-state economies. Competitive equilibrium of the urban system as a whole is also discussed and compared to Pareto optimum allocation. Some of the results that characterize the resource allocation under static analysis are reestablished for the dynamic case as well. But others that follow from comparative statics are shown to be incorrect under the dynamic analysis.

IN AN EARLIER PAPER1 I derived conditions for the stability of equilibrium in monopolistic competition for two competitors. The extension of that analysis to cover more than two competitors is by no means obvious, and it is a matter of importance to know how the number of competitors affects the question of stability. It is therefore to the solution of the problem for n competitors that the present paper will be devoted. Although the economic problem will be limited to the question of monopolistic price competition, the methods employed can be used to test the stability of any equilibrium determined by the-solution of a system of linear equations. In Section I we shall formulate a demand function for n competitors in an imperfect market, and also their cost functions. In Section II we shall derive the conditions for the existence of equilibrium and in Section III we shall determine the conditions for its stability in the cases both of noncontinuous and continuous adjustment on the basis of a given set of expectations. In Section IV we shall consider the implications of our results for the general cases of two, three, and n competitors, while in Section V we solve the problem completely for n identical competitors. Finally in Section VI, we shall adumbrate the problems involved when the stability of the expectations themselves is brought into question.

In an atomless economy any allocation that is not blocked by "small" coalitions is in the core. Hence, in such an economy, a competitive equilibrium is characterized by the blocking power of part of the coalitions which excludes all "big" coalitions. THE CORE of an economy consists of all the allocations that are not blocked by any coalition. In this note we prove that for any positive number 8, the core of an atomless economy coincides with the set of allocations that are not blocked by any coalition of measure less than s. This result implies that even if the "large" coalitions cannot be formed, any unblocked allocation is still in equilibrium with respect to some price system. In particular, the formation of the coalition of all traders or any "large" coalition is not needed to insure the Pareto-optimality of final allocation. We prove here directly that the core is equal to the set of allocations that are not blocked by small coalitions. The same result can also be obtained by proving that Aumann's [1], Vind's [5], or Hildenbrand's [2] equivalence theorems hold with the additional restriction on the measure of the blocking coalitions. In any case the proof is a simple application of Liapunov's convexity theorem [3 and 4] (for the statement of the theorem see also [2, Appendix, p. 451]).

THE THEORY of the general competitive equilibrium, as developed by Leon Walras [64], has recently been reformulated in terms of fairly advanced mathematical methods. The first problem studied extensively was concerned with the conditions under which a competitive equilibrium exists for a model of general equilibrium. Among contributions to the existence problem, Wald [63], Arrow and Debreu [5], McKenzie [35], Nikaido [48], Gale [18], and Negishi [44] may be cited. The optimality problem of a competitive equilibrium was also investigated by, e.g., Arrow [2], Debreu [15, 16], and, in

It is well known, and easy to establish, that there exist markets that do not have competitive equilibria, provided the traders do not have convex preferences--that is, that the set of commodity bundles preferred or indifferent to a given bundle is not always convex. It is proved, nevertheless, that in a market consisting of a continuum of traders, each one individually insignificant, there is always a competitive equilibrium, even when the preferences are not convex. (Author)

It is suggested that the most natural mathematical model for a market with "perfect competition" is one in which there is a continuum of traders (like the continuum of points on a line). It is shown that the core of such a market coincides with the set of its "equilibrium allocations", i.e., allocations which constitute a competitive equilibrium when combined with an appropriate price structure.

THE PROBLEMS of distribution in an economic system may be analysed either by means of the behavioral assumptions of a competitive model or by the more flexible techniques of n person game theory. In the competitive model, consumers are assumed to respond to a set of prices by maximizing utility subject to a budget constraint and producers by maximizing profit. Consistent production decisions and an allocation of commodities are obtained by the determination of a set of prices at which all markets are in equilibrium. The analysis of these problems by means of n person game theory requires us to specify the production and distribution activities that are available to an arbitrary coalition of economic agents. It is frequently sufficient to summarize the detailed strategic possibilities open to a coalition by the set of possible utility vectors that can be achieved by the coalition. For example, in a pure exchange economy each coalition will have associated with it the collection of all utility vectors that can be obtained by arbitrary redistributions of the resources of that coalition. The core of an n person game is a generalization of Edgeworth's contract curve. A vector of utility levels is suggested which is feasible for all of the players acting collectively, and an arbitrary coalition is examined to see whether it can provide higher utility levels for all of its members. If this is possible, the utility vector which was originally suggested is said to be blocked by the coalition. The core of the n person game consists of those utility vectors which are feasible for the entire group of players and which can be blocked by no coalition. As we have seen during the last several years, there is an intimate connection between these two methods of analysis. If the conventional assumptions of the competitive model are made, such as convexity of preferences and convexity and constant returns to scale for the production set, then there will be a price system at which all markets are in equilibrium and a resulting assignment of commodity bundles to consumers. The utility vector associated with this competitive equilibrium may be shown to be in the core. Even further, if the number of consumers tends

This paper is a sequel to "On the Stability of the Competitive Equilibrium, I," by K. J. Arrow and L. Hurwicz. It extends the results of "I" in several directions. In particular, it provides a proof of stability in the large (and not merely locally) when all goods are gross substitutes; this result is found to be valid for processes where the price adjustment rate is a continuous sign-preserving, but not necessarily proportionate, function of excess demand. The paper deals both with systems where one of the commodities plays the role of numeraire and with systems where all commodities are treated symmetrically.