Competitive and core allocations in large economies with differential information

Summary. We study the core and competitive allocations in exchange economies with a continuum of traders and differential information. We show that if the economy is “irreducible”, then a competitive equilibrium, in the sense of Radner (1968, 1982), exists. Moreover, the set of competitive equilibrium allocations coincides with the “private core” (Yannelis, 1991). We also show that the “weak fine core” of an economy coincides with the set of competitive allocations of an associated symmetric information economy in which the traders information is the joint information of all the traders in the original economy. Received March 22, 2000; revised version: May 1, 2000     

Efficiency, monotonicity and rationality in public goods economies

Summary. In economies with public goods, we identify a necessary and sufficient condition for the existence of cost monotonic, Pareto optimal and individually rational mechanisms. These exist if and only if the preferences of the agents satisfy what we call the equal ordering property. We also show that when this condition holds the egalitarian equivalent correspondence is the only cost monotonic selection from the core of the economy. Furthermore, it is unambiguous in the sense that the agents are indifferent among all the allocations in it. Received: February 26, 1996; revised version: January 31, 1997     

Characterization and incentive compatibility of Walrasian expectations equilibrium in infinite dimensional commodity spaces

Summary. We consider a differential information economy with infinitely many commodities and analyze the veto power of the grand coalition with respect the ability of blocking non-Walrasian expectations equilibrium allocations. We provide two different Walrasian expectations equilibrium equivalence results. First by perturbing the initial endowments in a precise direction we show that an allocation is a Walrasian expectations equilibrium if and only if it is not privately dominated by the grand coalition. The second characterization deals with the fuzzy veto in the sense of Aubin but within a differential information setting. This second equivalence result provides a different characterization for the Walrasian expectations equilibrium and shows that the grand coalition privately blocks in the sense of Aubin any non Walrasian expectations equilibrium allocation with endowment participation rate arbitrarily close to the total initial endowment participation for every individual. Finally, we show that any no free disposal Walrasian expectations equilibria is coalitional Bayesian incentive compatible. Since the deterministic Arrow-Debreu-McKenzie model is a special case of the differential information economy model, one derives new characterizations of the Walrasian equilibria in economies with infinitely many commodities.Received: 29 October 2003, Revised: 24 February 2004, JEL Classification Numbers: D51, D82, D11. Correspondence to: Emma Moreno-GarcíaThe authors are grateful to an anonymous referee for his/her careful reading and helpful comments and suggestions.C. Hervés and E. Moreno acknowledge support by Research Grant BEC2000-1388-C04-01 (Ministerio de Ciencia y Tecnología and FEDER); and support by the Research Grant SA091/02 (Junta de Castilla y León).     

The bargaining set of a large economy with differential information

Summary. We study the Mas-Colell bargaining set of an exchange economy with differential information and a continuum of traders. We established the equivalence of the private bargaining set and the set of Radner competitive equilibrium allocations. As for the weak fine bargaining set, we show that it contains the set of competitive equilibrium allocations of an associated symmetric information economy in which each trader has the “joint information” of all the traders in the original economy, but unlike the weak fine core and the set of fine value allocations, it may also contain allocations which are not competitive in the associated economy. Received: February 15, 1999; revised version: August 9, 1999     

Continuous choice functions and the strong axiom of revealed preference

Summary In this note, it is shown that a result due to Uzawa showing that the weak axiom of revealed preference together with a regularity condition implies the strong axiom of revealed preference can be derived from the regularity conditionalone, if the choice function is continuous in prices and in income. The observation that the weak axiom of revealed preference is not needed suggests that this regularity condition is remarkably strong. The main result of the paper is quite general in the sense that the only additional assumption on the choice function is that the whole budget be spent.I thank Charles Blackorby for his comments on an earlier version.     

Egalitarian equivalence and strategyproofness in the queueing problem

We investigate the implications of egalitarian equivalence (Pazner and Schmeidler in Q J Econ 92:671–687, 1978) together with queue efficiency and strategyproofness in the context of queueing problems. We completely characterize the class of mechanisms satisfying the three requirements. Though there is no mechanism in this class satisfying budget balance, feasible mechanisms exist and we characterize the set of all such mechanisms. We also show that it is impossible to find a mechanism satisfying queue efficiency, egalitarian equivalence, and a stronger notion of strategyproofness called weak group strategyproofness. In addition, we show that generically there is no mechanism satisfying both egalitarian equivalence and no-envy.     

Edgeworth's conjecture with infinitely many commodities: commodity differentiation

Summary. Convergence of the cores of finite economies to the set of Walrasian allocations as the number of agents grows has long been taken as one of the basic tests of perfect competition. The present paper examines this test in the most natural model of commodity differentiation: the commodity space is the space of nonnegative measures, endowed with the topology of weak convergence. In Anderson and Zame [12], we gave counterexamples to core convergence in L ¹, a space in which core convergence holds for replica economies and core equivalence holds for continuum economies; in addition, we gave a core convergence theorem under the assumption that traders' utility functions exhibit uniformly vanishing marginal utility at infinity. In this paper, we provide two core convergence results for the commodity differentiation model. A key technical virtue of this space is that relatively large sets (in particular, closed norm-bounded sets) are compact. This permits us to invoke a version of the Shapley-Folkman Theorem for compact subsets of an infinite-dimensional space. We show that, for sufficiently large economies in which endowments come from a norm bounded set, preferences satisfy an equidesirability condition, and either (i) preferences exhibit uniformly bounded marginal rates of substitution or (ii) endowments come from an order-bounded set, core allocations can be approximately decentralized by prices. Received: July 29, 1996; revised version: January 14, 1997     

Finite allocation mechanisms: approximate Walrasian versus approximate Direct Revelation

Summary. This paper obtains finite counterparts of previous results that showed the informational efficiency of the Walrasian mechanism among all mechanisms yielding Pareto-optimal individually rational trades in exchange economies while using a continuum of possible messages. In particular, we develop finite counterparts of the superiority, with respect to message-space dimension, of the Walrasian mechanism over Direct Revelation (DR). We measure a finite mechanism's cost by the number of its (equilibrium) messages. Our two main results are as follows: (1) For exchange economies we find that the overall (maximum) error of a (sufficiently fine) approximate Walrasian mechanism is less than the overall error of a not-more-costly approximate DR mechanism whose equilibrium outcomes are trades that are (approximately) Pareto optimal and individually rational; more generally, approximate Walrasian mechanisms are superior, in the same sense, to approximations of any continuum mechanism whose outcomes are Pareto optimal individ ually rational trades and whose message space has higher dimension than that of the Walrasian mechanism. (2) As we increase without limit the dimension of the set of environments (characteristics) defining our class of exchange economies, the extra cost of DR approximations relative to Walrasian approximations, when both achieve the same overall error, also grows without limit. Thus the informational superiority of the Walrasian mechanism emerges again when we approximate it and take the finite number of messages in the approximation as our cost measure. Received: June 16, 2002; revised version: July 22, 2002 RID="*" ID="*" The second author is grateful for support from National Science Foundation grant #IIS-0118600. Correspondence to: T. Marschak     

The measure of blocking coalitions in differential information economies

We provide an evaluation of the measure of privately blocking coalitions in differential information economies. In the case of atomless economies, it is proved that for a Pareto optimal allocation that is not a Walrasian expectations equilibrium, to any symmetric profile there corresponds a ball such that "almost half” of the profiles it contains are privately blocking. Analogous results are proved in the case of finite differential information economies for generalized coalitions and social coalition structures. From a different point of view, the paper can be considered as a contribution showing private core equivalence theorems under restrictions on coalition formation. We thank an anonymous referee for observations and comments improving an earlier version of the present paper.     

Cournot-Nash and Lindahl equilibria in pure public “bad” economies

Summary. Using a general equilibrium framework, this paper analyzes the equilibrium provision of a pure public bad commodity (for example pollution). Considering a finite economy with one desired private good and one pure public “bad” we explicitly introduce the concept of Lindahl equilibrium and the Lindahl prices into a pure public bad economy. Then, the Lindahl provision is analyzed and compared with the Cournot-Nash provision. The main results for economies with heterogeneous agents state that the asymptotic Lindahl allocation of the pure public bad is the null allocation. In contrast, the asymptotic Cournot-Nash provision of the public bad might approach infinity. Other results were obtained in concert with the broad analysis of the large finite economies with pure public bad commodities. Received: July 26, 2001; revised version: March 12, 2002 RID="*" ID="*" We are indebt to Nicholas Yannelis and anonymous referee for their valuable comments and suggestions. Correspondence to: B. Shitovitz     

Bargaining, coalitions and competition

We study a Gale-like matching model in a large exchange economy, in which trade takes place through non-cooperative bargaining in coalitions of finite size. Under essentially the same conditions of core equivalence, we show that the strategic equilibrium outcomes of our model coincide with the Walrasian allocations of the economy. Our method of proof makes use of the theory of the core. With respect to previous work, our positive implementation result applies to a substantially larger class of economies: the model relaxes differentiability and convexity of preferences, and also admits an arbitrary number of divisible and indivisible goods.     

FAIR ALLOCATION OF DISPUTED PROPERTIES

We model problems of allocating disputed properties as generalized exchange economies. Therein, agents have preferences and claims over multiple goods, and the social endowment of each good may not be sufficient to satisfy all individual claims. We focus on market‐based allocation rules that impose a two‐step procedure: assignment of rights based on claims first and voluntary exchange based on the assigned rights afterward. We characterize three focal egalitarian rights‐assignment rules that guarantee that the allocation rules are fair. We apply our results to problems of greenhouse gas emissions and contested water rights.     

Core equivalence with congested public goods

Summary This paper examines a model of an infinite production economy with a finite number of types of agents andsemi- public goods, which are subjected to crowding and exclusion. The utility of an agent depends not only on the vector of public commodities produced by the coalition to which she belongs, but also on the mass of agents of her type who are the members of this coalition. The main purpose of the paper is to derive necessary and sufficient conditions on the local degrees of congestion which would guarantee the equivalence between the core and the set of equal treatment Lindahl equilibria. We prove that this equivalence holdsif and only if there are constant returns to group size for each type of agents. It implies that linearity of each agent's congestion function with respect to the mass of the agents of her own type is necessary for the core equivalence to hold.The final version of this paper was written while Shlomo Weber was visiting the Technical University of Dresden as the Fellow of the Alexander von Humboldt Foundation, whose support is gratefully acknowledged. The authors are grateful to Peter Meyer, Nicholas Yannelis and the anonymous referees for useful remarks and suggestions.     

Asset trading volume in a production economy

Judd et al. (J Finance 63: 2203–2217, 2003) show that the stationary Lucas tree model cannot generate nontrivial asset trading: Heterogenous agents will optimally choose a fixed portfolio after initial rebalancing. This paper explores asset trading volume in production economies with heterogeneous agents and dynamically complete market structures. We establish a recursive version of the Negishi approach to prove the existence of a competitive equilibrium. Furthermore, we develop a general method to solve for equilibrium portfolios in production economies within a fairly general set of complete market structures. We thus establish the theoretical reasons why production economies in general generate a nontrivial volume of asset trading even if heterogeneity of the agents is kept to a minimum. We would like to thank W. Brock, D. DeJong and, especially, H. Ennis for comments and suggestions. We also thank seminar participants at Di Tella and San Andrés Universities (Argentina), the Institute for Advanced Studies (Austria), SED Meetings 2005 (Budapest) and SAET Conference 2005 (Vigo).     

Core and Walrasian equilibria when agents' characteristics are extremely dispersed

Summary. It is shown that core-Walras equivalence fails whenever the commodity space is a non-separable Banach space. The interpretation is that a large number of agents guarantees core-Walras equivalence only if there is actually a large number of agents relative to the size of the commodity space. Otherwise a large number of agents means that agents' characteristics may be extremely dispersed, so that the standard theory of perfect competition fails. Supplementing the core-Walras non-equivalence result, it is shown that in the framework of economies with weakly compact consumption sets – as developed by Khan and Yannelis (1991) – the core is always non-empty, even if consumption sets are non-separable. December 12, 2001; revised version: December 6, 2002 RID="*" ID="*" Thanks to E. Dierker, M. Nermuth, R. Tourky, and N. C. Yannelis for helpful discussions and suggestions, and thanks to a referee for comments which helped to improve the final version.     

Existence and optimality of oligopoly equilibria in linear exchange economies

Summary. We consider a linear exchange economy and its successive replicas. We study the notion of Cournot-Walras equilibrium in which the consumers use the quantities of commodities put on the market as strategic variables. We prove that, generically, if the number of replications is large enough but finite, the competitive behaviour is an oligopoly equilibrium. Then, under a mild condition, which may be interpreted in terms of market regulation and/or market activity, we show that any sequence of oligopoly equilibria of successive replica economies converges to the Walrasian outcome and furthermore that every oligopoly equilibrium of large, but finite, replica is Pareto optimal. Consequently, under the same assumptions on the fundamentals of the economy, one has an asymptotic result on the convergence of oligopoly equilibria to the Walras equilibrium together with a generic existence result for the Cournot-Walras. Received: June 20, 2002; revised version: November 20, 2002 RID="*" ID="*" Part of this paper was written while the second author was visiting the Universidad de Vigo. The support of the department of mathematics is gratefully acknowledged. Correspondence to: J.M. Bonnisseau     

Lotteries, Sunspots, and Incentive Constraints

We study a prototypical class of exchange economies with private information and indivisibilities. We establish an equivalence between lottery equilibria and sunspot equilibria and show that the welfare and existence theorems hold. To establish these results, we introduce the concept of the stand-in consumer economy, which is a standard, convex, finite consumer, finite good, pure exchange economy. With decreasing absolute risk aversion and no indivisibilities, we prove that no lotteries are actually used in equilibrium. We provide a simple numerical example with increasing absolute risk aversion in which lotteries are necessarily used in equilibrium. We also show how the equilibrium allocation in this example can be implemented in a sunspot equilibrium. Journal of Economic Literature Classification Numbers: D11, D50, D82.     

An ascending double auction

We show why the failure of the affiliation assumption prevents the double auction from achieving efficient outcomes when values are interdependent. This motivates the study of an ascending price version of the double auction. It is shown that when there is a sufficiently large, but still finite, number of sellers, this mechanism has an approximate perfect Bayesian equilibrium in which traders continue bidding if and only if their true estimates of the ‘value’ of the object being traded exceed the current price. This equilibrium is ex post efficient and has a rational expectations property in the sense that along the equilibrium path traders appear to have made the best possible trades conditional on information revealed by the trading process. We thank two anonymous referees and Dan Kovenock, the Editor, whose detailed comments and suggestions have allowed us to substantially improve the paper. We also thank seminar participants at University of Toronto, University of Wisconsin-Madison, Summer 2003 North American Meetings of the Econometric Society, 2004 NSF Decentralization Conference for their comments.     

Endogenous market segmentation with heterogeneous agents

Summary. An explanation is provided for the evolution of segmented marketplaces in a pairwise exchange economy. Large traders operating in a pairwise exchange market prefer to meet other similar traders, because this enables them to trade their endowments in a smaller number of encounters. Large and small traders, however, cannot be distinguished a priori, and the existence of the small traders imposes a negative externality on the large traders. We show that, under conditions which are not very restrictive, establishing a separate market (perhaps with an entry fee) designated for the large traders induces the two types of traders to segment themselves. However, this segmentation is not necessarily welfare improving. Received: January 12, 2001; revised version: July 17, 2002 RID="*" ID="*" I wish to thank the participants in the Friday Theory Workshop at the University of Sydney, and the participants at the 17th Australian Theory Workshop at the University of Melbourne for comments and discussion. John Hillas and Stephen King pointed out an omission in an earlier version, and Catherine de Fontenay and Hodaka Morita made extensive comments on earlier drafts. This work was initiated while I was a short-term visitor at the University of Southern California.     

Left-side Relatively Weak Increases in Risk and Their Comparative Statics

This paper proposes a new concept, a left-side relatively weak increase in risk (L-RWIR) order, that extends the definition of a relatively weak increase in risk (RWIR) order. We show that, for the class of linear payoffs, one can obtain an appealing comparative statics result for L-RWIR shifts imposing additional restrictions on risk preferences of a risk-averse decision maker.JEL classification: D81.revised version received October 10, 2003Acknowledgements The authors would like to thank an anonymous referee for insightful comments and useful suggestions.