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     

Private Value Allocations in Large Economies with Differential Information

We consider a Radner-type (e.g., Radner, 1968, Econometrica36, 31–58) pure exchange economy with differential information and a continuum of agents. We show that under appropriate assumptions the set of Aumann–Shapley private value allocations in such an economy coincides with the set of Radner competitive equilibrium allocations. Journal of Economic Literature Classification Numbers: C71, D51, D82.     

The efficiency of competitive mechanisms under private information

We consider the efficiency properties of exchange economies where privately informed traders behave strategically. Specifically, a competitive mechanism is any mapping of traders’ reports about their types to an equilibrium price vector and allocation of the reported economy. In our model, some traders may have non-vanishing impact on prices and allocations regardless of the size of the economy. Although truthful reporting by all traders cannot be achieved, we show that, given any desired level of approximation, there is such that any Bayesian-Nash equilibrium of any competitive mechanism of any private information economy with or more traders leads, with high probability, to prices and allocations that are close to a competitive equilibrium of the true economy. In particular, allocations are approximately efficient. A key assumption is that there is small probability that traders behave non-strategically.     

Large economies with differential information and without free disposal

We consider exchange economies with a continuum of agents and differential information about finitely many states of nature. It was proved in Einy et al. (Econ Theory 18, 321–332, 2001) that if we allow for free disposal in the market clearing (feasibility) constraints then an irreducible economy has a competitive (or Walrasian expectations) equilibrium, and moreover, the set of competitive equilibrium allocations coincides with the private core. However when feasibility is defined with free disposal, competitive equilibrium allocations may not be incentive compatible and contracts may not be enforceable (see e.g. Glycopantis et al. in Econ Theory 21, 495–526, 2002). This is the main motivation for considering equilibrium solutions with exact feasibility. We first prove that the results in Einy et al. (Econ Theory 18, 321–332, 2001) are still valid without free-disposal. Then, motivated by the issue of contracts’ execution, we adapt the incentive compatibility property introduced in Krasa and Yannelis (Econometrica 62, 881–900, 1994) and we prove that every Pareto optimal exact feasible allocation is incentive compatible, implying that contracts of competitive or core allocations are enforceable. We would like to thank two anonymous reviewers and the Associate Editor for their valuable suggestions and remarks. This work was partially done while V.F. Martins-da-Rocha was visiting the Dipartimento di Matematica e Informatica of the Università degli Studi di Perugia. We thank the audience of the First General Equilibrium Workshop at Rio. Section 6 dealing with contract enforcement and coalitional incentive compatibility has benefited from discussions with J. Correia-da-Silva, W. Daher, F. Forges, C. Hervès-Beloso, E. Moreno-García, K. Podczeck, Y. Vailakis and N.C. Yannelis.     

OLIGOPOLY AND COST SHARING IN ECONOMIES WITH PUBLIC GOODS

We study economies that involve both small and large traders as well as the choice of a public project. Within this framework, we establish two sufficient conditions under which the set of competitive allocations coincides with the core. Our first core equivalence result holds under the assumption that there is a countably infinite set of large traders similar to each other. The second result, independent of the number of large traders, requires the existence of a coalition of small traders with the same characteristics of the large traders. Finally, we show how the generalized Aubin approach to cooperation may dispense with both conditions.     

Communication in financial markets with several informed traders

This paper investigates the incentives for informed traders in financial markets to reveal their information truthfully to the public. In the model, a subset of traders receive noisy signals about the value of a risky asset. The signals are composed of a directional component (“high” vs. “low”) as well as a precision component that represents the quality of the directional component. Between trading periods, the informed agents make public announcements to the uninformed traders. With a sufficiently large number of informed traders, an equilibrium exists in which the directional components are credibly revealed, but not the precision components. Even though the informed traders retain some of their rivate information, the post-communication estimate of the asset value converges in probability to the full-information estimate as the number of informed traders increases. The paper is based on a chapter of my Ph.D. thesis at the University of Western Ontario and was circulated previously under the title “Public Communication Devices in Financial Markets.” I thank my dissertation committee Arthur Robson, Hari Govindan, and Al Slivinski for their guidance and support. I also thank Murali Agastya, Roland Benabou, Philippe Grégoire, Rick Harbaugh, Mike Peters, an anonymous referee and an associate editor, and seminar participants at various universities and conferences at which this paper was presented.     

The bargaining set in strategic market games

This paper presents a hybrid equilibrium notion that blends together the ‘cooperative’ and the ‘noncooperative’ theories of competition. In particular, the Mas-Colell bargaining set has been modified in order to accommodate the features of strategic market games. In other words, allocations, objections and counter objections of the standard bargaining set theory are described for an economy, where trades among groups of individuals are conducted via the Shapley–Shubik mechanism. In the main part of the paper, it is proved that in atomless economies the allocations resulting from this equilibrium notion are competitive.     

Non-computability of competitive equilibrium

Summary. We provide a “computable counterexample” to the Arrow-Debreu competitive equilibrium existence theorem [2]. In particular, we find an exchange economy in which all components are (Turing) computable, but in which no competitive equilibrium is computable. This result can be interpreted as an impossibility result in both computability-bounded rationality (cf. Binmore [5], Richter and Wong [35]) and computational economics (cf. Scarf [39]). To prove the theorem, we establish a “computable counterexample” to Brouwer's Fixed Point Theorem (similar to Orevkov [32]) and a computable analogue of a characterization of excess demand functions (cf. Mas-Colell [26], Geanakoplos [16], Wong [50]). Received: September 9, 1997; revised version: December 17, 1997     

The Arrow-Debreu model: How math can hide a fatal conceptual error

I argue that math, like love, can cover a multitude of sins, and I use the neoclassical object of adoration, the Arrow-Debreu model, as the case in point. It is commonplace that the Arrow-Debreu (AD) model of general equilibrium does not describe the real world, but it is equally commonplace to accept it as representing the pure logic of the competitive capitalist economy in an idealized world free of transactions costs. I show that the AD model fails even as an idealized model; it actually mistakes the logic of pure capitalism. Unlike McKenzie’s model of idealized general equilibrium under constant returns to scale, Arrow and Debreu claim to have shown the existence of competitive equilibrium under decreasing returns to scale and positive pure profits. The AD model (again unlike the McKinzie model) needs to assign the profits to individuals and this is done using the notion of “ownership of the production set.” But this notion suffers from a fatal ambiguity. If Arrow and Debreu interpret it to mean “ownership of a corporation” then a simple argument in the form “labor can hire capital or capital can hire labor” defeats the alleged necessity of assigning residual claimancy to the corporation. A given corporation may or may not end up exploiting a set of production opportunities (represented by a production set) depending on whether it hires in labor and undertakes production or hires out its capital to others (all by assumption at the parametrically given prices). In the latter case, residual claimancy is elsewhere. There is no such property right as “ownership of a production set” in a private property market economy. The legal party which purchases or already owns all the inputs used up in production has the defensible legal claim on the outputs: there is no need to also “purchase the production set.” At any set of prices that allow positive pure profits, anyone in the idealized AD model could bid up the price of the inputs and thus try to reap a smaller but still positive profit. Therefore,pace Arrow and Debreu, there could be no equilibrium with positive pure profits. In the Appendix, the property rights fallacy that afflicts the AD model is shown to also afflict orthodox capital theory and corporate finance theory. World Bank The findings, interpretations and conclusions expressed in this paper are entirely those of the author and should not be attributed in any manner to the World Bank, to its affiliated organizations or to the members of its Board of Directors or the countries they represent.     

EDGAR on the internet: The welfare effects of wider information distribution in an experimental market for risky assets

Policies such as the SEC’s Fair Disclosure Rule, and technologies such as SEC EDGAR, aim to disseminate corporate disclosures to a wider audience of investors in risky assets. In this study, we adopt an experimental approach to measure whether this wider disclosure is beneficial to these investors. Price-clearing equilibrium models based on utility maximization and non-revealing and fully-revealing prices predict that in a pure exchange economy, an arbitrary trader would prefer that no investors are informed rather than all are informed; non-revealing theory further predicts that an arbitrary trader would prefer a situation in which all traders are informed rather than half the traders are informed. These predictions can be summarized as “None > All > Half”. A laboratory study was conducted to test these predictions. Where previous studies have largely focused on information dissemination and its effects on equilibrium price and insider profits, we focus instead on traders’ expected utility, as measured by their preferences for markets in which none, half, or all traders are informed. Our experimental result contradicts the prediction and indicates “Half > None > All”, i.e. subjects favor a situation where a random half is informed. The implication is that in addition to testing predictions of price equilibrium, experiments should also be used to verify analytical welfare predictions of expected utility under different policy choices. JEL Classification D82, D53, G14, L86 This work was largely completed while this author was at The Hong Kong University of Science and Technology.     

Retrading in market games

When agents are not price takers, they typically cannot obtain an efficient real location of resources in one round of trade. This paper presents a non-cooperative model of imperfect competition where agents can retrade allocations, consistent with Edgeworth's idea of recontracting. We show (a) there are Pareto optimal allocations, including competitive equilibrium allocations, that can be approximated arbitrarily closely when trade is myopic, i.e., when agents play a static Nash equilibrium at every round of retrading; (b) any converging sequence of allocations generated by myopic retrading can be supported along some retrade-proof subgame perfect equilibrium path when traders anticipate future rounds of trading.     

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 equilibrium of the bargaining game and core convergence theorem on an exchange economy with limited traders

Debreu and Scarf (1963), Hildenbrand (1974), Aumann (1964), Dierker (1975), Bewley (1973), and others have shown that the core of an exchange economy with infinitely many or finitely many traders converges. However, an exchange economy does not always consist of infinitely many or finitely many traders. This note provides proof of the core convergence theorem on an exchange economy with limited traders by a bargaining game methodology. The main contribution of this note is to innovate the equilibrium solution to the bargaining game in the exchange economy. In this note, the concept of common payoff is introduced; in the bargaining game of a coalition on its common payoff, all coalition members will get the same distribution, thus the distribution scheme of the cooperation surplus of the exchange economy is determined. This note shows that the bargaining game among the traders on the distribution of the cooperation surplus will make the pure exchange economy with limited traders converge to the Walrasian equilibrium, all the allocations other than the Walrasian equilibrium will be eliminated from the core of this economy.     

Growth in economies with non convexities: sunspots and lottery equilibria

Summary. We investigate the relation between lotteries and sunspot allocations in a dynamic economy where the utility functions are not concave. In an intertemporal competitive economy, the household consumption set is identified with the set of lotteries, while in the intertemporal sunspot economy it is the set of measurable allocations in the given probability space of sunspots. Sunspot intertemporal equilibria whenever they exist are efficient, independently of the sunspot space specification. If feasibility is, at each point in time, a restriction over the average value of the lotteries, competitive equilibrium prices are linear in basic commodities and intertemporal sunspot and competitive equilibria are equivalent. Two models have this feature: Large economies and economies with semi-linear technologies. We provide examples showing that in general, intertemporal competitive equilibrium prices are non-linear in basic commodities and, hence, intertemporal sunspot equilibria do not exist. The competitive static equilibrium allocations are stationary, intertemporal equilibrium allocations, but the static sunspot equilibria need not to be stationary, intertemporal sunspot equilibria. We construct examples of non-convex economies with indeterminate and Pareto ranked static sunspot equilibrium allocations associated to distinct specifications of the sunspot probability space.Received: 25 August 2003, Revised: 16 March 2004, JEL Classification Numbers: D84, D90.Correspondence to: Paolo SiconolfiWe thank Herakles Polemarchakis for helpful conversations on the topic. The research of Aldo Rustichini was supported by the NSF grant NSF/SES-0136556.     

Fair welfare maximization

We consider the general problem of finding fair constrained resource allocations. As a criterion for fairness we propose an inequality index, termed “fairness ratio,” the maximization of which produces Lorenz-undominated, Pareto-optimal allocations. The fairness ratio does not depend on the choice of any particular social welfare function, and hence it can be used for an a priori evaluation of any given feasible resource allocation. The fairness ratio for an allocation provides a bound on the discrepancy between this allocation and any other feasible allocation with respect to a large class of social welfare functions. We provide a simple representation of the fairness ratio as well as a general method that can be used to directly determine optimal fair allocations. For general convex environments, we provide a fundamental lower bound for the optimal fairness ratio and show that as the population size increases, the optimal fairness ratio decreases at most logarithmically in what we call the “inhomogeneity” of the problem. Our method yields a unique and “balanced” fair optimum for an important class of problems with linear budget constraints.     

Evolutionary choice of markets

We consider an economy where a finite set of agents can trade on one of two asset markets. Due to endogenous participation the markets may differ in the liquidity they provide. Traders have idiosyncratic preferences for the markets, e.g.due to differential time preferences for maturity dates of futures contracts. For a broad range of parameters we find that no trade, trade on both markets (individualization) as well as trade on one market only (standardization) is supported by a Nash equilibrium. By contrast, whenever the number of traders becomes large, the evolutionary process selects a unique stochastically stable state which corresponds to the equilibrium with two active markets and coincides with the welfare maximizing market structure. We are grateful to Thorsten Hens, Fernando Vega-Redondo and a referee for valuable comments. We also thank seminar participants at the University of Zurich, the CES research seminar at the University of Munich, the Koc University in Istanbul as well as conference participants at the SAET conference in Ischia, the ESEM in Lausanne and the ESF workshop on Behavioural Models in Economics and Finance in Vienna. A first version of the paper was written while Marc Oliver Bettzüge was visiting the Institute for Empirical Research in Economics at the University of Zurich. Financial Support by the Swiss Banking Institute and by the National Centre of Competence in Research “Financial Valuation and Risk Management” (NCCR FINRISK) is gratefully acknowledged. The NCCR FINRISK is a research program supported by the Swiss National Science Foundation.     

The evolution of portfolio rules and the capital asset pricing model

The aim of this paper is to test the performance of capital asset pricing model (CAPM) in an evolutionary framework. We model an economy where a heterogeneous population of long-lived agents invest their wealth according to different portfolio rules, and prove that traders who either “believe” in CAPM and use it as a rule of thumb, or are endowed with genuine mean-variance preferences, under some very weak conditions, vanish in the long run.We show that a sufficient condition to drive CAPM or mean-variance traders’ wealth shares to zero is that an investor endowed with a logarithmic utility function enters the market.     

Complete and incomplete markets with short-sale constraints

Summary. This paper argues that the introduction of a short-sale constraint in the Arrow-Radner framework invalidates standard definitions of complete and incomplete markets. Two threshold values with familiar properties arise in this constrained set-up. If short sales are not allowed on some security, then financial markets will be incomplete in the standard sense. Beyond a particular level of the short-sale bound, financial markets are “complete”, since the short-sale constraint is not effective. For intermediate bounds the distinction between complete and incomplete financial markets is blurred. Although some technical definitions hold, agents can not fully transfer wealth among states. These intermediate cases, called “technically incomplete markets”, exhibit interesting welfare properties. For instance, the resulting equilibrium allocations may not be Pareto-dominated by those of the non-restricted complete markets equilibrium. Received: November 28, 2000; / revised version: November 9, 2001     

Multiple solutions under quasi-exponential discounting

Infinite horizon dynamic optimization problems with non-exponential time preferences may not only exhibit time inconsistency but may also have multiple solutions with distinct payoffs. We here show that such multiplicity is generic in the sense that it occurs in an open set of such decision problems, even with small state- and action-spaces. Non-exponential discounting allows for an “addictive” equilibrium alongside a “virtuous” equilibrium. We also provide a sufficient condition for uniqueness in infinitely repeated decision problems with general action spaces. Authors thank Philippe Jehiel, Maria Saez-Marti and Bill Sandholm for comments to an earlier version, and the Knut and Alice Wallenberg Research Foundation for financial support.     

Asset price volatility in a nonconvex general equilibrium model

Summary. Asset prices and returns are known to vary significantly more than␣output or aggregate consumption growth, and an order of magnitude in excess of what is justified by innovations to fundamentals. We study excess price volatility in a lifecycle economy with two assets (claims on capital and␣a public debt bubble), heterogeneous agents, and increasing returns to financial intermediation. We show that a relatively modest nonconvexity generates a set valued equilibrium correspondence in asset prices, with two␣stable branches. Price volatility is the outcome of an equilibrium selection mechanism, which mixes adaptive learning with “noise”, and alternates stochastically between the two stable branches of the price correspondence. Received: March 19, 1998; revised version: June 2, 1998