Comparing finite mechanisms

Summary. This paper obtains finite analogues to propositions that a previous literature obtained about the informational efficiency of mechanisms whose possible messages form a continuum. Upon reaching an equilibrium message, to which all persons “agree”, a mechanism obtains an action appropriate to the organization's environment. Each person's privately observed characteristic (a part of the organization's environment) enters her agreement rule. An example is the Walrasian mechanism in an exchange economy. There a message specifies a proposed trade vector for each trader as well as a price for each non-numeraire commodity. A trader agrees if the price of each non-numeraire commodity equals her marginal utility for that commodity (at the proposed trades) divided by her marginal utility for the numeraire. At an equilibrium message, the mechanism's action consists of the trades specified in that message, and (for classic economies) those trades are Pareto-optimal and individually rational. Even though the space of environments (characteristics) is a continuum, mechanisms with a continuum of possible messages are unrealistic, since transmitting every point of a continuum is impossible. In reality, messages have to be rounded off and the number of possible messages has to be finite. Moreover, reaching a continuum mechanism's equilibrium message typically requires infinite time and that difficulty is absent if the number of possible messages is finite. The question therefore arises whether results about continuum mechanisms have finite counterparts. If we measure a continuum mechanism's communication cost by its message-space dimension, then our corresponding cost measure for a finite mechanism is the (finite) number of possible equilibrium messages. We find that if two continuum mechanisms yield the same action but the first has higher message-space dimension, then a sufficiently fine finite approximation of the first has larger error than an approximation of the second if the cost of the first approximation is no higher than the cost of the second approximation. An approximation's “error” is the largest distance between the continuum mechanism's action and the approximation's action. We obtain bounds on error. We also study the performance of Direct Revelation (DR) mechanisms relative to “indirect” mechanisms, both yielding the same action, when the environment set grows. We find that as the environment-set dimension goes to infinity, so does the extra cost of the DR approximation, if the error of the DR approximation is at least as small as the error of the indirect approximation. While the paper deals with information-processing costs and not incentives, it is related to the incentive literature, since the Revelation Principle is central to much of that literature and one of our main results is the informational inefficiency of finite Direct Revelation mechanisms. Received: May 21, 2001; revised version: December 14, 2001 RID="*" ID="*" Earlier versions of this paper were presented at the Decentralization Conference, Washington University, St Louis, April 2000 and at the Eighth World Congress of the Econometric Society, August 2000. We are grateful for comments received on those occasions. The second author gratefully acknowledges support from National Science Foundation grant #IIS9712131. Correspondence to: T. Marschak     

Simple Strategy-Proof Approximately Walrasian Mechanisms

This note provides an indirect analysis of the incentive properties of the Walrasian mechanism. It presents mechanisms under which truth-telling is a dominant strategy in finite exchange economies (in contrast to the Walrasian mechanism) and whose outcomes (generically) approximate Walrasian ones for large economies. These mechanisms provide new insights on the well-know trade-off between efficiency and incentive compatibility in finite economies. Journal of Economic Literature Classification Numbers: D50, D71, D78.     

A Unique Informationally Efficient Allocation Mechanism In Economies With Consumption Externalities*

This article investigates the informational requirements of resource allocation processes in pure exchange economies with consumption externalities. It is shown that the distributive Lindahl mechanism has a minimal informational size of the message space, and thus it is informationally the most efficient allocation process that is informationally decentralized and realizes Pareto‐efficient allocations over the class of economies that include nonmalevolent economies. Furthermore, it is shown that the distributive Lindahl mechanism is the unique informationally efficient decentralized mechanism that realizes Pareto‐efficient and individually rational allocations over a certain class of nonmalevolent economies.     

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     

Discrete Walrasian exchange process

In an exchange economy, we define a discrete exchange process, which is Walrasian, since the trades are determined by the equilibrium allocation of the local equilibrium. We prove that this process attains a Pareto optimal allocation after a finite number of steps and the local equilibrium price then supports the Pareto optimal allocation. Furthermore, along the process, the allocation remains feasible and the utility of each consumer is non-decreasing.     

Characterizing Natural Implementability: The Fair and Walrasian Correspondences

We characterize natural implementability in exchange economies when a social choice correspondence possibly recommends a Pareto efficient allocation where the common marginal rate of substitution is not determined uniquely. We find that the no-envy and efficient (or fair) correspondence cannot be implemented by any natural mechanism with a price and a quantity announcement, but can be implemented by a natural mechanism with a price and two-quantity announcement. On the other hand, the constrained Walrasian correspondence can be implemented by a natural price-quantity mechanism with budget balance. Journal of Economic Literature Classification Numbers: C72, D51, D63, D78.     

Rate-of-return dominance and efficiency in an experimental economy

Summary. One of the main challenges for monetary economics is to explain the use of assets that are dominated in rate-of-return as media of exchange. We use experimental methods to study how a fiat money might come to be used in transactions when an identically marketable, dividend-bearing asset, a consol, is also available. Our experimental economies, which have an overlapping generations structure, have the property that the only stationary rational expectations equilibria (SREE) require exclusive use of the consol as the medium of exchange. In a baseline treatment, agents use the consol exclusively, as would occur in an SREE. However, in other treatments, we observe episodes of rate-of-return dominance,with consistent use of fiat money as a medium of exchange. The results show that two properties of our economies are associated with the rate of return dominance anomaly. The first is a history of trading with fiat money, prior to the introduction of the consol. The second is the timing of the dividend payment; when the dividend payment follows the execution of trades between generations, hoarding of the consol occurs on the part of the old, who earn dividends by hoarding. In our economies, settling transactions with a dividend-bearing asset does not improve allocations over those resulting from trading with fiat money. Received: July 11, 2002; revised version: July 25, 2002 RID="*" ID="*"We thank Anne Villamil, participants in the 2000 Purdue University Conference on Monetary Economics, the Summer 2000 meetings of the Economic Science Association, and a referee, for very helpful comments. We thank the Krannert School of Management and the Purdue University Center for International Business, Education and Research for financial support and Vivian Lei for research assistance. We also thank Ron Michener for referring us to the historical account of the early introduction of money into the American colonies, as reported by Benjamin Franklin. Correspondence to: G. Camera     

The notion of private information in a modern perspective: a reappraisal of Hayek's contribution

Hayek's critical attitude towards Walrasian modelling was based on informational considerations. In his view, a meaningful notion of equilbrium has to deal with the consistency of agents' plans which information is dispersed throughout the economy. He emphasized that only in this context could the role of market prices as aggregators of information be correctly analysed. Recent developments in general equilbrium theory with rational expectations have taken up the issue formally. In this paper, Haek's main ideas about the competiitive mechanism are compared with these recent results. It is argued that Hayek's notion of private information is different from that used in modern equilibrium theory, not only because of its dynamic content, as many critics observed ex post, as is usually in a general equilibrium framework. This issue of whether modern notions of private information, namel that used in conract theory, are good substitutes for Hayek' notion of personal knowledge is also examined in detail. From this analysis, it is possible to conclude that Hayek's notion of equilibrium neither entails Pareto optimality prperties nor full informational efficiency.     

On Nash-Implementation in the Presence of Withholding

In this paper we construct a completely feasible and continuous mechanism whose Nash allocations coincide with Lindahl allocations when both preferences and initial endowments are unknown to the designer and unreported endowments are withheld. This mechanism extends the mechanism of Hong by allowing for semi-positivity of endowments for private goods economies and the mechanism of Tian by allowing for any number of private goods for public goods economies. Thus our mechanism improves all the existing mechanisms that implement Walrasian or Lindahl allocations. Journal of Economic Literature Classification Numbers: C72, D6l, D78, H41.     

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.     

A limit theorem for an approximate core of a large but finite economy

We present a formulation of an approximate core that can be sustained as an approximate equilibrium for a ‘large enough’ finite exchange economy whose traders need not have transitive, continuous or convex preferences.     

Evolution of Walrasian equilibrium in an exchange economy

We study the stochastic stability of a dynamic trading process in an exchange economy. We use a simplified version of a trading model à la Shapley and Shubik (J Polit Econ 85:937–968, 1977). Two types of agents equipped with Leontief preferences trade goods in markets by offering endowments, and actual trades occur at market clearing prices. Better behavior tends to spread through the same type of agents by imitation, and agents also make mistakes occasionally. We provide a sufficient condition for the perturbed dynamic process to have a unique stochastically stable state that is a Walrasian equilibrium allocation. In this sense, we give a rationale for Walrasian behavior.     

Search cost,trading strategies and optimal market structure

The theory of Walrasian equilibrium yields a set of prices at which the aggregate competitive demand for each commodity equals its aggregate competitive supply. However, even at equilibrium prices the theory of competitive equilibrium does not explicitly offer explanation regarding the manner in which trades are actually executed. This paper considers a model where trade takes place in a decentralized fashion and examines in a dynamic game-theoretic framework, the role of social institution of money and markets in facilitating exchange. The steady state Nash equilibrium derived in the paper demonstrates how, depending on the level of transaction costs associated with a market setup (synonymously, trading posts to exchange possible pairs of goods) appropriate monetary trade emerges, which like a hub and spoke network (Starr and Stinchcombe, 1999) makes some markets non-functioning and in equilibrium only the markets having trade through the medium of exchange continue to exist. However, despite the obvious advantages of a market setup in reducing search costs, pure random search for a complementary trading partner (as considered by Ostroy and Starr, 1974; Kiyotaki and Wright , 1989; and others) prevails in many economies, especially, in many developing economies. This paper models this feature of developing economies by introducing differences in transaction costs across agents and shows why sustainable equilibria might exist exhibiting random search for certain commodities even in the presence of established markets.     

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     

Theorems on correspondences and stability of the core

In this note two theorems strengthening Grodal's (1971) Theorem on correspondences are proved. The first drops the convexity assumption. The second strengthens that theorem further for the case when the range is the positive orthant. In this case, the conclusion of Grodal's Theorem - the intersection of the integral with the interior of the range being open- is modified to read as the integral being a relative open subset of the positive orthant. An example is provided to show that, such a strengthening is not valid in general. This allows us to dispense with the requirment of convexity of preferences in Grodal's (1971) theorems on the closedness of the set of Pareto optimal allocations, the core, and the continuity of the core correspondence for pure exchange economies. We apply this result to show that blocking coalitions in a large economy are stable. Received: September 30, 1998; revised version: September 18, 2001 RID="*" ID="*" The author is grateful to an anonymous referee for helpful comments. The usual disclaimer applies.     

The existence of fully rational expectations approximate equilibria with noisy price observations

Nonrevealing fully rational expectations approximate equilibria exist in microeconomic pure exchange economies in which uninformed agents have suitably dispersed noisy price observations. Such traders maximize a state-dependent expected utility conditional on the price vector they observe, the distributions of noisy price observations, and the correct equilibrium relationship between states of the world and prices. In equilibrium, aggregate excess demand is small with high probability in every state of the world (and its expectation is also small); this magnitude diminishes as the noisy price observations become more accurate. Equilibria are obtained by applying a fixed point argument to state-dependent excess demand functions which are smooth because of the noisy price observations.     

Nominal uniqueness and money non-neutrality in the limit-price exchange process

We define continuous-time dynamics for exchange economies with fiat money. Traders have locally rational expectations, face a cash-in-advance constraint, and continuously adjust their short-run dominant strategy in a monetary strategic market game involving a double-auction with limit-price orders. Money has a positive value except on optimal rest-points where it becomes a ??veil?? and trade vanishes. Typically, there is a piecewise globally unique trade-and-price curve both in real and in nominal variables. Money is not neutral, either in the short-run or long-run and a localized version of the quantity theory of money holds in the short-run. An optimal money growth rate is derived, which enables monetary trade curves to converge towards Pareto optimal rest-points. Below this growth rate, the economy enters a (sub- optimal) liquidity trap where monetary policy is ineffective; above this threshold inflation rises. Finally, market liquidity, measured through the speed of real trades, can be linked to gains-to-trade, households?? expectations, and the quantity of circulating money.     

Core-equivalence for the Nash bargaining solution

Summary. Core equivalence and shrinking of the core results are well known for economies. The present paper establishes counterparts for bargaining economies, a specific class of production economies (finite and infinite) representing standard two-person bargaining games and their continuum counterparts as coalition production economies. Thereby we get core equivalence of the Nash solution. The results reconfirm the Walrasian approach to Nash bargaining of Trockel (1996). Moreover we establish the same speed of convergence as is known from Debreu (1975) and Grodal (1975) for replicated pure exchange economies and for regular purely competitive sequences of economies, respectively.Received: 13 June 2003, Revised: 13 January 2004, JEL Classification Numbers: C71, C78, D51.This article is dedicated to Birgit Grodal, a friend since 30 years.Financial support of the DFG under grant #444 USA 111/2/03 is gratefully acknowledged.     

Perfect competition in asymmetric information economies: compatibility of efficiency and incentives

The idea of perfect competition for an economy with asymmetric information is formalized via an idiosyncratic signal process in which the private signals of almost every individual agent can influence only a negligible group of agents, and the individual agents’ relevant signals are essentially pairwise independent conditioned on the true states of nature. Thus, there is no incentive for an individual agent to manipulate her private information. The existence of incentive compatible, ex post Walrasian allocations is shown for such a perfectly competitive asymmetric information economy with or without “common values”. Consequently, the conflict between incentive compatibility and Pareto efficiency is resolved exactly, and its asymptotic version is derived for a sequence of large, but finite private information economies.     

Balancedness and the core in economies with asymmetric information

Summary. A condition is given that is equivalent to balancedness of all NTU-games derived from an exchange economy with asymmetric information when endowments are variable. The condition is applicable to the ex-ante model with expected utilities, but also to the more general model of Arrow-Radner type economies without subjective probabilities. Differences in the interpretation of measurability assumptions between these two models are discussed, and another model with information consistent utility functions is developed in which the result would also hold. Received: December 12, 2001; revised version: November 1, 2002 RID="*" ID="*"I thank two anonymous referees whose comments led to an improvement of the paper.