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     

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.     

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.     

Core convergence with asymmetric information

We analyze the ex ante incentive compatible core for replicated private information economies. We show that any allocation in the core when the economy is replicated sufficiently often is approximately Walrasian for the associated Arrow–Debreu economy.     

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.     

Double implementation in economies with production technologies unknown to the designer

Summary. This paper considers double implementation of Walrasian allocations and Lindahl allocations in Nash and strong Nash equilibria for both private and public goods economies when preferences, initial endowments, production technologies, and coalition patterns are all unknown to the designer. It will be noted that the mechanisms presented here are feasible and continuous. In addition, unlike most mechanisms proposed in the literature, our mechanism works not only for three or more agents, but also for two-agent economies, and thus it is a unified mechanism which is irrespective of the number of agents. Received: March 12, 1998; revised version: March 12, 1998     

The comparative statics of cumulative distribution function changes for the class of risk averse agents

We examine the connection between Walrasian equilibria of a limit economy (with infinitesimal firms) and the noncooperative (Cournot) equilibria of approximating finite economies (with significant firms). Following earlier work of Novshek and Sonnenschein we allow for set-up cost and permit a minimal form of mixed strategies. We depart from them by requiring that the aggregate production set exhibits some degree (however small) of decreasing returns. Contrasting with their results, it is shown that a (regular) Walrasian equilibrium of a limit economy can always be approximated by a sequence of noncooperative equilibria for the tail of the approximating (finite) economies. Thus, there is a surprising qualitative discontinuity when one passes from the Novshek-Sonnenschein case of aggregate constant returns to scale of the decreasing returns case of this paper.     

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).     

Walrasian equilibria as limits of noncooperative equilibria. Part II: Pure strategies

We examine the connection between Walrasian equilibria of a limit economy (with infinitesimal firms) and noncooperative (Cournot) equilibria of approximating finite economies (with significant firms). Nonconvex production sets, decreasing returns in the aggregate, and endogenous determination of the number of active firms are allowed. A Walrasian equilibrium is a limit of pure strategy noncooperative equilibria only if a condition (loosely analogous to downward sloping demand in the partial equilibrium constant returns to scale case) holds. The condition is also sufficient to guarantee the existence of a robust sequence of pure strategy noncooperative equilibria which converges to the Walrasian equilibrium.     

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.     

More on economies with a finite set of Walrasian equilibria

Proposition. The graph of the Walrasian equilibrium correspondence is a piecewise continuously differentiable manifold. Furthermore, there exists an open dense set of economies, Ω, such that for all w in Ω (a) the number of equilibria of the economy w is finite; and (b) there exists a neighborhood V(w) on which the set of equilibria is represented by a finite family of piecewise continously differentiable functions.     

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     

Incentive Mechanism Design for Production Economies with Both Private and Public Ownerships

In this paper we deal with the problem of incentive mechanism design which yields efficient allocations for general mixed ownership economies when preferences, individual endowments, and coalition patterns among individuals are unknown to the planner. We do so by doubly implementing the proportional solution for economies with any number of private sector and public sector commodities and any number of individuals as well as the coexistence of privately and publicly owned firms with general convex production possibility sets. Furthermore, the mechanisms work not only for three or more agents, but also for two-agent economies. Journal of Economic Literature Classification Numbers: C72, D61, D71, D82.     

Walrasian General Equilibrium Allocations and Dynamics in Two-Sector Growth Models

This paper analyzes and solves miniature Walrasian general equilibrium systems of momentary and moving equilibria. The Walrasian framework encompasses the fundamental neoclassical and classical two‐sector growth models; the families of solutions of steady‐state and persistent growth per capita in various competitive two‐sector economies are parametrically characterized. Moreover, the endogenous behavior of relative prices and the sectoral allocation of primary factors are analyzed in detail. The technology parameters of the capital good industry are decisive for obtaining long‐run per capita growth in closed (global) economies. A review of the literature complements the theorems on the general equilibrium allocations, dynamic systems, and the time paths of Walrasian two‐sector economies.     

Nash implementation in production economies

Summary This paper provides a general way to incorporate private ownership production economies into the implementation of the Walrasian correspondence. We present two mechanisms, both of which permit agents to behave strategically with respect to their initial endowments, preferences, and production possibility sets. The first mechanism deals with the case of endowment destruction, the second deals with the case of endowment withholding. We show that each mechanism Nash implements the Walrasian correspondence. In addition, both mechanisms are individually feasible, balanced, continuous and only require the transmission of prices and quantities of goods as messages.This paper is based on Chapters 2 and 3 of my dissertation submitted to the University of Minnesota. I want to thank E. Green, J. Jordan, M. Richter, H. Weinberger and especially L. Hurwicz for their many helpful suggestions and comments.     

Ex ante efficiency implies incentive compatibility

We show that when agents become informationally negligible in a large economy with asymmetric information, every ex ante efficient allocation must be incentive compatible. This means that any ex ante core or Walrasian allocation is incentive compatible. The corresponding result is false for fixed finite-agent economies with asymmetric information. An example is also constructed to show that the ex post version of the result does not hold. Furthermore, we show that the result is sharp in the sense that it will fail to hold if one relaxes any of the main assumptions, namely, strong conditional independence on the information structure, strict concavity on the utility functions, type independence on the utility functions and endowments.      

On persistent disequilibrium

Consider a standard model of a Walrasian economy where the time derivative of price change is a sign preserving function of excess demand. The only steady states of such a system are the Walrasian equilibria. For if the system is not at a Walrasian equilibrium then there must be excess demands or supplies in some market and thus those prices must be changing. Walras' law implies that all prices cannot change in the same direction, and thus relative prices must change.However, it seems that in real world economies there have been states of persistent disequilibrium. How can this be? How can there be stationary states of an economy that are not Walrasian equilibria?The answer presented in this paper goes something like this: the demands actually presented to the market, i.e., the demands that affect price movements, are not the Walrasian demands. Rather, these demands, the effective demands, are a function of both price and quantity signals. There is no reason why there cannot be equilibria of the effective demand system that are not equilibria of the Walrasian demand system.In this paper, I present a simple abstract model of effective demand systems and given a condition for this system to have non-Walrasian equilibria. I also present a simple example of this phenomena. Finally I discuss the real world implications of this model. Related concepts of effective demand are discussed in [A1H], [2], [3], [L5], and [L6].     

Time inconsistency of benevolent government economies

A dynamic Walrasian economy is said to exhibit inconsistency if the competitive equilibrium path resulting from government reoptimization at some time τ>0 is not a continuation of the competitive equilibrium path resulting from the initial government optimization at time 0. The present paper establishes necessary and sufficient conditions for consistency for a general class of dynamic Walrasian economies. It is seen, for example, that reliance on nondistortionary policy instruments is neither necessary nor sufficient for consistency. It is also shown that Pareto optimal paths can be supported as optimal competitive equilibrium paths only if consistency prevails. However, consistent optimal competitive equilibrium paths need not be Pareto optimal.     

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.     

On the existence of equilibria in economies with infinitely many commodities and without ordered preferences

The present paper deals with the existence of equilibria in economies whose commodity space is L_∞(M, M, μ) and where the agents' preferences need not be complete or transitive. Applying a fixed point theorem of Browder, an equilibrium existence theorem for abstract economies (generalized qualitative games) is proven where each agent's choice set is contained in an arbitrary topological vector space. With the help of this theorem the existence of Walrasian general equilibrium for a suitably specified economic model is obtained. The final result is a generalization of T. F. Bewley's (J. Econ. Theory4 (1972), 514–540) equilibrium existence theorem to the case of non-ordered preferences.