Walrasian Allocation Research Articles

Asymptotic equivalence between Cournot–Nash and Walras equilibria in exchange economies with atoms and an atomless part

In this paper, we consider an exchange economy a la Shitovitz (Econometrica 41:467–501, 1973), with atoms and an atomless set. We associate with it a strategic market game of the kind first proposed by Lloyd S. Shapley, known as the Shapley window model. We analyze the relationship between the set of the Cournot–Nash allocations of the strategic market game and the Walras allocations of the exchange economy with which it is associated. We show, with an example, that even when atoms are countably infinite, any Cournot–Nash allocation of the game is not a Walras allocation of the underlying exchange economy. Accordingly, in the original spirit of Cournot (Recherches sur les principes mathematiques de la theorie des richesses. Hachette, Paris, 1838), we partially replicate the mixed exchange economy by increasing the number of atoms, without affecting the atomless part, and ensuring that the measure space of agents remains finite. Our main theorem shows that any sequence of Cournot–Nash allocations of the strategic market games associated with the partial replications of the exchange economy has a limit point for each trader and that the assignment determined by these limit points is a Walrasian allocation of the original economy.

Let them cheat!

We consider the problem of fairly allocating a bundle of infinitely divisible goods among a group of agents with “classical” preferences. We propose to measure an agentʼs “sacrifice” at an allocation by the size of the set of feasible bundles that the agent prefers to her consumption. As a solution, we select the allocations at which sacrifices are equal across agents and this common sacrifice is minimal. We then turn to the manipulability of this solution. In the tradition of Hurwicz (1972), we identify, under some mild assumptions on preferences, the equilibrium allocations of the manipulation game associated with this solution when all commodities are normal: for each preference profile, each equal-division constrained Walrasian allocation is an equilibrium allocation; conversely, each equilibrium allocation is equal-division constrained Walrasian. Furthermore, we show that if normality of goods is dropped, then equilibrium allocations may not be equal-division constrained Walrasian.

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 a 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.

A Note on the Private Core and Coalitional Fairness under Asymmetric Information

We study relations among Walrasian expectations allocations, coalitional fair allocations and the private core of economies with uncertainty and asymmetric information. Our analysis covers finite exchange models, as well as models of mixed markets consisting of some large traders and an ocean of small traders. The adopted notion of coalitional fairness requires that: 1. Under a “c-fair” allocation, no coalition could benefit from achieving the net trade of some other disjoint coalition; 2. Coalition bargaining takes place without information sharing among agents. We introduce a notion of restricted Walrasian expectations allocation and examine its relations with c-fairness.

Manipulation games in economies with indivisible goods

In this paper we study the strategic aspects of the No-Envy solution for the problem of allocating a finite set of indivisible goods among a group of agents when monetary compensations are possible. In the first part of the paper we consider the case where each agent receives, at most, one indivisible good. We prove that the set of equilibrium allocations of any direct revelation game associated with a subsolution of the No-Envy solution coincides with the set of envy-free allocations for the true preferences. Under manipulation all the subsolutions of the No-Envy solution are equivalent. In the second part of the paper, we allow each agent to receive more than one indivisible good. In this situation the above characterization does not hold any more. We prove that any Equal Income Walrasian allocation for the true preferences can be supported as an equilibrium allocation of any direct revelation game associated with subsolutions of the No-Envy solution, but also non-efficient allocations can be supported.

Core allocations and small income transfers

We show that, given any allocation f in the core of an exchange economy, we can find small income transfers and a Walrasian allocation f ˜ relative to the transfers such that most agents are indifferent between f and f ˜ . In addition, we can find small income transfers and an approximate Walrasian allocation f ˆ relative to the transfers such that all agents are indifferent between f ˆ and f.

Mistakes in Cooperation: The Stochastic Stability of Edgeworth's Recontracting

In an exchange economywith a finite number of indivisible goods, we analyze a dynamic trading process of coalitional recontracting where agents may make mistakes with small probability. We show first that the recurrent classes of the unperturbed (mistake- free) process consist of (i) all core allocations as absorbing states, and (ii) cycles of non-core allocations. Next, we introduce a perturbed process, where the resistance of each transition is a function of the number of agents that make mistakes -do not improve- in the transition and of the seriousness of each mistake. If preferences are always strict, we show that the unique stochasticallystable state of the perturbed process is the Walrasian allocation. In economies with indifferences, non-core cycles are sometimes stochastically stable, while some core allocations are not.

Competitive Bargaining Equilibrium

We propose a simple bargaining procedure, the equilibrium of which converges to the Walrasian allocation as the agents become increasingly patient. We thus establish that the competitive outcome obtains even if agents have market power and are not price-takers. Moreover, where in other bargaining protocols the final outcome depends on bargaining power or relative impatience, the outcome here is determinate and depends only on preferences and endowments. This procedure has therefore important implications for policy applications compared to standard bargaining rules.

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.

Stabilizing sunspots

This paper analyzes stabilization of sunspot equilibria in an incomplete financial market framework. Indexation of nominal bonds or introduction of real securities can eliminate the sunspot equilibria. However, we show that the utility of one type of consumer may be minimized at a Walrasian allocation relative to other sunspot equilibrium allocations. Thus, if considering stabilization policies from the status quo of incomplete financial markets, there may be no consensus on implementing them.

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

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.

Mistakes in Cooperation: The Stochastic Stability of Edgeworth's Recontracting

In an exchange economyw ith a finite number of indivisible goods, we analyze a dynamic trading process of coalitional recontracting where agents may make mistakes with small probability. We show first that the recurrent classes of the unperturbed (mistakefree) process consist of (i) all core allocations as absorbing states, and (ii) cycles of non-core allocations. Next, we introduce a perturbed process, where the resistance of each transition is a function of the number of agents that make mistakes - do not improve - in the transition and of the seriousness of each mistake. If preferences are always strict, we show that the unique stochasticallystable state of the perturbed process is the Walrasian allocation. In economies with indifferences, non-core cycles are sometimes stochastically stable, while some core allocations are not.

Liquidity and the Simple IO of Stock Exchanges

It is usually thought that network externalities, which are inherent to liquidity, make it desirable to concentrate transactions in one stock exchange. This paper shows that when the value of liquidity stems from the ability to potentially reach as many traders as possible, the market is integrated when every broker meets every other broker in at least one exchange. Thus, fragmentation is not about trades being executed in different exchanges but of connectedness among brokers. An implication of this distinction is that in an integrated market the network externality created by liquidity becomes pecuniary and the optimal number of exchanges depends only on the shape of the (physical) technology to execute trades - whether it exhibits increasing, constant or decreasing returns to scale - as in any standard industry. We characterize the planner's allocation and compare it with that reached by a monopoly. It is shown that when exchanges are natural monopolies a particular ownership structure of the exchange and allocation of voting rights over the exchange fee achieve the planner's optimum. With decreasing returns to scale the Walrasian allocation is efficient, provided that the market is integrated. Nevertheless, with few exchanges the price-taking assumption is suspect. If exchanges are not price takers, there are many other equilibria, all of them inefficient. Moreover, there are reasons to doubt that the market will become integrated. Fragmentation softens price competition between exchanges and may help a monopolist exchange to erect a barrier to entry even when he has no cost advantage.

Implementation in Economies with a Continuum of Agents

We study a general implementation problem for exchange economies with a continuum of players and private information, and test the robustness of the results for sequences of approximating finite economies. Assuming that the designer knows the distribution of the characteristics in the economy we consider continuous and unique implementation in both its equilibrium and dominant strategies versions and obtain results for general self-selective (first- and second-best) allocations. An upper hemicontinuity property of Bayesian equilibria of approximating economies for continuous mechanisms is demonstrated. Using this, we can, for example, conclude that if a given continuous mechanism implements uniquely a Walrasian allocation in the continuum economy then all Bayesian equilibria of large approximation economies will (with probability close to one) yield an allocation which is almost ex-post Pareto optimal.

Existence of equilibrium for walrasian endowment games

This work deals with no-destruction Walrasian endowment games, where agents can manipulate by declaring false initial endowments. The existence theorem of non-trivial Nash equilibrium (NE) for large economies is the main purpose of the paper. This theorem also implies (1) the existence of nontrivial NE for Walrasian preference games, and (2) the existence of NE allocations which are almost efficient. It is also shown that under some regularity conditions, and when the economy gets large, every sequence of NE allocations converges to a true Walrasian allocation.

Catastrophic Walrasian Equilibrium from the Non-Walrasian Viewpoint: A Three-Good Macroeconomic Example

A Walrasian equilibrium is catastrophic from the non-Walrasian viewpoint if some small deviation from the Walrasian prices produces only non-Walrasian equilibrium allocations a long way from the Walrasian allocation. A precise account of this phenomenon is given for a class of three good, single household private ownership production economies.

On the Dynamics of Disequilibrium: A Simple Remark

Recent works in the field of disequilibrium analysis have increased our understanding of the functioning of the economic system when transactions take place at prices that may not achieve the equilibrium of supply and demand as the classics understood it. The works of Benassy [2] and Dreze [4] show us a way of representing the allocation process when prices are assumed to be rigid in the short run, in the context of the fixed price method of Hicks [5]. The foregoing studies allow us to look at the dynamics of an economy where prices would be fixed at each period, but possibly moving from period to period. In this context, one must search for the dynamic forces which are underlying the disequilibrium allocations. A question of interest is then to ask whether these forces are bound to drive the system towards a Walrasian equilibrium. This note brings a first answer to a part of this general problem. A simple model of an exchange economy with two non-durable commodities is considered. In a neighbourhood of a locally stable Walrasian equilibrium, a result in comparative statics is established: the agents on the short side of the market always prefer the Walrasian allocation to the disequilibrium allocation, while, if trade actually takes place, there exists a disequilibrium allocation which is strictly preferred to the Walrasian allocation by all the agents on the long side of the market. This suggests the following interpretation: if the agents in the economy have a perfect foresight of the dynamic behaviour of the market, in a neighbourhood of any stable Walrasian equilibrium, the dynamics of the system should take the form of a struggle between the buyers and the sellers for the settling of the price. The paper is divided into two sections. In the first one, the results are stated and their economic interpretation is discussed. The second section contains the proofs.