Walrasian Outcome Research Articles

Cournot vs. Walras: A reappraisal through simulations

Best-reply behavior in Cournot oligopolies generally leads to Cournot-Nash equilibrium, but imitative behavior selects the Walrasian equilibrium as the unique stochastically stable state. Previous work (Alós-Ferrer, 2004) showed that in the presence of memory, imitative behavior leads to a non-trivial dynamics selecting all quantities between the Cournot and Walrasian outcomes. However, the scope of previous results was limited to specific assumptions on demand and cost functions, and did not provide information on the shape of the distribution of outcomes. We use computational simulations to address these limitations. We show that the selection result for non-trivial memory holds beyond the set of well-behaved Cournot games previously analyzed. Further, we find that, in Cournot games, the limit distribution of long-run outcomes is highly skewed towards the Walrasian quantity. Although longer memory increases the importance of the Cournot equilibrium, the competitive outcome remains the dominant prediction.

Designing a strategyproof spot market mechanism with many traders: twenty-two steps to Walrasian equilibrium

To prove their Walrasian equilibrium existence theorem, Arrow and Debreu (Econometrica 22(3):265–290, 1954) devised an abstract economy that Shapley and Shubik (J Polit Econ 85:937–968, 1977) criticized as a market game because, especially with untrustworthy traders, it fails to determine a credible outcome away from equilibrium. All this earlier work also postulated a Walrasian auctioneer with complete information about traders’ preferences and endowments. To ensure credible outcomes, even in disequilibrium, warehousing is introduced into a multistage market game. To achieve Walrasian outcomes in a large economy with incomplete information, even about traders’ endowments, a strategyproof demand revelation mechanism is considered and then extended to include warehousing.

Markets, correlation, and regret-matching

Inspired by the existing work on correlated equilibria and regret-based dynamics in games, we carry out a first exploration of the links between equilibria and dynamics in (exchange) economies. The leading equilibrium concept is Walrasian equilibrium, and the dynamics (specifically, regret-matching dynamics) apply to trading games that fit the economic structure and whose pure Nash equilibria implement the Walrasian outcomes. Interestingly, in the case of quasilinear utilities (or “transferable utility”), all the concepts essentially coincide, and we get simple deterministic dynamics converging to Walrasian outcomes. Connections to sunspot equilibria are also studied.

The Second Welfare Theorem in Economies with Non‐Walrasian Markets

AbstractThe standard version of the second welfare theorem assumes that market operations produce Walrasian outcomes. Therefore, if there are individuals who can manipulate prices, the conclusion of the second welfare theorem is questionable. In this paper, we address the decentralization of a Pareto‐optimal allocation, when markets are non‐Walrasian. Our objective in this paper is to develop a game which can implement Pareto‐optimal allocations as Nash equilibria of strategic exchange in markets. In this way, we develop a version of the second welfare theorem for economies where markets are strategic.

Investments and the holdup problem in a matching market

This paper studies the properties of the steady state equilibrium in a bilateral matching market with ex ante investments at the market entry stage. Investment incentives depend on search frictions because both parties in a match are partially locked-in when they bargain over the joint surplus from their sunk investments. The associated holdup problem is more important for the long side of the market. In the extreme case of perfectly substitutable investments only the agents on the short side make investments. When market frictions become negligible, the market equilibrium approaches the Walrasian outcome.

Competitive behavior in market games: Evidence and theory

We explore whether competitive outcomes arise in an experimental implementation of a market game, introduced by Shubik (1972). Market games obtain Pareto inferior (strict) Nash equilibria, in which some markets are closed. We find that subjects do not coordinate on autarkic Nash equilibria, but favor more efficient Nash equilibria in which all markets are open. As the number of subjects participating in the market game increases, the Nash equilibrium they achieve approximates the associated Walrasian equilibrium of the underlying economy. Motivated by these findings, we investigate theoretically whether evolutionary forces lead to Walrasian outcomes in market games. We introduce a strong version of evolutionary stable strategies (SESS )f orfinite populations. Our concept requires stability against deviations by coalitions of agents. A small coalition of trading agents is sufficient for Pareto-improving trade to be generated. In addition, provided that agents lack market power, Nash equilibria corresponding to approximate competitive outcomes constitute the only approximate SESS.

Imitation and the evolution of Walrasian behavior: Theoretically fragile but behaviorally robust

A well-known result by Vega-Redondo (1997) [18] implies that in symmetric Cournot oligopolies, imitation leads to the Walrasian outcome. We show that this result is not robust to the slightest asymmetry in costs, since every outcome where agents choose identical actions will be played some fraction of the time in the long run. We then conduct experiments to check this fragility. We obtain that, contrary to the theoretical prediction, the Walrasian outcome is a good predictor of market outcomes. Finally, we suggest a new theory based on a mix of imitation and other learning processes that explains subjects' behavior fairly well.

Efficiency of Simultaneous Search

This paper presents an equilibrium labor search model in which workers can simultaneously apply to multiple firms to increase their search intensity. They observe firms’ wage postings before choosing where to apply. Owing to coordination frictions, a firm may not receive any applications; otherwise it is able to hire unless all its applicants have better offers. It is shown that the equilibrium converges to the efficient Walrasian outcome as application costs vanish. Even for nonnegligible application costs, the entry of firms, the search intensity, and the number of filled vacancies are constrained efficient. Wage dispersion is essential for constrained efficiency.

Imitation and the Evolution of Walrasian Behavior: Theoretically Fragile but Behaviorally Robust

A well-known result by Vega-Redondo (1997) implies that in symmetric Cournot oligopoly, imitation leads to the Walrasian outcome where price equals marginal cost. In this paper, we show that this result is not robust to the slightest asymmetry in fixed costs. Instead of obtaining the Walrasian outcome as unique prediction, every outcome where agents choose identical actions will be played some fraction of the time in the long run. We then conduct experiments to check this fragility. We obtain that, contrary to the theoretical prediction, the Walrasian outcome is still a good predictor of behavior.

Price Setting and Competition with Search Frictions

This paper investigates price determination in a decentralized economy in which buyers' valuations are stochastic and unobservable. In such a market, each buyer's reservation utility depends both on the prevailing price and on the price he actually encounters. The buyer's willingness to trade is shown to be decreasing in the price, and this creates the trade-off for the sellers' price setting. Even though the sellers have incentives to manipulate the buyer's willingness to trade, the economy is not fully competitive; it does not converge to the Walrasian outcome as search frictions disappear. The model is used to study various market structures to explore the nature of market power in search equilibrium. It is shown that price dispersion arises as a result of search frictions and oligopolistic price setting.

24. Pricing in Bertrand competition with increasing marginal costs

Bertrand competition under decreasing returns involves a wide interval of pure strategy Nash equilibrium prices. We first present results of experiments in which two, three and four identical firms repeatedly interact in this environment. More firms lead to lower average prices. However, prices remain substantially above the Walrasian level. With more than two firms the predominant market price is 24, a price not predicted by conventional equilibrium theories. This phenomenon can be captured by a simple imitation model and by a focal point explanation. For the long run, the model predicts that prices converge to the Walrasian outcome. We then use data from three new treatments to properly test the influence of imitation and focality. We find that both forces are present, but that imitation dominates in large markets with a long interaction.

Competitive bargaining equilibrium

In a simple exchange economy we propose a bargaining procedure that leads to a Walrasian outcome as the agents become increasingly patient. The competitive outcome therefore 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. Our bargaining procedure involves bargaining over prices and maximum quantity constraints, and it guarantees convergence to a Walrasian outcome for any standard exchange economy. In contrast, without quantity constraints we show that equilibrium is generically inefficient.

Submodularity and the evolution of Walrasian behavior

Vega-Redondo (1997) showed that imitation leads to the Walrasian outcome in Cournot Oligopoly. We generalize his result to aggregative quasi-submodular games. Examples are the Cournot Oligopoly, Bertrand games with differentiated complementary products, Common-Pool Resource games, Rent-Seeking games and generalized Nash-Demand games.

Search and Bargaining in Large Markets With Homogeneous Traders

We study decentralized trade in dynamic markets with homogeneous, non-atomic, buyers and sellers that wish to exchange one unit. In the first part of the paper we characterize equilibrium in a bargaining model with two-sided time varying outside options. In the second part we analyze a market equilibrium model in which (i) buyers and sellers are randomly matched in pairs; (ii) each buyer-seller pair bargains over the price of a good; and (iii) each agent has the option of abandoning negotiations, in which case the value of returning to the pool of unmatched agents constitutes an outside option. The second part is therefore an application of the first part in which the values of the outside options are endogenous to the model. Conditions for uniqueness of the market equilibrium are given; when it is unique it converges to the Walrasian outcome as frictions vanish. To the extent that multiplicity of market equilibria may (under some conditions) persist as frictions are removed, the limit of some sequences of equilibrium prices may converge to non-Walrasian values.

Submodularity and the Evolution of Walrasian Behavior

Vega-Redondo (1997) showed that imitation leads to the Walrasian outcome in Cournot Oligopoly. We generalize his result to aggregative quasi-submodular games. Examples are the Cournot Oligopoly, Bertrand games with differentiated complementary products, Common-Pool Resource games, Rent-Seeking games and generalized Nash-Demand games.

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.

Two are few and four are many: number effects in experimental oligopolies

In this paper we investigate how the competitiveness of Cournot markets varies with the number of firms in an industry. We review previous Cournot experiments in the literature. Additionally, we conduct a new series of experiments studying oligopolies with two, three, four, and five firms in a unified frame. With two firms we find some collusion. Three-firm oligopolies tend to produce outputs at the Nash level. Markets with four or five firms are never collusive and typically settle at or above the Cournot outcome. Some of those markets are actually quite competitive with outputs close to the Walrasian outcome.

Cournot versus Walras in dynamic oligopolies with memory

This paper explores the impact of memory in Cournot oligopolies where firms learn through imitation of success as suggested in Alchian [J. Polit. Econ. 57 (1950) 211] and modeled in Vega-Redondo [Econometrica 65 (1997) 375]. As long as memory includes at least one period, the long-run outcome corresponds to any quantity between the Cournot one and the Walras one. The (evolutionary) stability of the walrasian outcome relies on interfirm comparisons of simultaneously observed profits (i.e., whether a firm earns more than others in a given period), whereas the stability of the Cournot–Nash equilibrium is derived from intertemporal comparisons of profits for each given firm (i.e., whether a deviation pays off for that firm).

Bargaining in a network of buyers and sellers

We analyze bargaining between buyers and sellers who are connected by an exogenously given network. Players can make repeated alternating public offers that may be accepted by any of the responders linked to each specific proposer. Our purpose is to find the conditions of the network which drive the price distribution in equilibrium. This paper uses graph theory tools to provide necessary and sufficient conditions regarding the architecture of networks for the subgame perfect equilibrium of the bargaining game to coincide with the Walrasian outcome.

Is Bargaining over Prices Efficient?

We consider the problem of two agents bargaining over the relative price of two goods they are endowed with. They alternatingly exchange price offers and the utilities are discounted. The recipient of an offer can either accept it and choose the quantities to be traded, or reject and counteroffer a different relative price. We study the set of equilibria as discounting frictions vanish and find that: (1) any generic economy has bargaining equilibria that are inefficient even as discounting frictions vanish; and (2) a bargaining equilibrium converging to a Walrasian outcome exists for some robust types of convergence of the discount factors, but it does not exist for other equally robust convergences. Moreover, in case there exists a bargaining equilibrium converging to a Walrasian outcome, then there is necessarily a multiplicity of them. As a consequence, unlike in Rubinstein's (1982) alternating-offer bargaining, the equilibrium outcome of this set-up is not generically unique and efficient.