Walrasian Equilibrium Price Research Articles

The ascending auction with flexible reporting

In the assignment problem where multiple heterogeneous indivisible items are assigned to unit-demand bidders, we introduce a novel ascending auction called the Flexible Reporting Ascending Auction to balance the privacy preservation and the speed of the ascending auction. Assuming bidders behave truthfully, it always results in a Walrasian equilibrium, along with the minimum Walrasian equilibrium price vector. If each bidder reports without a contradiction, bidders’ truthful behavior forms a Nash equilibrium. Our auction generalizes some well-known ascending auctions in literature, and we also provide simulation results to compare these auctions.

Strategy-proof multi-object mechanism design: Ex-post revenue maximization with non-quasilinear preferences

A seller is selling multiple objects to a set of agents, who can buy at most one object. Each agent's preference over (object, payment) pairs need not be quasilinear. The seller considers the following desiderata for her mechanism, which she terms desirable: (1) strategy-proofness, (2) ex-post individual rationality, (3) equal treatment of equals, (4) no wastage (every object is allocated to some agent). The minimum Walrasian equilibrium price (MWEP) mechanism is desirable. We show that at each preference profile, the MWEP mechanism generates more revenue for the seller than any desirable mechanism satisfying no subsidy. Our result works for the quasilinear domain, where the MWEP mechanism is the VCG mechanism, and for various non-quasilinear domains, some of which incorporate positive income effect of agents. We can relax no subsidy to no bankruptcy in our result for certain domains with positive income effect.

Price Convergence under a Probabilistic Double Auction

This paper uses a dual approach to study a class of quasi-linear exchange economies with indivisible or divisible goods in search for an equilibrium. Our model aims at an economy with a large scale and an agent’s individual demand or supply is contaminated with stochastic errors (noises). We study a probabilistic $$\alpha $$ -double auction and are interested in the convergence of a price process it generates, with weight $$\alpha $$ being a random variable with unknown distributions over [0, 1]. We show convergence results when the two step sizes are diminishing or probabilistically diminishing in the means. An error bound is estimated when the two step sizes are constant, bounded away from zero, while $$\alpha $$ remains a random variable. We provide conditions under which the double auction generates a price process that converges in mean square to the set of Walrasian equilibrium prices of the underlying economy.

How much market making does a market need?

AbstractWe consider a simple model for the evolution of a limit order book in which limit orders of unit size arrive according to independent Poisson processes. The frequencies of buy limit orders below a given price level, respectively sell limit orders above a given level, are described by fixed demand and supply functions. Buy (respectively, sell) limit orders that arrive above (respectively, below) the current ask (respectively, bid) price are converted into market orders. There is no cancellation of limit orders. This model has been independently reinvented by several authors, including Stigler (1964), and Luckock (2003), who calculated the equilibrium distribution of the bid and ask prices. We extend the model by introducing market makers that simultaneously place both a buy and sell limit order at the current bid and ask price. We show that introducing market makers reduces the spread, which in the original model was unrealistically large. In particular, we calculate the exact rate at which market makers need to place orders in order to close the spread completely. If this rate is exceeded, we show that the price settles at a random level that, in general, does not correspond to the Walrasian equilibrium price.

Equilibrium in the Assignment Market under Budget Constraints

We reexamine the well-known assignment market model in a more general and more practical environment where agents may be financially constrained. These constraints will be shown to have an important impact on the set of Walrasian equilibria. We prove that a price adjustment process will either find a unique minimal Walrasian equilibrium price vector, or exclusively validate the nonexistence of equilibrium.

Strategy-Proof Multi-Object Auction Design: Ex-Post Revenue Maximization with No Wastage

A seller is selling multiple objects to a set of agents, who can buy at most one object. Each agent's preference over (object, payment) pairs need not be quasilinear. The seller considers the following desiderata for her mechanism, which she terms desirable: (1) strategy-proofness, (2) ex-post individual rationality, (3) equal treatment of equals, (4) no wastage (every object is allocated to some agent). The minimum Walrasian equilibrium price (MWEP) mechanism is desirable. We show that at each preference profile, the MWEP mechanism generates more revenue for the seller than any desirable mechanism satisfying no subsidy. Our result works for the quasilinear domain, where the MWEP mechanism is the VCG mechanism, and for various non-quasilinear domains, some of which incorporate positive income effect of agents. We can relax no subsidy to no bankruptcy in our result for certain domains with positive income effect.

Do prices coordinate markets?

A Walrasian equilibrium outcome has a remarkable property: the induced allocation maximizes social welfare while each buyer receives a bundle that maximizes her individual surplus at the given prices. There are, however, two caveats. First, minimal Walrasian prices necessarily induce indifferences. Thus, without coordination, buyers may choose surplus maximizing bundles that conflict with each other. Accordingly, buyers may need to coordinate with one another to arrive at a socially optimal outcome---the prices alone are not sufficient to coordinate the market. Second, although natural auctions converge to Walrasian equilibrium prices on a fixed population, in practice buyers typically observe prices without participating in a price computation process. These prices are not perfect Walrasian equilibrium prices, but we may hope that they still encode distributional information about the market. To better understand the performance of Walrasian prices in light of these two problems, we give two results. First, we propose a mild genericity condition on valuations so that the minimal Walrasian equilibrium prices induce allocations resulting in low overdemand, no matter how the buyers break ties. In fact, under our condition the overdemand of any good can be bounded by 1, which is the best possible at the minimal prices. Second, we use techniques from learning theory to argue that the overdemand and welfare induced by a price vector converge to their expectations uniformly over the class of all price vectors, with sample complexity linear and quadratic in the number of goods in the market respectively. The latter results make no assumption on the form of the valuation functions.

An Extension of Ausubel's Auction for Heterogeneous Discrete Goods

ABSTRACTAusubel's dynamic private‐values auction for heterogeneous discrete goods, Ausubel (2006), yields an efficient equilibrium outcome but it is designed for a limited class of environments. If bidders’ values for bundles of goods are not integers, then Ausubel's auction may end without allocating goods if no information on bidders’ values is available. In this paper, I extend Ausubel's auction for heterogeneous discrete goods to real‐valued quasilinear utility functions. The mechanism I propose reaches a Walrasian equilibrium price vector in finite ‘steps’ without any additional information on bidders’ values. In the extension of Ausubel's auction, truthful bidding constitutes an efficient equilibrium.

A method for finding the maximal set in excess demand

We present a polynomial time method for identifying the maximal set in excess demand at a given payoff vector. This set can be used in “large” partnership formation problems to identify the minimum element in the set of individually rational payoff vectors at which there is no overdemanded set of agents. This minimum element corresponds to the minimum Walrasian equilibrium price vector in a special case of the partnership formation problem.

An Efficient and Incentive Compatible Dynamic Auction for Multiple Complements

This article proposes an efficient and incentive compatible dynamic auction for selling multiple complementary goods. The seller has reserve prices. The auctioneer announces a current price for every bundle of goods and a supply set of goods, every bidder responds with a set of goods demanded at these prices, and the auctioneer adjusts prices. We prove that even when bidders can exercise their market power strategically, this dynamic auction always induces them to bid truthfully, resulting in an efficient allocation, its supporting Walrasian equilibrium price for every bundle of goods, and a generalized Vickrey-Clarke-Groves payment for every bidder.

An Extension of Ausubel’s Auction for Heterogeneous Discrete Goods

Ausubel’s dynamic private-values auction for heterogeneous discrete goods, Ausubel (2006), yields an efficient equilibrium outcome but it is designed for a limited class of environments. If bidders’ values for bundles of goods are not integers, then Ausubel’s auction may end without allocating goods if no information on bidders’ values is available. In this paper, I extend Ausubel’s auction for heterogeneous discrete goods to real-valued quasilinear utility functions. The mechanism I propose reaches a Walrasian equilibrium price vector in finite “steps” without any additional information on bidders’ values. In the extension of Ausubel’s auction, truthful bidding constitutes an efficient equilibrium. JEL Category D44 ∗I would like to thank Ket Richter and Jan Werner for their continuous support, patience, and effort for my education. Their guidance has been enlightening. I would like to thank late Leo Hurwicz, and Andy McLennan for their contribution to my education. I benefited a lot from discussions with David Rahman and Itai Sher. I also would like to thank Pat Bajari, Tuba Inal, Ichiro Obara, Raj Singh, and the seminar participants at Central Michigan University, METU, the University of Minnesota, the Virginia Commonwealth University, Whitman College, the Games 2008, and the Society for Economic Design 2008 Conference for their valuable comments. All errors are mine. †http://hakaninal.weebly.com

Connected Substitutes and Invertibility of Demand

We consider the invertibility (injectivity) of a nonparametric nonseparable demand system. Invertibility of demand is important in several contexts, including identification of demand, estimation of demand, testing of revealed preference, and economic theory exploiting existence of an inverse demand function or (in an exchange economy) uniqueness of Walrasian equilibrium prices. We introduce the notion of and show that this structure is sufficient for invertibility. The connected substitutes conditions require weak substitution between all goods and sufficient strict substitution to necessitate treating them in a single demand system. The connected substitutes conditions have transparent economic interpretation, are easily checked, and are satisfied in many standard models. They need only hold under some transformation of demand and can accommodate many models in which goods are complements. They allow one to show invertibility without strict gross substitutes, functional form restrictions, smoothness assumptions, or strong domain restrictions. When the restriction to weak substitutes is maintained, our sufficient conditions are also nearly necessary for even local invertibility.

Walrasian prices in markets with tradable rights

In this pape,r we consider an exchange economy where there is an external restriction for the consumption of goods. This restriction is defined by both, a cap on consumption of certain commodities and the requirement of an amount of rights for the consumption of these commodities. The caps for consumption are imposed exogenously due to the negative effects that the consumption may produce. The consumption rights or licences are distributed among the agents. This fact leads to the possibility of establishing licence markets. These licences do not participate in agents’ preferences, however the individual’s budgetary constraint may be modified, leading to a reassignment of resources. Our aim is to show the existence of a Walrasian equilibrium price system linking tradable rights prices with commodity prices.

Walrasian Equilibrium and Reputation under Imperfect Public Monitoring

This paper examines a reputation-based mechanism that sustains the provision of high quality in the presence of competition among providers, where the incentive for high-quality production comes from a reputation premium: firms with higher reputations charge higher prices. The way we model the market highlights the fact that prices are not solely determined from consumers' willingness to pay as in the monopolistic setting studied in the previous literature. In effect, equilibrium prices are determined endogenously, from the interaction of the distribution of consumers' valuations for high quality and the distribution of firms' reputations—the demand and the supply sides of the market, respectively. This paper shows that: (i) there is a steady-state distribution of reputations, a result that allows the study of a stationary equilibrium; (ii) there are parameter configurations for which the existence of a high-quality equilibrium is guaranteed, and where the incentives for high quality production in the repeated game depend on the shape of the price function; and (iii) the Walrasian-equilibrium price function depends on the shape of the steady-state distribution of reputations, and the assignment of consumers to firms with different reputation levels in such an equilibrium is positively assortative if quality is a normal good.

Microeconomics-based resource allocation in overlay networks by using non-strategic behavior modeling

Behavior modeling has recently been investigated for designing self-organizing mechanisms in the context of communication networks in order to exploit the natural selfishness of the users with the goal of maximizing the overall utility. In strategic behavior modeling, the users of the network are assumed to be game players who seek to maximize their utility with taking into account the decisions that the other players might make. The essential difference between the aforementioned researches and this work is that it incorporates the non-strategic decisions in order to design the mechanism for the overlay network. In this solution concept, the decisions that a peer might make does not affect the actions of the other peers at all. The theory of consumer-firm developed in microeconomics is a model of the non-strategic behavior that we have adopted in our research. Based on it, we have presented distributed algorithms for peers’ “joining” and “leaving” operations. We have modeled the overlay network as a competitive economy in which the content provided by an origin server can be viewed as commodity and the origin server and the peers who multicast the content to their downside are considered as the firms. On the other hand, due to the dual role of the peers in the overlay network, they can be considered as the consumers as well. On joining to the overlay economy, each peer is provided with an income and tries to get hold of the service regardless to the behavior of the other peers. We have designed the scalable algorithms in such a way that the existence of equilibrium price (known as Walrasian equilibrium price) is guaranteed.

Characterization of the Walrasian equilibria of the assignment model

We study the assignment model where a collection of indivisible goods are sold to a set of buyers who want to buy at most one good. We characterize the extreme and interior points of the set of Walrasian equilibrium price vectors for this model. Our characterizations are in terms of demand sets of buyers. Using these characterizations, we also give a unique characterization of the minimum and the maximum Walrasian equilibrium price vectors. Also, necessary and sufficient conditions are given under which the interior of the set of Walrasian equilibrium price vectors is non-empty. Several of the results are derived by interpreting Walrasian equilibrium price vectors as potential functions of an appropriate directed graph.

Undecidability of Uzawa equivalence theorem and LLPO (Lesser limited principle of omniscience)

The Uzawa equivalence theorem [H. Uzawa, Walras’s Existence Theorem and Brouwer’s Fixed Point Theorem, Economic Studies Quarterly 8 (1962) 59–62] showed (classically) that the existence of Walrasian equilibrium in an economy with continuous excess demand functions is equivalent to Brouwer’s fixed point theorem, that is, the existence of a fixed point for any continuous function from an n -dimensional simplex to itself. We examine the Uzawa equivalence theorem from the point of view of constructive mathematics, and show that this theorem, properly speaking, the assumption of the existence of a Walrasian equilibrium price vector in this theorem, implies LLPO (Lesser limited principle of omniscience), and so it is non-constructive.

Overdemand and Underdemand in Economies with Invisible Goods and Unit Demands

We study an economy where a collection of indivisible goods are sold to a set of buyers who want to buy at most one good. We characterize the set of Walrasian equilibrium price vectors in such an economy using sets of overdemanded and underdemanded goods. Further, we give characterizations for the minimum and the maximum Walrasian equilibrium price vectors of this economy. As a consequence of these characterizations, we can say if overdemanded sets of goods and underdemanded sets of goods exist at a price vector given its position in the price vector space with respect to the minimum and the maximum Walrasian equilibrium price vectors. We also analyze what happens with the Walrasian equilibrium price vectors if one of the buyers or one of the goods is removed from the economy.

THE IMPACT OF RENT CONTROLS IN NON‐WALRASIAN MARKETS: AN AGENT‐BASED MODELING APPROACH

ABSTRACT We use agent‐based models to consider rent ceilings in non‐Walrasian housing markets, where bargaining between landlord and tenant leads to exchange at a range of prices. In the non‐Walrasian setting agents who would be extramarginal in the Walrasian setting frequently are successful in renting, and actually account for a significant share of the units rented. This has several implications. First, rent ceilings above the Walrasian equilibrium price (WEP) can affect the market outcome. Second, rent ceilings that reduce the number of units rented do not necessarily reduce total market surplus. Finally, the distributional impact of rent controls differs from the Walrasian setting.

An Efficient Dynamic Auction for Heterogeneous Commodities

This article proposes a new dynamic design for auctioning multiple heterogeneous commodities. An auctioneer wishes to allocate K types of commodities among n bidders. The auctioneer announces a vector of current prices, bidders report quantities demanded at these prices, and the auctioneer adjusts the prices. Units are credited to bidders at the current prices as their opponents' demands decline, and the process continues until every commodity market clears. Bidders, rather than being assumed to behave as price-takers, are permitted to strategically exercise their market power. Nevertheless, the proposed auction yields Walrasian equilibrium prices and, as from a Vickrey-Clarke-Groves mechanism, an efficient allocation.