Walrasian Price Research Articles

Monotonicity of extremal Walrasian prices in gross substitutes markets

We consider a market of indivisible items with all agents possessing gross substitutes valuations. In this setting, Walrasian prices are guaranteed to exist. Assume that some items become unsaleable, or some new gross substitutes agents enter the market. How does this influence the set of Walrasian prices? In this note, we demonstrate that minimal and maximal Walrasian prices exhibit monotonous behavior in response to changes in supply or demand in gross substitutes markets.

A flow‐based ascending auction to compute buyer‐optimal Walrasian prices

AbstractWe consider a market where a set of objects is sold to a set of buyers, each equipped with a valuation function for the objects. The goal of the auctioneer is to determine reasonable prices together with a stable allocation. One definition of “reasonable” and “stable” is a Walrasian equilibrium, which is a tuple consisting of a price vector together with an allocation satisfying the following desirable properties: (i) the allocation is market‐clearing in the sense that as much as possible is sold, and (ii) the allocation is stable in the sense that every buyer ends up with an optimal set with respect to the given prices. Moreover, “buyer‐optimal” means that the prices are smallest possible among all Walrasian prices. In this paper, we present a combinatorial network flow algorithm to compute buyer‐optimal Walrasian prices in a multi‐unit matching market with truncated additive valuation functions. The algorithm can be seen as a generalization of the classical housing market auction and mimics the very natural procedure of an ascending auction. We use our structural insights to prove monotonicity of the buyer‐optimal Walrasian prices with respect to changes in supply or demand.

Business cycles, sectoral price stabilization, and climate change mitigation: A model of multi-sector growth in the tradition of the Bielefeld disequilibrium approach

Sectoral price stabilization is critical in the pressing context of de-globalization and the climate crisis. Strategic price controls may be effective in reducing economic volatility, but fail to curb emissions, unlike quantity controls such as rationing consumption. In contrast, carbon pricing may accelerate decarbonization, but generate inflationary pressures. In order to evaluate the effectiveness of sector-oriented policies in stabilizing sectoral prices and accelerating the low-carbon transition at the same time, this paper employs a novel environmentally-extended, data-driven dynamic model of multi-sector growth based on the Bielefeld disequilibrium approach to the microeconomic stability of competitive economies. The general composite model allows to empirically characterize the network structure and intensity of dynamic price-quantity interactions at the sector level, which capture heterogeneous firm behavior by integrating long-run Walrasian price and classical quantity dynamics coupled with short-run Keynesian features in the form of target rate of return pricing and demand-led investment. A Bayesian hierarchical model on US BEA data empirically estimates the linear adjustment coefficients and stable combinations are computed. The US economy is shown to feature a highly hierarchical network structure of intermediate production that is particularly vulnerable to micro-economic shocks driven by wars and climate change. Tax-subsidy schemes are shown to be the most effective in stabilizing sectoral prices and economy-environment interactions.

Walrasian pricing in multi-unit auctions

Multi-unit auctions are a paradigmatic model of resource allocation, where a seller brings multiple units of a good to a set of buyers equipped with monetary budgets. It is well known that Walrasian equilibria do not always exist in this model, however compelling relaxations such as Walrasian envy-free pricing do.We design a best possible envy-free and prior-free mechanism for multi-unit auctions with budgets. When the market is even mildly competitive, the approximation ratios of this mechanism are small constants for both the revenue and welfare objectives, and in fact for welfare the approximation converges to 1 as the market becomes fully competitive. We also give an impossibility theorem, showing that truthfulness requires discarding resources and is thus incompatible with (Pareto) efficiency.

Asymptotically optimal prior-free asset market mechanisms

We develop a prior-free mechanism for an asset market that is dominant-strategy incentive compatible, ex post individually rational, constrained efficient, and asymptotically optimal—as the number of agents grows large, the designer's profit from using this mechanism approaches the profit it would optimally make if it knew the agents' type distribution at the outset. The direct implementation first identifies the agent whose value equals the Walrasian price. The second step can be described algorithmically as consisting of ascending and descending clock auctions that start from the Walrasian price, estimate virtual types, and stop eliminating trades when the estimated virtual value exceeds the estimated virtual cost. The mechanism permits partial clock auction implementation. Our approach accommodates heterogeneity among groups of traders and discrimination among these, provided heterogeneity is not too accentuated.

When Walras meets Vickrey

We consider general asset market environments in which agents with quasilinear payoffs are endowed with objects and have demands for other agents' objects. We show that if all agents have a maximum demand of one object and are endowed with at most one object, the VCG transfer of each agent is equal to the largest net Walrasian price of this agent. Consequently, the VCG deficit is equal to the sum of the largest net Walrasian prices over all agents. Generally, whenever Walrasian prices exist, the sum of the largest net Walrasian prices is a nonnegative lower bound for the deficit, implying that no dominant‐strategy mechanism runs a budget surplus while respecting agents' ex post individual rationality constraints.

Pricing algorithms in oligopoly with decreasing returns

Pricing algorithms are computerized procedures a seller may use to adapt instantaneously its price to market conditions, including to prices quoted by its rivals. These algorithms are related to the extensive use of web-collectors which contribute in many industries to identifying the best price. In such settings, price competition operates between algorithms, no longer between executives of brick and mortar companies. In this context, the question is to know how implicit forms of collusion may arise between the sellers. This paper is aimed at discussing this conceptual issue in a price-setting homogeneous product oligopoly with decreasing returns to scale where algorithms implement matching policies. Using fixed point argument, we find a family of equilibrium prices encompassing Cournot and Pareto efficient solutions, if matching is allowed upward and downward. Dynamical stability is studied in the linear demand constant return case. When matching operates only for price undercutting, this family is extended up to a bottom value of the market price, close to the Walrasian price. Pricing algorithms may solve the Bertrand–Edgeworth paradox.

Price discovery in a matching and bargaining market with aggregate uncertainty

We introduce aggregate uncertainty into a Rubinstein and Wolinsky (1985)-type dynamic matching and bilateral bargaining model. The market can be either in a high state, where there are more buyers than sellers, or in a low state, where there are more sellers than buyers. Traders do not know the state. They randomly meet each other and bargain by making take-it-or-leave-it offers. The only information transmitted in a meeting is the time a trader spent on the market. There are two kinds of search frictions: time discounting and exogenous exit. We find that as the search frictions vanish, the market discovers the competitive price quickly: the prices offered in equilibrium converge in expectation to the true-state Walrasian price at the rate linear in the total search friction. This rate is the same as it would be if the state were commonly known.

A speedy auction using approximated bidders\u2019 preferences

This paper presents a combinatorial auction, which is of particular interest when short completion times are of importance. It is based on a method for approximating the bidders’ preferences over two types of item when complementarity between the two may exist. The resulting approximated preference relation is shown to be complete and transitive at any given price vector. It is shown that an approximated Walrasian equilibrium always exists if all bidders either view the items as substitutes or complements. If the approximated preferences of the bidders comply with the gross substitutes condition, then the set of approximated Walrasian equilibrium prices forms a complete lattice. A process is proposed that is shown to always reach the smallest approximated Walrasian price vector. Simulation results suggest that the approximation procedure works well as the difference between the approximated and true minimal Walrasian prices is small.

Price Discovery in a Matching and Bargaining Market with Aggregate Uncertainty

We introduce aggregate uncertainty into a Rubinstein and Wolinsky (1985)-type dynamic matching and bilateral bargaining model. The market can be either in a high state, where there are more buyers than sellers, or in a low state, where there are more sellers than buyers. Traders do not know the state. They randomly meet each other and bargain by making take-it-or-leave-it offers. The only information transmitted in a meeting is the time a trader spent on the market. There are two kinds of search frictions: time discounting and exogenous exit. We find that as the search frictions vanish, the market discovers the competitive price quickly: the prices offered in equilibrium converge in expectation to the true-state Walrasian price at the rate linear in the total search friction. This rate is the same as it would be if the state were commonly known.

The deficit on each trade in a Vickrey double auction is at least as large as the Walrasian price gap

We prove that the deficit on each trade in a Vickrey double auction for a homogeneous good with multi-unit traders with multi-dimensional types is at least as large as the Walrasian price gap. We also show that as the number of traders grows large the aggregate deficit is bounded below by the ratio of the Walrasian price and the elasticity of excess supply at the Walrasian price.

Capacity Evaluation Strategies under Dynamic Linear Pricing

Traditionally, a fixed unit price is applied in examining capacity planning. However, using a fixed price to predict capacity requirements may not be appropriate for practical problems. The unit price will be varied due to the difference between quantity supplied and demanded, the market competition, and the consumer’s behavior. Also, different unit prices will affect the quantity demanded since increasing the unit price will decrease the quantity demanded. In this research, we examine the capacity requirements in multiple periods with Walrasian price adjustment constraints. A mathematical model is presented and the results indicate that industries might increase the unit price to reduce machine and inventory costs.

Computing Walrasian equilibria: fast algorithms and structural properties

We present the first polynomial time algorithm for computing Walrasian equilibrium in an economy with indivisible goods and general buyer valuations having only access to an aggregate demand oracle, i.e., an oracle that given prices on all goods, returns the aggregated demand over the entire population of buyers. For the important special case of gross substitute valuations, our algorithm queries the aggregate demand oracle {widetilde{O}}(n) times and takes {widetilde{O}}(n^3) time, where n is the number of goods and the {widetilde{O}}(cdot ) notation denotes an asymptotic bound up to polylogarithmic factors. Both algorithms are randomized. At the heart of our solution is a method for exactly minimizing certain convex functions which cannot be evaluated but for which the subgradients can be computed. We also give the fastest known algorithm for computing Walrasian equilibrium for gross substitute valuations in the value oracle model. Our algorithm has running time {widetilde{O}}((mn + n^3) T_V) where T_V is the cost of querying the value oracle. A key technical ingredient is to regularize a convex programming formulation of the problem in a way that subgradients are cheap to compute. En route, we give necessary and sufficient conditions for the existence of robust Walrasian prices, i.e., prices for which each agent has a unique demanded bundle and the demanded bundles clear the market. When such prices exist, the market can be perfectly coordinated by solely using prices.

Fast Convergence in the Double Oral Auction

A classical trading experiment consists of a set of unit demand buyers and unit supply sellers with identical items. Each agent’s value or opportunity cost for the item is his private information, and preferences are quasilinear. Trade between agents employs a double oral auction (DOA) in which both buyers and sellers call out bids or offers that an auctioneer recognizes. Transactions resulting from accepted bids and offers are recorded. This continues until there are no more acceptable bids or offers. Remarkably, the experiment consistently terminates in a Walrasian price. The main result of this article is a mechanism in the spirit of the DOA that converges to a Walrasian equilibrium in a polynomial number of steps, thus providing a theoretical basis for the empirical phenomenon described previously. It is well known that computation of a Walrasian equilibrium for this market corresponds to solving a maximum weight bipartite matching problem. The uncoordinated but mildly rational responses of agents thus solve in a distributed fashion a maximum weight bipartite matching problem that is encoded by their private valuations. We show, furthermore, that every Walrasian equilibrium is reachable by some sequence of responses. This is in contrast to the well-known auction algorithms for this problem that only allow one side to make offers and thus essentially choose an equilibrium that maximizes the surplus for the side making offers. Our results extend to the setting where not every agent pair is allowed to trade with each other.

Walrasian's characterization and a universal ascending auction

We introduce a novel characterization of all Walrasian price vectors in terms of over- and under demanded sets for monotone gross substitute combinatorial auctions.For ascending and descending auctions we suggest a universal framework for finding the minimum or maximum Walrasian price vectors for monotone gross substitute combinatorial auctions. An ascending (descending) auction is guaranteed to find the minimum (maximum) Walrasian if and only if it follows the suggested framework.

Dynamic Stability of Post-Keynesian Pricing

AbstractConventional economic theory assumes a Walrasian pricing mechanism that is known to pose theoretical difficulties. Less well-known is that conventional price theory conflicts with empirical studies of price-setting in industrial firms. Post-Keynesian theory, which assumes mark-up pricing on normal costs and infrequent price changes, is consistent with observation, and we show in this paper that post-Keynesian pricing, unlike conventional pricing, features stable dynamics. We focus on the short run, because post-Keynesian theory posits complex and historically-contingent long-term price dynamics. Specifically, we show that under very general conditions, prices converge to a unique equilibrium price vector.

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.

Sequential pairwise trading: convergence and welfare implications

This paper characterises the out-of-equilibrium dynamics of a symmetric, pure-exchange economy with two goods and N agents with uniformly distributed preferences and identical endowments. Relaxing the auctioneer assumption, but maintaining a global price rule, sequential random pairwise trading at out-of-equilibrium prices is allowed. Initial mispricing implies rationing, determining excess demand (supply) fading away only at convergence, when the price of the initially cheaper (more expensive) good becomes more expensive (cheaper) than the Walrasian one. The system converges when the sequential price reaches the Walrasian price evaluated at current updated endowments. A perfectly symmetric setting, by initial mispricing and consequent rationed trading, creates asymmetric allocations even at convergence, where welfare is just slightly lower than auctioneer Pareto one. This model sketches a basis for price over-reaction micro-foundations and captures endogenous 'wealth divide' among the population, induced by whether trading is dominated by preferences over goods or by speculation around their prices.

Does Initial Mispricing imply Equilibrium Price Overreaction and Wealth Divide?

We discover that letting agents pairwise sequentially exchange at wrong prices has a robust effect on prices at convergence. If the initial relative price for a good is cheaper than the equilibrium walrasian price due to initial endowments, the initial excess demand effect pushes resource allocation.This paper characterizes the out-of-equilibrium dynamics of a symmetric, pure exchange economy with two goods and N agents with uniformly distributed preferences and identical endowments. Relaxing the auctioneer assumption, but maintaining a global price rule, sequentially random pairwise trading at out-of-equilibrium prices is allowed. Initial mispricing implies rationing, determining excess demand (supply) fading away only at convergence, when the price of the initially cheaper (more expensive) good becomes more expensive (cheaper) than the walrasian one. The system converges when the sequential price reaches the walrasian price evaluated at current updated endowments. A perfectly symmetric setting, by initial mispricing and consequent rationed trading, creates asymmetric resource allocations even at convergence, where welfare is less than a standardized 1% lower than the auctioneer Pareto one.This model sketches a possible basis for price over-reaction microfoundation and captures endogenous wealth divide among the population, induced by whether agent trading is dominated by good preferences or just by speculation around their prices.

First Price Package Auction with Many Traders

This paper studies a first price package auction with many sellers in addition to many buyers in a decentralized networked market. We first show that the set of equilibrium payoffs with profit-target strategies is equal to a bidder optimal core. We then show that for buyers, the equilibrium payoff vector is Pareto-dominated by the VCG payoff vector, but Pareto-dominates theWalrasian competitive equilibrium payoff vector. This contrasts with a package auction with a single seller, in which it is shown that the unique equilibrium payoff vector is equal to the VCG payoff vector supported by a Walrasian pricing if goods are substitute. Finally, we discuss coalition-proofness of equilibria and stability of networks.