Competitive Equilibrium Price Research Articles

Solving Strong-Substitutes Product-Mix Auctions

This paper develops algorithms to solve strong-substitutes product-mix auctions: it finds competitive equilibrium prices and quantities for agents who use this auction’s bidding language to truthfully express their strong-substitutes preferences over an arbitrary number of goods, each of which is available in multiple discrete units. Our use of the bidding language and the information it provides contrasts with existing algorithms that rely on access to a valuation or demand oracle. We compute market-clearing prices using algorithms that apply existing submodular minimization methods. Allocating the supply among the bidders at these prices then requires solving a novel constrained matching problem. Our algorithm iteratively simplifies the allocation problem, perturbing bids and prices in a way that resolves tie-breaking choices created by bids that can be accepted on more than one good. We provide practical running time bounds on both price finding and allocation and illustrate experimentally that our allocation mechanism is practical. Funding: E. Baldwin and P. Klemperer were supported by the Economic and Social Research Council [Grant ES/L003058/1]. P. W. Goldberg and E. Lock were supported by a JP Morgan faculty fellowship during the work on the final version of the paper. Supplemental Material: The online companion is available at https://doi.org/10.1287/moor.2019.0248 .

Local Energy Markets*

In current power markets, the bulk of electricity is sold wholesale and transported to consumers via long‐distance transmission lines. Recently, decentralized local energy markets have evolved, often as isolated networks based on solar generation. We analyze strategic pricing, investment, and welfare in local energy markets. We show that local energy markets yield competitive equilibrium prices and provide efficient investment incentives. Yet, we find that strategic behavior leads to allocative inefficiency. We propose a clearing mechanism that induces truth‐telling behavior and restores first‐best welfare.

How to trade thirty thousand products: A wholesale market design for road capacity

One of the core reasons for urban traffic congestion is the mispricing of the traffic’s main resource, the scarce road capacity. Recent proposals suggest a two-stage market for road capacity similar to other major utilities markets such as those for electricity. In such markets an Independent System Operator (ISO) provides road infrastructure which multiple Service Providers (SP) then resell to end consumers. The size of such markets in terms of the number of traded items, road segments per hour, is exceptional. A road network in a city consists of tens of thousands different road segments and it is unclear how a wholesale market for service providers should be designed to keep it tractable for buyers and sellers. Three problems arise in this context: First, how can a service provider express preferences for so many products. Second, the optimization problem is very large and it is far from obvious that problems of this size can be solved. Finally, the allocation problem is non-convex and finding competitive equilibrium prices on such markets is impossible in general. We suggest a parsimonious bid language based on origin–destination pairs and introduce a mixed-integer optimization problem to maximize welfare on such markets. Based on the MATSim traffic simulator, we show that one can solve realistic problems based on traffic data for the city of Berlin to optimality within 15 minutes. Importantly, we approximate competitive equilibrium prices which require only negligible side-payments as the allocation problem is “almost” convex. Overall, the paper provides a tangible proposal for wholesale markets for road capacity.

Strong substitutes: structural properties, and a new algorithm for competitive equilibrium prices

We show the strong substitutes product-mix auction bidding language provides an intuitive and geometric interpretation of strong substitutes as Minkowski differences between sets that are easy to identify. We prove that competitive equilibrium prices for agents with strong substitutes preferences can be computed by minimizing the difference between two linear programs for the positive and the negative bids with suitably relaxed resource constraints. This also leads to a new algorithm for computing competitive equilibrium prices which is competitive with standard steepest descent algorithms in extensive experiments.

MONEY METRICS IN APPLIED WELFARE ANALYSIS: A SADDLEPOINT REHABILITATION

AbstractOnce a popular tool to estimate welfare changes, the money metric of McKenzie‐Samuelson gradually faded from use after welfare theorists and practitioners argued that it led to inegalitarian recommendations. We prove that, at a competitive equilibrium price, any associated competitive allocation maximizes the money‐metric sum; and, as is well understood, competitive allocations can be egalitarian or inegalitarian. The result applies to economies in which individual demand is not rationalizable by any binary relation, let alone a binary relation representable by a utility function, a behavioral setting considered, for example, in Bernheim–Rangel.

Pricing models of two-sided markets incorporating service quality

Purpose This paper aims to study the pricing problems on the two-sided market for cases of monopoly and duopoly competition, specifically investigating the impact of platform service quality on the market. Theoretical analysis and computational studies are conducted to investigate the impact of different parameters on the system outcomes. Design/methodology/approach Mathematical formulations are proposed for cases of monopoly and duopoly competition. For monopolistic market, the optimal pricing and service quality strategies are obtained using mathematical programming method. For duopolistic market, the equilibrium outcomes are derived by game theory. Sensitivity analysis and numerical studies are also adopted to investigate the impact of different parameters. Findings For monopolistic market, the platform will provide a low service quality when the service cost parameter is large. However, when the cost parameter is small, the platform provides a higher service quality and higher registration prices. Furthermore, the sum of the optimal prices is proportional to the service quality and inversely proportional to the user price sensitivity. For duopolistic market, the competitive equilibrium prices exist under a certain condition. The determinants of equilibrium prices are the gap between the service qualities of two platforms and the cross-group externalities. Originality/value For monopolistic market, this paper specifies the role of platform service quality in determining the platform’s pricing strategy. For duopolistic market, this paper presents a market sharing mechanism between two platforms and explores the equilibrium pricing strategies for platforms with different service quality level.

Are Truthful Bidders Paying too Much? Efficiency and Revenue in Display Ad Auctions

Display ad auctions have become the predominant means to allocate user impressions on a website to advertisers. These auctions are conducted in milliseconds online, whenever a user visits a website. The impressions are typically priced via a simple second-price rule. For single-item auctions, this Vickrey payment rule is known to be incentive-compatible. However, it is unclear whether bidders should still bid truthful in an online auction where impressions (or items) arrive dynamically over time and their valuations are not separable, as is the case with campaign targets or budgets. The allocation process might not maximize welfare and the payments can differ substantially from those paid in an offline auction with a Vickrey-Clarke-Groves (VCG) payment rule or also competitive equilibrium prices. We study the properties of the offline problem and model it as a mathematical program. In numerical experiments, we find that the welfare achieved in the online auction process with truthful bidders is high compared to the theoretical worst-case efficiency, but that the bidders pay significantly more on average compared to what they would need to pay in a corresponding offline auction in thin markets with up to four bidders. However, incentives for bid shading in these second-price auctions decrease quickly with additional competition and bidders risk losing.

A Matter of Equality: Linear Pricing in Combinatorial Exchanges

Combinatorial exchanges that allow for package offers to address nonconvexities in demand or supply typically employ linear and anonymous prices because they are simple, tractable, and fair. Despite their prevalence, linear anonymous prices do not necessarily correspond to Walrasian competitive equilibrium prices in such settings, and their impact is not well understood. This paper is the first to analyze the effect of different pricing rules on the efficiency of combinatorial exchanges, using both analytic methods and numerical experiments. Our analysis is motivated by a combinatorial fishery-rights exchange designed to reform the fishing industry in New South Wales (NSW), Australia. We find that when linearity and anonymity are required for only one side of the market, the average efficiency loss is negligible. In contrast, with a single linear price vector for both sides, the efficiency loss is substantial, especially when the market is small. In a formal model, we show that efficiency losses decrease when the number of buyers grows or the size of the submitted packages decreases. Besides the reform of the NSW fishing industry, our results have important implications for other cap-and-trade programs as well as other industries where demand or cost complementarities play a role.

A simple model of production and trade in an oligopolistic market: back to basics

Abstract We provide a two good model of oligopolistic production and trade with one good being commodity money. There is the usual demand function of the consumers for the produced good that producer-sellers face. Each seller is a budget constrained preference maximizer and derives utility (or satisfaction) from consuming bundles comprising commodity money and the produced good. We define a competitive equilibrium strategy profile and a Cournotian equilibrium and show that under our assumptions both exist. We further show that at a competitive equilibrium strategy profile, each seller maximizes profits given his own consumption of the produced good and the price of the produced good, the latter being determined by the inverse demand function. Similarly we show that at a Cournotian the sellers are at a Cournot equilibrium given their own consumption of the produced good. Assuming sufficient differentiability of the cost functions we show that at a competitive equilibrium each seller either sets price equal to marginal cost or exhausts his capacity of production; at a Cournotian equilibrium each seller either sets marginal revenue equal to marginal cost or exhausts his capacity of production. We also study the evolution of Cournotian strategies as the sellers and buyers are replicated. As the number of buyers and sellers go to infinity any sequence of interior symmetric Cournotian equilibrium strategies admits a convergent subsequence, which converges to an interior symmetric competitive equilibrium strategy. In a final section we discuss the Bertrand Edgeworth price setting game and show that a Bertrand Edgeworth equilibrium must be a derived from a competitive equilibrium price. Here we show that if at a symmetric competitive equilibrium, the sellers consume positive quantities of the produced good then the competitive equilibrium cannot be a Bertrand Edgeworth equilibrium. Thus, if at all symmetric competitive equilibria the sellers consume positive amounts of the produced good, then a Bertrand Edgeworth equilibrium simply does not exist.

One‐seller assignment markets with multi‐unit demands: Core and competitive equilibrium

AbstractWe consider an assignment market with one seller who owns several indivisible heterogeneous goods and many buyers each willing to buy up to a given capacity. In this market, the core contains the Vickrey payoff vector. However, core allocations may not be supported by competitive equilibrium prices, even in a finite replication of the market. We first characterize the convexity of the associated coalitional game and we show that it is a sufficient condition so that the buyers‐optimal core allocation is competitive. With respect to the seller‐optimal core allocation, we provide a characterization of competitiveness by means of buyers’ valuations. In addition, we characterize in terms of the valuation matrix the coincidence between the core and the set of competitive equilibrium payoff vectors.

Line Balancing in Parallel M/M/1 Lines and Loss Systems as Cooperative Games

We consider production and service systems that consist of parallel lines of two types: (i) M/ M/1 lines and (ii) lines that have no buffers (loss systems). Each line is assumed to be controlled by a dedicated supervisor. The management measures the effectiveness of the supervisors by the long run expected cost of their line. Unbalanced lines cause congestion and bottlenecks, large variation in output, unnecessary wastes and, ultimately, high operating costs. Thus, the supervisors are expected to join forces and reduce the cost of the whole system by applying line‐balancing techniques, possibly combined with either strategic outsourcing or capacity reduction practices. By solving appropriate mathematical programming formulations, the policy that minimizes the long run expected cost of each of the parallel‐lines system, is identified. The next question to be asked is how to allocate the new total cost of each system among the lines' supervisors so that the cooperation's stability is preserved. For that sake, we associate a cooperative game to each system and we investigate its core. We show that the cooperative games are reducible to market games and therefore they are totally balanced, that is, their core and the core of their subgames are non‐empty. For each game a core cost allocation based on competitive equilibrium prices is identified.

Coarse Competitive Equilibrium and Extreme Prices

We introduce a notion of coarse competitive equilibrium, to study agents’ inability to tailor their consumption to prices. Our goal is to incorporate limited cognitive ability (in particular limited attention, memory, and complexity) into the analysis of competitive equilibrium. Compared to standard competitive equilibrium, our concept yields more extreme prices and, when all agents have the same endowment, riskier allocations. We provide a tractable model suitable for general equilibrium analysis as well as asset pricing. (JEL D11, D51, D91, G10)

On the behavior of commodity prices when speculative storage is bounded

This paper investigates the implications of bounded speculative storage, storage bounded from below at zero and above at a capacity, on commodity prices. Binding capacity mirrors the non-negativity constraint on storage and leads to negative price spiking and higher volatility when the market is in deep contango, i.e. low current prices at high stock levels. With bounded storage there is no need to restrict storage to be costly to ensure a rational expectations equilibrium. This allows the model to cover a wide range of storage technologies, including free and productive storage. We also provide an alternative expression for speculative prices that highlights the key role of the storage boundaries. The competitive equilibrium price is the sum of discounted future probability weighted boundary prices. The boundary prices can be viewed as dividends on commodities in storage reflecting the realization of economic profits from storage.

Why do restaurants serve lunch cheaper than dinner?

The article explores a question why restaurants serve lunch meals cheaper than dinner meals. Departing from traditional demand curves derived from utility functions, the article uses a distribution function of reservation prices of patrons to represent a market demand curve. Given identical convex market demand curves in two distinct markets isolated by the time, restaurants exploit the convexity of the market demand curves by setting two prices whose average price is higher than the competitive market equilibrium price. This exploitation opportunity induces restaurants to adopt oligopolistic pricing behaviours.

Stability and competitive equilibria in multi-unit trading networks with discrete concave utility functions

Hatfield, Kominers, Nichifor, Ostrovsky, and Westkamp showed the existence of stable outcomes and competitive equilibria in a model of trading networks under the assumption that all agents’ preferences satisfy a condition called the full substitutes condition. In this paper, we extend their model by using discrete concave utility functions called twisted \(\hbox {M}^{\natural }\)-concave functions. We show that a valuation function of an agent is twisted \(\hbox {M}^{\natural }\)-concave if and only if the agent’s preference satisfies the generalized variant of the full substitutes condition. We also show that under the generalized full substitutes condition, there exist stable outcomes and competitive equilibria in the extended model and the set of competitive equilibrium price vectors forms a lattice. In addition, we discuss the connection among competitive equilibria, stability, and efficiency. Finally, we investigate the relationship among stability, strong group stability, and chain stability and verify these three stability concepts are equivalent as long as valuation functions of all agents are twisted \(\hbox {M}^{\natural }\)-concave.

Competitive economy as a ranking device over networks

We propose a novel approach to generating a ranking of items in a network (e.g., of web pages connected by links or of articles connected by citations). We transform the network into an exchange economy, and use the resulting competitive equilibrium prices of the network nodes as their ranking. The widely used Google's PageRank comes as a special case when the nodes are represented by Cobb–Douglas utility maximizers. We further use the economic metaphor to combine between the Citation Count and PageRank by imposing a redistributive taxing scheme.

Price Reaction to Information with Heterogeneous Beliefs and Wealth Effects: Underreaction, Momentum, and Reversal

This paper analyzes how asset prices in a binary market react to information when traders have heterogeneous prior beliefs. We show that the competitive equilibrium price underreacts to information when there is a bound to the amount of money traders are allowed to invest. Underreaction is more pronounced when prior beliefs are more heterogeneous. Even in the absence of exogenous bounds on the amount that traders can invest, prices underreact to information provided that traders become less risk averse as their wealth increases. In a dynamic setting, underreaction results in initial momentum and then reversal in the long run. (JEL D83, D84, G11, G12, G14)

Rational Bubble on Interest-Bearing Assets

This paper compares fiat money and a Lucas' tree in an overlap- ping generations model. A Lucas' tree with a positive dividend has a unique competitive equilibrium price. Moreover, the price converges to the monetary equilibrium value of fiat money as the dividend goes to zero in the limit. Thus, the value of liquidity represented by a ra- tional bubble is part of the fundamental price of a standard interest- bearing asset. A Lucas' tree has multiple equilibrium prices if the dividend vanishes permanently with some probability. This case may be applicable to public debt, but not to stock or urban real estate.

DESIGNED TO FAIL: THE MEDICARE AUCTION FOR DURABLE MEDICAL EQUIPMENT

We examine the theoretical properties of the auction for Medicare Durable Medical Equipment. Two unique features of the Medicare auction are (1) winners are paid the median winning bid and (2) bids are nonbinding. We show that median pricing results in allocation inefficiencies as some high‐cost firms potentially displace low‐cost firms as winners. Further, the auction may leave demand unfulfilled as some winners refuse to supply because the price is set below their cost. We also introduce a model of nonbinding bids that establishes the rationality of a lowball bid strategy employed by many bidders in the actual Medicare auctions and recently replicated in Caltech experiments. We contrast the median‐price auction with the standard clearing‐price auction where each firm bids true costs as a dominant strategy, resulting in competitive equilibrium prices and full efficiency. (JEL D44, I11, H57)

A Simple Cournotian Model of Production and Trade

We provide a two good model of oligopolistic production and trade with one good being commodity money. There is the usual demand function of the consumers for the produced good that producer-sellers face. Each seller is a budget constrained preference maximizer and derives utility (or satisfaction) from consuming bundles comprising commodity money and the produced good. We define a competitive equilibrium strategy profile and a Cournotian equilibrium and show that under our assumptions both exist. We further show that at a competitive equilibrium strategy profile, each seller maximizes profits given his own consumption of the produced good and the price of the produced good, the latter being determined by the inverse demand function. Similarly we show that at a Cournotian the sellers are at a Cournot equilibrium given their own consumption of the produced good. Assuming sufficient differentiability of the cost functions we show that at a competitive equilibrium each seller either sets price equal to marginal cost or exhausts his capacity of production; at a Cournotian equilibrium each seller either sets marginal revenue equal to marginal cost or exhausts his capacity of production. We also study the evolution of Cournotian strategies as the sellers and buyers are replicated. As the number of buyers and sellers go to infinity any sequence of interior symmetric Cournotian equilibrium strategies admits a convergent subsequence, which converges to an interior symmetric competitive equilibrium strategy. In a final section we discuss the Bertrand Edgeworth price setting game and show that a Bertrand Edgeworth equilibrium must be a derived from a competitive equilibrium price. Here we show that if at a symmetric competitive equilibrium, the sellers consume positive quantities of the produced good then the competitive equilibrium cannot be a Bertrand Edgeworth equilibrium. Thus, if at all symmetric competitive equilibria the sellers consume positive amounts of the produced good, then a Bertrand Edgeworth equilibrium simply does not exist.