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This paper considers a chief interface between mathematical programming and economics, namely: money-based trade of perfectly divisible and transferable goods. Three important and related features are singled out here: first, convexity enters via acceptable payments, second, convolution of monetary criteria secures Pareto efficiency, and third, competitive equilibrium obtains when agents' subdifferentials intersect. [ABSTRACT FROM AUTHOR]

The refinement and formalization of equilibrium concepts mark the establishment of game theory as a distinct discipline. The development of game theory has been centered around the fundamental properties of various equilibrium concepts. It is generally accepted that the nonexistence of equilibrium is seen as a negative outcome, impeding the advancement of equilibrium research. This holds true for economic research as well. This paper illustrates, through two examples from the literature on non-cooperative games and perfectly competitive markets, that sometimes valuable interpretations can be provided for the nonexistence of equilibrium. The first example examines the evolution of fashion phenomena through a network game based on matching pennies, where the nonexistence of equilibrium is used to interpret the emergence of fashion cycles. The second example discusses the matching problem between companies and workers in a perfectly competitive labor market, where the nonexistence of equilibrium is used to explain the phenomenon of early contracting. Additionally, we briefly introduce Shapley's insightful interpretation regarding the empty core in transferable utility cooperative games. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Can order markets lead participants towards price-taking equilibrium? Viewing market sessions as steps of iterative algorithms, this paper indicates positive prospects for convergence. Mathematical arguments turn on convolution, efficiency and generalized gradients. Economic arguments revolve around reservation costs, derived from indifference or threshold payments for quantities supplied or demanded. [ABSTRACT FROM AUTHOR]

Platforms competition affects both the structure of the whole industry and the fate of the individual companies within it. The extant studies either simply divide the participants of a platforms war into suppliers and consumers or adopt unbounded rationality assumptions. This paper constructs a game model that includes platforms, suppliers, and the final consumers. We analyze the condition and stability of each equilibrium and the impacts of the characteristics of the core and complementary products on the equilibrium by using the method of evolutionary game. Our results show the following: (1) If the two platforms initially have separate advantages in terms of the number of suppliers or consumers, there will be an equilibrium of the coexistence; otherwise, there will be an equilibrium of single platform dominating. (2) The equilibrium of a single platform dominating is more stable than that of the coexistence of multiple platforms. (3) Incidental shocks to the number of suppliers are more likely to break the coexistence equilibrium than consumers. (4) Platforms with poor core product quality and a high price for core and complementary products can coexist with other superior platforms only if the former has more consumers and suppliers at the beginning of the competition. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

It is widely considered that a (perfectly) competitive equilibrium cannot survive to the existence of fixed costs because firms generate losses in equilibrium. In this theoretical and methodological article, I demonstrate that this statement is not valid by developing some counter-examples. In particular, I clearly show that a competitive equilibrium and fixed costs are tenable, depending on the ownership structure of models. I then delimit the role of fixed costs in macroeconomic models. Notably, I find that fixed costs can improve the level of aggregate output in the long run. [ABSTRACT FROM AUTHOR]

The global macroeconomic shocks of the last decade entail the restructuring of national production networks and induce processes of input substitution. We suggest mathematical tools of Young duality for variational inequalities for studying these processes. Based on the tools we provide, a new mathematical model of a production network with several final consumers is created. The model is formulated as a pair of conjugated problems: a complementarity problem for optimal resource allocation with neoclassical production functions and the Young dual problem for equilibrium price indices on network products. The solution of these problems gives an equilibrium point in the space of network inter-industry flows and price indices on goods. Based on our previous results, we suggest an algorithm for model identification with an official economic statistic in the case of constant elasticity of substitution production functions. We give an explicit solution to the complementarity problems in this case and develop the algorithm of the inter-industry flows scenario projection. Since the algorithm needs the scenario projection of final sales structure as its input, we suggest a modified methodology that allows the calculation of scenario shifts in final consumer spending. To do this, we employ the generalized nonparametric method of demand analysis. As a result, we develop new technology for scenario calculation of a national input–output table, including shifts in final consumer spending. The technology takes into account a substitution of inputs in the network and is based on officially published national statistics data. The application of the methodology to study tax collection scenarios for Kazakhstan's production network is demonstrated. [ABSTRACT FROM AUTHOR]

Input–output analysis finds widespread application in estimating the shock effects on production networks within both local and global economies. We are developing a new technology for intersectoral analysis that takes into account the substitution of production factors within a complex supply network triggered by external or internal shocks. This technology is based on the explicit solution of a pair of convex programming problems: the resource allocation problem under the assumption of Constant Elasticity of Substitution (CES) technologies and the special dual Young problem. Solving these problems, we can ascertain the equilibrium inputs and price indexes of goods within the production network. In this paper, we apply this technology to analyze the economy of Kazakhstan in the context of the COVID-19 pandemic. Our calculations provide us with the means to discuss the macroeconomic responses of the multi-sectoral production network in Kazakhstan to both external and internal shocks stemming from the pandemic. [ABSTRACT FROM AUTHOR]

Summaries by Nobel laureate Kenneth J. Arrow of his 1962 lectures on general equilibrium theory at Northwestern University.   These summaries were widely circulated but unpublished.  Reissued here, edited with updated notation. With permission of the Arrow family.

We study the fair division problem of allocating a mixed manna under additively separable piecewise linear concave (SPLC) utilities. A mixed manna contains goods that everyone likes and bads (chores) that everyone dislikes as well as items that some like and others dislike. The seminal work of Bogomolnaia et al. argues why allocating a mixed manna is genuinely more complicated than a good or a bad manna and why competitive equilibrium is the best mechanism. It also provides the existence of equilibrium and establishes its distinctive properties (e.g., nonconvex and disconnected set of equilibria even under linear utilities) but leaves the problem of computing an equilibrium open. Our main results are a linear complementarity problem formulation that captures all competitive equilibria of a mixed manna under SPLC utilities (a strict generalization of linear) and a complementary pivot algorithm based on Lemke's scheme for finding one. Experimental results on randomly generated instances suggest that our algorithm is fast in practice. Given the PPAD -hardness of the problem, designing such an algorithm is the only non–brute force (nonenumerative) option known; for example, the classic Lemke–Howson algorithm for computing a Nash equilibrium in a two-player game is still one of the most widely used algorithms in practice. Our algorithm also yields several new structural properties as simple corollaries. We obtain a (constructive) proof of existence for a far more general setting, membership of the problem in PPAD , a rational-valued solution, and an odd number of solutions property. The last property also settles the conjecture of Bogomolnaia et al. in the affirmative. Furthermore, we show that, if the number of either agents or items is a constant, then the number of pivots in our algorithm is strongly polynomial when the mixed manna contains all bads. Funding: Financial support from the Division of Computing and Communication Foundations, National Science Foundation (NSF) [Grants 1942321, 1750436] is gratefully acknowledged. [ABSTRACT FROM AUTHOR]

We introduce a novel simulated certainty equivalent approximation (SCEQ) method for solving dynamic stochastic problems. Our examples show that SCEQ can quickly solve high‐dimensional finite‐ or infinite‐horizon, stationary or nonstationary dynamic stochastic problems with hundreds of state variables, a wide state space, and occasionally binding constraints. With the SCEQ method, a desktop computer will suffice for large problems, but it can also use parallel tools efficiently. The SCEQ method is simple, stable, and can utilize any solver, making it suitable for solving complex economic problems that cannot be solved by other algorithms. [ABSTRACT FROM AUTHOR]

This paper studies matching markets in the presence of middlemen. In our framework, a buyer–seller pair may either trade directly or use the services of a middleman; and a middleman may serve multiple buyer–seller pairs. For each such market, we examine the associated TU game. We first show that, in our context, an optimal matching can be obtained by considering the two-sided assignment market where each buyer–seller pair is allowed to use the mediation services of any middleman free of charge. Second, we prove that matching markets with middlemen are totally balanced: in particular, we show the existence of a buyer-optimal (seller-optimal) core allocation where each buyer (seller) receives her marginal contribution to the grand coalition. In general, the core does not exhibit a middleman-optimal allocation, not even when there are only two buyers and two sellers. However, we prove that in these small markets the maximum core payoff to each middleman is her marginal contribution. Finally, we establish the coincidence between the core and the set of competitive equilibrium payoff vectors. [ABSTRACT FROM AUTHOR]

Electricity markets are characterized by large price changes between peak periods with large consumption and night periods with low consumption. In this situation, an energy storage can play a useful role for the consumer: at low prices, the consumer stores energy in it, and uses it at high prices; in those periods when he has a high need for energy. In this work, a mathematical model has been developed for calculating the optimal volumes of introducing different types of storages, as well as the optimal expansion of the power grid, based on the given structure of capacities and demand for electricity. A two-level market model is studied: the first level is intended for the analysis of short-term decision-making tasks by economic agents; the second level is for calculating the optimal infrastructure parameters. The parameters of storages and the transmission capacities are chosen based on maximizing the average values of the profits of corresponding economic agents. These profits are computed based on the first-level model, taking into account the costs of construction and the expected operating periods of infrastructure facilities. [ABSTRACT FROM AUTHOR]

In this paper, we prove the non-emptiness of the fuzzy core in a finite coalition production economy under some standard assumptions which generalizes the results by Boehm [Rev Econ Stud (1974) 41: 429-436] and Predtetchinski [Economic Theory (2005) 26: 717-724], where we allow the utility functions to be possibly discontinuous. We also derive the non-emptiness of α -cores for production economies with possibly discontinuous payoff functions which implies the well-known α -core existence theorem by Scarf [Journal of Economic Theory (1971) 3: 169-181], and derive the non-emptiness of fuzzy α -cores for production economies with continuous payoff functions. [ABSTRACT FROM AUTHOR]

For an exchange economy with a continuum of traders and a finite-dimensional commodity space under some standard assumptions, Aumann showed that the core and the set of competitive allocations (two most important solutions) coincide and Mas-Colell proved that the bargaining set and the set of competitive allocations coincide. However, in the case of exchange economies with a finite number of traders, it is well-known that the set of competitive allocations could be a strict subset of the core which can also be a strict subset of the bargaining set. In this paper, we establish the equivalence of the fuzzy core, the fuzzy bargaining set (or Aubin bargaining set), and the set of competitive allocations in a finite coalition production economy with an infinite-dimensional commodity space under some standard assumptions. We first derive a continuous equivalence theorem and then discretize it to obtain the desired equivalence in finite economies. [ABSTRACT FROM AUTHOR]

Abstract By the first welfare theorem, competitive market equilibria belong to the core and hence are Pareto optimal. Letting money be a commodity, this paper turns these two inclusions around. More precisely, by generalizing the second welfare theorem we show that the said solutions may coincide as a common fixed point for one and the same system. Mathematical arguments invoke conjugation, convolution, and generalized gradients. Convexity is merely needed via subdifferentiablity of aggregate “cost”, and at one point only. Economic arguments hinge on idealized market mechanisms. Construed as algorithms, each stops, and a steady state prevails if and only if price-taking markets clear and value added is nil.

In their seminal 1954 paper on the existence of competitive equilibrium, Arrow and Debreu state what they call an "important principle", namely that it is necessary for the existence of equilibrium that every consumer has some asset or can supply some labour service which has a positive price at equilibrium. It does not seem to have been noticed that this claim is incorrect. We provide a very simple model of a private ownership economy with three goods where a competitive equilibrium exists, but consumers who have nothing to sell but their labour end up with zero wealth in equilibrium. As zero wealth must be taken to mean non-survival, and the Arrow–Debreu model is frequently interpreted as assuming that all consumers can survive without trade, we also briefly discuss the issue of non-survival in equilibrium. We finally point out that our example illustrates the possibility that technological progress may result in a situation where the value of work becomes negligible. [ABSTRACT FROM AUTHOR]

This paper studies optimal index design to facilitate both hedging and alleviate illegal manipulation in a competitive equilibrium paradigm, modified to deal with manipulation. Specifically, a large trader is trading both derivatives and assets that effectively hides her trades behind the competitive market clearing mechanism. Unlike the strategic game paradigm, a volume-weighted average pricing (VWAP) index introduces basis risk and encourages manipulation because of the additional randomness in volume weight and the greater price impact enjoyed by the large trader. In contrast, an equal-weighted average pricing (EWAP) index preserves market completeness and discourages manipulation. [ABSTRACT FROM AUTHOR]

It is well known that linear prices supporting a competitive equilibrium exist in the case of convex markets, however, in the presence of integralities this is open and hard to decide in general. We present necessary and sufficient conditions for the existence of such prices for decentralized market problems where market participants have integral decision variables and their feasible sets are given in complete linear description. We utilize total unimodularity and the aforementioned conditions to show that such linear prices exist and present some applications. Furthermore, we compute competitive equilibria for two classes of decentralized market problems arising in energy markets and show that competitive equilibria may exist regardless of integralities. [ABSTRACT FROM AUTHOR]

Several structural results for the set of competitive equilibria in trading networks with frictions are established: The lattice theorem, the rural hospitals theorem, the existence of side‐optimal equilibria, and a group‐incentive‐compatibility result hold with imperfectly transferable utility and in the presence of frictions. While our results are developed in a trading network model, they also imply analogous (and new) results for exchange economies with combinatorial demand and for two‐sided matching markets with transfers. [ABSTRACT FROM AUTHOR]

We provide a version of the Coase theorem within a general equilibrium framework. We consider an economy with other-regarding preferences, and where rights, licenses, or permissions are required to use, consume or transform some specific commodities. These permissions are initially allocated among consumers and, as the commodities, can be costlessly traded. In this scenario, we define different veto mechanisms and the corresponding core solutions that, naturally, result in the same set of efficient allocations. Our final result sets sufficient conditions on preferences and the requirement of rights to ensure that any equilibrium allocation belongs to the core and, in particular, is efficient. [ABSTRACT FROM AUTHOR]

Due to the small number of participants in realistic cases, the Nash equilibrium generated by an oligopolistic market is more appropriate for depicting the actual energy market than the competitive equilibrium generated by a competitive market. However, the market efficiency decreases when the Nash equilibrium is used rather than the competitive equilibrium because of the strategic behaviors of the participants. In this paper, we propose two new mechanisms (subsidization and punishment) for each generator, both of which drive the Nash equilibrium to the competitive equilibrium in heterogeneous oligopolistic energy markets with a uniform pricing mechanism. Under the two proposed mechanisms, the equivalence of the Nash equilibrium and the competitive equilibrium is strictly proven. Furthermore, we apply a distributed algorithm to numerically confirm that both mechanisms perfectly eliminate the market efficiency loss. Finally, these two mechanisms are applied to the energy market under three different stages of development. Simulation results illustrate that the additional costs caused by the mechanisms converge to a low value and that the subsidization mechanism slightly outperforms the punishment mechanism.

We give the statement of the real-time adaptive enterprise resource management problem and provide examples of contemporary adaptive resource management problems in various fields of application, showing the dimension and other features of the problems being solved. Requirements for the solution of the problems under consideration are stated, and general principles improving the resource management adaptability are proposed. Existing approaches are briefly analyzed, and their limitations are highlighted. A new methodology for solving the problems under consideration is proposed, an overview of developments is given, and the first experience of applying this methodology to the creation of intelligent resource management systems is discussed. The possibility of solving extremely complex resource management problems is shown by means of a modified ontology-based concept of the network of needs and capacities. In the framework of this concept, the schedule is constructed as a "competitive equilibrium" in the virtual market of a unified multiagent system, tuned to a specific enterprise with the use of applied ontologies. An example of developing a prototype of an intelligent resource management system for a multisatellite Earth remote sensing constellation and the results of its experimental study are given. An approach to assessing the performance and efficiency of adaptive scheduling models and methods developed for real-time resource management is proposed. Prospects for further development of the approach to solving complex adaptive resource management problems are shown. [ABSTRACT FROM AUTHOR]

We show that debt is sustainable at a competitive equilibrium based solely on the reputation for repayment; that is, even without collateral or legal sanctions available to creditors. In an incomplete asset market, when the rate of interest falls recurrently below the rate of growth of the economy, self‐insurance is more costly than borrowing, and repayments on loans are enforced by the implicit threat of loss of the risk‐sharing advantages of debt contracts. Private debt credibly circulates as a form of inside money, and it is not valued as a speculative bubble. Competitive equilibria with self‐enforcing debt exist under a suitable hypothesis of gains from trade. [ABSTRACT FROM AUTHOR]

Motivated by the observed beef and livestock market impacts of coronavirus-caused meatpacking plant shutdowns, this paper constructs a theoretical model to study the effects of closures among downstream producers on both consumers and upstream producers. It analyzes the pre-shock long-run equilibrium, the effects of firm shutdowns among the downstream producers, the effects of government subsidies, and the new long-run equilibrium if the underlying cause for firm shutdowns persists. [ABSTRACT FROM AUTHOR]