Economic Equilibrium Problems Research Articles

Negative electricity prices as a signal for lacking flexibility? On the effects of demand flexibility on electricity prices

PurposeThe purpose of this paper is to examine how active consumers, i.e. consumers that can inter-temporally shift their load, can influence electricity prices. As demonstrated in this paper, inter-temporal load shifting can induce negative electricity prices, a recurring phenomenon on power exchanges.Design/methodology/approachThe paper presents a novel electricity-market model assuming a nodal-pricing, energy-only spot market with active consumers. This study formulates an economic equilibrium problem as a linear program and uses an established six-node case study to compare equilibrium prices of a model with inflexible demand to a model with flexible demand of active consumers.FindingsThis study illustrates that temporal coupling of hourly market clearing through load shifting of active consumers can cause negative electricity prices that are not observed in a model with ceteris paribus inflexible demand. In such situations, where compared to the case of inflexible demand more flexibility is available in the system, negative electricity prices signal lower total system costs. These negative prices result from the use of demand flexibility, which, however, cannot be fully exploited due to limited transmission capacities, respectively, loop-flow restrictions.Originality/valueLiterature indicates that negative electricity prices result from lacking flexibility. The results illustrate that active consumers and their additional flexibility can lead to negative electricity prices in temporally coupled markets, which in general contributes to increased system efficiency as well as increased use of renewable energy sources. These findings extend existing research in both the area of energy flexibility and causes for negative electricity prices. Therefore, policymakers should be aware of such (temporal coupling) effects and, e.g. continue to allow negative electricity prices in the future that can serve as investment signals for active consumers.

Infinite dimensional tensor variational inequalities with applications to an economic equilibrium problem

In this paper, we present a general oligopolistic market equilibrium model in which each firm produces several commodities in a time interval. To this aim, we introduce tensor variational inequalities in Hilbert spaces which are a powerful tool to analyse the model. Indeed we characterize the equilibrium condition by means of a suitable time-dependent tensor variational inequality. In addition, we prove some existence and regularity results and a numerical scheme to compute the solution. Finally we provide a numerical example.

Quasivariational inequalities for dynamic competitive economic equilibrium problems in discrete time

Quasivariational inequalities for dynamic competitive economic equilibrium problems in discrete time

Polyhedral complementarity problem with quasimonotone decreasing mappings

The fixed point problem of piecewise constant mappings in Rn is investigated. This is a polyhedral complementarity problem, which is a generalization of the linear complementarity problem. Such mappings arose in the author?s research on the problem of economic equilibrium in exchange models, where mappings were considered on the price simplex. The author proposed an original approach of polyhedral complementarity, which made it possible to obtain simple algorithms for solving the problem. The present study is a generalization of linear complementarity methods to related problems of a more general nature and reveals a close relationship between linear complementarity and polyhedral complementarity. The investigated method is an analogue of the well-known Lemke method for linear complementarity problems. A class of mappings is described for which the process is monotone, as it is for the linear complementarity problems with positive principal minors of the constraint matrix (class P). It is shown that such a mapping has always unique fixed point.

Decomposition approach for solving large-scale spatially disaggregated economic equilibrium problems

This paper employs the Dantzig-Wolfe decomposition procedure to solve large-scale economic equilibrium problems formulated as nonlinear programs with block-diagonal linear constraints, where each block characterizes the supply possibilities in a region, and a set of unifying constraints that characterize the supply-demand balances. We derive lower and upper bounds for the value of the objective function at each step of the decomposition procedure, and use the percentage deviation between the two bounds as guidance for terminating the iterations to obtain an approximation of the equilibrium solution. Our computational results with moderate-size problems show that the decomposition procedure can reduce the solution time substantially compared to the direct solution approach without using decomposition. We present a large-scale empirical application where the impacts of the US biofuel mandates on agricultural and transportation fuel sectors were analyzed. Two powerful optimization solvers could not handle the problem due to the sheer size of the model and nonlinearity involved in the objective function, whereas we could solve the economic equilibrium successfully using the decomposition approach.

A stochastic variational approach to study economic equilibrium problems under uncertainty

This paper focuses on the analysis of an economic equilibrium model under time and uncertainty by using a stochastic variational inequality approach. Such an approach allows to capture, in a finite set of stages, the evolutionary aspects of the problem in response to an increasing level of information.

Stochastic variational formulation for a general random time-dependent economic equilibrium problem

In the paper, in a Hilbert space setting, a random time-dependent oligopolistic market equilibrium problem in presence of both production and demand excesses is studied and the random time-dependent Cournot–Nash equilibrium principle by means of a stochastic variational inequality is characterized. Then, some existence results to such problem are established and the stochastic continuity of the equilibrium solution is proved. Moreover a simple numerical example illustrates the theoretical results.

Optimal stochastic extragradient schemes for pseudomonotone stochastic variational inequality problems and their variants

We consider the stochastic variational inequality problem in which the map is expectation-valued in a component-wise sense. Much of the available convergence theory and rate statements for stochastic approximation schemes are limited to monotone maps. However, non-monotone stochastic variational inequality problems are not uncommon and are seen to arise from product pricing, fractional optimization problems, and subclasses of economic equilibrium problems. Motivated by the need to address a broader class of maps, we make the following contributions: (i) We present an extragradient-based stochastic approximation scheme and prove that the iterates converge to a solution of the original problem under either pseudomonotonicity requirements or a suitably defined acute angle condition. Such statements are shown to be generalizable to the stochastic mirror-prox framework; (ii) Under strong pseudomonotonicity, we show that the mean-squared error in the solution iterates produced by the extragradient SA scheme converges at the optimal rate of O(1/k) statements that were hitherto unavailable K in this regime. Notably, we optimize the initial steplength by obtaining an {\epsilon}-infimum of a discontinuous nonconvex function. Similar statements are derived for mirror-prox generalizations and can accommodate monotone SVIs under a weak-sharpness requirement. Finally, both the asymptotics and the empirical rates of the schemes are studied on a set of variational problems where it is seen that the theoretically specified initial steplength leads to significant performance benefits.

ECONOMIC EQUILIBRIUM AND MONETARY POLICY

The problem of economic equilibrium as a theoretical basis of monetary policy has been considered. It has been noted, that it is necessary to take into account the state of imbalance in the economy and the factors of market failure. The conclusion has been made, that it is important to focus on hybrid forms of monetary policy, which means the use of different combinations of target reference points. The need is implied from this for a more flexible approach to the formation of modern policy of the Bank of Russia, which would not absolutize the regime of inflation targeting.

Solving policy design problems: Alternating direction method of multipliers-based methods for structured inverse variational inequalities

Inverse variational inequalities have broad applications in various disciplines, and some of them have very appealing structures. There are several algorithms (e.g., proximal point algorithms and projection-type algorithms) for solving the inverse variational inequalities in general settings, while few of them have fully exploited the special structures. In this paper, we consider a class of inverse variational inequalities that has a separable structure and linear constraints, which has its root in spatial economic equilibrium problems. To design an efficient algorithm, we develop an alternating direction method of multipliers (ADMM) based method by utilizing the separable structure. Under some mild assumptions, we prove its global convergence. We propose an improved variant that makes the subproblems much easier and derive the convergence result under the same conditions. Finally, we present the preliminary numerical results to show the capability and efficiency of the proposed methods.

On the study of the economic equilibrium problem through preference relations

In this paper we consider a competitive economic equilibrium problem where preferences of consumers are expressed by means of a binary relation. The aim is to find a suitable quasi-variational inequality which characterizes the equilibria and, by using tools of variational theory, to study such equilibria. The novelty of this paper consists in the study of an economic equilibrium problem by a variational approach without the need of representing the consumer's preferences by a utility function.

About the timing of replacing transport means

Quality demands imposed to transportation by beneficiaries might be reached in the situation that the transport means that get obsolete with time, are replaced with others to satisfy the need for mobility addressed to the system. Establishing the moment of transport means replacement is an economic equilibrium problem. As time passes, obsolescence arises and maintenance and operating costs rise. They increase until they exceed the acquisition cost of a new vehicle. The research methodology consists in establishing the most appropriate maintenance and operating costs evolution function so that, by finding its minimum and considering the costs of acquiring a new transport vehicle to determine the minimum value of both cost categories. To take into account real situations, calculation encloses an updated value of costs in time. The theoretical considerations are supported by a case study where the economic moment of replacing a transport vehicle is determined for both cases, with and without updating.

Evolutionary quasi-variational inequality for a production economy

This paper is focused on the analysis of a time-dependent general equilibrium problem for a production economy by using the variational inequality approach. More precisely, the time-dependent allocation-price equilibrium consists in searching the optimal allocation by households and firms and in searching the market clearing condition. Firstly, a new formulation of the economic equilibrium problem by means of an evolutionary quasi-variational inequality is proven in the general context of semistrictly quasiconcave utility functions. Finally, an existence result for equilibrium solutions is given.

On the existence of solutions to stochastic quasi-variational inequality and complementarity problems

Variational inequality problems allow for capturing an expansive class of problems, including convex optimization problems, convex Nash games and economic equilibrium problems, amongst others. Yet in most practical settings, such problems are complicated by uncertainty, motivating the examination of a stochastic generalization of the variational inequality problem and its extensions in which the components of the mapping contain expectations. When the associated sets are unbounded, ascertaining existence requires having access to analytical forms of the expectations. Naturally, in practical settings, such expressions are often difficult to derive, severely limiting the applicability of such an approach. Consequently, our goal lies in developing techniques that obviate the need for integration and our emphasis lies in developing tractable and verifiable sufficiency conditions for claiming existence. We begin by recapping almost-sure sufficiency conditions for stochastic variational inequality problems with single-valued maps provided in our prior work Ravat and Shanbhag (in: Proceedings of the American Control Conference (ACC), 2010), Ravat and Shanbhag (SIAM J Optim 21: 1168–1199, 2011) and provide extensions to multi-valued mappings. Next, we extend these statements to quasi-variational regimes where maps can be either single or set-valued. Finally, we refine the obtained results to accommodate stochastic complementarity problems where the maps are either general or co-coercive. The applicability of our results is demonstrated on practically occuring instances of stochastic quasi-variational inequality problems and stochastic complementarity problems, arising as nonsmooth generalized Nash-Cournot games and power markets, respectively.

Influence of Consumer Preferences on the Equilibrium in an Open Economy

This investigation describes possible equilibrium states of an open economy under the presence of monopolies. Walrasian equilibrium is considered. Each subject of the economy is an insatiable consumer. The structure of consumption in the economy depends on volumes of production of goods. An algorithm is proposed to solve the problem of economic equilibrium. The presented constraints on model characteristics provide the existence of equilibrium in the economy. The possibility of staying the economy in an equilibrium state with specified intervals of values of selected characteristics is revealed. The achievement of a desirable equilibrium state is provided by the choice of levels of taxation of monopolies.

Solutions of parametric complementarity problems monotone with respect to parameters

In many applied problems (such as, e.g., elasto-hydrodynamic lubrication problem, some economic equilibrium problems, etc.), one of the important question is if certain complementarity problem's solution is monotone with respect to parameters. Our paper investigates this question and provides several sufficient conditions that guarantee such a monotonicity of the solutions to linear and nonlinear complementarity problems with parameters. In the majority of cases, it is required that the principal mapping of the complementarity problem be monotone by decision variables and, vice versa, antitone with respect to parameters.

Selective bi-coordinate variations for resource allocation type problems

We suggest a modification of the coordinate descent methods for resource allocation problems, which keeps the basic convergence properties of the gradient ones, but enables one to reduce the total computational expenses and to provide all the computations in a distributed manner. We describe applications to economic (auction) equilibrium problems and give preliminary results of computational tests.

Selective Bi-Coordinate Variations for Resource Allocation Type Problems

We suggest a modification of the coordinate descent methods for resource allocation problems, which keeps the basic convergence properties of the gradient ones, but enables one to reduce the total computational expenses and to provide all the computations in a distributed manner. We describe applications to economic (auction) equilibrium problems and give some preliminary results of computational tests.

Weak and strong convergence of proximal penalization and proximal splitting algorithms for two-level hierarchical Ky Fan minimax inequalities

The theory of Ky Fan minimax inequalities provides a powerful general framework for the study of convex programming, variational inequalities and economic equilibrium problems. One of the fundamental methods for finding a solution of Ky Fan minimax inequalities is the proximal point algorithm, where a lot of papers have been dedicated to this subject. In this paper, a general class of two-level hierarchical Ky Fan minimax inequalities is introduced in real Hilbert spaces. For a wide class of Bregman functions, an association of inexact implicit Bregman-penalization proximal and Bregman-splitting proximal algorithms are suggested and analysed. Weak and strong convergences are proved under essentially weaker conditions. We conclude this paper with a hierarchical minimization problem and a numerical example.

Theory and Algorithms of Variational Inequality and Equilibrium Problems, and Their Applications

The variational inequality problem is a general problem formulation that encompasses many mathematical problems, among others, including nonlinear equations, optimization problems, complementarity problems, and fixed point problems. Variational inequality is developed as a tool for the study of certain classes of partial deferential equations, economic equilibrium problems, and the pricing model of the