Walrasian equilibrium theory with and without free-disposal: theorems and counterexamples in an infinite-agent context

This paper provides four theorems on the existence of a free-disposal equilibrium in a Walrasian economy: the first with an arbitrary set of agents with compact consumption sets, the next highlighting the trade-offs involved in the relaxation of the compactness assumption, and the last two with a countable set of agents endowed with a weighting structure. The results generalize theorems in the antecedent literature pioneered by Shafer–Sonnenschein in 1975, and currently in the form taken in He–Yannelis 2016. The paper also provides counterexamples to the existence of non-free-disposal equilibrium in cases of both a countable set of agents and an atomless measure space of agents. One of the examples is related to one Chiaki Hara presented in 2005. The examples are of interest because they satisfy all the hypotheses of Shafer’s 1976 result on the existence of a non-free-disposal equilibrium, except for the assumption of a finite set of agents. The work builds on recent work of the authors on abstract economies, and contributes to the ongoing discussion on the modelling of “large” societies.

Walrasian equilibrium behavior in nature

The interaction between land plants and mycorrhizal fungi (MF) forms perhaps the world’s most prevalent biological market. Most plants participate in such markets, in which MF collect nutrients from the soil and trade them with host plants in exchange for carbon. In a recent study, M. D. Whiteside et al. [Curr. Biol. 29, 2043–2050.e8 (2019)] conducted experiments that allowed them to quantify the behavior of arbuscular MF when trading phosphorus with their host roots. Their experimental techniques enabled the researchers to infer the quantities traded under multiple scenarios involving different amounts of phosphorus resources initially held by different MF patches. We use these observations to confirm a revealed preference hypothesis, which characterizes behavior in Walrasian equilibrium, a centerpiece of general economic equilibrium theory.

New Clearing Model to Mitigate the Non-Convexity in European Day-ahead Electricity Market

Nowadays, the payment scheme of European Day-Ahead Market is based on the market clearing price by running the Pan-European Hybrid Electricity Market Integration Algorithm. However, this conventional payment scheme is challenging because of the non-convexity and the short computation time requirement. Thus, the aim of this work is to propose a new clearing model in order to mitigate this challenge. The model is based on make-whole payment mechanism and it includes two major steps: (i) maximizing social welfare and (ii) achieving a Walrasian equilibrium by the “minimum-uplift approach”. The proposed model is validated and investigated by two case studies: one is an artificially created Day-Ahead Market session containing all type of bids encountered in Europe and containing a very large number of bids to stress the algorithm and the other is a reduced, but realistic, model of European market where real data from February to December of 2017 were considered. The tests show a consistent improvement of the numerical performances of the proposed model with respect to the conventional one while the economic performance is not altered, but is slightly improved. Moreover, because the tests are based on real data during a long period of time, the results show that proposed model is very promising for the real application.

A Proportional-Switch Adjustment Model towards Mixed Equilibrium with Multiroute Choice Behaviour Criterion

Based on the price-quantity adjustment behaviour principle of the non-Walrasian equilibrium theory, this paper adopted a new QUE (quantity adjustment user equilibrium) criterion to formulate the route comfort choice behaviour. The purpose of the present paper is to establish a proportional-switch adjustment model which aims to reflect the route adjustment behaviour interaction between the traditional UE (user equilibrium) travellers and the QUE travellers and converge to a mixed equilibrium state. It is assumed that a group of road network travellers follow the UE criteria by choosing the travel route with the purpose of minimizing their route travel time (travel cost). In addition, the other group of travellers follow the QUE criteria by selecting the route with the largest residual capacity to achieve a more comfortable travel experience. The travel route adjustment behaviour of the two group travellers generates the dynamic traffic flow evolution towards the mixed equilibrium, and the route adjusting flow is proportional to the difference of traveller decision-making variable among the alternative routes. Simple illustrative examples are used to evaluate the performance of the proposed model, and the uniqueness and stability of the solution are demonstrated by applying the variational inequality and Lyapunov stability theorem.

Pure exchange and general equilibrium

This chapter focuses on situations of pure exchange, where consumers wish to exchange bundles of goods they hold at the outset for other bundles they will subsequently consume. It uses this setting to introduce the theory of price-mediated market transactions and, more particularly, the theory of general equilibrium, in which all markets in all goods are considered simultaneously. Following in the footsteps of generations of classical microeconomists, the chapter makes the assertion that in many situations of pure exchange, the consumer will wind up at the consumption allocation part of some Walrasian equilibrium for the economy, and insofar as there are markets in these goods, prices will correspond to equilibrium prices. One thing that the concept of a Walrasian equilibrium does not provide is any sense of how market operates. There is no model here of who sets prices, or what gets exchanged for what, when, and where.

Using mathematical models in a disequilibrium economy with offsetting demand

When modeling a nonequilibrium economy, the behavior of participants is described by the same optimization problems, including the criterion and internal technological and budgetary constraints, as in the theory of Walrasian equilibrium. They are only supplemented by external restrictions on the purchase (or sale) of scarce (slow-moving) products. Various principles are known for establishing these boundaries. They can be fixed (a rigid scheme of rationing) and not depend directly on the decisions of the participant, or be determined by the demand expressed by them (flexible scheme). The presented demand for rationable products, as a rule, does not coincide with the Walrasian one. We will call it an order. In well-known models, the order is considered equal to active demand. The concept of active demand has been successfully used in price control models. However, it is not the object of the choice of participants aimed at optimizing their criteria. Meanwhile, it seems natural that manufacturers and consumers, seeking to maximize utility, are free to choose order sizes at their own discretion. Modeling of the situation arising with this approach is the goal of the present work and is based on a modification of the rationing scheme proposed by J.P. Benassi The work also considers equilibrium models at fixed prices, in which participants, when forming demand, take into account the scarcity of products and the level of satisfaction of orders. Models are used to assess the impact of taxes, government spending, and other macro-regulators on employment and national income. The paper provides an overview of literary sources in the subject area, as well as an economic interpretation of the results.