A quasi-variational approach to a competitive economic equilibrium problem without strong monotonicity assumption

This paper is focused on the investigation of the Walrasian economic equilibrium problem involving utility functions. The equilibrium problem is here reformulated by means of a quasi-variational inequality problem. Our goal is to give an existence result without assuming strong monotonicity conditions. To this end, we make use of a perturbation procedure. In particular, we will consider suitable perturbed utility functions whose gradient satisfies a strong monotonicity condition and whose associated equilibrium problem admits a solution. Then, we will prove that the limit solution solves the unperturbed problem. We stress out that our result allows us to consider a wide class of utility functions in which the Walrasian equilibrium problem may be solved.     

A Variational Approach to the Maximization of Preferences Without Numerical Representation

Journal of Optimization Theory and Applications - We introduce a variational approach to study a maximization problem of preferences that cannot be represented by a utility function. In such...     

Incomplete financial markets model with nominal assets: Variational approach

We deal with the analysis of the general equilibrium model with incomplete financial markets and nominal assets. We assume that there are 2 periods of time, say today and tomorrow. We define a consumption, portfolio holding, commodity and asset price vector as an equilibrium vector associated with a given economy if at those prices and economies households maximize utility under a budget constraints and markets clear. While the path breaking proofs of existence by Cass [6] and Werner [25] use a fixed point argument, we provide an independent existence proof in terms of variational inequalities (about the variational approach for the analysis of general equilibrium models see for example [9] and [10]). The analysis presented in this paper indicates that the variational inequality approach promises to be applicable in many specifications of the incomplete market model.     

An existence result of a quasi-variational inequality associated to an equilibrium problem

In this paper we consider a Walrasian pure exchange economy with utility function which is a particular case of a general economic equilibrium problem, without production. We assume that each agent is endowed with at least of a commodity, his preferences are expressed by an utility function and it prevails a competitive behaviour: each agent regards the prices payed and received as independent of his own choices. The Walrasian equilibrium can be characterized as a solution to a quasi-variational inequality. By using this variational approach, our goal is to prove an existence result of equilibrium solutions.     

Existence Results for Quasi-variational Inequalities with Applications to Radner Equilibrium Problems

Set-Valued and Variational Analysis - A quasi-variational inequality corresponds to a variational inequality in which the constraint set depends on the current value of the variable....     

Variational Problem,Generalized Convexity,and Application to a Competitive Equilibrium Problem

The main purpose of this article is to present a new formulation of a competitive equilibrium in terms of a suitable quasivariational inequality involving multivalued maps. More precisely, a pure exchange economy is considered where the consumer's preferences are represented by utility functions that we assume to be generalized concave and non-differentiable. In the concave context, we have characterized the equilibrium by means of a variational problem involving the subdifferential. Now, by relaxing concavity and differentiability assumptions on utility functions, the subdifferential operator of the utility function is replaced by a suitable multimap involving a new concept, recently introduced in [1 D. Aussel and N. Hadjisavvas ( 2005 ). Adjusted sublevel sets, normal operator and quasiconvex programming . SIAM J. Optim. 16 : 358 – 367 .[Crossref], [Web of Science ®] , [Google Scholar]]: the normal operator to the adjusted sublevel sets. Thanks to this variational formulation we are able to achieve the existence of equilibrium points by using arguments of the set-valued analysis. Finally, we provide some example of utility functions which verify our assumptions.     

Walrasian Equilibrium Problem with Memory Term

The aim of this paper is to study the Walrasian equilibrium problem when the data are time-dependent. In order to have a more realistic model, the excess demand function depends on the current price and on previous events of the market. Hence, a memory term is introduced; it describes the precedent states of the equilibrium. This model is reformulated as an evolutionary variational inequality in the Lebesgue space L ²([0,T],ℝ), and, thanks to this characterization, existence and qualitative results on equilibrium solution are given.     

Lagrangean variables in infinite dimensional spaces for a dynamic economic equilibrium problem

This paper is focused on the study of a dynamic competitive equilibrium by using Lagrangean multipliers. This mathematical formulation allows us the improve the Walrasian model by considering the common possibility of an uncharged delayed payment in a given time (for example, by using a credit card). Firstly the economic equilibrium problem is reformulated as an evolutionary variational problem; then the Lagrangean theory in infinite dimensional spaces is applied. Thanks to the application of this theory we obtain the existence of Lagrangean multipliers, which allows us to give a computational procedure for the equilibrium solutions.     

A new contribution to a dynamic competitive equilibrium problem

The aim of this paper is to study the Walrasian equilibrium problem when the data are time dependent. For this model an existence result is provided using the variational inequality theory in infinite dimensional spaces. Our results are the generalization of some of the results obtained by several authors in the static case (see e.g. Donato et al. (2008) [5], Donato et al. (2008) [4] and Mordukhovich (2006) [11], Nagurney (1993) [2] and the references therein).