Some remarks on the Walras equilibrium problem in Lebesgue spaces

Abstract

We introduce a parametric variational inequality in order to model the time dependent Walras economic equilibrium and discuss its relation with an integral formulation in the spaces (L ∞ , L 1 ). The role of monotonicity is analysed and, as a classical example, we study the Walras problem using the Cobb-Douglas functions in this new functional setting.

Quasi-variational analysis of a dynamic Walrasian competitive exchange economy

“…Remark 3.3. Condition (2) has been derived from variational inequality (6). It is natural to ask oneself whether it is possible to derive a point-wise, non-production condition from the same, that is n a=1 (x (j) a (t) − e (j) a (t)) ≤ 0 for all j = 1, 2, and for almost all t ∈ T .…”

“…Instead, the finality of the strategy pursued by agents is finalized toward maximizing, under the constrains of budget, the mean value of the utility in the given period. Therefore, in respect of the above economic justification, the principal activities will be considered in terms of mean values and in force of condition 3), which permit us to justify that the non-production in mean value does not imply the non-production in any instant (fact inaccurately contradicted in the recent paper [17] and long before mentioned in [5,6,13,14,15,16] for supporting the existence of a dynamic equilibrium for a pure exchange economy over time and currently proved in [32]), we will define a new dynamic competitive equilibrium. Successively in Proposition 3.1, we will prove that such equilibrium, under some assumptions, satisfies Walras' law.…”

On a competitive economic equilibrium referred to a continuous time period

“…Vitanza et al in [10,12,13,14] and Causa et al in [5,6] suggest a evolutionary definition without suitable economical and mathematical motivations and propose contextually partial results.…”