Finitely additive exchange economies with properness allocation
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Finitely additive exchange economies with properness allocation Anna Martellotti1 Received: 13 May 2020 / Accepted: 18 September 2020 © Unione Matematica Italiana 2020
Abstract This is a short and compact survey of finitely additive models in Equilibrium theory; starting from the classical finite-dimensional countably additive model as investigated by Aumann we present the further developments and the core-walras eqivalences that have been proven in finitely additive coalitional finitely dimensional models as well as those in infinite dimensional settings, both in the countably additive case as well as in the finitely additive one. Keywords Coalitional economies · Coalitional preferences · Core-walras equivalence · Finitely additive measures Mathematics Subject Classification D5 · D51
1 Introduction: the classical model After Aumann, the representation of large scale economies, namely markets where all the agents are supposed to be price takers, is obtained by assuming that the grand coalition is a non-atomic probability space, (Ω, Σ, P). In this setting, the subsets of the σ -algebra Σ are called coalitions. The probability P measures somehow the influence power of each coalition, and its non atomicity ensures that singletons have therefore null power; in other words individuals (a synonimous for agents) do not influence the mechanism of determining prices for the goods in the economies, unless they join significant coalitions. The classical model assumes that the commodity space is a finite dimensional vector space, usually represented by an euclidean space Rn ; n is the number of goods treated by the model. More openly, and since we shall be concerned with pure exchange economies without
Caro Mimmo, avrei voluto dedicarti un articolo scientifico vero e proprio: purtroppo, nonostante i tanti tentativi, non sono riuscita a dimostrare il risutato che volevo dedicarti; se tu ci fossi ancora quasi sicuramente ci saresti riuscito tu. Ma io non sono te, perció mi sono accontentata di dedicarti un breve e semplice survey di alcuni risultati nei quali ho applicato tante delle cose che abbiamo dimostrato assieme. So che dove sei badi alla sostanza delle cose, che leggerai nel mio cuore e che ti basteranno gli sforzi che ho fatto per non sfigurare in questo omaggio corale
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Anna Martellotti [email protected] Via Seneca, 5, 06121 Perugia , Italy
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production, what agents exchange are commodities, namely vectors x = (x 1 , . . . , xn ) where each component of x represents the amount of good 1, good 2 and so on it contains. How do agents exchange goods on the market? Each agent is supposed to be provided with two things: an initial bundle and a preference criterion. If every ω ∈ Ω is assigned an initial bundle in the positive orthant of Rn , a map e : Ω → Rn+ is automatically determined. What is usually assumed for this map, called initial allocation, is e ∈ L 1 (P). n Every Σ-measurable R+ -valued application is called an allocation and allocations fulfilling the condition
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