Set-valued Mapping, Gneral Equilibrium And On The Definition Of The Best Possibilities Frontier Of Utility

In some popular in western economics literature, about the so-called " the bestpossibility frontier of utility " exists in the definition of ambiguity.This paper first analyzes the two literature about the best possibility frontier ofutility demarcation. One kind is the special definition in paper [9], and points out thatthere is serious error in its mathematical expression,which can cause the fact that theset of given utility possibility may conclude utility value which can not beachieved,and does not have the nature of Pareto optimal; Another kind is the articlerelevant results in [9], from its nature of the established mathematical model, therelated conclusion involves hyperspace set-valued mapping fixed point existenceproblem, so in the fourth section of the article, according to the problem typicallyfully double heads model are discussed, and according to the hyperspace set-valuedmapping fixed point of the relevant conclusions, conclusions in the literature from theway of pure mathematics, analysis the cause of the error. Counter examples given inthe paper show that, as a result of paper [6,10] defined in the lack of necessaryassumptions, therefore cannot guarantee that defined the best possibility frontier ofutility exists.In this paper,we draw the following conclusions:1. Even in the double head model, the corresponding fully satisfies thehyperspace conditions of the fixed point theorem of set-valued mappings, there is noguarantee that everywhere is the fixed point, called the best possibility frontier ofutility doesn’t exist, its assumptions those everywhere on the production possibilityfrontier overall production and exchange Pareto optimality is also does not exist;2. Because do not meet certain strict conditions, is likely to be even a reasonablefixed point does not exist, it is double head model suggests that there is no to achievePareto optimal production and exchange of Walrasian equilibrium. This conclusionrevealed throughout the western economics of general equilibrium and Paretooptimality theory of welfare economics logic defects, is worthy of in-depthdiscussion.