The General Economics With Partial Preorder Or Non-ordered Peferences

At the end of 19th century, the economists represented by Walras and Pareto established the mathematical economics, and presented the general economical equilibrium theory which strictly proved by Arrow and Debreu in 1954. The preferences of consumer were always assumed to be complete ordering relations and could be represented by an utility function. Arrow and Debreu demonstrated the existence of the equilibrium with some assumptions of utility function. But in practice, these assumptions of the preferences were rather harsh. In view of this, we posited some assumptions of the preferences to appropriate the actuality.In the first part, we supposed that the preferences were partial preorder relations, namely, the preferences were reflexive, transitive and incomplete. And we supposed that the preferences were lower semicontinuous only. Therefore the preferences could not be represented by an utility function. We proved the existence of the equilibrium by the fixed-point theorem and compared the compactness condition in this existence result with other compactness conditions.In the second part, we posited monotonicity, locally non-saturation assumptions on the preferences. This kind of preferences were reputed as non-ordered preference relations. Furthermore, we described the boundary behavior of the preferences. And then we proved the existences of the equilibrium by the mapping degree theory. Thus, the exchange economic with non-ordered preferences had Walras equilibrium.