Analysis To A Special Kind Of Structure Of Market Excess Demand Function

Analysis of general equilibrium is defined to be a fundamental research, which offers key elementary support for efficiency and stability in market mechanism that modern economics depicts, as well for logical base on which to develop miroeconomics and macroeconomics, As is widely acknowledged, market mechanism of dispersive decisions advances to accellerate the efficiency in resources allocating. In the meanwhile, it lays a deep foundation for breakthroughs and innovations occuring to modern economics theory and then provides a mathematic provement with strictness and integrity to illustrate numerous traditional arguments rooted in people's mind(see[1,2,3]).Why do economists show a great interest in the theory of general equilibrium? The reason lies in what we hope for is the drive powered by economy including supply and demand reinforces to reach an ideal state, i.e. equilibrium, through a series of resource allocatings in consistency with a variety of pricings. The ideal situation is a very anticipated termination led by a systematic economy movement, or in the right path to the desirable outcome. Sequently, equilibrium is a principle tool with which a market economist describes phenomenon in maths and thus predicts orientation and consequence of economic development in future. Furthermore, the nature in market economy efficiency is dependent of equilibrium in a sense, more safe to say. to ensure it move in the right direction(see[4,5,6,7,8]).Several main and traditional issues on equilibrium lies in its existence, uniqueness, stability, efficiency and bargaining, which arouses major concerns by the mass. However, excess demand functions of general equilibrium doesn't deserve any attention it should. As a matter of fact, the functions perform a key role to express equilibrium with more brevity, to bear out its existence in mathematic rigidity and bring about more than a scientific but simple approach to uniqueness efficiency and stabilization(see[19,10,11]). Homogeneous of degree zero in all prices ,Walras' Law together with continuity gurantee that excess demand functions must have a zero . or in more familiar terminology ,that equilibrium prices must exist(see[12,13]).In 1954, it was the first time that Arrow(Nobel-laureates in economy 1972) and Debreu (Nobel-laureates in economy 1983) showed the stricter description and proof of existence to general equilibrium with market excess demand funtions.But can an arbitrary continuous funtions, defined on a compact subset C of the interior of a positive orthant, be an excess demand funtion for some commodity in a general equlib-rium economy? A linear aggregate excess demand is theoretical possible? To answer these questions meant studying of structure of market excess demand functions. From 1954 to 1972, people didn't pay more attention to these , and analysis to structure of market excess demand functions was just statistical and empirical. The first answer to these questions was given by Sonnenschein and he obtained important results like theorem 4.2.1 (see[13]) and theorem 4.2.2(see[13]). Then Mantel and other peoples made the further study for it,including the decomposition of excess demand functions on Banach spaces(see[14]). In 1999, Liu xinge and Cai haitao gained important results such as theorem 4.2.4(see[15]) on the basis of the front studying achievments. There is no doubt that the advancement of studying of market excess demand functions enriched and advocated the development and studying of general equilibrium. With matrix formed by market excess demand funtions' differential ceofficient to price and wealth, people effectively analyze price effects ,wealth effects, substition effects, stability of equilibrium(Hicksian stable or dynamically stable) and some important results such as: if the economy is regular, then the number of normalized equilibrium price vector is finite. It's concluded that the research on excess demand functions, especially the structure of that, is not only necessary but significant. The work consists of five sections. First, preword, development of general equilibrium in brief. Second, the paper sum up the theory of customers and shows the relationship among Walras demand function, preference of customers as well as utility function. In the third section, there is something about Walras equilibrium. Fourth, it gives analysis of excess demand function with which it proved the existence of equilibrium, and in the pure exchange economy it gives colollary4.2.1.3. What's more important, on the basis of the theorem4.2.1, theorem4.2.2(see[13])and4.2.4(see[15]), it deduced the structure of excess demand functions of a special kind of economy ,that is, theorem4.2.3 and corollary4.2.3.1 as well as theorem4.2.5.The main results are as follows :Theorem 4.2.1'13] If f has a continuous derivative on [8,1/8] for some 6, 0 < 8 < 1, then there exist two consumers (I/1, to1) and (l/2,^2) such that / = E{ on [6,1/6] , J = {1,2}.Corollary 4.2.1.3 If f has a continuous derivative on [8,1/8] for some 6, 0 < 8 < 1, then there exist two utility functions U1 and U2, such that h{pi) = x\(p\) + x^lpi).Theorem 4.2.2(13i If g is a polynomial defined on 11(8), andO < 6 < 1, then thereexists a collection of individuals {[/>'},ie J, such that Ej = g on U(S).Theorem 4.2.3 Let 0 < 6 < 1, and Eu E^,.. .,ï¿¡n_i be n-1 polynomials defined on 11(5) satisfying ^pf-=^0-,(s,t — 1,2, ??? , n - l).Then there exists an n commodity economy {U\ w*}, i G J , such that (E{, Ei, ? ? ? , ^₁)=(ï¿¡1, E2, â–  â–  ? , En^) on II(<5).Corollary 4.2.3.1 Let 0 < 6 < 1, and Ei,E2,.. .,^n-i be n-1 polynomials denned on II((5) satisfying ^-jif=^^,(s, i = 1,2, ? ? ? , n — 1). Then there exists an n commodity economy {[/\w{},i e J,such that (ï¿¡f,ï¿¡2J,- ? ? ,EJn_l,-pjï¿¡J-J ï¿¡?/) = jB on n(<5).Theorem 4.2.4t15' Let 0 < S < 1, and Ei,E2,.. .,ï¿¡n-i be n — 1 polynomials defined on U{6) satisfying ^ = A(s,*)fg,(A(s,t)A(t,s) = 1, (s,t = 1,2, ? ? ? ,n-l)), Then there exists an n commodity economy {f/1,^1},! G J, such that (E^/E^, ? ? ? , ï¿¡'^₁)=(jE:i, ï¿¡2, ??-,ï¿¡?_!) on II(i be n - 1 polynomials defined on li(<5) satisfying f^=A(s,t)^,(A(s,t)A(t,s) = l,(s,f = 1,2,- ? ? ,n - 1), such that (E{,Ei,---,EZ₁)=(E1,E2,---'En-l) on 11(5).