离散空间中的一般均衡理论(英文)

第27卷第6期吉首大学学报(自然科学版)Vol.27No.6 2006年11月J ournal of J ishou University(Natural Science Edi ti on)Nov.2006
Article:1007-2985(2006)06-0006-03
A General Equilibrium Theory in Discrete Spaces
LI Chun-lin,QU B o-yun
(M aths&Stat.School,Hebei University of Economics&Business,Shijiazhuang050061,China)
Abstract:This paper generalizes the general equilibrium theore m to the case of indivisible markets where preferences is strictly convex,c ontinuous and strongly monotone.The authors thus give a sufficient condition on the existence of dis-crete equilibrium.
Key words:fixed point theorem;discrete spaces;general equilibrium
CLC number:O177Document code:A
1Introduction
The application of fixed point theore m in economics is far more widely than it is in mathematics.In microeconom-ics,fixed point theorem is a general solution to the existence of equilibrium,particularly from Brouwer.s and Kakutani. s theorems.
Brouwer.s theorem states that a continuous func tion from a non-empty compact convex set into itself has a fixed point.But in economics,a correspondence is much more general than a function,so Kakutani.s theore m,which states that a compact and convex-valued correspondence from a c ompact convex set into itself has a fixed point,if it is upper hem-i continuous,i.e.if its image does not explode rapidly,is more widely used.It is also well known that a compact and c onvex-valued correspondence from a compact convex set into itself has a fixed point,if it.s lower he m-i continuous[1-2],i.e.if its image does not shrink rapidly.Unfortunately,those theorems,respectively,carry over to the situation where goods and markets are indivisible.For example,in the real world,most goods can only be identified by integer.We can buy one apple,but what.s the meaning of0.01apples?If we consider goods in inte ger spaces,the original fixed point theore ms fail to hold.This paper atte mpts to close this gap.
Many economists contribute a lot to the existence and uniqueness of general equilibrium through the centuries.For e xample,in the middle of20th century,Debreu G[3]gave the first proof of existence of equilibrium by calculus founda-tions.Then based upon n goods and n prices,Smale S[4-6]found another solution in terms of Global Newton Method. After that,Debreu G[3]wrote of the radical change of mathematical tools from calculus to convexity and algebraic topo-l ogy.Marc Oliver Bettzuge[7]has pointed out tha t the uniqueness can be derived via the Mitjushin-Polterovich theorem in inc omple te market.Almost at the same time,Nash brought about Kakutani.s theorem in the existence of equilibrium in game theory,which has flourished the application of fixed point theory in general equilibrium theory.
Some work has already been done to establish the fixed point theorem in discrete set.References[8-10]have
X Received date:2006-06-26
Biography:LI Chun-lin(1963-),male,was born in Xingtai city,Hebei Province,doctor,professor,master adviser of Maths&Stat.
School,Hebei University of Economics&Business;research areas are functional analysis and mathematical modeling,quantity eco-nomics and statistic analysis.
given definition of /discrete convexity 0in their own way,respectively.But their definitions are still in argument.This paper atte mpts to give a more general case of discrete equilibrium,thus to generalize the Walrasian equilibrium theorem in discrete set.
In this paper,the important Kakutani .s fixed point theorem was generalized to discrete set.We have derived a suf -ficient condition on the preferences,endowments and feasible allocations.
2 Model and Definitions
Suppose there is an ec onomy with I >0consumers,J >0fir ms,in which there are L indivisible commodities.
Each consumer i =1,,,I is characterized by a consumption set X i <Z L and a preference relation L i defined on
X i .We assume that the preferences are rational (plete and transitive),the property is the constraint of contin -uous space in discrete spaces.Each firm j =1,,,J is characterized by a technology,or produc tion set,Y j <Z L .We assume that every Y j is nonempty and closed.The initial resources of commodities in the economy,that is ,the econo -my .s endowments are given to us by a vector X =( X 1,,, X L )I Z L .Thus,the basic data on preferences,technolo -gies,and resources for this economy are summarized by ({(X i ,L i )}I i =1,{Y j }J j =1, X ).
An allocation (x ,y )=(x 1,,,x I ,y 1,,,y J )is a specification of a consumption vector x i I X i for each con -sumer i =1,,,I ,and a production vector y j I Y j for each firm j =1,,,J .An allocation (x ,y )is feasible if
6i x li
[ X l +6j y lj for every commodity l .That is,if 6i x i [ X +6j
y j .We denote the set of feasible allocation by A ={(x ,y )I X 1@,@X I @Y 1@,@Y J :6i x i [ X +6j y j }<Z L (I +J ).
The notion of a socially desirable outcome that we focus on is that of a Pareto optimal allocation.
D efinition 1 A feasible allocation (x ,y )is Pareto optimal if there is no other allocation (x c ,y c )I A that Pareto dominates it,that is,if there is no feasible allocation (x c ,y c )such that x c i L i x i for all i and x c i :i x i for some i .
Let .s study the properties of competitive private ownership economies.Consumer i has an initial endowment vec -tor of commodities X i I R L and a claim to a share H ij I [0,1]of the profits of firm j ,where
X =6i X i and 6i H ij =1for every firm j .Thus,the basic preference,technological,resource,and ownership data of a private o wnership econo -my are summarized by ({(X i ,L i )}I i =1,{Y j }J j =1,{(X i ,H i 1,,,H iJ )}I i =1).
Definition 2 Given a private o wnership economy specified by ({(X i ,L i )}I i =1,{Y j }J j =1,{(X i ,H i 1,,,H iJ )}I i =1),an allocation (x *,y *)and a price vector p =(p 1,,,p L )c
onstitute a Walrasian (or competitive)equ-i librium if:
( )For every j ,y *j maximizes profits in Y j ,that is,p #y j [p #y *j for all P y j I Y j ;
( )For every i ,x *i is maximal for L i in the budget set {x i I X i :p #x i [p #X i +
6j H ij p #y *j };( )6i x *i = X +6j y *
j .3 Existence of Equilibrium
No w let .s consider a pure e xchange economy,the analysis is almost the same as the prior section.Suppose J =1,Y 1=-Z L +,X i =Z L +,the preferences L i is continuous,strongly monotone and strictly convex.
6i X i m 0.Then
there e xists a Walrasian equilibrium,
E =(x 1,,,x I )=((x 11,,,x L 1),,,(x 1I ,,,x LI )).
Define P z I R N ,+z +=sup i
|z i |,we denote the set of nearest feasible allocation of E by C (E )={x I X :+x -E +[1}.7第6期 李春林,等:离散空间中的一般均衡理论
Proposition1In a indivisible pure exchange ec onomy,if the preferences L i are strictly c onvex,continuous and strongly monotone,there exists an equilibrium allocation in C(E).
In order to prove this proposition,we first need to prove the following lemmas.
Lemma1x*is a Pareto improvement of initial endowment X if there is an allocation x*=(x*1,x*2)in an indivisible pure e xchange economy such that x*I C(E).
Proof At point E,every consumer maximize his utility in consumption set.Based upon continuity and conve x-i ty,there exists a linear allocation of E and X such that it is the X.s Pareto improvement of every consumer.Because of the continuity of preferences L i,there is an allocation indifferent with the points in C(E).This allocation is X.s Pa-reto improve ment.
Lemma2x*I C(E)if and only if there is no x c I C(E)to be the Pareto improve ment of x*.
Proof Because the points in C(E)are substitution of each other,when we adjust from one point to another,the consumption vectors are also substitution of each other.For stric tly increasing preferences L i,it becomes the utility substitution,that is,there must be one consumer.s utility decreasing.Thus there is no x c I C(E)to be the Pareto im-provement of x*.
Lemma3If x I C(E)is X.s Pareto improve ment,then there is no x.s Pareto improvement y if y|C(E).
Proof Since E is the optimal allocation of budget set,every allocation outside C(E)must be in the budget set for every consumer,so the utility of this point is lower than E.Because of the conve xity and continuity of utility func-tion,if y is x.s Pare to improvement outside C(E),then there must be an allocation vector indifferent with points in C(E)and better than y.Contradict to y is x.s Pareto improvement.
Proposition1holds obviously.
4Conclusion
In this paper,the important Kakutani.s fixed point theorem was generalized to discrete set.We have derived a suf-ficient condition on the preferences,endowments and feasible allocations.The question for a sufficient condition of the uniqueness of equilibrium,however,remains unsolved.We leave its investigation for further researc h.
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[5]SMALE S,HURWICZ L.The Stabili ty of the Competitive Equilibrium I[J].Econometrica,1958,26:522-552.
[6]SMALE S BLOCK,HURWICZ L.The Stability of Competitive Equilibrium II[J].Econometrica,1959,27:82-109.
[7]MARC OLIVER BE TTZUGE.An Extension of a Theorem by Mitjushin and Polterovich to Incomplete Markets[J].Journal of M ath-
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又因为由(7)式可得
[A n]+3=2(u(2n)+1)n I Z+.(8)所以由(1),(3),(4)和(8)式可知
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1
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参考文献:
[1]SZ#SZ R.Proposed Problem9761[J].Octogon.M ath.Mag.,2006,14(1):371.
[2]华罗庚.数论导引[M].北京:科学出版社,1979.
Integral Part of Powers of Real Qu adratic Units
LE Mao-hua
(Department of Mathematics,Zhanjiang Normal College,Zhanjiang524048,Guangdong China)
Abstract:Let D be a positive integer which is not a perfec t square,and let A=(u+v D)2,where(u,v)is positive integer solution of the Pell equation u2-Dv2=1.It is proved that,for any positive integer n,[A n]is an odd integer and[A n]+3is a perfect square,where[A n]is the integral part of A n.
Key words:real quadratic field;unit;po wer;integral part
(责任编辑向阳洁) (上接第8页)
离散空间中的一般均衡理论
李春林,屈驳韵
(河北经贸大学数学与统计学学院,河北石家庄050061)
摘要:在消费者偏好函数是强凸、连续和严格单调的条件下,给出了不可分市场的一般均衡存在定理,因而也给出了离散空间中一般均衡存在的一个充分条件.
关键词:不动点定理;离散空间;一般均衡
中图分类号:O177文献标识码:A
(责任编辑向阳洁)。