2024年5月2日发(作者：)

第27卷 第1期

2006年1月

吉首大学学报(自然科学版)

JournalofJishouUniversity(NaturalScienceEdition)

Vol.27 No.1

Jan.2006

ArticleID:1007-2985(2006)01-0023-04

ASufficientConditionforGeneralEquilibriumin

X

IndivisibleMarkets

LIChun-lin,LIXiu-jun,YAORan-ting

122

(1..,HebeiUniversityofEconomicsandBusiness,Shijiazhuang050061,China;

mentofMaths,HebeiUniversityofTechnology,Tianjin300130,China)

Abstract:Thispapergeneralizesthegeneralequilibriumtheoremtothecaseofindivisiblemarketswherepreferencesisstrictlyconvex,con-

tinuousandstronglymonotonous.

Keywords:fixedpointtheorem;indivisiblemarkets;generalequilibrium

CLCnumber:O177 Documentcode:A

1 Introduction

Theapplicationoffixedoeco-

nomics,fixedpointtheoremisageneralsolutiontotheexistenceofequilibrium,ku-

emstatesthat/acontinuousfunctionfromanon-emptycompactconvexsetintoitself

conomics,acorrespondenceismuchmoregeneralthanafunction,em,

whichstatesthat/acompactandconvex-valuedcorrespondencefromacompactconvexsetintoitselfhasafixedpoint,

ifitisupperhem-icontinuous0,magedoesnotexploderapidly,sowellknown

that/acompactandconvex-valuedcorrespondencefromacompactconvexsetintoitselfhasafixedpoint,

hem-icontinuous0

[1-2]

,unately,thosetheorems,respectively,carryover

mple,intherealworld,mostgoodscanonlybeident-i

uyoneapple,aningof0.01apples?Ifweconsidergoodsinintegerspaces,

perattemptstoclosethisgap.

Manyeconomistscontributealottot

th[3]

example,inthemiddleof20century,Debreugavethefirstproofofexistenceofequilibriumbycalculusfoundations.

Thenbaseduponngoodsandnprices,hat,De-

breuwroteoftheradicalchanggehas

pointedoutthattheuniquenesscat

thesametime,emintheexistenceofequilibriumingametheory,whichhasflour-

vetalhavegivendefinitionof/discrete

convexity0intheirownway,respectively.

Thispapergivesasufficientconditionontheindivisiblegoodsmarketsandpricevectors,whichallowsgenera-l

emtothecasewherethepricemightnotbespannedinthecompactconvexset.

X

[6-8]

[5]

[3]

[4]

Receiveddate:2005-10-17

Biography:LIChun-lin(1963-),male,wasborninXingtaiCity,HebeiProvince,professor,materadviserofHebeiUniversityof

EconomicsandBusiness;researchareaarefunctionalanalysisandmathematicalmodeling,quantityeconomicsandstatisticsanalysis.

24

吉首大学学报(自然科学版)第27卷

2 EdgeworthEconomy

First,egoodsareindivisible,wedefineacon-

sumptionsetbyZ

+

,andpreferences<

i

owmentvectorX

i

=(X

1i

,X

2i

),X

i

IZ

+

.Theto-

talendowmentofeachlisX

l

=X

l1

+X

l2

,X

li

IZ

+

.Theallocationinthiseconomyisanassignmentofanonnegative

consumptionvectortoeachconsumer:

4

x=(x

1

,x

2

)=((x

11

,x

21

),(x

12

,x

22

)) xIZ

+

.

Wesayanallocationisfeasiblefortheeconomyif

x

l1

+x

l2

[X

l

forl=1,2.

thatinthisnotionoffeasibility,weareimplicitlyassumingthatthereisfreedisposalofcommodities).

Thefeasibleallocationforwhichequalityholdsin

(1)tefulfeasiblea-l

locationscanbedepictedbymeansofanEdgeworthbox,

showninfig.1.

IntheEdgeworthbox,itiesare

measuredintheusualway,withthesouthwestcorneras

rast,itiesaremea-

h

consumers,theverticaldimensionmeasuresquantitiesof

good2,andthehorizontaldimensionmeasuresquantities

gthoftheboxisX

1

,-

tisX

2

,theeconomy.s

ntintheboxrepresents

mple,fig.1de-

pictstheendowmentvectorX=((X

11

,X

21

),(X

12

,X

22

))ofthetwoconsumers.

Asischaracteristicofgeneralequilibriumtheory,,for

anypricesp=(p

1

,p

2

),hequalsthemarketvalueofhisendowmentsofcommodities,p#X

i

=

p

1

X

1i

=p

2

X

2i

.,-

mentvectorX

i

,hisbudgetsetcanbeviewedsolelyasafunctionofprices:

B

i

(p)={x

i

IZ

+

:p#x

i

[p#X

i

}.

Contrasttocontinuouscommodityspaces,althoughwecanalsodrawabudgetline,throughtheendowmentpointX

withslope-(p

1

/p

2

),asshowninfig.2,ethatonlyallocationsonthe

budgetlineareaffordabletobothconsumerssimultaneouslyatprices(p

1

,p

2

).Fig.3illustrateshowtheconsumption

,theconsumerdemandshismost

preferredpointinB

1

(p),whichcanbeexpressedusinghisdemandfunctionasx

1

(p,p#X

1

).Indiscretemarket,the

optimalallocationmightnotbeunique,theallocationwillbedeterminedbythemarket,asible,theprice

vectorpvaries,thebudgetlinepivotsaroundtheendowmentpointX,andthedemandedconsumptionstraceouta

curve,denotedbyOC

1

,asitshowsinfig.4,atthiscurvepasses

eateveryptheendowmentvectorw

1

=(X

11

,X

21

)isaffordabletoconsumer1,

itfollowsthatthisconsumermu-

ketequilibriumwhereconsumerstakepricesasgiven,,theconsumersshouldfulfill

2

2

22

(1)

Thatis,gateendowmentofit(Note

Fig.1 AnEdgeworthBox