高级微观经济学所需的数学知识

condition.Becomingarichcountryalsodependsonotherfactorssuchaspoliticalsystem,socialinfrastructures,andculture.Additionally,noexampleofacountrycanbefoundsofarthatitisrichinthelongrun,thatisnotamarketeconomy.Apositivestatementstatefactswhilenormativestatementgiveopinionsorvaluejudgments.Distinguishingthesetwostatementscanvoidmanyunnecessarydebates.1.2LanguageandMethodsofMathematicsThissectionreviewssomebasicmathematicsresultssuchas:continuityandconcavityoffunctions,SeparatingHyperplaneTheorem,optimization,correspondences(pointtosetmappings),�xedpointtheorems,KKMlemma,maximumtheorem,etc,whichwillbeusedtoprovesomeresultsinthelecturenotes.Forgoodreferencesaboutthematerialsdiscussedinthissection,seeappendixesinHildenbrandandKirman(1988),Mas-Colell(1995),andVarian(1992).1.2.1FunctionsLetXandYbetwosubsetsofEuclidianspaces.Inthistext,vectorinequalities,=,�,and>,arede�nedasfollows:Leta;b2Rn.Thena=bmeansas=bsforalls=1;:::;n;a�bmeansa=bbuta6=b;a>bmeansas>bsforalls=1;:::;n.De�nition1.2.1Afunctionf:X!Rissaidtobecontinuousifatpointx02X,limx!x0f(x)=f(x0);orequivalently,forany�>0,thereisa�>0suchthatforanyx2Xsatisfyingjx�x0j<�,wehavejf(x)�f(x0)j<�Afunctionf:X!RissaidtobecontinuousonXiffiscontinuousateverypointx2X.Theideaofcontinuityisprettystraightforward:Thereisnodisconnectedpointifwedrawafunctionasacurve.Afunctioniscontinuousif\small"changesinxproduces\small"changesinf(x).12Theso-calleduppersemi-continuityandlowersemi-continuitycontinuitiesareweakerthancontinuity.Evenweakconditionsoncontinuityaretransfercontinuitywhichchar-acterizemanyoptimizationproblemsandcanbefoundinTian(1992,1993,1994)andTianandZhou(1995),andZhouandTian(1992).De�nition1.2.2Afunctionf:X!Rissaidtobeuppersemi-continuousifatpointx02X,wehavelimsupx!x0f(x)5f(x0);orequivalently,forany�>0,thereisa�>0suchthatforanyx2Xsatisfyingjx�x0j<�,wehavef(x)<f(x0)+�:Althoughallthethreede�nitionsontheuppersemi-continuityatx0areequivalent,thesecondoneiseasiertobeversi�ed.Afunctionf:X!Rissaidtobeuppersemi-continuousonXiffisuppersemi-continuousateverypointx2X.De�nition1.2.3Afunctionf:X!Rissaidtobelowersemi-continuousonXif�fisuppersemi-continuous.Itisclearthatafunctionf:X!RiscontinuousonXifandonlyifitisbothupperandlowersemi-continuous,orequivalently,forallx2X,theuppercontoursetU(x)�fx02X:f(x0)=f(x)gandthelowercontoursetL(x)�fx02X:f(x0)5f(x)gareclosedsubsetsofX.LetfbeafunctiononRkwithcontinuouspartialderivatives.Wede�nethegradientofftobethevectorDf(x)=�@f(x)@x1;@f(x)@x2;:::;@f(x)@xk�:Supposefhascontinuoussecondorderpartialderivatives.Wede�netheHessianoffatxtobethen�nmatrixdenotedbyD2f(x)asD2f(x)=�@2f(x)@xi@xj�;whichissymmetricsince@2f(x)@xi@xj=@2f(x)@xj@xi:13De�nition1.2.4Afunctionf:X!Rissaidtobehomogeneousofdegreekiff(tx)=tkf(x)Animportantresultconcerninghomogeneousfunctionisthefollowing:Theorem1.2.1(Euler'sTheorem)Ifafunctionf:Rn!Rishomogeneousofdegreekifandonlyifkf(x)=nXi=1@f(x)@xixi:1.2.2SeparatingHyperplaneTheoremAsetX�Rnissaidtobecompactifitisboundedandclosed.AsetXissaidtobeconvexifforanytwopointsx;x02X,thepointtx+(1�t)x02Xforall05t51.Geometricallytheconvexsetmeanseverypointonthelinesegmentjoininganytwopointsinthesetisalsointheset.Theorem1.2.2(SeparatingHyperplaneTheorem)SupposethatA;B�RmareconvexandA\B=;.Then,thereisavectorp2Rmwithp6=0,andavaluec2Rsuchthatpx5c5py8x2A&y2B:Furthermore,supposethatB�Rmisconvexandclosed,A�Rmisconvexandcompact,andA\B=;.Then,thereisavectorp2Rmwithp6=0,andavaluec2Rsuchthatpx<c<py8x2A&y2B:1.2.3ConcaveandConvexFunctionsConcave,convex,andquasi-concavefunctionsarisefrequentlyinmicroeconomicsandhavestrongeconomicmeanings.Theyalsohaveaspecialroleinoptimizationproblems.De�nition1.2.5LetXbeaconvexset.Afunctionf:X!RissaidtobeconcaveonXifforanyx;x02Xandanytwith05t51,wehavef(tx+(1�t)x0)=tf(x)+(1�t)f(x0)ThefunctionfissaidtobestrictlyconcaveonXiff(tx+(1�t)x0)>tf(x)+(1�t)f(x0)14forallx6=x02Xan0<t<1.Afunctionf:X!Rissaidtobe(strictly)convexonXif�fis(strictly)concaveonX.Remark1.2.1Alinearfunctionisbothconcaveandconvex.Thesumoftwoconcave(convex)functionsisaconcave(convex)function.Remark1.2.2Whenafunctionfde�nedonaconvexsetXhascontinuoussecondpartialderivatives,itisconcave(convex)ifandonlyiftheHessianmatrixD2f(x)isnegative(positive)semi-de�niteonX.Itisitisstrictlyconcave(strictlyconvex)iftheHessianmatrixD2f(x)isnegative(positive)de�niteonX.Remark1.2.3Thestrictconcavityoff(x)canbecheckedbyverifyingiftheleadingprincipalminorsoftheHessianmustalternateinsign,i.e.,������f11f12f21f22������>0;���������f11f12f13f21f22f23f31f32f33���������<0;andsoon,wherefij=@2f@xi@xj.Thisalgebraicconditionisusefulforcheckingsecond-orderconditions.Ineconomictheoryquasi-concavefunctionsareusedfrequently,especiallyfortherepresentationofutilityfunctions.Quasi-concaveissomewhatweakerthanconcavity.De�nition1.2.6LetXbeaconvexset.Afunctionf:X!Rissaidtobequasi-concaveonXifthesetfx2X:f(x)=cgisconvexforallrealnumbersc.Itisstrictlyquasi-concaveonXiffx2X:f(x)>cgisconvexforallrealnumbersc.Afunctionf:X!Rissaidtobe(strictly)quasi-convexonXif�fis(strictly)quasi-concaveonX.15Remark1.2.4Thesumoftwoquasi-concavefunctionsingeneralisnotaquasi-concavefunction.Anymonotonicfunctionde�nedonasubsetoftheonedimensionalrealspaceisbothquasi-concaveandquasi-convex.Remark1.2.5Whenafunctionfde�nedonaconvexsetXhascontinuoussecondpartialderivatives,itisstrictlyquasi-concave(convex)ifthenaturallyorderedprincipalminorsoftheborderedHessianmatrix�H(x)alternateinsign,i.e.,���������0f1f2f1f11f12f2f21f22���������>0;������������0f1f2f3f1f11f12f13f2f21f22f23f3f31f32f33������������<0;andsoon.1.2.4OptimizationOptimizationisafundamentaltoolforthedevelopmentofmodernmicroeconomicsanal-ysis.Mosteconomicmodelsarebasedonthethesolutionofoptimizationproblems.Resultsofthissubsectionareusedthroughoutthetext.Thebasicoptimizationproblemisthatofmaximizingorminimizingafunctiononsomeset.ThebasicandcentralresultistheexistencetheoremofWeierstrass.Theorem1.2.3(WeierstrassTheorem)Anyupper(lower)semicontinuousfunctionreachesitsmaximum(minimum)onacompactset,andthesetofmaximumiscompact.EQUALITYCONSTRAINEDOPTIMIZATION16Anoptimizationproblemwithequalityconstraintshastheformmaxf(x)suchthath1(x)=d1h2(x)=d2...hk(x)=dk;wheref,h1,...,hkaredi�erentiablefunctionsde�nedonRnandk<nandd1;:::;dkareconstants.ThemostimportantresultforconstrainedoptimizationproblemsistheLagrangemultipliertheorem,givingnecessaryconditionsforapointtobeasolution.De�netheLagrangefunction:L(x;�)=f(x)+kXi=1�i[di�hi(x)];where�1;:::;�karecalledtheLagrangemultipliers.Thenecessaryconditionsforxtosolvethemaximizationproblemisthatthereare�1;:::;�ksuchthatthe�rst-orderconditions(FOC)areheld:L(x;�)@xi=@f(x)@xi�kXl=1�l@hl(x)@xi=0i=1;2;:::;n:INEQUALITYCONSTRAINEDOPTIMIZATIONConsideranoptimizationproblemwithinequalityconstraints:maxf(x)suchthatgi(x)5dii=1;2;:::;k:Apointxmakingallconstraintsheldwithequality(i.e.,gi(x)=diforalli)issaidtosatisfytheconstrainedquali�cationconditionifthegradientvectors,Dg1(x);Dg2(x);:::;Dgk(x)arelinearlyindependent.Theorem1.2.4(Kuhn-TuckerTheorem)Supposexsolvestheinequalityconstrainedoptimizationproblemandsatis�estheconstrainedquali�cationcondition.Then,thereareasetofKuhn-Tuckermultipliers�i=0,i=1;:::;k,suchthatDf(x)=kXi=1�iDgi(x):17Furthermore,wehavethecomplementaryslacknessconditions:�i=0foralli=1;2;:::;k�i=0ifgi(x)<Di:ComparingtheKuhn-TuckertheoremtotheLagrangemultipliersintheequalitycon-strainedoptimizationproblem,weseethatthemajordi�erenceisthatthesignsoftheKuhn-TuckermultipliersarenonnegativewhilethesignsoftheLagrangemultiplierscanbeanything.Thisadditionalinformationcanoccasionallybeveryuseful.TheKuhn-Tuckertheoremonlyprovidesanecessaryconditionforamaximum.Thefollowingtheoremstatesconditionsthatguaranteetheabove�rst-orderconditionsaresu�cient.Theorem1.2.5(Kuhn-TuckerSu�ciency)Supposefisconcaveandeachgiiscon-vex.Ifxsatis�estheKuhn-Tucker�rst-orderconditionsspeci�edintheabovetheorem,thenxisaglobalsolutiontotheconstrainedoptimizationproblem.Wecanweakentheconditionsintheabovetheoremwhenthereisonlyoneconstraint.LetC=fx2Rn:g(x)5dg.Proposition1.2.1Supposefisquasi-concaveandthesetCisconvex(thisistrueifgisquasi-convex).Ifxsatis�estheKuhn-Tucker�rst-orderconditions,thenxisaglobalsolutiontotheconstrainedoptimizationproblem.Sometimeswerequirextobenonnegative.Supposewehadoptimizationproblem:maxf(x)suchthatgi(x)5dii=1;2;:::;kx=0:ThentheLagrangefunctioninthiscaseisgivenbyL(x;�)=f(x)+kXl=1�l[dl�hl(x)]+nXj=1�jxj;18where�1;:::;�karethemultipliersassociatedwithconstraintsxj=0.The�rst-orderconditionsareL(x;�)@xi=@f(x)@xi�kXl=1�l@gl(x)@xi+�i=0i=1;2;:::;n�l=0l=1;2;:::;k�l=0ifgl(x)<dl�i=0i=1;2;:::;n�i=0ifxi>0:Eliminating�i,wecanequivalentlywritetheabove�rst-orderconditionswithnonnegativechoicevariablesasL(x;�)@xi=@f(x)@xi�kXi=1�l@gi(x)@xi50withequalityifxi>0i=1;2;:::;n;orinmatrixnotation,Df��Dg50x[Df��Dg]=0wherewehavewrittentheproductoftwovectorxandyastheinnerproduction,i.e.,xy=Pni=1xiyi.Thus,ifweareataninterioroptimum(i.e.,xi>0foralli),wehaveDf(x)=�Dg:1.2.5TheEnvelopeTheoremConsideranarbitrarymaximizationproblemwheretheobjectivefunctiondependsonsomeparametera:M(a)=maxxf(x;a):ThefunctionM(a)givesthemaximizedvalueoftheobjectivefunctionasafunctionoftheparametera.Letx(a)bethevalueofxthatsolvesthemaximizationproblem.ThenwecanalsowriteM(a)=f(x(a);a).ItisoftenofinteresttoknowhowM(a)changesasachanges.19Theenvelopetheoremtellsustheanswer:dM(a)da=@f(x;a)@a������x=x(a):ThisexpressionsaysthatthederivativeofMwithrespecttoaisgivenbythepartialderivativeoffwithrespecttoa,holdingx�xedattheoptimalchoice.Thisisthemeaningoftheverticalbartotherightofthederivative.Theproofoftheenvelopetheoremisarelativelystraightforwardcalculation.NowconsideramoregeneralparameterizedconstrainedmaximizationproblemoftheformM(a)=maxx1;x2g(x1;x2;a)suchthath(x1;x2;a)=0:TheLagrangianforthisproblemisL=g(x1;x2;a)��h(x1;x2;a);andthe�rst-orderconditionsare@g@x1��@h@x1=0(1.1)@g@x2��@h@x2=0h(x1;x2;a)=0:Theseconditionsdeterminetheoptimalchoicefunctions(x1(a);x2(a);a),whichinturndeterminethemaximumvaluefunctionM(a)�g(x1(a);x2(a)):(1.2)Theenvelopetheoremgivesusaformulaforthederivativeofthevaluefunctionwithrespecttoaparameterinthemaximizationproblem.Speci�cally,theformulaisdM(a)da=@L(x;a)@a������x=x(a)=@g(x1;x2;a)@a������xi=xi(a)��@h(x1;x2;a)@a������xi=xi(a)20Asbefore,theinterpretationofthepartialderivativesneedsspecialcare:theyarethederivativesofgandhwithrespecttoaholdingx1andx2�xedattheiroptimalvalues.1.2.6Point-to-SetMappingsWhenamappingisnotasingle-valuedfunction,butisapoint-to-setmapping,itiscalledacorrespondence,ormulti-valuedfunctions.Thatis,acorrespondenceFmapspointxinthedomainX�RnintosetsintherangeY�Rm,anditisdenotedbyF:X!2Y.WealsouseF:X!!YtodenotethemappingF:X!2Yinthislecturenotes.De�nition1.2.7AcorrespondenceF:X!2Yis:(1)non-emptyvaluedifthesetF(x)isnon-emptyforallx2X;(2)convexvaluedifthesetF(x)isaconvexsetforallx2X;(3)closedvaluedifthesetF(x)isaclosedsetforallx2X;(4)compactvaluedifthesetF(x)isacompactsetforallx2X.Intuitively,acorrespondenceiscontinuousifsmallchangesinxproducesmallchangesinthesetF(x).Unfortunately,givingaformalde�nitionofcontinuityforcorrespondencesisnotsosimple.Figure1.1showsacontinuouscorrespondence.Figure1.1:AContinuouscorrespondence.Thenotionsofhemi-continuityareusuallyde�nedintermsofsequences(seeDebreu(1959)andMask-Collelletal.(1995)),but,althoughtheyarerelativelyeasytoverify,they21arenotintuitiveanddependonontheassumptionthatacorrespondenceiscompacted-valued.Thefollowingde�nitionsaremoreformal(see,Border,1988).De�nition1.2.8AcorrespondenceF:X!2Yisupperhemi-continuousatxifforeachopensetUcontainingF(x),thereisanopensetN(x)containingxsuchthatifx02N(x),thenF(x0)�U.AcorrespondenceF:X!2Yisupperhemi-continuousifitisupperhemi-continuousateveryx2X,orequivalently,ifthesetfx2X:F(x)�VgisopeninXforeveryopensetsubsetVofY.Remark1.2.6Upperhemi-continuitycapturestheideathatF(x)willnot\suddenlycontainnewpoints"justaswemovepastsomepointx,inotherwords,F(x)doesnotsuddenlybecomesmuchlargerifonechangestheargumentxslightly.Thatis,ifonestartsatapointxandmovesalittlewaytox0,upperhemi-continuityatximpliesthattherewillbenopointinF(x0)thatisnotclosetosomepointinF(x).De�nition1.2.9AcorrespondenceF:X!2Yissaidtobelowerhemi-continuousatxifforeveryopensetVwithF(x)\V6=;,thereexistsaneighborhoodN(x)ofxsuchthatF(x0)\V6=;forallx02N(x).AcorrespondenceF:X!2Yislowerhemi-continuousifitislowerhemi-continuousateveryx2X,orequivalently,thesetfx2X:F(x)\V6=;gisopeninXforeveryopensetVofY.Remark1.2.7Lowerhemi-continuitycapturestheideathatanyelementinF(x)canbe\approached"fromalldirections,inotherwords,F(x)doesnotsuddenlybecomesmuchsmallerifonechangestheargumentxslightly.Thatis,ifonestartsatsomepointxandsomepointy2F(x),lowerhemi-continuityatximpliesthatifonemovesalittlewayfromxtox0,therewillbesomey02F(x0)thatisclosetoy.Remark1.2.8Basedonthefollowingtwofacts,bothnotionsofhemi-continuitycanbecharacterizedbysequences.(a)IfacorrespondenceF:X!2Yiscompacted-valued,thenitisupperhemi-continuousifandonlyifforanyfxkgwithxk!xandfykgwithyn2F(xk),thereexistsaconvergingsubsequencefykmgoffykg,ykm!y,suchthaty2F(x).22(b)AcorrespondenceF:X!2Yissaidtobelowerhemi-continuousatxifandonlyifforanyfxkgwithxk!xandy2F(x),thenthereisasequencefykgwithyk!yandyn2F(xk).De�nition1.2.10AcorrespondenceF:X!2Yissaidtobeclosedatxifforanyfxkgwithxk!xandfykgwithyk!yandyn2F(xk)impliesy2F(x).FissaidtobeclosedifFisclosedforallx2XorequivalentlyGr(F)=f(x;y)2X�Y:y2F(x)gisclosed:Remark1.2.9Regardingtherelationshipbetweenupperhemi-continuityandclosedgraph,thefollowingfactscanbeproved.(i)IfYiscompactandF:X!2Yisclosed-valued,thenFhasclosedgraphimpliesitisupperhemi-continuous.(ii)IfXandYareclosedandF:X!2Yisclosed-valued,thenFisupperhemi-continuousimpliesthatithasclosedgraph.Becauseoffact(i),acorrespondencewithclosedgraphissometimescalledupperhemi-continuityintheliterature.Butoneshouldkeepinmindthattheyarenotthesameingeneral.Forexample,letF:R+!2Rbede�nedbyF(x)=8<:f1xgifx>0f0gifx=0:Thecorrespondenceisclosedbutnotupperhemi-continuous.Also,de�neF:R+!2RbyF(x)=(0;1).ThenFisupperhemi-continuousbutnotclosed.Figure1.2showsthecorrespondenceisupperhemi-continuous,butnotlowerhemi-continuous.Toseewhyitisupperhemi-continuous,imagineanopenintervalUthatencompassesF(x).Nowconsidermovingalittletotheleftofxtoapointx0.ClearlyF(x0)=fy^gisintheinterval.Similarly,ifwemovetoapointx0alittletotherightofx,thenF(x)willinsidetheintervalsolongasx0issu�cientlyclosetox.Soitisupperhemi-continuous.Ontheotherhand,thecorrespondenceitnotlowerhemi-continuous.Toseethis,considerthepointy2F(x),andletUbeaverysmallintervalaroundythat23doesnotincludey^.IfwetakeanyopensetN(x)containingx,thenitwillcontainsomepointx0totheleftofx.ButthenF(x0)=fy^gwillcontainnopointsneary,i.e.,itwillnotinterestU.Figure1.2:Acorrespondencethatisupperhemi-continuous,butnotlowerhemi-continuous.Figure1.3showsthecorrespondenceislowerhemi-continuous,butnotupperhemi-continuous.Toseewhyitislowerhemi-continuous.Forany05x05x,notethatF(x0)=fy^g.Letxn=x0�1=nandletyn=y^.Thenxn>0forsu�cientlylargen,xn!x0,yn!y^,andyn2F(xn)=fy^g.Soitislowerhemi-continuous.Itisclearlylowerhemi-continuousforxi>x.Thus,itislowerhemi-continuousonX.Ontheotherhand,thecorrespondenceitnotupperhemi-continuous.IfwestartatxbynotingthatF(x)=fy^g,andmakeasmallmovetotherighttoapointx0,thenF(x0)suddenlycontainsmaypointsthatarenotclosetoy^.Sothiscorrespondencefailstobeupperhemi-continuous.Combiningupperandlowerhemi-continuity,wecande�nethecontinuityofacorre-spondence.De�nition1.2.11AcorrespondenceF:X!2Yatx2Xissaidtobecontinuousifitisbothupperhemi-continuousandlowerhemi-continuousatx2X.AcorrespondenceF:X!2Yissaidtobecontinuousifitisbothupperhemi-continuousandlowerhemi-continuous.24Figure1.3:Acorrespondencethatislowerhemi-continuous,butnotupperhemi-continuous.Remark1.2.10Asitturnsout,thenotionsofupperandhemi-continuouscorrespon-dencebothreducetothestandardnotionofcontinuityforafunctionifF(�)isasingle-valuedcorrespondence,i.e.,afunction.Thatis,F(�)isasingle-valuedupper(orlower)hemi-continuouscorrespondenceifandonlyifitisacontinuousfunction.De�nition1.2.12AcorrespondenceF:X!2YsaidtobeopenififsgraphGr(F)=f(x;y)2X�Y:y2F(x)gisopen:De�nition1.2.13AcorrespondenceF:X!2YsaidtohaveupperopensectionsifF(x)isopenforallx2X.AcorrespondenceF:X!2YsaidtohaveloweropensectionsifitsinversesetF�1(y)=fx2X:y2F(x)gisopen.Remark1.2.11IfacorrespondenceF:X!2Yhasanopengraph,thenithasupperandloweropensections.IfacorrespondenceF:X!2Yhasloweropensections,thenitmustbelowerhemi-continuous.1.2.7ContinuityofaMaximumInmanyplaces,weneedtocheckifanoptimalsolutioniscontinuousinparameters,say,tocheckthecontinuityofthedemandfunction.Wecanapplytheso-calledMaximumTheorem.25Theorem1.2.6(Berg'sMaximumTheorem)Supposef(x;a)isacontinuousfunc-tionmappingfromA�X!R,andtheconstraintsetF:A!!Xisacontinuouscorrespondencewithnon-emptycompactvalues.Then,theoptimalvaluedfunction(alsocalledmarginalfunction):M(a)=maxx2F(a)f(x;a)isacontinuousfunction,andtheoptimalsolution:�(a)=argmaxx2F(a)f(x;a)isaupperhemi-continuouscorrespondence.1.2.8FixedPointTheoremsToshowtheexistenceofacompetitiveequilibriumforthecontinuousaggregateexcessdemandfunction,wewillusethefollowing�xed-pointtheorem.ThegeneralizationofBrouwer's�xedtheoremcanbefoundinTian(1991)thatgivesnecessaryandsu�cientconditionsforafunctiontohavea�xedpoint.Theorem1.2.7(Brouwer'sFixedTheorem)LetXbeanon-empty,compact,andconvexsubsetofRm.Ifafunctionf:X!XiscontinuousonX,thenfhasa�xedpoint,i.e.,thereisapointx�2Xsuchthatf(x�)=x�.Figure1.4:Fixedpointsaregivenbytheintersectionsofthe450lineandthecurveofthefunction.Therearethree�xedpointsinthecasedepicted.26Example1.2.1f:[0;1]![0;1]iscontinuous,thenfhasa�xedpoint(x).Toseethis,letg(x)=f(x)�x.Then,wehaveg(0)=f(0)=0g(1)=f(1)�150.Fromthemean-valuetheorem,thereisapointx�2[0;1]suchthatg(x�)=f(x�)�x�=0.Whenamappingisacorrespondence,wehavethefollowingversionof�xedpointtheorem.Theorem1.2.8(Kakutani'sFixedPointTheorem)LetXbeanon-empty,com-pact,andconvexsubsetofRm.IfacorrespondenceF:X!2Xisaupperhemi-continuouscorrespondencewithnon-emptycompactandconvexvaluesonX,thenFhasa�xedpoint,i.e.,thereisapointx�2Xsuchthatx�2F(x�).TheKnaster-Kuratowski-Mazurkiewicz(KKM)lemmaisquitebasicandinsomewaysmoreusefulthanBrouwer's�xedpointtheorem.ThefollowingisageneralizedversionofKKMlemmaduetoKyFan(1984).Theorem1.2.9(FKKMTheorem)LetYbeaconvexsetand;6�X�Y.SupposeF:X!2Yisacorrespondencesuchthat(1)F(x)isclosedforallx2X;(2)F(x0)iscompactforsomex02X;(3)FisFS-convex,i.e,foranyx1;:::;xm2Xanditsconvexcombinationx�=Pmi=1�ixi,wehavex�2[mi=1F(xi).Then\x2XF(x)6=;.Here,ThetermFSisforFan(1984)andSonnenschein(1971),whointroducedthenotionofFS-convexity.ThevariouscharacterizationresultsonKakutani'sFixedPointTheorem,KKMLem-ma,andMaximumTheoremcanbefoundinTian(1991,1992,1994)andTianandZhou(1992).27ReferenceBorder,K.C.,FixedPointTheoremswithAppplicationstoEconomicsandGameThe-ory,Cambridge:CambridgeUniversityPress,1985.Debreu,G.(1959),TheoryofValue,(Wiley,NewYork).Fan,K.,\SomePropertiesofConvexSetsRelatedtoFixedPointTheorem,"Mathe-maticsAnnuls,266(1984),519-537.Hildenbrand,W.,A.P.Kirman,EquilibriumAnalysis:VariationsontheThemsbyEdgeworthandWalras,North-Holland:NewYork,1988.Jehle,G.A.,andP.Reny,AdvancedMicroeconomicTheory,Addison-Wesley,1998,Chapters1-2.Luenberger,D.,MicroeconomicTheory,McGraw-Hill,1995,AppendixesA-D.MasColell,A.,M.D.Whinston,andJ.Green,MicroeconomicTheory,OxfordUniversityPress,1995,MathematicalAppendix.Nessah,R.andG.Tian,\ExistenceofSolutionofMinimaxInequalities,EquilibriainGamesandFixedPointswithoutConvexityandCompactnessAssumptions,"(withRabiaNessah),JournalofOptimizationTheoryandApplications,forthcoming.Qian,Y.\UnderstandingModernEconomics,"EconomicandSocialSystemCompari-son,2(2002).RubinsteinAriel,LectureNotesinMicroeconomics(modelingtheeconomicagent),PrincetonUniveristyPress,2005.Takayama,A.MathematicalEconomics,thesecondedition,Cambridge:CambridgeUniversityPress,1985,Chapters1-3.Tian,G.,TheBasicAnalyticalFrameworkandMethodologiesinModernEconomics,2004(inChinese).http://econweb.tamu.edu/tian/chinese.htm.Tian,G.,\FixedPointsTheoremsforMappingswithNon-CompactandNon-ConvexDomains,&qu