高级微观经济学所需的数学知识.pdf

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1、condition.Becomingarichcountryalsodependsonotherfactorssuchaspoliticalsystem,socialinfrastructures,andculture.Additionally,noexampleofacountrycanbefoundsofarthatitisrichinthelongrun,thatisnotamarketeconomy.Apositivestatementstatefactswhilenormativestatementgiveopini

2、onsorvaluejudgments.Distinguishingthesetwostatementscanvoidmanyunnecessarydebates.1.2LanguageandMethodsofMathematicsThissectionreviewssomebasicmathematicsresultssuchas:continuityandconcavityoffunctions,SeparatingHyperplaneTheorem,optimization,correspondences(pointto

3、setmappings),fixedpointtheorems,KKMlemma,maximumtheorem,etc,whichwillbeusedtoprovesomeresultsinthelecturenotes.Forgoodreferencesaboutthematerialsdiscussedinthissection,seeappendixesinHildenbrandandKirman(1988),Mas-Colell(1995),andVarian(1992).1.2.1FunctionsLetXandYbe

4、twosubsetsofEuclidianspaces.Inthistext,vectorinequalities,=,≥,and>,aredefinedasfollows:Leta,b∈Rn.Thena=bmeansa=bforalls=1,...,n;ssa≥bmeansa=bbuta̸=b;a>bmeansas>bsforalls=1,...,n.Definition1.2.1Afunctionf:X→Rissaidtobecontinuousifatpointx0∈X,limf(x)=f(x0),x→x0orequival

5、ently,foranyϵ>0,thereisaδ>0suchthatforanyx∈Xsatisfying

6、x−x0

7、<δ,wehave

8、f(x)−f(x0)

9、<ϵAfunctionf:X→RissaidtobecontinuousonXiffiscontinuousateverypointx∈X.Theideaofcontinuityisprettystraightforward:Thereisnodisconnectedpointifwedrawafunctionasacurve.Afunctioniscontinuou

10、sif“small”changesinxproduces“small”changesinf(x).12Theso-calleduppersemi-continuityandlowersemi-continuitycontinuitiesareweakerthancontinuity.Evenweakconditionsoncontinuityaretransfercontinuitywhichchar-acterizemanyoptimizationproblemsandcanbefoundinTian(1992,1993,1

11、994)andTianandZhou(1995),andZhouandTian(1992).Definition1.2.2Afunctionf:X→Rissaidtobeuppersemi-continuousifatpointx0∈X,wehavelimsupf(x)5f(x0),x→x0orequivalently,foranyϵ>0,thereisaδ>0suchthatforanyx∈Xsatisfying

12、x−x0

13、<δ,wehavef(x)

14、ntheuppersemi-continuityatx0areequivalent,thesecondoneiseasiertobeversified.Afunctionf:X→Rissaidtobeuppersemi-continuousonXiffisuppersemi-c