微观经济学的数学方法(1)

Mathematicalmethodsforeconomictheory:atutorialbyMartinJ.Osborne1TableofcontentsIntroductionandinstructions1.Reviewofsomebasiclogic,matrixalgebra,andcalculuso1.1Logico1.2Matricesandsolutionsofsystemsofsimultaneousequationso1.3Intervalsandfunctionso1.4Calculus:onevariableo1.5Calculus:manyvariableso1.6Graphicalrepresentationoffunctions2.Topicsinmultivariatecalculuso2.1Introductiono2.2Thechainruleo2.3Derivativesoffunctionsdefinedimplicitlyo2.4Differentialsandcomparativestaticso2.5Homogeneousfunctions3.Concavityandconvexityo3.1Concaveandconvexfunctionsofasinglevariableo3.2Quadraticforms3.2.1Definitions3.2.2Conditionsfordefiniteness3.2.3Conditionsforsemidefinitenesso3.3Concaveandconvexfunctionsofmanyvariableso3.4Quasiconcavityandquasiconvexity4.Optimizationo4.1Introductiono4.2Definitionso4.3Existenceofanoptimum5.Optimization:interioroptimao5.1Necessaryconditionsforaninterioroptimumo5.2Sufficientconditionsforalocaloptimumo5.3Conditionsunderwhichastationarypointisaglobaloptimum6.Optimization:equalityconstraintso6.1Twovariables,oneconstraint6.1.1Necessaryconditionsforanoptimum6.1.2InterpretationofLagrangemultiplier6.1.3Sufficientconditionsforalocaloptimum6.1.4Conditionsunderwhichastationarypointisaglobaloptimumo6.2nvariables,mconstraintso6.3Envelopetheorem7.Optimization:theKuhn-Tuckerconditionsforproblemswithinequalityconstraints2o7.1TheKuhn-Tuckerconditionso7.2WhenaretheKuhn-Tuckerconditionsnecessary?o7.3WhenaretheKuhn-Tuckerconditionssufficient?o7.4Nonnegativityconstraintso7.5Summaryofconditionsunderwhichfirst-orderconditionsarenecessaryandsufficient8.Differentialequationso8.1Introductiono8.2First-orderdifferentialequations:existenceofasolutiono8.3Separablefirst-orderdifferentialequationso8.4Linearfirst-orderdifferentialequationso8.5Phasediagramsforautonomousequationso8.6Second-orderdifferentialequationso8.7Systemsoffirst-orderlineardifferentialequations9.Differenceequationso9.1First-orderequationso9.2Second-orderequations3Mathematicalmethodsforeconomictheory:atutorialbyMartinJ.OsborneCopyright©1997-2003MartinJ.Osborne.Version:2003/12/28.THISTUTORIALUSESCHARACTERSFROMASYMBOLFONT.IfyouroperatingsystemisnotWindowsoryouthinkyoumayhavedeletedyoursymbolfont,pleasegiveyoursystemacharactercheckbeforeusingthetutorial.Ifyousystemdoesnotpassthetest,seethepageoftechnicalinformation.(Note,inparticular,thatifyourbrowserisNetscapeNavigatorversion6orlater,orMozilla,youneedtomakeasmallchangeinthebrowsersetuptoaccessthesymbolfont:here'show.)IntroductionThistutorialisahypertextversionofmylecturenotesforasecond-yearundergraduatecourse.Itcoversthebasicmathematicaltoolsusedineconomictheory.Knowledgeofelementarycalculusisassumed;someoftheprerequisitematerialisreviewedinthefirstsection.Themaintopicsaremultivariatecalculus,concavityandconvexity,optimizationtheory,differentialequations,anddifferenceequations.Theemphasisthroughoutisontechniquesratherthanabstracttheory.However,theconditionsunderwhicheachtechniqueisapplicablearestatedprecisely.Aguidingprincipleisaccessibleprecision.Severalbooksprovideadditionalexamples,discussion,andproofs.ThelevelofMathematicsforeconomicanalysisbyKnutSysdaeterandPeterJ.Hammond(Prentice-Hall,1995)isroughlythesameasthatofthetutorial.MathematicsforeconomistsbyCarlP.SimonandLawrenceBlumeispitchedataslightlyhigherlevel,andFoundationsofmathematicaleconomicsbyMichaelCarterismoreadvancedstill.Theonlywaytolearnthematerialistodotheexercises!Iwelcomecommentsandsuggestions.Pleaseletmeknowoferrorsandconfusions.Theentiretutorialiscopyrighted,butyouarewelcometoprovidealinktothetutorialfromyoursite.(Ifyouwouldliketotranslatethetutorial,pleasewritetome.)Acknowledgments:Ihaveconsultedmanysources,includingthebooksbySydsaeterandHammond,SimonandBlume,andCartermentionedabove,Mathematicalanalysis(2ed)byTomM.Apostol,Elementarydifferentialequationsandboundaryvalueproblems(2ed)byWilliamE.BoyceandRichardC.DiPrima,andDifferentialequations,dynamicalsystems,andlinearalgebrabyMorrisW.HirschandStephenSmale.Ihavetakenexamplesandexercisesfromseveralofthesesources.4InstructionsThetutorialisacollectionofmainpages,withcross-referencestoeachother,andlinkstopagesofexercises(whichinturnhavecross-referencesandlinkstopagesofsolutions).Themainpagesarelistedinthetableofcontents,whichyoucangotoatanypointbypressingthebuttonontheleftmarkedContents.Eachpagehasnavigationalbuttonsontheleft-handside,whichyoucanusetomakeyourwaythroughthemainpages.Themeaningofeachbuttondisplaysinyourbrowser'sstatusbox(atthebottomofthescreenforNetscapeNavigator)whenyouputthemouseoverthatbutton.Onmostpagestherearetenbuttons(thoughonthisinitialpagethereareonlysix),withthefollowingmeanings.oGotothenextmainpage.oGotothenexttop-levelsection.oGobacktothepreviousmainpage.oGobacktotheprevioustop-levelsection.oGotothemainpage(text)forthissection.oGototheexercisesforthissection.oGotothesolutionstotheexercisesforthissection.oGotothetableofcontents.oSearchthroughallpagesofthetutorialforastring.oViewtechnicalinformationaboutviewingandprintingpages.Ifyou'dliketotry