微观经济学的数学方法(1)

Mathematicalmethodsforeconomictheory:atutorialbyMartinJ.OsborneTableofcontentsIntroductionandinstructions1.Reviewofsomebasiclogic,matrixalgebra,andcalculus1.1Logic1.2Matricesandsolutionsofsystemsofsimultaneousequations1.3Intervalsandfunctions1.4Calculus:onevariable1.5Calculus:manyvariables1.6Graphicalrepresentationoffunctions2.Topicsinmultivariatecalculus2.1Introduction2.2Thechainrule2.3Derivativesoffunctionsdefinedimplicitly2.4Differentialsandcomparativestatics2.5Homogeneousfunctions3.Concavityandconvexity3.1Concaveandconvexfunctionsofasinglevariable3.2Quadraticforms3.2.1Definitions3.2.2Conditionsfordefiniteness3.2.3Conditionsforsemidefiniteness3.3Concaveandconvexfunctionsofmanyvariables3.4Quasiconcavityandquasiconvexity4.Optimization4.1Introduction4.2Definitions4.3Existenceofanoptimum5.Optimization:interioroptima5.1Necessaryconditionsforaninterioroptimum5.2Sufficientconditionsforalocaloptimum5.3Conditionsunderwhichastationarypointisaglobaloptimum6.Optimization:equalityconstraints6.1Twovariables,oneconstraint6.1.1Necessaryconditionsforanoptimum6.1.2InterpretationofLagrangemultiplier6.1.3Sufficientconditionsforalocaloptimum6.1.4Conditionsunderwhichastationarypointisaglobaloptimum6.2nvariables,mconstraints6.3Envelopetheorem7.Optimization:theKuhn-Tuckerconditionsforproblemswithinequalityconstraints7.1TheKuhn-Tuckerconditions7.2WhenaretheKuhn-Tuckerconditionsnecessary?7.3WhenaretheKuhn-Tuckerconditionssufficient?7.4Nonnegativityconstraints7.5Summaryofconditionsunderwhichfirst-orderconditionsarenecessaryandsufficient8.Differentialequations8.1Introduction8.2First-orderdifferentialequations:existenceofasolution8.3Separablefirst-orderdifferentialequations8.4Linearfirst-orderdifferentialequations8.5Phasediagramsforautonomousequations8.6Second-orderdifferentialequations8.7Systemsoffirst-orderlineardifferentialequations9.Differenceequations9.1First-orderequations9.2Second-orderequationsMathematicalmethodsforeconomictheory: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.Aguidingprincipleis"accessibleprecision".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.InstructionsThetutorialisacollectionof"main"pages,withcross-referencestoeachother,andlinkstopagesofexercises(whichinturnhavecross-referencesandlinkstopagesofsolutions).Themainpagesarelistedinthetableofcontents,whichyoucangotoatanypointbypressingthebuttonontheleftmarked"Contents".Eachpagehasnavigationalbuttonsontheleft-handside,whichyoucanusetomakeyourwaythroughthemainpages.Themeaningofeachbuttondisplaysinyourbrowser'sstatusbox(atthebottomofthescreenforNetscapeNavigator)whenyouputthemouseoverthatbutton.Onmostpagestherearetenbuttons(thoughonthisinitialpagethereareonlysix),withthefollowingmeanings.Gotothenextmainpage.Gotothenexttop-levelsection.Gobacktothepreviousmainpage.Gobacktotheprevioustop-levelsection.Gotothemainpage("text")forthissection.Gototheexercisesforthissection.Gotothesolutionstotheexercisesforthissection.Gotothetableofcontents.Searchthroughallpagesofthetutorialforastring.Viewtechnicalinformationaboutviewingandprintingpages.Ifyou'dliketotryusingthebuttonsnow,presstheblackright-pointingarrow(onayellowbackground),whichwilltakeyoutothenextmainpage;tocomebackhereafterwards,presstheblackleft-pointingarrowonthatpage.Afteryoufollowalinkonamainpage,pressthewhite"Text"buttontoreturntothepageifyouwishtodosobeforegoingtothenextmainpage.Tohelpyouknowwhereyouare,anabbreviatedtitleforthemainpagetowhichthebuttonsontheleftcorrespondisgivenatthetopofthelightyellowpanel.(Forthispage,forexample,theabbreviatedtitleis"Introduction".)Pagesofexamplesandsolutionstoexerciseshaveorangebackgroundstomakeiteasiertoknowwhereyouare.Ifyougetlost,pressthe"Text"buttonor"Contents"button.TechnicalitiesThetutorialuses"frames"extensively.Ifyourbrowserdoesn'tsupportframes,I'mnotsurewhatyou'llsee;IsuggestyougetarecentversionofNetscapeNavigator.(OtherfeaturesthatIusemayalsonotbesupportedbyotherbrowsers.)Someveryoldbrowsersthatsupportframesdonothandlethe"Back"and"Forward"buttonscorrectlyinframes.HTMLhasnotagstodisplaymath.Ihave"faked"themathbyusingtextitalicfontsforromanletters,theWindowssymbolfontformostsymbols(gifsforothers),smallfontsforsubscriptsandsuperscripts,andtablesforalignments.TheresultisreasonableusingNetscapeNavigatorwitha12or14pointbasefontandarelativelyhighresolutionmonitor,butmaynotbesogreatunderothercircumstances.Ifwhatyouseeonyourscreenlooksawful,letmeknowandI'llseeifIcandoanythingaboutit.MathML,avariantofHTML,hasextensivecapabilitiesforbeautifullydisplayingmath,butiscurrentlysupportedonlybyNetscapeNavigator7.1anditscousins(e.g.Mozilla).IamworkingonaMathMLversionofthetutorial.1.Reviewofsomebasiclogic,matrixalgebra,andcalculus1.1LogicBasicsWhenmakingprecisearguments,weoftenneedtomakeconditionalstatements,likeifthepriceofoutputincreasesthenacompetitivefirmincreasesitsoutputorifthedemandforagoodisadecreasingfunctionofthepriceofthegoodandthesupplyofthegoodisanincreasingfunctionofthepricethenanincreaseinsupplyateverypricedecreasestheequilibriumprice.ThesestatementsareinstancesofthestatementifAthenB,whereAandBstandforanystatements.WealternativelywritethisgeneralstatementasAimpliesB,or,usingasymbol,asAB.YettwomorewaysinwhichwemaywritethesamestatementareAisasufficientconditionforB,andBisanecessaryconditionforA.(NotethatBcomesfirstinthesecondofthesetwostatements!!)Importantnote:ThestatementABdoesnotmakeanyclaimaboutwhetherBistrueifAisNOTtrue!ItsaysonlythatifAistrue,thenBistrue.Whilethispointmayseemobvious,itissometimesasourceoferror,partlybecausewedonotalwaysapplytherulesoflogicineverydaycommunication.Forexample,whenwesay"ifit'sfinetomorrowthenlet'splaytennis"weprobablymeanboth"ifit'sfinetomorrowthenlet'splaytennis"and"ifit'snotfinetomorrowthenlet'snotplaytennis"(andmaybealso"ifit'snotclearwhethertheweatherisgoodenoughtoplaytennistomorrowthenI'llcallyou").Whenwesay"ifyoulistentotheradioat8o'clockthenyou'llknowtheweatherforecast",ontheotherhand,wedonotmeanalso"ifyoudon'tlistentotheradioat8o'clockthenyouwon'tknowtheweatherforecast",becauseyoumightlistentotheradioat9o'clockorcheckontheweb,forexample.Thepointisthattherulesweusetoattachmeaningtostatementsineverydaylanguageareverysubtle,whiletherulesweuseinlogicalargumentsareabsolutelyclear:whenwemakethelogicalstatement"ifAthenB",that'sexactlywhatwemean---nomore,noless.Wemayalsousethesymbol""tomean"onlyif"or"isimpliedby".ThusBAisequivalenttoAB.Finally,thesymbol""means"impliesandisimpliedby",or"ifandonlyif".ThusABisequivalenttoABandBA.IfAisastatement,wewritetheclaimthatAisnottrueasnot(A).IfAandBarestatements,andbotharetrue,wewriteAandB,andifatleastoneofthemistruewewriteAorB.Note,inparticular,thatwriting"AorB"includesthepossibilitythatbothstatementsaretrue.TworulesRule1IfthestatementABistrue,thensotooisthestatement(notB)(notA).ThefirststatementsaysthatwheneverAistrue,Bistrue.ThusifBisfalse,Amustbefalse---hencethesecondstatement.Rule2Thestatementnot(AandB)isequivalenttothestatement(notA)or(notB).Notethe"or"inthesecondstatement!IfitisnotthecasethatbothAistrueandBistrue(thefirststatement),theneitherAisnottrueorBisnottrue.QuantifiersWesometimeswishtomakeastatementthatistrueforallvaluesofavariable.Forexample,lettingD(p)bethetotaldemandfortomatoesatthepricep,itmightbetruethatD(p)>100foreverypricepinthesetS.Inthisstatement,"foreveryprice"isaquantifier.Importantnote:Wemayuseanysymbolforthepriceinthisstatement:"p"isadummyvariable.AfterhavingdefinedD(p)tobethetotaldemandfortomatoesatthepricep,forexample,wecouldwriteD(z)>100foreverypricezinthesetS.Giventhatwejustusedthenotationpforaprice,switchingtozinthisstatementisalittleodd,BUTthereisabsolutelynothingwrongwithdoingso!Inthissimpleexample,thereisnoreasontoswitchnotation,butsometimesinmorecomplicatedcasesaswitchisunavoidable(becauseofaclashwithothernotation)orconvenient.ThepointisthatinanystatementoftheformA(x)foreveryxinthesetYwemaylegitimatelyuseanysymbolinsteadof"x".AnothertypeofstatementwesometimesneedtomakeisA(x)forsomexinthesetY,or,equivalently,thereexistsxinthesetYsuchthatA(x)."Forsomex"(alternatively"thereexistsx")isanotherquantifier,like"foreveryx";mycommentsaboutnotationapplytoit.Exercises1.1ExercisesonlogicA,B,andCarestatements.Thefollowingtheoremistrue:ifAistrueandBisnottruethenCistrue.Whichofthefollowingstatementsfollowfromthistheorem?IfAistruethenCistrue.IfAisnottrueandBistruethenCisnottrue.IfeitherAisnottrueorBistrue(orboth)thenCisnottrue.IfCisnottruethenAisnottrueandBistrue.IfCisnottruetheneitherAisnottrueorBistrue(orboth).AandBarestatements.Thefollowingtheoremistrue:AistrueifandonlyifBistrue.Whichofthefollowingstatementsfollowfromthistheorem?IfAistruethenBistrue.IfBistruethenAistrue.IfAisnottruethenBisnottrue.IfBisnottruethenAisnottrue.LetGbeagroupofpeople.AssumethatforeverypersonAinG,thereisapersonBinGsuchthatAknowsafriendofB.IsittruethatforeverypersonBinG,thereisapersonAinGsuchthatBknowsafriendofA?[Solutions]1.1SolutionstoexercisesonlogicOnly(e)followsfromthetheorem.Allfourstatementsfollowfromthetheorem.Yes.InthefirststatementAandBarevariablesthatcanstandforanyperson;theymaybereplacedbyanyothertwosymbols---forexample,AcanbereplacedbyB,andBbyA,whichgivesusthesecondstatement.1.2MatricesandsolutionsofsystemsofsimultaneousequationsMatricesIassumethatyouarefamiliarwithvectorsandmatricesandknow,inparticular,howtomultiplythemtogether.(Dothefirstfewexercisestocheckyourknowledge.)Thedeterminantofthe2  2matrix a  b  c  d isad bc.Ifad bc 0,thematrixisnonsingular;inthiscaseitsinverseis1ad bc d  b  c  a .(Youcancheckthattheproductofthematrixanditsinverseistheidentitymatrix.)Thedeterminantofthe3  3matrix a  b  c  d  e   f   g  h  i is =a(ei h f ) b(di g f ) +c(dh eg).If 0thematrixisnonsingular;inthiscaseitsinverseis1 D11  D12  D13  D21  D22  D23  D31  D32  D33 whereDijisthedeterminantofthe2  2matrixobtainedbydeletingtheithcolumnandjthrowoftheoriginal3  3matrix.Thatis,D11 =ei h f ,D12 =bi ch,D13 =b f  ec,D21 =di  f g,D22 =ai cg,D23 =a f  dc,D31 =dh eg,D32 =ah bg,andD33 =ae db.(Again,youcancheckthattheproductofthematrixanditsinverseistheidentitymatrix.)ThedeterminantofthematrixAisdenotedA.SolutionsofsystemsofsimultaneousequationsConsiderasystemoftwoequationsintwovariablesxandy:ax +by =ucx +dy =v.Herearethreewaystosolveforxandy.Isolateoneofthevariablesinoneoftheequationsandsubstitutetheresultintotheotherequation.Forexample,fromthesecondequationwehavey =(v cx)/d.Substitutingthisexpressionforyintothefirstequationyieldsax +b(v cx)/d =u,whichwecanwriteas(a bc/d)x +bv/d =u,sothatx = u bv/da bc/dorx = ud bvad bc.Tofindywenowusethefactthaty =(v cx)/d,togety = va cuad bc.UseCramer'srule(discoveredbyGabrielCramer,1704-1752).Writethetwoequationsinmatrixformas a  b  c  d    x  y    = u  v   (*)ByCramer'srule,thesolutionsaregivenbyx =  u  b  v  d   ad bcandy =  a  u  c  v   ad bc(wheread bcisthedeterminantofthematrixintheexpressionoftheequationsinmatrixform,andthematrixinthenumeratorofeachexpressionisobtainedbyreplacingthecolumninthematrixontheleftof(*)thatcorrespondstothevariablebeingsolvedforwiththecolumnvectorontherightof(*)),orx =ud bvad bcandy =va cuad bc.Writethetwoequationsinmatrixformas a  b  c  d    x  y    = u  v   (aswhenusingCramer'srule)andsolvebyinvertingthematrixonthelefthandside.Theinverseofthismatrixis1ad bc d  b  c  a   sowehave x  y    =1ad bc d  b  c  a    u  v   sothatx =ud bvad bcandy =va cuad bc.Whichofthesethreemethodsisbestdependsonseveralfactors,includingyourabilitytorememberCramer'sruleand/orhowtoinvertamatrix.Ifyouhavetosolveforonlyoneofthevariables,Cramer'sruleisparticularlyconvenient(ifyoucanrememberit).Forasystemofmorethantwovariables,thesamethreemethodsareavailable.Thefirstmethod,however,isprettymessy,andunlessyouareadeptatinvertingmatrices,Cramer'sruleisprobablyyourbestbet.1.2ExercisesonmatrixalgebraandsolvingsimultaneousequationsLetA = 4  1  6  9   andB = 0  3  3  2   Find(i)A +B,(ii)2A B,(iii)AB,(iv)BA,and(v) A(thetransposeofA).LetA = 4  1  6  9  2  3   andB = 0  3  3  2   (i)IsABdefined?Ifso,findit.(ii)IsBAdefined?Ifso,findit.Givenu =(5,2,3),findu·u(thescalarproduct,orinnerproduct).Youbuynitemsinthequantitiesq1,...,qnatthepricesp1,...,pn.Expressyourexpenditureusing(i) notation,(ii) vectornotation.Letx =(x1,x2)andletA = a11  a12  a21  a22   FindxAx.FindthedeterminantsofthematricesAandBinProblem 1.FindtheinverseofthematrixAinProblem 1,andverifythatitisindeedtheinverse.Findthedeterminantsofthefollowingtwomatrices.A = 8  1  3  4  0  1  6  0  3   andB = a  b  c  b  c  a  c  a  b   AreeitherofthematricesAandBinthepreviousproblemnonsingular?FindtheinverseofthematrixAinProblem 8.UseCramer'sruletofindthevaluesofxandythatsolvethefollowingtwoequationssimultaneously.3x 2y= 112x + y= 12Solvethetwoequationsinthepreviousproblembyusingmatrixinversion.UseCramer'sruletofindthevaluesofx,y,andzthatsolvethefollowingthreeequationssimultaneously.4x +3y 2z= 7x + y= 53x + z = 4Solvethethreeequationsinthepreviousproblembyusingmatrixinversion.1.2Solutionstoexercisesonmatrixalgebraandsolvingsimultaneousequations 4  2  9  7    8  5  9  20    3  14  27  0    18  27  0  21    4  6  1  9   Yes; 3  14  27  0  9  0   Nou·u =38.(i) i=1npiqi;(ii) p·qwherep =(p1,...,pn)andq =(q1,...,qn).a11x12 +(a12 +a21)x1x2 +a22x22.A =42.B =9.A1 =(1/42) 9  1  6  4   ;A1A =I.A =6;B =3abca3 b3 c3.A:yes;B:ifandonlyif3abc a3 +b3 +c3.A1 = 0  1/2  1/6  1  1  2/3  0  1  2/3   Wehavex = 11  2  12  1   / 3  2  2  1    =5andy = 3  11  2  12   / 3  2  2  1    =2Solutionis(1/7) 1  2  2  3    11  12    = 5  2   Wehavex = 7  3  2  5  1  0  4  0  1   / 4  3  2  1  1  0  3  0  1    =0andy = 4  7  2  1  5  0  3  4  1   / 4  3  2  1  1  0  3  0  1    =5andy = 4  3  7  1  1  5  3  0  4   / 4  3  2  1  1  0  3  0  1    =4Solutionis(1/7) 1  3  2  1  10  2  3  9  1    7  5  4    = 0  5  4   1.3IntervalsandfunctionsIntervalsAnintervalisasetof(real)numbersbetween,andpossiblyincluding,twonumbers.Theintervalfromatobisdenotedasfollows:[a, b]ifaandbareincluded(i.e.[a, b] ={x:a x b})(a, b)ifneitheranorbisincluded(i.e.(a, b) ={x:a