高级微观经济学 黄有光 AdMicroL1MathTools.docx

高级微观经济学 黄有光 AdMicroL1MathTools.docx

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高级微观经济学黄有光AdMicroL1MathTools

ECC5650MicroeconomicTheory

Topic1:

Set,Topology,RealAnalysisandOptimization

（withthecomplementofAssociateProf.MichaelLi）

Readings:

JR-Chapters1&2,supplementedbyDL-AppendixC

1.1Introduction

Inthislecture,wewillquicklygothroughsomebasicmathematicalconceptsandtoolsthatwillbeusedthroughouttherestofthecourse.Asthisisareviewsession,theattentionwillbemainlyonrefreshingonthelanguage,styleandrigorofmathematicalreasoning.Ineconomicanalysis,especiallymicroeconomicanalysis,mathematicsisalwaystreatedasatool,nevertheend.Ontheotherhand,byintegratingeconomicswithrigorousmathematics,wewillbeabletodevelopthetheoreticalexpositionsinasoundandlogicalmanner,whichiswhyeconomicsisalsoknownaseconomicscience.Notmanyothertraditionallyknownassocialsciencefieldsmanagetopassthiscriticalstage.Butitisimportanttorememberthatasaneconomist,wemustgobeyondthenormalmathematicaltreatmentandtheunderlyingeconomicsandtheirpolicyimplicationsarefarmoreimportantandinteresting.

Theplanofthislecturegoeslikethis.First,wewillreviewthebasicsettheory.Wethenmoveontoabitoftopology.Afterreviewingbasicelementsofrealanalysis,wewillcoversomekeyresultsinoptimization.

1.2BasicsofSetTheory

1.2.1BasicConcepts

∙set:

acollectionofelements

∙setsoperations:

union,intersection

∙realsets:

（thenotionofvectors）

∙:

forany;:

thereexists;:

suchthat;:

belongsto;isanelementof

1.2.2Convexity&Relations

ConvexSet:

∙AsetSRnisconvexif

∙Intuitively,asetisaconvexsetifandonlyif（iff）wecanconnectanytwopointsinastraightlinethatliesentirelywithintheset.

∙Convexsethasnoholes,nobreaks,noawkwardcurvaturesontheboundaries;theyareconsideredas“nicesets”.

∙Theintersectionoftwoconvexsetsremainsconvex.

Relations

∙Foranytwogivensets,SandT,abinaryrelationRbetweenSandTisacollectionoforderedpairs（s,t）withsSandtT.

∙ItisclearthatRisasubsetofST:

（s,t）RorsRt.

PropertiesofRelations:

∙Completeness

∙RSSiscompleteiffforallxandy（xy）inS,xRyoryRx.

∙Reflexivity

∙RSSisreflexiveifforallxinS,xRx.

∙Transitivity

∙RSSistransitiveifforallx,y,zinS,xRyandyRzimpliesxRz.

1.3Topology

Topologyattemptstostudythefundamentalpropertiesofsetsandmappings.OurdiscussionwillbemainlyontherealspaceRn.

∙Arealtopologicalspaceisnormallydenotedas（Rn,d）,wheredisthemetricdefinedontherealspace.Intuitivelyspeaking,disadistancemeasurebetweentwopointsintherealspace.

∙EuclideanspacesarespecialrealtopologicalspacesassociatedwiththeEuclideanmetricdefinedasfollows:

1.3.1SetsonaRealTopologicalSpace

-Balls

∙Open-Ballforapointx0:

for>0,

∙Closed-Ballforapointx0:

for>0,

OpenSets

∙SRnisopensetif,xS,>0suchthat（）B（x）S.

∙PropertiesofOpenSets:

∙Theemptysetandthewholesetareopenset

∙Unionofopensetsisopen;intersectionofopensetsisopentoo.

∙Anyopensetcanberepresentedasaunionofopenballs:

where

.

ClosedSets

∙Sisaclosedsetifitscomplement,Sc,isanopenset.

∙ApointxSisaninteriorpointifthereissome-ballcenteredatxthatisentirelycontainedinS.ThecollectionofallinteriorpointsofSisdenotedbyintS,knownastheinteriorofS.

∙PropertiesofClosedSets:

∙Theemptysetandthewholesetareclosed;

∙Unionofanyfinitecollectionofclosedsetsisaclosedset;

∙Intersectionofclosedsetsisaclosedset.

CompactSets

∙AsetSisboundedif>0suchthat（）:

SB（x）forsomexS.

∙AsetinRnthatisclosedandboundediscalledacompactset.

1.3.2Functions/MappingsonRn

∙LetDRm,f:

DRn.Wesayfiscontinuousatthepointx0Dif

>0,>0

∙SpecialCase:

DR,f:

DR.fiscontinuousatx0Dif

>0,>0

PropertiesofContinuousMappings:

∙LetDRm,f:

DRn.Then

∙fiscontinuousforallopenballBRn,f--1（B）isopeninD

forallopensetSRn,f--1（S）isopeninD

∙IfSDiscompact（closedandbounded）,thenitsimagef（S）iscompactinRn.

1.3.3WeierstrassTheorem&TheBrouwerFixed-PointTheorem

Thesetwotheorems,knownasexistencetheorems,areveryimportantinmicroeconomictheory.“Anexistencetheorem”specifiesconditionsthat,ifmet,somethingexists.Inthemeantime,pleasekeepinmindthattheconditionsintheexistencetheoremsarenormallysufficientconditions,meaningthatiftherequiredconditionsareNOTmet,itdoesnotmeanthenonexistenceofsomething–itmaystillexist.Theexistencetheoremssayverylittleaboutexactlocationofthissomething.Inotherwords,existencetheoremsarepowerfultoolsforshowingthatsomethingisthere;butitisnotsufficientinactuallyfindingtheequilibrium.

WeierstrassTheorem–ExistenceofExtremeValues

∙Thisisafundamentalresultinoptimizationtheory.

∙（WeierstrassTheorem）Letf:

SRbeacontinuousreal-valuedmappingwhereSisanonemptycompactsubsetofRn.Thenaglobalmaximumandaglobalminimumexist,namely,

TheBrouwerFixed-PointTheorem

Manyprofoundquestionsaboutthefundamentalconsistencyofmicroeconomicsystemshavebeenansweredbyreformulatingthequestionasoneoftheexistenceofafixedpoint.Examplesinclude:

∙Theviewofacompetitiveeconomyasasystemofinterrelatedmarketsislogicallyconsistentwiththissetting;

∙Thewell-knownMinimaxTheoremingametheory

∙（BrouwerFixed-PointTheorem）LetSRnbeanonemptycompactandconvexset.Letf:

SSbecontinuousmapping.ThenthereexistsatleastonefixedpointoffinS.Thatis,x*Ssuchthatx*=f（x*）.

1.4Real-ValuedFunctions

∙Bydefinition,areal-valuedfunctionisamappingfromanarbitrarysetD（domainset）ofRntoasubsetRofthereallineR（rangeset）.

∙f:

DR,withDRn&RR.

Increasing/DecreasingFunctions:

∙Increasingfunction:

f（x0）f（x1）wheneverx0x1;

∙Strictlyincreasingfunction:

f（x0）>f（x1）wheneverx0>x1;

∙Stronglyincreasingfunction:

f（x0）>f（x1）wheneverx0x1andx0x1

∙Similarly,wecandefinethethreetypesofdecreasingfunctions.

ConcavityofReal-ValuedFunctions

∙Assumptionf:

DR,withDRnisconvexsubsetofRn&RR.

∙f:

DRisconcaveifforallx1,x2D,

∙Intuitivelyspeaking,afunctionisconcaveiffforeverypairofpointsonitsgraph,thechordbetweenthemliesonorbelowthegraph.

∙f:

DRisstrictconcaveifforallx1x2inD,

∙f:

DRisquasiconcaveifforallx1,x2D,

∙f:

DRisstrictlyquasiconcaveifforallx1x2inD,

ConvexityofReal-ValuedFunctions

∙Afterthediscussionofconcavefunctions,wecantakecareoftheconvexfunctionsbytakingthenegativeofaconcavefunction.

∙f:

DRisconvexifforallx1,x2D,

∙f:

DRisstrictconvexifforallx1x2inD,

∙f:

DRisquasiconvexifforallx1,x2D,

∙f:

DRisstrictlyquasiconvexifforallx1x2inD,

PropertiesofConcave/ConvexFunctions

∙f:

DRisconcavethesetofpointsbeneaththegraph,i.e.,{（x,y）|xD,f（x）y}isaconvexset.

∙f:

DRisconvexthesetofpointsabovethegraph,i.e.,{（x,y）|xD,f（x）y}isaconvexset.

∙f:

DRisquasiconcavesuperiorsets,i.e.,{x|xD,f（x）y}areconvexforallyR.

∙f:

DRisquasiconvexinferiorsets,i.e.,{x|xD,f（x）y}areconvexforallyR.

∙Iffisconcave/convexfisquasiconcave/quasiconvex;

∙f（strictly）concave/quasiconcave-f（strictly）convex/quasiconvex.

∙Letfbeareal-valuedfunctiondefinedonaconvexsubsetDofRnwithanonemptyinterioronwhichfisatwicedifferentiablefunction,thenthefollowingstatementsareequivalent:

∙Iffisconcave.

∙TheHessianmatrixH（x）isnegativesemidefiniteforallxinD.

∙Forallx0D,f（x）f（x0）+f（x0）（x–x0）,xD.

HomogeneousFunctions

∙Areal-valuedfunctionf（x）iscalledhomogeneousofdegreekif

.

PropertiesofHomogeneousFunctions:

∙fishomogeneousofdegreek,itspartialderivativesarehomogeneousofdegreek–1.

∙（Euler’sTheorem）f（x）ishomogeneousofdegreekiff

1.5IntroductiontoOptimization

∙Wewillfocusonreal-valuedfunctionsonly.

MainConceptsofOptima

∙Localminimum/maximum

∙Globalminimum/maximum

∙Interiormaxima,boundarymaxima

1.5.1UnconstrainedOptimization

First-Order（Necessary）ConditionforLocalInteriorOptima

∙Ifthedifferentiablefunctionf（x）reachesonalocalinteriormaximumorminimumatx*,thenx*solvesthesystemofsimultaneousequations:

f（x*）=0.

Second-Order（Necessary）ConditionforLocalInteriorOptima

Letf（x）betwicedifferentiable.

1.Iff（x）reachesalocalinteriormaximumatx*,thenH（x*）isnegativesemidefinite.

2.Iff（x）reachesalocalinteriorminimumatx*,thenH（x*）ispositivesemidefinite.

Notes:

∙Thereisasimplemethodincheckingwhetheramatrixisanegative（positive）semidefinte,whichistoexaminethesignsofthedeterminantsoftheprincipleminorsforthegivenmatrix.

Local-GlobalOptimizationTheorem

∙Foratwicecontinuouslydifferentiablereal-valuedconcavefunctionfonD,thefollowingthreestatementsareequivalent,wherex*isaninteriorpointofD:

1.f（x*）=0.

2.fachievesalocalmaximumatx*.

3.fachievesaglobalmaximumatx*.

StrictConcavity/ConvexityandUniquenessofGlobalOptima

∙Ifx*maximizesthestrictlyconcave（convex）functionf,thenx*istheuniqueglobalmaximizer（minimizer）.

1.5.2ConstrainedOptimi