高级微观经济学英文ppt课件.ppt

Microeconomics AdvancedMicroeconomics lecture1 productiontheoryI QinZhouProfessorSchoolofEconomics ManagementSoutheastUniversityPhone 86 25 52090731 Email Office Rm 315 A BuildingSchoolofE M Summary textbook Varian HalR 1992 MicroeconomicsAnalysis 3rded Mas Colell A M Whinston andJ Green 1995 MicroeconomicsTheory assignments Oncetwoweeks Teamwork Deliverontheclass examinations Final term 80 ofthequestionscomingfromtheassignments ScoreFinal term 50 Assignments 30 SameScoreintheteam Homework 20 TheBasicFrameworkofMicroeconomics Technology Preference Choice production Choice purchase Supply Demand MarketEquilibrium Technology Contain production possibilities set PS and productionfunction Propertiesofthe PS Technicalrateofsubstitution Returnstoscale 1 Productionset Productionplan productionvector orinput outputvector ProductionsetY alltechnologicalfeasibley RestrictedproductionsetY z Someofyiinyarerestrictedonz Short runproductionset 1 Productionset Inputrequirementset Allyiinyarenegative letthembe x thenxispositive andtherestyjtobeq So and andtheinputrequirementsetis 1 Productionset Transformationfunction andsatisfiedifandonlyifyistechnologicallyefficient then y2 y1 1 Productionset Productionfunction IfO 1 andI n 1 then means Isoquant Question1 calculatethePS RS TF PF IsoquantofCobb DouglastechnologyandLeontieftechnology 2 PropertiesofPS Yisnonempty wehavesomethingtodo Yisclose Ycontainit sboundary Nofreelunch Seethefig Freedisposal Seethefig 2 PropertiesofPS Additive freeentrance Convexity Seethefig Proposition1 ifYisconvex soisV q Proposition2 ifV q isconvex f x isquasi concave 3 Technicalrateofsubstitution Marginalrateoftransformation MRT ofgoodjforgoodiatyandT y 0 T y 0 wegot Seethefig 3 Technicalrateofsubstitution oneproduction themarginalratetechnicalofsubstitutionatx Asxchanges wegottechnicalrateofsubstitution 3 Technicalrateofsubstitution Theelasticityofsubstitution thecurvatureoftheisoquant Question2 calculatethe TRS andthe elasticityofsubstitution ofCobb DouglastechnologyandCEStechnology 4 Returnstoscale Nonincreasingreturnstoscale Nondecreasingreturnstoscale Constantreturntoscale 4 Returnstoscale Proposition3 YisconstantreturnstoscaleifYisboth additive and convexity Proposition4 singleproduction ifandonlyiff ishomogenousofdegree1 Yisconstantreturnstoscale 4 Returnstoscale WhyweassumethatYisnon increasing ordecreasingforusual returnstoscale SupposeYisdecreasingreturnstoscale andit sPFf x nowweintroducenewinputz anddifineanewPF F ishomogenousofdegree1 4 Returnstoscale Homogeneousandhomothetictech Homogeneousofdegreek Homotheticfunction Elasticityofscale Assignment Textbook ex 1 1 ex 1 3 ex 1 7 ex 1 9 PropertiesofPS y2 y1 Freelunch PropertiesofPS PropertiesofPS y2 y1 Technicalrateofsubstitution THANKYOU AdvancedMicroeconomics lecture2 productiontheoryII profitmaximization Contain SomedefinitionIsoprofitcurveDemandfunctionandit spropertiessupplyfunctionandit spropertiesProfitfunctionandit sproperties 1 Somedefinition Basicassumption Yischangeless notechnicalimprovement Thefirmisthepricetaking Profitfunction Whenitisoneproduction 2 Isoprofitcurve Theprofit Wethengottheisoprpfitcurve Thenandisnegativesemidifinite q x x 2 Isoprofitcurve WeakAxiomofProfitMaximization WAPM ifys ytareinY andchoicebyfirmunderpricepsandpt then wecanget or 2 Isoprofitcurve 2 Isoprofitcurve y1 y2 x q y1 y2 x q YI YO 3 Demandfunction factordemandfunction set proposition5 it shomogenousofdegreezero 3 Demandfunction Oneproduction regularpas1 letx w betheprofitmaximizationchoice functionmeansxissinglepointunderw ofinputfactorvectorunderfactorpricew thenmusthold 3 Demandfunction When differentiatewithrespecttow weget orSubstitutionmatrixisasymmetricnegativematrix 1 2 3 Demandfunction Aretheprofitmaximizationgiveusthesufficientinformationabouttheproductionbehavior Suchas 1 factordemandfunction and 2 yeild supply function Proposition6 demandforfactoriis or 4 supplyfunction Supplyset Proposition7 y ishomogenousofdegree1 Proposition8 IfYisconvexforanyp thenisy p ifYisstrictlyconvex y p isasinglepoint 4 Supplyfunction Proposition9 Hotelling slemma y p issinglepoint then recallprop 6 Thesupplysubstitutionmatrixissymmetricpositivematrix And why seethefig 5 Profitfunction Proposition10 ishomogenousofdegree1 Proposition11 isconvex 2 5profitfunction LeChatelierPrinciple somethingislimitedinshort term butnotlong term definiteasz and let if Assignment Textbook ex 1 7 ex 2 2 ex 2 5 ex 3 4 Supplyfunction y1 y2 y p AdvancedMicroeconomics LECTURE3 productiontheoryIII CostMinimization Content DefinitionsPropertiesofcostfunctionWACMSomeformsofcostfunctions 1 Definitions Oneproduction costfunctionisTheoptimalsolutionx w q istheconditionalfactordemandfunction Question1 calculatetheconditionalfactordemandfunctionofC Dtech andCEStech 1 Definitions Recallthecostminimizationcondition letx 0 thensetLagrangian Wegot seethefig Whatis 2 Propertiesofcostfunction Proposition1 c w q ishomogeneousofdegree1inw andnon decreasinginq Proposition2 c w q isconcavefunctionofw Proposition3 x w q ishomogeneousofdegree0inw Proposition4 ifV q isconvex thenisx ifV q isstrictlyconvex x issinglepoint 2 Propertiesofcostfunction Proposition5 Shephard slemma ifx w q issinglepoint thenProposition6 issymmetricnegativesemi definite andseethefig Proposition7 iff isHD1 c andx isHD1too iff isconcave c isconvexinq 3 WACM WeakAxiomofCostMinimization WACM ifxs xtareinY andchoicebyfirmunderpricewsandwt then wecanget 3 WACM x2 xA x1 xB 3 WACM x2 xA x1 xB VI VO 4 Someformsofcostfunctions If correspondingprice andisthelimitedfactor fixedassets whileisthevariablefactor Thetotalcostis 4 Someformsofcostfunctions Envelopetheorem anobjectivefunctiondependonsomeparameter Then seetheprove 4 Someformsofcostfunctions LACandSAC SoSeethefig 4 Duality Giventhetechnology wecanobtainthecostfunction arethecostfunctioncontainsthesameinformationofthetechnology productionfunction Iftheansweris yes thenthecostminimizationbehaviorwillindicatethetechnologyofthefirm Assignment Textbook ex 5 3 ex 5 8 ex 5 11 ex 5 12 ex 5 17 Definitions x2 x1 x w q w Propertiesofcostfunction x2 x w q w Proveofenvelopetheorem Letbethesolutionoftheproblem thenDifferentiatingbothsides wehave Andx maximizef soThen LACandSAC q AC AC q q AdvancedEconomics lecture4 Duality ZhouQin 64 Duality Giventhetechnology wecanobtainthecostfunction arethecostfunctioncontainsthesameinformationofthetechnology productionfunction Iftheansweris yes thenthecostminimizationbehaviorwillindicatethetechnologyofthefirm 65 Duality Content DualityinmathematicsSufficientconditionforcostfunctionFactordemandfunctionGeometryofduality 66 1 Dualityinmathematics Someconceptsandproperties Half spaces Normalvector Hyperplane Kisconvexclosure Kisconcave K istheclosedconvexhull 67 1 Dualityinmathematics Supportfunction infimum givenanalternativedescriptionforK Proposition8 isHD1andconcave Seethefig 68 1 Dualityinmathematics Dualitytheorem Kisnonemptyclosure anditssupportfunctionisdifferentiableat thenthereisonlyone thatSeethefig 69 2 S C forcostfunction Differentiablefunctionsatisfied HD1ofwConcaveofwNon decreasingofwNonnegativefor Thenisthecostfunctiondefinitebythetech of 70 3 whataboutdemandfunction DualityindicatethatHD1andconcaveiswhattheconvextechneedforthecost Whatabouttheotherfunction suchasfactordemandfunction isHD0andissymmetricnegativesemidefinite Thenitistheconditionalfactordemandfunctionofacertaintech 71 3 whataboutdemandfunction Example givenacostfunction thenwhat sthecorrespondedtech andfactordemandfunction Proposition9 elasticityofscale 72 4 Geometryofduality 73 4 Geometryofduality 74 4 Geometryofduality 75 4 Geometryofduality 76 Supportfunction p K 77 Dualitytheorem K x1 x2 p2isdecreasing 78 Dualitytheorem K p2 79 AdvancedEconomics lecture5 consumptiontheoryI ZhouQin PreferenceandChoice Content PreferenceandrationalityTheruleofchoiceBudgetDemandfunctionWAanddemandedlaw 1 Preference An acceptable alternativesset oraconsumptionset A weak preferencerelation abinaryrelationshiponX forany means xisatleastasgoodasy Astrictpreference Anindifferencerelation 1 Preference Definition preferenceonXisrationalif It scomplete It sreflexive It stransitive 1 Preference Propersition1 ifisrational then isirreflexiveandtransitiveisreflexiveandtransitiveandsymmetricIf 2 Theruleofchoice Wecanonlyobservewhatamanchoiceunderacertainconstrain Istheman spreferencerationalwhenweobservedseveralchoiceofhim Orifhispreferencewasrational whatruleswillhischoicetake 2 Theruleofchoice Choiceconstruct achoiceundercertainbudget Budgetset and isabudgetfamily representedalimitedlistofallpossiblechoiceundercertainsystem nature technologyandothersocialenvironmentsthatthedecision makermusttake Choicerule maybecontainmorethanoneelements 2 Theruleofchoice WeakAxiomofRevealedPreference WARPorWA ifforcertain andforand thenmustRevealedPreference 2 Theruleofchoice ChoicesetProposition2 ifarerational thenchoiceconstructsatisfiedWA Prop 2indicatethatifthedecision makerhaverationalpreference thanhischoicesatisfiedWA IshispreferencerationalifhischoiceisobservedsatisfyWA 2 Theruleofchoice Definition for wesay rationalpreferencerationalizationPropersition3 ifsatisfiedWAandcontainallsubsetsofXthatisternaryandlessthanternary example X x y z B x y y z x z C x y x C y z y C x z z B C satisfiedWARP butthereisnorationalizationpreferencerelationoverit 3 Competivebudget ConsumptionsetXandit spricewiththewealthw 0 Definition Walras competive budgetsetProposition4 ifXisconcave thenis 4 Demandfunction Foranypandwarestrictlypositive thecorrespondingdemandsetx p w arenonempty ifx p w issinglepoint wecallittheWalrasian Marshallian demandfunction x p w ishomogenousofdegreezeroandsatisfiedWalras law Thatispx wforall wewillprovethemnextlecture 4 Demandfunction Wealtheffects Engelfunction givenprice functionoverwealthWealthexpansionpath Wealtheffects 4 Demandfunction Wealtheffects CommoditylisnormalgoodsifCommoditylisinferiorgoodsifThedemandisnormalifSeethefig 4 Demandfunction Priceeffects Supplycurve Priceeffects Giffengood Seethefig 4 Demandfunction CournotAggregation ifWalrasiandemandfunctionx p w isHD0andsatisfiedWalrasLaw then EngelAggregation ifWalrasiandemandfunctionx p w satisfiedWalrasLaw then 5 WAanddemandlaw Walrasiandemandfunctionx p w satisfiedWAifforanywehave Seethefig 5 WAanddemandlaw Changinginpricewillchangewealthtoo Buthowcanwetellthedemandchangingbypricechangingfromwealthchanging Givenachangingfrom andpeoplewillnotgetworsethatisherewealthchanging compensation wascalled Slutskywealthcompensation and Slutsky compensatedpricechanging 5 WAanddemandlaw Proposition5 x p w satisfiedWAifandonlyif andwhen Prop 5indicates orthat scalled demandlaw or compensationdemandlaw 5 WAanddemandlaw Slutskymatrix substitutionmatrix Substitutioneffectsisn s dGiffengoodisnecessaryinferiorgood Assignment IfThenForthenIfLaspeyresquantityindexislessthan1 then andifPaaschequantityindexislargethan1 then Consumptionset Consumptionset Consumptionset expansionpath x2 x1 Wealtheffects luxurygoods Supplycurve x2 x1 Giffengood x2 x1 5 WAanddemandlaw x2 x1 x2 x1 5 WAanddemandlaw x2 x1 x2 x1 ThankYou 111 AdvancedMicroeconomicslecture5 consumptiontheoryII ZhouQin 112 Utilitymaximization Content UtilityfunctionUtilitymaximizationExpenditureminimizationRelationshipbetweenUMPandEMPIntegrability 113 1 Frompreferencestoutility Definition isautilityfunctionofpreference ifCanwealwaysfindautilityfunctionof MaybeIfXisfinite therealwaysexistutilityfunction Proposition1 onlyrationalcanberepresentedbyautilityfunction N CnotS C Lexicographicpreference rationalbutnoutilityfunctionexist 114 1 Frompreferencestoutility Continuity then orxuppercontoursetsandlowercontoursetsareclosure Proposition2 Ifiscontinuous thenexistacontinuousutilityfunctionrepresenting 115 1 Frompreferencestoutility Desirability preferenceisdesirableifismonotone islocalnon satiation thereisaproposition3 isstrongmonotone thenit smonotone ismonotone it slocalnon satiation 116 1 Frompreferencestoutility Convexity xuppercontoursetsareconvex Decreasinginmarginalrateofsubstitution peoplelikevariety Preferencesareconvexmeansutilityfunctionisquasi concave Note quasi concavedoesnotcontainconcave 117 1 Frompreferencestoutility If thenarehomothetic Proposition4 Thearehomotheticifandonlyifit sutilityfunctionisHD1 Ifthenarequasi linearofcommodity1 it scalledstandardcommodity Proposition5 arequasi linearifandonlyifit sutilityfunctionisGormanform 118 1 Frompreferencestoutility Preference Utilityfunction rationality exist Xfinite Desirability increasing Continuity continuity Convexity Quasi concave Homothetic HD1 Quasi linear Gormanform 119 2 Utilitymaximization Consumer sproblem UMP ThesolutionsarecalledWalras DemandCorrespondence orfunctionifit ssinglepoint 120 2 Utilitymaximization PropertiesofHD0SatisfiedWalras LawIfareconcave areconcavetoo ifarestrictlyconcave issinglepoint 121 2 Utilitymaximization Ifthesolution thenHereistheshadowprice themarginalutilityofoptimalconsumption Thevaluefunctioniscalled indirectutilityfunction 122 3 Expenditureminimization EMP Thesolutionwascalled Hicks orcompensation demandcorrespondence singlepoint function It sHD0ofp It sconvex asis andsinglepointwhenarestrictlyconvex 123 3 Expenditureminimization Thevaluefunction expenditurefunction moneymetricuntility So whenpricechanging ifthewealthofconsumerchangecorrespondentlytokeepthesameutilitylevel thenisthedemandchanging HereHickswealthcompensationisRecalltheSlutskywealthcompensation 124 4 Relationship Proposition5 Duality u x isacontinuousutilityfunctionof w 0 x isthesolutionofUMP whenit srequiredthat x isthesolutionofEMP andvalueofEMPisw Iftherequiredutility x isthesolutionofEMP whentheexpenditureisp x x isthesolutionofUMP andvalueofUMPisu 125 4 Relationship Hicksdemandandexpenditurefunction iss n s d HicksandWalrasdemand Slutskyequation Walrasdemandandindirectutilityfunction Roy sidentity 126 4 Relationship UMP EMP x p w v p w h p u e p u duality Slutskyequation Roy sidentity 127 5 Integrability Demandfunctionx p w c d isHD0 satisfiedWalrasLawandhaveasubstitutionmatrixS p w iss n s d forany p w ifit sdeducedbyrationalpreference Andifweobservedanx p w satisfiedsuchconditions canwefindapreferencetorationalizationx p w Thattheintegrabilityproblem 128 5 Integrability expenditurefunction preference Proposition6 differentiablee p u istheexpenditurefunctionofsets Weneedtoprovee p u isthesupportfunctionofV u thatisseethefig 129 5 Integrability Demand expenditurefunction Partialdifferentialequation Theexistenceofsolutionmeanssubstitutionmatrixissymmetric 130 Assignment ex 7 4 ex 8 7 ex 8 14 131 quasi concaveandconcave x u x x1 x2 132 SlutskyandHickscompensation 133 expenditurefunction preference x1 x2 x2 x1 134 AdvancedMicroeconomicslecture6 consumptiontheoryIII ZhouQin Utilitymaximization Content RelationshipbetweenUMPandEMPIntegrabilityAggregationacrossgoods 1 Relationship Proposition5 Duality u x isacontinuousutilityfunctionof w 0 x isthesolutionofUMP whenit srequiredthat x isthesolutionofEMP andvalueofEMPisw Iftherequiredutility x isthesolutionofEMP whentheexpenditureisp x x isthesolutionofUMP andvalueofUMPisu 1 Relationship Hicksdemandandexpenditurefunction iss n s d HicksandWalrasdemand Slutskyequation Walrasdemandandindirectutilityfunction Roy sidentity 1 Relationship UMP EMP x p w v p w h p u e p u duality Slutskyequation Roy sidentity 2 Integrability Demandfunctionx p w c d isHD0 satisfiedWalrasLawandhaveasubstitutionmatrixS p w iss n s d forany p w ifit sdeducedbyrationalpreference Andifweobservedanx p w satisfiedsuchconditions canwefindapreferencetorationalizationx p w Thattheintegrabilityproblem 2 Integrability expenditurefunction preference Proposition6 differentiablee p u istheexpenditurefunctionofsets Weneedtoprovee p u isthesupportfunctionofV u thatisseethefig 2 Integrability Demand expenditurefunction Partialdifferentialequation Theexistenceofsolutionmeanssubstitutionmatrixissymmetric 3 Aggregationacrossgoods Whylocalanalysisisrational Thatis it srationaltomodelconsumerchoicebypartialmaximization What stherestrictionofthepreferencethatwecandolikethat Separability partitioningconsumptionbundleintotwo sub bundles x z andpricevector p q 3 Aggregationacrossgoods UMP Let andset Solution howcanwegetit 3 Aggregationacrossgoods Twoways Aggregatepricesatfirst andthenmaximizeUonthebudget thisiscalledHicksianseparability Maximizeuonbudgetatfirst thenaggregatequantitiestoget it scalledfunctionalseparability 3 Aggregationacrossgoods Hicksianseparability norelativepricechange so let Definetheindirectutilityfunction AsRoy sidentityshow 3 Aggregationacrossgoods Constructiondirectutilityfunctionhasthepropertythat MaximizeUmeansmaximizeu 3 Aggregationacrossgoods Functionalseparability IndependencepropertyWeaklyseparableutilityfunction Thentheproblemis 3 Aggregationacrossgoods Letbetheexpenditurefunction thentheproblemis Ifislineras vishomothetic letand then 3 Aggregationacrossgoods wegetthesamesolutionasproblem Localanalysis Assignment Textbook Ex 9 10 Ex 9 11 Ex 10 1 SlutskyandHickscompensation expenditurefunction preference x1 x2 x2 x1 155 AdvancedMicroeconomicslecture7 consumptiontheoryIV ZhouQin 156 AggregationandWelfare Content Consumer ssurplusAggregationRepresentationconsumer 157 1 Consumer ssurplus Moneymetricutilityfunctionmeasurethewelfareofconsumerineconomychanging Definition equivalentvariation EV andcompensatingVariation CV seethefig Let 158 1 Consumer ssurplus Considerachangingonlyaccurseinpriceofcommodity1 159 1 Consumer ssurplus p1 x1 160 1 Consumer ssurplus Commoditytaxation Thelossofthetaxation 1 2 161 1 Consumer ssurplus 162 1 Consumer ssurplus x2 x p1 w x p0 w u1 u0 L1 x1 L2 163 2 Aggregationacrossconsumers Canasocietybehavelikearationman Ifasocietybehavelikearationman cantheutilityofthis man representedthewelfareofsociety Aggregatedemand WealthdistributionAndtotalwealth 164 2 Aggregationacrossconsumers Step1 thatmeansaggregatedemandareindependentonthedistributionofthewealth Proposition1 ifandonlyifeachconsumerhasaindirectutilityfunctionofGorman saggregatedemandsa