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1、ECC5650–MicroeconomicTheoryTopic4:TopicsinConsumerTheoryPrimaryReadings:·JR–Chapter4:4.2,.4.3&4.4·DL–Chapters4,5&11:4.7,5.8,11.1-11.5AdditionalReferences(attheend)Inthislecture,wewilldiscussafewspecialtopicsthathaveimportantimplicationsintheconsumertheory.Thefirstqu
2、estionisrelatedtowhethertheprocessofderivingconsumer’sdemandfunctionisreversible,i.e.,foragivendemandfunction,dowehaveautilityfunctionthatisconsistentwiththedemandfunction.Secondly,wewilldiscusstheconsumerpreferencesunderuncertainty.Wewillestablishtheexistenceofthee
3、xpectedutilityfunctionbasedonasetofaxioms.Theissueofrisk(asaresultofuncertainty),intheparticular,themeasureandimplicationoftheriskaversion,willbeaddressedtoo.Thirdly,wediscussanextensionofconsumertheory,theconceptofdiamondgoodswherethevalue,ratherthanjustthequantity
4、,ofagoodenterstheutilityfunction.Interestingly,wemaythenhaveupward-slopingcompensateddemandcurvesandburden-free(notjustexcess-burdenfree)andnegative-burdentaxes.4.1IntegrabilityProblemGivenademandfunctionx(p,y),canwerecovertheunderlyingutilityfunction?Thisistheso-ca
5、lledintegrabilityproblem.Technicallyspeaking,thisisaproblemofsolvingasystemofpartialdifferentialequations.Recallthatfromthediscussionsinthepreviouslectures,weknowthatawell-behaveddemandfunctionx(p,y)satisfiesthefollowingconditions:1.Homogeneityofdegree0in(p,y)::x(tp
6、,ty)=x(p,y)forallt³0;2.Walras’Law(Budgetbalancedness):p×x(p,y)=y;3.SlutskymatrixSissymmetric4.SlutskymatrixSisnegativesemidefinite.FollowingresultsdemonstratesthattheCondition1isredundant:itisaconsequenceofWalras’sLawandthesymmetryoftheSlutskymatrix.Proposition:Ifth
7、edemandfunctionx(p,y)satisfiestheWalras’sLawanditsSlutskymatrixissymmetric,thenitishomogeneousofdegreezeroinpandy.Proof.WeknowthatfromtheWalras’Law:p×x(p,y)=y,wewillgetthefollowingidentities:Definefi(t)=xi(tp,ty)forallt>0.Weneedtoprovethatfi(t)isconstant,i.e.,fi’(t)
8、=0.Now,8Thisprovestheresult.Theimplicationoftheaboveresultisremarkable:·Thethreeconditions,namelyWalras’Law,SymmetryoftheSlutskymatrix,and
