伦敦经济学院高微讲义-消费者理论

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1、Theconsumeraimstomaximiseutility...increasingpreferencesubjecttothebudgetconstraintx2x1ConstraintsetThisisknownastheprimalproblem...subjecttoU(x)maxSpixi£Mni=1Solutiontotheprimalproblemx*Butthere'sanotherwayatlookingatallthisContoursofobjectivefunctionreducingcostAlternatively...x2x

2、1theconsumeraimstominimisecost......subjecttoautilityconstraintConstraintsetuThisalternativeapproachisknownasthedualproblemThesolutiontothedualproblemsubjecttoSpxni=1iiU(x)³uminBut...x*Butwherehaveweseenthedualproblembefore…??z1z2z*Q_reducingcostcostminimisationbythefirmContoursofob

3、jectivefunctionp1x1+p2x2=constantComparingcost-minimisationforthefirmLet'srunthroughtheformalitiesofthesolution...z1z2z*Q_reducingcostux2x1reducingcostx*andfortheconsumer......thedifferenceisonlyinnotationsotheirsolutionfunctionsandresponsefunctionsmustbethesameUsetheFOCAtx*wefind..

4、.xi*=Hi(p,u)C(p,u)Problem......solutionresponse:minxåpixisubjecttoU(x)³uin=1Cost-minimisation:theconsumerCost-minproblemsareidenticalzi*=Hi(w,Q)C(w,Q)minzåwizisubjecttoG(z)³Qim=1Problem......solutionresponse:lU1(x*)=p1lU2(x*)=p2.........lUn(x*)=pnu=U(x*)LagrangemultiplierüýþgoodsMRS

5、=priceratio_______=Ui(x)piUj(x)pjIfbothiandjarebought...C(p,u):=vectorofgoodspricesSpecifiedutilitylevelminåpixi{U(x)³u}Thesolutionistheconsumer’scostfunctionorexpenditurefunctionExample:Cobb-DouglasutilitytwogoodsSolveforconditionaldemandsC(p,u)=C(p,u)=Non-decreasingineveryprice.In

6、creasinginatleastonepriceIncreasinginutilityu.ConcaveinpHomogeneousofdegree1inallpricesp.Shephard'slemma.PropertiesofTheCostFunction(Checkthatallthisisjusta“translation”fromthelecture2onthefirm)x2x1increasingpreferencex*ux2x1reducingcostx*sowecanlinkuptheirsolutionfunctionsandrespon

7、sefunctionsandthedualproblemareequivalentLet'srunthroughtheprimalproblemAgainusetheFOCAtx*wefind...TheprimalproblemHowtosolvetheproblemNowthatweknowtheconstraintsontheconsumerandtheobjectivefunctionTheobviousthingtodomightbetosetupastandardLagrangeanandsolveit:ObjectivefunctionConst

8、raintparameterparam