indirect utility funcion

《indirect utility funcion》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、ECO305—FALL2003—September25INDIRECTUTILITYFUNCTION∗U(Px,Py,M)=max{U(x,y)

2、Pxx+Pyy≤M}∗∗=U(x,y)=U(Dx(Px,Py,M),Dy(Px,Py,M))PROPERTIESOFU∗:(1)Nomoneyillusion—Homogeneousdegreezero:∗∗U(kPx,kPy,kM)=U(Px,Py,M)(2)Asmoneyincomechanges:¯¯∂U∗∂U¯∂x∗∂U¯∂y∗¯¯=¯+¯∂M∂x¯∂M∂y¯∂M"∗

3、∗#∂x∗∂y∗=λPx+Py∂M∂M∂M=λ=λ∂M(3)Aspricechanges:¯¯∂U∗∂U¯∂x∗∂U¯∂y∗¯¯=¯+¯∂Px∂x¯∗∂Px∂y¯∗∂Px1"#∂x∗∂y∗=λPx+Py∂Px∂Px∗∗=−λx(justlikeM↓byx)(Laststep:differentiateadding-upidentityw.r.t.Px:∗∗Pxx+Pyy=M∂x∗∂y∗∗x+Px+Py=0)∂Px∂PxDivideprice-andincome-changeequations:∂U∗/∂P∗xRoy’sI

4、dentity:x=−∂U∗/∂M(4)ContoursofU∗in(P,P)spacewithMfixed:xy(LiketheaterwithstageatNEcorner)2EXPENDITUREFUNCTIONSolvetheindirectutilityfunctionforincome:∗∗u=U(Px,Py,M)⇐⇒M=M(Px,Py,u)∗M(Px,Py,u)=min{Pxx+Pyy

5、U(x,y)≥u}“Dual”ormirrorimageofutilitymaximizationproblem.Econ

6、omics—incomecompensationforpricechangesOptimumquantities—CompensatedorHicksiandemands∗H∗Hx=Dx(Px,Py,u),y=Dy(Px,Py,u)PROPERTIESOFM∗:(1)Homogeneousdegree1in(Px,Py)holdingufixed:∗∗M(kPx,kPy,u)=kM(Px,Py,u)(2)Hotelling’sorShepherd’sLemma—Compensateddemandspartialderiv

7、ativesw.r.t.prices:H∗H∗Dx(Px,Py,u)=∂M/∂Px,Dy(Px,Py,u)=∂M/∂PyProof:M∗=PDH+PDH,u=U(DH,DH).Soxxyyxy∗HHH∂M/∂Px=Dx+Px∂Dx/∂Px+Py∂Dy/∂PxHH0=Ux∂Dx/∂Px+Uy∂Dy/∂PxHH=λ[Px∂Dx/∂Px+Py∂Dy/∂Px]3(3)“Weakly”concavein(Px,Py)holdingufixed.Cobb-Douglasexample:(P)1/3(P)2/3xyPROPERTIES

8、OFHICKSIANDEMANDFUNCTIONS:(1)Ownsubstitutioneffectnegative:¯∂x¯∂DH∂2M∗¯x¯==≤0∂Px¯∂Px∂P2u=constx(2)Symmetryofcross-priceeffects:∂DH∂2M∗∂DHxy==∂Py∂Px∂Py∂Px(Net)substitutesif>0,complementsif<0Generalconcept:Comparativestatics4COBB-DOUGLASEXAMPLE(Direct)UTILITYFUNCTIO

9、N:U(x,y)=αln(x)+βln(y),α+β=1∗∗x=αM/Px,y=βM/PyINDIRECTUTILITYFUNCTION∗U(Px,Py,M)=α[ln(α)+ln(M)−ln(Px)]+β[ln(β)+ln(M)−ln(Py)]=junk+ln(M)−αln(Px)−βln(Py)Roy’sIdentity:∂U∗/∂P−α/PαMxx∗−=−==x∂U∗/∂M1/MPxEXPENDITUREFUNCTION∗∗uαβM=M(Px,Py,u)=e(Px)(Py)Hicksiandemandfuncti

10、onsHuα−1βHuαβ−1x=αe(Px)(Py),y=βe(Px)(Py)5SLUTSKYEQUATIONLinkbetweenMarshallianandHicksiandemandsEqualifu=U∗(P,P,M),M=M∗(P,P,u).xyxyForgoodiwhereimaybeeitherxory,HM∗Di