Slutsky-Decomposition斯勒茨基分解

SlutskyEquation,Slutskydecomposition,TheSlutskydecompositionisananalyticaltoolthatallowsustounderstandhowdemandchangeswhenapricechanges.Itdoesthisbybreakingthetotalchangeindemandupintosmallerpieces:substitutioneffect:changeindemandduetochangeinrelativepriceskeepingthepurchasingpowerconstantordinaryincomeeffect:changeindemandduetochangerealincomekeepingtherelativepricesconstantThesignoftheoveralleffectdependsonthesignofthepieces,butthesignofthepiecesiseasiertodetermine.,M.Lombardi,2,Slutskydecomposition,Changeofpriceonegood:Oldpriceofgood1:p1Newpriceofgood1:p1(income,m,andpriceofgood2arekeptconstant)Changeofpriceofgood1:p1=p1-p1Thetotalchangeinthedemandofgood1duetothechangeinprice,holdingincome,m,andpriceofgood2,p2,constant,isx1=x1(p1,p2,m)-x1(p1,p2,m),M.Lombardi,3,Slutskydecomposition,Theoptimalquantitiesdemandedattheoldpricesarex1=x1(p1,p2,m)andx2=x2(p1,p2,m)Thebudgetlineatthosequantitiesism=p1x1+p2x2Toadjustmoneyincomesoastokeeptheoriginalpurchasingpoweroftheconsumerconstantweneedtoknowtheamountofincomemthatwilljustmaketheoptimaldemandsforgoods1&2(x1,x2)beforethepricechangeofgood1justaffordable.Thatis,m=p1x1+p2x2Thechangeinincomenecessarytomakethebundle(x1,x2)justaffordableis(m-m)=m=x1p1Notethatthechangeinincomeandthechangeinpricewillalwaysmoveinthesamedirection,M.Lombardi,4,Slutskydecomposition,Thedemandofgood1whenitspricechangeandweadjusttheincomemtokeeptheoriginaldemandjustaffordableisdenotedbyx1(p1,p2,m)Thetotalchangeinthedemandofgood1duetothechangeinprice,holdingincome,m,andpriceofgood2,p2,constant,isx1=x1(p1,p2,m)-x1(p1,p2,m)+x1(p1,p2,m)-x1(p1,p2,m)=x1(p1,p2,m)-x1(p1,p2,m)+x1(p1,p2,m)-x1(p1,p2,m)=SubstitutionEffect+(Ordinary)IncomeEffect,M.Lombardi,5,Slutskydecomposition,SE:isthechangeinthedemandofgood1whenthepriceofgood1changefromp1top1and,atthesametime,moneyincomechangetomSE=x1(p1,p2,m)-x1(p1,p2,m)IE:isthechangeinthedemandofgood1whenincomechangefrommtom,holdingthepriceofgood1fixedatp1IE=x1(p1,p2,m)-x1(p1,p2,m),Slutskydecomposition:example,Supposethatthedemandfunctionofgood1is10+(m/10p1)Supposethattheinitialsituationischaracterisedbyp1=3andm=120 x1(p1,p2,m)=10+(120/30)=14Supposethatthenewpriceofgood1decreases,p1=2.Then,x1(p1,p2,m)=10+(120/20)=16Thetotalchangeindemandofgood1isx1=16-14=+2Thechangeinincomenecessarytomakethebundle(x1,x2)justaffordableis(m-m)=m=x1p1=-14.Thenm=m+m=106Thedemandofgood1whenthepriceofgood1changefromp1top1and,atthesametime,moneyincomechangetomisx1(p1,p2,m)=10+(m/10p1)=15.3Therefore,SE=15.3-14=1.3andIE=16-15.3=0.7,ThesignsofSEandIE,Thesubstitutioneffectmovesoppositetothepricemovement,wesayitisalwaysnegative:Ifthepricegoesdown,thequantitydemandedviathesubstitutioneffectisnon-negative,SE0.Ifthepricegoesup,thequantitydemandedviathesubstitutioneffectisnon-positive,SE0.Foranormalgood,theordinaryincomeeffectalsomovesoppositetothepricemovement,wesayitisnegative:Ifthepricegoesdown,thequantitydemandedviatheordinaryincomeeffectisnon-negative,IE0.Ifthepricegoesup,thequantitydemandedviatheordinaryincomeeffectisnon-positive,IE0.,ThesignsofSEandIE,Thetotalchangeindemandduetoapricechangeforanormalgood:isalwaysnon-negativeifthepricegoesdown,x0.isalwaysnon-positiveifthepricegoesup,x0.,ThesignsofSEandIE,Thetotalchangeindemandduetoapricechangeforaninferiorgoodisambiguous:ThegoodisaGiffengoodiftheordinaryincomeeffectIEoutweighsthesubstitutioneffectSE,thatis:x0ifthepricegoesupx0ifthepricegoesdown,ThesignsofSEandIE,Foraninferiorgood,thesubstitutioneffectisalwaysnegative.Foraninferiorgood,theordinaryincomeeffectmovesinthesamedirectionofthepricemovement:Ifthepric