Slutsky-Decomposition斯勒茨基分解

SlutskyEquation1M.LombardiSlutskydecompositionTheSlutskydecompositionisananalyticaltoolthatallowsustounderstandhowdemandchangeswhenapricechanges.Itdoesthisbybreakingthetotalchangeindemandupintosmallerpieces:–substitutioneffect:changeindemandduetochangeinrelativepriceskeepingthepurchasingpowerconstant–ordinaryincomeeffect:changeindemandduetochangerealincomekeepingtherelativepricesconstantThesignoftheoveralleffectdependsonthesignofthepieces,butthesignofthepiecesiseasiertodetermine.M.Lombardi2SlutskydecompositionChangeofpriceonegood:•Oldpriceofgood1:p1•Newpriceofgood1:p’1(income,m,andpriceofgood2arekeptconstant)•Changeofpriceofgood1:∆p1=p’1-p1•Thetotalchangeinthedemandofgood1duetothechangeinprice,holdingincome,m,andpriceofgood2,p2,constant,is∆x1=x1(p’1,p2,m)-x1(p1,p2,m)M.Lombardi3Slutskydecomposition•Theoptimalquantitiesdemandedattheoldpricesarex1=x1(p1,p2,m)andx2=x2(p1,p2,m)•Thebudgetlineatthosequantitiesism=p1x1+p2x2•Toadjustmoneyincomesoastokeeptheoriginalpurchasingpoweroftheconsumerconstantweneedtoknowtheamountofincomem’thatwilljustmaketheoptimaldemandsforgoods1&2(x1,x2)beforethepricechangeofgood1justaffordable.Thatis,m’=p’1x1+p2x2•Thechangeinincomenecessarytomakethebundle(x1,x2)justaffordableis(m’-m)=∆m=x1∆p1–NotethatthechangeinincomeandthechangeinpricewillalwaysmoveinthesamedirectionM.Lombardi4Slutskydecomposition•Thedemandofgood1whenitspricechangeandweadjusttheincomemtokeeptheoriginaldemandjustaffordableisdenotedbyx1(p’1,p2,m’)•Thetotalchangeinthedemandofgood1duetothechangeinprice,holdingincome,m,andpriceofgood2,p2,constant,is∆x1=x1(p’1,p2,m)-x1(p1,p2,m)+[x1(p’1,p2,m’)-x1(p’1,p2,m’)]=[x1(p’1,p2,m’)-x1(p1,p2,m)]+[x1(p’1,p2,m)-x1(p’1,p2,m’)]=SubstitutionEffect+(Ordinary)IncomeEffectM.Lombardi5Slutskydecomposition•SE:isthechangeinthedemandofgood1whenthepriceofgood1changefromp1top’1and,atthesametime,moneyincomechangetom’SE=x1(p’1,p2,m’)-x1(p1,p2,m)•IE:isthechangeinthedemandofgood1whenincomechangefromm’tom,holdingthepriceofgood1fixedatp’1IE=x1(p’1,p2,m)-x1(p’1,p2,m’)6Slutskydecomposition:exampleSupposethatthedemandfunctionofgood1is10+(m/10p1)•Supposethattheinitialsituationischaracterisedbyp1=3andm=120x1(p1,p2,m)=10+(120/30)=14•Supposethatthenewpriceofgood1decreases,p’1=2.Then,x1(p’1,p2,m)=10+(120/20)=16•Thetotalchangeindemandofgood1is∆x1=16-14=+2•Thechangeinincomenecessarytomakethebundle(x1,x2)justaffordableis(m’-m)=∆m=x1∆p1=-14.Thenm’=m+∆m=106•Thedemandofgood1whenthepriceofgood1changefromp1top’1and,atthesametime,moneyincomechangetom’isx1(p’1,p2,m’)=10+(m’/10p’1)=15.3•Therefore,SE=15.3-14=1.3andIE=16-15.3=0.77ThesignsofSEandIEThesubstitutioneffectmovesoppositetothepricemovement,wesayitisalwaysnegative:•Ifthepricegoesdown,thequantitydemandedviathesubstitutioneffectisnon-negative,SE≥0.•Ifthepricegoesup,thequantitydemandedviathesubstitutioneffectisnon-positive,SE≤0.•Foranormalgood,theordinaryincomeeffectalsomovesoppositetothepricemovement,wesayitisnegative:–Ifthepricegoesdown,thequantitydemandedviatheordinaryincomeeffectisnon-negative,IE≥0.–Ifthepricegoesup,thequantitydemandedviatheordinaryincomeeffectisnon-positive,IE≤0.8ThesignsofSEandIEThetotalchangeindemandduetoapricechangeforanormalgood:•isalwaysnon-negativeifthepricegoesdown,Δx₁≥0.•isalwaysnon-positiveifthepricegoesup,Δx₁≤0.9ThesignsofSEandIEThetotalchangeindemandduetoapricechangeforaninferiorgoodisambiguous:•ThegoodisaGiffengoodiftheordinaryincomeeffectIEoutweighsthesubstitutioneffectSE,thatis:–Δx₁0ifthepricegoesup–Δx₁0ifthepricegoesdown10ThesignsofSEandIE•Foraninferiorgood,thesubstitutioneffectisalwaysnegative.•Foraninferiorgood,theordinaryincomeeffectmovesinthesamedirectionofthepricemovement:–Ifthepricegoesdown,thequantitydemandedviatheordinaryincomeeffectisnon-positive,IE≤0–Ifthepricegoesup,thequantitydemandedviatheordinaryincomeeffectisnon-negative,IE≥0.11Slutskydecomposition:examples12