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1、SlutskyEquationM.Lombardi1SlutskydecompositionTheSlutskydecompositionisananalyticaltoolthatallowsustounderstandhowdemandchangeswhenapricechanges.Itdoesthisbybreakingthetotalchangeindemandupintosmallerpieces:substitutioneffect:changeindemandduetochangeinrelativepriceskeepingthepurchasingpowerconst
2、antordinaryincomeeffect:changeindemandduetochangerealincomekeepingtherelativepricesconstantThesignoftheoveralleffectdependsonthesignofthepieces,butthesignofthepiecesiseasiertodetermine.M.Lombardi2SlutskydecompositionChangeofpriceonegood:Oldpriceofgood1:p1Newpriceofgood1:p’1(income,m,andpriceofgoo
3、d2arekeptconstant)Changeofpriceofgood1:∆p1=p’1-p1Thetotalchangeinthedemandofgood1duetothechangeinprice,holdingincome,m,andpriceofgood2,p2,constant,is∆x1=x1(p’1,p2,m)-x1(p1,p2,m)M.Lombardi3SlutskydecompositionTheoptimalquantitiesdemandedattheoldpricesarex1=x1(p1,p2,m)andx2=x2(p1,p2,m)Thebudgetline
4、atthosequantitiesism=p1x1+p2x2Toadjustmoneyincomesoastokeeptheoriginalpurchasingpoweroftheconsumerconstantweneedtoknowtheamountofincomem’thatwilljustmaketheoptimaldemandsforgoods1&2(x1,x2)beforethepricechangeofgood1justaffordable.Thatis,m’=p’1x1+p2x2Thechangeinincomenecessarytomakethebundle(x1,x2
5、)justaffordableis(m’-m)=∆m=x1∆p1NotethatthechangeinincomeandthechangeinpricewillalwaysmoveinthesamedirectionM.Lombardi4SlutskydecompositionThedemandofgood1whenitspricechangeandweadjusttheincomemtokeeptheoriginaldemandjustaffordableisdenotedbyx1(p’1,p2,m’)Thetotalchangeinthedemandofgood1duetothech
6、angeinprice,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.Lombardi5SlutskydecompositionSE:isthechangeinthedemandofgood1whenthepriceofgood1
7、changefromp1top’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’)6Slutskydecompo
