ccer双学位中微第20章课件

ChapterTwentyCostMinimization成本最小化ChapterTwentyCostMinimizatio1StructureThecostminimizationproblemAveragecostsReturnstoscaleandtotalandaveragecostsShortrunandlongruncostsStructureThecostminimization2CostMinimizationAfirmisacost-minimizerifitproducesanygivenoutputlevely³0atsmallestpossibletotalcost.c(y)denotesthefirm’ssmallestpossibletotalcostforproducingyunitsofoutput.c(y)isthefirm’stotalcostfunction（总成本函数）.CostMinimizationAfirmisac3CostMinimizationWhenthefirmfacesgiveninputpricesw=(w1,w2,…,wn)thetotalcostfunctionwillbewrittenas

c(w1,…,wn,y).CostMinimizationWhenthefirm4TheCost-MinimizationProblemConsiderafirmusingtwoinputstomakeoneoutput.Theproductionfunctionis

y=f(x1,x2).Taketheoutputlevely³0asgiven.Giventheinputpricesw1andw2,thecostofaninputbundle(x1,x2)isw1x1+w2x2.TheCost-MinimizationProblemC5TheCost-MinimizationProblemForgivenw1,w2andy,thefirm’scost-minimizationproblemistosolve

subjecttoTheCost-MinimizationProblemF6TheCost-MinimizationProblemThelevelsx1*(w1,w2,y)andx1*(w1,w2,y)intheleast-costlyinputbundlearethefirm’sconditionaldemandsforinputs1and2（条件要素需求）.The(smallestpossible)totalcostforproducingyoutputunitsisthereforeTheCost-MinimizationProblemT7ConditionalInputDemandsGivenw1,w2andy,howistheleastcostlyinputbundlelocated?Andhowisthetotalcostfunction（成本函数）computed?ConditionalInputDemandsGiven8Iso-costLines（等成本线）Acurvethatcontainsalloftheinputbundlesthatcostthesameamountisaniso-costcurve.E.g.,givenw1andw2,the$100iso-costlinehastheequationIso-costLines（等成本线）Acurvet9Iso-costLinesGenerally,givenw1andw2,theequationofthe$ciso-costlineis

i.e.

Slopeis-w1/w2.Iso-costLinesGenerally,given10Iso-costLinesc’ºw1x1+w2x2c”ºw1x1+w2x2c’<c”x1x2Slopes=-w1/w2.Iso-costLinesc’ºw1x1+w2x2c”11They’-OutputUnitIsoquantx1x2Allinputbundlesyieldingy’units

ofoutput.Whichisthecheapest?f(x1,x2)ºy’They’-OutputUnitIsoquantx1x12TheCost-MinimizationProblemx1x2Allinputbundlesyieldingy’units

ofoutput.Whichisthecheapest?f(x1,x2)ºy’TheCost-MinimizationProblemx13TheCost-MinimizationProblemx1x2Allinputbundlesyieldingy’units

ofoutput.Whichisthecheapest?f(x1,x2)ºy’x1*x2*TheCost-MinimizationProblemx14TheCost-MinimizationProblemx1x2f(x1,x2)ºy’x1*x2*Ataninteriorcost-mininputbundle:

(a)TheCost-MinimizationProblemx15TheCost-MinimizationProblemx1x2f(x1,x2)ºy’x1*x2*Ataninteriorcost-mininputbundle:

(a)and

(b)slopeofisocost=slopeof

isoquant;i.e.TheCost-MinimizationProblemx16ACobb-DouglasExampleofCostMinimizationAfirm’sCobb-Douglasproductionfunctionis

Inputpricesarew1andw2.Whatarethefirm’sconditionalinputdemandfunctions?ACobb-DouglasExampleofCost17ACobb-DouglasExampleofCostMinimizationAttheinputbundle(x1*,x2*)whichminimizes

thecostofproducingyoutputunits:(a)

(b)andACobb-DouglasExampleofCost18ACobb-DouglasExampleofCostMinimization(a)(b)From(b),Nowsubstituteinto(a)togetSoisthefirm’sconditional

demandforinput1.ACobb-DouglasExampleofCost19ACobb-DouglasExampleofCostMinimizationisthefirm’sconditionaldemandforinput2.SinceandACobb-DouglasExampleofCost20ACobb-DouglasExampleofCostMinimizationSothecheapestinputbundleyieldingy

outputunitsisACobb-DouglasExampleofCost21Fixedw1andw2.ConditionalInputDemandCurvesFixedw1andw2.ConditionalIn22Fixedw1andw2.ConditionalInputDemandCurvesFixedw1andw2.ConditionalIn23Fixedw1andw2.ConditionalInputDemandCurvesFixedw1andw2.ConditionalIn24Fixedw1andw2.ConditionalInputDemandCurvesFixedw1andw2.ConditionalIn25Fixedw1andw2.ConditionalInputDemandCurvesoutput

expansion

pathCond.demand

for

input2

Cond.

demand

forinput1Fixedw1andw2.ConditionalIn26ACobb-DouglasExampleofCostMinimizationFortheproductionfunctionthecheapestinputbundleyieldingyoutput

unitsisACobb-DouglasExampleofCost27ACobb-DouglasExampleofCostMinimizationSothefirm’stotalcostfunctionisACobb-DouglasExampleofCost28APerfectComplementsExampleofCostMinimizationThefirm’sproductionfunctionis

Inputpricesw1andw2aregiven.Whatarethefirm’sconditionaldemandsforinputs1and2?Whatisthefirm’stotalcostfunction?APerfectComplementsExample29APerfectComplementsExampleofCostMinimizationx1x2x1*=y/4x2*=y4x1=x2min{4x1,x2}ºy’Whereistheleastcostlyinputbundleyieldingy’outputunits?APerfectComplementsExample30APerfectComplementsExampleofCostMinimizationThefirm’sproductionfunctionisandtheconditionalinputdemandsareandSothefirm’stotalcostfunctionisAPerfectComplementsExample31AverageTotalProductionCostsForpositiveoutputlevelsy,afirm’saveragetotalcostofproducingyunitsisAverageTotalProductionCosts32Returns-to-ScaleandAv.TotalCostsThereturns-to-scalepropertiesofafirm’stechnologydeterminehowaverageproductioncostschangewithoutputlevel.Ourfirmispresentlyproducingy’outputunits.Howdoesthefirm’saverageproductioncostchangeifitinsteadproduces2y’unitsofoutput?Returns-to-ScaleandAv.Total33ConstantReturns-to-ScaleandAverageTotalCostsIfafirm’stechnologyexhibitsconstantreturns-to-scalethendoublingitsoutputlevelfromy’to2y’requiresdoublingallinputlevels.Totalproductioncostdoubles.Averageproductioncostdoesnotchange.ConstantReturns-to-Scaleand34DecreasingReturns-to-ScaleandAverageTotalCostsIfafirm’stechnologyexhibitsdecreasingreturns-to-scalethendoublingitsoutputlevelfromy’to2y’requiresmorethandoublingallinputlevels.Totalproductioncostmorethandoubles.Averageproductioncostincreases.DecreasingReturns-to-Scalean35IncreasingReturns-to-ScaleandAverageTotalCostsIfafirm’stechnologyexhibitsincreasingreturns-to-scalethendoublingitsoutputlevelfromy’to2y’requireslessthandoublingallinputlevels.Totalproductioncostlessthandoubles.Averageproductioncostdecreases.IncreasingReturns-to-Scalean36Returns-to-ScaleandAv.TotalCostsy$/outputunitconstantr.t.s.decreasingr.t.s.increasingr.t.s.AC(y)Returns-to-ScaleandAv.Total37Returns-to-ScaleandTotalCostsWhatdoesthisimplyfortheshapesoftotalcostfunctions?Returns-to-ScaleandTotalCos38Returns-to-ScaleandTotalCostsy$c(y)y’2y’c(y’)c(2y’)Slope=c(2y’)/2y’=AC(2y’).Slope=c(y’)/y’=AC(y’).Av.costincreaseswithyifthefirm’s

technologyexhibitsdecreasingr.t.s.Returns-to-ScaleandTotalCos39Returns-to-ScaleandTotalCostsy$c(y)y’2y’c(y’)c(2y’)Slope=c(2y’)/2y’=AC(2y’).Slope=c(y’)/y’=AC(y’).Av.costdecreaseswithyifthefirm’s

technologyexhibitsincreasingr.t.s.Returns-to-ScaleandTotalCos40Returns-to-ScaleandTotalCostsy$c(y)y’2y’c(y’)c(2y’)=2c(y’)Slope=c(2y’)/2y’=2c(y’)/2y’=c(y’)/y’soAC(y’)=AC(2y’).Av.costisconstantwhenthefirm’s

technologyexhibitsconstantr.t.s.Returns-to-ScaleandTotalCos41Short-Run&Long-RunTotalCostsInthelong-runafirmcanvaryallofitsinputlevels.Considerafirmthatcannotchangeitsinput2levelfromx2’units.Howdoestheshort-runtotalcostofproducingyoutputunitscomparetothelong-runtotalcostofproducingyunitsofoutput?Short-Run&Long-RunTotalCos42Short-Run&Long-RunTotalCostsThelong-runcost-minimizationproblemis

Theshort-runcost-minimizationproblemissubjecttosubjecttoShort-Run&Long-RunTotalCos43Short-Run&Long-RunTotalCostsTblemisthelong-runproblemsubjecttotheextraconstraintthatx2=x2’.Ifthelong-runchoiceforx2wasx2’thentheextraconstraintx2=x2’isnotreallyaconstraintatallandsothelong-runandshort-runtotalcostsofproducingyoutputunitsarethesame.Short-Run&Long-RunTotalCos44Short-Run&Long-RunTotalCostsBut,ifthelong-runchoiceforx2

¹x2”thentheextraconstraintx2=x2”preventsthefirminthisshort-runfromachievingitslong-runproductioncost,causingtheshort-runtotalcosttoexceedthelong-runtotalcostofproducingyoutputunits.Short-Run&Long-RunTotalCos45Short-Run&Long-RunTotalCostsx1x2Considerthreeoutputlevels.Short-Run&Long-RunTotalCos46Short-Run&Long-RunTotalCostsx1x2Inthelong-runwhenthefirm

isfreetochoosebothx1and

x2,theleast-costlyinput

bundlesare...Short-Run&Long-RunTotalCos47Short-Run&Long-RunTotalCostsx1x2Long-run

output

expansion

pathShort-Run&Long-RunTotalCos48Short-Run&Long-RunTotalCostsx1x2Long-run

output

expansion

pathLong-runcostsare:Short-Run&Long-RunTotalCos49Short-Run&Long-RunTotalCostsNowsupposethefirmbecomessubjecttotheshort-runconstraintthatx2=x2”.Short-Run&Long-RunTotalCos50Short-Run&Long-RunTotalCostsx1x2Short-run

output

expansion

pathLong-runcostsare:Short-Run&Long-RunTotalCos51Short-Run&Long-RunTotalCostsx1x2Short-run

output

expansion

pathLong-runcostsare:Short-runcostsare:Short-Run&Long-RunTotalCos52Short-Run&Long-RunTotalCostsx1x2Short-run

output

expansion

pathLong-runcostsare:Short-runcostsare:Short-Run&Long-RunTotalCos53Short-Run&Long-RunTotalCostsx1x2Short-run

output

expansion

pathLong-runcostsare:Short-runcostsare:Short-Run&Long-RunTotalCos54Short-Run&Long-RunTotalCostsShort-runtotalcostexceedslong-runtotalcostexceptfortheoutputlevelwheretheshort-runinputlevelrestrictionisthelong-runinputlevelchoice.Thissaysthatthelong-runtotalcostcurvealwayshasonepointincommonwithanyparticularshort-runtotalcostcurve.Short-Run&Long-RunTotalCos55Short-Run&Long-RunTotalCostsy$c(y)cs(y)Ashort-runtotalcostcurvealwayshas

onepointincommonwiththelong-run

totalcostcurve,andiselsewherehigher

thanthelong-runtotalcostcurve.Short-Run&Long-RunTotalCos56ChapterTwentyCostMinimization成本最小化ChapterTwentyCostMinimizatio57StructureThecostminimizationproblemAveragecostsReturnstoscaleandtotalandaveragecostsShortrunandlongruncostsStructureThecostminimization58CostMinimizationAfirmisacost-minimizerifitproducesanygivenoutputlevely³0atsmallestpossibletotalcost.c(y)denotesthefirm’ssmallestpossibletotalcostforproducingyunitsofoutput.c(y)isthefirm’stotalcostfunction（总成本函数）.CostMinimizationAfirmisac59CostMinimizationWhenthefirmfacesgiveninputpricesw=(w1,w2,…,wn)thetotalcostfunctionwillbewrittenas

c(w1,…,wn,y).CostMinimizationWhenthefirm60TheCost-MinimizationProblemConsiderafirmusingtwoinputstomakeoneoutput.Theproductionfunctionis

y=f(x1,x2).Taketheoutputlevely³0asgiven.Giventheinputpricesw1andw2,thecostofaninputbundle(x1,x2)isw1x1+w2x2.TheCost-MinimizationProblemC61TheCost-MinimizationProblemForgivenw1,w2andy,thefirm’scost-minimizationproblemistosolve

subjecttoTheCost-MinimizationProblemF62TheCost-MinimizationProblemThelevelsx1*(w1,w2,y)andx1*(w1,w2,y)intheleast-costlyinputbundlearethefirm’sconditionaldemandsforinputs1and2（条件要素需求）.The(smallestpossible)totalcostforproducingyoutputunitsisthereforeTheCost-MinimizationProblemT63ConditionalInputDemandsGivenw1,w2andy,howistheleastcostlyinputbundlelocated?Andhowisthetotalcostfunction（成本函数）computed?ConditionalInputDemandsGiven64Iso-costLines（等成本线）Acurvethatcontainsalloftheinputbundlesthatcostthesameamountisaniso-costcurve.E.g.,givenw1andw2,the$100iso-costlinehastheequationIso-costLines（等成本线）Acurvet65Iso-costLinesGenerally,givenw1andw2,theequationofthe$ciso-costlineis

i.e.

Slopeis-w1/w2.Iso-costLinesGenerally,given66Iso-costLinesc’ºw1x1+w2x2c”ºw1x1+w2x2c’<c”x1x2Slopes=-w1/w2.Iso-costLinesc’ºw1x1+w2x2c”67They’-OutputUnitIsoquantx1x2Allinputbundlesyieldingy’units

ofoutput.Whichisthecheapest?f(x1,x2)ºy’They’-OutputUnitIsoquantx1x68TheCost-MinimizationProblemx1x2Allinputbundlesyieldingy’units

ofoutput.Whichisthecheapest?f(x1,x2)ºy’TheCost-MinimizationProblemx69TheCost-MinimizationProblemx1x2Allinputbundlesyieldingy’units

ofoutput.Whichisthecheapest?f(x1,x2)ºy’x1*x2*TheCost-MinimizationProblemx70TheCost-MinimizationProblemx1x2f(x1,x2)ºy’x1*x2*Ataninteriorcost-mininputbundle:

(a)TheCost-MinimizationProblemx71TheCost-MinimizationProblemx1x2f(x1,x2)ºy’x1*x2*Ataninteriorcost-mininputbundle:

(a)and

(b)slopeofisocost=slopeof

isoquant;i.e.TheCost-MinimizationProblemx72ACobb-DouglasExampleofCostMinimizationAfirm’sCobb-Douglasproductionfunctionis

Inputpricesarew1andw2.Whatarethefirm’sconditionalinputdemandfunctions?ACobb-DouglasExampleofCost73ACobb-DouglasExampleofCostMinimizationAttheinputbundle(x1*,x2*)whichminimizes

thecostofproducingyoutputunits:(a)

(b)andACobb-DouglasExampleofCost74ACobb-DouglasExampleofCostMinimization(a)(b)From(b),Nowsubstituteinto(a)togetSoisthefirm’sconditional

demandforinput1.ACobb-DouglasExampleofCost75ACobb-DouglasExampleofCostMinimizationisthefirm’sconditionaldemandforinput2.SinceandACobb-DouglasExampleofCost76ACobb-DouglasExampleofCostMinimizationSothecheapestinputbundleyieldingy

outputunitsisACobb-DouglasExampleofCost77Fixedw1andw2.ConditionalInputDemandCurvesFixedw1andw2.ConditionalIn78Fixedw1andw2.ConditionalInputDemandCurvesFixedw1andw2.ConditionalIn79Fixedw1andw2.ConditionalInputDemandCurvesFixedw1andw2.ConditionalIn80Fixedw1andw2.ConditionalInputDemandCurvesFixedw1andw2.ConditionalIn81Fixedw1andw2.ConditionalInputDemandCurvesoutput

expansion

pathCond.demand

for

input2

Cond.

demand

forinput1Fixedw1andw2.ConditionalIn82ACobb-DouglasExampleofCostMinimizationFortheproductionfunctionthecheapestinputbundleyieldingyoutput

unitsisACobb-DouglasExampleofCost83ACobb-DouglasExampleofCostMinimizationSothefirm’stotalcostfunctionisACobb-DouglasExampleofCost84APerfectComplementsExampleofCostMinimizationThefirm’sproductionfunctionis

Inputpricesw1andw2aregiven.Whatarethefirm’sconditionaldemandsforinputs1and2?Whatisthefirm’stotalcostfunction?APerfectComplementsExample85APerfectComplementsExampleofCostMinimizationx1x2x1*=y/4x2*=y4x1=x2min{4x1,x2}ºy’Whereistheleastcostlyinputbundleyieldingy’outputunits?APerfectComplementsExample86APerfectComplementsExampleofCostMinimizationThefirm’sproductionfunctionisandtheconditionalinputdemandsareandSothefirm’stotalcostfunctionisAPerfectComplementsExample87AverageTotalProductionCostsForpositiveoutputlevelsy,afirm’saveragetotalcostofproducingyunitsisAverageTotalProductionCosts88Returns-to-ScaleandAv.TotalCostsThereturns-to-scalepropertiesofafirm’stechnologydeterminehowaverageproductioncostschangewithoutputlevel.Ourfirmispresentlyproducingy’outputunits.Howdoesthefirm’saverageproductioncostchangeifitinsteadproduces2y’unitsofoutput?Returns-to-ScaleandAv.Total89ConstantReturns-to-ScaleandAverageTotalCostsIfafirm’stechnologyexhibitsconstantreturns-to-scalethendoublingitsoutputlevelfromy’to2y’requiresdoublingallinputlevels.Totalproductioncostdoubles.Averageproductioncostdoesnotchange.ConstantReturns-to-Scaleand90DecreasingReturns-to-ScaleandAverageTotalCostsIfafirm’stechnologyexhibitsdecreasingreturns-to-scalethendoublingitsoutputlevelfromy’to2y’requiresmorethandoublingallinputlevels.Totalproductioncostmorethandoubles.Averageproductioncostincreases.DecreasingReturns-to-Scalean91IncreasingReturns-to-ScaleandAverageTotalCostsIfafirm’stechnologyexhibitsincreasingreturns-to-scalethendoublingitsoutputlevelfromy’to2y’requireslessthandoublingallinputlevels.Totalproductioncostlessthandoubles.Averageproductioncostdecreases.IncreasingReturns-to-Scalean92Returns-to-ScaleandAv.TotalCostsy$/outputunitconstantr.t.s.decreasingr.t.s.increasingr.t.s.AC(y)Returns-to-ScaleandAv.Total93Returns-to-ScaleandTotalCostsWhatdoesthisimplyfortheshapesoftotalcostfunctions?Returns-to-ScaleandTotalCos94Returns-to-ScaleandTotalCostsy$c(y)y’2y’c(y’)c(2y’)Slope=c(2y’)/2y’=AC(2y’).Slope=c(y’)/y’=AC(y’).Av.costincreaseswithyifthefirm’s

technologyexhibitsdecreasingr.t.s.Returns-to-ScaleandTotalCos95Returns-to-ScaleandTotalCostsy$c(y)y’2y’c(y’)c(2y’)Slope=c(2y’)/2y’=AC(2y’).Slope=c(y’)/y’=AC(y’).Av.costdecreaseswithyifthefirm’s

technologyexhibitsincreasingr.t.s.Returns-to-ScaleandTotalCos96Returns-to-ScaleandTotalCostsy$c(y)y’2y’c(y’)c(2y’)=2c(y’)Slope=c(2y’)/2y’=2c(y’)/2y’=c(y’)/y’soAC(y’)=AC(2y’).Av.costisconstantwhenthefirm’s

technologyexhibitsconstantr.t.s.Returns-to-ScaleandTotalCos97Short-Run&Long-RunTotalCostsInthelong-runafirmcanvaryallofitsinputlevels.Considerafirmthatcannotchangeitsinput2levelfromx2’units.Howdoestheshort-runtotalcostofproducingyoutputunitscomparetothelong-runtotalcostofproducingyunitsofoutput?Short-Run&Long-RunTotalCos98Short-Run&Long-RunTotalCostsThelong-runcost-minimizationproblemis

Theshort-runcost-minimizationproblemissubjecttosubjecttoShort-Run&Long-RunTotalCos99Short-Run&Long-RunTotalCostsTblemisthelong-runproblemsubjecttotheextraconstraintthatx2=x2’.Ifthelong-runchoiceforx2wasx2’thentheextraconstraintx2=x2’isnotreallyaconstraintatallandsothelong-runandshort-runtotalcostsofproducingyoutputunitsarethesame.Short-Run&Long-RunTot