Ch21-Cost-Curves-中级微观经济学

ChapterTwenty-OneCostCurvesTypesofCostCurvesAtotalcostcurveisthegraphofafirm’stotalcostfunction.Avariablecostcurveisthegraphofafirm’svariablecostfunction.Anaveragetotalcostcurveisthegraphofafirm’saveragetotalcostfunction.TypesofCostCurvesAnaveragevariablecostcurveisthegraphofafirm’saveragevariablecostfunction.Anaveragefixedcostcurveisthegraphofafirm’saveragefixedcostfunction.Amarginalcostcurveisthegraphofafirm’smarginalcostfunction.TypesofCostCurvesHowarethesecostcurvesrelatedtoeachother?Howareafirm’slong-runandshort-runcostcurvesrelated?Fixed,Variable&TotalCostFunctionsFisthetotalcosttoafirmofitsshort-runfixedinputs.F,thefirm’sfixedcost,doesnotvarywiththefirm’soutputlevel.cv(y)isthetotalcosttoafirmofitsvariableinputswhenproducingyoutputunits.cv(y)isthefirm’svariablecostfunction.cv(y)dependsuponthelevelsofthefixedinputs.Fixed,Variable&TotalCostFunctionsc(y)isthetotalcostofallinputs,fixedandvariable,whenproducingyoutputunits.c(y)isthefirm’stotalcostfunction;y$Fcv(y)y$Fcv(y)c(y)FAv.Fixed,Av.Variable&Av.TotalCostCurvesThefirm’stotalcostfunctionis

Fory>0,thefirm’saveragetotalcostfunctionisAv.Fixed,Av.Variable&Av.TotalCostCurvesWhatdoesanaveragefixedcostcurvelooklike?

AFC(y)isarectangularhyperbolasoitsgraphlookslike...$/outputunitAFC(y)y0AFC(y)®0asy®¥Av.Fixed,Av.Variable&Av.TotalCostCurvesInashort-runwithafixedamountofatleastoneinput,theLawofDiminishing(Marginal)Returnsmustapply,causingthefirm’saveragevariablecostofproductiontoincreaseeventually.$/outputunitAVC(y)y0$/outputunitAFC(y)AVC(y)y0Av.Fixed,Av.Variable&Av.TotalCostCurvesAndATC(y)=AFC(y)+AVC(y)$/outputunitAFC(y)AVC(y)ATC(y)y0ATC(y)=AFC(y)+AVC(y)$/outputunitAFC(y)AVC(y)ATC(y)y0AFC(y)=ATC(y)-AVC(y)AFC$/outputunitAFC(y)AVC(y)ATC(y)y0SinceAFC(y)®0asy®¥,

ATC(y)®AVC(y)asy®¥.AFC$/outputunitAFC(y)AVC(y)ATC(y)y0SinceAFC(y)®0asy®¥,

ATC(y)®AVC(y)asy®¥.Andsinceshort-runAVC(y)must

eventuallyincrease,ATC(y)musteventuallyincreaseinashort-run.MarginalCostFunctionMarginalcostistherate-of-changeofvariableproductioncostastheoutputlevelchanges.Thatis,MarginalCostFunctionThefirm’stotalcostfunctionis

andthefixedcostFdoesnotchangewiththeoutputlevely,so

MCistheslopeofboththevariablecostandthetotalcostfunctions.MarginalandVariableCostFunctionsSinceMC(y)isthederivativeofcv(y),cv(y)mustbetheintegralofMC(y).Thatis,MarginalandVariableCostFunctionsMC(y)y0Areaisthevariable

costofmakingy’units$/outputunitMarginal&AverageCostFunctionsHowismarginalcostrelatedtoaveragevariablecost?Marginal&AverageCostFunctionsSinceTherefore,asas$/outputunityAVC(y)MC(y)$/outputunityAVC(y)MC(y)$/outputunityAVC(y)MC(y)$/outputunityAVC(y)MC(y)$/outputunityAVC(y)MC(y)Theshort-runMCcurveintersects

theshort-runAVCcurvefrom

belowattheAVCcurve’sminimum.Marginal&AverageCostFunctionsSimilarly,sinceTherefore,asas$/outputunityMC(y)ATC(y)asMarginal&AverageCostFunctionsTheshort-runMCcurveintersectstheshort-runAVCcurvefrombelowattheAVCcurve’sminimum.And,similarly,theshort-runMCcurveintersectstheshort-runATCcurvefrombelowattheATCcurve’sminimum.$/outputunityAVC(y)MC(y)ATC(y)Short-Run&Long-RunTotalCostCurvesAfirmhasadifferentshort-runtotalcostcurveforeachpossibleshort-runcircumstance.Supposethefirmcanbeinoneofjustthreeshort-runs;

x2=x2¢

or x2=x2¢¢x2¢<x2¢¢<x2¢¢¢.

or x2=x2¢¢¢.y0F¢=w2x2¢F¢cs(y;x2¢)$yF¢0F¢=w2x2¢F¢¢F¢¢=w2x2¢¢cs(y;x2¢)cs(y;x2¢¢)$yF¢0F¢=w2x2¢F¢¢=w2x2¢¢Alargeramountofthefixed

inputincreasesthefirm’s

fixedcost.cs(y;x2¢)cs(y;x2¢¢)$F¢¢yF¢0F¢=w2x2¢F¢¢=w2x2¢¢Alargeramountofthefixed

inputincreasesthefirm’s

fixedcost.Whydoes

alargeramountof

thefixedinputreducetheslopeofthefirm’stotalcostcurve?cs(y;x2¢)cs(y;x2¢¢)$F¢¢MP1isthemarginalphysicalproductivityofthevariableinput1,sooneextraunitofinput1givesMP1extraoutputunits.Therefore,theextraamountofinput1neededfor1extraoutputunitisShort-Run&Long-RunTotalCostCurvesunitsofinput1.Eachunitofinput1costsw1,sothefirm’sextracostfromproducingoneextraunitofoutputisShort-Run&Long-RunTotalCostCurvesistheslopeofthefirm’stotalcostcurve.Ifinput2isacomplementtoinput1then

MP1ishigherforhigherx2.Hence,MCislowerforhigherx2.Thatis,ashort-runtotalcostcurvestarts

higherandhasalowerslopeifx2islarger.yF¢0F¢=w2x2¢F¢¢=w2x2¢¢F¢¢¢F¢¢¢=w2x2¢¢¢cs(y;x2¢¢¢)cs(y;x2¢)cs(y;x2¢¢)$F¢¢Short-Run&Long-RunTotalCostCurvesThefirmhasthreeshort-runtotalcostcurves.Inthelong-runthefirmisfreetochooseamongstthesethreesinceitisfreetoselectx2equaltoanyofx2¢,x2¢¢,orx2¢¢¢.Howdoesthefirmmakethischoice?yF¢0F¢¢¢y¢y¢¢For0£y£y¢,choosex2=?cs(y;x2¢¢¢)cs(y;x2¢)cs(y;x2¢¢)$F¢¢yF¢0F¢¢¢y¢y¢¢For0£y£y¢,choosex2=x2¢.cs(y;x2¢¢¢)cs(y;x2¢)cs(y;x2¢¢)$F¢¢yF¢0F¢¢¢y¢y¢¢For0£y£y¢,choosex2=x2¢.Fory¢

£y£y¢¢,choosex2=?cs(y;x2¢¢¢)cs(y;x2¢)cs(y;x2¢¢)$F¢¢yF¢0F¢¢¢y¢y¢¢For0£y£y¢,choosex2=x2¢.Fory¢

£y£y¢¢,choosex2=x2¢¢.cs(y;x2¢¢¢)cs(y;x2¢)cs(y;x2¢¢)$F¢¢yF¢0F¢¢¢y¢y¢¢For0£y£y¢,choosex2=x2¢.Fory¢

£y£y¢¢,choosex2=x2¢¢.Fory¢¢

<y,choosex2=?cs(y;x2¢¢¢)cs(y;x2¢)cs(y;x2¢¢)$F¢¢yF¢0F¢¢¢cs(y;x2¢¢¢)y¢y¢¢For0£y£y¢,choosex2=x2¢.Fory¢

£y£y¢¢,choosex2=x2¢¢.Fory¢¢

<y,choosex2=x2¢¢¢.cs(y;x2¢)cs(y;x2¢¢)$F¢¢yF¢0cs(y;x2¢)cs(y;x2¢¢)F¢¢¢cs(y;x2¢¢¢)y¢y¢¢For0£y£y¢,choosex2=x2¢.Fory¢

£y£y¢¢,choosex2=x2¢¢.Fory¢¢

<y,choosex2=x2¢¢¢.c(y),the

firm’slong-

runtotal

costcurve.$F¢¢Short-Run&Long-RunTotalCostCurvesThefirm’slong-runtotalcostcurveconsistsofthelowestpartsoftheshort-runtotalcostcurves.Thelong-runtotalcostcurveisthelowerenvelopeoftheshort-runtotalcostcurves.Short-Run&Long-RunTotalCostCurvesIfinput2isavailableincontinuousamountsthenthereisaninfinityofshort-runtotalcostcurvesbutthelong-runtotalcostcurveisstillthelowerenvelopeofalloftheshort-runtotalcostcurves.$yF¢0F¢¢¢cs(y;x2¢)cs(y;x2¢¢)cs(y;x2¢¢¢)c(y)F¢¢Short-Run&Long-RunAverageTotalCostCurvesForanyoutputlevely,thelong-runtotalcostcurvealwaysgivesthelowestpossibletotalproductioncost.Therefore,thelong-runav.totalcostcurvemustalwaysgivethelowestpossibleav.totalproductioncost.Thelong-runav.totalcostcurvemustbethelowerenvelopeofallofthefirm’sshort-runav.totalcostcurves.

Short-Run&Long-RunAverageTotalCostCurvesE.g.supposeagainthatthefirmcanbeinoneofjustthreeshort-runs;

x2=x2¢

or x2=x2¢¢(x2¢<x2¢¢<x2¢¢¢)

or x2=x2¢¢¢

thenthefirm’sthreeshort-runaveragetotalcostcurvesare...y$/outputunitACs(y;x2¢¢¢)ACs(y;x2¢¢)ACs(y;x2¢)Short-Run&Long-RunAverageTotalCostCurvesThefirm’slong-runaveragetotalcostcurveisthelowerenvelopeoftheshort-runaveragetotalcostcurves...y$/outputunitACs(y;x2¢¢¢)ACs(y;x2¢¢)ACs(y;x2¢)AC(y)Thelong-runav.totalcost

curveisthelowerenvelope

oftheshort-runav.totalcostcurves.Short-Run&Long-RunMarginalCostCurvesQ:Isthelong-runmarginalcostcurvethelowerenvelopeofthefirm’sshort-runmarginalcostcurves?A:No.Short-Run&Long-RunMarginalCostCurvesThefirm’sthreeshort-runaveragetotalcostcurvesare...y$/outputunitACs(y;x2¢¢¢)ACs(y;x2¢¢)ACs(y;x2¢)y$/outputunitACs(y;x2¢¢¢)ACs(y;x2¢¢)ACs(y;x2¢)MCs(y;x2¢)MCs(y;x2¢¢)MCs(y;x2¢¢¢)y$/outputunitACs(y;x2¢¢¢)ACs(y;x2¢¢)ACs(y;x2¢)MCs(y;x2¢)MCs(y;x2¢¢)MCs(y;x2¢¢¢)AC(y)y$/outputunitACs(y;x2¢¢¢)ACs(y;x2¢¢)ACs(y;x2¢)MCs(y;x2¢)MCs(y;x2¢¢)MCs(y;x2¢¢¢)AC(y)y$/outputunitACs(y;x2¢¢¢)ACs(y;x2¢¢)ACs(y;x2¢)MCs(y;x2¢)MCs(y;x2¢¢)MCs(y;x2¢¢¢)MC(y),thelong-runmarginal

costcurve.Short-Run&Long-RunMarginalCostCurvesForanyoutputlevely>0,thelong-runmarginalcostofproductionisthemarginalcostofproductionfortheshort-runchose