ChapterNineteenProfit-MaximizationEconomicProfitAfirmusesinputsj=1…,mtomakeproductsi=1,…n.Outputlevelsarey1,…,yn.Inputlevelsarex1,…,xm.Productpricesarep1,…,pn.Inputpricesarew1,…,wm.

TheCompetitiveFirmThecompetitivefirmtakesalloutputpricesp1,…,pnandallinputpricesw1,…,wmasgivenconstants.EconomicProfitTheeconomicprofitgeneratedbytheproductionplan(x1,…,xm,y1,…,yn)isEconomicProfitOutputandinputlevelsaretypicallyflows.(流量）E.g.x1mightbethenumberoflaborunitsusedperhour.（每小時勞動投入量）Andy3mightbethenumberofcarsproducedperhour.（每小時產(chǎn)出量）Consequently,profitistypicallyaflowalso;e.g.thenumberofdollarsofprofitearnedperhour.EconomicProfitHowdowevalueafirm?如何為一個公司訂價？Supposethefirm’sstreamofperiodiceconomicprofitsisP0,P1,P2,…andristherateofinterest.Thenthepresent-valueofthefirm’seconomicprofitstreamisEconomicProfitAcompetitivefirmseekstomaximizeitspresent-value.一個競爭性企業(yè)為最大化利潤流現(xiàn)值。How?EconomicProfitSupposethefirmisinashort-runcircumstanceinwhich（假定公司處于短期的生產(chǎn)環(huán)境）

Itsshort-runproductionfunctionisEconomicProfitSupposethefirmisinashort-runcircumstanceinwhichItsshort-runproductionfunctionis

Thefirm’sfixedcostis

anditsprofitfunctionisShort-RunIso-ProfitLines

短期等利潤線A$P

iso-profitline

（等利潤線）containsalltheproductionplansthatprovideaprofitlevel$P.A$Piso-profitline’sequationis

Short-RunIso-ProfitLinesA$P

iso-profitlinecontainsalltheproductionplansthatyieldaprofitlevelof$P.Theequationofa$Piso-profitlineis

I.e.Short-RunIso-ProfitLines短期等利潤線hasaslopeofandaverticalinterceptofShort-RunIso-ProfitLinesIncreasing

profityx1Short-RunProfit-Maximization

短期公司利潤最大化Thefirm’sproblemistolocatetheproductionplanthatattainsthehighestpossibleiso-profitline,giventhefirm’sconstraintonchoicesofproductionplans.（給定公司的生產(chǎn)計劃約束，使生產(chǎn)計劃盡可能地達(dá)到最高的等利潤線上。）Q:Whatisthisconstraint?（生產(chǎn)計劃約束？）Short-RunProfit-MaximizationThefirm’sproblemistolocatetheproductionplanthatattainsthehighestpossibleiso-profitline,giventhefirm’sconstraintonchoicesofproductionplans.（生產(chǎn)計劃的選擇約束）Q:Whatisthisconstraint?（技術(shù)約束）A:Theproductionfunction.（生產(chǎn)函數(shù)）Short-RunProfit-Maximizationx1Technically

inefficient

plansyTheshort-runproductionfunctionand

technologysetforShort-RunProfit-Maximizationx1Increasing

profityShort-RunProfit-Maximizationx1yShort-RunProfit-Maximizationx1yGivenp,w1andtheshort-run

profit-maximizingplanisShort-RunProfit-Maximizationx1yGivenp,w1andtheshort-run

profit-maximizingplanis

Andthemaximum

possibleprofit

isShort-RunProfit-Maximizationx1yAttheshort-runprofit-maximizingplan,theslopesoftheshort-runproductionfunctionandthemaximal

iso-profitlineare

equal.Short-RunProfit-Maximizationx1yAttheshort-runprofit-maximizingplan,theslopesoftheshort-runproductionfunctionandthemaximal

iso-profitlineare

equal.Short-RunProfit-Maximization

isthemarginalrevenueproductof

input1,therateatwhichrevenueincreases

withtheamountusedofinput1（要素1的邊際產(chǎn)品收益）.Ifthenprofitincreaseswithx1.Ifthenprofitdecreaseswithx1.

Short-RunProfit-Maximization;ACobb-DouglasExampleSupposetheshort-runproduction

functionisThemarginalproductofthevariable

input1isTheprofit-maximizingconditionisShort-RunProfit-Maximization;ACobb-DouglasExampleSolvingforx1givesShort-RunProfit-Maximization;ACobb-DouglasExampleSolvingforx1givesThatis,Short-RunProfit-Maximization;ACobb-DouglasExampleSolvingforx1givesThatis,soShort-RunProfit-Maximization;ACobb-DouglasExampleisthefirm’s

short-rundemandforinput1whenthelevelofinput2isfixedatunits.Short-RunProfit-Maximization;ACobb-DouglasExampleisthefirm’s

short-rundemandforinput1whenthelevelofinput2isfixedatunits.

Thefirm’sshort-runoutputlevelisthusComparativeStaticsofShort-RunProfit-Maximization比較靜態(tài)分析Whathappenstotheshort-runprofit-maximizingproductionplanastheoutputpricepchanges?ComparativeStaticsofShort-RunProfit-MaximizationTheequationofashort-runiso-profitline

issoanincreaseinpcauses

--areductionintheslope,and--areductionintheverticalintercept.ComparativeStaticsofShort-RunProfit-Maximizationx1yComparativeStaticsofShort-RunProfit-Maximizationx1yComparativeStaticsofShort-RunProfit-Maximizationx1yComparativeStaticsofShort-RunProfit-MaximizationAnincreaseinp,thepriceofthefirm’soutput,causesanincreaseinthefirm’soutputlevel(thefirm’ssupplycurveslopesupward),andanincreaseinthelevelofthefirm’svariableinput(thefirm’sdemandcurveforitsvariableinputshiftsoutward).ComparativeStaticsofShort-RunProfit-MaximizationTheCobb-Douglasexample:When

thenthefirm’sshort-run

demandforitsvariableinput1isanditsshort-run

supplyisComparativeStaticsofShort-RunProfit-MaximizationTheCobb-Douglasexample:When

thenthefirm’sshort-run

demandforitsvariableinput1isincreasesaspincreases.anditsshort-run

supplyisComparativeStaticsofShort-RunProfit-MaximizationTheCobb-Douglasexample:When

thenthefirm’sshort-run

demandforitsvariableinput1isincreasesaspincreases.anditsshort-run

supplyisincreasesaspincreases.ComparativeStaticsofShort-RunProfit-MaximizationWhathappenstotheshort-runprofit-maximizingproductionplanasthevariableinputpricew1changes?ComparativeStaticsofShort-RunProfit-MaximizationTheequationofashort-runiso-profitline

issoanincreaseinw1causes

--anincreaseintheslope,and--nochangetotheverticalintercept.ComparativeStaticsofShort-RunProfit-Maximizationx1yComparativeStaticsofShort-RunProfit-Maximizationx1yComparativeStaticsofShort-RunProfit-Maximizationx1yComparativeStaticsofShort-RunProfit-MaximizationAnincreaseinw1,thepriceofthefirm’svariableinput,causesadecreaseinthefirm’soutputlevel(thefirm’ssupplycurveshiftsinward),andadecreaseinthelevelofthefirm’svariableinput(thefirm’sdemandcurveforitsvariableinputslopesdownward).ComparativeStaticsofShort-RunProfit-MaximizationTheCobb-Douglasexample:When

thenthefirm’sshort-run

demandforitsvariableinput1isanditsshort-run

supplyisComparativeStaticsofShort-RunProfit-MaximizationTheCobb-Douglasexample:When

thenthefirm’sshort-run

demandforitsvariableinput1isdecreasesasw1increases.anditsshort-run

supplyisComparativeStaticsofShort-RunProfit-MaximizationTheCobb-Douglasexample:When

thenthefirm’sshort-run

demandforitsvariableinput1isdecreasesasw1increases.decreasesasw1increases.anditsshort-run

supplyisLong-RunProfit-Maximization

長期利潤最大化Nowallowthefirmtovarybothinputlevels.Sincenoinputlevelisfixed,therearenofixedcosts.Long-RunProfit-MaximizationBothx1andx2arevariable.Thinkofthefirmaschoosingtheproductionplanthatmaximizesprofitsforagivenvalueofx2,andthenvaryingx2tofindthelargestpossibleprofitlevel.Long-RunProfit-MaximizationTheequationofalong-runiso-profitline

issoanincreaseinx2causes

--nochangetotheslope,and--anincreaseintheverticalintercept.Long-RunProfit-Maximizationx1yLong-RunProfit-Maximizationx1yLargerlevelsofinput2increasethe

productivityofinput1.Long-RunProfit-Maximizationx1yLargerlevelsofinput2increasethe

productivityofinput1.Themarginalproduct

ofinput2is

diminishing.Long-RunProfit-Maximizationx1yLargerlevelsofinput2increasethe

productivityofinput1.Themarginalproduct

ofinput2is

diminishing.Long-RunProfit-Maximizationx1y

foreachshort-run

productionplan.Long-RunProfit-Maximizationx1yThemarginalproduct

ofinput2is

diminishingso...

foreachshort-run

productionplan.Long-RunProfit-Maximizationx1ythemarginalprofit

ofinput2is

diminishing.

foreachshort-run

productionplan.Long-RunProfit-MaximizationProfitwillincreaseasx2increasessolongasthemarginalprofitofinput2

Theprofit-maximizinglevelofinput2thereforesatisfiesLong-RunProfit-MaximizationProfitwillincreaseasx2increasessolongasthemarginalprofitofinput2

Theprofit-maximizinglevelofinput2thereforesatisfies

Andissatisfiedinanyshort-run,so...Long-RunProfit-MaximizationTheinputlevelsofthelong-runprofit-maximizingplansatisfy

Thatis,marginalrevenueequalsmarginalcostforallinputs.andLong-RunProfit-MaximizationTheCobb-Douglasexample:When

thenthefirm’sshort-run

demandforitsvariableinput1isanditsshort-run

supplyisShort-runprofitistherefore…Long-RunProfit-MaximizationLong-RunProfit-MaximizationLong-RunProfit-MaximizationLong-RunProfit-MaximizationLong-RunProfit-MaximizationWhatisthelong-runprofit-maximizing

levelofinput2?SolvetogetLong-RunProfit-MaximizationWhatisthelong-runprofit-maximizing

input1level?SubstituteintotogetLong-RunProfit-MaximizationWhatisthelong-runprofit-maximizing

input1level?SubstituteintotogetLong-RunProfit-MaximizationWhatisthelong-runprofit-maximizing

outputlevel?SubstituteintotogetLong-RunProfit-MaximizationWhatisthelong-runprofit-maximizing

outputlevel?SubstituteintotogetLong-RunProfit-MaximizationSogiventhepricesp,w1andw2,and

theproductionfunctionthelong-runprofit-maximizingproduction

planisReturns-to-ScaleandProfit-MaximizationIfacompetitivefirm’stechnologyexhibitsdecreasingreturns-to-scalethenthefirmhasasinglelong-runprofit-maximizingproductionplan.（該競爭性企業(yè)若存在規(guī)模報酬遞減的技術(shù)，則會存在唯一一個長期利潤最大化的產(chǎn)出計劃）Returns-toScaleandProfit-Maximizationxyy*x*Decreasing

returns-to-scaleReturns-to-ScaleandProfit-MaximizationIfacompetitivefirm’stechnologyexhibitsexhibitsincreasingreturns-to-scalethenthefirmdoesnothaveaprofit-maximizingplan.Returns-toScaleandProfit-Maximizationxyy”x’Increasing

returns-to-scaley’x”Increasing

profitReturns-to-ScaleandProfit-MaximizationSoanincreasingreturns-to-scaletechnologyisinconsistentwithfirmsbeingperfectlycompetitive.（因此，規(guī)模報酬遞增的技術(shù)假設(shè)與競爭性企業(yè)假定是矛盾的）Returns-to-ScaleandProfit-MaximizationWhatifthecompetitivefirm’stechnologyexhibitsconstantreturns-to-scale?Returns-toScaleandProfit-Maximizationxyy”x’Constant

returns-to-scaley’x”Increasing

profitReturns-toScaleandProfit-MaximizationSoifanyproductionplanearnsapositiveprofit,thefirmcandoubleupallinputstoproducetwicetheoriginaloutputandearntwicetheoriginalprofit.Returns-toScaleandProfit-MaximizationTherefore,whenafirm’stechnologyexhibitsconstantreturns-to-scale,earningapositiveeconomicprofitisinconsistentwithfirmsbeingperfectlycompetitive.Henceconstantreturns-to-scalerequiresthatcompetitivefirmsearneconomicprofitsofzero.Returns-toScaleandProfit-Maximizationxyy”x’Constant

returns-to-scaley’x”P=0RevealedProfitability

顯示利潤Consideracompetitivefirmwithatechnologythatexhibitsdecreasingreturns-to-scale.Foravarietyofoutputandinputpricesweobservethefirm’schoicesofproductionplans.Whatcanwelearnfromourobservations?RevealedProfitabilityIfaproductionplan(x’,y’)ischosenatprices(w’,p’)wededucethattheplan(x’,y’)isrevealedtobeprofit-maximizingfortheprices(w’,p’).RevealedProfitabilityxy

ischosenatpricesRevealedProfitabilityxy

ischosenatpricesso

isprofit-maximizingattheseprices.RevealedProfitabilityxy

ischosenatpricesso

isprofit-maximizingattheseprices.

wouldgivehigher

profits,sowhyisitnot

chosen?RevealedProfitabilityxy

ischosenatpricesso

isprofit-maximizingattheseprices.

wouldgivehigher

profits,sowhyisitnot

chosen?Becauseitis

notafeasibleplan.RevealedProfitabilityxy

ischosenatpricesso

isprofit-maximizingattheseprices.

wouldgivehigher

profits,sowhyisitnot

chosen?Becauseitis

notafeasibleplan.Sothefirm’stechnologysetmustlieunderthe

iso-profitline.RevealedProfitabilityxy

ischosenatpricesso

isprofit-maximizingattheseprices.Sothefirm’stechnologysetmustlieunderthe

iso-profitline.Thetechnology

setissomewhere

inhereRevealedProfitabilityxy

ischosenatpricesso

maximizesprofitattheseprices.

wouldprovidehigher

profitbutitisnotchosenRevealedProfitabilityxy

ischosenatpricesso

maximizesprofitattheseprices.

wouldprovidehigher

profitbutitisnotchosen

becauseitisnotfeasibleRevealedProfitabilityxy

ischosenatpricesso

maximizesprofitattheseprices.

wouldprovidehigher

profitbutitisnotchosen

becauseitisnotfeasibleso

thetechnologysetliesunder

theiso-profitline.RevealedProfitabilityxy

ischosenatpricesso

maximizesprofitattheseprices.Thetechnologysetis

alsosomewherein

here.RevealedProfitabilityxyThefirm’stechnologysetmustlieunder

bothiso-profitlinesRevealedProfitabilityxyTh