Chapter9PROFITMAXIMIZATIONCopyright?2005bySouth-Western,adivisionofThomsonLearning.Allrightsreserved.1Chapter9PROFITMAXIMIZATIONCoTheNatureofFirmsAfirmisanassociationofindividualswhohaveorganizedthemselvesforthepurposeofturninginputsintooutputsDifferentindividualswillprovidedifferenttypesofinputsthenatureofthecontractualrelationshipbetweentheprovidersofinputstoafirmmaybequitecomplicated2TheNatureofFirmsAfirmisaContractualRelationshipsSomecontractsbetweenprovidersofinputsmaybeexplicitmayspecifyhours,workdetails,orcompensationOtherarrangementswillbemoreimplicitinnaturedecision-makingauthorityorsharingoftasks3ContractualRelationshipsSomeModelingFirms’BehaviorMosteconomiststreatthefirmasasingledecision-makingunitthedecisionsaremadebyasingledictatorialmanagerwhorationallypursuessomegoalusuallyprofit-maximization4ModelingFirms’BehaviorMosteProfitMaximizationAprofit-maximizingfirmchoosesbothitsinputsanditsoutputswiththesolegoalofachievingmaximumeconomicprofitsseekstomaximizethedifferencebetweentotalrevenueandtotaleconomiccosts5ProfitMaximizationAprofit-maProfitMaximizationIffirmsarestrictlyprofitmaximizers,theywillmakedecisionsina“marginal”wayexaminethemarginalprofitobtainablefromproducingonemoreunitofhiringoneadditionallaborer6ProfitMaximizationIffirmsarOutputChoiceTotalrevenueforafirmisgivenbyR(q)=p(q)qIntheproductionofq,certaineconomiccostsareincurred[C(q)]Economicprofits()arethedifferencebetweentotalrevenueandtotalcosts(q)=R(q)–C(q)=p(q)q–C(q)7OutputChoiceTotalrevenueforOutputChoiceThenecessaryconditionforchoosingthelevelofqthatmaximizesprofitscanbefoundbysettingthederivativeofthefunctionwithrespecttoqequaltozero8OutputChoiceThenecessaryconOutputChoiceTomaximizeeconomicprofits,thefirmshouldchoosetheoutputforwhichmarginalrevenueisequaltomarginalcost9OutputChoiceTomaximizeeconoSecond-OrderConditionsMR=MCisonlyanecessaryconditionforprofitmaximizationForsufficiency,itisalsorequiredthat“marginal”profitmustbedecreasingattheoptimallevelofq10Second-OrderConditionsMR=MCProfitMaximizationoutputrevenues&costsRCq*ProfitsaremaximizedwhentheslopeoftherevenuefunctionisequaltotheslopeofthecostfunctionThesecond-orderconditionpreventsusfrommistakingq0asamaximumq011ProfitMaximizationoutputrevenMarginalRevenueIfafirmcansellallitwisheswithouthavinganyeffectonmarketprice,marginalrevenuewillbeequaltopriceIfafirmfacesadownward-slopingdemandcurve,moreoutputcanonlybesoldifthefirmreducesthegood’sprice12MarginalRevenueIfafirmcanMarginalRevenueIfafirmfacesadownward-slopingdemandcurve,marginalrevenuewillbeafunctionofoutputIfpricefallsasafirmincreasesoutput,marginalrevenuewillbelessthanprice13MarginalRevenueIfafirmfaceMarginalRevenueSupposethatthedemandcurveforasubsandwichisq=100–10pSolvingforprice,wegetp=-q/10+10ThismeansthattotalrevenueisR=pq=-q2/10+10qMarginalrevenuewillbegivenbyMR=dR/dq=-q/5+1014MarginalRevenueSupposethattProfitMaximizationTodeterminetheprofit-maximizingoutput,wemustknowthefirm’scostsIfsubscanbeproducedataconstantaverageandmarginalcostof$4,thenMR=MC-q/5+10=4q=3015ProfitMaximizationTodeterminMarginalRevenueandElasticityTheconceptofmarginalrevenueisdirectlyrelatedtotheelasticityofthedemandcurvefacingthefirmThepriceelasticityofdemandisequaltothepercentagechangeinquantitythatresultsfromaonepercentchangeinprice16MarginalRevenueandElasticitMarginalRevenueandElasticityThismeansthatifthedemandcurveslopesdownward,eq,p<0andMR<pifthedemandiselastic,eq,p<-1andmarginalrevenuewillbepositiveifthedemandisinfinitelyelastic,eq,p=-andmarginalrevenuewillequalprice17MarginalRevenueandElasticitMarginalRevenueandElasticityeq,p<-1MR>0eq,p=-1MR=0eq,p>-1MR<018MarginalRevenueandElasticitTheInverseElasticityRuleBecauseMR=MCwhenthefirmmaximizesprofit,wecanseethatThegapbetweenpriceandmarginalcostwillfallasthedemandcurvefacingthefirmbecomesmoreelastic19TheInverseElasticityRuleBecTheInverseElasticityRuleIfeq,p>-1,MC<0Thismeansthatfirmswillchoosetooperateonlyatpointsonthedemandcurvewheredemandiselastic20TheInverseElasticityRuleIfAverageRevenueCurveIfweassumethatthefirmmustsellallitsoutputatoneprice,wecanthinkofthedemandcurvefacingthefirmasitsaveragerevenuecurveshowstherevenueperunityieldedbyalternativeoutputchoices21AverageRevenueCurveIfweassMarginalRevenueCurveThemarginalrevenuecurveshowstheextrarevenueprovidedbythelastunitsoldInthecaseofadownward-slopingdemandcurve,themarginalrevenuecurvewillliebelowthedemandcurve22MarginalRevenueCurveThemargMarginalRevenueCurveoutputpriceD(averagerevenue)MRq1p1Asoutputincreasesfrom0toq1,totalrevenueincreasessoMR>0Asoutputincreasesbeyondq1,totalrevenuedecreasessoMR<023MarginalRevenueCurveoutputprMarginalRevenueCurveWhenthedemandcurveshifts,itsassociatedmarginalrevenuecurveshiftsaswellamarginalrevenuecurvecannotbecalculatedwithoutreferringtoaspecificdemandcurve24MarginalRevenueCurveWhentheTheConstantElasticityCaseWeshowed(inChapter5)thatademandfunctionoftheformq=apb

hasaconstantpriceelasticityofdemandequaltobSolvingthisequationforp,wegetp=(1/a)1/bq1/b=kq1/b

wherek=(1/a)1/b25TheConstantElasticityCaseWeTheConstantElasticityCaseThismeansthatR=pq=kq(1+b)/b

andMR=dr/dq=[(1+b)/b]kq1/b=[(1+b)/b]pThisimpliesthatMRisproportionaltoprice26TheConstantElasticityCaseThShort-RunSupplybyaPrice-TakingFirmoutputpriceSMCSACSAVCp*=MRq*Maximumprofitoccurswherep=SMC27Short-RunSupplybyaPrice-TaShort-RunSupplybyaPrice-TakingFirmoutputpriceSMCSACSAVCp*=MRq*Sincep>SAC,profit>028Short-RunSupplybyaPrice-TaShort-RunSupplybyaPrice-TakingFirmoutputpriceSMCSACSAVCp*=MRq*Ifthepricerisestop**,thefirmwillproduceq**and>0q**p**29Short-RunSupplybyaPrice-TaShort-RunSupplybyaPrice-TakingFirmoutputpriceSMCSACSAVCp*=MRq*Ifthepricefallstop***,thefirmwillproduceq***q***p***Profitmaximizationrequiresthatp=SMCandthatSMCisupward-sloping<030Short-RunSupplybyaPrice-TaShort-RunSupplybyaPrice-TakingFirmThepositively-slopedportionoftheshort-runmarginalcostcurveistheshort-runsupplycurveforaprice-takingfirmitshowshowmuchthefirmwillproduceateverypossiblemarketpricefirmswillonlyoperateintheshortrunaslongastotalrevenuecoversvariablecostthefirmwillproducenooutputifp<SAVC31Short-RunSupplybyaPrice-TaShort-RunSupplybyaPrice-TakingFirmThus,theprice-takingfirm’sshort-runsupplycurveisthepositively-slopedportionofthefirm’sshort-runmarginalcostcurveabovethepointofminimumaveragevariablecostforpricesbelowthislevel,thefirm’sprofit-maximizingdecisionistoshutdownandproducenooutput32Short-RunSupplybyaPrice-TaShort-RunSupplybyaPrice-TakingFirmoutputpriceSMCSACSAVCThefirm’sshort-runsupplycurveistheSMCcurvethatisaboveSAVC33Short-RunSupplybyaPrice-TaShort-RunSupplySupposethatthefirm’sshort-runtotalcostcurveisSC(v,w,q,k)=vk1+wq1/k1-/

wherek1isthelevelofcapitalheldconstantintheshortrunShort-runmarginalcostis34Short-RunSupplySupposethattShort-RunSupplyTheprice-takingfirmwillmaximizeprofitwherep=SMCTherefore,quantitysuppliedwillbe35Short-RunSupplyTheprice-takiShort-RunSupplyTofindthefirm’sshut-downprice,weneedtosolveforSAVCSVC=wq1/k1-/SAVC=SVC/q=wq(1-)/k1-/SAVC<SMCforallvaluesof<1thereisnopricelowenoughthatthefirmwillwanttoshutdown36Short-RunSupplyTofindthefiProfitFunctionsAfirm’seconomicprofitcanbeexpressedasafunctionofinputs=pq-C(q)=pf(k,l)-vk-wlOnlythevariableskandlareunderthefirm’scontrolthefirmchooseslevelsoftheseinputsinordertomaximizeprofitstreatsp,v,andwasfixedparametersinitsdecisions37ProfitFunctionsAfirm’seconoProfitFunctionsAfirm’sprofitfunctionshowsitsmaximalprofitsasafunctionofthepricesthatthefirmfaces38ProfitFunctionsAfirm’sprofiPropertiesoftheProfitFunctionHomogeneitytheprofitfunctionishomogeneousofdegreeoneinallpriceswithpureinflation,afirmwillnotchangeitsproductionplansanditslevelofprofitswillkeepupwiththatinflation39PropertiesoftheProfitFunctPropertiesoftheProfitFunctionNondecreasinginoutputpriceafirmcouldalwaysrespondtoariseinthepriceofitsoutputbynotchangingitsinputoroutputplansprofitsmustrise40PropertiesoftheProfitFunctPropertiesoftheProfitFunctionNonincreasingininputpricesifthefirmrespondedtoanincreaseinaninputpricebynotchangingthelevelofthatinput,itscostswouldriseprofitswouldfall41PropertiesoftheProfitFunctPropertiesoftheProfitFunctionConvexinoutputpricestheprofitsobtainablebyaveragingthosefromtwodifferentoutputpriceswillbeatleastaslargeasthoseobtainablefromtheaverageofthetwoprices42PropertiesoftheProfitFunctEnvelopeResultsWecanapplytheenvelopetheoremtoseehowprofitsrespondtochangesinoutputandinputprices43EnvelopeResultsWecanapplytProducerSurplusintheShortRunBecausetheprofitfunctionisnondecreasinginoutputprices,weknowthatifp2>p1(p2,…)(p1,…)Thewelfaregaintothefirmofthispriceincreasecanbemeasuredbywelfaregain=(p2,…)-(p1,…)

44ProducerSurplusintheShortProducerSurplusintheShortRunoutputpriceSMCp1q1Ifthemarketpriceisp1,thefirmwillproduceq1Ifthemarketpricerisestop2,thefirmwillproduceq2p2q245ProducerSurplusintheShortThefirm’sprofitsrisebytheshadedareaProducerSurplusintheShortRunoutputpriceSMCp1q1p2q246Thefirm’sprofitsrisebytheProducerSurplusintheShortRunMathematically,wecanusetheenvelopetheoremresults47ProducerSurplusintheShortProducerSurplusintheShortRunWecanmeasurehowmuchthefirmvaluestherighttoproduceattheprevailingpricerelativetoasituationwhereitwouldproducenooutput48ProducerSurplusintheShortProducerSurplusintheShortRunoutputpriceSMCp1q1Supposethatthefirm’sshutdownpriceisp0p049ProducerSurplusintheShortProducerSurplusintheShortRunTheextraprofitsavailablefromfacingapriceofp1aredefinedtobeproducersurplus50ProducerSurplusintheShortProducersurplusatamarketpriceofp1istheshadedareaProducerSurplusintheShortRunoutputpriceSMCp1q1p051ProducersurplusatamarketpProducerSurplusintheShortRunProducersurplusistheextrareturnthatproducersmakebymakingtransactionsatthemarketpriceoverandabovewhattheywouldearnifnothingwasproducedtheareabelowthemarketpriceandabovethesupplycurve52ProducerSurplusintheShortProducerSurplusintheShortRunBecausethefirmproducesnooutputattheshutdownprice,(p0,…)=-vk1profitsattheshutdownpriceareequaltothefirm’sfixedcostsThisimpliesthatproducersurplus=(p1,…)-(p0,…)=(p1,…)–(-vk1)=(p1,…)+vk1producersurplusisequaltocurrentprofitsplusshort-runfixedcosts53ProducerSurplusintheShortProfitMaximizationandInputDemandAfirm’soutputisdeterminedbytheamountofinputsitchoosestoemploytherelationshipbetweeninputsandoutputsissummarizedbytheproductionfunctionAfirm’seconomicprofitcanalsobeexpressedasafunctionofinputs(k,l)=pq–C(q)=pf(k,l)–(vk+wl)54ProfitMaximizationandInputProfitMaximizationandInputDemandThefirst-orderconditionsforamaximumare/k=p[f/k]–v=0/l=p[f/l]–w=0Aprofit-maximizingfirmshouldhireanyinputuptothepointatwhichitsmarginalcontributiontorevenuesisequaltothemarginalcostofhiringtheinput55ProfitMaximizationandInputProfitMaximizationandInputDemandThesefirst-orderconditionsforprofitmaximizationalsoimplycostminimizationtheyimplythatRTS=w/v56ProfitMaximizationandInputProfitMaximizationandInputDemandToensureatruemaximum,second-orderconditionsrequirethatkk=fkk<0ll=fll<0kkll-kl2=fkkfll–fkl2>0capitalandlabormustexhibitsufficientlydiminishingmarginalproductivitiessothatmarginalcostsriseasoutputexpands57ProfitMaximizationandInputInputDemandFunctionsInprinciple,thefirst-orderconditionscanbesolvedtoyieldinputdemandfunctionsCapitalDemand=k(p,v,w)LaborDemand=l(p,v,w)Thesedemandfunctionsareunconditionaltheyimplicitlyallowthefirmtoadjustitsoutputtochangingprices58InputDemandFunctionsInprincSingle-InputCaseWeexpectl/w0diminishingmarginalproductivityoflaborThefirstorderconditionforprofitmaximizationwas/l=p[f/l]–w=0Takingthetotaldifferential,weget59Single-InputCaseWeexpectl/Single-InputCaseThisreducestoSolvingfurther,wegetSincefll

0,l/w060Single-InputCaseThisreducesTwo-InputCaseForthecaseoftwo(ormoreinputs),thestoryismorecomplexifthereisadecreaseinw,therewillnotonlybeachangeinlbutalsoachangeinkasanewcost-minimizingcombinationofinputsischosenwhenkchanges,theentireflfunctionchangesBut,eveninthiscase,l/w061Two-InputCaseForthecaseofTwo-InputCaseWhenwfalls,twoeffectsoccursubstitutioneffectifoutputisheldconstant,therewillbeatendencyforthefirmtowanttosubstitutelforkintheproductionprocessoutputeffectachangeinwwillshiftthefirm’sexpansionpaththefirm’scostcurveswillshiftandadifferentoutputlevelwillbechosen62Two-InputCaseWhenwfalls,twSubstitutionEffectq0lperperiodkperperiodIfoutputisheldconstantatq0andwfalls,thefirmwillsubstitutelforkintheproductionprocessBecauseofdiminishingRTSalonganisoquant,thesubstitutioneffectwillalwaysbenegative63SubstitutionEffectq0lperperOutputEffectOutputPriceAdeclineinwwilllowerthefirm’sMCMCMC’Consequently,thefirmwillchooseanewlevelofoutputthatishigherPq0q164OutputEffectOutputPriceAdeclOutputEffectq0lperperiodkperperiodThus,theoutputeffectalsoimpliesanegativerelationshipbetweenlandwOutputwillrisetoq1q165OutputEffectq0lperperiodkpCross-PriceEffectsNodefinitestatementcanbemadeabouthowcapitalusagerespondstoawagechangeafallinthewagewillleadthefirmtosubstituteawayfromcapitaltheoutputeffectwillcausemorecapitaltobedemandedasthefirmexpandsproduction66Cross-PriceEffectsNodefiniteSubstitutionandOutputEffectsWehavetwoconceptsofdemandforanyinputtheconditionaldemandforlabor,lc(v,w,q)theunconditionaldemandforlabor,l(p,v,w)Attheprofit-maximizinglevelofoutputlc(v,w,q)=l(p,v,w)67SubstitutionandOutputEffectSubstitutionandOutputEffectsDifferentiationwithrespecttowyieldssubstitutioneffectoutputeffecttotaleffect68SubstitutionandOutputEffectImportantPointstoNote:Inordertomaximizeprofits,thefirmshouldchoosetoproducethatoutputlevelforwhichthemarginalrevenueisequaltothemarginalcost69ImportantPointstoNote:InorImportantPointstoNote:Ifafirmisapricetaker,itsoutputdecisionsdonotaffectthepriceofitsoutputmarginalrevenueisequaltopriceIfthefirmfacesadownward-slopingdemandforitsoutput,marginalrevenuewillbelessthanprice70ImportantPointstoNote:IfaImportantPointstoNote:Marginalrevenueandthepriceelasticityofdemandarerelatedbytheformula71ImportantPointstoNote:MargiImportantPointstoNote:Thesupplycurveforaprice-taking,profit-maximizingfirmisgivenbythepositivelyslopedportionofitsmarginalcostcurveabovethepointofminimumaveragevariablecost(AVC)ifpricefallsbelowminimumAVC,thefirm’sprofit-maximizingchoiceistoshutdownandproducenothing72ImportantPointstoNote:ThesImportantPointstoNote:Thefirm’sreactionstothevariouspricesitfacescanbejudgedthroughuseofitsprofitfunctionshowsmaximumprofitsforthefirmgiventhepriceofitsoutput,thepricesofitsinputs,andtheproductiontechnology73ImportantPointstoNote:ThefImportantPointstoNote:Thefirm’sprofitfunctionyieldsparticularlyusefulenveloperesultsdifferentiationwithrespecttomarketpriceyieldsthesupplyfunctiondifferentiationwithrespecttoanyinputpriceyieldsthe(inverseof)thedemandfunctionforthatinput74ImportantPointstoNote:ThefImportantPointstoNote:Short-runchangesinmarketpriceresultinchangesinthefirm’sshort-runprofitabilitythesecanbemeasuredgraphicallybychangesinthesizeofproducersurplustheprofitfunctioncanalsobeusedtocalculatechangesinproducersurplus75ImportantPointstoNote:ShortImportantPointstoNote:Profitmaximizationprovidesatheoryofthefirm’sderiveddemandforinputsthefirmwillhireanyinputuptothepointatwhichthevalueofitsmarginalproductisjustequaltoitsper-unitmarketpriceincreasesinthepriceofaninputwillinducesubstitutionandoutputeffectsthatcausethefirmtoreducehiringofthatinput76ImportantPointstoNote:ProfiChapter9PROFITMAXIMIZATIONCopyright?2005bySouth-Western,adivisionofThomsonLearning.Allrightsreserved.77Chapter9PROFITMAXIMIZATIONCoTheNatureofFirmsAfirmisanassociationofindividualswhohaveorganizedthemselvesforthepurposeofturninginputsintooutputsDifferentindividualswillprovidedifferenttypesofinputsthenatureofthecontractualrelationshipbetweentheprovidersofinputstoafirmmaybequitecomplicated78TheNatureofFirmsAfirmisaContractualRelationshipsSomecontractsbetweenprovidersofinputsmaybeexplicitmayspecifyhours,workdetails,orcompensationOtherarrangementswillbemoreimplicitinnaturedecision-makingauthorityorsharingoftasks79ContractualRelationshipsSomeModelingFirms’BehaviorMosteconomiststreatthefirmasasingledecision-makingunitthedecisionsaremadebyasingledictatorialmanagerwhorationallypursuessomegoalusuallyprofit-maximization80ModelingFirms’BehaviorMosteProfitMaximizationAprofit-maximizingfirmchoosesbothitsinputsanditsoutputswiththesolegoalofachievingmaximumeconomicprofitsseekstomaximizethedifferencebetweentotalrevenueandtotaleconomiccosts81ProfitMaximizationAprofit-maProfitMaximizationIffirmsarestrictlyprofitmaximizers,theywillmakedecisionsina“marginal”wayexaminethemarginalprofitobtainablefromproducingonemoreunitofhiringoneadditionallaborer82ProfitMaximizationIffirmsarOutputChoiceTotalrevenueforafirmisgivenbyR(q)=p(q)qIntheproductionofq,certaineconomiccostsareincurred[C(q)]Economicprofits()arethedifferencebetweentotalrevenueandtotalcosts(q)=R(q)–C(q)=p(q)q–C(q)83OutputChoiceTotalrevenueforOutputChoiceThenecessaryconditionforchoosingthelevelofqthatmaximizesprofitscanbefoundbysettingthederivativeofthefunctionwithrespecttoqequaltozero84OutputChoiceThenecessaryconOutputChoiceTomaximizeeconomicprofits,thefirmshouldchoosetheoutputforwhichmarginalrevenueisequaltomarginalcost85OutputChoiceTomaximizeeconoSecond-OrderConditionsMR=MCisonlyanecessaryconditionforprofitmaximizationForsufficiency,itisalsorequiredthat“marginal”profitmustbedecreasingattheoptimallevelofq86Second-OrderConditionsMR=MCProfitMaximizationoutputrevenues&costsRCq*ProfitsaremaximizedwhentheslopeoftherevenuefunctionisequaltotheslopeofthecostfunctionThesecond-orderconditionpreventsusfrommistakingq0asamaximumq087ProfitMaximizationoutputrevenMarginalRevenueIfafirmcansellallitwisheswithouthavinganyeffectonmarketprice,marginalrevenuewillbeequaltopriceIfafirmfacesadownward-slopingdemandcurve,moreoutputcanonlybesoldifthefirmreducesthegood’sprice88MarginalRevenueIfafirmcanMarginalRevenueIfafirmfacesadownward-slopingdemandcurve,marginalrevenuewillbeafunctionofoutputIfpricefallsasafirmincreasesoutput,marginalrevenuewillbelessthanprice89MarginalRevenueIfafirmfaceMarginalRevenueSupposethatthedemandcurveforasubsandwichisq=100–10pSolvingforprice,wegetp=-q/10+10ThismeansthattotalrevenueisR=pq=-q2/10+10qMarginalrevenuewillbegivenbyMR=dR/dq=-q/5+1090MarginalRevenueSupposethattProfitMaximizationTodeterminetheprofit-maximizingoutput,wemustknowthefirm’scostsIfsubscanbeproducedataconstantaverageandmarginalcostof$4,thenMR=MC-q/5+10=4q=3091ProfitMaximizationTodeterminMarginalRevenueandElasticityTheconceptofmarginalrevenueisdirectlyrelatedtotheelasticityofthedemandcurvefacingthefirmThepriceelasticityofdemandisequaltothepercentagechangeinquantitythatresultsfromaonepercentchangeinprice92MarginalRevenueandElasticitMarginalRevenueandElasticityThismeansthatifthedemandcurveslopesdownward,eq,p<0andMR<pifthedemandiselastic,eq,p<-1andmarginalrevenuewillbepositiveifthedemandisinfinitelyelastic,eq,p=-andmarginalrevenuewillequalprice93MarginalRevenueandElasticitMarginalRevenueandElasticityeq,p<-1MR>0eq,p=-1MR=0eq,p>-1MR<094MarginalRevenueandElasticitTheInverseElasticityRuleBecauseMR=MCwhenthefirmmaximizesprofit,wecanseethatThegapbetweenpriceandmarginalcostwillfallasthedemandcurvefacingthefirmbecomesmoreelastic95TheInverseElasticityRuleBecTheInverseElasticityRuleIfeq,p>-1,MC<0Thismeansthatfirmswillchoosetooperateonlyatpointsonthedemandcurvewheredemandiselastic96TheInverseElasticityRuleIfAverageRevenueCurveIfweassumethatthefirmmustsellallitsoutputatoneprice,wecanthinkofthedemandcurvefacingthefirmasitsaveragerevenuecurveshowstherevenueperunityieldedbyalternativeoutputchoices97AverageRevenueCurveIfweassMarginalRevenueCurveThemarginalrevenuecurveshowstheextrarevenueprovidedbythelastunitsoldInthecaseofadownward-slopingdemandcurve,themarginalrevenuecurvewillliebelowthedemandcurve98MarginalRevenueCurveThemargMarginalRevenueCurveoutputpriceD(averagerevenue)MRq1p1Asoutputincreasesfrom0toq1,totalrevenueincreasessoMR>0Asoutputincreasesbeyondq1,totalrevenuedecreasessoMR<099MarginalRevenueCurveoutputprMarginalRevenueCurveWhenthedemandcurveshifts,itsassociatedmarginalrevenuecurveshiftsaswellamarginalrevenuecurvecannotbecalculatedwithoutreferringtoaspecificdemandcurve100MarginalRevenueCurveWhentheTheConstantElasticityCaseWeshowed(inChapter5)thatademandfunctionoftheformq=apb

hasaconstantpriceelasticityofdemandequaltobSolvingthisequationforp,wegetp=(1/a)1/bq1/b=kq1/b

wherek=(1/a)1/b101TheConstantElasticityCaseWeTheConstantElasticityCaseThismeansthatR=pq=kq(1+b)/b

andMR=dr/dq=[(1+b)/b]kq1/b=[(1+b)/b]pThisimpliesthatMRisproportionaltoprice102TheConstantElasticityCaseThShort-RunSupplybyaPrice-TakingFirmoutputpriceSMCSACSAVCp*=MRq*Maximumprofitoccurswherep=SMC103Short-RunSupplybyaPrice-TaShort-RunSupplybyaPrice-TakingFirmoutputpriceSMCSACSAVCp*=MRq*Sincep>SAC,profit>0104Short-RunSupplybyaPrice-TaShort-RunSupplybyaPrice-TakingFirmoutputpriceSMCSACSAVCp*=MRq*Ifthepricerisestop**,thefirmwillproduceq**and>0q**p**105Short-RunSupplybyaPrice-TaShort-RunSupplybyaPrice-TakingFirmoutputpriceSMCSACSAVCp*=MRq*Ifthepricefallstop***,thefirmwillproduceq***q***p***Profitmaximizationrequiresthatp=SMCandthatSMCisupward-sloping<0106Short-RunSupplybyaPrice-TaShort-RunSupplybyaPrice-TakingFirmThepositively-slopedportionoftheshort-runmarginalcostcurveistheshort-runsupplycurveforaprice-takingfirmitshowshowmuchthefirmwillproduceateverypossiblemarketpricefirmswillonlyoperateintheshortrunaslongastotalrevenuecoversvariablecostthefirmwillproducenooutputifp<SAVC107Short-RunSupplybyaPrice-TaShort-RunSupplybyaPrice-TakingFirmThus,theprice-takingfirm’sshort-runsupplycurveisthepositively-slopedportionofthefirm’sshort-runmarginalcostcurveabovethepointofminimumaveragevariablecostforpricesbelowthislevel,thefirm’sprofit-maximizingdecisionistoshutdownandproducenooutput108Short-RunSupplybyaPrice-TaShort-RunSupplybyaPrice-TakingFirmoutputpriceSMCSACSAVCThefirm’sshort-runsupplycurveistheSMCcurvethatisaboveSAVC109Short-RunSupplybyaPrice-