nicholson微观经济理论（答案+ppt+习题库）-110-03-033593-0

PRODUCTIONFUNCTIONSTHEPRINCIPALACTIVITYOFANYRMISTOTURNINPUTSINTOOUTPUTSBECAUSEECONOMISTSAREINTERESTEDINTHECHOICESTHERMMAKESINACCOMPLISHINGTHISGOAL,BUTWISHTOAVOIDDISCUSSINGMANYOFTHEENGINEERINGINTRICACIESINVOLVED,THEYHAVECHOSENTOCONSTRUCTANABSTRACTMODELOFPRODUCTIONINTHISMODELTHERELATIONSHIPBETWEENINPUTSANDOUTPUTSISFORMALIZEDBYAPRODUCTIONFUNCTIONOFTHEFORMQH11005FK,L,M,111WHEREQREPRESENTSTHERMSOUTPUTOFAPARTICULARGOODDURINGAPERIOD,1KREPRESENTSTHEMACHINETHATIS,CAPITALUSAGEDURINGTHEPERIOD,LREPRESENTSHOURSOFLABORINPUT,MREPRESENTSRAWMATERIALSUSED,2ANDTHENOTATIONINDICATESTHEPOSSIBILITYOFOTHERVARIABLESAFFECTINGTHEPRODUCTIONPROCESSEQUATION111ISASSUMEDTOPROVIDE,FORANYCONCEIVABLESETOFINPUTS,THEENGINEERSSOLUTIONTOTHEPROBLEMOFHOWBESTTOCOMBINETHOSEINPUTSTOGETOUTPUT11CHAPTER1HEREWEUSEALOWERCASEQTOREPRESENTONERMSOUTPUTWERESERVETHEUPPERCASEQTOREPRESENTTOTALOUTPUTINAMARKETGENERALLY,WEASSUMETHATARMPRODUCESONLYONEOUTPUTISSUESTHATARISEINMULTIPRODUCTRMSAREDISCUSSEDINAFEWFOOTNOTESANDPROBLEMS2SOMETIMESRAWMATERIALINPUTSAREDISREGARDEDANDOUTPUT,Q,ISMEASUREDINTERMSOF“VALUEADDED”MARGINALPRODUCTIVITYINTHISSECTIONWESHALLSTUDYTHECHANGEINOUTPUTBROUGHTABOUTBYACHANGEINONEOFTHEPRODUCTIVEINPUTSFORTHEPURPOSESOFTHISEXAMINATIONANDINDEEDFORMOSTOFTHEPURPOSESOFTHISBOOK,ITWILLBEMORECONVENIENTTOUSEASIMPLIEDPRODUCTIONFUNCTIONDENEDASFOLLOWSPRODUCTIONFUNCTIONTHERMSPRODUCTIONFUNCTIONFORAPARTICULARGOOD,Q,QH11005FK,L,112SHOWSTHEMAXIMUMAMOUNTOFTHEGOODTHATCANBEPRODUCEDUSINGALTERNATIVECOMBINATIONSOFCAPITALKANDLABORLOFCOURSE,MOSTOFOURANALYSISWILLHOLDFORANYTWOINPUTSTOTHEPRODUCTIONPROCESSWEMIGHTWISHTOEXAMINETHETERMSCAPITALANDLABORAREUSEDONLYFORCONVENIENCESIMILARLY,ITWOULDBEASIMPLEMATTERTOGENERALIZEOURDISCUSSIONTOCASESINVOLVINGMORETHANTWOINPUTSOCCASIONALLY,WEWILLDOSOFORTHEMOSTPART,HOWEVER,LIMITINGOURDISCUSSIONTOTWOINPUTSWILLBEQUITEHELPFULBECAUSEWECANSHOWTHESEINPUTSONTWODIMENSIONALGRAPHSMARGINALPHYSICALPRODUCTTOSTUDYVARIATIONINASINGLEINPUT,WEDENEMARGINALPHYSICALPRODUCTASFOLLOWSMARGINALPHYSICALPRODUCTTHEMARGINALPHYSICALPRODUCTOFANINPUTISTHEADDITIONALOUTPUTTHATCANBEPRODUCEDBYEMPLOYINGONEMOREUNITOFTHATINPUTWHILEHOLDINGALLOTHERINPUTSCONSTANTMATHEMATICALLY,MARGINALPHYSICALPRODUCTOFCAPITALH11005MPKH11005H11005FK113MARGINALPHYSICALPRODUCTOFLABORH11005MPLH11005H11005FLNOTICETHATTHEMATHEMATICALDENITIONSOFMARGINALPRODUCTUSEPARTIALDERIVATIVES,THEREBYPROPERLYREECTINGTHEFACTTHATALLOTHERINPUTUSAGEISHELDCONSTANTWHILETHEINPUTOFINTERESTISBEINGVARIEDFORANEXAMPLE,CONSIDERAFARMERHIRINGONEMORELABORERTOHARVESTTHECROPBUTHOLDINGALLOTHERINPUTSCONSTANTTHEEXTRAOUTPUTTHISLABORERPRODUCESISTHATFARMHANDSMARGINALPHYSICALPRODUCT,MEASUREDINPHYSICALQUANTITIES,SUCHASBUSHELSOFWHEAT,CRATESOFORANGES,ORHEADSOFLETTUCEWEMIGHTOBSERVE,FOREXAMPLE,THAT50WORKERSONAFARMAREABLETOPRODUCE100BUSHELSOFWHEATPERYEAR,WHEREAS51WORKERS,WITHTHESAMELANDANDEQUIPMENT,CANPRODUCE102BUSHELSTHEMARGINALPHYSICALPRODUCTOFTHEFTYRSTWORKERISTHEN2BUSHELSPERYEARH11128QH5007H11128LH11128QH5007H11128KDEFINITIONDEFINITION268PARTIVPRODUCTIONANDSUPPLYDIMINISHINGMARGINALPRODUCTIVITYWEMIGHTEXPECTTHATTHEMARGINALPHYSICALPRODUCTOFANINPUTDEPENDSONHOWMUCHOFTHATINPUTISUSEDLABOR,FOREXAMPLE,CANNOTBEADDEDINDENITELYTOAGIVENELDWHILEKEEPINGTHEAMOUNTOFEQUIPMENT,FERTILIZER,ANDSOFORTHXEDWITHOUTEVENTUALLYEXHIBITINGSOMEDETERIORATIONINITSPRODUCTIVITYMATHEMATICALLY,THEASSUMPTIONOFDIMINISHINGMARGINALPHYSICALPRODUCTIVITYISANASSUMPTIONABOUTTHESECONDORDERPARTIALDERIVATIVESOFTHEPRODUCTIONFUNCTIONH11005H11005FKKH110210114H11005H11005FLLH110210THEASSUMPTIONOFDIMINISHINGMARGINALPRODUCTIVITYWASORIGINALLYPROPOUNDEDBYTHENINETEENTHCENTURYECONOMISTTHOMASMALTHUS,WHOWORRIEDABOUTRAPIDINCREASESINPOPULATIONRESULTINGINLOWEREDLABORPRODUCTIVITYHISGLOOMYPREDICTIONSFORTHEFUTUREOFHUMANITYLEDECONOMICSTOBECALLEDTHE“DISMALSCIENCE”BUTTHEMATHEMATICSOFTHEPRODUCTIONFUNCTIONSUGGESTSTHATSUCHGLOOMMAYBEMISPLACEDCHANGESINTHEMARGINALPRODUCTIVITYOFLABOROVERTIMEDEPENDNOTONLYONHOWLABORINPUTISGROWING,BUTALSOONCHANGESINOTHERINPUTS,SUCHASCAPITALTHATIS,WEMUSTALSOBECONCERNEDWITHH11128MPL/H11128KH11005FLKINMOSTCASES,FLKH110220,SODECLININGLABORPRODUCTIVITYASBOTHLANDKINCREASEISNOTAFOREGONECONCLUSIONINDEED,ITAPPEARSTHATLABORPRODUCTIVITYHASRISENSIGNICANTLYSINCEMALTHUSTIMEPRIMARILYBECAUSEMAJORINCREASESINCAPITALINPUTSHAVEOFFSETTHEIMPACTOFDIMINISHINGMARGINALPRODUCTIVITYALONEAVERAGEPHYSICALPRODUCTIVITYINCOMMONUSAGETHETERMLABORPRODUCTIVITYOFTENMEANSAVERAGEPRODUCTIVITYWHENITISSAIDTHATACERTAININDUSTRYHASEXPERIENCEDPRODUCTIVITYINCREASES,THISISTAKENTOMEANTHATOUTPUTPERUNITOFLABORINPUTHASINCREASEDALTHOUGHTHECONCEPTOFAVERAGEPRODUCTIVITYISNOTNEARLYASIMPORTANTINTHEORETICALECONOMICDISCUSSIONSASMARGINALPRODUCTIVITYIS,ITRECEIVESAGREATDEALOFATTENTIONINEMPIRICALDISCUSSIONSBECAUSEAVERAGEPRODUCTIVITYISEASILYMEASUREDSAY,ASSOMANYBUSHELSOFWHEATPERLABORHOURINPUT,ITISOFTENUSEDASAMEASUREOFEFCIENCYWEDENETHEAVERAGEPRODUCTOFLABORAPLTOBEAPLH11005H11005H11005115NOTICETHATAPLALSODEPENDSONTHELEVELOFCAPITALEMPLOYEDTHISOBSERVATIONWILLPROVETOBEQUITEIMPORTANTWHENWEEXAMINETHEMEASUREMENTOFTECHNICALPROGRESSATTHEENDOFTHISCHAPTEREXAMPLE111ATWOINPUTPRODUCTIONFUNCTIONSUPPOSETHEPRODUCTIONFUNCTIONFORYSWATTERSDURINGAPARTICULARPERIODCANBEREPRESENTEDBYQH11005FK,LH11005600K2L2H11002K3L3116FK,LH5007LQH5007LOUTPUTH5007H5007LABORINPUTH111282QH5007H11128L2H11128MPLH5007H11128LH111282QH5007H11128K2H11128MPKH5007H11128KCHAPTER11PRODUCTIONFUNCTIONS269TOCONSTRUCTTHEMARGINALANDAVERAGEPRODUCTIVITYRELATIONSOFLABORLFORTHISFUNCTION,WEMUSTASSUMEAPARTICULARVALUEFORTHEOTHERINPUT,CAPITALKSUPPOSEKH1100510THENTHEPRODUCTIONFUNCTIONISGIVENBYQH1100560,000L2H110021000L3117MARGINALPRODUCTTHEMARGINALPRODUCTIVITYFUNCTIONISGIVENBYMPLH11005H11005120,000LH110023000L2,118WHICHDIMINISHESASLINCREASES,EVENTUALLYBECOMINGNEGATIVETHISIMPLIESTHATQREACHESAMAXIMUMVALUESETTINGMPLEQUALTO0,120,000LH110023000L2H110050119YIELDS40LH11005L21110ORLH11005401111ASTHEPOINTATWHICHQREACHESITSMAXIMUMVALUELABORINPUTBEYOND40UNITSPERPERIODACTUALLYREDUCESTOTALOUTPUTFOREXAMPLE,WHENLH1100540,EQUATION117SHOWSTHATQH1100532MILLIONYSWATTERS,WHEREASWHENLH1100550,PRODUCTIONOFYSWATTERSAMOUNTSTOONLY25MILLIONAVERAGEPRODUCTTONDTHEAVERAGEPRODUCTIVITYOFLABORINYSWATTERPRODUCTION,WEDIVIDEQBYL,STILLHOLDINGKH1100510APLH11005H1100560,000LH110021000L21112AGAIN,THISISANINVERTEDPARABOLATHATREACHESITSMAXIMUMVALUEWHENH1100560,000H110022000LH110050,1113WHICHOCCURSWHENLH1100530ATTHISVALUEFORLABORINPUT,EQUATION1112SHOWSTHATAPLH11005900,000,ANDEQUATION118SHOWSTHATMPLISALSO900,000WHENAPLISATAMAXIMUM,AVERAGEANDMARGINALPRODUCTIVITIESOFLABORAREEQUAL3NOTICETHERELATIONSHIPBETWEENTOTALOUTPUTANDAVERAGEPRODUCTIVITYTHATISILLUSTRATEDBYTHISEXAMPLEEVENTHOUGHTOTALPRODUCTIONOFYSWATTERSISGREATERWITH40WORKERS32MILLIONTHANWITH30WORKERS27MILLION,OUTPUTPERWORKERISHIGHERINTHESECONDCASEWITH40WORKERS,EACHWORKERPRODUCES800,000YSWATTERSPERPERIOD,WHEREASWITH30WORKERSEACHWORKERPRODUCES900,000BECAUSECAPITALINPUTYSWATTERPRESSESISHELDCONSTANT,INTHISEXAMPLETHEH11128APLH5007H11128LQH5007LH11128QH5007H11128L270PARTIVPRODUCTIONANDSUPPLY3THISRESULTISQUITEGENERALBECAUSEH11005,ATAMAXIMUMLH11080MPLH11005QORMPLH11005APLLH11080MPLH11002QH5007H5007L2H11128APLH5007H11128LDIMINISHINGMARGINALPRODUCTIVITYOFLABOREVENTUALLYRESULTSINADECLININGLEVELOFOUTPUTPERWORKERQUERYHOWWOULDANINCREASEINKFROM10TO11AFFECTTHEMPLANDAPLFUNCTIONSHEREEXPLAINYOURRESULTSINTUITIVELYISOQUANTMAPSANDTHERATEOFTECHNICALSUBSTITUTIONTOILLUSTRATEPOSSIBLESUBSTITUTIONOFONEINPUTFORANOTHERINAPRODUCTIONFUNCTION,WEUSEITSISOQUANTMAPAGAIN,WESTUDYAPRODUCTIONFUNCTIONOFTHEFORMQH11005FK,L,WITHTHEUNDERSTANDINGTHAT“CAPITAL”AND“LABOR”ARESIMPLYCONVENIENTEXAMPLESOFANYTWOINPUTSTHATMIGHTHAPPENTOBEOFINTERESTANISOQUANTFROMISO,MEANING“EQUAL”RECORDSTHOSECOMBINATIONSOFKANDLTHATAREABLETOPRODUCEAGIVENLEVELOFOUTPUTFOREXAMPLE,ALLTHOSECOMBINATIONSOFKANDLTHATFALLONTHECURVELABELED“QH1100510”INFIGURE111ARECAPABLEOFPRODUCINGTENUNITSOFOUTPUTPERPERIODTHISISOQUANTTHENRECORDSTHEFACTTHATTHEREAREMANYALTERNATIVEWAYSOFPRODUCINGTENUNITSOFOUTPUTONEWAYMIGHTBEREPRESENTEDBYPOINTAWEWOULDUSELAANDKATOPRODUCETENUNITSOFOUTPUTALTERNATIVELY,WEMIGHTPREFERTOUSERELATIVELYLESSCAPITALANDMORELABORANDTHEREFOREWOULDCHOOSEAPOINTSUCHASBHENCE,WEMAYDENEANISOQUANTASFOLLOWSISOQUANTANISOQUANTSHOWSTHOSECOMBINATIONSOFKANDLTHATCANPRODUCEAGIVENLEVELOFOUTPUTSAY,Q0MATHEMATICALLY,ANISOQUANTRECORDSTHESETOFKANDLTHATSATISESFK,LH11005Q01114THEREAREMANYISOQUANTSINTHEKH11002LPLANEEACHISOQUANTREPRESENTSADIFFERENTLEVELOFOUTPUTISOQUANTSRECORDSUCCESSIVELYHIGHERLEVELSOFOUTPUTASWEMOVEINANORTHEASTERLYDIRECTIONPRESUMABLY,USINGMOREOFEACHOFTHEINPUTSWILLPERMITOUTPUTTOINCREASETWOOTHERISOQUANTSFORQH1100520ANDQH1100530ARESHOWNINFIGURE111YOUWILLNOTICETHESIMILARITYBETWEENANISOQUANTMAPANDTHEINDIVIDUALSINDIFFERENCECURVEMAPDISCUSSEDINPARTSIIANDIIITHEYAREINDEEDSIMILARCONCEPTS,BECAUSEBOTHREPRESENT“CONTOUR”MAPSOFAPARTICULARFUNCTIONFORISOQUANTS,HOWEVER,THELABELINGOFTHECURVESISMEASURABLEANOUTPUTOFTENUNITSPERPERIODHASAQUANTIABLEMEANINGECONOMISTSARETHEREFOREMOREINTERESTEDINSTUDYINGTHESHAPEOFPRODUCTIONFUNCTIONSTHANINEXAMININGTHEEXACTSHAPEOFUTILITYFUNCTIONSTHEMARGINALRATEOFTECHNICALSUBSTITUTIONRTSTHESLOPEOFANISOQUANTSHOWSHOWONEINPUTCANBETRADEDFORANOTHERWHILEHOLDINGOUTPUTCONSTANTEXAMININGTHESLOPEPROVIDESINFORMATIONABOUTTHETECHNICALPOSSIBILITYOFSUBSTITUTINGLABORFORCAPITALAFORMALDENITIONISPROVIDEDBYDEFINITIONCHAPTER11PRODUCTIONFUNCTIONS271MARGINALRATEOFTECHNICALSUBSTITUTIONTHEMARGINALRATEOFTECHNICALSUBSTITUTIONRTSSHOWSTHERATEATWHICHLABORCANBESUBSTITUTEDFORCAPITALWHILEHOLDINGOUTPUTCONSTANTALONGANISOQUANTINMATHEMATICALTERMS,RTSLFORKH11005H20841QH11005Q01115INTHISDENITION,THENOTATIONISINTENDEDASAREMINDERTHATOUTPUTISTOBEHELDCONSTANTASLISSUBSTITUTEDFORKTHEPARTICULARVALUEOFTHISTRADEOFFRATEWILLDEPENDNOTONLYONTHELEVELOFOUTPUTBUTALSOONTHEQUANTITIESOFCAPITALANDLABORBEINGUSEDITSVALUEDEPENDSONTHEPOINTONTHEISOQUANTMAPATWHICHTHESLOPEISTOBEMEASUREDRTSANDMARGINALPRODUCTIVITIESTOEXAMINETHESHAPEOFPRODUCTIONFUNCTIONISOQUANTS,ITISUSEFULTOPROVETHEFOLLOWINGRESULTTHERTSOFLFORKISEQUALTOTHERATIOOFTHEMARGINALPHYSICALPRODUCTIVITYOFLABORMPLTOTHEMARGINALPHYSICALPRODUCTIVITYOFCAPITALMPKTHEH11002DKH5007DLDEFINITION272PARTIVPRODUCTIONANDSUPPLYANISOQUANTMAPISOQUANTSRECORDTHEALTERNATIVECOMBINATIONSOFINPUTSTHATCANBEUSEDTOPRODUCEAGIVENLEVELOFOUTPUTTHESLOPEOFTHESECURVESSHOWSTHERATEATWHICHLCANBESUBSTITUTEDFORKWHILEKEEPINGOUTPUTCONSTANTTHENEGATIVEOFTHISSLOPEISCALLEDTHEMARGINALRATEOFTECHNICALSUBSTITUTIONRTSINTHEGURE,THERTSISPOSITIVEANDDIMINISHINGFORINCREASINGINPUTSOFLABORKPERPERIODLPERPERIODKALALBKBABQH1100530QH1100520QH1100510FIGURE111PROOFOFTHISASSERTIONPROCEEDSBYSETTINGUPTHETOTALDIFFERENTIALOFTHEPRODUCTIONFUNCTIONDQH11005H11080DLH11001H11080DKH11005MPLH11080DLH11001MPKH11080DK,1116WHICHRECORDSHOWSMALLCHANGESINLANDKAFFECTOUTPUTALONGANISOQUANT,DQH110050OUTPUTISCONSTANT,SOMPLH11080DLH11005H11002MPKH11080DK1117THISSAYSTHATALONGANISOQUANT,THEGAININOUTPUTFROMINCREASINGLSLIGHTLYISEXACTLYBALANCEDBYTHELOSSINOUTPUTFROMSUITABLYDECREASINGKREARRANGINGTERMSABITGIVESH11002H20841QH11005Q0H11005RTSLFORKH110051118HENCETHERTSISGIVENBYTHERATIOOFTHEINPUTSMARGINALPRODUCTIVITIESWECANUSETHERESULTOFEQUATION1118TOSEETHATTHOSEISOQUANTSTHATWEOBSERVEMUSTBENEGATIVELYSLOPEDBECAUSEBOTHMPLANDMPKWILLBENONNEGATIVENORMWOULDCHOOSETOUSEACOSTLYINPUTTHATREDUCEDOUTPUT,THERTSALSOWILLBEPOSITIVEORPERHAPSZEROBECAUSETHESLOPEOFANISOQUANTISTHENEGATIVEOFTHERTS,ANYRMWEOBSERVEWILLNOTBEOPERATINGONTHEPOSITIVELYSLOPEDPORTIONOFANISOQUANTALTHOUGHITISMATHEMATICALLYPOSSIBLETODEVISEPRODUCTIONFUNCTIONSWHOSEISOQUANTSHAVEPOSITIVESLOPESATSOMEPOINTS,ITWOULDNOTMAKEECONOMICSENSEFORARMTOOPERATEATTHOSEINPUTCHOICESREASONSFORADIMINISHINGRTSTHEISOQUANTSINFIGURE111AREDRAWNNOTONLYWITHANEGATIVESLOPEASTHEYSHOULDBEBUTALSOASCONVEXCURVESALONGANYONEOFTHECURVES,THERTSISDIMINISHINGFORHIGHRATIOSOFKTOL,THERTSISALARGEPOSITIVENUMBER,INDICATINGTHATAGREATDEALOFCAPITALCANBEGIVENUPIFONEMOREUNITOFLABORBECOMESAVAILABLEONTHEOTHERHAND,WHENALOTOFLABORISALREADYBEINGUSED,THERTSISLOW,SIGNIFYINGTHATONLYASMALLAMOUNTOFCAPITALCANBETRADEDFORANADDITIONALUNITOFLABORIFOUTPUTISTOBEHELDCONSTANTTHISASSUMPTIONWOULDSEEMTOHAVESOMERELATIONSHIPTOTHEASSUMPTIONOFDIMINISHINGMARGINALPRODUCTIVITYAHASTYUSEOFEQUATION1118MIGHTLEADONETOCONCLUDETHATARISEINLACCOMPANIEDBYAFALLINKWOULDRESULTINARISEINMPK,AFALLINMPL,AND,THEREFORE,AFALLINTHERTSTHEPROBLEMWITHTHISQUICK“PROOF”ISTHATTHEMARGINALPRODUCTIVITYOFANINPUTDEPENDSONTHELEVELOFBOTHINPUTSCHANGESINLAFFECTMPKANDVICEVERSAITISGENERALLYNOTPOSSIBLETODERIVEADIMINISHINGRTSFROMTHEASSUMPTIONOFDIMINISHINGMARGINALPRODUCTIVITYALONETOSEEWHYTHISISSOMATHEMATICALLY,ASSUMETHATQH11005FK,LANDTHATFKANDFLAREPOSITIVETHATIS,THEMARGINALPRODUCTIVITIESAREPOSITIVEASSUMEALSOTHATFKKH110210ANDFLLH110210THATTHEMARGINALPRODUCTIVITIESAREDIMINISHINGTOSHOWTHATISOQUANTSARECONVEX,WEWOULDLIKETOSHOWTHATDRTS/DLH110210SINCERTSH11005FL/FK,WEHAVEH110051119DFL/FKH5007DLDRTSH5007DLMPLH5007MPKDKH5007DLH11128FH5007H11128KH11128FH5007H11128LCHAPTER11PRODUCTIONFUNCTIONS273BECAUSEFLANDFKAREFUNCTIONSOFBOTHKANDL,WEMUSTBECAREFULINTAKINGTHEDERIVATIVEOFTHISEXPRESSIONH110051120USINGTHEFACTTHATDK/DLH11005H11002FL/FKALONGANISOQUANTANDYOUNGSTHEOREMFKLH11005FLK,WEHAVEH110051121BECAUSEWEHAVEASSUMEDFKH110220,THEDENOMINATOROFTHISFUNCTIONISPOSITIVEHENCETHEWHOLEFRACTIONWILLBENEGATIVEIFTHENUMERATORISNEGATIVEBECAUSEFLLANDFKKAREBOTHASSUMEDTOBENEGATIVE,THENUMERATORDENITELYWILLBENEGATIVEIFFKLISPOSITIVEIFWECANASSUMETHIS,WEHAVESHOWNTHATDRTS/DLH110210THATTHEISOQUANTSARECONVEX4IMPORTANCEOFCROSSPRODUCTIVITYEFFECTSINTUITIVELY,ITSEEMSREASONABLETHATTHECROSSPARTIALDERIVATIVEFKLH11005FLKSHOULDBEPOSITIVEITSEEMSPLAUSIBLETHATIFWORKERSHADMORECAPITAL,THEYWOULDHAVEHIGHERMARGINALPRODUCTIVITIESBUT,ALTHOUGHTHISISPROBABLYTHEMOSTPREVALENTCASE,ITDOESNOTNECESSARILYHAVETOBESOSOMEPRODUCTIONFUNCTIONSHAVEFKLH110210,ATLEASTFORARANGEOFINPUTVALUESWHENWEASSUMEADIMINISHINGRTSASWEWILLTHROUGHOUTMOSTOFOURDISCUSSION,WEARETHEREFOREMAKINGASTRONGERASSUMPTIONTHANSIMPLYDIMINISHINGMARGINALPRODUCTIVITIESFOREACHINPUTSPECICALLY,WEAREASSUMINGTHATMARGINALPRODUCTIVITIESDIMINISH“RAPIDLYENOUGH”TOCOMPENSATEFORANYPOSSIBLENEGATIVECROSSPRODUCTIVITYEFFECTSEXAMPLE112ADIMINISHINGRTSINEXAMPLE111,THEPRODUCTIONFUNCTIONFORYSWATTERSWASGIVENBYQH11005FK,LH11005600K2L2H11002K3L31122GENERALMARGINALPRODUCTIVITYFUNCTIONSFORTHISPRODUCTIONFUNCTIONAREMPLH11005FLH11005H110051200K2LH110023K3L21123MPKH11005FKH11005H110051200KL2H110023K2L3NOTICETHATEACHOFTHESEDEPENDSONTHEVALUESOFBOTHINPUTSSIMPLEFACTORINGSHOWSTHATTHESEMARGINALPRODUCTIVITIESWILLBEPOSITIVEFORVALUESOFKANDLFORWHICHKLH11021400H11128QH5007H11128KH11128QH5007H11128LF2KFLLH110022FKFLFKLH11001F2LFKKH5007H5007H5007FK3DRTSH5007DLFKFLLH11001FLKH11080DK/DLH11002FLFKLH11001FKKH11080DK/DLH5007H5007H5007H5007H5007FK2DRTSH5007DL274PARTIVPRODUCTIONANDSUPPLY4ASWEPOINTEDOUTINCHAPTER2,FUNCTIONSFORWHICHTHENUMERATORINEQUATION1121ISNEGATIVEARECALLEDSTRICTLYQUASICONCAVEFUNCTIONSBECAUSEFLLH110051200K2H110026K3LANDFKKH110051200L2H110026KL3,1124ITISCLEARTHATTHISFUNCTIONEXHIBITSDIMINISHINGMARGINALPRODUCTIVITIESFORSUFCIENTLYLARGEVALUESOFKANDLINDEED,AGAINBYFACTORINGEACHEXPRESSION,ITISEASYTOSHOWTHATFLL,FKKH110210IFKLH11022200EVENWITHINTHERANGE200H11021KLH11021400WHERETHEMARGINALPRODUCTIVITYRELATIONSFORTHISFUNCTIONBEHAVE“NORMALLY,”HOWEVER,THISPRODUCTIONFUNCTIONMAYNOTNECESSARILYHAVEADIMINISHINGRTSCROSSDIFFERENTIATIONOFEITHEROFTHEMARGINALPRODUCTIVITYFUNCTIONSEQUATION1123YIELDSFKLH11005FLKH110052400KLH110029K2L2,1125WHICHISPOSITIVEONLYFORKLH11021266THENUMERATOROFEQUATION1121WILLTHEREFOREDENITELYBENEGATIVEFOR200H11021KLH11021266,BUTFORLARGERSCALEYSWATTERFACTORIESTHECASEISNOTSOCLEAR,BECAUSEFKLISNEGATIVEWHENFKLISNEGATIVE,INCREASESINLABORINPUTREDUCETHEMARGINALPRODUCTIVITYOFCAPITALHENCE,THEINTUITIVEARGUMENTTHATTHEASSUMPTIONOFDIMINISHINGMARGINALPRODUCTIVITIESYIELDSANUNAMBIGUOUSPREDICTIONABOUTWHATWILLHAPPENTOTHERTSH11005FL/FKASLINCREASESANDKFALLSISINCORRECTITALLDEPENDSONTHERELATIVEEFFECTSONMARGINALPRODUCTIVITIESOFDIMINISHINGMARGINALPRODUCTIVITIESWHICHTENDTOREDUCEFLANDINCREASEFKANDTHECONTRARYEFFECTSONCROSSMARGINALPRODUCTIVITIESWHICHTENDTOINCREASEFLANDREDUCEFKSTILL,FORTHISYSWATTERCASE,ITISTRUETHATTHERTSISDIMINISHINGTHROUGHOUTTHERANGEOFKANDL,WHEREMARGINALPRODUCTIVITIESAREPOSITIVEFORHIGHERVALUESOFKANDL,THEDIMINISHINGMARGINALPRODUCTIVITIESEXHIBITEDBYTHEFUNCTIONARESUFCIENTTOOVERCOMETHEINUENCEOFANEGATIVEVALUEFORFKLONTHECONVEXITYOFISOQUANTSQUERYFORCASESWHEREKH11005L,WHATCANBESAIDABOUTTHEMARGINALPRODUCTIVITIESOFTHISPRODUCTIONFUNCTIONHOWWOULDTHISSIMPLIFYTHENUMERATORFOREQUATION1121HOWDOESTHISPERMITYOUTOMOREEASILYEVALUATETHISEXPRESSIONFORSOMELARGERVALUESOFKANDLRETURNSTOSCALEWENOWPROCEEDTOCHARACTERIZEPRODUCTIONFUNCTIONSTHERSTIMPORTANTQUESTIONTHATMIGHTBEASKEDABOUTTHEMISHOWOUTPUTRESPONDSTOINCREASESINALLINPUTSTOGETHERFOREXAMPLE,SUPPOSETHATALLINPUTSWEREDOUBLEDWOULDOUTPUTDOUBLEORWOULDTHERELATIONSHIPNOTBEQUITESOSIMPLETHISISAQUESTIONOFTHERETURNSTOSCALEEXHIBITEDBYTHEPRODUCTIONFUNCTIONTHATHASBEENOFINTERESTTOECONOMISTSEVERSINCEADAMSMITHINTENSIVELYSTUDIEDTHEPRODUCTIONOFPINSSMITHIDENTIEDTWOFORCESTHATCAMEINTOOPERATIONWHENTHECONCEPTUALEXPERIMENTOFDOUBLINGALLINPUTSWASPERFORMEDFIRST,ADOUBLINGOFSCALEPERMITSAGREATERDIVISIONOFLABORANDSPECIALIZATIONOFFUNCTIONHENCE,THEREISSOMEPRESUMPTIONTHATEFCIENCYMIGHTINCREASEPRODUCTIONMIGHTMORETHANDOUBLESECOND,DOUBLINGOFTHEINPUTSALSOENTAILSSOMELOSSINEFCIENCYBECAUSEMANAGERIALOVERSEEINGMAYCHAPTER11PRODUCTIONFUNCTIONS275BECOMEMOREDIFCULTGIVENTHELARGERSCALEOFTHERMWHICHOFTHESETWOTENDENCIESWILLHAVEAGREATEREFFECTISANIMPORTANTEMPIRICALQUESTIONPRESENTINGATECHNICALDENITIONOFTHESECONCEPTSISMISLEADINGLYSIMPLERETURNSTOSCALEIFTHEPRODUCTIONFUNCTIONISGIVENBYQH11005FK,LANDALLINPUTSAREMULTIPLIEDBYTHESAMEPOSITIVECONSTANT,MWHEREMH110221,WECLASSIFYTHERETURNSTOSCALEOFTHEPRODUCTIONFUNCTIONBYEFFECTONOUTPUTRETURNSTOSCALEIFMK,MLH11005MFK,LH11005MQCONSTANTIIFMK,MLH11021MFK,LH11005MQDECREASINGIIIFMK,MLH11022MFK,LH11005MQINCREASINGININTUITIVETERMS,IFAPROPORTIONATEINCREASEININPUTSINCREASESOUTPUTBYTHESAMEPROPORTION,THEPRODUCTIONFUNCTIONEXHIBITSCONSTANTRETURNSTOSCALE5IFOUTPUTINCREASESLESSTHANPROPORTIONATELY,THEFUNCTIONEXHIBITSDIMINISHINGRETURNSTOSCALEANDIFOUTPUTINCREASESMORETHANPROPORTIONATELY,THEREAREINCREASINGRETURNSTOSCALEITISTHEORETICALLYPOSSIBLEFORAFUNCTIONTOEXHIBITCONSTANTRETU