10.2 Economies of Scale and Scope

Scope economies work like scale economies, except that they account for advantages of producing multiple products, where scale economies involve an advantage of ... PreviousSection TableofContents NextSection 10.2EconomiesofScaleandScope LearningObjectives Whenfirmsgetbigger,whendoaveragecostsriseorfall? Howdoessizerelatetoprofit? AneconomyofscaleSituationthatexistswhenlargerscalelowersaveragecost.—thatlargerscalelowerscost—ariseswhenanincreaseinoutputreducesaveragecosts.Wemeteconomiesofscaleanditsopposite,diseconomiesofscale,intheprevioussection,withanexamplewherelong-runaveragetotalcostinitiallyfellandthenrose,asquantitywasincreased. Whatmakesforaneconomyofscale?Largervolumesofproductionspermitthemanufactureofmorespecializedequipment.IfIamproducingamillionidenticalautomotivetaillights,Icanspend$50,000onanautomatedplasticstampingmachineandonlyaffectmycostsby5centseach.Incontrast,ifIamproducing50,000units,thestampingmachineincreasesmycostsbyadollareachandismuchlesseconomical. Indeed,itissomewhatmoreofapuzzletodeterminewhatproducesadiseconomyofscale.Animportantsourceofdiseconomiesismanagerialinnature—organizingalarge,complexenterpriseisachallenge,andlargerorganizationstendtodevotealargerpercentageoftheirrevenuestomanagementoftheoperation.Abookstorecanberunbyacoupleofindividualswhorarely,ifever,engageinmanagementactivities,whereagiantchainofbookstoresneedsfinance,humanresource,riskmanagement,andother“overhead”typeexpensesjustinordertofunction.Informaloperationofsmallenterprisesisreplacedbyformalproceduralrulesinlargeorganizations.Thisideaofmanagerialdiseconomiesofscaleisreflectedintheaphorism“Aplatypusisaduckdesignedbyacommittee.” Inhisinfluential1975bookTheMythicalMan-Month,IBMsoftwaremanagerFredBooksdescribesaparticularlyseverediseconomyofscale.Addingsoftwareengineerstoaprojectincreasesthenumberofconversationsnecessarybetweenpairsofindividuals.Iftherearenengineers,thereare½n(n–1)pairs,sothatcommunicationcostsriseatthesquareoftheprojectsize.ThisispithilysummarizedinBrooks’Law:“Addingmanpowertoalatesoftwareprojectmakesitlater.” Anotherrelatedsourceofdiseconomiesofscaleinvolvessystemslack.Inessence,itiseasiertohideincompetenceandlazinessinalargeorganizationthaninasmallone.Therearealotoffamiliarexamplesofthisinsight,startingwiththePeterPrinciple,whichstatesthatpeopleriseinorganizationstothepointoftheirownincompetence,meaningthateventuallypeopleceasetodothejobsthattheydowell.LaurenceJohnstonPeter(1919–1990).Thenotionthatslackgrowsasanorganizationgrowsimpliesadiseconomyofscale. Generally,formanytypesofproducts,economiesofscalefromproductiontechnologytendtoreduceaveragecost,uptoapointwheretheoperationbecomesdifficulttomanage.Herethediseconomiestendtopreventthefirmfromeconomicallygettinglarger.Underthisview,improvementsininformationtechnologiesoverthepast20yearshavepermittedfirmstogetlargerandlarger.Whilethisseemslogical,infactfirmsaren’tgettingthatmuchlargerthantheyusedtobe;andtheshareofoutputproducedbythetop1,000firmshasbeenrelativelysteady;thatis,thegrowthinthelargestfirmsjustmirrorsworldoutputgrowth. RelatedtoaneconomyofscaleisaneconomyofscopeSituationthatexistswhenproducingmorerelatedgoodslowersaveragecost..Aneconomyofscopeisareductionincostassociatedwithproducingseveraldistinctgoods.Forexample,Boeing,whichproducesbothcommercialandmilitaryjets,canamortizesomeofitsresearchanddevelopment(R&D)costsoverbothtypesofaircraft,therebyreducingtheaveragecostsofeach.Scopeeconomiesworklikescaleeconomies,exceptthattheyaccountforadvantagesofproducingmultipleproducts,wherescaleeconomiesinvolveanadvantageofmultipleunitsofthesameproduct. Economiesofscalecanoperateattheleveloftheindividualfirmbutcanalsooperateatanindustrylevel.Supposethereisaneconomyofscaleintheproductionofaninput.Forexample,thereisaneconomyofscaleintheproductionofdiskdrivesforpersonalcomputers.ThismeansthatanincreaseintheproductionofPCswilltendtolowerthepriceofdiskdrives,reducingthecostofPCs,whichisascaleeconomy.Inthiscase,itdoesn’tmattertothescaleeconomywhetheronefirmormanyfirmsareresponsiblefortheincreasedproduction.ThisisknownasanexternaleconomyofscaleAneconomyofscalethatoperatesattheindustrylevel,nottheindividualfirmlevel.,oranindustryeconomyofscale,becausethescaleeconomyoperatesattheleveloftheindustryratherthanintheindividualfirm.Thus,thelong-runaveragecostofindividualfirmsmaybeflat,whilethelong-runaveragecostoftheindustryslopesdownward. Eveninthepresenceofanexternaleconomyofscale,theremaybediseconomiesofscaleatthelevelofthefirm.Insuchasituation,thesizeofanyindividualfirmislimitedbythediseconomyofscale,butnonethelesstheaveragecostofproductionisdecreasinginthetotaloutputoftheindustry,throughtheentryofadditionalfirms.Generallythereisanexternaldiseconomyofscaleifalargerindustrydrivesupinputprices;forexample,increasinglandcosts.Increasingtheproductionofsoybeanssignificantlyrequiresusinglandthatisn’tsowellsuitedforthem,tendingtoincreasetheaveragecostofproduction.Suchadiseconomyisanexternaldiseconomyratherthanoperatingattheindividualfarmerlevel.Second,thereisanexternaleconomyifanincreaseinoutputpermitsthecreationofmorespecializedtechniquesandagreatereffortinR&Dismadetolowercosts.Thus,ifanincreaseinoutputincreasesthedevelopmentofspecializedmachinetoolsandotherproductioninputs,anexternaleconomywillbepresent. Aneconomyofscaleariseswhentotalaveragecostfallsasthenumberofunitsproducedrises.Howdoesthisrelatetoproductionfunctions?Welety=f(x1,x2,…,xn)betheoutputwhentheninputsx1,x2,…,xnareused.Arescalingoftheinputsinvolvesincreasingtheinputsbyafixedpercentage;e.g.,multiplyingallofthembytheconstantλ(theGreekletter“lambda”),whereλ>1.Whatdoesthisdotooutput?Ifoutputgoesupbymorethanλ,wehaveaneconomyofscale(alsoknownasincreasingreturnstoscaleSituationthatexistswhenincreasingallinputsbythesamescalarfactorincreasesoutputbymorethanthescalarfactor.):Scalingupproductionincreasesoutputproportionatelymore.Ifoutputgoesupbylessthanλ,wehaveadiseconomyofscale,ordecreasingreturnstoscaleSituationthatexistswhenincreasingallinputsbythesamescalarfactorincreasesoutputbylessthanthescalarfactor..Andfinally,ifoutputrisesbyexactlyλ,wehaveconstantreturnstoscaleSituationthatexistswhenincreasingallinputsbythesamescalarfactorincreasesoutputbythatscalarfactor..Howdoesthisrelatetoaveragecost?Formally,wehaveaneconomyofscaleif f(λ x 1 ,λ x 2 ,…,λ x n )>λf( x 1 , x 2 ,…, x n ) ifλ>1. Thiscorrespondstodecreasingaveragecost.Letw1bethepriceofinputone,w2thepriceofinputtwo,andsoon.Thentheaveragecostofproducingy=f(x1,x2,…,xn)is AVC= w 1 x 1 + w 2 x 2 +…+ w n x n f( x 1 , x 2 ,…, x n ) . Whathappenstoaveragecostaswescaleupproductionbyλ>1?CallthisAVC(λ). AVC(λ) =  w 1 λ x 1 + w 2 λ x 2 +…+ w n λ x n f(λ x 1 ,λ x 2 ,…,λ x n ) =λ w 1 x 1 + w 2 x 2 +…+ w n x n f(λ x 1 ,λ x 2 ,…,λ x n ) = λf( x 1 , x 2 ,…, x n ) f(λ x 1 ,λ x 2 ,…,λ x n ) AVC(1) Thus,averagecostfallsifthereisaneconomyofscaleandrisesifthereisadiseconomyofscale. Anotherinsightaboutthereturnstoscaleconcernsthevalueofthemarginalproductofinputs.Notethatifthereareconstantreturnstoscale,then x 1 ∂f ∂ x 1 + x 2 ∂f ∂ x 2 +…+ x n ∂f ∂ x n = d dλ f(λ x 1 ,λ x 2 ,…,λ x n ) | λ→1 = lim λ→1   f(λ x 1 ,λ x 2 ,…,λ x n )−f( x 1 , x 2 ,…, x n ) λ−1 =f( x 1 , x 2 ,…, x n ) . Thevalue ∂f ∂ x 1 isthemarginalproductofinputx1,andsimilarly ∂f ∂ x 2 isthemarginalproductofthesecondinput,andsoon.Consequently,iftheproductionfunctionexhibitsconstantreturnstoscale,itispossibletodivideupoutputinsuchawaythateachinputreceivesthevalueofthemarginalproduct.Thatis,wecangive x 1 ∂f ∂ x 1 tothesuppliersofinputone, x 2 ∂f ∂ x 2 tothesuppliersofinputtwo,andsoon;andthisexactlyusesupalloftheoutput.Thisisknownas“payingthemarginalproduct,”becauseeachsupplierispaidthemarginalproductassociatedwiththeinput. Ifthereisadiseconomyofscale,thenpayingthemarginalproductisfeasible;butthereisgenerallysomethingleftover,too.Ifthereareincreasingreturnstoscale(aneconomyofscale),thenitisnotpossibletopayalltheinputstheirmarginalproduct;thatis, x 1 ∂f ∂ x 1 + x 2 ∂f ∂ x 2 +…+ x n ∂f ∂ x n >f( x 1 , x 2 ,…, x n ). KeyTakeaways Aneconomyofscaleariseswhenanincreaseinoutputreducesaveragecosts. Specializationmayproduceeconomiesofscale. Animportantsourceofdiseconomiesismanagerialinnature—organizingalarge,complexenterpriseisachallenge,andlargerorganizationstendtodevotealargerpercentageoftheirrevenuestomanagementoftheoperation. Aneconomyofscopeisareductionincostassociatedwithproducingseveralrelatedgoods. Economiesofscalecanoperateattheleveloftheindividualfirmbutcanalsooperateatanindustrylevel.Attheindustrylevel,scaleeconomiesareknownasanexternaleconomiesofscaleoranindustryeconomiesofscale. Thelong-runaveragecostofindividualfirmsmaybeflat,whilethelong-runaveragecostoftheindustryslopesdownward. Generallythereisanexternaldiseconomyofscaleifalargerindustrydrivesupinputprices.ThereisanexternaleconomyifanincreaseinoutputpermitsthecreationofmorespecializedtechniquesandagreatereffortinR&Dismadetolowercosts. Aproductionfunctionhasincreasingreturnstoscaleifanincreaseinallinputsbyaconstantfactorλincreasesoutputbymorethanλ. Aproductionfunctionhasdecreasingreturnstoscaleifanincreaseinallinputsbyaconstantfactorλincreasesoutputbylessthanλ. Theproductionfunctionexhibitsincreasingreturnstoscaleifandonlyifthecostfunctionhasaneconomyofscale. Whenthereisaneconomyofscale,thesumofthevaluesofthemarginalproductexceedsthetotaloutput.Consequently,itisnotpossibletopayallinputstheirmarginalproduct. Whenthereisadiseconomyofscale,thesumofthevaluesofthemarginalproductislessthanthetotaloutput.Consequently,itispossibletopayallinputstheirmarginalproductandhavesomethingleftoverfortheentrepreneur. Exercises GiventheCobb-Douglasproductionfunction f( x 1 , x 2 ,…, x n )= x 1 a 1   x 2 a 2  …  x n a n , showthatthereisconstantreturnstoscaleif a 1 + a 2 +…+ a n =1, increasingreturnstoscaleif a 1 + a 2 +…+ a n >1 ,anddecreasingreturnstoscaleif a 1 + a 2 +…+ a n <1. Supposeacompanyhastotalcostgivenby rK+ q 2 2K , wherecapitalKcanbeadjustedinthelongrun.Doesthiscompanyhaveaneconomyofscale,diseconomyofscale,orconstantreturnstoscaleinthelongrun? Aproductionfunctionfishomogeneousofdegreerif f(λ x 1 ,λ x 2 ,…,λ x n )= λ r f( x 1 , x 2 ,…, x n ). Considerafirmwithaproductionfunctionthatishomogeneousofdegreer.Supposefurtherthatthefirmpaysthevalueofmarginalproductforallofitsinputs.Showthattheportionofrevenueleftoveris1–r. PreviousSection TableofContents NextSection