The Monopolist's Optimal R&D Portfolio

Lambertini, Luca Working Paper The Monopolist's Optimal R&D Portfolio Quaderni - Working Paper DSE, No. 391 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca (2000) : The Monopolist's Optimal R&D Portfolio, Quaderni - Working Paper DSE, No. 391, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4910 This Version is available at: https://hdl.handle.net/10419/159232 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ TheMonopolist’sOptimalR&DPortfolio1 LucaLambertini DipartimentodiScienze Economiche StradaMaggiore45 40125 Bologna,Italy fax +39-051-2092664 e-mail [email protected] October3,2000 1IthankGianpaoloRossiniand Dan Sasakiforveryhelpfulcommentsand discussion.Theusualdisclaimerapplies. Abstract Themonopolist’sincentivestowardsproductand process innovationsare evaluatedagainst thesocialoptimum.Themain …ndingsarethat (i) the incentivetoinvestincost-reducingR&Disinverselyrelatedtothenumber ofvarietiesbeingsuppliedatequilibrium,underbothregimes;(ii)distortions obtain undermonopoly,w.r.t.boththenumberofvarietiesand thetechnology.Withsubstitutes(respectively,complements),themonopolist’sproduct rangeis smaller(respectively, larger) than undersocialplanning.Forany given numberofgoods,themonopolistoperatesatahighermarginalcost thantheplannerdoes. JELClassi…cation: L12, O31 Keywords:multiproduct…rm,process innovation,productinnovation 1Introduction Firms’ investmentinaparticulartechnologyisusuallylinkedtothekind and numberofproductstheywant tosupply.Indeed,casualobservationsuggests that…rmsactivateR&Dportfoliosincludingseveralinnovationprojects.Surprisinglyenough,thisproblemhasreceivedrelativelyscantyattention(see BhattacharyaandMookherjee,1986;DasguptaandMaskin,1987).R&Dcan be carriedoutalongtwodi¤erentdirections,namely,process and product innovation.Tothebestofmy knowledge,theliteratureusuallytreatsthe twokindsofinnovationseparately,anoteworthyexception beingRosenkranz (1996).Bonanno and Haworth(1998)consideramarketforverticallydi¤erentiatedgoodswhere…rmscaninvesteithertoreduce marginalcostorto introduce anewvariety.However,theydonotinvestigatethepossibilityfor …rmstopursuebothinnovationsat thesametime. There existsawideliterature concerningmultiproduct…rms.1The early studiesinthisdirectionjusti…edthesupplyofproductlinesonthegroundsof productioncosts.Inthetheoryofcontestablemarkets(Baumol, Panzarand Willig,1982;Panzar,1989),the existence ofmultiproduct…rmsisjusti…ed byeconomiesofscope. Theroleofdemand-sidefactorsinin‡uencingthe…rms’optimalproduct rangehasreceivedmuchless attention,thereason beingthat theanalysisof thisissue can be complicated bythe externalitiesthatmultiproduct…rmstry tointernalise.Thatis,a…rmhastocontrolcompetition between products belongingtothesameline,so astolimitcannibalisationasmuchaspossible. Branderand Eaton(1984)focusupontheinterplaybetweenconsumer’sdemandfordi¤erentiatedgoodsononeside,and thestrategicand technological e¤ectsa¤ecting…rms’behaviour,ontheotherside.Relying onatheoreticalmodelwheretheanalysiscon…nestoCournotcompetition,Branderand Eatonverifythat…rms’strategicdecisionsastoproductrangeand output levelmayleadtomarketequilibriawhere…rms supplyproductrangescharacterised byahigh degree ofsubstitutability.Thisresultisderived under theassumptionthateach …rm’sproductrange consistsina given numberof varieties,and isthereforesubject toafairlynaturalcritique,namely,that …rmsmayendogenouslyalterthespanoftheirproductrangeforstrategic reasons.Thisincentiveisinvestigated by Wernerfelt (1986),…ndingthat the 1Foranexhaustiveoverviewofthetheoryofmultiproduct…rmsinoligopolisticor perfectlycompetitive environments,see MacDonaldand Slivinsky (1987), Okuguchiand Szidarovsky (1990)and DeFraja(1994). 1 drivingforcesaretheheterogeneityofconsumertastesononesideand the costofproductproliferationontheother.Thenestedlogitapproachtothe sameproblemrevealsthat, inafree entryequilibrium,therearetoo many …rmsbut too fewvarietiesper…rm,and thetotalnumberofvarietiesistoo small, comparedtothesocialoptimum(see Andersonand dePalma,1992; Anderson,dePalma and Thisse,1992). Thebehaviourof…rmsinchoosing optimalproductlineshasbeenwidely investigatedintheaddress modelsofproductdi¤erentiation,underboth monopolyandoligopoly(seeMussaandRosen,1978;MaskinandRiley,1984; Gabszewicz etal.,1986;Bonanno,1987;Champsaurand Rochet,1989,inter alia).Themainissueatstakeinthis strand ofresearchisthemonopolist’ss incentivetodistortqualityand quantityascomparedtothesocialoptimum, and hisassociatedattemptatdiscriminating amongcustomerswith di¤erent willingness topayforquality.Afurtherissueisproductproliferationasan entry-deterringdevice (see, in particular,Judd,1985;and Bonanno 1987). Thetheoryofmultiproduct…rmshasevolvedalongseveral linesofresearch,arelevantonebeingdriven bytheideathatconsumersmaybear switchingcosts,either realorperceivedas such,relatedtothepurchaseof productlines.Consequently,consumers’brand loyaltycan besohighthat theypurchasegoodsfromone…rmonly,and…rms’pricingbehaviourbecomes quasi-collusive(see Klemperer,1992,1995; KlempereranPadilla,1997,inter alia). Here,Iproposeinamodelwhichcombinestechnologicaland demand factors.Myaimistoinvestigateproductand process innovationsjointly, in amonopolymodelwheretherepresentative consumerischaracterised bya preference forvariety,and goodsmaybe eithersubstitutesorcomplements. Thepossibilityforthetechnologytoexhibiteconomies(ordiseconomies)of scopeisconsidered,andthe…rm’sresearchportfolioisdrivenbytheinterplay betweenscope economiesand productsubstitutability(orcomplementarity). Themonopolyoptimumisevaluatedagainst thesocialoptimum.Themain …ndingscan besummarisedasfollows.Underbothmonopolyand social planning,(i)ifgoodsaresubstitutes(respectively,complements) the…rm …ndsitoptimaltoproduce morethanonevarietyifproductinnovationcosts arelower(respectively,higher) thanacriticalthreshold;and (ii)irrespective ofwhethergoodsaresubstitutesorcomplements,theincentivetoreduce marginalproductioncostisinverselyrelatedwiththenumberofvarieties. Theintuition behind thisresultisthat the…rm mayincreaseherownability toextractsurpluseitherbyreducingmarginalcostfora given numberofva2 rieties,orbyexpandingtheproductrangefora givenmarginalcost.Hence, cost-reducingR&Dactivitiesand productinnovationareusedas substitutes foroneanother.Finally,(iii)foranyproductrange,themonopolistinvestmentin process innovationistoo littleascomparedtothesocialoptimum. Conversely,foranygivenmarginalcost,themonopolistdistort theproduct rangerespect tothesocialoptimum.Therefore,themarketsu¤ersfrom distortionalongboth dimensionsoftheinnovationactivity.2 Theremainderofthepaperisorganisedasfollows.Section2introduces demand and technology.TheoptimalR&Dportfolio ofthemonopolistis evaluatedinsection3.Section4discussesthesociallyoptimalproduction plan.Theplanner’soptimalR&Dportfolioisassessedinsection4.Section 5providesconcludingcomments. 2Themodel Considerthefollowingmonopolysetting.The…rmsuppliesn¸1products,eachvarietyi=1;2;3:::nbeingcharacterised bythefollowinginverse demand function:3 pi=max 8 < :0;®¡qi¡°X j6=iqj 9 = ;;(1) where°2[¡1;1]measure complementarity/substitutabilitybetweenany twoproducts.If°2[¡1;0)(respectively,°2(0;1])goodsaredemand complements(respectively,substitutes).If°=0;goodsareindependent ofeachotherand the…rmisamonopolistonnindependentmarkets,so that the ensuing analysisreducestoverifyingthepro…tabilityofeachsingle productonitsownmarket. Forthesakeofsimplicity,Iassumethat theoutputofR&Dactivityis certain.Thetotalcostbornebythen-goodmonopolistare: 2Asimilarapproachisadopted byLambertiniand Orsini(2000)inamonopolymodel withverticaldi¤erentiation.Alsointhiscase,there emergesituationswherethemonopolisto¤erstoo fewproductsatahighermarginalcostcomparedtosocialplanning. 3Thisdemand structurewasintroduced byBowley(1924),and ithasbeen used,more recently,byseveralauthors(see Spence,1976;Dixit,1979;Singhand Vives,1984;inter alia). 3 ²ifn>1; C(n)=c(k)n X i=1qi+µnF+»k2;µ>0;»>0(2) ²ifn=1;C(1)=c(k)q+F+»k2;»>0(3) where: ²c(k)isaconstantmarginalcostwhichcan beloweredthroughanR&D e¤ortkin process innovation.The costofsuchactivityis»k2;which entailsthatprocess innovationtakesplace atdecreasingreturns,with c0(k)´@c(k) @k<0;c00(k)´@2c(k) @k2>0;c(0)=c; lim k!1 c(k)=0(4) ²µnFisthe costassociatedtoproductinnovationupton>1varieties.Ifµ2[0;1)(resp., µ>1)wehave economiesofscope(resp., diseconomiesofscope).Notice thathereproductinnovationtranslates intoproductproliferation,whileleavinguna¤ectedthedegree ofsubstitutabilityorcomplementarity°. Therefore,therelevantpro…tfunctionis: ¦M=n X i=1 2 4®¡qi¡°X j6=iqj¡c(k) 3 5qi¡µnF¡»k2;(5) tobemaximisedw.r.t.thevector-´fq1;:::qi;:::qn;k;ng:Iassumethat themonopolistdoesnotpractice anyformofprice discriminationamong consumers.Tosimplifycalculations,withoutloss ofgeneralityIsolve…rst themarketproblem.Then,Iwill proceedtocharacterisethemonopoly optimumw.r.t.theR&D choices. The…rstordercondition(FOC)forpro…tmaximisationw.r.t.varietyi isthefollowing: @¦M @qi=®¡2 0 @qi+°X j6=iqj 1 A¡c(k)=0:(6) 4 Introducingthesymmetryconditionqi=qj=q;theaboveFOCrewrites: @¦M @qi=®¡2[1+°(n¡1)]q¡c(k)=0;(7) yielding q¤=®¡c(k) 2[1+°(n¡1)](8) astheoptimaloutputlevelper-variety.Theoverall monopolyoutputin equilibriumisthenQ¤=n[®¡c(k)]=f2[1+°(n¡1)]g:Equilibriumprofits,giventhemarginalcostand theproductline,are: ¦M(n;k)=n[®¡c(k)]2 4[1+°(n¡1)]¡µnF¡»k2:(9) Thesolutiontothepro…tmaximisation problemofthemonopolistisclosely characterisedinthenextsection. 3TheoptimalR&Dportfolioundermonopoly Maximisationof¦M(k;n)withrespect tofk;ngyieldsthefollowingresult. Proposition1Inequilibrium,themonopolist’s R&Dportfolioisgivenby n¤=°¡1 °+[®¡c(k¤)]qµF(1¡°) 2µF° andc0(k¤)=¡4»k¤[1+°(n¤¡1)] n¤[®¡c(k¤)] providedthat c00(k¤)¸4»[1+°(n¤¡1)] n¤[®¡c(k¤)]¢"4»(k¤)2 n¤[®¡c(k¤)]2¡1# and F2"0;[®¡c(k¤)]2(1¡°) 4µ #8°2(0;1] ; (10) F¸[®¡c(k¤)]2(1¡°) 4µ8°2[¡1;0):(11) 5 Proof.See theAppendix. Notethateither(10)or(11)can besatis…ed.Beingtheymutuallyexclusive,Proposition1entailsthat, ifthemonopolist…ndsitoptimaltoproduce morethanonevarietyinthe caseofsubstitutes,hewill necessarily…nd it optimaltosupplyasinglegoodinthe caseofcomplements,and vice versa. Asacomplement to Proposition1,thefollowingCorollarycan bestated: Corollary1Underperfectsubstitutability, i.e., °=1;themonopolistproducesatmostone variety. Proof.Toprovetheaboveresult, itsu¢cestocheckfrom(9) that lim °!1¦M(n;k)=[®¡c(k)]2 4¡µnF¡»k2:(12) Namely,when productsarehomogeneous,themostpro…tablestrategy consistsinoperatingasingleplant,reducingthustoaminimumtheamount ofsunkcosts.Inconnectionwiththisfairlyintuitiveresult,Iamnowina positiontocharacterisethein‡uence oftherelevantdemand and costparametersfF;µ;°gupontheoptimalproductrangen¤:Thisis summarisedin thefollowing: Proposition2Considern¤>1;i.e., either F2h0;b Fi8°2(0;1] orF¸b F8°2[¡1;0); whereb F=[®¡c(k¤)]2(1¡°) 4µ:Thenwehave @n¤ @F<0and@n¤ @µ <08°2(0;1] ; @n¤ @F>0and@n¤ @µ >08°2[¡1;0): 6 3, itisnevertheless truethat theplanner’sincentivetowardsinvestmentin process innovationislargerthanthemonopolist’s. Summingup,wemayexpect themonopolist todistortinvestmentin both process and productinnovationaswell asoutputat thesametime,and these distortionsobviouslyinteractwithoneanother.Inturn,thisentailsthat, if aregulatorintroducesapolicyaimedatcorrecting oneofthesedistortions, thiswill creates someundesirablefeedbackontheremainingvariables.For instance,subsidisingproductinnovation(inthe caseofsubstitutes)would causeareductionofthemonopolist’sinvestmentin process innovation, inducingthenanoutputreduction. 6Concludingremarks Ihave evaluatedthemonopolist’sbehaviouralongtwodimensionsofhis innovation portfolio, i.e., theR&Dactivitiestowardsproductand process innovation.Then,Ihaveassessedthemonopolyequilibriumagainst the socialoptimum. Theforegoing analysisrevealsthat, inadditiontothewell knowoutput distortion usuallyassociatedwiththemonopolist’soptimalmarketingdecisions,themarketalsosu¤ersfromdistortionsalongthetwodimensionsof innovation,at thesametime.thatis,themonopolistso¤erstoo many(or too few)varieties,producedatalargermarginalcost,thantheplanner.The di¤erentincentivestowardsinnovationcharacterisingthemonopolistand the plannerentailsthatanycomparativeassessmentofthetworegimes should takeinto account thatboth productrangeand technologycan be expected todi¤eracross regimes.Consequently,thetaskfortheregulatorismore involvedthanwewereleadtothinkonthebasisofthepreviousliterature, in that thedistortionsalongboth dimensionsofthemonopolist’sR&Dactivity add totheusualoutputdistortion duetomonopolypricing. 13 Appendix ProofofProposition1 Ilookforthesolutionw.r.t.fk;ngofthefollowingsystem: @¦M(n;k) @k=¡n[®¡c(k)]c0(k) 2[1+°(n¡1)]¡2»k=0;(33) @¦M(n;k) @n=[®¡c(k)]2 4[1+°(n¡1)]¢(1¡°n [1+°(n¡1)] )¡µF=0;(34) wherec0(k)´@c(k)=@k:Second orderconditionsfor(33-34) toyieldan internalsolutionrequirethat theHessianmatrixH[¦M(n;k)]·0;thatis: @2¦M(n;k) @n2=°(°¡1)[®¡c(k)]2 2[1+°(n¡1)]3·0;(35) @2¦M(n;k) @k2=nn[c0(k)]2¡[®¡c(k)]c00(k)o¡4»[1+°(n¡1)] 2[1+°(n¡1)]·0;(36) wherec00(k)´@2c(k)=@k2;and @2¦M(n;k) @n2¢@2¦M(n;k) @k2¸@2¦M(n;k) @k@n¢@2¦M(n;k) @n@k=Ã@2¦M(n;k) @k@n !2: (37) Consider…rst (33)inisolation.Foranyn,theoptimalR&D e¤ortin process innovation,k¤;isimplicitlygiven by: c0(k¤)=¡4»k¤[1+°(n¡1)] n[®¡c(k¤)]:(38) Nowconsider(34).ThisFOChastwocriticalpoints: n1=°¡1 °¡[®¡c(k)]qµF(1¡°) 2µF°(39) n2=°¡1 °+[®¡c(k)]qµF(1¡°) 2µF°(40) ofwhichonlyn2satis…es(35)forall fc(k);°g.Therefore,n2candidatesas theoptimalproductrange. 14 Consider(38).Substitutingthe expressionforc0(k¤)into(36)and simplifying,Iobtain: @2¦M(n;k) @k2=4»[1+°(n¤¡1)]"4»(k¤)2 n¤[®¡c(k¤)]2¡1#+(41) ¡n¤[®¡c(k¤)]c00(k¤)·0; yielding c00(k¤)¸4»[1+°(n¤¡1)] n¤[®¡c(k¤)]¢"4»(k¤)2 n¤[®¡c(k¤)]2¡1#:(42) TheSOCconcerningtheoptimalnumberofgoodscan bequicklydealtwith, byobservingthat°(°¡1)[®¡c(k)]2 2[1+°(n¤¡1)]3·0 is satis…edforall °2[¡1;1]: Nowlookat the condition(37).Thiscan bewrittenas: @2¦M(n;k) @n2¢@2¦M(n;k) @k2¡Ã@2¦M(n;k) @k@n !2=(43) =°(1¡°)[®¡c(k¤)] 4[1+°(n¤¡1)]¢ 8 < :4»[1+°(n¤¡1)]hn¤[®¡c(k¤)]2¡4»(k¤)2i n¤[®¡c(k¤)]2+ +n¤[®¡c(k¤)]c00(k¤)o¡4¢([»(1¡°)k¤] n¤[1+°(n¤¡1)] )2¸0: Theaboveinequalityis satis…edfor c00(k¤)¸4»[1+°(n¤¡1)] °fn¤[®¡c(k¤)]g3¢n4»(k¤)2h(1¡°)2+°n¤(2¡2°+°n¤)i+ (44) ¡°n¤[®¡c(k¤)]2o: Finally,notethat 4»[1+°(n¤¡1)] n¤[®¡c(k¤)]¢"4»(k¤)2 n¤[®¡c(k¤)]2¡1#(45) 15 islargerthan 4»[1+°(n¤¡1)]n4»(k¤)2[(1¡°)2+°n¤(2¡2°+°n¤)]+¡°n¤[®¡c(k¤)]2o °fn¤[®¡c(k¤)]g3 (46) overthewholeadmissiblerangeofparameters.Therefore,(42)isalsosu¢- cient toensureglobaloptimality. Fortheabovesolutiontobe economicallyacceptable, itmustalsobethat n¤¸1;i.e., [®¡c(k¤)]qµF(1¡°)¡2µF 2µF°¸0(47) yielding F2"0;[®¡c(k¤)]2(1¡°) 4µ #8°2(0;1] ; (48) F¸[®¡c(k¤)]2(1¡°) 4µ8°2[¡1;0):(49) Thisconcludestheproof. ProofofProposition4 TheproofproceedsalongthesamelinesasforProposition2.The e¤ects onnSPofavariationinForµaredescribed by: @nSP @F=¡[®¡c(k¤)]p1¡° 2F°p2µF(50) and @nSP @µ =¡[®¡c(k¤)]p1¡° 2µ°p2µF:(51) Thesignofboth(50)and (51)isnegative(respectively,positive)forall °2(0;1](respectively,°2[¡1;0)). 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