Dynamic Hotelling Monopoly with Product Development

Lambertini, Luca Working Paper Dynamic Hotelling Monopoly with Product Development Quaderni - Working Paper DSE, No. 399 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca (2001) : Dynamic Hotelling Monopoly with Product Development, Quaderni - Working Paper DSE, No. 399, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4902 This Version is available at: https://hdl.handle.net/10419/159240 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/ DynamicHotellingMonopolywith ProductDevelopment1 LucaLambertini DepartmentofEconomics UniversityofBologna StradaMaggiore45 40125 Bologna,Italy fax: +39-051-2092664 e-mail: [email protected] January9,2001 1IthankRobertoCelliniand Dan Sasakiforveryinsightfulcommentsand suggestions.Theusualdisclaimerapplies. Abstract IcharacteriseR&Dinvestmentin productinnovationofapro…t-seekingmonopolistversusthatofasocialplannerinaspatialmarket,undereither partialorfull marketcoverage.Underpartialcoverage,thesteadystate productdesignistheoutcomeofthetradeo¤betweentheincentivetolocate ascloseaspossibletothemiddleofthepreference space,and theincentive tosaveuponR&D costs.Theplannerdoesnotproduce thevarietypreferred bytheaverage consumer, insituationswheretheR&Dinvestmentis too costly.Thisresultisreinforced underfull marketcoverage,wherethe planner’sincentivetoinnovateisalwaysweakerthanthemonopolist’s,and theplannerproducestheaverage(and median)consumer’spreferredvariety ifand onlyiftherentalprice ofcapital isnil. Keywords:horizontaldi¤erentiation,R&D,steadystate,saddlepoint JELClassi…cation: D24,L12, O31 1Introduction Theanalysisofdynamicmonopolydatesbackto Evans(1924)and Tintner (1937),who analysedthepricingbehaviourofa…rmsubject to a U-shaped variable costcurve.1Theanalysisofintertemporalcapitalaccumulation appearedlateron(see Eisnerand Strotz,1963,interalia). The existingliteratureinvestigates severalfeaturesofmonopolymarkets, in particularseveralformsofdiscrimination,eitherthrough(intertemporal) pricing(see Stokey,1981;Bulow,1982;Gul, Sonnenscheinand Wilson,1986) orthrough productproliferation(see Mussa and Rosen,1978;Maskinand Riley,1984;Gabszewicz,Shaked,Suttonand Thisse,1986;Bonanno,1987). Adynamicmodelofdurable-goodmonopolywithcapitalaccumulationin continuoustimeisinKamienand Schwartz(1979). Anotherdynamictoolwhich hasreceivedaconsiderableamountofattentionisadvertising,eversince Vidaleand Wolfe(1957)and Nerloveand Arrow(1962).2Ataxonomyintroduced bySethi(1977)distinguishesbetweenadvertisingcapitalmodelsand sales-advertisingresponsemodels.The …rstcategoryconsidersadvertising asaninvestmentinastockofgoodwill, àlaNerlove-Arrow.Thesecond categorygathersmodelswherethere exists adirectrelationship betweentherateofchangeinsalesand advertising,àla Vidale-Wolfe. 1See Chiang(1992)forarecentexpositionoftheoriginalmodelbyEvans,aswell as laterdevelopments. 2Forexhaustivesurveys,see Sethi(1977);Jørgensen(1982);Feichtingerand Jørgensen (1983);Erickson(1991);Feichtinger,Hartland Sethi(1994).Forduopolymodelswith dynamicpricing and advertising,see in particularLeitmann and Schmitendorf(1978),and Feichtinger(1983). 1 Awideattention hasbeen devotedtotheissueofoptimalproductdesign instaticmodelsofproductdi¤erentiation,bethateitherhorizontal(Bonanno,1987)orvertical(Spence,1975;Mussa and Rosen,1978;Maskinand Riley,1984;Gabszewicz,Shaked,Suttonand Thisse,1986).Accordingto these contributions,apro…t-seekingmonopolistdesignstheproduct to…t thetasteofthemarginalconsumer,whileabenevolentplanneraiming at themaximizationofsocialwelfaretakesinto account thetasteoftheaverage consumer.Therefore,wheneverthemarginalwillingness topayoftheaverage consumerishigher(lower) thanthemarginalconsumer’s,weobservea downward(upward)distortioninthe equilibriumdesignoftheproduct.A situationwherethisdoesnothappenisthehorizontaldi¤erentiationmodel àlaHotelling(1929)investigated by Bonanno(1987),whereall consumers havethesamegross surplus.Consequently,the…rmlocatesinthemiddleof theproductspace irrespectiveofherobjectivefunction. Inthispaper,Iproposeamonopolymodelwherethe…rmlocatesthe productinaspatialmarketrepresentingthespace ofconsumerpreferences, asinHotelling(1929).Thelocationoftheproductinthespace ofconsumer preferencesdependsupontheR&Dinvestmentcarriedoutbythe…rmover time,accordingto a technologycharacterised bydecreasingreturns. Theanalysisiscarriedoutunderthealternativeassumptionsofpartial and full marketcoverage.Inthe…rstcase,thesteadystateproductdesignis theoutcomeofthetradeo¤betweentheincentivetolocateascloseaspossible tothemiddleofthepreference space,and theincentivetosaveuponR&D costs.Theplannerdoesnotproduce thevarietypreferred bytheaverage consumer, insituationswheretheR&Dinvestmentistoo costly. 2 Thisconclusionemergesevenmore clearlyfromthefull coverageframework,wheretheplanner’sincentivetoinvestin productinnovationisalways weakerthanthemonopolist’s,and theplannerproducestheaverage(and median)consumer’spreferredvarietyifand onlyiftherentalprice ofcapital isnil. Hence, ingeneral, therearesituationswheretheR&D e¤ortneededto meet thetastesoftheaverage consumeris sociallytoo expensivewhile,on the contrary,themonopolist…ndsitconvenient tolocateherproductin themiddleofthepreference space inordertomaximisethe extractionof surplusfromconsumers.Anancillarybutrelevantcorollaryisthat,whenever themonopolist’svarietyisclosertotheaverage consumertastethanthe planner’s,the extentofmarketcoverageat thesteadystateislargerunder monopolythan undersocialplanning.Therefore,whileitiscertainlytrue thatwelfareishigherunderplanningthan undermonopoly,thishappens becauseofasmallerR&D e¤ortin productinnovationratherthanalarger productionand/oramore e¢cientlocationoftheproductintheadmissible spectrumofvarieties. Theremainderofthepaperis structuredasfollows.Thesetup islaidout insection2.Thebehaviouroftheplannerand themonopolistunderpartial marketcoverageisillustratedinsection3.Section4containsacomparative evaluationoftheirperformancesinsteadystate.Thefull marketcoverage settingisinvestigatedinsection5,and the comparativeassessmentofsocialplanning and monopolyisinsection6.Section7containsconcluding remarks. 3 2Themodel Thesetup sharesitsbasicfeatureswithBonanno(1987).Iconsideramarket forhorizontallydi¤erentiated productswhere consumersareuniformlydistributedwith unitdensityalongtheunitinterval[0;1].Let themarketexist overt2[0;1):Themarketis served bya…rmsellingasinglegood, located atx(t)2[1=2;1]:Thisassumptionisjusti…ed bythesymmetryofthemodel around 1/2.Theinitialconditionisx(0)=1;and the…rm maymodifyits locationovertimeaccordingtothefollowingdynamicsofthestatevariable:3 @x(t) @t=¡k(t) 1+k(t)¢x(t);(1) wherek(t)istheR&D e¤ortcarriedoutbythe…rmat timet:Observethat technology(1)exhibitsdecreasingreturnstoscale. Thegeneric consumerlocatedata(t)2[0;1]buysoneunitofthegood, ifnetsurplusfrompurchaseisnon-negative: U(t)=s¡p(t)¡[x(t)¡a(t)]2¸0;i=1;2;(2) wherep(t)isthe…rm’smill price,and sisgross consumersurplus,thatis, thereservation price thata generic consumeriswillingtopayforthegood. Therefore,scan be consideredasapreference parameterwhich,together withthedisutilityoftransportation,yieldsameasureofconsumers’tastefor thegood. Two alternativesituationsmayarise: 3AstochasticR&Drace forproductinnovationinaHotellingduopolywithquadratic disutilityoftransportationisinvestigated byHarter(1993). 4 [1]Partial market coverage (pmc):Themillpriceissuchthatmarginal consumerlocatedatm(t)2(0;x(t))enjoyszerosurplus,thatis, p(t)=s¡[x(t)¡m(t)]2(3) Thisgivesrisetoamarketdemand equalto: y(t)=1¡m(t):(4) [2]Full market coverage (fmc):Thereservation price sis su¢ciently highto allowforthe…rmtoserveall consumers,sothatdemand is y(t)=1:Insuchacase,forall x(t)2·1 2;1¸;themonopolyprice is p(t)=s¡[x(t)]2: Iassumethatthe…rmoperatesatconstantmarginalproductioncost,and, forthesakeofsimplicity,Inormaliseit tozero.4Therefore, instantaneous pro…tsare: ¼(t)=p(t)y(t)= 8 < : £s¡(x(t)¡m(t))2¤[1¡m(t)]¡½k(t)underpmc s¡[x(t)]2¡½k(t)underfmc(5) where½istherentalprice ofcapital, assumedtobe equaltothediscount rate.Instantaneousconsumersurplusis: CS(t)=Z1 m(t) £s¡p(t)¡(x(t)¡a(t))2¤da(t)=(6) =£1¡(m(t))2¤[3x(t)m(t)¡2m(t)¡1] 3 4Thepropertiesofcapitalaccumulationforbothadvertising and productioninasimilar monopolysetting areinvestigatedinLambertini(2000). 5 underpartialmarketcoverage.Instantaneous socialwelfareunderpartial coverageamountsto SW(t)=¼(t)+CS(t):(7) Underfull marketcoverage,the(inverse)measureofsocialwelfareisgiven bytheintegraloftransportationcostsoverthepopulationofconsumers: TC(t)=Z1 0[x(t)¡a(t)]2da(t)=[x(t)]2¡x(t)+1 3:(8) Notice that,onthebasisof(8),thesocialplannercansetanyprice p(t)2 £0;s¡(x(t))2¤: 3Productdevelopmentunderpartialmarket coverage 3.1SociallyoptimalR&D Inscenario 1,theobjectiveofabenevolentsocialplanneris max a(t) Z1 0e¡½tSW(t)dt=Z1 0e¡½t (£1¡(m(t))2¤[3x(t)m(t)¡2m(t)¡1] 3+ +£s¡(x(t)¡m(t))2¤[1¡m(t)]¡½k(t)ªdt(9) s:t:@x(t) @t=¡k(t) 1+k(t)¢x(t)(10) where½denotestimediscounting.Inchoosingtheoptimal locationofthe marginalconsumer(s)atanyt;theplannerindeedmaximisesdiscounted socialwelfarew.r.t.output (or,alternatively,price).The corresponding Hamiltonianfunctionis: H(t)=e¡½t¢ (£1¡(m(t))2¤[3x(t)m(t)¡2m(t)¡1] 3+(11) 6 Thenegativerootcanobviouslybedisregarded,sothat theoptimalR&D e¤ortinsteadystateis: kss M=¡1+p2x(t)[x(t)¡m(t)][1¡m(t)] ½;(40) wheresubscriptMstandsformonopoly.8Moreover, kss M=0,x(t)= m(t)§ s [m(t)]2+2½2 [1¡m(t)] 2(41) whichobviouslyentails: xss M= m(t)+ s [m(t)]2+2½2 [1¡m(t)] 2:(42) By using(36)and (40),thefollowingpropertiescan beshowntohold: ²Atx(0)=1;kss M=¡1+1 ½ r2 3s>0forall s > 3 2½2: ²kss M=0atxss M=m(t)+ s [m(t)]2+2½2 [1¡m(t)] 22µ1 2;1¸forall s2 "3 2½2;min (5 4;24½2+1¡p16½2+1 8 )! : ²kss M=0atxss M=1 2,fors=min (5 4;24½2+1¡p16½2+1 8 ) : ²kss M=¡1+1 6½ p12s¡1+p12s+1atx(t)=1 2,forall s > min (5 4;24½2+1¡p16½2+1 8 ) : 8Theoptimalcapital levelundermonopoly,kss M;can berewrittenasafunctionof fx(t);s;½gonly.However,thisexpressioniscumbersomeand thereforeitisomitted. 13 Theabove can besummarised by: Proposition2Forall s2 "3 2½2;min (5 4;24½2+1¡p16½2+1 8 )! ;the monopolyproblemadmitsauniquesteadystate equilibriumwherekss M=0at xss M2µ1 2;1¸: Inordertocharacterisethe extentofmarketcoverageinsteadystate, observethat,from(36),m(t)>0forall s2µ0;5 4 ¶;with 24½2+1¡p16½2+1 8<5 4forall ½2 " 0;p2 2 ! ; 24½2+1¡p16½2+1 8¸5 4forall ½¸p2 2: (43) Therefore,wehavethefollowingCorollaryto Proposition2: Corollary2Forall ½2 " 0;p2 2 ! ands2 "3 2½2;24½2+1¡p16½2+1 8 ! ; kss M=0atxss M2µ1 2;1¸;andmss M2(0;xss M):Partialmarketcoverage obtains. Forall ½2 " 0;p2 2 ! ands2 "24½2+1¡p16½2+1 8;5 4 ! ;kss M>0at xss M=1 2;andmss M2(0;xss M):Partialmarketcoverage obtains. Forall ½2 " 0;p2 2 ! ands¸5 4;kss M>0atxss M=1 2;mss M=0:Full marketcoverage obtains. For½=p2 2ands=5 4;insteadystate½kss SP=0;xss SP=1 2;mss SP=0¾: Full marketcoverage obtainsasaninternalsolution. Forall ½>p2 2ands > 5 4;insteadystatekss SP>0;withx=1 2and m=0:Full marketcoverage obtainsasacornersolution. 14 Thedynamicanalysisinthespace fx(t);k(t)gshowsthat,whenever xss M2µ1 2;1¸;thenitisasaddle.Thephasediagramisqualitativelyequivalent to…gure1. 4Planningvsmonopolyunderpartialmarketcoverage Theaboveanalysiscan besummarisedso asto giveacomparative evaluation oftheplanner’sand themonopolist’sbehaviour,asfollows.Firstofall, the followingresultstemsfromthestraightforwardcomparison betweenkss Mand kss SPatx(0)=1: Lemma2Att=0;theoptimalinstantaneousinvestmentislargerunder socialplanningthanundermonopoly, forall admissible valuesoffs;½g: Thatis,thesocial incentivetowardsinvestmentin productdevelopment isinitiallylargerthanthemonopolist’s.Moreover: I.Forall s2[0;½2];neithertheplannernorthemonopolistinvestin productinnovation,and xSP(t)=xM(t)=1forever. II.Forall s2µ½2;3 2½2¸;theplannercarriesoutR&Dforproduct innovation,therebyobtainingxss SP2µ1 2;1¸;whilethemonopolistdoesnot invest. III.Forall s2 Ã3 2½2;min (8½2+1 4;24½2+1¡p16½2+1 8;5 4 )# ; wehavethefollowingcases: 15 A]If½2 " 0;p2 2 ! ;thenmin (8½2+1 4;24½2+1¡p16½2+1 8;5 4 ) = 24½2+1¡p16½2+1 8and xss i2µ1 2;1¸;mss i2(0;xss i);i=M;SP: Partialmarketcoverageobtainsunderbothregimes. Moreover, if½2[0;0:3067);then 1 2<xss M<xss SP<1forall s2 Ã3 2½2;24½2+1¡p16½2+1 8 # :(44) Thatis,there existsasubsetofparameterswherethemonopolistlocatesclosertothemiddleoflinearcitythantheplannerdoes. B]If½¸p2 2;thenmin (8½2+1 4;24½2+1¡p16½2+1 8;5 4 ) =5 4and 1 2<xss SP<xss M<1forall s2µ3 2½2;5 4 ¸and all ½¸p2 2:(45) Thatis, inthisparameter range,theplannerworksoutaproduct designthat…tsthetastesofthemedian(and average)consumerbetter thantheproductsupplied bythemonopolist. IV.If s2 " min (8½2+1 4;24½2+1¡p16½2+1 8;5 4 ) ; max (8½2+1 4;24½2+1¡p16½2+1 8;5 4 )! ; thefollowingcasesarise: 16 A]If½2 " 0;p2 2 ! ;thens2 "24½2+1¡p16½2+1 8;5 4 ! and½xss M=1 2;kss M>0¾; ½xss SP2µ1 2;1¸;kss SP=0¾:This situationisdescribedin …gure2. Figure2:Comparativedynamicsinthespace (x(t);k(t)) , ½2 " 0;p2 2 ! : 6 -x(t) 0 1/2 k(t) x(0)=1xss SP kss SP(0) kss M(0) B]If½¸p2 2;thens2 "5 4;24½2+1¡p16½2+1 8 ! and½xss M2µ1 2;1¸;kss M=0¾; ½xss SP=1 2;kss SP>0¾:This situationisdescribedin …gure3. 17 Figure3:Comparativedynamicsinthespace (x(t);k(t)) , ½¸p2 2: 6 -x(t) 0 1/2 k(t) x(0)=1xss M kss M(0) kss SP(0) V.Ifs¸max (8½2+1 4;24½2+1¡p16½2+1 8;5 4 ) ;thenacornersolutionobtainsinbothregimes,with½xss i=1 2;kss SP¸kss M>0¾;i=M;SP: Asamatterofcuriosity, itiseasytoverifythat,when ( s=5 4;½=p2 2 ) ; theplanning optimumand themonopolyoptimumcoincideat thesaddle point½kss i=0;xss i=1 2 ¾: Theabovediscussioncan besummarised bysayingthatneithertheplannernorthepro…t-seekingmonopolistdoesnecessarilyaimatdevelopingthe productvarietypreferred bytheaverage(and median)consumerlocatedat 1/2.Ingeneral, theybothreach1/2 forsu¢cientlyhighlevelsofthegross 18 surpluss(caseV). Otherwise,thesteadystatelocationoftheproductdependsuponthetradeo¤betweentheincentivetolocateascloseaspossible tothemiddleofthepreference space,so asto(i)extractasmuchsurplusas possible, inthemonopolycase,and (ii)minimisetotaltransportation disutility,undersocialplanning;and theincentivetosaveonR&D costs.In particular,the conventionalwisdomassociatedwithstaticmodels(Spence, 1975;Bonanno,1987),maintainingthat thebenevolentplannercaresabout theaverage consumer,doesnotcarryovertothepresentdynamicanalysis. Indeed,theplannermaynotproduce thevarietypreferred bytheaverage consumer,aslong asmarketcoverageis su¢cientlylarge.Thatis,there aresituationswheretheR&Dinvestmentneededtoreach1/2 is sociallytoo costly.Thisisthe caseif ½2 " 0;p2 2 ! and min (8½2+1 4;24½2+1¡p16½2+1 8;5 4 ) =24½2+1¡p16½2+1 8: Onthe contrary,underthesame conditions,themonopolist…ndsitconvenient tolocateat1/2 so astomaximisethe extractionofsurplusfrom consumers.Thisalsoimpliesthefollowingcorollary: Corollary3Wheneverxss M<xss SP;the extentofmarketcoverage at the steadystateislargerundermonopolythanundersocialplanning. Nevertheless, itisobviouslytruethatwelfareishigherunderplanning than undermonopoly,becauseofasmallerR&D e¤ortinproductinnovation. 19 5Productdevelopmentunderfullmarketcoverage 5.1SociallyoptimalR&D Underfull marketcoverage,thebenevolentplanneraimsatminimisingthe integraloftransportationcostsnetofR&D costs,underthekinematicsof locationasin(10).Accordingly,therelevantHamiltonianis: H(t)=e¡½t¢½x(t)¡[x(t)]2¡1 3¡½k(t)¡¸(t)k(t) 1+k(t)¢x(t)¾;(46) withthefollowing optimalityconditions: @H(t) @k(t)=¡½¡¸(t) [1+k(t)]2¢x(t)=0;(47) ¡@H(t) @x(t)=@¯(t) @t)@¸(t) @t=·½+k(t) 1+k(t) ¸¸(t)¡1+2x(t);(48) lim t!1 ¯(t)¢x(t)=0:(49) From(47),Icanestablishthat ¸(t)=¡½[1+k(t)]2 x(t);(50) k(t)=¡1+1 p½ p¡¸(t)x(t);(51) whichentails:@k(t) @t_¡@¸(t) @tx(t)¡@x(t) @t¸(t):(52) Using(48)and (50),(52)rewritesas: @k(t) @t_½2[1+k(t)]2¡x(t)[2x(t)¡1]:(53) 20 Therootsofther.h.s.expressionin(53)are: k(t)=¡1§px(t)[2x(t)¡1] ½:(54) ObviouslyIcan disregardthenegativesolution,thesteadystateinvestment being: kss SP=¡1+px(t)[2x(t)¡1] ½ > = < 0forx(t) > = < 1+p8½+1 4:(55) Notice that1+p8½+1 42·1 2;1¶forall ½2[0;1):(56) Thisentailsthat, if½¸1;theR&Dinvestmentin productinnovationis sociallytoo costlyfortheplannertoundertakeitatall. Indeed,atx(0)=1; wehavekss SP=¡1+1=½<0forall ½¸1: Thephasediagramisqualitativelythesameasin …gure1,and therefore itisomitted.Theabovediscussionleadstothefollowing:9 Proposition3Underfull marketcoverage,thesocialplanner reachesa steadystateat xss SP=1+p8½+1 42·1 2;1¶forall ½2[0;1): Forall ½¸1;theplannerdoesnotinvestandremainsatx(0)=1forever. 9Thestabilityanalysisisomitted hereaswell asinthenextsection.In bothcases, equilibria aresaddles. 21 5.2OptimalR&Dinapro…t-seekingmonopoly Themonopolistsmaximisesnetdiscounted pro…ts,underthedynamic constraint (10).Accordingly,themonopolist’sHamiltonianis: H(t)=e¡½t¢½s¡[x(t)]2¡½k(t)¡¸(t)k(t) 1+k(t)¢x(t)¾;(57) withthefollowing optimalityconditions: @H(t) @k(t)=¡½¡¸(t) [1+k(t)]2¢x(t);(58) ¡@H(t) @x(t)=@¯(t) @t)@¸(t) @t=·½+k(t) 1+k(t) ¸¸(t)+2x(t);(59) lim t!1 ¯(t)¢x(t)=0:(60) Asthemonopolist’sproblemlargelyreplicatestheplanner’s,Icanquickly outlinethesolution.Theoptimalvalueof¸(t)isde…nedasin(50),while thestatevariable evolvesaccordingtothesamedynamicsasin(52),which inthiscase can bewrittenasfollows: @k(t) @t_½2[1+k(t)]2¡2[x(t)]2:(61) Therootsofther.h.s.expressionin(61)are: k(t)=¡1§x(t)p2 ½:(62) Asintheplanner’scase,thenegativesolutioncan bedisregarded,thesteady stateinvestmentbeing: kss M=¡1+x(t)p2 ½ > = < 0forx(t) > = < ½p2 2:(63) 22 [17]Leitmann,G.and W.E.Schmitendorf(1978),“Pro…tMaximization throughAdvertising:A NonzeroSumDi¤erentialGameApproach”, IEEE TransactionsonAutomaticControl,23,646-650. 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