Research and Development Cooperatives and Market Collusion: A Global Dynamic Approach

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1 J Optim Theory Appl (7) 74:567 6 DOI.7/s Researh and Development Cooperatives and Market Collusion: A Global Dynami Approah Jeroen Hinloopen, Grega Smrkolj 3 Florian Wagener 4 Reeived: 9 August 6 / Aepted: 6 June 7 / Published online: 5 July 7 The Author(s) 7. This artile is an open aess publiation Abstrat We present a ontinuous-time generalization o the seminal researh and development model o d Aspremont and Jaquemin (Am Eon Rev 78(5):33 37, 988) to examine the trade-o between the beneits o allowing irms to ooperate in researh and the orresponding inreased potential or produt market ollusion. We show the existene o a solution to the optimal investment problem using a ombination o results rom visosity theory and the theory o planar dynamial systems. In partiular, we show that there is a ritial level o marginal ost at whih irms are indierent between doing nothing and starting to develop the tehnology. We ind that olluding irms develop urther a wider range o initial tehnologies, pursue innovations more quikly, and are less likely to abandon a tehnology. Produt market ollusion ould thus yield higher total surplus. Keywords Antitrust poliy Biurations Collusion R&D ooperatives Spillovers Communiated by David G. Luenberger. B Grega Smrkolj Grega.Smrkolj@newastle.a.uk Jeroen Hinloopen J.Hinloopen@uva.nl Florian Wagener F.O.O.Wagener@uva.nl CPB Netherlands Bureau or Eonomi Poliy Analysis, The Hague, The Netherlands Amsterdam Shool o Eonomis, University o Amsterdam, Amsterdam, The Netherlands 3 Newastle University Business Shool, 5 Barrak Road, Newastle upon Tyne NE 4SE, UK 4 CeNDEF, Amsterdam Shool o Eonomis, University o Amsterdam, Amsterdam, The Netherlands

2 568 J Optim Theory Appl (7) 74:567 6 Mathematis Subjet Classiiation 9B38 49N9 Introdution An important reason or allowing irms to set up researh and development (R&D) ooperatives is that these organizations, jointly ontrolled by at least two partiipating entities, whose primary purpose is to engage in ooperative R&D [] internalize tehnologial spillovers the ree low o knowledge rom the knowledge reator to its ompetitors. Indeed, Bloom et al. [] estimate that a % inrease in a ompetitor s R&D is assoiated with up to a 3.8% inrease in a irm s own market value. The exemption or R&D ooperatives in anti-artel legislation is thus pereived to diminish the ailure o the market or R&D. However, as Sherer [3] observes: the most egregious prie ixing shemes in Amerian history were brought about by R&D ooperatives, an observation that onstitutes the lassi ounterargument to a permissive antitrust treatment o R&D markets [4 6]. For instane, Goeree and Helland [7] ind that in the US the probability that irms join an R&D ooperative has dropped due to a revision o antitrust lenieny poliy in 993. This revision is pereived as making ollusion less attrative. They onlude that Our results are onsistent with RJVs [researh joint ventures] serving, at least in part, a ollusive untion. The laboratory experiments o Suetens [8] also show that members o an RJV are more likely to ollude on prie. At the same time, it is quite well established that the prospet o uture market power enhanes a irm s inentives to invest in R&D [9]. As Greenspan [] puts it: No one will ever know what new produts, proesses, mahines, and ost-saving mergers ailed to ome into existene, killed by the Sherman At beore they were born. No one an ever ompute the prie that all o us have paid or that At whih, by induing less eetive use o apital, has kept our standard o living lower than would otherwise have been possible. In this paper, we develop a dynami model o R&D that onsiders expliitly the ost o new proesses that ailed to ome into existene beore they were born beause o the ban on prie-ixing agreements. The hannels through whih ooperation in R&D ailitates produt market ollusion have been examined in a number o theoretial studies [ 5]. Aording to Fisher [6, p. 94]: [irms] ooperating in R&D will tend to talk about other orms o ooperation. Furthermore, in learning how other irms reat and adjust in living with eah other, eah ooperating irm will get better at oordination. Hene, ompetition in the produt market is likely to be harmed. In the short run, the redued intensity o produt market ompetition is likely to hurt onsumers. At the same time, it ould enhane the untioning o an R&D ooperative. For instane, Geroski [7] argues that it is the eedbak rom produt markets that direts researh toward proitable traks and that, thereore, or an innovation to be ommerially suessul, there must be strong ties between marketing and development o new produts. And Jaquemin [8] puts orward that R&D ooperatives

3 J Optim Theory Appl (7) 74: are ragile and unstable. He reasons that when there is no ooperation in the produt market, there exists a ontinuous ear that one partner in the R&D ooperative may be strengthened in suh a way that it will beome too strong a ompetitor in the produt market. Preventing irms rom ollaborating in the produt market may thereore destabilize R&D ooperatives or prevent their ormation in the irst plae. Our ous is on private inentives to develop ost-saving tehnologies over time. In partiular, we show that i irms ollude in the produt market, a wider range o tehnologies is ully developed. We also show that irms ompeting in the produt market realize an inerior produtive eiieny. We thus identiy situations where produt market ollusion inreases total surplus. Dynami models o R&D were irst introdued to study patent raes whereby suessul innovators apture the entire market. This literature starts with Loury [9] and Lee and Wilde [] ([] surveys the early ontributions). Patent rae models examine, in essene, the time it takes or a ost-saving innovation to be ompleted. R&D investments redue this ompletion period. Beause in these models the R&D proess itsel annot ail, the R&D-investment deision is transormed into a stati one. Meanwhile, a large literature has developed on the relation between intelletual property rights and antitrust poliies. For instane, Quirmbah [] inds that there is an optimal level o ollusion that is in-between peret ompetition and ull ollusion. And Green and Sothmer [3] show that it is optimal to allow or ollusion through sequential liensing in ase the next innovation is a truly new appliation o existing patents. More reently, another strand o dynami R&D models has developed: ontinuous-time generalizations o strategi R&D models. These models allow or smoothing the investment eorts over a long time [4], a type o investment behavior that is observed in pratie and that onstitutes a key eature o ontinuoustime models. Cellini and Lambertini [4] is the irst ontinuous-time generalization o the seminal analysis o d Aspremont and Jaquemin [5]. In the duopoly game o d Aspremont and Jaquemin [5], irms irst invest in ost-reduing R&D and then play a Cournot game in the produt market. In the ontinuous-time version o Cellini and Lambertini [4], both irms start rom an initial tehnology (that is, a level o marginal ost) and invest ontinuously in R&D. This gradually redues the initial level o marginal ost toward the steady-state level. In ontrast to the stati generalization o d Aspremont and Jaquemin [5] by Hinloopen [6], Cellini and Lambertini [4] ind that the aggregate level o R&D is monotonially inreasing in the number o independent ompetitors. We also onsider a ontinuous-time generalization o d Aspremont and Jaquemin [5]. There are two distinguishing eatures o our analysis. First, we onsider all possible initial marginal ost levels, inluding those exeeding the hoke prie (the lowest prie or whih there is no demand). Espeially in the early stages o development, it is quite likely that the ost o a new tehnology (the ost, say, to develop a prototype) exeeds the highest willingness to pay in the market. We haraterize situations where suh initial tehnologies are only developed i irms ollude in the produt market. Indeed, exluding initial marginal osts that are above the hoke prie ignores new proesses [that] ailed to ome into existene, [as they are] killed by the Sherman At beore they were born. These instanes onstitute a diret welare gain o produt market ollusion.

4 57 J Optim Theory Appl (7) 74:567 6 Seond, in addition to near-equilibrium paths, we onsider all trajetories that are andidates or an optimal solution. This global analysis yields a biuration diagram that indiates or every possible parameter ombination the qualitative eatures o any market equilibrium as well as o the transient dynamis toward it. We thus identiy ritial parameter values: points in parameter spae at whih the optimal investment untion hanges qualitatively. In partiular, we determine the value o marginal osts or whih R&D investments are terminated, and or whih they are not initiated at all. We prove that these ritial ost levels are aeted by irm ondut. Thereore, extending the R&D ooperative to produt market ollusion an lead to qualitatively dierent long-run solutions, in spite o starting rom an idential initial tehnology. The related literature [4,7 3] has not onsidered initial marginal ost levels that exeed the hoke prie nor has it arried out a global analysis. In all these papers, any o the initial (permissible) tehnologies will be developed to ull materialization; tehnologies that are only developed under speii regimes (i.e., produt market ollusion) remain hidden. The only exeption is Hinloopen et al. [3], who haraterize the equilibria o a ontinuous-time dynami monopoly with R&D investments. We expand their analysis in three diretions. First, we onsider a duopoly rather than a monopoly. Seond, we examine two dierent senarios: one in whih irms ooperate in R&D and ompete in the produt market (labeled partial ollusion ), and one in whih irms ooperate both in R&D and in setting prie (labeled ull ollusion ). Indeed, omparing the two senarios allows us to examine the eets o extending ooperation in R&D toward ollusion in the produt market. And third, rather than relying on numerial simulations, we prove a set o propositions that haraterize the dynamis o the model throughout the entire parameter spae. Our ramework yields our possible outomes or any initial draw o a new tehnology (. [3]). First o all, a Promising Tehnology arrives, whereby the initial tehnology is developed through ontinuous R&D investments. This an our or initial ost levels both below and above the hoke prie. In the latter ase, prodution starts only ater some time, beause early R&D eorts have to bring down marginal ost below the hoke prie. Seond, a Strained Market arises: initial marginal ost is below the hoke prie and irms invest in R&D, but the tehnology is not likely to be developed to ull materialization. In ase o an Unertain Future, the third situation, it is not immediately lear whether the long-run steady state will be reahed, or that it is optimal to gradually leave the market. Only time will tell. Fourth, an Obsolete Tehnology an emerge: whatever the initial marginal ost, the tehnology is either not developed or developed only to be taken o the market in due time. The long-run steady state will not be reahed in either ase. All our tehnologies an emerge under both partial ollusion and ull ollusion. Comparing the two senarios throughout the entire parameter spae, we ind that i irms ollude in the produt market (i) it is more likely that an initial tehnology qualiies as a Promising Tehnology, and i so, that it is more likely to be developed urther, (ii) it is less likely that an initial tehnology qualiies as an Obsolete Tehnology, and i so, it is more likely that irms invest in R&D, albeit temporarily, and (iii) i an initial tehnology auses a Strained Market or i it indues an Unertain Future, it is less likely that it will be taken o the market in due time. Put dierently, due to produt market ollusion it is more likely that irms invest

5 J Optim Theory Appl (7) 74: in R&D, and that these investments eventually lead to a steady state with positive prodution. Our analysis qualiies the per se prohibition o ollusion in produt markets or high-teh industries. A higher total surplus obtains i olluding irms develop an initial tehnology and arrive at the saddle-point steady state while irms that ompete in the produt market would not develop the tehnology at all. We show that this is more likely to happen i new tehnologies arrive in irumstanes that oer a high proit potential (that is, large markets and eiient R&D proesses). Under these irumstanes, produt market ollusion an also yield higher total surplus i ompeting irms would develop the new tehnology as well, be it to take it o the market in due time, or to arrive at the saddle-point steady state. And in so ar, higher R&D investments as suh are desirable (as suggested in the endogenous growth literature; see, e.g., [3,33]) the ase or prohibiting ollusion per se is urther weakened. On the other hand, olluding irms tend to hold on longer to tehnologies that are destined to leave the market. This is not desirable rom a soial welare point o view i that prevents the development o new, superior tehnologies. A partiularly diiult situation arises when the initial tehnology is above the hoke prie and i it will be developed only i irms ollude in the produt market. The welare ost o prohibiting irms to ollude then remains hidden beause no prodution is aeted by this prohibition. There is no prodution yet, and beause ollusion is prohibited, there will be no prodution in the uture. Put dierently, no prodution will be taken o the market i irms are prohibited to ollude in the produt market, leaving the welare ost unnotied. Our analysis thus oers a irst glane at new proesses [that] ailed to ome into existene, killed by the Sherman At beore they were born. The remainder o the paper is organized as ollows. The basis o the model are introdued in Set.. In Set. 3, the neessary onditions or optimal prodution and investment shedules are derived under partial ollusion and ull ollusion. The orresponding biuration diagrams are derived in Set. 4 and the two senarios are ompared in Set. 5. Setion 6 onludes. Appendies ontain the proos o all propositions. The Model Our present model is an extension o the global monopoly ramework o Hinloopen et al. [3] to two irms, and it builds on Cellini and Lambertini [7]. Do note that Smrkolj and Wagener [34] show that the equilibrium onsidered in [7] is not subgame peret. Time t is ontinuous: t [, [. There are two a priori ully symmetri irms that both produe a homogeneous good at onstant marginal osts i (t). At every instant, the market prie p(t) is given as p(t) = Ā Q(t), () where Q(t) = q (t) + q (t), with q i (t) the quantity produed by irm i at time t, and where Ā is the hoke prie.

6 57 J Optim Theory Appl (7) 74:567 6 Eah irm i an redue its marginal ost i (t) by investing in R&D. In partiular, when irm i exerts R&D eort k i (t), its marginal ost evolves as d i dt (t) ċ i(t) = i (t) ( k i (t) βk j (t) + δ ), () where k j (t) is the R&D eort exerted by its rival and where β [, ] measures the degree o spillover. Note that eiieny o prodution is assumed to derease at a onstant rate, as aptured by δ >. This depreiation is due to (exogenous) aging o tehnology and organizational orgetting [8,35]. As Benkard [36, p. 59] observes: an airrat produer s stok o prodution experiene is onstantly being eroded by turnover, lay os and simple losses o proiieny at seldom repeated tasks. When produers ut bak output, this erosion an even outpae learning, ausing the stok o experiene to derease. In our model, R&D investment yields know-how gains but the logi o the argument is the same. For instane, omplementary inputs that are typially purhased also onstitute a ration o prodution ost. Inorporating these inputs beomes ever more ostly due to their inherent evolution over time, espeially or irms that are relatively sluggish in R&D, as R&D eorts also determine any irm s absorptive apaity [37]. A non-positive depreiation rate yields trivial equilibria. Every initial tehnology will be developed in ase δ is negative, as there is an exogenous redution in marginal ost over time. For δ =, onsider δ to be marginally positive. In that ase, the value o initial marginal ost that would make it optimal not to invest in R&D is ar above the hoke prie beause only an ininitesimally small investment in R&D is then needed to redue marginal ost over time. Both irms are endowed with a given idential initial tehnology i () = j () =, whih represents the state o the tehnology at the moment o the invention o the produt. Per unit o time, the osts o R&D eorts are Γ i (k i ) = bk i, (3) where b > is inversely related to the ost-eiieny o the R&D proess. The R&D proess is thus assumed to exhibit dereasing returns to sale ([38]; see also the disussion in [3]). Both irms disount the uture with the same onstant rate ρ >. Either irm s instantaneous proit thereore equals π i (q i, Q, k i, i ) = (Ā Q i )q i bk i, (4) with orresponding total disounted proit Π i (q i, Q, k i, i ) = π i (q i, Q, k i, i )e ρt dt. (5) The model has ive parameters: Ā, β, b, δ, and ρ. To simpliy the analysis, we resale the model suh that it has only three parameters. Resaling is done by hoosing

7 J Optim Theory Appl (7) 74: natural units or the problem; it does not involve making speial parameter hoies. Rather, eah hoie o parameters in the original model orresponds to a hoie o parameters in the resaled model. The omplexity redution obtained by the saling is a onsequene o the at that in the original parameters, many hoies give rise to mathematially equivalent models. In mathematial terms, we embed the given iveparameter amily o models in a six-parameter amily. We then show that the saling transormations we onsider allow us to hoose three parameter values to be equal to, eetively reduing the number o parameters to three. Lemma. The ollowing equations deine new variables t = t δ, i = Ā i, q i = Ā q i, Q = Ā Q, k i = Ā b k i, π i = Ā π i, Π i = Ā δ Π i, and new parameters φ = Ā/( δ b) and ρ = δ ρ. In the new variables, the model takes the orm: π i ( q i, Q, k i, i ) = ( Q i ) q i k i, (6) Π i ( q i, Q, k i, i ) = π i ( q i, Q, k i, i )e ρ t d t, (7) )) i = i ( φ ( k i + β k j, i () =, (8) where i, and with the ontrol restritions q i and k i. The proo o the lemma is given in Appendix A. Remark. Resaling the model as in Lemma. introdues a new parameter: φ. It is one-to-one related to the proit potential o a tehnology. Higher potential revenues ome with a higher Ā, and eah unit o R&D eort osts more i b inreases, while it redues marginal ost by less the higher is δ. In sum, a lower (higher) φ orresponds to a lower (higher) proit potential. Remark. In mathematial terms, the original model is a speimen o the sixparameter model given by ċ i = i ( δ φ(ki + βk j ) ), Π i = with parameters values (A, b,δ,β,φ,ρ)= (Ā, b, δ, β,, ρ). ( ) (A Q i )q i bki e ρt dt,

8 574 J Optim Theory Appl (7) 74:567 6 The equivalent model in new variables is an instane o the same six-parameter model, but with parameters values (A, b,δ,β,φ,ρ)= ( Ā,,, β, δ b, ρ δ ). We an, and will, without loss o generality drop the tildes rom the new variables, the bars rom the parameters, and take A =, b = and δ =. To illustrate the useulness o Lemma., onsider two models with dierent original parameterizations: (i) Ā =, b =, δ =., ρ =., β =.5, (ii) Ā =, b = 4, δ =., ρ =., β =.5. Both models orrespond to the same resaled model with φ = Ā/ δ b = 5, ρ = ρ/ δ =.5, β =.5 and are thereore mathematially equivalent in the sense that they have the same solution in resaled variables. 3 Partial Collusion and Full Collusion In this setion, we derive the neessary onditions or optimal prodution and investment shedules in ase irms ooperate in R&D but ompete in the produt market (Set. 3.), and in ase irms ooperate in R&D and ollude in the produt market (Set. 3.). 3. Partial Collusion Both irms operate their own R&D laboratory and prodution aility. They selet their output levels non-ooperatively and adopt a stritly ooperative behavior in determining their R&D eorts so as to maximize joint proits. These assumptions amount to imposing a priori the symmetry ondition k i (t) = k j (t) = k(t). i () = j () = implies that i (t) = j (t) = (t). Equation (8) thus reads as ċ = ( ( + β)φk). (9) It may seem reasonable to assume that when irms ooperate in R&D, they also ully share inormation, that is, to assume the level o spillover to be at its maximum (β = ; see [39]). For the sake o generality, we do not aprioriix the value o β at its maximal value. There are also intuitive arguments or not doing so as there might still be some ex post dupliation and/or substitutability in R&D outputs i irms operate separate laboratories (see the disussion in [4]). The instantaneous proit o irm i is π i (q i, Q, k, ) = ( Q )q i k, () with Q = q + q, yielding its total disounted proit over time

9 J Optim Theory Appl (7) 74: Π i (q i, Q, k, ) = π i (q i, Q, k, )e ρt dt. () As irms jointly deide on their R&D eorts, the only independent deisions are those o prodution. However, as quantity variables do not appear in the equation or the state variable (9), prodution eedbak strategies o a dynami game are simply stati Cournot Nash strategies o eah orresponding instantaneous game. Maximizing π i over q i gives us standard Cournot best-response untions or the produt market q i (q j ) = { ( q j ), i q j <,, i q j. () Note that the onstraint q i is binding when q j. Solving or Cournot Nash prodution levels, we obtain q N = { 3 ( ), i <,, i. (3) Consequently, the instantaneous proit o eah irm is π(, k) = { 9 ( ) k, i <, k, i. (4) We assume that irms ae no inanial onstraints; they an invest in R&D prior to prodution. Indeed, redit rationing would impose an upper limit on the value o an indierene point; qualitatively it would not hange our onlusions, however. Also, or a sample o Italian manuaturing irms, Piga and Atzeni [4] ind that redit onstraints are negligible or R&D intensive irms. And Bond et al. [4] ind no signiiant relationship between the level o R&D investments and ash low or German and UK irms, while Harho [43] inds a weak but statistially signiiant relationship or both small and large German irms. The sensitivity o R&D investments to ash low lutuations seems to be stronger or US irms (e.g., [44,45]), but by and large, the literature on the importane o inanial onstraints or R&D investment is inonlusive (see [46] or an overview). The dynami optimization problem o the R&D ooperative boils down to inding an R&D eort shedule k or either irm that maximizes the total disounted joint proit, taking into aount the state Eq. (9), the initial ondition () =, and the ontrol onstraint k(t) whih must hold at all times. Note that aording to (9), i >, then (t) > or all t. The natural state spae o this problem would be the interval ], [ o positive marginal ost levels, but or mathematial onveniene, we extend this to R by speiying that π i (q i, Q, k, ) = i <. In order to lose the model, we have to speiy the set o admissible eort shedules k(t).

10 576 J Optim Theory Appl (7) 74:567 6 Deinition 3. An R&D eort shedule is admissible i it is a bounded nonnegative measurable untion. To solve this problem, we introdue the urrent-value Pontryagin untion (also alled the un-maximized Hamilton or pre-hamilton untion), whereby we omit a ator or joint proits to obtain the solution expressed in per-irm values (due to symmetry, maximizing per-irm total proit orresponds to maximizing joint total proit) P(, k, λ) = { 9 ( ) k + λ( ( + β)φk), i <, k + λ( ( + β)φk), i, (5) where λ is the urrent-value ostate variable o a irm in the R&D ooperative. The ostate (or shadow value) measures the marginal worth o the inrement in the state or eah irm at time t when moving along the optimal path. As we expet an inrease o the marginal osts to entail lower proits or the irm, we expet the shadow value to be non-positive that is λ(t) along optimal trajetories. We use Pontryagin s maximum priniple to obtain the solution to our optimization problem. Maximizing over the ontrol k yields k = max {, } λ( + β)φ. (6) The maximum priniple states urther that the optimizing trajetory neessarily orresponds to the trajetory o the state ostate system ċ = P λ, λ = ρλ P, (7) where k is replaed by its maximizing value. For λ, relation (6) gives a oneto-one orrespondene between the ostate λ and the ontrol k. We use this relation to transorm the state-ostate system into a state-ontrol system whih an optimizing trajetory has to satisy neessarily as well. This system onsists o two regimes (ollowing the two part omposition o the Pontryagin untion). The irst one orresponds to < and positive prodution (q = ( )/3). The seond one orresponds to and zero prodution. Note that in the non-resaled model, the analogous onditions or positive and zero prodution are (t) <Ā and (t) Ā, respetively. The state-ontrol system with positive prodution onsists o the ollowing two dierential equations: ( + β)φ ċ = ( ( + β)φk), k = ρk ( ). (8) 9 The state-ontrol system with zero prodution is given by ċ = ( ( + β)φk), k = ρk. (9)

11 J Optim Theory Appl (7) 74: Full Collusion Under ull ollusion, irms determine jointly their R&D eorts and their output levels. This amounts to imposing a priori the symmetry onditions k i (t) = k j (t) = k(t) and q i (t) = q j (t) = q(t). Equation (8) reads again as Eq. (9). The proit o eah irm at every instant is π(q, k, ) = ( q )q k, () with orresponding total disounted proit Π(q, k, ) = π(q, k, )e ρt dt. () The optimal ontrol problem o the two olluding irms is to ind ontrols q and k that maximize the proit untional Π subjet to the state Eq. (9), the initial ondition () =, and two ontrol onstraints that must hold at all times: q and k. Notie, again, that aording to (9), i >, then (t) > or all t. The urrent-value Pontryagin untion in ase o ull ollusion reads as: P(, q, k,λ)= ( q ) q k + λ ( ( + β)φk), () where λ is the urrent-value ostate variable. It now measures the marginal worth at time t o an inrement in the state or a olluding irm when moving along the optimal path. The neessary onditions or the solution to the dynami optimization problem onsist again o a state-ontrol system whih has two regimes. As in the partial ollusion ase, the irst regime orresponds to < and positive prodution (q = ( )/4), while the seond regime orresponds to and zero prodution. The state-ontrol system in the region with positive prodution reads as ċ = ( ( + β)φk), k = ρk ( + β)φ ( ), (3) 8 whereas the state-ontrol system with zero prodution is ċ = ( ( + β)φk), k = ρk. (4) 4 Analysis Consider the system ċ = ( ( + β)φk), k = ρk αφ( + β)( )χ(), (5) where χ() = i< < and χ() = i (or ). Systems (8) (9) and (3) (4) are instanes o system (5), with α = /9 or the partial ollusion

12 578 J Optim Theory Appl (7) 74:567 6 senario and α = /8 or the ull ollusion senario. Indeed, the monopoly system in Hinloopen et al. [3] is also a speial ase o system (5), with α = /4. The irst result gives the properties o the steady states o the state-ontrol system (see Appendix B or the proo). Proposition 4. Let D = 4 ρ α( + β) φ. Depending on the value o D, there are three dierent situations. A. I D >, the state-ontrol system with positive prodution (3) has three steady states: i. ( e, k e ) = ((, ) is an unstable node, ii. ( e, k e ) = + D, and ( iii. ( e, k e ) = D, (+β)φ (+β)φ ) is either an unstable node or an unstable ous, ) is a saddle-point steady state. B. At D =, there are two steady states: i. ( e, k e ) = ((, ), whih is an unstable node, and ii. ( e, k e ) =, (+β)φ ), whih is a semi-stable steady state. C. I D <, the origin ( e, k e ) = (, ) is the unique steady state o the state-ontrol system with positive prodution, whih is unstable. The system onsequently exhibits a saddle-node biuration at D =. Remark 4. The stable maniold o the saddle-point steady state is one o the andidates or an optimal solution. However, as neither the Mangasarian nor the Arrow onavity onditions are satisied, the stable maniold is not neessarily optimal. Proposition 4. already implies that there should be other andidates or optimality as there is a parameter region or whih there is no saddle point and hene no stable maniold to it. The ollowing result lariies ( Appendix C ontains the proo). Proposition 4. The set o andidates or an optimal solution onsists o the stable paths W s o the saddle-point steady state and the trajetory E through the point (, k) = (, ). Proposition 4. is illustrated in Fig.. The thik blak lines W s and E indiate optimal solutions. The dotted vertial line = separates the region with zero prodution rom the region with positive prodution. We label the trajetory E the exit trajetory, as ollowing this trajetory implies that irms eventually leave the region with positive prodution. Proposition 4. only redues the set o trajetories by applying neessary onditions or optimality, but there is no guarantee that an optimal solution exists. The next proposition summarizes when an optimal solution exists. Proposition 4.3 For all admissible values o the parameters, and all initial points, the optimal ontrol problem has at least one solution, whih is among the andidates

13 J Optim Theory Appl (7) 74: Fig. Candidate maximizing trajetories W s and E in the state-ontrol spae k = q> q = k W s e + ċ = (+β)φ e E e.5.5 speiied in Proposition 4.. Moreover, there is at most one initial state ĉ suh that there are two optimizing trajetories starting at ĉ. The proo is in Appendix D. To assess the dependene o the solution struture on the model parameters, we arry out a biuration analysis. This onsists o identiying those parameter values or whih the qualitative struture o the optimal dynamis hanges. These biurating values bound open parameter regions suh that the optimal dynamis are qualitatively idential or all parameter values in a region (see [47,48]). Put dierently, or all points in a region, a suiiently small hange in parameter values will not lead to a qualitative hange o the optimal dynamis; regions haraterize stable types o dynamis. System (5) has our distint stable dynamis types (. [3]). These are illustrated in Fig. in ase o partial ollusion. Note that the same types emerge under ull ollusion (the stable dynamis types are ompared aross senarios in Set. 5). The irst type is a Promising Tehnology. In this ase, there exists an initial tehnology ĉ > that is an indierene threshold: a point in state spae where the deision maker is indierent between two optimal trajetories that have distint long-term limit behavior. In partiular, or < ĉ it is optimal to start developing the initial tehnology, ending up in the saddle-point steady state in the region with positive prodution. I < ĉ, initially irms invest only in R&D; prodution begins whenever (t) <. I ĉ, it is optimal not to initiate R&D eorts; potential uture proits do not suie to ompensate or losses that would be inurred in the initial periods during whih irms would invest in R&D but would not produe yet (note that or =ĉ, there are two distint R&D investment trajetories, whih are, nevertheless, both optimal; see also Proposition 4.3). The seond type orresponds to a Strained Market, where there is an indierene threshold below the hoke prie (that is, in the region with positive prodution): < ĉ <. In this ase, i < < ĉ, the initial tehnology will be developed toward the saddle-point steady state. I ĉ < <, the exit trajetory applies; R&D investments only serve to slow down the tehnologial deay. In a small part o the parameter spae, the third type arises: an Unertain Future. Initial tehnologies (states) are now divided by a repelling steady state (rather than an indierene point). I the system starts exatly at the repelling point R, it stays there

14 58 J Optim Theory Appl (7) 74: e - e + k.5. e - k (a) (b).8 e - e k.4. k R () Fig. R&D investment trajetories or the our stable dynamis types o system (5) with α = /9 (partial ollusion). Parameters: a (β, ρ, φ) = (,.5, 4), b (β, ρ, φ) = (,, 3.5), (β, ρ, φ) = (, 4, 6.), d (β, ρ, φ) = (,,.5). a Promising Tehnology, b Strained Market, Unertain Future, d Obsolete Tehnology (d) indeinitely; when it starts lose to it, it stays there or a long period o time, ater whih it onverges to one o the two attrators: the steady state or the exit trajetory. The ourth type typiies the dynamis o an Obsolete Tehnology. Whatever the initial tehnology, (eventually) the irms leave the market; R&D investments are only used to slow down the tehnial deay. The our dierent dynamis types are grouped onveniently in a biuration diagram (see Fig. 3): the graph that indiates or every possible parameter ombination the qualitative eatures o any market equilibrium as well as the transient dynamis toward them. In Fig. 3, the uppermost urve represents parameter values or whih the indierene point is exatly at =. At the saddle-node urve (SN), an optimal repeller and an optimal attrator ollide and disappear. The urve SN orresponds to saddle-node biurations in the state-ontrol system that do not orrespond to optimal dynamis. At the indierene-attrator biurations (IA), an indierene point ollides with an optimal attrator and both disappear. Finally, at an indierene-repeller biuration (IR), an indierene point turns into an optimal repeller. The entral indierenesaddle-node (ISN) biuration point at (ρ, φ( + β)) (.4, 8.78) organizes the biuration diagram. The urve representing indierene points at = obtains a value o φ( + β).998 or ρ = 5.

15 J Optim Theory Appl (7) 74: I Promising Tehnology II Strained Market φ( + β ) 5 ISN IR SN III Unertain Future 5 IA SN IV Obsolete Tehnology Fig. 3 Biuration diagram (partial ollusion) ρ 5 Collusion and the Inentives to Innovate In this setion, we ompare the global optimum o the two senarios. For a welare omparison, we introdue total disounted values o proits (Π), onsumer surplus (CS), and total surplus (TS): Π = π(t)e ρt dt, (6) CS = ( p(t))q(t)e ρt dt = q(t) e ρt dt, (7) TS = Π + CS, (8) where at time t = irms start with and then invest along the optimal trajetory γ(t) = ((t), k(t)) as t. We irst ormally establish that the two senarios yield dierent (optimal) trajetories.in Fig. 4, the biuration diagrams o both senarios are superimposed. There are signiiant quantitative dierenes between the two diagrams, as releted by the dierent loations o the urves that divide the parameter spae. Let I i, II i,...,i =, denote regions I, I I,... under senario i, with i = () orresponding to partial (ull) ollusion. The ollowing then holds (see Appendix E or the proo). Proposition 5. The ollowing inlusions hold: I I, I II I II, I II III I II III. The irst inlusion o Proposition 5. implies that the Promising Tehnology region is larger i irms ollude in the produt market; due to ollusion, the situation

16 58 J Optim Theory Appl (7) 74: I II φ( + β ) 5 SN IR III ISN 5 IA IV SN Fig. 4 Superimposed biuration diagrams. Partial ollusion urves are gray, ull ollusion ones are blak ρ where irms irst invest in R&D, and only ater some initial development period start produing, is more likely to our. From the third inlusion ollows that the Obsolete Tehnology region is smaller i irms ollude; irms that ollude are less likely either not to develop an initial tehnology or to invest in R&D only to abandon the tehnology in time. 5. R&D Investment Inentives In line with muh o the related literature [9], Proposition 5. suggests that olluding irms have in general a stronger inentive to invest in R&D. This turns out to be the ase, as the next proposition ormally shows (see Appendix F. or the proo). Proposition 5. Investment in R&D in the ull ollusion senario is always at least as high as in the orresponding partial ollusion senario. Proposition 5. implies the ollowing. First o all, whenever both senarios lead to the saddle-point steady state, marginal osts in the ull ollusion senario are lower than in ase o partial ollusion, beause ully olluding irms have invested more in ost-reduing R&D to arrive at the long-run equilibrium. Put dierently, produt market ollusion yields a higher prodution eiieny. Seond, i the initial tehnology leads to prodution ater some initial development period only, olluding irms will enter this prodution phase more quikly beause at every instant o the pre-prodution phase they invest more in R&D in order to bring the level o marginal osts below the hoke prie.

17 J Optim Theory Appl (7) 74: k CS (a) () Π (b) Fig. 5 State-ontrol spae (a), total disounted proit (b), onsumer surplus (), and total surplus (d), when the exit trajetory is an optimal solution. Parameters: (β, ρ, φ) = (,, ). Curves o the partial (ull) ollusion senario are gray (blak) TS (d) Third, irms that ollude in the produt market abandon obsolete tehnologies at a lower pae. This impliation has a similar vein as the argument o Arrow [49], that a monopolist has less inentive to invest in R&D than an otherwise idential but peretly ompetitive market, beause by doing so the monopolist replaes urrent monopoly proits by uture (higher) monopoly proits. Here, the alternative or olluding irms is to exit the market more quikly (rather than staying in the market as a monopolist, as in Arrow [49]), an alternative that or them is not optimal (see Fig. 5). The dierene in R&D intensity aross the two senarios is also releted in the type o trajetories that irms selet. To haraterize this dierene or all possible situations, it is onvenient to have deined the threshold level o initial marginal ost ĉ between eventual exit and eventual positive prodution. Formally, set ĉ = in the Obsolete Tehnology region and let ĉ and ĉ denote the threshold level or the partial ollusion and the ull ollusion senarios, respetively. We an then state the ollowing (see Appendix F. or the proo). Proposition 5.3 For all parameter values, either ĉ < ĉ or ĉ =ĉ =. The impliations o Proposition 5.3 are twoold. First, i irms ollude in the produt market, the set o initial tehnologies that are developed toward the saddle-point steady state is larger (see Fig. 6). In partiular, i the initial tehnology alls in the nonempty interval ]ĉ, ĉ [, it will only be brought to ull materialization i irms ollude in the produt market.

18 584 J Optim Theory Appl (7) 74: (a).9 (b) k.5 Π ĉ ĉ.5 () ĉ ĉ (d) CS TS ĉ ĉ ĉ ĉ Fig. 6 State-ontrol spae (a), total disounted proit (b), onsumer surplus (), and total surplus (d), when the indierene point is in the region with zero prodution. Parameters: (β, ρ, φ) = (,.,.5).Curves o the partial (ull) ollusion senario are gray (blak) Seond, the set o initial tehnologies that triggers no investment in R&D at all or that indues irms to selet the exit trajetory is smaller i irms ollude in the produt market. Figure 7 illustrates this or a Strained Market. The strained investment irumstanes indue partially olluding irms to exit the market in due time or all > ĉ. In ontrast, ully olluding irms exit the market only or > ĉ. Initial tehnologies in the interval ]ĉ, ĉ [ are thereore only brought to ull maturation by irms that ollude in the produt market. We an onlude that due to ollusion in the produt market (i) it is more likely that an initial tehnology qualiies as a Promising Tehnology, and i so, that it is more likely to be developed urther, (ii) it is less likely that an initial tehnology qualiies as an Obsolete Tehnology, and i so, it is more likely that irms invest in R&D, albeit temporarily, and (iii) i an initial tehnology auses a Strained Market or i it indues an Unertain Future, it is less likely that it will be taken o the market in due time. Put dierently, due to produt market ollusion it is more likely that irms invest in R&D, and that these investments eventually lead to a steady state with positive prodution. 5. Total Surplus We next onsider the eet o produt market ollusion on total surplus. Obviously, ollusion in the produt market yields higher total surplus i olluding irms develop an

19 J Optim Theory Appl (7) 74: (a). (b) k.5. Π CS ĉ ĉ () TS (d) Fig. 7 State-ontrol spae (a), total disounted proit (b), onsumer surplus (), and total surplus (d), when the indierene point is within the region with positive prodution. Parameters: (β, ρ, φ) = (,., ). Curves o the partial (ull) ollusion senario are gray (blak); urves o the stable path (exit trajetory) are solid (dotted). Dots indiate the saddle-point steady state initial tehnology and arrive at the saddle-point steady state while irms that ompete in the produt market would not develop the tehnology at all. Formally: Proposition 5.4 Whenever both senarios have an indierene point above the hoke prie, the ull ollusion senario yields higher onsumer surplus and total surplus than the partial ollusion senario or all initial tehnologies in-between the two indierene points. Proo The proo ollows trivially rom the at that i) or all values o above the indierene point in the region where, both q = and k = or all t [, [, and ii) or all values o below the indierene point, Π>and or some inite time T also q > or all t > T as t. Figure 6 illustrates Proposition 5.4: or all ]ĉ, ĉ [, ollusion in the produt market yields a higher total surplus. Figure 8 illustrates some omparative statis o the indierene points in this ase. Indeed, these points are positively related to market size and R&D eiieny. Note, however, that also ĉ =ĉ ĉ inreases i the R&D proess beomes more eiient and/or i the market size beomes larger, the more so the lower is the disount rate (in Fig. 8, a lower disount rate orresponds to a larger slope o the onvex urves). Beause uture mark-ups are positively related to both market size and R&D eiieny, an inrease in either o these two has a larger

20 586 J Optim Theory Appl (7) 74: ρ =. 8 ρ = ĉ 5 Δ ĉ ρ = 4 C 3 Δ ĉ C φ( +β ) Fig. 8 Dependene o the indierene point ĉ on model parameters. Curves o the partial (ull) ollusion senario are dotted (solid) (positive) eet on uture proits i irms ollude in the produt market. And these uture beneits eature more prominently in total disounted proits i the disount rate is lower. Thereore, indierene points orrespond to lower marginal ost values i the disount rate goes up, all else equal (. the relative loation o C and C in Fig. 8). Produt market ollusion an also yield higher total surplus i olluding irms arrive at the saddle-point steady state while irms that ompete in the produt market would selet the exit trajetory. In these ases, irms that ompete in the produt market temporarily produe more. This is oset by the added beneits o sustained R&D investments under ull ollusion i the disount rate is suiiently small (see Fig. 7). Finally, ollusion in the produt market an also yield a higher total surplus i under both senarios irms would selet the trajetory toward the saddle-point steady state: in Fig. 9, or all ], ĉ [, total surplus is higher i irms ollude in the produt market. In this example, the disount rate is high: ρ =, whih orresponds, or instane, to the non-resaled variables δ =. and ρ =.. Also, the initial marginal osts are suiiently high. In suh an environment, the higher R&D investments and the redued importane that is attahed to uture surplus work in avor o produt market ollusion as under this senario irms will reah the prodution stage more quikly, a beneit that more than osets the welare loss o uture inreased mark-ups. That is, a higher (resaled) disount rate ρ = ρ/ δ implies either a higher disount rate ρ or a lower δ. With a lower δ, ost redutions take longer, suh that the time dierene in reahing the prodution stage between the two senarios beomes more pronouned.

21 J Optim Theory Appl (7) 74: (a).35.3 TS x 3 (b) 3.5 TS ĉ ĉ Fig. 9 Total surplus when the indierene point is in the region with zero prodution. Parameters: (β, ρ, φ) = (,, 5). Gray urves orrespond to partial ollusion, whereas the blak ones orrespond to ull ollusion. 3.6, ĉ 4., ĉ For all ], ĉ [, total surplus is higher i irms ollude in the produt market 6 Conlusions Shumpeter s amous observation ontinues to hallenge the design o optimal ompetition poliies or high-teh setors. The lassi rationale or ompetition poliies is rooted in their eet on total surplus. Typially, produt market ollusion transers onsumer surplus to irm proits, resulting in a net loss o total surplus. To date, the

22 588 J Optim Theory Appl (7) 74:567 6 literature onsiders this result to be robust to the inreased inentive to invest in R&D that omes with ollusion in the produt market. Our analysis shows that it atually ails this robustness hek i the phase o development prior to prodution is taken into aount and/or i all possible R&D investment trajetories are onsidered. Aording to our analysis, extending an R&D ooperative agreement to ollusion in the produt market is welare enhaning i the market size is large and/or the R&D proess is eiient, given a relatively modest disount rate. The proit potential o a new tehnology is then relatively large. As a result, irms that ollude in the produt market bring more initial tehnologies to ull materialization. A partiularly disturbing situation arises when the initial draw out o ]ĉ, ĉ [ is above the hoke prie ( > ). The welare ost o prohibiting irms to ollude in the produt market then remains hidden beause no prodution is aeted by this prohibition. There is no prodution yet, and beause ollusion is prohibited, there will be no prodution in the uture. Put dierently, no prodution will be taken o the market i irms are prohibited to ollude in the produt market. Our analysis thus signals a potential problem or antitrust poliy as it shows that prohibiting ollusion in the produt market per se is not univoally welare enhaning. It also shows that the assoiated welare osts might not surae. Further researh is needed to substantiate our qualiiation o prohibiting ollusion per se, inluding the development o riher models that allow or learning by doing, stohasti R&D, and asymmetries between irms. Aknowledgements Thanks are due to Bernd Ebersberger, Morten Hviid, Maurizio Iaopetta, Corinne Langinier, Brue Lyons, Stephen Martin, Jo Seldeslahts, Nan Yang, and to seminar partiipants at the Tinbergen Institute (Amsterdam, June ), at EARIE (Stokholm, September ), at CeNDEF, University o Amsterdam (Amsterdam, April ), at the Centre or Competition Poliy at the University o East Anglia (Norwih, April ), at the th Viennese Workshop on Optimal Control, Dynami Games and Nonlinear Dynamis (Vienna, June ), at the 4th International Shumpeter Soiety Conerene (Brisbane, July ), at the Netherlands Eonomists Day (Amsterdam, Otober ), at OFCE, Skema Business Shool (Sophia Antipolis, May 3), at IIOC 3 (Boston, May 3), and at CEA 6 (Ottawa, June 6) or many valuable omments and disussions. The onstrutive omments o the editor and two anonymous reerees are also grateully aknowledged. Open Aess This artile is distributed under the terms o the Creative Commons Attribution 4. International Liense ( whih permits unrestrited use, distribution, and reprodution in any medium, provided you give appropriate redit to the original author(s) and the soure, provide a link to the Creative Commons liense, and indiate i hanges were made. Appendix A: Proo o Lemma. We shall reer to the original variables t, q i,... as the old variables, and to the variables t, q i,...as the new variables. In the new variables, the let- and right-hand sides o equation () take the orm d i dt = Ā δ d i d t, i( k i βk j + δ) = Ā i ( Ā k i β b Ā b ) k j + δ.

23 J Optim Theory Appl (7) 74: Equation () then simpliies to d i d t ( = i Ā ) ( k ) ( )) δ i + β k j = i φ ( k i + β k j, b as laimed in the lemma. Writing the total disounted proit in the new variables yields Π i = = = Ā δ ( ) (Ā Q i )q i bki e ρt dt ( ( (Ā Ā Q Ā i )Ā q i b Ā b ( ( Q i ) q i k i ) k i ) ) e ρ t d t = Ā δ e δ ρ( t/ δ) d t δ Π i. Appendix B: Proo o Proposition 4. Seond resaling o the problem Reall the dynami optimization problem: to maximize ( Π = α( ) χ() k ) e ρt dt, subjet to the dynami restrition ċ = ( φ( + β)k). This problem is rewritten by introduing onstants K = φ( + β) and μ = αφ ( + β), (9) 4ρ and the variable u = k/k. It is then seen to be equivalent to the problem to maximize V = Π K = subjet to the dynami restrition ( 4ρμ( ) χ() u ) e ρt dt, (3) ċ = ( u) (3) and the ontrol restrition u. The Pontryagin untion o this problem is P = 4ρμ( ) χ() u + λ( u),

24 59 J Optim Theory Appl (7) 74:567 6 whih is maximized at u = max {, } λ. (3) This yields the Hamilton untion H = 4ρμ( ) χ() + λ + { (λ) 4, i λ,, i λ>. (33) I λ, the assoiated state-ostate equations read as ċ = H λ = λ +, λ = ρλ H = ρλ + 8ρμ( )χ() λ λ, (34) whereas i λ>, they simpliy to ċ =, λ = (ρ )λ + 8ρμ( )χ(). (35) For λ, relation (3) deines a variable transormation that puts the system into state-ontrol orm ċ = F (, u) = ( u), u = F (, u) = ρ (u 4μ( )χ()). (36) Note that this system is only valid or u, as or λ>, the relation between u and λ ails to be one-to-one. For later use, we note that in (, u) variables, the Hamilton untion takes the orm H ontrol (, u) = 4ρμ( ) χ() + u u. (37) B. Steady States To determine the steady states o the state-ontrol system (36), we solve the equations ċ =, u =. It is immediate that this system has no solutions i >. I, the equation ċ = is satisied i = oru =. Substitution into u = o the ormer yields the steady state (, u) = (, ). Substitution o the latter leads to the quadrati equation + /(4μ) =, whih an be written as ( ) D = with D = 4 ( ). (38) μ Note that D < 4, as all parameters are assumed to have positive values. For D >, the quadrati equation has two real solutions ± = ± D = ± /μ,