Citation for published version (APA): Smrkolj, G. (2013). Dynamic models of research and development Amsterdam: Rozenberg

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1 UvA-DARE (Digital Aademi Repository) Dynami models of researh and development Smrkolj, G. Link to publiation Citation for published version (APA): Smrkolj, G. (23). Dynami models of researh and development Amsterdam: Rozenberg General rights It is not permitted to download or to forward/distribute the text or part of it without the onsent of the author(s) and/or opyright holder(s), other than for stritly personal, individual use, unless the work is under an open ontent liense (like Creative Commons). Dislaimer/Complaints regulations If you believe that digital publiation of ertain material infringes any of your rights or (privay) interests, please let the Library know, stating your reasons. In ase of a legitimate omplaint, the Library will make the material inaessible and/or remove it from the website. Please Ask the Library: or a letter to: Library of the University of Amsterdam, Seretariat, Singel 425, 2 WP Amsterdam, The Netherlands. You will be ontated as soon as possible. UvA-DARE is a servie provided by the library of the University of Amsterdam ( Download date: 3 Jan 29

2 Chapter 3 Cartels and Innovation 3. Introdution There are ompelling reasons for rival firms to set up R&D ooperatives. These organizations, jointly ontrolled by at least two partiipating entities, whose primary purpose is to engage in ooperative R&D (Caloghirou et al., 23) allow risks to be spread, seure better aess to finanial markets, and pool resoures suh that eonomies of sale and sope in both researh and development are better realized. In the words of John Kenneth Galbraith (952, pp , emphasis added): Most of the heap and simple innovations have, to put it bluntly and unpersuasively, been made. Not only is development now sophistiated and ostly but it must be on a suffiient sale so that suess and failures will in some measure average out. Moreover, R&D ooperatives internalize tehnologial spillovers - the free flow of knowledge from the knowledge reator to its ompetitors. Sustaining R&D ooperatives is thus pereived to diminish the failure of the market for R&D. 2 However, as Sherer (98) observes: the most egregious prie fixing shemes in Amerian history were brought about by R&D ooperatives, an observation that onfirms a widely-aired suspiion (see, e.g., Pfeffer and Nowak (976), Grossman and Shapiro (986), Bloom et al. (27) estimate that a % inrease in a ompetitor s R&D is assoiated with up to a 2.4% inrease in a firm s own market value. Not surprisingly, internalizing tehnologial spillovers is one of the prime reasons for firms to join an R&D ooperative (Hernan et al., 23; see also Roeller et al., 27). 2 This motivates in partiular why independent firms are allowed to ooperate in R&D. See Martin (997) for an overview of the poliy treatment of R&D ooperatives in the E.U., the U.S., and Japan.

3 and Brodley, 99). 3 The hannels through whih ooperation in R&D an failitate produt market ollusion have been examined in a number of theoretial studies (see, e.g., Martin (995), Greenlee and Cassiman (999), Cabral (2), Lambertini et al. (22) and Miyagiwa, 29). As Martin (995) puts it: ommon assets reate ommon interests, and ommon interests make it more likely that firms will non-ooperatively refrain from rivalrous behavior. (Martin, 995, p. 74). 4 While prie fixing may lead to a redution of standard surplus measures, in this hapter we hallenge the view that extending ooperative behavior to the produt market neessarily diminishes onsumer surplus and total surplus. Geroski (992) argues that it is the feedbak from produt markets that direts researh towards profitable traks and that, therefore, for an innovation to be ommerially suessful there must be strong ties between marketing and development of new produts. Jaquemin (988) observes that R&D ooperatives are fragile and unstable. He reasons that when there is no ooperation in the produt market, there exists a ontinuous fear 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 firms from ollaborating in the produt market may therefore destabilize R&D ooperatives, or prevent their reation in the first plae. Our fous is on the inentives to develop further an initial tehnology ( ideas ). In general, we find that produt market ollusion fosters R&D investment inentives beause more of the ensuing eonomi rents an be appropriated by the investing firms. As a result, if firms ollude, they will bring more initial tehnologies to full maturation. And this is unambiguously welfare enhaning. Stati models of R&D predit total surplus to go down if members of an R&D ooperative 3 Goeree and Helland (28) find that in the U.S. the probability that firms join an R&D ooperative has gone down due to a revision of antitrust lenieny poliy in 993. This revision is pereived as making ollusion less attrative. Goeree and Helland (28) onlude that Our results are onsistent with RJVs [researh joint ventures] serving, at least in part, a ollusive funtion. Related evidene is reported by Duso et al. (2). They find that the ombined market share delines if partners in an RJV ompete on the same produt market ( horizontal RJVs ), while it inreases if members of the RJV are not diret rivals ( vertial RJVs ). The laboratory experiments of Suetens (28) show diretly that members of an RJV are more likely to ollude on prie. 4 In a similar vein, Fisher (99, p. 94) onludes that...[firms] ooperating in R&D will tend to talk about other forms of ooperation. Furthermore, in learning how other firms reat and adjust in living with eah other, eah ooperating firm will get better at oordination. Hene, ompetition in the produt market is likely to be harmed. 54

4 ollude in the produt market. 5 But a stati view of the world neessarily ignores an important aspet of R&D: time. It takes time for an initial idea to be developed towards a marketable produt; ontinuous proess innovations gradually redue prodution osts (Utterbak, 994). In this hapter, therefore, we develop a dynami model of R&D to examine the welfare impliations of produt market ollusion by firms of an R&D ooperative. This analysis builds upon our disussion in Chapter 2, where we developed a global framework for an innovating monopolist. Stati models of R&D also predit that the marginal benefit of any R&D investment inreases if firms ollude in the produt market. That is, firms are willing to spend more resoures on R&D if the intensity of produt market ompetition is diminished through some ollusive agreement. 6 This suggests that any initial idea (that is, any initial level of marginal osts) is more likely to be developed further if firms ollude in the produt market. Therefore, in a formal analysis, no level of initial marginal osts should be exluded from the analysis, in partiular marginal osts that exeed the hoke prie (that is, the lowest prie at whih the quantity sold is zero). Moreover, requiring marginal osts to be below the hoke prie at all times impliitly imposes R&D ativity and prodution to oexist at all times. Surely this assumption is quite unlikely to hold for new tehnologies at their early stages of development. Researh starts long before a prototype sees the light; development begins long before the launh of a new produt. To properly assess the welfare impliations of produt market ollusion indued by an R&D ooperative, this development phase should be inluded in the analysis. Therefore, a distinguishing feature of our approah is that we provide a global analysis. That is, we onsider all possible values of initial marginal osts, inluding those above the 5 d Aspremont and Jaquemin (988) are the first to show that a senario where firms ooperate in R&D and ollude in the ensuing produt market yields a lower total surplus than the situation where firms ooperate in R&D only. 6 Again, d Aspremont and Jaquemin (988) are the first to show this formally. This touhes upon the debate between Shumpeter Mark I (...new ombinations are, as a rule, embodied, as it were, in new firms whih generally do not arise out of the old ones but start produing beside them;...in general it is not the owner of stage-oahes who builds railways ; Shumpeter, 934, p. 66) and Shumpeter Mark II ( As soon as we go into the details and inquire into the individual items in whih progress was most onspiuous, the trail leads not to the doors of those firms that work under onditions of omparatively free ompetition but preisely to the doors of the large onerns...and a shoking suspiion dawns upon us that big business may have had more to do with reating that standard of living than with keeping it down ; Shumpeter, 943, p. 82). 55

5 hoke prie. Hene, we allow researh efforts to preede prodution. 7 Also, we do not limit ourselves to an analysis of equilibrium paths but we onsider all trajetories that are andidates for an optimal solution. This enables us to determine the loation of ritial points - points at whih the optimal investment funtion qualitatively hanges. In partiular, we determine the value of marginal osts for whih R&D investments are terminated, and for whih they are not initiated at all. The size of these ritial ost levels is affeted by firm ondut. Extending the R&D ooperative agreement to produt market ollusion an lead to qualitatively different long-run solutions, despite starting from an idential initial tehnology. For a global analysis we have to use bifuration theory. This gives us a bifuration diagram that indiates for every possible parameter ombination the qualitative features of any market equilibrium. Like in the monopoly ase disussed in Chapter 2, it yields four distint possibilities: (i) initial marginal osts are above the hoke prie and the R&D proess is initiated; after some time prodution starts and marginal osts ontinue to fall with subsequent R&D investments; (ii) initial marginal osts are above the hoke prie and the R&D proess is not initiated, yielding no ativity at all; (iii) initial marginal osts are below the hoke prie and the R&D proess is initiated; prodution starts immediately, marginal osts ontinue to fall over time, and the steady-state is reahed that is haraterized by ontinuous R&D investments, and (iv) initial marginal osts are below the hoke prie and the initiated R&D proess is progressively saled down; prodution starts immediately but the tehnology (and prodution) will die out over time; the firms leave the market. To date, the literature has onsidered possibility (iii) only, and only partially so. We then ompare two different senarios aross these possibilities. In the first senario, labeled ompetition, firms ooperate in R&D and ompete on the onomitant produt market. In the seond senario, labeled ollusion, ooperation in R&D is extended to ollusion in the produt market. We then ompare the qualitative properties of these two senarios in order to assess the potential set-bak of R&D ooperatives in that they an serve 7 Here we deviate from the related literature that, with no exeption, restrits the analysis to initial levels of marginal osts that are below the hoke prie (f. Petit and Tolwinski (999), Cellini and Lambertini (29), Lambertini and Mantovani (29), and Kova et al. (2)). As will beome lear below, this restrition exludes a ruial part of the parameter spae. 56

6 as a platform to oordinate pries. Our analysis yields three key findings: (i) if firms ollude, the range of initial marginal osts that leads to the reation of a new market is larger, (ii) ollusion in the produt market aelerates the speed with whih new tehnologies enter the produt market, and (iii) the set of initial marginal osts that indues firms to abandon the tehnology in time is larger if firms do not ollude in the produt market. Related, we show that there are parameter onfigurations whereby ollusion in the produt market yields higher total surplus. We thus qualify the onlusion of Petit and Tolwinski (999, p. 26) that [ollusion] is soially inferior to other forms of industrial strutures, a onlusion that is based on a loal analysis and whih an be seen as representative for the related literature. Our results are not without poliy impliations. When designing antitrust poliies, it is important to understand that these poliies not only affet urrent markets, but also markets that are not yet visible. Preventing firms from olluding in the produt market redues the number of potential R&D trajetories that suessfully lead to new markets. In itself this onstitutes a welfare loss. However, beause not developing further an initial tehnology does not surfae as a diret surplus loss, this welfare loss remains hidden. It is left for future researh to assess empirially the size of this hidden ost. 3.2 The model Time t is ontinuous: t [, ). There are two a priori fully symmetri firms whih both produe a homogenous good at onstant marginal osts. In every instant, market demand is: p(t) = A Q(t), (3.) where Q(t) = q (t) + q 2 (t), with q i (t) the quantity produed by firm i at time t, and where p(t) and A are respetively the market prie at time t and the hoke prie. Eah firm an redue its marginal ost by investing in R&D. In partiular, firm i exerts R&D effort k i (t) and as a onsequene of these investments, its marginal ost evolves over 57

7 time as follows: d i dt (t) ċ i(t) = i (t) ( k i (t) βk j (t) + δ), (3.2) where k j (t) is the R&D effort exerted by its rival and where β [, ] measures the degree of spillover. The parameter δ > is the onstant rate of derease in effiieny due to the ageing of tehnology and organizational forgetting. 8 Both firms have an idential initial tehnology i () = j () =, whih is drawn by Nature. The ost of R&D efforts per unit of time Γ i (k i (t)) takes the form Γ i (k i ) = bk 2 i, (3.3) where b > is inversely related to the ost-effiieny of the R&D proess. Hene, the R&D proess exhibits dereasing returns to sale (Shwartzman (976); see also the disussion in Chapter 2). Both firms disount the future with the same onstant rate ρ >. Either firms instantaneous profit therefore equals: 9 π i (q i, Q, k i, i ) = (A Q i )q i bk 2 i, (3.4) yielding total disounted profit: Π i (q i, Q, k i, i ) = π i (q i, Q, k i, i )e ρt dt. (3.5) The model has five parameters: A, β, b, δ, and ρ. The analysis an be simplified by onsidering a resaled version of the model whih, as defined in Lemma 9, arries only three parameters: β, φ, and ρ (see Appendix 2.A in Chapter 2 for the proof). Lemma 9. By hoosing the units of t, q i, q j, i, j, k i, and k j appropriately, we an assume 8 Assuming an exogenous depreiation rate is ommon in the literature. See footnote 5 in Chapter 2 for a disussion. 9 Impliitly we assume here that firms know market demand in advane of prodution. Relaxing this assumption would make the analysis more omplex, while it would not alter any of our onlusions as to the omparison of the ompetitive and ollusive senario, provided that under both senarios firms have idential expetations about future demand. 58

8 A =, b =, and δ =. This yields the following resaled version of the model: π i ( q i, Q, k i, i ) = ( Q i ) q i k 2 i, (3.6) i = i ( Π i ( q i, Q, k i, i ) = π i ( q i, Q, k i, i )e ρ t d t (3.7) ) ) ( ki + β k j φ, i () =, i [, ) t [, ) (3.8) q i, ki (3.9) ρ >, φ > (3.) with onversion rules: q i = A q i, q j = A q j, k i = A b ki, k j = A b kj, i = A i, j = A j, π i = A 2 π i, π j = A 2 π j, φ = A δ b, t = t δ, ρ = ρ δ. The resaled version of the model introdues a new parameter φ, whih aptures the profit potential of a tehnology in hand: a higher (lower) A implies higher (lower) potential sales revenue, a higher (lower) b implies that eah unit of R&D effort osts the firm more (less), whereas a higher (lower) δ implies that eah unit of R&D effort redues the marginal ost by less (more). Therefore, a higher (lower) φ orresponds to a higher (lower) profit potential of a tehnology. For notational onveniene, we heneforth omit tildes. 3.3 Competition versus Collusion This setion derives the neessary onditions for optimal prodution and investment shedules in ase firms ooperate in R&D, but ompete in the produt market (a senario labelled ompetition ), and in ase firms ooperate in R&D and ollude in the produt market (a senario labelled ollusion ) Competition Both firms operate their own R&D laboratory and prodution faility, and while they selet their output levels non-ooperatively, they adopt a stritly ooperative behavior in determining 59

9 their R&D efforts so as to maximize joint profits. These assumptions amount to imposing a priori the symmetry ondition k i (t) = k j (t) = k(t). As i () = j () =, this implies that i (t) = j (t) = (t). Equation (3.8) thus reads as: ċ = ( ( + β)φk). (3.) It may seem reasonable to assume that when firms ooperate in R&D, they also fully share information, that is, to assume the level of spillover to be at its maximum (β = ; see Kamien et al., 992). For the sake of generality, we do not a priori fix the value of β at its maximal value. There are also intuitive arguments for not doing so as there might still be some ex post dupliation and/or substitutability in R&D outputs if firms operate separate laboratories (see also the disussion in Hinloopen, 23). The instantaneous profit of firm i is π i (q i, Q, k, ) = ( Q )q i k 2, (3.2) with Q = q + q 2, yielding its total disounted profit over time: Π i (q i, Q, k, ) = π i (q i, Q, k, )e ρt dt. (3.3) As firms ooperatively deide on their R&D efforts, the only independent deisions are those of prodution levels. However, as quantity variables do not appear in the equation for the state variable (3.), prodution feedbak strategies of a dynami game are simply stati Cournot-Nash strategies of eah orresponding instantaneous game. Maximising π i over q i gives us standard Cournot best-response funtions for the produt market: q i (q j ) = ( q 2 j) if q j <, if q j. (3.4) We onsider symmetri equilibria only. See Salant and Shaffer (998) for a speifi example of a stati model of R&D in whih it is optimal for firms in an R&D ooperative to make unequal investments. 6

10 This expresses the fat that the onstraint q i is binding when q j. Solving for Cournot-Nash prodution levels, we obtain q N = ( ) if <, 3 if. (3.5) Consequently, the instantaneous profit of eah firm is π(, k) = ( 9 )2 k 2 if <, k 2 if. (3.6) The dynami optimization problem of the R&D ooperative redues to finding an R&D effort shedule k for eah firm that maximizes the total disounted joint profit of the two firms, taking into aount the state equation (3.), the initial ondition () =, and the boundary ondition k(t) whih must hold at all times. Note that aording to (3.), if >, then (t) > for all t. The state spae of this problem is the interval [, ) of marginal ost levels. To solve this problem, we introdue the urrent-value Pontryagin funtion (also alled pre-hamilton or un-maximized Hamilton funtion) P (, k, λ) = ( 9 )2 k 2 + λ( ( + β)φk) if <, k 2 + λ( ( + β)φk) if, (3.7) where λ is the urrent-value o-state variable of a firm in the R&D ooperative. The o-state, or shadow value, measures the marginal worth of the inrement in the state for eah firm in the artel at time t when moving along the optimal path. As marginal ost is a bad, we expet λ(t) along optimal trajetories. We use Pontryagin s maximum priniple to obtain the solution to our optimization problem. We omit a fator of 2 for ombined profits to obtain the solution expressed in per-firm values. Due to symmetry, maximizing the per-firm total profit orresponds to maximizing the two firms ombined total profit. 6

11 Maximising over the ontrol k yields k = λ( + β)φ, (3.8) 2 whenever the value on the right hand side is nonnegative, and k = otherwise. The maximum priniple states further that the optimizing trajetory neessarily orresponds to the trajetory of the state-ostate system ċ = P λ, λ = ρλ P, where k is replaed by its maximizing value. For λ, relation (3.8) gives a one-to-one orrespondene between the o-state λ and the ontrol k. We use this relation to transform the state-ostate system into a state-ontrol system whih an optimizing trajetory has to satisfy neessarily as well. This system onsists of two regimes (following the two part omposition of the Pontryagin funtion). The first one orresponds to < and positive prodution (q = ( )/3). The seond one orresponds to and zero prodution. 2 The state-ontrol system with positive prodution onsists of the following two differential equations: 3 k = ρk (+β)φ 9 ( ), ċ = ( ( + β)φk). (3.9) 2 Reall from Lemma 9 that A = in the resaled model. In the non-resaled model, the analogous onditions for positive and zero prodution are (t) < A and (t) A, respetively. 3 Our losed-loop solution differs from that of Cellini and Lambertini (29), who onsider the ase when marginal ost is always lower than the hoke prie. This is so beause their proof that the open-loop and losed-loop solutions oinide is flawed by the fat that in their derivation of the losed-loop solution, players output hoies are not properly treated as funtions of the state variable. The derivations of the authors impliitly assume that if marginal ost within the R&D ooperative hanges, the opponent s quantity does not hange, whih is the violation of the feedbak priniple underlying the losed-loop solution. It is also ounterintuitive as firms in the R&D ooperative are supposed to jointly deide on their R&D efforts taking into aount that marginal ost in any period affets the ensuing Nash-equilibrium profits. Our alulations also show that the solution of Cellini and Lambertini (29) yields situations in whih per-firm profits in the ompetitive senario exeed those of the ollusive senarios, whih obviously ontradits the notion that in ase of ollusion firms maximize joint profits. 62

12 The state-ontrol system with zero prodution is given by k = ρk, ċ = ( ( + β)φk). (3.2) Collusion If firms ollude, they adopt a stritly ooperative behavior in determining both their R&D efforts and output levels. These assumptions amount 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 (3.8) reads therefore as: ċ = ( ( + β)φk). (3.2) The profit of eah firm in every instant is: π(q, k, ) = ( 2q )q k 2, (3.22) yielding its total disounted profit over time: Π(q, k, ) = π(q, k, )e ρt dt. (3.23) The optimal ontrol problem of the two firms is to find ontrols q and k that maximize the profit funtional Π subjet to the state equation (3.2), the initial ondition () =, and two boundary onditions whih must hold at all times: q and k. 4 Notie again that aording to (3.2), if >, then (t) > for all t. The urrent-value Pontryagin funtion now reads as: P (, q, k, λ) = ( 2q ) q k 2 + λ ( ( + β)φk), (3.24) where λ is the urrent-value o-state variable. It now measures the marginal worth at time t 4 Again, due to symmetry, maximizing per-firm total profit is the same as maximizing the two firms ombined total profit. 63

13 of an inrement in the state for a olluding firm when moving along the optimal path. The neessary onditions for the solution to the dynami optimization problem onsist again of a state-ontrol system whih has two regimes. As in the ompetitive ase, the first regime orresponds to < and positive prodution (q = ( )/4), while the seond orresponds to and zero prodution. The state-ontrol system in the region with positive prodution reads as k = ρk (+β)φ 8 ( ), ċ = ( ( + β)φk), (3.25) whereas the state-ontrol system with zero prodution is k = ρk, ċ = ( ( + β)φk). (3.26) 3.4 Analysis The systems (3.9) (3.2) and (3.25) (3.26) have the same struture as the state-ontrol system for the monopoly problem onsidered in Chapter 2. Indeed, onsider the system ċ = ( ( + β)φk), (3.27) k = ρk α( + β)φ( )χ [,] (), (3.28) where χ A () = if A and χ A () = if A. The monopoly system in Chapter 2 as well as systems (3.9) (3.2) and (3.25) (3.26) are instanes of the system (3.27) (3.28), with α = /9 for the ompetitive senario, α = /8 for the ollusion senario, and α = /4 for the monopoly studied in Chapter 2. Therefore, the analysis of system (3.27) (3.28) orresponds to the analysis of the system in Chapter 2. Aordingly, we onfine ourselves to stating the prinipal results of system (3.27) (3.28), inluding its bifuration diagrams. All proofs are in the appendies of Chapter 2. The first result gives the properties of the steady states of the state-ontrol system. 64

14 Proposition 2. Let D = 4 ρ α( + β) 2 φ 2. Depending on the value of D, there are three different situations.. If D >, the state-ontrol system with positive prodution (3.25) has three steady states: i. ( C, k C ) = (, ) is an unstable node, ( ii. ( C, k C ) = + ) D, is either an unstable node or an unstable fous, 2 (+β)φ and iii. ( C, k C ) = ( ) D, is a saddle-point steady state. 2 (+β)φ 2. At D =, there are two steady states: i. ( C, k C ) = (, ), whih is an unstable node, and ( ) ii. ( C, k C ) =,, whih is a semi-stable steady state. 2 (+β)φ 3. If D <, the origin ( C, k C ) = (, ) is the unique steady state of the state-ontrol system with positive prodution, whih is unstable. The system onsequently exhibits a saddle-node bifuration at D =. The stable manifold of the saddle-point steady state is one of the andidates for an optimal solution. As neither the Mangasarian nor the Arrow onavity onditions are satisfied, the stable manifold is not neessarily optimal. Note that Proposition 2 already implies that there should be other andidates for optimality as there is a parameter region for whih there is no saddle point, and hene no stable manifold to it. We have the following result. Corollary 2. The set of andidates for an optimal solution onsist of the stable path of the saddle-point steady state and the trajetory through the point (, k) = (, ). The thik blak lines L and L 2 in Figure 3. indiate these two andidates. In this figure, the dotted vertial line = separates the region with zero prodution from the region of 65

15 q > q = k = L k L2 φ S 2 L3 S 3 ċ = S.5.5 Figure 3.: Candidate trajetories in the state-ontrol spae. positive prodution. We label the trajetory L 2 the exit trajetory, as both firms eventually leave the region < < of positive prodution when following this trajetory. To assess the dependene of the solution struture on the model parameters, we arry out a bifuration analysis. This onsists in identifying the parameter values at whih the qualitative struture of the optimal dynamis hanges. These bifurating values bound open parameter regions suh that the optimal dynamis are qualitatively equal for all parameter values in the region (see Wagener (23), Kiseleva & Wagener, 2, 2). As for any point in the region, a suffiiently small hange in the parameter value does not lead to a qualitative hange of the dynamis, these regions are said to haraterize stable types of dynamis. In Chapter 2, we identified four distint stable types. Figure 3.2 illustrates these types; Figure 3.3 shows the orresponding bifuration diagram. The first type is one of a Promising Tehnology, where there is an indifferene threshold 5 in the region of no prodution. In an optimal ontrol problem, an indifferene threshold is a point in state spae where the deision maker is indifferent between two optimal trajetories that have distint long-term limit behavior. In ase of a Promising Tehnology, there is a point ĉ >, suh that for < ĉ, it is optimal to start developing the initial tehnology, 5 Also known as Skiba, Dehert-Nishimura-Skiba or DNSS point; see Grass et al. (28). 66

16 e + S 2 k.5 k. S (a) Promising Tehnology (b) Strained Market S 2 S k.5 k R () Unertain future (d) Obsolete tehnology Figure 3.2: The four stable types of dynamis. ending up in the saddle-point steady state in the region of positive prodution. In partiular, this happens for initial values of that are in the no-prodution region. If ĉ, it is optimal not to initiate R&D efforts as in this ase potential future profits do not suffie to ompensate for losses that would be inurred in the initial periods during whih firms would invest in R&D but would not produe yet. Note that for = ĉ, there are two entirely different R&D investment poliies, whih are, nevertheless, both optimal. The seond type orresponds to a Strained Market, where there is an indifferene threshold in the region of positive prodution: < ĉ <. The new feature here is that for ĉ <, the firm eventually leaves the market. It is optimal however to invest in R&D to slow down the tehnologial deay. We label the third type, whih ours only in a small part of the parameter spae, the Unertain Future. Instead of an indifferene point, here a repelling steady state divides the 67

17 25 2 I Promising tehnology II Strained market 5 A(+β) δ b ISN IR SN III Unert. fut. 5 IA SN IV Obsolete tehnology ρ/δ Figure 3.3: Bifuration diagram for β =. initial states that optimally onverge to the steady state with positive prodution and those with no prodution region. If the system starts exatly at the repelling point, it stays there indefinitely; when it starts lose to it, it stays there for a long period of time, after whih it onverges to one of the steady states. The fourth type typifies the dynamis of an Obsolete Tehnology. Whatever the initial state, the firms let the tehnology deay and eventually leave the market. In the region of positive prodution, the deay is again slowed down by R&D investments. In the bifuration diagram, the uppermost urve represents parameter values for whih the indifferene 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 bifurations in the state-ontrol system that do not orrespond to optimal dynamis. At the indifferene-attrator bifurations (IA), 6 an indifferene point ollides with an optimal attrator and both disappear. Finally, at an indifferene-repeller bifuration, an indifferene 6 For the terminology, see Kiseleva & Wagener (2, 2). 68

18 25 2 I II 5 A(+β) δ b ISN SN IR III 5 IA IV SN ρ/δ Figure 3.4: Bifuration diagram. The plot depits the bifuration urves for the ompetitive senario (grey) together with those of the ollusive senario (blak). Throughout, the ollusive urves pass through lower values of A(+β)/(δ b) (non-resaled variables) when ompared at equal values of ρ/δ (non-resaled). point turns into an optimal repeller. The entral indifferene-saddle-node (ISN) bifuration point at ( ρ, φ) (2.4, 8.78) organizes the bifuration diagram. The urve representing indifferene points at = obtains a value of φ for ρ = Collusion and the inentives to innovate Having haraterized the global optimum of both the ompetitive and the ollusive senario, we an ompare their respetive bifuration diagrams. These are superimposed in Figure 3.4. Qualitatively, there is no differene between the diagrams. There are, however, important quantitative differenes whih the following observation summarizes: Numerial observation. Over the entire parameter spae we observe that if firms ollude, the bifuration urves lie below the onomitant urves in ase firms ompete. 69

19 This observation has two orollaries. First, the Promising Tehnology region is larger if firms ollude. Put differently, if firms ollude, the situation where firms first invest in R&D, and only after some initial development period start produing, is more likely to our. Seond, if firms ollude, the Obsolete Market region is smaller. That is, due to ollusion, it is less likely that firms either do not develop further an initial tehnology, or that they invest in R&D only to abandon the tehnology in time. Numerial observation 2. Over the entire parameter spae we observe that whenever a threshold value of initial marginal osts exists in both senarios (be it an indifferene point or a repeller), it is larger if firms ollude k.25 Π (a) ĉ ĉ (b) ĉ ĉ CS TS ĉ ĉ () ĉ ĉ (d) Figure 3.5: State-ontrol spae (a), total disounted profit (b), onsumer surplus (), and total surplus (d), when the indifferene point is in the region with zero prodution. Parameters: (β, ρ, φ) = (,., 2.25). Grey urves orrespond to ompetition, whereas the blak ones orrespond to olllusion. For all (ĉ, ĉ 2 ), the ollusive senario brings about higher onsumer surplus and total surplus than the ompetitive senario. The impliations of this observation are twofold. First, if firms ollude, the set of initial 7

20 tehnologies that are developed further and that lead to the saddle-point steady state is larger. Figure 3.5 illustrates this impliation. If the initial tehnology draw falls in the non-empty interval (ĉ, ĉ 2 ), the firms will develop the tehnology and this will eventually give rise to a new market, but only if firms ollude. If they ompete, neither firm will develop the tehnology. Note that a higher value of initial marginal ost implies larger early-stage losses beause there is no profitable prodution yet. Obviously, these losses are more quikly off-set by future profits if firms ollude, due to higher mark-ups. Therefore, under ollusion, firms an afford to invest more in R&D prior to prodution, and thereby to bring down over time a higher initial level of marginal ost. 9 ρ/δ =. 8 ρ/δ = ĉ 5 2 ĉ ρ/δ = 4 C 3 2 ĉ C A(+β) δ b Figure 3.6: Dependene of the indifferene point ĉ on model parameters. Curves are drawn for three fixed values of ρ. Curves for ompetition (dotted) lie below the urves for ollusion (full). For this situation, Figure 3.6 illustrates some omparative statis of the indifferene points. Obviously, these points are positively related to market size and R&D effiieny. Note however that also the differene ĉ between ĉ ompetitive and ĉ ollusive inreases if the R&D 7

21 proess beomes more effiient and/or if the market size beomes larger, the more so the lower the disount rate is. This inelastiity orresponds in Figure 3.6 to a larger slope of the onvex urves. Beause future mark-ups are positively related to both market size and R&D effiieny, an inrease in either one of these has a larger (positive) effet on future profits if firms ollude. And these future benefits feature more prominently in total disounted profits if the disount rate is lower. Put differently, indifferene points our at smaller values if the disount rate goes up, all else equal (f. the relative loation of C and C 2 in Figure 3.6). A partiular situation arises when the indifferene point under ollusion is above the hoke prie, while it is below the hoke prie if firms ompete. This is the ase for all points in Figure 3.4 in between the two bifuration urves that separate the Promising Tehnology region from the Strained Market region. In any suh a situation, only olluding firms may develop further a tehnology whih requires investments in advane of prodution. Competing firms never develop it further as for them the exit trajetory is optimal for all initial osts above the hoke prie. Obviously, the latter senario yields a lower total surplus. 7 Seond, if firms ollude, the set of initial tehnologies that triggers no investment in R&D at all or that indues firms to selet the exit trajetory is smaller. Figure 3.7 illustrates this for the Strained Market region. The strained investment irumstanes, in the sense of a high depreiation rate in omparison to the market size and R&D effiieny, indue ompeting firms to exit the market in due time for all > ĉ. In ontrast to this, olluding firms exit the market only for > ĉ 2, whih is again due to larger mark-ups in the produt market. Initial tehnologies in the interval (ĉ, ĉ 2 ) are therefore brought to full maturation only by olluding firms, whih leads to a diret welfare gain of ollusion. So far we an onlude that due to ollusion (i) it is more likely that we have a Promising Tehnology, and if so, that it is more likely to be developed further, (ii) it is less likely that we have an Obsolete Tehnology, and if so, it is more likely that firms invest in R&D, albeit temporarily, and (iii) if the tehnology auses a Strained Market or if it indues an Unertain Future, it is less likely that it will be taken of the market in due time. In sum, due to ollusion it is more likely that firms invest in R&D, and that these investments eventually lead to a 7 In the next setion, we disuss what our analysis implies for ompetition poliies. 72

22 k Π ĉ ĉ 2 (a) (b) CS.5 TS () (d) Figure 3.7: State-ontrol spae (a), total disounted profit (b), onsumer surplus (), and total surplus (d), when the indifferene point is within the region with positive prodution. Parameters: (β, ρ, φ) = (,., 2). Grey urves orrespond to ompetition, whereas the blak ones orrespond to ollusion. Full urves orrespond to the stable path, whereas the dotted ones to the exit trajetory. Dots indiate the saddle-point steady state. For all (ĉ, ĉ 2 ), the ollusive senario brings about higher onsumer surplus and total surplus than the ompetitive senario. steady state with positive prodution. The next observation is about the intensity of the R&D proess as suh. Numerial observation 3. Over the entire parameter spae we observe that whenever both senarios trigger either the exit trajetory or the stable path towards the saddle-point steady state, the trajetory of the ollusive senario lies above that of the ompetitive senario. This observation implies the following. First, whenever both senarios lead to the saddlepoint steady state, marginal osts in the ollusive senario are lower than in ase of ompetition, beause olluding firms have invested more in ost-reduing R&D to arrive at the long-run 73

23 equilibrium. Put differently, ollusion yields a higher prodution effiieny. Seond, if the initial tehnology leads to prodution after some initial development period only, olluding firms will enter this prodution phase more quikly. That is, at every instant of the preprodution phase, olluding firms invest more in R&D in order to bring some initial level of marginal osts below the hoke prie. As a result, less favorable initial tehnologies will be brought to the market if firms ollude. Third, olluding firms abandon obsolete tehnologies at a lower pae. This impliation, that a monopolist holds on longer to a tehnology that is bound to leave the market, has a similar vein as the argument of Arrow (962), that a monopolist has less inentive to invest in R&D than an otherwise idential but perfetly ompetitive market, beause by doing so the monopolist replaes urrent monopoly profits by future (higher) monopoly profits. Here, of ourse, the alternative for the olluding firms is to exit the market more quikly (rather than staying in the market as a monopolist, as in Arrow, 962), an alternative that for them is not optimal (see Figure 3.8). 3.6 Competition poliies Summarizing the results of the previous setion, we have found that the ollusive senario is more R&D intensive: R&D investment levels are higher and the set of initial tehnologies that is developed further is larger. The prie to be paid for this inreased innovation intensity is the higher mark-up in the produt market. It should therefore not ome as a surprise that the welfare omparison between the two senarios yields a mixed piture. First, as alluded to in the previous setion, if the firms develop an initial tehnology in suh a way that this leads to a positive prodution steady state, then this always yields a higher total surplus over the alternative of no R&D investment at all. Indeed, in Figure 3.5, for all (ĉ, ĉ 2 ), the ollusive senario is the better alternative. Numerial observation 4. Over the entire parameter spae we observe that whenever both senarios have an indifferene point above the hoke prie, the ollusive senario yields higher onsumer surplus and total surplus than the ompetitive senario for all initial tehnologies in between the two indifferene points. 74

24 k.5 Π (a) (b) CS.5. TS () (d) Figure 3.8: State-ontrol spae (a), total disounted profit (b), onsumer surplus (), and total surplus (d), when the exit trajetory is an optimal solution. Parameters: (β, ρ, φ) = (,, 2). Grey urves orrespond to ompetition, whereas the blak ones orrespond to ollusion. The ompetitive senario brings about higher onsumer and produer surplus than the ollusive senario for all. This observation qualifies the argument that R&D ooperatives make it easier for firms to ollude in the onomitant produt market and that this is neessarily welfare reduing. Obviously, this fails to be the ase for all in the interval (ĉ, ĉ 2 ). It is also not neessarily valid in situations where ollusion indues firms to selet the stable path while ompetition indues them to exit the market (reall Figure 3.7). For ompetition authorities, a partiularly diffiult situation arises when the initial draw out of (ĉ, ĉ 2 ) is above the hoke prie ( > ). Then the welfare osts of prohibiting firms to ollude in the produt market do not surfae beause no prodution is affeted by this prohibition. There is no prodution yet, and beause ollusion is prohibited, there will be no prodution in the future. Yet, in this ase, prohibiting firms of an R&D ooperative to ollude 75

25 .45 4 x TS TS (a) ĉ (b) ĉ2 Figure 3.9: Total surplus when the indifferene point is in the region with zero prodution. Parameters: (β, ρ, φ) = (,, 5). Grey urves orrespond to ompetition, whereas the blak ones orrespond to ollusion. 3.6, ĉ 4., ĉ For all (, ĉ 2 ), total surplus is higher if firms ollude in the produt market. in the produt market is welfare reduing. To the extent that ompetition poliies are designed to enhane total surplus, a general prohibition of produt market ollusion is not first-best per se. At the same time, and more in line with traditional views, Figures 3.5 and 3.7 suggest that if both senarios indue firms to selet the stable path towards the saddle-point steady state, the ompetitive senario yields a higher total surplus (Figure 3.8 ontains a similar suggestion in ase both senarios indue firms to selet the exit trajetory). 8 However, this is not neessarily the ase, as Figure 3.9 illustrates. Although both senarios would indue firms to selet the trajetory towards the saddle-point steady state, for all (, ĉ 2 ), total surplus is higher if firms ollude in the produt market. In this example, the disount rate is high: ρ =, whih orresponds, for instane, to δ =. and ρ =.. Also, the initial marginal osts have to be high for the ollusive senario to outperform the ompetitive senario in terms of onsumer surplus and total surplus. In suh an environment, the higher R&D investments and the redued importane that is attahed to future surplus are favorable for the ollusive senario: if firms ollude, they reah the prodution stage more quikly, a benefit that more than off-sets the onomitant welfare loss of inreased mark-ups in the future. 9 To 8 As noted above, over the entire trajetory, ollusion yields more R&D investments. Insofar higher investment levels as suh are desirable, the ase for prohibiting ollusion in the produt market is weakened. 9 More preisely, a higher disount rate ρ = ρ/δ implies either a higher disount rate ρ or a lower δ. With a lower δ, any ost redution takes longer, suh that whenever future benefits are disounted, the time differene in reahing the prodution stage between the senarios beomes more pronouned. 76

26 Π.2 LI.5. π.8.6 p t (a) t (b) Figure 3.: Marginal ost, prie, and Lerner Index (a); total disounted profit and instantaneous profit (b), for ollusion. Parameters: (β, ρ, φ) = (,., 2.25) and starting point = 2. illustrate further what diffiulties ompetition authorities fae, onsider Figure 3.. Among others, it shows the development of the Lerner index over time towards its long-run level of.92 for the parameter onfiguration of Figure 3.5, where the initial draw = 2 is from the interval (ĉ, ĉ 2 ). This ase illustrates what has been alluded to by Lindenberg and Ross (98, p. 28): [The Lerner index] does not reognize that some deviation of P from MC omes from... the need to over fixed osts and does not ontribute to market value in exess of replaement ost. 2 The high value of the Lerner index is due to ollusion, whih, in this ase, is welfare enhaning. Indeed, this example suggests that the ourt was right in its ruling of US vs. Eastman Kodak (995) when it onluded that Kodak s film business is subjet to enormous expenses that are not refleted in its short-run marginal osts. More generally, it illustrates the diffiulty in designing optimal ompetition poliies for high-teh industries. This is illustrated further if one onsiders instantaneous profits and total disounted profits, as in Panel (b) of Figure 3.. Clearly, after a while, the former are muh larger than the latter. But the high instantaneous mark-ups should not be onsidered as a signal of potential welfare losses, beause if it had not been for these mark-ups, in the long run there would have been no market at all. 2 See Elzinga and Mills (2) for a ritial assessment of the use of the Lerner index; see also Armentano (999). 77

27 3.7 Conluding remarks We present an analysis of R&D ooperatives whereby the phase prior to prodution is taken into aount, beause it is well known that ollusion triggers the inentives to invest in R&D. Our global analysis shows that if firms ollude in the produt market, the set of initial tehnologies that is developed further inreases, and that, in partiular, more initial tehnologies are brought to full maturation. This is a diret welfare gain of produt market ollusion. Also, the probability that an initial tehnology indues firms to leave the market altogether is redued, whih again is welfare enhaning. Our analysis presents a problem for ompetition poliy beause it shows that prohibiting ollusion in the produt market per se is not univoally welfare enhaning. It also shows that the assoiated welfare osts might not surfae beause a prohibition of produt market ollusion affets R&D investment deisions prior to the prodution phase. Any deision not to develop further some initial tehnology does not materialize as a welfare ost beause no prodution is affeted (yet). 78