Minimum quality standards and novelty requirements in a one-shot development race

Prokop, Jacek; Regibeau, Pierre; Rockett, Katharine Working Paper Minimum quality standards and novelty requirements in a one-shot development race Economics Discussion Papers, No. 2009-33 Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Prokop, Jacek; Regibeau, Pierre; Rockett, Katharine (2009) : Minimum quality standards and novelty requirements in a one-shot development race, Economics Discussion Papers, No. 2009-33, Kiel Institute for the World Economy (IfW), Kiel This Version is available at: https://hdl.handle.net/10419/27731 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc/2.0/de/deed.en Discussion Paper Nr. 2009-33 | July 10, 2009 | http://www.economics-ejournal.org/economics/discussionpapers/2009-33 Minimum Quality Standards and Novelty Requirements in a One-Shot Development Race Jacek Prokop, Pierre Regibeau and Katharine Rockett Warsaw School of Economics; University of Essex and CEPR; University of Essex and CEPR Abstract The authors examine the timing and quality of product introduction in an R&D stopping game, where they allow for horizontal and vertical differentiation in the product market. They observe that discontinuous changes in introduction dates can occur as firms’ abilities as researchers change. Further, the authors observe differences in the social optimality of entry patterns depending on the underlying research abilities of the firms. Minimum quality standards and novelty requirements can play a role in correcting these suboptimal patterns of entry. The authors find that increasing the novelty requirement does not necessarily increase either the profits or, consequently, the investment levels of the initial innovator, contrary to much of the cumulative innovation literature. When the research abilities of the firms differ, either the high ability firm or the low ability firm may be the first mover. Policy interventions have much more ambiguous welfare effects in this asymmetric case, as they can change the order of entry. Paper submitted to the special issue The Knowledge-Based Society: Transition, Geography, and Competition Policy JEL: L15, L16, O31, O33, O34 Keywords: Innovation; minimum quality standards; novelty requirements; stopping game Correspondence Katharine Rockett, Department of Economics, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, England; e-mail address: [email protected] The authors would like to thank Steve Garber, Kai-Uwe Kuhn and Carmen Matutes for helpful comments. Pierre Regibeau and Katherine Rockett acknowledge support of the Spanish DGICYT under grant number PB-92-1138, and Katherine Rockett acknowledges sponsorship by the Research Council of Norway (Project 172603/V10). © Author(s) 2009. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany 1. Introduction In a large number of industries, regulations on the conditions of entry, including product standards, have significant and long-lasting effects on industry behaviour and structure. Riordan (1992) analyses such effects in a theoretical model capturing features of cable TV and telephone technologies, while Gruber and Verboven (2001) find empirical evidence for long-lasting effects in mobile telecommunications. Boom (1995), Herguera and Lutz (1998) and Motta and Thisse (1993) examine the consequences of a variety of regulatory measures, such as product safety and environmental standards in an international context. This type of control of “product quality” is still topical, as shown by Manski (2009) recent an opinion piece advocating a change in FDA procedures to allow limited diffusion of new drug entities before they have been fully investigated according current “quality control” procedures. Theoretical treatments initially suggested that minimum quality standards were socially very desirable. For example Ronnen (1991), in a seminal paper, shows that, in a Shaked and Sutton framework with endogenous (costly) quality, a minimum quality standard unambiguously raises consumer surplus, the low quality producer’s profit, and industry surplus while only harming the high quality producer. By forcing the low quality seller to improve its product a binding minimum quality standard also leads the high-quality sellers to raise its own quality in an effort to alleviate price competition1. While this analysis focuses on static efficiency effects, Riordan’s (1992) work, analyses the dynamic effects of regulation on the timing of adoption of 1 This result has been shown to be sensitive to the underlying assumptions on cost (Crampes and Hollander, 1995), preference (Kuhn, 2007) and the number of competitors in the market (Scarpa, 1998). 2 quality-improving technologies. Riordan’s interest is in the dynamic consequences of price regulation and protection from entry. In his model, regulation is used to correct social inefficiencies in the timing of entry. Our paper is very much in the spirit of Riordan (1992) since we also consider a framework where firms strategically choose their timing of entry. However, we examine the effect and optimal uses of two different policy tools, namely minimum quality standards and novelty requirements. Novelty requirements, seen as a minimum “quality increment” required of follow-up products, have already been analysed in papers such as Scotchmer and Green(1990), and Matutes, Régibeau and Rockett (1996) for a two-stage sequence of innovations, and O’Donoghue (1998) and Hopenhayn and Mitchell (2001, 2006) in quality ladder models. More precisely, we model a one-shot research and development stopping game between two firms where the payoffs depend on the difference in the qualities of the introduced products. Firms may improve their products during a waiting period before entry. The longer they wait, the higher the quality of their product. Once the entry decision is taken, the product design is frozen. Introduction can occur only once. The profits of the entrants depend on both the quality level of the final product and on the difference in product qualities when both firms are active in the market. In contrast to earlier uses of this sort of timing model, such as Dutta, Lach and Rustichini (1990) or Régibeau and Rockett (1996 and 2005), we allow firms to have different levels of “skill” in innovation. Indeed the derivation of equilibria for such an asymmetric “stopping game” is itself of some independent interest. 3 Dutta, Lach and Rustichini (1990) show that, with symmetric firms two types of equilibria can arise within this type of stopping game. Both equilibria are characterised by staggered introduction, where one firm leads with a low quality product on which it earns temporary monopoly rents, while the other enters later with a higher quality product. In the first type of equilibrium, the second mover earns higher lifetime profit than the first mover and, given the expected interval between the two product entries, the first mover maximises its total discounted profits. There is no rent dissipation since the two firms do not compete to be first to market. We refer to this case as a “stand-alone” equilibrium. There are also pre-emption equilibria, where there is rent dissipation and rent equalisation as firms “race” to move forward their entry date up to the point where they are indifferent between moving second or first.2 As we show, because small changes in our parameters can change the type of equilibrium that prevails, there can be abrupt changes in the equilibrium entry times and profitability of entering firms for small changes in their (identical) research ability. Indeed, while the firms “race”-- and so dissipate profits -- when they are both highly skilled, less-skilled firms tend to settle into an equilibrium profit without preemption. When we allow firms to have asymmetric skill levels, we find no necessary correlation between skill and either quality of the final product introduced or the order of entry. This result, which is broadly similar to that of Riordan (1992), stands in contrast to Quint and Einav (2005) who adopt a war of attrition model to determine the order of entry. The crucial difference is that these two authors do not allow the 2 This is the type of equilibria studied in the seminal paper by Fudenberg and Tirole (1985). 4 quality (and hence the profitability) of an entrant to improve with waiting: while a cost is sunk each period before entry, no gain accrues in exchange for this cost. In our framework, the cost incurred during the waiting period results in an improved product that will eventually be offered on the market. Our result that the “better” firm needs not enter first is also related to Argenziano and Dengler (2008) who study entry behaviour of firms that differ in production cost (or, equivalently, flow profits upon entry). They find that, for the same entry cost, the firm with the higher flow profits always enters first in a two-firm waiting game model like ours but that this result cannot be generalised to three firms. While the more efficient firm has a stronger incentive to enter than a less efficient firm, all else equal, the first mover must also take into account how long it will be before the rival enters. If a less efficient firm tends to follow more closely than a more efficient firm, then the more efficient firm may decline the leadership position. Indeed, in their three-firm case, a less efficient firm always moves first. In our case, the firms do not have exogenously assigned flow profit levels, but rather choose these levels endogenously by means of the timing game. Still, the basic intuition for our order of entry result is similar: while the more skilled firm has a stronger incentive to enter, all else equal, it must balance this against the fact that the less skilled firm may still enter in the future with a high quality product and this will reduce future profits. If the lower skilled firm tends to follow relatively quickly, then the high skilled firm may postpone entry, leaving the first mover position to the low-skilled firm. The motivation for introducing policy instruments into our framework is that the privately optimal and the socially optimal timing of entry need not be the same in a 5 stopping game such as ours, even when the skill levels of the two firms are the same. Riordan’s (1992) work concentrates on a case where the private incentives to enter are much too great. Hence, he focuses on policy instruments such as entry control that can effectively retard entry. In our framework, the leader may enter too early or too late, and the follower may enter with either more or less than the socially optimal delay. The pattern of entry tends to be related to the size of the innovative step. When firms are highly skilled in research, so that quality increments come very cheaply, the leader tends to introduce socially too early, while the follower tends to introduce socially too late. By contrast, when research skill is very low, both the leader and the follower move socially too fast to market. For intermediate ranges, the follower tends to enter too quickly, while the leader may lag or lead the socially optimal introduction date. As a result, minimum quality standards, which effectively prevent entry into the market before a given date, may or may not improve welfare. When research skill is symmetric and either very high or quite low, there always exists a minimum quality standard that improves welfare. For intermediate ranges, this need not be the case. When we allow for research abilities to differ, the pattern is even more complex. We observe first that a minimum quality standard can actually change the order of entry. As this can generate earlier participation by a high ability firm, this can improve welfare. Unfortunately, and contrary to the symmetric ability case, the minimum quality standard can also affect the delay between first and second product introductions. More precisely, when the more able firm moves first, a minimum quality standard can also constrain the date of entry of the second mover as the lowerskilled firm will only be able to satisfy the standard significantly later than the more 6 able first-mover. As the difference in skill levels gets more pronounced, this effect becomes larger. Hence, the welfare effect of the minimum quality standard depends crucially on the spread of research abilities of the firms involved. Since welfare is affected by both the date of first introduction and the delay that elapses between first and second entry, a novelty requirement is a natural instrument to introduce into this setting. By imposing a minimum quality difference between first and second mover, a novelty requirement effectively increases the time gap between the two dates of entry. Novelty requirements have been studied in quality ladder models in a series of papers, including Hopenhayn and Mitchell (2001). That paper takes a mechanism design approach, where the innovator’s type determines the entry timing of the follower and is private information of the innovator. Our focus is different at several levels: we assume full information on type, and instead concentrate on deriving the timing of entry endogenously in a stopping time framework. Further, our modelling allows for discontinuous change in the type of entry equilibrium. Importantly, while our model includes quality improvements, it is not the case that the higher quality product must “stand on the shoulders” of the lower quality product. Our model of quality increase is not a model of cumulative innovation, where a follower firm’s innovation requires the existence of the lead firm’s innovation to be placed on the market. Instead, the research paths of the two firms can be developed independently of each other. We limit our analysis to the case of firms with identical research abilities. We find that, like a minimum quality standard, the novelty requirement improves welfare for high or low research abilities, with no necessary improvement over an intermediate 7 range. We get some rather counter-intuitive results. For example, when the equilibrium is pre-emptive, a binding novelty requirement decreases the profits of both firms, so that stronger patent protection is associated with lower profits for all firms in the industry. The intuition for this result is very different from the recent literature on the negative effects of strong patent protection. In our case, the novelty requirement lowers the follower’s profit, and so reduces the “opportunity cost” of moving early for the leader. As a result, the leader enters too fast with a very low quality product as part of pre-emptive behaviour. In other words, by worsening the prospects of the second mover, a stronger novelty requirement intensifies the race for the first innovation, dissipating rents. Welfare can also move quite discontinuously as a function of the novelty requirement. For example, as we pass from the range of abilities for which pre-emptive behaviour occurs to that where stand alone behaviour prevails, we observe a discontinuous jump in the welfare benefit of the novelty requirement. The rest of the paper is organised as follows. Section 2 presents the model and some preliminary results on the types of equilibria we observe in our model. Further, we show that the non-cooperative choice of introduction dates does not generally maximise welfare in our model. We move on quickly from these, as our main interest is not in the baseline equilibria but the effect of two policy instruments, minimum quality standards and novelty requirements, on these equilibria. We consider the effect of the policy instruments in Section 3. Section 4 considers how these results change when asymmetries in skills are introduced to the model. Section 5 concludes the paper. 8 second. This shown in figure 1.b., where the two curves intersect to the right of the maximand ts. Hence, in equilibrium, one firm moves at ts and the other follows after the optimal delay specified in lemma 1. We call this type of equilibrium a “stand alone equilibrium”5 . In such an equilibrium, the follower makes higher profits than the leader. Hence, we obtain the following characterisation of the equilibria of this game: Proposition 1: If the research abilities of the two firms are limited, (i.e. c r c8.15.1 <≤ θ ), then there are two stand-alone subgame perfect equilibria in pure strategies where one firm introduces at time ] 3 1[ 1 21 max rcY Ycc r tj θ θθ − − −+= and the other firm introduces after the additional period defined in Lemma 1. For higher research abilities ( r θ ≥ 1.8c) there are two pre-emptive subgame perfect equilibrium outcomes, where one firm moves first at ] 3 2 1[ 1 2 rcY Ycc tp θ θθ − − −= and the other firm introduces after the additional period defined in Lemma 1. Proof: See Appendix 2.2 Welfare It is straightforward to determine the level of social surplus generated at each date. When firm A is a monopolist, introducing at time tA, we have: 5 Dutta, Lach and Rustichini call it a “maturation equilibrium”. 15 c q SS A M 8 32 = if cqA2 ≤ 2 c qA− if cqA2> and the entire market is served if and only if . We will focus on this case, below. cqA2≥ Arbitrarily assuming that firm A introduces first in the duopolistic equilibrium, we have: ] 36 )( [5 4 2 c qq c w qq qSS BABA B D− +− − += if cqq AB 3 ≤ − 2 c qB− if cqq AB 3> − where both firms have positive market shares if cqq AB 3 ≤ − and firm A is a limit pricing monopolist if . cqq AB 3>− Total discounted social surplus is given by: ])([ 1ABA B B A rt D rtrt M t rtD t t rtM eSSeeSS r dteSSdteSS −−− ∞ −− +−=+ ∫∫ . Let be the social surplus maximising stopping dates for A and B, respectively. ),( s B s Att The firms’ non-cooperative choice of introduction dates will not generally maximise welfare. Indeed, the inefficiency may come at the level of either the first or second 16 mover. Given the date of introduction of the first product, whether the follower moves too quickly or too slowly depends on the research “ability” of the industry. Specifically, if the ability is high enough that the follower enters with a drastic innovation then the follower tends to introduce too late compared to the socially optimal date. On the other hand, if ability is low enough that the follower only enters with an incremental innovation, then the follower will tend to introduce socially too quickly. This is stated formally in proposition 2. Proposition 2: Assume that θA= θB= θ. Given an initial date of introduction, tBA, the interval before the second product is introduced can be either socially too short (for r θ < 2.55c) or socially too long (for c r55.2≥ θ ). Proof: see Appendix The claims in proposition 2 are due to two opposing effects. On the one hand, waiting increases the quality of firm B’s product. While this raises both social surplus and firm B’s profits from time tB on, the increase in B’s profits is smaller than the increase in social surplus as long as firm B’s rate of improvement is small enough that both firms remain in the market following B’s entry. This is because competition from the lower quality level places a limit on the surplus that B can extract from consumers and so allows some consumer surplus to remain with purchasers. Hence, firm B’s waiting time tends to be too small compared to the social optimum (t B 6 BB 6 For c r55.2≥ θ , it is equal to the increase in social surplus. 17 smaller than tBs) because the reward to its research falls short of the full social benefit it generates. On the other hand, waiting also postpones the introduction of B’s product. This cost of waiting for firm B is smaller than or equal to the cost of waiting for society because, as before, firm B cannot usually appropriate the full social benefit of an increase in the quality of its product7. This effect tends to make firm B’s waiting time too large compared to the social optimum (tB greater than tBBs). When research ability is high enough compared to time preference, firm B serves the whole market as a monopolist as soon as its good is introduced . In this case, firm B can capture all the social benefits of a given quality increase since individual consumer demands are inelastic. This means that the first of the two effects discussed above disappears and the privately chosen waiting time of the follower is socially excessive when the second introduction represents such a leap in quality that it effectively eliminates the first product from contention. For lower values of research ability compared to time preference, on the other hand, the first effect actually dominates so that t 8 BB is smaller than the social optimum. Overall, then, incremental follow-up innovations tend to be introduced too quickly, while drastic ones tend to be introduced too9. A second source of inefficiency is that, given the second mover’s optimal reaction, the first innovation can be introduced too early or too late so that tA can be greater or smaller than tAs 7 In other words, )(),(0 B M BA DD BtSSttSS −≤− π 8 High rates of research ability compared to time preference refers in this case to the range c r55.2≥ θ 9 Other models, cited in our introduction, of cumulative innovation have not exhibited this feature because of restrictions on the nature of competition, either restrictions on the parameter ranges considered or restrictions on the dimensions of differentiation allowed. Our model encompasses sufficient generality to allow this case to emerge. 18 Proposition 3: Assume that θA= θB= θ. The initial date of introduction, tBA, will be too early compared to the social optimum if the speed of learning is either small (i.e, c r c8.15.1 <≤ θ ) or large enough that the second mover enters with a drastic innovation, (i.e. c r55.2≥ θ ). The first introduction always occurs too early in a standalone equilibrium. For intermediate research abilities, the first product is introduced too late. Proof: See Appendix When the rate of research progress (or “ability”) is high enough that drastic innovation will follow, the leader’s date of introduction tends to be too early. Combining this with our earlier welfare results on the follower, we observe a pattern of sequential monopoly for this parameter range, with the first product introduced socially too early and the follower’s product introduced socially too late. On the other hand, when research progress is very slow, both the leader and the follower move socially too fast to market, each with a small improvement. For intermediate ranges, the follower moves socially too quickly, given the leader’s introduction date, but the leader may move too quickly or too slowly. 3. Policy Instruments We have observed that there can be a deviation between the socially and the privately optimal entry date of both the leader and the follower. We now consider two policy instruments that might help reduce this discrepancy. The first instrument, a minimum quality standard, can only retard the date of first entry. It can therefore only be useful 19 over parameter ranges where the leader enters too soon in equilibrium. The second instrument, a novelty requirement, effectively increases the time that elapsed between first and second entry. As such it is useful when the second entrant moves too soon. However, the welfare analysis of the two instruments is more complex than this. By delaying first entry, a minimum quality standard also delays the date of second entry. Similarly, a novelty requirement directly changes the behaviour of the follower, but this in turn also affects the behaviour of the leader. 3.1 Minimum Quality Standards A binding minimum quality standard forces the first entrant to introduce its product later than it would have wished to. The impact of such a policy over the range where we have a stand-alone equilibrium is straightforward for two reasons. Firstly, a binding minimum quality cannot change the nature of the equilibrium. By pushing back the profit-maximising date of first introduction, the policy further decreases the profits of the leader. Since these were already lower than those of the follower, the equilibrium remains of the stand alone variety. Secondly, we already know from proposition 3 that, over the range where stand alone equilibria prevail, the first product is introduced to early. We can therefore conclude that there always is a minimum quality standard that would strictly increase welfare. We now turn to the parameter range for which we have pre-emptive equilibria. This creates an additional technical difficulty as the policy can itself change the nature of the equilibrium: by decreasing the profits of the first mover it can turn a pre-emptive equilibrium into a stand-alone equilibrium. Taking this potential switch into account we find that there exists a binding quality standard that increases welfare if research 20 abilities are high enough. In particular, welfare increases when abilities are such that the follower would enter with a drastic innovation. These results are summarised in proposition 4. Proposition 4: Assume that θA= θB= θ. There is a binding minimum quality standard that raises welfare if research abilities are limited (1.5c < B r θ < 1.8c) or when they are large enough that the follower would enter with a drastic innovation ( c r55.2≥ θ ). For intermediate values of research ability, our result depends on the level of research ability: in the range [1.8c< r θ <2.19c] the minimum quality standards cannot be used to improve welfare, while for range [2.19c < r θ < 2.55c], the minimum quality standard can improve welfare. Proof: See Appendix 3.2 Novelty Requirements Following Scotchmer and Green (1990) and O’Donoghue (1998) we interpret a novelty requirement as a restriction on the vertical scope of patents. In other words, a follower must demonstrate a minimum improvement over the initial (patented) product to be allowed to exploit its own product commercially. However, our approach departs from the existing literature in that we do not assume that the innovations of the two firms are “cumulative” in the traditional sense of the term. Cumulative innovation refers to situations where the second innovation or “improvement” would not be possible without the prior development of the first 21 innovation. There is therefore an exogenously determined sequence of investment, with investment on the follow-up innovation starting only once the initial innovation has been obtained. By contrast, in our model both firms “race” from the beginning and the timing of both introductions as well as the identity of first and second innovators are determined endogenously. This is still consistent with a novelty requirement as an interpretation, but it means that the order of entry is not “set” beforehand. As we will see, this leads to rather different conclusions. A binding novelty requirement has several effects on the equilibrium timing of entry. Firstly, it delays the introduction of the second product. When we have very high research ability in the industry, this effect does not improve welfare as the follower already waits socially too long to introduce (proposition 2). This occurs over the range where the second innovation is drastic. In other words, the incentive to improve quality to the point of dominating the industry is so powerful that a novelty requirement is not necessary. Indeed, as incentives to wait are already excessive, any binding novelty requirement would in fact decrease both the profits of the second mover and welfare. It would however increase the profits of the initial innovator since it increases the length of its monopoly period. For lower research abilities, we know that the second product is introduced to quickly so that a novelty requirement can be used to increase welfare. By delaying second entry beyond the profit-maximising date, a binding novelty requirement decreases the profits of the second mover. By the same token, it also increases the profit of the first mover, for any given date of first introduction, since it extends the length of its monopoly period. These effects have quite different 22 consequences depending on the type of equilibrium involved. In a stand-alone equilibrium a binding novelty requirement makes the leader wait longer as the value of quality improvements can be enjoyed over a longer monopoly phase. Since initial entry occurs to early over this range (proposition 3), this effect improves welfare. It also ensures that – as in the previous literature – a binding novelty requirement increases the profits of the initial innovator and decreases those of the follower. The situation is quite different over the range where pre-emptive equilibria arise. Precisely because it increases the profit of the leader and decrease those of the follower for a given date of first introduction, the novelty requirement leads to faster introduction by the leader: starting from an initial equilibrium where the profits of the two firms are equal, firms will compete to introduce even earlier in order to eliminate the discrepancy between leader and follower’s profits resulting from the novelty requirement. This increases welfare for intermediate values of the research ability parameter but decreases as soon as we hit the range where θ/r is large enough to guarantee that the follower enters with a drastic improvement. More notably, though, in a pre-emption equilibrium10, a binding novelty requirement ends up decreasing the profits of both the follower and the initial inventor. Putting these effects together give us the results presented in proposition 5. Proposition 5: Assume that θA= θB= θ. If Bc r55.2≥ θ then any binding novelty requirement decreases welfare. For 1.5c < r θ <1.8c , there exists a binding novelty 10 Starting from a pre-emption equilibrium the game will remain in a pre-emption equilibrium for any binding novelty requirement. On the other hand, a binding novelty requirement can change a stand alone equilibrium into a pre-emptive equilibrium. This technical difficulty is taken into account when deriving the results in proposition 5. 23 requirement which improves welfare. For c r8.1> θ , any binding novelty requirement decreases the profits of both the follower and the initial inventor and speeds up the date of initial entry. Proof: See appendix The last part of the proposition implies, perhaps counter-intuitively, that for the range over which pre-emptive equilibria arise, a binding socially optimal novelty requirement decreases both the R&D investment and the profits of the first mover. Making (patent) protection of the first innovation “stronger” does not therefore necessarily make the first mover better off, nor does it necessarily increase its R&D investment. The intuition for the desirability of “weak” patents is that the novelty requirement lowers the “opportunity cost” of being a follower and so allows for the leader to pre-empt earlier with a lower-quality product. This ends up being bad for the leader, but as the alternative of moving later is also less attractive, this worse alternative can still be the optimal choice for the firm. This argument is quite distinct from previous arguments against strong patent in the burgeoning literature on weak patents11. Another interesting feature of this analysis is that it indicates that welfare can move quite discontinuously with policy due to discontinuities in the behaviour of the following firm. 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Riordan, M., (1992). “Regulation and Pre-emptive Technology”, Rand Journal of Economics 23(3), 334-349. 32 Ronnen, U. (1991). “Minimum Quality Standards, Fixed Costs and Competition”, RAND Journal of Economics 22(4), 490-504. Scarpa, C. (1998). “Minimum Quality Standards with More than Two Firms”, International Journal of Industrial Organization 16(5), 665-676. Scotchmer, S., and J. Green (1990). “Novelty and Disclosure in Patent Law”, Rand Journal of Economics 21(1), 131-146. Scotchmer, S., (1996). “Protecting Early Innovators: Should Second Generation Products be Patentable?” Rand Journal of Economics, 27(2), 322-331. Vickers, J. (1996). “The Evolution of Market Structure when there is a Sequence of Innovations” Journal of Industrial Economics 35(1), 1-12. See http://www2.dse.unibo.it/wp/235.pdf and http://www2.dse.unibo.it/wp/427.pdf http://www2.dse.unibo.it/wp/532.pdf for more references also http://aic.ucdavis.edu/publications/MQSandWelfare.pdf 33 Full Appendix Sketch of Proofs for Preliminary results on which Propositions will build: First, it can be shown that all profit functions have a unique maximum as a function of the time of introduction. This is important to the results that follow. Second, we will assume in our methodology for the proofs that follow that we are working with a finite, but arbitrarily fine time grid. It will also simplify notation considerably to choose a grid such that all relevant critical points fall precisely on the grid. Sketch of Proof of Lemma 1: Deriving the time of entry of the follower Define: 2 ] 3 1[ 2 ,c c r e tt rt iaj f i θϕ πϕ +≡−≡ − and )( c r ert ib −≡ − θϕπ . In other words, the best response following lag will be denoted φ, while the profits denoted a and b are the profits of the follower, assuming that she shares the market or appropriates the entire market, respectively . Note that argmax θ ϕπ c r aia 32 −=≡ and argmax θ ϕπ c r bib +=≡ 1. Hence, the date that maximises the follower’s profit differs, depending on whether we assume she waits long enough to appropriate the entire market or not. As ia π is the correct profit function only for c a3 ≤ θϕ , we can substitute for φa to assert that φa is a candidate best response only if c r3< θ . Similarly, ib π is the correct profit function only for c b3> θϕ so that, also by substitution, φb is a candidate best response function only if c r2≥ θ . It is also the case that if c r5.1< θ , the follower is never willing to wait once the first product is introduced. Since φa φ≥b if and only if c r4≥ θ , we must consider four cases. Case 1: c r4≥ θ . As θ ϕ c a 3 >, the maximum of ia π over the range for which this is a candidate best response, θ ϕ c3 ≤, is θ ϕ c3 =, while the maximum of ib π over θ ϕ c3 ≥is b ϕ . Since ) 3 () 3 ()( θ π θ πϕπ cc iaibbib => , the best response is bi ϕ ϕ =. Hence, for this parameter range the follower chooses to wait so as to develop a drastic product innovation. Case 2: c r c43 <≤ θ is the same as case 1. 34 Case 3: c r c32 <≤ θ . As θ ϕ c a 3 <, a ϕ maximises ia π over θ ϕ c3 ≤while b ϕ maximises ib π over θ ϕ c2 >. To determine the follower’s best response, we must compare )( aia ϕ π and )(bib ϕ π . Substituting and solving, we have )()( aiabib ϕ π ϕ π >if and only if θ θ rc e c r 4 2 9− >or .553.2 c r> θ In other words, for c r553.2> θ it is best for the follower to wait and appropriate the entire market rather than share. In other words, in this case the follower (endogenously) chooses to develop a drastic innovation. For parameter ranges below this, the follower chooses to develop an incremental innovation. Case 4: c r c2 2 3<< θ . Over this range, a ϕ maximises ia π over θ ϕ c3 ≤, while the maximum of ib π over θ ϕ c3 <occurs at θ ϕ c3 =. Since ) 3 () 3 ()( θ π θ πϕπ cc ibiaaia => , the best response is ai ϕ ϕ =. In other words, the follower chooses to develop an incremental innovation over this range. ■ Summarising, then, we have a range over which the follower chooses to wait to develop a drastic innovation, c r553.2> θ , and a range over which the follower endogenously chooses to be an incrementalist c r c553.25.1 ≤< θ . Lemma 2: (Deriving the entry time that maximises the discounted payoffs, without pre-emption) For the symmetric case, we have: θ c r+ 1 if c r55.2≥ θ tjmax = 2 ] 3 1[ 1 21 rcY Ycc r θ θθ − − −+ if c r c55.25.1 << θ where θ rc eeY 3 2− = Lemma 3: (Deriving the pre-emption date) For the symmetric case, we have: θ θ θ rc rc ee ee r c − − − − − + 1 1 1 1 if c r55.2≥ θ tp = ]1 3 2 [ 1 2− − +rcY Ycc θ θθ if c r c55.25.1 << θ Sketch of Proof of lemma 2 and lemma 3 ( for the symmetric case) 35 Four cases must be analysed since the profit function can take one of two forms during the monopoly period and one of two forms during the duopoly period. Case 1: c r55.2≥ θ and θ c tj 2 >. In this case, θ c r tj+= 1 max which satisfies the assumption that θ c tj 2 >. Setting )1()()( 1 e e e r ct t rc rt j jj j θ θ π − −− − =equal to e e e rr rc rt f i i θ θ π − − =1yields the pre-emption time ] 1 [ 1 1 1 − − − − − += θ θ θ rc rc p e e r c t. Also, since pj tt > max 2 e e rc < − θ for c r3≥ θ . Case 2 : c r55.2≥ θ and θ c tj 2 ≤. This case cannot arise in equilibrium because timax still is greater than tp and 2 1 1 1 ]] 1 [[ 2 − − − − − = θ θ θ θ rc rc p e e r ct > θ c2. Case 3: c r c55.22 ≤≤ θ and θ c tj 2 >. In this case, )1/() 3 1()(2 12 max Y rc Y cc r tj−−−+= θ θθ , where Y = θ rc ee 3 2 −. Setting ]) 3 1(2)1)([( 121 rc cYYcte rj rt j i θ θπ −+−−= − equal to j rt f iYe rcr c− =2 ] 3 [2 θ π yields ].1 3 2 [ 1 2− − += rcY Ycc tp θ θθ Also, if and only if pj tt > max 2 9 ] 1 [c Y Y r< − θ . To see that this condition is satisfied for the range we consider in this case, notice that for ]3,2[ cc r∈ θ , ]541.1,582.0[ 1∈ −Y Y. Indeed, note that for ]2, 2 3 [c c r∈ θ , [,541.1[ 1∞∈ −Y Y, so that tjmax must be smaller than tp for low enough values of r θ . Numerical computations show that this occurs for 804.1≤ r θ . Case 4: c r c55.22 ≤≤ θ and θ c tj 2 ≤. This case cannot occur because tp<tjmax and 2 1 ]}1 3 2 [ 1 2{ 2− − =rcY Yc tp θ θ , which is greater than θ c2because, for c r c55.22 ≤≤ θ , 1]1 3 2 [ 1 2>− −rcY Y θ . 36 Lemma 4: For the asymmetric case, we have expressions for the behaviour of the follower: l j i ij i l j l j f it c r ttt θ θ θ θ − ++=− 1 )( for c r i55.2≥ θ = l j ij i t i c r θ θ θ θ − +− 32 for c r i55.2< θ As the proof of this case differs trivially from the proof of lemma 1, the proof will be omitted (but is available from the authors upon request). Notice that the lag in introduction dates now depends on the difference in ability between the two firms. Sketch of Proof of Proposition 1 -- the equilibrium entry dates -- including elements of proposition 6. Let (x,y) denote the decisions of each of the players at any time, t, where 0 indicates a decision to stop and 1 indicates a decision to continue development. Furthermore, let G0 denote the full game and Gt denote the subgame that starts at time t. For a moment, consider the full asymmetric case, where θA < θB. Four critical points will be important to the proofs that follow. First, t B pA represents the time where the payoff to firm A from stopping -- at some time, t -- before firm B is equal to its payoff from stopping after firm B has stopped – at some time, t. In other words, A would be willing to pre-empt back to this date, but at no earlier date. Second, point tpB represents the time when the payoff to firm B from stopping before firm A is equal to its payoff from stopping after firm A has stopped. In other words, this is the analogous earliest pre-emption date for firm B. Third, point tAmax represents the time when the payoff to firm A from stopping first is maximised. Finally, point tBmax represents the time when the payoff to firm B from stopping first is maximised. Our first case, which we will fully develop, is the case where θA < θBB and tpA < tpB<tAmax<tBmax. Indeed, the argument for this case is completely analogous to the symmetric case of tp < tmax. We will make the argument for the full asymmetric case, as this can also serve to form the basis for proposition 6, but keep in mind that the argument is the same as for the analogous symmetric case. Of course, for the asymmetric case, we can have tpB > tAmax so that we have tpA < tAmax<tpB<tBmax as well when θA < θB . We consider this latter case more briefly, below. B To anticipate our conclusions, and again using the phrasing for the asymmetric case, if tpA<tpB<tAmax<tBmax we will show that the following outcome is a unique subgame perfect equilibrium outcome for the full game, G0: both firms wait until time tpB. Then, firm A stops the R&D phase at tpB. Firm B stops its research later, at tB*(tpB). For the corresponding symmetric case where tp < tmax, there are two such outcomes that differ only in the identity of the firm that moves first. In other words one firm, firm A (firm B) stops the R&D phase at tp, while the other, firm B (firm A) stops its research later at t*(tp). These are the “pre-emption equilibrium” equilibria in the text. We can also have the case where tAmax<tBmax<tpB<tpA. In this case, the unique subgame perfect equilibrium outcome for the full game, G0 is that firm A moves first 37 at tAmax if and only if while firm B moves first at t )()( maxmax B f AA l Att ππ ≥Bmax if and only if . In the analogous symmetric case where t )()( maxmax B f AA l Att ππ <max is smaller than tp, we will show that there are two subgame perfect equilibrium outcomes for the full game, G0. In each of these, both firms wait until the time tmax. One of the two firms, firm A (firm B), stops the R&D phase at tmax while the other, firm B (firm A), stops its research later at tf(tmax). These are the “stand alone” equilibria in the text. Case I: tAmax > tpB The profit function of the leader has a unique maximum and the profit function of the follower is decreasing in the date of introduction by the leader. Necessary conditions: First, we show that if the strategy combination is a subgame perfect equilibrium for G0, then the following local strategy combinations must be a part of it: (1,1) for t=0,…..,tpB-1 =),( t B t Ass (i) (0,1) for t=tpB. Consider the behaviour of the firms in periods 0,…,tAmax. At least one firm must stop its research not later than tAmax. Indeed, if firm B did not stop, then it would be the best response for firm A to stop its research at tAmax simply from the definition of tAmax as the time that maximises A’s discounted payoffs when it is the first to stop. Thus (1,1) for t=(0,…,tAmax) cannot be part of the equilibrium local strategies for G0. Similarly, the local strategy (0,0) at any time maxA tt ≤ cannot be a part of a subgame perfect equilibrium for G0, because firm A can increase its payoff from the subgame Gt by changing its local strategy to 1. Thus, in a subgame perfect equilibrium for G0, the local strategies (0,1) or (1,0) are played at some time, t, where . Denote by t )0( maxA tt ≤≤ * the smallest t for which (0,1) or (1,0) is an equilibrium local strategy. We show that t* = tpB. It is a dominant strategy for firm B not to stop first before time tpB since tpB is defined as the time at which first mover and second mover profits are equal. Since the first mover payoff to firm A is increasing in the interval [0, tpB], the best response of firm A is to stop not earlier than tpB. Thus, . Suppose that . We consider two possible cases: pB tt ≥* pB tt >* Case 1: (0,1) is played at t*. Again, from the definition of tpB, firm B would do better by stopping one period before t*, which is greater than or equal to tpB . Thus, the local strategy (0,1) at t*> tpB is not a part of a subgame perfect equilibrium. Case 2: (1,0) is played at t*. In this case firm A would do better by stopping one period earlier, at t* - 1, which is greater than or equal to tpA. Again, this follows from the definition of tpA. Thus, the local strategy (1,0) at t* is not a part of a subgame perfect equilibrium. 38 Hence, t* must be equal to tpB. Finally, the pair (1,0) cannot be an equilibrium local strategy combination at t* = tpB. Suppose, to the contrary, that (1,0) at t* = tpB is played in a subgame perfect equilibrium. Then firm A can do better by playing (0,1) at tpB – 1, which is greater than or equal to tpA. (In other words, rather than simply letting firm B pre-empt, it would be better for firm A to be the first mover. As we have not “rolled back the game” to A’s earliest pre-emption date, A should prefer this.) Hence, the local strategy combination given by (i) must be a part of any subgame perfect equilibrium for G0. Sufficient Conditions Now, we show that the following strategy combination is a subgame perfect equilibrium: (1,1) for t=0,…..,tpB-1 =),( t B t Ass (0,1) for t=tpB, tpB+2, … (ii) (1,0) for t=tpB+1, tpB+3,…. Consider a subgame Gt for t=tpB+1, tpB + 3, …. Firm A’s payoff from the subgame Gt under the strategy combination listed above in (ii) is . In other words, it is the maximum payoff firm A can obtain given that firm B stopped at time t. Firm B’s payoff from the subgame G )),(( *tttAA π t under the strategy combination (ii) is . In other words, it is the maximum payoff firm B can obtain given that firm A stops at time t+1 if firm B would fail to stop at t. Thus the strategy combination (ii) induces a Nash equilibrium on every subgame of G ))(,( *ttt AB π t for t=tpB+1 , tpB+3…. Similarly, the strategy combination (ii) induces a Nash equilibrium on every subgame of Gt for t=tpB, tpB+2…. Now consider a subgame Gt for pB tt ≤ ≤ 0. Firm A’s payoff from the subgame Gt under the strategy combination (ii) is . In other words, it is the payoff from stopping no sooner than t ))(,( * pBBpBA ttt π pB and inducing the optimal response of firm B. No greater payoff can be obtained, for two reasons. First, stopping earlier would only bring a lower payoff for firm A since its payoffs increase over this range of times as a first mover. In other words, A can pre-empt successfully before time tpB, given that tpA<tpB, but it would prefer to pre-empt as late as possible. Second, given that firm B stops at time tpB+1 if firm A would fail to stop at tpB, firm A would be transformed into a follower and would earn a lower payoff. Firm B’s payoff from the subgame Gt under the strategy combination (ii) is . This cannot be increased given that firm A stops at time t ))(,( * pBBpBB ttt π pB. If firm B stops earlier it would get only a lower payoff, as t* is the optimal following time (and stopping at the same time is never optimal). We have shown that the strategy combination (ii) induces a Nash equilibrium on every subgame of G0, thus it is a subgame perfect equilibrium. 39 For the symmetric case, we make the same argument but eliminate the “A” and “B” designations from the subscripts. In other words, we could specify the strategy combination (iii), below for the symmetric case: (1,1) for t=0,…..,tp-1 (1,1) for t=0, …, tp-1 =),( t B t Ass (0,1) for t=tp, tp+2, … or… (1,0) for t=tp, tp+2, … (iii) (1,0) for t=tp+1, tp+3,…. (0,1) for t=tp+1,tp+3… Either firm could be the first one to pre-empt, with the other firm following at a later date. Hence, for the symmetric case we have two equilibria, characterised by one firm stopping first at tp while the other continues on and stops at a later date, t*(tp). Case II: tAmax < tpB As was stated above, this case will be dealt with much more briefly than case I. As in case I, this case has analogous arguments for the symmetric and asymmetric cases. Hence, let θA < θB and the ranking tAmax<tBmax<tpB<tpA prevail. Then note the following: Step 1: The two firms never stop at the same date. This follows immediately from lemma 4. Step 2: tBmax < tpB. If the game reaches tBmax, then we know that B stops from the definition of tBmax. Notice that from tpB on, firm B’s dominant strategy is to stop since the leader’s profit is falling and the follower’s profit is less than the leader’s. Hence, if the game ever reaches tpB, B stops as well. Given this, and the assumed ranking of dates, if consider a fine but discrete grid, at tpB – 1, A waits and B stops. Further, and following similar reasoning, at tpB – 2, A waits and B stops. This argument can be repeated so that the game unfolds backwards until we reach tBmax. Step 3: For all t < tBmax, B’s dominant strategy is to wait. Step 4: Given firm B’s behaviour, firm A must essentially choose between moving first, in which case it maximises its payoffs by introducing at tAmax or moving second, in which case it will follow B’s entry date of tBmax. In other words, A’s best response to B’s strategy is to move first at tAmax if and to let B introduce first at t )()( maxmax B f AA l Att ππ ≥ Bmax otherwise. An abbreviated argument for the analogous symmetric ability case is the following: Necessary conditions If a strategy combination is a subgame perfect equilibrium for G0, then one of the two following local strategy combinations must be part of it: (1,1) for t=0,…..,tmax-1 (1,1) for t=0,…tmax-1 =),( t B t Ass … or… (0,1) for t=tmax (1,0) for t=tmax In other words, no firm stops before tmax, at which point one of the two stops. The other continues. 40 =),( t B t Ass (0,1) for t=tc,…,t1, tpA, tpA+1, tpA+2… (1,0) for all tc-1, tc-1-1,…t1-1, tpA-1 This concludes the derivation of the equilibrium. Now, we move on to consideration of the minimum quality standard. The equilibrium we have derived is one where the more efficient firm, firm B, moves first ( ). The unconstrained equilibrium is such that firm B moves first at t pBc tt ≥ c-1 . As such, any minimum quality standard q pB t≥min θ ≤Atc is ineffective since it does not prevent firm A from credibly threatening to introduce first at tc. For 1min − ≤ <cAcA tqt θ θ , firm A can only credibly threaten to stop at tc-1 so that firm B introduces first at tc-1-1. Following the same reasoning, one can see that greater values of qmin induce later dates of first introduction by B in a stepwise fashion. Now let maxmin BApAA tqt θ θ ≥ < . Firm A cannot introduce before tpA (since the minimum quality standard does not allow it). This induces firm B to move first at 1 min − A q θ . If maxmin BAtq θ > then firm A cannot introduce before tBmax (since the minimum quality standard does not allow it). Hence, we have a stand alone equilibrium where firm A introduces first at tBmax. On the other hand, we have that when tc< tpB, firm A introduces first at tAmax in the unconstrained equilibrium. Define as the largest t * i ti which is still smaller than tpB and as the smallest t * 1− i ti which is larger than tpB. The earliest date at which firm B will want to introduce first is . Therefore, any minimum quality requirement such that just pushes back the date of firm A’s introduction to 1 * 1− −i t * minmax iAAA tqt θθ ≤< A q θ min . For , however, firm A prefers to let firm B introduce first at and the minimum quality requirement reverses the order of introduction. For even larger minimum quality requirements, the analysis is analogous to that of the previous paragraph. * 1min * − ≤< iAiA tqt θθ * 1− i t Sketch of analytical results used to establish results from numerical simulations To perform the numerical computations we must determine timax, tp and for l s t ]2,5.1] cc r∈ θ . 1. timax Assume first that θ c ti 2 max >so that the whole market is served during the initial monopoly period. Under this assumption we have: 2 max ) 3 1( 1 2 1 rcx xcc r ti θ θθ − − −+= 47 Where θ rc eex − − =1. Numerical computations show that this is indeed greater than θ c2 for c r804.1≥ θ . Let us now assume that θ c ti 2 max <. Under this assumption, it follows that: 2 1 3 2 3 2 max ] )1(9 8 1[1( 1 θ θ rc rc i ee ee r t − − − −+= ) Which numerical computations show to be larger than θ c2. This is a contradiction. Hence, it follows that θ c ti 2 max = must hold for the range for which stand alone equilibria prevail. Numerical computation show that this range is [804.1,5.1] cc r∈ θ . 2. tp Assuming that tp θ c2 ≥ we get )1 3 2 ( 1 2− − += rcY Ycc tp θ θθ where θ rc eeY 3 2− =. Computations show tp to be always greater than θ c2for all c r5.1> θ . 3. l s t Assuming that θ c tl s 2 ≥, one gets ) 3 5 2( 32 1 crr Yc r tl s θ θ −++= where θ rc eeY 3 2− = Which is indeed greater than θ c2 if and only if ) 1 3 2 ( 5 9 Y c r+< θ 48 which numerical computations show to be satisfied for all c r5.1> θ . 4. Comparison of the different stopping times Based on steps 1, 2 and 3, above, for c r804.1< θ , we know that θ c tp 2 > = timax so that the equilibrium must be “stand alone”. Since max 2 i l st c t=> θ , a minimum quality standard of is called for in this range. For l s t θ c r c2804.1 ≤≤ θ , tp<timax for c r804.1> θ so that we have a pre-emptive equilibrium over this range if and only if tp<timax : )1 3 2 ( 1 2− − += rcY Ycc tp θ θθ < timax Numerical computations show that this holds, indeed, for this range. Hence, we have a pre-emption equilibrium over this range. Furthermore, computations show that the pre-emptive equilibrium occurs earlier than tsl -- so that a minimum quality standard is desirable -- if and only if c r19.2≤ θ . This is clearly the case for the parameter range we are considering. Brief description of numerical simulations for footnotes General methodology: The simulations were run by using MathCad 4.0 for Windows. The general methodology of these programmes is as follows: 1. Iterate on the introduction date of the leader. 2. The optimal behaviour of the follower is determined by our lemma 1, above, or by a novelty requirement if it is binding. 3. The profits of the leader and follower as well as social surplus are obtained, both with A as leader and with B as leader. 4. The equilibrium is determined by inspection of these four elements. In other words, one determines which of the six rankings we listed at the beginning of the proof for propositions 6 and 7 actually prevails. Since the programme is run again for each combination of parameters, the size of the iteration grid can be adjusted to each case in order to remove any possible ambiguity as to which ranking occurs. 5. Simulations were performed for a large number of combinations of rc θ (and for an even larger number when θA and θB were not equal). It is important to notice that two special problems arise in finding the equilibrium through simulations when rankings 3, 4 or 5 prevail. First, the precise value of tc and 49 its position relative to tpB depend on the grid used. In other words, a change in the size of the grid can affect the equilibrium outcome. To minimise the effect, simulations for these cases were run on a grid up to 10,000 times finer than simulations used for rankings 1, 2 and 6. The second problem is the multiplicity of equilibria resulting from the existence of two pure strategy equilibria at tpA -1, t1-1, …. In the text, we derived the subgame perfect equilibrium outcome of G0 when the Pareto superior equilibrium is believed to arise at each of these points. Other selection rules would not affect the nature of the argument presented in the text but they could, for a given grid, lead to rather different outcomes. Fortunately, the use of an extremely fine grid in out simulations also helps alleviate this problem because all these equilibria converge to the same outcome in the continuous limit. 50 Note: We do not show in proposition 4 (analytically) that stand alone equilibrium is less than social optimum. We only show this for theta > 2.55rc. Lower range follows from numerical simulations only. We’d have to assume step 2’s condition. Note: Numerical simulations allow us to describe the regions for which minimum quality standards improve welfare – or not. Indeed, we find that there is an intermediate region ( )19.2804.1 c r c<< θ where quality standards cannot improve welfare, while a welfare-improving standard can be found for lower values of research ability. 51 Please note: You are most sincerely encouraged to participate in the open assessment of this discussion paper. You can do so by posting your comments. Please go to: http://www.economics-ejournal.org/economics/discussionpapers/2009-33 The Editor © Author(s) 2009. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany