Spillovers, Licensing and Welfare

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1 Spillovers, Licensing and Welfare Shuai Niu The University of New South Wales This version: March 19, 01 Abstract In this paper, we introduce two methods of technology transfer between asymmetric competitors, knowledge spillovers and licensing. In the previous studies, these two methods are usually considered as either-or alternatives and the links between them are rarely of concern to researchers. We contribute to the literature by including both knowledge spillovers and the possibility of licensing in an asymmetric duopoly model, analyzing the interactions between them, and incorporating this aspect into the policy making process. JEL classification: L1, L4, L40, L50 Keywords: spillovers, licensing, R&D, welfare, policy School of Economics, Australian School of Business, The University of New South Wales, Sydney, NSW 05, Australia. I wish to thank Hodaka Morita and Arghya Ghosh for their kindly encouragements and valuable comments.

2 1 Introduction In reality, there are two common ways of technology diffusion between firms of different productivity, knowledge spillovers and licensing. The main objective of this paper is to develop the appropriate public policies towards these two methods. Two policy tools will be discussed, the patent policy and the licensing policy. Regarding patent policy, we mean the government adjusts the strength of patent protection to impact the extent of knowledge spillovers. As for the licensing policy, we mean the government intervenes in technology agreements between asymmetric firms by just saying yes or not to a potential licensing deal. Based on these two policy tools, a optimal policy system will be developed, under which the patent policy and the licensing policy will coordinate with each other to improve social welfare. In the previous studies, the analysis of knowledge spillovers and that of licensing are usually conducted independently, i.e. when talking about knowledge spillovers the possibility of post R&D licensing is usually not allowed and when talking about licensing the knowledge spillover rate is usually set to be zero. If we follow this tradition, there are two problems cannot be resolved: 1) which technology diffusion method is better from the standpoint of social welfare? ) are there any interactions between knowledge spillovers and licensing and how these interactions will impact the development of optimal public policies? In this paper, we contribute to the literature by giving answers to the above two questions. The whole analysis is based on a homogenous product asymmetric duopoly model. Among the two firms, only firm 1 has the ability to conduct cost reducing R&D. Firm does not have R&D capability, but its production efficiency can be raised by knowledge spillovers or licensing. Initially, both firms produce with the constant marginal production cost c, to reduce it by x firm one will incur costs kx. 1 If there is no technology diffusion, the two firms will interact in a two stage game: firm 1 first chooses the innovation level and then competes with firm in the final product market à la Cournot. Incorporating knowledge spillovers into the R&D stage, we can discuss the optimal patent policy without licensing. In this paper, θ [0, 1] is used to stand for the spillover rate and it is supposed to be a declining function of the strength of patent protection. The value of θ will impact not only the equilibrium market structure but also the conducts 1 k can be considered as an indicator of the innovator s R&D efficiency. As the rise of k, firm 1 s efficiency in research will decline. This two stage game is a common framework applied in the analysis of firms R&D activities. To the best of our knowledge, it is first introduced by Brander and Spencer (198). In that paper, the authors analyze the process innovation in a duopoly and model the interactions between the two firms like this: firms first determine the R&D investments non-cooperatively and then engage in quantity competition in the final product market. In the R&D stage, each firm has two choices, they can use R&D for strategic purpose (s1) or simply to minimize production costs (s). It is proved that, s1 is a dominant strategy and both firms will use R&D strategically in equilibrium. 1

3 of the incumbent firms (see Lemma 1). That creates a correlation between knowledge spillovers and welfare, based on which we can determine the socially optimal spillover rate and formulate the corresponding policy recommendation: if licensing is not feasible, to maximize social welfare the patent protection strength should first be intensified and then be relaxed as the continue decline of the innovating firm s R&D efficiency (the optimal patent policy without licensing, see Proposition 1). Incorporating the possibility of licensing after the R&D stage and before the output stage, we can discuss the optimal licensing policy without knowledge spillovers. A licensing contract between asymmetric competitors has two effects on social welfare: on the one hand, it helps to save on the overall production costs via technology transfer (the cost saving effect, which is always positive), and on the other hand, it impacts the competitive behaviors via the new created pecuniary correlation between competitors (the strategic effect, which can be negative). It will be proved that, without spillovers if k < 0.69 the strategic effect is negative, it dominates the cost saving effect, and the introduction of licensing is welfare reducing (see Lemma 7). Suppose the government intervenes in technology agreements by just saying yes or no to a certain licensing contract, then the policy implication should be like this a potential licensing contract should be permitted only if the innovating firm s R&D efficiency is not too high (the optimal licensing policy without knowledge spillovers, see Proposition 4) Compare the welfare level realized under the optimal patent policy without licensing with that realized under licensing without knowledge spillovers, we can conclude that as long as licensing is welfare improving it will be a better way to transfer technology between asymmetric competitors from the standpoint of social welfare (see Proposition 5). This answers the first question pointed out above. The above ananlysis is still based on the traditional paradigm. That is, as two technology diffusion methods knowledge spillovers and licensing are considered as either-or alternatives. Incorporating knowledge spillovers into the R&D stage and allowing the possibility of post R&D licensing at the same time, we can explore the interactions between these two methods and explain how the coexistence of spillovers and licensing will influence the development of optimal policies. To achieve this purpose, three steps will be taken. In the first step, we re-derive the optimal patent policy under the premise that licensing is feasible. In this case, the government does not intervene in the licensing deal and whether or not to license is totally up to the involved firms decisions. Under licensing, the passive technology diffusion role used to be played by knowledge spillovers will become superfluous, and now knowledge spillovers can only influence social welfare indirectly via its impact on the licensing contract terms (which will influence the equilibrium output and innovation levels). This points to a new relationship between knowledge spillovers and social welfare. Based on this, we can derive the optimal spillover rate and

4 get the corresponding policy implications: if licensing is feasible, the government should intensify patent protection when the innovating firm s R&D efficiency falls (the optimal patent policy under licensing, see Proposition ). In the second step, we re-derive the optimal licensing policy under positive knowledge spillovers. In this case, the knowledge spillover rate is set to be exogenously determined and it can take any value between zero and one. The introduction of positive knowledge spillovers will influence the relative importance of the cost saving effect and the strategic effect. Consequently, the optimal policy suggestion becomes that as long as the knowledge spillover rate is not too small or the innovator s R&D efficiency is not too high, the government should always take a permissive attitude towards royalty licensing (the optimal licensing policy with spillovers, see Proposition ). In the last two steps, we have analyzed the individual adjustment of each policy tool, i.e. the possibility of post R&D licensing is considered in the development of patent protection policy and knowledge spillovers are incorporated into the licensing policy making process. In the final step, we turn to the coordination between different policy tools. Given the above analysis, we can conclude that to optimally intervene in technology transfers between asymmetric competitors the government has two choices: it can either forbid licensing and adopt the optimal patent policy without licensing or permit licensing and adopt the optimal patent policy under licensing. It can be proved that the first choice is better than the second one if and only if k < 0.61 (see Proposition 6). This conclusion indicates a new optimal policy system emphasizing the interactions between knowledge spillovers and licensing. And, that answers the second question pointed out above. The remainder of this paper is organized as follows. The related literature is reviewed in the next section. In section, we introduce the basic model settings of this paper. In section 4, the optimal patent protection policy is analyzed. In this section, we first specify the optimal patent policy without licensing, then turn to the case with licensing, and finally explain the changes brought about by the introduction of licensing. The licensing policy is discussed in section 5. In this section, how the impact of licensing on social welfare will be influenced by knowledge spillovers is explained. In section 6, we focus on the coordination between patent policy and licensing policy and examine the extent to which the new constructed policy system can improve social welfare. The main results of this paper are summarized in the concluding section.

5 Related literature The characteristics of knowledge spillovers and licensing, their welfare implications, and the corresponding policy recommendations have been widely discussed in the previous literature. The spillover effect is an important feature of the R&D activities. Due to the existence of knowledge spillovers, part of the input or output of one firm s R&D will diffuse into its competitors without any reward. That makes R&D partially appropriable and at the same time intensifies market competition. The consequence is that firms incentives to conduct research are reduced and the economy ends at a lower equilibrium innovation level compared to the case without knowledge spillovers (Katz, 1986). From the standpoint of social welfare, knowledge spillover encourages technology diffusion and helps to save on the overall production costs, which is welfare improving, but at the same time, it depresses the R&D incentives, which can be welfare reducing. Based on this tradeoff, the socially optimal spillover rate is discussed in some studies. Under constant elasticity demand, Spence (1984) analyzes input spillovers and concludes that the absence of spillovers should not be the socially optimal choice. For the linear demand case, Stepanova (009) demonstrates that social welfare is maximized under an intermediate to low level of spillovers (between 0 and 0.5). De Bondt et al. (199) consider output spillovers and prove that social welfare achieves the maximum under an intermediate to high spillover rate (between 0.5 and 1). It can be seen that, depending on how the knowledge spillovers are modeled the socially optimal rate derived can be very different. Amir (000) gives a comparison between the input spillover model and the output spillover model. Licensing is another way to transfer technology between asymmetric firms. Different from spillovers, it is an active dissemination of knowledge and is rewardable. In practice, there are three familiar licensing forms: fixed fee licensing, royalty licensing and the two-part tariff licensing. In an early survey, Rostoker (198) summarizes the utilization frequency of each contract form. For the recent evidence, the work by Macho-Stadler et al. (1996) can be referred. Given a certain cost gap, technology sharing between asymmetric competitors has two effects on joint profits. 4 On the one hand, it raises the less efficient firms productivity and brings about cost savings (the cost saving effect, which raises joint profits), but on the other hand, it may remove the initial cost gap and intensify competition (the Another important licensing contract form is the profit sharing licensing (or ad valorem scheme). Under this contract, the patentee use its superior technology in exchange of the licensee s partial financial interest. For detail analysis, the work by Mukhopadhyay et al. (1999) and Martín and Saracho (010) can be referred. Profit sharing contract is not covered in this paper. 4 In this paper, we only deal with the case where the patentee is an incumbent firm. For the outsider patentee case, Kamien (199) provides a survey of the related researches. 4

6 reinforced competition effect, which reduces joint profits). For fixed fee licensing, the cost saving effect dominates only if the initial cost gap is sufficiently small. So, this kind of contract is likely to occur between firms which are reasonably close in terms of their initial technologies (Marjit, 1990). As for the royalty licensing, Rockett (1990) proves that the equilibrium royalty rate is equal to the initial cost difference. That means, the reinforced competition effect will be zero and this kind of contract is always profitable. In a homogenous product duopoly model with Cournot competition, Wang (1998) compares between these two contract forms and concludes that royalty licensing is superior to fixed fee licensing for the licensor. 5 In a differentiated product model, Faulí-Oller and Sandonis (00) analyze the welfare effect of licensing and find that under Cournot competition the introduction of licensing is always welfare improving. 6 In the above analysis, the cost gap is set to be exogenously determined. However, in reality the possibility of post R&D licensing will certainly influence the innovator s R&D incentives. In a situation where two firms bid for a certain innovation, Katz and Shapiro (1985) point out two effects of licensing: it increases the rewards to the winner and it increases the rewards to the loser if the licensee enjoys some of the joint profits from licensing. The net influence of licensing on each firm s willingness to bid will depend on which effect dominates. In a process innovation model with uncertainty, Gallini and Winter (1985) prove that suppose the low-cost firm enjoys all the profits from licensing, then royalty licensing will encourage research if the firms initial production technologies are relatively symmetric. Incorporating the endogenous innovation into the welfare analysis, Mukherjee and Mukherjee (00) conclude that social welfare is non-decreasing in the presence of fixed fee licensing. In Chang et al. (010), an asymmetric duopoly game is introduced, in which only one of the two firms has the ability to conduct cost reducing R&D and license the innovation, and the equilibrium indicates that R&D investment under royalty licensing is higher than that under fixed fee licensing, fixed fee licensing is always welfare improving, but royalty licensing can be welfare reducing. In the literature reviewed above, as two technology diffusion methods knowledge spillovers and licensing are considered as either-or alternatives, i.e. when talking about knowledge spillovers the possibility of licensing is usually not allowed and when talking about licensing knowledge spillovers are usually set to be zero. As in reality these two 5 This theoretical conclusion leads to a contradiction, as in reality royalty licensing is obviously not the only contract form utilized in technology agreement. To compensate for the gap between theory and reality, some new elements are introduced into the basic model, like product differentiation in Cournot competition (Mukherjee and Balasubramanian, 001; Wang, 00; Erkal, 005), varied degree of competitive behavior (Saracho, 005), cost difference before innovation (Poddar and Sinha, 010; Mukherjee, 007), and more than one backward firms (Marjit et al., 000; Kamien and Tauman, 00; Sen and Tauman, 007). Under these modified model settings, the rationality of each contract form can be theoretically explained. 6 Under Bertrand competition, the introduction of licensing can be welfare reducing for close enough substitutes and large enough but not drastic innovation. 5

7 methods do coexist and there are important interactions between them, the above conclusions can be misleading. Aware of this, some researchers have made some targeted effort. For example, the possibility of licensing is incorporated into the analysis of optimal intellectual property rights protection. Depending on the specific modeling method, related studies can be divided into two sets. The first set adopts deterministic process innovation model. In a North-South trading environment, Chin and Grossman (1988) compare two polar cases, complete protection (in this case the possibility of fixed fee licensing is considered) and complete violation, and conclude that patent protection enhances (reduces) world efficiency when R&D productivity is high (low). Žigić (1998) extends Chin and Grossman s work by assuming that the strength of patent protection can attain many different levels, but the possibility of licensing is not allowed. Bagchi and Mukherjee (010) analyze the impact of licensing on optimal patent policy and find that depending on the innovation cost, the introduction of licensing may either increase or reduce the optimal strength of patent protection. However, in their model only drastic innovation is considered (if there is complete protection, the innovator will be a unconstrained monopoly) and the innovation level is exogenously given. The second set describes R&D as an uncertain process. In a non-tournament model, Mukherjee (00) proves that with fixed fee licensing, welfare is higher under strong (weak) patent protection whenever licensing is profitable under both patent systems (under only weak patent system); with royalty licensing, welfare tends to be higher under weak patent protection if cost reduction is not very small. In a R&D racing model with the possibility of licensing, Martin (00) distinguishes the impacts of input and output spillovers and demonstrates that for not very high input spillovers, social welfare is maximized at an intermediate output spillover rate, whereas for very high input spillovers, the zero output spillover rate is most socially desirable. In this paper, we depart from the above work in three ways. Firstly, we adopt a deterministic innovation model, consider different strength of patent protection and allow for post R&D licensing. So, technically it can be considered as an extension to the first literature set and a supplement to the second one. Secondly, how the existence of knowledge spillovers will influence the welfare implication of licensing is analyzed. Finally, we develop a new policy system based on the coordination between patent policy and licensing policy. To the best of our knowledge, the last two points are completely new and they are not covered by either set. 6

8 Basic model setup Suppose there are two firms in the market, firm 1 and firm, producing homogenous product. The representative consumer s utility function is set to be of the form U(q 1, q ) = a(q 1 + q ) 1(q 1 + q ), where a is a positive constant and q i is the quantities of products bought from firm i = 1,. Given this, the optimal consumption behavior will lead to the following inverse demand curve p = a q 1 q. 7 Initially, firm 1 and firm both produce at the constant marginal production cost c. By investing in cost-reducing R&D, firm 1 can improve its productility. Specifically, to reduce the marginal cost by x firm 1 will incur costs kx, where k is a positive constant indicating R&D efficiency. When the value of k rises, firm 1 s efficiency in research will decline. To simplify analysis, without loss of generality we add one constraint on the initial production cost level and firm 1 s R&D efficiency: Assumption 1: a > c and k > 4. 9 Firm does not have R&D capacity, but it can partly benefit from firm 1 s innovation through knowledge spillovers. In this paper, we focus on output spillovers and use θ [0, 1] to stand for the spillover rate. Apart from the passive technology diffusion realized by knowledge spillovers, the innovator may find it profitable to transfer its advanced production technology actively via licensing. Here, we consider a two-part tariff licensing contract. Under this contract, given the output level of the licensee q, the licensor s licensing benefits can be expressed as L = rq + T, where r is the royalty rate and T is the lump sum payment. 8 To avoid the possibility that the licensing contract be strategically used by the involved firms to facilitate colluding, following Shapiro (1985) we put a constraint on the two-part tariff contract: Assumption : Lump sum payment from the licensor to the licensee is prohibited, i.e. T should not be negative. As for the profits division in the technology agreement, in this paper we consider a simple case and assume: Assumption : The licensor has a complete bargaining power and enjoys all the profits from licensing. The licensee will accept the contract as long as the licensing does not make it worse off. Two firms are supposed to interact in a three stage game. In the first stage game, firm 1 determines the innovation level x. Given a certain spillover rate θ, that will determine the two firms cost structures, c 1 = c x and 7 See Singh and Vives (1984) for a detail analysis. 8 This is a relatively flexible contract form, by letting r = 0 we can get the fixed fee licensing contract, and by letting T = 0 we can get the pure royalty licensing. 7

9 c = c θx. In the second stage game, the two firms negotiate on whether or not to engage in licensing. If agreement is reached, the optimal contract terms to the innovator will be chosen and the cost structures will become c 1 = c x and c = c x. In the third stage game, they compete with each other in the final product market à la Cournot given the decisions made in the first two stages. As in this paper our main interest lies in the policy analysis towards technology transfer between asymmetric competitors, some words need to be said on the two policy tools. Assumption 4: The value of θ is a declining function of the strength of patent protection. By adjusting the patent policy properly, the government can realize any knowledge spillover rate in [0,1]. Assumption 5: Regarding licensing policy, we mean the government intervenes in technology agreements between asymmetric competitors by just saying yes or no to a potential licensing deal. Based on the equilibrium of the above game and the knowledge about how it will be influenced by government interventions, we can develop the corresponding policy recommendations. Summing up all the above settings, the theoretical framework to be used in the following analysis can be expressed as follows. R&D stage spillovers patent policy production stage licensing licensing policy welfare Figure 1: A framework for the analysis of public policies towards R&D 8

10 4 The optimal patent policy In this section, we will first discuss the optimal patent protection policy without the possibility of post R&D licensing and that under licensing, and then explain the changes brought about by the introduction of licensing. 4.1 The case without licensing If licensing is not feasible, firm 1 and firm will interact in a two stage game: firm 1 first chooses the innovation level, and then compete with firm in the final product market. In this case, the two firms profit functions can be expressed as: π 1 = (a c + x q 1 q )q 1 kx, π = (a c + θx q 1 q )q (1) Based on this, the above two stage game can be solved backwardly. And, the equilibrium innovation and output levels are: Lemma 1: If k + θ 1 0, x = a c, q 4k 1 1 = a c+x, q = 0; If k + θ 1 > 0 and k (1 θ)( θ) < 0, x = a c, q 1 θ 1 = a c+x, q = 0; If k (1 θ)( θ) 0, x = ( θ)(a c), q 9k ( θ) 1 = a c+( θ)x, q = a c+(θ 1)x. Proof: see the appendix. Three implications can be derived from the above result. Firstly, depending on the specific values of k and θ, the equilibrium market structure can be a monopoly or a duopoly. Specifically, for a large k and a high knowledge spillover rate, in equilibrium both firms will stay in the market (the duopoly). For the intermediate values of k and θ, to maintain the monopoly position firm 1 will use R&D strategically to force firm leave the market (the constrained monopoly). If R&D efficiency is very high and the spillovers are very limited, there will be no potential threat from firm and firm 1 will enjoy a full monopoly power (the unconstrained monopoly). Secondly, the equilibrium innovation level will be influenced by the extent of spillovers and the characteristics of the influence are very different under different market structures. In the duopoly case, if we raise the spillover rate the equilibrium innovation level will decline accordingly. 9 The reason is that knowledge spillover is not rewardable and it strengthes competitors, so the increase of it will dampen the innovator s incentives to conduct research (Katz, 1986). Whereas, if there is only one firm in the market and that firm uses R&D strategically to maintain the monopoly position (the constrained monopoly case), the increase of knowledge spillovers will motivate innovation. 10 The story behind this is that to maintain a monopoly position in the final product market 9 x θ = [9k+( θ) ](a c) [9k ( θ) ] < x θ = (a c) (1 θ) > 0. 9

11 firm 1 should adjust the innovation level to meet (1 θ)x a c. 11 If we raise θ marginally, to make sure this inequality still stands x should be raised accordingly. In the unconstrained monopoly case, there is no potential threat from firm and firm 1 enjoys a full monopoly power, so the marginal change of spillovers will have no effect on the equilibrium innovation level. Finally, in a duopoly given a certain innovation level the equilibrium outputs of the two firms will be influenced by the extent of knowledge spillovers. A marginal increase of the spillover rate will narrow the cost gap between two competitors. Under strategic interactions, that will induce firm 1 to reduce output and firm to increase output. As the output expansion of firm is larger than the output contraction of firm 1, the net effect on industry output is positive. 1 The above analysis indicates that, knowledge spillovers can not only influence the equilibrium market structure but also will impact the conducts of the incumbent firms (the innovation and output choices). That creates a correlation between knowledge spillovers and the equilibrium social welfare. Given that by adjusting the strength of patent protection the government can influence the extent of knowledge spillovers, a natural question is what is the socially optimal spillover rate? Suppose social welfare is comprised of industry profits and consumer surplus. Then, given the representative consumer s utility function and the two firms production cost structures, the correlation between knowledge spillovers and the equilibrium social welfare can be expressed as: W (θ, x, q 1, q ) = (a c + x)q 1 + (a c + θx)q 1 (q 1 + q ) kx () To explicitly illustrate through what intermediate mechanism the change of knowledge spillovers can exert influence on social welfare, we totally differentiate W (θ, x, q 1, q ) with respect to θ and get dw dθ = W + } θ {{} KDE PE {}}{ [ W q 1 q 1 θ + W q q θ ] + [ W q 1 q 1 x + W q q x + W x ] x θ }{{} IE () It can be seen that, as the change of knowledge spillovers three effects on social welfare can be identified: the knowledge diffusion effect (KDE), the production effect (PE), and the innovation effect (IE). In other words, the variation of knowledge spillovers will influence social welfare by impacting the extent of technology diffusion, the industry outputs, and the equilibrium innovation level. The sign of each effect will depend on the 11 The derivation of this inequality can be found in the proof to Lemma 1. Given k + θ 1 > 0 and k (1 θ)( θ) < 0, we can get 1 θ > 0. 1 q 1 θ = x q < 0, θ = x (q1+q) > 0, θ = x > 0. 10

12 specific market structure realized in equilibrium, and the main results are summarized in the following lemma: Lemma : In the unconstrained monopoly case, KDE = PE = IE = 0; In the constrained monopoly case, KDE = PE = 0 and IE > 0; In the duopoly case, KDE > 0, PE > 0 if and only if k (1 θ)( θ) > 0, and IE < 0. Proof: see the appendix. Balancing between these effects, the optimal θ in each possible equilibrium market structure can be derived. Next, by comparing the potential welfare level that can be realized in different market structures we can determine the socially optimal knowledge spillover rate: Proposition 1: If licensing is not feasible, to maximize social welfare the government should adjust the patent protection strength appropriately to maintain the knowledge spillover rate at θ = with f(0.98) 0.41 and f ( ) > 0. Proof: see the appendix. 1k+1 if 4 9 k 0 if k 0.98 f(k) if k 0.98 In order to give a more intuitive impression of the above conclusion, the socially optimal knowledge spillover rate is described in the following graph: 1 Θ Figure : The optimal strength of patent protection without licensing The rationale behind figure is very straightforward. When R&D efficiency is very high, i.e. k, it is optimal for the government to maintain a monopoly market structure and at the same time raise the spillover rate as high as possible to make use of 1 Due to the multidimensional effects of spillovers on social welfare, the optimal spillover rate displays a discontinuous characteristic at k At this point, the maximum welfare can be realized at either θ = 0 (under which, the innovation is large and the spillover is small) or θ 0.41 (under which, the innovation is small and the spillover is large). k 11

13 the potential competition from firm (which will inspire firm 1 s R&D investments) to increase social welfare. Because the maximum spillover rate that can sustain monopoly will decline as the rise of k, the socially optimal spillover rate will fall as the decline of firm 1 s R&D efficiency. When k, for any spillover rate in equilibrium both firms will stay in the market. In the duopoly case, when firm 1 s R&D efficiency is relatively high, the innovation effect dominates and to maximize social welfare the government should maintain a very strict patent protection policy. As the rise of k, the relative importance of the knowledge diffusion effect will increase. And, now it is better for the government to relax patent protection to a certain extent and let the follower firm enjoy some benefits from firm 1 s innovation. 14 In all, without licensing to maximize social welfare the optimal patent policy should be like this the protection strength should first be intensified and then be relaxed as the continue decline of the innovating firm s R&D efficiency. 4. The case with licensing If licensing is feasible, firm 1 and firm will interact in a three stage game. In the first stage game, firm 1 determines the innovation level. In the second stage game, firm 1 decides whether or not to transfer the advanced technology to firm. If yes, the optimal licensing contract form to the innovator will be chosen. In the final stage game, the two firms engage in quantity competition. In this case, the profit functions can be expressed as: π 1 = (a c + x q 1 q )q 1 + rq + T kx, π = (a c + x q 1 q )q rq T (4) Based on this, the backward induction method can be applied to solve the three stage game and get the sub-game perfect Nash equilibrium: Lemma : If k + θ 1 0, in equilibrium licensing will not happen and the innovation and output levels are x = a c, q 4k 1 1 = a c+x, q = 0; If k+θ 1 > 0, pure royalty licensing will take place in equilibrium with royalty rate r = (1 θ)x, and the innovation (7 5θ)(a c) and output levels are x =, q [9k ( θ) (1 θ)(θ 1)] 1 = a c+( θ)x, q = a c+(θ 1)x. Proof: see the appendix. A comparison between the equilibrium of this three stage game and that of the two stage game analyzed in last subsection indicates that, as the royalty rate is just equal to the cost gap before licensing, whether there is licensing or not will not impact the equilibrium output given a certain θ and x. 14 In the duopoly case, the influence of the production effect is relatively small. To determine the optimal spillover rate, the main tradeoff is between the knowledge diffusion effect and the innovation effect. 1

14 However, there are also two changes caused by the introduction of licensing. Firstly, in the area where k + θ 1 > 0 and k (1 θ)( θ) 0 firm 1 will choose to license its advanced technology to its competitor rather than use innovation strategically to monopolize the market. So, in some cases the introduction of licensing will change the equilibrium market structure. Secondly, depending on the feasibility of licensing the impact of knowledge spillovers on innovation can be very different. We have already known that, without licensing the increase of knowledge spillovers will always reduce the innovator s R&D incentives if the equilibrium market structure is a duopoly. However, under licensing the marginal increase of spillovers may promote innovation. To understand why knowledge spillovers can encourage innovation in duopoly under licensing, we divide the licensor s revenue into two parts, the indoor revenue and the revenue from licensing. The increase of spillovers will reduce the marginal indoor revenue, but it may increase the marginal licensing revenue due to the output expansion of the licensee. When the latter dominates, knowledge spillovers will increase the marginal rate of return on R&D investment and promote innovation. Specifically, Lemma 4: Under (royalty) licensing, the marginal increase of knowledge spillovers will promote innovation as long as θ 7 5k+1 5. Proof: see the appendix. It can be seen that, the introduction of licensing will impact not only the equilibrium market structure but also the innovating firm s R&D incentives. Naturally, the optimal patent protection policy needs to be reconsidered. Given Lemma, to determine the socially optimal knowledge spillover rate under licensing two cases should be considered. If the government maintains a relatively strong patent protection policy and the R&D efficiency is high, i.e. k + θ 1 0, firm 1 will enjoy an unconstrained monopoly position. For this case, social welfare will be fixed at k(6k 1)(a c) (4k 1) independent of the knowledge spillover rate. 15 On the other hand, if k + θ 1 > 0, licensing will take place in equilibrium and the social welfare can be expressed as W (x, q 1, q ) = (a c + x)(q 1 + q ) 1 (q 1 + q ) kx (5) Totally differentiate W (x, q 1, q ) with respect to θ, we can get dw dθ = [ W q 1 q 1 θ + W q q θ ] + }{{} PE IE {}}{ [ W q 1 q 1 x + W q q x + W x ] x θ (6) 15 Substitute the innovation and output levels, x = a c 4k 1, q 1 = a c+x, q = 0, into Eq. (), we can get the social welfare level in this case. 1

15 It can be seen that, under licensing if we change the knowledge spillover rate marginally two effects on social welfare can be identified, the production effect and the innovation effect, noted as PE and IE, respectively. Regarding the sign of each effect, we have the following conclusion: Lemma 5: PE > 0, and IE > 0 if and only if θ < 7 5k+1 5. Proof: see the appendix. Balancing between the production effect and the innovation effect, we can get the optimal spillover rate under licensing. Compare with the monopoly case, the optimal patent protection policy with the possibility of licensing should be: Proposition : When licensing is feasible, the government should adjust the patent protection strength to maintain the knowledge spillover rate at with g( 4 9 ) 0.4 and g ( ) < 0. Proof: see the appendix. θ = g(k) Graphically, the socially optimal knowledge spillover rate can be described as follows Θ k Figure : The optimal strength of patent protection under licensing With the possibility of licensing, it will always be in the social interest to maintain a competitive rather than a monopoly market structure regardless of firm 1 s R&D efficiency. The explanation to the downward trend of the socially optimal spillover rate following the increase of k is that in duopoly the marginal change of knowledge spillovers has two effects on equilibrium social welfare, the production effect and the innovation effect. When the innovating firm s R&D efficiency is very high, the government can relax patent protection to some extent and use the production effect to increase welfare. Whereas, if R&D efficiency is low, the innovation becomes so scarce that the government should strengthen patent protection to protect the R&D incentives. 14

16 4. The impact of licensing In this subsection, we will first give a brief comparison between the two patent protection schemes discussed above, and then explain the potential welfare loss if the existence of the post R&D licensing is ignored in the patent protection policy making process. For explanation clarity, in what follows we name the optimal protection policy without licensing (see Proposition 1) the patent scheme 1 and that under licensing (see Proposition ) the patent scheme. Without licensing, to derive the socially optimal spillover rate the main tradeoff is between the knowledge diffusion effect and the innovation effect. Whereas, if we allow for licensing the new tradeoff becomes between the production effect and the innovation effect. 16 This difference leads to very distinct policy implications. Compare between figure and figure, we can tell that depending on R&D efficiency three possible differences between the two patent schemes can be identified. Firstly, if k, the ideal market structures under the two patent schemes will be different. To maximize social welfare the patent policy ignoring licensing requires a monopoly structure, whereas duopoly is more desirable to the policy considering the possibility of licensing. Secondly, for an intermediate efficiency level, i.e. k 0.98, to encourage innovation patent scheme 1 insists a very strict patent protection, whereas patent scheme advises relaxing patent protection to raise industry outputs. Finally, if firm 1 s R&D efficiency is low, following scheme the government will strength patent protection to inspire innovation, but under scheme 1 a loose patent policy will be adopted to promote knowledge diffusion. If licensing is feasible in reality, among the two patent policies scheme should be the reasonable policy choice and the resulting equilibrium is more socially desirable. However, suppose the government adopts scheme 1 ignoring the possibility of licensing, then what is the consequence on social welfare? From Lemma, we know that if feasible, licensing will take place in equilibrium as long as k + θ 1 > 0. That means, if k < licensing will happen under patent scheme 1 and the equilibrium market structure is a duopoly, which is inconsistent to the government s expectation. If k >, the market structures expected by the two policies coincide with each other. However, compared to patent scheme, patent scheme 1 sets a too strict patent protection strength when k 0.98 and a too loose protection strength when k The consequence is very straightforward: when firm 1 s R&D efficiency is high the equilibrium innovation level under patent scheme 1 will be higher than the optimal (the equilibrium innovation level under scheme ), whereas for a low efficiency level it will be lower than optimal. This is the potential welfare loss if the possibility of licensing is not considered in the development of the patent protection policy. 16 If licensing happens, the passive technology diffusion role used to be played by knowledge spillovers will become superfluous, and now knowledge spillovers can only influence social welfare indirectly via its impact on the equilibrium royalty rate (and then the output and innovation levels). 15

17 5 The optimal licensing policy In this section, we will turn to the welfare analysis of licensing. One discovery is that, depending on whether there are spillovers or not the welfare effect of a certain licensing deal can be very different. That means, knowledge spillovers should play an important role in the formulation of the reasonable licensing policy. Based on the equilibrium of the second stage game (see Lemma 1) and Eq. (), we can get the equilibrium welfare without licensing. The equilibrium social welfare with the possibility of licensing can be derived similarly from Lemma and Eq. (5). Compare between these two cases, we have the following result: Lemma 6: With knowledge spillovers, royalty licensing is welfare improving if and only if k > K(θ), where K(0) 0.69, K(0.8) 4 9 and K ( ) < 0. By the welfare function, we can give an intuitive explanation to Lemma 6. Without licensing, the social welfare function is of the form W = (a c + x)q 1 + (a c + θx)q 1 (q 1 + q ) kx, whereas under licensing it becomes W = W + (1 θ)xq. The new item (1 θ)xq represents the backward firm s cost saving under technology agreement, we call it the cost saving effect of licensing, noted as CSE. The possibility of licensing will change the R&D incentives and then the equilibrium output levels. That means, the equilibrium value of W will also be different in these two cases. We call this part of the welfare change the strategic effect of licensing, noted as SE. It can be proved that, Lemma 7: CSE > 0, SE < 0 if and only if k < ( θ)(4 5θ) 9, and the overall effect is positive if and only if k > K(θ). Proof: see the appendix. Given the above results, suppose the government intervenes in the technology agreement by just saying yes or no to a certain licensing contract, then the optimal licensing policy considering the existence of knowledge spillovers should be like this: Proposition : As long as the knowledge spillover rate is not too small or the innovator s R&D efficiency is not too high, the government should always take a permissive attitude towards licensing between asymmetric competitors. As a comparison, by letting θ = 0, we can get the optimal licensing policy without knowledge spillovers: 17 Proposition 4: If θ = 0, the introduction of (royalty) licensing is welfare improving if and only if k > So, licensing taking place in the range where k < 0.69 should be forbidden by the government. A comparison between Proposition and 4 tells us that if knowledge spillovers in R&D are ignored, the derived policy recommendation can be misleading. For example, if the 17 Without knowledge spillovers, the same result regarding the welfare effect of licensing is derived in Chang et al. (010). 16

18 innovator s R&D efficiency is high and the spillover rate is not too small, i.e. k < 0.69 and K(θ) < k, licensing will be forbidden according to Proposition 4 even if actually it is welfare improving. This is the potential welfare loss if the existence of knowledge spillovers is not incorporated into the licensing policy making process. Before ending the analysis on licensing, we need to explain more about the definition of the licensing policy adopted in this paper. In the previous literature, the only recommended constraint on licensing contract is that lump sum payment from the licenser to the licensee should be forbidden. Under this traditional rule, the introducing of licensing can be welfare reducing if the innovator s R&D efficiency is high. To avoid this situation, in this paper we take a step forward by breaking the whole policy process into two stages: in the first stage, the government intervenes in technology transfer agreements between asymmetric competitors by just saying yes or no to a potential licensing deal; in the second stage, once permitted the involved firms are free to choose the contract form ensuring that there is no lump sum payment from the licenser to the licensee. Anticipating that the contract form to be chosen in the second stage will be pure royalty licensing, in the first stage the government s best choice is to approve licensing only in the parameter range where royalty licensing is welfare improving. Of course, we can continue to add constraints on licensing, like intervening in the choices between different contract forms. However, for two reasons this possibility is not considered in this paper. On the one hand, that will greatly complicate our analysis, and on the other hand, including it will have little effect on our main point (that is, the introducing of spillovers may change the welfare implication of a certain licensing deal). 17

19 6 The optimal policy system In the above analysis, we have analyzed two technology diffusion methods between asymmetric competitors, knowledge spillovers and licensing, and discussed the corresponding policy implications. In the previous literature, these two methods are usually considered as either-or alternatives, i.e. when talking about knowledge spillovers the possibility of post R&D licensing is usually not allowed and when talking about licensing the knowledge spillover rate is usually set to be zero. Following this tradition, a question is still to be answered: among these two methods, which one is better from the standpoint of social welfare? Based on the equilibrium of the second stage game (see Lemma 1), the social welfare function without licensing (see Eq. ()), and the optimal knowledge spillover rate specified by Proposition 1, we can determine the maximum social welfare that can be realized by knowledge spillovers. By letting θ = 0, from Lemma and Eq. (5) we can get the equilibrium social welfare achieved under licensing. Compare between these two social welfare levels, we can reach the following conclusion: Proposition 5: As two technology transfer methods, licensing is more socially desirable than knowledge spillovers if and only if k > Proof: see the appendix. In reality, knowledge spillovers and licensing coexist with each other and there are important interactions between them. So, the real problem faced by the government is how to coordinate between these two methods rather than how to choose between them. So far, we have pointed out two potential improvements existing in the current policies towards R&D. 18 To achieve the first one, the possibility of ex post R&D licensing should be considered in the development of patent protection policy. To realize the second one, knowledge spillovers should be incorporated into the licensing policy making process. Both improvements involve with the individual adjustment of each policy tool. Next, we will introduce another potential improvement of the current policies. To achieve this third one, the coordination between different policy tools should be emphasized. In the above analysis, we have already proved that patent scheme 1 (see Proposition 1) is the optimal patent protection policy if licensing is not feasible, and patent scheme (see Proposition ) is the optimal choice if licensing is feasible. Based on this, to optimally intervene in the R&D related activities the government has two choices: it can either forbid licensing and adopt patent scheme 1 or permit licensing and adopt patent scheme. Compare the social welfare realized under these two choices for different levels of the innovating firm s R&D efficiency, we have the following suggestion: Proposition 6: To maximize social welfare, the government should forbid licensing 18 Proposition is an improvement of Proposition 1. And, Proposition is an improvement of Proposition 4. 18

20 and adopt patent scheme 1 when k < 0.61, and permit licensing and adopt scheme for the other cases. Proof: see the appendix. The significant difference between this new proposed policy system and the old one (specified by Proposition 1 and ) is that in the new system the important interactions between knowledge spillovers and licensing are incorporated into the policy making process, whereas in the old one it is not. In order to show why this difference can be important, the next we will give an estimation of the potential welfare gains if we switch between these two policy systems. Lemma 9: Suppose the R&D efficiency indicator k is normally distributed in the range [ 4, 5 ] with mean and variance 1 and the post R&D licensing is feasible, then 9 6 roughly speaking by switching from the old system to the more reasonably new one the expected social welfare can be raised by about.1 percents. Proof: see the appendix. As this is not a negligible benefit, reconsidering the current policies towards R&D is beneficial to the government. 19

21 7 Conclusion In this paper, we explore an asymmetric duopoly model to develop the optimal policy system towards technology transfer between competitors. Two policy tools are mentioned, the patent protection policy and the licensing policy. The former tool impacts social welfare by manipulating the extent of knowledge spillovers, and the latter one does that by controlling the feasibility of licensing. As two technology diffusion methods, knowledge spillovers and licensing are usually considered as either-or alternatives in past studies. That is, when talking about knowledge spillovers the possibility of post R&D licensing is usually not allowed and when talking about licensing the knowledge spillover rate is usually set to be zero. Following this tradition, we discuss the traditional policy recommendations towards these two methods: the strength of patent protection should first be intensified and then be relaxed as the continue decline of the innovating firm s R&D efficiency (the optimal patent policy without licensing); a potential licensing deal should be forbidden if the innovator s R&D efficiency satisfies k > 0.69 (the optimal licensing policy without spillovers). Compare the social welfare levels realized under these two policies, we can conclude that licensing is a better way to transfer technology between asymmetric competitors as long as it is welfare improving. In reality, knowledge spillovers and licensing coexist with each other and there are important interactions between them: the feasibility of licensing will significantly influence the socially optimal spillover rate and the introduction of positive knowledge spillovers may change the welfare implication of a certain licensing deal. Given this, the optimal public policies should be revised as follows: if licensing is feasible, the strength of patent protection should always be intensified when the innovating firm s R&D efficiency falls (the optimal patent policy under licensing); even the innovator is of high R&D efficiency, a potential licensing deal can still be welfare improving and should be permitted by the government as long as the knowledge spillover rate is not too small (the optimal licensing policy with spillovers). That points out two potential improvements existing in the current policies. Another potential improvement can be achieved through the coordination between different policy tools. In summary, the optimal policy system emphasizing the interactions between knowledge spillovers and licensing should be like this the government should forbid licensing and adopt the optimal patent policy without licensing when k < 0.61, and permit licensing and adopt the optimal patent policy under licensing for the other cases. As a rough estimation, suppose R&D efficiency is normally distributed and licensing is feasible, then the new proposed policy system can raise the expected social welfare by about.1 percents compared to the traditional one. 0

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