Patent renewals and R&D incentives

RAND Jounal of Economics Vol. 30, No., Summe 999 pp. 97 3 Patent enewals and R&D incentives Fancesca Conelli* and Mak Schankeman** In a model with moal hazad and asymmetic infomation, we show that it can be welfae impoving to diffeentiate patent lives when fims have diffeent R&D poductivities. A unifom patent life povides too much R&D incentive to low-poductivity fims and too little to high-poductivity ones. The optimally diffeentiated patent scheme can be implemented though a menu of patent lives (o enewals) and associated fees. We chaacteize the optimal mechanism and use simulation analysis to compae it with existing patent enewal systems and to illustate the potential welfae gains fom the optimal policy.. Intoduction Most patent systems equie payment of a seies of enewal fees to maintain patent potection up to the statutoy patent life. Typically, moe than half of all patents ae voluntaily cancelled by nonpayment within ten yeas of the date of patent application. Thus, even though all counties impose a unifom statutoy patent life, thee is de facto diffeentiation in patent lives. Econometic studies have confimed that enewal fees influence the decision to patent and that moe valuable patents ae held longe (Pakes, 986; Schankeman and Pakes, 986; Schankeman, 998; Lanjouw, 998). Howeve, in pactice patent enewal fees ae used to finance patent offices, and thee is no eason to believe that the existing patten of de facto patent lives induced by these fees impoves welfae. The cental idea in this aticle is that patent fees can be used as an incentive device to implement a policy of optimally diffeentiated patent lives (and, moe geneally, diffeentiated patent potection). To demonstate the potential benefits of using patent enewal fees in this way, we show how diffeentiated patent lives can be bette, in tems of social welfae, than a unifom patent life. * London Business School and CEPR; fconelli@lbs.ac.uk. ** London School of Economics, EBRD, and CEPR; schankem@ebd.com. We ae gateful to two anonymous efeees and the Edito, Rob Pote, fo vey constuctive comments. We also thank Leonado Felli, Nancy Gallini, Bonwyn Hall, Matin Hellwig, Haizhou Huang, Paul Klempee, Jenny Lanjouw, Eic Maskin, Aiel Pakes, Suzanne Scotchme, Jean Tiole, and paticipants at vaious seminas fo comments, and Oli Aav fo eseach assistance. All emaining eos ae ous. Fancesca Conelli thanks CREST-LEI, Tel Aviv Univesity, and the Cente fo Economic Studies in Munich fo hospitality duing pat of this wok. An ealie vesion of this aticle was ciculated unde the title Optimal Patent Renewals (LSE, June 996). Copyight 999, RAND 97

98 / THE RAND JOURNAL OF ECONOMICS The use of patents as a policy instument to povide R&D incentives makes sense only if thee is pivate infomation about the cost o value of inventions (Wight, 983). We develop a static model of innovation that incopoates both asymmetic infomation on cost (R&D poductivity) and moal hazad on the R&D effot undetaken by the fim. Ou basic intuition is that diffeentiated patent lives can be welfae impoving because of an incentive effect : allowing fims with high R&D capabilities to choose longe patent lives gives these fims an incentive to invest moe R&D esouces. Any unifom patent life will povide too much incentive to low R&D-poductivity fims and too little incentive to high ones. This geneates both a suboptimal level and distibution of R&D. We believe that this basic point will cay ove to a dynamic famewok involving sequential innovation, but this is not exploed hee. Diffeentiating patent lives can impove social welfae when thee is ex post heteogeneity in the value of inventions that the govenment cannot obseve. This can aise fom ex ante heteogeneity in R&D poductivity o fom uncetainty in the eseach pocess. The optimal scheme involves the govenment offeing fims an incentive-compatible menu of patent lives and associated lump-sum patent fees. Each fim chooses its pefeed patent life. This menu of patent lives is equivalent to the govenment offeing a schedule of annual enewal fees, if thee is no postpatent leaning by the fim. Howeve, we show that if the fim leans about the value of its invention afte patenting, the enewal scheme is supeio in tems of social welfae. We use simulations to illustate what the optimal diffeentiated patent scheme might look like and to compae it to existing patent systems. We find fou stiking chaacteistics: thee is a minimum patent life even fo small inventions; fo most inventions the ange of optimal patent lives is quite naow; optimal patent lives ae much longe fo paticulaly good inventions (much geate than existing statutoy maximum lives); and patent fees should ise much moe shaply with patent life elative to the existing fee schedules. In contast to this aticle, most of the taditional patent-design liteatue has focused on the optimal unifom patent length and, moe ecently, on othe dimensions of patent policy such as beadth (Nodhaus, 969; Klempee, 990; Gilbet and Shapio, 990; Geen and Scotchme, 995; Scotchme, 996; O Donoghue, Scotchme, and Thisse, 998). Thee ae only two ecent aticles that study diffeentiated patent potection in diffeent famewoks. Scotchme (999) analyzes a static model with pivate infomation on the cost and value of inventions, but no moal hazad (the fim chooses which ideas to develop, but not how much R&D to do). She shows that asymmety of infomation is sufficient to justify the use of patents to povide R&D incentives, and that any diect mechanism can be implemented using a enewal mechanism. de Laat (997) analyzes a patent ace in which the imitation delay is pivate infomation and studies optimal diffeentiation of patent length and beadth. The aticle is oganized as follows. Section pesents the model. Section 3 chaacteizes the second-best patent policy and the optimal unifom length, which seve as benchmaks fo the welfae analysis. Section 4 pesents the main esult: the conditions unde which it is optimal to diffeentiate patent lives, and the chaacteization of the optimal scheme. In Section 5 we simulate the optimal mechanism and the associated welfae gains, and we compae the optimal patent lives and fees to existing patent systems in Fance, Gemany, and the United Kingdom. In Section 6 we biefly discuss thee extensions of the model: stochastic R&D outcomes, postpatent leaning, and the appopiability envionment. In Section 7 concluding emaks summaize the key findings and suggest diections fo futue eseach. RAND 999.

CORNELLI AND SCHANKERMAN / 99. The model The timing of the model is as follows: at the beginning of the fist peiod, fims decide how much to invest in R&D, which yields an invention at the end of the peiod. In the second peiod, fims choose a patent life of length T. The invention continues to be used indefinitely afte patent expiation. Let denote the size of the invention, which affects the level of demand, Q( p ), with Q 0 (subscipts denote patial deivatives). We assume that the innovating fim chages a unifom monopoly pice while the patent is in foce, and that the competitive pice pevails afte the patent expies. These assumptions simplify the analysis, but the agument also holds fo moe complex appopiability envionments, such as allowing fo pice discimination and licensing (see Section 6). Without loss of geneality we set maginal cost to zeo. The fim with the patent sets pice p to maximize flow evenue pq( p ). Let p*() agmax p pq( p ) and flow pofits duing the patent life be () p*()q( p*() ). Since by the envelope theoem d/d p*()q 0, () is monotonic inceasing and we can expess as a function of the level of (maximized) pofits: v(). The appopiability assumptions ae embedded in the function v(). Moe effective appopiation by the fim would shift the v() function down since it allows a smalle invention to geneate the same pofit. We efe to as the output of the R&D, but it should be intepeted as a summay statistic fo, given the capacity of the fim to appopiate the suplus, which detemines the elationship between and. All that mattes fo the subsequent analysis is that we can expess pofits as a monotonic function of the size of the invention. The fim maximizes total pofit given by T (, T) et dt ( e T). () Flow welfae (pofits plus consume suplus) duing the patent life is 0 W() Q(p v()) dp. () p*(v()) Afte the patent expies, pofits ae zeo and flow welfae is B() Q(p v()) dp. (3) 0 Note that B() W() D(), whee p*(v()) D() Q(p v()) dp (4) 0 is the deadweight loss fom the patent. It is easy to show that D () cannot be signed, so that lage inventions may geneate eithe lage o smalle deadweight loss. Howeve, The analysis also holds fo a pocess invention. In that case, if is unit cost savings, is monotonic in, but the level of demand Q is unaffected (given p). RAND 999.

00 / THE RAND JOURNAL OF ECONOMICS we assume that lage inventions geneate geate social benefits afte the patent expies, in the sense that B () 0 Q ( p v()) dp 0, since we have assumed Q 0. The govenment s objective function is W() D() t t T 0 T S(, T) W()e dt D()e dt e. (5) The fist tem epesents pofits and consume suplus fo the (infinite) life of the invention; the second tem is the additional gain that accues afte the patent expies (fomely deadweight loss). We tun next to the pocess geneating pofits. This involves, fist, the poduction of an innovation and, second, the appopiation of the suplus by the fim in the fom of pofits. Let z denote R&D input, and the paamete eflect both the capacity of the fim to geneate an innovation and to appopiate the suplus it geneates (heeafte, the R&D poductivity paamete ). The potential suplus fom the innovation is indexed by, which depends on R&D input and the fim s R&D poductivity paamete, i.e., (, z). Pofits depend on this suplus and the ability of the fim to appopiate these ents, i.e., h(, z). Theefoe, the maginal poductivity of R&D, h z,isa function of : diffeent fims have diffeent maginal poductivity in the R&D pocess. We assume that h 0, h z 0, and h zz 0. We also make the natual assumption that the maginal poductivity of R&D is nondeceasing in : h z 0. This ensues that the optimal diffeentiated patent policy is implementable (see Section 4). To simplify the analysis we assume quadatic costs of R&D effot, (z) z /, but the qualitative esults cay ove fo any nonconcave cost function. We will also study the special case z, whee fims have a constant maginal poductivity, which can be intepeted as the atio of R&D to pofit. The model teats the R&D pocess as deteministic (the stochastic R&D case is discussed biefly in Section 6). Given its pivate infomation on, the fim chooses the size of the invention (hence ) by setting R&D input z. The govenment does not know the value of the invention (equivalently, ), but it knows that is dawn fom the distibution function G(), defined ove the inteval [0, ], with density g(). 3. Benchmaks If the govenment obseved and R&D effot, it could enfoce the fist-best level of R&D effot without esot to patents. This is not achievable because of asymmetic infomation, thus we study the case whee the govenment uses patents to povide R&D incentives. 3 In this section we chaacteize the full-infomation, second-best patent policy, whee the govenment sets the optimal patent length fo each value of, and the optimal unifom patent life when is pivate infomation. These will seve as benchmaks fo the subsequent welfae compaisons. In both cases, the govenment has to take into account that given a patent length T, the fim sets R&D to maximize pofits. Given R&D effot z and poductivity, the fim eans pofit net of esouce costs: The fist-best level of R&D is defined by z FB () agmax z {[B(h(, z))]/ z /}. 3 See Scotchme (999) fo a geneal agument that asymmetic infomation is sufficient to justify the use of patents. RAND 999.

CORNELLI AND SCHANKERMAN / 0 T z z h(, z)et dt h(, z)( e T ). (6) 0 The fist-ode condition of the maximization of (6) is T ( e )hz z 0. Since h(, z) is concave, the second-ode conditions ae satisfied, and by the implicit function theoem thee exists a function such that z* (, T), with T 0 and sign[ ] sign[h z ]. Define the optimized pofit as H(, T) h(, (, T)), with H T 0 and H 0 povided h z 0. Fo efeence, in the linea case z we obtain z*(, T) (/)( e T ). Poposition. The full-infomation, second-best policy, T**(), satisfies the following equation at each value of : RAND 999. [W etd ]H z*z* et T T D(). (7) Poof. Maximizing S(, T) in (5) subject to the constaint H(, T) yields the fist-ode conditions in Poposition. Q.E.D. The policy T**() equates the maginal social benefit and cost of extending T. The left-hand side of (7) is the maginal benefit fom extending the patent life T. This eflects both the incentive effect on R&D and thus on the size of the invention (pofits), and the maginal social value of lage inventions. The ight-hand side is the maginal social cost, made up of the additional R&D cost and the discounted value of the additional peiod s deadweight loss associated with an incease in T. Now conside the case whee is pivate infomation, and suppose the govenment sets the optimal unifom patent length. This involves maximizing 0 S(, T) dg() subject to H(, T). The optimal unifom length is descibed in Poposition. Poposition. The optimal unifom patent length, T U, satisfies the following equation: T T T T 0 0 [W e D ]H dg() {z*z* e D()} dg(). (8) Poof. As in Poposition, using the objective function 0 S(, T) dg(). Q.E.D. Note that the govenment must know the distibution of the R&D paamete G() to set the optimal unifom patent length. As Section 4 shows, this infomation is not needed to set optimal diffeentiated patent lives. 4. Optimal diffeentiated patent policy When the fim s R&D poductivity is pivate infomation, the govenment may want to povide an incentive stuctue that shifts the distibution of R&D effot towad the high- fims. With the optimal unifom length, the high- fim aleady obtains geate pofit fom its moe valuable invention. But it is not sufficient because, in setting T U, the govenment has aveaged acoss (see Poposition ) and thus povides too little incentive to high- fims and too much to low- fims. The consequence is that

0 / THE RAND JOURNAL OF ECONOMICS the social cost of poducing inventions is not minimized. In this section we use a mechanism design appoach to deive the optimal patent policy T*(), and show that unde cetain conditions it is indeed diffeentiated. When the optimal patent scheme is diffeentiated, we chaacteize two indiect mechanisms to implement this policy. Fist, the govenment can offe a menu of patent lengths and associated upfont fees, and fims self-select when they apply fo a patent. If the fim does not lean about the value of the invention afte it patents, this mechanism is equivalent to a second one, a enewal scheme whee the fim makes a sequence of decisions to extend the patent by payment of enewal fees. In Section 6 we intoduce ex post leaning in a simplified vesion of the model and show that the enewal scheme is supeio to a menu with upfont fees. By the evelation pinciple, we estict attention to diect mechanisms whee the fim announces ˆ and the govenment detemines the patent length and the fee as a function of the announced value ˆ : {T( ˆ ), f ( ˆ )}. 4 Facing this schedule, the fim chooses R&D effot z to maximize pofits in (6), which yields z* (, T( ˆ )). Its payoff is theefoe given by RAND 999. T(ˆ ) 0 t U(, ˆ ) h(, z*)e dt (z*) f (ˆ ). (9) The welfae-maximization poblem becomes subject to and [ ] W(h(, z*)) D(h(, z*)) max et() z* dg(), (0) T,f 0 IR constaint: U(, ) 0, () IC constaint: agmax U(, ˆ ),, ˆ. () ˆ This is a standad mechanism design poblem, except that the fees that implement the optimal policy do not ente the objective function. To solve it, we fist use the individual-ationality (IR) and incentive-compatibility (IC) constaints to obtain the fee schedule f (), and then deive T() fom the unconstained maximization of (0). The following poposition chaacteizes the optimal patent mechanism. Poposition 3. An optimal diffeentiated patent policy, { f *(), T*()}, has to satisfy the following necessay conditions fo each : (i) R(T, ) [W etd ]H z*z* et T T D() 0 (3) s 0 (ii) f *() H(, T)A() (z*) h (s, z)a(s) ds, (4) whee z* (, T()) and A() [/]( e T() ). 4 Fo simplicity, we will assume that T is diffeentiable. Howeve, the agument can be extended to the case in which T is piecewise diffeentiable. See Scotchme (999) fo a discussion in a elated setup.

CORNELLI AND SCHANKERMAN / 03 Poof. See the Appendix. In the Appendix we deive the fist-ode conditions of the maximization implied by constaint () and, following Myeson (98), obtain the fee schedule that guaantees incentive compatibility, given in equation (4). The fist-ode conditions of the maximization of the objective function in (0) give equation (3). We also deive the sufficient conditions fo { f *(), T*()} to be optimal i.e., the second-ode conditions fo the maximization of (0) and the maximization implied by (). This last condition implies that the optimal patent length schedule is stictly inceasing: T * () 0. Theefoe, we must check that the patent policy defined by (4) is inceasing. If it is not, the optimal diffeentiated patent policy is not incentive compatible and the best the govenment can do is to set T constant. 5 Remak. Since the fist-ode condition in (3) is the same as in Poposition, the optimal diffeentiated patent schedule implements the full-infomation, second-best policy. Remak. Since welfae in (0) is maximized pointwise, the optimal patent schedule T*() does not depend on G(). By contast, the optimal unifom patent length does depend on this distibution. Remak 3. The patent fees in (4) equal the pesent value of maximized pofits net of R&D costs minus the infomation ent that must be left to the fim to induce evelation of. In the Appendix we show that the fees ae defined up to a constant. In (4) this constant is set equal to zeo, so that the fees ae as high as possible consistent with the individual-ationality constaint fo all types. Thus, the govenment is extacting all the infamaginal ent fom the fim except the infomational ent needed fo incentive compatibility. Scaling down all fees by a constant, howeve, would not change the optimal patent schedule because patent fees ae pue tansfes that do not affect social welfae in this model. If instead we teated them as a substitute fo costly public funds (as in Laffont and Tiole, 993), optimal patent policy would be closely tied to the shadow pice of such funds. It would be staightfowad to extend the model in this way. Howeve, in this aticle we want to emphasize the R&D incentive aspect of patent policy, since we do not think it is pactical to tie patent policy to fiscal conditions. Note that even when optimal patent fees ae the highest level possible consistent with the individual-ationality and incentive-compatibility constaints, they could be negative fo low- fims. Since incentive compatibility equies T() to incease in, to avoid vey long patents it may be optimal fo the govenment to set vey shot lives and to subsidize the R&D fo low- fims. But this would involve costly public funds, and monitoing costs to ensue that the subsidized fims actually innovate. These costs would have to be taken into account in the optimal design if patent fees enteed the objective function. In that setting, the optimal policy could imply that all the fees ae scaled up, to ensue they ae non-negative fo all s. In this case, fims with low values may not undetake any R&D at all. This would be equivalent to imposing a minimum standad fo patentability. With a deteministic R&D pocess, constaining fees to be nonnegative implies that some low- fims will choose not to do R&D, since it will not geneate sufficient pofit to cove the minimum patent fee. Howeve, since the govenment can only diffeentiate patent potection on the basis of R&D outcomes, if these ae stochastic then 5 If (4) implies a nonmonotonic schedule T(), T may be constant only fo some intevals. See Guesneie and Laffont (984). RAND 999.

04 / THE RAND JOURNAL OF ECONOMICS some high- fims may also poduce innovations that fall below the theshold. But this will occu less fequently than fo low- fims (in the stochastic dominance sense). Thee will be second-best, ex ante efficiency, but ex post inefficiency will aise due to asymmetic infomation. We now chaacteize two special cases to highlight the intuition and the key factos shaping the optimal design. Coollay. If z, then z* [/]( e T() ) and the optimal diffeentiated patent policy {T*(), f*()} satisfies the following necessay and sufficient conditions fo each : T T() (i) [W e D ( e )] D() 0 (ii) s f *() ( e T() ) ( e T(s) ) ds D() (iii) D [W et D ] D. 0 Conditions (i) and (ii) ae just the conditions given in Poposition 3 fo the case z. To deive the inequalities in (iii), we substitute the fist-ode condition given in (i) into the two sufficient conditions deived in the Appendix (poof of Poposition 3, point (iii)). The left-hand side guaantees that T * () 0, and the ight-hand side is the second-ode condition fo the welfae maximization. The inequalities in (iii) depend on how welfae and deadweight loss vay with the level of pofit. This undelines the fact that optimal patent policy depends on how the distibution of pivate and social benefits vay with the size of the invention, and thus on any othe policies that affect this elationship (such as competition policy constaints on licensing and othe foms of appopiation). If B() is convex, it is moe likely that the left-hand inequality (incentive compatibility) is satisfied. This may aise fom two souces. Fist, lage innovations may be moe likely to geneate R&D spilloves than maginal ones. This emains an unesolved empiical issue. 6 Second, the demand elasticity fo poducts deived fom lage inventions may be lowe than fo moe maginal inventions. In the absence of full appopiation by the invento, this can geneate a convex welfae function. Fo example, in the phamaceutical industy an impotant new dug tageted at a lage maket may be socially moe valuable than many smalle impovements on existing dugs that geneate the same pivate etuns to the fim. By allowing longe patents and hence moe than popotional inceases in pofits fo lage inventions, a diffeentiated patent policy can induce fims to tilt thei R&D activity towad poducing such inventions. Coollay. If welfae and deadweight loss ae popotional to pofits, W() and D(), and z, then it is optimal to set a unifom patent life equal to 7 6 The empiical liteatue documents R&D spilloves (e.g., Jaffe, 986; Benstein and Nadii, 989; Jaffe, Tajtenbeg, and Hendeson, 993), but thee is no evidence on whethe the wedge between social and pivate etuns is elated to the level of pivate etuns. 7 The linea specification equies, i.e., the maginal deadweight loss must exceed the gain in consume suplus fom an incease in patent life. Othewise the optimal unifom length is infinite. RAND 999.

CORNELLI AND SCHANKERMAN / 05 T U [ln( ) ln( )]. In this special case, T () 0, and since the second-ode conditions fo welfae * maximization ae met, the optimal patent policy is constant. Howeve, note that fo a geneic concave pofit function h(, z) it is not geneally optimal to set a unifom patent length even when welfae and deadweight loss ae popotional to pofits. We have chaacteized the optimal diect mechanism. We now show it can be implemented by using two altenative indiect mechanisms, which ae obseved in pactice: an upfont menu of patent lengths and fees, and a enewal scheme. With the upfont menu, a fim chooses among diffeent patent lengths T and pays the coesponding fee F(T). To find the fee schedule that implements the optimal diect mechanism, note that T*() is monotone stictly inceasing, so we can invet it and obtain (T*). Substituting this into the fee schedule of the optimal diect mechanism, f *(), deived in Poposition 3, we obtain the lump-sum payment associated with each patent length, F(T) f *((T)). The maximum patent life is given by T T*(). To find the enewal mechanism, we need the fee R(t) that a fim must pay in each peiod t if it wants to enew its patent fo that peiod. The minimum patent life is given by T*(0) 0, which is associated with a fee R(0) f *(0). All the othe enewal fees fo each peiod T*() ae such that T*() f *() R(t)et dt. 0 The elationship between the fees of these two indiect mechanisms is given by df df *((T)) R(T)e T. dt dt That is, the gadient of the optimal upfont payment schedule is given by the pesent value of the enewal fee, fo each patent length. To establish equivalence between the diect mechanism and the enewal scheme, we need to show that the incentive-compatibility constaint is satisfied and the fim chooses the same level of R&D unde the enewal scheme. In the diect mechanism the fim s optimization and incentive-compatibility constaint in (6) and () ae expessed as functions of. Since thee is a one-to-one elationship between T and, the solutions to these maximization poblems will be the same if they ae expessed in tems of T, as in the indiect mechanism. 8 5. Simulating the optimal mechanism In this section we use simulation analysis to illustate the key featues of the optimal diffeentiated patent mechanism and to compute the welfae gains elative to an optimal unifom patent system. In addition, we compae the patent lives and fees 8 If the patent schedule T*() is discontinuous, this agument equies efinement. Scotchme (999) shows that, in both he model and ous, any R&D plan implementable with a diect mechanism is implementable with a enewal mechanism, even when T*() is discontinuous. This would cove, fo example, cases whee fims ae segmented into two goups (low and high poductivity) with diffeent fixed patent lives. RAND 999.

06 / THE RAND JOURNAL OF ECONOMICS fom the optimal mechanism with existing statutoy patent lives and fees in Fance, Gemany, and the United Kingdom. To conduct the simulations, we use a linea R&D pocess, z, and specify flow welfae and deadweight loss as W() and D(). We un the simulations fo a vaiety of (,, ) values. 9 Fo each value of, we compute the optimal patent length T() fom equation (3) and check that the second-ode and incentivecompatibility conditions ae met. The optimal fee schedule f () is computed fom (4) and welfae S(, T) fom (0). The distibution of is calibated to be boadly consistent with the obseved distibution of the atio of pofits to R&D in U.S. manufactuing. The distibution is assumed to be skewed to the ight, eflecting a tail of highly poductive R&D-pefoming fims, with a mean value of 3.75. This is vey close to the (weighted) aveage atio of cash flow to R&D in U.S. manufactuing fo the peiod 983 987 of 3.7, based on Compustat data. 0 Figues 3 summaize the esults fo selected paamete values, but the featues epoted hee ae obust to paamete vaiations within the ange examined. We pesent the optimal patent schedule T(), the optimal patent fees F(T), and the atio of F(T) to maximized pofits fo each T (the equivalent tax ate on patent etuns), which we denote by q(t). We also povide the optimal unifom length, T U, and the pecentage gain in welfae fom intoducing the optimal mechanism, elative to the optimal unifom length, denoted by S. As Figue shows, the simulations indicate an optimal unifom patent length of between 5 and 9 yeas, which is vey close to the statutoy lifespans in most counties. Thee ae thee stiking and obust featues of the optimal diffeentiated patent schedules in Figue. Fist, thee is a minimum length (about 7 yeas), even fo vey low values of. This eflects the fact that while the social value fo such small patents is low, so too is the deadweight loss, and the R&D incentive effect justifies the potection. Second, fo the bulk of the distibution of, the ange of optimal patent lives is quite naow, typically between 8 and 5 yeas. Howeve, the thid featue is that the optimal lives fo vey high values of ae much longe than existing statutoy limits (0 yeas o less). And although thee may be elatively few patents that waant longe patent lives, these ae the patents with the geatest contibution to welfae. The optimal mechanism aises aggegate welfae by 7%, as compaed to the optimal unifom length. The welfae gain depends on two featues of the undelying welfae and deadweight-loss functions. Fist, the welfae gain ises with the convexity of the welfae function,. Second, the gain is lage when the atio of deadweight loss to welfae declines moe quickly with the size of the innovation (pofits), i.e., when is highe. The welfae gain fom the optimal mechanism exceeds 0% fo some paamete values examined. 9 Expeiments showed that the second-ode and incentive-compatibility conditions ae moe commonly satisfied when,, and is sufficiently lage. We examined the paamete space (.0,.0), (, 0), and (.5,.0), which allows fo vaiations in the convexity of the welfae function (), the atio of the deadweight loss to pofits ( and ), and the way in which this atio vaies with pofits (). 0 Two points should be noted. Fist, cash flow is defined hee as opeating income plus depeciation minus taxes. Fo details see Hall (99). Second, we epesent G() by a seies of five unifom distibutions ove the ange (0, 30): 0% of the mass between. and, 55% between and 4, 0% between 4 and 6, and the emaining 5% between 6 and 30. Simulations ae conducted ove the gid of at intevals of.. Unde the ecent Wold Tade Oganization Ageement, signatoy counties have hamonized thei statutoy patent life at 0 yeas (but not thei enewal fees). If becomes too high, the benefits fom lage inventions incease vey apidly and the optimal patent length associated with the highest s becomes infinite. RAND 999.

CORNELLI AND SCHANKERMAN / 07 FIGURE OPTIMAL DIFFERENTIATED PATENT LENGTHS Figue shows that the optimal (upfont) patent fees ise shaply with patent life. The gadient of the cuves coesponds to the annual enewal fee, which also ises fo patent lengths up to about 0. This featue is qualitatively consistent with existing statutoy enewal fee schedules (Schankeman and Pakes, 986). The optimal patent fees ise moe apidly than the associated pofits fom the patent, so that the equivalent tax, q(t), is pogessive, as shown in Figue 3. This impotant featue of the optimal mechanism is violated by existing enewal fee schedules. To make the compaison, we use the estimates of the value of patent ights fom Schankeman and Pakes (986) to deive the atio between actual cumulative enewal fees and the pofits fom patent potection in Fance, Gemany, and the United Kingdom. We use thei paamete estimates of the distibution of initial etuns to patent potection. We take 500 andom daws fom this distibution and, using thei estimated depeciation ate and the obseved enewal fees, compute fo each daw the optimal cancellation date (o the statutoy maximum, whicheve is ealie). We then compute the atio between the pesent value of the enewal fees and the etuns fom patent potection until that cancellation date. (Note that enewal fees ae equied fo patent ages 0 in Fance, 3 8 in FIGURE OPTIMAL PATENT FEE SCHEDULES RAND 999.

08 / THE RAND JOURNAL OF ECONOMICS FIGURE 3 EQUIVALENT TAX RATES FROM THE OPTIMAL PATENT MECHANISM Gemany, and 5 6 in the United Kingdom.) The value-weighted aveage of this atio fo all patents enewed to each patent length is epoted in Figue 4. The figue shows that fo all thee counties enewal fees constitute a egessive tax on pofits fom patents, declining fom about 50% fo patents cancelled at ealy ages to less than % fo those enewed until the statutoy limit. In shot, the simulation analysis indicates that optimal patent lives should extend beyond the typical statutoy maximum (o the 0-yea life equied by the Wold Tade Oganization Ageement), and that the optimal enewal fees should ise much moe with patent length than existing fee schedules. 6. Extensions In this section we biefly intoduce thee extensions of the model: stochastic R&D outcomes, postpatent leaning, and appopiability. The pupose is to illustate how the model can be adapted to incopoate a iche desciption of the R&D pocess and postinvention competition. A complete teatment of these issues is left fo futue wok. FIGURE 4 EQUIVALENT TAX RATES IMPLIED BY STATUTORY PATENT FEES IN DIFFERENT COUNTRIES RAND 999.

CORNELLI AND SCHANKERMAN / 09 Stochastic R&D outcomes. To intoduce a stochastic R&D pocess, we conside a isk-neutal fim with R&D technology h(, z), whee has distibution V() and is obseved only afte the fim chooses its R&D. Unde the diect mechanism, the fim announces ˆ and obtains a patent of length T( ˆ ). The pofit-maximizing level of R&D, z*, is given by [ ] h z(, z*) ( e T(h(,z*)) ) z* dv() 0. Then the welfae-maximization poblem becomes [ ] W(h(, z*) ) D(h(, z*) ) max et() z* dg() dv(). T,f 0 Given a distibution V(), standad simulation techniques can be used to solve this poblem. While the chaacteistics of the optimal patent schedule will depend on V(), thee will again be conditions unde which it is optimal to diffeentiate patent lives. Moeove, when R&D outcomes ae stochastic, thee is an additional eason fo the govenment to diffeentiate. Since the fim s net pofit in (6) is convex in T, offeing a convex schedule of patent lives will incease the incentives to undetake R&D. In the exteme case whee all fims have the same R&D poductivity that they lean only afte innovating (i.e., R&D outcomes ae puely stochastic), it may be optimal to offe a zeo patent fo low values of and an infinite life fo high values. (Details of the analysis ae available on equest.) Postpatent leaning. We have assumed that fims know the value of thei innovation when they apply fo a patent. In eality, howeve, fims only have a pio distibution on thei etuns, which they update as they lean duing the ealy life of the patent. Econometic studies have documented such postpatent leaning and shown that it is lagely completed within fou o five yeas afte the patent application date (Pakes, 986; Lanjouw, 998). We conside a simplified model with postpatent leaning and show that, in this case, a patent enewal scheme is welfae supeio to an upfont menu of lengths and fees. We believe that the qualitative esult extends to moe complex models of leaning, but we have not fomally poved this conjectue. 3 We modify the timing of the model in the following way. In peiod fims apply fo the patent knowing only the expected pofit pe peiod,. Afte a peiod of length, fims lean with cetainty the value of the pofit pe peiod, which can take on a value of eithe zeo o with equal pobability. We call the peiod stating afte peiod, and assume that z. 4 The evelation pinciple also holds in this setting (Townsend, 98), so we can focus on the diect mechanism. The govenment sets up a mechanism in which each fim announces ˆ in peiod and ˆ in peiod. The mechanism specifies a fee f ( ˆ ) 3 The intuition also applies to the case of pue obsolescence, whee thee is some pobability each peiod that the fim s etuns fom the invention fall to zeo. See Lanjouw (998) fo paametic estimates of obsolescence, using patent enewal data. 4 A moe geneal setup would model second-peiod etuns as coelated with accoding to a conditional distibution function G(, ) that satisfies the fist-ode stochastic dominance, G 0. Then the optimal patent schedule and fees would depend on the natue of the leaning pocess, as eflected in the function G (Conelli and Schankeman, 996). A geneal teatment, applied to the case of pollution pemits, can be found in Laffont and Tiole (994). RAND 999.

0 / THE RAND JOURNAL OF ECONOMICS fo a patent length T ( ˆ ) in the fist peiod, and a fee f (, ˆ ˆ ) fo length T (, ˆ ˆ ) in the second peiod. Fo simplicity we estict the mechanism so that T ( ˆ ) : the fim can eithe choose a length shote than in which case enewal neve aises o it has to choose whethe to enew at date. Rathe than fully chaacteize the optimal mechanism fo this special case, we focus on whethe it is bette fo the govenment to allow fims to abandon the patent athe than to equie the full patent fees to be paid upfont. Poposition 4. It is optimal fo the govenment to use a patent enewal scheme athe than an ex ante payment scheme. Poof. See the Appendix. The intuition fo this esult is that expected pofits ae convex in the level of ex post pofit (i.e., in the andom component associated with leaning), so the isk-neutal fim pefes the option of taking out a longe patent life if ex post pofits tun out to be high. Pofits ae convex because the optimal patent scheme involves giving longe patent lives to moe pofitable inventions (fo incentive compatibility). Appopiability envionment. The optimal patent policy depends on the appopiability envionment, including patent licensing ules and othe aspects of antitust policy (Gallini and Tebilcock, 998; Gilbet and Shapio, 997). The eason is that the flow welfae and deadweight loss will depend on the degee of appopiability, as well as the size of the invention. We biefly sketch how the model can be extended to incopoate this inteaction. Let denote the degee of appopiability (the faction of the invention s social benefits eceived by the fim: /B). As in Section, we can invet (, ) and wite flow welfae and deadweight loss as W(, ) and D(, ). To incopoate appopiability in the R&D pocess, we wite h(z,, ), whee all deivatives ae positive. Note that total social benefits B() do not depend on, so that W D. It is impotant to distinguish between two types (o souces) of appopiability, which diffe in tems of how they ae likely to affect flow welfae and deadweight loss. The fist aises fom factos that enhance the patentee s ability to appopiate consume suplus (e.g., moe feedom to pice disciminate). This gives the fim geate ability to extact infamaginal ents and should be associated with inceased flow welfae and deceased deadweight loss (given the size of the invention). The second type aises fom geate maket powe due, fo example, to it being moe difficult to imitate the invention fo eithe legal o technical easons. This gives the fim geate ent by shifting the magin and should be associated with deceased flow welfae and inceased deadweight loss. Once the welfae and deadweight loss functions ae specified, the optimal patent policy, obtained using the methods in this aticle, will be a function of the appopiability paamete. It is staightfowad to deive the compaative statics of a change in on the optimal patent menu. The implications will depend, among othe things, on whethe the change in appopiability is associated with moe lenient antitust policy o geate maket powe (i.e., on how it affects flow welfae and deadweight loss). Thus, while no geneal conclusions can be eached without futhe specification, the famewok can be used to study this issue. 7. Conclusions This aticle shows how patent enewal fees can be used to diffeentiate patent lives and theeby solve poblems of asymmetic infomation between R&D fims and policy RAND 999.

CORNELLI AND SCHANKERMAN / makes. This allows the govenment to povide R&D incentives moe efficiently. Ou appoach emphasizes how heteogeneity among fims is cucial in detemining the optimal use of patent fees. We also illustate with simulation analysis how to implement the optimal patent mechanism and compae the key featues of the simulated optimal mechanism with existing patent schemes. Ou model has been delibeately simplified, but the easoning undelying the deivation of the optimally diffeentiated patent mechanism should apply to iche specifications. The application of optimal egulation unde vaious foms of asymmetic infomation can be extended to policy design in many othe contexts (see Laffont and Tiole (994) fo an application to pollution egulation). The eseach can be extended in seveal useful diections. Fist, postpatent leaning is an empiically impotant featue, and moe detailed study of this issue is waanted. The simulation analysis in Section 5 can be extended to study how postpatent leaning and R&D uncetainty affect the optimal patent mechanism. Second, appopiability conditions ae an impotant deteminant of R&D incentives and hence of the optimal patent mechanism, and they equie futhe modelling. Finally, the analysis should be extended to a dynamic famewok with sequential innovative activity and stategic inteaction among fims. Appendix Poofs of Popositions 3 and 4 follow. Poof of Poposition 3. (i) Define U() max U(, ˆ ). ˆ By the envelope theoem, Reintegating this equation we obtain du ( e T() )h. d T(s) U() h s(s, z)( e ) ds K, 0 whee K is a constant of integation. Given that the fees do not ente the govenment s objective function, K is not uniquely defined. Howeve, the minimum K will satisfy the individual-ationality constaint with stict equality: U(0) 0. If we set K 0 and equate U() tou(, ˆ ), defined in (9) and evaluated at ˆ, we obtain (4). This yields the maximum fees consistent with individual ationality and incentive compatibility in this model. (ii) Maximizing (0) pointwise on [0, ] yields the fist-ode conditions (3). (iii) The sufficient conditions ae given by the second-ode conditions fo the maximization of (0) and fo the incentive compatibility to hold. The fist of these is given by RAND 999. [W etd ]H [W etd ]H (z*) z*z* etd() et TT T T TT D HT 0. The second-ode condition fo incentive compatibility the maximization in () is U T ()[h e T A()h z z *] 0. ˆ This implies that T () 0. To guaantee T () 0, note that by implicit function theoem T () has the same sign as R (), so the sufficient condition will be satisfied if R () 0. Q.E.D. Poof of Poposition 4. A fim chooses z to maximize net pofits. Conside the decision poblem fo a fim with a high enough so that it suvives at least to (the othe case is identical to the analysis in Section 4): T

/ THE RAND JOURNAL OF ECONOMICS This yields [ ] T (,) t t z 0 max ze dt f () f (, 0) e dt f (, ) z T (,) z( e ) f () [ f (, 0) f (, )] z. T (,) z* ( e ). (A) The govenment maximizes its objective function subject to the individual-ationality constaint of the fist and second peiod T(,) ˆ EU(, ˆ ) ( e ) f (ˆ ) [ f (ˆ, 0) f (ˆ, )] 0 (A) and T (, ) [e e ] f (, ) 0. (A3) Thee is also the fist-peiod incentive-compatibility constaint, which equies that the fim not misepesent given that it anticipates epoting tuthfully in peiod, agmax EU(, ˆ ) ˆ (A4) and the second-peiod incentive-compatibility constaints and f (,) ˆ f (,0) ˆ ˆ (A5) T (,0) ˆ T (,) ˆ [e e ] f (ˆ, ) f (ˆ, 0) ˆ. (A6) Fo 0, (A) and (A3), and the fact that T ( ˆ, 0) does not have any incentive effect as it is clea fom equation (A) imply that T (,0) ˆ 0 and f (,0) ˆ 0. That is, implementation of this mechanism equies that the govenment leave fims the option to abandon thei patent in case the invention tuns out to be low-valued. Note that this poof elies only on the individual-ationality constaints fo the two peiods. The incentive-compatibility constaints ae given only fo completeness. Q.E.D. Refeences BERNSTEIN, J.I. AND NADIRI, M.I. Reseach and Development and Inta-industy Spilloves: An Empiical Application of Dynamic Duality. Review of Economic Studies, Vol. 56 (989), pp. 49 67. CORNELLI, F. AND SCHANKERMAN, M. Optimal Patent Renewals. Woking Pape no. EI/3, London School of Economics, 996. DE LAAT, E. Patent Policy and the Timing of Imitation. Mimeo, Tinbegen Institute, Easmus Univesity, Rottedam, 997. GALLINI, N. AND TREBILCOCK, M. Intellectual Popety Rights and Competition Policy: A Famewok fo the Analysis of Economic and Legal Issues. In R.D. Andeson and N.T. Gallini, eds., Competition Policy and Intellectual Popety Rights in the Knowledge-Based Economy. Calgay: Univesity of Calgay Pess, 998. GILBERT, R. AND SHAPIRO, C. Optimal Patent Length and Beadth. RAND Jounal of Economics, Vol. (990), pp. 06. AND. Antitust Issues in the Licensing of Intellectual Popety: The Nine No-No s Meet the Nineties. Bookings Micoeconomic Papes, (997), pp. 83 349. RAND 999.

CORNELLI AND SCHANKERMAN / 3 GREEN, J.R. AND SCOTCHMER, S. On the Division of Pofit in Sequential Innovation. RAND Jounal of Economics, Vol. 6 (995), pp. 0 33. GUESNERIE, R. AND LAFFONT, J.-J. A Complete Solution to a Class of Pincipal-Agent Poblems with an Application to the Contol of a Self-managed Fim. Jounal of Public Economics, Vol. 5 (984), pp. 39 369. HALL, B. Investment and Reseach and Development at the Fim Level: Does the Souce of Financing Matte? NBER Woking Pape no. 4096, 99. JAFFE, A. Technological Oppotunity and Spilloves of R&D: Evidence fom Fims Patents, Pofits and Maket Value. Ameican Economic Review, Vol. 76 (986), pp. 984 00., TRAJTENBERG, M., AND HENDERSON, R. Geogaphical Localization of Knowledge Spilloves as Evidenced by Patent Citations. Quately Jounal of Economics, Vol. 08 (993), pp. 577 598. KLEMPERER, P. How Boad Should the Scope of Patent Potection Be? RAND Jounal of Economics, Vol. (990), pp. 3 30. LAFFONT, J.-J. AND TIROLE, J. A Theoy of Incentives in Pocuement and Regulation. Cambidge, Mass.: MIT Pess, 993. AND. Pollution Pemits and Compliance Stategies. Woking Pape no. 39, IDEI, Toulouse, 994. LANJOUW, J. Patent Value in the Shadow of Infingement: Simulation Estimations of Patent Value. Review of Economic Studies, Vol. 65 (998), pp. 67 70. MYERSON, R.B. Optimal Auction Design. Mathematics of Opeations Reseach, Vol. 6 (98), pp. 69 63. NORDHAUS, W. Invention Gowth and Welfae. Cambidge, Mass.: MIT Pess, 969. O DONOGHUE, T., SCOTCHMER, S., AND THISSE, J.-F. Patent Beadth, Patent Life, and the Pace of Technological Pogess. Jounal of Economics and Management Stategy, Vol. 7 (998), pp. 3. PAKES, A.S. Patents as Options: Some Estimates of the Value of Holding Euopean Patent Stocks. Econometica, Vol. 54 (986), pp. 755 784. SCHANKERMAN, M. How Valuable is Patent Potection? Estimates by Technology Field. RAND Jounal of Economics, Vol. 9 (998), pp. 77 07. AND PAKES, A. Estimates of the Value of Patent Rights in Euopean Counties Duing the Post-950 Peiod. Economic Jounal, Vol. 96 (986), pp. 05 077. SCOTCHMER, S. Potecting Ealy Innovatos: Should Second Geneation Poducts Be Patentable? RAND Jounal of Economics, Vol. 7 (996), pp. 3 33.. On the Optimality of the Patent Renewal System. RAND Jounal of Economics, Vol. 30 (999), pp. 8 96. TOWNSEND, R.M. Optimal Multipeiod Contacts and the Gain fom Enduing Relationships Unde Pivate Infomation. Jounal of Political Economy, Vol. 90 (98), pp. 66 86. WRIGHT, B.D. The Economics of Invention Incentives: Patents, Pizes, and Reseach Contacts. Ameican Economic Review, Vol. 73 (983), pp. 69 707. RAND 999.