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The optimal life of a patent when the timing of innovation is stochastic
International Journal of Industrial Organization 17 (1999) 827–846
The optimal life of a patent when the timing of innovation is stochastic Vincenzo Denicolo` * Dipartimento di Scienze Economiche, Universita` di Bologna, Piazza Scaravilli 2, I-40125 Bologna ( Bo), Italy Accepted 2 September 1997
Abstract I re-examine the problem of optimal patent life within a model where the timing of innovations is stochastic. It is assumed that innovations occur according to a Poisson stochastic process where the hazard function is linear in R&D effort. Firms performing R&D have to pay a fixed cost, and there is free entry in the R&D industry. Among the comparative statics results, I show that small innovations should be protected more than larger ones. Also, when the patent life is set optimally, there is under-investment in R&D. 1999 Elsevier Science B.V. All rights reserved.
1. Introduction The aim of the patent system is to remunerate innovative activity and stimulate investment in R&D. However, patents entail a social cost because of the monopoly privileges accorded to the patentees. Thus, in deciding the lifetime of a patent, society must balance the gains accruing from faster technological progress against the welfare loss associated with the temporary monopoly in the use of the new technology. The classic analysis of the optimal patent life is based on the hypothesis of a deterministic R&D technology–see Nordhaus (1969), (1972) and
* E-mail address: [email protected] (V. Denicolo) ` 0167-7187 / 99 / $ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S0167-7187( 97 )00061-1
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Scherer (1972). In Nordhaus’ model, investment in R&D leads, with probability one, to an innovation whose size, measured as the percentage cost reduction that the innovation brings about, is an increasing function of the R&D expenditure. Though the assumption of deterministic innovations was standard at the time of Nordhaus’ report, its limits as a model of innovative activity have since been widely recognized. That firms can instantaneously ‘‘buy’’ a reduction of their variable production costs seems more appropriate as a description of investment in some fixed production factor than as a model of R&D.1 Starting with Loury (1979) and Lee and Wilde (1980), the standard approach to R&D competition assumes that the date of the innovation is uncertain (see Reinganum (1989) for a survey). In this paper, I re-examine the issue of the optimal lifetime of a patent within a standard model of patent races.2 Thus, the present paper fills a gap in the literature on the optimal patent life and provides a new framework where issues of patent policy can be analyzed. Besides allowing for uncertainty in the timing of innovation, I generalize Nordhaus’ analysis and the subsequent literature in other ways. In particular, my description of the product market is more general than in most of the previous literature. Among the results I obtain, two are especially noteworthy. First, I show that, generally speaking, at the optimal patent life, there is under-investment in R&D. Second, I prove that with perfect competition in the product market, small innovations should be protected more than large ones. This problem cannot be addressed using a Nordhaus-type model, where the size of the innovation depends continuously on R&D investment rather than being an exogenous parameter. The layout of the paper is as follows. In Section 2, I describe the basic assumptions of the model. The patent race is analyzed in Section 3. Section 4 states the social problem, characterizes the optimal patent life, and derives several comparative statics results. I then show in Section 5 that, when the length of the patent is set optimally, there is always under-investment in R&D with respect to
1
Kamien and Schwartz (1974) were the first to criticize Nordhaus’ model on these grounds. They analyze the effect of varying the patent’s length on the equilibrium R&D effort in a model that explicitly incorporates uncertainty on the timing of the innovation. However, they do not analyze the full equilibrium in the R&D industry nor do they address the issue of the optimum patent length. 2 Gilbert and Shapiro (1990) claim that the emphasis on the optimum life of patents has been misplaced, for the relevant policy dimension is patent breadth, not length. They have found a sufficient condition for the optimal patent length to be infinite, with patent’s breadth adjusted so as to provide the patentee with the appropriate incentive to perform R&D. However, conditions can be found under which optimal patents should confer the widest protection of the invention, and their lifetime should be ` 1996). adjusted to stimulate investment in R&D (see Klemperer, 1990; Gallini, 1992; Denicolo, Moreover, it is likely that the same factors which affect the optimal length of patents also determine the optimal patent breadth when the latter is the relevant policy instrument.
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the social optimum. Finally, Section 6 analyzes the issue of small versus large innovations, assuming perfect competition in the product market.
2. Basic assumptions For the sake of simplicity, and also to facilitate a comparison with the earlier literature, I retain some of the original hypotheses of Nordhaus (1969). Thus I consider a single cost reducing innovation,3 and assume that the private and social discount rates coincide. Also, I postulate a sharp distinction between the invention industry and the product market. Firms which operate in the invention industry are pure laboratories that produce inventions and innovations. They race to innovate first and the winner patents the innovation which will then be licensed to users operating in the downstream product market. However, I depart from Nordhaus (1969) in a number of ways. I assume that the size of the innovation is exogenously given, but its timing stochastically depends on R&D investment. In modelling the R&D technology, I follow Loury (1979) and Dasgupta and Stiglitz (1980). They assume a Poisson discovery process and postulate that R&D costs are paid at the beginning of the race (contractual costs). In particular, Dasgupta and Stiglitz (1980) assume that each R&D project is characterized by an optimal scale, but that each firm can operate many R&D projects simultaneously. Ignoring the integer constraint, this is tantamount to assuming a linear (in R&D costs) hazard function. This hypothesis, though somewhat restrictive, seems justified because there is no strong evidence of the existence of economies of scale in R&D. Moreover, it greatly simplifies the analysis 4 and so will be maintained throughout the paper. Concerning the structure of the invention industry, I analyze an oligopolistic equilibrium with a fixed R&D cost and free entry. The analysis is especially simple in the limit case where the fixed cost tends to zero, which loosely speaking corresponds to perfect competition in the invention industry.
3
However, I allow for drastic innovations. In a sense, a product innovation is equivalent to a drastic cost reducing innovation, for one can imagine that the new product was already known but could be produced only at prohibitively high costs before the innovation. 4 The hypothesis that R&D costs are paid at the beginning of the race is made for analytical convenience. Indeed, under this assumption on R&D costs, a linear hazard function yields an interior solution to the firm’s profit maximization problem. If instead one modeled R&D costs as in the alternative specification of Lee and Wilde (1980), according to which they are flow costs that are paid until the innovation occurs, one would have to assume a concave hazard function (at least over the relevant range) to obtain an interior solution. For a comparison of these two approaches to modelling R&D costs see Reinganum (1989).
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In most of my analysis, I black-box the product market, which is described by the following three parameters: – H, the patentee’s profits during the lifetime of the patent; – J, the further increase (if any) in social welfare that the innovation brings about during the lifetime of the patent (this increase in social welfare is not captured by the patentee; it may be enjoyed by consumers or by other firms); – K, the deadweight loss from the patent, that is, the potential increase in social welfare which becomes available to society only when the patent expires (during the lifetime of the patent, it is lost due to the monopolistic distortions created by the patent). H, K and J are stationary flows; in other words, there is no dynamics in the product market apart from that generated by the innovation. This description of the product market may be compatible with several alternative assumptions on the type of competition and the nature of the innovation. The initial structure of the product market determines the optimal licensing strategy of the innovator and the resulting changes in the market equilibrium; thus, it determines the values of H, K and J. In particular, my description of the product market allows for the case of imperfect patent protection, so that the patentee’s competitors may imitate the innovation or invent around the patent without infringing it. Obviously, H is always positive. Though one may easily construct nonpathological examples where K and J are negative, to simplify the exposition I shall assume J,K > 0. In the special case K 5 J 5 0, an infinite patent life would guarantee that the social and private value of the innovation coincide (see Loury, 1979; Lee and Wilde, 1980). Another special case that has been paid much attention to in the literature is J 5 0; this case arises, for instance, with perfect competition in the product market when the innovation is not drastic and there is perfect patent protection (as in Nordhaus’ model). The case of perfect competition in the product market may be used as an illustrative example. (It will also be used in Section 6 to address the issue of small vs. large innovations.) Suppose initially all firms have access to the same technology and can produce with constant marginal cost c, and the innovation reduces production costs to c 2 d. Let Q( p) denote the demand function. If the innovation is not drastic and there is perfect patent protection, so that the innovation cannot be imitated, the patentee will license the new technology for a fee d per unit of output. Thus, in the post-innovation equilibrium output will remain at the pre-innovation level Q 0 5 Q(c) and the patentee’s profits will be equal to the cost reduction d times the pre-innovation output. This is the area H in Fig. 1. After the patent expires, there is free access to the new technology and the output level rises to the point Q 1 5 Q(c 2 d), where the price equals the new
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Fig. 1. Non-drastic innovation.
marginal cost. The patentee’s profits are then driven to zero. The deadweight loss created by the patent is given by the triangle K in Fig. 1. In the case depicted in Fig. 1, during the lifetime of the patent the patentee reaps the entire social benefit from the innovation (J 5 0). But now consider the case of a drastic innovation.5 In this case, the patentee will license the new technology to a single firm for a fixed fee equal to the area H in Fig. 2. The main difference to the
Fig. 2. Drastic innovation.
5
An innovation is drastic if the monopoly price associated with the new marginal cost c 2 d is lower than the old marginal cost c.
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case of non-drastic innovation is that now part of the increase in social welfare which society enjoys before the patent expires is not captured by the patentee. This corresponds to the area J in Fig. 2.6 At time T, the equilibrium price falls to c 2 d and social welfare is further increased by an amount K.
3. The patent race In this section I analyze the invention industry. I consider a race for a patentable innovation between n symmetric R&D firms (laboratories). I model the R&D technology as in Loury (1979) and Dasgupta and Stiglitz (1980). At time t 5 0, each firm i decides its R&D effort x i and pays a lump-sum amount a x i , where a is the constant marginal cost of research effort. The R&D effort determines the expected time of successful completion of the R&D project. Assuming that the date of innovation by firm i is exponentially distributed, and is independent of the date of innovation of other firms, the probability of firm i being successful at or prior to date t is 1 2 e 2x i t . The payoff function of firm i is the present value of expected profits, net of R&D costs, i.e.: `
EHexpF 2SO x 1 rDGt x VJdt 2 ax 2 F n
pi 5
j
0
x iV 5 ]]]] 2 a x i 2 F, n j 51 x j 1 r
O
i
i
j 51
(1)
where r is the interest rate, x i is i’s R&D effort and also i’s hazard rate, so that exp[2(o jn51 x j )t] is the probability that no firm has innovated by time t, V is the present value of the profits accruing to the winner, F is a fixed R&D cost, and a x i is the variable cost of R&D. Both fixed and variable costs are paid at time t 5 0. The prize to the winner V is given by the discounted profits from the patent:
6 A similar picture emerges in the case of imperfect patent protection. If the patent has a very broad scope, the new production process cannot be imitated and therefore the non-innovating firms will remain at their pre-innovation cost c. But if the patent is more narrowly defined, one can imagine that even the non-innovating firms can develop similar processes without infringing the patent and therefore reduce their costs to a certain extent – see Nordhaus (1972). One can measure the breadth of the patent by the fraction of the cost reduction that does not spill out as freely available technology to the non-innovating firms. Thus, denoting by (1 2 g ) the breadth of the patent, the non-innovating firms will have marginal costs equal to (c 2 g d). As a result J . 0, reflecting the difference between the social and the private benefit from the innovation during the life of the patent.
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833
T
E
zH V 5 H e 2rt dt 5 ], r
(2)
0
where z 5 1 2 e 2rT
(3)
denotes the fraction of the overall discounted profits H /r that is captured by the patentee. Each firm chooses its R&D investment to maximize its expected profits (1). The first order condition for a maximum is: (X2i 1 r)V ]]]] 5 a, (r 1 X2i 1 x i )2
(4)
n
where X2i 5 o j ±i x j . It may easily be checked that the second order condition holds. Since all firms are identical, I look for a symmetric equilibrium where x i 5 x for all i. Condition (4) then becomes: [(n 2 1)x 1 r]V ]]]]] 5 a. (r 1 nx)2
(5)
In the following I shall assume that there is free entry in the R&D industry. Thus, the number of operating firms n is endogenous and is determined by the zero-profit condition: 7 xV ]] 2 a x 2 F 5 0. nx 1 r
(6)
Eqs. (5) and (6) determine the equilibrium number of firms n and the individual R&D effort x. They can be explicitly solved yielding: 8 Œ] FV 2 F x 5 ]]], (7) a ] V ra n 5 ] 2 ]]] . (8) F Œ] FV 2 F
œ
7 8
I ignore the integer problem treating n as a continuous variable. From Eqs. (5) and (6) one can obtain: x 2V (nx 1 r)2 5 ]. F
Substituting this expression into Eq. (6), Eq. (7) follows.
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Clearly, the equilibrium number of firms is a decreasing function of the fixed R&D cost F. When F goes to 0, the invention industry tends to become perfectly competitive.9 The aggregate R&D effort X 5 nx is (using Eq. (2)): ]] 1 zH zFH X 5 ] ] 2 ]] 2 r. (9) a r r
S œ D
If, even granting maximum patent protection, the policy-maker could not induce a positive R&D effort, patent policy would be ineffective. Thus I assume that an infinite patent life leads to X . 0. More precisely, denote by T¯ the minimum patent 2rT¯ length that induces a positive equilibrium R&D effort, and let z¯ 5 1 2 e , so that: ] ]] 2 r F F ] 1 ] 1 ar . z¯ 5 ] (10) H 4 4 Then, I shall assume that z¯ , 1 (or, equivalently, that T¯ is finite). This requires: ] ]] 2 H F F ]. ] 1 ] 1 ar . (11) r 4 4
Sœ œ
D
Sœ œ
D
In the limiting case F 5 0, this inequality reduces to ] H ] . r. a
œ
(12)
4. The optimal patent life The government chooses the value of the patent grant’s duration T that maximizes social welfare, that is the sum of consumers’ surplus and profits, net of R&D costs. I assume that the social rate of discount equals the interest rate r. Normalizing to zero the flow of social welfare before the innovation, the expected discounted social welfare is: `
E
W 5 XS e 2(X 1r)t dt 2 a X 2 nF,
(13)
0
9
The case of free entry in the R&D industry was first described, in the context of Nordhaus-type models, by Stiglitz (1969) and has been subsequently analyzed by Berkowitz and Kotowitz (1982) and De Brock (1985). Nordhaus’ own analysis considered only the case of monopoly in the invention industry. The present model can also be used to analyze the monopoly case (n 5 1)–see Denicolo` (1995) for details. Most of the results of the paper carry over to the case of monopoly in the invention industry.
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where S is the social value of the innovation. That is, S is the overall discounted social benefit from the new technology, which equals (H 1 J) before the patent expires and (H 1 K 1 J) thereafter: `
T
E
S 5 (H 1 J)e 0
2rt
E
dt 1 (H 1 J 1 K)e 2rt dt.
(14)
T
From Eqs. (13) and (14) one obtains:
F
G
X H1J K W 5 ]] ]] 1 (1 2 z)] 2 a X 2 nF. X 1r r r
(15)
To interpret this expression, notice that if the innovation process was completed instantaneously, the discounted social benefit would be (H 1 J) /r plus a fraction (1 2 z) of K /r. The factor X /(X 1 r) is a further discount factor that accounts for the (uncertain) delay in the occurrence of the innovation. If X is large, the expected date of innovation is small and therefore this discount factor is close to 1. As X decreases, society on average must wait longer and longer for the innovation and therefore the discount factor falls. Given the zero-profit condition (6), the policy-maker’s objective function may also be rewritten as:
F
G
X J H 1K W 5 ]] ] 1 (1 2 z)]] . X 1r r r
(16)
The social problem may now be stated as follows: Choose T so as to maximize W, given that X depends on T according to Eq. (9). Notice that W depends on T only through z, and (given r) there is a one-to-one correspondence between z and T. Thus the social problem is to choose z so as to maximize W. Differentiation of Eq. (16) yields the first order condition of the social problem, which applies when the solution is interior:
F
G
K 1H r J H 1 K dX 2 X ]] 1 ]] ] 1 (1 2 z)]] ] 5 0. r r dz (X 1 r) r
(17)
The first term on the right-hand side of Eq. (17) is the direct effect of a change in z on social welfare. When z increases, society can appropriate only a smaller fraction of the potential discounted gain K /r. In other words, when T increases society must wait longer and longer to appropriate the welfare triangle K. Moreover, since the R&D industry profits vanish, the private reward to the innovator zH /r is dissipated in the patent race and this also increases with z. Clearly, the direct effect of an increase in z on social welfare is always strictly negative provided X . 0 and vanishes at X 5 0. It measures the marginal social cost (MSC) of longer patent terms. The second term is the indirect effect of z on W through X and measures the
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marginal social benefit (MSB) of increasing the patent’s life. The net marginal social gain from a longer patent life is proportional to the sum of three components: J /r and a fraction (1 2 z) of H /r and K /r. The reason that only a fraction (1 2 z) of H /r and K /r enters the marginal social benefit is the following: zH /r is offset by the change in R&D costs (by the zero-profit condition) and zK /r is lost due to monopolistic distortions during the patent’s lifetime. Finally, the term r /(X 1 r) reflects the change of the ‘‘discount factor’’ X /(X 1 r) due to a change in X. Since ] ] dX H 1 FH X 1 r 1 FH ] 5 ] 2 ] ] 5 ]] 1 ] ] . 0, (18) dz ra 2a zr z 2a zr
œ
œ
the marginal social benefit of increasing the patent life is always positive. From the observation that the social cost of increasing the patent life vanishes at X 5 0 (that is, at T 5 T¯ or z 5 z¯ ) whereas the social benefit is always positive, it follows that the optimal life of a patent T * is positive and is associated with a positive aggregate R&D effort; that is, T * . T¯ . 0. Then, two cases are possible. If MSB is always greater than MSC, then social welfare is always increasing in the patent length and therefore the optimal patent life is infinite. If instead at z 5 1 the inequality MSB,MSC holds, then there will be a finite optimal patent life. Upon substitution, the marginal social cost becomes: ]] K 1 H zH 1 zHF ]] ] ] ]] 2 r ; MSC 5 2 r ra a r
S
œ
D
(19)
¯ Using Eqs. (9) and (18), the marginal it is increasing in z and vanishes at z 5 z. social benefit becomes: ] Œ] zH 2 ]12 ŒrF (H 1 K 1 J) MSB 5 ]]]] 2 (H 1 K) ]]]] (20) ] . Œ] z zH 2ŒrF
GS
F
D
MSB is always positive and decreasing in z. Evaluating MSC and MSB at z 5 1, it follows immediately that the optimal life of a patent is finite if and only if: ] (H 1 K) H 1 HF ]]] ] ] ]2r J, 2 r ra a r
S
œ
] Œ] ŒH 2 rF ]]]] ] . Œ] H 2 ]12 ŒrF
DS
D
(21)
This inequality is always satisfied when J 5 0. When condition (21) holds so that the optimal patent life is finite, T * is given by the unique solution to the first order condition: ] ]] Œ] zH 2 ]12 ŒrF (H 1 K 1 J) 1 zH 1 zHF ]]]] 2 1 ]]]] ] ] ] ]] 2 r 5 0, 2 2 (22) ] Œ] ŒzH r ra a r z(H 1 K) 2 rF
F
GS
D S
œ
D
¯ which lies in the interval (z,1). Fig. 3 presents the curves corresponding to the marginal social cost and the
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Fig. 3. Optimal patent length.
marginal social benefit of increasing the patent duration. The intersection of the two curves determines z * , which corresponds to the optimal patent life T * . Fig. 3 depicts the case where z * , 1 and therefore T * is finite. If the marginal social benefit curve always lies above the marginal social cost curve, the optimal patent life is infinite. I now turn to the comparative statics. Proposition 1. If the optimal patent length is finite, then it is: ( i) increasing in J, the difference between the social and private benefit from the innovation during the lifetime of the patent grant; ( ii) increasing in the variable cost of R& D a; ( iii) increasing in the fixed R& D cost F; ( iv) decreasing in the deadweight loss of the patent K, if J . 0; (v) decreasing in the patentee’ s profits H. Proof. Let C denote the left-hand side of Eq. (22). One can easily check that Cz , 0. This implies that for any arbitrary parameter m that influences z * , the sign of ≠z * / ≠m equals the sign of Cm . Since: ] Œ] zH 2 ]12 ŒrF 1 CJ 5 ]]] ]]]] ] . 0, z(H 1 K) Œ] zH 2ŒrF
S
D
]] 1 zH 1 zHF Ca 5 ] ] 2 ] ]] . 0, a r ra a r
S
œ D
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] ] ŒrzH 1 (H 1 K 1 J) 1 zH ] . 0, CF 5 ]] 2 1 ]]]] ] ]]]] ] Œ] 2 1 ] Œ Œ 2r a r z(H 1 K) 4 F ( zH 2 rF )
F
G
œ
results (i) to (iii) immediately follow. Concerning K, ] Œ] zH 2 ]12 ŒrF J CK 5 2 ]]]2 ]]]] ] , 0, Œ] z(H 1 K) zH 2ŒrF
S
D
if J . 0.10 Lastly, consider the effect of H. Differentiating Eq. (22) one obtains: ] Œ]] 1 (H 1 K 1 J) rzF /H 1 z 1 zF ] ]]]] ]]]] ] ] ] ] CH 5 CK 2 2 1 Œ] Œ] 2 2 2 4 r ra 2a rH z(H 1 K) s zH 2 rFd , 0. j
F
G
S
œ D
The reason why the optimal life of the patent increases with J is obvious, since J is the social benefit from the innovation which is not captured by the patentee and is therefore a positive externality of the R&D activity. The effect of a on T * is only slightly less intuitive. On the one hand, an increase in a leads to a fall of X and this reduces the marginal social cost of increasing the patent’s life. On the other hand, a change in a has two offsetting effects on the marginal social benefit: both dX / dz and (X 1 r) decrease with a, but their ratio is independent of a, so that the marginal social benefit is unaffected by a change in a. Then it is clear that an increase in a must lead to an increase in z * , and hence in the optimal patent’s lifetime T * . When F decreases, the aggregate R&D effort X increases and this allows society to shorten the patent grant’s term. Since the equilibrium number of firms n increases when F decreases, F may be taken as a measure of the degree of
10
In the special case J 5 0, social welfare becomes: H 1 K (1 2 z)X W 5 ]] ]]. r X 1r
Thus the social problem reduces to maximizing: (1 2 z)X L 5 ]]. X 1r This special form of the social welfare function may be explained as follows. When J 5 0, society in fact does not benefit from the innovation until the patent expires, for under free entry the private reward from the innovation is exactly matched by R&D costs. Thus social welfare equals (H 1 K) /r discounted at rate r for a period equal to the expected date of innovation plus the length of the patent. The corresponding discount factor is given by L. The patent’s length affects social welfare only through this discount factor. Since L does not depend on K neither directly nor through X, the optimal patent length does not depend on K.
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competition in R&D. Proposition 1 then states that more competition in the invention industry entails a shorter patent life.11 The optimal patent life is decreasing in H and K. As far as K is concerned, it is obvious that it has an adverse effect on the optimum life of the patent, for K measures the social cost of patent protection. Concerning H, it is clear that a higher H enhances the private incentive to perform R&D, but it also makes R&D investment more desirable from a social point of view. As it turns out, X is enhanced more than is socially desirable, so that the policy-maker reacts by lowering the patent duration. A partial explanation is that when H increases (and K and J stay constant), a larger fraction of the social benefit from the innovation becomes privately appropriable, reducing the need to stimulate R&D. However, it can be checked that if H, K, and J all increase in the same proportion, the optimal patent life still decreases. I suspect that this particular result depends on the specific functional forms that are implied by the exponential distribution of the date of the innovation, and may be reversed with different distributions. I have been unable to determine the effect of a change in the interest rate r on T * . It can be shown 12 that z * is increasing in r, but this is not sufficient to sign dT * / dr. A special case of the model with free entry arises when the fixed R&D cost F goes to zero. In this case it turns out that the equilibrium number of firms and the individual R&D effort are indeterminate. However, the aggregate R&D effort X is determinate and is given by: 13
11 At first, one might think that since the case of monopoly in the invention industry corresponds to n 5 1, and the optimal patent life is decreasing in n, T * must be larger under monopoly than under free entry. However, the comparative statics of the free entry model is not the same as that of a model with a fixed number of firms. Thus, even if the fixed cost F is set at a level which induces only one firm to enter the R&D industry, the corresponding optimal patent life need not equal the optimal life under monopoly and for certain parameter values may indeed be longer. However, it can be shown that in the limiting case F 5 0 (see below), the optimal patent life is shorter than under monopoly (see Denicolo` (1995) for details). This parallels results obtained within the context of a Nordhaus-type model by Berkowitz and Kotowitz (1982) and De Brock (1985), but is different from the effect of a change in F. 12 It is easy to show that Cr . 0. An increase in r shifts the marginal social cost curve downward and the marginal social benefit curve upward. 13 Interestingly, the equilibrium condition (23) may be justified in a different way. Following Dasgupta and Stiglitz (1980), one may define perfect competition in the R&D industry as a state where each firm maximizes its profits taking the expected date of the innovation as given. In other words, firms do not take into account the effect of their R&D investment on the aggregate R&D effort X. Under my assumptions, this implies that the profit function perceived by a representative firm i is linear in x, and its derivative is:
≠pi V ] 5 ]] 2 a. ≠x i X 1 r It follows that if there is an equilibrium in the invention industry with positive R&D effort, it must satisfy Eq. (23). This argument shows that the assumption of Dasgupta and Stiglitz (1980) of a perfectly competitive invention industry corresponds in my model to F 5 0.
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zH XC 5 ] 2 r. ra
(23)
Substituting Eq. (23) into Eq. (16) I obtain the following expression for social welfare (neglecting constants): H 1K ra 1 W 5 2 ]]z 2 ](H 1 K 1 J)]. r H z
(24)
The solution to the social problem is then straightforward and can be calculated explicitly: ]]] ] a H 1K 1 J z * 5 min 1,r ] ]]] . (25) H H 1K
S œœ
D
From Eq. (25) all the comparative statics results of Proposition 1 can immediately be confirmed. In this case, it is also possible to sign the effect of a change in r on the optimal patent life. Proposition 2. Under perfect competition in the invention industry, if the optimal lifetime of the patent is finite, then it is increasing in the rate of interest r. Proof. Using the definition of z, given by Eq. (3), I obtain
F
G
≠T 1 ≠z 1 1 ] 5 ] ] ]] 1 ]log(1 2 z) . ≠r r ≠r 1 2 z r From Eq. (25) it follows that ≠z / ≠r 5 z /r, whence: ≠T * 1 z ]] 5 ]2 ]] 1 log(1 2 z) . 0. j ≠r r (1 2 z)
F
G
5. Over-investment in R&D? It is well known that with free entry in the invention industry there may be over-investment in R&D, due to a socially wasteful duplication of efforts. I now ask whether over-investment can still occur once the life of the patent is optimally set in the way analyzed in the previous sections. Since this question involves a comparison between the socially optimal level of investment in R&D and the equilibrium level, to avoid spurious effects I rule out fixed R&D costs.14 Thus I limit my analysis to the case of perfect competition.
14 Since there are constant returns to scale in R&D apart from the fixed cost, society would obviously pay the fixed cost F only once; this would give a socially managed R&D industry a technological advantage over an oligopolistic one.
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To begin with, I consider the constrained social optimum, that is I assume that the policy-maker fixes the level of R&D investment taking as given the life of the patent, and therefore taking into account that society must wait until T to get the welfare triangle K. From Eq. (15), it is clear that the constrained socially optimal R&D investment is: ]]]]] H 1 J 1 (1 2 z)K XS 5 ]]]]] 2 r. (26) a
œ
Of course, the unconstrained socially optimal level of investment corresponds to z 5 0 and is therefore higher than the constrained one. I then compare XS and the equilibrium R&D effort under perfect competition XC , given that z is set optimally according to Eq. (25). Proposition 3. With perfect competition in the invention industry, if the patent life is set optimally, there is under-investment in R& D unless K 5 0 and J < H(H / a r 2 2 1); in this case, the equilibrium aggregate R& D investment coincides with the constrained socially optimal one. Proof. I must distinguish between the case when the optimal patent life is finite and the case when it is infinite. If the optimal patent life is finite, z * , 1, and from Eq. (26) I have: ]]]]] H 1 K 1 J 2 z *K XS 1 r 5 ]]]]], a
œ
whereas the aggregate equilibrium R&D effort with perfect competition satisfies: ] ]]] H H 1K 1 J XC 1 r 5 ] ]]]. a H 1K
œ œ
Squaring and using Eq. (25) I can conclude that XS > XC provided: ]]] ] H 1K 1 J a ]]] 2 r ] > 0. H 1K H
Sœ
K
œ D
Since the first term inside the parentheses is greater than 1 while the second term is strictly less than 1 by condition (12), the inequality always holds and is strict if K . 0. If K 5 0 then XS 5 XC ; note that when K 5 0, for z * , 1 to hold it must be that J , H(H /a r 2 2 1). When z * 5 1, i.e. when H 1K 1 J H ]]] > ]2 , H 1K ar I have:
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]]]]]] ]] H1J H2 K H H ]2 2 1 > ] 5 XC 1 r. XS 1 r 5 ]] > ]] 2 2 1] a a a r a r ar
œ
œ
S
D
The second inequality is strict if K . 0 and the first one is strict if J . H(H /a r 2 2 1). Thus, only if K 5 0 and J 5 H(H /a r 2 2 1) does one have XS 5 XC . j I have thus shown that over-investment cannot occur once the life of the patent is optimally set; on the contrary, there will generally be under-investment in R&D. Of course, since the unconstrained social optimum calls for an even greater R&D investment than XS , the under-investment result holds also with respect to this latter benchmark. One implication of Proposition 3 is that, unless K 5 0, the optimal patent life cannot be found by simply equating the equilibrium R&D effort to the socially optimal one. The intuition is pretty clear. If there is over-investment in R&D, i.e. XS , XC , reducing the patent life will both reduce the deadweight loss associated with monopolistic use of the innovation and bring aggregate R&D effort closer to the socially optimal level. Thus there is a definite gain from reducing T. When XS 5 XC , the loss from reducing the aggregate R&D level below the socially optimal level is second order, so if K . 0 the government will still find it optimal to reduce the patent length. In other words, since K . 0 means that stimulating R&D activity by prolonging the patent duration is socially costly, the government will stop increasing T before reaching the socially optimal R&D effort. This conclusion implies that patent policy needs to be supplemented by other instruments in regulating the R&D activity. Once the patent life has been set optimally, thus solving the duplication of efforts problem, there is still room to use R&D subsidies to close the gap between the socially optimal and the equilibrium level of R&D investment.
6. Small and large innovations While some of the comparative statics results derived in the previous sections parallel those obtained in Nordhaus-type models, the set of exogenous parameters in my model is different from that of the Nordhaus tradition and it is not always possible to draw analogies. For instance, my model can be used to ask whether large innovations should be protected more or less than small ones, a problem that in Nordhaus-type models would be meaningless because the size of the innovation is endogenous. To address this issue, I focus on the case of perfect competition in the product market. I assume that initially production is carried out at constant unit cost c, and that the innovation reduces the production cost to (c 2 d). I want to analyze the effect of a change in the cost improvement d on the optimal patent life.
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Proposition 4. With perfect competition in the product market, if the innovation is non-drastic, the optimal lifetime of the patent is decreasing in the size of the innovation d, and becomes infinite for sufficiently small inventions. Proof. Clearly, an increase in d affects both H and K positively, and therefore, as far as the invention is non-drastic (so that J 5 0), it leads to a reduction of the socially optimal life of the patent by Proposition 1. To see that the optimal patent ¯ But Eq. life becomes infinite for sufficiently small innovations, recall that z * . z. (10) implies that z¯ tends to 1 if H becomes sufficiently small relative to r, F and a. Now, since H 5 Q 0 d (where Q 0 denotes the pre-innovation equilibrium output), it is clear that for sufficiently small innovations the optimal patent length is infinite. More precisely, define d¯ as an invention so small that condition (11) holds as an equality. Then, it is clear that when d tends to d¯ from the above, the optimal life of the patent tends to infinity.15 j Proposition 4 provides an answer in the negative to the question whether large innovations should be protected more than smaller ones. Small innovations should be protected for a longer period, for two reasons: first, large innovations are associated with large deadweight losses; second, small innovations require long patent duration to persuade firms in the invention industry to make a positive R&D effort. Obviously, very small innovations (i.e. d , d¯ ) will never command a positive R&D effort and therefore need not be protected. Proposition 4 shows that T * is negatively related to d, as long as the invention is non-drastic. I now ask what happens when the innovation is so large as to become drastic. For drastic innovations, J is positive. Moreover, as long as the marginal revenue curve is downward sloping, J increases with the size of the innovation. This observation led Nordhaus (1972) to conjecture that the optimal patent life is longer for drastic and product inventions. But this is not necessarily true, because H continues to increase with the size of the innovation even when the innovation is drastic, and the same may be true of K; these countervailing effects may outweigh the effect of a positive and increasing J. In order to bring these points out more clearly it will prove useful to consider the case of a product invention with a linear demand function. Assume that the demand function is p 5 a 2 Q, where p is price and Q is industry output and that the initial cost is c 5 a, so that before the innovation the good is not produced.16 Proposition 5. With perfect competition in the product market and a linear
15
Gilbert and Shapiro (1990) report a similar result. By continuity, the same conclusion holds when the size of the market before the innovation, Q 0 , is positive but small. 16
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demand function, if the initial size of the market is zero, the optimal patent life decreases as the size of the innovation increases. Proof. In this case, H 5 ]41 d 2 and K 5 J 5 ]21 H. The marginal social cost becomes: ] 3 d 2 1 zd 2 d zF MSC 5 ] ] ] ] 2 ] ] 2 r , 16 r 2 ra a r
S
D
œ
and the marginal social benefit reduces to: ] d2 4 dŒ]z 2ŒrF MSB 5 ] ] 2 3 ]]]] ] . 8 z dŒ]z 2 2ŒrF
DS
S
D
¯ both MSC and MSB are positive; in particular, for z . z¯ the Recall that for z . z, ] inequality dŒ]z . 2ŒrF holds. The first order condition for the social problem is: ] ] 3 1 zd 2 d zF 4 dŒ]z 2ŒrF Q 5 2 ] ] ] 2 ] ] 2 r 1 ] 2 3 ]]]] ] 5 0. 2r 2 ra a r z dŒ]z 2 2ŒrF
S
D S DS
œ
D
Since: ] Œ] 3 zd 1 zF 4 zrF Qd 5 2 ] ] 2 ] ] 2 ] 2 3 ]]]]] ] ,0 ] 2r ra a r z Œ (d z 2 2ŒrF )2
S
œ D S
D
and ] d 4 dŒz] 2ŒrF Qz 5 ]Qd 2 ]2 ]]]] ] , 0, 2z z dŒ]z 2 2ŒrF
S
D
it is clear by implicit differentiation that the optimal patent life decreases as d increases. j
7. Concluding remarks In this paper, I have developed a simple model of patent races where issues of patent policy can be analyzed. There are several interesting extensions of the basic model that are left for future research. First, while I have taken the size of the innovation as exogenous, it is possible to assume that firms choose the size of their target innovation. Denoting by D the size of the innovation, one could postulate that the hazard rate is x i f (Di ), where f (D) is a decreasing function, and let firms choose both x i and Di . This would allow us to integrate Nordhaus’ original analysis into the framework of the present paper. Second, other policy instruments could be used along with patent life. Patent breadth is an example; R&D subsidies is another example. Proposition 3 suggests that when the patent’s life is set optimally, there is still a need of stimulating R&D investment, and R&D subsidies could be used for that purpose.
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Finally, I have modeled a one-shot race. Many interesting issues of patent policy can be comprehended only in models with a sequence of innovations–see, for instance, Green and Scotchmer (1995) and Chang (1995). When there is a sequence of innovations, increasing patent life can actually lead firms to reduce their R&D effort, as shown by Cadot and Lippman (1996). In their model, when the innovator can lose his dominant position, R&D investment is not monotone in the length of the patent because short patents have a disciplining effect of inducing early innovators to continue to invest in R&D. This suggests that one-shot models may tend to over-estimate the optimal patent life.
Acknowledgements The comments of two anonymous referees and seminar participants at Bocconi University were extremely helpful. Financial support from MURST is gratefully acknowledged.
References Berkowitz, M.K., Kotowitz, Y., 1982. Patent policy in an open economy. Canadian Journal of Economics 15, 1–17. Cadot, O., Lippman, S.A., 1996. Barriers to imitation and the incentive to innovate. mimeo, UCLA. Chang, H.F., 1995. Patent scope, antitrust policy, and cumulative innovation. Rand Journal of Economics 26, 34–57. Dasgupta, P., Stiglitz, J., 1980. Uncertainty, industrial structure and the speed of R&D. Bell Journal of Economics 11, 1–28. De Brock, L.M., 1985. Market structure, innovations and optimal patent life. Journal of Law and Economics 28, 223–244. ` V., 1995. The optimal life of a patent when the timing of innovations is stochastic. Denicolo, Department of Economics Discussion Paper No. 189, University of Bologna. ` V., 1996. Patent races and optimal patent breadth and length. Journal of Industrial Economics Denicolo, 44, 249–265. Gallini, N., 1992. Patent policy and costly imitation. Rand Journal of Economics 23, 52–63. Gilbert, R., Shapiro, C., 1990. Optimal patent length and breadth. Rand Journal of Economics 21, 106–112. Green, J., Scotchmer, S., 1995. On the division of profit in sequential innovation. Rand Journal of Economics 26, 20–33. Kamien, M.I., Schwartz, N.L., 1974. Patent life and R&D rivalry. American Economic Review 64, 183–187. Klemperer, P., 1990. How broad should the scope of patent protection be? Rand Journal of Economics 21, 113–130. Lee, T., Wilde, L., 1980. Market structure and innovation: A reformulation. Quarterly Journal of Economics 94, 429–436. Loury, G., 1979. Market structure and innovation. Quarterly Journal of Economics 93, 395–410. Nordhaus, W., 1969. Invention, Growth and Welfare: A Theoretical Treatment of Technological Change. MIT Press, Cambridge.
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Nordhaus, W., 1972. The optimal life of a patent: Reply. American Economic Review 62, 428–431. Reinganum, J., 1989. The timing of innovations. In: Schmalensee, R., Willig, R. (Eds.), Handbook of Industrial Organization. North-Holland, Amsterdam, Ch. 14. Scherer, F.M., 1972. Nordhaus’ theory of optimal patent life: A geometric reinterpretation. American Economic Review 62, 422–427. Stiglitz, J., 1969. Theory of innovation: Discussion. American Economic Review Papers and Proceedings 59, 46–49.
The optimal life of a patent when the timing of innovation is stochastic Vincenzo Denicolo` * Dipartimento di Scienze Economiche, Universita` di Bologna, Piazza Scaravilli 2, I-40125 Bologna ( Bo), Italy Accepted 2 September 1997
Abstract I re-examine the problem of optimal patent life within a model where the timing of innovations is stochastic. It is assumed that innovations occur according to a Poisson stochastic process where the hazard function is linear in R&D effort. Firms performing R&D have to pay a fixed cost, and there is free entry in the R&D industry. Among the comparative statics results, I show that small innovations should be protected more than larger ones. Also, when the patent life is set optimally, there is under-investment in R&D. 1999 Elsevier Science B.V. All rights reserved.
1. Introduction The aim of the patent system is to remunerate innovative activity and stimulate investment in R&D. However, patents entail a social cost because of the monopoly privileges accorded to the patentees. Thus, in deciding the lifetime of a patent, society must balance the gains accruing from faster technological progress against the welfare loss associated with the temporary monopoly in the use of the new technology. The classic analysis of the optimal patent life is based on the hypothesis of a deterministic R&D technology–see Nordhaus (1969), (1972) and
* E-mail address: [email protected] (V. Denicolo) ` 0167-7187 / 99 / $ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S0167-7187( 97 )00061-1
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Scherer (1972). In Nordhaus’ model, investment in R&D leads, with probability one, to an innovation whose size, measured as the percentage cost reduction that the innovation brings about, is an increasing function of the R&D expenditure. Though the assumption of deterministic innovations was standard at the time of Nordhaus’ report, its limits as a model of innovative activity have since been widely recognized. That firms can instantaneously ‘‘buy’’ a reduction of their variable production costs seems more appropriate as a description of investment in some fixed production factor than as a model of R&D.1 Starting with Loury (1979) and Lee and Wilde (1980), the standard approach to R&D competition assumes that the date of the innovation is uncertain (see Reinganum (1989) for a survey). In this paper, I re-examine the issue of the optimal lifetime of a patent within a standard model of patent races.2 Thus, the present paper fills a gap in the literature on the optimal patent life and provides a new framework where issues of patent policy can be analyzed. Besides allowing for uncertainty in the timing of innovation, I generalize Nordhaus’ analysis and the subsequent literature in other ways. In particular, my description of the product market is more general than in most of the previous literature. Among the results I obtain, two are especially noteworthy. First, I show that, generally speaking, at the optimal patent life, there is under-investment in R&D. Second, I prove that with perfect competition in the product market, small innovations should be protected more than large ones. This problem cannot be addressed using a Nordhaus-type model, where the size of the innovation depends continuously on R&D investment rather than being an exogenous parameter. The layout of the paper is as follows. In Section 2, I describe the basic assumptions of the model. The patent race is analyzed in Section 3. Section 4 states the social problem, characterizes the optimal patent life, and derives several comparative statics results. I then show in Section 5 that, when the length of the patent is set optimally, there is always under-investment in R&D with respect to
1
Kamien and Schwartz (1974) were the first to criticize Nordhaus’ model on these grounds. They analyze the effect of varying the patent’s length on the equilibrium R&D effort in a model that explicitly incorporates uncertainty on the timing of the innovation. However, they do not analyze the full equilibrium in the R&D industry nor do they address the issue of the optimum patent length. 2 Gilbert and Shapiro (1990) claim that the emphasis on the optimum life of patents has been misplaced, for the relevant policy dimension is patent breadth, not length. They have found a sufficient condition for the optimal patent length to be infinite, with patent’s breadth adjusted so as to provide the patentee with the appropriate incentive to perform R&D. However, conditions can be found under which optimal patents should confer the widest protection of the invention, and their lifetime should be ` 1996). adjusted to stimulate investment in R&D (see Klemperer, 1990; Gallini, 1992; Denicolo, Moreover, it is likely that the same factors which affect the optimal length of patents also determine the optimal patent breadth when the latter is the relevant policy instrument.
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the social optimum. Finally, Section 6 analyzes the issue of small versus large innovations, assuming perfect competition in the product market.
2. Basic assumptions For the sake of simplicity, and also to facilitate a comparison with the earlier literature, I retain some of the original hypotheses of Nordhaus (1969). Thus I consider a single cost reducing innovation,3 and assume that the private and social discount rates coincide. Also, I postulate a sharp distinction between the invention industry and the product market. Firms which operate in the invention industry are pure laboratories that produce inventions and innovations. They race to innovate first and the winner patents the innovation which will then be licensed to users operating in the downstream product market. However, I depart from Nordhaus (1969) in a number of ways. I assume that the size of the innovation is exogenously given, but its timing stochastically depends on R&D investment. In modelling the R&D technology, I follow Loury (1979) and Dasgupta and Stiglitz (1980). They assume a Poisson discovery process and postulate that R&D costs are paid at the beginning of the race (contractual costs). In particular, Dasgupta and Stiglitz (1980) assume that each R&D project is characterized by an optimal scale, but that each firm can operate many R&D projects simultaneously. Ignoring the integer constraint, this is tantamount to assuming a linear (in R&D costs) hazard function. This hypothesis, though somewhat restrictive, seems justified because there is no strong evidence of the existence of economies of scale in R&D. Moreover, it greatly simplifies the analysis 4 and so will be maintained throughout the paper. Concerning the structure of the invention industry, I analyze an oligopolistic equilibrium with a fixed R&D cost and free entry. The analysis is especially simple in the limit case where the fixed cost tends to zero, which loosely speaking corresponds to perfect competition in the invention industry.
3
However, I allow for drastic innovations. In a sense, a product innovation is equivalent to a drastic cost reducing innovation, for one can imagine that the new product was already known but could be produced only at prohibitively high costs before the innovation. 4 The hypothesis that R&D costs are paid at the beginning of the race is made for analytical convenience. Indeed, under this assumption on R&D costs, a linear hazard function yields an interior solution to the firm’s profit maximization problem. If instead one modeled R&D costs as in the alternative specification of Lee and Wilde (1980), according to which they are flow costs that are paid until the innovation occurs, one would have to assume a concave hazard function (at least over the relevant range) to obtain an interior solution. For a comparison of these two approaches to modelling R&D costs see Reinganum (1989).
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In most of my analysis, I black-box the product market, which is described by the following three parameters: – H, the patentee’s profits during the lifetime of the patent; – J, the further increase (if any) in social welfare that the innovation brings about during the lifetime of the patent (this increase in social welfare is not captured by the patentee; it may be enjoyed by consumers or by other firms); – K, the deadweight loss from the patent, that is, the potential increase in social welfare which becomes available to society only when the patent expires (during the lifetime of the patent, it is lost due to the monopolistic distortions created by the patent). H, K and J are stationary flows; in other words, there is no dynamics in the product market apart from that generated by the innovation. This description of the product market may be compatible with several alternative assumptions on the type of competition and the nature of the innovation. The initial structure of the product market determines the optimal licensing strategy of the innovator and the resulting changes in the market equilibrium; thus, it determines the values of H, K and J. In particular, my description of the product market allows for the case of imperfect patent protection, so that the patentee’s competitors may imitate the innovation or invent around the patent without infringing it. Obviously, H is always positive. Though one may easily construct nonpathological examples where K and J are negative, to simplify the exposition I shall assume J,K > 0. In the special case K 5 J 5 0, an infinite patent life would guarantee that the social and private value of the innovation coincide (see Loury, 1979; Lee and Wilde, 1980). Another special case that has been paid much attention to in the literature is J 5 0; this case arises, for instance, with perfect competition in the product market when the innovation is not drastic and there is perfect patent protection (as in Nordhaus’ model). The case of perfect competition in the product market may be used as an illustrative example. (It will also be used in Section 6 to address the issue of small vs. large innovations.) Suppose initially all firms have access to the same technology and can produce with constant marginal cost c, and the innovation reduces production costs to c 2 d. Let Q( p) denote the demand function. If the innovation is not drastic and there is perfect patent protection, so that the innovation cannot be imitated, the patentee will license the new technology for a fee d per unit of output. Thus, in the post-innovation equilibrium output will remain at the pre-innovation level Q 0 5 Q(c) and the patentee’s profits will be equal to the cost reduction d times the pre-innovation output. This is the area H in Fig. 1. After the patent expires, there is free access to the new technology and the output level rises to the point Q 1 5 Q(c 2 d), where the price equals the new
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Fig. 1. Non-drastic innovation.
marginal cost. The patentee’s profits are then driven to zero. The deadweight loss created by the patent is given by the triangle K in Fig. 1. In the case depicted in Fig. 1, during the lifetime of the patent the patentee reaps the entire social benefit from the innovation (J 5 0). But now consider the case of a drastic innovation.5 In this case, the patentee will license the new technology to a single firm for a fixed fee equal to the area H in Fig. 2. The main difference to the
Fig. 2. Drastic innovation.
5
An innovation is drastic if the monopoly price associated with the new marginal cost c 2 d is lower than the old marginal cost c.
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case of non-drastic innovation is that now part of the increase in social welfare which society enjoys before the patent expires is not captured by the patentee. This corresponds to the area J in Fig. 2.6 At time T, the equilibrium price falls to c 2 d and social welfare is further increased by an amount K.
3. The patent race In this section I analyze the invention industry. I consider a race for a patentable innovation between n symmetric R&D firms (laboratories). I model the R&D technology as in Loury (1979) and Dasgupta and Stiglitz (1980). At time t 5 0, each firm i decides its R&D effort x i and pays a lump-sum amount a x i , where a is the constant marginal cost of research effort. The R&D effort determines the expected time of successful completion of the R&D project. Assuming that the date of innovation by firm i is exponentially distributed, and is independent of the date of innovation of other firms, the probability of firm i being successful at or prior to date t is 1 2 e 2x i t . The payoff function of firm i is the present value of expected profits, net of R&D costs, i.e.: `
EHexpF 2SO x 1 rDGt x VJdt 2 ax 2 F n
pi 5
j
0
x iV 5 ]]]] 2 a x i 2 F, n j 51 x j 1 r
O
i
i
j 51
(1)
where r is the interest rate, x i is i’s R&D effort and also i’s hazard rate, so that exp[2(o jn51 x j )t] is the probability that no firm has innovated by time t, V is the present value of the profits accruing to the winner, F is a fixed R&D cost, and a x i is the variable cost of R&D. Both fixed and variable costs are paid at time t 5 0. The prize to the winner V is given by the discounted profits from the patent:
6 A similar picture emerges in the case of imperfect patent protection. If the patent has a very broad scope, the new production process cannot be imitated and therefore the non-innovating firms will remain at their pre-innovation cost c. But if the patent is more narrowly defined, one can imagine that even the non-innovating firms can develop similar processes without infringing the patent and therefore reduce their costs to a certain extent – see Nordhaus (1972). One can measure the breadth of the patent by the fraction of the cost reduction that does not spill out as freely available technology to the non-innovating firms. Thus, denoting by (1 2 g ) the breadth of the patent, the non-innovating firms will have marginal costs equal to (c 2 g d). As a result J . 0, reflecting the difference between the social and the private benefit from the innovation during the life of the patent.
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T
E
zH V 5 H e 2rt dt 5 ], r
(2)
0
where z 5 1 2 e 2rT
(3)
denotes the fraction of the overall discounted profits H /r that is captured by the patentee. Each firm chooses its R&D investment to maximize its expected profits (1). The first order condition for a maximum is: (X2i 1 r)V ]]]] 5 a, (r 1 X2i 1 x i )2
(4)
n
where X2i 5 o j ±i x j . It may easily be checked that the second order condition holds. Since all firms are identical, I look for a symmetric equilibrium where x i 5 x for all i. Condition (4) then becomes: [(n 2 1)x 1 r]V ]]]]] 5 a. (r 1 nx)2
(5)
In the following I shall assume that there is free entry in the R&D industry. Thus, the number of operating firms n is endogenous and is determined by the zero-profit condition: 7 xV ]] 2 a x 2 F 5 0. nx 1 r
(6)
Eqs. (5) and (6) determine the equilibrium number of firms n and the individual R&D effort x. They can be explicitly solved yielding: 8 Œ] FV 2 F x 5 ]]], (7) a ] V ra n 5 ] 2 ]]] . (8) F Œ] FV 2 F
œ
7 8
I ignore the integer problem treating n as a continuous variable. From Eqs. (5) and (6) one can obtain: x 2V (nx 1 r)2 5 ]. F
Substituting this expression into Eq. (6), Eq. (7) follows.
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Clearly, the equilibrium number of firms is a decreasing function of the fixed R&D cost F. When F goes to 0, the invention industry tends to become perfectly competitive.9 The aggregate R&D effort X 5 nx is (using Eq. (2)): ]] 1 zH zFH X 5 ] ] 2 ]] 2 r. (9) a r r
S œ D
If, even granting maximum patent protection, the policy-maker could not induce a positive R&D effort, patent policy would be ineffective. Thus I assume that an infinite patent life leads to X . 0. More precisely, denote by T¯ the minimum patent 2rT¯ length that induces a positive equilibrium R&D effort, and let z¯ 5 1 2 e , so that: ] ]] 2 r F F ] 1 ] 1 ar . z¯ 5 ] (10) H 4 4 Then, I shall assume that z¯ , 1 (or, equivalently, that T¯ is finite). This requires: ] ]] 2 H F F ]. ] 1 ] 1 ar . (11) r 4 4
Sœ œ
D
Sœ œ
D
In the limiting case F 5 0, this inequality reduces to ] H ] . r. a
œ
(12)
4. The optimal patent life The government chooses the value of the patent grant’s duration T that maximizes social welfare, that is the sum of consumers’ surplus and profits, net of R&D costs. I assume that the social rate of discount equals the interest rate r. Normalizing to zero the flow of social welfare before the innovation, the expected discounted social welfare is: `
E
W 5 XS e 2(X 1r)t dt 2 a X 2 nF,
(13)
0
9
The case of free entry in the R&D industry was first described, in the context of Nordhaus-type models, by Stiglitz (1969) and has been subsequently analyzed by Berkowitz and Kotowitz (1982) and De Brock (1985). Nordhaus’ own analysis considered only the case of monopoly in the invention industry. The present model can also be used to analyze the monopoly case (n 5 1)–see Denicolo` (1995) for details. Most of the results of the paper carry over to the case of monopoly in the invention industry.
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where S is the social value of the innovation. That is, S is the overall discounted social benefit from the new technology, which equals (H 1 J) before the patent expires and (H 1 K 1 J) thereafter: `
T
E
S 5 (H 1 J)e 0
2rt
E
dt 1 (H 1 J 1 K)e 2rt dt.
(14)
T
From Eqs. (13) and (14) one obtains:
F
G
X H1J K W 5 ]] ]] 1 (1 2 z)] 2 a X 2 nF. X 1r r r
(15)
To interpret this expression, notice that if the innovation process was completed instantaneously, the discounted social benefit would be (H 1 J) /r plus a fraction (1 2 z) of K /r. The factor X /(X 1 r) is a further discount factor that accounts for the (uncertain) delay in the occurrence of the innovation. If X is large, the expected date of innovation is small and therefore this discount factor is close to 1. As X decreases, society on average must wait longer and longer for the innovation and therefore the discount factor falls. Given the zero-profit condition (6), the policy-maker’s objective function may also be rewritten as:
F
G
X J H 1K W 5 ]] ] 1 (1 2 z)]] . X 1r r r
(16)
The social problem may now be stated as follows: Choose T so as to maximize W, given that X depends on T according to Eq. (9). Notice that W depends on T only through z, and (given r) there is a one-to-one correspondence between z and T. Thus the social problem is to choose z so as to maximize W. Differentiation of Eq. (16) yields the first order condition of the social problem, which applies when the solution is interior:
F
G
K 1H r J H 1 K dX 2 X ]] 1 ]] ] 1 (1 2 z)]] ] 5 0. r r dz (X 1 r) r
(17)
The first term on the right-hand side of Eq. (17) is the direct effect of a change in z on social welfare. When z increases, society can appropriate only a smaller fraction of the potential discounted gain K /r. In other words, when T increases society must wait longer and longer to appropriate the welfare triangle K. Moreover, since the R&D industry profits vanish, the private reward to the innovator zH /r is dissipated in the patent race and this also increases with z. Clearly, the direct effect of an increase in z on social welfare is always strictly negative provided X . 0 and vanishes at X 5 0. It measures the marginal social cost (MSC) of longer patent terms. The second term is the indirect effect of z on W through X and measures the
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marginal social benefit (MSB) of increasing the patent’s life. The net marginal social gain from a longer patent life is proportional to the sum of three components: J /r and a fraction (1 2 z) of H /r and K /r. The reason that only a fraction (1 2 z) of H /r and K /r enters the marginal social benefit is the following: zH /r is offset by the change in R&D costs (by the zero-profit condition) and zK /r is lost due to monopolistic distortions during the patent’s lifetime. Finally, the term r /(X 1 r) reflects the change of the ‘‘discount factor’’ X /(X 1 r) due to a change in X. Since ] ] dX H 1 FH X 1 r 1 FH ] 5 ] 2 ] ] 5 ]] 1 ] ] . 0, (18) dz ra 2a zr z 2a zr
œ
œ
the marginal social benefit of increasing the patent life is always positive. From the observation that the social cost of increasing the patent life vanishes at X 5 0 (that is, at T 5 T¯ or z 5 z¯ ) whereas the social benefit is always positive, it follows that the optimal life of a patent T * is positive and is associated with a positive aggregate R&D effort; that is, T * . T¯ . 0. Then, two cases are possible. If MSB is always greater than MSC, then social welfare is always increasing in the patent length and therefore the optimal patent life is infinite. If instead at z 5 1 the inequality MSB,MSC holds, then there will be a finite optimal patent life. Upon substitution, the marginal social cost becomes: ]] K 1 H zH 1 zHF ]] ] ] ]] 2 r ; MSC 5 2 r ra a r
S
œ
D
(19)
¯ Using Eqs. (9) and (18), the marginal it is increasing in z and vanishes at z 5 z. social benefit becomes: ] Œ] zH 2 ]12 ŒrF (H 1 K 1 J) MSB 5 ]]]] 2 (H 1 K) ]]]] (20) ] . Œ] z zH 2ŒrF
GS
F
D
MSB is always positive and decreasing in z. Evaluating MSC and MSB at z 5 1, it follows immediately that the optimal life of a patent is finite if and only if: ] (H 1 K) H 1 HF ]]] ] ] ]2r J, 2 r ra a r
S
œ
] Œ] ŒH 2 rF ]]]] ] . Œ] H 2 ]12 ŒrF
DS
D
(21)
This inequality is always satisfied when J 5 0. When condition (21) holds so that the optimal patent life is finite, T * is given by the unique solution to the first order condition: ] ]] Œ] zH 2 ]12 ŒrF (H 1 K 1 J) 1 zH 1 zHF ]]]] 2 1 ]]]] ] ] ] ]] 2 r 5 0, 2 2 (22) ] Œ] ŒzH r ra a r z(H 1 K) 2 rF
F
GS
D S
œ
D
¯ which lies in the interval (z,1). Fig. 3 presents the curves corresponding to the marginal social cost and the
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Fig. 3. Optimal patent length.
marginal social benefit of increasing the patent duration. The intersection of the two curves determines z * , which corresponds to the optimal patent life T * . Fig. 3 depicts the case where z * , 1 and therefore T * is finite. If the marginal social benefit curve always lies above the marginal social cost curve, the optimal patent life is infinite. I now turn to the comparative statics. Proposition 1. If the optimal patent length is finite, then it is: ( i) increasing in J, the difference between the social and private benefit from the innovation during the lifetime of the patent grant; ( ii) increasing in the variable cost of R& D a; ( iii) increasing in the fixed R& D cost F; ( iv) decreasing in the deadweight loss of the patent K, if J . 0; (v) decreasing in the patentee’ s profits H. Proof. Let C denote the left-hand side of Eq. (22). One can easily check that Cz , 0. This implies that for any arbitrary parameter m that influences z * , the sign of ≠z * / ≠m equals the sign of Cm . Since: ] Œ] zH 2 ]12 ŒrF 1 CJ 5 ]]] ]]]] ] . 0, z(H 1 K) Œ] zH 2ŒrF
S
D
]] 1 zH 1 zHF Ca 5 ] ] 2 ] ]] . 0, a r ra a r
S
œ D
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] ] ŒrzH 1 (H 1 K 1 J) 1 zH ] . 0, CF 5 ]] 2 1 ]]]] ] ]]]] ] Œ] 2 1 ] Œ Œ 2r a r z(H 1 K) 4 F ( zH 2 rF )
F
G
œ
results (i) to (iii) immediately follow. Concerning K, ] Œ] zH 2 ]12 ŒrF J CK 5 2 ]]]2 ]]]] ] , 0, Œ] z(H 1 K) zH 2ŒrF
S
D
if J . 0.10 Lastly, consider the effect of H. Differentiating Eq. (22) one obtains: ] Œ]] 1 (H 1 K 1 J) rzF /H 1 z 1 zF ] ]]]] ]]]] ] ] ] ] CH 5 CK 2 2 1 Œ] Œ] 2 2 2 4 r ra 2a rH z(H 1 K) s zH 2 rFd , 0. j
F
G
S
œ D
The reason why the optimal life of the patent increases with J is obvious, since J is the social benefit from the innovation which is not captured by the patentee and is therefore a positive externality of the R&D activity. The effect of a on T * is only slightly less intuitive. On the one hand, an increase in a leads to a fall of X and this reduces the marginal social cost of increasing the patent’s life. On the other hand, a change in a has two offsetting effects on the marginal social benefit: both dX / dz and (X 1 r) decrease with a, but their ratio is independent of a, so that the marginal social benefit is unaffected by a change in a. Then it is clear that an increase in a must lead to an increase in z * , and hence in the optimal patent’s lifetime T * . When F decreases, the aggregate R&D effort X increases and this allows society to shorten the patent grant’s term. Since the equilibrium number of firms n increases when F decreases, F may be taken as a measure of the degree of
10
In the special case J 5 0, social welfare becomes: H 1 K (1 2 z)X W 5 ]] ]]. r X 1r
Thus the social problem reduces to maximizing: (1 2 z)X L 5 ]]. X 1r This special form of the social welfare function may be explained as follows. When J 5 0, society in fact does not benefit from the innovation until the patent expires, for under free entry the private reward from the innovation is exactly matched by R&D costs. Thus social welfare equals (H 1 K) /r discounted at rate r for a period equal to the expected date of innovation plus the length of the patent. The corresponding discount factor is given by L. The patent’s length affects social welfare only through this discount factor. Since L does not depend on K neither directly nor through X, the optimal patent length does not depend on K.
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competition in R&D. Proposition 1 then states that more competition in the invention industry entails a shorter patent life.11 The optimal patent life is decreasing in H and K. As far as K is concerned, it is obvious that it has an adverse effect on the optimum life of the patent, for K measures the social cost of patent protection. Concerning H, it is clear that a higher H enhances the private incentive to perform R&D, but it also makes R&D investment more desirable from a social point of view. As it turns out, X is enhanced more than is socially desirable, so that the policy-maker reacts by lowering the patent duration. A partial explanation is that when H increases (and K and J stay constant), a larger fraction of the social benefit from the innovation becomes privately appropriable, reducing the need to stimulate R&D. However, it can be checked that if H, K, and J all increase in the same proportion, the optimal patent life still decreases. I suspect that this particular result depends on the specific functional forms that are implied by the exponential distribution of the date of the innovation, and may be reversed with different distributions. I have been unable to determine the effect of a change in the interest rate r on T * . It can be shown 12 that z * is increasing in r, but this is not sufficient to sign dT * / dr. A special case of the model with free entry arises when the fixed R&D cost F goes to zero. In this case it turns out that the equilibrium number of firms and the individual R&D effort are indeterminate. However, the aggregate R&D effort X is determinate and is given by: 13
11 At first, one might think that since the case of monopoly in the invention industry corresponds to n 5 1, and the optimal patent life is decreasing in n, T * must be larger under monopoly than under free entry. However, the comparative statics of the free entry model is not the same as that of a model with a fixed number of firms. Thus, even if the fixed cost F is set at a level which induces only one firm to enter the R&D industry, the corresponding optimal patent life need not equal the optimal life under monopoly and for certain parameter values may indeed be longer. However, it can be shown that in the limiting case F 5 0 (see below), the optimal patent life is shorter than under monopoly (see Denicolo` (1995) for details). This parallels results obtained within the context of a Nordhaus-type model by Berkowitz and Kotowitz (1982) and De Brock (1985), but is different from the effect of a change in F. 12 It is easy to show that Cr . 0. An increase in r shifts the marginal social cost curve downward and the marginal social benefit curve upward. 13 Interestingly, the equilibrium condition (23) may be justified in a different way. Following Dasgupta and Stiglitz (1980), one may define perfect competition in the R&D industry as a state where each firm maximizes its profits taking the expected date of the innovation as given. In other words, firms do not take into account the effect of their R&D investment on the aggregate R&D effort X. Under my assumptions, this implies that the profit function perceived by a representative firm i is linear in x, and its derivative is:
≠pi V ] 5 ]] 2 a. ≠x i X 1 r It follows that if there is an equilibrium in the invention industry with positive R&D effort, it must satisfy Eq. (23). This argument shows that the assumption of Dasgupta and Stiglitz (1980) of a perfectly competitive invention industry corresponds in my model to F 5 0.
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zH XC 5 ] 2 r. ra
(23)
Substituting Eq. (23) into Eq. (16) I obtain the following expression for social welfare (neglecting constants): H 1K ra 1 W 5 2 ]]z 2 ](H 1 K 1 J)]. r H z
(24)
The solution to the social problem is then straightforward and can be calculated explicitly: ]]] ] a H 1K 1 J z * 5 min 1,r ] ]]] . (25) H H 1K
S œœ
D
From Eq. (25) all the comparative statics results of Proposition 1 can immediately be confirmed. In this case, it is also possible to sign the effect of a change in r on the optimal patent life. Proposition 2. Under perfect competition in the invention industry, if the optimal lifetime of the patent is finite, then it is increasing in the rate of interest r. Proof. Using the definition of z, given by Eq. (3), I obtain
F
G
≠T 1 ≠z 1 1 ] 5 ] ] ]] 1 ]log(1 2 z) . ≠r r ≠r 1 2 z r From Eq. (25) it follows that ≠z / ≠r 5 z /r, whence: ≠T * 1 z ]] 5 ]2 ]] 1 log(1 2 z) . 0. j ≠r r (1 2 z)
F
G
5. Over-investment in R&D? It is well known that with free entry in the invention industry there may be over-investment in R&D, due to a socially wasteful duplication of efforts. I now ask whether over-investment can still occur once the life of the patent is optimally set in the way analyzed in the previous sections. Since this question involves a comparison between the socially optimal level of investment in R&D and the equilibrium level, to avoid spurious effects I rule out fixed R&D costs.14 Thus I limit my analysis to the case of perfect competition.
14 Since there are constant returns to scale in R&D apart from the fixed cost, society would obviously pay the fixed cost F only once; this would give a socially managed R&D industry a technological advantage over an oligopolistic one.
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To begin with, I consider the constrained social optimum, that is I assume that the policy-maker fixes the level of R&D investment taking as given the life of the patent, and therefore taking into account that society must wait until T to get the welfare triangle K. From Eq. (15), it is clear that the constrained socially optimal R&D investment is: ]]]]] H 1 J 1 (1 2 z)K XS 5 ]]]]] 2 r. (26) a
œ
Of course, the unconstrained socially optimal level of investment corresponds to z 5 0 and is therefore higher than the constrained one. I then compare XS and the equilibrium R&D effort under perfect competition XC , given that z is set optimally according to Eq. (25). Proposition 3. With perfect competition in the invention industry, if the patent life is set optimally, there is under-investment in R& D unless K 5 0 and J < H(H / a r 2 2 1); in this case, the equilibrium aggregate R& D investment coincides with the constrained socially optimal one. Proof. I must distinguish between the case when the optimal patent life is finite and the case when it is infinite. If the optimal patent life is finite, z * , 1, and from Eq. (26) I have: ]]]]] H 1 K 1 J 2 z *K XS 1 r 5 ]]]]], a
œ
whereas the aggregate equilibrium R&D effort with perfect competition satisfies: ] ]]] H H 1K 1 J XC 1 r 5 ] ]]]. a H 1K
œ œ
Squaring and using Eq. (25) I can conclude that XS > XC provided: ]]] ] H 1K 1 J a ]]] 2 r ] > 0. H 1K H
Sœ
K
œ D
Since the first term inside the parentheses is greater than 1 while the second term is strictly less than 1 by condition (12), the inequality always holds and is strict if K . 0. If K 5 0 then XS 5 XC ; note that when K 5 0, for z * , 1 to hold it must be that J , H(H /a r 2 2 1). When z * 5 1, i.e. when H 1K 1 J H ]]] > ]2 , H 1K ar I have:
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]]]]]] ]] H1J H2 K H H ]2 2 1 > ] 5 XC 1 r. XS 1 r 5 ]] > ]] 2 2 1] a a a r a r ar
œ
œ
S
D
The second inequality is strict if K . 0 and the first one is strict if J . H(H /a r 2 2 1). Thus, only if K 5 0 and J 5 H(H /a r 2 2 1) does one have XS 5 XC . j I have thus shown that over-investment cannot occur once the life of the patent is optimally set; on the contrary, there will generally be under-investment in R&D. Of course, since the unconstrained social optimum calls for an even greater R&D investment than XS , the under-investment result holds also with respect to this latter benchmark. One implication of Proposition 3 is that, unless K 5 0, the optimal patent life cannot be found by simply equating the equilibrium R&D effort to the socially optimal one. The intuition is pretty clear. If there is over-investment in R&D, i.e. XS , XC , reducing the patent life will both reduce the deadweight loss associated with monopolistic use of the innovation and bring aggregate R&D effort closer to the socially optimal level. Thus there is a definite gain from reducing T. When XS 5 XC , the loss from reducing the aggregate R&D level below the socially optimal level is second order, so if K . 0 the government will still find it optimal to reduce the patent length. In other words, since K . 0 means that stimulating R&D activity by prolonging the patent duration is socially costly, the government will stop increasing T before reaching the socially optimal R&D effort. This conclusion implies that patent policy needs to be supplemented by other instruments in regulating the R&D activity. Once the patent life has been set optimally, thus solving the duplication of efforts problem, there is still room to use R&D subsidies to close the gap between the socially optimal and the equilibrium level of R&D investment.
6. Small and large innovations While some of the comparative statics results derived in the previous sections parallel those obtained in Nordhaus-type models, the set of exogenous parameters in my model is different from that of the Nordhaus tradition and it is not always possible to draw analogies. For instance, my model can be used to ask whether large innovations should be protected more or less than small ones, a problem that in Nordhaus-type models would be meaningless because the size of the innovation is endogenous. To address this issue, I focus on the case of perfect competition in the product market. I assume that initially production is carried out at constant unit cost c, and that the innovation reduces the production cost to (c 2 d). I want to analyze the effect of a change in the cost improvement d on the optimal patent life.
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Proposition 4. With perfect competition in the product market, if the innovation is non-drastic, the optimal lifetime of the patent is decreasing in the size of the innovation d, and becomes infinite for sufficiently small inventions. Proof. Clearly, an increase in d affects both H and K positively, and therefore, as far as the invention is non-drastic (so that J 5 0), it leads to a reduction of the socially optimal life of the patent by Proposition 1. To see that the optimal patent ¯ But Eq. life becomes infinite for sufficiently small innovations, recall that z * . z. (10) implies that z¯ tends to 1 if H becomes sufficiently small relative to r, F and a. Now, since H 5 Q 0 d (where Q 0 denotes the pre-innovation equilibrium output), it is clear that for sufficiently small innovations the optimal patent length is infinite. More precisely, define d¯ as an invention so small that condition (11) holds as an equality. Then, it is clear that when d tends to d¯ from the above, the optimal life of the patent tends to infinity.15 j Proposition 4 provides an answer in the negative to the question whether large innovations should be protected more than smaller ones. Small innovations should be protected for a longer period, for two reasons: first, large innovations are associated with large deadweight losses; second, small innovations require long patent duration to persuade firms in the invention industry to make a positive R&D effort. Obviously, very small innovations (i.e. d , d¯ ) will never command a positive R&D effort and therefore need not be protected. Proposition 4 shows that T * is negatively related to d, as long as the invention is non-drastic. I now ask what happens when the innovation is so large as to become drastic. For drastic innovations, J is positive. Moreover, as long as the marginal revenue curve is downward sloping, J increases with the size of the innovation. This observation led Nordhaus (1972) to conjecture that the optimal patent life is longer for drastic and product inventions. But this is not necessarily true, because H continues to increase with the size of the innovation even when the innovation is drastic, and the same may be true of K; these countervailing effects may outweigh the effect of a positive and increasing J. In order to bring these points out more clearly it will prove useful to consider the case of a product invention with a linear demand function. Assume that the demand function is p 5 a 2 Q, where p is price and Q is industry output and that the initial cost is c 5 a, so that before the innovation the good is not produced.16 Proposition 5. With perfect competition in the product market and a linear
15
Gilbert and Shapiro (1990) report a similar result. By continuity, the same conclusion holds when the size of the market before the innovation, Q 0 , is positive but small. 16
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demand function, if the initial size of the market is zero, the optimal patent life decreases as the size of the innovation increases. Proof. In this case, H 5 ]41 d 2 and K 5 J 5 ]21 H. The marginal social cost becomes: ] 3 d 2 1 zd 2 d zF MSC 5 ] ] ] ] 2 ] ] 2 r , 16 r 2 ra a r
S
D
œ
and the marginal social benefit reduces to: ] d2 4 dŒ]z 2ŒrF MSB 5 ] ] 2 3 ]]]] ] . 8 z dŒ]z 2 2ŒrF
DS
S
D
¯ both MSC and MSB are positive; in particular, for z . z¯ the Recall that for z . z, ] inequality dŒ]z . 2ŒrF holds. The first order condition for the social problem is: ] ] 3 1 zd 2 d zF 4 dŒ]z 2ŒrF Q 5 2 ] ] ] 2 ] ] 2 r 1 ] 2 3 ]]]] ] 5 0. 2r 2 ra a r z dŒ]z 2 2ŒrF
S
D S DS
œ
D
Since: ] Œ] 3 zd 1 zF 4 zrF Qd 5 2 ] ] 2 ] ] 2 ] 2 3 ]]]]] ] ,0 ] 2r ra a r z Œ (d z 2 2ŒrF )2
S
œ D S
D
and ] d 4 dŒz] 2ŒrF Qz 5 ]Qd 2 ]2 ]]]] ] , 0, 2z z dŒ]z 2 2ŒrF
S
D
it is clear by implicit differentiation that the optimal patent life decreases as d increases. j
7. Concluding remarks In this paper, I have developed a simple model of patent races where issues of patent policy can be analyzed. There are several interesting extensions of the basic model that are left for future research. First, while I have taken the size of the innovation as exogenous, it is possible to assume that firms choose the size of their target innovation. Denoting by D the size of the innovation, one could postulate that the hazard rate is x i f (Di ), where f (D) is a decreasing function, and let firms choose both x i and Di . This would allow us to integrate Nordhaus’ original analysis into the framework of the present paper. Second, other policy instruments could be used along with patent life. Patent breadth is an example; R&D subsidies is another example. Proposition 3 suggests that when the patent’s life is set optimally, there is still a need of stimulating R&D investment, and R&D subsidies could be used for that purpose.
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Finally, I have modeled a one-shot race. Many interesting issues of patent policy can be comprehended only in models with a sequence of innovations–see, for instance, Green and Scotchmer (1995) and Chang (1995). When there is a sequence of innovations, increasing patent life can actually lead firms to reduce their R&D effort, as shown by Cadot and Lippman (1996). In their model, when the innovator can lose his dominant position, R&D investment is not monotone in the length of the patent because short patents have a disciplining effect of inducing early innovators to continue to invest in R&D. This suggests that one-shot models may tend to over-estimate the optimal patent life.
Acknowledgements The comments of two anonymous referees and seminar participants at Bocconi University were extremely helpful. Financial support from MURST is gratefully acknowledged.
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Nordhaus, W., 1972. The optimal life of a patent: Reply. American Economic Review 62, 428–431. Reinganum, J., 1989. The timing of innovations. In: Schmalensee, R., Willig, R. (Eds.), Handbook of Industrial Organization. North-Holland, Amsterdam, Ch. 14. Scherer, F.M., 1972. Nordhaus’ theory of optimal patent life: A geometric reinterpretation. American Economic Review 62, 422–427. Stiglitz, J., 1969. Theory of innovation: Discussion. American Economic Review Papers and Proceedings 59, 46–49.