Empirica (2008) 35:233–240 DOI 10.1007/s10663-007-9059-7 ORIGINAL PAPER

Optimal patent length and height Patrick F. E. Beschorner

Published online: 12 January 2008 Ó Springer Science+Business Media, LLC. 2008

Abstract Patent breadth and length have been discussed extensively in innovation literature. In this article, I analyze the optimal degree of novelty of patent protection and its tradeoff with patent length. In the context of subsequent innovations each innovation builds on the previous one. The degree of novelty necessary for a noninfringing patent is crucial for the firms’ incentive to innovate. One of the findings is that a monopolist’s optimal degree of novelty is lesser than would be socially desirable. Furthermore, there exists a finite optimal patent length. Competitors introducing an improved technology cause uncertainty which may be compensated by extending patent length. Keywords

Innovation
Patent policy
Patent height
Patent length

JEL Classification

O31
L15
D61

1 Introduction The patent system forces innovators to disclose their technology and, in turn, grants temporary market power to reward innovators, which entails a welfare loss. The magnitude of the loss is determined by the scope of patent protection defined by breadth, length, and height, which are specified by the patent office. Patent height delimits the extent of protection against improvements while breadth refers to

P. F. E. Beschorner (&) Centre for European Economic Research (ZEW), L7, 1, 68161 Mannheim, Germany e-mail: [email protected]

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horizontal differentiation.1 I address the question of specifying patent protection to optimal length and height. The tradeoff between length and breadth has been analyzed by several authors.2 In Gilbert and Shapiro (1990) patents with infinitesimal breadth and infinite length are optimal because breadth only affects the patentee’s ability to exploit the patent monopoly while it does not reduce the availability of product variants. In Klemperer (1990) the social costs of patents consist either of the consumers’ switching or transportation costs to less preferred (horizontal) product variants or from withdrawing from the entire product class. When the first type prevails because the consumers differ in their preference for the entire product class rather than in their switching costs, then he advocates a narrow and infinitely long patent. Conversely, when consumers differ by their willingness to switch between product variants while they have the same commitment to the product class, an infinitely broad patent prevents the availability of other variants. Then no transportation costs occur and the patentee undercuts the consumers’ reservation price such that no loss from restrained demand occurs. O’Donoghue et al. (1998) analyze the tradeoff between patent height and length in an infinite sequence of innovations. They assume an exogenous flow of exclusive and firm-specific ideas with stochastic degree of novelty and constant cost per innovation. In the monopoly case they find out that for a given rate of innovation a patent policy with infinite length and finite height minimizes R&D costs while a policy with infinite height and finite length minimizes the cost from delayed diffusion. In the oligopoly case with vertical differentiation and limited entry as in Shaked and Sutton (1983), the policy with infinite height may outperform. O’Donoghue (1998) obtains similar results in an oligopoly model with patent races. In the present vertical differentiation model, I adopt that ideas are private knowledge but—in contrast to O’Donoghue et al. (1998)—their cost of realization varies according to the size of the innovative step. Furthermore, I restrict the analysis to a one-time innovation, thus having the patentee become a monopolist. Using a setting analogous to Gilbert and Shapiro (1990) and Klemperer (1990) allows comparison with their findings. In the model, optimal patent policy results in a finite patent length. Moreover, uncertainty about the effective patent length and premature obsolescence of a technology due to possible market entry by a non-infringing innovator can be compensated by adjusting the statutory patent length. This is in line with Gallini (1992). Section 2 presents the model, Sect. 3 introduces potential entry by a subsequent innovation during the patent lifetime and Sect. 4 concludes. 2 The model In this section, first I derive the monopolist’s profit maximizing strategy and the first best patent policy implying a negative profit for the innovator. I compare both 1

The term ‘patent height’ is used by van Dijk (1996). This is equivalent to ‘minimum inventive step’ in Gallini and Scotchmer (2002, p. 66) or Scotchmer (2005) or to ‘leading breadth’ in O’Donoghue et al. (1998). O’Donoghue (1998, p. 657) makes a further distinction to ‘patentability requirement’ which does not protect from a patent on an improved technology in contrast to an improved product that is marketed.

2

See Gilbert and Shapiro (1990) and Klemperer (1990), and Gallini (1992).

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scenarios with the second best where the monopolist’s participation constraint is binding. The model combines vertical differentiation as in Shaked and Sutton (1983) and unit demand with the notation from Gilbert and Shapiro (1990). A continuum of consumers differs in a utility index m which is uniformly distributed on the interval ½0; 1. U ¼ m
h  p is the flow of utility from quality h at the price p; all consumers with m  p=h purchase one unit. Assume that all firms produce at zero marginal cost and that the quality h is offered by a competitive industry at price zero. An innovative firm conducts research and raises quality by the inventive step d at cost d2 and becomes a monopolist. Furthermore, the innovation is assumed to be patentable, i.e., the new technology does not infringe upon the old one.3 Consumers purchase the high-quality product if m
ðh þ dÞ  p  m
h. This is equivalent to m  p=d and the demand is 1  p=d. For a monopolist, the value of a patent on a quality improvement d is Z T

p p 1  erT  d2 V¼ 1  p ert dt  d2 ¼ 1  p ð1Þ d d r 0 where r is the interest rate for a period of unit length and T is the patent length. After patent expiration the new technology is publicly available and the quality h þ d is offered at the competitive price p = 0. Maximizing the private value of the patent yields the first-order condition p¼

1 d 2

ð2Þ

and 1 1  erT 1 1  erT ; dm ¼ : 16 8 r r The social welfare for patent length T, inventive step d, and price p is 0 1 pm ¼

C ZT BZ p=d Z 1 B C rt B W¼ B m
h dm þ mðh þ dÞ dm C Ce dt 0 p=d @|fflfflfflfflfflfflfflfflffl A ffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 0 ðiÞ

þ

Z1Z

ð3Þ

ð4Þ

ðiiÞ

 mðh þ dÞ dm ert dt  d2 :

1

ð5Þ

0

T |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ðiiiÞ

2

¼

rT

hþd p 1e  2r d 2r

 d2 :

ð6Þ

3

See Denicolo` (2000, p. 495). He analyzes welfare effects of patent races with different modes of innovation. These may or may not be patentable or infringing. The highest social gain is achieved when innovations are not infringing or patentable.

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It consists (i) of the benefit of the consumers with m  p=d from the low quality during patent lifetime, (ii) of the benefit of the customers m  p=d from the high quality during patent lifetime, (iii) of offering high quality to all consumers after this time, and of the costs of the innovation. The social welfare is strictly decreasing in T. In the first best scenario a minimum patent length T ! 0 is socially desirable. It offers no patent protection because intellectual property is a public good. Making a new technology immediately available to all firms does not allow an innovator to cover his R&D costs. In the second best scenario the social planner prescribes a minimum inventive step d and a patent length T given the price-setting firm’s participation constraint V C 0. Proposition 1 In the second best scenario there exists a unique welfare maximizing patent length T  2 ð0; 1Þ and the firms’ participation constraint is binding, V ¼ 0. Proof Showing that the participation constraint is binding simplifies the maximization problem. The firm profit V being an increasing function of T and the social welfare being a strictly decreasing function of T imply that the firm profit is zero, V ¼ 0. Combining (1) and (2) amounts to 1 1  erT d¼
 d0 ðTÞ: 4 r

ð7Þ

This implicit function defines combinations of the patent length T and the degree of novelty d0 such that the maximum profit of a price-setting firm is zero. Inserting the participation constraint d0 ðTÞ into max W T;d

s:t:

V¼0

ð8Þ

simplifies to  2 h 1 1  erT 1 1  erT 1  erT 1  erT þ max   , max T T 2r 2r 4r 4 4r 2r 4r rT 2rT 16rh þ 1 þ 2e  3e  X ðT Þ 32r 2 where limT!1 X ðTÞ [ X ð0Þ [ ð0Þ: oX ðT Þ 3e ¼ oT completes the proof.

2rT

e 16r

rT

8 < [0 ¼0 : \0

for for for

T\T  T ¼ ðln 3Þ=r  T  T [ T

ð9Þ (

Proposition 2 A social planner would impose a higher minimum inventive step ds [ dm than a monopolist would choose and this implies a higher price ps [ pm for the innovative product.

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Proof From (7) we know that the patent length determines the minimum inventive step. Inserting T  into (7) yields ds ¼

1 1 1  erT [ ¼ dm : 6r 8 r

ð10Þ

From (2) and (3) follows ps [ pm . ( The intuition for this result is that the social planner considers the social welfare which comprises the consumer surplus in contrast to a monopolist who cares only about profit. The social welfare takes into account that after patent expiration all consumers benefit from an invention. A small interest rate gives more weight to future consumer surplus and this allows to engage in more costly R&D. This implies a higher minimum inventive step. 3 Subsequent innovation One limitation of the model is that the flow of profit during patent lifetime T may cease. Patents do not protect from imitation or other forms of infringement4 and they do not protect from a competitor’s non-infringing improved product before the expiration of the patent.5 I analyze the optimal policy when the timing of the introduction of a higher-valued non-infringing product is stochastic while patent policy considers no premature obsolescence. The social planner would consider that the patent holder’s incentive to innovate is smaller than that of an outside firm.6 Here, I assume that there are a large number of firms, so the patentee is unlikely to be his own successor.7 Suppose that a competitor B introduces a further improvement on firm A’s product that makes the latter’s product obsolete and its profit drop to zero. Let s denote the timing of this invention, which may occur either during firm A’s patent lifetime T or thereafter. Firm A’s flow of profit lasts until patent expiration or until s, whichever occurs first. The expected value of the patent for firm A is ZT
p V¼ 1  p Prðt  sÞert dt  d2 d

ð11Þ

0

Rx and the price is set according to (2). Let f ðsÞ and FðxÞ ¼ 0 f ðtÞ dt be the probability density function and the cumulative distribution function of s, respectively. Assume that the functions are continuous and Fð0Þ ¼ 0. Then the participation constraint of firm A transforms into ZT ZT d 1 V¼ ½1  FðtÞert dt  d2  0 , d  ½1  FðtÞert dt: ð12Þ 4 4 0

4

0

See van Dijk (1996), Hussinger (2004), Lemley and Shapiro (2004) and Takalo (1998).

5

See O’Donoghue et al. (1998, pp 4, 8) who distinguish the effective patent life which may be shorter than the statutory patent life which I denote by T.

6

See Tirole (1992, p 392f) for a discussion of the efficiency and the replacement effect.

7

This is analogous to O’Donoghue et al. (1998).

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The revenue during T is weighted by the probability that no outperforming product is marketed. Increasing probability of entry during T is expressed by a shift of probability mass from high values of s to small values such that the new density function f~ðsÞ crosses f ðsÞ exactly once, f~ðsÞ [ rless f ðsÞ for s 7 s0 . This is a particular case of first-order stochastic dominance. Proposition 3 Increasing uncertainty according to f~ðsÞ [ rless f ðsÞ for s 7 s0 requires a longer patent protection. Proof maxd;T W s:t: V^  0 determines optimal policy. For the same argument as in Proposition 1 (12) is binding. Combining with (6) yields W¼

h 1 þ 2r 8r 2



1 4 16

ZT 0

ZT

½1  FðtÞert dt 

1 ð1  ert Þ 32r

ZT

½1  FðtÞert dt

0

32

½1  FðtÞert dt5

0

and—after rearranging—the first-order condition 1  FðTÞ 3 þ erT
LðTÞ  ¼ 5  4FðTÞ r

ZT

½1  FðtÞerT dt  RðTÞ:

ð13Þ

0

Lð0Þ ¼ 4=5r; LðT ! 1Þ; Rð0Þ ¼ 0; and RðT ! 1Þ 20; 1=r constitute the existence of an optimal length T  because both functions are continuous. Uniqueness is given by the monotonicity of L and R. oLðTÞ=oT\0 because the numerator in the first factor of L falls at a higher rate than the denominator as T increases. Without uncertainty, F ¼ 0, (13) corresponds to (9). Increasing uncertainty corresponds to a cut of F for any given value of T. This reduces both functions L and R, but R drops by a higher proportion. For a given value of T, the first factor of L— which puts more weight on the effect of the denominator—falls to a lower extent than R. Therefore, the patent length has to be extended to meet the first-order condition. ( Uncertainty reduces patent value because the flow of profit may cease before patent expiration. Extending the patent length compensates for this possible revenue loss. For the optimal patent policy, the design of patent protection should take the likelihood and the timing of subsequent innovations into account. However, the extension of T does not imply that d must be increased. T and F in (12) exert opposing forces making the overall effect on d ambiguous. Further insight by O’Donoghue (1998) suggests that a higher inventive step hampers and retards innovation while Gallini (1992) focuses on possible inventing around and imitation, which become more likely the longer patent protection is. Certainly, both instruments, patent length and minimum inventive step, affect the probability of market entry and, thus, the innovators’ ability to recover his R&D cost.

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Depending on model specification, the instruments can raise or reduce the competitors’ incentive for a costly innovation. The underlying utility function of the vertical differentiation model allows to valuate a quality improvement and catches the effect that higher quality raises the willingness to pay in a robust way. Also, the convex cost function reflects that an innovation becomes increasingly costly the higher the innovative step is. Other assumptions on market size and the shape of the functions are arbitrary and further specification would not result in further insights for optimal patent policy. Currently, patent length is uniformly 20 years for all technologies and industries in most countries. The value of patents is also affected by the technology to which they may be classified.8 This would call for an industry-specific adjustment of the patent length to meet the participation constraint of innovators. For example, software patents are granted for the same length as in chemicals or pharmaceuticals while they significantly differ in the cost of development. If software patents were subject to quick further improvements, optimal patent policy would suggest to set lower patentability requirements and to grant a longer patent protection.

4 Conclusion I have explored the tradeoff between patent length and patent height. This adds to the discussion by Gilbert and Shapiro (1990) and Klemperer (1990) that patent protection should have a finite length. Furthermore, patent protection has to be lengthened when premature obsolescence of a patent due to a subsequent innovation is likely. Optimal policy has to consider industry and demand specifics and it should rely on information which is accessible to the patent authority. While patent length can be set precisely, this cannot be done with patent height and breadth. Since the latter instruments are not quantifiable they are—as already being done—employed at the discretion of the authorities. I have considered stochastic entry by a superior product. Infringements and law suits are other sources of uncertainty. Also repeated subsequent innovations and adding breadth as a third instrument are still open questions. Acknowledgements Financial support from the Deutsche Forschungsgemeinschaft through SFB/TR 15 is gratefully acknowledged. I am grateful to Gerhard Clemenz and Uwe Dulleck at NOeG2005, Konrad Stahl, the seminar participants at INNO-tec, and two anonymous referees for very helpful comments.

References Denicolo` V (2000) Two-stage patent races and patent policy. RAND J Econ 31(3):488–501 Gallini NT (1992) Patent policy and costly imitation. RAND J Econ 23(1):52–63 Gallini NT, Scotchmer S (2002) Intellectual property: when is it the best incentive mechanism? In: Jaffe A, Lerner J, Stern S (eds) Innovation policy and the economy, vol 2. MIT Press, Cambridge, pp 51–78 8

See Gambardella et al. (2005).

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Gambardella A, Harhoff D, Verspagen B (2005) The value of patents, mimeo Gilbert R, Shapiro C (1990) Optimal patent length and breadth. RAND J Econ 21(1):106–112 Hussinger K (2004) Is silence golden? Patents versus secrecy at the firm level, ZEW discussion paper no. 04–78 Klemperer P (1990) How broad should the scope of patent protection be? RAND J Econ 21(1):113–130 Lemley MA, Shapiro C (2004) Probabilistic patents. J Econ Perspect 19(2):75–98 O’Donoghue T (1998) A patentability requirement for sequential innovation. RAND J Econ 28(4): 654–679 O’Donoghue T, Scotchmer S, Thisse J-F (1998) Patent breadth, patent life, and the pace of technological progress. J Econ Manage Strat 7:1–32 Scotchmer S (2005) Innovation and incentives. MIT Press, Cambridge Shaked A, Sutton J (1983) Natural oligopolies. Econometrica 51(5):1469–1483 Takalo T (1998) Innovation and imitation under imperfect patent protection. J Econ 67(3):229–241 Tirole J (1992) The theory of industrial organization. MIT Press, Cambridge van Dijk TW (1996) Patent height and competition in product improvements. J Industr Econ 44:151–167

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