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Optimal Patent Length and Breadth Author(s): Richard Gilbert and Carl Shapiro Reviewed work(s): Source: The RAND Journal of Economics, Vol. 21, No. 1 (Spring, 1990), pp. 106-112 Published by: Blackwell Publishing on behalf of The RAND Corporation Stable URL: http://www.jstor.org/stable/2555497 . Accessed: 18/01/2012 17:55 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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RAND Journal of Economics Vol. 21, No. 1, Spring 1990

Optimal patent length and breadth Richard Gilbert* and Carl Shapiro* *

In providing rewards to innovators, there is a tradeoff between patent length and breadth. This article provides conditions under which the optimal patent policy involves infinitelylived patents, with patent breadth adjusting to provide the required rewardfor innovation. 1. Introduction * The primarypurpose of the patent system is to rewardinnovators. Unfortunately, because these rewardsare based on the creation of market power, they necessitate some welfare loss. Much of the debate about patent policy had focused on this tradeoff between the dynamic benefits associated with innovation and the static costs of patent monopoly power. This debate has been cast in terms of the optimal lifetime for patents.' Patent policy can be decomposed into two parts: first, a choice of how much to reward each patent; and second, how to structure each given reward. While the question of how much to reward patentees necessarily requires some estimate of the elasticity of supply of inventions, the efficient way in which to structure a reward of given size does not. It is this latter question that we address here. In particular, we examine the socially optimal mix between patent length and patent breadth, for a given size of the patentee's prize. In posing this question we begin with a very general definition of patent breadth: we simply identify the breadth of a patent with the flow rate of profit available to the patentee while the patent is in force. Our work suggests that the conventional analysis of optimal patent length, based on the tradeoff between the incentives for innovation and the extent of static monopoly deadweight loss, has been misplaced, or at least takes too limited a view of the instruments that make up "patent policy." When patent policy is viewed to be a choice of patent breadth as well as patent length, we find that the optimal length may easily be infinite. The appropriate margin on which patent policy should operate may not be patent length, but rather patent breadth. * The University of California, Berkeley. * * Princeton University. We thank William Baxter, John Vickers, Robert Willig, and a referee for valuable comments. C. Shapiro acknowledges the financial support of the National Science Foundation and the John M. Olin Foundation. ' See, for example, Nordhaus ( 1969) and ( 1972) and Scherer ( 1972).

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107

We pose the following policy question: What is the optimal mix between patent length and breadth as instruments to reward innovation? Our reduced-form analysis in Section 2 addressesthis problem quite generally. There we provide a general condition for the optimal length of the patent grant to be infinite. Our subsequent analysis in Section 3 is in the context of a homogenous product, where we interpret breadth as the ability of the patentee to raise price. In independent work, Klemperer ( 1990) considers the patent breadth problem in a model of spatial product differentiation. A larger patent breadth in his model corresponds to a larger region of the product space that is included in the patent grant. Thus the focus of Klemperer's analysis is on the optimal scope of patent protection, while we emphasize the extent to which a patentee may exploit the patent monopoly for a given coverage of the patent grant. We show that in a homogeneous-good market, our general condition from Section 2 is typically met, so the socially optimal way to rewardinnovation involves patents of infinite length. This is in contrast to results in the traditional literature on the optimal patent life, which assumes that the degree of patent protection is fixed. A special case of our result appears in Tandon (1982). He shows, in an example using linear demand and constant marginal costs, that optimal patent lifetimes are infinite when patent policy consists of a patent lifetime and a royalty rate for compulsory licensing. Klemperer ( 1990) also provides an example of customer tastes for which a very narrow, but infinitely long, patent grant is optimal. Why are infinitely-lived patents optimal? Increasing the breadth of the patent typically is increasingly costly, in terms of deadweight loss, as the patentee's market power grows. When increasing the length of the patent, by contrast, there is a constant tradeoff between the additional reward to the patentee and the increment to deadweight loss, at least if the underlying environment is stationary. So, in many circumstances, the socially cost-effective way to achieve a given reward to innovators is to have infinitely-lived patents with the minimum market power necessary to attain the required reward level.

2. The general

result

* We study the socially cost-effective way in which to achieve any given reward, V, for an innovation. This cost-effectiveness problem is a necessary piece of optimal patent policy. There are two instruments available to achieve the desired reward: the length of the patent and its breadth. The length is simply the lifetime of the patent grant, which we denote by T. Patent breadth is less straightforward, since "breadth" can mean many different things. We discuss different interpretations of patent breadth below. But any definition of breadth involves the idea that a broader patent allows the innovator to earn a higher flow rate of profits during the lifetime of the patent. So we begin with a reduced-formspecification in which we simply identify breadth with the flow rate of profits, ir, available to the patentee while the patent is in force. Optimal patent policy consists of choosing T and ir to maximize social welfare, W, which equals the sum of consumer surplus and profits, subject to achieving the given reward V for the patentee. The key tradeoff is that between social welfare and profits, W(ir) on a flow basis. The assumption that W'(ir) < 0 reflects the idea that broader patents confer greater market power and associated deadweight loss. Once the patent has expired, flow profits decline to ir and flow social welfare rises to W = W(7-0. Suppose that the underlying environment is stationary and predictable.2 Discounted social welfare is given by

2

importance of these assumptions,and the consequences of relaxingthem, are discussedin the conclusion.

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+

W(r)e-rtdt

Q(T, 7r)

f

We-rtdt,

and the present value of the patentee's profits are 0r

rT

V( T, r)

irertdt +

=

7ertdt.

f

The optimal mix between length and breadth involves maximizing Q(T, V( T, r) ? V. Our main result is found in the following proposition.

ir)

subject to

Proposition 1. Suppose that W"(ir) < 0 at all ir, i.e., patent breadth is increasingly costly in terms of deadweight loss. Then optimal patent policy calls for infinitely-lived patents. Proof. Define O(T) as the flow of profits required in order to achieve a total reward to the patentee of V if the lifetime of the patent is T. By definition, T 00 1 - e~~~~~~~-rT e--rT X(1) v fJ5o(T)e-rtdt + f e -rtdt = 5(T) r r Differentiating ( 1 ) with respect to T gives 0=

(0(T)

r)e-rT+

-

0/(T)

1-

(2)

r

Now consider the total welfare that is achieved if the patent lifetime is set at T and the breadth of the patent at X( T). Total welfare is Q(T, 0( T)). Differentiating with respect to T we have dT Since OQ/OT= (W(r) -

-

W)e-rTandOQ/Oir

dQ

dT

aOT aOr

= (W(O(T))

- W)e

-rT

= W'(r)(1 + W'(0( T))

-

e-rT)/r,

we have

e -rT I ~~~~~~~r

Substituting from (2) for 0'( T) gives dQ = (W(q(T))

-

W)e-rT-

(q(T)

- r)W'(q(T))e-rT

If W"(ir) < 0 on [or, O(T)], then -W'(O(

Hence d is optimal.

T)), >

W(O(T))

> ?0 as required. Increasing T always raises welfare, so an infinitely-lived patent Q.E.D.

The key condition for Proposition 1 is that increasing the patentee's rewardson a flow basis is increasingly costly in terms of social welfare, i.e., W"(ir) < 0. We next explore conditions under which this condition is met, looking more closely at the meaning of the patent breadth variable.

3. Optimal patent and antitrust policy * Both patent and antitrust policies can constrain the price set by a patentee. Policies that determine the scope of patent protection affect pricing by defining the products that can substitute for a patented good. Greater protection from infringement-greater patent scope

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109

in the sense of Klemperer ( 1990) -permits a higher optimal price. Other policies affect the ability to exploit the patent grant for any given scope of the grant. For example, attempts by the patentee to set price above some level may requirepracticessuch as exclusive territories, price-restrictedlicenses, tying arrangements, or other vertical restraints, that may call forth antitrust suits, either from private parties or the government. In addition, the patentee may be subject to compulsory licensing at "reasonable fees" which again imposes a price ceiling. Alternatively, patent protection may determine the costs of imitation and thus the price that the patentee can charge without facing such imitation. In all of these cases patent breadthtranslatesinto a maximum price the patentee can charge,or equivalently,a minimum quantity that he must sell. We analyze optimal policy here for this particularinterpretation of patent breadth: the ability of the patentee to raise the price for the single product that embodies the innovation. Our approach in this section should be distinguished from that in Klemperer's ( 1990) spatial model. In his model, increasing the scope of the patent grant makes noninfringing substitute products less attractive to consumers. With the definition used in this section, patent breadth has no effect on the set of substitute products that are offered to consumers; breadthonly affectsthe price that the patentee can charge.In Klemperer'smodel, substitution away from the patented product can actually be reducedby widening the scope of the patent grant. Klemperer assumes that within the scope of the patent grant, the patentholder offers product varieties to serve every customer's tastes. Therefore, no customer whose mostpreferredvariety is within the scope of the patent grant need suffer a disutility (in product characteristics) from purchase of the patented good. If the patent grant is very wide in scope, nearly all customers would prefer to buy the patented product (despite a high price) rather than substitute a lower-priced unpatented good with less desirable characteristics. If, in addition, the amount of the product that consumers purchase is not very sensitive to the price, there may be little deadweight loss associated with patents that are very wide in scope. Thus, in Klemperer's model, patents that are wide in scope and short in duration can be preferred to patents that are narrow in scope and long in duration. In our model, by contrast, the extent of substitution away from the patented product, and the deadweight loss associated with that substitution, always rises with the breadth of the patent. It should be noted that when pricing control is included as a policy instrument along with the scope of patent protection in Klemperer's spatial model, the optimal policy calls for a wide scope of patent protection along with price controls to achieve no more than the desired reward for innovation. Consider then a process or product innovation for which the (inverse) demand is given by p(x). Welfareis w(x) = B(x) - C(x), whereB(x) fo p(z)dz is the total benefit function and C(x) is the patentee's cost function. Profits are p0(x)= xp(x) - C(x). Call xm the monopoly output and x* the welfare-maximizing output. In the relevant range, xm < x < x*, we make the weak assumptions that 0. Define g(ir) as the inverse function of sp(x), i.e., sp(g(r)) = r. Then W(r) = w(g(ir)). Proposition 2. If profits and welfare are both concave in output, then welfare is concave in the patentee's profits, so the optimal patent lifetime is infinite. Proof. Taking derivatives of W(ir) twice, we have WJ"(_r)= w'g" + w"(g')2.

Since w' > 0 and w" < 0, W"(r) < 0 if g" < 0. But g'(ir) -

=

1/ p'(g(r)), so

X

g (X) = _

g "(ir)

(3)

_ _ _ _ so(g(7r))2

_

_

_

With (p"< 0, and since g' < 0 in the relevant range, we indeed have g" < 0.

Q.E.D.

Remark.The conditionsof Proposition2 are met if the demand and marginalrevenue

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curves slope down and the marginal cost curve slopes up. Even weaker conditions, however, will sufficeto establishthe concavity of welfarein the patentee'sprofits.Suppose that marginal costs do not decrease with output, c" 2 0. Define E 3-p(x)/xp'(x) as the elasticity of demand and let 0 be the elasticity of E with respect to price. Then direct calculations demonstratethat W"(ir) < 0 if 0 > -( - m)/m wherem (p - c')/p is the markup.If the elasticity of demand is constant or increasing in price, this condition is surely met. =

Even if the weak conditions of Proposition 2 or the preceding remark are not met, we still know that infinite lifetimes are optimal for "small" patents. Proposition 3. For small values of the patentee's reward, V, the optimal policy involves an infinitely-lived patent. Proof. First we show that for prices sufficiently close to marginal cost, welfare is concave in the patentee's profits. Since welfare is maximized when price equals marginal costs, i.e., when x = x*, we know that w'(x*) = 0 and wI'(x* ) < 0. For prices close to marginal cost, w' is close to zero and w" is still negative. Therefore, from equation ( 3), W' < 0 at such prices. Consider any candidate optimum policy with a finite lifetime, (r, T). If W"(r) < 0 on the interval [r, 7r, then we know from the proof of Proposition 1 that a slight decrease in ir, with the necessary increase in T to leave V unchanged, will increase welfare. If ( 7r,T) is to be an optimum, we must therefore have W"(ir) > 0 for at least some ir on the interval [r, 7r]. Differentiation of Q as in Proposition 1 reveals, however, that for (r, T) to be a local optimum requires W"(7r) <. If(7r, W) is as drawn in Figure 1, any small change in T away from T will lower welfare. For small values of V, however, we can always find a nonmarginal change that raises welfare. Refer to Figure 1. From the properties we have established in the previous two paragraphs,there must exist a 7r < 7rwith the following two properties:(a) the ratio of flow deadweight loss to flow profits is the same at 7-ras at r; and (b) W(r) is concave on the interval (or, 7) .

FIGURE 1 (r, W) IS NOT OPTIMAL W

W(ir)

W j

wL

i~~~~~~~~~~~~~~~~~~~~~~~~~i

X

itM

X VM~~~~~~~7

GILBERT AND SHAPIRO

/

11 1

Now consider the alternative policy of allowing a flow of profits ir over a long enough time period to give a total reward of V. For V small enough, this is always possible, since the flow rate of profitsneed only be r V. Since this new policy has the same ratio of deadweight loss to "excess profits" (r - 7) as does the candidate optimum, and the total excess profits are the same, the total deadweight loss must also be equal. In other words, the new policy is just as good as the candidate optimum. But the new policy can be improved upon with a marginal change to reduce flow profits and increase patent length, since W"(ir) < 0 on the interval [or, ir]. We can conclude that the original candidate optimum was not in fact optimal. Q.E.D. The optimal policy for small values of V calls for T = so and ir = r V. With price close to marginal cost, the deadweight loss is proportional to (p - c')2, so the ratio of deadweight loss to patentee profits approaches zero as V does so. This is a local version of our general theme, that the social costs of patent prizes are minimized by keeping prices as close as possible to marginal cost, i.e., with narrow but lengthy patents. Propositions 2 and 3 suggestto us that when we consider several instruments of optimal patent policy, i.e., the scope of protection from imitation as well as the length of the patent grant, then optimal policy may well call for infinitely-lived patents. In this sense, previous emphasis on the optimal length of patents seems misplaced. If one takes a broader view of patent policy, either to include the antitrust treatment of intellectual property or protection from imitation, the policy margin of patent length is not a useful one on which to operate. The optimality of infinitely-lived patents carries over to a patentee offering a range of products relying on the same patent. In this case the optimal policy involves infinitely-lived patents along with Ramsey pricing, since Ramsey pricing is the (static) solution to the problem of achieving a given profit level at least social cost. In the case of products with independent demands, the constant of proportionality between the markup of each good and its elasticity of demand is an increasing function of the overall required profit level, V.

4. Conclusions * This short article reports a simple but general result in the design of optimal patent policy. If one interpretspatent policy broadlyenough to include at least one policy instrument that affects the flow of profits from the sale of the patented product, then optimal policy calls for infinitely-lived patents whenever patent breadth is increasingly costly in terms of deadweightloss. We have shown this typically to be the case for the conventional deadweight loss stemming from monopoly pricing of a single patented product. Klemperer ( 1990) demonstrates that broad, short-lived patents can be optimal if wider patents discourage substitution away from the patented product by making the noninfringing alternatives less attractive to buyers. Given the overall level of rewardsto innovators, our analysis suggests that appropriate treatment of intellectual property calls for longer patent lives combined with more careful antitrust treatment of patent practices, such as the provisions of licensing contracts. Of course, if the current level of rewardsto innovators is viewed to be inadequate, then it may be appropriate to give stronger protection from infringement even as patent lifetimes are extended. Our point is that longer patent lifetimes are optimal, whatever one believes about the overall level of rewardsto innovators, so long as patent breadth is increasingly costly in terms of deadweight loss. On the other hand, we must express a warning about the policy-relevance of our finding here. One limitation is clear from the fact that Klemperer obtains a very different result, namely the optimality of very short, very broad patents, if substitution to alternativeproducts is the main source of deadweight loss, rather than substitution out of the product class altogether.

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Another limitation in our analysis is due to our assumption that the underlying environment is predictable. Suppose instead that there is uncertainty about future demand and cost conditions. Such uncertainty would have no effect on our results if the firm is risk neutral. If the firm is risk averse, however, efficient risk sharing then calls for broader, shorter patents than in the risk-neutral case. So long as the uncertainty about profits is larger for profits earned in the more distant future, the efficient way to provide the innovator with a given level of expected utility will involve declining (expected) profits over time. If the available policy instrument is patent lifetime, a finite lifetime will be optimal. The most important limitation of our analysis stems from our assumption that the underlying environment is stationary. We made this assumption to focus attention on a single innovation. In practice, however, inventions build on each other, and a long patent grant may have deleterious effects on the incentives of other firms to engage in related research, for fear that they will be at the mercy of the original patentee.3 What would have happened in telecommunications, for example, if the telephone were still patented?4 In particular, there appears to be a danger that an overly-long patent would retard subsequent innovation by establishing monopoly rights to an entire line of research. If this is the case, the tradeoff between deadweight loss and profits at the margin would no longer be constant as the patent lifetime increases. Rather, there might be increasing social costs in comparison to patentee profits as the patent grant is extended in time. Further modelling of markets with a sequence of related innovations will be required to characterize optimal patent policy in such settings. References KLEMPERER,P. "How Broad should the Scope of Patent Protection Be?" The RAND Journal of Economics, Vol.

21 (1990), pp. 113-130. W. Invention, Growth and Welfare:A Theoretical Treatment of Technological Change. Cambridge, MA: MIT Press, 1969. "The Optimum Life of a Patent: Reply." American Economic Review, Vol. 62, (1972), pp. 428-431. SCHERER,F. M. "Nordhaus' Theory of Optimal Patent Life: A Geometric Reinterpretation."American Economic Review, Vol. 62, (1972), pp. 422-427. TANDON, P. "Optimal Patents with Compulsory Licensing." Journal of Political Economy, Vol. 90, (1982), pp. 470-486.

NORDHAUS,