"BARRIERS TO IMITATION AND THE INCENTIVETO INNOVATE"

by

0. CADOT*and

S. A. LIPPMAN**

95/23/EPS

* Assistant Professor of Economics, at INSEAD, Boulevard de Constance, 77305Fontainebleau Cede; France.

** Professor of Economics at the John E. Anderson Graduate School of Management,UCLA, Los Angeles, CA 90024-1481, USA.

A working paper in the 1NSEAD Working Paper Series is intended as a means whereby afaculty researcher's thoughts and findings may be communicated to interested readers. Thepaper should be considered preliminary in nature and may require revision.

Printed at INSEAD, Fontainebleau, France

Barriers to Imitation and the Incentive toInnovate

Olivier Cadot and Steven A. Lippman 1

February 1995

Abstract

When innovation is followed by imitator entry, the degree to which the innovatorcan appropriate the rents induced by its innovations influences the rate of innovativeactivity. Our interest focuses upon the interaction between the rate of innovativeactivity and the length of the delay between the innovation and imitation, in a modelin which innovative activity generates a sequence of new innovations in the face ofmarket saturation and discounting. The optimal rate of innovation depends upon fourdistinct economic forces: the appropriability effect stressed in the literature, fightingmarket saturation, a competitive motivation (to maintain the monopoly position),and a strategic motivation (to deter entry). The goal of our analysis is to elicit thecircumstances in which each force dominates. Because of these countervailing forces,the optimal rate of innovation may not be monotone in the delay 1; furtermore, amore easily saturated market can benefit the innovator.

JEL classification numbers: 030, 031.Keywords: Innovation, Imitation, Entry deterrence.

'Respectively INSEAD, bd de Constance, 77305 Fontainebleau, France; and Anderson School ofManagement, UCLA, Los Angeles, CA 90024-1481, USA. This research was supported in part byINSEAD and the John E. Olin Center, UCLA. We are grateful to Richard Rumelt and BernardSinclair-Desgagne for useful conversations and comments.

1 IntroductionModern growth theory (going back to the writings of the Austrian school) as well as

a long list of empirical studies, discussed in Scherer (1980, Chapter 15), stress theimportance of technological progress to dynamic economic performance (in contrastto the static efficiency of perfect competition). In order for innovation to be profitable,it is clear that innovators must be able to reap, at least temporarily, sufficient profitsto compensate for its substantial cost, risk, and creativity. But a myriad of well-known examples — including the introduction of diet cola by Royal Crown Cola, thepersonal computer by Apple, the (Beta format) VCR by Sony, and the CAT scannerby EMI — evidence the fact that imitation can follow closely on the heels of successfulinnovation, eroding the rents accruing to the initial innovator. Some protection, inthe form of either legal rights or economic barriers to imitation, is thus necessary toensure an adequate rate of technological progress. Statistical evidence indeed supportsthe conclusion (Scherer, 1980, p. 438) that "A bit of monopoly power in the formof structural concentration is conducive to invention and innovation." What is notclear is just what level of protection is needed to power the socially optimal rate ofinnovation; furthermore, it is not certain that more protection always increases theincentive to innovate. Levin et al (1987), for instance, note (p. 788) that "Becausetechnological advance is often an interactive, cumulative process, strong protection ofindividual achievements may slow the general advance"; they cite the semiconductorindustry as an example in which the rapid progress experienced in the 1950s and1960s would have been impeded under a regime of strong protection. 1

Empirical reality and history not withstanding, it has been a commonplace inthe economics literature on innovation (going back at least to Schumpeter in 1911)that more appropriability, usually in the form of a longer patent life, would increasemonotonically the rate of innovative activity. The basic tradeoff relevant to patentpolicy was thus, in the early patent literature, 2 between the dynamic incentive effectsof appropriability and the static welfare costs of monopoly power. More recent work

'In addition to a deliberate policy of the US Department of Defense of ensuring (through second-sourcing and other means) a wide dissemination of technological advances throughout the industry,Levin et al stress (p. 94) that "Indeed, but for two accidents of history - the invention of the transistorby AT&T rather than by a firm less willing to disseminate its knowledge and the simultaneousoccurrence of complementary product and process innovations at TI and Fairchild in the late 1950s- the patent system might have retarded the advance of semiconductor technology."

2See for instance Arrow (1962); Nordhaus (1969, 1972), or Scherer (1972).

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has focused on slightly different questions, such as how governments can use simul-taneously patent length and compulsory licensing fees to maximize welfare (see forinstance Tandon, 1983), or how patent length and 'breadth' (or scope) ought to betraded off for a given level of appropriability (Gilbert and Shapiro, 1990; Klemperer,1990). 3 A recent paper, by Gallini (1992), also showed that a longer patent lengthmay encourage imitators to 'invent around' the patent, thus failing to provide moreappropriability. But the basic assumption, that more appropriability is conduciveto more innovative effort, has by and large not been questioned. Yet Levin et al

warn (p. 787) that "it should not be taken for granted that [...] better protectionnecessarily leads to more innovation." Mindful of this, the present paper revisits thequestion of the incentive effects of appropriability on innovative effort. Specifically,we focus, in a model of repeated innovation, on the interaction between the rate Aof innovative activity and the exogenous delay £ between the introduction of a newproduct or process and its imitation. As suggested by the case of the semiconductorindustry, our analysis (see Theorem 1) produces a non-monotonic relationship be-tween A and £: intermediate levels of protection in the face of imitation engender thegreatest amount of innovative activity. The reason is that, unlike one-shot innovation,repeated innovation serves not only to create a rent initially, but also, later on, todefend it against imitator entry. Ex post product-market competition 4 can thus havefavorable incentive effects on the pace of innovation, because it forces innovators tofight for their position through (stochastically) more frequent product improvements.

1.1 Delays to Imitation: Patents and Isolating Mechanisms

Prior to undertaking an innovative activity, the innovating firm (hereafter called thedeveloper) must believe not only that the (expected) rents produced by the forthcom-ing innovation will be sufficient to cover the expense attributable to the innovativeactivity but also that these rents will flow to the developer and not be competed awayby imitation.

In theory, a patent confers perfect appropriability of a length of time £ duringwhich the new invention is legally protected from imitation. However, the study by

3Patent scope is also considered by Matutes, Regibeau, and Rockett (1991) among others.4Such competition

is different from the ex ante rivalry between firms racing for the patent. The latter is, of course,always favorable to more innovative effort - in fact, such rivalry can lead to socially wasteful over-spending in R&D; see Reinganum (1989) for a survey of the vast literature on patent races.

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Levin et al. found (p. 810) that typical patented products in 34% of the 129 linesof business studied were imitated within 1 year of introduction (and 56% of majorpatented products were imitated within 3 years of introduction). 5 The study by Mans-field et al. (1981) provides further evidence of the imperfection of patent protection:60% of patented successful innovations were imitated within 4 years of their introduc-tion (p. 913). Both studies found that patents do not matter very much except in thechemical industries (and via the cumulative impact of cross-licensing arrangementsin semiconductors). 6 Because design information embodied in a product is subjectto discovery via reverse engineering and other transfer mechanisms, patents can be(necessary and) effective in protecting product innovations. When it comes to processinnovations, however, patents are the least effective means of appropriation due tothe direct leakage of information as well as the demonstration effect 7 : informationcontained in patent documents leads many firms to "refrain from patenting processesto avoid disclosing either the fact or the details of an innovation." (Levin et al., p.795).

Fortunately, however, there are a number of factors, called isolating mechanismsby Rumelt (1987) and entry barriers by economists, which inhibit imitation. Inaddition to patents, these isolating mechanisms come in a variety of first-mover ad-vantages including causal ambiguity regarding the sources of efficiency (see Lippman

5Teece (1987) relates the case of the CAT scanner developed by EMI in which the embeddedtechnology was easy to imitate and the patents were easily circumvented. The superior marketingabilities of GE and Technicare, two technologically capable imitators with strong reputations forproducing high quality medical equipment, enabled them rather than EMI to capture the rents.

6 Mansfield et al. found (p. 916) that "patents are regarded as particularly important in the drugindustry"; the study also reports (p. 915), "Excluding drug innovations, the lack of patent protectionwould have affected less than one-fourth of the patented innovations in our sample." Similarly, Levinet al. found (p. 802) that "the three industries in which product patents were viewed as most effective[were] organic chemicals, pesticides, and drugs." Taylor and Silbertson (1973) found that 64% ofpharmaceutical R&D expenditures are dependent upon patent proctection. Evidently, the discretenature of molecular structures imbues the enforcement of chemical patents against infringementswith an easy to verify argument.

7The lack of protection afforded by patents is particularly strong as regards process innovationsand also product innovations in electronics and machinery. As reported in Mansfield (p. 913), "Themedian estimated increase in [the cost of imitation] due to patent protection was ... about 7% inelectronics and machinery." For many of these innovations "patents would not add a great deal toimitation cost (or time)." and (p. 914) "In the bulk of the cases, the new product could have beenimitated in 2 years or less even if the imitator carried out the project at the most leisurely pace."This is a far cry from the 17 years of protection afforded, in principle, by a patent.

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and Rumelt (1982)), the ability to tie up the market in ancillary specialized assets(say via long-term contracts with suppliers), and other factors common to the indus-trial organization literature (e.g., economies of scale, economies of scope, customerloyalty, customer switching costs, 8 reputation embodied in brand name capita?).

After patents, the most important isolating mechanism emanates from lead timesor lags. These lags leading to delay in imitation can originate from the time re-quired for competitors to recognize the market success of the new innovation, thetime required to reverse engineer the new product, bottlenecks in obtaining the useof specialized marketing and distribution channels, 1° or delays in manufacturing (e.g.,the time required to (a) learn how to manufacture a high quality product, (b) acquireand ready specialized equipment necessary for mass production, (c) line up suppliersand distributors).

Tacit knowledge, causal ambiguity clothed in its less extreme form, is particularlyimportant for retarding the imitation of entrepreneurial and process innovation. By itsvery nature tacit knowledge tends to be non-observable (though it could be revealedby hiring the developer's employees) and difficult to transmit. Thus, lack of employeemobilityll and secrecy (as it is in guarding the formula for Coca-Cola) are ofteneffective in retarding imitation when tacit knowledge is involved.

Whether it arises from a patent, a first-mover advantage, bottlenecks in gaining

8Nintendo created very large customer switching costs by making their game cartridges incom-patible with other game systems.

9Teece (1987, p. 208 - 209) points out that the IBM PC was a huge success even though itsarchitecture was ordinary and its components were standard off-the-shelf parts available from outsidevendors. One key to its success was "The reputation behind the letters I, B, M." Similarly, the G.D.Searle's trade names NutraSweet and Equal for aspartame "will become essential assets when thepatents on aspartame expire."

1°Teece (1987) notes that large firms are more likely to possess the specialized assets which enablethem to gain lead time.

"Ziegler (1985) attests to the fact that the success of an entrepreneur facilitates the entry of otherentrepreneurs and firms not only by bringing forth the new innovation which can then be imitatedbut also by demonstrating to these would-be entrants that profits are more than a mere possibility.Furthermore, Ziegler (1985, p. 119) notes that non-competition clauses and contingent benefits maynot be effective in preventing employee defections. Thus, employees who leave to found their ownfirms is an important source of mobility. The most noteworthy example of entry via the appropriationof design know-how of key employees leaving the parent firm is the lineage from Bell Labs to ShockleyTransistor to Fairchild Semiconductor to Intel. That part of the semiconductor industry located inthe Silicon Valley is also well known for its high mobility of scientific and engineering personnel andthe concomitant "free exchange of technical information." Levin (1982, p. 27)

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use of a complementary specialized asset, or tacit knowledge, the delay time £ untilthe developer's innovation is imitated is an important determinant of the developer'srate A of innovative activity.

1.2 Model and Results

We consider the interaction between a single innovator whose innovative activity gen-erates a sequence of new inventions and a single imitator whose ability to appropriatethe rents generated by the new inventions is restricted by his inability to enter themarket for a period t after the introduction of each new invention. This paper ex-plores the tension between the innovator's rate A of innovative activity and its abilityto appropriate the value bestowed by the new inventions. This ability is measured bythe delay £ until the imitator enters the market for each new invention.

Henceforth, to facilitate communication we refer to the delay £ as the patentlength. Of course, the delay t can, as discussed at length above, emanate from othersources. We begin with a brief consideration of a 'one-shot' model of innovationand imitation in which the force of appropriability acts (as anticipated) to accelerateinnovation: A increases in £. After this, the R&D activity is modeled as generat-ing a sequence of innovations beginning first with a model in which the imitator iscommitted followed in section 3 by a model in which it is possible to deter entry.

We demonstrate that the optimal rate of innovation depends upon four distincteconomic forces: the appropriability effect stressed in the literature, the fight againstobsolescence (or market saturation), the competitive motivation (to maintain themonopoly position), and the strategic motivation (to deter entry by an imitator).Theorem 1, our main result, establishes the (surprising) result that a decrease in ap-propriability can in fact induce more innovative activity. The driving force behindthis result is the dynamics introduced by the foreseeable sequence of innovations inwhich the developer is induced to increase his rate of innovation in order to maintainhis (monopolistic) position as the only firm with the right to market the currentlydominant product. Whereas appropriability increases with the delay £, the competi-tive pressure dwindles. Theorem 1 also shows that the net effect of these two opposingforces first increases and then decreases in t: the optimal rate of innovative activityis not monotone. Theorems 2 and 3 describe the set of Nash equilibria when entrydeterrence is possible; moreover, it is shown in Theorem 3 that entry deterrence canentice the developer to overinvest in R&D.

As our analysis progresses, the four forces are often seen to work in opposition;

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consequently, non-monotonicities in of the developer's optimal rate A of innovation,as exhibited in Theorems 1-3, are to be expected. In the same vein, Theorem 4verifies the existence of circumstances in which more obsolescence actually benefitsthe innovator. The paper concludes with a discussion of policy implications.

2 Patent Length and the Incentive to Innovate

2.1 The Basic Appropriability Effect

The `appropriability effect' of patent length on innovation stressed in the literaturecan be captured in a simple one-shot model of innovation and imitation. 12 At timet = 0, the innovating (or developing) firm, labelled firm D, chooses an intensity level\ of its R&D effort: A is a surrogate for the size of firm D's R&D department. The(random) arrival time r of the innovation is distributed exponentially with rate A. 13

In order to have an R&D department of size A, firm D incurs a flow cost of c(A)per unit time until time r when the discovery takes place. 14 We assume that thefunction c is convex, continuously differentiable, and c(0) 0. Upon discovery, thenew product is costlessly brought to market; from then on, the instantaneous profitrate decreases exponentially at rate (> 0) as the market saturates. 15 Firm D's new

12The one-shot models of Gilbert and Shapiro (1990), Klemperer (1990), and Gallini (1992) analyzethe trade-off between patent width and patent breadth. Gilbert and Shapiro (1990, p. 106) "simplyidentify the breadth of a patent with the flow rate of profit available to the patentee while thepatent is in force." They provide simple conditions under which (socially) optimal patent policy sets

= oo. Klemperer's (1990, p. 116) definition of patent breadth "is the region of (differentiated)product space covered" so that (p. 115) "Wider patents reduce consumers' freedom to substitutecompetitively provided, unpatented varieties of the product." As the importance of the welfare losswhen a less-preferred variety is purchased increases relative to the welfare loss when no variety withinthe product class is purchased, the optimal patent becomes wider and shorter. Unlike our paper,neither of these two papers is concerned with the supply of inventive activity. In Gallini's (1992)model patent breadth is the cost of imitation. In her model with competitive imitation, an increasein imparts a greater incentive to imitators to invent around the patent; consequently, an increasein I can induce the developer to curtail his inventive activities.

13A constant hazard rate is not necessarily the best formulation because it amounts to assumingthat past effort has no bearing on the instantaneous probability of success, but it simplifies theanalysis considerably.

14In effect, the firm's R&D department is fired at t, but one could equivalently suppose that thefirm must employ its R&D department forever without affecting the result.

15This assumption is not needed at this stage but is introduced here to ensure uniform treatmentof the problem between this subsection and the rest of the paper.

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product is protected from imitation by a patent of length £. During the protectionperiod (7 < t < T + £), firm D collects a rent (net of all production costs, but notof R&D costs) at a flow rate of rme-13t . At time t = T + t, the patent lapses andthe imitator introduces a substitute for firm D's product, reducing firm D's rent torye-pt , with rc < rm . All costs and rents are discounted at rate a. Taking account ofmarket saturation and discounting, the value v(s) of the prize awarded to innovationis given by

i 00v(t) = f r„,,e-("+'5)s ds + i rce-('+43)s ds =

o I[r, — (r,, — rc)e-(a+0)e] I (a+ le). (1)

Both intuition and (1) reveal that v'(t) > 0 : imitation washes away the developer'sprize, but patent protection lessens this erosion.

Because the prize v(1) is received at time 7, firm D's net expected discountedprofits 71-D over an infinite horizon is given by (recall that Ecal" = Aga + A))

7rD = E [e-"v(t) — for c(A)C" ds] = [A v (€) — c(a)]/(a + A). (2)

The first-order condition for profit maximization (by choice of A) is

tav(t) + c(A)]/(a + A) = ci(A), (3)

where the left-hand and right-hand side of the equation are, respectively, the marginalbenefit and marginal cost of R&D effort. Let A*(t) be the solution to (3). Under thesecond-order condition (which is easily shown to hold), the sign of the comparative-statics effect of a change in patent length £ on optimal effort A* is given by the signof the cross-partial derivative 027rD /49Aat, where

821DiaAat = at/(t)/(a + A) 2 = a(rm — rc)e-(a+'3)4/(a + A) 2 > 0.

Thus,dA*(t)/dt > 0.

Firm D's R&D effort unambiguously rises in response to a longer patent. Wheninnovative activity is a one-shot affair, it cannot be used by the developer to fend offimitation.

We turn now to a slightly different setting where repeated innovation serves notonly to create a rent but also to defend it.

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2.2 Repeated Innovation with a Precommitted Imitator

We modify here the setting of subsection 2.1 only in allowing firm D to sequentiallyintroduce new generations of the product. Each new generation renders the previousone obsolete: sales of the old generation fall to zero as soon as the new generationis introduced. 16 Each generation is protected by a patent of duration 1; as soon asthe patent expires, imitation begins and, as per section 2.1, the developer's revenueflow falls by a factor of rc/rm . The imitator is committed to staying in the marketwithout regard to the size of A. (We consider imitator exit in the next section.) FirmD chooses the size A of its R&D department at time t = 0; its choice cannot berevised thereafter.' As before, the flow cost associated with a department of size A isc(A). The random additional times required to produce each successive generation areindependent and have the same distribution as r; as in section 2.1 r is exponentiallydistributed with parameter A. We say that the ith 'cycle' begins with the introductionof the ith generation and ends with the introduction of the i + 1 st generation. Theexpected discounted cycle length -y satisfies -y .--:- Eccer = A/(a + A). Let r be firmD's expected discounted revenue during a cycle and let R be its expected discountedrevenue over the infinite horizon so that

m1}

ax-er,t}r = E [Imill{.7.' –(a+43)s ds]+ ELIrni e rye-(a+°3)* c/s1

o i= {rn, — (r,, — Tc)e—(12+.8+1 /(a + 0 + A),

R = rsy + r•y2 + • • • = r-/(1 — -y) = .1r/a,

and rD , firm D's net expected discounted profits over an infinite horizon, satisfies

co+ ,8 + A) ] frn, _ (rm _ rc)e-(a+o+A)/ _ c, .A

16We ignore time-consistency and intertemporal pricing issues [see Bulow (1982), Stokey (1981),Waldman (1993)].

17This assumption is tantamount to assuming that adjustment costs associated with changes in Aare very large, or, in the words of Robert Reich (quoted in Cohen and Levinthal (1994)), "once off thetechnologicalescalator it's difficult to get back on". That R&D is necessarily a continuous activityis supported by Cohen and Levinthal's statement (p. 237), "to integrate complex technologicalknowledge successfully into the firm's activities, the firm requires an internal staff of technologistsand scientists who are both competent in their fields and familiar with the firm's idiosyncratic needsand capabilities."

irD -=[ (4)

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The first-order condition is

1 (a + )3)r„, a + 13 + ( At (r, — r,)e = e(A), (5)+ )5 -FA [a±fi+A a -I- )3 + A

and the second-order condition holds. Let Ain (t) denote the (unique) solution of(5). Again, under the second-order condition, the effect of a change in patent lengthon optimal R&D effort )in (t) is given by the sign of the cross-partial derivative.Straightforward calculations produce 5 2 7rD /OAN = (1 — tA)(rm — rc)e-(Q4-'3+A)t/a.Thus, Ain (.e) satisfies

chin /d.€ > 0 if and only if tA in (t) < 1. (6)

A useful benchmark is the optimal level G out of firm D's R&D effort when there is noimitator. Define Rout by Aout Ain (t).

FIGURE 1 HERE

THEOREM 1 With repeated innovation, there exist two critical values 4 < 4 suchthat (i) Ain (t) > Acrut whenever £ > 4 and (ii) )in (t) is increasing for £ < 4 anddecreasing for t >

PROOF See appendix.

Theorem 1 shows that the appropriability effect identified in section 2.1 andstressed by the literature -- namely, the value v(t) of the patent increases withthe patent length £ -- is just one part of the story. When each new innovation isprotected by a patent of finite duration, it behooves the innovator to continue R&Dbecause each new product introduction "refreshes" the market and forces the imita-tor to incur yet another delay £. 18 Thus, innovation is based on a tactical motivation(fight market saturation) and a competitive motivation (temporarily throw the in-novator out of the market and become a monopolist). Even if there is no marketsaturation (,3 = 0), in which case innovation is exclusively driven by the developer's

18This result is not due to the deterministic nature of the delay Suppose that the imitationdelay is due not to patent protection but rather to the inherent difficulties in "reverse-engineering" anew product, so that the arrival time of the imitation is stochastic. Provided that the instantaneousprobability of success is increasing in past effort spent on that particular product (which is not truefor the exponential distribution), the reasoning goes through.

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competitive motivation, the reasoning is unchanged. When the duration of patentprotection increases, two forces work against each other in the determination of Ain (t)-On the one hand, the size v(s) of the prize (appropriability) increases so it pays toincrease R&D; on the other hand, the competitive pressure from imitation is reducedso the need to defend that rent with new product introductions diminishes. Theorem1 reveals that the second effect dominates beyond 4. This result, intermediate levelsof protection (I) in the face of imitation engender the greatest amount of innovativeactivity (A), comports with the vast empirical literature studying the relationshipbetween concentration and technological progress. Scherer (1980, p. 438) concludesthat "the most favorable climate for rapid technological change ... is a subtle blendof competition and monopoly."

When £ < to, the imitator's ease of entry hampers the developer's ability to defendhis monopoly rents severely enough to overwhelm the competitive force. In this casethe force of appropriability dominates, and the developer innovates at a rate below)out •

This basic non-monotonicity result was established in a context where the imita-tor's behavior is entirely passive. We turn now to the strategic interaction betweenthe innovator's R&D effort and the imitator's entry decision.

3 Innovation and Entry DeterrenceThe cost of "reverse-engineering" is k per product imitated, and the imitator now facesan entry decision at time t = 0. Clearly the imitator's decision will be affected by theremaining length T - l of the product cycle over which the imitation cost k can berecouped. The innovator understands this channel of influence on the imitator's entrydecision. The obvious question is whether the possibility of deterring the imitatorfrom entering can induce the developer to overinvest in R&D. Theorem 3 demonstratesthat overinvestment can occur. However, the imitator's lack of commitment can alsospur a reduced level of innovative activity for large (and some other) values of Weconsider two games where the innovator chooses A while the imitator chooses in orout. If the imitator selects the strategy q from his strategy space [0, 1], he has selectedthe action in with probability q and out with probability 1 — q. When q = 1 [q = 0],we shall say that he has selected in [out]. Using the same reasoning as in section 2,

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the imitator's expected discounted profit 7r1 over one cycle19 is given by

](A ; t) E i

riemax{

'Mri -(a+o). ds _ 1{.7.>1} &e e—cr

--(.+0-FA)t / (a #rie ke-(c'e+A)t,

where r1 is the imitator's rate of profit while selling in the market. Define A c (t) byAc(t) = (r1/k)e-131 — a — # (see figure 2). By construction, ri(Ac(1),€)-a:- 0: A c is the imitator's zero-profit curve. When the developer's rate A of inno-vative activity exceeds Mt), the imitator's expected profit is strictly negative and hechooses not to enter. When the inequality is reversed, he enters. The first game hassimultaneous decisions; the second is a Stackelberg game with the developer movingfirst.

3.1 Simultaneous Game

Let A(q; £) and Q(A; 1) denote, respectively, the developer's and the imitator's best re-sponse correspondences, and let A*(t) and Q*(j) denote the (set of) Nash equilibriumrate of innovation and entry status (they need not be unique).

Theorem 2 tells us that for each I 'the' equilibrium rate A*(t) of innovative activityis restricted to A in (i), Aout , or (when Mt) lies between A in (t) and A0,,t ) Ac(e), a convexcombination of these two numbers. Let q(i) satisfy equation (8) in the appendix.

THEOREM 2 In the simultaneous entry game with k > 0, the following is an ex-haustive description of all Nash equilibria:(i) if Ac (i) < Ain (t) < Aout , then (Aout , out) is the unique equilibrium;(ii) if Ac (i) < Aout < Ain, (t), then (Aout , out) is the unique equilibrium;(iii) if Ac (t) > max fa in (t), Gout }, then (A in (i), in) is the unique equilibrium;(iv) if Ain (i) > A c (t) > Aout , then (Mt), q(t)) is the unique equilibrium.(v) if Aout > Ac (t) > then the game has three equilibria: (Ain (i), (Aout, out),and (Mt), q(t))•

PROOF See appendix.

FIGURES 2A AND 2B HERE

19The imitator's discounted profit over the infinite horizon is simply Airilcr. Thus, it suffices toexamine 21 in order to determine the imitator's action.

(7)

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Note that there is no overinvestment in the simultaneous game: there is no suchthat A*(t) exceeds both A in (t) and Acid . Theorem 2 states that when the imitatorfaces a positive imitation cost, the possibility of keeping the imitator permanentlyout of the market (rather than for only one product cycle) constitutes a fourth force,distinct from the three forces identified in section 2, affecting firm D's incentive toinnovate. This fourth force affects the market outcome in three ways.

First, in the first region listed in Theorem 2, where A c (i) < Ain (i) < Aout , in orderto keep the imitator out the developer chooses a higher rate of innovation than hewould if the imitator were committed to being in the market (see figure 2a). Strategicentry deterrence also happens in one of the three equilibria in the fifth region listedin Theorem 2. Second, an increase in patent duration can cause A* to decrease in aregion of I where Ain is increasing (in figure 2b this is the region where £2 < L < Li).

Third, with patent duration fixed at a sufficiently high level (the second region inTheorem 2: t > 4 in figure 2a and > 4 in figure 2b), the non-committed imitatorchooses to stay out, relieving firm D from the competitive pressure of imitation; as aresult, the rate of innovation is lower than in the committed-imitator case. The reasonis simple: as discussed in section 2, when patent duration is longer than a certaincritical value (namely to), the presence of a committed-imitator provides a positiveincentive to innovate. Therefore, when the non-committed imitator chooses to stayout, firm D reduces its innovative effort compared to the committed-imitator case. Insum, strategic interaction between the developer and a non-committed imitator in asimultaneous game affects the incentive to innovate, but it does so in a non-monotoneway.

3.2 Sequential Game

In the sequential game the developer moves first: he chooses a rate A of innova-tion. After observing A, the imitator, cognizant of the reverse-engineering cost k perproduct as well as the developer's choice of A, 20 decides whether or not to enter. Inorder to avoid non-existence of equilibrium, we assume that the imitator elects notto enter when his expected profit is zero. With this assumption, there is a unique

"In regard to the entrant's knowledge of A, Cohen and Levinthal (1994, p. 228) remark that"the incumbent's endogenous expectations of future technical advances diffuse to the entrant." Thisdiffusion of the incumbent's expectations provides some (albeit small) support for the sequentialmodel vis-a-vis the simultaneous model.

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subgame-perfect equilibrium.21 If MO) > Aout, we obtain, for some values of anew equilibrium which was absent from the simultaneous game, namely (Mt), out).

Theorem 3 states that there exist critical regions of in which (Mt), out) is theunique equilibrium — and it entails `overinvestment'. Just as is true for a great manymodels in the patent race literature that "competition may lead to greater levels ofinvestment as the incumbent makes strategic investments to dampen the entrant'sinvestment incentives" (Cohen and Levinthal (1994, p. 228), it is also true thatimitation can lead to overinvestment by the developer.

THEOREM 3 Consider the sequential entry game with k > 0. If MO) < Aout , the

unique equilibrium of the game is (Rout , out) for any £. If Ac (0) > )gout , then the

unique equilibrium of the game varies as follows (see figure 3):

(i) (Rout, out) when Ac(t) < )out,

(k(t), out) when Ain (.e) < Mt) < Rout,

(Ain (t), in) when < Ac(t) < Aout < Ain(e),

(iv) either (Ain (t), in) or Pc(t), out) when .bout < Ain(t) < Mt).

Moreover, there exists at least one interval (a, b) such that A*(t) > max { Ain (i), A0,4

for a < t < b.

PROOF See appendix.

FIGURE 3 HERE

Thus, in a sequential game, strategic entry deterrence can induce the developerto choose a rate of innovation that is higher than the maximum he would ever choosewith a committed imitator. The reason can be made clear by using an analogy withinvestment in plant capacity: consider A, the rate of innovative intensity chosen bythe innovator at time t = 0, as an investment in 'innovative capacity'. (Such aninterpretation is consistent with the idea that A is a surrogate for the size of firmD's R&D department.) Following Tirole (1988), let us define `overinvestment' in thesequential game as a level of investment in excess of that chosen by the innovator in theequilibrium of the simultaneous game. 22 (In ranges of .e where the simultaneous gamehas multiple equilibria, we take the maximum value of A*(1) as the benchmark.) We

21 1n a continuous time version of this game in which the rate A is selected at each moment intime, the equilibrium we obtain in the game in which the decision is made once and for all remainsan equilibrium in the more complicated game.

22See Tirole (1988), p. 325, footnote 40.

13

now ask, is there overinvestment (thus defined) in the sequential game? The answeris yes (for a <t < b), and the reason is the following. First, notice that as A is chosenat time t = 0 and cannot be revised afterwards, it acts as a commitment. Second, itis easily checked, by inspection of figure 2 or of equation (7), that in a neighborhoodof 7r1 = 0, arvaA < 0. Suppose that, were the innovator to accomodate entry,the optimal value of A would be slightly lower than Mt), the critical value of Awhich makes the imitator's profits zero. By raising A slightly, to Mt), the innovatorcan then reduce the attractiveness of entry by just enough to keep the imitator out.Theorem 3 states that, when a < < b, the loss due to a higher-than-optimal levelof A (the `overinvestment') is more than compensated by the benefit of keeping theimitator out. In this range of £, the innovator's strategy can be loosely compared towhat Fudenberg and Tirole (1984) called a "top dog" strategy.23

3.3 Imitation and Market Saturation

We now consider the effect of changes in the rate /3 of market saturation (equivalently,obsolescence) on the equilibrium rate of innovation in the simultaneous game with

= 0 (i.e., no barriers to imitation). The assumption = 0 simplifies the algebra.Theorem 4 shows that faster obsolescence can lead to an increase in the developer'sequilibrium profit.

Evaluating (5) at £ = 0 yields the first-order conditions (oz-F0)r, = (a+0+A) 2 c'(A)if the imitator is in and (a -I- 0)r„, = (a + A)2 c'(A) if he is out. Let A in (0) andRout (/3)denote the solution to these two equations respectively; Ain ($) < A out (0) (seefigure 4). It is clear from (7) that the imitator's zero-profit curve in (0, A) space isgiven by Ac(0) = (ri/k)— /3.

FIGURE 4 HERE

THEOREM 4 In a simultaneous game with positive imitation cost (k > 0) and no

barriers to imitation = 0), there exists at least one critical value 13" of 13 such that,

23This is somewhat stretching Fudenberg and Tirole's terminology, as the game described here isdifferent from – and simpler than – the standard two-stage capacity-production game. In our model,the scale at which firms operate in the second stage (production) is unaffected by the innovator'sfirst-stage capacity decision. Therefore the point of the innovator's choice of A is not to shift reactionfunctions in the second stage, but rather to deter entry altogether.

14

for e small, rD[A*(4 + E), Q*(/ + 6), Q + 6] > 7D[A*(fl),Q*(4),S]•

PROOF See appendix.

The intuition of Theorem 4 is closely related to that of Theorem 2. When is smallenough, the unique equilibrium is (Ain (0), in). As /3 increases to a first critical point0_, two additional equilibria appear: (a °,40), out) and a mixed-strategy equilibriumsimilar to the one we constructed in Theorem 2 (see appendix). As increases furtherpast a second critical point /3+ (> 0....), the only equilibrium that remains is (A out , out).Therefore, as /3 increases, at some point between /3_ and 0+ , the equilibrium levelof innovation has to jump from Ain to Aoiit . This upward jump in A, associated withthe imitator's change of status (from in to out), increases the developer's equilibriumprofit.

Thus, although faster obsolescence (equivalently, a smaller market) always reducesthe developer's profit given the status of the imitator (in or out), in a simultaneousgame obsolescence can act as a barrier to imitation. When this barrier is sufficientto preclude the imitator's entry, increased obsolescence plays to the advantage of thedeveloper by enabling him to operate as a monopoly.

4 Concluding Remarks

Scotchmer (1991) stresses the importance of the cumulative nature of innovations:"almost all technical progress builds on a foundation provided by earlier innova-tors." 24 A major shortcoming of the theoretical economics literature on patents andR&D is its failure to embrace this cumulative aspect of the innovation process: thisfailure strips the dynamism (and life) out of the process. 25 As a consequence, anymodelling effort which fails to account for more than one generation of an inventionmust be viewed with some suspicion, especially as regards the supply of innovations.Our paper begins to address this defect by positing an innovator whose innovative

24She proceeds by examining instruments to protect the incentives for cumulative R&D and focusesupon the problem of 'double marginalisation'. Proper "incentives to find fundamental technologiesmay require that the first patent holder earn profit from the second generation products that fol-low." (p. 30.) However, in order to "give the second innovator an incentive to invest wheneversocial benefits exceed R&D costs, the second innovator must earn the entire social surplus of hisinnovation." (p. 34.)

25There are exceptions (e.g. Balcer and Lippman, 1984, Cauley and Lippman, 1994, Reinganum,1985, and Vickers, 1984), but most of the literature uses a one-shot model.

activities generate a sequence of innovations, but our effort falls short in that eachinnovation does not build upon its predecessor: neither the incremental value northe interarrival time of the i 1" innovation depends upon any aspect of the ithinnovation.

Whereas the importance of patents is arguable and varies across industries (seeLevin et al.), imitation presents a serious impediment to perfect appropriability inevery industry. Various isolating mechanisms, or entry barriers, facilitate the appro-priation of rents generated by innovation; we summarize them by a nonnegative scalarvariable £, the prototypical barrier being a patent. We show how the introduction ofsuccessive generations of a new product can be used by an innovating firm to pursueseveral distinct objectives simultaneously (creating a rent and subsequently defendingit against the combined effects of market saturation and imitator entry); our resultson the incentive effects of imitation delays derive from the interplay of these motiva-tions. When imitation delays are sufficiently long (€ > 4), the presence of an imitatorincreases the incentive to innovate, i.e. an innovator faced with entry will choose alevel of innovative effort higher than a monopolist would. However, the intensity ofinnovative effort is non-monotone in the length of the entry delay: increasing thatdelay beyond a threshold 4 (> £) reduces the incentive to innovate. 26 The rea-son is that for values of t beyond ti , the longer delay relieves the innovator of thecompetitive pressure of imitation which, alongside with market saturation, drives hisincentive to introduce new generations of the product. In sum, the incentive effect ofimitation on innovation is negative only for sufficiently small values of the entry delayt and, once positive, is largest for intermediate values of t. The policy implications ofthese results depend on whether imitation is socially optimal or not. If imitator entryis efficient, i.e. if the increase in social welfare stemming from competition betweenthe innovator and the imitator outweighs the cost of imitation, 27 then for values of €greater than 4 both the incentive effect and the direct effect of longer patent durationon welfare are negative, so that optimal patent length is finite (as in Gallini, 1992 –albeit for different reasons) and is less than If, on the other hand, imitator entry

26This result vanishes under certainty; if r (the interarrival time of new generations) is determin-istic, the optimal innovative effort rises monotonically in the entry delay up to = r and remainsconstant afterwards.

"This needs not be true. If the demand curve is very inelastic, monopoly entails little deadweightloss, so that imitator entry redistributes rents without creating much surplus; entry can then beprivately optimal (if the rent redistribution is greater than the imitation cost) without being sociallyoptimal.

16

is inefficient, optimal patent length is clearly infinite.The paper also shows how the innovator's capacity to produce sequentially new

generations of a product with (stochastically) short interarrival times, by acting as acommitment, can result in strategic entry deterrence. If the imitator's entry decisionis simultaneous with the innovator's intensity decision, a sufficiently long imitationdelay results in what Bain (1956) called 'blockaded entry': in that case the innovator'seffort is at its monopoly level and is strictly less than what it would be with acommitted imitator. The level of innovative effort in the simultaneous game providesa benchmark against which we define `overinvestment' in a sequential game where theimitator decides whether or not to enter only after having observed the innovator'sdecision. We show that the perfect equilibrium of the sequential game can entailoverinvestment; the attractiveness of such a strategy derives from the fact that moreinnovative capacity, by reducing the average length of time over which the imitatorcan recoup his fixed costs, depresses the expected profitability of entry. If moreinnovation is socially beneficial, imitation is thus welfare-enhancing even in ranges of£ where it does not take place in equilibrium (i.e. where the equilibrium outcome isentry deterrence rather than accommodation). 28

Our results were derived from an extremely simple model, which needs to berefined along several lines. For instance, 'market saturation' should be derived froma full model of intertemporal pricing and consumer behaviour; barriers to imitationshould be variable not only in terms of delays (patent length) but also in terms ofproduct positioning (patent scope), and so forth. Nevertheless, as they stand, theseresults do suggest that the consequences of moving from a one-shot to a repeatedmodel of innovation are nontrivial for the analysis of patent protection.

281n general, the welfare effects of overinvestment in capacity depend on whether the excess ca-pacity built as entry deterrence is used or not. Dixit (1980) showed that in a perfect equilibriumthe incumbent never wants to build idle capacity. Here, the situation is simpler: if one interprets Aas the innovator's 'capacity' to innovate, the variable cost of innovation is zero up to A and infiniteafterwards; so the innovator will clearly always want to use all his capacity, which implies thatsociety will benefit from it.

17

References

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[3] Baumol, William B., John Panzar, and Robert Willig (1982), Contestable Mar-

kets and the Theory of Market Structure, Harcourt Brace Jovanovitch.

[4] Bulow, Jeremy I. (1982), "Durable-Goods Monopolists," Journal of Political

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[5] Cauley, Fattaneh G., and Steven A. Lippman (1994), "Myopia andR&D/Production Complimentarities," Economic Theory 4, 437-451.

[6] Cohen, Wesley M., and Daniel A. Levinthal (1994), "Fortune Favors the PreparedFirm," Management Science 40, 227-251.

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[13] Kaufer, Erich (1989), The Economics of the Patent System, Harwood, Chur,Switzerland.

18

[14] Klemperer, Paul (1990), "How Broad Should the Scope of Patent ProtectionBe?," RAND Journal of Economics 21, 113-130.

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Economics 13, 418 - 438.

[18] Mansfield, Edwin, Mark Schwartz, and Samuel Wagner (1981), "Imitation Costsand Patents: An Empirical Study," Economic Journal 91, 907-918.

[19] Nordhaus, William D. (1969), Invention, Growth, and Welfare: A Theoretical

Treatment of Technological Change, Cambridge, MA: MIT press.

[20] Nordhaus, William D. (1972), "The optimum Life of a Patent: Reply", American

Economic Review 62, 428-431.

[21] Reinganum, Jennifer (1989), "The Timing of Innovation: Research, Developmentand Diffusion", in R. Schmalensee and R. Willig (eds.), Handbook of Industrial

Organization, Vol. 1, Elsevier, pp 849-908.

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20

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21

Appendix

Proof of Theorem 1 To begin, observe from (5) that Ain is a continuously dif-ferentiable function, whence g(t) tAin is continuously differentiable. Furthermore,g(0) = 0 and g(t) --+ oo as £ oo, as lime.. Ain(t) = Aout > 0. Thus, there is atleast one solution to g(t) = 1. Let 4 be a solution to g(t) = 1.

To establish that 4 is the unique solution to g(t) 1, define h by h(t) Ain(t)111, so that h1 (1) = clAin (1)1d1 + 1/12 , h(ti ) = 0, and h' is continuous on (0, oo).Suppose that h(s) = 0; then (6) implies clAin (s)1 = 0, whence h' (s) = 1/s2 > 0.Because h' is continuous, h(t) > 0 for some neighborhood of s: h crosses 0 from belowonly. Therefore, h(t) = 0 has a unique solution. By (6) and the definition of 4, Ain(1)is strictly increasing on [0, 4) and strictly decreasing on (4, co]. As per (5), Ain (0) andAout solve (a+,8)7.,= (a-1-13+))2c'()) and (cr-143)r,„ = A)2c'(A), respectively.By assumption, c' is nondecreasing so that these equations admit but one solutioneach; in addition rc < rm implies that Ain(0) < A.A . Thus, lime.. Ain(t) = Aout >A in (0) and Ain is decreasing on [4, oo}: Ain(ii) > Aout . Because Ain is increasing on[0,4), it follows also that there is an to < 4 such that Ain (t) > A out whenever 1 > 4.29

0

Proof of Theorem 2 Let Lout = {t : Aout > Ac(t)}, Lin = {t : Ain (t) < Mtn,£2 = min Lout , so Ac(4) = Aout , and 4 = min ft : Ain (t) > Ac (t)}. Finally, letgo(A,i) and g1 (Are) stand for (A, q, t) = arD (A, q, .1)1 OA evaluated at q = 0 andq = 1 respectively. Fix £ E Lin so that A in (l) < Mt). Clearly, (Aio (t), 1) is anequilibrium. We claim it is unique in L iu \Lout . To see this, consider any A > Mt).Then Q(A; 1) = 0 so A(0; 1) = Aout < Ac(t), a contradiction.

If E Lout , Aout > Ac(1) so by the argument of the previous paragraph, (A out , 0)is an equilibrium and it is unique in Lout\Lin. If E Lout n Lin , the game has threeequilibria: (A in (i), 1); (A*(co), 0), and a mixed-strategy equilibrium (Mt), q(t)). Inorder to construct the mixed-strategy equilibrium, let q(t) satisfy

q(t) gi [A c (t), I)] + [1 — q(1)] go[A,(t),11 = 0. (8)

If such a q(t) exists, A(q, t) = Ac(t) and Q[Ac(i); ti = q(t), as 71 [Ao(t), 1,1} =i[Mt), 0,11 = 0 implies that q(t) is a weak best response to A c (t).) We need to

29In general, tAin(i) is maximized at a value £2 > 11•

22

show that q(t) exists and is in [0, 1]. If Lout n Lin 0 0, £2 < 4, so t E Lost n Lin 4#*

£2 < < £3 . Suppose that £ = £3 , then Mt) = Ain (t) by continuity of these functions.As gi [Ain (t), 1] 0, it follows that gi [Ac(t), = 0; so, setting q(t3 ) = 1 satisfies (8).Suppose now that £ = £2 , Ao (t) = Aout so by a similar argument, setting q(t2 ) = 0satisfies (8). Next, fix any £ E (t2, 4). By monotonicity of k(t), A in(t) < k(e) < Aout.The last inequality implies that, under the second-order condition, 9 1 [Ac(t),t] > 0.The first inequality similarly implies that go[A c(t), < 0. So q(t) gi[Ac(t), + [1 -q(t)] go[Ac(t), £] = 0 q(t) E (0, 1). Finally, if £ E out U Lin); then (Mt), q(t))is the unique equilibrium of the game. To see this, take any convex subintervalL C R+\(Lout U Lin), and denote by 1_ and 4 its min and max respectively. Byconstruction of Lout and Lin, £ E L implies that Acrut < Mt) < Ain (1). Monotonicityof Ac (t) implies that A*(t_) = Ac(L) and A*(4) = A c (t+ ). Then q(L) = 1 andq(4) = 0 satisfy (8). Next, fix any £ E (t_, 4). By the second-order condition,g1[Ac(1),1] > 0 and go[Ac(t), < 0; so q(1) gi [Ac(1),t] + [1 - q(t)] go[Ao (t),t] = 0implies that q(t) E (0,1). For uniqueness, consider any A > Mt). Then Q(A, £) = 0,but A(0, £) = Aout < Mt), a contradiction; a similar argument holds for any A < Ac(t).

0

Proof of Theorem 3 Let 1 min {t2 , t3} and (5 = 7rp [Ac(1),Q[Ac(1),i], -7rD [Ain (1), Q[Ain(i),1],1]. We claim that S > 0. To see this, note that Q[Ain (1),1] = 1whereas Q [A c (1) , 1] = 0. Thus ir D[Ain(i) Q[Ain(i), = 7rD[Ain(i), 1 , 1] < lrD[Ain (e),o , .e] < 7rD [Aoist , 0, = 7rDPtc(i) , = 7rD [Ac Q [Ac(i), . Next, note that7rD(A, q,i) is continuous in A and £, while Ain (.e) and Mt) are both continuous in £;so 1- D (Ain (i), 1, £) and 7D (Ao (t), 0, £) are both continuous in £. Therefore there existsan e > 0 such that for £ > - E, 7rD [Ac(i) , Q[Ac (i), 1] , 1] > 7rD [Ain (e), Q[Ain,(1) , 1] , 1]By continuity, one can find an interval [a, 1)] where A*(t) > max {A in (t), A"t } fora < t < b.

0

Proof of Theorem 4 Let o_ and '3+ satisfy respectively Aout(0-) = A( ii..) andAin (4) = Ac (3+ ). Consider j3 such that ( A* (g), Q* (0)) = (Ain (3), in) and (A*(/3 +

Q*(0 + e)) = (A0, (,3 + e), out) for any e > 0. As (A in (,3), in) is the unique equilib-rium to the left of [0_, 0+] and (A0,43+ e), out) is the unique equilibrium to the rightof [0_„ 8+] 0 exists and is in [0-, 04 Now, notice that lrD[A(in; 0), Q[Ain(0), fib =Irn[A(in, 0), in, 0] < rD[A(in; 0), out, 9] 7rD [A(out; 0), out, = 7rD[A(out ; 15),

23

Q [Aout(p),8], 8]. Next, observe that 7rD is continuous in A and p, and that Ain(0)and A.ut(s) are continuous in j3, so 7D[Ain(0), in, )3] and rip [Aout(P), out, 0] are contin-uous in Q. Therefore there exists an e > 0 such that 7rD[A*(,8 + 6), Q*(/3 + e), 0 + el >7 r D[Ain(0), Q* (P), Pl.

0

24

ft.

Ow 4.MI. -__

- At= 1

Figure 1

A

■

Aout

£1

A

Figure 2a

/ t3 \

£1

£2 to

Figure 2b

flout

£0 £2

Q3

£1

Figure 3

/ a \bto

Figure 4

a + 0

£1