Cumulative Innovation, Growth, and the Protection of Ideas

Transcription

1 Cumuative Innovation, Growth, and the Protection of Ideas Davin Chor y Edwin L.-C. Lai z This draft: 3 December 204 Abstract We construct a tractabe genera equiibrium mode of cumuative innovation and growth, in which new ideas stricty improve upon frontier technoogies, and productivity improvements are drawn in a stochastic manner. The presence of positive knowedge spiovers impies that the decentraized equiibrium features an aocation of abor to R&D activity that is stricty ower than the socia panner s benchmark, which suggests a roe for patent poicy. We focus on a non-infringing inventive step requirement (or patent breadth ), which stipuates the minimum improvement to the best patented technoogy that a new idea needs to make for it to be patentabe and non-infringing. We estabish that there exists a nite patent breadth that maximizes the rate of innovation, as we as a separate optima breadth that maximizes wefare, with the former being stricty greater than the atter. These concusions are robust even when aowing for an additiona instrument in the form of patent ength poicy. Keywords: innovation, growth, inteectua property protection JEL Cassi cation: 03, 034, 040 We thank Costas Arkoakis, Angus Chu, Jonathan Eaton, Gino Gancia, Gene Grossman, Erzo G.J. Luttmer, Kaina Manova, Marc Meitz, Giacomo Ponzetto, Stephen Redding, Esteban Rossi-Hansberg, and Jonathan Voge for hepfu comments, as we as audiences at Mebourne, Penn State, the Phiadephia Fed, Princeton, UC San Diego, Seou Nationa University, Stanford, the Barceona GSE Summer Forum, the SED Annua Conference (Seou), the Asian Meeting of the Econometric Society (Singapore), and the AEA meetings (Phiadephia), HKUST Conference on Internationa Economics, University of Internationa Business and Economics in Beijing. The work in this paper has been partiay supported by the Hong Kong Research Grants Counci (Genera Research Funds Project No ). Davin thanks the Internationa Economics Section at Princeton for their hospitaity. This paper previousy circuated under the tite: Cumuative Innovation, Growth, and Wefare-Improving Patent Poicy. A errors are our own. y Department of Economics, Nationa University of Singapore. Address: Arts Link, AS2 #06-02, Singapore E-mai: davinchor@nus.edu.sg z (Corresponding author) Department of Economics, Schoo of Business and Management, Hong Kong University of Science and Technoogy, Cear Water Bay, Kowoon, Hong Kong. Emai: eai@ust.hk and edwin..ai@gmai.com

2 Introduction Long run growth continues to be an important topic of research for economists, despite the onset of the Great Recession in 2008 and the attention this has drawn to business cyce uctuations. It is widey recognized that ong run growth is utimatey driven by innovation. It foows that innovation-reevant poicies such as the protection of inteectua property rights (IPR), antitrust aws and institutions of property rights protection in genera can cruciay a ect ong run growth outcomes (e.g., North and Thomas 973; Romer 990). This view is supported by a number of empirica studies which have found that the protection of inteectua property has contributed greaty to growth, particuary in such key industries as pharmaceuticas, chemicas, data processing equipment and biotechnoogy (Goud and Gruben 996; Maskus and McDanie 999). The economic anaysis of IPR protection has thus focused on the e cacy of patents as an instrument for promoting research, growth and wefare. The buk of this iterature has arguaby focused on patent ength the duration of a patent s vaidity as the key poicy of interest. In this regard, there is now a we-deveoped argument for the existence of an optima patent ength in a broad cass of modes where innovation expands upon the set of products (e.g., Nordhaus 969; Tiroe 988; Grossman and Lai 2004): An increase in the patent ength induces a higher rate of innovation by extending the duration of the innovator s monopoy power (the dynamic gains), and this is traded o at the margin against the consumer surpus that is conceded (the static osses). In practice, however, there are many dimensions to patent protection. Innovation often aso takes the form of improvements, where new technoogies continuay buid upon od ones in a cumuative fashion, rather than create a radicay new product. In order to provide su cient incentive for innovation, the IPR regime thus has to protect patent-hoders from minor, incrementa ideas. Toward this end, patent systems typicay contain causes that require future inventions to demonstrate a arge enough improvement in order for them to be non-infringing and patentabe. In this paper, we deveop a mode of endogenous growth with cumuative productivity/quaity improvements, and appy it to investigate the e ects of instituting a patent breadth (or a inventive step requirement) to protect inteectua property in the above spirit, namey by reguating how di cut it wi be for a new idea to be considered non-infringing and patentabe. Is the patent breadth in fact a reevant and active poicy margin? We woud argue that the answer is a rmative. Ordover (99) and Maskus and McDanie (999) describe how Japan adopted a poicy of narrow patent breadth during , in order to encourage more innovation and faciitate its catch-up with more advanced countries at that time. More recenty, despite the harmonization of patent Bodrin and Levine (2008) have argued that the patent system as currenty conceived and impemented, cedes too much power to incumbent patent-hoders to the extent that it has discouraged innovation e ort instead.

3 ength among WTO signatories under the TRIPS agreement, individua countries sti have discretion to impement varying patent breadth standards, as re ected in di erences in what constitutes infringement. For exampe, in its 24 August 202 judgement on the Appe v. Samsung case, a US court found Samsung to be guity of infringing on Appe s patents. But just a few days ater on 3 August, a Japanese court reached the opposite concusion. 2 This outcome con rms the view of many that the US tends to a ord reativey broad patent protection compared to some other deveoped countries. Exampes of the speci c means by which the authorities can reduce the patent breadth incude imiting the extent to which the doctrine of equivaents is invoked and imiting the product/technoogy scope of patent caims. 3 As a baseine, we rst construct in Section 2 a genera equiibrium mode of growth driven by cumuative innovation, without yet introducing a binding patent breadth. We draw on the industry structure and Poisson arriva process for ideas from Kortum (997) and Eaton and Kortum (200), but adapt their setup to a speci cation where: (i) new ideas stricty improve upon existing frontier technoogies, as in the modes of Grossman and Hepman (99) and Aghion and Howitt (992); and (ii) the size of the productivity improvement that each new idea makes is an independent draw from an underying Pareto distribution. We focus on a steady state in which a constant share of abor is engaged in innovation, with the remaining workers aocated to production activity. It turns out that this steady state sustains a positive rate of innovation and hence growth, even with a constant workforce size. Of note, this decentraized equiibrium features a stricty smaer aocation of abor to R&D than that which a socia panner woud optimay choose. We moreover argue that under the Pareto speci cation, this wedge arises soey from the fact that individua researchers do not internaize the positive knowedge spiovers of their R&D e ort on future innovators when ideas buid cumuativey on each other. The above resut that the decentraized equiibrium features under-investment in R&D suggests that there is room for poicies to improve upon wefare outcomes. Given the innovation process in our mode, a natura poicy to consider is a requirement on how much of an improvement a new idea needs to make in order to be considered patentabe and non-infringing. Speci cay, if z is the productivity associated with the current best patent for a given product, a new idea woud need to deiver a productivity of at east Bz for it to be patentabe, where B is a poicy parameter set by the patent authority. For simpicity, we wi further assume that a ideas that are patentabe wi aso be deemed to be non-infringing on the scope of a other existing patents (and vice versa), so that goods made using the new idea can be marketed without fear of ega action from incumbent patent-hoders. 4 We wi therefore 2 See in particuar The Economist (202) and The New York Times (202). 3 The doctrine of equivaents is a ega rue that hods a party iabe for patent infringement even though the infringing device or process does not fa within the itera scope of a patent caim, but nevertheess is deemed equivaent to the caimed invention. The US is often considered one of the most ibera invokers of the doctrine of equivaents. See Raston (2007). 4 In the patenting iterature, this is otherwise known as the eading breadth, namey the extent to which a new innovation needs to improve upon an existing patent to be considered non-infringing on the atter s patent rights. If the new invention is deemed to be infringing, the innovator woud need to pay a royaty to the incumbent patent-hoder in order to egay 2

4 refer more precisey to this instrument, B, as a non-infringing inventive step (NIS) requirement; within the context of our mode, this wi aso serve to capture the patent breadth. 5 We buid this patent instrument into our growth mode in Section 3 and unpack its various e ects. First, a higher B raises the pro ts of patent-hoders as it aows them to exercise their monopoy power for a onger expected duration, by protecting them against future innovations that are too incrementa ( pro t e ect). At the same time, a more stringent NIS requirement raises the bar that ideas have to cear to quaify for a new patent ( hurde e ect), this being the key additiona compication reative to anayses of patent ength poicy. These two e ects ceary exert forces in opposite directions on the incentives to undertake research. We nd that as ong as the capacity of the economy to generate ideas is su cienty arge, the pro t e ect wi dominate when B is sma, so that research incentives improve when the NIS requirement is initiay raised above. However, the hurde e ect necessariy dominates when B is arge, as an excessivey high bar woud ceary discourage research. The reationship between equiibrium R&D e ort and the patent breadth parameter thus takes on an inverted U-shape, and there is a unique vaue of B (denoted by B v ) that maximizes the innovation rate. We then turn to the impications of this NIS requirement for wefare. The margina socia cost of setting a higher B is that it hurts consumers by conferring more monopoy power to patent-hoders, this being the famiiar static oss arising from patent protection. Nevertheess, a higher NIS requirement raises innovation when B is su cienty sma, which transates into faster reductions in prices for consumers (the dynamic gain). Our key resut estabishes that there is indeed a unique wefare-maximizing patent breadth (denoted by B w ), which is in genera binding (i.e., stricty bigger than ). This optima tradeo between dynamic gains and static osses occurs at a vaue of B where research e ort is sti increasing in B, impying that the NIS requirement that maximizes the innovation rate is necessariy arger than that which maximizes wefare (B w < B v ). We extend our mode in Section 4 to incorporate patent ength poicy, given its common use in practice. Interestingy, when both patent ength and NIS instruments are avaiabe to the patent authority, we nd that the wefare-maximizing poicy entais setting an in nite patent ength in tandem with a nite patent breadth of B = B w, this being the same B w derived earier in Section 3. This in nite patent ength resut arises because there are no diminishing returns to innovation e ort in our mode the Poisson arriva rate of ideas is proportiona to the aggregate number of research workers at each date which provides strong incentives to raise the patent ength to promote innovation. These dynamic gains from extending the patent ength are aways arger than the static consumer surpus osses, so ong as the parameters market the new product. The patentabiity requirement (to quaify for a new patent) and the eading breadth (to avoid infringement) are thus cosey reated but distinct concepts, athough we wi not emphasize this distinction for the purposes of our anaysis. See O Donoghue (998) and Scotchmer (2004), Chapter 3, for a review of these issues. 5 For our purposes, we wi thus use the terms NIS requirement, inventive step requirement, patentabiity requirement, and patent breadth interchangeaby. 3

5 that govern the innovative capacity of the economy are arge enough to begin with to ensure a positive eve of R&D activity in the steady state. In contrast, the e ect of the NIS requirement on the rate of innovation is dampened by the hurde e ect, which gets stronger as B increases, hence ensuring that B w is nite. We provide further discussion in this section of how the optima NIS requirement woud respond to the patent ength, if the atter were nevertheess set at a nite vaue for exogenous reasons. In terms of its structure, our framework fas within the cass of endogenous growth modes in which innovation occurs aong quaity (or productivity) adders (Grossman and Hepman 99; Aghion and Howitt 992). Firms compete by investing in R&D and successfu innovation aows them to cimb onto the next rung of the adder, a process resembing the patent race in Reinganum (985). We however augment our mode with features drawn from Kortum (997) and Eaton and Kortum (200), so that the size of each innovation step is stochastic rather than deterministic. Reative to Kortum (997), our approach speci es productivity improvements, rather than productivity eves, to be the stochastic outcomes of R&D e ort, so that each new idea buids on its predecessor in a stricty cumuative fashion and steady-state growth emerges endogenousy. 6 Our motivation for modeing this cumuative innovation process stems from our inherent interest in exporing the scope for patent poicy intervention in the presence of knowedge spiovers, which are absent in the baseine mode of Kortum (997). We shoud stress that the mode we deveop is very tractabe for anayzing the research questions at hand. When each productivity improvement is independenty drawn from an underying Pareto distribution, the og productivity eve of the best idea inherits a Gamma distribution, an observation that faciitates expicit expressions for wefare and the growth rate. Convenienty, a steady-state outcomes in our mode end up depending ony on three deep parameters, namey the rate of time preference, the Pareto dispersion parameter, and what we sha term the innovative capacity of the economy. This parsimony and tractabiity aows us to perform a cear decomposition of the e ects of patent poicy on the innovation rate and wefare in a genera equiibrium setting. Whie severa other papers have aso worked with this speci cation of Pareto productivity improvements in an endogenous growth setting (Koéda 2004; Minniti et a. 20), our approach di ers in the extent of the cosed-form anaytics that we are abe to pursue. Our paper is naturay reated to existing work that has studied various cosey-reated poicy instruments. In particuar, O Donoghue (998) concudes that a patentabiity requirement can raise socia wefare, whie Hunt (2004) nds an inverted-u shape reationship between the rate of innovation and the strength of this requirement. 7 Whie these echo severa of our ndings, most resuts in this prior 6 This di ers from the approach in severa recent contributions where steady state growth arises instead through the earning or di usion of ideas from high to ow productivity rms; see for exampe, Luttmer (2007), Avarez et a. (2008), Lucas and Mo (204), and Pera and Tonetti (204). 7 Green and Scotchmer (995) tacke a simiar probem, but focus on the eading breadth (the minimum required inventive step to avoid infringement of existing patents), as opposed to the patentabiity requirement, as their poicy instrument of 4

6 work are derived from partia equiibrium settings. An exception to this is O Donoghue and Zweimüer (2004), who embed such patentabiity considerations into a quaity-adder endogenous growth mode, though their anaysis focuses more on the e ects of such poicies on innovation and ess on wefare. 8 The question of the optima combination of patent ength and patent breadth has aso been expored (see for exampe, Gibert and Shapiro 990, Kemperer 990, Gaini 992, Denicoò 996), but as we sha make cear in Section 4, the various concepts of the patent breadth adopted in this iterature di er from the NIS instrument that we consider. The protection of ideas can go beyond patent protection. Not a industries use patent protection as the primary too to protect the idea of the incumbent. very ong imitation ags or it is very hard to reverse-engineer a technoogy. For exampe, some industries may have In those industries, the more innovation-reevant poicy may be antitrust aws. On that note, our resuts share some features with Sega and Whinston (2007), if we interpret the NIS more broady as (the inverse of) the eve of protection that new entrants receive against an incumbent s anti-competitive behavior. In particuar, Sega and Whinston (2007) nd that the eve of antitrust protection can have both positive and negative e ects on the rate of innovation, when the atter is ongoing and cumuative. From an empirica perspective, it has been observed that increased patent protection does not necessariy transate into a faster rate of innovation; see for exampe Sakakibara and Branstetter (200) on Japan, Bessen and Maskin (2009) on the US software industry, and the extensive review in Bodrin and Levine (2008, Chapter 8). These patterns can be rationaized if the NIS requirement has been set too high, namey at a eve of B above B v. In this range, the hurde e ect dominates and owers the ex-ante chance of an idea being patentabe and non-infringing, eading to ess innovation when B is raised. 9 The idea that patents can impede the accumuation of knowedge aso nds strong causa support in Gaasso and Schankerman (204), a nding we view as consistent once again with the presence of a hurde e ect. The rest of the paper proceeds as foows. Section 2 deveops our baseine mode of cumuative innovation and anayzes the wefare properties of the decentraized equiibrium. We incorporate a binding NIS requirement in Section 3 and derive our key resuts on the scope for this patent instrument to raise innovation and wefare. We extend the mode to incorporate patent ength poicy in Section 4. Section 5 concudes. Detaied proofs are presented in the Appendix. interest. O Donoghue et a. (998) consider both eading and agging breadth poicies, where the atter serves to protect patent-hoders against imitators. 8 On a reated note, Li (200) anayzes the e ect of agging patent breadth (protection against imitators) in a quaityadder growth setting. Kwan and Lai (2003) incorporate patent ength considerations into an endogenous growth mode in the vein of Romer (990). 9 Bessen and Maskin (2009) provide an aternative expanation, namey that the rate of innovation can decine in the strength of patent protection when innovations are sequentia and di erent ines of research e ort compement each other. 5

7 2 Baseine Mode with No Patent Breadth Poicy We rst buid and sove our baseine mode of cumuative innovation and growth, in order to famiiarize readers with the key features of the setup. Accordingy, in this section, innovators wi not face a binding patent breadth for their ideas to be both patentabe and non-infringing (i.e., marketabe). For now, we sha aso assume that patents do not expire, namey that the patent ength is in nite, so that incumbent patent-hoders ose their monopoy power ony when superseded by a new innovation. This wi pace the focus on the innovation process in our mode, which we proceed to describe next. We wi then cose the mode in genera equiibrium and discuss its properties. 2. Mode setup: The innovation process Consider an economy composed of one industry, in which a continuum of di erentiated varieties indexed by j 2 [0; ] is produced. 0 The economy is endowed with L units of abor, which is the ony factor of production. A of this abor is ineasticay suppied at the wage, w, where indexes time. Firms in the economy are sma, in the sense that each rm produces ony one variety, whie taking the prevaiing wage as given. (In the aggregate, however, w wi be an outcome of the genera equiibrium of the mode.) Each unit of abor (or simpy worker ) can engage in one of two activities, namey either in the production of di erentiated varieties or in R&D activity. With regard to the former, production takes pace under a constant returns-to-scae technoogy. Let Z (j) denote the abor productivity associated with the best avaiabe idea for producing variety j at time ; =Z (j) units of abor are thus required to produce each unit of this variety. Then, the unit cost faced by the rm that produces this variety is simpy: w =Z (j). On the other hand, the objective of R&D activity is to generate ideas to improve upon existing technoogies. Each idea spes out a technoogy (equivaenty, a abor productivity eve) for a speci c di erentiated variety. We mode the generation of these ideas as a Poisson process with a constant arriva rate of for each R&D worker. Foowing Kortum (997) and Kette and Kortum (2004), conditiona on receiving a new idea, the identity of the variety to which the idea appies is determined by a random draw from a uniform distribution on the unit interva. 2 We specify a setting in which the innovation process is stricty cumuative. In particuar, knowedge about production technoogies di uses immediatey as soon as the good in question is marketed, so that the underying knowhow becomes avaiabe to a agents in the economy. For exampe, this coud be 0 It is straightforward to extend the mode to incude a non-innovating outside sector, whose output can then pay the roe of the numeraire. In other words, the probabiity that an individua worker wi receive a new idea during a sma time interva is given by. Moreover, each R&D worker can receive ony one idea at any instant in time. 2 As in these preceding papers, this rues out the possibiity that innovation e ort can be directed toward the production of speci c varieties. 6

8 because it is easy to reverse-engineer the technoogy after observing a physica sampe of the good. As the current best technoogy for producing each marketed good is widey-known, subsequent innovation e ort stricty buids upon this knowedge to generate productivity improvements. In equiibrium, the best patented technoogy for each di erentiated variety wi indeed be used in production, with the good being marketed, and hence each subsequent arriving idea wi aways improve upon the frontier patented technoogy for the variety to which it appies. 3 In the absence of IPR protection, the di use nature of knowedge woud provide itte incentive for private agents to undertake R&D. We therefore require that an IPR regime be in pace that aows any new idea to be patented at negigibe cost. By patenting, the rm in possession of the new idea gains excusive rights to produce and market the variety (say, variety j) with the new technoogy, and wi indeed have the entire market for j to itsef as it is now the most productive manufacturer of j. This monopoy power ony expires when the next idea that improves upon the technoogy for j arrives. 4 Ideas that are not patented but which are marketed can immediatey be egay imitated by other rms, which woud compete away the pro ts accruing to the innovator. It foows that rms wi immediatey patent any new ideas that they receive, so that no goods wi be marketed without rst being patented. Having spet out the arriva process for ideas, we now describe what governs the productivity eves associated with these ideas. To initiaize the innovation process, we assume that at the start of time ( = 0), there is a baseine technoogy for each variety that is freey avaiabe to a rms. We normaize the productivity of this baseine technoogy to be for a varieties, namey: Z 0 (j) = for a j 2 [0; ]. Now, de ne Z (k) (j) to be the productivity associated with the k-th idea to arrive (after time 0) for variety j, where k is a non-negative integer. Thus, fz (0) (j); Z () (j); Z (2) (j); : : :g form a sequence of the successive best technoogies for producing this variety. To describe how this frontier technoogy evoves, de ne (k+) (j) Z (k+) (j)=z (k) (j) to be the productivity improvement associated with the next idea to arrive. We specify (k+) (j) to be a random variabe that is an independent draw from the foowing standardized Pareto distribution with shape parameter > : Pr (k+) (j) < z = z, where z 2 [; ), for a k 0. () Note that a ower impies a more fat-taied distribution which paces greater weight on drawing reativey arge productivity improvements. For simpicity, the distribution in () does not depend on j, so that the underying innovation process is symmetric across varieties. Moving forward, we wi thus write Z (k) (j) simpy as Z (k), since the distribution of the productivity eve of the k-th idea to arrive wi be identica 3 Aternativey, the innovation process can be set up as one entaiing improvements aong a quaity dimension, where each arriving idea yieds a higher utiity to consumers with no change in the good s production cost (and hence market price). 4 Given the continuous measure of varieties, there is a zero probabiity that the same agent wi consecutivey receive two ideas for producing the same variety. 7

9 for a varieties. 5 The expression in () embodies the notion of cumuative innovation, since the ower bound of the support of the distribution of productivity improvements is. In e ect, after the k-th idea has arrived, the productivity Z (k) associated with that idea becomes the new knowedge frontier which the (k + )-th idea wi improve upon. Note aso that we have assumed that the distribution in () does not depend on how many ideas have aready arrived (k) or on the productivity eve of the ast drawn idea (Z (k) ). In sum, this means that conditiona on the reaized vaue of Z (k), the next arriving idea Z (k+) can be viewed as a draw from a Pareto distribution with the same shape parameter but with a ower bound of Z (k). 6 At this juncture, it is usefu to discuss the reationship between the innovation process that we have just described and that advanced in Kortum (997) and Eaton and Kortum (200). In the notation that we have adopted, the anaogue of their speci cation for the (stationary) distribution that governs innovation is: Pr(Z (k+) < z) = z, where z 2 [; ), for a k 0. (2) Thus, in this earier work, ideas that arrive may or may not surpass the current state-of-the-art technoogy, Z (k) ; those ideas that fa short of the frontier are not competitive enough to survive in the market. As more ideas accumuate over time in their economy, it becomes ess ikey that a new idea wi surpass the current frontier. In contrast, our interest ies in understanding an innovation process in which each new idea stricty improves upon Z (k). The two approaches therefore represent two opposite ends of the spectrum: Whie Kortum (997) and Eaton and Kortum (200) adopt a non-cumuative formuation, we instead expore a situation where innovation is fuy cumuative, this being motivated by our interest in anayzing the externaities that arise from R&D activity in this atter setting. 2.2 Genera equiibrium We now embed the above cumuative innovation process in a genera equiibrium setting. Utiity: The utiity function of the representative consumer as of date 0 is given by: U 0 = Z 0 e n u d. (3) Here, is the rate of time preference (a parameter), whie u aggregates the instantaneous utiity from 5 This Pareto speci cation for each productivity improvement is aso adopted by Koéda (2004), Minniti et a. (20), and Desmet and Rossi-Hansberg (202). In particuar, Minniti et a. (20) provide descriptive evidence of: (i) substantia cross- rm heterogeneity in the usefuness of innovations (as captured by patent citations), and (ii) the Pareto providing a reasonabe t to the distribution of the vaue of patents especiay in its right-tai. 6 Reca that if a Pareto distribution is truncated from the eft, the resuting distribution remains Pareto with the same shape parameter, but with the eft truncation vaue serving as the new ower bound of its support. 8

10 the consumption of di erentiated varieties at time. Speci cay, u is given by: Z u = exp n x (j) dj, (4) where x (j) denotes the quantity of variety j consumed at time. 0 The representative consumer chooses fx (j)g =0 in order to maximize (3), subject to the intertempora budget constraint: Z 0 e r X d b(0), (5) where X = R 0 p (j) x (j) dj denotes the ow of consumption spending at time, with p (j) being the corresponding price of variety j at that time (which the consumer takes as given). r is the prevaiing interest rate, which wi be pinned down by the rate of return earned on owning a patent, this being the ony asset in our economy. Finay, b(0) denotes the present vaue of the future stream of wage income that wi be earned by each consumer pus the vaue of her initia asset hodings at date 0. It is we-known (see for exampe, Grossman and Hepman, 99) that the soution to this dynamic optimization probem yieds: where 0.) X r = + _ X, (6) X _ is the time derivative of X. (We wi omit the time subscript for equations that hod for a Thus, the rate of growth of consumption spending shoud equa the di erence between the market interest rate and one s private rate of time preference. It wi now be convenient to set aggregate consumption expenditure E LX as the numeraire for each. Since E = and L is constant over time, it foows from (6) that r =. Moreover, the expenditure on each variety wi be constant and equa to at each date, since we have a unit measure of varieties. As the expenditure on each j is invariant to its price, the price easticity of demand for each variety is. Market structure and pro ts: Firms compete by setting prices. If no ideas have yet arrived for variety j by time, then that variety is priced at margina cost (w ), since the baseine technoogy is freey accessibe to a potentia producers. On the other hand, if at east one idea has arrived for variety j, then rms compete under Bertrand competition. The rm possessing the most productive idea wi set a imit price that is just enough to keep the second most productive rm (and by impication, a other rms) out of the market. Therefore, the equiibrium price for variety j at time when precisey k ideas have arrived is equa to: p (j) = w =Z k (j). The price markup that the most productive rm sets is simpy: Z k (j)=z k (j), so that it inherits the Pareto distribution from (). To be more expicit, denoting (m) as the cdf of the price markup m, we have: (m) = m for m. The expected ow pro ts earned by a rm which hods the patent for the best technoogy for a given variety can now be computed as: Z m = E d(m) = m +. (7) m= 9

11 Figure : Circuar Fow in the Economy The above makes use of the fact that E is aso the expenditure per variety, since we have a unit measure of di erentiated varieties. 7 Savings and investment: Let v 2 [0; ] denote the share of the abor endowment L that is hired by rms to engage in R&D activity. (The remaining fraction, v, works in the production of varieties.) We assume that rms need to obtain nancing in order to hire R&D workers. This spending on R&D is the ony form of investment in our mode, in the sense that the innovation is undertaken to generate ideas that yied a future stream of pro ts. In exchange for the nancing they obtain, rms issue caims on the ow of pro ts from their patents. These caims (which we can think of as equity) are the ony assets in the economy, and the tota vaue of these assets at time is denoted by A. (Note that A is aso equa to the vaue of each patent, given that we have a unit measure of varieties and ony one active patent for each variety.) The above investment in R&D activity is nanced through the savings of workers. This coud take pace with savers directy owning the equity of rms, or with the nancing channeed through an intermediary such as a bank. Figure summarizes this circuar ow of funds in the economy between workers, rms and the nancia intermediary sector (or capita market). Bear in mind that workers have two sources of income, namey their abor income and the return on assets. spending, wl + ra The aggregate savings in the economy are thus equa to tota income net of consumption E. Equating this with aggregate net investment ( _ A) at each point in time, we have: wl + ra = _ A. (8) Research incentives: Since vl workers are empoyed in R&D, the Poisson arriva rate for new ideas in the economy as a whoe is equa to vl. The rate of return r for owning an asset (namey, a 7 To be cear, is equa to pro ts for a variety conditiona on at east one idea having arrived for the variety in question. In particuar, is not equa to aggregate pro ts in the economy. 0

12 patent) must equa the ow pro t rate minus the probabiity of su ering a compete capita oss due to the arriva of a new idea that supersedes the existing patent. This impies: r = A vl. (9) As r =, one can rewrite (9) as A = = ( + vl), which is an expression for the expected present discounted vaue of each patent. Intuitivey, this is equa to the present vaue of ow pro ts, discounted by the rate of time preference pus the hazard rate of osing the market to a subsequent innovator. Labor market equiibrium: We consider an equiibrium in which a positive amount of production (and hence consumption) takes pace in each time period. This impies that the wage of a production worker needs to weaky exceed the vaue of the margina product of being an R&D worker. The atter is given by the ow rate of ideas that each R&D worker can generate mutipied by the vaue of each idea, namey A. We thus have: A w. (0) Equation (0) wi hod with equaity when some innovation activity takes pace, namey when v ies in the interior of [0; ]. Workers woud then be indi erent between being empoyed in R&D and production. Steady state: The ve equations (6), (7), (8), (9) and (0) de ne a system in the ve unknowns, A, w, r, and v, which pins down the steady state of our mode. In what foows, we focus on a steady state in which some innovation occurs, and in which the share of the abor force empoyed in R&D is constant over time. In particuar, this means that (0) wi hod as an equaity. A quick inspection of our system of equations then impies that the vaue of a patent (A), the return on assets (r), and the return to abor (w) wi a be constant in this steady state. Moreover, a famiiar set of arguments can be appied to show that this steady state is one to which the economy immediatey jumps. To see this, (8) and (0) together impy that: _ A=A = L + (=A). If L + were to exceed =A at any time aong the transition path, _ A=A woud be positive, and the subsequent increase in A woud further widen the gap between L + and =A on the right-hand side. Aso, the arger is A, the faster is the rate of increase in A, so that the vaue of a patent woud continue to increase to in nity. However, (9) paces an upper bound of = on the vaue of a patent, so that the expectation that A wi increase inde nitey cannot be met. A simiar argument can be used to rue out the reverse case where L + fas short of =A on the transition path. Thus, expectations about the vaue of a patent can ony be fu ed if the economy jumps immediatey to a situation where _ A=A = 0. It is now straightforward to sove the system of ve equations after setting _ X = _ A = 0. This yieds in particuar the foowing expression for the market aocation of abor to R&D activities, v eqm : v eqm = L L ( + ) ()

13 Note from the above that v eqm is ceary ess than. To further ensure that v eqm > 0, we need to impose the foowing: Assumption : L >. Intuitivey, for there to be a positive amount of R&D in the steady state, we require that: (i) the innovative capacity of the economy (captured by L) be su cienty high; (ii) the dispersion of the ideas distribution be arge ( sma); and/or (iii) consumers be su cienty patient ( ow). Using (), one can verify directy that the research intensity of the economy varies naturay with the underying parameters of the mode. Firms hire a greater share of the workforce in R&D when the arriva rate of ideas is higher (dv eqm =d > 0), or when those ideas are drawn from a Pareto distribution with a fatter right-tai (dv eqm =d < 0). If agents are more patient when vauing future reative to current consumption, this aso raises R&D e ort in the steady state (dv eqm =d < 0). 8 Our mode exhibits a scae e ect in that rate of innovation is increasing in L. However, we can prove that our resuts are robust to the remova of the scae e ect. We show this in an onine appendix Wefare We turn next to the task of evauating country wefare, in order to faciitate our ater anaysis of the e cacy of patent poicy. The utiity speci cation in (3) and (4) impies that wefare depends on the rea n R o wage in each period, as: u = w = exp 0 n p (j)dj. (Reca in particuar that the economy jumps immediatey to its steady state.) Since a varieties are ex ante symmetric and we have a unit measure of these varieties, the aw of arge numbers impies that the idea price index in the denominator is equa to: exp fe[n P ]g, where P is a random variabe whose reaization is the price of a variety at time ; the expectation operator is taken over this price distribution. We therefore need to understand how prices evove over time. Due to the Poisson nature of the innovation process, the probabiity that exacty k ideas have arrived by time when vl units of abor are engaged in R&D at each date is: (vl)k e vl, where k is a non-negative integer. When k = 0, the variety in question wi be priced at w (its margina cost). On the other hand, when k, under the imit-pricing rue, the price of a variety wi instead be a random variabe that inherits the distribution of w =Z (k ). The expected og price of a variety at time is thus: E [n P ] = (vl)0 e vl n w + 0! X (vl) k k= e vl n w E[n Z (k ) ]. (2) 8 Athough our mode exhibits a scae e ect in that the rate of innovation is increasing in L, we want to stress that a economic outcomes of interest in our mode, such as v, growth rates and wefare, ony depend on the product L, and not on the speci c vaues of and L separatey. What is more important is therefore not the size of the economy as measured by L, but its innovative capacity as captured by L. 9 The URL of the onine appendix is ect.pdf 2

14 Note that the rst term and each term in the summation in (2) is equa to the probabiity that k ideas have arrived between times 0 and, mutipied by the og price at time when there have indeed been exacty k ideas (where k = 0; ; 2; :::; ). We show in the Appendix how to evauate (2) expicity. The key to this is to recognize that in the underying innovation process, the random variabe Z (k ) = Z (k ) =Z (0) is the product of k independent reaizations from the standardized Pareto distribution given earier in (). (Reca that Z (0) =.) Buiding o this observation, one can show that n Z (k ) is a random variabe from a Gamma distribution with mean E[n Z (k ) ] = (k )= (see the Appendix). 20 The expected og productivity of the k-th idea to arrive thus increases ineary in k, whie increasing aso in the thickness of the right-tai of the Pareto distribution from which the productivity improvements are drawn. Substituting this expression for E[n Z (k ) ] into (2) and simpifying, one then obtains: E [n P ] = n w + vl e vl. 2 It foows that per-period utiity (the rea wage) is given by: u = exp vl e vl. 22 De ning the growth rate of the rea wage to be g d n (u ) =d, we have: g = vl e vl, (3) which is ceary positive when v > 0. Athough the economy jumps immediatey to a steady state in which A,, and w (the nomina wage) are constant, the rea wage nevertheess rises over time as varieties are on average becoming cheaper when there is a positive amount of R&D. In other words, Assumption which guarantees that v eqm > 0 aso ensures that g > 0 for a 0. Substituting in the expression for v eqm from (), one can further verify that: dg =d > 0 and dg =d < 0. Thus, a higher arriva rate of ideas (higher ) and a arger average productivity improvement (smaer ) both raise the growth rate of the rea wage at each date. From (3), one can moreover see that the growth rate of the rea wage rises over time (dg =d > 0): From an initia vaue of g = 0, this asymptotes toward a maximum growth rate of vl=. This property derives from the fact that as time progresses, the baseine technoogy is shed from use for a greater and greater share of varieties. As the rst idea arrives for successive varieties, the innovation process gets jump-started for a greater measure of varieties in the unit interva, hence causing the overa growth rate to rise over time. However, this e ect peters out, as the rst idea eventuay arrives in expectation for a varieties. It is instructive here to compare the above against the properties of the modes in Kortum (997) and Eaton and Kortum (200), which aso focus on a steady state in which the share of the workforce 20 To be absoutey precise, this statement about the distribution of n Z (k ) hods ony for k 2. Nevertheess, when k =, we have that E[n Z (0) ] = 0, so that the formua E[n Z (k ) ] = (k )= is aso vaid for k =. 2 Much work has been done documenting the t of the Pareto distribution for rm size distributions (e.g., Axte 200; Luttmer 2007; Arkoakis 20). Interestingy, the Gamma distribution aso features a thick right-tai, athough it matches the empirica distribution of US rms ess we for the argest rm sizes (Luttmer 2007). 22 The nomina wage can aso be soved for expicity from the system of ve equations that pin down the steady state. This is given by: w = =( + L). 3

15 empoyed in R&D is constant. In these preceding papers, innovation is not cumuative in nature, and perpetua growth in rea wages is sustained instead by a growing R&D workforce (vl), which grows at the same exogenous rate as the abor force (L). Thus, more ideas are drawn in each period by the ever-growing number of R&D workers, overcoming the fact that it gets harder and harder for each idea drawn from the stationary distribution in (2) to surpass the technoogica frontier. In contrast, the mode which we have just presented generates steady-state growth in rea wages through the cumuative nature of innovation new ideas aways stricty improve on the technoogica frontier without requiring that the abor force grow over time. Finay, the expected wefare of the representative consumer is obtained by substituting the expression for u into (3) and evauating the associated integra. After some agebraic simpi cation, this yieds: U 0 = (vl) 2 2 ( + vl) = (L ) 2 2 ( + )( + L), (4) where the ast equaity foows from repacing v by the expression for v eqm from (). One can show via straightforward di erentiation that so ong as Assumption hods, (4) is increasing in and decreasing in. Wefare therefore rises either as innovations arrive more frequenty or as the average productivity improvement increases. 2.4 Contrast with the socia optimum To understand the e ciency properties of the steady state which we have just soved for, it is instructive to compare the above market equiibrium with the outcomes under a benign socia panner. Conceptuay, this socia panner s probem can be formuated as a abor aocation decision over the share of abor to empoy in R&D, as we as the vaue of L p (j) for each j 2 [0; ], namey the amount of abor assigned to the production of variety j at each point in time. Formay, the socia panner sets out to sove: max v;fl p (j)g j=0 s.t. Z 0 U 0 L p (j) dj = L( v) for a 0, (5) and Lx (j) = L p (j)z (j) for a j 2 [0; ] and 0. (6) The rst constraint (5) is a abor market-cearing condition that states that a abor not engaged in R&D must be empoyed in production. On the other hand, the second constraint (6) sets the quantity demanded of variety j equa to the quantity produced at each period in time. As we show in the Appendix, the soution to this socia panner s probem features an equa aocation of abor to the production of each variety. In other words, given the choice of v, we have: L p (j) = L( v). Using this property, the panner s probem can then be simpi ed to the foowing unconstrained 4

16 maximization probem over v: max v U 0 = n( v) + vl 2. The above maximand is a concave function in v and thus yieds a unique optima aocation of abor between research and production activities. This socia panner s aocation, denoted by v SP, is given by: v SP = L L. (7) This ies stricty in the interior of [0; ] if L >, namey if Assumption hods. Moreover, comparing this with the aocation that woud emerge in the market equiibrium from (), one immediatey has the foowing resut: Proposition The share of abor that a socia panner woud aocate to research is stricty arger than that which is observed in the market equiibrium, namey v SP > v eqm. The decentraized equiibrium in our mode therefore unambiguousy yieds ess investment in R&D e ort reative to the sociay-optima eve. One can moreover see that the reative extent to which v eqm fas short of v SP, namey (v SP v eqm )=v SP is increasing in. Intuitivey, the ess fat-taied is the Pareto distribution of productivity improvement draws, the ess attractive are the potentia private returns (pro ts) from R&D, and hence the greater the extent of under-investment in R&D in the market equiibrium reative to the socia optimum. The iterature on endogenous growth in the presence of knowedge spiovers has highighted severa externaities that drive a wedge between the market and socia-panner outcomes (e.g., Grossman and Hepman, 99; Aghion and Howitt, 992), and these forces are present too in our mode. First, there is an intertempora spiover e ect arising from the cumuative nature of innovation: Firms appy a higher e ective discount rate when evauating the vaue of a patent because they do not internaize the positive knowedge spiovers from their innovation on future productivity improvements. Second, there is an appropriabiity e ect, in that the private pro ts which rms earn are in genera smaer than the fu gains to consumer surpus that each innovation generates. Third, a business-steaing e ect is at pay, since innovation e ort erodes the pro ts of preceding innovators in a way that a socia-panner woud want to fuy internaize. The rst two of these e ects tend to decrease R&D in the market equiibrium reative to the panner s probem, whie the ast e ect pushes rms toward over-investing in R&D. Proposition impies that in our mode, the former two e ects must dominate the atter businesssteaing mechanism Note that the presence of monopoy-pricing power per se does not distort abor aocations in our mode. The reason is that a rms charge the same markup in expectation (drawn from the standardized Pareto distribution, (m)), so that the aocation of production abor across varieties cannot be improved upon ex ante. See the reated discussion in Grossman and Hepman (99), p.70. 5

17 We can in fact make a more precise statement concerning the reative importance of these three externaities. By rearranging (), observe that the market aocation of abor to research activity is determined as the soution to: ( v)l (+vl) panner s probem impies that v SP soves: ( =. On the other hand, the rst-order condition of the socia v)l =. Thus, the ony wedge between the two soutions arises from the di erent discount rates that are respectivey appied: In the market equiibrium, rms use a higher discount rate of + vl, which takes into account the ow probabiity of su ering a compete pro t oss to a new innovation, on top of the socia discount rate. The ony externaity that is reevant in our mode is thus the intertempora spiover e ect; evidenty, the appropriabiity and business-steaing e ects must o set each other exacty. 3 Patent Breadth Poicy The under-investment in R&D activity which our baseine mode features gives rise to the possibiity of wefare-improving poicy interventions. We turn next to anayze a poicy instrument that can potentiay achieve this, namey a patent breadth as impemented through a non-infringing inventive step (NIS) requirement. (We continue to maintain the assumption that patents do not expire, but wi return to incorporate a nite patent ength ater in Section 4.) 3. Poicy setup: The patenting environment We retain the cumuative innovation process described in Section 2., where each successive productivity improvement is an independent draw from the standardized Pareto distribution. Even though patenting was necessary in the baseine mode to confer a successfu innovator with short-term monopoy power, the patent authority there payed a reativey passive roe as a arriving ideas woud automaticay improve upon the frontier technoogy and hence quaify for a patent. Suppose now that the government announces at date = 0 that there wi be a non-infringing inventive step requirement equa to B with immediate e ect: A new patent wi be granted if and ony if the k-th idea to arrive for a given variety improves upon the productivity of the (k )-th idea by at east B. More formay, this k-th idea is not eigibe for a patent if Z (k) 2 [Z (k ) ; BZ (k ) ), an event that occurs with probabiity B, based on the Pareto distribution in (). In this situation, the rm in possession of this new idea woud have no incentive to produce the good even though it embodies an incrementay better technoogy, given that it woud have no ega right to market the good. Consequenty, non-patentabe ideas, which aso infringe, are not marketed; the underying knowedge does not spread to the rest of society and hence cannot be buit upon by subsequent innovators. This serves to capture the idea that the pubic discosure of technica knowhow that each patent appication requires is a crucia patform for faciitating knowedge di usion. 6