Understanding the Link between Patent Value and Citations:
`Creative Destruction or Defensive Disruption?
`
`David S. Abrams
`University of Pennsylvania
`
`Ufuk Akcigit
`University of Pennsylvania & NBER
`
`Jillian Popadak
`University of Pennsylvania
`
`April 8, 2013
`
`Abstract
`
`The patent system is the leading legal mechanism for protecting new inventions and as
`such, patents are used in a host of research to proxy for innovative activity. Understanding
`how new products and processes are created and how to value them is critical to fields
`as diverse as industrial organization, endogenous growth theory, and intellectual property
`law. In this paper we provide the first evidence that much of the work in these literatures is
`based on an erroneous assumption: that the value of innovation is proportional to citation-
`weighted patent counts. Using a proprietary dataset with patent-specific revenues, we find
`that there is an inverted-U relationship between patent value and citations. We attempt
`to explain this relationship using a simple model of firms, allowing for both productive and
`defensive patents. Simulations from the model match the empirical regularity that some
`very high-value patents receive substantially fewer citations than less valuable patents.
`Further, we find evidence of greater use of defensive patenting along the dimensions where
`it is predicted. These findings have important implications for our basic understanding of
`growth, innovation, and intellectual property policy.
`
`JEL Codes: O3, L2, K1.
`
`Keywords: Productive innovation, Defensive innovation, Patents, Creative Destruc-
`tion, Citations, Patent Value, Competition, Intellectual Property, Entrepreneurship.
`
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`1 Introduction
`
`One of the core questions of economics, both at the micro and macro level, is what leads
`
`to productivity gains.
`
`In order to understand what policies impact innovative activity and
`
`ultimately productivity, it is crucial to start with a good metric to value innovation. While
`
`the importance of such a metric has long been recognized (Scherer 1956; Grilliches 1990) so
`
`too have the inadequacies of the proxies for value that are in widespread use (Schankerman
`
`and Pakes 1986; Hall and Harhoff 2012).
`
`Over the last 30 years, two primary metrics have been used to proxy for the value of
`
`innovation, patent counts and citation-weighted patent counts. The intuition is simple: fields
`
`with greater innovative activity will have more value to protect and will do so by applying
`for more patents. Weighting patent counts by forward citations1 is a natural augmentation
`to simple patent counts, given the well-known fact that patents vary tremendously in value2.
`This metric, however, is based on the assumption that a larger number of citations corresponds
`
`to higher value.
`
`Figure 1: LIFETIME FORWARD CITATIONS VS. REVENUE
`Notes: Data is normalized so that the mean annual revenue is $10,000.
`
`Yet, the history of science and economics is replete with theories that did not bear up
`
`1Forward citations is the number of citations received by a particular patent by subsequent patents.
`2Fewer than 10 percent of patents are worth the money spent to secure them (Allison, Lemley, Moore,
`Trunkey 2009), but the most valuable ones are thought to be worth hundreds of millions of dollars (Hall, Jaffe,
`and Trajtenberg 2005).
`
`1
`
`Mean Citations vs Lifetime Revenue
`
`40
`
`30
`
`Mean Citations
`
`20
`
`10
`
`0
`
`0
`
`200
`100
`Lifetime Revenue ($000s)
`
`300
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`under empirical scrutiny and until now there has been no good way to test this assumption.
`
`In order to say anything convincing about innovation we need a credible measure of its value.
`
`In Figure 1 we present strong evidence that the main approach to valuing innovation is fatally
`
`flawed. The relationship between citations and patents is not only non-linear, it is not even
`
`monotonic. This striking finding calls for a deeper understanding of the process of innovation,
`
`patenting, and citations, which we explore empirically and theoretically in this paper.
`
`The citation-value relationship revealed in Figure 1 is extremely surprising relative to what
`
`has previously been assumed. Prior empirical study of the relationship was quite limited
`
`due to several problems: companies are reluctant to share proprietary patent data, single
`
`firm portfolios tend to have limited technological breadth and small sample size, and almost
`
`no companies allocate revenues to specific patents. This paper is only possible by virtue of
`
`access to a very large, diversified patent portfolio owned by non-practicing entities (NPEs)
`
`that calculate patent-specific revenues. We discuss details of the data set and its advantages
`
`for academic inquiry further in Section II.
`
`We introduce a theoretical model that suggests that the inverted-U shape is the result of
`
`two types of innovative effort, which we characterize as productive and defensive. Productive
`
`innovative effort leads to the traditional increasing relationship between patent value and
`
`citations; defensive innovative effort, however, leads to a negative relationship between patent
`
`value and citations. In an economy that exhibits both of these types of innovative effort, the
`
`link between patent value and citations will be the inverted-U that we observe empirically.
`
`We test several predictions of the model, besides the overall inverted-U shape. Defen-
`
`sive patenting should be more prevalent among larger entities, for divisional and continuation
`
`patents, for newer patents, and in technology classes with rapid growth. Each of these predic-
`
`tions is borne out in the data and we find evidence that defensive patenting is more prominent
`
`in these categories.
`
`This is certainly not the first paper that has attempted to examine the relationship between
`
`patent value and citations, but it is the first not severely constrained, for the reasons mentioned
`
`above. Trajtenberg (1990) is perhaps the leading prior work on the subject, but he had access
`
`to a data set several orders of magnitude smaller than in this paper. In addition, all patents
`
`were in a single narrow field (computed tomography or CT) and values were imputed from
`
`a structural model of the CT device. Harhoff, Scherer, and Vopel (2003), obtain categorical
`
`measures of value on 772 patents from a survey of German patents with 1977 priority that
`
`were renewed to full term. Several excellent studies examine the patent value distribution
`
`using the renewal decision to infer value (Pakes 1986; Schankerman and Pakes 1986; Bessen
`
`2008). These papers make use of the contingent claim valuation method pioneered by Pakes
`
`and Schankerman. Since a renewal decision can only convey an upper or lower bound on value,
`
`this approach is not useful for learning more about the citation-value relationship.
`
`In the legal literature, defensive patenting has received a great deal of attention in re-
`
`2
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`cent years as allowable subject matter has widened to include software and business methods
`
`patents. As the number of patents granted has increased, technological progress has led to
`
`devices that implicate thousands of separate patents. Some have argued that we have arrived
`
`at a point where the patent system is actually detrimental to innovation (Bessen and Meurer
`
`2008; Boldrin and Levine 2012). We capture these observations and intuitions by modeling
`
`defensive patents as ones which do not lead to substantial further work in a field and in fact
`
`may stifle it (blocking patents). Thus, there may be extremely valuable defensive patents that
`
`receive very few citations, leading to a null or negative relationship between forward citations
`
`and revenue.
`
`A single figure is not enough to convince one of the correctness of a theory, or even of the
`
`robustness of the empirical findings. We aim to tackle both of these tasks in the balance of the
`
`paper, but we take the unusual step of including this striking figure in the beginning because
`
`it immediately conveys our central contribution. In Section II we provide substantial detail
`
`about incentives to patent and cite, the business models of NPEs and further description of the
`
`data. Section III introduces our model which we believe captures some of the key elements of
`
`the patenting and citing processes. In Section IV we present the main empirical results and a
`
`discussion of them. Section V concludes and makes the point that the goal of this work is not
`
`to undermine the large body of work on innovation that has relied on widely-held assumptions
`
`about the patent value-citations relationship. Rather, we hope that this will help build a more
`
`robust literature that informs some of the central economic issues of our time.
`
`2 Background
`
`Since the major limitation of previous studies of patent value is due to the lack of available
`
`data on individual patent revenues, it is worth discussing the data source and characteristics
`
`in some detail. The data in this paper was provided by large non-practicing entities (NPEs),
`
`with focuses in the technology sectors. NPEs are firms whose revenue primarily derives not
`
`from producing products based on patented technology, but from licensing patents. These
`
`companies employ a range of different business models ranging from aggressive litigators to
`
`passive licensors, and the number of patents held by NPEs continues to grow rapidly.
`
`This is fortunate for those interested in learning about innovation as NPEs function as an
`
`excellent data source in many ways, and when compared to traditional patent holding firms,
`
`NPEs have several advantages as an object of study. Their portfolios can be substantially
`
`larger than practicing firms, since their capital is almost exclusively employed in assembly and
`
`licensing, rather than production. NPEs are more diversified than practicing firms as well,
`
`since it is often easier to acquire the breadth of expertise necessary to acquire and license
`
`patents in a large array of fields, rather than to practice them. The data available from NPEs
`
`is also likely to be substantially more useful for researchers, as they tend to determine patent-
`
`3
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`specific revenues. This is something that almost no practicing firms do, unless licensing is a
`
`major part of their business. This should not be surprising since ultimately most firms care
`
`about overall profit from innovation, not specifically from which patent the profit derives.
`
`Table I reports variables definitions and summary statistics for the primary patent and
`
`assignee characteristics analyzed in this paper. After dropping design and plant patents, we
`
`observe 46,891 regular, utility patents. The average lifetime patent value is $204,212, but
`
`the standard deviation is $1.9 million. The mean number of forward citations is 13, but the
`
`median is 0. This degree of skewness in the distributions of patent value and forward citations
`
`is similar to that reported by Trajtenberg (1990); Harhoff, Scherer, and Vopel (2003); and
`
`Bessen (2008).
`
`The heterogeneity in the underlying patent characteristics and assignees is extensive. The
`
`patents are licensed to and acquired from a broad range of intellectual property sources includ-
`
`ing individual inventors, small firms, large firms, universities, hospitals, and government agen-
`
`cies. The dataset represent patents originated in 89 different countries, and patents granted
`
`in the United States represent just less than the majority at 46 percent. Individual inventors
`
`account for 58% of the patents, and the average patent has 2 inventors that make 20 claims, of
`
`which 16 are dependent claims. On average, backward citations are not concentrated in very
`
`recent patents with only 20% in the three years prior to application.
`
`Table II describes the diverse range of technologies that are patented. Our sample covers
`
`267 unique primary technology classifications, which we have grouped into 10 broad technology
`
`categories. The technology categories include: internet and software, wireless communications,
`
`circuits, network communications, computer architecture, peripheral devices, semiconductors,
`
`electromechanical, optical networking, and nanotechnology.
`
`In our subsequent theoretical and empirical analyses, where we attempt to provide a theo-
`
`retical foundation for the inverted U-shape in the data, we focus on a few variables characterized
`
`by productive and defensive innovations. While building our theoretical model, we rely on the
`
`Schumpeterian theory of creative destruction (see the recent survey by Aghion, Akcigit and
`
`Howitt (2013) for more on this topic), where each new innovation builds on previous tech-
`
`nologies, but also makes them obsolete by introducing a better one. This tension between
`
`the incumbent technology owner’s wish to defend its monopoly power and the future innova-
`
`tor’s wish to utilize the spillovers generated by the current incumbent help us rationalize the
`
`non-monotonic relationship between patent value and subsequent entry, identified by forward
`
`citations. Moreover, models presented by Farrell and Shapiro (2008) emphasize the ability of
`
`patent holders, even of weak or less productive patents, to hold up firms through the threat of
`
`infringement. Similarly, our model emphasizes the decision to innovate productively or defen-
`
`sively. Intuitively, this suggests that non-original and less productive patent applications with
`
`a higher concentration of backward citations in recent years are more likely to be strategic or
`
`4
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`defensive patents. Around 16% of the patents in our sample are non-original3 and only 20%
`of the backward citations are in the recent past.
`
`Since a major contribution of this paper is a better understanding of the relationship
`
`between patent value and citations, it is important to clearly define how those are calculated
`
`in this paper. The NPEs from which the data are derived purchase or entering into revenue
`
`sharing agreements with patent owners. Revenue is generated by licensing the patents in the
`
`entire NPE’s portfolio or a subset of a NPE’s portfolio. Revenue is allocated on a patent-year-
`
`customer level based on the prominence the patent played in negotiations with the customer.
`
`This allocation scheme is disciplined by competing interests on two sides. Patent owners who
`
`are due a share of future revenues seek to maximize the revenue allocated, while the incentive
`
`of shareholders in the NPE is for larger revenue allocation to patents in which they have a stake
`
`and less to others, since total revenue allocation is a zero-sum game. We aggregate revenues
`
`to the patent-year level and then compute the mean revenue profile over the life of a patent
`
`separately for each of the 10 primary technology categories. We estimate lifetime revenue for
`
`each patent by inflating the observed revenue by the ratio of lifetime revenue to the mean of
`
`the years we observe for each patent. We then normalize all revenue amounts so that mean
`
`annual revenue is $10,000 in order to maintain the confidentiality of the revenue data.
`
`Lifetime citations are computed in a similar manner. We obtain data on forward citations,
`
`defined as the total number of times a patent has subsequently been cited. By definition, newer
`
`patents will have less time to acquire citations than old ones and this must be accounted for.
`
`We define “lifetime citations” as the total number of citations we expect a patent to have by
`
`its expiration. We compute this by first producing the forward citation- patent age profile for
`
`each of our ten technology categories. Figure 2 presents the incremental patent citation profile
`
`and an associated revenue profile on aggregate. There is substantial variation by technology
`
`class; therefore, we create separate revenue and citation profiles for each technology class. We
`
`calculate lifetime citations by inflating the total citations already received by the ratio of the
`
`total mean citations divided by the mean for the average patent of the same age as the one
`
`in question. One small flaw in this procedure is that it will understate the number of lifetime
`
`citations for any patent that has zero in our dataset, but the mean number of lifetime cites
`
`should still be correct.
`
`3Within the intellectual property legal framework, an original patent is an application that establishes its
`own filing date and does not have an effective filing date based upon another previously filed application. If
`an ”original” application is then used to establish an effective filing date of a later filed application, it becomes
`known as a parent application and the later filings are either divisions or continuations. There can be many
`strategic advantages to non-original patents if the first-to-file is important or if one desires to prolong the original
`patents disclosure.
`
`5
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`Figure 2: INCREMENTAL FORWARD CITATIONS AND REVENUE BY PATENT AGE
`Notes: Data is normalized so that the mean annual revenue is $10,000.
`
`3 Theory of Patent Valuations and Citations
`
`In the previous section, we provided a striking new empirical finding which is at odds with the
`
`received wisdom about the link between patent value and citations. How can we reconcile the
`
`two and account for the inverted-U? In this section, we offer a new model of innovation, patents,
`
`and citations. Our purpose is to develop a better understanding of the underlying reasons for
`
`the observed inverted-U relationship between citations and patent value. We embed intuitive
`
`assumptions into a structural model, and show that the model fits the observed pattern well.
`
`Our model features two distinct types of innovation efforts – productive and defensive.
`
`The intuition for productive innovation follows the traditional economic view that patents
`
`are offered as a contract between society and the inventor. In return for a limited period of
`
`exclusivity, the inventor agrees to make his invention public rather than keeping it secret. This
`
`institutional arrangement promotes the diffusion of ideas and economic growth. However, this
`
`is likely not the full story. Therefore, we also introduce the notion of the defensive innovation, a
`
`type of destructive creation. This idea seeks to capture the fact that when firms and individuals
`
`are endowed with a complex legal instrument, they may use it strategically in ways that do
`
`not serve the original intent of the legislation that created the instrument in the first place.
`
`To help put some structure on these two types of innovative effort, we develop a model.
`
`For reasons that we explain below, our model predicts that the link between patent value and
`
`citations are positive for productive innovation efforts and negative for destructive innovation
`
`6
`
`30000
`
`Profile of Citations & Revenue by Patent Age
`
`20000
`
`10000
`
`Annual Revenue
`
`5
`
`10
`patent_age
`
`15
`
`Incremental Citations
`
`Incremental Revenue
`
`0
`
`20
`
`2
`
`1.5
`Forward Citations
`
`1
`
`0
`
`.5
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`efforts. The combination of the two cases generates the inverted-U relationship that is so
`
`prominent in the data. One of the reasons for approaching this problem from a structural
`
`paradigm is that it will allow us to quantify a number of crucial moments such as the size
`
`of the creative production and non-creative destruction. Further, given the properties of the
`
`decentralized market that we embed in the model, we will be able to make welfare statements
`
`such as what the impact of a counterfactual innovation policy may be. Thus, this type of
`
`exercise leads to practical findings for researchers, practitioners, and policymakers alike.
`
`3.1 The Case of Productive Innovations
`
`In this section, we introduce a continuous-time model with a representative household. The
`
`household consumes a basket of goods, each of which is produced by a different incumbent
`
`monopolist. The economy features a large number of outside entrepreneurs who invest in
`
`productive innovations. These productive innovations enable the entrepreneurs to innovate, to
`
`replace existing incumbents, and to obtain market share. In the first model with productive
`
`innovations, we abstract from incumbent innovations and focus only on entrants’ innovations.
`
`This assumption is relaxed in the subsequent model where we allow incumbent firms to create
`
`defensive innovations, which protect their valuable productive patents and market share.
`
`The key feature of the productive innovation model that relates to citations is how new
`
`innovations arrive. Specifically, we assume that new innovations and innovative efforts arrive
`
`in clusters and that each new patent cites the prior art within the same technology cluster.
`
`Intuitively, certain markets become hot and attract all the top talent to invest their innovative
`
`efforts in that market. This simple logic leads to clustering of innovations by technology sector
`
`over time . Although this is an assumption, it is also consistent with empirical evidence (Jaffe
`
`and Lerner 2004). In terms of the model, what follows from this logic is an endogenous-citation
`
`dynamic.
`
`The link between the citations and patent value comes from the fact that more novel
`
`innovations will have larger mark-ups due to their originality, denoted by the step size of a new
`
`innovation. In the model, this then translates into larger patent values. Thus, the first simple
`
`model of productive innovation effort leads to the traditional conclusion of a positive correlation
`
`between patent citations and patent value. At the same time, more novel innovations will
`
`generate larger spillovers for the subsequent innovations, which will encourage new innovations
`
`by outside entrepreneurs. With more entrepreneurs entering the market, a natural cluster of
`
`innovative effort over time by technology is created. Since a new innovation must cite the
`
`previous related patents upon which it builds, more novel patents receive more citations on
`
`average. Given the intuition and logic underlying this first model of productive innovation, we
`
`now turn to the details.
`
`7
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`Basic Environment Consider the following continuous time economy that admits a repre-
`sentative household. The household consumes a unique consumption basket Ct that consists
`of large set of varieties indexed by j ∈ [0, 1] as follows:
`
`(cid:90) 1
`
`Ct = exp
`
`ln cjtdj,
`
`(1)
`
`0
`
`In this expresssion, cjt is the quantity of variety j at time t. We normalize the price of the
`final good Ct to be 1 in every period without loss of generality. The consumption basket is
`produced in a perfectly competitive market.
`
`Each variety j is produced by a monopolist who owns the latest innovation (patent) in
`
`sector j. The monopolist’s production function takes the following simple form
`
`cjt = qjtljt
`
`(2)
`
`where ljt is the labor employed for production and qjt is the variety-specific labor productivity.
`In what follows, new innovations will improve labor productivity, which leads to an aggregate
`
`growth in this economy. The linear production function implies that the marginal cost of
`producing 1 unit of cjt is simply
`
`Mjt =
`
`wt
`qjt
`
`where wt is the market wage rate which is taken as given by the firm. Note that all monopolists
`hire from the same labor market in the economy, hence every monopolist faces the same wage
`rate wt.
`Labor productivity qjt is improved through subsequent innovations in each product line
`j. Innovations belong to technology clusters. Let n index the order of an innovation in a
`
`technology cluster such that the very first patent that starts a new technology class has n = 0,
`
`the first follow-on innovation in the same technology cluster is indexed by n = 1, the second
`
`follow-on innovation by n = 2, and so on. Each innovation by a new entrant into j improves
`the previous incumbent’s technology by a factor of (1 + ηn) which is only a function of the
`order n of the patent in the technology class and remains constant as long as the same firm is
`in charge of production. Consider a product line where productivity at time t is qjt and a new
`innovation of step size ηn is received during (t, t + ∆t) . Then the labor productivity evolves
`as:
`
`qjt+∆t = (1 + ηn) qjt.
`
`(3)
`
`When a new firm innovates and enters into j as the new market leader, the latest innovator
`
`and the previous incumbent compete in prices `a la Bertrand.
`
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`3.1.1 Static Equilibrium: Production, Pricing and Profits
`
`It is useful to solve the static production and pricing decisions before we describe the innovation
`
`technology. Consider the consumption basket in (1) . Because the consumption basket has a
`Cobb-Douglas form with respect to all varieties, the household will spend the same amount Ct
`on each variety j. Hence the demand for each variety j can be expressed as
`
`cjt =
`
`Ct
`pjt
`
`(4)
`
`where pjt is the price charged by the monopolist j. Note that the Bertrand competition between
`the new monopolist and the previous incumbent, together with the unit elastic demand curve
`
`in (4) implies that the monopolist will follow limit pricing and charge a price that is equal to
`
`the marginal cost of the previous incumbent. If the productivity of the current monopolist in
`j is qjt and the size of her innovation was ηn, then the marginal cost of the previous incumbent
`is simply (1 + ηn) wt/qjt, which implies that the current monopolist’s price is simply
`
`pjt =
`
`(1 + ηn) wt
`qjt
`
`.
`
`Therefore we can express the equilibrium profit of the monopolist j as
`
`πt (qjt) = [pjt − Mjt] cjt
`= πnCt
`
`where we define πn ≡ ηn
`as the normalized profit (= πt (qjt) /Ct). This is the first step in
`1+ηn
`establising the value of an innovation. Because a new innovation grants a patent protection
`
`until another new innovation makes it obsolete through creative destruction, the value of an
`
`innovation (patent) will be the expected sum of future monopoly profits that will be generated
`
`by this innovation.
`The following lemma summarizes the rest of the static equilibrium variables Ct and wt.
`
`Lemma 1 The aggregate consumption in this economy is equal to
`
`Ct = Qt
`
`where Qt is defined as a productivity index
`
`(cid:20)(cid:90) 1
`
`0
`
`Qt ≡
`
`−1 dj
`(1 + ηj)
`
`(cid:21)−1
`
`(cid:90) 1
`
`0
`
`exp
`
`ln
`
`qjt
`1 + ηj
`
`dj.
`
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`Moreover, the wage rate is equal to
`
`wt = Qt
`
`(cid:90) 1
`
`0
`
`−1 dj.
`(1 + ηj)
`
`3.1.2 R&D and Productive Innovations
`
`The economy has a measure of outside entrepreneurs who try to innovate and replace the exist-
`
`ing incumbents. Outside entrepreneurs invest in R&D to produce a new innovation stochasti-
`
`cally. When they are successful, they improve the latest quality as in (3) . However productive
`
`innovations come in clusters as in Akcigit and Kerr (2010). In particular, new entrants invest
`
`in two types of innovations:
`
`1. radical innovations,
`
`2. follow-on innovations.
`
`When a new radical innovation occurs, it re-starts a new technology cluster with a step size
`η0 = η > 0. Alternatively, if a new follow-on innovation occurs, it directly builds on the existing
`technology and the marginal contribution of this new innovation depends on how exploited the
`
`technologies are within the same technology cluster. In other words, follow-on innovations run
`into dimishing returns within the cluster such that the nth follow-up innovation has a step size
`of ηn = ηαn where α ∈ (0, 1). For mathematical convenience, we assume that after a certain
`number of follow-on innovations (n > n∗), the step size becomes a constant value ηn = ηαn∗
`.
`In summary, the step size of the n+1st patent in a given technology cluster can be summarized
`as follows:4
`
`
`
`ηn =
`
`η if radical innovation
`ηαn if follow-on innovation and n < n∗
`if follow-on innovation and n ≥ n∗
`ηαn∗
`
`.
`
`Since innovations come in technology clusters and that each new innovation utilizes the spillover
`
`from the previous patents from the same technology class, our model generates a natural
`
`interpretation of citations. When there is a major innovation in a technology class with a step
`
`size η, it produces spillovers for the subsequent innovations since the follow-on step size becomes
`
`ηα which encourages new entry into the field. Innovations must cite previous innovations
`
`within the same technology cluster, acknowledging that the patents are technologically related.
`
`Therefore, patents from the same technology cluster will cite the initial major patent that
`
`opened the field. The following example will elaborate this structure further.
`
`4Note that in principle, we can allow the step size ηj to be a function of the sector j. This would not have
`any major impact on the inverted-U relationship that our model predicts.
`
`10
`
`PMC Exhibit 2220
`Apple v. PMC
`IPR2016-01520
`Page 11
`
`

`

`Example 1 This example is provided to show the connection between our model and the data.
`
`In particular, we describe how technology clusters emerge and who cites who in those clusters.
`
`The following chart illustrates an example of some innovation patterns in a single product line:
`
`η
`P10
`
`...
`
`(cid:124)(cid:123)(cid:122)(cid:125)
`
`|||
`
`ηα
`P8
`
`(cid:123)(cid:122)
`
`ηα2
`P9
`
`(cid:125)
`
`(cid:124)
`
`η P
`
`7
`
`|||
`
`(cid:124) (cid:123)(cid:122) (cid:125)
`
`ηα
`P6
`
`η P
`
`5
`
`|||
`
`ηα
`P2
`
`ηα2
`P3
`
`(cid:123)(cid:122)
`
`ηα3
`P4
`
`(cid:125)
`
`(cid:124)
`
`η P
`
`1
`
`|||
`
`Tech Cluster 4
`Tech Cluster 3
`Tech Cluster 2
`Tech Cluster 1
`An example of a sequence of innovations in a product line
`
`Example starts with a radical innovation P1 which has a step size η. Then innovation P2 follows
`on P1 with a step size ηα. Since P3 is the second follow-on innovation in cluster 1, it has a
`step size ηα2 and so on. Note that P5, P7 and P10 turn out to be a radical innovations which
`start new technology clusters; therefore their step sizes are η. As a result, innovation step sizes
`
`follow cycles. Finally, the citing-cited pairs can be summarized as follows:
`
`Cited Citing
`P1 :
`P2, P3, P4
`P2 :
`P3, P4
`P3 :
`P4
`P4 :
`none
`P5 :
`P6
`
`Cited Citing
`P6 :
`P7 :
`P8 :
`P9 :
`P10 :
`
`none
`P8, P9
`P9
`none
`
`...
`
`Consider P2, for instance. Since it builds only on P1, P2 cites only P1. However, there are two
`patents (P3, P4) in the cluster that are building on P2. Hence, P2 receives two citations from
`them.
`
`Now we can turn to the value of an innovation. Consider an innovation of step size ηn = ηαn.
`Let the aggregate innovation arrival rate of the next follow-on innovation be denoted by ¯zn+1
`and the next radical innovation by ¯z0. Then the steady-state value of the nth innovation is
`summarized by the following continuous time Hamilton-Jacobi-Bellman (HJB) equation
`(¯z0∆t + ¯zn+1∆t) × 0
`+ (1 − ¯z0∆t − ¯zn+1∆t) Vnt+∆t
`
`(cid:34)
`
`Vnt =
`
`ηn
`1 + ηn
`
`Ct∆t + (1 − r∆t)
`
`(cid:35)
`
`.
`
`This expression is intuitive. During a small ∆t, nth innovation in a cluster delivers a profit
`Ct∆t to its owner. The future period is discounted by (1 − r∆t). After ∆t, with
`ηn
`of
`1+ηn
`probability ¯zn+1∆t there is a new follow-on entry, and with probability ¯z0∆t there is a radical
`entry. In both cases, the incumbent exits the market becuase she is replaced by a new entrant
`and her firm value decreases to 0. With the remaining probability (1 − ¯zn+1∆t − ¯z0∆t) , the
`
`11
`
`PMC Exhibit 2220
`Apple v. PMC
`IPR2016-01520
`Page 12
`
`

`

`incumbent survives the threat of entry and receives the continuation value Vt+∆t of being the
`incumbent. Subtracting (Vnt+∆t − r∆tVnt) from both sides, dividing through ∆t, and taking
`the limit ∆t → 0 leads to the following HJB equation:
`
`rVn − ˙Vn = πnCt − (¯zn+1 + ¯z0) Vn.
`
`where πn ≡ ηn
`
`1+ηn
`
`. The following lemma provides the exact form of the value function.
`
`Lemma 2 The normalized value of the nth follow-on innovation at time t is equal to
`
`vn ≡ Vnt
`Ct
`
`=
`
`πn
`ρ + ¯zn+1 + ¯z0
`
`(5)
`
`(6)
`
`1+ηn
`
`where πn ≡ ηn
`.
`Proof. This result follows from using the household’s Euler equation r − g = ρ in (5)
`This expression simply says that the value of an innovation depends mainly on four fac-
`tors: First, a larger step size ηn implies larger mark-up and therefore higher innovation value.
`Second, if the aggregate consumption Ct is larger, each variety will receive a larger demand
`and hence generate higher per-period profit and innovation value. Third, present discounted
`value of future profits depends on growth rate adjusted interest rate r − g, which boils down
`to the discount rate ρ through the household problem. Finally, the rate of creative destruction
`of the next follow-on innovation ¯zn+1 or radical innovation ¯z0 lowers the value of the current
`innovation due to shorter expected duration of monopoly