The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier

Transcription

1 Workin Paper No. 54 The Growth Dynamics of Innovation, Diffusion, and the Technoloy Frontier Jess Benhabib Jesse Perla Christopher Tonetti May 5

2 The Growth Dynamics of Innovation, Di usion, and the Technoloy Frontier Jess Benhabib NYU Jesse Perla UBC May 8, 5 Draft Version: 33 Christopher Tonetti Stanford GSB VERY PRELIMINARY, ROUGH, AND INCOMPLETE [Download Newest Version] [Download Technical Appendix] Abstract The recent literature on idea flows studies technoloy di usion in isolation, in environments without the eneration of new ideas. Without new ideas, rowth cannot continue forever if there is a finite technoloy frontier. In an economy in which firms choose to innovate, adopt technoloy, or keep producin with their existin technoloy, we study how innovation and di usion interact to endoenously determine the productivity distribution with a finite but expandin frontier. There is a tension in the determination of the productivity distribution innovation tends to stretch the distribution, while di usion compresses it. Finally, we analyze the deree to which innovation and technoloy di usion at the firm level contribute to areate economic rowth and can lead to hysteresis. Keywords: Endoenous Growth, Technoloy Di usion, Innovation, Imitation, R&D, Technoloy Frontier JEL Codes: O4, O3, O3, O33, O4 Introduction The productivity distribution plays a critical role in many studies in international trade (e.., Eaton and Kortum () and Melitz (3)), macroeconomics (e.., Hsieh and Klenow (9)), industrial oranization (e.., Hopenhayn (99), Foster, Haltiwaner, and Syverson (8)), and other areas of economics. In much of this literature, the productivity of firms evolves exoenously accordin to some shock process, and thus key determinants of this essential object are not studied. There is a theoretical literature that does focus on productivity rowth, as pioneered by Romer (986,

3 99), Seerstrom, Anant, and Dinopoulos (99), Rivera-Batiz and Romer (99), Grossman and Helpman (99, 993), and Ahion and Howitt (99). The key forces that enerate productivity rowth in these papers are innovation and imitation. Followin Kortum (997) recent papers such as Perla and Tonetti (4) and Lucas and Moll (4) used search theory to develop a new microfoundations for technoloy di usion. In these papers all firms are alike except for their initial productivity and have no ex ante comparative advantae in innovation or imitation. However, these papers abstract away from innovation, so rowth cannot continue forever if the existin technoloy distribution has a finite frontier, or even if it has infinite support but thin tails. Thus lon-run rowth in these models relies on the counter-factual assumption that at all times there are firms producin with arbitrarily lare productivities, described by a distribution with infinite support. In this paper, we build on this new microfoundation of technoloy di usion by introducin endoenous innovation to explain economic rowth throuh within-firm productivity improvements. The shape of the distribution of active technoloies defines the opportunities for adoption. Innovation and adoption interact to determine the shape of the distribution of productivities, which in turn determines the incentives to adopt and to innovate. One of the aims of this paper is to model the evolution of productivity distributions with a frontier that is finite for all times, t<. For rowth to continue forever, the frontier must row throuh innovation. A second important aim of this paper is to model the interaction of adoption and innovation decisions. For simplicity, we start by modelin a deterministic innovation process, and continue by introducin exoenous stochastic innovation throuh eometric Brownian motion or discrete-state Markov chains, and finally model stochastic innovations that are subject to firm choice. A common interestin feature of this class of models with initial distributions that have fat tails is the existence of a continuum of stationary distributions, i.e., hysteresis. We explore the conditions under which multiple stationary distributions occur in the various models that we consider. In a number of the models considered, the shape of the stationary distribution depends on the initial distribution (e.., Perla and Tonetti (4)), the properties of the exoenous shock process, or both (e.., Luttmer (7)). However, when there is an innovation decision, the shape of the stationary distribution is endoenous and depends on the parameters of the model and the optimal adoption and innovation choices of aents. In Appendix D we develop a model of stochastic technoloy adoption and deterministic innovation. Technoloy adoption takes the form of draws from the distribution of existin technoloy in use, while innovation is simply modeled as exoenous multiplicative rowth for all aents. We characterize the full dynamic path of the economy startin from arbitrary initial productivity distributions. An important result is that with a finite technoloy frontier or a thin tailed initial productivity distribution, eventually all adoption stops and lon-run rowth is entirely driven by the exoenous innovation process. Furthermore, the economy exhibits hysteresis: there exists a continuum of stationary distributions parameterized by the tail index of the initial distribution. If innovation is stochastic, incentives to adopt are renewed as successful firms pull away and unsuccessful firms fall behind. We find that there are three distinct sources of rowth in these classes of models, which this paper will decompose rowth into contributions from: () firm level research decisions, i.e., innovation ; () incentives for the relatively unproductive to catchup to the areate distribution, i.e., catchup di usion ; and (3) firms receivin a sequence of bad shocks relative to the rowin distribution, i.e., stochastic di usion. We find catch-up di usion can only occur durin transition dynamics, or in the lon-run with a thick-tailed productivity distribution. Stochastic di usion can only occur in models with risky innovation. The forces of di usion and innovation may interact when firms endoenously choose both, as innovation investment chanes with the internalized option value of future technoloy di usion For a formal demonstration that rowth cannot continue forever if the existin technoloy distribution has infinite support but thin tails, see the section on Thin and Fat Tailed Distributions in Technical Appendix D.4.

4 opportunities. A novel element of this model is that since firms internalize the value of di usion, there are interestin trade-o s between innovation and di usion, which can a ect the optimal rowth rates. This is in contrast to papers like Luttmer (7), where di usion e ects incumbents by increasin their fixed costs relative to profits and forcin more exit e ectively, stochastic di usion. Those models provide a di erent, and more directly Schumpeterian, mechanism. In Sections. and. we describe the basic structure of our model, includin the stochastic processes for innovation and technoloy di usion, the adoption decisions by firms, and the resultin law of motion for the productivity distribution. In Section we assume that the firms that are in the innovative state all row at the same exoenous rate. In Section.3 we study and characterize the stationary distribution of this model when the initial productivity distribution has finite support. If the initial distribution has finite support, it will maintain a finite support at all finite times. However, even thouh the support remains finite, the normalized stationary distribution will be unbounded. That is, the ratio of the frontier productivity to the mean productivity, z, will row towards infinity as time proresses. We obtain a unique stationary distribution with lon-run adoption, and since the frontier remains finite 8 t <, the lon-run rowth rate is equal to the exoenous innovation rate. Lookin at data, however, we do not observe a perpetual spreadin of the productivity distribution (see, e.., Luttmer (), a sinificant shortcomin of this specification of the model. To achieve a stationary distribution that is both finite and bounded, in that the ratio of frontier productivity to the lowest productivity converes to a finite constant z, in Section.4 we modify the innovation process to allow some aents to leap-fro to the frontier with a positive probability. This is a continuous time analo of the traditional quality-ladder model, in which successful innovators jump to the technoloy frontier. In this case, we aain have a unique stationary distribution, where the lon-run rowth rate is equal to the exoenous innovation rate, while firms below the frontier continue to adopt. The leapfroin/quality ladder process creates a locomotive e ect where laards do not perpetually remain behind, so z is finite in the stationary distribution. For completeness in Section.5 we also study the case, without leap-froin, where the initial productivity distribution has infinite support. Unlike the cases where the initial distribution has finite support and remains finite for all t, there is no requirement that the innovation rate equals the rowth rate, as rowth can also be driven by adoption from the unbounded tail. Proposition 3 characterizes the stationary equilibria. Unlike the previous cases, now there exists a continuum of stationary equilibria, as in Perla and Tonetti (4). Thus far, we have considered innovation processes that were exoenous. In Section 3., we eneralize the model to allow firms to choose their innovation rowth rate at some cost. Firms optimally choose to either invest in adoption or innovation, with the result that innovation rowth rates are increasin with a firm s productivity. Startin with a finite initial distribution, if firms cannot leap-fro to the frontier, the unique stationary distribution is unbounded. In the lon run, the rowth rate of the economy is equal to the innovation rowth rate chosen by the frontier firm. However, if we also allow some firms to leap-fro to the frontier, there now exists a continuum of stationary distributions that are bounded in relative terms (i.e., the ratio of most to least productive does not divere). (See Section 3..) The rowth rate of the economy is endoenous and equal to the innovation rowth rate chosen by the firm at the frontier. However, in contrast to all previous cases, even with finite initial distributions, there exists a continuum of stationary distributions. This is because there is an important interaction between the incentives to adopt and to innovate that enerates a self-sustainin feedback. We can index the stationary distributions by the relative A side-e ect of introducin stochastic innovation throuh Geometric Brownian Motion (GBM) is that the support of the productivity distribution becomes infinite instantly. A desirable model property is that at any point in time the technoloies in use for production and available for adoption are characterized by a distribution with a finite frontier. To achieve this, in Section, we depart from GBM and instead model stochastic innovation as a discrete Markov process. We bein our analysis studyin an exoenous uncontrolled innovation process. 3

5 frontier, z. Even with a finite frontier, a distribution that has more weiht in hiher productivities produces stroner incentives to adopt. The optimal innovation policy is increasin in productivity, as opposed to the exoenous innovation policy that was flat. This innovation policy enerates more mass in the riht tail, and thus enerates more incentives to adopt, as adopters internalize the option value of innovation. An interestin feature of the class of models with initial distributions that have fat tails is the existence of a continuum of stationary distributions, i.e., hysteresis. In the model considered in Section.5, the shape of the stationary distribution depends on the initial distribution (e.., Perla and Tonetti (4)), the properties of the exoenous shock process, or both (e.., Luttmer (7)). However, when there is an innovation decision, the shape of the stationary distribution is endoenous and depends on the parameters of the model and the optimal adoption and innovation choices of aents. We summarize our results on hysteresis in Section 3.3. While the baseline model is written with an exoenous number of firms and linear profits in productivity, see Appendix D.6 in Appendix D.6 for a qualitatively similar version of the model with monopolistic competition, free-entry, all costs denoted in labor, and Geometric Brownian motion for firm dynamics.. Recent Literature Related papers in this class of models include Lucas and Moll (4), Kortum (997), Luttmer (7, 4) and Sampson (4) which emphasize selection from optimal entry/exit, and Alvarez, Buera, and Lucas (8, 3), which emphasizes di usion as an arrival rate of ideas from the productivity distribution. Buera and Oberfield (4) is a related semi-endoenous rowth model of international di usion of technoloy and its connection to trade. Another approach, taken in Jovanovic and Rob (989), is to add positive spillovers from the di usion process itself, which can create balanced rowth. While Perla and Tonetti (4) isolated the role of rowth throuh catch-up di usion, in some sense the role of stochastic di usion is isolated in Luttmer (7). The catchup di usion e ect is not present in Luttmer (7) in the same sense, as the incumbent firms lower in the productivity distribution ain no benefit from rowth. However, in our model, stochastic di usion is di erent from Luttmer (7), as firms internalize the value of an uprade rather than bein driven into un-profitability and exit from GE e ects. In this paper we are considerin process innovation rather than new product innovation. Smaller firms may be especially innovative in comin up with new products as in Klette and Kortum (4) or Acemolu, Akciit, Bloom, and Kerr (3), but this is not considered in our innovation technoloy, as all firms have one product and the number of products in equilibrium is kept fixed for simplicity. Other papers emphasizin the role of an endoenous innovation choice include Atkeson and Burstein () and Stokey (4). Acemolu, Ahion, and Zilibotti (6), Köni, Lorenz, and Zilibotti (), Chu, Cozzi, and Galli (4), Stokey (4), and Benhabib, Perla, and Tonetti (4) also explore the relationship between innovation and di usion from di erent perspectives. The crucial element that enables the interestin trade-o between innovation and technoloy di usion in our model is that the incumbents internalize some of the value from the evolvin distribution of technoloies, distortin their innovation choices. We describe this as an option value of di usion, where incumbents take into account the possibility of future improvements in their productivity throuh jumps from technoloy di usion. The lower the relative productivity of a firm, the hiher the expected benefit of adoption via a jump to a superior technoloy, and the sooner the expected time to execute the adoption option. Therefore, low productivity firms have hih option values of di usion, while very hih productivity firms may have an asymptotically irrelevant contribution from technoloy di usion. 4

6 This tension between innovation and technoloy di usion explored here has a di erent emphasis in Luttmer (7,, 4), where the enerator of di usion is entry/exit in equilibrium, and only new entrants can internalize the benefits of technoloy di usion. The main similarity is that in both papers, some firms sample from the existin distribution of productivity. In particular, Luttmer (7) is interested in the role of technoloy di usion throuh entry, so it is the entrants who ain the benefits of a rowin economy. As incumbents pay a fixed cost that rows with the scale of the economy, entry can spur more exit. Therefore neative profits that results in exit leads to entry and to technoloy di usion. In our model incumbents, or operatin firms, choose when to exploit the incentives to adopt a new technoloy. 3 The di erence between whether incumbents or entrants internalize the value of a rowin economy leads to very di erent implications for technoloy di usion. Baseline Model with Exoenous, Stochastic Innovation. Basic Setup Consider a discrete two-state Markov process drivin the exoenous rowth rate of an operatin firm. In the hih state, the firm is innovatin and increasin its productivity deterministically. In the low state, its productivity does not row throuh innovation. This captures the concept that some times firms have ood ideas or projects that enerate rowth and some times firms are just producin usin their existin technoloy. This innovation status chanes accordin to a continuous time Markov chain. 4,5 Firm Heteroeneity and Choices Assume firms producin a homoeneous product are heteroeneous over their productivity, Z, and over their deree of innovation, i {`, h}. The mass of firms of productivity less than Z in innovation state i is defined as i(t, Z) (i.e., an unnormalized CDF). Define the technoloy frontier as the maximum productivity, B(t) sup {support { i (t, )}} apple, and normalize the mass of firms to so that `(t, B(t)) + h (t, B(t)) =. At any point in time, the minimum of the support of the distribution will be an endoenously determined M i (t), so that i(t, M i (t)) =. Define the distribution unconditional on type as (t, Z) `(t, Z)+ h (t, Z). AfirmwithproductivityZ can choose to continue producin with its existin technoloy, in which case it would row stochastically, or it can choose to adopt a new technoloy instantaneously. As we will show in equilibrium, all firms choose an identical threshold, M(t), above which they will continue operatin with their existin technoloy (i.e., a firm with Z apple M(t) chooses to adopt a new technoloy). As draws are instantaneous, this endoenous M(t) becomes the evolvin minimum 3 Mechanically, these di erences manifest themselves in the option value of the Bellman equation. In Luttmer (7), incumbents are only a ected neatively by rowth and have a zero option value of technoloy di usion, whereas in our model incumbents have a positive option value of di usion as they can always adopt from by takin a draw from the existin distribution. 4 Luttmer () also emphasizes the need for fast rowin firms driven by di erences in the quality of blueprints for size expansion to account for the size distribution of firms. In his model, firms will stochastically slow down eventually, where here will assume that firms can jump back and forth between the states. In other work, Luttmer (4) emphasizes the role of a stochastic shock as experimentation as distinct from deterministic innovation, and important in the eneration of endoenous tail parameters. 5 Lucas and Moll (4) provides an extension of their baseline model with the addition of exoenous innovators in the distribution in order to discuss finite support. 5

7 of the i(t, Z) distribution. 6,7 The cost of adoption scales with the economy, and for simplicity is proportional to the endoenous scale of the economy, M(t). 8 If a firm adopts a new technoloy, then it immediately chanes its productivity to a draw from a distortion of the i(t, Z) distribution. 9 Assume that an adoptin firm draws a (i, Z) from distributions ˆ `(t, Z) and ˆ h(t, Z), both of which will be determined by the equilibrium `(t, Z) and h(t, Z). Assume that this ives a proper cdf, so ˆ `(t, ) = ˆ h(t, ) = and ˆ `(t, B(t)) + ˆ h(t, B(t)) =. Stochastic Process for Innovation The jump intensity from low to hih is ` > and from hih to low is h >.Since the Markov chain has no absorbin states, and there is a strictly positive flow between the states for all Z, the support of the distribution conditional on ` or h is the same (except, perhaps, exactly at an initial condition). Recall that support { (t, )} [M(t),B(t)). The rowth rate of the upper and lower bounds of the support are defined as (t) M (t)/m (t) and B (t) B (t)/b(t) ifb(t) <. Value Functions and the Growth Rate of the Frontier While i = h, firms row at an exoenous innovation rate >, and, without loss of enerality, do not row if i = `., The continuation value functions are V i (t, Z) and include the drift in a hih state as well as the intensity of jumps between i. For the two discrete states, the Bellman equations in the continuation reion are, rv`(t, Z) =Z + ` (V h (t, Z) V`(t, Z)) {z } t V`(t, Z) {z } Jump to h Capital Gains rv h (t, Z) =Z + Z@ Z V h (t, Z) + {z } h (V`(t, Z) V h (t, Z)) +@ {z } t V h (t, Z) () Exoenous Innovation Jump to ` From this process, if B() <, thenb(t) will remain finite for all t, as it evolves from the innovation of firms in the interval infinitesimally close to B(t); that is, B (t)/b(t) = if h(t, B(t)) h(t, B(t) ) >, for all >. With the continuum of firms and the memoryless Poisson arrival of chanes in i, there will always be some h firms that have not jumped to the low state for any t, so the rowth rate of the frontier is always. 6 The technical appendix is located at [Download Technical Appendix]. 7 To show that the minimum of support is the endoenous threshold, assume a Poisson arrival rate of draw opportunities approachin infinity. In any positive time interval firms would ain an acceptable draw with probability, so that Z>M(t) almost surely. Because of the immediacy of draws, the stationary equilibrium does not depend on whether draws are from the unconditional distribution or are from the distribution conditional on bein above the current adoption threshold. This is the same as the small time limit of Perla and Tonetti (4), which solves both versions of the model. The derivation of the cost function as the limit of the arrival rate of unconditional draws is in Technical Appendix C.. 8 In Technical Appendix B, a more elaborate version of this is derived in eneral equilibrium where is the quantity of labor required for adoption, but it ends up bein qualitatively equivalent. An alternative is to have the cost scale with the firm s Z, which introduces a less convenient smooth pastin condition, but remains otherwise tractable. 9 See Technical Appendix C. for a proof that the ability for a firm to recall its last productivity doesn t chane the equilibrium conditions, and Technical Appendix C. for a derivation of this where adoption is not instantaneous. In Section 3, the rowth rate will become a control variable for a firm, with the choice subject to a convex cost. For notational simplicity, define the di erential such z When is univariate, derivatives will be denoted as v (z) dv(z). dz apple Orderin the states as {l, h}, the infinitesimal enerator for this continuous time Markov chain is Q = ` `, with adjoint operator Q. The KFE and Bellman equations can be formally derived usin these h h operators and the drift process. () 6

8 Technoloy Di usion Firms upradin their technoloy throuh adoption receive a new i type and a draw of Z from the productivity distribution. The exact specification typically does not a ect the qualitative results, so we will write the process fairly enerally and then analyze specific cases. While the draw from ˆ i(t, Z) is left eneral, we are maintainin a simplification that the ross value of adoption is independent of an aent s current type. In principle, there may be adopters hittin the adoption threshold with either innovation type. Assume that h and ` types have the same adoption threshold M(t), to be proved later. A flow S i (t) of firms cross into the adoption reion at time t and choose to adopt a new technoloy. Denote the total flow of adoptin firms as S(t) S`(t)+S h (t). Law of Motion The Kolmoorov Forward Equations (KFEs) in CDFs include the drift and jumps between innovation t `(t, Z) = ` `(t, Z)+ h h (t, Z) +(S`(t)+S {z } h (t)) ˆ {z } `(t, Z) {z } Net Flow from Jumps Flow Adopters Draw apple t h (t, Z) = Z@ Z Z h (t, Z) {z } Innovation S`(t) {z } Adopt h h(t, Z)+ ` `(t, Z)+(S`(t)+S h (t))ˆh(t, Z) S h (t) (4) Reconizin that the i jumps are of measure when calculatin how many firms cross the boundary in any infinitesimal time period, the flow of adopters comes from the flux across the movin M(t) boundary, 3 S`(t) M (t)@ Z `(t, M(t)) (6) S h (t) M (t) M(t) {z } Relative Speed of Z h (t, M(t)) {z } PDF at boundary Adoption Decision Firms choose thresholds, below which they adopt a new technoloy throuh the technoloy di usion process. 4 Necessary conditions for the optimal stoppin problem include value matchin and smooth pastin conditions at the endoenously chosen adoption boundary, M(t), V i (t, M(t)) = {z } Value at Threshold Z B(t) M(t) V`(t, Z)dˆ`(t, Z)+ Z B(t) M(t) V h (t, Z)dˆh(t, Z) {z } Gross Adoption Value M(t) {z } Adoption Z V`(t, M(t)) =, if M (t) > Z V h (t, M(t)) =, if M (t) M(t) > (). Normalization and Stationarity To find a balanced rowth path (BGP), it is convenient to transform this system to a stationary set of equations by normalizin variables relative to the endoenous boundary M(t). Define the 3 This is consistent with the solution to the ODEs in (3) and (4) atz = M(t), and is clear in the normalized (6) and (7). 4 While the threshold could depend on the type i, see Appendix A. for a proof that ` and h aents choose the same threshold, M(t), if the net value of adoption is independent of the current innovation type. (3) (5) (7) (8) 7

9 chane of variables, normalized distribution, and normalized value functions as, z lo(z/m(t)) () F i (t, z) =F i (t, lo(z/m(t))) i (t, Z) () v i (t, z) =v i (t, lo(z/m(t))) V i(t, Z) M(t) The adoption threshold was chosen to be normalized to lo(m(t)/(m(t))) =, and the relative technoloy frontier is z(t) lo(b(t)/m (t)) apple. See Fiure for a comparison of the normalized and unnormalized distributions. From this, F`(t, ) = F h (t, ) = and F`(t, z(t)) + F h (t, z(t)) =. Denote the unconditional, normalized distribution with F () = and F ( z) = as F (z) F`(z)+ F h (z). Z (Z) Unnormalized z F (z) Normalized PDF M (t) M (t) M(t) M(t) B(t) Z z(t) z Fiure : Normalized vs. Unnormalized Distributions With the above normalizations, the value function, productivity distribution, and rowth rates can be stationary and independent of time. 5 When the distribution is not time varyin, let F (z) denote the probability density function. Summary of Stationary Equilibrium A full derivation of the normalization is done in Appendix A., which leads to the followin normalized set of stationary equations for the evolution of 5 An important example is when (t, Z) is Pareto with minimum of support M(t) and tail parameter : (t, Z) = M(t), for M(t) apple Z (4) Z Then F (t, z) is independent of M and t: F (t, z) = e z, for apple z<. (5) This is the cdf of an exponential distribution, with parameter >. From a chane of variables, if X Exp( ), then e X Pareto(, ). Hence, is the tail index of the unnormalized Pareto distribution for Z. 8

10 the distribution, =F `(z) `F`(z)+ h F h (z)+(s` + S h ) ˆF`(z) S` (6) =( )F h (z) hf h (z)+ `F`(z)+(S` + S h ) ˆF h (z) S h (7) =F`() = F h () (8) =F`( z)+f h ( z) (9) S` = F `() () S h =( ) Fh () () The summary of necessary conditions for the firm s problem are (r )v`(z) =e z v `(z)+ ` (v h (z) v`(z)) () (r )v h (z) =e z ( )v h (z)+ h (v`(z) v h (z)) (3) v`() = v h () = Z z v`(z)d ˆF`(z)+ Z z v h (z)d ˆF h (z) (4) v `() =, if > (5) vh () =, if > (6) To interpret: (6) to(9) are the stationary KFE with initial conditions and boundary values. S` in () is the flow of ` aents movin backwards at a relative speed of across the barrier, while S h in () is the flow of h aents movin backwards at the slower relative speed of across the barrier. The ˆF i (z) specification is some function of the equilibrium F i (z), and will be analyzed further in Sections.3 to.5. () and (3) are the Bellman Equations in the continuum reion, where (4) is the value matchin condition between the continuation and technoloy adoption reions. The smooth pastin conditions in (5) and (6) are only necessary if the firms of a particular i are driftin backwards relative to the adoption threshold. See Fiure for a visualization of the normalized Bellman equations. v h (z) v i (z) h v`(z) v i () ` z z Fiure : Normalized, Stationary Value Functions In the normalized setup, z(t) lo(b(t)/m (t)), and a necessary condition for a stationary equilibrium with sup { z(t) 8t} < is that = B =. This is necessary because if < B,then z diveres, while if > B, the minimum of the support would eventually be strictly reater than the maximum of the support. 9

11 F (z) Stochastic innovation spreads Adoption compresses z z Fiure 3: Tension between Stochastic Innovation and Adoption Terminoloy for Various Cases of the Normalized Support There are three possibilities for the stationary z that we will analyze separately. The first is if z =, whichwewill call infinite support, which happens for any initial condition that starts with B() = (i.e., sup support {F (, )} = ). The second case is when B() < (which implies B(t) < ), but where lim t! z(t) =. We label this case finite, unbounded support. The final case is when the initial condition has finite support, and lim t! z(t) <, which we will refer to as finite, bounded support. An important question will be whether the unbounded and infinite support examples have the same stationary equilibrium. It will turn out that this is not the case, suestin that caution should be used when usin an infinite support as an approximation of a finite, but ultimately unbounded, empirical distribution. An important question is whether a stationary equilibrium with bounded finite support can even exist for a iven version of the model. This will be discussed further in Propositions and..3 Stationary BGP with Finite Initial Support In this section we study the stationary distribution, the BGP, when the initial distribution (,Z) has finite support. Consider for simplicity that the process of adoptin new technoloies is disruptive to R&D, so the firm starts in the ` type reardless of its former type, and the Z is drawn from a distortion of the unconditional distribution. 6 The distortion, representin the deree of imperfect mobility, is indexed by apple> where the aent draws its Z from the cdf (t, Z) apple. Note that for hiher apple, the probability of a better draw increases. As (t, B(t)) apple = and (t, M(t)) apple =, for all apple>, this is a valid probability distribution. While apple is exoenous here, in Section 3., we solve a version of the model with directed technoloy di usion where apple is endoenous. ˆ `(t, Z) ( `(t, Z)+ h (t, Z)) apple (7) Normalizin to a stationary draw distribution and then usin the definition of the unconditional normalized distribution, F (z), yields adoption distribution ˆF`(z) =(F`(z)+F h (z)) apple = F (z) apple (8) We write F (z) apple for the normalized draw process (for which, iven the assumptions, all firms end up in the low state). 6 Unlike the infinite support case in Section.3, the equilibrium are not sensitive to the deree of correlation in the draws, and we have simply chosen the most convenient.

12 Due to the bounded rowth rates of the Markov process, if the support of (,z)isfinite, then it remains finite as it converes to a stationary distribution. With an exoenous and a finite frontier, a necessary requirement for non-deeneracy of F i (z) isthen =. Hence, in the stationary equilibrium there are no h type aents hittin the adoption threshold, and the smooth pastin condition for h firms is not a necessary condition. Necessary conditions for a stationary equilibrium with a finite initial frontier are v`(z), v h (z), F`(z), F h (z), S, such that, (r )v`(z) =e z v `(z)+ `(v h (z) v`(z)) (9) (r )v h (z) =e z + h (v`(z) v h (z)) (3) v`() = Z z v`(z)df (z) apple (3) v `() = (3) =F `(z)+sf(z)apple + h F h (z) `F`(z) S (33) = `F`(z) hf h (z) (34) =F`() = F h () (35) =F h ( z)+f`( z) (36) S = F `() (37) Define the constants, ˆ `, ` h r + h +. The followin characterizes the equilibrium, Proposition (Stationary Equilibrium with Continuous Draws and Finite, Unbounded Support). There does not exist an equilibrium with finite and bounded support (for any apple>). There exists auniqueequilibrium,with = and z!. In the case of apple =,theuniquestationarydistributionis, where is the tail index of the power law distribution: and F `() is determined by model parameters: F `() = h r(r+ h + `) The firm value functions are, F`(z) = +ˆ e z (38) F h (z) =ˆF`(z), (39) ( + ˆ)F `(), (4) (4) p ((4 +r )( +r+ h ) +( +( r)r )( r h) `+( r) `)+ + ( )(3r+ h + `) ( h + `)( r h). (4) v`(z) = +(r ) ez + (r )( + ) e z (43) v h (z) = ez + h v`(z), (44) r + h where >, therateatwhichtheoptionvalueisdiscounted,isivenby (r ). (45)

13 Proof. See Appendix B.. So far, while the distributions of productivities z with initial distributions that have finite support also have finite support for t<, stationary distributions all have asymptotically infinite relative support, that is z!: the ratio of frontier to lowest productivity oes to infinity. In the ` state the rowth rate is zero and z stays put, but in the h state the rowth rate of z is positive. Given the Markov process for ` and h, there will be some aents who hit lucky streaks more than others, escape from the pack, and break away. Given a fixed barrier, the loic is similar to the linear or asymptotically linear Kesten processes bounded away from zero with a ne terms, for which, under appropriate conditions, the asymptotic tail index can be explicitly computed in terms of the stationary distribution induced by the Markov process. In our model however, the adoption process introduces non-linear jumps that are not multiplicative in productivities, and which do not permit the simple characterization of the tail index. The endoenous absorbin adoption barrier (which acts like a reflectin barrier with stochastic jumps) complicates the analoy since one miht think if the frontier is rowin rapidly, the endoenous barrier could also move rapidly to keep up with the frontier. However, the incentives for adoption, which drive the speed of the movin barrier, are driven by the mean draw in productivity. Therefore, if the frontier diveres to infinity but the mean doesn t keep rowin at the same rate, the frontier technoloy will divere. As we will see in Section.4, if we strenthen the adoption process by allowin a positive fraction of adopters to leapfro to the frontier, the multiplicative jump process eneratin the escape to infinity in relative productivities may in fact be contained. This stationary equilibrium is unique and independent of the initial distribution (which will contrast the case of infinite initial support discussed in Proposition 3 which featured hysteresis and a continuum of stationary solutions). Fiure 4 provides an example where =.,r =.6, ` =., h =.3, and = v`(z) v h (z).4. F `(z) F h (z) vi(z) 5 F i (z) z z Fiure 4: Normalized, Stationary Value Functions and PDFs for the Unbounded Cases

14 .4 Stationary BGP with Bounded Relative Support Proposition shows that while the frontier remains finite for t <, the ratio of the frontier productivity to the mean of the distribution is finite, but tends to infinity as t!. 7 This is because a diminishin, yet strictly positive number of firms keep ettin lucky and row at forever, but as the mass of aents with extremely hih z is thinnin out, it doesn t stronly e ect the di usion incentives and adoption probabilities of those with a low z (i.e., the mean expands slower than the frontier). Since there is no leapfroin of these perpetually lucky firms, this process will continue forever. An alternative way to model within-firm productivity chane is to assume that firms can leapfro to the frontier with some probability. Such leapfroin is a continuous time version of a quality-ladders model that keeps the frontier bounded. 8 The jumps can occur either for firms adoptin throuh di usion, or for innovatin firms that succesfully leapfro to the frontier, which can be viewed as positive spillovers from the frontier to innovators. We model leapfroin as an innovation that propels firms to the frontier of the productivity distribution. 9 This major innovation with spillovers from the frontier can be achieved by all firms operatin their existin technoloy, i.e., both i = ` and h types. However, since such an innovation is potentially disruptive, those firms that jump to the frontier become ` types and must wait for the Markov transition to h before they become innovators aain. To accomodate firms jumpin to the frontier, we modify the model presented in Proposition by addin an arrival rate for operatin firms of jumps to the frontier,. See Fiure 5 for a visualization of the stationary value functions. To consider the case where it is adopters rather than just innovatin firms who jump to the frontier, see Section 3. where an endoenous jump probability is chosen by adoptin firms. There may be a jump discontinuity in the riht continuous cdf at z. Due to riht continuity of the cdf, the mass at the discontinuity z = z is: ` =lim! (F`( z) F`( z )) (46) h =lim! (F h ( z) F h ( z )) (47) Set apple = for simplicity, and define H (z) as the Heaviside operator. The followin characterizes 7 This result is robust to variations in the di usion specification includin assumin that adoptin aents draw from the F h (z) distribution and start with a h type with apple>, which adds the maximum possible incentives to increase di usion and compress the distribution. 8 In a model of leapfroin arrivals and a multiplicative step above the frontier, in continuous time the frontier would become infinite immediately. Alternatively, it could be recast as a step-by-step innovation model in the spirit of Ahion, Akciit, and Howitt (3) with the same qualitative results. 9 Rather than bein the autarkic process improvement of the rowth, this is leap-froin and may be viewed as a meldin of innovation and di usion, as the jump is a function of the existin productivity distribution. The intuition here is that while the stochastic, continuous rowth of innovators is process improvement, these would be the sorts of innovations that are captured as new patents citin the prior-art. Note that here, unlike quality ladder models, their cannot be a multiplicative jump, or the absolute frontier would divere to infinity as their would be some aent with an arbitrarily lare number of jump arrivals in any positive time period. The assumption of a jump to the ` state at the frontier is only for analytical convenience, and this assumption can be chaned with no qualitative di erences. If some firms jumped to the h state at the frontier instead, then a riht discontinuity in F h (z) would exist, h >, and more care is necessary in solvin the KF and interatin the value matchin condition. With this specification, a possible downside is that v`( z) <v h ( z ) for some set of small, and those firm would rather keep the lower z rather than innovate. Intuitively, the idea with this specification is similar to the notion of neative shocks to productivity due to experimentation, and that innovation can be disruptive to a firm. This simplification helps ensure that the values of jumps to the frontier remains identical for both aents, and hence all types have the same adoption threshold as demonstrated in Appendix A.. If it is adopters, as nested by the endoenous probability in Section 3., who jump, this doesn t occur. 3

15 v h (z) v i (z) v`(z) v i () h ` z z Fiure 5: Normalized, Stationary Value Functions with Bounded Support the necessary conditions for a stationary equilibrium, (r )v`(z) =e z v `(z)+ `(v h (z) v`(z))) + (v`( z) v`(z)) (48) (r )v h (z) =e z + h (v`(z) v h (z)) + (v`( z) v h (z)) (49) v`() = Z z v`(z)(f `(z)+f h (z))dz (5) v `() = (5) =F `(z)+ hf h (z) `F`(z) F`(z)+ H (z z)+sf(z) S (5) = `F`(z) hf h (z) F h (z) (53) =F`() = F h () (54) =F h ( z)+f`( z) (55) S = F `() (56) Let r> and define the constants, ˆ + ` h, r + `+ h r + h, and = r +. Furthermore, assume that the value of F `() that solves (6) is larer than /. Proposition (Stationary Equilibrium with a Bounded Frontier). With the maintained assumptions, a unique equilibrium with z < exists with = where the stationary distribution is, where F`(z) = F `() (F `() / )( + ˆ) ( e z ) (57) F h (z) =ˆF`(z), (58) ( + ˆ)(F `() / ) (59) lo( F`()/ ) z =. (6) The equilibrium F `() solves the followin implicit equation substitutin for and z, + F `() e z ( +e z ) ( +r) + e z (e z ) ( r) + e ( + ) z + ( + ) + e z z + = r ( F `() )( + ). (6) 4

16 The value functions for the firm are, v`(z) = e z + ( + ) e z + e z + r e z (6) v h (z) = ez +( h )v`(z)+ v`( z). (63) r + h Proof. See Appendix B.. In the above, note that F i (z) are continuous, and ` = h =. This is because leapfroin firms become type `, at which point they immediately fall back in relative terms. As those firms fallin back also jump back and forth to the h type, the F`(z) and F h (z) distribution smoothly mix, ensurin continuity. If such firms were added as h aents, there would be a jump discontinuity in the cdf exactly at z. The Bellman equations, such as (6), now contain the value of production in perpetuity and the option value for both the current z and the frontier z. The above is an empirical tail index that can be estimated from a discrete set of data points. An example for r =.6, ` =., h =.3, = 5, =., = =. is iven in Fiure 6. With these parameters, the frontier is z =.47, or convertin from los, the frontier firm is approximately times as e cient as the least productive firm at adoption threshold. The computed from this example is., close to the empirical Zipf s law. 3 5 v`(z) v h (z).8 F `(z) F h (z) z z Fiure 6: Exoenous v i (z), and Fi (z) with a Bounded Frontier As noted at the end of previous section, leapfroin to the frontier by a positive mass of aents can contain the escape in relative productivities by lucky firms who et streaks of lon sojourns in the hih rowth roup h. Firms which had leapfroed to the frontier may row fast until they et a draw that slows them down by puttin them back in the ` state. They will be overtaken by others who leapfro to the frontier from anywhere in the productivity distribution and replenish it. This leapfroin/quality ladder process prevents laards from remainin as laards forever. The distribution of relative productivities then remains bounded as the frontier acts as a locomotive in a relay race. Note that this locomotive process is similar to models of technoloy di usion where the rowth rate of adopters is an increasin function the distance to the frontier, unlike innovators with multiplicative rowth in their productivity level (see Benhabib, Perla, and Tonetti (4)). As 5

17 adopters fall behind, their rowth rate increases to match the rowth rate of innovators, so relative productivities remain bounded. Comparative statics on the z and are shown in Fiure 7 for chanes in,,, and h. For example, hiher rowth rates of innovators leads to a more distant technoloy frontier, but also to thinner tails. Alternatively, a hiher cost of adoption leads to a more distant frontier, and thicker tails z z z.4.6 z h Fiure 7: Comparative Statics with a Bounded Frontier From (6), a relationship can be found that determines the rane of the productivity distribution for any particular F `(): z = + h lo(f `()) lo( / ) + ` + h F `() /. (64).5 Stationary BGP with Infinite Support For completeness, if (,Z) has infinite support, (t, Z) will convere to a stationary distribution as t!. A continuum of stationary distributions, each with its associated areate rowth rates, are possible from di erent initial conditions. The intuition for this hysteresis is identical to that discussed in Appendix D and Perla and Tonetti (4). This section introduces an important di erence from the setup used in Sections.3 and.4: the adoption technoloy will instead have firms copyin both the type and productivity of the draw, rather than always startin in the ` state. The normalized adoption distributions are then, ˆF`(z) F`(z) and ˆF h (z) F h (z), which can be verified to yield a proper CDF: ˆF`()+ ˆF h () = Uniqueness of related models with Geometric Brownian Motion is discussed in Luttmer (). While an exactly correlated draw of the type and the productivity is not necessary here, see Technical Appendix C.3 for a proof that independent draws of Z and the innovation type for adopters has only deenerate stationary distributions in equilibrium. 6

18 F`()+F h () =. There can be no jumps to the frontier, or else the problem is undefined (i.e., require = ). As B(t) = for all t, unlike in the examples with finite support, there is no requirement that = to ensure a stationary non-deenerate productivity distribution. However, is neccesary to ensure that S h. Summarizin the stationary equations, (r )v`(z) =e z v `(z)+ ` (v h (z) v`(z)) (65) (r )v h (z) =e z ( )vh (z)+ h(v`(z) v h (z)) (66) Z Z v`() = v h () = r = v`(z)df`(z)+ v h (z)df h (z) (67) {z } Adopt both i and Z of draw v `() = (68) vh () = (69) =F `(z) `F`(z)+ h F h (z)+(s` + S h )F`(z) S` (7) =( )Fh (z) hf h (z)+ `F`(z)+(S` + S h )F h (z) S h (7) S` = F `() (7) S h =( )Fh () (73) =F`() = F h () (74) =F`()+F h () (75) To characterize the continuum of stationary distributions, parameterize the set of solutions by a scalar. Define the followin as a function of the parameter with an accompanyin rowth rate,, apple apple F`(z) v`(z) ~F (z) v(z) (76) F h (z) v h (z) # # A " B " r+ ` h ` r+ h ' p ( h ) + l ( + h )+ l (78) # C D " + + h l +' l " h(( h+' l)+ l) ( h+'+ l ) ( h+' l)+ l ( ) h + h+ l +' ( ) # Proposition 3 (Stationary Equilibrium with Infinite Support). There exists a continuum of equilibria parameterized by > for ( ) that satisfies Z h (I r + = + B) e Iz + e Bz B A T e Di Cz dz (8) and the parameter restrictions iven in (B.54) to (B.57). Thestationarydistributionsandthevalue functions are iven by: (77) (79) (8) ~F (z) = I e Cz C D (8) ~F (z) =e Cz D (83) v(z) =(I + B) e Iz + e Bz B A (84) By construction, is also the tail index of the unconditional distribution F (z) F`(z)+F h (z). 7

19 Proof. See Appendix B.3 The proof in Appendix B.3 provides (B.54) to(b.57) as a complicated set of parameter restrictions to ensure that r>and that the eienvalues of B and C are positive. Positive eienvalues of B ensure that value matchin is defined, and the option value of di usion asymptomatically oes to for lare z. See Fiures 8 and 9 for an example of infinite support with =.,r =.5, ` =.4, h =.3, and =6.. In this equilibrium, =.9, =.5 and F`() =.988. The relationship between and is shown in Fiure 9. The rowth rate is decreasin as the tail becomes thinner, and the total number of aents in the h state increases. 5 4 v`(z) v h (z) 3.5 F `(z) F h (z) vi(z) 3 Fi(z) z 3 z Fiure 8: Exoenous v i (z), and Fi (z) with a Infinite Frontier Note the distinction between Propositions and 3: the stationary equilibrium associated with initially finite vs. initially infinite support are di erent, even thouh the initially finite support case of Proposition also has an asymptotically unbounded relative support. 8

20 .4 ( ). F h () S( ) Fiure 9: Growth rate as a Function of with an Infinite Frontier 3 Endoenous, Stochastic Innovation This section introduces endoenous innovation into the stochastic model with finite support. We assume that firms can control the drift of their innovation process, as in Atkeson and Burstein () and Stokey (4). At first we assume that the arrival rate of jumps to the frontier,, is zero in order to analyze the unbounded case with z!. Then in Section 3. we move to the model with endoenously chosen innovation rates and jump probabilities and show that the frontier z is finite. We are modelin the innovation choice with no direct spillovers to ensure that it is as simple as possible, and orthoonal to the technoloy di usion process. The interactions are between the tradeo s in the firm s choices, rather than a coupled innovation and adoption technoloy Continuous Choice with the Finite, Unbounded, Frontier Assume that, with a convex cost proportional to its current Z, a firm in the innovative state can choose its own rowth rate. Let > be the productivity of their R&D technoloy, and the cost quadratic in the rowth rate. Adaptin the equations in Section.3 after normalizin the 3 This is in contrast to approaches such as Chor and Lai (3), where they are interested in the direct interaction with a dependent innovation process, with areate spillovers of knowlede. 9

21 innovation cost, and usin the result that z!in the absence of jumps to the frontier, (r )v`(z) =e z v `(z)+ `(v h (z) v`(z)) (85) 8 9 >< >= (r )v h (z) = max >: ez e z {z } v i () v`() = v h () = R&D cost Z ( ) vh {z } (z)+ h(v`(z) v h (z)) >; Drift (86) v`(z)df (z) apple (87) v `() = v h () = (88) =F `(z)+(s` + S h )F (z) apple + h F h (z) `F`(z) S` (89) =( (z)) Fh {z } `F`(z) hf h (z) S h (9) Drift =F`() = F h () (9) =F h ()+F`() (9) S` = F `() (93) S h = Fh () (94) Define the constant, r + ` + h r + ` (95) With this setup, instead of all firms rowin at rate exoenously, h-type firms are choosin a rowth rate that is a function of their current productivity level, z. As the choice of is increasin in z in equilibrium, aents in the h state will end up crossin the endoenous adoption threshold, as shown in Appendix A., and thus the smooth pastin condition for h types is now necessary. Proposition 4 (Stationary q Equilibrium with Continuous Endoenous Innovation and Finite, Unbounded Support). If r>,thenauniqueequilibriumexistswitharowthrateof, apple = r The value function of the firm solves the followin system of non-linear ODEs, r r. (96) (r )v h (z) =e z v h (z)+ 4 e z v h (z) + h (v`(z) v h (z)) (97) (r )v`(z) =e z v `(z)+ `(v h (z) v`(z)) (98) v`() = v h () = r. (99) Given a solution to this system, the endoenous innovation choice is such that, and () =, lim z! (z) = With this (z), F i (z) solves the KFEs in (89) to (9). (z) = e z vh (z). () Proof. See Appendix C.. A numerical method to compute the equilibrium is described in Technical Appendix E.3