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

Transcription

1 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier 1 / 31 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Jess Benhabib, Jesse Perla, and Christopher Tonetti NYU, UBC, and Stanford GSB September 22nd, 2015

2 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Overview 2 / 31 Quick Motivation and Broad Generalizations Innovation Long-run growth rates Aggregate growth rates (usually Shumpeterian) Model of frontier agents Silicon valley, pharma v.s. Technology Diffusion Medium-run (> 50 years?) Distribution shape from aggregate growth rates Model of less productive Coffee shops, agriculture

3 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Overview 2 / 31 Quick Motivation and Broad Generalizations Innovation Long-run growth rates Aggregate growth rates (usually Shumpeterian) Model of frontier agents Silicon valley, pharma v.s. Technology Diffusion Medium-run (> 50 years?) Distribution shape from aggregate growth rates Model of less productive Coffee shops, agriculture If analyzed in isolation: No diffusion = aggregate growth rate = average innovation rate No innovation = aggregate growth can eventually stop (e.g. finite support with:?,?, and? with no Brownian Motion)

4 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Overview 2 / 31 Quick Motivation and Broad Generalizations Innovation Long-run growth rates Aggregate growth rates (usually Shumpeterian) Model of frontier agents Silicon valley, pharma v.s. Technology Diffusion Medium-run (> 50 years?) Distribution shape from aggregate growth rates Model of less productive Coffee shops, agriculture Innovation AND Technology Diffusion Are these models entirely orthogonal? How do they interact? Role of market power? Partial excludability due to frictions? Subsidize innovation vs. adoption? Growth and inequality consequences, and long vs. short-run effects?

5 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Overview 3 / 31 Broad Question How do technology adoption and innovation jointly determine (1) the aggregate growth rate, (2) the shape of the productivity distribution, and (3) the technology frontier?

6 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Overview 4 / 31 Agenda: Build Full Model to Investigate Forces Style: add minimal elements to technology diffusion mechanism: 1 Technology diffusion/adoption process, orthogonal to innovation 2 Exogenous stochastic innovation process to maintain finite frontier 3 Add non-orthogonal leap-frogging to the frontier jumps to maintain a bounded frontier (in relative terms ) 4 Endogenize innovation intensity. Technology frontier grows with endogenous choice of the highest productivity agent Simplest? Paper has discussion of innovation processes that don t work. (1,2,3) are semi-endogenous growth with endogenous distribution Paper and Tech Appendix: GBM, Monopolistic Competition, and Free Entry

7 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Overview 5 / 31 Perla and Tonetti (2014) in Discrete Time V t (z) = { max z + βv t+1 (z), β {produce,adopt} 0 } V t+1 (z )dφ t (z ) Adopters in period t Φ t (Z) M t M t+1 Z Adopters in period t + 1 Φ t+1 (Z) M t M t+1 M t+2 Z

8 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Exogenous Innovation 6 / 31 Model Summary with Exogenous, Stochastic Innovation Notation: Partial Derivative Operator: t t, or Univariate F (z) Idiosyncratic firm innovation state i {l, h} df (z) dz Idiosyncratic firm productivity Z with CDF Φ i (t, Z), PDF Z Φ i (t, Z) Unconditional distribution: Φ(t, Z) i Φ i(t, Z) support {Φ i (t, )} = [M(t), B(t)). Technology frontier = B(t) Orthogonal Sources of Firm Dynamics: 1 Innovation is a stochastic process for Z (conditional on i) 2 Technology diffusion is an immediate draw of a new productivity Incumbent can draw a productivity from the existing distribution in the economy, which will evolve endogenously Assume draw from the distorted unconditional CDF: Φ(t, Z) κ for κ > 0 Proportional adoption cost: ζ > 0

9 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Markov Chain 7 / 31 Stochastic Innovation with a Finite Frontier Need stochastic innovation process, but GBM has infinite frontier Necessary: Without stochastic innovation (or experimentation ), the distribution compresses and technology diffusion eventually stops Assume a Markov chain of innovation states, i l: firm is stagnant, waiting to be innovative or adopt technology h: firm innovative. Z grows at rate γ > 0 (exogenous for now) l h jump intensity = λ l. h l jump intensity = λ h Technology diffusion Assume start in l state (for simplicity) If adopting a technology, immediately gets a Z draw (i.e., infinitesimal time spent innovating prior to adopting new technology) Random search is easy: others have decentralized teaching, congestions Partial excludability from diffusion information friction. Positive incentives to innovate (see monopolistic competition/free entry)

10 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Markov Chain 8 / 31 Optimal Stopping and Endogneous Minimum of Support Assume a decision rule where: Z < M(t) means adopt Optimal stopping problem for endogenous threshold M(t) Continuation value of firm V i (t, Z) Value matching condition at M(t), the value of technology diffusion Can prove l and h firms choose same M(t) Define S i (t) as the endogenous flow of adopters crossing M(t) Paths of Z are continuous, so those hitting M(t) boundary immediately adopt if they draw Z > M(t). If they didn t draw above M(t), they would draw again immediately Continuous time: certain draw = min support {Φ(t, )} = M(t) (Could relax for a non-immediate arrival rate. Same qualitative results)

11 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Markov Chain 9 / 31 Firm s Problem Bellman Equations: rv l (t, Z) = Z Flow Profits +λ l (V h (t, Z) V l (t, Z)) Jump to h + t V l (t, Z) Capital Gains rv h (t, Z) = Z + γz Z V h (t, Z) +λ h (V l (t, Z) V h (t, Z)) + t V h (t, Z) Exogenous Innovation Value Matching + Smooth Pasting: V i (t, M(t)) = Value at Threshold B(t) M(t) Z V i (t, M(t)) = 0 Cost Z V l (t, Z )dφ(t, Z ) κ } {{ } Gross Adoption Value ζm(t) Adoption Cost

12 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Markov Chain 10 / 31 Evolution of the Distribution Kolmogorov Forward Equations: t Φ l (t, Z) = λ l Φ l (t, Z) + λ h Φ h (t, Z) + (S l (t) + S h (t)) Φ(t, Z) κ S l (t) Net Flow from Jumps Flow Adopters Draw Z Adopt t Φ h (t, Z) = γz Z Φ h (t, Z) λ h Φ h (t, Z) + λ l Φ l (t, Z) S h (t) Innovation 0 = Φ i (t, M(t)) 1 = Φ l (t, ) + Φ h (t, ) Evolution of the Technology Frontier and Adoption Threshold: t B(t) =γ B(t) t M(t) g(t) M(t)

13 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Markov Chain 11 / 31 Normalization and Stationarity z log(z/m(t)) z(t) = log(b(t)/m(t)) log(m(t)/m(t)) = 0 F i (t, z) Φ i (t, Z) v i (t, z) V i (t, Z)/M(t) (Relative Frontier) (Adoption Threshold) Z Φ(t, Z) Unnormalized PDF z F (t,z) Normalized PDF γ M(t) M (t) B(t) Z 0 M (t) M(t) γ z(t) z

14 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Markov Chain 12 / 31 Tension between Stochastic Innovation and Adoption z F (t, z) Stochastic innovation spreads 0 Adoption compresses z(t) z Need possibility of relative (not necessarily absolute) bad luck Adoption value > 0 + increasing dispersion = technology diffusion t z(t) = γ g(t), from h firms infinitesimally close to the frontier

15 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Markov Chain 13 / 31 Infinite vs. Unbounded vs. Bounded Frontier 1 Infinite Support: If B(0) = z(t) = for all t g > γ, both l and h agents fall back. Hysteresis, continuum of stationary equilibrium {α, g}. See paper. 2 Finite, Unbounded Support: If B(0) <, and If z(t) < but lim t z(t) = g = γ, unique solution infinite support! With innovation, infinite support approximation is not innocuous 3 Finite, Bounded Support: If B(0) <, and If lim t z(t) < Can compression from adoption balance the spread from stochastic innovation?

16 A BGP with finite support must have g = γ. Relative to the frontier, h agents don t move and l agents fall back at rate g. The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Markov Chain 14 / 31 Normalized, Stationary Value Functions v h (z) v i (z) λ h v l (z) v s λ l 0 z z

17 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Markov Chain 15 / 31 Stationary, Normalized Firm s Problem with Finite Support (r g)v l (z) = e z gv l }{{ (z) } + λ l (v h (z) v l (z)) Fall Backwards Jump to h (r g)v h (z) = e z + λ h (v l (z) v h (z)) v l (0) = v h (0) = v l (0) = 0 Jump to l z v l (z)df (z) κ ζ = 1/(r g) 0 } {{ } Start l type M(t) same: use smooth pasting and value matching in bellman Note absence of drift and smooth pasting condition for h types Robust to adoption technology variations (e.g. could start h type)

18 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Markov Chain 16 / 31 KFE for the Stationary Distribution with Finite Support 0 = gf l (z) Drift for l + λ h F h (z) λ l F l (z) Net Flow 0 = λ l F l (z) λ h F h (z) 0 = F l (0) = F h (0) 1 = F h ( z) + F l ( z) S = gf l (0) Only l cross + SF (z) κ S Draw < z Can show z for all κ. No bounded equilibrium exist. Define ˆλ λ l λ h, λ λ l r γ+λ h + 1. Changes with model variations

19 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Markov Chain 17 / 31 Equilibrium for κ = 1 F l (z) = 1 1+ˆλ e αz F h (z) λ v l (z) = γ + (r γ) λ ez Production in Perpetuity + 1 (r γ)(ν + 1) e νz Option Value of Diffusion Where α is the tail index of the asymptotic power law α (1 + ˆλ)F l (0) And ν is the rate the diffusion option value decreases ν = (r γ) λ γ And an implicit equation for F l (0). Unique Stationary Equilibrium

20 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Markov Chain 18 / 31 Example Unbounded Equilibrium v l (z) v h (z) F l (z) F h (z) vi (z) F i (z) z z While finite support, empirically F (z) is a power-law

21 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Bounded Support 19 / 31 Stochastic Innovation with a Bounded Frontier Fundamental issue: Positive measure of firms have been lucky forever Probability of adopting their technologies drops to 0 Alternatively: Assume there is some sort of chance for leap-frogging (analogy is non-multiplicative version of quality ladders ) Either innovation or adoption could have a probability of jump 1 Innovation attempts gain a major insight and jump to the frontier 2 Adopters firm gaining a huge spillover and jump to the frontier Assume arrival rate η > 0 for jumps to z for operating firms For simplicity, assume disruptive jump sets firm to l F l (z) and F h (z) could be discontinuous (but won t be in this setup) (Paper solves for endogenous jump probability of adopters as well)

22 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Bounded Support 20 / 31 Bounded Frontier Due to Leap-Frogging v h (z) v i (z) η v l (z) v s λ h λl η 0 z z

23 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Bounded Support 21 / 31 Bellman Equation and KFE Add jumps to z < to Bellman equations, (r g)v l (z) = e z gv l (z) + λ l(v h (z) v l (z))) + η(v l ( z) v l (z)) To Frontier (r g)v h (z) = e z + λ h (v l (z) v h (z)) + η(v l ( z) v h (z)) To Frontier KFE has insertion at z with a Heaviside: H (z z) 0 = gf l (z) + λ hf h (z) λ l F l (z) ηf l (z) Jump Out 0 = λ l F l (z) λ h F h (z) ηf h (z) Jump Out + ηh (z z) Jump to l + SF (z) κ S Same Adoption

24 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Bounded Support 22 / 31 Equilibrium for κ = 1 F l F l (z) = (0) (F l (0) η/γ)(1 + ˆλ) (1 e αz ) F h (z) ( λ v l (z) = e z + 1 γ(1 + ν) ν e νz + η (e z + 1ν )) e ν z r γ Where α is the tail index of the asymptotic power law α (1 + ˆλ)(F l (0) η/γ) And ν the rate of diffusion option value decrease And z < where ν r γ + η λ γ z = log(γf l (0)/η) α And an implicit equation for F l (0). Unique Equilibrium

25 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Bounded Support 23 / 31 Example Bounded Equilibrium v l (z) v h (z) F l F h (z) (z) B B Empirically a truncated power-law distribution

26 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Bounded Support 24 / 31 Example Bounded Comparative Statics η z α z α γ z α z α ζ λ h

27 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Endogenous Innovation 25 / 31 Endogenous Innovation Let innovating firms choose γ(t, Z) 0. Pay a proportional convex cost 1 χ γ2 Z The h agents cross the diffusion threshold if γ(0) < g Add the endogenous γ choice to the HJBE and make stationary (r g)v h (z) = max γ 0 {ez 1 χ ez γ 2 (g γ)v }{{ h(z) } R&D cost Control Drift + λ h (v l (z) v h (z)) + η(v l ( z) v h (z))} The KFE depends on the endogenous γ(z) for the h types, 0 = (g γ(z))f h(z) +λ l F l (z) λ h F h (z) ηf h (z) S h Controlled Drift

28 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Endogenous Innovation 26 / 31 Endogeneity and γ Can show that the optimal choice is, γ(z) = χ 2 e z v h(z) 0 z z γ(0) = 0 and g γ( z) Furthermore, (with λ from parameters) the maximum equilibrium g is, [ g max = λ(r + η) 1 1 ] χ λ(r + η) 2 The value of adoption now includes jumps to the frontier, v s 1 + ηv l( z) r g + η This creates feedback between endogenous z and v s

29 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Endogenous Innovation 27 / 31 Uniqueness of Stationary Equilibrium If η = 0 or η 0, then z is unbounded, and equilibrium is unique Option value of diffusion is asymptotically 0 [ g lim g max (η) = λr 1 1 χ ] η 0 λr 2 If η > 0, then z is bounded, and there is hysteresis Continuum of g g max with accompanying z and F i (z) If z <, then option value of diffusion is always positive. Different shapes and z provide different adoption incentives, which provide different γ choices, which can fulfill the shape Define α(z) F (z)/(1 F (z)) as a local tail index i.e., negative of local slope in a log-log plot of productivity distribution Constant and equal to the tail index for a Pareto distribution

30 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Endogenous Innovation 28 / 31 Example Unbounded Endogenous γ(z) with η = γ(z) v l (z) v h (z) z F l (z) F h (z) α(z) z z z

31 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Endogenous Innovation 29 / 31 Example Bounded Endogenous γ(z) v l (z) v h (z) 3 2 F l (z) F h (z) γ(z) z z z z α(z)

32 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Endogenous Innovation 30 / 31 Endogenous Jumps and Directed Technology Adoption Adopting firms choose θ [0, 1) and κ > 0 (i.e., directed adoption) Proportional quadratic costs are 1 ς θ2 and 1 ϑ κ2 Value of technology adoption (i.e. value matching condition) v i (0) = { z } max (1 θ) v l (z)df (z) κ + θv l ( z) ζ 1 0 θ<1, κ>0 ς θ2 1 ϑ κ2 0 Can show optimal choices at adoption time are, θ = 1 1 ς(v l ( z) v l (0) ζ) κ = ϑ(1 θ) 2 z 0 v l (z) log(f (z))f (z)κ dz

33 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier Conclusion 31 / 31 Conclusion To model innovation and technology diffusion with a frontier need: Stochastic productivity + Markov-chain If a bounded frontier is empirically necessary need jumps to the frontier Asymptotically unbounded infinite support Feedback between option value of diffusion and technology frontier the key to innovation and diffusion interaction Growth rate of frontier is that of the frontier agent TODO: policy analysis

34 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier 1 / 7 Geometric Brownian Motion Back Start with a standard baseline: exogenous GBM One i type, idiosyncratic Z Will use to understand role of stochastics, decompose growth rates Exogenous GBM for innovation: dz = (γ + σ 2 /2)Zdt + σzdw W is standard Brownian motion Mean growth rate of Z is γ If σ > 0, then B(t) = due to Brownian shocks Solve as an optimal stopping problem for endogenous threshold M(t) Continuation value of firm V (t, Z) Value matching condition at M(t), the value of technology diffusion Smooth pasting conditions necessary at M(t)

35 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier 2 / 7 Firm s Problem Bellman Equation + Value Matching + Smooth Pasting rv (t, Z) = V (t, M(t)) = Value at Threshold Z + (γ + σ 2 /2)Z Z V (t, Z) + σ2 2 Z 2 ZZ V (t, Z) Flow Profits Exogenous Innovation + t V (t, Z) Capital Gains Z V (t, M(t)) = 0 Cost Z V (t, Z )dφ(t, Z ) κ } {{ } Gross Adoption Value ζm(t) Adoption Cost

36 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier 3 / 7 Evolution of the Distribution The Kolmogorov Forward Equation (for the CDF Φ(t, Z)) t Φ(t, Z) = (γ + σ 2 /2) Z ZΦ(t, Z) + σ2 2 ZZ Z 2 Φ(t, Z) Deterministic Drift Brownian Motion + S(t)Φ(t, Z) κ S(t), for M(t) Z B(t) Firm draws - Adopters Φ(t, M(t)) = 0 Φ(t, ) = 1 S(t) the endogenous flow of adopters crossing M(t) If σ > 0, frontier is infinite immediately If σ = 0 and B(0) <, frontier grows at rate γ

37 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier 4 / 7 Stationary Equilibrium for GBM (if κ = 1) The continuum of equilibria are indexed by α > 1, where g = γ + 1 (α 1)ζ (r γ) (α 1) 2 ζ } {{ } Catch-up Diffusion ( ( ) ) α α(α 1) r γ σ2 2 ζ ( ( ) ) (α 1) (α 1) r γ σ2 2 ζ 1 Stochastic Diffusion + σ2 2 Where σ also affects expected time to execute diffusion option F (z) = 1 e αz v(z) = 1 r γ ez + 1 ν(r γ) e νz ν = g γ σ 2 + (g γ σ 2 ) 2 + r g σ 2 /2 > 0

38 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier 5 / 7 Dynamics with Deterministic Innovation Proposition (Dynamic Solution with Deterministic Innovation) Define g(m) = M (M)/M, the growth rate of M at state M. For an arbitrary initial condition Φ(0, Z), κ = 1, and r > γ 1 M M g(m) = γ + ZdΦ M(Z) (1 + ζ(r γ)) ζmφ M (Z) (1) where Φ M (Z) Φ(0,Z) Φ(0,M) 1 Φ(0,M), is a truncation of the initial condition. Note: given M(0) can use g(m) to construct M(t)

39 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier 6 / 7 Asymptotics with Deterministic Innovation Proposition Let g(t) M (t)/m(t) = output growth on a BGP 1 Φ(0, Z) not power law = lim t g(t) = γ. B(t) 2 Φ(0, Z) not power law = lim t M(t) > 1(i.e., doesn t collapse) 3 Φ(0, Z) power law with tail parameter α > 0 and κ = 1 1 ζ(r γ)(α 1) lim g(t) = γ + t Z lim V (t, Z) = t r γ Perpetuity ζα(α 1) (2) + Option Value of Diffusion {}}{ C 1 e (r γ)t M(t) Z r g g γ Z given t (3) BGP Summary: not power law = no catchup diffusion = require stochastic innovation for asymptotic technology diffusion

40 The Growth Dynamics of Innovation, Diffusion, and the Technology Frontier 7 / 7 References I