Growth without scale effects due to entropy

Transcription

1 Growth without scale effects due to entropy Tiago Sequeira* Pedro Gil** Oscar Afonso** *University of Beira Interior and **FEP - University of Porto sequeira@ubi.pt June 28, 2016

2 Our Contribution On the Asymptotic Properties of the Knowledge Function Ȧ = F (A, L) (1) In order to obtain endogenous growth, the derivative of the knowledge function (Ȧ) in order to stock of knowledge (A) must be constant and positive in the steady-state;

3 Our Contribution On the Asymptotic Properties of the Knowledge Function Ȧ = F (A, L) (1) In order to obtain endogenous growth, the derivative of the knowledge function (Ȧ) in order to stock of knowledge (A) must be constant and positive in the steady-state; if the knowledge function is linear in the stock of knowledge, there are no transitional dynamics and there are scale-effects;

4 Our Contribution On the Asymptotic Properties of the Knowledge Function Ȧ = F (A, L) (1) In order to obtain endogenous growth, the derivative of the knowledge function (Ȧ) in order to stock of knowledge (A) must be constant and positive in the steady-state; if the knowledge function is linear in the stock of knowledge, there are no transitional dynamics and there are scale-effects; if there are DRS in the stock of knowledge, there are transitional dynamics, no scale-effects but semi-endogenous growth;

5 Our Contribution On the Asymptotic Properties of the Knowledge Function Ȧ = F (A, L) (1) In order to obtain endogenous growth, the derivative of the knowledge function (Ȧ) in order to stock of knowledge (A) must be constant and positive in the steady-state; if the knowledge function is linear in the stock of knowledge, there are no transitional dynamics and there are scale-effects; if there are DRS in the stock of knowledge, there are transitional dynamics, no scale-effects but semi-endogenous growth; if there are difficulty effects, elimination of the scale-effects requires that difficulty effects grow at the same rate as population;

6 Our Contribution Our Contribution Ȧ = F (A, L) (2) We contribute to the literature including a mechanism that simultaneously allow for transitional dynamics and endogenous growth (a constant and positive derivative of the knowledge function (Ȧ) in order to stock of knowledge (A), as t +.);

7 Our Contribution Our Contribution Ȧ = F (A, L) (2) We contribute to the literature including a mechanism that simultaneously allow for transitional dynamics and endogenous growth (a constant and positive derivative of the knowledge function (Ȧ) in order to stock of knowledge (A), as t +.); That mechanism is based on the first principle of entropy.

8 Our Contribution Overview Our Contribution Motivation Complexity Effects A time-varying Complexity Effect due to Entropy Empirical Motivation and Calibration of the Complexity Function The Model Main Features Results Contrafactual Exercise Results with International Spillovers The very long-run Conclusions

9 Motivation Complexity Effects Complexity Effects and Scale Effects Market complexity effect in R&D activities as a mechanism for the elimination of the scale effects on growth (e.g., Dinopoulos and Thompson 1999, 2000; Barro and Sala-i-Martin 2004; Etro, 2008; Gil et al., 2013);

10 Motivation Complexity Effects Complexity Effects and Scale Effects Market complexity effect in R&D activities as a mechanism for the elimination of the scale effects on growth (e.g., Dinopoulos and Thompson 1999, 2000; Barro and Sala-i-Martin 2004; Etro, 2008; Gil et al., 2013); These complexity costs offset the positive effect of scale on the (expected) profits of the successful innovator. In this way, scale variables like the level of the population do not influence the BGP growth.

11 Motivation Complexity Effects Are there scale effects? Modern evidence against scale effects on growth (e.g., Jones, 1995; Dinopoulos and Thompson, 2000; Zachariadis, 2003; Laincz and Peretto, 2006; Ha and Howitt, 2007; Madsen, 2008; Sedgley and Elmslie, 2010; Ang and Madsen, 2015);

12 Motivation Complexity Effects Are there scale effects? Modern evidence against scale effects on growth (e.g., Jones, 1995; Dinopoulos and Thompson, 2000; Zachariadis, 2003; Laincz and Peretto, 2006; Ha and Howitt, 2007; Madsen, 2008; Sedgley and Elmslie, 2010; Ang and Madsen, 2015); there are compelling arguments for scale effects from historical evidence (e.g., Kremer, 1993).

13 Motivation A time-varying Complexity Effect due to Entropy A time-varying complexity effect due to entropy Entropy Although the concept of entropy originated in Thermodynamics commonly understood as a measure of molecular disorder within a macroscopic system and its statistical definition was developed in Statistical Mechanics, it has been adapted and extended by other fields of study, including Information Theory, Biology, and Economics.

14 Motivation A time-varying Complexity Effect due to Entropy A time-varying complexity effect due to entropy Entropy Although the concept of entropy originated in Thermodynamics commonly understood as a measure of molecular disorder within a macroscopic system and its statistical definition was developed in Statistical Mechanics, it has been adapted and extended by other fields of study, including Information Theory, Biology, and Economics. Tsallis Index of Entropy A widely used family of entropy indices based on the contributions by Tsallis, and Patil and Taillie, is usually written in the form S q = { 1 W i=1 pq i q 1, q 1 W i=1 p ilnp i, q = 1, (3)

15 Motivation A time-varying Complexity Effect due to Entropy Tsallis Index of Entropy with uniform distribution If the distribution is uniform, such that, e.g., all the messages in the message space or the biological varieties (species) in the variety space are equiprobable, p = 1/W, the entropy is maximized and S q becomes { 1 W 1 q q 1, q 1 S q = lnw, q = 1, (4)

16 Motivation A time-varying Complexity Effect due to Entropy A Complexity Effect due to Entropy Theoretical Foundation We introduce a state-dependent time-varying complexity effect due to entropy in growth theory.

17 Motivation A time-varying Complexity Effect due to Entropy A Complexity Effect due to Entropy Theoretical Foundation We introduce a state-dependent time-varying complexity effect due to entropy in growth theory. R&D Function (1 + ψ)(a t+1 A t ) = δ (A t ) L A t L χ(at) t, (5) χ(a t ) = max { 0, S q = { 1 A 1 q t } q 1, q 1 lna t, q = 1 (6)

18 Motivation A time-varying Complexity Effect due to Entropy Lemma With q > 1, then χ(a t ) converges to b q 1. Thus for b = q 1, χ(a t ) converges to 1 as A goes to infinity. With q 1 then χ(a t ) converges to +. Proof. Calculate lim A + χ(a t ) for q > 1 and q 1 in equation S q, respectively.

19 Motivation A time-varying Complexity Effect due to Entropy Χ(A) q 1 1 q> 1 A

20 Motivation Empirical Motivation and Calibration of the Complexity Function Empirical Motivation and Calibration of the Complexity Function Empirical Evaluation of the Complexity Effect Let ΔA t+1 = A t+1 A t in (5). By applying logs and solving for χ, we have the recursive equation: χ = lnδ ln(1 + ψ) + lna t + lnl A t ln(δa t+1 ) lnl t To obtain estimates for χ over time, we consider the calibrated values of δ and the time series data for A, L A and L.

21 Motivation Empirical Motivation and Calibration of the Complexity Function [σ = 0.196, q = 2.15 and b = 1.08] Figure: Comparison between empirical series for χ (blue series) and entropy series for χ(a) (red series)

22 Motivation Empirical Motivation and Calibration of the Complexity Function [σ = 0.64, q = 1.29 and b = 0.38] Figure: Comparison between empirical series for χ (blue series) and entropy series for χ(a) (red series)

23 The Model Main Features The Model: Main Features OLG model with young individuals maximizing intertemporal utility such that s t = β 1 β w t. (7)

24 The Model Main Features The Model: Main Features OLG model with young individuals maximizing intertemporal utility such that s t = β 1 β w t. (7) Population grows exogenously at the rate n;

25 The Model Main Features The Model: Main Features OLG model with young individuals maximizing intertemporal utility such that s t = β 1 β w t. (7) Population grows exogenously at the rate n; A continuum of size 1 of competitive firms produces a homogeneous output using a Cobb-Douglas technology and employing physical capital and labor;

26 The Model Main Features The Model: Main Features OLG model with young individuals maximizing intertemporal utility such that s t = β 1 β w t. (7) Population grows exogenously at the rate n; A continuum of size 1 of competitive firms produces a homogeneous output using a Cobb-Douglas technology and employing physical capital and labor; The composite capital good K is a CES aggregate of quantities, x i, of specialized capital goods;

27 The Model Main Features The Model: Main Features OLG model with young individuals maximizing intertemporal utility such that s t = β 1 β w t. (7) Population grows exogenously at the rate n; A continuum of size 1 of competitive firms produces a homogeneous output using a Cobb-Douglas technology and employing physical capital and labor; The composite capital good K is a CES aggregate of quantities, x i, of specialized capital goods; R&D sector with state-dependent time-variant complexity effect due to entropy.

28 The Model Main Features Theorem We derive the two dynamic equations that describe growth dynamics in this model: and Δk t = ā (A t) σ L α(1 χ(at)) t kt α 1 + n k t (8) ( (1 + ψ)δa t = δ (A t ) L 1 χ(at) t 1 ). (9) αδ

29 The Model Main Features Theorem We derive the two dynamic equations that describe growth dynamics in this model: and Δk t = ā (A t) σ L α(1 χ(at)) t kt α 1 + n k t (8) ( (1 + ψ)δa t = δ (A t ) L 1 χ(at) t 1 ). (9) αδ Theorem In the steady state with increasing population, it must be that χ(a t ) = 1; consequently: ΔA t A t = 1 Δkt 1+ψ (δ 1/α), k t = σ 1 1 α 1+ψ (δ 1/α) and there is a feasible steady state with endogenous growth if and only if δα > 1.

30 The Model Results Results Figure: Evolution of the main model series (blue line series) and comparison with data series (σ = 0.196, q = 2.15 and b = 1.08, k 0 = 0.073, L 0 = 0.8, A 0 = 0.5).

31 The Model Contrafactual Exercise Figure: Evolution of the main model series (blue line series) and comparison with data TFP series (σ = 0.196, imposed χ(a), k 0 = 0.073, L 0 = 0.8, A 0 = 0.5).

32 The Model Results with International Spillovers Domestic and International Spillovers Figure: Evolution of the main model series (blue line series) and comparison with data series (σ = 0.196, q = 2.15 and b = 1.08, φ = 0.864, μ = , k 0 = , L 0 = 0.8, A 0 = 0.5).

33 The Model The very long-run The Economy in the very long-run: the model Maintain the state dependent time-varying complexity effects derived from entropy and that eliminate scale-effects asymptotically.

34 The Model The very long-run The Economy in the very long-run: the model Maintain the state dependent time-varying complexity effects derived from entropy and that eliminate scale-effects asymptotically. Add a state-dependent time-varying spillover effect (due to networks) that is derived from the small world theory (Strulik, 2014).

35 The Model The very long-run The Economy in the very long-run: the model Maintain the state dependent time-varying complexity effects derived from entropy and that eliminate scale-effects asymptotically. Add a state-dependent time-varying spillover effect (due to networks) that is derived from the small world theory (Strulik, 2014). Allow a replication of a very long period of stagnation before the take-off which mimics the evolution of the world in the very-long run.

36 The Model The very long-run The Economy in the very long-run Figure: Evolution of the main model series (blue line series) and comparison with data series (σ = 0.169, q = 1 and b = 0.21, k 0 = 4.2, L 0 = 0.1, A R0 = 1, Ā = 1000, ω = ).

37 Conclusions Conclusions We introduce the concept of entropy into the endogenous growth theory to describe the complexity effect that tends to dilute the market-size effect in fully developed economies;

38 Conclusions Conclusions We introduce the concept of entropy into the endogenous growth theory to describe the complexity effect that tends to dilute the market-size effect in fully developed economies; The concept of entropy fits well into a complexity effect that is state-dependent and time-varying. We relate the complexity function to the number of varieties that firms in the industrialized world have to deal with;

39 Conclusions Conclusions We introduce the concept of entropy into the endogenous growth theory to describe the complexity effect that tends to dilute the market-size effect in fully developed economies; The concept of entropy fits well into a complexity effect that is state-dependent and time-varying. We relate the complexity function to the number of varieties that firms in the industrialized world have to deal with; We note that the state-dependent time-varying effect linked with the concept of entropy is essential to replicate the post-industrial Revolution evolution of the economy, both in levels and in growth rates;

40 Conclusions We also devise a model that intends to replicate the very long-run evolution of the economy from A.D. 1 to A.D. 2000;

41 Conclusions We also devise a model that intends to replicate the very long-run evolution of the economy from A.D. 1 to A.D. 2000; Whatever the scenario in the future, the state-dependent time-varying complexity effect seems to contribute to the explanation of the growth slowdown that happened in the current generation.

42 Conclusions The End