Transitional Dynamics in an Endogenous Growth Model with Physical Capital, Human Capital and R&D (with Spillovers)

Transcription

1 Transitional Dynamics in an Endogenous Growth Model with Physical Capital, Human Capital and R&D (with Spillovers) Tiago Neves Sequeira Abstract The dynamics of the most prominent growth models are well understood. There is a family of models with Physical, Human capital accumulation and R&D for which convergence properties have been recently discussed (Funke and Strulik, 2000; Arnold, 2000, Gómez, 2005). However, spillovers in R&D have been ignored in this context. We introduce spillovers in this model and derive its steady-state and stability properties. This new feature in the model implies a fourthorder differential equation reduced system and provides one unique Balanced Growth Path along with a two dimensional stable manifold. Moreover, transition is oscillatory towards the steady-state for plausible values of parameters. JEL Classification: C62, O11, O41. Key-Words: convergence; stability; spillovers; R&D. 1 Introduction Arnold (1998, 2000a) introduced a model with Physical Capital, Human Capital and R&D and studied its convergence properties. Funke and Strulik (2000) integrated three models as different stages of economic development and presented a model with Physical Capital, Human Capital and Departamento de Gestão e Economia, Universidade da Beira Interior and INOVA, Faculdade de Economia, Universidade Nova de Lisboa. ( sequeira@ubi.pt). I thank Ana Balcão Reis for useful comments and financial support from POCTI/FCT - Fundação para a Ciência e Tecnologia. The usual disclaimer applies. 1

2 R&D as a developed country stage. Gómez (2005) showed that the convergence features of the model change dramatically when accounting for the predetermination of r and he has fully characterize the models convergence properties. In particular, the model showed oscillatory dynamics towards the steady-state and it converges through a two-dimensional stable manifold, describing a unique Balanced Growth Path. Before these contributions, human capital based growth models and R&D-based growth models were studied separately. Benhabib and Perli (1994) showed that a endogenous growth model with externalities in human capital accumulation yield indeterminacy of equilibria. Ladrón-de-Guevara et al. (1997) studied the equilibrium dynamics of two extensions of the Uzawa-Lucas framework, in which they discovered multiple equilibrium in a model with leisure, but without externalities. Arnold (2000b,c) demonstrates that the steady-state of the Romer s (1990) model of R&D is globally saddle-point stable. Eicher and Turnovsky (2001) showed that a two-sector R&D-based non-scale growth model are represented by a two-dimensional stable saddle-path. This model showed oscillatory transition. Arnold (2006) studied the stability properties of the Jones (1995) model and concluded for an unique Balanced Growth Path on a two-dimensional stable manifold. These two last articles showed that the system of differential equations to be analyzed in non-scale R&D growth models are of order four. We add spillovers in R&D to the Arnold s (2000a) model and derive the model equilibrium and stability properties. It is worth noting that the existence and high magnitude of spillovers have been empirically reached by a recent and vast literature (e.g. Grilliches, 1992; Porter and Stern, 2000 and Engelbrecht, 1997; Barrio-Castro, 2002 in specifications that included human capital) but models that jointly consider human capital accumulation and R&D have not taken them into account until now. The consideration of spillovers in the model provides a determinate Balanced Growth Path that converges to an unique feasible steady-state. In comparison with Gómez (2005), the consideration of spillovers increases the order of differential equations system to four (as in the non-scale R&D growth models) and in terms of results, it maintains the oscillatory convergence for typical calibrations. 1 In the present article, the predetermined variables uniquely determine the starting point in the transition path. In comparison with the Jones (1995) model, 2 the introduction of accumulation 1 This also allows for a lower and more reasonable value for the markup to fulfill the stability conditions, contrary to what happened in Gómez (2005). 2 The stability properties have been studied in Arnold (2006). 2

3 of human capital does not change the main stability properties. In section 2, we present the model. In Section 3, we present the Balanced Growth Equilibrium and we present the stability analysis of the steady-state. In section 4, we calculate a growth path for typical calibration values and we discuss the results. In section 5, we conclude. 2 The Model This section recapitulates the Arnold s (1998, 2000a) - Funke and Strulik s (2000) - model, assuming a different function for the production of new ideas, which accounts for the existence of spillovers in R&D. 2.1 Setup of the Model Consider a closed economy inhabited by a constant population, normalized to one, of identical infinitely-lived households that maximize the intertemporal utility function: 0 C 1 θ 1 θ e ρt dt, ρ > 0, θ > 0, (1) where C denotes consumption, subject to the budget constraint and the knowledge accumulation technology. Human Capital, H, can be devoted to production (H Y ), education (H H ) and R&D (H n ), respectively: and is calculated according to: H = H Y + H H + H n, (2) H = ξh H. (3) The accumulation of human capital is a non-market activity. R&D technology is equal to that in Jones (1995), with constant returns to scale in human capital, given by n = ɛh n n φ, (4) where 0 < φ < 1 is the parameter that governs spillovers and n is the number of available varieties. This process is possible because of monopolistic competition in differentiated goods, as we will describe later. The budget constraint faced by the household is 3

4 W = rw + w(h H H ) C. (5) where r is the return per unit of aggregate wealth, W, and w the wage per unit of employed human capital, H H H. Let g z = z denote the growth z rate of any variable z. The first order conditions for maximization of (1), using (5) and (3) as constraints give: g C = (r ρ)/θ, (6) g w = r ξ. (7) A single homogeneous final good Y is produced with Cobb-Douglas technology Y = A 1 K β D η H 1 β η Y, A 1 > 0, β > 0, η > 0, β + η < 1, (8) where K is physical capital, H Y is human capital allocated to final good production and D is an index of differentiated goods, [ n 1/α D = x α i di], 0 < α < 1, (9) 0 where x i is the amount used for each one of the n intermediate goods. The market or the final good is perfectly competitive and its price is normalized to one and α governs the substitutivity between varieties. Profit maximization gives inverse factor demands: r = βy K, (10) and w = P D = ηy D, (11) (1 β η)y H Y, (12) where P D represents the price index for intermediates. Each firm in the differentiated-goods sector owns a patent for selling its variety x i. Let v t denote the expected value of innovation, defined by 4

5 v t = t e [R(τ) R(t)] π(τ)dτ, where R(t) = t 0 r(τ)dτ. (13) Taking into account the cost of innovation as implied by (4), free entry conditions in R&D are defined as follows: w/ɛ > vn φ if ṅ = 0 (H n = 0) or (14) w/ɛ = vn φ if ṅ > 0 (H n > 0). (15) where w is the wage paid to human capital. Finally, no-arbitrage requires that the valorization of the patent plus profits is equal to investing resources in the riskless asset: v + π = rv v = r π/v. (16) v Producers act under monopolistic competition and maximize operating profits π i = (P xi 1)x i. (17) The variable P xi denotes the price of an intermediate and 1 is the unit cost of Y. From profit maximization in the intermediate-goods sector, each firm charges a price P xi = 1/α. (18) With identical technologies and symmetric demand, the quantity supplied is the same for all goods, x i = x. Hence, equation (9) simplifies to D = n 1/α x. (19) From P D D = pxn together with equations (18) and (19) we obtain the total quantity of intermediates employed as X = xn = αηy. (20) After insertion of equations (18) and (20) into (17), profits can be rewritten as a function of aggregate output and the number of existing varieties: π = (1 α)ηy/n. (21) 5

6 Before we proceed with the analysis we compute some equations that will be useful. Insertion of equation (20) in the resource constraint K = Y n x 0 idi C simplifies it to K = (1 αη)y C (22) and insertion of (19) and (20) in the production function (8) gives: Y 1 η = A 1 (αη) η K β n η 1 α α (u1 H) 1 β η, (23) where u 1 = H Y H is the proportion of human capital employed in the final good production. Similarly, we also denote u 2 = Hn as the share of human H capital allocated to research and u 3 = H H H as the share of human capital allocated to human capital accumulation. After time-differentiation the previous function we obtain the output growth rate (1 η)g Y = βg K + [ 1 α α ]ηg n + (1 β η)(g u1 + g H ). (24) Log-differentiation of equations (10) and (12) provides g r = g Y g K, (25) g w = g Y (g u1 + g H ). (26) To avoid excessive length we will concentrate in the innovative economy description. In fact, as the change we introduce is in the R&D production process, the previous stage would remain equal to the developing economy stage in Gómez (2005) and in Funke and Strulik (2000). 2.2 The Dynamic of the Economy The fully industrialized economy is characterized by the presence of both human capital accumulation ( H > 0) and R&D ( n > 0). The following system of differential equations describes the dynamics of the fully industrialized economy: g χ = ( 1 θ 1 αη ) r + χ ρ β θ, (27) g r = 1 β η (r ξ) + η β β 6 1 α α g n, (28)

7 where χ = C/K. Inserting the growth rate version of (15), (7), (12) and (21) into (16), we reach u 1 = (1 β η)(ξ + φg n). (29) ɛ(1 α)ηψ Using this last equation, eq. (3) and the definition ψ = H/n 1 φ (noting that u 1 + u 2 + u 3 = 1), we reach g ψ = ξ ( [ ] ) (1 β η)(ξ + φgn ) g n (1 φ)g n. (30) (1 α)η ɛψ We note that g n is no longer given as a function of χ and r, that could be directly substituted in previous equations, as in previous contributions (e.g. Gómez, 2005:5). This implies, from (29), that g u1 = (1 φ)g n g H + φ g n. (ξ+φg n) Then, we note that g u1 = g Y g w g H is verified by (26). Then, using (25), g K from (22) and (7), we reach the equation that describes the evolution of g n : g n = ξ + φg [ n g r + 1 αη ] r χ (r ξ) (1 φ)g n. (31) φ β The system (27), (28), (30) and (31) describe the evolution of the economy. We can further simplify eq. (31), substituting g r from (28): g n = ξ + φg [ n (1 α)η r χ + 1 η ( ) ] η φ β β ξ + 1 α (1 φ) g n. (32) β α 3 Balanced Growth Equilibrium In this section, we present the equations that describe the steady-state of the model. Theorem 1 Let ξ > ρ and θ > 1. There is one positive steady-state of the model given by (r, χ, ψ, g n). 7

8 ( (1 β η) η ( (1 β η) ) α θ (1 φ) + 1 ξ ρ r 1 α = ), (33) α θ (1 φ) + (θ 1) η 1 α ( 1 αη χ = 1 ) r + ρ β θ θ, (34) ψ = ξ [g n(1 α)η + (1 β η)(ξ + φgn)], (35) (1 α)ηɛ(ξ (1 φ)gn) gn = r (1 θ) + θξ ρ. (36) θ(1 φ) Proof. The shares of human capital to different sectors must be constant for an interior steady-state solution. In particular the constancy of u 1 and u 2 together guarantee that gn is constant. Then, the fact that the share in human capital accumulation u 3 is also constant and the constancy of gn implies by (3) gh is constant. With u 2 and gn constant, ψ must be constant, by (4). Thus gh = (1 φ)g n. Constancy of gc implies by (6) that r is constant. Finally, as r is constant, so is (Y/K), by (10). This implies that gy = g K. This equality, equation (24) and the constancy of gn, gh and u 1 imply that gy and g K are constant. Thus χ = (C/K) is constant (to see this divide (22) by K). We now derive necessary and sufficient conditions for positivity. For r 1 > 0 ξ > ρ, where A 2 A 3 = 1 β η α η 1 α = ( θ 1 αη 1 A β 2) ( 1 αη β 1) (1 φ). For χ > θ(1+a 2 ) 0, we have (A 2 + 1)θξ > A 3 ρ, where. 3 For ψ > 0 A 4 ξ > A 5 ρ (always verified), with A 4 = (A 2+1)θξ ; A θa 2 +(θ 1) 5 = 1 1. Finally (θ 1) θa+(θ 1) g n > 0 ξ > ρ. This condition together with θ 1 imply r > 0, ψ > 0 and χ > 0. Positivity of ψ is directly implied by the transversality condition on H. Transversality condition on human capital may be written as lim t e ρt λ 2 (t)h(t) = 0 (37) ( ) λ (λ 2 is the co-state of H), which converts on lim t ρ + 2 λ 2 + g H < 0. As λ 2 λ 2 = ρ ξ and g H = (1 φ)g n - given by the constancy of (35) and (36), the transversality condition is equivalent to ξ (1 φ)g n > 0. 3 By initial assumptions on parameters 1 αη > β. 8

9 3.1 Stability We will now analyze the dynamics of the model in the neighborhood of the steady-state. As we can assume, as usually, that the stocks of physical, human and the number of varieties move sluggishly, so that K(0), H(0) and n(0) are given by their historical values. Thus ψ is predetermined. The analysis of the linearized system around the steady-state will establish that, for most reasonable values, the system has two eigenvalues have negative real parts. We also show that initial conditions K(0), H(0) and n(0) are sufficient to determine the initial point in the two-dimensional stable manifold, thus the Balanced Growth Path is uniquely determined. Linearizing the system (27), (28), (30) and (31) around its steady-state (r, χ, ψ, gn) gives the following fourth-order system: r 1 β η r η 1 α 0 0 β β α r ( ) χ ψ = 1 1 αη χ χ 0 0 θ β 0 0 ξ (1 φ)gn B 1 (1 φ)ψ ξ+φgn g (1 α)η n ξ+φg n 0 B φ β φ 2 gn ξ B 3 φgn (1 β η) 1 where B 1 = ξφ (1 α)η ɛ + ξ ( ) η ɛ ; B 1 α 2 = (1 φ) ; β α ( (1 α)η B 3 = r χ + 1 η β β ξ r r χ χ ψ ψ g n g n, ), (38) or x = J(x x ), where J is the Jacobian in (38). To demonstrate the conditions under which the system is completely stable and unstable we state the following theorem. Theorem 2 In the conditions of Theorem 1 and of Lemma 2, there exists a unique steady-state for which there converges a unique path. The proof makes use of three lemmas. The first two demonstrate that, under certain conditions, there are two stable roots, e.g., the stable manifold is two dimensional. The third demonstrates that the balanced growth path is uniquely determined by the state variables in the original model. We also state the conditions according to which the system will be unstable. 4 4 Eicher and Turnovsky (2001:95) also could not rule out instability, with the case of four positive roots. 9

10 Lemma 1 There is an even number of stable roots, zero (completely unstable) or two (saddle-path stable). 5 Proof. This implies that the determinant of (38) is positive. This is, [ ] [ B 2 gn ξ 1 β η B φgn β θ + 1 η β ] η 1 α ξ + φgn > 0. β β α φ (39) We provide a proof. Using the fact that, by (32), in steady-state B 3 = B 2 gn, and simplifying terms the expression (39) turns out to be equal to ( (1 β η) θ η α (1 φ) 1 α ) + (θ 1) > 0, (40) which is equivalent to the denominator of the r (see eq. (33)). Thus, the system has an even number of stable roots. Lemma 2 There are not zero stable roots. Thus, by Lemma 1, there are two negative roots. Proof. Let δ i (i = 0,...3) denote the coefficient of the roots q in the characteristic equation: f(q) = δ 0 + δ 1 q + δ 2 q 2 + δ 3 q 3 + q 4. Developing the determinant in the characteristic equation gives, δ 0 = Det(J) and δ 3 = T r(j). We state coefficients δ 2 and δ 3 depending on parameters in the model δ 1 = B 4 (ξ (1 φ)gn) δ 0 /(ξ (1 φ)gn), (41) [( ) δ 2 = B 4 + B 2 gn ξ B 3 + χ 1 β η ] r (ξ (1 φ)g φgn β n), (42) where B 4 = ξ + [ ( ) φg n (1 α) η 1 β η (1 α)η 1 β η ] (1 φ) r. φ α β β β β Letting q 1,...q 4 denoting the characteristic root, the characteristic equation can also be written as f(q) = 4 i=1 (q i q). This yields Viéta s formulae: δ 0 = q 1 q 2 q 3 q 4 ; δ 1 = (q 1 q 2 q 3 + q 1 q 2 q 4 + q 1 q 3 q 4 + q 2 q 3 q 4 ); δ 2 = q 1 q 2 + q 1 q 3 + q 1 q 4 + q 2 q 3 + q 2 q 4 + q 3 q 4 ; δ 3 = (q 1 + q 2 + q 3 + q 4 ). Suppose that there are zero stable roots. Thus with all roots with positive real 5 The case with four negative roots is excluded as there is a root (e 1 = ξ + φg n g n), that by the transversality condition on H, is always positive. 10

11 part, δ 1 < 0 and δ 2 > 0. If one of these conditions fail, there must be at least one negative root. However, by Lemma 1, the unique alternative possible root solution is one with two stable roots. After re-arranging expression in (42), grouping terms with negative and indeterminate sign and noting again that B 3 = B 2 g n in the steady-state, we reach a sufficient condition to obtain δ 2 < 0 which is the condition under which the term with indeterminate sign is negative: (ξ (1 φ)g n) B 2 ξ + φg n φ This expression is verified for 1+ φ ξ g n > α β 1 α η < 0. (43). Thus under this condition the α β 1 α η system has exactly two stable roots. Then 1 + φ ξ g n < is one necessary condition for δ 2 > 0 which is a necessary condition to obtain instability. With usual calibration parameters, we reach two complex conjugate stable eigenvalues and two unstable roots, as the conditions stated in Lemmas 1 and 2 are easily verified. Lemma 3 The state variables K(0), H(0) and n(0) uniquely determine the starting point in the stable manifold. Proof. The initial values X(0) satisfy: X(0) X = 2 Ω i b i, (44) where b i are the eigenvectors corresponding to the two negative (real parts) eigenvalues, and Ω i are constants that can be determined. (44) constitutes a system of 4 equations into 5 unknowns (C/K(0), r(0), g n (0), Ω 1 (0), Ω 2 (0)). Equations (10), (23) and (29) may be re-written as: i=1 r(0) = A 1β η β η 1 α 1 η K(0) 1 η n(0) 1 η α ( (1 β η)(ξ+φg n (0)) ɛ(1 α)η K(0) n(0)) 1 β η 1 η (45) With this additional equation, we reach a system with 5 equations into 5 unknowns. This means that the stable variables select a specific starting point in the transition path. This theorem, establishes, that there is an unique transition path to the steady-state, under some reasonable conditions. This is a result similar to that in Arnold (2006) and Eicher and Turnovsky (1996), but in a model with human capital accumulation. 11

12 r 4 Adjustment Paths In this section, we present the calibration and compute the adjustment paths for typical values of parameters. To integrate the fourth-order differential equations system we use the method of backward integration described by Brunner and Strulik (2002). Most parameters for calibration are taken from Goméz (2005). The additional parameter φ was set equal to 0.4, using evidence from Barrio- Castro (2002). Then, the calibration assumes the following values: β = 0.36; η = 0.36; α = 0.4; ξ = 0.05; ρ = 0.023; θ = 2; δ = 0.1; A 1 = 1 and φ = 0.4. Eigenvalues are: ; ; i and i. Figures 1 to 3 show adjustment paths of the growth rate of output, the interest rate, the shares of human capital allocated to each of the sectors, the human capital and varieties growth rates and the human capital-varieties ratio g K u u These figures can be compared with Figure 4 in Gómez (2005). 12

13 u H x H/n g H 10 g n Figure 1: Transition Paths for Representative Variables Figures show oscillatory adjustment until the steady-state. Intuitively, the presence of spillovers increase the investment in R&D and the growth rates of per capita output, when comparing to Gómez (2005) results. Overall, there is an overshooting effect in the beginning of the transition path. However, the economy stabilizes within the first 200 years. As a sensivity analysis exercise we have considered different values for the substitutivity between varieties (which also governs the markup) and for the spillovers. While variations in the value of spillovers maintain the saddle-path property and the oscillatory pattern, extreme markups values makes the difference in what determinacy and monoticity are concerned. For very high substitutivity (low markup) - α 0.94, the balanced-growth path comes out to be determinate, but without the transition oscillatory pattern (two negative real roots). For very low substitutivity - α 0.2 (high markup), the steady-state turns out to be completely unstable (the four roots come out to be positive). 7 It is worth noting, however, that for reasonable values of markups (between 1.2 and 1.8) - contrary to what happened in Gómez (2005) - the oscillatory pattern through a two-dimensional stable manifold arises, being similar to the depicted in Figure 1. 7 Results are available from the author if requested. 13

14 5 Conclusion We introduce R&D spillovers into the growth model with physical capital, human capital an new varieties described by Arnold (1998, 2000a) and Funke and Strulik (2000). The dynamics of the model is characterized by the behavior of a system of four differential equations. The system can either be unstable or saddle-path stable. Within mild conditions it proves to be characterized by a two-dimensional stable manifold and converges through an uniquely determined balanced growth path to a steady-state. Thus, when compared to the model without spillovers, its presence increases the order of differential equations system to four, also providing a two-dimensional stable manifold. In this case, however, initial predetermined variables uniquely determine the starting point in the path to the steady-state. This model maintains the oscillatory convergence and allows for more reasonable markup values to fulfil stability conditions. When compared with a model without human capital (e.g. Jones, 1995 model), the stability properties are very similar. References [1] Arnold, L. (1998), Growth, welfare and trade in an integrated model of human-capital accumulation and research, Journal of Macroeconomics, 20(1), [2] Arnold, L. (2000a), Endogenous Growth with Physical capital, Human capital and Product variety: A comment, European Economic Review, 44, [3] Arnold, L. (2000b), Endogenous Technological Change: A note on stability, Economic Theory, 16, [4] Arnold, L. (2000c), Stability of the Market Equilibrium in Romer s Model of Endogenous Technological Change: A complete characterization, Journal of Macroeconomics, 22(1), [5] Arnold, L. (2006), The Jones R&D Growth Model: Comment on Stability, Review of Economic Dynamics, 9(1), [6] Barrio-Castro, T., López-Bazo, E. and Serrano-Domingo, G. (2002), New Evidence on international R&D spillovers, human capital and productivity in the OECD, Economics Letters, 77,

15 [7] Benhabib, J. and R. Perli (1994), Uniqueness and Indeterminacy: On the dynamics of Endogenous Growth, Journal of Economic Theory, 63, [8] Brunner, M. and H. Strulik (2002), Solution of Perfect Foresight Saddlepoint Problems: A simple method and aplications, Journal of Economics Dynamics and Control, 25(5): , May. [9] Eicher, T. and S. Turnovsky (2001), Transitional Dynamics in a Two- Sector non-scale Growth Model, Journal of Economic Dynamics and Control, 25, [10] Engelbrecht, H. (1997), International R&D spillovers, human capital and productibity in OECD countries: An Empirical investigation, European Economic Review, 41, [11] Funke, M. and H. Strulik (2000), On Endogenous growth with physical capital, human capital and product variety, European Economic Review, 44, [12] Gómez, M. (2005), Transitional Dynamics in an Endogenous growth model with physical capital, human capital and R&D, Studies in Nonlinear Dynamics & Econometrics, 9, 1, Article 5. [13] Griliches, Z. (1992), The search for R&D Spillovers, Scandinavian Journal of Economics, 94 (Supplement), [14] Jones, C. (1995), R&D-based models of endogenous growth, Journal of Political Economy, vol. 103, n.4, [15] Ladrón-de-Guevara, S. Ortigueira and M. Santos (1997), Equilibrium Dynamics in Two-Sector Models of Endogenous Growth, Journal of Economic Dynamics and Control, 21, [16] Porter, M. and S. Stern (2000), Measuring the Ideas Production Function: evidence from international patent output, NBER Working Paper