Dynamical System of a Multi-Capital Growth Model

Transcription

1 Applied Mathematical Sciences, Vol. 9, 2015, no. 83, HIKARI Ltd, Dynamical System of a Multi-Capital Growth Model Eva Brestovanská Department of Economics and Finance Faculty of Management, Comenius University Odbojárov str Bratislava, Slovakia Milan Medveď Department of Mathematical Analysis Faculty of Mathematics, Physics and Informatics, Comenius University Mlynská dolina, Bratislava, Slovakia Copyright c 2015 Eva Brestovanská and Milan Medveď. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we define a multi-capital model of the Solow type and derive a system of differential equations corresponding to a production function depending on several capitals, labor and technological progress. Variables of this system of differential equations are technological progress, labor and capitals not capitals per labor as in the classical Solow models. A stability analysis of the system near its steady-state solution is given. The obtained result is applied to a system with production function of the Cobb-Douglas type depending on several capitals. Mathematics Subject Classification: primary 91B55, 91B62; secondary 34D20 Keywords: multi-capital, Solow model, dynamical system, stability, Cobb- Douglas production function

2 4104 Eva Brestovanská and Milan Medveď 1 Introduction The classical Solow differential equations [4], [5] describe a dynamics of the ratio: the capital per labor. In the paper by F. Fabião [1] a new type of dynamic equations, describing the dynamics of the capital, not capital per labor. A condition for the existence of steady-states is formulated and a result on their stability is proved there. We extend this result to a model where the production function depends on several capitals. Our dynamic equations for the labor and technical progress have quadratic right-hand sides. We prove a stability results for steady-states of the system. We apply these results to a model corresponding to the Cobb-Douglas production function depending on several capitals. 2 Dynamic Equations of a Multi-Capital Solow Type Model We define a growth model of the Solow-type modeled by the following production function, dynamic equations for m capitals K 1, K 2,..., K m, equations for investments I 1, I 2,..., I m, dynamic equations for technological progress A and for labor L : 1. Production function: t = AtF K 1 t, K 2 t,..., K m t, Lt Dynamic equations for capitals: K it = s i t δ i K i, i = 1, 2,..., m, 2 where s i, δ i, i = 1, 2,..., m are positive constants; 3. Equations for investments: I it = s i t, i = 1, 2,..., m; 3 4. Dynamic equations for technological progress: where a, b R are positive constants; 5. Dynamic equations for labor: Deriving the equation 1 in order to time we obtain A t = GAt, 4 L t = HLt, 5 = A m F + A K i + K i L L. 6

3 Dynamical system of a multi-capital growth model 4105 Dividing both sides of 1 by and using the equality A = F we have = A A + 1 m K i + F K i L L. 7 The functions GA and HL are assumed to be linear in many Solow many generalized Solow models. However under this assumption the system with the dynamic equation 7 does not have steady-state solutions. In the paper [1] another condition see 26 is assumed. However then steady-states of the system are not generic see [2]. We will assume that the functions GA, HL are quadratic and under this assumption steady-states of the system exist and they are generic. 3 Stability Analysis of System with Cobb- Douglas Multi-Capital Production Function In this section we study the system DS with the production function In this case F = t = At K i t α i L 1 m α i m, α i < 1. 8 K i t α i L 1 m α i, = α k K α k 1 k m L = 1 1 α j j=1 L m j=1 α j These equalities and the equation 20 yield 1 F K k = α k s k, Substituting these equalities into 6 we have If the condition,i k K i t α i L 1 m α i, K α i i. 1 F L L = 1 A m = A + α k s k + 1 A A + 1 m is satisfied then we have the system j=1 j=1 α j L L. α j L. 9 L α i L L 0 10

4 4106 Eva Brestovanská and Milan Medveď K i = s i δ i K i, i = 1, 2,..., m, 11 m α k = s k, 12 for which the set of steady-state solutions are defied by the equations K i = δ i s i, i = 1, 2,..., m; 13 The follwing theorem is a generalization of Proposition 4 from the paper [1]. Theorem 3.1 If the condition 10 is satisfied and s i α i δ i K i < 1, 14 then any steady-state solution of the system 11, 12 is asymptotically stable. Proof. Let K 0, 0 where K 0 = K 0 1, K 0 2,..., K 0 m with K 0 i = s i δ i 0 > 0, i = 1, 2,..., m, i. e. K 0, 0 is a steady-state solution of the system. Define the function V K 1, K 2,..., K m, = s k Obviously V K 0, 0 = 0 and V K, > 0 for K, V K 0, 0 and the derivative of V along solutions of the system is V K, = 2 = 2 s k 2 s k We have the equality V K, = s k s k = 16 s k s k s k α k s k s kα k 2 s 2 1 α k s k. s k α k From this equality it follows that V K, < 0 for all K, 0, 0 if the condition 14 is satisfied and from the Lyapunov stability theorem see [3, Theorem 3, page 131] we obtain the assertion of the theorem.. =

5 Dynamical system of a multi-capital growth model 4107 Under the condition 10 the steady-state solutions of the system 11, 12 are non-generic, non-hyperbolic because the set of all steady-state solutions is an one-dimensional manifold defined by the equation K 1, K 2,..., K m = δ 1 s 1, δ 2 s 2,..., δm s m. However by the Kupka-Smale theorem see [4, page 267] the set of all steady-states of systems of autonomous differential equations consists generically of isolated points and they are all hyperbolic. Now let us consider the system = a ba + A = a baa, a, b > 0, 17 L = c dll, c, d > 0, 18 α k s k + 1 j=1 α j c dl. 19 K it = s i t δ i K i, i = 1, 2,..., m, 20 The condition 10 is satisfied at the point A, L = A 0, L 0 = a, c only and b d the point K 0, A 0, L 0, 0, where K 0 = s1 0, s 2 0,..., s m 0, A 0 = a δ 1 δ 2 δ m b, L 0 = c d, 21 0 = A 0 F K 0, L 0 22 is the steady-state solution of the system. Since F K, L = K i t α i L 1 m α i the equation 22 has the solution 0 = c d a b 1 β m 23 si d α i β, 24 δ i c where β = 1 m α i. We have obtained that the steady-state solution of the system 17, 18, 20, 19 is A 0, L 0, K 10, K 20,..., K m0, 0 = where 0 is given by the formula 24. a b, c d, s 1 0, s 2 0,..., s m 0, 0, 25 δ 1 δ 2 δ m Theorem 3.2 The steady-state 25 of the system 17, 18, 20, 19 is asymptotically stable.

6 4108 Eva Brestovanská and Milan Medveď Proof. Define the function V A, L, K 1, K 2,..., K m, = s k 2 + a ba 2 + c dl Obviously V A 0, L 0, K 10, K 20,..., K m0, 0 = 0, V A, L, K 1, K 2,..., K m, > 0 for A, L, K 1, K 2,..., K m, A 0, L 0, K 10, K 20,..., K m0, 0 and using the procedure from the proof of Theorem 3.1 one can prove that 2 s 2 1 V A, L, K 1, K 2,..., K m, = s k α k if the condition 14 is satisfied. Since s i α i 0 δ i K i0 = 2ba ba 2 A 2dc dll < 0, s i α i 0 δ i s i 0 δ i = α i < 1, the condition 14 is satisfied for K 1,..., K m, sufficiently close to the steady-state K 10, K 20,..., K m0, 0 and the assertion of the theorem follows from the Lyapunov stability theorem see [3, Theorem 3, page 131]. Acknowledgements. Research of the both authors was supported by the Slovak Grant Agency VEGA-MŠ, project No.1/0071/14. References [1] F. Fabião, Solow model, an economical dynamical systems,applied Mathematica Sciences, , [2] M. Medveď, Fundamentals of Dynamical Systems and Bifurcation Theory, Adam Hilger, Bristol, Philadelphia and New ork, [3] L. Perko, Differential Equations and Dynamical Systems, Text in Aplied Mathematics 7, Springer, New ork, Berlin, [4] R. M. Solow, A contribution to the theory of economic growth, Quarterly Journal of Economic, , [5] R. M. Solow, Technical change and the aggregate production function, Review of Economics and Statistics, , Received: April 7, 2015; Published: May 29, 2015