Another Proof for the Stability of a Modified Solow Model

Transcription

1 Applied Mathematical Sciences, Vol 5, 2011, no 25, Another Proof for the Stability of a Modified Solow Model Massimiliano Ferrara Department SSGES Mediterranean University of Reggio Calabria, Italy massimilianoferrara@unircit Abstract On the basis of Poincarè-Bendixson Theorem we derive another proof for the global asymptotic stability result of the Solow model with Bertalanffy population growth law introduced by Brida and Limas Maldonado [6] Mathematics Subject Classification: 91B62 Keywords: Solow model; stability; Bertalanffy 1 Introduction The neoclassical growth models of Ramsey [20] and Solow [21] provide the basic framework for much of modern macroeconomics The Solow model explains economic growth with accumulation of labor, capital, and other production factors with diminishing returns to scale The economy converges towards a steady state equilibrium where the level of per capita income is determined by savings and investment, depreciation, and population growth, but there is no permanent income growth The Ramsey model instead explicitly models the choice of consumption at a point in time and so it endogenizes the saving rate As a result, unlike in the Solow model, the saving rate may not be constant along the transition to the long run steady state A standard assumption in the neoclassical models is that population grows at a positive given rate, yielding that population increases to infinity in the long run This fact, which seems to be not a realistic assumption, led some authors to explore the consequences of investigating the Solow model under a different and more realistic population growth law (see, eg, Accinelli and Brida [1-2], Bucci and Guerrini [7], Ferrara and Guerrini [8-10], Germanà and L Guerrini [11], Guerrini [12-19]) Recently, Brida and Limas Maldonado [6] have studied the Solow model by modelling the

2 1230 M Ferrara population growth rate via the von Bertalanffy law [3] Within this framework, they have solved the model in closed-form and proved its global asymptotic stability The goal of the present paper is to give a different proof of this last statement using the Poincarè-Bendixson Theorem 2 The model We start by considering the Solow model with von Bertalanffy population growth law as introduced by Brida and Limas Maldonado [6] There is a closed economy with an homogeneous good produced according to a Cobb-Douglas production function Y = K α L 1 α, 0 <α<1, where K is physical capital, L denotes labor/population, and Y is the flow of output, which is assumed to be used for consumption or for investment I in physical capital Since a constant fraction δ of the capital stock depreciates every period, the net increase in capital at any moment in time equals gross investment less capital depreciation, ie K = I δk The output of the economy equals total income, whereas investments equal savings Additionally, we suppose savings to evolve over time as a constant fraction s of output If we define a new variable, the capital-labor k = K/L, then taking derivatives with respect to time in the definition of k yields k = d(k/l)/dt = K/L ( L/L)k Consequently, we get k = sk α (δ + L/L)k Contrary to standard Solow model [17], population growth rate is not constant, but given by the von Bertalanffy growth law [3], L = r(l L), where L is a theoretical maximum asymptote size of the labor force, and r is a constant that determines the speed at which the labor force reaches the asymptote With the inclusion of a von Bertalanffy population growth law, the economy of this modified Solow model is thus described by a system of two differential equations k = sk α ( δ + L L ) k, L = r(l L) (1) Brida and Limas Maldonado [6] have analyzed this model and they have, in particular, proved its global asymptotic stability In the next section, we will provide a new proof of this result based on the Poincarè-Bendixson Theorem 3 Stability We start studying the steady states of our model, where a steady state is defined as a situation in which the growth rate of the per-capita physical capital stock and the growth rate of labor are both equal to zero In studying the steady states, we will confine our analysis to interior steady states only

3 Stability of a modified Solow model 1231 Lemma 31 There exists a unique steady state equilibrium (k,l ) of (1), where ( s ) 1/(1 α) k =, L = L δ Proof Immediate from being steady states characterized by k =0, L =0 Proposition 32 The steady state equilibrium is a stable node Hence, Proof The Jacobian matrix J associated to the equilibrium point (k,l )is [ ] (1 α)δ J r/l = 0 r The trace of this matrix is negative and its determinant is positive This leads to the statement of our Proposition (see, eg, Blume and Simon [4]) Theorem 33 (k,l ) is globally asymptotically stable Proof According to the Proposition 32, (k,l ) is a locally stable equilibrium To prove that it is globally asymptotically stable, we first note that any orbit starting in R 2 + is bounded In other words, there is a compact in which this orbit stays closed In such a compact, we now show that the trajectory can converge only to the equilibrium point Using the Poincarè-Bendixson theorem (see, eg, Boyce and DiPrima [5]), and because the compact considered contains only one singular point, there can be only two kinds of behaviour: the ω-limit set ω(k 0,L 0 ), namely the set of accumulation points of the trajectory, is a single point, which is an equilibrium point, or it is a periodic orbit Now, an easy computation yields (k 1 L 1 k) + (k 1 L 1 L) = (1 α)sl 1 k α 2 rl L 2 k 1 < 0 k L Hence, Dulac criterion (see, eg, Boyce and DiPrima [5]) implies that there cannot be any periodic orbit Consequently, ω(k 0,L 0 ) is reduced to a single point and we reach the conclusion that all trajectories, no matter where the initial point is, will be drawn to the steady state References [1] E Accinelli and JG Brida, The Ramsey model with logistic population growth, Economics Bulletin, 3 (2007), 1-8 [2] E Accinelli and JG Brida, The dynamics of the Ramsey economic growth model with the von Bertalanffy population growth law, Applied Mathematical Sciences, 1 (2007),

4 1232 M Ferrara [3] L von Bertalanffy, A quantitative theory of organic growth (Inquiries on growth laws II), Human Biology, 10 (1938), [4] LE Blume and CP Simon, Mathematics for Economists, Norton, New York, 1994 [5] WE Boyce and R DiPrima, Elementary Differential Equations and Boundary Value Problems, John Wiley and Sons, 1995 [6] JG Brida and EJ Limas Maldonado, Closed form solutions to a generalization of the Solow growth model, Applied Mathematical Sciences, 1 (2010), [7] A Bucci and L Guerrini, Transitional dynamics in the Solow-Swan growth model with AK technology and logistic population change, BE Journal of Macroeconomics, 9 (2009), 1-16 [8] M Ferrara and L Guerrini, On the dynamics of a three sector growth model, International Review of Economics, 55 (2008), [9] M Ferrara and L Guerrini, The Ramsey model with logistic population growth and Benthamite felicity function revisited, WSEAS Transactions on Mathematics, 8 (2009), [10] M Ferrara and L Guerrini, More on the Green Solow model with logistic population change, WSEAS Transactions on Mathematics, 8 (2009), [11] C Germanà, and L Guerrini, On the closed-form solution of the improved labor augmented Solow-Swan model, Applied Sciences, 7 (2005), [12] L Guerrini, The Solow-Swan model with a bounded population growth rate, Journal of Mathematical Economics, 42 (2006), [13] L Guerrini, The Ramsey model with a bounded population growth rate, Journal of Macroeconomics, 32 (2010), [14] L Guerrini, A closed-form solution to the Ramsey model with logistic population growth, Economic Modelling, 27 (2010), [15] L Guerrini, The Ramsey model with AK technology and a bounded population growth rate, Journal of Macroeconomics, 32 (2010), [16] L Guerrini, Transitional dynamics in the Ramsey model with AK technology and logistic population change, Economics Letters, 109 (2010), 17-19

5 Stability of a modified Solow model 1233 [17] L Guerrini, The AK Ramsey model with von Bertalanffy population law and Benthamite function, Far East Journal of Mathematical Sciences, 45 (2010), [18] L Guerrini, A note on the Ramsey growth model with the von Bertalanffy population law, Applied Mathematical Sciences, 4 (2010), [19] L Guerrini, The AK Ramsey growth model with the von Bertalanffy population law, Applied Mathematical Sciences, 4 (2010), [20] FP Ramsey, A mathematical theory of savings, Economic Journal, 38 (1928), [21] RM Solow, A contribution to the theory of economic growth, Quarterly Journal of Economics, 70 (1956), Received: October, 2010