2 The model with Kaldor-Pasinetti saving

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1 Applied Mathematical Sciences, Vol 6, 2012, no 72, The Solow-Swan Model with Kaldor-Pasinetti Saving and Time Delay Luca Guerrini Department of Mathematics for Economic and Social Sciences University of Bologna, Italy Abstract In this paper, we analyze the model associated to an economic growth process with a Kaldor-Pasinetti saving and a delay in production We demonstrate that the steady state may exhibit Hopf cycles Mathematics Subject Classification: 34K18, 91B62 Keywords: Solow-Swan model; Kaldor-Pasinetti; delay; Hopf bifurcation 1 Introduction The analysis of the fundamental issues in dynamical macroeconomics usually begins with the study of the Solow-Swan growth models [11-12], which is a one-dimensional dynamical system able to generate monotonic convergence to a steady growth equilibrium As shown by Asea and Zak [1], when production occurs with a delay, the resulting model is able to generate chaotic dynamics In this paper, we study the dynamics of a Solow-Swan growth model modified by introducing different but constant saving propensities attached to factor shares (Kaldor-Pasinetti saving) and a time delay in the productive operation of installed capital The length of delay which preserves the stability of the positive equilibrium is estimated, and the existence of Hopf bifurcation when the delay crosses through a critical value is investigated For future research it would be interesting to extend this line of research to other economic growth models (eg, Ferrara and Guerrini [2-3], Guerrini [4-8]) 2 The model with Kaldor-Pasinetti saving Let us consider the standard Solow-Swan growth model [11-12] in the tradition of Kaldor [9] and Pasinetti [10], where two types of agents, workers and shareholders, have different but constant saving rates The production function is

2 3570 L Guerrini assumed to have the Cobb-Douglas form f(k t )=k α t,α (0, 1), where k t is capital per worker As a consequence, capital accumulation is described by the following equation: k t =[s w + α(s c s w )]k α t nk t, (1) where s w (0, 1) and s c (0, 1), s w <s c, denote the constant saving rates for workers and shareholders, respectively Here n>0 is the constant population growth rate For simplicity, there is no capital depreciation First, we investigate the question of existence and uniqueness of non-trivial steady states for Eq (1) The equilibrium points are all the solutions of k t =0 Since shareholders do not save less than workers, there is only one positive steady state k = {[s w + α(s c s w )]/n} 1/(1 α) Furthermore, a graphical analysis makes it clear that whenever k t <k, k t > 0 and whenever k t >k, k t < 0 Thus, the capital-effective labor ratio monotonically converges to the steady state value k implying the model s asymptotic stability 3 The model with Kaldor-Pasinetti saving and time delay We modify the model by assuming a time lag T 0 in the production technology As a result, Eq (2) is replaced by a differential equation with a delay parameter: k t =[s w + α(s c s w )]k α t T nk t, (2) for some initial function k t = φ t,t [ T,0] The equilibria of Eq (2) coincide with those of Eq (1) In order to investigate the stability of k we analyze the linearized version of (2) about the point k Linearization and introduction of the new variable z t = k t k reduce Eq (2) to The corresponding characteristic equation is z t = αnz t T nz t (3) λ + n αne λt =0 (4) We recall that the stability of the positive steady state and local Hopf bifurcations can be determined by the distribution of the roots of Eq (4), and the positive steady state k of Eq (2) is stable if and only if the zero steady state of Eq (4) is stable Moreover, to obtain the values T such that the equilibrium point changes from local asymptotic stability to instability or viceversa, we need to find the imaginary solutions of Eq (4)

3 The Solow-Swan model 3571 Proposition 31 1 Let α 1/ 1+n 2 Then all roots of the characteristic Eq (4) have negative real parts 2 Let α>1/ 1+n 2, then the following hold (a) For T = T j = cos 1 (1/α)+2jπ α2 (1 + n 2 ) 1 (j =0, 1, 2, ), Eq (4) has a pair of imaginary roots ±iω 0, where ω 0 = α 2 (1 + n 2 ) 1 (b) For T [0,T 0 ), all roots of Eq (4) have negative real parts; for T = T 0, all roots of Eq (4), except ±iω 0, have negative real parts; for T (T j,t j+1 ], Eq (4) has 2(j +1) roots with positive real parts Proof We start examining the imaginary solutions of Eq (4) Let λ = ±iω be these solutions and without loss of generality let us assume ω>0 It is easy to see that Eq (4) has a pure imaginary solution iω if and only if iω + n αne iωt =0 Separating the real and imaginary parts yields leading to α cos ωt = 1 and ω = αn sin ωt, (5) ω 2 = α 2 (1 + n 2 ) 1 Therefore, ω = α 2 (1 + n 2 ) 1ifα 2 (1 + n 2 ) 1 > 0, ie α>1/ 1+n 2 We can conclude that the characteristic equation (4) has a pair of imaginary roots ±ω 0 i if α>1/ 1+n 2 and T = T j Theorem 32 If α 1/ 1+n 2, the steady state k is asymptotically stable for all T 0 If α>1/ 1+n 2, we have that k is asymptotically stable when T [0,T 0 ) and unstable when T>T 0 Moreover, the economy undergoes a Hopf bifurcation at k if T = T j,j=0, 1, 2,

4 3572 L Guerrini Proof Let λ(t )=μ(t)+iω(t ) be a root of Eq (4) near T = T j such that μ(t j )=0,ω(T j )=ω 0,j=0, 1, 2, By substituting λ(t ) into Eq (4) and differentiating both sides of the equation with respect to T, we obtain dλ dt (1 α)ne λt ( T dλ dt + λ ) =0, so that ( ) 1 dλ 1 = dt λ(λ + n) T λ Hence, it follows that ( ) 1 dλ 1 Re dt ω0 2 + n > 0, ie d (Reλ) 2 dt > 0, j=0, 1, 2, λ=iω T =T j = From the previous Proposition and the above discussion we get the statement References [1] PK Asea, and P Zak, Time-to-build and cycles, Journal of Economic Dynamics and Control, 23 (1999), [2] M Ferrara and L Guerrini, The Ramsey model with logistic population growth and Benthamite felicity function revisited, WSEAS Transactions on Mathematics, 8 (2009), [3] M Ferrara and L Guerrini, A note on the Uzawa-Lucas model with unskilled labor, Applied Sciences, 12 (2010), [4] L Guerrini, Logistic population change and the Mankiw-Romer-Weil model, Applied Sciences, 12 (2010), [5] L Guerrini, The Ramsey model with AK technology and a bounded population growth rate, Journal of Macroeconomics, 32 (2010), [6] L Guerrini, A note on the Ramsey growth model with the von Bertalanffy population law, Applied Mathematical Sciences, 4 (2010), [7] L Guerrini, The AK Ramsey growth model with the von Bertalanffy population law, Applied Mathematical Sciences, 4 (2010), [8] L Guerrini, The AK Ramsey model with von bertalanffy population law and Benthamite function, Far East Journal of Mathematical Sciences, 45 (2010),

5 The Solow-Swan model 3573 [9] N Kaldor, A model of economic growth, Economic Journal, 67 (1957), [10] L Pasinetti, Rate of profit and income distribution in relation to the rate of economic growth, Review of Economic Studies, 29 (1962), [11] RM Solow, A contribution to the theory of economic growth, Quarterly Journal of Economics, 70 (1956), [12] TW Swan, Economic growth and capital accumulation, Economic Record, 32 (1956), Received: March, 2012