The Pasinetti-Solow Growth Model With Optimal Saving Behaviour: A Local Bifurcation Analysis Pasquale Commendatore 1 and Cesare Palmisani 2 1 Dipartimento di Teoria Economica e Applicazioni Università di Napoli Federico II via Rodinò 22, I-80138 Naples, Italy (e-mail: commenda@unina.it) 2 Dipartimento di Matematica e Statistica Università di Napoli Federico II via Cinthia 26, I-80126, Naples, Italy (e-mail: cpalmisani@unina.it) Abstract. We present a discrete time version of the Pasinetti-Solow economic growth model. Workers and capitalists are assumed to save on the basis of rational choices. Workers face a finite time horizon and base their consumption choices on a life-cycle motive, whereas capitalists behave like an infinitely-lived dynasty. The accumulation of both capitalists and workers wealth through time is reduced to a two-dimensional map whose local asymptotic stability properties are studied. Various types of bifurcation emerge (flip, Neimark-Sacker, saddle-node and transcritical): a precondition for chaotic dynamics. Keywords: Economic growth, optimal choice, wealth distribution, dynamical systems, bifurcation analysis. 1 Introduction In this paper we put forward a two-dimensional economic growth model close to the one developed by Samuelson and Modigliani [7]. Following Pasinetti s [6] suggestion, these authors extended the well-known Solow [8] model by distinguishing in the economy two types of agents or classes, capitalists and workers, characterised by different saving behaviour. Our model departs from the standard Pasinetti-Solow model for two crucial assumptions. First, capitalists and workers saving decisions are based on optimal behaviour. To model such behaviour we follow Michl [4] and [5] assuming that an infinitely lived dynasty of capitalists saves on the basis of an altruistic motive, whereas workers are driven by a life-cycle motive. Second, we assume discrete time. Accordingly, various types of bifurcations occur. In particular period-doubling or Neimark-Saker bifurcations, a precondition for the emergence of chaotic dynamics, can occur by reducing the elasticity of substitution between the factors of production and by varying suitably workers and capitalists time discount factors. We show how these bifurcations can occur via a diagrammatical tool known as stability triangle (Azariadis [1]).
2 Commendatore and Palmisani 2 The model We frame the analysis in discrete time denoting by t the time unit. Moreover, we express all the relevant variables in terms of labour units. We consider a single good economy. Production involves only two factors, capital and labour, and a production function with constant elasticity of substitution (CES) between capital and labour: f(k t ) = [α + (1 α)k ρ t ] 1 ρ = [αk ρ t + (1 α)] 1 ρ kt (1) where k t is the capital/labour ratio, 0 < α < 1 the distribution coefficient, < ρ < 1 (ρ 0) the substitution coefficient and (1 ρ) 1 the elasticity of substitution. The only sources of income in the economy are wages and profits and capital the only asset. For each short-run equilibrium, perfectly competitive labour and capital markets ensure equality between the wage rate w t and the marginal product of labour f(k t ) f (k t )k t and between the profit rate r t and the marginal product of capital f (k t ). That is, w t = f(k t ) f (k t )k t (2) r t = f (k t ) (3) There are two distinct groups of agents, capitalists and workers. Both save and accumulate capital, k c,t and k w,t representing respectively capitalists and workers capital, where 0 k c,t k t, 0 k w,t k t and k t = k c,t + k w,t. We represent capitalists as a single dynasty with an infinite time horizon. Each generation cares about its offspring and saves for a bequest motive behaving like an infinitely-lived household (Barro [2]). Given initial wealth k c,0 and time discounting factor 0 < β c < 1, at beginning of period 0 capitalists choose consumption quantities (c c,0, c c,1, c c,2,..., c c,t,...) = (c c,t ) 0 to solve the following constrained utility maximisation problem: t= max(1 β c ) βc t ln c c,t subject to c c,t + k c,t+1 (1 + r t δ)k c,t ; where k c,t represents capitalists wealth and δ the rate of capital depreciation. Solutions satisfy the condition c c,t+1 = (1 β c )(1 + r t δ)k c,t. The difference between profits, capitalists only source of income, and consumption gives capitalists saving: t=0 s c,t = [β c (1 + r t δ) (1 δ)]k c,t (4) The population of workers has an overlapping generation structure. Each generation of workers faces a finite time horizon composed of two periods. Each individual is active, working and earning a wage, only when young,
The Pasinetti-Solow Growth Model With Optimal Saving Behaviour 3 i.e., in the first period of her life; while she is in retirement when old, i.e., in the second period of life. Each young worker inelastically supplies one unit of labour at the wage w t. In contrast to capitalists, workers save out of wages only to be able to consume during retirement according to a lifecycle motive. There is no intergenerational transfer of wealth (Diamond [3]). Given the time discount factor 0 < β w < 1, at time t a young worker chooses the quantities of current consumption c w,t and future consumption c w,t+1 in order to solve the following constrained utility maximisation problem: max(1 β w ) ln c w,t + β w ln c w,t+1 subject to c w,t + c w,t+1 1+r t δ w t. The solution is c w,t = (1 β w )w t, from which the individual worker saving can be easily deduced: s w,t = w t c w,t = β w w t (5) Assuming that the labour force grows at the constant rate n, capitalists and workers accumulate capital according to the rules k c,t+1 = 1 1 + n (1 δ)k c,t + i c,t (6) k w,t+1 = 1 1 + n i w,t (7) where i c,t and i w,t represent, respectively, capitalists and workers gross investment. In a short-run equilibrium, gross investment equals saving. Using (2), (3), (4), (5) and discarding the time subscripts, (6) and (7) become G c (k c, k w ) = 1 1 + n β c(1 δ + f (k))k c (8) G w (k c, k w ) = 1 1 + n β w(f(k) f (k)k) (9) Recalling that k = k c + k w, the two-dimensional dynamical system (8) and (9) fully describes the time evolution of workers and capitalists wealth. 3 Existence and properties of long run equilibria The long run (or steady growth) equilibria correspond to the fixed points of the two-dimensional map (8) and (9) obtained by imposing G c (k c, k w ) = k c and G w (k c, k w ) = k w. There exist three different types of equilibria: a Pasinetti equilibrium involves capitalists owing a positive share of capital; a dual equilibrium, instead, allows only workers to own capital; finally, in a trivial equilibrium, the overall capital is zero. Denoting by e f (k) =
4 Commendatore and Palmisani f (k)k/f(k) = (1 α)(αk ρ + 1 α) 1 the capital-output elasticity, where 0 e f (k) < 1, the three types of equilibria, can be expressed as: Pasinetti equilibrium: (k c, k w ) where k c = k k w, kw k = βwr 1+n ( 1 e f (k) e f (k) A Pasinetti equilibrium exists iff: 0 < Dual equilibrium: ( k c, k w ) ) and f (k) = r 1+n β c (1 δ). β w r 1 + n + β w r < e f (k) < 1 (10) where k c = 0, k w = k and f( k) 1+n =. k β w(1 e f ( k)) Considering (1), a dual equilibrium solves ( 1 α ) 1 ρ = 1+n. The e f ( k) β w (1 e f ( k)) latter equation can be transformed into x (1 x) ρ h(x) = (1 α)( 1 + n β w ) ρ (11) where h(x) = and x = e f ( k). The shape of h(x) depends on the sign of ρ. When 0 < ρ < 1, h(x) is strictly increasing and convex for all 0 x < 1. By the intermediate value theorem, there exists a unique x which solves (11). Instead, when ρ < 0, h(x) is strictly increasing for 0 x < x 1 1 ρ ; and strictly decreasing for x < x < 1. By the intermediate value theorem, iff (1 α)( 1+n β w dual equilibria (strict inequality). ) ρ h(x ) = ( ρ) ρ (1 ρ) ρ 1 there exist one (equality) or two Trivial equilibrium: (k 0 c, k 0 w) = (0, 0) The trivial equilibrium exists and is unique iff f(0) = 0. 4 Bifurcation analysis To investigate the bifurcation process involved in the system (8) and (9), we use the geometrical method based on the stability triangle (see Azariadis [1]). Briefly, a fixed point (y, z) is locally stable if the trace and the determinant of the Jacobian matrix J(y, z) evaluated at (y, z), that is, T (y, z) and D(y, z), lie within the boundaries of the isosceles triangle ABC shown in Figure 1. That is, if the conditions (i) 1 + T (y, z) + D(y, z) > 0, (ii)d(y, z) < 1 and (iii) 1 T (y, z) + D(y, z) > 0 hold. The system loses stability if, varying a parameter, trace or determinant move outside the triangle. According to which side of the triangle is crossed, a particular type of local bifurcation occurs. If the interior of the segment AB is crossed, the system undergoes a flip or period doubling bifurcation. Along this segment, the smallest eigenvalue
The Pasinetti-Solow Growth Model With Optimal Saving Behaviour 5 of J(y, z) is smaller than 1. If the segment BC is crossed, the system undergoes a Neimark-Sacker bifurcation. Along this segment, the two eigenvalues are complex conjugates with modulus equal to 1. Finally, if the segment AC is crossed, the system may undergo a saddle-node or, alternatively, a transcritical bifurcation. Along this segment, the largest eigenvalue is equal to 1. Applying the linearization theorem, the Jacobian evaluated at the Pasinetti equilibrium (k c, k w ) is ( ) r J(k c, k w ) = r+1 δ e kc f (k) 1 + r k r+1 δ e kc f (k) k β wr 1+n e f (k) β wr 1+n e f (k) The corresponding trace and the determinant are: T (k c, k w ) = 1 + D(k c, k w ) + r r + 1 δ e f (k)k c k (12) D(k c, k w ) = β wr 1 + n e f (k) (13) where e f (k) = f (k)k/f (k) = α(1 ρ)[α + (1 α)k ρ ] 1 is the marginal productivity elasticity with respect to capital. e f (k) < 0 measures the curvature of the production function, the closer is this elasticity to 0 ( ), the larger (smaller) is the degree of substitutability between capital and labour. This variable has a crucial importance in determining the local dynamics properties of the system. Indeed, using (12) and (13), conditions (i) to (iii) can be rewritten as (i) e f (k) > Ω 2 2βw r 1+n r r+1 δ kc k if Ω < 0. It is always satisfied otherwise. Note that for 0 < k c < k, we have 1+n β w r < Ω < or < Ω < 1 1+n r β c 2β w < 1, it follows that e f (k) < 1 is a necessary condition for a flip bifurcation to occur. (ii) e f (k) > 1+n β w r. Note that if the discount rate of capitalists is larger or not too smaller than the one of workers, i.e. β w /β c < 1 + 1 δ 1+n, from (ii) it follows that e f (k) < 1 represents a necessary condition for a Neimark-Saker bifurcation to occur; β wr 1+n+β w r. (iii) e f (k) > Condition (iii) is included in condition (10). The violation of condition (iii) involves a transcritical bifurcation characterised by two coexisting equilibra exchanging stability: that is, the Pasinetti equilibrium loses stability in favour of a dual equilibrium. For the bifurcation analysis we choose the worker s discount rate β w as control parameter. It is easy to verify that trace and determinant in (12)
6 Commendatore and Palmisani and (13), redefined as T (β w ) and D(β w ) respectively, depend linearly on β w. More precisely, varying β w within the interval [0, 1] allows to plot in the Trace-Determinant plane a segment S with length λ and slope µ, where 0 < µ [r+1 δ]e f (k) < 1 and λ (T (1) T (0)) 2 + (D(1) D(0)) 2 and r+(1 δ)e f (k) where (T (0), D(0)) and (T (1), D(1)) represent the start and end points of S. The following proposition spells out the local stability properties of the Pasinetti equilibrium and clarifies the conditions that should hold for the emerging of a specific type of bifurcation: Proposition 1: Given (T (0), D(0)) and (T (1), D(1)) in the Trace- Determinant plane, the segment S satisfies the following properties: Case 1: when 1 < T (0) < 1 it never intersects the AB-side; it intersects the BC-side if µ µ < 1 and λ 10 ; it intersects the AC-side if 0 < µ µ 1 2 T (0) < 1 and λ 10. Case 2: when 3 T (0) 1 it intersects the AB-side if µ > 0 and λ 2 ; it intersects the BC-side if µ µ < 1 and λ 17 ; it intersects the AC-side if 0 < µ µ and λ 17. Case 3: when T (0) < 3 it intersects the AB-side if 0 < µ µ 1 2+T (0) and λ λ (T (0) + 1) ; it intersects the BC-side if µ µ µ < 1 and λ λ 1 + (T (0) 2) 2 ; it intersects the AC-side if 0 < µ µ and λ λ. r Figure 1 illustrates Proposition 1. Since T (0) = 1 + e f (k) r+1 δ < 1 and D(0) = 0, S starts on the horizontal axis, on the left of 1. S 1 and S 2 exemplify case 1. Both start at 1 < T (0) < 1 and have sufficient length to exit the stability triangle ABC. S 1 satisfies the inequality µ < µ while the inequality µ > µ holds for S 2. As the parameter β w is increased from 0 to 1, moving along S 1 the Pasinetti equilibrium loses stability via a transcritical bifurcation; instead, along S 2 stability loss occurs via a Neimark- Saker bifurcation. S 1 and S 2 exemplify case 2. Both start at 3 T (0) 1 and have sufficient length to cut through ABC. µ < µ holds for S 1 and µ > µ for S 2. As the parameter β w is increased from 0 to 1, S 1 or S 2 enter the triangle ABC and the Pasinetti equilibrium gains stability via a (reversed) flip bifurcation; as β w is further increased stability loss may occur via a transcritical bifurcation, along S 1 ; or via a Neimark-Saker bifurcation, along S 2.
The Pasinetti-Solow Growth Model With Optimal Saving Behaviour 7 Finally, exemplifying case 3, S 1 and S 2 start at T (0) < 3. Their behaviour and properties are similar to those of S 1 and S 2. Fig. 1. Stability Triangle. We conclude this section discussing briefly the stability conditions for dual and trivial equilibria. The Jacobian evaluated at a dual equilibrium ( k c, k w ) = (0, k w ) is ( ) J( k c, k r+1 δ w ) = r+1 δ 0 βwr 1+n e f ( k) βwr 1+n e f ( k) where r = f ( k). Defining by Λ 1 and Λ 2, the eigenvalues of the lower triangular matrix J( k c, k w ), trace and determinant can be expressed as T ( k c, k w ) = Λ 1 + Λ 2 and D( k c, k w ) = Λ 1 Λ 2, where Λ 1 r+1 δ r+1 δ and Λ 2 βwr 1+n e f ( k) are both real and positive. A dual equilibrium is stable as long as Λ 1 < 1 and Λ 2 < 1. The violation of the first inequality leads to a transcritical bifurcation involving an exchange of stability in favour of the Pasinetti equilibrium. Indeed, it is possible to show that this inequality corresponds to the reversed of the central inequality in (10). Moreover, the violation of the second inequality leads to a saddle-node bifurcation involving the collision and disappearance of two dual equilibria, one stable and the other unstable.
8 Commendatore and Palmisani Finally, it is straightforward to show that the trivial equilibrium is always unstable as long as k > 0. 5 Conclusions In this paper we presented a two-dimensional growth model in which two groups of economic agents, capitalists and workers, follow different saving rules. This model represents a discrete time version of the one developed in various stages by Solow [8], Pasinetti [6] and Samuelson and Modigliani [7]. A crucial difference is the assumption of optimal saving behaviour. We identified three different types of long-run equilibria and verified their local stability properties. We verified also the emergence of various types of local bifurcations. In particular, flip and Neimark-Saker bifurcations can occur by reducing the degree of substitutability between productive factors and choosing suitably capitalists and workers time discount factors. Simulations, not presented here, show that the time evolution of capitalists and workers wealth accumulation may undertake several types of complex behaviour. References 1.C. Azariadis. Intertemporal Macroeconomics. Cambridge Massachusetts, 1993, Blackwell Publisher. 2.R.J. Barro. Are Government Bonds Net Wealth? The Journal of Political Economy, 82: 1095-1117, 1974. 3.P. Diamond. National Debt in a Neoclassical Growth Model. American Economic Review, 55:1126-1150, 1965. 4.T.R. Michl. Capitalists, Workers, and the Burden of Debt. Review of Political Economy, 18: 449-467, 2006. 5.T.R. Michl. Capitalist, Workers, and Social Security. Metroeconomica, 58: 244-268, 2007. 6.L. Pasinetti. Rate of Profit and Income Distribution in Relation to the Rate of Economic Growth. The Review of Economic Studies, 34: 267-279, 1962. 7.P.A. Samuelson and F. Modigliani. The Pasinetti-Paradox in Neoclassical and more General Models. Quaterly Journal of Economics, 33: 269-301, 1966. 8.R. Solow. A Contribution to the Theory of Economic Growth. Quaterly Journal of Economics, 70: 65-94, 1956.