Coexistence of attractors and homoclinic loops in a kaldor-like business cycle model

Coexistence of attractors is often a characteristic feature of economic models represented by nonlinear dynamic systems [see, among others, Agliari et al (2002), Bischi & Kopel (2001), Dieci et al (2001), Agliari et al (2000)]. Generally speaking, when multiple attractors coexist in the phase-space for a particular choice of the parameters of the model, a crucial question is about the role played by the initial conditions in determining the asymptotic behavior of the system. Moreover, in order to perform a proper bifurcation analysis with respect to some specific parameters it is necessary to take into account that parameter variations determine in general both qualitative changes (including appearance/disappearance) of the attractors, and structural changes of the basins of attraction of the coexisting attractors. The latter point has been less emphasized in the economic literature. In general, typical features of such qualitative changes of the basins are the following: (a) they are due to global bifurcations (not associated with the eigenvalues of the linearized system around a particular steady state) and (b) they may bring about a kind of "complexity" which is different from the one usually reported in the literature (associated with "strange attractors", and "sensitivity to initial conditions"): Namely, simple attractors (steady states, cycles of low period, attracting closed curves) may have basins with complex structures. In recent years, several studies have pointed out particular mechanisms of basin bifurcations, which are associated with contacts between basin boundaries and "critical sets", in the case of dynamical systems represented by the iteration of noninvertible maps [Mira et al (1996), Agliari et al (2002), Agliari (2001)]. Other possible mechanisms, which may occur in the case of invertible maps as well, are associated with homoclinic tangencies of the stable and unstable manifolds of saddles. The present Chapter illustrates the latter type of phenomena, in situations of coexisting attractors that arise from a particular version of Kaldor's business cycle model in discrete time, described by a nonlinear two-dimensional dynamical system. The particular Kaldor-like model at hand, where consumption is modelled as an S-shaped function of income, and investment is a linear increasing function of output (and a linear decreasing function of capital), has been developed in Herrmann (1985), and studied also in Lorenz (1992, 1993), Dohtani et al (1996), mainly in order to prove the emergence of chaotic dynamics in Kaldor-like models under extreme values of the output adjustment parameter. However, the particular parameter constellation which is assumed within the present Chapter (under which multiple equilibria exist) has been excluded from the analysis carried out in earlier work, though it corresponds to economically meaningful situations. We will show that for this choice of parameters, business fluctuations along a stable closed curve (which typically arise in Kaldor model), coexist with alternative dynamic outcomes (stable steady states, or stable periodic orbits of low period), which the system may reach in the long-run depending on the initial state. Furthermore, we will explain the bifurcation mechanisms which determine such situations of coexistence, the appearance or disappearance of attractors and the qualitative changes of the basins of attraction. The global dynamic phenomena which are detected in this Chapter are described in Chapter 1 and have also been detected in a different version of the Kaldor model in discrete-time [see Bischi et al (2001) and Agliari et al (2005b)], where investment is an increasing S-shaped function of output (and depends negatively on capital stock) and savings depend linearly on income. Therefore such dynamic phenomena seem to be very persistent ones, and their occurrence seems to be ultimately related to the following basic assumptions: (i) investment or consumption have sigmoid shaped graphs, in a way that the marginal propensity to invest is larger (smaller) than the marginal propensity to save for normal (extreme) levels of income, and (ii) the investment schedule shifts downwards (upwards) as output increases (decreases) as a result of a negative dependence on accumulated stock of capital. Both these assumptions are essential qualitative features of Kaldor's original model. On the other hand, very similar dynamic phenomena have been detected also in Agliari et al (2005a), where a two-dimensional map with a "minimal" structure qualitatively similar to that in Agliari et al (2005b), and to the one being studied here, has been analyzed in details. Further examples are shown in this book. Chapters 9 and 11. The Chapter is organized as follows. In Section 8.2 we present the business cycle model, perform useful changes of coordinates, and reduce it to a twodimensional map. Section 8.3 presents some general properties of the map, namely the symmetry, the steady states and local asymptotic stability conditions, and the conditions for invertibility. Section 8.4 focuses on particular global bifurcations, involving qualitative changes of the basins of attraction, occurring in a particular regime of parameters where three equilibria exist, and relates these phenomena to the behavior of the stable and unstable manifolds of saddles. © Springer-Verlag BerHn Heidelberg 2006.