Sunspots and Hopf bifurcations in continuous time endogenous growth models

Abstract

First we treat a three-dimensional continuous time abstract stationary model that includes one predetermined variable and two non-predetermined variables. We construct stationary sunspot equilibria in this model under the following two alternative conditions: (i) a steady state has two stable roots and one unstable root; and (ii) A closed orbit has a two-dimensional manifold on which it is asymptotically stable. Next, we apply these results to the models due to Lucas and Romer that undergo Hopf bifurcations for some parameter values. We construct sunspot equilibria in these models.

Bifurcation and sunspots in the continuous time equilibrium model with capacity utilization

“…The deterministic equilibrium dynamics is then given by ≈ 1.730757162505650786. 9 The same stochastic process is also used by Shigoka (1994), Drugeon and Wigniolle (1996) and Nishimura and Shigoka (2006). 10 See Doob (1953, II.2) for the concept of separability.…”

“…2 2 See Shigoka (1994), Drugeon and Wigniolle (1996) and Nishimura and Shigoka (2006) for a different kind of sunspot equilibria in continuous time models. …”

“…SeeShigoka (1994),Drugeon and Wigniolle (1996) andNishimura and Shigoka (2006) for a different kind of sunspot equilibria in continuous time models.3Wen (1998, p. 14) has assumed for the sake of simplicity that A t in his model is constant, which is equivalent to assuming that b = 0 and A = 0 in our specification of the positive external effects. As a result, a steady state is unique in his model.…”

Global analysis and indeterminacy in a two‐sector growth model with human capital

“…Very few authors have engaged in the investigation of global indeterminacy in continuous-time two-sector models. In a context in which a unique balanced growth path exists, Mattana and Venturi (1999), Mattana (2004), Nishimura and Shigoka (2006), and Slobodyan (2007) obtain a global indeterminacy result proving the existence of periodic orbits arising via a Hopf bifurcation. In a context in which two balanced growth paths coexist, Mino (2004) shows that, in a Lucas-Uzawa framework, both balanced growth paths may be selected starting from the same initial values of the state variables.…”

“…However, a fast-growing number of contributions suggest caution in drawing predictions on the future evolution of the economy based exclusively on local analysis; in fact, local stability analysis refers to a small neighborhood of an equilibrium point, whereas the initial values of the jumping variables do not have to belong to such a neighborhood (see, for example, M atsuyama 1991; Raurich-Puigdevall 2000; Boldrin et al 2001); Benhabib and Eusepi 2005;Benhabib et al 2008;Coury and Wen 2009;M attana et al 2009) These works stress the relevance of global analysis in order to get satisfactory information about the equilibrium selection process: in fact, global analysis allows us to highlight more complex contexts in which equilibrium selection is not unequivocally determined by the initial values of the state variables. 3 The indeterminacy of equilibrium selection occurs, for example, when there exists an attracting limit cycle around a non-attractive equilibrium point (see, for example, M attana and Venturi, 1999;Nishimura and Shigoka, 2006;Slobodyan, 2007). Another context in which global analysis techniques allow us to detect cases of indeterminacy is that in which multiple equilibrium trajectories exist, starting from the same initial values of state variables, approaching different equilibrium points or, in general, different -limit sets (see, for example, M atsuyama, 1991(see, for example, M atsuyama, , Benhabib et al 2008M attana et al 2009;Antoci et al 2011;, Antoci et al 2014;Carboni and Russu, 2013).…”