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

Abstract

In the present paper, we construct a continuous time model of economic growth with positive externalities and with variable capacity utilization, and study the global equilibrium paths in this model. If there is a homoclinic orbit or a periodic solution in this model, equilibrium is globally indeterminate. We show that positive externalities can yield multiple steady states, a one‐parameter family of homoclinic orbits, and a two‐parameter family of periodic solutions. It is also shown that there exists a sunspot equilibrium in this model.

Poverty trap and global indeterminacy in a growth model with open-access natural resources

“…Therefore, if one is able to detect, in a significant region of the phase space, a separatrix between two sets of points whose trajectories show different behavior, that may lead to relevant information on the manifold of interest. We observe that this approach, which in some contexts, as the present one, can prove useful, differs from others in global indeterminacy literature, based on bifurcation techniques; see, for example, Matsuyama [35], Pintus et al [45], Benhabib and Eusepi [11], Benhabib et al [12], Mattana et al [36]. However it must be stressed that gaining information on separatrices is paramount to any global analysis: in particular it can lead to information on size and/or shape of attractive basins (which several authors consider the main goal of global indeterminacy analysis).…”

“…12 Notice that, in our model, multiplicity of stationary states may occur also in a context of social constant returns to scale, α + β + γ = 1, whereas it is ruled out if the elasticity γ of the production function with respect to natural capital E is relatively high, that is if α + γ 1. Now, let P * = (K * , E * , L * ) be a stationary state of (6) and consider the Jacobian matrix of system (6) evaluated at P * Fig.…”

“…In this context, local stability analysis may be misleading, in that it refers to a neighborhood of a stationary state, whereas the initial values of jumping variables do not have to belong to such a neighborhood. In the words of Matsuyama [35, p. Although some works on indeterminacy focus on global dynamics and stress the relevance of global analysis (see, among the others, Pintus et al [45], Raurich-Puigdevall [46], Benhabib and Eusepi [11], Karp and Paul [30], Pérez and Ruiz [43], Benhabib et al [12], Mattana et al [36], Coury and Wen [22], Brito and Venditti [17], Antoci et al [4]), the literature on indeterminacy is almost exclusively based on local analysis, due to the fact that dynamic models exhibiting indeterminacy are often highly nonlinear and difficult to be analyzed globally. 1 Few papers study global indeterminacy in environmental dynamics; Karp and Paul [30] study (two-dimensional) dynamics of labor migration and environmental change in an economy in which there are two productive sectors, both generating pollution but only one affected by it; Pérez and Ruiz [43] find that global indeterminacy can occur in a two-dimensional endogenous growth model with pollution and public abatement activities; Antoci et al [4], in an overlapping generations context, show that the consumption of private goods as substitutes for free-access environmental goods can produce global indeterminacy and chaos.…”

Endogenous Discounting and Global Indeterminacy

“…This paper shows that the growth model with endogenous discounting proposed in [27] presents global indeterminacy of the equilibrium in the full onset of the original ℝ 3 structure. In detail, a study of the properties of the steady state in the vicinity of a codimension 2 pitchfork-Hopf interaction, allows us to demonstrate that global indeterminacy can arise from plausible values of the parameters in correspondence of the emergence of a trapping region with an invariant torus quasi-periodic dynamics.…”

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

“…2 The relative simplicity of local analysis explains why many works in the literature focus on local indeterminacy issues. 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).…”