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

Abstract

The purpose of the present paper is to highlight some features of global dynamics of the two‐sector growth model with accumulation of human and physical capital analyzed by Brito and Venditti, which is a specialization of the model proposed by Mulligan and Sala‐i‐Martín. In particular, our analysis focuses on the context in which the Brito–Venditti system admits two balanced growth paths, each corresponding, after a change of variables, to an equilibrium point of a three‐dimensional system, and proves the possible existence of points trueP‾ such that in any neighborhood of trueP‾ lying on the plane corresponding to fixed values of the state variables there exist points trueQ‾ whose positive trajectories tend to either equilibrium point. This implies that equilibrium selection in the Brito–Venditti system may depend on the expectations of economic agents rather than on the history of the economy. That is, economies with identical technologies and preferences, starting from the same initial values of the state variables (history), may follow rather different equilibrium trajectories according to the economic agents’ choices of the initial values of the jumping variables (expectations). Moreover, we prove that the basins of attraction (two or three‐dimensional) of locally indeterminate equilibrium points may be very large, as they can extend up to the boundary of the system phase space.

Globally Indeterminate Growth Paths in the Lucas Model of Endogenous Growth

“…Global indeterminacy is more difficult to detect since it requires more complicate mathematical techniques. For the case of the two-sector, continuoustime, endogenous growth model with infinitely lived agents, the property of global indeterminacy has been proved in connection with: (i) the existence of a Hopf cycle around a unique rest point in a well-located bidimensional manifold [Nishimura and Shigoka (2006)]; (ii) the existence of a homoclinic orbit connecting a saddle point to itself and enclosing a source or a sink, again in a well-located bidimensional manifold [Mino (2004), Mattana et al (2009), and Bella and Mattana 2014]; (iii) different ω-limit sets [Brito and Venditti (2010) and Antoci et al 2014]; and (iv) a chaotic attractor which appears after the rupture of a Shilnikov homoclinic orbit doubly asymptotic to a saddle-focus in R 3 [Bella et al (2017)].…”

Linear production function, externalities and indeterminacy in a capital-resource growth model

“…They discuss jointly, the roles of the elasticity of intertemporal substitution in consumption and the elasticity of the labour supply on the local indeterminacy properties of the long-run equilibrium. Assuming constant returns to scale at the private and social levels, Brito and Venditti (2010) and Antoci et al (2012) consider a two-sector endogenous growth model where the productions of the final good and human capital require economy-wide external effects and analyse the existence of local and global indeterminacy simultaneously. Finally, considering a one-sector growth model with social increasing returns, Itaya (2008) shows how pollution may affect indeterminacy.…”

Multiple equilibria and global indeterminacy in an endogenous growth model with congestible public goods

“…Therefore, multiple equilibrium trajectories, departing from a given initial condition, in the vicinity of an (apparently) saddle‐path stable equilibrium, may approach a different rest point, namely an attracting focus (i.e., a trapping region) in a larger dynamic picture. This result is commonly referred to as global indeterminacy of the equilibrium (see Antoci, Galeotti, & Russu, ; Bella & Mattana, ; Brito & Venditti, ; Mattana, Nishimura, & Shigoka, ; Mattana & Venturi, ; Raurich‐Puigdevall, ; Xie, ).…”