Existence and implications of a pitchfork-Hopf bifurcation in a continuous-time two-sector growth model

Giovanni Bella,Paolo Mattana,Beatrice Venturi

doi:10.1007/s11403-021-00331-8

Giovanni Bella, Paolo Mattana + Show 1 more

Open Access

https://doi.org/10.1007/s11403-021-00331-8

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Abstract

This paper shows that the dynamics of the Lucas (J Monet Econ, 22:3–42, 1988) endogenous growth model with flow externalities may give rise to a 2-torus, a compact three-dimensional manifold enclosed by a two-dimensional surface. The implications of this result are relevant for many fields of economic theory. It is first of all clear that if we choose to initialize the dynamics in the basin of attraction of this trapping region, a continuum of perfect foresight solutions may be observed. A simple econometric exercise, linking the physical-to-human capital ratio (state variable) to the 5-years forward variance of the growth rate of an unbalanced sample of 183 countries, seems to provide empirical backing for the phenomenon. Other important consequences, relevant from the point of view of endogenous cycles theory, are also scrutinized in the paper.

Highlights

A large body of literature has clearly established that the equilibrium conditions of twosector continuous-time endogenous growth models with market imperfections do not always determine a unique perfect foresight path
The opposite occurs for human capital and working time, whose fluctuations are more rapid
This paper shows that the Lucas (1988) growth model with flow externalities à la Chamley (1993) may give rise to global indeterminacy of the equilibrium for reasons different than the existence of closed loops in well-located two-dimensional manifolds

Summary

Introduction

A large body of literature has clearly established that the equilibrium conditions of twosector continuous-time endogenous growth models with market imperfections do not always determine a unique perfect foresight path. The properties of a singularity of type (ii) have been used by Mattana et al (2009) and by Bella and Mattana (2014) to achieve global indeterminacy results for a variant of the model with = 0 In the former contribution, the Kopell and Howard (1975) theorem is used to show that there are regions of the parameter space in which there exists either a closed loop or a homoclinic orbit in a welllocated reduced manifold. 2, by exploiting the Gavrilov-Guckenheimer bifurcation theorem, we construct a correspondence with a simple topological-equivalent truncated planar system in cylindrical coordinates

The onset of a 2‐torus

Implications of toroidal motion for business cycles

A global indeterminacy result in R3

Conclusions