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

In this paper we use global analysis techniques to analyze an economic growth model with environmental negative externalities, giving rise to a three-dimensional dynamic system (the framework is the one introduced by Wirl (1997) [53]). The dynamics of our model admits a locally attracting stationary state P1⁎, which is, in fact, a poverty trap  , coexisting with another stationary state P2⁎ possessing saddle-point stability. Global dynamical analysis shows that, under some conditions on the parameters, if the initial values of the state variables are close enough to the coordinates of P1⁎, then there exists a continuum of equilibrium trajectories approaching P1⁎ and one trajectory approaching P2⁎. Therefore, our model exhibits global indeterminacy, since either P1⁎ or P2⁎ can be selected according to agent expectations. Moreover, we prove that conditions guaranteeing the attractivity of P1⁎ also imply the saddle-point stability of P2⁎. However, when P1⁎ is not attractive, numerical simulations show the possible existence of one or two limit cycles: an attractive one surrounding P1⁎ and one endowed with a two-dimensional stable manifold surrounding P2⁎.