Fractal attractors and singular invariant measures in two-sector growth models with random factor shares

Abstract

We analyze a multi-sector growth model subject to random shocks affecting the two sector-specific production functions twofold: the evolution of both productivity and factor shares is the result of such exogenous shocks. We determine the optimal dynamics via Euler-Lagrange equations, and show how these dynamics can be described in terms of an iterated function system with probability. We also provide conditions that imply the singularity of the invariant measure associated with the fractal attractor. Numerical examples show how specific parameter configurations might generate distorted copies of the Barnsley's fern attractor.

Fractal attractors in economic growth models with random pollution externalities

“…Also new human capital is produced according to a Cobb-Douglas technology, which however employs only human capital (Lucas, 1988 andRebelo, 1991). As in the study by La Torre et al (2018), the production technologies of the final good and new human capital are directly affected by exogenous shocks which take both a multiplicative form through coefficients z t and g t , respectively, and an exponential form affecting the factor shares in both production functions; therefore, output is given by y t ¼ z t A t k a t t ðu t h t Þ c t , where a t and c t denote the random physical and human capital shares of income, respectively, while human capital is given by h tþ1 ¼ g t B t ½ð1 À u t Þh t / t , with / t denoting the random human capital share of human capital. In this formulation, A t and B t denote the pollution-induced productivity levels in the final and human capital sectors, respectively.…”

“…La Torre, D., Marsiglio, S. & Privileggi, F. (2018 We analyze a discrete time two-sector economic growth model where the production technologies in the final and human capital sectors are affected by random shocks both directly (via productivity and factor shares) and indirectly (via a pollution externality). We determine the optimal dynamics in the decentralized economy and show how these dynamics can be described in terms of a twodimensional affine iterated function system with probability.…”

“…Such a peculiarity of our framework, due to the presence of the pollution externality, allows us to identify a suitable parameter configuration capable of generating exactly the classical Barnsley's fern as the attractor of the log-linearized optimal dynamical system. To the best of our knowledge, no other macroeconomic model has ever been able to give rise to the Barnsley's fern [since the coefficients in their iterated function system can only take on positive values, La Torre et al (2018) model is able to generate only distorted copies of the fern, which ultimately do not even remotely resemble a fern] for some specific-but fully admissible-parameter configuration.…”

“…Most of these works analyze discrete time stochastic economic growth models with logarithmic utility and CobbDouglas production, either in a one-sector or two-sector framework, showing that through appropriate logtransformations their optimal dynamics can be converted into an affine iterated function system converging to a singular measure supported on some fractal set; in the case of unidimensional iterated function systems such an attractor can be the Cantor set (Montrucchio and Privileggi, 1999;Mitra et al, 2003;Mitra and Privileggi, 2004;2006;2009;Marsiglio, 2012;Privileggi and Marsiglio, 2013;and La Torre et al, 2015), while in the case of two-dimensional iterated function systems it can be either the Sierpinski gasket or distorted-copies of the Barnsley's fern (La Torre et al, 2011;2015;. With the exception of La Torre et al (2018) who considered also shocks affecting factor shares, in all these works randomness affects economic activities through the productivity channel, following the Brock and Mirman (1972) tradition. Some of these works also identify specific parameter configurations under which the invariant probability measure turns out to be either singular or absolutely continuous (Mitra et al, 2003 andLa Torre et al, 2015;.…”

“…Specifically, we build on the model by La Torre et al (2018) in which such direct effects have already been accounted for in order to allow for pollution to be an additional and indirect source of randomness. To the best of our knowledge, none of the extant works has ever considered how the presence of externalities complicates aggregate macroeconomic dynamics and what this may imply for the attractor of the associated iterated function system.…”