Fractals and Self-Similarity in Economics: The Case of a Stochastic Two-Sector Growth Model

Abstract

We study a stochastic, discrete-time, two-sector optimal growth model in which the production of the homogeneous consumption good uses a Cobb-Douglas technology, combining physical capital and an endogenously determined share of human capital. Education is intensive in human capital as in Lucas (1988), but the marginal returns of the share of human capital employed in education are decreasing, as suggested by Rebelo (1991). Assuming that the exogenous shocks are i.i.d. and affect both physical and human capital, we build specific configurations for the primitives of the model so that the optimal dynamics for the state variables can be converted, through an appropriate log-transformation, into an Iterated Function System converging to an invariant distribution supported on a generalized Sierpinski gasket.

Self-similar measures in multi-sector endogenous growth models

“…The results highlighted in Proposition 1 are pretty standard in the literature (see Bethmann, 2007Bethmann, , 2013La Torre et al, 2011). It is also very well-known that the Uzawa-Lucas (1988) framework, because of the linearity in the production of (new) human capital, may generate sustained long-run growth.…”

“…In all examples we keep constant the discount factor, β = 0.96, and the random shocks' values on the final consumption good production, q 1 = 0.2 and q 2 = 0.6; moreover, we set b = 1/ (1 − u − v)+0.01, where u and v are defined in (20) and (21) respectively, so to have always sustained growth. The first three examples cover the general setting envisaged by Proposition 5 for which we keep constant the random shock value on knowledge production, r = 0.5, while the last three satisfy the restrictions of Corollary 2-so that α < φ must hold and the r value is constrained to be given by (44)-and thus the IFS (34) produces generalized Sierpinski gaskets as attractors.…”

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

“…We contribute to this branch of literature by extending the analysis to a framework in which factor shares evolve randomly, in order to understand what this might imply for macroeconomic dynamics. For the sake of simplicity, we focus on the standard optimal growth model under uncertainty discussed in La Torre et al (2011) in which the social planner seeks to maximize the representative household's infinite discounted sum of instantaneous utility functions -which are assumed to be logarithmic -subject to the laws of motion of physical, k t , and human, h t , capital. At each time t, the planner chooses consumption, c t , and the share of human capital, u t , to allocate into production of the unique homogeneous consumption good which uses a Cobb-Douglas technology combining physical and human capital.…”

“…We wish to contribute to this literature by extending the analysis of two-sector random growth models and their relation with fractal steady states in order to allow for the random shock to affect not only the productivity level of the sector-specific production functions (La Torre et al, 2011Torre et al, , 2015, but also their factor shares. To the best of our knowledge, the possibility of exogenous shocks on factor shares thus far have been considered only in the onesector growth model by Mirman and Zilcha (1975), which has recently been extended to the case of learning by Mirman et al (2016).…”

“…Nonetheless, it is an interesting generalization of the traditional setup both from the economic and mathematical point of view; indeed, variable factor shares may describe the change in the structure of economic activities which we have observed in industrialized economies over the last decades (Nickell et al, 2008;Marsiglio et al, 2016), and also imply that the optimal economic dynamics may be characterized by an IFS with variable coefficients which makes the analysis of convergence and invariant probability properties not trivial at all. In order to look at this in the simplest possible setup we build on the model discussed in La Torre et al (2011) in which endogenous growth is ruled out (see La Torre et al, 2015, for a discussion of how results may differ in a framework with endogenous growth), and show that through an appropriate log-transformation the optimal nonlinear dynamic system can be converted into a topologically equivalent linear IFS, although such a transformation requires us to impose a substantial number of restrictions on the model's parameters. We can however show that the system converges to a singular measure supported on some fractal set, which (because of the imposed restrictions) turns out to be a distorted copy of Barnley's fern.…”