On Lipschitz continuity of the iterated function system in a stochastic optimal growth model

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

“…We can thus conclude that (19) and (20) are definitely the optimal policies for human and physical capital respectively. Note that, not surprisingly, such optimal policies simplify into those found in La Torre et al (2011) whenever factor shares are completely deterministic.…”

“…A growing number of studies has recently focused on characterizing the conditions under which this might occur by relying on the iterated function system (IFS) literature (Hutchinson, 1981;Vrscay, 1991;Barnsley, 1993). Most of the existing works dealing with economic growth and IFS analyze the traditional discrete time one-sector growth model with logarithmic utility and Cobb-Douglas production, in either its simplest setup or in slightly 1 extended formulations; such a basic model through an appropriate transformation can be converted into a one-dimensional IFS, and it is thus possible to show that its optimal dynamic path may converge to a singular measure supported on a Cantor set, and also that the invariant probability may be either singular or absolute continuous according to specific parameter configurations (Montrucchio and Privileggi, 1999;Mitra et al, 2003;Mitra and Privileggi, 2004, 2006, 2009Marsiglio, 2012;Privileggi and Marsiglio, 2013). Very few are those that instead consider more sophisticated two-sector growth models giving rise to a two-dimensional IFS; the analysis in this framework is clearly more complicated but it is still possible to show that the optimal dynamic path may converge to a singular measure supported on some fractal set, like the Sierpinski gasket, and to eventually characterize singularity versus absolute continuity of the invariant probability (La Torre et al, 2011; La Torre et al, 2015).…”

Environmental shocks and sustainability in a basic economy-environment model

“…Mitra et al (2003) analysed a one-sector growth model with two random shocks whose optimal trajectory may converge to a singular distribution supported on a Cantor set, also characterising singularity versus absolute continuity of the invariant probability measure. Mitra and Privileggi (2004, 2006, 2009 further generalised that result by establishing conditions on the fundamentals under which the invariant probability happens to be supported on a Cantor set, and provided an estimate of the Lipschitz constant for the optimal policy. La and Marsiglio (2012) showed that the optimal path of a two-sector growth model subject to random shocks may converge to an invariant distribution supported on some fractal set.…”

“…Hence, at each iteration of the IFS (5) a hole appear in all components of the pre-fractal, so that in the limit a Cantor-like set appears as unique invariant set for the system [see, e.g., Falconer (1997Falconer ( , 2003, Mitra et al (2003), and Mitra and Privileggi (2004, 2006, 2009 for details]. ■ By asymmetric Cantor set here we mean a Cantor-like set in which the components in the left tail are smaller than those in the right tail; this is due to the different slopes of the linear maps φ i in (5), the slope of the lower map φ 0 being smaller than that of the upper map φ 1 , as illustrated in Figure 1 further on.…”

Self-similar measures in multi-sector endogenous growth models

“…In the latter case, whenever 0 < α < 1/2 the attractor A * of (13) is a generalized topological Cantor set contained in (σq) 1/(1−α) , σ 1/(1−α) . (13) is the same as the no-overlap property 0 ≤ α < 1/2 for the IFS (15): as both maps f 0 , f 1 are strictly increasing, the former property (see Mitra and Privileggi, 2009) can be written as…”