Cantor type attractors in stochastic growth models

Mitra, Tapan; Privileggi, Fabio

doi:10.1016/j.chaos.2005.08.094

Cited by 12 publications

(13 citation statements)

References 14 publications

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“…Note that the condition α < φ required in Corollary 2 precludes the possibility of generating the standard, symmetric Sierpinski triangle with vertices(0, 0), (0, 1) and (1, 1) through (35), as its construction postulates that α = φ = 1/2 must hold. Hence, the attractor of (35) must always be a generalized Sierpinski gasket, that is, a Sierpinski gasket-like set whose prefractals 8 are composed by triangles that do not overlap for smaller values of α and φ, while they tend to overlap for larger values of either α or φ (see the examples in Section 5).…”

Section: Generalized Sierpinski Gaskets As Attractorsmentioning

confidence: 99%

“…9 However, by applying the main result of Section 2, Theorem 5, to the IFS (35) it is possible to establish the following sufficient condition for the singularity of the invariant measure. As there are three maps in the IFS (34), we can set only two out of the three probabilities associated to each map w i , say p 0 and p 1 , as the third must complement to 1:…”

Section: Singular Self-similar Measuresmentioning

confidence: 99%

“…In other words, it acts as a correction factor affecting variable ψ t in (35) in such a way so to balance the effects induced by all model parameters (factor shares and exogenous shocks' values), letting the log-transformation always work, for any feasible parameters' values. Whenever, as in Corollary 2, one puts some restrictions on the δ value, the range of applicability of Proposition 5 dramatically narrows, as heavy restrictions on parameters immediately become necessary in order to maintain the result of Proposition 5.…”

Section: Generalized Sierpinski Gaskets As Attractorsmentioning

confidence: 99%

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Self-similar measures in multi-sector endogenous growth models

Torre¹,

Marsiglio²,

Mendivil³

et al. 2015

Chaos, Solitons & Fractals

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We analyze two types of stochastic discrete time multi-sector endogenous growth models, namely a basic Uzawa-Lucas (1965, 1988 model and an extended three-sector version as in La Torre and Marsiglio (2010). As in the case of sustained growth the optimal dynamics of the state variables are not stationary, we focus on the dynamics of the capital ratio variables, and we show that, through appropriate log-transformations, they can be converted into affine iterated function systems converging to an invariant distribution supported on some (possibly fractal) compact set. This proves that also the steady state of endogenous growth models-i.e., the stochastic balanced growth path equilibrium-might have a fractal nature. We also provide some sufficient conditions under which the associated self-similar measures turn out to be either singular or absolutely continuous (for the three-sector model we only consider the singularity). AbstractWe analyze two types of stochastic discrete time multi-sector endogenous growth models, namely a basic Uzawa-Lucas (1988) model and an extended three-sector version as in La Torre and Marsiglio (2010). As in the case of sustained growth the optimal dynamics of the state variables are not stationary, we focus on the dynamics of the capital ratio variables, and we show that, through appropriate log-transformations, they can be converted into affine iterated function systems converging to an invariant distribution supported on some (possibly fractal) compact set. This proves that also the steady state of endogenous growth modelsi.e., the stochastic balanced growth path equilibrium-might have a fractal nature. We also provide some sufficient conditions under which the associated self-similar measures turn out to be either singular or absolutely continuous (for the three-sector model we only consider the singularity).

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Section: Generalized Sierpinski Gaskets As Attractorsmentioning

confidence: 99%

Section: Singular Self-similar Measuresmentioning

confidence: 99%

Section: Generalized Sierpinski Gaskets As Attractorsmentioning

confidence: 99%

See 1 more Smart Citation

Self-similar measures in multi-sector endogenous growth models

Torre¹,

Marsiglio²,

Mendivil³

et al. 2015

Chaos, Solitons & Fractals

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“…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.…”

Section: The Modelmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

Environmental shocks and sustainability in a basic economy-environment model

Privileggi

Marsiglio

2013

IJANS

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Abstract:We study a stochastic, discrete-time, economy-environment integrated model, where human activity affects the evolution of pollution over time. We assume that exogenous i.i.d. environmental shocks determine the rate of pollution transfer. We show that the pollution to capital ratio dynamics can be read as an iterated function system converging to an invariant distribution supported on a (asymmetric) Cantor set, and that human intervention aiming at offsetting the environmental impact of economic activities is needed to ensure sustainability.

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A stochastic economic growth model with health capital and state-dependent probabilities

Torre

Marsiglio

Mendivil

et al. 2019

Chaos, Solitons & Fractals

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We analyze a simple stochastic model of economic growth in which physical and health capital accumulation jointly contribute to determine long run economic growth. Health capital is subject to random shocks via the effects of behavioral changes: unpredictable changes in individuals' attitude toward healthy behaviors may reduce the effectiveness of health services provision; this in turn, by reducing the production of new health capital, lowers economic production activities negatively affecting economic growth. Unlike the extant literature, we assume that the probability with which such random shocks occur is not constant but state-dependent. Specifically, the probability that behavioral changes will negatively impact on health capital and economic growth depends on the level of economic development, proxied by the relative abundance of health capital with respect to physical capital. We show that our model's dynamics can be converted into an iterated function system with state-dependent probabilities which converges to an invariant self-similar measure supported on a (possibly fractal) compact attractor. We develop a numerical method to approximate the invariant distribution to illustrate its features in specific model's parametrizations, exemplifying thus the effects of state-dependent probabilities on the model's steady state.

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Cantor type attractors in stochastic growth models

Cited by 12 publications

References 14 publications

Self-similar measures in multi-sector endogenous growth models

Self-similar measures in multi-sector endogenous growth models

Environmental shocks and sustainability in a basic economy-environment model

A stochastic economic growth model with health capital and state-dependent probabilities

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