Article Contents

2021, Volume 26, Issue 11: 5873-5903. Doi: 10.3934/dcdsb.2021070

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Public debt dynamics under ambiguity by means of iterated function systems on density functions

1.
SKEMA Business School and Université Côte d'Azur, Sophia Antipolis, France
2.
Department of Economics and Management, University of Pisa, Pisa, Italy
3.
Department of Mathematics and Statistics, Acadia University, Wolfville, Canada
4.
Department of Economics and Statistics "Cognetti de Martiis", University of Turin, 10153 Torino, Italy

^* Corresponding author: Fabio Privileggi
^* Corresponding author: Fabio Privileggi

Received: September 2020

Revised: November 2020

Early access: March 2021

Published: November 2021

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Abstract

We analyze a purely dynamic model of public debt stabilization under ambiguity. We assume that the debt to GDP ratio is described by a random variable, and thus it can be characterized by investigating the evolution of its density function through iteration function systems on mappings. Ambiguity is associated with parameter uncertainty which requires policymakers to respond to such an additional layer of uncertainty according to their ambiguity attitude. We describe ambiguity attitude through a simple heuristic rule in which policymakers adjust the available vague information (captured by the empirical distribution of the debt ratio) with a measure of their ignorance (captured by the uniform distribution). We show that such a model generates fractal-type objects that can be characterized as fixed-point solutions of iterated function systems on mappings. Ambiguity is a source of unpredictability in the long run outcome since it introduces some singularity features in the steady state distribution of the debt ratio. However, the presence of some ambiguity aversion removes such unpredictability by smoothing out the singularities in the steady state distribution.

Keywords:

Dynamical systems,
global attractor,
generalized fractal transform,
fixed point equation,
iterated function systems on density functions,
ambiguity,
public debt.

Mathematics Subject Classification: Primary: 28A80; Secondary: 91B64.

Citation:

Full Text(HTML)

Figure 1. First $ 7 $ iterations of operator $ T^{*} $ defined in (11) for the only map $ w\left(x\right) = \frac{1}{2}x+\frac{1}{4} $ together with the only greyscale map $ \phi\left(y\right) = 2y $ starting from $ u_{0}\left(x\right) = 12\left(x-\frac{1}{2}\right)^2 $

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Figure 2. cumulative distribution functions associated to the densities $ u_{t} $ in Figure 1

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Figure 3. First $ 7 $ iterations of operator $ T^{*}_{2} $ defined in (16) for the maps $ w_{1}\left(x\right) = \frac{1}{4}x $ and $ w_{2}\left(x\right) = \frac{3}{4}x+\frac{1}{4} $ together with greyscale maps $ \phi_{1}\left(y\right) = 2y $ and $ \phi_{2}\left(y\right) = \frac{2}{3}y $ starting from $ u_{0}\left(x\right)\equiv1 $

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Figure 4. cumulative distribution functions associated to the densities $ u_{t} $ in Figure 3

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Figure 5. First $ 7 $ iterations of operator $ T^{*}_{2} $ defined in (16) for the maps $ w_{1}\left(x\right) = \frac{1}{4}x $ and $ w_{2}\left(x\right) = \frac{3}{4}x+\frac{1}{4} $ together with greyscale maps $ \phi_{1}\left(y\right) = 2y $ and $ \phi_{2}\left(y\right) = \frac{2}{3}y $ starting from $ u_{0}\left(x\right) = 3.3852e^{-\left (6x-3\right)^2} $

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Figure 6. cumulative distribution functions associated to the densities $ u_{t} $ in Figure 5

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Figure 7. a) $ 5^{th} $ and b) $ 7^{th} $ iteration of operator $ T^{*} _{2} $ defined in (16) for the wavelets maps $ w_{1}\left(x\right ) = \frac{1}{2}x $ and $ w_{2}\left(x\right) = \frac{1}{2}x+\frac{1}{2} $ together with greyscale maps $ \phi_{1}\left(y\right) = \phi_{2}\left(y\right) = y $ starting from $ u_{0}\left(x\right) = 3x^{2} $, c) and d) cumulative distribution functions associated to the densities $ u_{5} $ and $ u_{7} $

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Figure 8. a) $ 5^{th} $ iteration of operator $ T^{*}_{3} $ defined in (16) for the maps $ w_{1}\left(x\right) = \frac{1}{8}x $, $ w_{2}\left(x\right) = \frac{3}{8}x+\frac{1}{8} $ and $ w_{3}\left(x\right) = \frac{1}{2}x+\frac{1}{2} $ together with greyscale maps $ \phi_{1}\left(y\right) = \frac{8}{3}y $, $ \phi_{2}\left(y\right) = \frac{8}{9}y $ and $ \phi_{3}\left(y\right) = \frac{2}{3}y $ starting from $ u_{0}\left(x\right)\equiv 1 $, b) its associated cumulative distribution function

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Figure 9. First $ 7 $ iterations of operator $ T $ defined in (19) for the maps $ w_{1}\left(x\right) = \frac{1}{4}x $ and $ w_{2}\left(x\right) = \frac{3}{4}x+\frac{1}{4} $ together with greyscale maps $ \phi_{1}\left(y\right) = \frac{2}{5}y+\frac{8}{5} $ and $ \phi_{2}\left(y\right) = \frac{2}{3}y $ starting from $ u_{0}\left(x\right) \equiv1 $

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Figure 10. cumulative distribution functions associated to the densities $ u_{t} $ in Figure 9

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Figure 11. First $ 7 $ iterations of operator $ T $ defined in (19) for the maps $ w_{1}\left(x\right) = \frac{1}{4}x $ and $ w_{2}\left(x\right) = \frac{3}{4}x+\frac{1}{4} $ together with greyscale maps $ \phi_{1}\left(y\right) = \frac{2}{5}y+\frac{8}{5} $ and $ \phi_{2}\left(y\right) = \frac{2}{3}y $ starting from $ u_{0}\left(x\right) = 3.3852e^{-\left(6x-3\right)^2} $

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Figure 12. cumulative distribution functions associated to the densities $ u_{t} $ in Figure 11

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Figure 13. a) $ 7^{th} $ iteration of operator $ T $ defined in (19) for the maps $ w_{1}\left(x\right) = \frac{1}{4}x $ and $ w_{2}\left(x\right) = \frac{3}{4}x+\frac{1}{4} $ together with greyscale maps $ \phi_{1}\left(y\right) = y+1 $ and $ \phi_{2}\left(y\right) = \frac{1}{3}y+\frac{1}{3} $ starting from $ u_{0}\left(x\right)\equiv 1 $, b) its associated cumulative distribution function

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Figure 14. a) $ 7^{th} $ iteration of operator $ T $ defined in (19) for the maps $ w_{1}\left(x\right) = \frac{1}{2}x $ and $ w_{2}\left(x\right) = \frac{1}{2}x+\frac{1}{2} $ together with greyscale maps $ \phi_{1}\left(y\right) = \frac{2}{5}y+\frac{3}{5} $ and $ \phi_{2}\left(y\right) = \frac{4}{5}y+\frac{1}{5} $ starting from $ u_{0}\left(x\right) = 12\left(x-\frac{1}{2}\right)^2 $, b) its associated cumulative distribution function

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Figure 15. a) $ 7^{th} $ iteration of operator $ T $ defined in (19) for the maps $ w_{1}\left(x\right) = \frac{1}{2}x $ and $ w_{2}\left(x\right) = \frac{1}{2} x+\frac{1}{2} $ together with greyscale maps $ \phi_{1}\left(y\right) = \phi_{2}\left(y\right) = \frac{1}{2}y+\frac{1}{2} $ starting from $ u_{0}\left(x\right) = 12\left (x-\frac{1}{2}\right)^2 $, b) its associated cumulative distribution function

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Figure 16. a) $ 5^{th} $ iteration of operator $ T $ defined in (19) for the maps $ w_{1}\left(x\right) = \frac{1}{8}x $, $ w_{2}\left(x\right) = \frac{3}{8}x+\frac {1}{8} $ and $ w_{3}\left(x\right) = \frac{1}{2}x+\frac{1}{2} $ together with greyscale maps $ \phi_{1}\left(y\right) = \frac{8}{15}y+\frac {32}{15} $, $ \phi_{2}\left(y\right) = \frac{16}{45}y+\frac{8}{15} $ and $ \phi_{3} \left(y\right) = \frac{2}{3}y $ starting from $ u_{0}\left(x\right)\equiv1 $, b) its associated cumulative distribution function

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Figure 17. a) $ 5^{th} $ iteration of operator $ T $ defined in (19) for the maps $ w_{1}\left(x\right) = \frac{1}{8}x $, $ w_{2}\left(x\right) = \frac{3}{8}x+\frac {1}{8} $ and $ w_{3}\left(x\right) = \frac{1}{2}x+\frac{1}{2} $ together with greyscale maps $ \phi_{1}\left(y\right) = \frac{32}{15}y+\frac {8}{15} $, $ \phi_{2}\left(y\right) = \frac{4}{45}y+\frac{4}{5} $ and $ \phi_{3}\left(y\right) = \frac{1}{15}y+\frac{3}{5} $ starting from $ u_{0}\left(x\right)\equiv1 $, b) its associated cumulative distribution function

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