doi: 10.3934/dcdsb.2021070

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

Received  September 2020 Revised  November 2020 Published  March 2021

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.

Citation: Davide La Torre, Simone Marsiglio, Franklin Mendivil, Fabio Privileggi. Public debt dynamics under ambiguity by means of iterated function systems on density functions. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021070
References:
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S. R. BakerN. Bloom and S.J. Davis, Measuring economic policy uncertainty, Quarterly Journal of Economics, 131 (2015), 1593-1636.  doi: 10.3386/w21633.  Google Scholar

[2]

S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équationsintégrales, Fundamenta Mathematicae, 3 (1922), 133-181.  doi: 10.4064/fm-3-1-133-181.  Google Scholar

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W. Brainard, Uncertainty and the effectiveness of policy, American Economic Review, 57 (1967), 411-425.   Google Scholar

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W. A. Brock and S. Durlauf, Macroeconomics and Model Incertainty, in D. Colander (Ed.), Post Walrasian Macroeconomics: Beyond the Dynamic Stochastic General Equilibrium Model, Cambridge University Press, Cambridge, 2006. Google Scholar

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F. Caprioli, Optimal fiscal policy under learning, Jouirnal of Economics Dynamics & Control, 58 (2015), 101-124.  doi: 10.1016/j.jedc.2015.05.008.  Google Scholar

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G. Cozzi and P. E. Giordani, Ambiguity attitude, R & D investments and economic growth, Journal of Evolutionary Economics, 21 (2011), 303-319.  doi: 10.1007/s00191-010-0217-x.  Google Scholar

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D. Ellsberg, Risk, ambiguity and the savage axioms, Quarterly Journal of Economics, 75 (1961), 643-669.  doi: 10.2307/1884324.  Google Scholar

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J. EtnerM. Jeleva and J.-M. Tallon, Decision theory under ambiguity, Journal of Economic Surveys, 26 (2012), 234-270.  doi: 10.1111/j.1467-6419.2010.00641.x.  Google Scholar

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B. Forte and E. R. Vrscay, Solving the inverse problem for function and image approximation using iterated function systems, Dynamics of Continuous, Discrete and Impulsive Systems, 1 (1995), 177-232.   Google Scholar

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B. Forte and E. R. Vrscay, Theory of Generalized Fractal Transforms, in Y. Fisher (Ed.), Fractal Image Encoding and Analysis, NATO ASI Series F, Springer Verlag, New York, 1998. Google Scholar

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P. GhirardatoF. Maccheroni and M. Marinacci, Differentiating ambiguity and ambiguity attitude, Journal of Economic Theory, 118 (2004), 133-173.  doi: 10.1016/j.jet.2003.12.004.  Google Scholar

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J. Hollmayr and C. Matthes, Learning about fiscal policy and the effects of policy uncertainty, Journal of Economic Dynamics & Control, 59 (2015), 142-162.  doi: 10.1016/j.jedc.2015.08.002.  Google Scholar

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A. G. Karantounias, Managing pessimistic expectations and fiscal policy, Theorertical Economics, 8 (2013), 193-231.  doi: 10.3982/TE899.  Google Scholar

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H. Kunze, D. La Torre, F. Mendivil and E. R. Vrscay, Fractal-Based Methods in Analysis, Springer, New York, 2012. doi: 10.1007/978-1-4614-1891-7.  Google Scholar

[21]

D. La TorreS. Marsiglio and F. Privileggi, Fractals and self-similarity in economics: the case of a stochastic two-sector growth model, Image Analysys and Stereology, 30 (2011), 143-151.  doi: 10.5566/ias.v30.p143-151.  Google Scholar

[22]

D. La TorreS. MarsiglioF. Mendivil and F. Privileggi, Self-similar measures in multi-sector endogenous growth models, Chaos, Solitons and Fractals, 79 (2015), 40-56.  doi: 10.1016/j.chaos.2015.05.019.  Google Scholar

[23]

D. La Torre, S. Marsiglio and F. Privileggi, Fractal attractors in economic growth models with random pollution externalities, Chaos, 28 (2018), 055916, 12 pp. doi: 10.1063/1.5023782.  Google Scholar

[24]

D. La TorreS. MarsiglioF. Mendivil and F. Privileggi, Fractal attractors and singular invariant measures in two-sector growth models with random factor shares, Communications in Nonlinear Science and Numerical Simulation, 58 (2018), 185-201.  doi: 10.1016/j.cnsns.2017.07.008.  Google Scholar

[25]

D. La Torre and S. Marsiglio, A note on optimal debt reduction policies, Macroeconomic Dynamics, 24 (2020), 1850-1860.  doi: 10.1017/S1365100519000014.  Google Scholar

[26]

T. MitraL. Montrucchio and F. Privileggi, The nature of the steady state in models of optimal growth under uncertainty, Economic Theory, 23 (2004), 39-71.  doi: 10.1007/s00199-002-0340-5.  Google Scholar

[27]

T. Mitra and F. Privileggi, Cantor Type Invariant Distributions in the Theory of Optimal Growth under Uncertainty, Journal of Difference Equations and Applications, 10 (2004), 489-500.  doi: 10.1080/1023619042000193649.  Google Scholar

[28]

T. Mitra and F. Privileggi, Cantor type attractors in stochastic growth models, Chaos, Solitons and Fractals, 29 (2006), 626-637.  doi: 10.1016/j.chaos.2005.08.094.  Google Scholar

[29]

T. Mitra and F. Privileggi, On Lipschitz continuity of the iterated function system in a stochastic optimal growth model, Journal of Mathematical Economics, 45 (2009), 185-198.  doi: 10.1016/j.jmateco.2008.08.003.  Google Scholar

[30]

L. Montrucchio and F. Privileggi, Fractal steady states in stochastic optimal control models, Annals of Operations Research, 88 (1999), 183-197.  doi: 10.1023/A:1018978213041.  Google Scholar

[31]

L. J. Olson and S. Roy, Theory of stochastic optimal economic growth, in R. A. Dana, C. Le Van, T. Mitra and K. Nishimura (Eds.), Handbook on optimal growth 1: discrete time, Springer, New York (2005), 297–335. doi: 10.1007/3-540-32310-4_11.  Google Scholar

[32]

F. Privileggi and S. Marsiglio, Environmental shocks and sustainability in a basic economy-environment model, International Journal of Applied Nonlinear Science, 1 (2013), 67-75.  doi: 10.1504/IJANS.2013.052755.  Google Scholar

[33]

D. Rodrik, Policy uncertainty and private investment, Journal of Development Economics, 36 (1991), 229-242.   Google Scholar

show all references

References:
[1]

S. R. BakerN. Bloom and S.J. Davis, Measuring economic policy uncertainty, Quarterly Journal of Economics, 131 (2015), 1593-1636.  doi: 10.3386/w21633.  Google Scholar

[2]

S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équationsintégrales, Fundamenta Mathematicae, 3 (1922), 133-181.  doi: 10.4064/fm-3-1-133-181.  Google Scholar

[3] M. F. Barnsley, Fractals Everywhere, Academic Press, New York, 1988.   Google Scholar
[4]

B. Born and J. Pfeifer, Policy risk and the business cycle, Journal of Monetary Economics, 68 (2014), 68-85.  doi: 10.1016/j.jmoneco.2014.07.012.  Google Scholar

[5]

W. Brainard, Uncertainty and the effectiveness of policy, American Economic Review, 57 (1967), 411-425.   Google Scholar

[6]

W. A. Brock and S. Durlauf, Macroeconomics and Model Incertainty, in D. Colander (Ed.), Post Walrasian Macroeconomics: Beyond the Dynamic Stochastic General Equilibrium Model, Cambridge University Press, Cambridge, 2006. Google Scholar

[7]

W. A. Brock and L. J. Mirman, Optimal economic growth and uncertainty: The discounted case, Journal of Economic Theory, 4 (1972), 479-513.  doi: 10.1016/0022-0531(72)90135-4.  Google Scholar

[8]

C. Camerer and M. Weber, Recent developments in modeling preferences: uncertainty and ambiguity, Journal of Risk and Uncertainty, 5 (1992), 325-370.  doi: 10.1007/BF00122575.  Google Scholar

[9]

F. Caprioli, Optimal fiscal policy under learning, Jouirnal of Economics Dynamics & Control, 58 (2015), 101-124.  doi: 10.1016/j.jedc.2015.05.008.  Google Scholar

[10]

G. Cozzi and P. E. Giordani, Ambiguity attitude, R & D investments and economic growth, Journal of Evolutionary Economics, 21 (2011), 303-319.  doi: 10.1007/s00191-010-0217-x.  Google Scholar

[11]

D. Ellsberg, Risk, ambiguity and the savage axioms, Quarterly Journal of Economics, 75 (1961), 643-669.  doi: 10.2307/1884324.  Google Scholar

[12]

J. EtnerM. Jeleva and J.-M. Tallon, Decision theory under ambiguity, Journal of Economic Surveys, 26 (2012), 234-270.  doi: 10.1111/j.1467-6419.2010.00641.x.  Google Scholar

[13]

B. Forte and E. R. Vrscay, Solving the inverse problem for function and image approximation using iterated function systems, Dynamics of Continuous, Discrete and Impulsive Systems, 1 (1995), 177-232.   Google Scholar

[14]

B. Forte and E. R. Vrscay, Theory of Generalized Fractal Transforms, in Y. Fisher (Ed.), Fractal Image Encoding and Analysis, NATO ASI Series F, Springer Verlag, New York, 1998. Google Scholar

[15]

D. Frisch and J. Baron, Ambiguity and rationality, Journal of Behavioral Decision Making, 1 (1988), 149-157.  doi: 10.1002/bdm.3960010303.  Google Scholar

[16]

P. GhirardatoF. Maccheroni and M. Marinacci, Differentiating ambiguity and ambiguity attitude, Journal of Economic Theory, 118 (2004), 133-173.  doi: 10.1016/j.jet.2003.12.004.  Google Scholar

[17] L. P. Hansen and T. J. Sargent, Robustness, Princeton University Press, Princeton, 2008.  doi: 10.1515/9781400829385.  Google Scholar
[18]

J. Hollmayr and C. Matthes, Learning about fiscal policy and the effects of policy uncertainty, Journal of Economic Dynamics & Control, 59 (2015), 142-162.  doi: 10.1016/j.jedc.2015.08.002.  Google Scholar

[19]

A. G. Karantounias, Managing pessimistic expectations and fiscal policy, Theorertical Economics, 8 (2013), 193-231.  doi: 10.3982/TE899.  Google Scholar

[20]

H. Kunze, D. La Torre, F. Mendivil and E. R. Vrscay, Fractal-Based Methods in Analysis, Springer, New York, 2012. doi: 10.1007/978-1-4614-1891-7.  Google Scholar

[21]

D. La TorreS. Marsiglio and F. Privileggi, Fractals and self-similarity in economics: the case of a stochastic two-sector growth model, Image Analysys and Stereology, 30 (2011), 143-151.  doi: 10.5566/ias.v30.p143-151.  Google Scholar

[22]

D. La TorreS. MarsiglioF. Mendivil and F. Privileggi, Self-similar measures in multi-sector endogenous growth models, Chaos, Solitons and Fractals, 79 (2015), 40-56.  doi: 10.1016/j.chaos.2015.05.019.  Google Scholar

[23]

D. La Torre, S. Marsiglio and F. Privileggi, Fractal attractors in economic growth models with random pollution externalities, Chaos, 28 (2018), 055916, 12 pp. doi: 10.1063/1.5023782.  Google Scholar

[24]

D. La TorreS. MarsiglioF. Mendivil and F. Privileggi, Fractal attractors and singular invariant measures in two-sector growth models with random factor shares, Communications in Nonlinear Science and Numerical Simulation, 58 (2018), 185-201.  doi: 10.1016/j.cnsns.2017.07.008.  Google Scholar

[25]

D. La Torre and S. Marsiglio, A note on optimal debt reduction policies, Macroeconomic Dynamics, 24 (2020), 1850-1860.  doi: 10.1017/S1365100519000014.  Google Scholar

[26]

T. MitraL. Montrucchio and F. Privileggi, The nature of the steady state in models of optimal growth under uncertainty, Economic Theory, 23 (2004), 39-71.  doi: 10.1007/s00199-002-0340-5.  Google Scholar

[27]

T. Mitra and F. Privileggi, Cantor Type Invariant Distributions in the Theory of Optimal Growth under Uncertainty, Journal of Difference Equations and Applications, 10 (2004), 489-500.  doi: 10.1080/1023619042000193649.  Google Scholar

[28]

T. Mitra and F. Privileggi, Cantor type attractors in stochastic growth models, Chaos, Solitons and Fractals, 29 (2006), 626-637.  doi: 10.1016/j.chaos.2005.08.094.  Google Scholar

[29]

T. Mitra and F. Privileggi, On Lipschitz continuity of the iterated function system in a stochastic optimal growth model, Journal of Mathematical Economics, 45 (2009), 185-198.  doi: 10.1016/j.jmateco.2008.08.003.  Google Scholar

[30]

L. Montrucchio and F. Privileggi, Fractal steady states in stochastic optimal control models, Annals of Operations Research, 88 (1999), 183-197.  doi: 10.1023/A:1018978213041.  Google Scholar

[31]

L. J. Olson and S. Roy, Theory of stochastic optimal economic growth, in R. A. Dana, C. Le Van, T. Mitra and K. Nishimura (Eds.), Handbook on optimal growth 1: discrete time, Springer, New York (2005), 297–335. doi: 10.1007/3-540-32310-4_11.  Google Scholar

[32]

F. Privileggi and S. Marsiglio, Environmental shocks and sustainability in a basic economy-environment model, International Journal of Applied Nonlinear Science, 1 (2013), 67-75.  doi: 10.1504/IJANS.2013.052755.  Google Scholar

[33]

D. Rodrik, Policy uncertainty and private investment, Journal of Development Economics, 36 (1991), 229-242.   Google Scholar

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 $
Figure 1">Figure 2.  cumulative distribution functions associated to the densities $ u_{t} $ in Figure 1
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 $
Figure 3">Figure 4.  cumulative distribution functions associated to the densities $ u_{t} $ in Figure 3
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} $
Figure 5">Figure 6.  cumulative distribution functions associated to the densities $ u_{t} $ in Figure 5
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} $
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
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 $
Figure 9">Figure 10.  cumulative distribution functions associated to the densities $ u_{t} $ in Figure 9
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} $
Figure 11">Figure 12.  cumulative distribution functions associated to the densities $ u_{t} $ in Figure 11
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
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
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
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
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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