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

• * Corresponding author: Fabio Privileggi
• 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.

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

 Citation: • • 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 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 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 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 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 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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