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A predator-prey economic system of tax evasion and corrupt behavior

  • * Corresponding author: Elvio Accinelli

    * Corresponding author: Elvio Accinelli 
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  • The goal of this research is to study an economic system characterized by the coexistence of tax evading citizens with corrupt auditors (public officials), that is a system of predators-preys, i.e. corrupts and not, evaders and not. We show that citizens choose non-corrupt behavior when their utility exceeds a threshold level with respect to an optimal progressive tax rate. On the other hand, auditors choose non-corrupt behavior if their utility is higher than a certain threshold level depending on the fines and probabilities of being sanctioned. Being aware of this situation, a government or central authority must study a dynamic evolutionary economic system (a generalized predator-prey model) by analyzing all possible steady states of this economic system. The stability of the steady states depends on the crucial parameters of the model, namely: i) a socially negative impact exerted by corrupt public officials or auditors on the evolution of the economic society, ii) the impact of the social weight of the citizens who pay taxes on corrupt auditors, iii) the behavior of imitation of the agents when perceiving the possibilities of being sanctioned or not, and iv) the inertia and feedback of the corrupt system (dishonest behavior and tax evasion). We conclude by highlighting the importance of bifurcations or qualitative changes in the parameters of this economic system to fight corruption and evasion.

    Mathematics Subject Classification: MSC2020 codes: 91-10, 91-11, 91A22, 91B15.


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  • Figure 1.  Perceived / expected risk of being detected. Source: Special Eurobarometer 402: Undeclared work in the European Union. How would you describe the risk of being detected in the economy https://data.europa.eu/euodp/en/data/dataset/S1080_79_2_402

    Figure 2.  Share of firms experiencing at least one bribe payment request (2017 or latest available). Source: World Bank Enterprise Survey. https://doi.org/10.1787/80ebe6e6-en

    Figure 3.  The interior equilibrium, $ (\alpha^*, \delta^*) $, into the square $ C = \left\{ (\alpha, \delta): 0 \leq \alpha \leq 1, \; 0 \leq \delta \leq 1\right\} $. The blue line is the nullcline $ \dot{\alpha} = \alpha (0.8+2.5\alpha-2.8\delta) = 0 $, and the dashed blue line is the nullcline $ \dot{\delta} = \delta(0.4-2\alpha+0.7\delta) = 0 $, the slopes satify $ \frac{b}{c} = \frac{2.5}{2.8}<\frac{e}{f} = \frac{2}{0.7} $. In the figure on the right we can notice that the vertex equilibrium $ (0, 0) $ is a source, but also the upper-border equilibrium $ (\frac{c-a}{b}, 1) = (0.88, 1) $ is unstable, and the other equilibria flee outside the unit square

    Figure 4.  The interior equilibrium, $ (\alpha^*, \delta^*) $, into the square $ C = \left\{ (\alpha, \delta): 0 \leq \alpha \leq 1, \; 0 \leq \delta \leq 1\right\} $. The blue line is the nullcline $ \dot{\alpha} = \alpha (1.05+5.5\alpha-4\delta) = 0 $, and the dashed blue line is the nullcline $ \dot{\delta} = \delta(1.05-14.5\alpha+4.5\delta) = 0 $, such that now the basin of attraction towards the socially good state equilibrium $ (1, 0) $ corresponding to taxpaying citizens and non-corrupt auditors, has been enlarged expanded

    Figure 5.  The nullclines do not intersect. The blue line is the nullcline $ \dot{\alpha} = \alpha(1.05+4.5 \alpha - 4 \delta) = 0 $, and the dashed blue line is the nullcline $ \dot{\delta} = \delta(1.05-3.5\alpha+4.5 \delta) = 0 $

    Figure 6.  The nullclines do not intersect. The blue line is the nullcline $ \dot{\alpha} = \alpha(1.05+5.5 \alpha - 4 \delta) = 0 $, and the dashed blue line is the nullcline $ \dot{\delta} = \delta(1.05-7.5\alpha+4.5 \delta) = 0 $

    Figure 7.  The nullclines do not intersect, and specifically the nullcline $ \nu $ is outside the unitary square. The blue line is the nullcline $ \dot{\alpha} = \alpha(1.05+2 \alpha - 17 \delta) = 0 $

    Figure 8.  The phase portrait on the left shows the transcritical and Saddle-node bifurcation in $ (\alpha^*, \delta^*) = (\alpha_0, \delta_0) = (0, 0) $, the blue line is the nullcline $ \dot{\alpha} = \alpha( \alpha - 2 \delta) = 0 $, and the dashed blue line is the nullcline $ \dot{\delta} = \delta(-3\alpha+2 \delta) = 0 $. The phase portrait on the right shows the transcritical and Saddle-node bifurcation in $ (\alpha^*, \delta^*) = (\alpha_1, \delta_1) = (1, 1) $, the blue line is the nullcline $ \dot{\alpha} = \alpha(1+ 1 \alpha - 2 \delta) = 0 $, and the dashed blue line is the nullcline $ \dot{\delta} = \delta(1-3\alpha+2 \delta) = 0 $

    Figure 9.  Saddle-node bifurcation in $ (\alpha^*, \delta^*) = (\alpha_0, \delta_1) = (0, 1) $. The phase portrait on the left shows how the interior equilibrium approaches the vertex equilibrium $ (0, 1) $, where its blue line is the nullcline $ \dot{\alpha} = \alpha(1.4+ 2.7 \alpha - 2 \delta) = 0 $, and its blue dashed line is the nullcline $ \dot{\delta} = \delta(0.8-10 \alpha+1 \delta) = 0 $. While in the two phase portraits on the right the bifurcation conditions are satisfied. There, the blue line is nullcline $ \dot{\alpha} = \alpha(1+ 1 \alpha - 1 \delta) = 0 $, and the dashed blue line is the nullcline $ \dot{\delta} = \delta ( 0.001-8\alpha+0.01\delta) = 0 $

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