# American Institute of Mathematical Sciences

November  2021, 26(11): 5807-5825. doi: 10.3934/dcdsb.2021116

## Dynamical analysis of a banking duopoly model with capital regulation and asymmetric costs

 1 Polytechnic University of Marche, Department of Management, Piazzale R. Martelli 8, 60121, Ancona (AN), Italy 2 University of Modena and Reggio Emilia, Department of Economics Marco Biagi, Via Jacopo Berengario 51, 41121, Modena, Italy

* Corresponding author: giovanni.campisi@unimore.it

Received  October 2020 Revised  February 2021 Published  November 2021 Early access  April 2021

It is well known that regulation and efficiency are two important issues on banking literature. The goal of the paper is to analyse them through a banking duopoly model with heterogeneous expectations. To this purpose, we consider two scenarios. In the first one, we focus on regulation effects. In particular, empirical literature on Italian banks finds evidence on the asymmetry of the costs of regulation that penalize small banks with respect to the large ones. In this direction, we analyse a duopoly model where small banks and large banks have different forecasting rules and we capture the differences of the regulations' effects assuming asymmetry in the cost functions. We introduce linear cost function for small banks and quadratic cost function for large banks. In the second scenario, we study the relation between regulation and bank efficiency highlighting empirical results showing that large banks register higher level of inefficiency than small banks. Moreover, in order to stress new evidences and to confirm empirical results on banking regulation and efficiency, we conduct an analytical and numerical analysis.

Citation: Serena Brianzoni, Giovanni Campisi. Dynamical analysis of a banking duopoly model with capital regulation and asymmetric costs. Discrete & Continuous Dynamical Systems - B, 2021, 26 (11) : 5807-5825. doi: 10.3934/dcdsb.2021116
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In the $(c_1,c_2)-$plane the set of points between the red and blue curve ensures positive equilibrium loans, while the yellow curve represents the parameter values $(c_1,c_2)$ for which a flip bifurcation occurs. In Panel (a) the parameters are: $a = 3.2$, $\alpha = 1.5$, $b = 1.5$, $\gamma = 0.7$, $r_k = 2.8$. In panel (b) $\gamma = 0.1$ while the other parameters as in Panel (a)
Two interesting economic scenarios through bifurcation diagrams. In Panel (a) the parameters are: $a = 3$, $\alpha = 1.38$, $b = 0.12$, $\gamma = 0.2$, $r_k = 2.8$, $c_{1} = 0.15$ and $c_{2}\in (0.01,0.6)$. In Panel (b) $c_{1} = 0.1$ and $c_{2}\in (0.01,0.6)$ while the other parameters as in panel (a)
On the left is depicted the attractor of the 4-cycle for $\alpha = 1.25$, $\gamma = 0.35$, $r_{k} = 2.8$, $a = 3.2$, $b = 0.1$, $c_{1} = 0.08$, $c_{2} = 0.7$. On the right panel a bifurcation diagram for $c_{1}\in (1.4,2.1)$ showing the period-doubling of the 2-cycle for $\alpha = 2$, $\gamma = 0.25$, $r_{k} = 2.3$, $a = 3.2$, $b = 0.4$, $c_{2} = 1.1$
On the left, basins of attraction of the two-cycle, given the following values of the parameters, $\alpha = 1.35$, $\gamma = 0.7$, $r_{k} = 1.5$, $a = 3$, $b = 1.9$, $c_{1} = 0.01$, $c_{2} = 0.27$. On the right, basins of attraction showing the birth and the stability of a 4-cycle for $a = 3.27$ and the other parameters as in the left panel
In (a), four pieces chaotic attractor for $\alpha = 1.38$, $\gamma = 0.35$, $r_{k} = 2.8$, $a = 3$, $b = 0.12$, $c_{1} = 0.13$, $c_{2} = 0.78$. In (b), two pieces chaotic attractor for $c_{2} = 0.8$ and the other parameters as in Panel (a)
In (a), connected chaotic attractor for $c_{2} = 0.84$ and the other parameters as in Panel (a) of Figure 5. On the right, cycle cartogram in the ($c_{1},c_{2}$)-plane for parameter values as in Panel (a) of Figure 5
The time series of the loans of large (Panel (a)) and small (Panel (b)) Italian banks. The real time series are depicted in blue, while in red the time series of the loans resulting from the stochastic model
Monte Carlo simulations for analyse differences in loans' demand of the two banks for several combinations of costs and regulation parameters. In (a), $c_{1},c_{2}\in [0.01,0.4]$ and the other parameters as in Table 1. In (b) and (c), joint effect of the costs and regulation keeping fixed all other parameters as in Table 1. In (b), the loans' demand are plotted with respect to the costs of the two banks when $c_{1},c_{2}\in [0.01,0.3]$ and $\gamma\in[0.05,0.4]$. In (c) the loans' demand are plotted with respect to the parameter of regulation $\gamma$ when $c_{1},c_{2}\in [0.01,0.3]$ and $\gamma\in[0.05,0.4]$. In (d), the effects of regulation on the loans' demand when $\gamma\in [0.05,0.4]$ and the other parameters as in Table 1
Parameter setting and initial values
 $\alpha$ $\gamma$ $r_{k}$ $a$ $b$ $c_{1}$ $c_{2}$ $\sigma_{1}$ $\sigma_{1}$ 1.04 0.18 3.6 1.68 0.043 0.08 0.38 0.12 0.06
 $\alpha$ $\gamma$ $r_{k}$ $a$ $b$ $c_{1}$ $c_{2}$ $\sigma_{1}$ $\sigma_{1}$ 1.04 0.18 3.6 1.68 0.043 0.08 0.38 0.12 0.06
Summary statistics of banks' loans including mean, standard deviation (sd), skewness, minimum and maximum value for real time series of the loans of large banks ($L_{1}^{real}$), real time series of the loans of small banks ($L_{2}^{real}$) and the simulated stochastic time series of the loans for large ($L_{1}^{SS}$) and small ($L_{2}^{SS}$) banks
 Mean sd Min Max Skewness Kurtosis $L_{1}^{real}$ 291, 440, 000 64, 858, 000 192, 020, 000 356, 490, 000 -0.4471 1.6507 $L_{2}^{real}$ 224, 430, 000 25, 598, 000 186, 240, 000 263, 480, 000 -0.0867 1.7266 $L_{1}^{SS}$ 345, 330, 000 7, 315, 000 331, 320, 000 355, 280, 000 -0.4075 2.2085 $L_{2}^{SS}$ 224, 080, 000 3, 652, 000 216, 660, 000 230, 890, 000 -0.1410 2.9930
 Mean sd Min Max Skewness Kurtosis $L_{1}^{real}$ 291, 440, 000 64, 858, 000 192, 020, 000 356, 490, 000 -0.4471 1.6507 $L_{2}^{real}$ 224, 430, 000 25, 598, 000 186, 240, 000 263, 480, 000 -0.0867 1.7266 $L_{1}^{SS}$ 345, 330, 000 7, 315, 000 331, 320, 000 355, 280, 000 -0.4075 2.2085 $L_{2}^{SS}$ 224, 080, 000 3, 652, 000 216, 660, 000 230, 890, 000 -0.1410 2.9930
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