## Main Article Content

## Abstract

A loan benchmark interest rate policy always becomes a challenging problem in the banking industry since it has a role in controlling bank loan expansion, especially when there is competition between two banks. This paper aims to assess the influence of the loan benchmark interest rate on the expansion of loans between two banks. We present a banking duopoly model in the form of two-dimensional difference equations which is constructed from heterogeneous expectation, where one of the banks sets its optimal loan volume based on the other bank’s rational expectation. The model’s equilibrium is investigated, and its stability is analyzed using the Jury stability condition. Investigation indicates that to ensure the stability of the banking loan equilibrium, it is advisable to establish a loan benchmark interest rate that is lower than the flip bifurcation value. Some numerical simulations, such as the bifurcation diagram, Lyapunov exponent, and chaotic attractor, are presented to confirm the analytical findings.

## Keywords

## Article Details

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

*How to Cite*

*Journal of the Indonesian Mathematical Society*,

*30*(2), 205–217. https://doi.org/10.22342/jims.30.2.1779.205-217

* * References

- Q. Duan, Y. Wei, and Z. Chen, “Relationship between the benchmark interest rate and a macroeconomic indicator,” Economic Modelling, vol. 38, pp. 220–226, 2014.
- M. Ulate, “Going negative at the zero lower bound: the effects of negative nominal interest rates,” Am. Econ. Rev., vol. 111, pp. 1–40, 2021.
- B. Xin and K. Jiang, “Economic uncertainty, central bank digital currency, and negative interest rate policy,” Journal of Management Science and Engineering, vol. 8, pp. 430–452, 2023.
- L. Brubakk, S. ter Ellen, and H. Xu, “Central bank communication through interest rate projections,” Journal of Banking & Finance, vol. 124, p. 106044, 2021.
- A. Rai, R. Seth, and N. White, “Central bank target rates and term structure of interest rates: A study of six asia-pacific countries,” IIMB Management Review, vol. 31, pp. 223–237, 2019.
- L. Fanti, “The dynamics of a banking duopoly with capital regulations,” Economic Modelling, vol. 37, pp. 340–349, 2014.
- S. Brianzoni and G. Campisi, “Dynamical analysis of a banking duopoly model with capital regulation and asymmetric costs,” Discrete and Continuous Dynamical Systems - Series B, vol. 26, pp. 5807–5825, 2021.
- S. Brianzoni, G. Campisi, and A. Colasante, “Nonlinear banking duopoly model with capital regulation: The case of italy,” Chaos, Solitons & Fractals, vol. 160, p. 112209, 2022.
- A. Elsadany, “Dynamics of a delayed duopoly game with bounded rationality,” Mathematical and Computer Modelling, vol. 52, pp. 1479–1489, 2010.
- A. Elsadany, “Dynamics of a cournot duopoly game with bounded rationality based on relative profit maximization,” Applied Mathematics and Computation, vol. 294, pp. 253–263, 2017.
- Y. Peng, Y. Xiao, Q. Lu, X. Wu, and Y. Zhao, “Chaotic dynamics in cournot duopoly model with bounded rationality based on relative profit delegation maximization,” Physica A: Statistical Mechanics and its Applications, vol. 560, p. 125174, 2020.
- X. Zhang, D. Sun, S. Ma, and S. Zhang, “The dynamics of a quantum bertrand duopoly with differentiated products and heterogeneous expectations,” Physica A: Statistical Mechanics and its Applications, vol. 557, p. 124878, 2020.
- S. Askar, “On the dynamics of cournot duopoly game with private firms: Investigations and analysis,” Applied Mathematics and Computation, vol. 432, p. 127354, 2022.
- J. Ren, H. Sun, G. Xu, and D. Hou, “Convergence of output dynamics in duopoly co-opetition model with incomplete information,” Mathematics and Computers in Simulation, vol. 207, pp. 209–225, 2023.
- D. Zhou, H. Yang, J. Pi, and G. Yang, “The dynamics of a quantum cournot duopoly with asymmetric information and heterogeneous players,” Physics Letters A, vol. 483, p. 129033, 2023.
- M. Ansori, N. Sumarti, K. Sidarto, and I. Gunadi, “Analyzing a macroprudential instrument during the covid-19 pandemic using border collision bifurcation,” Rect@: Revista Electr´onica de Comunicaciones y Trabajos de ASEPUMA, vol. 22, pp. 91–103, 2021.
- M. Ansori and S. Hariyanto, “Analysis of banking deposit cost in the dynamics of loan: Bifurcation and chaos perspectives,” BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 16, pp. 1283–1292, 2022.
- M. Ansori and S. Khabibah, “The role of the cost of a loan in banking loan dynamics: Bi-furcation and chaos analysis,” BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 16,pp. 1031–1038, 2022.
- M. Ansori, G. Theotista, and Winson, “Difference equation-based banking loan dynamics with reserve requirement policy,” International Journal of Difference Equations, vol. 18, pp. 35–48, 2023.
- M. Ansori, G. Theotista, and M. Febe, “The influence of the amount of premium and membership of idic on banking loan procyclicality: A mathematical model,” Advances in Dynamical Systems and Applications, vol. 18, pp. 111–123, 2023.
- N. Ashar, M. Ansori, and H. Fata, “The effects of capital policy on banking loan dynamics: A difference equation approach,” International Journal of Difference Equations, vol. 18, pp. 267–279, 2023.
- H. Fata, N. Ashara, and M. Ansori, “Banking loan dynamics with dividend payments,” Advances in Dynamical Systems and Applications (ADSA), vol. 18, pp. 87–99, 2023.
- M. F. Ansori and F. H. G¨um¨u¸s, “A difference equation of banking loan with nonlinear deposit interest rate,” Journal of Mathematical Sciences and Modelling, vol. 7, no. 1, pp. 14–19, 2024.
- M. Klein, “A theory of the banking firm,” Journal of Money, Credit, and Banking, vol. 3, pp. 205–218, 1971.
- M. Monti, “Deposit, credit and interest rates determination under alternative objective functions,” in Mathematical methods in investment and finance (G. P. Szego and K. Shell, eds.), Amsterdam: North-Holland, 1972.
- X. Freixas and J. Rochet, Microeconomics of Banking. MIT Press, Cambridge, 1997. [27] A. Dixit, “Comparative statics for oligopoly,” International Economic Review, vol. 27, pp. 107–122, 1986.
- G. Gandolfo, Economic Dynamics: Methods and Models. Elsevier Science Publisher BV, Amsterdam, The Netherlands, 2nd ed., 1985.
- A. Medio, Chaotic Dynamics: Theory and Applications to Economics. Cambridge University Press, Cambridge, UK, 1992.
- G.-I. Bischi, C. Chiarella, M. Kopel, and F. Szidarovszky, Nonlinear Oligopolies: Stability and Bifurcations. Springer-Verlag, Berlin, 2010.
- P. van den Driessche, “Reproduction numbers of infectious disease models,” Infectious Disease Modelling, vol. 2, no. 3, pp. 288–303, 2017.

#### References

Q. Duan, Y. Wei, and Z. Chen, “Relationship between the benchmark interest rate and a macroeconomic indicator,” Economic Modelling, vol. 38, pp. 220–226, 2014.

M. Ulate, “Going negative at the zero lower bound: the effects of negative nominal interest rates,” Am. Econ. Rev., vol. 111, pp. 1–40, 2021.

B. Xin and K. Jiang, “Economic uncertainty, central bank digital currency, and negative interest rate policy,” Journal of Management Science and Engineering, vol. 8, pp. 430–452, 2023.

L. Brubakk, S. ter Ellen, and H. Xu, “Central bank communication through interest rate projections,” Journal of Banking & Finance, vol. 124, p. 106044, 2021.

A. Rai, R. Seth, and N. White, “Central bank target rates and term structure of interest rates: A study of six asia-pacific countries,” IIMB Management Review, vol. 31, pp. 223–237, 2019.

L. Fanti, “The dynamics of a banking duopoly with capital regulations,” Economic Modelling, vol. 37, pp. 340–349, 2014.

S. Brianzoni and G. Campisi, “Dynamical analysis of a banking duopoly model with capital regulation and asymmetric costs,” Discrete and Continuous Dynamical Systems - Series B, vol. 26, pp. 5807–5825, 2021.

S. Brianzoni, G. Campisi, and A. Colasante, “Nonlinear banking duopoly model with capital regulation: The case of italy,” Chaos, Solitons & Fractals, vol. 160, p. 112209, 2022.

A. Elsadany, “Dynamics of a delayed duopoly game with bounded rationality,” Mathematical and Computer Modelling, vol. 52, pp. 1479–1489, 2010.

A. Elsadany, “Dynamics of a cournot duopoly game with bounded rationality based on relative profit maximization,” Applied Mathematics and Computation, vol. 294, pp. 253–263, 2017.

Y. Peng, Y. Xiao, Q. Lu, X. Wu, and Y. Zhao, “Chaotic dynamics in cournot duopoly model with bounded rationality based on relative profit delegation maximization,” Physica A: Statistical Mechanics and its Applications, vol. 560, p. 125174, 2020.

X. Zhang, D. Sun, S. Ma, and S. Zhang, “The dynamics of a quantum bertrand duopoly with differentiated products and heterogeneous expectations,” Physica A: Statistical Mechanics and its Applications, vol. 557, p. 124878, 2020.

S. Askar, “On the dynamics of cournot duopoly game with private firms: Investigations and analysis,” Applied Mathematics and Computation, vol. 432, p. 127354, 2022.

J. Ren, H. Sun, G. Xu, and D. Hou, “Convergence of output dynamics in duopoly co-opetition model with incomplete information,” Mathematics and Computers in Simulation, vol. 207, pp. 209–225, 2023.

D. Zhou, H. Yang, J. Pi, and G. Yang, “The dynamics of a quantum cournot duopoly with asymmetric information and heterogeneous players,” Physics Letters A, vol. 483, p. 129033, 2023.

M. Ansori, N. Sumarti, K. Sidarto, and I. Gunadi, “Analyzing a macroprudential instrument during the covid-19 pandemic using border collision bifurcation,” Rect@: Revista Electr´onica de Comunicaciones y Trabajos de ASEPUMA, vol. 22, pp. 91–103, 2021.

M. Ansori and S. Hariyanto, “Analysis of banking deposit cost in the dynamics of loan: Bifurcation and chaos perspectives,” BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 16, pp. 1283–1292, 2022.

M. Ansori and S. Khabibah, “The role of the cost of a loan in banking loan dynamics: Bi-furcation and chaos analysis,” BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 16,pp. 1031–1038, 2022.

M. Ansori, G. Theotista, and Winson, “Difference equation-based banking loan dynamics with reserve requirement policy,” International Journal of Difference Equations, vol. 18, pp. 35–48, 2023.

M. Ansori, G. Theotista, and M. Febe, “The influence of the amount of premium and membership of idic on banking loan procyclicality: A mathematical model,” Advances in Dynamical Systems and Applications, vol. 18, pp. 111–123, 2023.

N. Ashar, M. Ansori, and H. Fata, “The effects of capital policy on banking loan dynamics: A difference equation approach,” International Journal of Difference Equations, vol. 18, pp. 267–279, 2023.

H. Fata, N. Ashara, and M. Ansori, “Banking loan dynamics with dividend payments,” Advances in Dynamical Systems and Applications (ADSA), vol. 18, pp. 87–99, 2023.

M. F. Ansori and F. H. G¨um¨u¸s, “A difference equation of banking loan with nonlinear deposit interest rate,” Journal of Mathematical Sciences and Modelling, vol. 7, no. 1, pp. 14–19, 2024.

M. Klein, “A theory of the banking firm,” Journal of Money, Credit, and Banking, vol. 3, pp. 205–218, 1971.

M. Monti, “Deposit, credit and interest rates determination under alternative objective functions,” in Mathematical methods in investment and finance (G. P. Szego and K. Shell, eds.), Amsterdam: North-Holland, 1972.

X. Freixas and J. Rochet, Microeconomics of Banking. MIT Press, Cambridge, 1997. [27] A. Dixit, “Comparative statics for oligopoly,” International Economic Review, vol. 27, pp. 107–122, 1986.

G. Gandolfo, Economic Dynamics: Methods and Models. Elsevier Science Publisher BV, Amsterdam, The Netherlands, 2nd ed., 1985.

A. Medio, Chaotic Dynamics: Theory and Applications to Economics. Cambridge University Press, Cambridge, UK, 1992.

G.-I. Bischi, C. Chiarella, M. Kopel, and F. Szidarovszky, Nonlinear Oligopolies: Stability and Bifurcations. Springer-Verlag, Berlin, 2010.

P. van den Driessche, “Reproduction numbers of infectious disease models,” Infectious Disease Modelling, vol. 2, no. 3, pp. 288–303, 2017.