Stability and Bifurcations in Banks and Small Enterprises—A Three-Dimensional Continuous-Time Dynamical System

Desogus, Marco and Venturi, Beatrice (2023): Stability and Bifurcations in Banks and Small Enterprises—A Three-Dimensional Continuous-Time Dynamical System. Published in: Journal of Risk and Financial Management , Vol. 16, No. 3: 171 (3 March 2023)

Abstract

Here, we discuss a three-dimensional continuous-time Lotka–Volterra dynamical system, which describes the role of government in interactions with banks and small enterprises. In Italy, during the COVID-19 emergency, the main objective of government economic intervention was to maintain the proper operation of the bank–enterprise system. We also review the effectiveness of measures introduced in response to the COVID-19 pandemic lockdowns to avoid a further credit crunch. By applying bifurcation theory to the system, we were able to produce evidence of the existence of Hopf and zero-Hopf bifurcating periodic solutions from a saddle focus in a special region of the parameter space, and we performed a numerical analysis.

• Item Type: MPRA Paper
• Language: English
• Keywords: Credit crunch; simulation; credit big data; nonlinear analysis; periodic solutions; stability; dynamical system; zero-Hopf bifurcation
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C62 - Existence and Stability Conditions of Equilibrium; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling; E - Macroeconomics and Monetary Economics > E3 - Prices, Business Fluctuations, and Cycles > E32 - Business Fluctuations ; Cycles; E - Macroeconomics and Monetary Economics > E5 - Monetary Policy, Central Banking, and the Supply of Money and Credit > E51 - Money Supply ; Credit ; Money Multipliers; G - Financial Economics > G2 - Financial Institutions and Services > G21 - Banks ; Depository Institutions ; Micro Finance Institutions ; Mortgages; G - Financial Economics > G2 - Financial Institutions and Services > G28 - Government Policy and Regulation

References

ABI-Cerved. 2018a. Outlook ABI-Cerved Sulle Sofferenze Delle Imprese. Tech. Rep., ABI. Available online: https://www.abi.it/DOC_Mercati/Analisi/Scenario-e-previsioni/Outlook-ABI-Cerved/Outlook%20Abi-Cerved%20su%20sofferenze_Numero%206%20feb%202018.pdf (accessed on 19 May 2022).

ABI-Cerved. 2018b. Outlook ABI-Cerved Sulle Sofferenze Delle Imprese. Tech. Rep., ABI. Available online: https://www.abi.it/DOC_Mercati/Analisi/Scenario-e-previsioni/Outlook-ABI-Cerved/Outlook%20Abi-Cerved%20sulle%20sofferenze%20delle%20imprese_Nr.%206%20dic%202018.pdf (accessed on 19 May 2022).

ABI-Cerved. 2019a. Outlook ABI-Cerved Sui Crediti Deteriorati Delle Imprese. Tech. Rep., ABI. Available online: https://www.abi.it/DOC_Mercati/Analisi/Scenario-e-previsioni/Outlook-ABI-Cerved/Outlook%20Abi-Cerved_Edizione%20dicembre%202019.pdf (accessed on 20 May 2022).

ABI-Cerved. 2019b. Outlook ABI-Cerved Sulle Sofferenze Delle Imprese. Tech. Rep., ABI. Available online: https://www.abi.it/DOC_Mercati/Analisi/Scenario-e-previsioni/Outlook-ABI-Cerved/Outlook%20Abi-Cerved%20sulle%20sofferenze%20delle%20imprese_Nr.%208%20luglio%202019.pdf (accessed on 20 May 2022).

Baldini, Andrea, and Marco Causi. 2020. Restoring credit market stability conditions in Italy: Evidences on Loan and Bad Loan dynamics. The European Journal of Finance 26: 746–73.

Bassetto, Marco, Marco Cagetti, and Mariacristina De Nardi. 2015. Credit crunches and credit allocation in a model of entrepreneurship. Review of Economic Dynamics 18: 53–76.

Bella, Giovanni, and Paolo Mattana. 2018. Bistability of equilibria and the 2-tori dynamics in an endogenous growth model undergoing the cusp–Hopf singularity. Nonlinear Analysis: Real World Applications 39: 185–201.

Bella, Giovanni, Paolo Mattana, and Beatrice Venturi. 2022. Existence and implications of a pitchfork-Hopf bifurcation in a continuous-time two-sector growth model. Journal of Economic Interaction and Coordination 17: 259–85.

Benhabib, Jess, and Takahiro Miyao. 1981. Some new results on the dynamics of the generalized Tobin model. International Economic Review 22: 589–96.

Bernanke, Ben Shalom, Mark Gertler, and Simon Gilchrist. 1994. The financial accelerator and the flight to quality. The Review of Economics and Statistics 78: 1–15.

Bischi, Gian Italo, and Fabio Tramontana. 2010. Three-dimensional discrete-time Lotka–Volterra models with an application to industrial clusters. Communications in Nonlinear Science and Numerical Simulation 15: 3000–14.

Bosi, Stefano, and David Desmarchelier. 2018. Pollution and infectious diseases. International Journal of Economic Theory 14: 351–72.

Buera, Francisco Javier, Roberto N. Fattal Jaef, and Yongseok Shin. 2015. Anatomy of a credit crunch: From capital to labor markets. Review of Economic Dynamics 18: 101–17.

Caggiano, Giovanni, Efrem Castelnuovo, and Giovanni Pellegrino. 2017. Estimating the real effects of uncertainty shocks at the zero lower bound. European Economic Review 100: 257–72.

Calcagnini, Giorgio, Germana Giombini, and Giuseppe Travaglini. 2019. A theoretical model of imperfect markets and investment. Structural Change and Economic Dynamics 50: 237–44.

Castellacci, Giuseppe, and Youngna Choi. 2014. Financial instability contagion: A dynamical systems approach. Quantitative Finance 14: 1243–55.

D’Ignazio Alessio, Ignazio, and Carlo Menon. 2020. Causal Effect of Credit Guarantees for Small-and Medium-Sized Enterprises: Evidence from Italy. The Scandinavian Journal of Economics 122: 191–218.

De Angelo, Harry, and Richard Roll. 2015. How stable are corporate capital structures? The Journal of Finance 70: 373–418.

Degryse, Hans, and Patrick Van Cayseele. 2000. Relationship lending within a bank-based system: Evidence from European small business data. Journal of financial Intermediation 9: 90–109.

Desogus, Marco, and Beatrice Venturi. 2019. Bank Crashes and Micro Enterprise Loans. International Journal of Business and Social Science 10: 191–211.

Desogus, Marco, and Elisa Casu. 2020a. A Contribution on Relationship Banking. Economic, Anthropological and Mathematical Reasoning, Empirical Evidence from Italy. International Research Journal of Finance and Economics 178: 25–49.

Desogus, Marco, and Elisa Casu. 2020b. What Are the Impacts of Credit Crunch on the Bank-Enterprise System? An Analysis Through Dynamic Modeling and an Italian Dataset. Applied Mathematical Sciences 14: 679–703.

Ditzen, Jan 2018. Cross-country convergence in a general Lotka–Volterra model. Spatial Economic Analysis 13: 191–211.

European Central Bank. 2017. Guidance to Banks on Non-Performing Loans. Tech. Rep., European Central Bank. Available online: https://www.bankingsupervision.europa.eu/ecb/pub/pdf/guidance_on_npl.en.pdf (accessed on 16 May 2022).

Ganong, Peter, and Jeffrey B. Liebman. 2018. The decline, rebound, and further rise in SNAP enrollment: Disentangling business cycle fluctuations and policy changes. American Economic Journal: Economic Policy 10: 153–76.

Hassan, Zara, Nauman Raza, Abdel-Haleem Abdel-Aty, Mohammed Zakarya, Riaz Ur Rahman, Adeela Yasmeen, Abdisalam Hassan, and Emad E. Mahmoud. 2022. New Fractal Soliton Solutions and Sensitivity Visualization for Double-Chain DNA Model. Journal of Function Spaces 2022: 2297866.

Hori, Takeo, and Koichi Futagami. 2019. A Non-unitary Discount Rate Model. Economica 86: 139–65.

Iyer Rajkamal, José-Luis Peydró, Samuel da-Rocha-Lopes, and Antoniette Schoar. 2014. Interbank liquidity crunch and the firm credit crunch: Evidence from the 2007–2009 crisis. The Review of Financial Studies 27: 347–72.

Kurkina, Elena S. 2017. Mathematical Models of Investment Cycles. Computational Mathematics and Modeling 28: 377–99.

Laeven, Luc, and Fabian Valencia. 2013. The real effects of financial sector interventions during crises. Journal of Money, Credit and Banking 45: 147–77.

Ledenyov, Dimitri O., and Viktor O. Ledenyov. 2012. On the new central bank strategy toward monetary and financial instabilities management in finances: Econophysical analysis of nonlinear dynamical financial systems. arXiv, arXiv:1211.1897.

Liu, Meng, and Meng Fan. 2017. Permanence of stochastic Lotka–Volterra systems. Journal of Nonlinear Science 27: 425–52.

Marasco, Addolorata, Antonella Picucci, and Alessandro Romano. 2016. Market share dynamics using Lotka–Volterra models. Technological Forecasting and Social Change 105: 49–62.

MISE—Ministero dello Sviluppo Economico. 2019. Parte la Riforma Del Fondo. Available online: https://www.fondidigaranzia.it/parte-la-riforma-del-fondo/ (accessed on 26 April 2022).

Neri, Umberto, and Beatrice Venturi. 2007. Stability and bifurcations in IS-LM economic models. International Review of Economics 54: 53–64.

Nishimura, Kazuo, and Tadashi Shigoka. 2019. Hopf bifurcation and the existence and stability of closed orbits in three-sector models of optimal endogenous growth. Studies in Nonlinear Dynamics and Econometrics 23: 1–21.

Petrosky-Nadeau, Nicolas. 2013. TFP during a Credit Crunch. Journal of Economic Theory 148: 1150–78.

Refaai, D.A., Mohammed M.A. El-Sheikh, Gamal A. Ismail, Mohammed Zakarya, Ghada AlNemer, and Haytham M. Rezk. 2022. Stability of Nonlinear Fractional Delay Differential Equations. Symmetry 14: 1606.

Rozendaal, Jeroen C., Yannick Malevergne, and Didier Sornette. 2016. Macroeconomic dynamics of assets, leverage and trust. International Journal of Bifurcation and Chaos 26: 1650133.

Tan, Brandon, Deniz O. Igan, Maria Soledad Martinez Peria, Nicola Pierri, and Andrea F. Presbitero. 2020. Government Intervention and Bank Market Power: Lessons from the Global Financial Crisis for the COVID-19 Crisis. IMF Working Paper. Available online: https://www.imf.org/en/Publications/WP/Issues/2020/12/11/Government-Intervention-and-Bank-Market-Power-Lessons-from-the-Global-Financial-Crisis-for-49934 (accessed on 5 July 2022).

Tsai, Bi-Huei. 2017. Predicting the competitive relationships of industrial production between Taiwan and China using Lotka–Volterra model. Applied Economics 49: 2428–42.

Wang, Sheng, Guixin Hu, and Linshan Wang. 2018. Stability in distribution of a stochastic competitive lotka-volterra system with S-type distributed time delays. Methodology and Computing in Applied Probability 20: 1241–57.

Wehinger, Gert. 2014. SMEs and the credit crunch: Current financing difficulties, policy measures and a review of literature. OECD Journal: Financial Market Trends 2013: 115–48.

Wei, Tie, Zhiwei Zhu, Yang Li, and Na Yao. 2018. The evolution of competition in innovation resource: A theoretical study based on Lotka–Volterra model. Technology Analysis and Strategic Management 30: 295–310.

Wong, Grace 2009. UK Unveils Second Bank Rescue Plan. CNN. Available online: https://money.cnn.com/2009/01/19/news/international/britain_bank_bailout/index.htm (accessed on 25 May 2022).

Xu, Changjin, Muhammad Farman, Ali Hasan, Ali Akgül, Mohammed Zakarya, Wedad Albalawi, and Choonkil Park. 2022. Lyapunov stability and wave analysis of COVID-19 omicron variant of real data with fractional operator. Alexandria Engineering Journal 61: 11787–802.

Zhao, Liming, and Zhipei Zhao. 2016. Stability and Hopf bifurcation analysis on a nonlinear business cycle model. Mathematical Problems in Engineering 2016: 2706719.