Resonant and non-resonant Double Hopf bifurcation in a 4D Goodwin model with Wage inequality

Cajas-Guijarro, John and Cajas-Guijarro, Daniel (2025): Resonant and non-resonant Double Hopf bifurcation in a 4D Goodwin model with Wage inequality.

Abstract

This paper presents a four-dimensional extension of the Goodwin model of endogenous cycles that integrates wage inequality and underemployment. The model distinguishes two classes of workers differentiated by productivity, wage levels, and bargaining strength, and endogenizes the underemployment rate through a simplified power-balance mechanism between capital and labor. We establish well-posedness of the system by proving existence–uniqueness of solutions, positivity, and forward invariance on a compact admissible set. The interior equilibrium is characterized in closed form and shown to generically undergo a double Hopf (Hopf–Hopf) bifurcation. Using center–manifold reduction and a third-order normal form, we derive the amplitude equations governing the interaction between two oscillatory modes (the Goodwin cycle and the underemployment cycle). The reduced dynamics predict the emergence of an invariant two-torus with quasi-periodic cycles and phase locking at low-order resonances ($1{:}1$, $1{:}2$, $1{:}3$). Numerical continuation and direct simulations corroborate the analytical predictions, documenting transitions between quasi-periodicity and resonant periodic orbits, and mapping the associated bifurcation structure in key parameters, such as the adjustment speed of the underemployment rate in response to deviations from steady-state equilibrium.

• Item Type: MPRA Paper
• Language: English
• Keywords: Goodwin model; Wage inequality; Stability; Double Hopf bifurcation; Normal form; Resonance
• Subjects: B - History of Economic Thought, Methodology, and Heterodox Approaches > B5 - Current Heterodox Approaches > B51 - Socialist ; Marxian ; Sraffian; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis; E - Macroeconomics and Monetary Economics > E3 - Prices, Business Fluctuations, and Cycles > E32 - Business Fluctuations ; Cycles; O - Economic Development, Innovation, Technological Change, and Growth > O4 - Economic Growth and Aggregate Productivity > O41 - One, Two, and Multisector Growth Models

References

Yu, P. (2005). Closed-form conditions of bifurcation points for general differential equations. International Journal of Bifurcation and Chaos, 15(4), 1467–1483. https://doi.org/10.1142/S0218127405012582

Parks, P. C. (1977). A new proof of Hermite’s stability criterion and a generalization of Orlando’s formula. International Journal of Control, 26(2), 197–206. https://doi.org/10.1080/00207177708922303

Kuznetsov, Y. A. (1999). Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODE’s. SIAM Journal on Numerical Analysis, 36(4), 1104–1124. https://doi.org/10.1137/S0036142998335005

Kuznetsov, Y. A. (2023). Elements of applied bifurcation theory (4th ed.). Applied Mathematical Sciences. Springer. https://doi.org/10.1007/978-3-031-22007-4

Wu, X. P. P., & Wang, L. (2019). Normal form of double-Hopf singularity with 1:1 resonance for delayed differential equations. Nonlinear Analysis: Modelling and Control, 24(2), 241–260. https://doi.org/10.15388/NA.2019.2.6

van Gils, S. A., Krupa, M., & Langford, W. F. (1990). Hopf bifurcation with non-semisimple 1:1 resonance. Nonlinearity, 3(3), 825–850. https://doi.org/10.1088/0951-7715/3/3/013

Goodwin, R. M. (1967). A growth cycle. In Socialism, capitalism and economic growth (pp. 54–58). Cambridge University Press. https://doi.org/10.1007/978-1-349-05504-3_12

Cajas Guijarro, J. (2024). An extended Goodwin model with endogenous technical change and labor supply. Structural Change and Economic Dynamics, 70, 699–710. https://doi.org/10.1016/j.strueco.2024.06.004

Stamegna, M. (2024). Wage inequality and induced innovation in a classical-Marxian growth model. Journal of Evolutionary Economics, 34(1), 127–168. https://doi.org/10.1007/s00191-024-00851-z

Dhooge, A., Govaerts, W., & Kuznetsov, Y. A. (2003). MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs. ACM Transactions on Mathematical Software, 29(2), 141–164. https://doi.org/10.1145/779359.779362

Dhooge, A., Govaerts, W., Kuznetsov, Y. A., Meijer, H. G. E., & Sautois, B. (2008). New features of the software MatCont for bifurcation analysis of dynamical systems. Mathematical and Computer Modelling of Dynamical Systems, 14(2), 147–175. https://doi.org/10.1080/13873950701742754

Broer, H., Hanßmann, H., & Wagener, F. (2021). Normal resonances in a double Hopf bifurcation. Indagationes Mathematicae, 32(1), 33–54. https://doi.org/10.1016/j.indag.2020.09.003

Knobloch, E., & Proctor, M. R. E. (1988). The double Hopf bifurcation with 2:1 resonance. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 415, 61–90. https://doi.org/10.1098/rspa.1988.0003

LeBlanc, V. G., & Langford, W. F. (1996). Classification and unfoldings of 1:2 resonant Hopf bifurcation. Archive for Rational Mechanics and Analysis, 136, 305–357. https://doi.org/10.1007/BF02206623

Caprino, S., Maffei, C., & Negrini, P. (1984). Hopf bifurcation at 1:1 resonance. Nonlinear Analysis: Theory, Methods & Applications, 8(9), 1011–1032. https://doi.org/10.1016/0362-546X(84)90096-8

Luongo, A., Paolone, A., & Di Egidio, A. (2003). Multiple timescales analysis for 1:2 and 1:3 resonant Hopf bifurcations. Nonlinear Dynamics, 34, 269–291. https://doi.org/10.1023/B:NODY.0000013508.50435.39

Shah, A., & Desai, M. (1981). Growth cycles with induced technical change. The Economic Journal, 91(364), 1006–1010. https://doi.org/10.2307/2232506

van der Ploeg, F. (1987). Growth cycles, induced technical change, and perpetual conflict over the distribution of income. Journal of Macroeconomics, 9(1), 1–12. https://doi.org/10.1016/S0164-0704(87)80002-2

Glombowski, J., & Krüger, M. (1988). A short-period growth cycle model. Recherches Économiques de Louvain/Louvain Economic Review, 54(4), 423–438. https://doi.org/10.2307/2526560

Skott, P. (1989). Effective demand, class struggle and cyclical growth. International Economic Review, 30(1), 231–247. https://doi.org/10.2307/2526560

Rada, C., Tavani, D., von Arnim, R., & Zamparelli, L. (2023). Classical and Keynesian models of inequality and stagnation. Journal of Economic Behavior & Organization, 211, 442–461. https://doi.org/10.1016/j.jebo.2023.05.015

Sato, Y. (1985). Marx–Goodwin growth cycles in a two-sector economy. Journal of Economics, 45(1), 21–34. https://www.jstor.org/stable/41796300

Cajas Guijarro, J. (2025). A classical-Marxian two-sector endogenous cycle model. Metroeconomica, 76(3), 384–404. https://doi.org/10.1111/meca.12487

Keen, S. (1995). Finance and economic breakdown: Modeling Minsky’s “Financial Instability Hypothesis.” Journal of Post Keynesian Economics, 17(4), 607–635. https://doi.org/10.1080/01603477.1995.11490053

Sordi, S., & Vercelli, A. (2014). Unemployment, income distribution and debt-financed investment in a growth cycle model. Journal of Economic Dynamics & Control, 48, 325–348. https://doi.org/10.1016/j.jedc.2014.09.030

Desai, M. (1973). Growth cycles and inflation in a model of the class struggle. Journal of Economic Theory, 6(6), 527–545. https://doi.org/10.1016/0022-0531(73)90074-4

Manfredi, P., & Fanti, L. (2006). Demography in macroeconomic models: When labour supply matters for economic cycles. Metroeconomica, 57(4), 536–563. https://doi.org/10.1111/j.1467-999X.2006.00255.x

Dávila-Fernández, M., & Sordi, S. (2019). Distributive cycles and endogenous technical change in a BoPC growth model. Economic Modelling, 77, 216–233. https://doi.org/10.1016/j.econmod.2018.09.002

Sportelli, M., & De Cesare, L. (2022). A Goodwin type cyclical growth model with two-time delays. Structural Change and Economic Dynamics, 61, 95–102. https://doi.org/10.1016/j.strueco.2022.02.002

Bella, G. (2025). Emergence of chaotic dynamics in the Goodwin model with disequilibrium in the goods market. Structural Change and Economic Dynamics, 73, 170–180. https://doi.org/10.1016/j.strueco.2025.01.005

Barrales-Ruiz, J., Mendieta-Muñoz, I., Rada, C., Tavani, D., & von Arnim, R. (2022). The distributive cycle: Evidence and current debates. Journal of Economic Surveys, 36(2), 468–503. https://doi.org/10.1111/joes.12432

Autor, D. (2019). Work of the past, work of the future. AEA Papers and Proceedings, 109, 1–32. https://doi.org/10.1257/pandp.20191110

Card, D., Lemieux, T., & Riddell, C. (2020). Unions and wage inequality: The roles of gender, skill and public sector employment. Canadian Journal of Economics, 53(1), 140–173. https://doi.org/10.1111/caje.12432

Koeniger, W., Leonardi, M., & Nunziata, L. (2007). Labor market institutions and wage inequality. Industrial and Labor Relations Review, 60(3), 340–356. https://doi.org/10.1177/001979390706000302

Card, D., & Lemieux, T. (2001). Can falling supply explain the rising return to college for younger men? A cohort-based analysis. The Quarterly Journal of Economics, 116(2), 705–746. https://doi.org/10.1162/00335530151144140

Acemoglu, D., & Restrepo, P. (2022). Tasks, automation, and the rise in U.S. wage inequality. Econometrica, 90(5), 1973–2016. https://doi.org/10.3982/ECTA19815

Palley, T. (2017). Wage- vs. profit-led growth: The role of the distribution of wages in determining regime character. Cambridge Journal of Economics, 41(1), 49–61. https://doi.org/10.1093/cje/bew004

Blecker, R., & Setterfield, M. (2019). Heterodox macroeconomics. Edward Elgar.

Palley, T. (2014). Capitalists, workers, and managers: Wage inequality and effective demand. Structural Change and Economic Dynamics, 30, 120–131. https://doi.org/10.1016/j.strueco.2014.05.001

Dutt, A. K., & Veneziani, R. (2019). Education and “human capitalists” in a classical-Marxian model of growth and distribution. Cambridge Journal of Economics, 43(2), 481–506. https://doi.org/10.1093/cje/bey025

Kennedy, C. (1964). Induced bias in innovation and the theory of distribution. The Economic Journal, 74(295), 541–547. https://doi.org/10.2307/2228295

Samuelson, P. (1965). A theory of induced innovation along Kennedy–Weisäcker lines. The Review of Economics and Statistics, 47(4), 343–356. https://doi.org/10.2307/1927763

Flaschel, P., & Greiner, A. (2011). Dual labor markets and the impact of minimum wages on atypical employment. Metroeconomica, 62(3), 512–531. https://doi.org/10.1111/j.1467-999X.2011.04122.x

Solow, R. (1956). A contribution to the theory of economic growth. The Quarterly Journal of Economics, 70(1), 65–94. https://doi.org/10.2307/1884513

Flaschel, P., Greiner, A., Logeay, C., & Proaño, C. (2012). Employment cycles, low income work and the dynamic impact of wage regulations: A macro perspective. Journal of Evolutionary Economics, 22, 235–250. https://doi.org/10.1007/s00191-011-0236-2

Flaschel, P., & Greiner, A. (2009). Employment cycles and minimum wages: A macro view. Structural Change and Economic Dynamics, 20, 279–287. https://doi.org/10.1016/j.strueco.2009.09.003

Dutt, A. K., Charles, S., & Lang, D. (2015). Employment flexibility, dual labour markets, growth, and distribution. Metroeconomica, 66(4), 771–807. https://doi.org/10.1111/meca.12093

Sasaki, H., Asada, Y., & Sonoda, R. (2015). Effects of minimum wage share and wage gap reduction on cyclical fluctuation: A Goodwin approach (MPRA Working Paper No. 121695) [Preprint]. MPRA. https://mpra.ub.uni-muenchen.de/121695/

Sasaki, H., Maruyama, J., & Sako, K. (2013). The macroeconomic effects of the wage gap between regular and non-regular employment and of minimum wages. Structural Change and Economic Dynamics, 26, 61–72. https://doi.org/10.1016/j.strueco.2013.06.001

Sasaki, H. (2016). Profit sharing and its effect on income distribution and output: A Kaleckian approach. Cambridge Journal of Economics, 40(2), 469–489. https://doi.org/10.1093/cje/beu087

Grasselli, M., & Maheshwari, A. (2018). Testing a Goodwin model with general capital accumulation rate. Metroeconomica, 69(3). https://doi.org/10.1111/meca.12204

Kalecki, M. (1943). Political aspects of full employment. The Political Quarterly, 14(4). https://doi.org/10.1111/j.1467-923X.1943.tb01016.x