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Dynamics of a business cycle model with two types of governmental expenditures: the role of border collision bifurcations

Author

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  • Mauro Gallegati

    (Polytechnical University of Marche)

  • Laura Gardini

    (University of Urbino)

  • Iryna Sushko

    (Institute of Mathematics, NASU, and Kyiv School of Economics)

Abstract

We reconsider the multiplier–accelerator model of business cycles, first introduced by Samuelson and then modified by many authors. The original simple model, besides damped oscillations, also leads to divergent oscillations. To avoid this, we introduce two different types of governmental expenditures leading a two-dimensional continuous piecewise linear map that can generate sustained oscillations (attracting cycles). The map is defined by three different linear functions in three different partitions of the phase plane, and this peculiarity influences the overall dynamics of the system. We show that, similar to the classical Samuelson model, there is a unique feasible equilibrium as well as converging oscillations. However, close to the center bifurcation value the attracting equilibrium coexists with attracting cycles of different periods, which lose stability via a center bifurcation simultaneously with the equilibrium. Moreover, we show that attracting cycles of particular type also exist when the equilibrium becomes an unstable focus. For several families of attracting cycles, by introducing the symbolic representation, we obtain boundaries of the related periodicity regions, associated with border collision bifurcations.

Suggested Citation

  • Mauro Gallegati & Laura Gardini & Iryna Sushko, 2021. "Dynamics of a business cycle model with two types of governmental expenditures: the role of border collision bifurcations," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 44(2), pages 613-639, December.
    • Handle: RePEc:spr:decfin:v:44:y:2021:i:2:d:10.1007_s10203-021-00333-y
    DOI: 10.1007/s10203-021-00333-y
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    References listed on IDEAS

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    1. Bischi Gian Italo & Lamantia Fabio, 2012. "Routes to Complexity Induced by Constraints in Cournot Oligopoly Games with Linear Reaction Functions," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 16(2), pages 1-30, April.
    2. Frank Westerhoff, 2006. "Samuelson's multiplier-accelerator model revisited," Applied Economics Letters, Taylor & Francis Journals, vol. 13(2), pages 89-92.
    3. Bischi, Gian Italo & Matsumoto, Akio & Carrera, Edgar J. Sanchez, 2020. "Foreword to the SCED special issue on “Nonlinear Social Dynamics”," Structural Change and Economic Dynamics, Elsevier, vol. 52(C), pages 236-237.
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    Cited by:

    1. Gardini, Laura & Radi, Davide & Schmitt, Noemi & Sushko, Iryna & Westerhoff, Frank, 2023. "Sentiment-driven business cycle dynamics: An elementary macroeconomic model with animal spirits," Journal of Economic Behavior & Organization, Elsevier, vol. 210(C), pages 342-359.

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