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

Mauro Gallegati, Laura Gardini () and Iryna Sushko ()
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Laura Gardini: University of Urbino
Iryna Sushko: Institute of Mathematics, NASU, and Kyiv School of Economics

Decisions in Economics and Finance, 2021, vol. 44, issue 2, No 7, 613-639

Abstract: 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.

Keywords: Multiplier–Accelerator model; Business cycle; Piecewise linear system; Border collision bifurcation; Two-dimensional piecewise smooth map (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10203-021-00333-y

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