Dynamics of Macroeconomics Models with Hysteresis Operators

We propose a class of DSGE macroeconomics models with the play hysteresis operator. The play operator introduces stickiness and path dependence into the response of economic agents to the inflation rate variation. The models can also be viewed as piecewise linear systems. We show that the systems have an entire line segment of equilibrium states. For a simpler variant of the model, we prove that when there is no exogenous noise, the line segment of equilibrium states is globally stable, and each trajectory converges to one of the states. For more complex versions of the model, other dynamics are possible. In order to understand those, we present a simple prototype 2-dimensional system and obtain a global bifurcation diagram, which includes transitions from the global stability to periodic dynamics, border collision bifurcations, and complex dynamics. We further perform numerical analysis of the DSGE models to study the effects of exogenous noise and shocks, the run-away inflation scenario, and sensitivity to parameters. We interpret our findings in economic context. Finally, we analyze further variations of the model, including the system with multiple agents modeled by the play operators with different thresholds; a model with the sticky Central Bank's policy; and, a hybrid model in which the inflation rate and output gap variables are continuous while the interest rate is updated at discrete times.