Endogenous Crisis Waves: Stochastic Model with Synchronized Collective Behavior

We propose a simple framework to understand commonly observed crisis waves in macroeconomic agent-based models, which is also relevant to a variety of other physical or biological situations where synchronization occurs. We compute exactly the phase diagram of the model and the location of the synchronization transition in parameter space. Many modifications and extensions can be studied, confirming that the synchronization transition is extremely robust against various sources of noise or imperfections.

Figure 1

Typical trajectories of the unemployment rate $u$ as a function of time for the macroeconomic agent-based model described in Ref. [7], for three different bankruptcy thresholds $\mathrm{\Theta }=1$, 1.5, and 3. As $\mathrm{\Theta }$ is increased, the economy evolves from a phase of residual unemployment (RU), endogenous crises (EC), and finally full employment (FE). In the EC phase one observes nearly periodic bursts of unemployment, corresponding to collective waves of bankruptcies. The period of these oscillations depends on the parameters of the model and can be on the order of several years in real units.

Figure 2

Numerical solutions of Eq. (12): for small values of $\phi$ there is only one linearly stable solution without oscillations (${\alpha }_1$); for intermediate values of $\phi$ a second linearly stable solution appears with oscillations (${\alpha }_2$); for high values of $\phi$ there is only a linearly unstable solution with oscillations corresponding to the appearance of the synchronized behavior. Inset: Typical trajectories of the fraction of bankrupt firms $1-\varphi$ as a function of time for different values of $\phi$. ${\varphi }_0$ is the theoretical value of stationary fraction of active firms, given by Eq. (6). For $\phi >{\phi }_c$ the synchronized behavior settles in. The plot corresponds to $z=0.002$ and $\beta =1.3$. Note that the time unit cannot be directly compared to that of Fig. 1.

Figure 3

Phase diagram of the model in Eq. (1) in the ($\beta$, Pe) plane as given by the solutions of Eq. (12) with $z=0.02$ (black lines) and $z=0.002$ (red lines). Dashed lines separate the region where the maximal solution has $\mathrm{Re}(\alpha )<0$ and $\mathrm{Im}(\alpha )=0$ (stable) from the region where $\mathrm{Re}(\alpha )<0$ and $\mathrm{Im}(\alpha )\ne 0$ (damped oscillations). Full lines separate the latter region from the one where $\mathrm{Re}(\alpha )>0$ (unstable). In the inset we plot the phase diagram in the ($\beta$, $z$) for $\mathrm{Pe}=0.5$.