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Synchronization in Sunspot Models

Patir, Assaf (2019): Synchronization in Sunspot Models.

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This note illustrates how agents' beliefs about economic outcomes can dynamically synchronize and de-synchronize to produce business-cycle-like fluctuations in a simple macroeconomic model. I consider a simple macroeconomic model with multiple equilibria, which are different ways that sunspots can forecast future output in a self-fulfilling manner. Agents are assumed to learn to use the sunspot variable through econometric learning. I show that if different agents have different interpretations of the sunspot, this leads to a complex nonlinear dynamic of synchronization of beliefs about the equilibrium being played. Depending on the extent of disagreement on the interpretation of the sunspot, the economy will be more or less volatile. The dispersion of the agents' beliefs is inversely related to volatility, since low dispersion implies that output is very sensitive to extrinsic noise (the sunspot). When disagreement crosses a critical threshold, the sunspot is practically ignored and the output is stable. The equation describing the evolution of the economy can be interpreted as a nonlinear-stochastic version of the Kuramoto model, a prototypical model of synchronization phenomena, and simulations confirm that the qualitative features of the model are in agreement with results from the Kuramoto literature.

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