Cycling in stochastic general equilibrium

By generalizing the measurements on the game experiments of mixed strategy Nash equilibrium, we study the dynamical pattern in a representative dynamic stochastic general equilibrium (DSGE). The DSGE model describes the entanglements of the three variables (output gap [$y$], inflation [$\pi$] and nominal interest rate [$r$]) which can be presented in 3D phase space. We find that, even though the trajectory of $\pi \text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}y\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}r$ in phase space appears highly stochastic, it can be visualized and quantified. It exhibits as clockwise cycles, counterclockwise cycles and weak cycles, respectively, when projected onto $\pi \text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}y$, $y\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}r$ and $r\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}\pi$ phase planes. We find also that empirical data of United State (1960-2013) significantly exhibit same cycles. The resemblance between the cycles in general equilibrium and the cycles in mixed strategy Nash equilibrium suggest that, there generally exists dynamical fine structures accompanying with equilibrium. The fine structure, describing the entanglement of the non-equilibrium (the constantly deviating from the equilibrium), displays as endless cycles.