Stability of a mean field game of production adjustment

Abstract

We propose a mean field game model to study the stability of production adjustment by extending a kind of linear quadratic Gaussian game in large population systems. First, we present the model's equilibrium solutions by a backward HJB equation and a forward FPK equation. Second, we employ the local perturbation method to transform the nonlinear problem into a linear one to study the solutions' stability. Third, we use the spectrum and contraction analyses to study the stationary solutions' linear stability. At last, we validate the results by the numerical simulation. The results show that the additional cost coefficient and noise intensity affect the system's stability. If the additional cost coefficient and the noise intensity are large enough for firms to change their outputs, then firms will try to reduce their production adjustment and remain stable long‐term.