The Cooperative Mean Field Game for Production Control with Sticky Price

We study a cooperative mean field game problem arising from the production control for multiple firms with price stickiness in the commodity market. The price dynamics for each firm is described as a controlled jump-diffusion process with mean-field interaction. Each firm aims to maximize the so-called social rewards which is defined by the average of individual rewards for all firms. By solving the limiting control problem for the representative firm and an associated fixed-point problem, we construct an explicit approximating optimal strategy when the number of firms grows large.