Equilibria in the Large-Scale Competition for Market Share in a Commodity with Resource-Buying

We study a mean field game model of Cournot/Bertrand competition between firms. Chan and Sircar introduced such a mean field model of competition in natural resource extraction. In their model, each firm has a finite reserve of a commodity and may choose to extract a positive quantity per unit time. We instead treat the situation in which firms compete to purchase raw materials, rather than produce the raw material. With this change, we arrive at the same nonlinear system of partial differential equations, but what corresponds to the positive rate of resource extraction in the Chan-Sircar model is instead negative in our setting. We prove existence of stationary solutions, using a Lyapunov-Schmidt decomposition and multiple applications of the implicit function theorem.