Dynamic Yardstick Competition as a Mean Field Game: Explicit Theory with OU, CIR, and GBM Cost Dynamics

This paper embeds yardstick competition in a dynamic mean field a mean game (MFG) framework to analyze the regulation of industries with a continuum of cost-reducing firms. Each firm faces a stochastic cost process -- modeled explicitly as Ornstein-Uhlenbeck (OU), Cox-Ingersoll-Ross(CIR), or Geometric Brownian Motion(GBM)-and chooses continuous-time effort to lower cost. A regulator sets each firm's price equal to the contemporaneous mean cost of the population. We formulate and solve the coupled Hamilton-Jacobi-Bellman and Fokker-Planck system describing this economy, prove existence and uniqueness of equilibrium using the Lasry-Lions monotonicity framework, and show that the yardstick mech anism achieves the social first best in closed form. For each diffusion process we derive fully explicit formulas for equilibrium effort, mean cost paths, adjustment speeds, and welfare, allowing complete comparative statics and shock analysis without numerical simulation. The model pro- vides a tractable and policy-ready extension of Shleifer's (1985) original insight, demonstrating that yardstick competition yields robust, efficient incentives even in large dynamic industries subject to stochastic shocks.