Wage Dispersion, On-the-Job Search, and Stochastic Match Productivity: A Mean Field Game Approach

Wage dispersion and job-to-job mobility are central features of modern labour markets, yet canonical equilibrium search models with exogenous job-offer ladders struggle to jointly account for these facts and the magnitude of frictional wage inequality. We develop a continuous-time equilibrium search model in which match surplus follows a diffusion process, workers choose on-the-job search and separation, firms post state-contingent wages, and the cross-sectional distribution of match states endogenously determines both outside options and the job ladder. On the theoretical side, we formulate the problem as a stationary mean field game with a one-dimensional surplus state, characterize stationary mean field equilibria, and show that equilibrium separation is governed by a free-boundary rule: matches continue if and only if surplus stays above an endogenous threshold. Under standard regularity and Lasry-Lions monotonicity conditions we prove existence and uniqueness of stationary equilibrium and obtain comparative statics for the separation boundary, wage schedules, and wage dispersion. On the quantitative side, we solve the coupled HJB and Kolmogorov system using monotone finite-difference methods and interpret the discretization as a finite-state mean field game. The model is calibrated to micro evidence on stochastic match productivity, job durations, tenure-dependent separation hazards, wage growth, and job-to-job mobility. The stationary equilibrium delivers a structural decomposition of wage dispersion into stochastic selection along job spells, equilibrium on-the-job search and the induced job ladder, and equilibrium wage policies with feedback through outside options. We use this framework to quantify how firing costs, search subsidies, and changes in match-productivity volatility jointly shape mobility, the job ladder, and the cross-sectional distribution of wages.