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 ladders struggle to account for both facts and the magnitude of frictional wage inequality. We develop a continuous-time equilibrium search model in which match surplus follows a diffusion, workers choose on-the-job search and separation, firms post state-contingent wages, and the cross-sectional distribution of match states endogenously pins down outside options and the job ladder. The problem is formulated as a stationary mean field game with a one-dimensional surplus state. We establish existence and uniqueness of stationary equilibrium under standard regularity and monotonicity conditions, and show that separation is governed by a free-boundary rule. Quantitatively, we solve the coupled Hamilton-Jacobi-Bellman & Kolmogorov system with monotone finite-difference methods, calibrate the model to micro evidence on match productivity and mobility, and use it to decompose wage dispersion and to study how firing costs, search subsidies, and volatility shape mobility, the job ladder, and the wage distribution.