The Analytic Theory of Frictional Labour Markets: A Generalized Hazard Function Approach

I develop an analytic framework for labour search-and-matching models with state-dependent wage rebargaining, endogenous separations, and a general menu of adjustment frictions, porting tools from the menu-cost pricing literature into the Diamond-Mortensen-Pissarides environment. My approach yields an exact spectral decomposition of the unemployment law of motion through a scalar Volterra renewal equation that sidesteps the cross-sectional density. The framework delivers the relationship between the spectrum of the system and its dynamics, exact and linearized impulse responses admitting a canonical five-channel decomposition that shows the importance of state dependency, the equilibrium hazard, and endogenizing the hazard, and a parameter-free $3/8$ eigenvalue spacing diagnostic for state dependence. This framework provides a complete set of tools to analyze labour market dynamics under rigidity, as well as demonstrating the importance of endogenous hazard and boundary responses to impulse control models.