Existence of a non-stationary equilibrium in search-and-matching models: tu and ntu

This paper proves the existence of a non-stationary equilibrium in the canonical search-and-matching model with heterogeneous agents. Non-stationarity entails that the number and characteristics of unmatched agents evolve endogenously over time. An equilibrium exists under minimal regularity conditions and for both paradigms considered in the literature: transferable and non-transferable utility. To address potential discontinuities in match opportunities across types, our analysis introduces a generalized Schauder fixed-point theorem suitable for models with discontinuous value functions.