Dynamic directed random matching

We develop a general and unified model in which a continuum of agents conduct directed random searches for counterparties. Our results provide the first probabilistic foundation for static and dynamic models of directed search (including the matching-function approach) that are common in search-based models of financial markets, monetary theory, and labor economics. The agents' types are shown to be independent discrete-time Markov processes that incorporate the effects of random mutation, random matching with match-induced type changes, and with the potential for enduring partnerships that may have randomly timed break-ups. The multi-period cross-sectional distribution of types is shown to be deterministic and is calculated using the exact law of large numbers.