Matching and Welfare in Large Markets – with an Application to Monitoring in Teams

We consider two-sided markets with a continuum of agents and a continuum of types. The single crossing condition (GID) introduced in Legros & Newman (2007) implies that equilib- ria matches are payoff equivalent to equilibria with positive assortative matching. We show the existence of a ``best equilibrium'' for each side of the market in the sense that for each agent on a given side, his payoff in this ``optimal'' equilibrium is the maximum over all possible equilibria. When the type distribution is positive and continuous, the marginal wage of an agent is equal to his marginal productivity, which, contrary to the TU case, cannot be defined independently of the equilibrium payoffs. The marginal wage function captures the overall distribution of marginal payoffs in the economy and is the solution to a system of differential equations. These equations capture the two effects in NTU models when GID holds: the stan- dard complementarity in types and the complementarity in types and payoffs. Because with non-transferabilities, equilibrium matches do not necessarily maximize total surplus, agents be- ing paid at their marginal productivity is neither a sufficient nor necessary condition for surplus efficiency. We illustrate this point in a principal-agent example with endogenous monitoring.