Constrained efficient allocation (CE) is characterized in a model of adverse selection and directed search (Guerrieri, Shimer, and Wright (2010)). CE is defined to be the allocation that maximizes welfare, the ex-ante utility of all agents, subject to the frictions of the environment. When equilibrium does not achieve the first best (the allocation that maximizes welfare under complete information), then welfare in the CE is strictly higher than welfare in the equilibrium allocation. That is, equilibrium is not constrained efficient. Under some conditions, welfare in the CE even attains welfare in the first best. Finally, sufficient conditions are provided under which equilibrium is not constrained Pareto efficient, either. Cross-subsidization is the key to all these results. In an asset market application, the first best is shown to be implementable through tax schedules that are monotone in the asset prices.