Surplus efficiency of ex ante investments in matching markets with nontransferabilities

Does a competitive equilibrium in a matching market provide adequate incentives for investments made before the market when utility is not perfectly transferable? In a one-sided market with a continuum of agents and finite types there is a constrained surplus efficient equilibrium, when a social planner can only affect investments but not payoffs nor matches, if an equal treatment property holds in equilibrium. Sufficient (but not full) utility transferability in a well defined sense implies this property. Ex post efficiency of payoffs (i.e., individual payoffs maximize the surplus in each match) alone is not sufficient to ensure that equilibrium investments maximize aggregate surplus.