Journal article: We study an economy where intermediaries compete over contracts in a nonexclusive insurance market affected by moral hazard. In this context, we show that, contrarily to what is commonly believed, market equilibria may fail to be efficient even if the planner is not allowed to enforce exclusivity of trades (third best inefficiency). Our setting is the same as that of Bisin and Guaitoli [Bisin, A., Guaitoli, D., 2004. Moral hazard with nonexclusive contracts. Rand Journal of Economics 2, 306-328]. We hence argue that some of the equilibrium conditions they imposed are not necessary, and we exhibit a set of equilibrium allocations which fail to satisfy them.