Team incentives under private contracting

We model a moral hazard in teams problem in which a profit-maximizing principal offers private contracts to multiple agents. Public contracts are common knowledge to all agents, but private contracts are known only by the principal and each individual agent. Public contracts can induce efficient outcomes but are subject to effort-reducing collusion between the principal and any given agent. Private contracts, by construction, are immune to such collusion but necessarily inefficient, as the principal is forced to make the team collectively the residual claimant (on margin), whereas efficiency requires that each individual agent be the residual claimant on his own.