We study the principal-agent(owner-manager) problem with moral hazard in continuous time with a Brownian filtration, recursive preferences, and multiple principals(one agent for each principal). Different principals' problems are connected, because the effort of each principal's agent affects the common probability measure, and therefore one agent's effort can impact the cash-flow drifts of all the principals. This could capture, for example, the impact of innovations by agents of one firm on the cash-flow prospects of competing firms. The externality of each agent's effort results in interdependence among the principals' optimal contracting problems. For the class of preferences we consider, solving the equilibrium reduces to computing a system of linked subjective cash-flow value processes, one for each principal. We show that the system has a closed-form solution, when each principal's cash flow is driven by an affine-yield state process. Each principal's optimal pay policy amounts to choosing the component of the subjective cash-flow volatility to transfer to the agent (that is, a volatility sharing rule). The optimal sharing rules are simple functions of each principal's own cash-flow volatility in the case when the impact of aggregate effort on drifts is additive, but are generally functions of all the principals' cash-flow volatilities when the impact of effort on the drift change is diminishing in aggregate effort. We provide a number of closed-form solutions to illustrate.
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