The Continuous-Time Principal-Agent Problem with Moral Hazard and Recursive Preferences

The Continuous-Time Principal-Agent Problem with Moral Hazard and Recursive Preferences

Abstract
We study the principal-agent problem with moral hazard in a continuous-time Brownian filtration with recursive preferences on the part of both principal and agent, and pay over the lifetime of the contract. Previous work has considered only additive utility, which, as is well known, arbitrarily links intertemporal substitution and risk aversion. Yet time-additivity offers essentially no advantage in tractability because agent optimality induces recursivity to the principal’s preferences even in the additive case. We show that the (necessary and sufficient) first-order conditions for the principal’s problem take the form of a forward-backward stochastic differential equation. If the agent’s first-order condition satisfies an invertibility condition, the principal’s problem can be rewritten with the agent’s utility satisfying a forward equation. The problem then becomes analogous to the optimal portfolio/consumption problem. Under translation-invariant preferences (a class that includes time-additive exponential utility) or scale-invariant (homothetic) preferences, the system uncouples and dramatically simplifies to the solution of a single backward stochastic differential equation. We obtain closed-form solutions for some parametric examples, including one with constant cash flow volatility and subjective beliefs that differ between principal and agent, and another with square-root cash-flow dynamics. Linear sharing rules are obtained only under very special conditions.

The Continuous-Time Principal-Agent Problem with Moral Hazard and Recursive Preferences

The Continuous-Time Principal-Agent Problem with Moral Hazard and Recursive Preferences

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