Is there a Golden Parachute in Sannikov's principal-agent problem?

Abstract

This paper provides a complete review of the continuous-time optimal contracting problem introduced by Sannikov [54], in the extended context allowing for possibly different discount rates of both parties. The agent's problem is to seek for optimal effort, given the compensation scheme proposed by the principal over a random horizon. Then, given the optimal agent's response, the principal determines the best compensation scheme in terms of running payment, retirement, and lump-sum payment at retirement.A Golden Parachute is a situation where the agent ceases any effort at some positive stopping time, and receives a payment afterwards, possibly under the form of a lump sum payment, or of a continuous stream of payments. We show that a Golden Parachute only exists in certain specific circumstances. This is in contrast with the results claimed by Sannikov [54], where the only requirement is a positive agent's marginal cost of effort at zero. Namely, we show that there is no Golden Parachute if this parameter is too large. Similarly, in the context of a concave marginal utility, there is no Golden Parachute if the agent's utility function has a too negative curvature at zero.In the general case, we provide a rigorous analysis of this problem, and we prove that an agent with positive reservation utility is either never retired by the principal, or retired above some given threshold (as in Sannikov's solution). In particular, different discount factors induce naturally a face-lifted utility function, which allows to reduce the whole analysis to a setting similar to the equal-discount rates one. Finally, we also confirm that an agent with small reservation utility does have an informational rent, meaning that the principal optimally offers him a contract with strictly higher utility value.

Moral Hazard, Dynamic Incentives, and Ambiguous Perceptions

“…In the analysis of the HJB equation (3.3), unlike Sannikov's approach we do not require an upper-boundary constraint to determines retirement with a smooth-pasting condition F ′ (W gp ) = F ′ 0 (W gp ). As illustrated by Possamaï and Touzi (2020), such a smoothpasting constraint as in Sannikov's approach can entail loss of generality. Instead, following Possamaï and Touzi (2020)'s dynamic programming approach and for simplicity, we model retirement with a finite random horizon for the contracting relationship.…”

“…On concavity of the principal's value function. In a model of dynamic moral hazard where there is no drift ambiguity Sannikov (2008) and Possamaï and Touzi (2020) show that the solution F to the HJB equation is concave. This property extends to the current model with drift-ambiguity.…”

“…We follow closely dynamic programming approach of Possamaï and Touzi (2020) to analyze dynamic principal agent problems in a classical setting of Sannikov (2008), we adopt it to allow for drift ambiguity. 13 In particular, the characterization of the optimal contract as a solution to the HJB equation follows closely the proof strategy used in the proof of Theorem 3.6 and Proposition 7.2 in Possamaï and Touzi (2020) for the case where the principal and the agent have a common discount factor. Indeed, the main new mathematical arguments will be due to possible heterogeneity in the worst-case drift scenarios perceived by the principal and the agent.…”

“…Possamaï and Touzi (2020) uses the general approach ofLin et al (2020) who develop formulation of the model where the agent's action can control both the drift and the volatility of output process.…”

Agency problem and mean field system of agents with moral hazard, synergistic effects and accidents

“…This problem has been investigated by the mathematical community in the last fifteen years. A particular extension to random horizon has been studied in [San08,PT20]. The recent article [CPT18] has proposed a comprehensive and rigorous mathematical method to solve this problem under integrability assumption for the contract ξ, in the Brownian model with controlled drift and volatility by using the theory of second-order backward stochastic differential equations to solve (IC) and classical verification result for solving (1.1).…”