Optimal Compensation with Hidden Action and Lump-Sum Payment in a Continuous-Time Model

Abstract

We consider a problem of finding optimal contracts in continuous time, when the agent's actions are unobservable by the principal, who pays the agent with a one-time payoff at the end of the contract. We fully solve the case of quadratic cost and separable utility, for general utility functions. The optimal contract is, in general, a nonlinear function of the final outcome only, while in the previously solved cases, for exponential and linear utility functions, the optimal contract is linear in the final output value. In a specific example we compute, the first-best principal's utility is infinite, while it becomes finite with hidden actions, which is increasing in value of the output. In the second part of the paper we formulate a general mathematical theory for the problem. We apply the stochastic maximum principle to give necessary conditions for optimal contracts. Sufficient conditions are hard to establish, but we suggest a way to check sufficiency using non-convex optimization.

A Solvable Time-Inconsistent Principal-Agent Problem

“…We will employ the stochastic maximum principle of the solution to the weak situation of the agent problem. The main idea is to apply random variational methods; the relative papers [8,20,26] used the similar approach.…”

“…Reference [5,6] uses the stochastic maximum principle to extended Holmstrom's model, and discuss the optimal solution of the agent with private information in the continuous-time model. Reference [7,8] uses the forward-backward stochastic differential equations to consider the optimal contract under moral hazard. Reference [9] systematically expands the problem of continuous-time principal-agent.…”

Dynamic managerial compensation: A variational approach

“…To the best of our knowledge, this approach is new to the dynamic mechanism design literature. Variational methods have been used in agency models with hidden actions only by Cvitanić et al (2009), Capponi et al (2012), and Sannikov (2014. For a general treatment of variational methods in optimization problem, see e.g.…”

Optimal contracting with moral hazard and behavioral preferences

“…Thus, in the concave domain in which U ′′ A (x) < 0, we have c ′ (x) < 1; but c ′ (x) is likely to be closer to one than when the principal is risk-neutral. 3 This means that, in the region in which the agent is risk-averse, the more risk-averse principal (higher risk aversion…”