Special Cases and Applications

We present here well-known examples and applications of continuous-time Principal–Agent models. The seminal work of Holmström and Milgrom (Econometrica 55:303–328,1987) is the first to use a continuous-time model, showing that doing that can, in fact, lead to simple, while realistic optimal contracts. In particular, if the principal and the agent maximize expected utility from terminal output value, and have non-separable cost of effort and exponential utilities, the optimal contract is linear in that value. With other utilities and separable cost of effort, the optimal contract is nonlinear in the terminal output value, obtained as a solution to a nonlinear equation that generalizes the first best Borch condition. In the case of the agent deriving utility from continuous contract payments on an infinite horizon, and if the principal is risk-neutral, the problem reduces to solving an ordinary differential equation for the principal’s expected utility process as a function of the agent’s expected utility process. That equation can then be solved numerically for various cases, including the case in which the agent can quit, or be replaced by another agent, or be trained and promoted. These cases are analyzed by studying the necessary conditions in terms of an FBSDE system for the agent’s problem, and, in Markovian models, by identifying sufficient conditions in terms of the HJB differential equation for the principal’s problem.