The First-Best Sharing Rule in the Continuous-Time Principal-Agent Model with Exponential Utility

The continuous-time principal-agent model with exponential utility developed by Holmström and Milgrom (1987) and generalized by Schättler and Sung (1993, 1996) and Sung (1995) admits a simple closed-form solution: The second-best sharing rule is linear in output. Unfortunately, the first-best sharing rule has never been derived. In this note, we show that the first-best sharing rule is also linear in output, which fits in nicely with an analogous result from static risk-sharing theory. In addition, we show that the slope is equal to the principal’s share of total absolute risk-aversion. This result is consistent with Borch’s (1962) fundamental theorem of Pareto-optimal risk-sharing.