2 Sharing with a Risk - Neutral Agent

In the classical principal–agent problem, a riskneutral agent bears all the risk. This particular solution, while acknowledged as a special case, is prominent in the minds of economists because the more general risk-averse case does not easily yield numerical results. For example, Jensen and Murphy (1990) find a divergence between the risk actually borne by chief executive officers and the risk-neutral solution, which seems too large to be accounted for by reasonable levels of risk aversion. Although the standard risk-neutral solution is correct, it is also misleading. Other solutions exist wherein the agent does not bear all the risk, and these may be considered more “natural,” since they are limits of the risk-averse case. Specifically, in Grossman and Hart’s (1983) principal–agent problem, where there is a finite number of actions and states, many optimal sharing rules exist; in only one does the agent bear all the risk.1 With a large enough stake in the project, the agent will not shirk—and with a finite number of states and actions, this stake need not be 100 percent. Once agents have some risk aversion, the principal– agent problem has a unique solution. For the two-state case, limits can be computed as risk aversion approaches zero. The risk-averse solutions do not converge to the classic risk-neutral solution, however, but to the solution with the lowest risk for the agent. Because less risk makes a risk-averse agent happier, he demands a lower risk premium, in turn making the principal happier. But exceptions occur. There are cases in which the optimal action discretely shifts with an infinitesimal increase in risk aversion. In this case, the sharing rule (and thus the risk borne by the agent) differs substantially when the principal wants to induce distinctly different actions. By increasing the number of actions, the results reduce to the standard continuous-action principal– agent models (see Holmstrom [1979]). Under reasonable conditions, the set of risk-neutral solutions shrinks to one. This should introduce a note of caution to applications of the principal–agent model. The simple risk-neutral solution is not a good approximation of the optimal contract, even for arbitrarily low risk aversion. It can be misleading to compare actual contracts in which risk aversion is important— executive compensation, for instance—with the predictions of the risk-neutral principal–agent model. Stated more positively, these results show how principal–agent theory implies the relatively flat sharing rules that are observed in practice.