Risk aversion, moral hazard, and the principal's loss

In their seminal paper on the principal-agent model with moral hazard, Grossman and Hart (1983) show that if the agent's utility function is $U(I,a)=-e^{-k(I-a)}$, then the loss to the principal from being unable to observe the agent's action is increasing in the agent's degree of absolute risk aversion. Their proof is restricted to the case where the number of observable outcomes is equal to two, and it uses an argument that is specific to that case. In this note, we provide an alternative proof that generalizes their result to any (finite) number of outcomes.