moral hazard with unobservable wealth and a continuum of actions

The main results are the following ones. First, the only consumption that can be part of an equilibrium is the one that optimizes the inter-temporal utility of the agent if he refuses the contract. Second, a necessary condition for an action to be implementable is that, at the optimal consumption level, the marginal utility of the first period equals the marginal utility of consumption in the second period, evaluated at the principal's desired action and at the optimal contract for that action. An implication of this condition is that, generically, at most a finite number of actions can be implemented. Third, if the agent's preferences exhibit IARA (increasing absolute risk aversion) or CARA, then only the lowest action is implementable in equilibrium. Fourth, if the cost of effort is sufficiently small, then only the lowest action is implementable in equilibrium. Fifth, I derive sufficient conditions on the primitives of the problem under which the principal can implement an action higher than the smallest one, unlike the case with a discrete number of actions analyzed in the literature.