Moral hazard and the spanning condition without the first-order approach

The “spanning condition”describes a situation where the agent’s e¤ort determines the weights placed on two distinct technologies. Su¢ cient conditions are known under which the …rst-order approach (FOA) is valid when the spanning condition holds. In this paper, a complete solution to the problem is provided. Thus, the problem is solved even in cases where the FOA is not valid. The solution has two main steps. The …rst step fully characterizes the set of actions that can be implemented. The second step establishes that the optimal contract can be found by applying the FOA to the newly identi…ed set of feasible actions. In short, a modi…ed, or two-step, FOA solves the problem. This procedure correctly solves a famous counterexample due to Mirrlees in which the standard FOA is not valid. A much simpler and economically more compelling counterexample is provided. Moreover, the model provides a tractable environment in which it is possible to study comparative statics even when the FOA is not justi…ed. An example involving threshold monitoring illustrates. Finally, the solution is generalized to the case where the agent determines the weights placed on an arbitrary number of technologies. The agent’s action is multi-dimensional in this case. JEL Classi…cation Numbers: D82, D86