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

The “spanning condition” describes a situation where the agent's effort determines the weights placed on two distinct technologies. Sufficient conditions are known under which the first-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 when the FOA is not valid. The solution has two main steps. The first step fully characterizes the set of implementable actions. The second step establishes that the optimal contract can be found by applying the FOA to the newly identified set of feasible actions. 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 justified.