Monotonicity of optimal contracts without the first-order approach

We develop a simple sufficient condition for an optimal contract of a moral hazard problem to be monotone in the output signal. Existing results on monotonicity require conditions on the output distribution (namely, the monotone likelihood ratio property (MLRP)) and additional conditions to guarantee that agent's decision is approachable via the first-order approach of replacing that problem with its first-order conditions. We know of no positive monotonicity results in the setting where the first-order approach does not apply. Indeed, it is well documented that when there are finitely many possible outputs, and the first-order approach does not apply, the MLRP alone is insufficient to guarantee monotonicity. However, we show that when there is an interval of possible output signals, the MLRP does suffice to establish monotonicity under additional technical assumptions that do not guarantee the validity of the first-order approach. To establish this result we examine necessary optimality conditions for moral hazard problems using a novel penalty function approach. We then manipulate these conditions and provide sufficient conditions for when they coincide with a simple version of the moral hazard problem with only two constraints. In this two-constraint problem, monotonicity is established directly via a strong characterization of its optimal solutions.