Monotonicity of Optimal Contracts Without the First-Order Approach
Bibliographic record
Abstract
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. The e-companion is available at https://doi.org/10.1287/opre.2018.1720 .
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.011 | 0.056 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.004 | 0.010 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.004 | 0.007 |
| Insufficient payload (model declined to judge) | 0.012 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".