Optimal Unit Sample Clauses

Optimal Unit. Contingent Contracts We first make the following observation based on Proposition 1 and Figure 2. ^ ^ Corollary 1 A unit-contingent contract (v(p), φ(p)) with penalty φ(p) < ∞ is strictly dominated by the contract (v(p), φ†(p)), where φ†(p) = 0 if φ(p) ≤ φ(p), and φ†(p) > φ(p) if φ(p) > φ(p). From Figure 2, it can be seen that (v(p), φ†(p)) defined in Corollary 1 will not change the plant’s expected utility, but increase the distributor’s expected utility. Intuitively, when the penalty is above the threshold, a higher penalty leads to higher truth-telling probability, benefiting the distributor. Corollary 1 implies that, without loss of optimality, we can restrict our attention to only two penalty values: zero and very high penalty. With the above observation, choosing a contract (v(p), φ(p)) is equivalent to choosing a contract price function v(p) and a price set L, within which the penalty is zero and beyond which the penalty is very high. Because there is no incentive for misreporting when p = v(p), we can exclude the set {p : p = v(p)} from L. Let L = Ll ∪ Lr, where Ll d=ef { p : p < v(p), φ(p) = 0 }, Lr d=ef { p : p > v(p), φ(p) = 0 }. ^ From Corollary 1, zero penalty is never optimal when the threshold φ(p) < 0. Thus, the two subsets above must be contained in the region where the threshold is non-negative: def def