3.5. THE OPTIMAL INCENTIVE SCHEME Erample: Suppose that there are two states s= 1, 2 and R(1)< R(2)and let p(a) denote the probability of success(s= 2). At an interior solution, the necessary condition derived above is equivalent to p(a)(R(2)-m(2)-V(R(1)-m(1))=0 R(2)-R(1)=(2)-v(1), assuming p(a)>0. This allocation will not satisfy the Borch conditions unless the agent is risk neutral on the interval w (1), w(2) Note that there may be no interior solution of the problem DP3 even under the usual Inada conditions. See Section 3.7.2 for a counter-example 3.5 The optimal incentive scheme In order to characterize the optimal incentive scheme more completely, we impose the following rstrictions The principal is risk neutral, which means that if two actions are equally costly to implement, he will always prefer the one that yields higher expected revenue. There is a finite number of states s=1..s and the revenue function R(s is increasing in s Monitone likelihood ratio property: There is a finite number of actions a=l,.., A and for any actions a b, the ratio p(b, s)/p(a, s) is non- decreasing in s. We also assume that the vectors p(b, and p(a are distinct, so for some states the ratio is increasing. The expected (a, s)R(s)is 1g In a. Now consider the modified DP4 of implementing a fixed value of a V""(a)=max p(a, s)V(R(s)-w(s) subject t a,s)U(0(s)-v(a)≥∑mb,sU((s)-v(b),wb<a ∑p(a,s)(m()-v(a)≥a s∈S3.5. THE OPTIMAL INCENTIVE SCHEME 7 Example: Suppose that there are two states s = 1, 2 and R(1) < R(2) and let p(a) denote the probability of success (s = 2). At an interior solution, the necessary condition derived above is equivalent to p0 (a) [V (R(2) − w(2)) − V (R(1) − w(1))] = 0 or R(2) − R(1) = w(2) − w(1), assuming p0 (a) > 0. This allocation will not satisfy the Borch conditions unless the agent is risk neutral on the interval [w(1), w(2)]. Note that there may be no interior solution of the problem DP3 even under the usual Inada conditions. See Section 3.7.2 for a counter-example. 3.5 The optimal incentive scheme In order to characterize the optimal incentive scheme more completely, we impose the following rstrictions: • The principal is risk neutral, which means that if two actions are equally costly to implement, he will always prefer the one that yields higher expected revenue. • There is a finite number of states s = 1, ..., S and the revenue function R(s) is increasing in s. • Monitone likelihood ratio property: There is a finite number of actions a = 1, ..., A and for any actions a<b, the ratio p(b, s)/p(a, s) is non￾decreasing in s. We also assume that the vectors p(b, ·) and p(a, ·) are distinct, so for some states the ratio is increasing. The expected revenue P s∈S pa(a, s)R(s) is increasing in a. Now consider the modified DP4 of implementing a fixed value of a: V ∗∗(a) = max w(·) X s∈S p(a, s)V (R(s) − w(s)) subject to X s∈S p(a, s)U(w(s)) − ψ(a) ≥ X s∈S p(b, s)U(w(s)) − ψ(b), ∀b < a, X s∈S p(a, s)U(w(s)) − ψ(a) ≥ u. ¯