CHAPTER 1. THE PRINCIPAL-AGENT PROBLEM for all s. In that case, the agents income is independent of his action, so in the hidden action case he would choose the cost-minimizing action hus the first best can be achieved with hidden actions only if the optimal action is cost-minimizing Suppose that the agent is risk neutral and the principal is(strictly)risk averse, i.e., V"(c)<0. Then the Borch conditions for the first best imply that the principals income R(s-w(s) is constant, as long as the solution is interior. This corresponds to the solution of "selling the firm to the agent but it works only as long as the agents non-negative consumption constraint inding In general, there is some constant y such that R(s-w(s)=min9, R(s) and (s)=max{R(s)-0,0} More generally, if we assume the first best is an interior solution and maintain the differentiability assumptions discussed above, the first-order condition for the first best ∑a(a,s)V(F(s)-(s)-(()+(a)=0 and the first-order(necessary) condition for the incentive-compatibility con straint is ∑p(a,s)(()-(a)=0 So the incentive-efficient and first-best contracts coincide only if Pa(a, s)V(R(s-w(s))=0 Note that there may be no interior solution of the problem DP3 even under the usual Inada conditions. See Section 1.7.2 for a counter-examp 1.5 The optimal incentive scheme In order to characterize the optimal incentive scheme more completely, we impose the following assumptions
6 CHAPTER 1. THE PRINCIPAL-AGENT PROBLEM for all s. In that case, the agent’s income is independent of his action, so in the hidden action case he would choose the cost-minimizing action. Thus, the first best can be achieved with hidden actions only if the optimal action is cost-minimizing. Suppose that the agent is risk neutral and the principal is (strictly) risk averse, i.e., V 00(c) < 0. Then the Borch conditions for the first best imply that the principal’s income R(s)−w(s) is constant, as long as the solution is interior. This corresponds to the solution of “selling the firm to the agent”, but it works only as long as the agent’s non-negative consumption constraint is not binding. In general, there is some constant y¯ such that R(s) − w(s) = min{y, R¯ (s)} and w(s) = max{R(s) − y, ¯ 0}. More generally, if we assume the first best is an interior solution and maintain the differentiability assumptions discussed above, the first-order condition for the first best is X s∈S pa(a, s) [V (R(s) − w(s)) − λU(w(s)] + λψ0 (a)=0. and the first-order (necessary) condition for the incentive-compatibility constraint is X s∈S pa(a, s) [U(w(s)] − ψ0 (a)=0. So the incentive-efficient and first-best contracts coincide only if X s∈S pa(a, s)V (R(s) − w(s)) = 0. Note that there may be no interior solution of the problem DP3 even under the usual Inada conditions. See Section 1.7.2 for a counter-example. 1.5 The optimal incentive scheme In order to characterize the optimal incentive scheme more completely, we impose the following assumptions:
1.5. THE OPTIMAL INCENTIVE SCHEME 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 revenue ses pa(a, s)R(s)is increasing in a Now consider the modified DP4 of implementing a fixed value of ∑p(a,s)(R()-m(s) subject to ∑p(a,s)U(m()-v()≥∑p(s)U((s)-v(b,b<a ∑p(a,s)(n(s)-v( The difference between DP4 and the original DP3 is that only the downward incentive constraints are included Obviously, V*(a)>v(a). Suppose that V**(a)>v(a). This means that the agent wants to choose a higher action than a in the modified problem But this is good for the principal, who will never choose a if he can get a better action for the same price. Thus max V*(a)=max V**(a) Thus, we can use the solution to the modified problem DP4 to characterize the optimal incentive scheme Theorem 3 Suppose that a E arg max V"(a). The incentive scheme w()is a solution of dPf if and only if it is a solution of DP3
1.5. THE OPTIMAL INCENTIVE SCHEME 7 • 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 nondecreasing 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. ¯ The difference between DP4 and the original DP3 is that only the downward incentive constraints are included. Obviously, V ∗∗(a) ≥ V ∗(a). Suppose that V ∗∗(a) > V ∗(a). This means that the agent wants to choose a higher action than a in the modified problem. But this is good for the principal, who will never choose a if he can get a better action for the same price. Thus, maxa V ∗ (a) = maxa V ∗∗(a). Thus, we can use the solution to the modified problem DP4 to characterize the optimal incentive scheme. Theorem 3 Suppose that a ∈ arg max V ∗(a). The incentive scheme w(·) is a solution of DP4 if and only if it is a solution of DP3