CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM Suppose, for example, that the principal is risk neutral and the agent strictly) risk averse, i. e, U(c)<0. The Borch conditions for an interior solution imply that w(s) is a constant 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 Risk neutral agent 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 impl 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, R(s) w(s)= maxR(s)-g,0J Both parties risk averse 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 ∑P2(a,s)[(R()-m(s)-MU((s)+'(a)=0. and the first-order(necessary) condition for the incentive-compatibility con- straint is ∑ma(a,s)U((s)-v()=0 So the incentive-efficient and first-best contracts coincide only if ∑p(a,sV(F(s)-t(s)=0 s∈S
6 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM Suppose, for example, that the principal is risk neutral and the agent is (strictly) risk averse, i.e., U00(c) < 0. The Borch conditions for an interior solution imply that w(s) is a constant 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. Risk neutral agent 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}. Both parties risk averse 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
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∈S
3.5. 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 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. ¯