《金融经济学》（英文版） Chapter 3 The principal-agent problem

Chapter 3 The principal-agent problem The principal-agent problem describes a class of interactions between two parties to a contract, an agent and a principal. The legal origin of these terms suggests that the principal engages the agent to act on his(the principal s behalf. In economic applications, the agent is not necessarily an employe of the principal. In fact, which of two individuals is regarded as the agent and which as the principal depends on the nature of the incentive problem Typically, the agent is the one who is in a position to gain some advantage by reneging on the agreement. The principal then has to provide the agent with incentives to abide by the terms of the contract We divide principal-agent problems into two classes: problems of hidden action and problems of hidden information. In hidden-action problems, the agent takes an action on behalf of the principal. The principal cannot observe the action directly, however, so he has to provide incentives for the agent to choose the action that is best for the principal. In hidden-information prob ems, the agent has some private information that is needed for some decision to be made by principal. Again, since the principal cannot observe the agent's information, he has to provide incentives for the agent to reveal the infor mation truthfully. We begin by looking at the hidden-action problem, also known as a moral hazard problem. 3.1 The model For concreteness, imagine that the principal and the agent undertake a risky venture together and agree to share the revenue. The agent takes some

Chapter 3 The principal-agent problem The principal-agent problem describes a class of interactions between two parties to a contract, an agent and a principal. The legal origin of these terms suggests that the principal engages the agent to act on his (the principal’s) behalf. In economic applications, the agent is not necessarily an employe of the principal. In fact, which of two individuals is regarded as the agent and which as the principal depends on the nature of the incentive problem. Typically, the agent is the one who is in a position to gain some advantage by reneging on the agreement. The principal then has to provide the agent with incentives to abide by the terms of the contract. We divide principal-agent problems into two classes: problems of hidden action and problems of hidden information. In hidden-action problems, the agent takes an action on behalf of the principal. The principal cannot observe the action directly, however, so he has to provide incentives for the agent to choose the action that is best for the principal. In hidden-information prob￾lems, the agent has some private information that is needed for some decision to be made by principal. Again, since the principal cannot observe the agent’s information, he has to provide incentives for the agent to reveal the infor￾mation truthfully. We begin by looking at the hidden-action problem, also known as a moral hazard problem. 3.1 The model For concreteness, imagine that the principal and the agent undertake a risky venture together and agree to share the revenue. The agent takes some 1

CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM action that affects the outcome of the project. The revenue from the venture is assumed to be a random function of the agents action. Let a denote the set of actions available to the agent with generic element a. Typically, A is either a finite set or an interval of real numbers. Let s denote a set of states with generic element s. For simplicity, we assume that the set S is finite. The probability of the state s conditional on the action a is denoted by p(a, s). The revenue in state s is denoted by R(s)20 The agent's utility depends on both the action chosen and the consump- tion he derives from his share of the revenue. The principals utility depend only on his consumption. We maintain the following assumptions about references · The agent' s utility function u:A×R+→ R is additively separable: u(a, c)=U(c-v(a) Further, the function U: R+-+R is C and satisfies U(c>0 and U"(c)≤0. The principals utility function V: R-R is C and satisfies V(c)>0 andW"(c)≤0. Notice that the agents consumption is assumed to be non-negative. This is interpreted as a liquidity constraint or limited liability. The principals consumption is not bounded below; in some contexts this is equivalent to assuming that the principal has large but finite wealth and non-negative consumption 3.2 Pareto efficiency The principal and the agent jointly choose a contract that specifies an action and a division of the revenue. A contract is an ordered pair(a, w()E Axw, where w is the set of incentive schemes and w(s)20 is the payment to the agent in state s Suppose that all variables are observable and verifiable. The principal and the agent will presumably choose a contract that is Pareto-efficient This leads us to consider the following decision problem(DP1) a∑>p(sV(F)-()

2 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM action that affects the outcome of the project. The revenue from the venture is assumed to be a random function of the agent’s action. Let A denote the set of actions available to the agent with generic element a. Typically, A is either a finite set or an interval of real numbers. Let S denote a set of states with generic element s. For simplicity, we assume that the set S is finite. The probability of the state s conditional on the action a is denoted by p(a, s). The revenue in state s is denoted by R(s) ≥ 0. The agent’s utility depends on both the action chosen and the consump￾tion he derives from his share of the revenue. The principal’s utility depends only on his consumption. We maintain the following assumptions about preferences: • The agent’s utility function u : A × R+ → R is additively separable: u(a, c) = U(c) − ψ(a). Further, the function U : R+ → R is C2 and satisfies U0 (c) > 0 and U00(c) ≤ 0. • The principal’s utility function V : R → R is C2 and satisfies V 0 (c) > 0 and V 00(c) ≤ 0. Notice that the agent’s consumption is assumed to be non-negative. This is interpreted as a liquidity constraint or limited liability. The principal’s consumption is not bounded below; in some contexts this is equivalent to assuming that the principal has large but finite wealth and non-negative consumption. 3.2 Pareto efficiency The principal and the agent jointly choose a contract that specifies an action and a division of the revenue. A contract is an ordered pair (a, w(·)) ∈ A×W, where W = {w : S → R+} is the set of incentive schemes and w(s) ≥ 0 is the payment to the agent in state s. Suppose that all variables are observable and verifiable. The principal and the agent will presumably choose a contract that is Pareto-efficient. This leads us to consider the following decision problem (DP1): max (a,w(·)) X s∈S p(a, s)V (R(s) − w(s))

3. 3. INCENTIVE EFFICIENCY ubject t ∑p(a,s((s)-v()≥ for some constant元 Proposition 1 Under the maintained assumptions, a contract(a, w())is Pareto-efficient if and only if it is a solution to the decision problem DP1 for sone u Suppose that(a, w()) is Pareto-efficient. Put i equal to the gent's pay off. By definition, the contract must maximize the principal's payoff subject to the constraint that the agent receive at least i. Conversely, suppose that the contract(a, w()is a solution to DPl for some value of i. If the contract is not Pareto-efficient, then there must be another contract that yields the same payoff to the principal and more to the agent. But then it must be possible to transfer wealth to the principal in some state, contradicting the optimality of (a,w() Suppose that the sharing rule satisfies w(s)>0 for all s. Then optimal risk sharing requires V(R(s)-(s)) 入 U((s) These are sometimes referred to as the borch conditions. If the action a belongs to the interior of A and if the functions p(a, s) and v(a) are differ entiable at a, then ∑m(a,s)(F(s)-(s)-X((s)+(a)=0 3.3 Incentive efficiency Now suppose that the agent's action is neither observable nor verifiable. In that case, the action specified by the contract must be consistent with the agent's incentives. A contract (a, w()) is incentive-compatible if it satisfies the constraint ∑p(a,s((s)-(a)≥∑p,s)(m()-v(b, s∈S s∈S

3.3. INCENTIVE EFFICIENCY 3 subject to X s∈S p(a, s)U(w(s)) − ψ(a) ≥ u, ¯ for some constant u¯. Proposition 1 Under the maintained assumptions, a contract (a, w(·)) is Pareto-efficient if and only if it is a solution to the decision problem DP1 for some u¯. Suppose that (a, w(·)) is Pareto-efficient. Put u¯ equal to the agent’s pay￾off. By definition, the contract must maximize the principal’s payoff subject to the constraint that the agent receive at least u¯. Conversely, suppose that the contract (a, w(·)) is a solution to DP1 for some value of u¯. If the contract is not Pareto-efficient, then there must be another contract that yields the same payoff to the principal and more to the agent. But then it must be possible to transfer wealth to the principal in some state, contradicting the optimality of (a, w(·)). Suppose that the sharing rule satisfies w(s) > 0 for all s. Then optimal risk sharing requires: V 0 (R(s) − w(s)) U0 (w(s)) = λ, ∀s. These are sometimes referred to as the Borch conditions. If the action a belongs to the interior of A and if the functions p(a, s) and ψ(a) are differ￾entiable at a, then X s∈S pa(a, s) [V (R(s) − w(s)) − λU(w(s)] + λψ0 (a)=0. 3.3 Incentive efficiency Now suppose that the agent’s action is neither observable nor verifiable. In that case, the action specified by the contract must be consistent with the agent’s incentives. A contract (a, w(·)) is incentive-compatible if it satisfies the constraint X s∈S p(a, s)U(w(s)) − ψ(a) ≥ X s∈S p(b, s)U(w(s)) − ψ(b), ∀b

CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM A contract is incentive-efficient if it is incentive-compatible and there does not exist another incentive-compatible contract that makes one party bet- ter off without making the other party worse off. We can characterize the incentive-efficient contracts using the following decision problem(DP2 max>p(a, s)V(R(s-w(s) (a,u() subject to ∑pa,s(m(s)-a)≥∑pb,s(m(s)-(), and ∑p(a,s)U(m(s)- Proposition 2 Under the maintained assumptions, a contract (a, w())is e-efficient only if it is a solution of DP2 for some constant i. A contract that solves DP2 is incentive-efficient if the participation constraint ng for every solution. The proof of the "only if"part is similar to the Pareto efficiency argument If (a, w() is a solution to DP2 and is not incentive-efficient, there exists an incentive-efficient contract that gives the principal the same payoff and the agent a higher payoff. But this contract must be a solution to DP2 that strictly satisfies the participation constraint The assumption of a uniformly binding participation constraint is restric tive: see Section 3. 7. 1 for a counter-example This DP can be solved in two stages. First, for any action a, compute the payoff V*(a) from choosing a and providing optimal incentives to the agent to choose a. Call this DP3 V"(a)=max)p(a, s)V(R(s)-w(s) s∈S b ∑p(a,s)(m(s)-v(a)≥∑p(b,)U((s)-v(b) ∑p(a,s)C(()-v(a)≥元 s∈S

4 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM A contract is incentive-efficient if it is incentive-compatible and there does not exist another incentive-compatible contract that makes one party bet￾ter off without making the other party worse off. We can characterize the incentive-efficient contracts using the following decision problem (DP2): max (a,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, and X s∈S p(a, s)U(w(s)) − ψ(a) ≥ u. ¯ Proposition 2 Under the maintained assumptions, a contract (a, w(·)) is incentive-efficient only if it is a solution of DP2 for some constant u¯. A contract that solves DP2 is incentive-efficient if the participation constraint is binding for every solution. The proof of the “only if” part is similar to the Pareto efficiency argument. If (a, w(·)) is a solution to DP2 and is not incentive-efficient, there exists an incentive-efficient contract that gives the principal the same payoff and the agent a higher payoff. But this contract must be a solution to DP2 that strictly satisfies the participation constraint. The assumption of a uniformly binding participation constraint is restric￾tive: see Section 3.7.1 for a counter-example. This DP can be solved in two stages. First, for any action a, compute the payoff V ∗(a) from choosing a and providing optimal incentives to the agent to choose a. Call this DP3 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, X s∈S p(a, s)U(w(s)) − ψ(a) ≥ u. ¯

3. 4. THE IMPACT OF INCENTIVE CONSTRAINTS Note that U( and V( are concave functions. A suitable transformation of this problem(see Section 3.10)is a convex programming problem for which the Kuhn-Tucker conditions are necessary and sufficient Once the function V* is determined, the optimal action is chosen to max nize the principals payoff a∈ arg max v(a) The advantage of the two-stage procedure is that it allows us to focus ol the problem of implementing a particular action. As we have seen, DP3 is (equivalent to)a convex programming problem and hence easier to"solve" and it turns out that many interesting properties can be derived from a study of DP3 without worrying about the optimal choice of action 3.3.1 Risk neutrality An interesting special case arises if the principal is risk neutral. In that case, maximization of the principals expected utility, taking a as given, is equivalent to minimizing the cost of the payments to the agent. Thus, DP3 can be re-written as nin∑p(a,s)(s) subject to p(a, s)U(w(s)-v(a)>>P(b, s)U(w(s))-v(b),Vb, ∑p(a,s)((s)-a)≥ 3.4 The impact of incentive constraint What is the impact of hidden actions? When does the imposition of incentive constraints affect the choice of contract? If one of the parties to the contract is risk neutral, it is particularly easy to check whether the first best can be achieved. that is. whether an incentive. efficient contract is also pareto-efficient Risk neutral principal

3.4. THE IMPACT OF INCENTIVE CONSTRAINTS 5 Note that U(·) and V (·) are concave functions. A suitable transformation of this problem (see Section 3.10) is a convex programming problem for which the Kuhn-Tucker conditions are necessary and sufficient. Once the function V ∗ is determined, the optimal action is chosen to max￾imize the principal’s payoff: a∗ ∈ arg max V ∗ (a). The advantage of the two-stage procedure is that it allows us to focus on the problem of implementing a particular action. As we have seen, DP3 is (equivalent to) a convex programming problem and hence easier to “solve” and it turns out that many interesting properties can be derived from a study of DP3 without worrying about the optimal choice of action. 3.3.1 Risk neutrality An interesting special case arises if the principal is risk neutral. In that case, maximization of the principal’s expected utility, taking a as given, is equivalent to minimizing the cost of the payments to the agent. Thus, DP3 can be re-written as min w(·) X s∈S p(a, 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, X s∈S p(a, s)U(w(s)) − ψ(a) ≥ u. ¯ 3.4 The impact of incentive constraints What is the impact of hidden actions? When does the imposition of incentive constraints affect the choice of contract? If one of the parties to the contract is risk neutral, it is particularly easy to check whether the first best can be achieved, that is, whether an incentive￾efficient contract is also Pareto-efficient. Risk neutral principal

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 con￾straint 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)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) 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. ¯

CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM The difference between DP4 and the original DP3 is that only the downward incentive constraints are included Obviously, V**(a)2 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 dP if and only if it is a solution of DPg 3.6 Monotonicity Many incentive schemes observed in practice reward the agent with higher rewards for higher outcomes, i. e, w(s) is increasing (or non-decreasing)in s It is interesting to see when this is a property of the theoretical optimal in- centive scheme. Assuming an interior solution, the Kuhn-Tucker (necessary) conditions are p(a, s)V(R(s)-w(s))-Ap(a, s)U(w(s))-ubP(a, s)-p(b, s))U(w(s))=0 V(B)-()=(x+ U(w(s)) ∑ 1-s P(a, s) By the mlrP, the right hand side is non-increasing in s, so the left hand side is non-increasing, which means that w(s) is non-decreasing 3. 7 Examples There are two outcomes s= 1, 2, where R(1)0. The agent's utilit

8 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM 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. 3.6 Monotonicity Many incentive schemes observed in practice reward the agent with higher rewards for higher outcomes, i.e., w(s) is increasing (or non-decreasing) in s. It is interesting to see when this is a property of the theoretical optimal in￾centive scheme. Assuming an interior solution, the Kuhn-Tucker (necessary) conditions are: p(a, s)V 0 (R(s)−w(s))−λp(a, s)U0 (w(s))− X b 0. The agent’s utility

3.7. EXAMPLES function U( is assumed to satisfy U(O)=0 and the reservation utility is 元=0. The inferior project can be implemented by setting w(s)=0 for s= 1, 2 Suppose the principal wants to implement a=2. The constraints can b written as (IC)(1-p(2,2)U((1)+p(2,2)U((2)-(2)≥ (1-p(1,2)U((1)+p(1,2)U(v(2) which simplifies to (p(2,2)-p(1,2)(U(2)-U(1))≥v(2) (IR)(1-p(2,2)U((1)+p(2,2)U((2)-v(2)≥0 In order to satisfy the (ir) constraint, consumption must be positive in at least one state. This implies that the expected utility from choosing low effort is strictly positive (1-p(1,2)U((1)+p(1,2)U((2)>0, so if the(IC) constraint is satisfied, the(IR) constraint must be strictly satisfied (1-p(2,2)U(v(1)+p(2,2U(v(2)-v(2)>0 Thus, if (w (1), w(2) is the solution to the optimal contract problem, the (IR) constraint does not bind. The principal's problem can then be written min(1-p(2,2)(1)+p(2,2)(2) st.(v(1),(2)≥0 (p(2,2)-p(1,2)(U((2)-U((1))≥v(2) Then it is clear that a necessary condition for an optimum is that w(1)=0 So the optimal contract for implementing a 2 is(0, w*(2)), where w*(2) solves the(IC) p(2,2)-p(1,2)U((2)=v(2) The payment w* (2) needed to give the necessary incentives to the manager

3.7. EXAMPLES 9 function U(·) is assumed to satisfy U(0) = 0 and the reservation utility is u¯ = 0. The inferior project can be implemented by setting w(s)=0 for s = 1, 2. Suppose the principal wants to implement a = 2. The constraints can be written as (IC) (1 − p(2, 2))U(w(1)) + p(2, 2)U(w(2)) − ψ(2) ≥ (1 − p(1, 2))U(w(1)) + p(1, 2)U(w(2)) which simplifies to (p(2, 2) − p(1, 2))(U(2) − U(1)) ≥ ψ(2) and (IR) (1 − p(2, 2))U(w(1)) + p(2, 2)U(w(2)) − ψ(2) ≥ 0. In order to satisfy the (IR) constraint, consumption must be positive in at least one state. This implies that the expected utility from choosing low effort is strictly positive: (1 − p(1, 2))U(w(1)) + p(1, 2)U(w(2)) > 0, so if the (IC) constraint is satisfied, the (IR) constraint must be strictly satisfied: (1 − p(2, 2))U(w(1)) + p(2, 2)U(w(2)) − ψ(2) > 0. Thus, if (w(1), w(2)) is the solution to the optimal contract problem, the (IR) constraint does not bind. The principal’s problem can then be written as: minw(1 − p(2, 2))w(1) + p(2, 2)w(2) s.t. (w(1), w(2) ≥ 0 (p(2, 2) − p(1, 2))(U(w(2)) − U(w(1))) ≥ ψ(2). Then it is clear that a necessary condition for an optimum is that w(1) = 0. So the optimal contract for implementing a = 2 is (0, w∗(2)), where w∗(2) solves the (IC): (p(2, 2) − p(1, 2))U(w∗ (2)) = ψ(2). The payment w∗(2) needed to give the necessary incentives to the manager will be higher:

CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM the higher the cost of effort v(2) the smaller the manager's risk tolerance(as measured by U(w(2)) U(0); the smaller the marginal productivity of effort(as measured by p(2, 2) p(1,2) To decide whether it is optimal to implement high or low effort, the prin- cipal compares the profit from optimally implementing each level of effort The maximum profit from low effort (1-p(1,2)R(2)+p(1,2)F(1) The maximum profit from high effort is (1-p(2,2)R(1)+p(2,2)R(2)-p(2,2)'(2) So high effort is optimal if and only if (p(2,2)-p(1,2)(R(2)-R(1)≥(2) that is, the increase in expected revenue is greater than the cost of providing managerial incentives 3.7.1 Optimality and incentive-efficiency Suppose there are two states s= 1, 2, two actions a= 1, 2 and the reservation utility is i=0. The principal and the agent are both risk neutral. The other parameters of the problem are given by R(1)=0<R(2) v(1)=0<(2) p(1,2)<p(2,2) The action a=l is optimally implemented by putting (s)=0,V The action a= 2 is optimally implemented by putting 0 if s= 1 w2( v(2)/(p(2,2)-p(1,2)ifs=2

10 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM • the higher the cost of effort ψ(2); • the smaller the manager’s risk tolerance (as measured by U(w(2)) − U(0)); • the smaller the marginal productivity of effort (as measured by p(2, 2)− p(1, 2)). To decide whether it is optimal to implement high or low effort, the prin￾cipal compares the profit from optimally implementing each level of effort. The maximum profit from low effort is (1 − p(1, 2))R(2) + p(1, 2)R(1). The maximum profit from high effort is (1 − p(2, 2))R(1) + p(2, 2)R(2) − p(2, 2)w∗ (2). So high effort is optimal if and only if (p(2, 2) − p(1, 2))(R(2) − R(1)) ≥ w∗ (2), that is, the increase in expected revenue is greater than the cost of providing managerial incentives. 3.7.1 Optimality and incentive-efficiency Suppose there are two states s = 1, 2, two actions a = 1, 2 and the reservation utility is u¯ = 0. The principal and the agent are both risk neutral. The other parameters of the problem are given by R(1) = 0 < R(2) ψ(1) = 0 < ψ(2) 0 < p(1, 2) < p(2, 2). The action a = 1 is optimally implemented by putting w1(s)=0, ∀s. The action a = 2 is optimally implemented by putting w2(s) = ½ 0 if s = 1 ψ(2)/(p(2, 2) − p(1, 2)) if s = 2