第三委托代理(纽约大学艾伦和盖尔金融经济学讲义)

Chapter3The principal-agent problemThe 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.3.1The modelFor concreteness,imagine that the principal and the agent undertake a risky venture together and agree to share the revenue.The agent takes some12CHAPTER3.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 afinite 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 isfinite.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 C2and satisfies U0(c)>0and U00(c)≤0.•The principal’s utility function V:R→R is C2and satisfies V0(c)>0 and V00(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 butfinite wealth and non-negative consumption.3.2Pareto efficiencyThe 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)≥0is 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 EFFICIENCY3 subject to X s∈S p(a,s)U(w(s))−ψ(a)≥¯u,for some constant¯u.Proposition1Under the maintained assumptions,a contract(a,w(·))is Pareto-efficient if and only if it is a solution to the decision problem DP1for 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 payoffsubject to the constraint that the agent receive at least¯u.Conversely,suppose that the contract(a,w(·))is a solution to DP1for some value of¯u.If the contract is not Pareto-efficient,then there must be another contract that yields the same payoffto 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)>0for all s.Then optimal risk sharing requires:V0(R(s)−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)andψ(a)are differ-entiable at a,thenX s∈S p a(a,s)[V(R(s)−w(s))−λU(w(s)]+λψ0(a)=0.3.3Incentive efficiencyNow 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 constraintX s∈S p(a,s)U(w(s))−ψ(a)≥X s∈S p(b,s)U(w(s))−ψ(b),∀b.4CHAPTER3.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 offwithout 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 toX 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.Proposition2Under the maintained assumptions,a contract(a,w(·))is incentive-efficient only if it is a solution of DP2for some constant¯u.A contract that solves DP2is 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 DP2and is not incentive-efficient,there exists an incentive-efficient contract that gives the principal the same payoffand the agent a higher payoff.But this contract must be a solution to DP2that strictly satisfies the participation constraint.The assumption of a uniformly binding participation constraint is restric-tive:see Section3.7.1for a counter-example.This DP can be solved in two stages.First,for any action a,compute the payoffV∗(a)from choosing a and providing optimal incentives to the agent to choose a.Call this DP3V∗(a)=maxw(·)X s∈S p(a,s)V(R(s)−w(s))subject toX 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 CONSTRAINTS5 Note that U(·)and V(·)are concave functions.A suitable transformation of this problem(see Section3.10)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,DP3is (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 DP3without worrying about the optimal choice of action.3.3.1Risk neutralityAn 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 asminw(·)X s∈S p(a,s)w(s))subject toX 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.3.4The impact of incentive constraintsWhat 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 thefirst best can be achieved,that is,whether an incentive-efficient contract is also Pareto-efficient.Risk neutral principal6CHAPTER3.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,thefirst best can be achieved with hidden actions only if the optimal action is cost-minimizing.Risk neutral agentSuppose that the agent is risk neutral and the principal is(strictly)risk averse,i.e.,V00(c)<0.Then the Borch conditions for thefirst 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 thefirm 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 thatR(s)−w(s)=min{¯y,R(s)}andw(s)=max{R(s)−¯y,0}.Both parties risk averseMore generally,if we assume thefirst best is an interior solution and maintain the differentiability assumptions discussed above,thefirst-order condition for thefirst best isX s∈S p a(a,s)[V(R(s)−w(s))−λU(w(s)]+λψ0(a)=0.and thefirst-order(necessary)condition for the incentive-compatibility con-straint is X s∈S p a(a,s)[U(w(s)]−ψ0(a)=0.So the incentive-efficient andfirst-best contracts coincide only ifX s∈S p a(a,s)V(R(s)−w(s))=0.3.5.THE OPTIMAL INCENTIVE SCHEME7 Example:Suppose that there are two states s=1,2and 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 top0(a)[V(R(2)−w(2))−V(R(1)−w(1))]=0orR(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 DP3even under the usual Inada conditions.See Section3.7.2for a counter-example.3.5The optimal incentive schemeIn 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 afinite number of states s=1,...,S and the revenue function R(s)is increasing in s.•Monitone likelihood ratio property:There is afinite 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 p a(a,s)R(s)is increasing in a.Now consider the modified DP4of implementing afixed value of a:V∗∗(a)=maxw(·)X s∈S p(a,s)V(R(s)−w(s))subject toX 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.8CHAPTER3.THE PRINCIPAL-AGENT PROBLEM The difference between DP4and the original DP3is 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)=maxaV∗∗(a).Thus,we can use the solution to the modified problem DP4to characterize the optimal incentive scheme.Theorem3Suppose that a∈arg max V∗(a).The incentive scheme w(·)is a solution of DP4if and only if it is a solution of DP3.3.6MonotonicityMany 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)V0(R(s)−w(s))−λp(a,s)U0(w(s))−X b<aµb{p(a,s)−p(b,s)}U0(w(s))=0orV0(R(s)−w(s))U0(w(s))=Ãλ+X b<aµb½1−p(b,s)p(a,s)¾!.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.3.7ExamplesThere are two outcomes s=1,2,where R(1)<R(2),and two projects a=1,2represented by the respective probabilities of success0<p(1,2)< p(2,2)<1.The costs of effort areψ(1)=0andψ(2)>0.The agent’s utility3.7.EXAMPLES9 function U(·)is assumed to satisfy U(0)=0and the reservation utility is ¯u=0.The inferior project can be implemented by setting w(s)=0for 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:min w(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=2is(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:10CHAPTER3.THE PRINCIPAL-AGENT PROBLEM•the higher the cost of effortψ(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.3.7.1Optimality and incentive-efficiencySuppose there are two states s=1,2,two actions a=1,2and the reservation utility is¯u=0.The principal and the agent are both risk neutral.The other parameters of the problem are given byR(1)=0<R(2)ψ(1)=0<ψ(2)0<p(1,2)<p(2,2).The action a=1is optimally implemented by puttingw1(s)=0,∀s.The action a=2is optimally implemented by puttingw2(s)=½0if s=1ψ(2)/(p(2,2)−p(1,2))if s=2.3.7.EXAMPLES11The payoffto the principal from each action isV∗(a)=½p(1,2)R(2)if a=1p(2,2)(R(2)−ψ(2)/(p(2,2)−p(1,2)))if a=2. Suppose the parameter values are chosen so that V∗(1)=V∗(2).Then the contract(a,w(·))=(1,w1(·))solves DP1for the reservation utility¯u=0 but is not incentive efficient,because the agent is better offwith the contract (a,w(·))=(2,w2(·)).3.7.2Boundary solutionsIn the preceding example,we note that the agent’s payoffis zero in state s=0 whichever action is implemented.It might be thought that this boundary solution is dependent on risk neutrality but in fact boundary solutions for optimal incentive scheme are possible even if U0(0)=∞,for example,for the utility function U(c)=cαwhere0<α<1.In this case,U(0)=0so, taking the other parameters from the previous example,the optimal incentive scheme for a=1is stillw1(s)=0,∀s.For a=2the optimal incentive scheme isw2(s)=½0if s=1U−1(ψ(2)/(p(2,2)−p(1,2)))if s=2.This example provides a good illustration of the dangers of simply assuming an interior solution.3.7.3Local incentive constraintsIn many problems,convexity implies that one only has to consider local deviations in order to characterize an optimum.The analogous principle in principal-agent problems is to check only local incentive constraints.For example,if a=1,...,A and it is desired to implement an action a then one would only check the neighboring constraints a−1and a+1(or in the case where only downward constraints are considered,one would look at the constraint between a and a−1only).There is in general no reason to think that this method will produce the right answer:there may well be non-local constraints that are binding at the optimum.For example,suppose that12CHAPTER3.THE PRINCIPAL-AGENT PROBLEM there are two states s=1,2and three actions a=1,2,3.The principal and the agent are both assumed to be risk neutral and the reservation utility is ¯u=0.The other parameters are as follows:R(1)=0<R(2)ψ(1)=0<ψ(2)=ψ(3)0<p(1,2)<p(2,2)<p(3,2).The optimal incentive scheme to implement a=3isw3(s)=½0if s=1(ψ(3)/(p(2,2)−p(1,2)))if s=2.Because a=2has the same cost but lower probability of success than a=3, the agent will never be tempted to choose a=2as long as the payment in state s=2is positive;but he may well be tempted to choose a=1if the payment in state s=2is too low.Thus,the incentive constraint between a=1and a=3will be binding but the incentive constraint between a=3 and a=2will not.To ensure that the local constraint was sufficient,we would need to impose the following inequality on the parameters:p(3,2)−p(2,2)ψ(3)−ψ(2)≤p(2,2)−p(1,2)ψ(2)−ψ(1).This is,in effect,an assumption of diminishing returns to scale:the marginal product of effort as measured by the ratio of the change in the probability of success to the change in cost is declining.In more general problems,stronger conditions are needed to ensure that only local incentive constraints bind. See,for example,the discussion of thefirst-order approach in Stole(2001).3.7.4Participation constraints3.8The value of informationThe principal may observe some information that is relevant to the agent’s action in addition to the revenue from the project.We can incorporate this possibility in the current setup by assuming that the state is an ordered pair s=(s1,s2)∈S1×S2and that the revenue is a function R(s1)of thefirst3.9.MECHANISM DESIGN13 component.Then s2is a pure signal of the action a.Thefirst-order condition for an interior solution to DP4isV0(R(s1,s2)−w(s1,s2)) U0(w(s1,s2))=Ãλ+X b<aµb½1−p(b,s1,s2)p(a,s1,s2)¾!The state s2gives information about the action of the principal if the like-lihood ratio p(b,s1,s2)/p(a,s1,s2)varies with s2for somefixed s1.In other words,all relevant information should be reflected in the agent’s payment.3.9Mechanism designThe principal-agent problem is a special case of the general problem of mech-anism design,that is,designing a game form that will implement a desired outcome as an equilibrium of the game.Suppose there is afinite number of agents i=1,...,I,each of whom has a typeθi∈Θi and chooses an action a i∈A i.There may also be a set of actions a0∈A0chosen by the mechanism designer.LetΘ=Q I i=1Θi and A=Q I i=0A i and denote elements ofΘand A byθand a respectively.An agent’s utility is given by u i(a,θ),that is, u i:A×Θ→R.An agent’s type is private information,but the distribution of types p(θ) is common knowledge,as are the setsΘi and A i and the utility functions u i.The mechanism designer faces two problems:how to get the agents to reveal their information truthfully and how to get them to choose the“right”actions.The general form of a mechanism contains two stages:in thefirst agents are asked to send messages to the planner and in the second the planner sends instructions to the agents.Let M i denote the space of messages available to agent i and let M=Q I i=1M i.Let M0denote the planner’s message space and f:M→M0denote the decision rule chosen by the planner.Then each agent has to choose a strategy(σi,αi),whereσi:Θi→M i andαi:M0×Θi→A i.Given f we have a well-defined game with players is i=1,...,I,strategy sets{Σi}I i=1and payofffunctions{U i}I i=1, where U i:Σ→R is defined byU i(σ,α,θ)=u i(α(f◦σ(θ),θ),θ).A Bayes-Nash equilibrium for this game is a strategy profile(σ∗,α∗)such that,for every agent i,E[U i(σ,α,θ)|θi]≥E[U i((σi,αi),(σ−i,αi),θ)|θi],∀θi,∀(σi,αi).14CHAPTER3.THE PRINCIPAL-AGENT PROBLEMA mechanism(f,M)is called a direct mechanism if M i=Θi for i= 1,...,I and M0=A.In other words,agents’messages are their types and the planner’s message is the vector of desired actions.For any agent i,the truthful communication strategy in a direct mechanism is a communication strategyσi such thatσi(θi)=θi,∀θi.Similarly,in a direct mechanism,an action strategyαi is truthful ifαi(a)=a i,∀a∈A.The Revelation Principle allows us to substitute direct mechanisms for gen-eral mechanisms and restrict attention to truthful strategies.Theorem4(RevelationP rinciple)Let(σ,α)be a Bayes-Nash equilibrium of the mechanism(f,M).Then there exists a Bayes-Nash equilibrium(ˆσ,ˆα) of the direct mechanism(ˆf,Θ)such that(ˆσi,ˆαi)are truthful for every i and the outcomes of the two equilibria are the same:α◦f◦σ=ˆα◦ˆf◦ˆσ.Proof.Putˆf=α◦f◦σ.Although the proof is trivial,this result offers a great simplification of the problem of characterizing implementable SCFs.A SCF is a function f:Θ→A that specifies an outcome for every state of natureθ.We can think of the SCF f as a collection of decision rules(f0,f1,...,f I),one for each agent i.The SCF f is incentive-compatible if,for every i,truth-telling is optimal and the decision rule f i is optimal,assuming that every other agent j tells the truth and follows the decision rule f j,that is,E[u i(f(θ),θ)|θi]≥E h u i³a i,f−i(ˆθi,θ−i),θ´|θi i.Theorem5The direct mechanism(f,Θ)has a truthful equilibrium if and only if f is an incentive-compatible SCF.Remark1The theorem suggests that we can“implement”f using a direct mechanism,but the direct mechanism may have other equilibria.Full im-plementation requires that every equilibrium of the mechanism used have the same outcome.For this it may be necessary to use a general mechanism. Most of the implementation literature is taken up with this problem of try-ing to eliminate unwanted equilibria,either by using fancier mechanisms or stronger solution concepts.3.10.NON-CONVEXITY AND LOTTERIES15 Remark2The principal-agent problem is a special kind of mechanism de-sign problem.In the problem described earlier,there are two“agents”(the principal and the agent both being economic agents in the eyes of the mech-anism designer).The agent chooses an action a∈A,the principal has no action to choose,and the mechanism designer chooses the incentive scheme w(·)∈W.Since there is no private information about types,the SCF is an incentive-efficient allocation f=(a,w(·))and the direct mechanism has a truthful equilibrium in which the agent chooses the correct value of a.Even in this simple context we can see the problem of multiple equilibria at work. Typically,the incentive scheme is chosen so that the agent is indifferent be-tween a and some other action b.It would be an equilibrium for the agent to choose b even though this would not be as good for the principal.We can use this example to illustrate how a more complex mechanism and a stronger equilibrium concept helps resolve this difficulty.Suppose that the principal is told to choose the incentive scheme and that the agent chooses his action after observing the incentive scheme.The appropriate solution concept here is subgame perfect equilibrium:the agent should choose the best response (action)to any incentive scheme and not simply the one that is chosen in equlibrium.Clearly,the truthful equilibrium of(a,w(·))remains a SPE of this game but(b,w(·))does not.If the principal anticipates that the agent will choose b under the incentive scheme w(·)and if the principal prefers a to b,then he will choose an alternativeˆw(·)which is very close to w(·)but makes the agent strictly prefer a to b.Thus,(b,w(·))is not a SPE. Remark3The sequential game described above,in which the principal of-fers an incentive scheme and the agent responds optimally to any scheme offered,is closer to the original formulation of the principal-agent problem than the decision problems analyzed above.We have taken an approach much closer to the Revelation Principal,in which we focus exclusively on the truth-ful equilibria.Within the context of mechanism design,we can see that both approaches are closely related.3.10Non-convexity and lotteriesThe principal-agent problem as stated earlier is not a convex programming problem because the feasible set defined by the incentive constraints is not16CHAPTER3.THE PRINCIPAL-AGENT PROBLEM convex:X s∈S p(a,s)U(w(s))−ψ(a)≥X s∈S p(b,s)U(w(s))−ψ(b),∀bThe concave function U(·)appears on both sides of the inequality.However, a simple transformation suggested by Grossman and Hart converts this into a convex programming problem.Let C(u)=U−1(u)for any number u.C(·)is convex because U(·)is concave and we can write the implementation problem equivalently asminu(·)X s∈S p(a,s)V(R(s)−C(u(s)))subject toX s∈S p(a,s)u(s)−ψ(a)≥X s∈S p(b,s)u(s)−ψ(b),∀b,X s∈S p(a,s)u(s))−ψ(a)≥¯u.Because the incentive scheme u(s)is written in terms of utility rather than consumption,the incentive constraints are linear in the choice variables and hence the feasible set is convex.This trick works because of the additive separability of the utility func-tion.In general,this will not work and we are stuck with a highly non-convex problem.One general solution to non-convexities is to introduce lotteries. Let the incentive scheme specify a probability distribution W(c,s)over non-negative consumption levels c conditional on the state s and let the utility function take the general form u(c,a).The incentive constraint is then writ-ten as X s∈S[p(a,s)−p(b,s)]u(c,a)W(dc,s)≥0,∀b.Expected utility is linear in probabilites,so once again the incentive con-straints define a convex feasible set of distributions W(·).Lotteries are not simply a solution to a technical problem(non-convexity).They can also increase welfare.Note that even if the implementation problem does not include non-convexities because of the additive separability of preferences,the global principal agent problem may do so because the cost functionψis non-convex. Although each action a can be implemented efficiently with a non-stochastic incentive scheme,there may be a gain from randomizing over the action a.。