Moral hazard with discrete soft information

Transcription

1 Moral hazard with discrete soft information September 2012 Abstract I study a simple model of moral hazard with soft information. The risk-averse agent takes an action and she alone observes the stochastic outcome; hence the principal faces a problem of ex post adverse selection. With limited instruments, the principal cannot solve these two problems independently. To accommodate ex post information revelation, he must distort the transfer schedule, as compared to the standard moral hazard problem. This is socially costly. These results are robust and suggest highpower contracts may have to be revisited when information is soft. Keywords: moral hazard, asymmetric information, soft information, contract, mechanism, audit. JEL Classification: D82. 1 Introduction In standard moral hazard problems the outcome of the agent s action is observable by the principal and may therefore (imperfectly) substitute itself for the non-observability of said action. Then a complete contract may be conditioned on the outcome. This is a convenient model that is used in many applications, ranging from labor contracts to corporate finance. But it does not necessarily fit many relevant situations. Actual performance may not observed at all: for example, an accounting report is not a direct observation of the state of an enterprise, but a message. 1

2 This paper studies exactly this problem: neither the action, nor the outcome are observable by the principal. He then faces a problem of ex ante moral hazard and ex post adverse selection. The information is said to be soft in that it is subject to manipulation on the part of the agent. Bar for the role of soft information, the model mirrors that of a standard moral hazard framework. A risk-neutral principal delegates production to a risk-averse agent. The agent s action a governs the distribution of a stochastic outcome, which she alone observes. That information must therefore be elicited; that is, it must satisfy some ex post incentive constraints. 1 Because the principal otherwise observes nothing, the contract must include an audit and some penalty. Applications of this model are broad-ranging. For example, after hiring the CEO, a board often asks of him (her) to report his (her) results while on the job. A defense contractor may be asked to reveal its production cost after investing in an uncertain technology. Kedia and Philippon (2006) develop and test a model of earnings management (a euphemism for fraudulent accounting), and document how pervasive the practice is. This work is closely related to Mookherjee and Png (1989, now MP), who show that with enough instruments, the twin problem of moral hazard and ex post adverse selection can be treated separately. That separability allows for ex post truthful revelation without any consequence on the incentives used to solve the ex ante moral hazard problem. Then the moral hazard problem can be solved in standard fashion, yielding standard results. That model is technically flawless but three objections may be raised. First, the real world does not accord with the results of MP; agents do mislead their principal. For example, Howie Hubler, a headstrong trader at Morgan Stanley single-handedly lost the firm $9 Bn after covering up his trades, was terminated and yet paid out past boni. Thus a model that systematically predicts truthful revelation has limited applicability. Second, the schemes suggested by MP require a reward for ex post truthful revelation; these are not observed in practice. A third objection is that the a reward that is necessary to induce information revelation may be arbitrarily large; it turns the principal into a source of money regardless 1 Throughout I will refer to incentive constraints as those addressing the adverse selection problem and moral hazard constraint as that dealing with the hidden action problem. 2

3 of the value of the productive relationship with the agent. This paper departs from MP first by letting the principal use a single transfer to address both moral hazard and ex post adverse selection; there is no bonus for not lying, so the principal must do with fewer instruments. This induces non-separability of the problems, which is the source of distortions. Second, the audit is imperfect. The technology is closer to one of sampling, which is what real audits do (such as financial audits), and has been modeled by Bushman and Kanodia (1996) or Demski and Dye (1999). This simple model delivers some important insights. First, the option to misreport induces an implicit lower bound on transfers because of the bounded penalty. 2 Transfers worse than this penalty cannot be implemented (for the agent is better-off simply lying). Second, any information revelation requires the agent s compensation to be distorted as compared to the standard moral hazard model. This is necessary to simultaneously satisfy any of the ex post incentive constraints and the moral hazard constraint. The optimal compensation schedule is flatter than in the standard model. A steep transfer scheme is usually good to induce effort, but here it also generates incentives to misreport ex post. The distortions accommodate the conflict between ex post incentives for information revelation and ex ante effort incentives. These distortions leave the agent s participation constraint slack and thus generate an ex ante rent. Then effort is more costly to the principal, and is therefore implemented for a smaller set of parameters than in the standard problem. The third insight is that steep contracts (those paying well for good performance) require an accurate audit technology. That is, it is because the principal can detect misreporting ex post that he can offer a high-power contract. This runs contrary to the standard results of the costly state verification literature (such as MP), which finds that audit and incentives are substitutes. It also departs from the standard moral hazard literature, which shows that monitoring and incentives are substitutes. Here they act like strategic complements. The work closest to this is MP, who combine a Grossman-Hart (1983) model with an 2 Bounded (small) penalties fit the Howie Hubler example. In practice courts only enforce penalties that are deemed reasonable (see Doornik, 2006). 3

4 ex post revelation mechanism. The agent s message conditions a payment to the principal and the probability of audit; that audit is perfect. Rewards (potentially unbounded) for truthful revelation are offered in equilibrium. 3 Gromb and Martimort (2007) use the same sequence of events as here, however they study the incentives of expert(s) to search and report information about others (an exogenous project), not themselves. To overcome the adverse selection problem, their incentive contract must be made state dependent although the expert(s) do(es) not exert any influence on it. A project s success is publicly observable, hence fully contractible. Levitt and Snyder (1997) develop a contracting model in which the agent receives an early (soft) signal about the likely success of the project, however the eventual outcome is fully observed by the principal, hence contractible. Here, information can only be observed, and reported, by the agent. To emphasize the point, the agent has no ex ante private information, which only emerges ex post. Green and Laffont (1986) study the principal-agent problem with partially verifiable information in the sense that the agent s message is constrained to lie in a subset M(θ) of the type space Θ, which varies with the true state in a publicly known fashion. M( )-implementable mechanisms exist and need not elicit truth-telling. I discuss this some more in Section 5. The balance of the paper is organised as follows. After introducing the model I present the problem of information revelation. Section 4 then analyses the optimal contract and Section 5 presents a discussion. I conclude in Section 6. 2 Model A principal delegates a task to an agent. At cost ψ(a) 0 the agent undertakes an action a {a, a} where ψ(a) = ψ > ψ(a) = 0. This action yields a stochastic outcome θ Θ {θ 1, θ 2, θ 3 }; there must be at least three states for information manipulation to not be trivial. I make the substantive (and later discussed) assumption that θ 3 θ 2 = θ 2 θ 1 = θ > 0. Let π i Pr(θ i a = a) and π i Pr(θ i a = a) denote the probabilities of each outcome 3 Mookherjee and Png s model yields a quirky byproduct: the agent strictly prefers being audited. 4

5 conditional on the agent s action; I suppose that π i ( a) satisfies the MLRP (i.e. π i /π i increases in θ). The agent s net utility is given by u(t, a) v(t i ) ψ(a), where v( ) is an increasing, concave function with v(0) = 0. The agent alone observes the outcome θ and reports a message m Θ to the principal, whereupon she receives the transfer t i (m). The principal can commit to the contract and his net payoff reads S(t; θ) = θ i t i. If the true state θ were observable by the principal, this construct would be a moot point and would collapse to the textbook moral hazard problem. By the MLRP, inducing effort requires t 3 t 2 t 1 with at least one strict inequality. Therefore it is immediate that absent any other instruments, the agent pools her messages to θ 3 regardless of the state (the principal entirely lacks ex post observability). A necessary element of any non-trivial contract is to restore at least some ex post observability, which I do by introducing an ex post audit. The audit technology is exogenously given in this model, costless (and therefore always run), but imperfect. With some probability p(m θ), the agent s deception is uncovered and she receives zero. Accepting that the penalty must be bounded, choosing zero is immaterial; this is discussed further in Section 5. 4 The function p( ) maps into [0, 1]; it is taken to be symmetric and such that p(0) = 0. 5 It is easy to show that p( ) must be increasing to be useful, which I therefore impose. I also let p(2x) 2p(x) (weak convexity). With this, the agent has ex post expected utility U (1 p(m θ))v(t(m)), which she seeks to maximise by choice of the message m Θ. The timing is almost standard: 1. The principal offers a contract C = t(m), Θ, p( ) consisting of a transfer, a message space and an (exogenous) audit technology; 2. The agent accepts or rejects the contract. If accepting, she also chooses an action a; 3. Action a generates an outcome θ Θ observed only by the agent; 4 With unbounded penalties truthful revelation necessarily obtains, as with unbounded rewards; that is, separability is restored. 5 This capture the idea that the audit is a sampling process, as real financial audits are. See the discussion. 5

6 4. The agent report a message m Θ; 5. Audit occurs; 6. Transfers are implemented and payoffs are realised. The solution of this problem in a subgame perfect equilibrium of the game just described. 3 Information revelation There always exists a trivial contract, in which the low action is sought from the agent. The principal needs only set t i = 0 i to implement a = a; this also elicits truth-telling ex post. I am interested in equilibria where effort is implemented. Two cases of interest arise; in the first one, truthful revelation is elicited ex post. In the second case, the principal may not seek to satisfy the ex post incentive constraint because they may be too costly. After she has taken some action a (now sunk), the agent maximises U by choice of a message m. The analysis begins with this problem. 3.1 Truthful revelation A mechanism is truthful if and only is the following constraints are satisfied:- v(t 1 ) (1 p( θ))v(t 2 ) (3.1) v(t 1 ) (1 p(2 θ))v(t 3 ) (3.2) v(t 2 ) (1 p( θ))v(t 3 ) (3.3) v(t 2 ) (1 p( θ))v(t 1 ) (3.4) v(t 3 ) (1 p(2 θ))v(t 1 ) (3.5) v(t 3 ) (1 p( θ))v(t 2 ) (3.6) These constraints do not yield the standard implementability condition, as can be verified by adding them up pairwise. For example, add (3.1) and (3.4) to find 1 (1 p( θ)), which is 6

7 trivially true and uninformative as to the shape of the transfer function. The system (3.1)- (3.6) forms the basis of the first claim. Because t 3 t 2 t 1 and p( ) 0, only (3.1)-(3.3) are relevant. Lemma 1 There exist transfers t 3 t 2 t 1 such that constraints (3.1)-(3.6) hold. Whenever the local constraints (3.1) and (3.3) are satisfied, the global constraint (3.2) is necessarily slack. Whenever the global constraint (3.2) binds at least one of the local constraints fails. This existence result remains silent as to optimality and does not imply that transfers satisfying (3.1)-(3.6) solve the principal s problem. Truthful revelation needs not be optimal, in particular because it necessarily generates an ex ante rent for the agent. Indeed, t 3 > 0 is required to induce participation with any effort but by (3.2), t 1 > 0 as well (and so t 2 > 0 too by (3.1)). However from Holmström (1979) we know that some t i must be negative for the participation constraint to bind at zero, hence the rent. 3.2 No truthful revelation Because combining ex ante effort incentives and ex post truthful revelation is costly, the principal may choose an alternative: truthful revelation may purposefully not be sought, which may make him better off. In this equilibrium, transfers are such that at least one of the ex post incentive constraints is violated, as in the arbitrary example of Figure 1. 6 Even in such a case at least some of the incentive constraints (3.1) to (3.3) must hold. If all constraints were to fail, the agent would only report θ 3, whereupon the principal would pay some transfer t 3 with probability Π = π 3 + (1 p)π 2 + (1 p) 2 π 1 (if effort is exerted) and zero with the balance of probabilities. The insurance-incentive trade-off immediately tells us this cannot be optimal; some type separation is always desirable to the principal. 7 6 The principal has committed to the contract and has no further move in the game, so information updating is a moot point. In particular, there is no renegotiation in this model. 7 This contract resembles the one-step bonus but can be improved upon. To do so, simply increase t 2 so that (3.3) binds. The total expected payment becomes ( Π π 2 1 p( θ)) ) t 3 +π 2 t 2 < Π t 3 ; but this contract delivers a higher expected utility to the agent at the same cost to the principal. That is, it gives room to 7

8 v(t i ) v(t 3 ) v(t 2 ) v(t 1 ) θ 1 θ 2 θ 3 Θ Figure 1: Arbitrary example of (3.3) failing, θ 2 pools with θ 3 (red dot). One must also note that since some incentive constraint will fail (by design), the global constraint (3.2) can no longer be ignored. However it is still true that there cannot be an equilibrium in which only (3.2) is violated, because (3.1) and (3.3) imply (3.2). Conversely, if (3.1) and (3.3) fail, it does not imply that (3.2) does. At face value there are many combinations of failing constraints to consider; fortunately the next lemma considerably reduces the set of cases to investigate. To conduct this analysis it is necessary to introduce the moral hazard problem, for it interacts with the ex post adverse selection one. To be accepted by the agent and induce effort, a contract must satisfy:- π i v i ψ (3.7) i π i v i ψ (3.8) i These two inequalities are the usual moral hazard and participation constraints. With this, Lemma 2 The principal does not offer a contract in which the agent exerts effort and any of the incentive constraints the principal to decrease t 2 and increase t 3. 8

9 1. (3.1) and (3.2); or 2. (3.1) and (3.3); or 3. (3.2) and (3.3); or 4. only (3.3); are violated. Thus the only viable case when the principal s contract is such that it does not induce the agent to truthfully reveal her information ex post requires (3.1) to be violated. Further, Lemma 3 There is no equilibrium such that the moral hazard constraint (3.7) is satisfied, the participation constraint (3.8) binds, and only the incentive constraint (3.1) is violated Lemmata 1 and 3 together imply that in any equilibrium the agent receives an ex ante rent. 4 The optimal contract Let ϕ v 1 ( ) denote the inverse function of the agent s utility, and v i = v(t i ) for some t i. The goal of the next two subsections is to compute the cost of either contract. The principal seeks to solve Problem 1 s.t. (3.1)-(3.3) and (3.7), (3.8). max v i 0 π i [θ i ϕ(v i )] i 4.1 Cost of the contract Truthful revelation Following Lemma 1, only (3.1) and (3.3) may bind in a truth-telling equilibrium. It also tells us that µ = 0 for constraints (3.1)-(3.3) to be satisfied. The next two lemmata inform us more precisely as to how these constraints conflict with the moral hazard problem. 9

10 Lemma 4 Suppose µ, λ > 0 (as in the standard moral hazard problem), then at least two of (3.1)-(3.3) must be violated. Hence there cannot be an equilibrium in which the standard solution of the moral hazard problem also accommodates the ex post information revelation problem. Further, for the constraints (3.1)-(3.3) to hold, either (3.7) fails or (3.8) must be slack. More precisely, Lemma 5 Suppose µ > 0 and (3.1), (3.3) are satisfied, then the moral hazard constraint (3.7) is violated. That is, either there is no truthful revelation ex post (by Lemma 4), or no effort can be induced without affording the agent a rent (by Lemma 5). Therefore attention can be restricted to the set of utilities v i such that the participation constraint (3.8) is slack. Taking (3.1) and (3.3) binding, the transfers must satisfy v 1 = (1 p( θ))v 2 and v 2 = (1 p( θ))v 3. Define further Π = π 3 + (1 p)π 2 + (1 p) 2 π 1. With this in hand, Proposition 1 The lowest-cost truth-telling equilibrium in which the agent is induced to exert effort entails v T 3 = ψ Π Π determined by a binding moral hazard constraint (3.7), and v T 1, v T 2 > 0 determined by (3.1) and (3.3), both binding No truthful revelation Following Lemmata 2 and 3, Problem 1 becomes Problem 2 s.t. (3.2),(3.3) and max v i 0 π i θ i {π 3 ϕ(v 3 ) + [π 2 + π 1 (1 p( θ))] ϕ(v 2 )} i π 3 v 3 + [ π 2 + π 1 (1 p( θ))] v 2 ψ (4.1) π 3 v 3 + [π 2 + π 1 (1 p( θ))] v 2 > 0 (4.2) 10

11 where (4.2) is the relevant form of the participation constraint (3.8). The problem is modified to reflect the fact that θ 1 is never reported, but that θ 2 is in its place. Then we have Proposition 2 The least-cost non-truthful equilibrium in which the agent is induced to exert effort entails v NT 3 = ψ Π Π ; (1 p( θ))v 2 > v NT 1 > (1 p(2 θ))v 3 determined by a binding moral hazard constraint (4.1), and v NT 2 determined by the binding incentive constraint (3.3). Note that v 1 exists. Propositions 1 and 2 lead to an immediate, but important, Corollary. Corollary 1 The principal is indifferent between truthful revelation and not. The truth-telling contract is just as costly as the one that is not truth-telling. In this simple framework it also induces the same effort and therefore the same success probabilities π i. To see why, notice that because at least some incentive constraint(s) bind(s), the contract is linear in the states θ i. When the agent lies and Constraint (3.1) fails, she receives either 0 or a weighted average of v 2 and v 3, where v 2 is determined by the binding constraint. In this case the weight on v 2 is π 2 + (1 p( θ))π 1. Under truthful revelation, the agent never receives 0 but a weighted sum of v i. Here the weights are π 1 on v 1 and π 2 on v 2. But because v 1 = v 2 (1 p( θ)) by (3.1), the effective weight on v 2 is π 2 + (1 p( θ))π 1 as well. This is the cost of incentive compatibility: to obtain truthful revelation, the principal must offer the agent exactly the same utility as if she were lying. Combined with the assumption θ 3 θ 2 = θ 2 θ 1 it generates an expected payoff that is linear in the states θ i. This is quite a special case; Section 5 discusses this in some more details. 4.2 Properties of the contract Corollary 1 implies that attention can be restricted to a truth-telling contract, the properties of which I would like to explore. The optimality conditions of Problem 1 are useful to expose 11

12 the next claim. To do so, attach multipliers γ 1, γ 2 to constraints (3.1) and (3.3), and λ and µ to each of (3.7) and (3.8), respectively. ϕ (v 1 ) γ 1 π 1 = µ + λ π 1 π 1 (4.3) ϕ (v 2 ) γ 2 γ 1 (1 p) π 2 = µ + λ π 2 π 2 (4.4) ϕ (v 3 ) + γ 2(1 p) π 3 = µ + λ π 3 π 3 (4.5) The MLRP ensures these conditions are not vacuous; the next claim follows immediately. Proposition 3 The schedule is flatter (than under the standard moral hazard problem). v T 1 solving (4.3) is distorted upwards, while v T 3 solving (4.5) is distorted downwards. v 2 is ambiguous. This result owes to the fundamental tension between ex post incentive compatibility, best satisfied with constant transfers, and ex ante effort incentives, best addressed with a compensation conditioned on performance. The distortions tilt the schedule and are such that (3.1) and (3.3) are just binding. The contract is low(er)-powered to accommodate ex post information manipulation. This is shown in Figure 2. The next Corollary speaks to the social cost of ex post information manipulation. Corollary 2 The principal induces costly effort if and only if i π iθ i ψ Π Π Π > ψ i.e. the agent receives an ex ante rent U T = ψ Π Π Π > 0. the proof of which is obvious and therefore omitted. Consequently the high action is implemented for a narrower set of parameters than under the standard moral hazard problem, which is socially costly. The principal overinsures the agent as compared to the standard problem (e.g. Holmström, 1979). Altering any of the technology ψ or information structure π i (. a) produces the same effects as in the standard moral hazard model. Of more interest are comparative statics with respect to the audit technology. Corollary 3 1. U T p < 0 12

13 v(t i ) v(t s 3) v(t 3 ) v(t s 2) v(t 2 ) v(t 1 ) v(t s 1) θ 1 θ 2 θ 3 Θ Figure 2: Optimal transfers are distorted (black dots). 2. vt 3 p > 0 Improving the audit technology tilts back the compensation schedule toward a steeper slope: the agent s utility in the good state increases. But it also eases the incentive constraints (3.1)- (3.3). The net effect is a decrease the agent s expected rent. Proposition 3 and Corollary 3 together suggest that high-powered contracts are not appropriate when information is manipulable. Furthermore, the steepness of the contract (the power of the incentives) depends on the audit technology. More precisely, a high-powered contract requires an accurate audit technology, which may sometimes come at a cost (when auditing is costly, for example). This is because a steep contract generates incentives for ex post manipulation of information. 5 Discussion Other penalties This paper purposefully departs from optimal penalties (or rewards) because they allow for the separation of the ex ante and ex post problems (as in Mookherjee and Png (1989)). Suppose however that some other bounded penalty l < 0 could be imposed on the agent, 13

14 then the constraints (3.1)-(3.3) would read: for i = 1, 2; v i [1 p( θ)]v i+1 +p( θ)v( l) and v 1 [1 p(2 θ)]v 3 + p(2 θ)v( l). These are easier to satisfy than the current constraints, for v( l) < 0, but as long as l is not too large, the problem remains in essence the same. It can also be verified that the Maximum Punishment Principle (Baron and Besanko (1984)) holds in this model because the audit does not return false negatives. Therefore nothing is gained (there are strict losses) from conditioning the magnitude of the penalty on the offence. Cost of effort and separation That truthful revelation obtains in equilibrium owes to the assumption that θ 3 θ 2 = θ 2 θ 3 = θ. When at least one incentive constraint binds this construction necessarily implies that the contract is linear in the state. This is trivially true when Constraint (3.1) fails because then only two states may ever be reported (θ 2 or θ 3 ). Satisfying (3.1) requires a further linear equality to hold, at the same cost as the deviation. In a separate paper (Roger, 2012b) I show that truthful revelation is never possible for at least a subset of the type space (at the bottom of the space). There the ex post incentive constraints (such as (3.1)) can only bind in one state; they are otherwise slack or fail. This is because continuous spaces allow for arbitrarily small misreporting and do not impose the discrete jump from θ i to θ i+1. Therefore it may be optimal for the principal to tolerate a small deviation for some types because the cost of seeking truthful revelation exceeds that of lying. Then, unlike, in Section 4, providing the agent with effort incentives may come at different costs when she is truthful and not. In fact truthful revelation is not essential for the provision of effort incentives; rather type separation is, because pooling stifles effort (see Roger, 2012a). While misreporting increases the cost of these ex ante incentives, pooling is worse. Participation fees To avoid leaving any rent to the agent the principal could consider imposing an ex ante participation fee φ, so that µ > 0. Then a contract includes a tariff (φ, t). Because φ is sunk, 14

15 the incentive constraints (3.1)-(3.6) remain essentially the same (up to the levels of v i ). So although the rent may be extracted from the agent, the transfers t i still have to be distorted to satisfy information revelation. The same welfare losses ensue. Relation to M-implementability (Green and Laffont (1986)) These authors study the implementability of a social choice function when the agent may report a message from a set M(θ) Θ, where M( ) is exogenous and publicly known. There is scope for misreporting in that the mapping m( ) is a correspondence. Green and Laffont (1986) provide a necessary and sufficient condition called the nested range condition (NRC) for the agent to report her information truthfully. The NRC does not hold in this model, although it corresponds to a game of of unidirectional distortions with an ordered space (to use their words) example a(2) in Green and Laffont. Indeed, the NRC requires M(θ 3 ) = {θ 3 }, M(θ 2 ) = {θ 2, θ 3 }, M(θ 1 ) = {θ 1, θ 2, θ 3 }. In contrast, Constraints (3.2), (3.3) holding and (3.1) failing imply M(θ 3 ) = {θ 3 }, M(θ 2 ) = {θ 2 }, M(θ 1 ) = {θ 2 }, whence θ 3 / M(θ 1 ). This violates the definition of the NRC. Notice further that the sets M(θ i ) in this paper are endogenous, unlike in Green and Laffont (1986). 6 Conclusion When the principal to a contract fraught with moral hazard also fails to observe any of the outcomes, he faces adverse selection ex post. With limited instruments, eliciting this private information requires a distortion of the compensation structure; it induces a flatter transfer scheme. This is a low(er)-power contract than in the standard moral hazard problem. This distortion has a bearing on the ex ante effort incentives. It is socially costly in that the high action can be implemented in fewer instances than in the standard case. These results obtain because of the fundamental tension between effort provision and information revelation, which require different instruments. For practitioners these results suggest that high-power contracts are not adequate when the information they depend on can be manipulated. In 15

16 this model, inducing any information revelation ex post also generates an ex ante rent to the agent. Truthful revelation obtains in the equilibrium of this game, as it does in the Mookherjee and Png (1989), however for very different reason and with different consequences. It can only be obtained with a distortion of the transfers, and therefore of the effort decision. In a more general model, truthful revelation may not obtain. This more general model should allow for at least two important modifications: (i) richer and more flexible messages for the agent (i.e. relaxing θ = θ 2 θ 1 = θ 3 θ 2 ) and (ii) afford the principal to choose his audit technology. This problem is explored in a series of companion papers (Roger, 2012a, 2012b), which show that as long as instruments are limited, the insights of this paper remain. The option to misreport ex post induces ex ante distortions that are socially costly, and when information is manipulable ex post, a high-power contract must be accompanied with an accurate audit. 7 Appendix: Proofs Proof of Lemma 1: Suppose (3.1) and (3.3) are satisfied (either strictly or with some slack), then v(t 1 ) (1 p( θ)) 2 v(t 3 ). Since (1 p( θ)) 2 > 1 p(2 θ), Condition (3.2) is necessarily slack. To show existence, take (3.1) and (3.3) binding. Then we have v(t 3 ) > (1 p)v(t 3 ) = v(t 2 ) > (1 p) 2 v(t 3 ) = v 1. For the last statement, take (3.2) binding. Then v(t 1 ) = (1 p(2 θ))v(t 3 ) (1 p( θ))v(t 2 ) by (3.1) (or (1 p( θ))v(t 1 ) by (3.3)). Either way it follows that 1 p(2 θ))v(t 3 ) (1 p( θ)) 2 v(t 3 ) for both (3.1) and (3.3) to hold. This contradicts (1 p( θ)) 2 > 1 p(2 θ). Proof of Lemma 2: 1. In the first case the agent reports θ 3 when observing θ 1. So Problem 1 becomes Problem 3 max v i 0 π i θ i {π 2 ϕ(v 2 ) + [π 3 + π 1 (1 p(2 θ))] ϕ(v 3 )} i 16

17 s.t. (3.3) and π 2 (v 2 v 3 ) π 1 (1 p(2 θ))v 3 ψ (7.1) π 2 v 2 + [π 3 + π 1 (1 p(2 θ))] v 3 ψ (7.2) where π 1 < 0 necessarily by the MLRP and π 2 may be either positive or negative. 2. Similarly, in the second case v 1 again never materialises because the agent reports θ 2 if observing θ 1 and θ 3 if observing θ 2 (and θ 3 if observing θ 3 ). The moral hazard constraint (7.1) becomes π 1 (v 2 v 3 ) p( θ)( π 2 v 3 + π 1 v 2 ) ψ. (7.3) 3. In this instance (Case 3), v 2 is never reported: the agent always prefers θ 3 instead of reporting θ 2. Then the moral hazard constraint reads π 1 (v 1 v 3 ) π 2 p( )v 3 ψ. (7.4) 4. In case 4 both (3.2) and (3.3) fail; the moral hazard constraint rewrites v 3 [ π 1 p(2 θ) + π 2 p( θ)] ψ (7.5) 5. In Case 5 only constraint (3.1) fails; then the moral hazard constraint becomes π 3 v 3 + v 2 ( π 2 + π 1 (1 p( θ))) ψ. (7.6) Which of these instance arises in equilibrium is endogenous; it depends on the contract chosen by the principal, that is, on the cost of implementing effort. Those costs are characterised by the constraints (7.1)-(7.4), which can be used to rank them. Pairwise comparisons of these constraints show that Case 1 features the lowest cost, so that it represents the only contract that may be offered by the principal. Indeed, simple inspection of the inequality π 1 (v 1 v 3 ) π 2 p( )v 3 π 1 (v 2 v 3 ) p( θ)( π 2 v 3 + π 1 v 2 ) shows that it holds. Similarly, inspection of the first and last equations shows that Case 1 17

18 is more expensive than Case 4. Next re-arrange π 2 (v 2 v 3 ) π 1 (1 p(2 θ))v 3 (< ) π 1 (v 2 v 3 ) p( θ)( π 2 v 3 + π 1 v 2 ) into π 2 (v 2 (1 p( θ))v 3 ) (<) π 1 (v 3 v 2 (1 p( θ))) + π 1 v 3 p(2 θ). Notice that π 1 (v 1 v 2 (1 p( θ))) < 0 because incentive constraint (3.1) fails (by assumption), the RHS can be bounded below by π 1 (v 1 (1 p(2 θ))v 3 ), which is positive because constraint (3.2) also fails (by assumption). Hence Case 2 is more expensive than Case 1, and therefore will never be offered either. Last, comparing (7.5) to 7.1 and re-arranging yields π 1 v 3 (1 p(2 θ)) < π 1 v 2 (1 p( θ)) by simple inspection of the incentive constraints (3.1) and (3.2), for π 1 < 0. Transitivity completes the proof. Proof of Lemma 3: Suppose (3.1) fails but (3.2) and (3.3) hold. θ 1 is never reported to the principal and the relevant moral hazard constraint is given by (7.6). Take the corresponding participation constraint π 3 v 3 + v 2 (π 2 + π 1 (1 p( θ))) binding at 0 then (7.6) is necessarily violated. Proof of Lemma 4: µ, λ > 0 i π iv i = ψ = i π iv i i π iv i = 0. Since v 3 v 2 v 1 by MLRP and i π iv i = ψ > 0, we must have v 3 > 0 > v 1. Therefore (3.2) cannot hold. Further, since v 2 v 1, (3.1) must also be violated regardless of whether v 2 0 or v 2 < 0. Clearly in the later case (3.3) also fails to hold. Proof of Lemma 5: µ > 0 i π iv i = ψ and when (3.7) holds, i π iv i ψ+ i π iv i. But but by (3.1)-(3.3) and MLRP, i π iv i > 0. Then (3.7) implies π i v i ψ + π i v i i i ψ ψ + i i π i v i π i v i > 0 which is an obvious contradiction. So (3.7) must be violated. 18

19 Proof of Proposition 1: Take the moral hazard constraint (3.7) and the ex post incentive constraints (3.1) and (3.3) binding to obtain v 3 (as in the text). To show this must be the lowest-cost contract, observe that v 3 must also solve Since ϕ ( ) is increasing, if γ 2 = 0 then v 3 > ϕ (v 3 ) = λ π 3 π 3 γ 2(1 p) π 3 (7.7) ψ and v Π Π 2 > (1 p)v 3 > (1 p) ψ Π Π is slack). It then follows that v 1 (1 p)v 2 > (1 p) 2 v 3 > (1 p) 2 can be constructed for the case γ 1 = 0 by using (4.4) instead. ψ Π Π (for (3.3). A similar argument Proof of Proposition 2: From Constraint (3.3) binding v NT 2 = (1 p)v NT 3. When (3.1) fails but (3.2) holds, the agent reports θ 2 in state θ 1 and receives v 2 with probability 1 p. Combining these two statements gives v NT 3 and v 2 as in the text. Proof of Corollary 1: In either case the cost of the contract is given by a binding moral hazard constraint (3.7) (or (4.1). Then Πv 3 = ψ, so that ( ) ψ C T = Π ϕ Π ( ) ψ C NT = Π ϕ Π Proof of Proposition 3: In the standard problem (4.3)-(4.5) read ϕ (v 1 ) = µ + λ π 1 π 1 (7.8) ϕ (v 2 ) = µ + λ π 2 π 2 (7.9) ϕ (v 3 ) = µ + λ π 3 π 3 (7.10) Increase v i i by some arbitrarily small amount ε > 0, so that µ = 0 as well but the cost of the contract comes within ε of the optimum (for ϕ is continuous and its derivative is bounded). Sum (4.3)-(4.5) to find E Θ [ϕ (v i )] + (γ 1 + γ 2 )p = 0 19

20 for µ = 0, so that at least one of γ 1, γ 2 is strictly positive. From the Proof of Proposition 1 we know that both are. Then compare each of (7.8)-(7.10) to (4.3)-(4.5), recalling that ϕ( ) is increasing convex. References [1] Baron D. and David Besanko Regulation, Asymmetric Information, and Auditing. The RAND Journal of Economics, Vol. 15, No. 4, pp [2] Bushman, R. and Chandra Kanodia (1996) A Note on Strategic Sampling in Agencies. Management Science, Vol.42, pp [3] Demski, J. and Ronald Dye (1999) Risk, Returns and Moral Hazard Journal of Accounting Research, Vol. 37, No. 1, pp [4] Green, J. and Jean-Jacques Laffont (1986) Partially verifiable information and mechanism design. The Review of Economic Studies, Vol. LIII, pp [5] D. Gromb and David Martimort (2007), Collusion and the organization of delegated expertise Journal of Economic Theory, Vol. 137 (1), pp [6] Grossman, S. and Oliver Hart (1983), An Analysis of the Principal-Agent Problem Econometrica, vol. 51(1), pages [7] Holmström, B. (1979) Moral hazard and observability. The Bell Journal of Economics, Vol. 10, pp [8] Holmström, B. and Paul Milgrom (1987) Aggregation and Linearity in the Provision of Intertemporal incentives. Econometrica, Vol. 55, pp [9] Kanodia, C. (1985) Stochastic Monitoring and Moral Hazard. Journal of Accounting Research, Vol.23, pp

21 [10] Kedia, S. and Thomas Philippon (2006) The economics of fraudulent accounting, NYU working paper [11] Levitt, S. and Christopher Snyder (1997 ) Is no News Bad News? Information Transmission and the Role of Early Warning in the Principal-Agent Model. Rand Journal of Economics, Vol. 28 (4), pp [12] Mookherjee D. and Ivan Png (1989) Optimal Auditing, Insurance, and Redistribution. The Quarterly Journal of Economics, vol. 104(2), pp [13] Roger, G. (2012a) Moral Hazard under Soft Information. mimeo, The University of NSW [14] Roger, G. (2012b) Optimal Contract under Moral Hazard and Soft Information. mimeo, The University of NSW 21