Ex ante moral hazard and ex post adverse selection with soft information

Transcription

1 Ex ante moral hazard and ex post adverse selection with soft information Guillaume Roger The University of New South Wales PRELIMINARY AND INCOMPLETE January 2010 Abstract We reverse the standard sequence of a complete contracting model: first the agent takes an action, then she alone observes the stochastic outcome and sends a message to the principal. Presuming of the validity of the Revelation Principle, the optimal (direct) mechanism with audit requires a two-part tariff to be offered to the agent, which weakens the ex ante incentives for effort. We then establish an equivalence result for the agent between a direct revelation mechanism and the general mechanism, and provide a necessary and sufficient condition for the Revelation Principle to hold. 1 Introduction Most contracting models of involving hidden information and hidden action study ex ante adverse selection and ex post moral hazard. This sequence prevails in complete contracting models of regulation (e.g. Laffont and Tirole (1986)), in studies of executive compensation (e.g. Inderst and Müller (2010)) or in incomplete contracting models (e.g. Holmström (1999)). While this timing of events may correspond to the problem of first selecting a CEO and then providing him with incentives to exert effort, it does not describe all situations of interest. For example, after hiring the CEO, a board of director may ask him for some reports on his actions while on the job. Or in School of Economics, UNSW. g.roger@unsw.edu.au 1

2 the context of regulation, a firm may be asked to reveal its production cost after some investment in an uncertain technology. This is the reverse sequence: a wealth-constrained agent takes an action that yields an outcome (stochastically) and is required to report that outcome. This paper sets out to study this very problem, when in addition the information is soft in that it is not ex post observable by the principal. 1 Applications of this model may include exactly the aforementioned examples or the problem of taxation, where the tax levi may affect occupational choices. Up to the reverse timing, the model is standard. However, this choice of sequence combined with the lack of ex post observability of the soft information introduces an additional cost to induce effort and a specific, and widely observed, structure of the incentive contract. This latter characteristic is our first substantive result (beyond the characterisation of the optimal incentive scheme). In the standard moral hazard model where the principal observe the outcome of the agent s action (Rogerson (1985), Jewitt (1988)), the only incentive problem is that of inducing the principal s desired action. Where effort monotonically governs the distribution of outcomes, implementing the principal s preferred action is achieved by conditioning the agent s compensation (monotonically) on that outcome. Here the principal must also elicit information ex post, and the agent evidently has every incentive to distort it. If the compensation is increasing in the private information, who would want to admit to a bad outcome? Consequently, (under a Direct Revelation Mechanism now DRM) the principal must commit to a state-independent lump-sum payment to the agent at the lower bound a two-part (but not affine) tariff. This weakens the ex ante effort incentives as it constitutes an insurance against failure, hence the optimal action is further distorted from the standard second-best. The optimal contract must accommodate a fundamental tension between ex post information revelation, which requires a constant transfer, and ex ante effort provision, which is best addressed with a state-contingent compensation. We refer to this contract as conditionally optimal, in the spirit of Laffont and Tirole (1986). Importantly, (i) it is a low(er)-powered incentive scheme and (ii) its two-part nature emerges endogenously in response to an informational problem, not to satisfy an exogenous participation constraint or by assumption on the part of the modeler. 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 1 Soft information is subject to manipulation on the part of the agent. 2

3 agent and may be audited ex post by the principal at some cost. The audit restores partial observability, and is therefore essential as compered to models of costly state-verification (Khalil (1997)), where it only assists in relaxing the incentive constraint of the agent. 2 For practitioners or policy-makers, these results suggest that uniformly increasing the steepness of incentive schemes may be counterproductive when information is soft. Our second set of results pertains more to methodology. A truthful DRM can be characterised. This however is not sufficient to speak to the validity of the Revelation Principle in this context. Indeed, because the lump-sum payment required for truthtelling in a DRM weakens the ex ante effort incentives, it is not immediate from the existence of the DRM that the principal wants to insist on costly truthtelling ex post that is, on adopting a DRM in the first place. In technical parlance, it is not obvious that the DRM implements all (or any) equilibrium allocations that the principal could obtain through a general mechanism. Still, we can (i) show that any optimal allocation implemented under a general mechanism with a broader message space M R is equivalent for the agent to that implemented by the DRM, and (ii) find a necessary and sufficient condition on the principal s payoff function for the Revelation Principle to hold. 3 Thus is it not the truthtelling requirement of the DRM that is costly to the principal, but the imperfect ex post observability of outcomes. Since its first incarnations (Green and Laffont (1977), Dasgupta, Hammond and Maskin (1979), Myerson (1979, 1981)), the Revelation Principle has been extended to suit the environment under scrutiny (e.g. Mookherjee and Reichelstein (1990), Bester and Strauz (2001), Martimort and Stole (2002), Kartasheva (2006)). In each of these cases however, the information is ex post publicly observable so that the principal s payoffs are naturally a function of the state only and not of the message. It is again the lack of (or limited) ex post observability that prevents us from extending the Revelation Principle completely, because here both the state and the message may enter the principal s payoffs. The works closest to ours are Gromb and Martimort (2007), Levitt and Snyder (1997) and Green and Laffont (1986). The present model departs from all by adopting soft information in a very strong sense: the principal never observes any outcome. 4 Gromb and Martimort (2007) 2 Here, without audit it would be impossible to specify a non-trivial incentive-compatible contract at the information revelation stage, and therefore impossible to implement any effort in the first place. 3 M is a superset of the type space : M R. 4 Gromb and Martimort s expert(s) receive(s) a soft signal, but whether a project is eventually successful is publicly 3

4 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 moral hazard problem, their incentive contract must be made state dependent although they do not exert any influence on it. In contrast, here a share of the agent s compensation must be made state independent to induce information revelation (and indirectly ex ante effort incentives). 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() of the type space, which varies with the true state in a publicly known fashion. M(.)-implementable mechanisms exist and be not elicit truthtelling. Their exogenous restriction to the set M() is achieved here endogenously through the introduction of the stochastic audit. Malcolmson (2009) studies a problem where, as in Gromb and Martimort( 2007), the agent acquires soft information and the return to the principal is publicly observable. That soft information may be used by the agent to make a decision yielding the verifiable outcome. The principal may have incentive to distort the decision rule away from the first-best to foster information acquisition. The balance of the paper starts by introducing the model; then the case is made for the use of ex post audits. Section 4 characterises the optimal contract presuming of the validity of the DRM. This is confirmed in Section 5. Some of the technical material is relegated to the Appendix. 2 Model A principal hires an agent. At some cost c(a) increasing and convex, the agent undertakes an action a R + that yields a stochastic outcome [, ] with conditional distribution F ( a) and corresponding density f( a) > 0. A higher action is better in the sense of first-order stochastic dominance: F ( a) < F ( a ) when a > a. The agent alone observes the outcome and reports a message m to the principal, whereupon she receives a transfer t. Because she is wealth-constrained, the limited liability constraint t 0 (2.1) applies throughout, and will therefore no longer be mentioned. Her net utility is given by (ex post) observable. u(t, a) = v(t) c(a). 4

5 where v(t) is a monotonically increasing, concave function of the transfer t. The principal receives a payoff that depends both on the message m and on the difference between the message m and the true state. Let z = m, then the principal s net payoff is S(m, t; ) = π(m, φ(z)) t, with π m > 0, π mm < 0, π φ < 0, π φφ < 0 and φ z > 0. The principal likes good news (m), but dislikes surprises (φ(m )). The mapping φ(z) captures the extent to which surprises can be damaging. For example, following a scandal, a board s tolerance for surprises announced ex post by the CEO is typically low, while principals may be more permissive in good times. Or φ(.) may include the (exogenous) probability that an outsider discovers and reveals discrepancies between the message and the state. 5 In an alternative interpretation, the principal may have to undertake some action based on the message and without ever observing the true state, the payoffs of which depend on said message and the true state. The timing is almost standard and corresponds to a complete contracting problem in which the principal is able to commit to the contract: 1. The principal offers a transfer t(m, ) 2. The agent accept or rejects the contract. If accepting, she also choose an action a 3. Action a generates an outcome 4. The agent report the message m M 5. Transfers are implemented and payoffs are realised. If the true state were observable by the principal, this construct would be a moot point in that it would collapse to the textbook moral hazard problem. 6 Rather, as described, the game is one of cheap talk with commitment, which differs from Crawford and Sobel (1982) where the receiver does not commit to a decision rule ex ante, but responds simultaneously to the sender s message. 5 The reader is invited to recall the unfolding of the Enron saga, ca Even if the state were not verifiable, a Maskin game of state-revelation can elicit truth-telling as a Nash equilibrium of the revelation game. 5

6 3 The need for audits For some action a and an outcome R, the agent receives a transfer t from the principal. At the stage of information revelation, effort is sunk so all that matters is the utility v(t) from the transfer t(m), m M R denoting a bounded, arbitrary message space. Given the monotonicity of v(t), for a function t(.) increasing, the message m = max m M, for t(.) decreasing, the message m = min m M and for t(.) constant the message can be anything. In the latter case we may want to break indifference in favour of truthful revelation, but there is no good reason to do that. In fact, given that the object is to elicit effort, constant transfers guarantee zero effort. Hence either we have an impossibility result all types pool to the same message m = min m or m = max m or a poor mechanism. Auditing can restore a measure of ex post observability. It is costly and therefore run with some probability α. It is evident that the transfer function t(m) must be increasing in the report. Suppose not, then m = min m and no effort is exerted, in which case t(m) 0 cannot be optimal. Therefore it is sufficient to limit ourselves to upwards deviations, denoted by z = m here, with impact φ(z). An audit is successful (uncovers a lie) with probability p(z) if it is performed. With this construct, the expected utility function of an agent at the revelation stage is and we have since z U = v(t(m)) [1 α(m)] + v(t(m))α(m) [p(z)0 + 1 p(z)] = v(t(m)) [1 α(m)p(z)] U t = v [1 α(m)p(z)] 0; 2 U t = z v αp z 0 = 1. This is a sorting condition on the expected utility of the agent. The principal pays k(α), increasing and convex, for an audit. 4 Analysis under a Direct Mechanism We can state the principal s problem wanting to implement some action a and seeking some ex post report m as Problem 1 max π(m, φ(z)) t(m)[1 α(m)p(z)]f(x a)dx k(α) a,t,α 6

7 s.t. and U(t, α; m, )f(x a)dx c(a) U(t, α; m, )f(x a)dx c(a) 0 (4.1) m arg max m M U(t, α; m, )f(x ã)dx c(ã) (4.2) U(t, α; m, ) (4.3) where the first two constraints are the standard participation and moral hazard constraints, and the last one is the information-revelation constraint. It is not an ex ante constraint as the agent knows the state at the time of revelation. 4.1 Information revelation Because auditing restores a measure of ex post observability, the mechanism t, α; a presents us with enough instrument to characterise a direct mechanism. In the Appendix we formalise this statement by showing that there is a unique optimal message m() in each state. Given the construction of U(m, α, t; ), the optimal message is continuously increasing in the type. Therefore the set of equilibrium messages is a compact interval of R, which can be mapped into the set of types. This is not sufficient to validate the Revelation Principle in the present setting in that this construction alone does not guarantee that the optimal allocation is implementable through a DRM. We postpone the validation of this approach until after the characterisation of the optimal contract under a DRM (Section 5) Characterising a truthful Direct Revelation Mechanism When the state is revealed to the agent, she faces the problem A message m solving max v(t(m)) [1 α(m)p(m )] m constitutes her best reply, with second-order condition v t (1 αp) v [ α p + αp z ] = 0 (4.4) [v (t ) 2 + v t ](1 αp) 2v t [ α p + αp z ] vαpzz 0 (4.5) 7

8 Therefore differentiating the FOC at m = yields [ ( dz [v (t ) 2 + v t ](1 αp) 2v t α p + αp z dm dz )] ( dz vαp zz d dm dz ) = 0 (4.6) d but since ( dz dm dz d ) = 0 and p(m ) = 0 as well, v (t ) 2 + v t = 0. That is, the marginal utility of inflating the report is 0. It follows that truthtelling requires 2v t [ α p + αp z ] + vαpzz 0 (4.7) from the SOC, for which it is sufficient to have α 0 and t 0. The law of motion of the utility function is given by [ ( U dz m m= = v t (1 αp) v α p + αp z dm dz )] d = v t 0 (4.8) A boundary condition for U() at remains to be determined. In a standard problem, this boundary condition is given by the participation condition at : U() u 0 whatever this outside option may be. In this problem, participation has already been committed to by the agent by the time we reach the reporting stage, so there is no such condition. Still, the boundary condition is essential to induce truthful revelation: it has to make the agent indifferent between reporting her private information and an alternative message m at, i.e. U() v(t( m)) [1 α( m)p( m )] > 0. where the second inequality follows from Lemma 3. Therefore the least-cost boundary that is necessary for truthtelling (not for participation) is a function U(t, α) such that U(t, α; ) v(t( m)) [1 α( m)p( m )] (4.9) where m solves (4.3) at. Using the envelop theorem, U t = v (1 αp) > 0 and U α = vp < 0. These results are summarised in our first Proposition, where the formal claim of existence can be found in the Appendix (Theorem 2), and follows directly from Lemmata 2 to 4 and Proposition 7 (also in the Appendix). 8

9 Proposition 1 A truthtelling Direct Revelation Mechanism exists. It is characterised by the (standard) conditions 1. U = v t 2. α 0, t 0 and the new 3. U(t, α; ) = v(t( m()) [1 α( m)p( m )] where m() solves max m v(t( m()) [1 α( m)p( m )] given t(m) and α(m). Thus the truthtelling requirement imposes some structure on the payment schedule beyond the standard monotonicity condition. Here the boundary condition (4.9) is a lump sum payment (given the schedules t, α). More precisely, define: h(.) v 1 ; we claim Proposition 2 The compensation schedule takes the form of a two-part tariff denoted t() = τ() + h(u),, with τ() = 0, τ 0. In addition, this contract affords the agent no information rent. Noticeably the second term of the tariff t() is state-independent. No information rent is meant in the sense that whatever rent the agent may obtain is therefore state-independent as well: U(t, α; ) is obviously invariant in the state. More intuitively, the agent s incentives do not change whether her private information is 1 or 2 > 1 : in any state the agent faces the same incentive to overstate her message according to the constraint (4.3). Proof: From the results of Proposition 1 we can construct a compensation scheme as follows: h(u), = ; t() = τ() + h(u), >. Fix the schedules τ, α; since v(t()) U, the cost to the principal of providing utility U to the agent is h(u). Therefore and t() = h(u) >, v(t) = v(τ() + h(u)) 9

10 with τ > 0 to satisfy Constraints (4.7) and (4.8). To show that the agent obtains no rent, from the law of motion and the boundary condition, we have Ud = v t d U() U = v(τ() + h(u) v(h(u) U() U = v(τ() + h(u)) U so that U() = v(τ() + h(u)). Accounting for the fact that the agent faces a DRM τ(m), U, α(m) that compels ex post truthtelling, the problem can be recast as Problem 2 s.t. max [π t] (1 αp)df ( a) k(α) a,t h(u),α [0,1] 2v t [ α ] p + αp z + vαpzz 0 v(t)(1 αp)df c(a) 0 v(t())(1 αp)f(x a)dx c(a) v(t())(1 αp)f(x ã)dx c(ã) (4.10) where the last two constraints pertain to the ex ante moral hazard problem. That is, up to the first constraint, Problem 2 is a standard formulation, conditional on the two-part tariff t = τ +h(u). No interim participation constraint is required as it is addressed by the ex ante IR condition. Instead, Equation (4.9) specifies the boundary condition necessary for truthtelling. The principal optimises ex ante with respect to t, α only as Constraint (4.9) arises from the ex post problem of truthful revelation. Once the solutions t, α are characterised, U(t, α; ) is determined at by (4.9); from there, t can be decomposed into τ and U. Next we characterise the optimal transfer schedule and probability of audit under the DRM Optimal compensation Adopting the first-order approach (see Jewitt (1988) for a validation), we can write Constraint (4.10) as a first order condition with respect to the action a chosen by the agent. v((τ + h(u)))(1 αp)df a ( a) = c (a) (4.11) 10

11 Then we claim: Proposition 3 The principal offers a contract made up of a conditionally optimal compensation schedule t () = τ () + h(u) satisfying the standard first-order condition of Problem 2, and decomposed by Condition (4.9), and an audit probability α > 0. The compensation schedule is characterised in the proof. It said to be conditionally optimal in the sense that the first-order condition is completely standard, up to the fact that the contract must accommodate the truthtelling requirement imposed by Condition (4.9). This is reminiscent of Laffont and Tirole (1986), where there is not first-order (direct) effort distortion but one that is only indirect (through the revelation problem). In their paper, type and effort are substitutes (but types are ex ante, hard information). Type revelation is obtained through ex post cost observability, while here truthtelling is achieved by a combination of the optimal U and the audit probability. Proof: Deriving the first-order conditions alone is not sufficient as the boundary U is endogenous to the transfer (the audit probability). To establish the result we need to construct an algorithm as follows. Recall that m() is the solution to the problem max m M v(t) [1 α(m)p(m )] for some α and some t. Now take an arbitrary transfer t(m()) 0 and (at least weakly increasing). Attaching multipliers µ and λ to constraints (4.1) and (4.11), the first-order conditions with respect to α read ( ) tpdf = k + λ v(t)pdf a + µ v(t)pdf where it is understood that p = p( m() ) > 0, where m() satisfies (4.12) m() = arg max m M v(t) [1 α (m)p(m )] (4.13) Ignore for now whether any of the multipliers are positive. Given α and the arbitrary t we can compute U(α, t) using (4.9). This is our initial condition on U. Next define the transfer t as a two-part tariff t = h(u(α, t)) + τ(m) as in Proposition 2 and optimise with respect to τ, with first-order condition (1 αp)df ( ) λ v (1 αp)df a + µ v (1 αp)df = 0 (4.14) using the envelop theorem (optimality of m for the agent). For τ solving (4.14), compute the agent s endogenous boundary condition U (α, h(u(α, t)) + τ ) using (4.9) again. 11 Re-evaluate

12 (4.14) using this new value and so on. Observe that U = (1 αp)v(h(u) + τ(m)) defines a fixed-point problem with a unique solution since the RHS is a contraction. To see why, differentiate with respect to U ( d (1 αp)v(h(u) + τ(m)) = (1 αp)v h + τ dm ) v(.)(α p + αp ) dm du du du = (1 αp)v h + dm ( v τ v(.)(α p + αp ) ) du = (1 αp) 1 binding only for m() =. The result follows from (v τ v(.)(α p + αp )) = 0 by the FOC (4.4) and h v 1. Last observe that to the exact argument of v (.), the optimality condition (4.14) is exactly the Jewitt condition (Jewitt (1988)) so we know that bothλ and µ are positive. ( 1 v = λ f ) a f + µ Condition (4.12) expresses the balance of the marginal benefit with the marginal cost of a change in α, which entails both a direct effect and an indirect one (through the constraints). The ( ) Jewitt condition 1 v = λ f a f + µ arises because given α and conditional on truthtelling (obtained through the two-part tariff), the ex ante problem remains standard: for the principal, τ and U are perfectly substitutable in any state Optimal action choice We are now in a position to complete the characterisation of the moral hazard problem under the DRM. This has become a standard problem, however with some more interesting results. While the substitutability of h(u) and τ is immaterial to the principal s objective function, it implies less powerful incentives for the agent lower marginal returns on effort. This is the object of the next Proposition, which is our second main result. Proposition 4 Imperfect observability undermines the provision of effort. Action a solving (4.11) is lower than in the standard moral hazard problem with perfect outcome observability. 12

13 Under the DRM, the truthtelling constraint imposes a non-trivial fixed component h(u) as part of the agent s compensation. This softens the incentives as the agent is guaranteed a fixed payment regardless of the state she reports. Proof: Since t = τ + h(u), U > 0 to satisfy truthtelling (Condition (4.9)). Take any (total) transfer t() = t() such that 0 h(ũ) < h(u); for any action a E [v(t) a] = v(t)df ( a) > v( t)df ( a) = E [ v( t) a ], by concavity of v(.). 7 Since E [v(t) a] is concave in a under the assumptions of the first-order approach (see Jewitt (1988)), v(t)df a ( a) < v( t)df a ( a), whence a < ã (where ã solves (4.11) under t) since c(a) is increasing convex. As will soon be more obvious, Proposition 4 is a statement about the cost of incompletely observable outcomes and the corresponding reliance on self-reporting, rather than about the optimality of the truthtelling DRM. That is, it compares the optimal effort level when acquiring information about the outcome ex post is costly (and requires the two-part tariff τ() + h(u)) to the standard case when the outcome is directly observable by the principal. Its direct consequence is our next results, which completes the characterisation of the DRM. Proposition 5 The principal selects a conditionally optimal, lower action in Problem 2 than he would absent the truthtelling constraint (4.9). That is, a low-powered incentive contract is offered to the agent. Proof: The first-order condition of Problem 2 with respect to the action a is the standard ( ) (π t) df a ( a SB ) + λ v(t)df aa ( a SB ) c (a SB ) = 0 (4.15) with the participation constraint binding since µ > 0. By Proposition 4, an action a solving (4.11) is more expensive than would be absent the truthtelling constraint. Given a pair of optimal t, α, the principal therefore selects a lower action a SB solving (4.15). 7 With complete ex post observability, the optimal U is zero. 13

14 5 Optimality of the direct revelation mechanism: the Revelation Principle It is now high time to enquire as to whether a DRM can really be an optimal mechanism for the principal in this problem. The reason is that the ex post compensation t(m) (in the form of t = τ + U under the DRM) is in tension with the ex ante effort incentives. Indeed, ex post information revelation calls for constant transfers, while effort provision requires state-dependent payments. Relying on a DRM turns out to be without much loss of generality, but not strictly so. To do so we proceed in two steps, first establishing an equivalence result for the agent, and then finding a condition on the principal s payoff for the Revelation Principle to hold. 5.1 Validity of direct revelation mechanisms: an equivalence result Two definitions need now be introduced. Let A be the space of allocation for a generic mechanism, and retain the notation M. Definition 1 Two mechanisms M, d(m), M, d (m) are payoff-equivalent when u(d(m)) = u(d (m )) for a generic payoff function u(.) and a generic allocation mapping d : M A. Definition 2 Two mechanisms M, d(m), M, d (m) are effort-equivalent when a SB = a, where a is the optimal action (solves (4.15)) under M, d (m). We can now claim: Proposition 6 The general mechanism M, α (m), t (m) and the direct mechanism, α (), U, τ() are 1. payoff-equivalent and; 2. effort-equivalent for the agent. Proof: We first show that the mechanisms are payoff-equivalent given some action a. They must therefore induce the same equilibrium action on the part of the agent. Fix a, the lowest transfer h(u) under the DRM is given by the boundary condition (4.9): U = v(t( m())) (1 α( m())p(( m()) )). 14

15 Of course, this is exactly the expected utility of the agent at under the general mechanism. Differentiating with respect to the type, we find d d m ( (v(t( m())) (1 α( m())p(( m()) ))) = v t (1 αp) v(.)(α p + αp z ) ) v(t( m))α( m)p z z d d = v(t( m))α( m())p z z because messages must be optimal and therefore satisfy the FOC (4.4) and the first-term collapses to zero (envelop theorem). Under the DRM, we have d d m ( (v(τ( m()) + U) (1 αp)) = v t (1 αp) v(.)(α p + αp z ) ) v(τ( m()) + U)α( m)p z z d d = v(τ( m()) + U)α( m)p z z = v(τ( m()) + U)α()p z z And since v(τ( m()) + U) = v(t( m)) by construction of the truth-inducing two-part tariff, their growth rates may only vary by according to differences in α( m()) and α() in any state. It can be verified that the principal s problem under the general mechanism M, α (m), t (m) Problem 3 s.t. max [π t] (1 αp)df ( a) k(α) a,t,α [0,1] v(t)(1 αp)df c(a) 0 v(t(m))(1 αp)f(x a)dx c(a) v(t(m))(1 αp)f(x ã)dx c(ã) admits first-order conditions with respect to α that are isomorphic to (4.12). Therefore, v(τ() + h(u))(1 αp)df ( a) = v(t( m))(1 αp)df ( a), the expected payoff under the DRM is the same as under the general mechanism, for some action a. It immediately follows that a d = a g (with obvious notation) the agent s first-order conditions with respect to the action are identical. Last, the principal faces isomorphic problems when selecting his preferred action, and therefore chooses the same effort level: a SB = a. This proves items 1 and 2. This equivalence result is silent as to the principal s incentives, and as such cannot yet be read as a statement about the Revelation Principle. (Whether it can be relied on in this model depends on the principal s payoffs.) 15

16 5.2 The Revelation Principle in this model Here we seek to extend the results of Proposition 6 by offering some condition for the Revelation Principle to hold. Theorem 1 The Revelation Principle holds if and only if π ( m, φ( m() )) df ( a ) π()df ( a SB ) where a is the optimal action under the general mechanism M, α (m), t (m). Notice that this necessary and sufficient condition pertains solely to the principal s payoffs under the optimal action. This is because a = a SB and both mechanisms are payoff equivalent to the agent so their cost is the same to the principal. This condition is less evident than may seem: it requires that there be no third mechanism other than the DRM or M, α (m), t (m) that the principal may want to implement. The only mechanism that is neither has to include a message space M that is a subset of of the optimal message space M else it does not constrain the agent. Proof: Take an arbitrary m < m(), then we can extend Lemma 4 (stated and proven in the Appendix) to the set of arbitrary messages: Lemma 1 Suppose the message space is M { m R m m < m() }, then there exists a threshold (t, α) such that m() = m,. The principal s problem becomes Problem 4 max [π( m) t( m)] (1 α( m)p( m))df ( a)+ [π(m) t(m)] (1 α(m)p(m))df ( a) k(α) a,t,α [0,1] s.t. v(t( m))(1 α( m)p( m))df + v(t( m))(1 α( m)p( m))df (x a) + v(t(m))(1 α(m)p(m))df c(a) 0 v(t(m))(1 α(m)p(m))df (x a) c(a) v(t( m))(1 α( m)p( m))df (x ã) + v(t(m))(1 α(m)p(m))df (x ã) c(ã) 16

17 the solution ã of which lies necessarily below a SB since the constraints bind (and the moral hazard constraint induces a lower action from the agent). Since ã generates a lower expected payoff for the principal, there is no alternative mechanism involving a truncated message space M that is preferred to either the DRM or the general mechanism. Corollary 1 A sufficient condition for the Revelation Principle to hold is where dφ dm m M, (π m + π φ ) df ( a) 0 = 1. This simply requires that the agent exaggerating the reports does not help the principal. This condition holds in the following examples. Example: Let π, then πdf ( a) = E[ a] and (π m + π φ ) df ( a) = 0 Example: Let π m κ(m ), then πdf ( a) = m(1 κ) + κdf ( a) and simply (π m + π φ ) df ( a) = (1 κ)df ( a) 0, κ 1. The sufficient condition hold for some values of m only in the next example. Example: Let π m (m )n n, so (π m + π φ ) df ( a) = ( 1 (m ) n 1 ) df ( a). And for n sufficiently large 1 (m ) n 1 0 for some m, where m is optimally chosen by the agent. That is, whether Corollary 1 can be used depends on the agent s payoff function v(t). 5.3 Discussion Proposition 4 may lead the analyst to conclude that the truthtelling requirement of the DRM is too costly, and to conjecture that there may be less onerous (non-truthful) mechanisms to avail. But it is poor ex post observability (rather than truthtelling) that is the source of the result captured by Proposition 4. Poor ex post information imposes either a non-trivial fixed component h(u) as part of the agent s compensation, or exposes the principal to a lie both at the same cost. Both equally soften the incentives as the agent is guaranteed either a fixed payment (under the DRM), or (optimally) overstates her private information m() under the general mechanism. Thus the rent arises from the lack of ex post observability, which is only imperfectly restored by the audit mechanism. 17

18 6 Conclusion When a principal cannot directly observe the outcome of his agent in a moral hazard framework and needs to elicit this information from that very agent, he face a problem of ex post adverse selection as well. This is costly to the principal in two ways: first, it requires paying a rent to the agent to obtain some information. Second, it distorts the ex ante effort incentives. The source of these distortions is not the truthtelling requirement imposed by the Direct Revelation Mechanism. Rather, it is the lack of ex post observability that introduces these new frictions. Under some conditions, there is no loss of generality in using a DRM that is, the Revelation Principle can be extended. This concern about the validity of the Revelation Principle arises too because of poor observability, in which case a principal s payoffs may be conditioned on messages rather than types. 18

19 7 Appendix: feasibility of DRMs In this section we characterise the set of equilibrium messages when the mechanism is not restricted to be a DRM. It is shown to be a compact interval in R (and therefore well ordered), which maps the set of types into the set of messages, uniquely for each type. To proceed, relax a DRM s requirement that m. Since R, let m M R but M. Redefine the transfer and audit probability as t(m) : M R R and α(m) : M [0, 1]. At the revelation stage, an agent selects a message m so as to solve max m M with almost identical first-order condition. v(t(m)) [1 α(m)p(z)] v t v(t( m)) = α p + αp z 1 α( m)p(z) (7.1) if the FOC binds. Let m(t, α; ) be the solution to this equation. For any transfer function t(m) and probability of audit α(m), it can be shown that m(t, α; ) varies continuously in (Theorem of the Maximum, Berge (1963)). That is, the agent s reporting behaviour generates a bounded set M defined as M = {m M m arg max U(t, α; ), }. Else, m = m for some (arbitrary) upper bound m < m() of M if v t v(t(m)) α p + αp z 1 α(m)p(z) where any derivative with respect to m is understood to be a left-hand derivative. The next claim will be useful throughout. Claim 1 Condition (7.1) admits a unique maximiser. Proof: To see why, rewrite it as d d dm ln v(t( m)) = dm ln(1 α( m)p(z)) and take z = m = m, with. Suppose m > m, then d ln v(t( m)) < d dm d d ln v(t(m)) = ln(1 α(m)p(m )) ln(1 α( m)p( m )) dm dm dm (7.2) This is not a surprise: recall that U satisfies the sorting condition and is concave in t, hence there exists an optimal message for each type. First off we want to characterise the set M. 19

20 Lemma 2 d m d 0 Proof: The solution of the first-order condition (7.1) is a message m(t, α; ). Continuity of the solution m(t, α; ) follows from the Theorem of the Maximum. Monotonicity can be easily established; jumping ahead and calling on the Theorem of Lebesgue, m(t, α; ) is a.e. differentiable. Differentiate with respect to and rearrange. Else m = m (by condition (7.2)) and the claim is trivially true. Directly from Condition (7.2), we also note Lemma 3 For any (weakly increasing) transfer function t(m) and audit probability α(m), m() >. Furthermore, d m dt < 0 and d m dα < 0 so that d m() dt < 0 and d m() dα < 0 as well. Proof: Take any (weakly increasing) monotone transfer function t(m) and any audit probability α(m). Since U t 0, min m(t, α, ) = m(t, α, ) m(). Suppose =, for m =, U m m= = v t > 0, so m() >. For the second set of statements, continuity and monotonicity can be easily shown, so that d m is a.e. differentiable with respect to t and α. Fix an arbitrary type such that the first-order condition (7.1) holds, differentiate it with respect to each of the variables and re-arrange. Using the fact that the SOC holds the claims follow. The results extend all the way to = from the right-hand side by (right-hand) continuity. Lemma 3 establishes that the message space is truncated by the agent s optimal reporting choice at the lower bound of the outcome space. In addition, Lemma 3 tells us that the set of reported messages increases there is less systematic over-reporting when the transfer increases and when the probability of audit increases. The latter claim is quite obvious, the first one less so: increasing t(m) renders lying more costly as there is more to lose for any actual state realisation. The transfer and the audit act as substitutes in the agent s information revelation problem. We now complete the description of the set of optimal messages M when M. Lemma 4 Take an arbitrary message space M and any t(m), α(m), then m() = min { m, m(t, α; ) } 20

21 where m(t, α; ) solves Condition (7.1) at. If m() = m, then there exists a threshold (t, α) such that m() =, (t, α) with d (t,α) dt > 0, d (t,α) dα > 0, (t, α) <. Proof: For the first statement, take any increasing monotone transfer function t(m) and audit probability α(m) defined on M. For any type and any message m, either FOC (7.2) or (4.4) holds. In the former case, the optimal message is exogenously bounded at m of M. Then the existence of (t, α) such that follows directly from FOC (7.2) and from Lemma 2. More precisely, at (t, α), the agent s utility U = t(m) [1 α(m)p(z)] is not differentiable; but the left-hand derivative is given by (7.1), while the right-hand derivative is zero (since m = ). When Condition (7.1 is relevant, the sorting condition ensures that the maximiser of U() is unique (Claim 1), and that max U() = U(). Therefore m(t, α; ) = max m(). The second set of statements follows directly from the existence of (t, α) and from Lemma 3. Indeed, d m dt < 0 wherever the FOC (7.1) holds. In particular, the left-hand derivative d m(α, t;) dt = (t,α) < 0 for some transfer t. Therefore m(α, t; (t, α)) < for any t > t. Hence by the first claim, there exist some other (t, α) > ( t, α) such that m =, (t, α). Since the threshold (t, α) is a monotonic function, it is a.e. differentiable except for finite set of points (Theorem of Lebesgue). As the true state approaches the upper bound of the support, the threat of audit does not have enough bite to prevent deviations all the way to the top. This is because p z z < 0. However, since over-reporting is dampened by increases in α, t, the threshold (t, α) above which the agent lies without restraint increases. So both the transfer and the audit probability improve information revelation, but at different costs. Collecting these preliminary results, we have Proposition 7 Fix t(m) and α(m), the reporting set is a compact interval: M [ m(), m() ]. Using this technical result we can finally claim Theorem 2 A Direct Revelation Mechanism exists. Proof: Since m() is unique for each and m() is monotonic, it is invertible hence, card M = card. That is, the mapping m() is a homeomorphism from to M. The general mechanism is a pair of functions α(m()), t(m()), which are both continuous and monotonic. Define the direct mechanism α(), t() as α α m and t t m. For any, there exists a unique pair α(), t() as well. 21

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