Optimal contract under adverse selection in a moral hazard model with a risk averse agent

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1 Optimal contract under adverse selection in a moral hazard model with a risk averse agent Lionel Thomas CRESE Université de Franche-Comté, IUT Besanon Vesoul, 30 avenue de l Observatoire, BP1559, Besançon, France Preliminary version. Do not quote. Abstract In this paper we study the optimal contract offered by a risk neutral principal to a risk averse agent when agent s efficiency and action both improve the probability of success of the project. Adding an adverse selection problem to the canonical moral hazard model has two consequences. On the one hand, the agency cost of adverse selection is added to the agency cost of moral hazard. On the other hand, it can reinforce or mitigate the insurance efficiency trade off. Moreover, these effects make existence of a fully separating contract difficult. Keywords Adverse selection, Moral hazard, Risk aversion. JEL Classification Numbers D82. Corresponding author. lionel.thomas@univ-fcomte.fr. 1

2 1 Introduction Since forty years, the theory of incentives has made considerable advances. The implications of pure adverse selection or pure moral hazard models are now well known. 1 However, there are many examples of contracts designed to solve simultaneously an adverse selection problem and a moral hazard problem. Insurance and financial contracts are particularly archetypical. In the former, a driver has private information on how good driver he is and also safely he drives. In the latter, a borrower has private information on how risky his project is and also hazardously he manages it. Paradoxically, despite the plethora of generalized agency (or mixed) models in the incentives literature, such problems where the principal s payoff depends intricately on both the private information and unobservable action of the agent have been little studied. In this paper we study the optimal contract offered by a risk neutral principal to a risk averse agent when efficiency and action both improve the probability of success of the project. Taking the complete information setting as a benchmark, we show the well known result that the contract offers full insurance: the agent only receives a fixed payment whether the project fails or successes. Then, considering that action becomes non observable to study the moral hazard framework. The contact offers high powered incentives: the fixed payment is increased by a positive bonus in the event of success. The agent bears some risk and receives a higher expected payment than with complete information. In other words, the principal pays a costly risk premium. A trade off between insurance and efficiency arises. It is optimal so to distort the efficient probability of success to take into account both high powered incentives and risk premium. Finally, we consider that efficiency is no more observable. Thus adverse selection is added to moral hazard and we obtain a generalized agency model. When the principal faces asymmetric information on the agent s type, she needs to elicit truthful information. To do this, she must give up an informational rent to the agent. Such a rent is costly and leads to an overall trade off between rent extraction, insurance and efficiency. This implies new distortions on the probability of success. The first one has for goal to reduce 1 See Laffont and Martimort (2002). 2

3 the informational rent. This is the standard effect due to the non observability of efficiency. So the agency cost of selection adverse is simply added to the agency cost of moral hazard. The second distortion is due to the fact that the informational rent modifies the insurance efficiency trade off. It follows that the adverse selection agency cost can reinforce or mitigate the moral hazard trade off. Finally, this overall trade off make existence of pooling likely. The paper is organized as followed. Section 2 presents the related literature. The model is stated in section 3. Sections 4 and 5 analyze complete information and moral hazard respectively. The generalized agency case is studied in section 6. 2 Related literature To be done... 3 The Model The basic data of the model follow Ollier and Thomas (2013). A principal contracts with an agent to produce a pecuniary output with random value x, x {x, x}. The high (resp. low) production x (resp. x) is associated with success (resp. failure). The realization of the high production needs that the agent exerts an action consisting in generating a probability of success ρ = Pr(x = x) (0, 1). But action is costly for the agent. He incurs an indirect disutility ψ(ρ, θ), with θ his efficiency. The principal does not observe the action nor the efficiency. But he knows that efficiency is drawn from a density f > 0 on [θ, θ] with cumulative F. The principal offers a contract a, b with a, a fixed payment, and b, a bonus in case of success. In other words, a is the non contingent component of the contract, and b, the contingent component. The principal. The principal is risk neutral. For a given probability of success and a given contract, his objective function is V = ( 1 ρ )( x a ) + ρ ( x (a + b) ) = x + ρ x a ρb, (1) 3

4 where x = x x > 0, is the value increase due to success. The agent. The agent is risk averse. The utility of the transfer in case of success (resp. failure) is u(a + b) (resp. u(a)). The agent s expected utility is U = ( 1 ρ ) u(a) + ρu(a + b) ψ(ρ, θ) = u(a) + ρ ( u(a + b) u(a) ) ψ(ρ, θ). (2) The properties of the functions u and ψ are the following. Property 1. The functions u and ψ verify u (.) > 0, u (.) < 0, u(0) = 0, and 2 ψ 1 (ρ, θ) > 0, ψ 11(ρ, θ) > 0, ψ 2 (ρ, θ) < 0, ψ 22(ρ, θ) > 0, ψ 12(ρ, θ) < 0. The function u reflects the risk aversion of the agent, the function ψ, standard assumptions on the probability of success and the disutility of effort. 3 For all types, the marginal indirect disutility is positive and increasing. In more productive states, the agent s disutility diminishes but at an increasing rate. Higher values of θ correspond to states in which a higher probability of success is less costly to generate. The problem. The principal s problem is to maximize the expectation of (1) with respect to {a, b}, subject to the participation and the incentive compatibility constraints. The former imply that the agent voluntary agrees to the contract. The latter require that the agent is honest and obedient. For the sake of clarity in resolution whatever the informational framework, it is useful to proceed to the following change of variables. Let u be the utility in case of failure, u the utility in case of success and u the increase in utility due to success. We obtain u = u(a), u = u(a + b), u = u u. 2 Subscript i denotes the partial derivative with the i th argument. 3 Indeed, an equivalent way to analyze this problem is to consider that the agent exerts an effort that, as far as his efficiency, increases the probability of success ρ(e, θ). But he incurs a disutility ϕ(e). See Ollier and Thomas (2013) for more details. 4 (3)

5 Denoting u 1 = w (so that w > 0, w > 0), we get the following lemma. Lemma 1. The payments are such that a = w(u), b = w(u) w(u). Proof. Straightforward. Using this lemma, the objective function (1) is V = x + ρ x w(u) ρ ( w(u) w(u) ). (4) From (3), the agent s utility (2) becomes U = u + ρ u ψ(ρ, θ). (5) Moreover, manipulating (5) and since u = u + u, we get u = U + ψ(ρ, θ) ρ u, ū = U + ψ(ρ, θ) + ( 1 ρ ) u. (6) Thus offering the contract U, u specifying the agent s expected utility, U, and the increase in utility due to success, u, is equivalent to offer the initial contract a, b. To get a well behaved model, we assume that the marginal indirect disutility is convex. Assumption 1. ψ is such that ψ The rest of the paper studies the optimal contract according to the following informational frameworks: complete information, moral hazard and adverse selection plus moral hazard. 4 Complete information When the information is complete, the principal observes the agent s efficiency and action. Only the participation constraint needs to be satisfied. Given (6), the problem is max (ρ, u,u) (4) subject to the participation constraint U 0. (PC ) We get the following lemma. 5

6 Lemma 2. The first best contract entails U F B (θ) = 0, u F B (θ) = 0, so u F B (θ) = ψ(ρ F B (θ), θ), with ρ F B (θ) given by x = w (u F B (θ))ψ 1 (ρ F B (θ), θ). (FB) Proof. See appendix 9.1. The interpretation is the following. To benefit from the value increase, x in (FB), the principal must increase the probability of success. Concomitantly, this increases the indirect disutility of the agent, measured by ψ 1. Thus, in order to ensure his participation, the principal must increase the expected payment, w(u) + ρ ( w( u + u) w(u) ). But it is increasing in u and u. It follows that U is costly (make use of (6)) and u is so. It is optimal to set U = u = 0. In other words, it is optimal to not give up a rent to the agent, U = 0, nor a contingent contract, u = 0. Instead, the agent receives a full insurance contract since u = 0 u = u. The first best payment is thus w(ψ), and the first best marginal cost of increasing the probability of success corresponds to the marginal payment w (u)ψ 1 in (FB). Finally, remark that the agent s net utility is null in each state of the nature, i.e u ψ = u ψ = 0. 5 Moral hazard In this framework, the agent s efficiency is still observable, but the action is not. The principal faces a moral hazard problem. The incentive question requires that the agent be obedient. The moral hazard incentive constraint is the following. Faced with an incentive contract U, u, the agent generates the probability p( u, θ) = arg max {u + ρ u ψ(ρ, θ)} ρ u = ψ 1 (p( u, θ), θ) > 0. (7) 6

7 Thus (6) becomes u = U + ψ(p( u, θ), θ) p( u, θ) u, ū = U + ψ(p( u, θ), θ) + ( 1 p( u, θ) ) u. (8) Given (8), the problem is max ( u,u) (4) subject to (PC ). We can state the following lemma. Lemma 3. The moral hazard contract entails U MH (θ) = 0, u MH (θ) strictly positive and such that x = ( w( u MH (θ) + u MH (θ)) w(u MH (θ)) ) + p( u MH (θ), θ) ( 1 p( u MH (θ), θ) ) ( w ( u MH (θ) + u MH (θ)) w (u MH (θ)) ) ψ 11 (p( u MH (θ), θ), θ), (MH ) with u MH (θ) = ψ(p( u MH (θ), θ), θ) p( u MH (θ), θ) u MH (θ). (9) Proof. See appendix 9.2 The comments are essentially the same than for complete information. The difference is in the marginal (expected) payment on the right hand side of (MH ). It is composed of two terms. To induce a positive action despite its non observability, u can no more be set to 0, i.e u > 0 (see (7)). This forces the principal to offer a contingent contract or high-powered incentives to the agent. Thus, the principal must give up the bonus to the agent, or the contingent component of the contract. This is the first term on the right-hand side of (MH ) (see Lemma 1). It represents the marginal cost due to high powered incentives. In parallel, as with complete information, the agent does not get a positive rent, U = 0, because it is costly for the principal. However, because the contract is contingent, the agent 7

8 bears some risk. It follows that the moral hazard expected payment is higher than the first best payment. We have w(u) + ρ ( w(u) w(u) ) = ( 1 ρ ) w(u) + ρw(u) > w((1 ρ)u + ρu), by Jensen s inequality = w(u + ρ u) = w(ψ), because U = 0. This arises because the principal has to pay a risk premium to the agent to ensure his participation. The second term on the right-hand side of (MH ) is the marginal cost due to the risk premium, given that u = ψ 1. All in all, the right hand side of (MH ) is the moral hazard marginal cost of increasing the probability of success. This marginal cost differs from the first best. So the first best action is distorted to take into account that a contingent contract is costly. To evaluate this distortion, let us write the limited development of (7). We get w( u + u) w(u) = w (u)ψ 1 + o(ψ1). 2 (10) Inserting (10) in (MH ), then adding and subtracting w (ψ)ψ 1, (MH ) becomes x ( w (u) w (ψ) ) ψ 1 o(ψ 2 1) p ( 1 p )( w (u) w (u) ) ψ 11 = w (ψ)ψ 1. (MH ) So, comparing (FB) and (MH ), two opposite distortions arise. Given that w (ψ)ψ 1 is increasing with the probability of success, 4 for a given probability and type, we have ( w (u) w (ψ) ) ψ 1 < 0, since u < ψ from (9), and w > 0. So, a negative net utility, u ψ, in case of failure, instead of null with complete information, pushes to an upward distortion, o(ψ1) 2 > 0, from (10). This reflects that the net utility in case of success, u ψ, is positive (recall that u < ψ and U = 0), instead of null with complete information. This implies a downward distortion, 4 Indeed, w ψ1 2 + w ψ 11 > 0. 8

9 p ( 1 p )( w ( u + u) w (u) ) ψ 11 > 0, because w > 0. The marginal cost due to the risk premium leads to a second downward distortion. These distortions illustrate the standard insurance efficiency trade off in moral hazard problems with a risk averse agent. To generate a positive probability of success, the principal must offer an incentive contract, which is costly. distort the efficient probability. This cost implies that it is optimal to 6 Generalized agency In this framework, the principal observes neither the agent s efficiency nor the agent s action. Following Myerson (1982), there is no loss of generality in focusing on direct revelation mechanisms. The contract offered by the principal is then U(ˆθ), u(ˆθ), where ˆθ is the agent s report on his efficiency. From the moral hazard section, we know that the agent chooses p( u(ˆθ), θ) = arg max{u(ˆθ) + ρ u(ˆθ) ψ(ρ, θ)} ρ u(ˆθ) = ψ 1 (p( u(ˆθ), θ), θ) > 0. (11) Let us denote by υ(ˆθ, θ) the indirect expected utility of an agent with efficiency θ that reports ˆθ. We have υ(ˆθ, θ) = u(ˆθ) + p( u(ˆθ), θ) u(ˆθ) ψ(p( u(ˆθ), θ), θ). Hence, the incentive constraint is ˆθ, θ Θ U(θ) = υ(θ, θ) υ(ˆθ, θ). (IC ) That is, the agent is better off reporting the truth on its efficiency. Let the reservation utility be normalized to 0, the participation constraint is θ Θ U(θ) 0. (PC ) The agent is not forced to accept the contract. 9

10 Moreover (8) becomes u(θ) = U(θ) + ψ(p( u(θ), θ), θ) p( u(θ), θ) u(θ), ū(θ) = U(θ) + ψ(p( u(θ), θ), θ) + ( 1 p( u(θ), θ) ) u(θ). (12) Given (12) and using (4), the principal s problem is { ( )} max x + p( u(θ), θ) x w(u(θ)) p( u(θ), θ) w(u(θ)) w(u(θ)) f(θ)dθ, (13) u(.),u(.) θ subject to (IC ) and (PC ). Let us begin the resolution of this problem by a reformulation. The following lemma characterizes necessary and sufficient conditions for (IC ). Lemma 4. (Ollier and Thomas (2013)). The allocation U(θ), u(θ) is incentive compatible if and only if, θ Θ U (θ) = ψ 2 (p( u(θ), θ), θ), (IC 1) u (θ) 0. (IC 2) To ensure revelation, the constraint (IC 1) tells us that the agent s expected utility, U(θ), must follow the path U (θ) = ψ 2 (p( u(θ), θ), θ). Thus, using Property 1, it is increasing with efficiency. Moreover, following (IC 2), she must ensure that the increase in utility due to success, u(θ), increases with the type. Then, we already know that U is costly for the principal. Since it is increasing with the agent s efficiency, it is optimal for her to not give up an informational rent to the most inefficient agent. That is U(θ) = 0. (PC 1) Finally, the principal s problem is max ( u(.),u(.)) (13) s.t. (IC 1), (IC 2) and (PC 1). This is an optimal control problem. Omitting arguments for clarity, U and u are state variables. But, in a first step, we assume complete sorting of types, i.e. u > 0. This allows us to consider u as a control. We associate the adjoint variable η with U. The Hamiltonian is H = { x + p x w(u + ψ p u) p ( w(u + ψ + (1 p) u) w(u + ψ p u) )} f ηψ 2. 10

11 Using (11) and (12), the maximum principle yields 5 as necessary conditions H u = { ( ) p 1 x + w (u)p p 1 w( u + u) w(u) as transversality conditions p ( w ( u + u)(1 p) + w (u)p )} f ηψ 12 p 1 = 0, (14) η = H U = { w (u) + p ( w ( u + u) w (u) )} f > 0; (15) η(θ) no condition, η(θ) = 0. (16) From (15) and (16), we have { η = w (u) + p ( w (u) w (u) )} f(τ)dτ < 0. (17) θ Moreover, using (IC 1) and (PC 1), we get We can present our first result. U = θ ψ 2 dτ. (18) Proposition 1. Assume complete sorting. The generalized agency contract entails U (θ) = θ ψ 2(p( u (τ), τ), τ)dτ, u (θ) strictly positive and such that x = ( w( u (θ) + u (θ)) w(u (θ)) ) + p( u (θ), θ) ( 1 p( u (θ), θ) ) ( w ( u (θ) + u (θ)) w (u (θ)) ) ψ 11 (p( u (θ), θ), θ) θ { w (u (τ)) + p( u (τ), τ) ( w ( u (τ) + u (τ)) w (u (τ)) )} f(τ)dτ f(θ) ψ 12 (p( u (θ), θ), θ), 5 See Seierstadt and Sydsaeter (1987). Sufficient conditions are presented in appendix 9.3. (GA) 11

12 with u (θ) = U (θ) + ψ(p( u (θ), θ), θ) p( u (θ), θ) u (θ). (19) When the principal does not observe the agent s type, she needs to elicit truthful information. To do this, she must give up an informational rent to the agent, U > 0, except at θ. Such a rent implies that the marginal cost of increasing the probability of success is modified compared to moral hazard only (i.e the right-hand side of (MH )). The first change is the presence of the last term in (GA). It reflects the marginal cost of the informational rent. It is composed of two terms. Because U is costly and must satisfy (IC 1), i.e U = ψ 2, it is optimal to influence this slope via the probability of success, to reduce the informational rent. This corresponds to ψ 12. This effect is usual in adverse selection problems. However, the second term differs from the canonical adverse selection model because the agent is not risk neutral. Indeed, to benefit from the participation of the type θ, the principal must increase the rent, or equivalently, the expected payment of all higher efficient agents. Thus, the shadow cost of U is no more θ neutrality, but θ {w (u) + p(w ( u + u) w (u))}fdτ. fdτ = 1 F, as with risk The second modification arises because u is increased by the informational rent (see (19)). This influences the moral hazard marginal cost in (MH ). This is measured by 6 ( w (u) w (u) ) + p ( 1 p )( w (u) w (u) ) ψ 11. (20) The sign of (20) is not clear cut. Indeed, the first term is positive. Because w > 0 from the agent s risk aversion, the informational rent implies a higher bonus, w(u) w(u), compared to moral hazard only. So adverse selection leads to an increase in the marginal cost due to high powered incentives. But the sign of the second is ambiguous. It depends on the sign of w, or equivalently, of u u + 3u 2. So, it depends on the level of u, that is, on the weight of the agent s prudence. If the agent is not prudent, u < 0 then u u + 3u 2 > 0, adverse selection leads to an increase in the marginal cost due to the risk premium. The same effect arises if the agent is weakly prudent, i.e u > 0 but u u + 3u 2 remains positive. By contrast, if the agent is strongly prudent, i.e u > 0 but u u + 3u 2 becomes 6 Combine the right hand side of (MH ) and (19). 12

13 negative, there is a reduction due to adverse selection. To summarize, adverse selection has the following consequences on the moral hazard marginal cost: a. with a non prudent agent, (20) is positive. Risk aversion and prudence contribute to increase the moral hazard marginal cost in presence of adverse selection, b. with a weakly prudent agent, (20) is still positive. But prudence alleviates the increase in moral hazard marginal cost, c. with a strongly prudent agent, (20) is either positive or negative. So, c1. in the first case, the same effect as b. occurs, but alleviation is reinforced, c2. in the second case, adverse selection reduces the moral hazard marginal cost because prudence offsets risk aversion. Altogether, the right hand side of (GA) is the generalized agency marginal cost of an increase in p. These modifications imply two distortions in the probability of success, compared with its moral hazard level. To see this, notice that (GA) is equivalent to x + θ { w (u) + p ( w (u) w (u) )} fdτ ψ 12 (p, θ) f(θ) = ( w(u) w(u) ) + p ( 1 p )( w (u) w (u) ) ψ 11 (p, θ). (GA ) We know that x + θ {w (u)+p(w ( u+u) w (u))}fdτ f ψ 12 x, with equality holding at θ. So comparing (MH ) and (GA ), the first modification pushes to a reduction in p because the moral hazard marginal cost is increasing in p. 7 This reduction is standard in adverse selection problems and reflects the usual rent extraction efficiency trade off: the probability of success for a given efficiency is distorted downward its reference level (here its moral hazard level) in order to reduce the more efficient agent s informational rent. Moreover, the second modification is due to the increase in u via U. As shown above, this can increase (cases a., b. and c1.) or reduce (case c2.) the moral hazard marginal 7 By concavity of the moral hazard problem, the moral hazard marginal cost increases in p. See equation (34) in appendix

14 cost. In the former case, adverse selection adds a second reduction in p because it makes the insurance efficiency trade off worse. In the latter case, adverse selection leads to an opposite effect and tends to increase p. In this case, the informational rent combined with the strong level of prudence softens the trade off due to moral hazard. These distortions reflect the overall trade off between rent extraction, insurance and efficiency. But, we have just identified that the agency cost of adverse selection follow two channels. The first distortion implies that it is added to the agency cost of moral hazard. The second distortion shows that it can reinforce or mitigate the moral hazard trade off. It follows that the nature of distortions is not equivalent among [θ, θ]. Indeed, when θ is low, we have U 0 and θ {w (u)+p(w ( u+u) w (u))}fdτ f ψ So the distortion associated with inefficient types is rather due to the rent extraction efficiency trade off. The principal is more concerned about the agency cost of adverse selection. The reverse is true for high θ since U 0 and θ {w (u)+p(w ( u+u) w (u))}fdτ f ψ The distortion associated with efficient types rather occurs from the insurance efficiency trade off, because the principal is more interested by the agency cost of moral hazard. It follows that in presence of adverse selection, each moral hazard probability of success is distorted. In particular, even if the marginal cost of the informational rent is null for the highest type, because η(θ) = 0, there is a distortion at the top. This contradicts the well known result of adverse selection only. The difference in the nature of distortions wonders whether the assumption of complete sorting in Proposition 1 is relevant. The following proposition answers this question. Proposition 2. The optimal adverse selection moral hazard contract is separating if ( ( )) ( )( p 12 x w(u) w(u) p2 1 2p w (u) w (u) ) ( ) η η + p 2 + p 22 f f Proof. See appendix 9.4. Before going into technical details, a first comment can be made. > 0. (21) This proposition highlights that pooling in this generalized agency framework can have two sources. The first one comes from the common value nature of the model. So non responsiveness can occur. That is, u MH (θ) does not satisfy the incentive constraint (IC 2). The two first terms in (21) reflect this phenomenon. But, because the u depends on U when there is adverse 14

15 selection, the conflict between the principal s preference and the monotonicity condition (IC 2) is somewhat modified compared with moral hazard only. The second source is due to the possible lack of monotonicity of the marginal cost of the informational rent. This concerns the two last terms of (21). This condition is well known in adverse selection models only. But, because the agent is risk averse, this condition does not need the standard condition of a monotone hazard rate, 1 F f non increasing in θ. Let us examine (21) in more details. First, since p 2 = ψ 12 ψ 11 > 0 and w > 0, a probability lesser than 1 2 contributes to the non responsiveness of the model. The reverse is true if p 1 2. Second, since x ( w(u) w(u) ) > 0, η > 0 and η < 0, (21) is more easily satisfied that p 12 > 0 and p 22 < 0. These conditions structure even more the function ψ than does Property 1 and Assumption 1. Indeed, we get p 12 = ψ 111ψ 12 ψ 112 ψ 11, (22) ψ11 3 p 22 = ψ 112ψ 12 ψ 212 ψ 11 ψ 2 11 ψ 12 ψ 111 ψ 12 ψ 112 ψ 11. (23) ψ 11 ψ 2 11 So a separating contract is conditioned by a combination of many second and third partial derivatives of the indirect disutility ψ. Thus, on can reasonably have doubts about the existence of a contract that strictly satisfies (IC 2) for all types. In this case, the monotonicity constraint is binding on some intervals. Well known dynamic optimization techniques in adverse selections models can be used to determine precisely these intervals. We state the following proposition in the case where there is pooling on a single interior interval. Proposition 3. Consider a single interior interval [θ 0, θ 1 ] where there is pooling. The generalized agency contract entails U (θ) = θ ψ 2(p( u (τ), τ), τ)dτ, if θ [θ, θ 0 ] [θ 1, θ], u (θ) = u (θ) 15

16 if θ [θ 0, θ 1 ], u (θ) = u k with 1 { (w( u x k + u (θ)) w(u (θ)) ) θ 0 p( u k, θ) ( 1 p( u k, θ) ) ( w ( u k + u (θ)) w (u k (θ)) ) ψ 11 (p( u k, θ), θ) + θ { w (u (τ)) + p( u k, θ) ( w ( u k + u (τ)) w (u (τ)) )} f(τ)dτ f(θ) } ψ 12 (p( u k, θ), θ) f(θ)dθ = 0 (24) and u (θ) = U (θ) + ψ(p( u (θ), θ), θ) p( u (θ), θ) u (θ). (25) Proof. Straightforward following Guesnerie and Laffont (1984). This proposition shows that when pooling arises, the optimal contract consists in verifying (GA) on average, and no more pointwise. This result is well known in adverse selection models only. In the light of Propositions 2 and 3, we observe that the presence of pooling in this contract is not so much a technical difficulty, but more a difficulty coming from the overall trade off between rent extraction, insurance and efficiency. This requires to manage different kind of incentives in a common value generalized agency model with risk averse agent. 7 An example: the DARA case Let us study the distortions in the simple case where the agent has a decreasing absolute risk aversion. Let x = u(y) = 2y, so that y = w(x) = x2 2, w (x) = x and w (x) = 1. Complete information. Since, U = u = 0 and u = ψ, (FB) becomes Moral hazard. We have x = ψ(ρ F B (θ), θ)ψ 1 (ρ F B (θ), θ). w( u + u) w(u) = ψ ψψ 1 pψ 2 1, w ( u + u) w (u) = ψ 1. 16

17 So (MH ) becomes x =ψ(p( u MH (θ), θ), θ)ψ 1 (p( u MH (θ), θ), θ) p( u MH (θ), θ)ψ 1 (p( u MH (θ), θ), θ) 2 + ψ 1(p( u MH (θ), θ), θ) 2 + p( u MH (θ), θ) ( 1 p( u MH (θ), θ) ) ψ 1 (p( u MH (θ), θ), θ)ψ 11 (p( u MH (θ), θ), θ). The distortion due to the insurance efficiency trade off is thus 1 2p ψ1 2 + p ( 1 p ) ψ 1 ψ So, if p < 1, the moral hazard problem implies two distortions reducing the efficient the 2 probability of success. Otherwise, the first term pushes to an upward distortion while the second pushes to a downward one. Generalized agency. We get So (AS-MH ) is such that w( u + u) w(u) = ψ ψψ 1 pψ ψ 1 U, w (u) + p ( w ( u + u) w (u) ) = U + ψ. 2 x =ψ(p( u (θ), θ), θ)ψ 1 (p( u (θ), θ), θ) p( u (θ), θ)ψ 1 (p( u (θ), θ), θ) 2 + ψ 1(p( u (θ), θ), θ) 2 + p( u (θ), θ) ( 1 p( u (θ), θ) ) ψ 1 (p( u (θ), θ), θ)ψ 11 (p( u (θ), θ), θ) + U (θ)ψ 1 (p( u (θ), θ), θ) { θ U (τ) + ψ(p( u (τ), τ), τ) } f(τ)dτ ψ 12 (p( u (θ), θ), θ). f(θ) In this example, adverse selection influences the rent efficiency trade off through the distortion θ and the insurance efficiency trade off through { } U + ψ fdτ ψ 12, f Uψ

18 Adverse selection leads to two distortions reducing the moral hazard probability of success. Notice that the agent s prudence does not influence the trade off between rent extraction and insurance because we have w = 0. 8 Conclusion To be done... 9 Appendix 9.1 Proof of Lemma 2 We denote µ the Kuhn and Tucker multiplier associated to (PC ). The Lagrangian is L = x + ρ x w(u + ψ ρ u) ρ(w(u + ψ + (1 ρ) u) w(u + ψ ρ u))) + µu. (26) Necessary conditions are L ρ = x w (u)(ψ 1 u) (w( u + u) w(u)) ρ(w ( u + u)(ψ 1 u) w (u)(ψ 1 u)) = 0 (27) L u = w (u)ρ ρ(w ( u + u)(1 ρ) + w (u)ρ) = 0 (28) L U = w (u) ρ(w ( u + u) w (u)) + µ = 0 (29) µ 0, µu = 0. (30) From (28), we have w ( u + u) = w (u) u = 0. Then, plugging this result in (29), we have µ = w (u) > 0 so, from (30), U = 0. After simplifications in (27), the probability ρ F B (θ) is given by (FB). 18

19 9.2 Proof of Lemma 3 The Lagrangian is similar as (26), L = x + p x w(u + ψ p u) p(w(u + ψ + (1 p) u) w(u + ψ p u)) + µu. Necessary conditions Given (7), necessary conditions are L u = p 1 x + w (u)p p 1 (w( u + u) w(u)) p(w ( u + u)(1 p) + w (u)p) = 0 (31) L U = w (u) p(w ( u + u) w (u)) + µ = 0 (32) µ 0, µu = 0. (33) From (32), we have µ > 0, since w > 0 by risk aversion and u > 0 by (7). Then U = 0 using (33). Moreover, collecting terms in (31), we get (MH ), since p 1 = 1 ψ 11. Sufficient conditions Since the constraint is linear in U, necessary conditions are sufficient if V in (4) is concave in ( u, U). We need to verify 2 V u =p ( ( )) 2 11 x w( u + u) w(u) p 1 ( w ( u + u)(1 p) + w (u)p ) p 1 (1 2p) ( w ( u + u) w (u) ) p(1 p) ( w ( u + u)(1 p) + w (u)p ) < 0. (34) Notice that p 11 = ψ from Assumption 1, and recall that, w > 0, w > 0 from ψ11 3 risk aversion, and p 1 = 1 ψ 11 > 0 from Property 1. Combining these elements with (MH ), (34) is verified, in particular, as long as p is not too higher than 1, if 1 2p < 0. 2 We need also which is negative. Finally, since ( if 2 V u 2 2 V U 2 2 V U 2 = pw (u) (1 p)w (u), (35) 2 V = p u U 1(w (u) w (u)) p(1 p)(w (u) w (u)), concavity is ensured ) 2 2 V u U 0. 19

20 9.3 Sufficient conditions for Proposition 1 Sufficient conditions require that H be concave in ( u, U). Let us compute 2 H u 2. Since p 1 = 1 ψ 11 and p 2 = ψ 12 ψ 11, (14) is equivalent to So we have H u = (p 1( x (w(u) w(u))) p(1 p)(w (u) w (u)))f + ηp 2 = 0. (36) 2 H u 2 = (p 11( x (w(u) w(u))) p 1 (w (u)(1 p) + w (u)p) p 1 (1 2p)(w (u) w (u)) p(1 p)(w (u)(1 p) + w (u)p)) f + ηp 12. (37) Following the second part of the proof of Lemma 3, and since η 0, a sufficient condition for 2 H u 2 is p 12 > 0 (see also equation (22)). Moreover, we have Finally, we get 2 H U 2 = pw (u) (1 p)w (u) < 0. (38) 2 H u U = p 1(w (u) w (u)) p(1 p)(w (u) w (u)). (39) and concavity is ensured if 2 H 2 H u 2 U 2 ( ) 2 2 H u U Proof of Proposition 2 Using the concavity of H in u, differentiating (36) and using (IC 1) and (11), we get (21). References [1] Guesnerie, R., Laffont J.-J.: A complete solution to a class of principal-agent problems with an application to the control of a self-managed firm. Journal of Public Economics 25, (1984) 20

21 [2] Laffont, J.-J., Martimort, D.: The Theory of Incentives: The Principal-Agent Model. Princeton University Press, Princeton and Oxford (2002) [3] Myerson, R.: Optimal coordination mechanisms in generalized principal-agent problems. Journal of Mathematical Economics 10, (1982) [4] Ollier, S., Thomas, L.: Ex post participation constraint in a principal-agent model with adverse selection and moral hazard. Journal of Economic Theory 148, (2013) [5] Seierstad, A., Sydsæter, K.: Optimal Control Theory with Economic Applications. North-Holland, Amsterdam (1987) 21