FUNDAÇÃO GETULIO VARGAS ESCOLA DE PÓS-GRADUAÇÃO EM ECONOMIA

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1 FUNDAÇÃO GETULIO VARGAS ESCOLA DE PÓS-GRADUAÇÃO EM ECONOMIA Henrique Brasiliense de Castro Pires Limited Liability and Non-responsiveness in Moral Hazard and Adverse Selection Problems Rio de Janeiro 15 de Outubro de 2015

2 Henrique Brasiliense de Castro Pires Limited Liability and Non-responsiveness in Moral Hazard and Adverse Selection Problems Dissertação submetida a Escola de Pós- Graduação em Economia como requisito parcial para a obtenção do grau de Mestre em Economia. Orientador: Humberto Luiz Ataíde Moreira Rio de Janeiro 15 de Outubro de 2015

3 Ficha catalográfica elaborada pela Biblioteca Mario Henrique Simonsen/FGV Pires, Henrique Brasiliense de Castro Limited liability and non-responsiveness in moral hazard and adverse selection problems / Henrique Brasiliense de Castro Pires f. Dissertação (mestrado) - Fundação Getulio Vargas, Escola de Pós-Graduação em Economia. Orientador: Humberto Luiz Ataíde Moreira. Inclui bibliografia. 1. Responsabilidade limitada. 2. Risco moral. 3. Seleção adversa (Seguro). 4. Informação assimétrica. I. Moreira, Humberto Ataíde. II. Fundação Getulio Vargas. Escola de Pós- Graduação em Economia. III. Título. CDD 330

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5 Abstract This work analyses the optimal menu of contracts offered by a risk neutral principal to a risk averse agent under moral hazard, adverse selection and limited liability. There are two output levels, whose probability of occurrence are given by agent s private information choice of effort. The agent s cost of effort is also private information. First, we show that without assumptions on the cost function, it is not possible to guarantee that the optimal contract menu is simple, when the agent is strictly risk averse. Then, we provide sufficient conditions over the cost function under which it is optimal to offer a single contract, independently of agent s risk aversion. Our full-pooling cases are caused by non-responsiveness, which is induced by the high cost of enforcing higher effort levels. Also, we show that limited liability generates non-responsiveness. KEYWORDS: Limited liability, non-responsiveness, pooling, moral hazard, adverse selection

6 Contents 1 Introduction 6 2 Related literature 8 3 The general model 11 4 The model with additional assumptions 15 5 Non-responsiveness 17 6 Examples 23 7 Limited liability 24 8 Full-pooling problems 26 9 Conclusion Appendix 32 6

7 List of Figures 1 Cost function Optimal solution to Pure Moral Hazard Problem

8 1 Introduction In the real-world we commonly observe contract menus much simpler than theory predicts. Standard models of adverse selection with a continuum of types usually prescribe an infinite number of contracts. In order to establish a better comprehension of what might explain this distinction, we focus on cases in which the optimal menu of contracts is simple. In this work we provide sufficient conditions for contract pooling in an environment with moral hazard, adverse selection and limited liability. We analyse the case with only two possible outputs, a risk neutral principal and a risk averse agent, who has private information over his type and effort choice. In our framework, the agent s type affects his cost function in an ordered way, in which higher types face lower costs. The goal of this model is to achieve a better comprehension of labor contracts. Usually, firms face asymmetric information with respect to both effort and skill of its worker. Additionally, minimum wages and requirements of minimum standards of working conditions act like limited liability constraints, by not allowing firms to pay bellow a certain level in any state of nature. Therefore, a model with simultaneous adverse selection and moral hazard under limited liability restrictions seems to be appropriated to describe labor relations. Throughout this paper, we are going to focus on the economic interpretations of this case. In spite of that, the same framework may be used to other situations, such as insurance, with similar interpretations. There is an extensive literature on pooling in adverse selection models. There are some reasons identified for it: the lack of single-crossing condition 1, the presence of countervailing incentives 2 and non-responsiveness 3. The force that generates pooling in our model is the last one, i.e., the incentive compatibility constraints for type report opposes what would be optimal to the principal in terms of generating incentives for effort. There are two states of nature, which might be used by the principal to create effort incentives to the agent. In order to do that, the principal provides a fixed 1 See Netzer and Scheuer (2010) and Araujo and Moreira (2010). 2 See Lewis and Sappington (1989) and Maggi and Rodríguez-Clare (1995). 3 See Guesnerie and Laffont (1984) and Morand and Thomas (2003). 6

9 payment and a bonus in case of high output. When the agent is risk neutral, as in Gottlieb and Moreira (2015), both - agent and principal - are only worried about expected return. Therefore, all contracts of a given menu must have the same expected value. In this case, the multiplicity of contracts generates gaming opportunities to the agent, while the principal is able to avoid it without reducing the expected output, by offering only the most powerful contract of the menu. In our case, with risk aversion, if the principal does that, she improves the incentives for effort, but worsens risk sharing. We also show that the result of contract simplicity for any cost function in the two possible outputs version of Gottlieb and Moreira (2015) s model does not hold for risk averse agents. Therefore, it is necessary to add assumptions on the cost function in order to guarantee optimality of a finite contract menu. In order to enforce effort, the principal necessarily needs to leave some risk to the agent, i.e., to generate more effort she may increase the bonus and/or reduce the fixed payment. When the principal faces limited liability constraints she cannot reduce the fixed payment below 0. Therefore, to enhance incentive for higher effort, she can only increase bonus, and the cost of sustaining high levels of effort becomes bigger. Due to this effect, the principal might prefer giving lower bonus to higher types instead of trying to sustain more elevated effort levels. In other words, the limited liability increases the cost of enhancing effort and - depending on the cost function shape - generates non-responsiveness, which causes pooling. In summary, the principal faces a trade-off between distorting effort choices and leaving high informational rents. Screening may cause big distortions on effort, while pooling leaves more informational rents to agents. We find conditions under which the distortions caused by screening surpasses the rents leaved by pooling. This paper unfolds as follows. Section 2 presents and discusses the related literature. Section 3 presents the most general version of the model and proves that simplicity of the optimal contract menu cannot be guaranteed with risk averse agents. Section 4 presents a more particular version of the model, supplies some auxiliary lemmas and a result about pooling at the top. Section 5 provides sufficient conditions for full pooling. In Section 6, we present two examples in which those conditions are satisfied. Section 7 analyses the role of limited liability in our results. In Section 8, we examine the problem of optimal pooling and Section 9 concludes. 7

10 2 Related literature Our paper tries to identify causes for the contract simplicity observed in reality. Theory suggests that the specific form of the contract menu has a huge impact on the principal s profit, i.e., incentives matter. Therefore, it has been a puzzle why we do not observe complex contract menus as predicted. Many possible explanations has emerged on literature, such as countervailing incentives - as Lewis and Sappington (1989) and Maggi and Rodríguez-Clare (1995) - or Spence-Mirlees condition violations, for example, Netzer and Scheuer (2010) and Araujo and Moreira (2010). None of these are the case of our model, which is more related to the literature that attributes pooling to non-responsiveness, such as Guesnerie and Laffont (1984). On the setting of Lewis and Sappington (1989), the agent has an incentive to understate his private information for some of its realizations, and to overstate it for others - which is called countervailing incentives. In their model, when this happens, the optimal contract menu is usually pooling. Maggi and Rodríguez-Clare (1995) clarify the relationship between pooling and countervailing incentives. Their main result is that the optimal contract menu will be pooling if the type dependent outside option of agents is concave, while it will be fully separating if it is convex. In our work, the outside option does not depend on agent s type and there never exists incentives to overstate their private information. Therefore, the reason for pooling in our model is completely different from theirs. An important argument of our paper is non-responsiveness as a force towards full-pooling. Guesnerie and Laffont (1984) were the first to point this possibility. The conflicting effects between the optimal provision of incentives for effort and incentive compatibility might result in simpler contract menus. Most of the models that analyse this issue - for example Caillaud et al. (1988)- share the features that if full information quantity is implementable, the asymmetric information is separating, otherwise it is pooling. Morand and Thomas (2003) clarify this discussion by showing that those results depend fundamentally on the assumption that the principal s marginal benefit from the contract with full information is not distorted by informational rents. The present paper is also related to the literature that analyses the impact of lim- 8

11 ited liability constraints, starting by the classic model of Innes (1990). Our work is even closer to those that include limited liability constraints on simultaneous moral hazard and adverse selection problems, such as Gottlieb and Moreira (2015). Innes (1990) and Poblete and Spulber (2012) analyse single task moral hazard with bilateral risk aversion. They show that the optimal contract takes the form of debt if the distribution of output satisfies the monotonicity of likelihood ratio condition. Gottlieb and Moreira (2015) introduce adverse selection and also allow for multidimensional effort choice. Their results show that offering a large menu of contracts creates gaming opportunities to agents and, therefore, reduces the principal s profits. If any incentive compatible menu is taken, it is possible to improve the principal s payoff by leaving only the most powerful contract. The agent s participation constraints will remain satisfied due to limited liability and the unique effect is an increase in effort, which augments profits. In their paper, they also solve the case with two output levels, which is very similar to our model. The crucial difference is that in our case, we have risk aversion which may jeopardize their results. By offering only the most powerful contract, the principal increases the incentives for effort, but she worsens risk sharing. Thus, the effect is ambiguous. This paper is also related to the sharecropping literature 4. There are several different explanations for the existence of sharecropping contracts and many of them are related to some features in our model. First, part of this literature analyses the role of risk sharing, such as Newbery (1977) and Newbery and Stiglitz (1979). Second, a group of papers also focus on the incentive provision, i.e., the landlord has a dual role as provider of insurance and land. Therefore, in the land owners perspective, the optimal contract comes from a trade-off between insurance and incentive provision. Stiglitz (1974) and, in a more general framework, Ross (1973) and Mirrlees (1974) are examples of that. Third, there are some papers that base their explanations on wealth constraints, such as Hurcwicz and Shapiro (1978). In this kind of models, a fixed rent contract would provide optimally incentives for effort if there was no limited liability. Although, as there are wealth constraints, the agent might not be able to pay his rent in bad states of nature, which allows sharecropping contracts to be optimal. Shetty (1988) also analyses heterogeneity in the wealth of tenants. Richer 4 See Singh (1987) for a survey 9

12 tenants have weaker liability restrictions, which implies that they are preferred by landlords. In this case, sharecropping is offered to poorer tenants and fixed rent contracts are designed to richer ones. Still in the sharecropping literature, Laffont and Matoussi (1995) emphasize the dual role of moral hazard in the provision of effort and financial constraints. In their model, they assume bilateral risk neutrality and study the relationship of moral hazard and limited liability. Despite many similarities of this literature with our model, none of these papers analyse simultaneously the interaction between risk aversion, moral hazard, adverse selection and limited liability. Another paper with a model similar to ours is Jullien, Salanié and Salanié (2007). They also have simultaneous moral hazard and adverse selection, but in their framework the private information of the agent is her risk aversion and not the cost of effort. They provide general properties of optimal contracts and also have some cases of pooling due to non-responsiveness. Despite those similarities, their model has some other crucial differences from ours, such as a finite number of types, while we use a continuum, and the absence of limited liability constraints. They also restrict themselves to CARA utility functions, while our results hold for any concave, monotonic and twice differentiable utility function. As stated before, the focus of this paper is describing and understanding labor contracts. Naturally, distinct activities and production sectors present different labor relations. Therefore, we observe a wide range of methods of remuneration, such as standard-rate pay or incentive pay. Our model is more related to sectors in which incentive pay are the most prevalent. There are some papers- such as Brown (1990) and MacLeod and Parent (1999), (2013)- that try to identify which characteristics are important for the firms choice of method of remuneration and in which sectors they are most common. They found that incentive pay are less likely in jobs with a variety of duties and more in those which the quality of work is easily monitored. Additionally, the existence of unions and the greater importance of teamwork reduce the prevalence of incentive pay. An example of sector that presents all the favorable characteristics to incentive pay is Sales. Also, it presents a high rate of workers under some kind of incentive remuneration, such as bonuses or commissions. 10

13 3 The general model We analyse a problem in which a risk neutral principal - referred as she - offers a contract to a risk averse agent - referred as he - in face of a binary output: high (x H ) or low (x L ), and restricted to weakly positive payments 5, i.e., a limited liability restriction. Also, we refer to x := x H x L > 0 as the incremental output. The agent chooses how much effort to exert, which changes the probability of success. The effort is unobservable to the principal and has a private cost to the agent. Additionally, the agent has a private information type that affects his cost of exerting effort. Therefore, the principal faces a moral hazard and an adverse selection problem simultaneously. Agent s preferences are represented by a strictly increasing, concave, and twice differentiable utility function u : R R. We assume that the agent s type belongs to a compact interval θ [θ; θ] R with a probability density function f(θ) > 0, θ [θ; θ]. The effort a is chosen in [0, 1] and determines the probability of high output. The cost of effort of each type θ is represented by a function C θ : [0, 1] R, for which min a {C θ (a)} 0 [θ, θ]. By the revelation principle, we can restrict our attention to direct mechanisms. A direct mechanism is a triple of B([θ, θ])-measurable functions (w, b, a) : [θ, θ] R 2 [0, 1], consisting of fixed payments w, bonuses b, and effort recommendations a. ( ) A type θ agent s contract is defined as a pair of payments w(θ), b(θ). An agent who reports type θ agrees to exert effort a(θ), receives w(θ) in case of failure and w(θ)+b(θ) in case of success. Given a mechanism, an agent of type θ gets the payoff: ( ) ( ) ( ) ( ) U(θ) := a(θ)u w(θ) + b(θ) + 1 a(θ) u w(θ) C θ a(θ). (1) The mechanism must satisfy incentive compatibility (IC), individual rationality (IR) and limited liability (LL) constraints. ( ) ( ) U(θ) âu w(ˆθ) + b(ˆθ) + (1 â)u w(ˆθ) C θ (â), θ, ˆθ, â. (IC) 5 As we are going to assume a strictly increasing utility function, this limited liability constraint over payments is equivalent to one over utility of payments in each state of nature. 11

14 U(θ) 0, θ. (IR) w(θ) 0, w(θ) + b(θ) 0 θ. (LL) The profit that the principal receives from type θ is given by [ ] π(θ) := x L w(θ) + a(θ) x b(θ). (2) An optimal mechanism maximizes the principal s expected payoff subjected to (IR),(IC) and (LL) θ [ ] max (w(θ),b(θ),a(θ)) θ x L + a(θ)[ x b(θ)] f(θ)dθ s.t. (IR), (IC) and (LL). (3) We assume that the utility of 0 is equal to 0 6 and that min a {C θ (a)} 0 θ [θ, θ]. Also we assume that the cost of effort is increasing on the type. This framework would be a special case of Moreira and Gottlieb (2015), except by the risk aversion of the agent. In their paper, there is bilateral risk neutrality. Their first result is the optimality of full-pooling, with only two possible output levels, for all cost functions that satisfy the condition above. We are going to show that this result does no longer holds for any strictly risk averse agent. In order to do that, first, we prove some auxiliary Lemmas. The first result is the redundancy of participation constraints when we have limited liability, which guarantees that there is no exclusion of any types in our problem. Lemma 1. Let (w,b,a) be a mechanism that satisfies incentive compatibility and limited liability. Then, it satisfies individual rationality. Proof. By the choice of effort: U(θ) u(w(θ) + b(θ))a + u(w(θ))(1 a) C θ (a), [0, 1]. By (LL): w(θ) 0 and w(θ) + b(θ) 0, which implies that U(θ) 0. a 6 This assumption would not be necessary and all results would persist if we had defined the limited liability constraint over utility levels in each state. 12

15 Now we define the agent s choice of effort. Given a contract (w, b) R 2, the type θ agent chooses the effort level given by: max a [0,1] au(w + b) + (1 a)u(w) C θ (a). (4) We define a : [θ, θ]xr 2 [0, 1] as the solution of problem (4) for a given contract ( ) w, b. Thus, a is a function of the triple (θ, w, b). To simplify the notation, we will suppress those terms whenever we believe it is necessary. Second, we show that the optimal effort level, a (θ, w, b), is decreasing in the fixed payment and increasing in the bonus. Lemma 2. The optimal effort level choice, a (θ, w, b), is decreasing in w and increasing in b. Proof. The marginal gain from effort is given by u(w+b) u(w). As u is strictly increasing and concave, this difference is increasing in b and decreasing in w. Therefore, the agent s effort choice level will be increasing in b and decreasing in w. Now we prove that there is no loss of generality in assuming that b(θ) [0, x]. Lemma 3. Suppose that min{c θ (a)} 0 and is attained at a = 0. Let (w,b) be the a optimal contract menu. Then, with no loss of generality, b(θ) [0, x] θ [θ, θ]. Proof. In the Appendix. Lemma 3 is intuitive. It states that it is never optimal to the principal to offer a contract with b(θ) > x, because it would induce the agent to overstate the benefits of his efforts and to exert an inefficiently high level of effort. Neither it is optimal to offer negative bonuses, because the agent would simply choose 0 effort. Finally, we show that without further assumptions on the cost function we cannot guarantee that full-pooling is going to be optimal with risk averse agents. This result is stated in the following proposition. Proposition 1. For any strictly concave utility function with u(0) = 0, there exists a cost function with min a {C θ (a)} 0 [θ, θ] for which the optimal contract menu has an infinite cardinality. Proof. In the Appendix. 13

16 The idea of the proof is to fix an arbitrary strictly concave utility function and construct a cost function for which the optimal menu has an infinite cardinality. We take a cost function under which even with zero bonus, lower types exert some positive amount of effort and that they do not respond to marginal increases on the bonus, while higher types do. Thus, the moral hazard problem remains for higher types and disappears for lower types. As only adverse selection remains for less efficient types, the principal s profit increases with the addition of an extra contract to any finite menu. For our proof, we use a cost function under which lower types do not respond to incentives near the effort level that minimizes costs, while higher types do. In other words, we use a cost function which, for lower types, presents a high marginal cost at the level of effort that minimizes costs. While, for higher types it presents zero marginal cost at the cost minimizing effort level. Figure 1 displays a function that satisfies these characteristics. For low levels of effort the cost is zero, and it remains zero longer for higher types. For high levels of effort lower types face higher marginal costs, while higher types face lower marginal costs. The marginal cost for lower types is sufficiently big to overcome the marginal gain of effort with the highest bonus the principal would be willing to pay, x. In other words, the marginal cost of effort for lower types is always bigger than the marginal benefit that comes from the bonus. In Figure 1, the lower types are represented by the green and the blue solid lines, while the higher types are represented by the red and the light blue solid lines. The dashed lines represent the maximum marginal benefit to the agent that comes from effort, i.e., u( x). Note that for lower types this marginal benefit is smaller than the marginal cost of effort, while for higher types the opposite occurs. 14

17 Figure 1: Cost function Proposition 1 shows that the result of Moreira and Gottlieb(2015) of contract simplicity when there are only two possible output levels only holds for a risk neutral agent, even if. Despite their big generality over the cost function, even if we restrict ourselves to convex cost functions their result would not hold. Additionally, the assumption of risk neutrality is very restrictive. Therefore, we search for conditions under which full-pooling would be optimal even in the presence of risk aversion. In order to relax the assumption over the utility function, we had to add more assumptions over the cost function. 4 The model with additional assumptions We have shown that without further assumptions on the cost function, it is not possible to assure a finite contract menu, which is a very weak definition of contract simplicity. From now on, we are going to search for sufficient conditions that guarantee the optimality of full-pooling, which may be considered the strongest definition of simplicity. In other words, we have shown that even under a weak definition, we cannot assure simplicity at the general model, and now we are going to provide sufficient conditions for simplicity even under the most restrictive definition. We assume that the cost function is three times differentiable with respect to a and θ. Also, we assume that costs are strictly increasing and convex with respect to effort, and that marginal costs are decreasing on agent s type. This last assumption 15

18 provides an ordering, in which the highest types are also the most efficient ones. Additionally, we normalize the cost of a = 0 to 0 and u(0) = 0. Formally, our assumptions, including those made on the previous section, are: Assumption 1. Cθ (a) θ Assumption 2. 2 C θ (a) a 2 < 0; Cθ (a) a > 0. > 0; 2 C θ (a) a θ 0. Assumption 3. C θ (0) = 0. Assumption 4. u (.) > 0; u (.) 0 and u(0) = 0. Assumption 5. u( x) > Cθ (0) a. Assumptions 1 and 2 are standard assumptions and refer to convexity of the cost function and to the single crossing condition. Assumption 3 guarantees the existence of a costless level of effort. u (.) = 0 is the case in which the agent is also risk neutral and it is analysed by Gottlieb and Moreira(2015). Assumption 5 is a technical assumption in order to avoid corner solutions on the agent s effort choice problem. Now we are going to prove that if a menu of contracts does not give maximum profit at θ, i.e., π(θ) does not reach its maximum at θ - then there is another contract menu with pooling at the top that strictly dominates this one. This result, implies that in the optimal contract menu, the most efficient type gives the maximum profit to the principal. To prove this result we need some auxiliary lemmas. Lemma 4. Given a contract (w, b) with b x, it is more profitable if chosen by higher types. Proof. In the Appendix. Lemma 4 is also intuitive. Given a contract, more efficient types exert more effort, which increases principal s profits if the bonus is lower than the extra output generated in case of success. Lemma 5. If a type θ prefers a more powerful contract (w, b) to another less powerful ( w, b)( i.e., w < w and b > b) then all types θ > θ will also prefer the more powerful one. Proof. In the Appendix. 16

19 Lemma 5 states that more efficient types are willing to accept more powerful contracts than less efficient ones. Therefore, it establishes that b(θ) being weakly increasing and w(θ) decreasing are necessary conditions for incentive compatibility. Proposition 2. For all contract menu that satisfies (IR),(IC) and (LL) in which the maximum profit is not reached at θ, then there exists an alternative menu with pooling at the top that gives a higher profit. Proof. Take an arbitrary menu of contracts that satisfies (IR),(IC) and (LL). Define the type in which the maximum π(θ) is reached as θ. Eliminate all contracts chosen by ( ) types θ > θ. By Lemma 5, all types θ > θ are going to choose contract w(θ ), b(θ ). By Lemma 4, principal s profit is going to increase. Proposition 2 comes from the elevated informational rents that must be left to agents with higher types in order to discriminate them. If this screening cost becomes high enough, the principal prefers to create pooling at the top. On one hand if the gain in efficiency with higher types surpasses the screening cost, then the profit is greater at those types. On the other hand if the screening cost is elevated, the principal will generate pooling at the top, in a way which the highest profit is going to be reached at the most efficient type. The proof of Proposition 2 follows the same approach used by Gottlieb and Moreira (2015). We take a menu that is candidate to optimality and construct an alternative menu which generates a higher profit point by point. In the bilateral risk neutral case, this approach is enough to guarantee optimality of full-pooling. With a risk averse agent, we can only assure that the profit function must be strictly increasing with respect to agent s type. Therefore, in order to find sufficient conditions for contract simplicity we must use stronger optimality conditions. 5 Non-responsiveness First, we show that limited liability constraints can be represented just by w(θ) 0, θ [θ, θ]. 17

20 Lemma 6. Subject to (LL), under (IC) or not, it is always optimal for the principal to offer contracts with b(θ) 0, θ [θ, θ]. Proof. In the appendix. Define (LL ) as w(θ) 0 θ. Then, by Lemma 6, there is no loss of generality in substituting (LL) by (LL ). Now we state the problem in which the principal knows the agent s type, but the effort still is unobservable. We call it pure moral hazard problem. θ max (w(.),b(.)) θ s.t. w(θ) 0. [x L w(θ) + ( x b(θ))a (θ, w(θ), b(θ))]f(θ)dθ (5) Remind that we defined a as the solution of problem (4) and that it is a function of (θ, w, b). Also, note that, due to Lemma 1, there is no loss of generality in suppressing (IR). If problem (4) has an interior solution, the function a is characterized by the firstorder condition: u(w + b) u(w) = Cθ (a ). (6) a As we have a convex cost function, the first-order condition above will be not only necessary but sufficient. We do not have to worry about corner solutions, because even if the agent chooses a = 1, the principal would choose (w, b) that assures that the first-order condition holds with equality, i.e., the Lagrangian multiplier of the restriction a 0 must be equal to 0. Otherwise, the principal could reduce the bonus and sustain the same level of effort. Additionally, Assumption 5 guarantees that, on the pure moral hazard problem, for every type, it is profitable to the principal to generate positive effort levels. By solving (5), we get w(θ) = 0, for all θ. Define b P M : [θ, θ] R + as solution of the pure moral hazard problem and let γ be the Lagrangian multiplier of the limited liability constraint. 18

21 The first-order conditions are: 1 + a ( x b(θ)) + γ = 0, (7) w ( x b(θ)) a b a = 0. (8) which implies γ > 0, w(θ) = 0 θ [θ, θ]. Lemma 5 implies that db(θ) dθ 0 and dw(θ) dθ 0 are necessary conditions for incentive compatibility. Thus, consider the following relaxed problem: θ max (w(.),b(.)) θ s.t. [x L w(θ) + ( x b(θ))a (θ, w(θ), b(θ))]f(θ)dθ db(θ) dθ µ(θ) 0 = µ(θ) (9) w(θ) 0. Note that differently from the usual way of solving optimal contract problems, we do not only relax the monotonicity condition, but also the envelope one. Instead of using the incentive compatibility constraints, we solve a relaxed problem that only uses a necessary condition for incentive compatibility. In the Appendix, we show that if there is a solution to problem (9), it is equivalent to the solution of the problem below with w(θ) = 0, θ [θ, θ]. max b(.) s.t. θ θ [x L + ( x b(θ))a (θ, 0, b(θ))]f(θ)dθ db(θ) dθ µ(θ) 0. = µ(θ) (10) 19

22 Define the Hamiltonian 7 : H[θ, b, µ, λ] = [x L + ( x b)a ]f(θ) + λµ. (11) By Pontryagin s maximum principle, the necessary conditions for an optimum ( b(θ), µ(θ), λ(θ)) are given by 1. H[θ, b, µ, λ] H[θ, b, µ, λ], (b, µ); 2. Except at points of discontinuity of b(θ), we have d λ(θ) dθ ( ) ( = [( x b(θ)) a θ, 0, b(θ) a θ, 0, b(θ)) ] f(θ); b 3. Transversality conditions: λ(θ) = λ(θ) = 0 are satisfied. In the Appendix, we show that H[θ, b, µ, λ] is a concave function of b and µ. As H[θ, b, µ, λ] is linear in µ, it is sufficient to check the concavity on b. Then, by Arrow s theorem, 8 we get sufficiency. Integrating the equation from item 2 of the necessary first-order condition we obtain θ λ(θ) = θ θ 0 = λ(θ) = λ(θ) = θ ) ) [( x b( θ)) ( θ, a 0, b( θ) a ( θ, ] 0, b( θ) f( θ)d θ, b ( ) ( [( x b(θ)) a θ, 0, b(θ) a θ, 0, b(θ)) ] f(θ)dθ. (12) b The condition H[θ, b, µ, λ] H[θ, b, µ, λ] requires that µ maximizes H[θ, b, µ, λ] subject to µ 0. This requirement implies that λ 0, or Whenever θ θ [( x ) ) a b( θ)) ( θ, 0, b( θ) a ( θ, ] 0, b( θ) f( θ)d θ 0. (13) b λ(θ) < 0 we must then have µ(θ) = d b(θ) dθ = 0. (14) 7 We will limit our study to bonuses profile functions, b : [θ, θ] R + which are piecewise continuously differentiable of class C 1. 8 For reference see Seierstad and Sydsaeter (1987), page

23 Thus, we get the following complementary slackness condition: d b(θ) dθ θ θ [( x ) ) a b( θ)) ( θ, 0, b( θ) a ( θ, ] 0, b( θ) f( θ)d θ = 0, θ [θ, θ]. (15) b Remind that we defined b P M as the solution of the pure moral hazard problem. Thus, if dbp M (θ) dθ d b(θ) dθ = 0. < 0, by the slackness condition we have that λ(θ) < 0, which implies Now, we define non-responsiveness. Definition 1 (Non-responsiveness). The problem presents non-responsiveness if the solution of the pure moral hazard problem is non-increasing, i.e., dbp M (θ) dθ 0 θ [θ, θ]. If b P M is decreasing for all θ, i.e., if the problem presents non-responsiveness, then the solution to the relaxed problem will be full-pooling, which satisfies the incentive compatibility constraints of problem (3). Hence, full-pooling will be optimal to the principal. The proposition below states this result. Proposition 3. If the solution of the pure moral hazard problem presents a decreasing bonus function, then full-pooling is optimal in problem (3). Now we are going to find sufficient conditions for dbp M (θ) dθ 0. From effort s choice problem (4), we have that u(w(θ) + b(θ)) u(w(θ)) = Cθ (a ) a ; as u(0) = 0 and w(θ) = 0, for all θ: u(b(θ)) = ) C (a θ (θ, 0, b(θ)) a. (16) Differentiating the expression above with respect of b(θ) we get u (b) = 2 C θ (a ) a a 2 b a b = From the pure moral hazard problem we know that: ( ) a (θ, 0, b P M (θ)) = ( x b P M (θ)) a θ, 0, b P M (θ) b u (b) 2 C θ (a ). (17) a 2 b P M (θ) = x a. (18) a b 21

24 Substituting equation (17) into (18) and taking derivatives with respect to θ, which is relegated to the Appendix, we get a sufficient condition for full pooling. 3 C θ (a ) a 3 0; 3 C θ (a ) a 2 θ + 3 C θ (a ) a a 3 θ 0, θ [θ, θ]. (19) The function a is endogenous, thus by differentiating (16) with respect to θ and substituting into (19) - also, shown in the Appendix - we get the following joint sufficient conditions over the cost function for full pooling. 3 C θ (a ) 2 C θ (a ) 3 C θ (a ) 2 C θ (a ), (20) a 2 θ a 2 a 3 a θ and 3 C θ (a ) a 3 0. (21) Note that our sufficient conditions are given by (20) and (21) holding at a. Therefore, holding for all points of the cost function are conditions that only depend on exogenous variables. To ease the interpretation of (20), suppose that the cost function is mutiplicatively separable: C θ (a) = g(θ)h(a), where g(θ) θ < 0, θ and h is three times differentiable and strictly convex. Then, condition (20) can be re-written as: 3 C θ (a) a 2 θ 2 C θ (a) a θ = h (a) h (a) (a) h (a) = a 3 2 C θ (a), (22) 2 which means that if the marginal cost of effort is high and the cost function is sufficiently increasingly convex, then we satisfy condition (20). We can interpret that as a high cost of enforcing elevated effort levels. Therefore, the principal prefers to pay a lower bonus to higher types instead of implementing bigger effort levels. These results can be summarized by the following Proposition: Proposition 4. If the cost function satisfies conditions (20) and (21), then the solution of the moral hazard problem has a decreasing bonus function, which implies optimality of full-pooling in problem (3). 22

25 The contract simplicity here is a consequence of the non-responsiveness observed at the pure moral hazard problem. The shape of the cost function cause an elevated cost to enforce high effort levels, such that it would be better for the principal to demand less effort from more efficient types at a lower bonus. Some types might be so efficient that the principal does not need to offer high bonuses in order to compensate them for their effort. Another interesting feature of Proposition 4 is that there are sufficient conditions on the cost function leading to full-pooling, regardless the agent s risk aversion. In principle, the gain from risk sharing is bigger with more risk averse agents, thus we should expect bigger gains from screening in these cases. On the other hand, a more risk averse agent will require a bigger bonus to chose a more powerful contract, which is more risky. Therefore, the cost of enforcing is also bigger. We found conditions under which the cost of screening surpasses its benefits. The principal faces a trade-off between providing incentives for exerting effort and leaving informational rents. When she screens, the principal distort effort choices, while when she offers a pooling contract menu, she leaves higher informational rents. In this section, we have provided sufficient conditions for the distortion on effort to be bigger than the informational rents leaved in case of pooling. 6 Examples In the previous section, we found conditions that guaranteed optimality of fullpooling. It is important to show that these conditions do not exclude all reasonable cost functions, i.e., that they are not satisfied only for very particular specifications of the cost of effort. First, we provide an example of a cost function that satisfies (20), i.e., the optimal contract menu will be full pooling for all concave utility functions. Example 1: The cost function C θ (a) = exp(βa) 1 θ satisfies conditions (20) and (21), for all β > 0. Indeed, 3 C θ (a ) 2 C θ (a ) a 2 θ a 2 = β4 β4 exp(2βa) θ3 θ exp(2βa) = C θ (a ) 2 C θ (a ) 3 3 a 3 a θ. 23

26 Therefore, with this cost specification, for all possible strictly increasing, concave and twice differentiable utility function, the optimal contract menu is full-pooling. db P M dθ Second, an example in which condition (20) is not satisfied, although we have 0, which implies full pooling at the original problem as well. Example 2: The utility and cost function are, respectively: where α > 0 and ρ > 1. The optimal effort given b and w = 0 is: u(y) = 1 exp( αy) and C θ (a) = aρ θ, ) a (θ, 0, b = [ θ ( ) ] 1 ρ 1 1 exp( αb). ρ Define the principal s profit at type θ, with fixed payment w and bonus b as P : [θ; θ] R 2 + R. Therefore, P(θ, 0, b(θ)) = x L + [ θ ( ) ] 1 ρ 1 1 exp( αb(θ)) ( x b(θ)). ρ From the first-order conditions of the pure moral hazard problem we obtain: ( x b P M (θ)) ( 1 exp( αb P M (θ)) ρ 1 ) 2 ρ ρ 1 αexp( αb P M (θ)) = [ ( 1 exp( αb P M (θ))) ] 1 ρ 1. Note that the expression above is independent of θ and only depends of the value of b P M (θ). Therefore, the optimal b P M (θ) will not be a function of θ. Hence, dbp M dθ = 0, θ [θ, θ]. By Proposition 3, we have that the optimal solution is characterized by full-pooling. Also, note that this result holds for any absolute risk aversion α > 0 and any ρ > 1. 7 Limited liability Until now, we have provided sufficient conditions on the cost functions for fullpooling. It is also important to understand what is the role of limited liability con- 24

27 straints in these results. The first immediate impact of LL is that it prevents exclusion of some types. This result comes directly from Lemma 1. By limiting principal s possible payments to be positive in all states, the limited liability constraints guarantee that any contract offered is better, to any agent, than the outside option. Limited liability, also, generates non-responsiveness. In order to create incentives to agent s effort, the principal must leave some risk to him. Therefore, the payments in both states must differ, and the greater this difference, the stronger are the incentives to exert effort. If there were no limited liability constraints, the principal could enlarge the difference between payments in the two states by reducing fixed payments even below zero. Under LL, principal s capability of reducing fixed payments is limited. Therefore, from the point which w reaches zero, the only tool left to generate risk is increasing bonuses. Enlarging bonuses is costlier than reducing fixed payments, thus limited liability is responsible for increasing the cost, to the principal, of sustaining high levels of effort. Due to that, there exist cases in which with LL there is non-responsiveness, while without it there is not. To illustrate this point, we present a numeric exercise based on Example 2. The utility and cost function are, respectively: u(y) = 1 exp( 2y), C θ (a) = a 3 2 θ. As shown before, in this case with limited liability, the pure moral hazard problem presents non-responsiveness. We did a numeric exercise 9, by solving the same problem without the limited liability restriction, with θ [0.5, 1] and x = 2. The optimal fixed payment and bonus functions for the pure moral hazard problem are displayed on Figures 2. As you can see, it does not present non-responsiveness. Therefore, this exercise highlights the role of limited liability for non-responsiveness. 9 The same qualitative results hold for a large range of parameters. 25

28 Figure 2: Optimal solution to Pure Moral Hazard Problem 8 Full-pooling problems Now we are going to characterize the optimal full-pooling contract. The problem of choosing the optimal pooling is: The first-order condition is: θ ( max [x L + ( x b)a θ, 0, b) ] f(θ)dθ. (23) b [0, x] θ θ [ x b] θ a θ (θ, 0, b)f(θ)dθ a (θ, 0, b)f(θ)dθ = 0. (24) b θ The first term of expression (24) can be interpreted as the expected additional rent received by the principal due to the increase on agents effort, i.e., the marginal benefit from increasing b. The second term is the additional cost that the principal has to pay if she marginally increases the bonus. The optimal choice occurs when those marginal effects are equal. If we evaluate (24) at b = 0, the second term vanishes and first one is bigger than zero. Therefore, it is not optimal to offer b = 0. Additionally, if we evaluate (24) at b = x the first term is zero and the second negative, thus b = x is not optimal. Therefore, the solution is necessarily interior. 26

29 Substituting equation (16) into (24) we have that: [ ] 1 θ 2 C θ (a ) θ u (b)[ x b] f(θ)dθ = a (θ, 0, b)f(θ)dθ. (25) θ a 2 θ The first immediate and intuitive interpretation of equation (25) is that an increase in the cost convexity implies in a lower optimal bonus. This result was expected, since a more convex cost function implies in a higher cost of enforcing elevated levels of[ effort. Note that 2 C θ (a ) a 2 with respect to θ we get: ] 1 is decreasing in θ if 2 C θ (a ) a 2 [ ] 2 C θ (a ) a 2 θ is increasing. Differentiating = 3 C θ (a ) a a 3 θ + 3 C θ (a ) a 2 θ. (26) Note that (26) being positive is the same condition which would guarantee full non-responsiveness (19). We now restrict our attention to such cases. Therefore, by (25), a distribution with more mass on higher types is going to generate a contract with lower bonus. That is: Proposition 5. If the optimal contract is full pooling, the optimal bonus function is: 1. increasing in x; 2. decreasing in the convexity of the cost function; 3. decreasing in the distribution function of types, when they are ordered by first-order stochastic dominance, if (19) is satisfied. Proof. In the Appendix. 9 Conclusion Adverse selection literature usually predicts large menu of contracts even infinite ones, while empirical observation points towards simpler mechanisms - with limited number of contracts, often a single one. In this paper we tried to provide some explanation to this mismatch between theory and reality. 27

30 In this work, we have examined the problem of simultaneous adverse selection and moral hazard with risk averse agents under limited liability constraints. We argue that limited liability increases the cost of enforcing high levels of effort. This feature associated with particular cost function shapes might generate a conflict between the optimal level of effort enforced and some necessary conditions for screening. In our problem, the principal has two instruments to perform two different tasks. She has a fixed payment and a bonus in order to provide incentives for effort and to screen the types. Limited liability reduces the range of possible fixed payments and, therefore, limits the effectiveness of this instrument. Hence, from the point which fixed payment achieves zero, the principal remains with only one instrument to accomplish two tasks. Under some conditions on the cost function, the optimal shape of the bonus function required by each of the tasks are opposites. The moral hazard problem asks for a bonus declining on type, while in order to screen the types is necessary an increasing bonus function. In our model, this conflict is what generates pooling. In our work, we showed that non-responsiveness for all types generates full-pooling. For future research, we would like to find out if non-responsiveness for regions of the type-space generates regions of pooling. Also, we showed that Gottlieb and Moreira (2015) s result of simplicity does no longer holds for any strictly risk averse utility function, without further assumptions over the cost function. We would like to learn whether their result would hold for low risk aversion levels with some regular assumption over the cost function. Additionally, we pursue a complete characterization of the solution of this problem in the presence of limited liability. 28

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34 10 Appendix Proof of the optimality of 0 fixed payments on Problem (9) Problem (9) can be described by the following Hamiltonian 10 : H[θ, w, b, µ, λ, κ] = [x L w(θ) + ( x b(θ))a ]f(θ) + λ(θ)µ(θ) + κ(θ)w(θ). From the necessary first-order conditions, we obtain: [ ] 1 + a f(θ) + γ(θ) = 0 w which implies γ(θ) > 0, w(θ) = 0, θ [θ, θ]. Then, if there is a solution to problem (9), it is equivalent to the solution of the problem below with w(θ) = 0, θ [θ, θ]. Proof of Lemma 3 Proof. Note that as the minimal cost of effort is given at the zero effort level, then any negative bonus would imply zero effort. Take any b < 0, then for all θ [θ, θ]: u(w) au(w + b) + (1 a)u(w) C θ (a) a [0, 1]. As the agent would choose a (θ) = 0, then his payoff would be the same with contract (w, b) and (w, 0. Therefore, there is no loss of generality in restricting the bonus to weakly positive values. Let Θ be an arbitrary non-empty subset of [θ, θ]. Take any contract menu that satisfies (IC),(IR), (LL) and in which b(θ) > x for θ Θ [θ, θ]. If the expected profit is lower than x L, then it would be better to offer a single contract (0, 0). If the expected profit is higher than x L, then the principal could increase her profits by dropping all contracts with b(θ) > x. Let w(θ), b(θ) be the ( ) ( original contract menu and w(θ), b(θ) ) the alternative one, constructed from the 10 We will limit our study to fixed payments and bonuses profile functions, w : [θ, θ] R + and b : [θ, θ] R + which are piecewise continuously differentiable of class C 1. 32