Supervision, Collusion, and Optimal Contract Design with Costly Information Acquisition

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1 Supervision, Collusion, and Optimal Contract Design with Costly Information Acquisition XiaoGang Che Guanxi Yi PRELIMINARY and INCOMPLETE November 2014 Abstract In this paper, we study the impacts of costly information acquisition on the optimal contract in the principal-supervisor-agent hierarchy, when information is hard for the supervisor alone but soft for the supervisor-agent coalition. In such a hierarchical environment, two types of corruptive behavior exist: bribery and extortion. Our novel findings show that first, supervision is still valuable and the optimal contract for the principal is collusion proof, where both bribery and extortion should be entirely prevented. This provides a support for fighting corruptive behavior in hierarchical organizations. Second, interestingly, collecting an additional report from the agent, i.e., the cross-checking mechanism, cannot further improve the optimal contract for the principal. This explains the observation of why most of organizations in reality only rely on the supervisors reporting when designing the incentive schemes. Keywords: Collusion, bribery, extortion, supervision, costly information acquisition, cross-checking mechanism. JEL codes: D82, L23. We are grateful to Kunal Sengupta for his insightful comments and suggestions. Business School, Durham University, U.K. xiaogang.che@durham.ac.uk. Economics Department, University of California, Los Angeles, U.S. guanxiyi@gmail.com.

2 1 Introduction In a principal-supervisor-agent hierarchy, the principal, whose objective function is to minimize its costs but induce the agent to work, offers a contract to both the supervisor and the agent; the agent is hired to produce some output for the principal, while the supervisor is employed to monitor the agent and report evidence of the effort level to the principal. A main concern in this hierarchical environment is the possibility of collusion between the supervisor and the agent, which would compromise the supervisor s usefulness 1. Collusion and its related issues, such as what incentive schemes the optimal contract should have for the principal, whether corruptive behavior should be allowed in the contract, have been broadly discussed in the literature 2. In this paper, we study a version of the three-level hierarchy and the related optimal contract design, where information is soft for the supervisor-agent coalition but hard for the supervisor alone 3. Under this information structure, there exists a tradeoff between bribery and extortion: when designing the contract schemes, the principal needs to reward the supervisor a wage payment to deter the collusion of the supervisor and the agent forging the evidence of the agent s effort. However, this wage payment induces the threat of extortion from the supervisor to become credible; she can extort to report non-conclusive evidence, when observing positive evidence of the agent s effort level. Khalil, Lawarrée, and Yun (2010), hereafter KLY, analyzed the case where information acquisition for the supervision is costless in the hierarchy, showing that although the least-cost-corruption-proof contract (the LCCP contract) can be utilized to deter both bribery and extortion, more interestingly and importantly, the principal benefits from allowing bribery between the supervisor and the agent in the contract, but extortion should always be prevented. In this paper, we are interested in examining the impacts of costly information acquisition on the optimal contract design for the principal in the hierarchy. In particular, whether or not the principal still benefits from allowing any corruptive behavior among the supervisor 1 In general, collusive behavior can be classified to two different types: bribery and extortion. Although both corruptions involve side-contracts between the supervisor and the agent, the main difference is that bribery occurs when both the supervisor and the agent are better off by misreporting to the principal, while extortion involves that the supervisor claims a payment from the agent by threatening a misreport to the principal, which hurts the agent. 2 See surveys from Tirole (1992), Bardhan (1997a), and others. 3 Hard information means that the signal reported by the supervisor is verifiable but hideable, while information is soft in the sense that the supervisor s signal is unverifiable. See Tirole (1986), Khalil, Lawarrée, and Yun (2010), etc. 1

3 and the agent, when acquiring the signal of the agent s effort level is costly. Before making the report, the supervisor may need to spend some effort, i.e, money and time, learning the signal. This can be interpreted as the cost of information acquisition 4. Clearly, the supervisor will only incur the cost to acquire the signal if it is in her interest to do so. Thus, in this environment, the contract design should have a feature that when a supervisor is hired, the expected payoff of the supervisor from incurring the cost and acquiring the signal is higher than the payoff when she is completely uninformed 5. Given this new constraint for the supervisor, we first characterize the least-cost-corruptionproof contract with costly information acquisition (the LCCP c contract) as the benchmark. In contrast to the one with no information acquisition cost, there are two interesting features in the LCCP c contract. First, to encourage the supervisor to incur the cost, the principal needs to reward her when she reports positive evidence of the agent s work level. Second, to deter the suppression of positive evidence - the threat of extortion from the supervisor, the principal needs to pay a higher reward to the agent under no conclusive evidence than under definitive evidence of work. We then investigate the optimal contract for the principal. Our first main result (Proposition 1) shows that the LCCP c contract provides the lower bound of the expected cost for the principal. The intuition is that since it is costly to acquire the informative signal, the principal needs to pay a positive reward to the supervisor when she reports a definitive signal of work. Therefore, unlike the case shown by KLY, distinguishing signals between informative and uninformative signals becomes valuable for the principal to design incentive schemes. This feature requires that in the optimization problem of the principal, any contract that induces the supervisor and the agent to work cannot violate the coalition incentive compatibility constraints. As a result, the LCCP c contract is optimal for the principal, indicating that when collecting signal is costly for the supervisor, supervision is still useful in the organization and the optimal contract is collusion proof; both bribery and extortion should not occur. So far we have only considered the case where the supervisor reports the signal to the principal in the hierarchy. However, in reality there should be no restrictions for the principal 4 Note that this cost is different from the collusion cost between the supervisor and the agent, like the cost of forging evidence, etc. See Yun (2012) for the analysis of the case where collusion formation is costly. 5 In this case, the tradeoff where fighting bribery makes the extortion threat from the supervisor become credible still exits. 2

4 to have an option of acquiring information from the agent, in other words, the agent can also send a report of his effort level to the principal. Therefore, it is of interest to examine whether there exists such an information-improving mechanism that by introducing the additional information resource, the tradeoff between bribery and extortion will not appear in the hierarchy, and further the LCCP c contract can be improved, lowering the expected cost of the principal. This is the second question in which we are interested. One may argue that since information is soft for the supervisor-agent coalition, it then appears that the agent can always collude with the supervisor and report a high level of effort in production, and thus collecting this additional report from the agent would not be helpful for the principal to design the incentive schemes in the contract. In the present paper, we introduce cross-checking mechanism into the hierarchy, which was first studied by Baliga (1999), and show that no matter whether or not the supervisor needs to pay the signal acquisition cost, collecting additional information resource from the agent makes the threat of extortion by the supervisor suppressing evidence become vacuous, although information is soft for the coalition. More importantly, by designing proper payments across the signals, there exists a pure-strategy (perfect Bayesian) equilibrium such that the supervisor and the agent always report their signals truthfully. Then we discuss whether the LCCP c contract can be improved under the cross-checking mechanism. We first discuss the benchmark case where there is no information acquisition cost for the supervisor, and show that the expected cost of the principal is the same as the cost when information is hard, which is strictly lower than that of the bribery contract characterized by KLY. However, this becomes not true with costly information acquisition. Our second main result (Proposition 2) shows that when the supervisor needs to pay a cost to acquire the information of the agent s work level, the LCCP c contract still provides the lower bound of the expected cost, regardless of whether or not the agent makes a report to the principal, demonstrating that additional information resource is useless for the principal to design incentive schemes in the contract, lowering the expected cost. We will discuss implications of the result in more details in the following literature review section. Related Literature. After the seminal work from Tirole (1986, 1992), corruption and the optimal contract design in hierarchical organizations has been discussed in the literature. The central finding Tirole obtains is that when information is hard, information from a 3

5 corruptible supervisor is still useful for the principal, and moreover by carefully designing the side payments, the optimal contract implemented by the principal is collusion-proof, where any bribery strategies between the supervisor and the agent can be deterred. Much of the following work extended Tirole s model and different results have been found. Kofman and Lawarree (1993), Mookherjee and Png (1995), and Kessler (2000) obtain the similar result that it is optimal for the principal to prevent bribery in the contract. However, several studies show that the principal may be better off from allowing bribery, even when information is hard 6. Additionally, research on the optimal contract design with soft information has also received some attention. Grimaud, Laffont, and Martimort (2003) consider a model where the monitoring environment is with soft information, and the supervisor and the agent collude under asymmetric information, showing that the optimal collusion-proof contract can still be implemented by delegating to the supervisor the task to contract with the agent. Angelucci and Russo (2012) show that with soft information, the supervisor is useful if and only if she is hired before the agent has chosen his action. Compared to bribery, extortion has received relatively little prior attention in the literature, as it becomes no longer relevant in the principal-supervisor-agent hierarchy, when information is either hard or soft. Banerjee (1997), and Guriev (2004) consider models where the principal delegates officials to impose red tape and extract money transfers from citizens. Hindriks, Keen, and Muthoo (1999) and Polinsky and Shavell (2000) indicate that incorruptible external agents can be utilized to deter extortion in organizations, although both studies focus on different model settings. Also, see other related studies on extortion by Bardhan (1997b), Auriol (2006), and Andrianova and Melissas (2009), etc. The present paper is closely related to the study by Khalil, Lawarrée, and Yun (2010). To the best of our knowledge, their model is the only one which examines both bribery and extortion in the principal-supervisor-agent organization, showing that a tradeoff between the two corruptive behavior induces the principle to be better off by allowing bribery. Given the same information structure, our paper analyzes an interesting case where the supervisor needs to incur a cost for signal collection in the three-level hierarchy and focus on whether allowing bribery is still optimal for the principal. Compared to their study, our contribution to the literature suggests that when information acquisition is costly for the supervisor, the supervision is still valuable and fighting corruptive behavior in the organizational hierarchy 6 See Cadot (1987), Che (1995), Kofman and Lawarrée (1996), Strausz (1997), Olsen and Torsvik (1998), Lambert-Mogiliansky (1998), Acemoglu and Verdier (2000), and Khalil and Lawarrée (2006), etc. 4

6 is optimal for the principal; both bribery and extortion should not be allowed in the optimal contract. The cross-checking mechanism in the paper is similar to the one considered by Baliga (1999), who shows that the supervisor and the agent will report truthfully, and the collusion possibility is fully deterred. As a result, the expected cost of the principal with soft information is equivalent to the cost with hard information. On the one hand, our findings reinforce the importance of additional information collection, showing that the principal by carefully designing the reward schemes under the cross-checking mechanism can eliminate not only bribery but also extortion without incurring any additional cost. On the other hand, we show that with costly information acquisition, collecting an additional information resource would not further help the principal lower the expected cost in the contract; the LCCP c contract is still optimal. This interesting result sheds lights on the observation of why in reality most of hierarchical organizations do not acquire any information from their agents and do cross-checking but only relies on information from supervisors, when designing the incentive schemes in the contract.this is our second contribution to the literature. The remainder of the article proceeds as follows. We describe the model setup in Section 2. We characterize the LCCP c contract in Section 3 and show that this contract provides the lower bound of the expected cost for the principal. Section 4 analyzes the optimal contract under cross-checking mechanism with and without costly information acquisition. We conclude this study in Section 5. Most of proofs are in the Appendix. 2 The basic Setup Following KLY, we consider a principal-supervisor-agent hierarchy, where the principal (it) is the owner of a firm, the supervisor (she) collects the output information for the principal, and the agent (he) is the productive unit in the firm. Let e {0, 1} and x {x H, x L } denote the agent s effort level and output level, where x H > x L > 0. If the agent chooses to work, i.e., e = 1, the output level x is either x H with probability π or x L with probability (1 π), where π (0, 1). If, however, he shirks, i.e., e = 0, the output level x is x L with probability one. All of three parties can observe the output level x, but the agent s effort level e is private information. (x H x L ) > 0 is sufficiently large such that the principal strictly prefers the 5

7 agent to work (e = 1). Given the output level x, the principal can hire a supervisor to collect evidence about whether or not the agent works. Clearly, if the output is x H, there is no need to employ the supervisor, as the principal knows e = 1 from the agent. Thus, the principal has an incentive to send the supervisor only if the low output x L is observed. Once the supervisor is used to monitor the agent, the true signal and the non-conclusive signal are found with probabilities p and 1 p, respectively, where p (0, 1). Let φ denote the non-conclusive signal, and then the signal σ observed by the supervisor will be one of the three values: {0, φ, 1}. After the information collection, the supervisor sends a report r {0, φ, 1} to the principal. According to report r, the principal collects the output x L and pays a wage transfer w r to the agent and a wage transfer s r to the supervisor, respectively, where w r [0, + ), and s r [0, + ). Therefore, wage transfers by the principal in the contract can be summarized as follows: If x H is observed, the principal does not hire the supervisor (with zero wage transfer) and pays w H to the agent. If, however, x L is observed, the principal pays w r and s r, depending on report r from the supervisor. The principal s objective function is to minimize the wages and induce e = 1. The agent is risk averse with a separable utility function U(w r, e) = u(w r ) ϕe and chooses whether or not to work, where ϕ > 0 denotes the dis-utility level of investing the effort in production; u(w r ) is concave and satisfies u(0) = 0, u (0) = +, and u (+ ) = 0. The supervisor is to maximize s r by strategically reporting r. The supervisor s information is hard. If σ = φ, she can only report r = φ. If σ = e, she can report r {e, φ}. The supervisor cannot forge information by acting alone but can decide to whether or not conceal it. However, if the supervisor and the agent cooperate as a coalition, then they can forge and report the information that the agent invests effort in work with no cost. The coalition can report r {0, φ, 1} regardless of signal σ. In short, the information structure is hard for the supervisor alone but soft for the supervisor-agent coalition. We also assume that the supervisor and the agent do not involve in collusion, when they are indifferent between not colluding and colluding. The supervisor does not extort when she is indifferent. Given the assumptions above, the timing of moves can be presented as follows: (1.) The principal offers a contract specifying the transfers to the agent as a function of output, and the supervisor s report; the transfers to the supervisor as a function of her report. 6

8 (2.) The agent and the supervisor accept/reject the contract. (3.) The agent decides whether to work (e = 1) or shirk (e = 0). (4.) Output x is realized. If the principal observes x L, it sends the supervisor. If it observes x H, the game moves to Step (8) 7. (5.) The supervisor and the agent observe the signal σ. (6.) The supervisor and the agent choose whether or not to make a side-contract. (7.) The supervisor makes report r to the principal. (8.) Wage transfers are implemented. Before exploring the complexity of collusion between the supervisor and the agent, we should mention that the principal can always choose to contract with the agent only and then entirely avoid the collusion possibility. In this case, the hierarchy only involves two levels: the principal and the agent; what the principal needs to do is to simply design the rewards to the agent according to the actual observed output levels. It is easy to check that the optimal contract is such that w s H = u 1 (ϕ/π) and w s L = 0. Following KLY, we refer to this contract in the absence of any supervision as the second-best contract. 3 Costly Information Collection for the Supervisor In this section, we study the impacts of costly information acquisition for the supervisor on the contract design in the hierarchy. Let c > 0 denote the information acquisition cost of the supervisor, and we modify step (5), the information discovery process of the supervisor, in the timing of the game as follows: The supervisor decides whether or not to incur the cost c for information collection. If the supervisor does not incur the cost, she is uninformed and observes no conclusive signal φ for sure. If she does so, she observes the true signal with probability p and no conclusive signal with 1 p, respectively. The agent can observe the signal σ with no cost. The principal faces a new constraint when designing the contract, that is, it should generate an incentive for the supervisor to incur the cost for collecting the informative signal σ. Given that in the contract the agent will choose to work along the equilibrium path, i.e., 7 The principal does not send the supervisor, the agent and the supervisor obtains w H and 0, respectively. 7

9 e = 1, then the constraint for the supervisor with costly collect information can be written as follows ps 1 + (1 p)s φ c s φ, indicating that the expected payoff of the supervisor from incurring the cost to collect information of the agent s work level cannot be less than that of remaining uninformed. Simplifying the equation above yields s 1 s φ c 0. For convenience, we call this constraint the p IC s constraint. In this section, we first characterize the optimal contract in the absence of any collusion (bribery and extortion) between the supervisor and the agent. This will help us shed light on how costly information acquisition affects the incentive schemes. Next, we examine the case with the presence of collusion possibility, and show that with soft information, supervision is still useful for the principal and the optimal contract is collusion proof. 3.1 Optimal contract in the absence of collusion Let us derive the optimal contract in the absence of collusion. In this case, once the IC s constraint is satisfied, the supervisor always reports truthfully what he observes. Then, the agent s individual rationality (denoted by IR a ) and incentive compatibility (denoted by IC a ) constraints are given by πu(w H ) + (1 π)[pu(w 1 ) + (1 p)u(w φ )] ϕ (IR a ), πu(w H ) + (1 π)[pu(w 1 ) + (1 p)u(w φ )]ϕ πu(w 0 ) + (1 π)u(w φ ) (IC a ). Obviously, the IR a constraint is also satisfied when the IC a constraint is satisfied. Therefore, the principal s optimization problem can be written as follows [ ] Min πw H + (1 π) p(w 1 + s 1 ) + (1 p)(w φ + s φ ), subject to the following constrains: πu(w H ) + (1 π)pu(w 1 ) π(1 p)u(w φ ) pu(w 0 ) ϕ (IC a ), s 1 s φ c p 0 (IC s), w H 0, w r 0, and s r 0, where r {0, φ, 1}. 8

10 The solution of the problem above is the optimal contract in the absence of collusion, when information collection is costly for the supervisor. We summarize the optimal contract as follows Lemma 1. When information collection is costly for the supervisor (c > 0) and the supervisor is incorruptible, (i) if w H hired; (ii) if w H > c, the principal offers the second-best contract where the supervisor is not 1 p c, the principal hires the supervisor and offers the contract which satisfies 1 p (1) for the agent, w H = w 1 > 0 = w 0 = w φ, (2) for the supervisor, s φ = 0, s 1 = c p, and s 0 0, (3) for the principal, the expected cost is given by π w H + (1 π)p( w 1 + s 1 ), where w H = w 1 = u 1 ( Proof. See Appendix A. ϕ ). π+(1 π)p In the absence of collusion between the supervisor and the agent, the optimal contract will reward the agent with definitive evidence of effort (r = 1) and moreover this reward is equal to the payment that high output is realized. In other cases (r = φ or 0), the reward is zero. Interestingly, in order to induce the supervisor to work, the principal needs to pay c/p when she reports a positive evidence of the agent s effort level, and zero when r = φ. 3.2 Optimal contract in the presence of collusion In this subsection, we first characterize the least-cost-corruption-proof (LCCP c ) contract when information acquisition for the supervisor is costly. As mentioned in the Introduction, this cost is used to capture the supervisor s effort to discover the signal of the agent s effort level. In the LCCP c contract, both bribery and extortion from the supervisor are deterred. We then examine whether or not the principal can benefit from allowing bribery. Our result will show that with the costly information acquisition the expected cost of any optimal contract cannot be lower than the expected cost of the LCCP c contract, and it is optimal for the principal to deter both types of corruption in the hierarchy. 9

11 In the hierarchical environment, since information is soft for the coalition, to prevent the bribery possibility of forging evidence between the supervisor and the agent, the principal should pay the supervisor a reward, when she reports φ, i.e., no evidence about the agent s effort level. However, this reward creates a potential problem that it becomes credible for the supervisor to suppress the evidence; she can require a transfer as extortion from the agent by threatening to misreport r = φ, even though she has observed σ = 1 8. This is the tradeoff between bribery and extortion, which the principal has to encounter. To deter both types of corruptive behavior between the supervisor and the agent, the principal needs to ensure that the contract satisfies two important constraints, beside the standard constraints of incentive compatibility (IC a ) and individual rationality (IR a ). The first is the coalition incentive compatibility constraints (the CIC constraints), which are used to prevent bribery, requiring that the aggregate transfers should be the same in every reporting state when the output is x L and the supervisor is hired; i.e., T 0 = T φ = T 1, where T r = w r + s r and r {0, φ, 1}; the second is the extortion deterrence constraints (the EF constraints), which are used to prevent extortion, requiring that s 1 s φ, denoted by EF 1, and s 0 s φ, denoted by EF 0. In the modified game above, we do not require that whether or not the supervisor incurs the cost and collects the information of the effort level is observable for all three parities in the hierarchy. There exists a possibility that the supervisor does not incur the cost, but colludes with the agent and misreport that the agent worked. We will discuss this collusion possibility in proof of Proposition (1) in Appendix B. Note that the EF 1 constraint is satisfied when the IC s constraint does. principal s minimization problem. Thus, the EF 1 constraint can be ignored in the Before investigating the features of the optimal contract with costly information collection, we first characterize the LCCP c contract, and it will serve as a benchmark for the following analysis. Given the constraints discussed above, the optimization problem for the 8 Note that if information is hard for the supervisor-agent coalition, then the coalition cannot forge the evidence and the threat of extortion discussed above becomes irrelevant, as the supervisor will not get a reward from reporting no conclusive evidence, see Tirole (1986), Kofman and Lawarree (1993), and Kessler (2000). 10

12 principal can be written as follows [ ] Min πw H + (1 π) p(w 1 + s 1 ) + (1 p)(w φ + s φ ), subject to the following constrains: πu(w H ) + (1 π)pu(w 1 ) π(1 p)u(w φ ) pu(w 0 ) ϕ (IC a ), s 1 s φ c p 0 (IC s), T 0 = T φ = T 1 (CIC), EF 0, w H 0, w r 0, and s r 0, where r {0, φ, 1}. The solution of the problem above is the least-cost-corruption-proof contract when information collection is costly for the supervisor, and both bribery and extortion are prevented in the contract. Let w H and w 1 denote the solutions of the two equations below: We then present the contract as follows. u (w1 + c ) p (1 π) = pu (w1) u (wh ) π(1 p), (1) πu(wh) + (1 π)pu(w1) π(1 p)u(w1 + c ) = ϕ. (2) p Lemma 2. When information collection is costly for the supervisor (c > 0), (i) if (wh w 1) c, the principal offers the second-best contract where the supervisor is p not hired; (ii) if (wh w 1) > c, the principal hires the supervisor and offers the least-cost-corruption- p proof (LCCP c ) contract which satisfies (1) for the agent, w H > w 1 > 0 = w 0, and w φ = w 1 + c p, (2) for the supervisor, s φ = 0 < s 0 = w φ, and s 1 = c p, (3) for the principal, the expected cost is given by πw H + (1 π)(w 1 + s 1). Proof. See Appendix B. 11

13 In contrast to the conditions of the LCCP contract with no cost, when information acquisition is costly for the supervision, whether or not supervision should be used does not only depend on the accurate level of the signal that she can provide, but crucially relies on the difference of the agent s wage payments between the high output, and the low output with definitive evidence of work (e = 1). More specifically, Lemma (2) indicates that the principal will still benefit from hiring a supervisor only if the difference between w H and w 1 is sufficiently large enough, i.e., greater than c/p; otherwise, it is optimal to use the second-best contract, where wh and w 1 are determined by Equations (1) and (2). Further, it is intuitive to see that when it becomes more costly for the supervisor to acquire evidence, i.e., a higher c, or the accurate level of the signal provided by the supervisor decreases, i.e., a lower p, it will be more likely for the principle not to choose any supervision. Closely looking at the LCCP c characterized in Lemma (2), there are two interesting features we should notice that first, like the LCCP contract, to deter both bribery and extortion, the principal needs to reward the agent a wage payment, even though there is no informative evidence that the agent chooses to work. Second, to ensure the incentive for the supervisor, the principal needs to reward the supervisor, when she incurs the cost of information collection and reports that the evidence of the agent s work is definitive. Let us comment further on the two features in LCCP c contract characterized above. Khalil, Lawarrée, and Yun (2010) stated that it is not the supervisor but the agent who benefits from the supervisor s ability to misreport information, as the only way to deter both corruptive behavior is to treat the reports r = 1 and r = φ in the same way. In Lemma (2), costly information acquisition from the supervisor further enhances this feature, in the sense that under the LCCP c contract, collusion proof requires a higher wage payment for the agent with non informative signal than with definitive signal of work, i.e., wφ > w 1. In this case, the supervisor cannot extract any rent from the threat of extortion, and the agent who is the potential victim, obtains a higher reward under no conclusive signal. In other words, it is more likely for the agent to receive a better wage payment when he chooses to shirk, rather than to work. In turn, this payment structure would encourage the agent to shirk, as shirking without being caught will be treated even better as if he worked. In the LCCP c contract, the supervisor obtains a reward of c/p when reporting r = 1 and zero when reporting r = φ to the principal. This payment structure captures the fact that preventing both corruptive behavior increases the incentive costs for the principal by 12

14 c/p. As a result, costly information acquisition reduces the usefulness of supervision; the supervision will only be valuable for the principal when the difference between wh and w 1 is sufficiently large. We next examine whether or not the principal can do better than the LCCP c contract characterized in Lemma (2). When a supervisor is hired, the principal prefers the supervisor to work, i.e., incurring the cost and collecting information. In this case, the threat of extortion from the supervisor is also deterred, as the EF 1 constraint is satisfied when the IC s constraint does. Therefore, in the following analysis, we only need to focus on whether or not allowing bribery gives a lower cost to the principal than that of the LCCP c contract. We here assume that when bribery is conducted in a side contract, the supervisor s and the agent s payoffs are determined by the standard Nash bargaining solution. Our result is stated as follows: Proposition 1. Given c > 0, the LCCP c contract provides the lower bound on expected cost for the principal, and therefore both bribery and extortion should not occur in the hierarchy. Proof. See Appendix C. We now provide the intuition of why allowing bribery cannot help the principal lower the expected cost. When the supervisor needs to incur a cost to acquire the signal of the agent s work level, a contract that induces both the supervisor and the agent to work requires that the supervisor is compensated positively when incurring the cost, and the agent obtains a higher payoff under the inconclusive signal of φ than under the definitive signal of 1. This two facts indicates that the principal strictly prefers to distinguish between signal 1 and signal φ. If the CIC constraint is violated and bribery is allowed, the supervisor s report becomes completely uninformative for the supervisor to separate the states between φ and 1; she reports r = 1 if T 1 > T φ, and she reports r = φ if T 1 < T φ. Thus, in order to design the correct incentive schemes in the contract, the CIC constraint should always hold. Further, given that the LCCP c contract is the one which satisfies all IC a, IC s, and CIC constraints with the minimum cost, it is optimal for the principal to prevent both bribery and extortion in the hierarchy. It is also worthwhile providing some comments on the difference between the LCCP c contract and the bribery contract found by KLY, where allowing bribery lowers the principal s expected cost. When it is not costly for the supervisor to collect evidence, the principal needs 13

15 to reward the agent the same payments across the states of 1 and φ in order to prevent the extortion threat from the supervisor. In this case, it is not necessary for the principal to distinguish the two states. By allowing bribery, it will lower the expected cost for the principal, as the agent s payoff becomes lower in the state of φ. However, as we discussed in the last paragraph, this argument becomes not true when the supervisor needs to incur a cost for the information collection; distinguishing the signals between φ and 1 becomes necessary for the principal to design the incentive schemes for inducing the supervisor and the agent to work. 4 Cross-Checking from the Supervisor and the Agent In this section, we will first provide a discussion of how cross-checking mechanism can prevent both bribery and extortion in the hierarchy and moreover induces a lower expected cost to the principal, when information is costless. This will serve as a benchmark case. Thereafter, we turn to examine whether the same effect remains, i.e., further improving the LCCP c contract for the principal, when information acquisition is costly for the supervisor. Note that although both corruptive behavior can be eliminated under the cross-checking mechanism, the designs of reward schemes are quite different with and without costly information acquisition. We will illustrate the differences in more details in the following analysis. 4.1 With no information acquisition cost In the cross-checking mechanism (Baliga, 1999), the principal requires both the supervisor and the agent to submit their reports, separately, and then decides the wage payments for them accordingly. Although the supervisor and the agent would jointly misreport their signals to obtain higher payoffs, this possibility can be easily prevented by designing different rewards across the different reporting signals. As mentioned above, under the information structure we consider here, it would also be interesting to examine whether or not the mechanism can eliminate both types of corruptive behavior and still results in the same expected cost for the principal with hard information, as has been shown in the Baliga s model. In this subsection, we consider the case where the supervisor does not pay any cost 14

16 for signal collection, and will show that a properly designed cross-checking mechanism can ensure that the supervisor and the agent truthfully submit their reports to the principal, and the extortion threat of the supervisor becomes not creditable by adding the new information resource from the agent. As a result, the expected cost of the principal is the same as the cost when information is hard 9, which is lower than the bribery contract characterized by KLY, and further lower than the LCCP contract. We here consider the same optimization problem of the principal described in Appendix D and show that how cross-checking mechanism can implement the optimal contract. In the following analysis we first introduce the cross-checking mechanism and the design of the rewards/wage transfers in the contract, according to both reports from the supervisor and the agent. Let r s {0, φ, 1} and r a {0, φ, 1} denote the reports from the supervisor and the agent to the principal, respectively, and then we modify step (7) in the timing of moves of KLY as follows: The supervisor and the agent simultaneously make reports r s and r a to the principal. Note that the contract payments offered by the principal to the agent and the supervisor specify the transfers as a function of output, the agent s report, and the supervisor s report. Moreover, we assume that under the modified game, the supervisor and the agent report truthfully when they are indifferent between truthful reporting and not truthful reporting. Then the implementation of the cross-checking mechanism is stated as follows: When the output is low (x = x L ), (a.) if the supervisor and the agent send different signals r s r a, the principal does not pay the supervisor and the agent; (b.) if both the supervisor and the agent announce r s = r a = 1, the principal pays the supervisor 0 and the agent w 1 ; (c.) if both the supervisor and the agent announce r s = r a = 0 or φ, the principal pays the supervisor w 1 ɛ and the agent ɛ, where ɛ is a real non-negative but sufficiently small payment See the characterization of the optimal contract with hard information in Appendix D. 10 In real life, the minimum currency unit always provides a lower bound for this small payment and thus serves as a focal point for the value of ɛ. 15

17 According to the cross-checking mechanism described above, we can construct the following payoff matrix for both the supervisor and the agent. Supervisor r s = 1 r s = φ r s = 0 r a = 1 (w 1, 0) (0, 0) (0, 0) Agent r a = φ (0, 0) (ɛ, w 1 ɛ) (0, 0) r a = 0 (0, 0) (0, 0) (ɛ, w 1 ɛ) In the table above both the supervisor and the agent reporting truthfully to the principal is a pure strategy (perfect Bayesian) equilibrium and moreover this equilibrium is collusionproof; there is no incentive for them to conduct any corruptive strategies. When the agent chooses to work, the threat from the supervisor to report r s = φ is vacuous under crosschecking, as the signals will not be matched and the supervisor still obtains no wage payment from the principal; misreporting the evidence cannot make her strictly better off. Therefore, when the principal designs the optimal contract, the EF 1 constraint becomes unnecessary to prevent extortion. The possibility of bribery between the supervisor and the agent is to jointly report 1 but the agent does not work, and then share w 1 by the standard Nash bargaining. Apparently, the supervisor cannot be better off by doing so, specially when ɛ is sufficiently small. Thus, this is not acceptable for the supervisor and the agent. When σ = φ, if the actual state is 1, i.e., the agent worked, then the supervisor earns almost an rent of w 1 in equilibrium, and therefore does not have any incentive to involve in any collusive strategy where both the agent and the supervisor report r a = 1 and r s = 1, respectively. Thus, there exists no acceptable collusion formation with σ = φ either. From the discussion above, we have seen that the cross-checking mechanism deters both types of corruption. It is also easy to check that the expected cost of the principal under the mechanism is the same as the cost under the optimal contract when information is hard. Therefore, summarizing the argument above gives the following result. Lemma 3. Given c = 0, under the cross-checking mechanism, bribery and the threat of extortion from the supervisor can be eliminated at no additional cost in the hierarchy, and the expected cost of the principal is the same as the cost with hard information. 16

18 Next we show that the bribery contract found in KLY can be further improved by adding the information resource from the agent 11. Lemma 4. The expected cost of the principal achieved by the cross-checking mechanism is strictly lower than the expected cost of the bribery contract. Proof. This proof is straightforward. Since the agent does not shirk along the equilibrium path, we only need to focus on the comparison of the principal s expected costs and IC constraints of the two contracts. Under the cross-checking mechanism, the contract is the same as the optimal contract with hard information, which can be characterized as follows: the principal s expected cost is πw H + (1 π)w 1, subject to IC constraint πu(w H ) + (1 π)pu(w 1) = ϕ. In the bribery contract characterized by KLY, the principal s expected cost is given by πw H + (1 π)w 1, subject to IC constraint πu(w H ) + (1 π)pu(w 1) π(1 p)u(w 1φ ) = ϕ. Let w H equal w H. Since 0 < w 1φ < w 1 for α (0, 1) and π(1 p)u(w 1φ ) > 0, the comparison of the two IC constraints shows that w 1 > w 1, suggesting that the cross-checking contract provides a lower expected cost to the principal. The intuition of Lemma (4) is that in the bribery contract characterized by KLY, when information is soft, bribery is used as a penalty on the agent when he chooses to shirk; agent has to share w 1 with the supervisor. Thus, allowing bribery improves the incentive for the agent to choose to work, which benefits the principal by reducing incentive costs in an indirect way. In contrast, the cross-checking mechanism utilizes a different way to improve the incentive of the agent; by introducing the second information resource and checking reports from both the supervisor and the agent, the principal can deter both bribery and extortion, in that the incentive of the agent choosing to work increases. As shown in Lemma (4), this gives a lower cost than that of allowing bribery. 4.2 With information acquisition cost Form the discussion above, we have seen that with no information acquisition cost, the optimal contract under cross-checking mechanism is lower than the bribery contract characterized by KLY, lower than the LCCP contract. In this subsection, we further examine 11 We provide the features of the bribery contract in Appendix E. More details in the contract can be referred to Khalil, Lawarrée, and Yun (2010). 17

19 whether the same property remains with costly information acquisition for the supervisor; that is, collecting additional report from the agent can improve the LCCP c contract for the principal. We still consider the optimization problem in Appendix D. However, in order to let the supervisor incur the information acquisition cost and provide informative signal, the IC s constraint should be included in the problem. Since c > 0, we modify steps (5) and (7) in the timing of the game following Sections (3) and (4.1), and the cross-checking mechanism accordingly as follows: When the output is low (x = x L ), (a.) if the supervisor and the agent send different signals r s r a, the principal does not pay the supervisor and the agent; (b.) if both the supervisor and the agent announce r s = r a = 1, the principal pays the supervisor c/p + ɛ and the agent w 1 ɛ; (c.) if both the supervisor and the agent announce r s = r a = φ, the principal pays the supervisor ɛ and the agent w 1 + c/p ɛ. (d.) if both the supervisor and the agent announce r s = r a = 0, the principal pays the supervisor w 1 + c/p ɛ and the agent ɛ, where ɛ is a real non-negative but sufficiently small payment 12. Again, according to the cross-checking mechanism described above, It is easy to construct a similar payoff matrix for the supervisor and the agent like the one in Section (4.2). Supervisor r s = 1 r s = φ r s = 0 r a = 1 (w 1 ɛ, c/p + ɛ) (0, 0) (0, 0) Agent r a = φ (0, 0) (w 1 + c/p ɛ, ɛ) (0, 0) r a = 0 (0, 0) (0, 0) (ɛ, w 1 + c/p ɛ) 12 In real life, the minimum currency unit always provides a lower bound for this small payment and thus serves as a focal point for the value of ɛ. 18

20 In the pure strategy (perfect Bayesian) equilibrium, the supervisor and the agent truthfully report their signals to the principal and moreover, the contract under the cross-checking mechanism is collusion-proof. Thus, with costly information acquisition for the supervisor, both possibilities of bribery and extortion can still be deterred with no additional cost. Further, it is clear to see that the optimization problem of the principal under the crosschecking mechanism is in fact exactly the same as the one in Section (3), and the optimal contract under the cross-checking mechanism provides the same expected cost for the principal as the LCCP c contract; in other words, inviting a report from the agent is useless for the principal to further reduce the expected cost, when the supervisor needs to incur a cost to collect the signal of the agent s effort level. Then, we can summarize this result as follows. Proposition 2. When information acquisition is costly c > 0, the cross-checking mechanism of collecting an additional information resource from the agent cannot further improve the contract for the principal, and the LCCP c contract is still optimal. Differing from what we have shown in Section (4.1), Proposition (2) indicates that with costly information acquisition in the hierarchy, the principal cannot lower the expected cost by collecting an additional report from the agent and cross-checking both reports. Why does the same effect not remain with costly signal collection? This is because even though cross-checking mechanism can eliminate both corruptive behavior without adding any additional cost and the EF 1 constraint becomes unnecessary in the optimization problem for the principal, the IC s constraint should still be satisfied in order to encourage the supervisor to provide informative signal. As shown in the above, this results in the same expected cost between the cross-checking contract and the LCCP c contract. Baliga (1999) showed that a principal by cross-checking reports can obtain the same expected cost, no matter whether information is hard or soft. However, in reality most of hierarchical organizations, where it is likely that information acquisition is costly, do not implement such mechanisms but only require reports from their supervisors. Our result above provides a novel explanation to those observations, that is, the reason why not collecting reports from agents is not because they will always send signals to favor themselves (conversely, reports from agents would be useful for the principal, lowering the expected cost as shown in Section (4.1)), but regardless of agents reports, the LCCP c contract is optimal for principals when information is costly. 19

21 5 Conclusion In this paper, we examined how costly information collection for the supervisor influences on the optimal contract choice in the principal-supervisor-agent hierarchy. In particular, we considered the information structure that information is soft for the supervisor-agent coalition but hard for the supervisor alone. We first characterize the LCCP c contract, where bother bribery and extortion are deterred, and then show that the principal cannot implement any optimal contract with a lower expected cost than that of the LCCP c contract. This implies that both types of corruption should not occur when information collection is costly for the supervisor. Our results suggest that hiring a supervisor is still useful for the principal when information is soft, and this study provides a further support for the argument that corruption should be completely deterred in organizational hierarchies. We further introduce the cross-checking mechanism into the hierarchy, where the principal decides the wage transfers, according to both the supervisor s and the agent s reports. By utilizing the second resource of information, extortion from the supervisor becomes not a credible threat, and in equilibrium both the supervisor and the agent report truthfully to the principal. When the information collection is costless, the optimal cross-checking contract, which is collusion-proof, gives a lower expected cost to the principal compared to the one where bribery is allowed. However, interestingly, with costly information acquisition, collecting a report from the agent becomes not useful for the principal, and the LCCP c contract is still optimal. This provides an explanation of why in some hierarchical organizations principals only acquire signals from supervisor. We suggest some possible extensions of the present analysis before we close. So far, we have assumed that the information collection cost is a fixed cost and once the supervisor incurs it, she will learn the true signal with probability p and no-conclusive signal with 1 p. By doing so, it simplifies the analysis and allows us to examine the impacts of the cost on the optimal contract design. However, in reality, the probability of learning the true signal and the cost would depend on how much effort the supervisor invests in the monitoring activities. If she exerts more efforts, it is more likely to observe the true signal, but a higher cost has to be paid in this case. Therefore, it would be interesting to examine whether our results still hold under this general case. In future research, it may also be interesting to examine whether or not both bribery and extortion should still be deterred, if we consider a more general production technology, 20

22 where the agent with a low effort may also yield a high output. Another interesting extension may be to investigate how the optimal contracts we characterized in Sections 3 and 4 should be modified, when the agent has a better outside option. As these possible extensions suggest, additional work related to corruption and the optimal contract design in hierarchical organizations remains to be done. References Acemoglu, D., and T. Verdier (2000): The choice between market failures and corruption, American Economic Review, 90(1), Andrianova, S., and N. Melissas (2009): Corruption, extortion, and the boundaries of the law, Journal of Law, Economics, and Organization, 25(2), Angelucci, C., and A. Russo (2012): Moral hazard in hierarchies and soft information, TSE Working Papers , Toulouse School of Economics (TSE). Auriol, E. (2006): Corruption in procurement and public purchase, International Journal of Industrial Organization, 24(5), Baliga, S. (1999): Monitoring and collusion with soft information, Journal of Law, Economics, and Organization, 15(2), Banerjee, A. V. (1997): A theory of misgovernance, The Quarterly Journal of Economics, 112(4), Bardhan, P. (1997a): Corruption and Development: A Review of Issues, Journal of Economic Literature, 35(3), (1997b): Corruption and development: A review of issues, Journal of Economic Literature, 35(3), Cadot, O. (1987): Corruption as a gamble, Journal of Public Economics, 33(2), Che, Y. K. (1995): Revolving doors and the optimal tolerance for agency collusion, RAND Journal of Economics, 26(3),