Optimal Contracts with Random Auditing

Transcription

1 Optimal Contract with Random Auditing Andrei Barbo Department of Economic, Univerity of South Florida, Tampa, FL. April 6, 205 Abtract In thi paper we tudy an optimal contract problem under moral hazard in a principal-agent framework where contract are implemented through random auditing. Thi monitoring intrument reveal the precie action taken by the agent with ome nondegenerate probability r, and otherwie reveal no information. We characterize optimal contract with random perfect monitoring under everal information tructure that allow for moral hazard and advere election. We evaluate the e ect of the intenity of monitoring, a meaured by r, on the value of the optimal contract. We how that more intene monitoring alway increae the value of a contract when the principal can commit to make payment even if the an evaluation reveal that the agent took an action not allowed by the term of the contract. When uch commitment i infeaible and in equilibrium the agent hirk under ome realization of hi type, the value of a contract may decreae in r. JEL Clai cation: D82, D86 Keyword: Optimal Contract, Random Auditing, Commitment. andreibarbo@gmail.com; Addre: 4202 Eat Fowler Ave, CMC 206, Tampa, FL ; Phone: ; Fax : ; Webite:

2 Introduction Previou literature that examined optimal contract problem under moral hazard conidered ituation where the principal oberve ome public ignal that i imperfectly correlated with the agent action (uually the e ort exerted). Thi ignal i oberved with probability and i employed in the contract deign to provide incentive to the agent. In thi paper we examine an alternative and, to the bet of our knowledge, novel cenario, where contract are implemented through random perfect monitoring. Thi monitoring intrument reveal the precie action taken by the agent with ome nondegenerate probability r, and otherwie reveal neither the agent action nor any ignal correlated with it, uch a a meaure of output. Numerou real-world contracting environment can be captured by thi peci cation of monitoring technology. In many ituation, it i cotly or infeaible for the employer of a large workforce to ae the contribution of each worker to the aggregate output or to it quality. Intead, the employer can provide incentive with a monitoring cheme that evaluate the output of randomly elected worker. Another typical example i that of an intitution providing a ervice whoe quality i determined by it agent action. In many uch ituation, the ervice provider may not have the capability to obtain feedback from all cutomer, but only from a ample of them. The contribution of thi paper i twofold. Firt, we characterize optimal contract with random auditing under everal tandard information tructure that allow for moral hazard and advere election. Second, we examine how the intenity of monitoring, a meaured by the probability r, impact the value of an optimal contract. We how that a higher value of r increae the value of a contract if the principal can credibly commit to not void the contract when the agent fail an audit (we ay that the agent fail an audit if the audit reveal that he exerted an action not allowed by the term of the contract; otherwie, the agent pae the audit). We how then that a higher value For ome real-world example of peci c rm employing thi type of monitoring, ee Rahman (202). 2

3 of r may decreae the value of a contract if the principal cannot make thi commitment and the contract induce hirking under ome realization of a tate variable. We invetigate thi optimal contracting problem in an agency framework built on everal modeling choice. Firt, the cot incurred by the agent from performing hi action (the e ort level) depend on a tate of nature which only the agent oberve. Thu, the model combine feature of moral hazard, determined by the hidden action of the agent, with advere election, determined by the private information the agent poee about hi cot type. Second, the contract i igned at an ex-ante tage, before the agent learn hi type. Contract with random auditing can alo be characterized with a more typical interim contracting aumption; the ex-ante peci cation we adopt allow examining the impact of the principal ability to partially inure the agent againt unfavorable draw of hi type that induce hirking, through a commitment to not void the contract when thi hirking i detected. Third, in our baeline peci cation of the model, we aume that the principal lack the ability to make uch a commitment. Finally, we aume away pre-play communication and examine eparately the optimal contract when communication i feaible. We characterize the optimal contract in thi etting and compare it with two benchmark, the rt-bet contract under full information, and the contract under pure moral hazard with no advere election. Under full information, monitoring play no role and the contract peci e a contant wage that perfectly inure the agent. If the agent i rik neutral over monetary tranfer (more preciely, if hi utility function, which i everywhere aumed eparable in monetary tranfer and cot of e ort, i quailinear in the tranfer), then the rt-bet contract can be implemented under aymmetric information acro all model peci cation. Thi implie that with a rik neutral agent, the intenity of monitoring ha no impact on the value of the optimal contract. On the other hand, if the agent i rik avere and there i moral hazard, the principal need to olve the uual trade-o between incentive and rik. Contract under moral hazard pecify a contant wage to be paid to the agent when no audit i performed, which we refer to a a alary, and type-or-action-contingent 3

4 wage for ituation when an audit i performed. When the agent cot type i alo obervable ex-pot with an audit, the action-and-type-contingent wage promied if the agent pae the audit i higher than the alary for ome type - thee type receive a reward when they pa an audit - while for the remaining type, it equal their alary. Thu, conditional on exerting e ort, the agent prefer being audited. 2 When the type i not obervable ex-pot, the action-contingent wage i higher than the alary for ome action allowed by the term of the contract, but lower than it for certain allowed but low level of e ort. Incentive proviion with a one-dimenional allocation pace in the preence of moral hazard and advere election may thu require that the agent be ometime penalized relative to hi alary even if he pae an audit. A econd objective of the paper i to examine how the intenity of monitoring impact the value of a contract and the role played in thi context by a credible commitment of the principal to make payment even when the agent fail an audit. While thi type of commitment hould be valuable, i.e., it hould reult in a weakly higher value of the contract, it i le clear a priori how it a ect the relationhip between the intenity of monitoring and the value of the contract. For all information tructure that we conider, if it i optimal to induce the agent to exert e ort under all potential cot type, implying that an audit i never failed, a higher value of r increae the value of the contract. On the other hand, when an optimal contract induce hirking for ome type, more frequent monitoring can reduce the value of thi contract if the principal cannot make that commitment. Thi occur when r i high and thu the agent i likely to be detected when hirking, requiring the principal to pay a large rik premium ex-ante. If the principal can commit, he avail of thi tool to reduce the diperion in the et of poible ex-pot wage faced by the agent and thu to lower the rik premium that need to be paid. In thi cae, the increaed power of incentive determined by the higher probability of monitoring render again the value of the contract be everywhere increaing in r. In many employment ituation, the cot of performing 2 Mookherjee and Png (989) conider a model that exhibit the ame preference of the agent for being audited. 4

5 obervationally identical tak by a worker may depend on circumtance which are not oberved by the employer or cannot be contracted upon. 3 It i well known that in uch cae, inuring the rik-avere worker againt high-cot realization may improve the value of the employment contract if the employer i approximately rik neutral. 4 Thi article ugget that when uch inurance i not feaible, and thu an employee cannot repond to high-cot realization by adjuting the e ort level exerted or hirking, a high frequency of monitoring may be uboptimal. A an extenion, we alo examine optimal contract with random auditing when pre-play communication i feaible. In uch ituation, the principal can require the agent to declare hi private information after he learn it, but before it i determined whether an audit i performed or not. Thi information can be employed to adjut the wage paid when an audit i not performed. Unlike the cae where communication i not feaible, the agent i never penalized when an audit i performed provided that he pae it. Intead, when an audit i paed, ome agent type receive a reward, while other type, for which the alary i u cient to provide incentive to exert e ort, receive only their alary. Similarly to the cae of pure moral hazard, the additional dimenion on which the contract term can be peci ed when communication i feaible allow providing incentive while limiting the audit rik that the agent i ubjected to. Our paper contribute to the literature on optimal contract by conidering a novel type of monitoring technology. The random nature of thi technology relate it cloet to the tream of literature that tudie the deign of optimal contract with cotly tate veri cation. The eminal paper in thi literature i Townend (979) who examined determinitic tate veri cation in an optimal inurance contract problem. Baiman and Demki (980) allowed for a potentially random acquiition of an additional informative ignal of the agent action conditional on any particular 3 For intance, the activity of a rm may require worker to perform tak which are obervationally identical to an outide party, but which may incur di erent cot on the worker depending on the peci c of the particular ituation in which that tak i performed. In other cae, the tak may be identical acro di erent ituation, but the worker ability may vary thu inducing a variance in the cot of performing that tak acro di erent worker. Finally, in other cae, the cot of performing a particular tak at a given time may depend on the phyical or mental tate of the worker at that time or on hi peronal opportunity cot of the time required for ful lling that tak. 4 See Knight (92) or Kihltrom and La ont (979). 5

6 oberved outcome. Border and Sobel (987) and Mookherjee and Png (989) conidered ituation where the tate veri cation i interpreted a an audit of a dicloure made by the agent regarding an outcome (hi income) which i determined in a tochatic manner by an unobervable action choen by the agent. The latter article how that optimal monitoring require random veri cation and, imilarly to a nding from our paper under certain information tructure, that the agent hould not be penalized when an audit reveal that he reported thruthfully. Strauz (2005) tudied the trategic e ect of the timing of veri cation in an agency model, ditinguihing between veri cation performed during and after the agent take hi action. 5 Our paper di er from thi literature in that we conider that the audit reveal the action taken by the agent rather than the tate. The framework i introduced in ection 2. In ection 3 we characterize the optimal contract with random auditing, including the two benchmark correponding to the cae of complete information and of pure moral hazard, repectively. Thi ection i alo where we examine the impact that the intenity of monitoring ha on the value of a contract and the role of commitment in thi context. In ection 4, we tudy optimal contract with communication. Section 5 conclude. 2 The Framework There are two player, a principal (P) and an agent (A). P own a rm and o er A a contract to work for thi rm in exchange for monetary compenation. A can accept or reject the contract. If A accept it and exert e ort e in ervice of the rm during the period of the contract, he produce an output whoe value i y(e), where y 0 () > 0 and y 00 () 0. Thi output i entirely appropriated by P. P i rik neutral, and thu hi payo when A exert e ort e and i paid a wage w i y (e) w. A preference are eparable in wage and e ort, and are repreented by a utility u(w) c(; e). The function u : R! R capture A preference over net monetary tranfer; we aume u 0 () > 0 5 More recent contribution to thi literature are Ben-Porath, Dekel, and Lipman (204) and Mylovanov and Zapechelnyuk (204) who tudy optimal allocation problem with tate veri cation when no tranfer are allowed. 6

7 and u 00 () 0, and normalize u(0) = 0. The cot for A of exerting e ort e i c(; e), where (i) i a random variable that take value in [; ] R, with a continuou denity function f () > 0, and (ii) c (; ) i a function with c (; 0) = 0, c e > 0, c ee > 0, c > 0 and c e > 0, for all 2 [; ] and e 0. In the following, a tandard in the literature, we frequently refer to a A type. A doe not know hi type at the time when he i preented with the contract, but upon accepting the contract, he oberve it before chooing the e ort level. The utility of A outide option i u. The function y (), u () and c (; ) are aumed to be twice continuouly di erentiable. We alo aume that the et of feaible e ort level i compact or otherwie that y() i bounded on R +. P doe not directly oberve the e ort e exerted by A or the output y(e). 6 Intead, he own a monitoring intrument which allow oberving e with probability r 2 (0; ). With probability r, P doe not oberve either e or any ignal correlated with e. We conider the value of r to be exogenou and public information. 7 Monitoring i random and A doe not know at the time when he chooe the e ort level whether or not P will oberve it. At the end of the contract period, it i public information whether or not P performed the audit, and the e ort level e when an audit wa performed. Given a et of allowed e ort level by a contract, if P acquire evidence through an audit that A e ort level i not in thi et, i.e., if A fail an audit, then P can void the contract and no tranfer are made. Unle peci ed otherwie, we aume that P cannot credibly promie ex-ante not to void the contract in uch circumtance. P can o er contract with wage chedule that are de ned contingent on all obervable. 8 More preciely, P can o er a contract of the form E; w n ; fw(e)g e2e, where (i) E R + i a et of allowed e ort level, (ii) w n i the wage paid to A if no audit i performed, and (iii) w(e) i the wage paid if an audit i performed and it reveal that A exerted e ort level e 2 E. One can think of 6 In line with the motivating example from Introduction, we aume that P employ a large number of agent, and that while he may oberve an aggregate output, thi carrie virtually no information of an individual contribution. 7 The value of r can be eaily endogenized by auming a cot of monitoring for the principal that depend on r. Since we do evaluate later the marginal increae in the value of a contract determined by an increae in r, the optimal value of r would then be determined by etting thi equal to the correponding marginal cot of increaing r. 8 Two aumption are made here. Firt, when an audit i performed, P obtain publicly veri able evidence of A e ort. Second, P can credibly promie di erent wage depending on whether or not an audit i performed. 7

8 w n a a alary, or bae wage, o ered to the worker a long a he i not caught hirking, and of the di erence w(e) w n a a wage adjutment implemented when an audit i performed and A pae it. A we how later, under certain information tructure, thi wage adjutment i nonnegative, i.e, it contitute a reward, but under other, it may be negative for low e ort level. 9 We complete the preentation of the framework with everal obervation. Firt, we note that the contract de ned above i deigned on contingencie determined trictly by ex-pot obervable outcome. Thi implicitly aume away pre-play communication. However, in principle, P could alo o er a contract of the type fe () ; w () ; w n ()g 2[;], which require an explicit dicloure of after A learn it, but before he i informed whether or not an audit i performed. P would then employ thi meage to adjut the wage paid when there i no audit and thu no obervable action. While ituation with pre-play communication do frequently emerge in real world, in many other employment ituation, random monitoring i ued preciely o a to reduce the adminitrative burden. In thi cae, requiring all worker to dicloe their private information, or equivalently to elect a contract out of a menu, may be adminitratively demanding and infeaible. We therefore focu the analyi on contract without communication, and characterize eparately in ection 4, a an extenion, the optimal contract when communication i feaible. 0 Second, we aumed for implicity that the lower bound i zero on the et of e ort level that A may exert and yet not be detected to be hirking in the abence of an audit. Thi modeling peci cation can be modi ed at the cot of adding ome light complication to have a poitive lower bound on thi et, and thu to allow capturing more realitic ituation where worker cannot "hirk in plain view". Finally, we aumed that P can perfectly meaure A e ort with an audit. Thi can be relaxed to aume that an audit only reveal a ignal correlated with the e ort, a in tandard moral hazard 9 While not explicitly modeled here, one can think of thi game a a tage play of a repeated game and of the wage de ned in the contract a promied continuation value to the agent under variou contingencie. We are currently working on a dynamic verion of thi model where thee peci cation are explicitly modelled. 0 See, for intance, Melumad and Reicheltein (989) for a dicuion on the value of communication in agencie. 8

9 problem. Thu, our model i a particular cae of a generic principal-agent model where P oberve a ignal informative of A action only with a nondegenerate probability. 3 Analyi A benchmark, we derive rt the optimal contract under two alternative cenario to the richer model introduced above. Firt, we elicit the e cient outcome in thi framework by examining the cae of full information, i.e., with no advere election or moral hazard, where both A type and action e are contractible upon. Second, we conider the cae of pure moral hazard, i.e., with no advere election, where A type i obervable ex-pot when an audit i performed and thu alo contractible upon. In both model we maintain our aumption of an ex-ante participation contraint for A. To implify the expoition, when tudying both benchmark we focu on the cae where it i pro table for P to induce all type of A to exert e ort. We then conider the general cae where thi aumption i dropped when tudying the full- edged model with moral hazard and advere election. 3. The Full-Information Benchmark When P oberve ex-pot both A type and the e ort he exerted, monitoring play no role. P thu o er a contract fe 0 (); w 0 ()g 2[;], where (i) e 0 () i the e ort required from type, and (ii) w 0 () i the wage promied to type in exchange. 2 The only contraint that P face i A participation Since random auditing play a role only under moral hazard, we forgo dicuing the le intereting benchmark with advere election but no moral hazard. 2 It i implicitly aumed here and in the ret of the paper that uch a contract i binding for A after he learn hi type, and thu we do not need to account for A participation contraint at an interim tage. 9

10 contraint, o the optimal contract under full information i the olution to the problem Z max fe()0;w()0g 2[;] Z.t. [y (e()) w()] f()d () [u (w()) c(; e())] f()d u (2) The next propoition, whoe proof i traightforward and thu omitted, elicit the condition de ning the correponding optimal contract when A i rik avere. 3 Propoition Aume u 00 (w) < 0 for all w. Alo, aume that it i optimal for P to induce all type of A to exert e ort. The optimal contract under full information i determined by w 0 () = w 0 for all 2 [; ] and ome w 0 2 R +, (2) ati ed with equality, and u 0 (w 0 ) c e(; e 0 ()) = y 0 (e 0 ()) (3) Since providing A with incentive to exert e ort or reveal information i unneceary, P inure A and o er a contant wage acro all tate. The condition in (3) equate the marginal cot and marginal bene t for P of implementing e ort. An additional amount of e ort e increae type cot by c e (; e())e; to compenate it, P ha to increae A wage by u 0 (w) c e(; e())e. Since the return for P from A additional e ort i y 0 (e()) e, P et e 0 () o a to atify (3). Note that ince e 0 () varie acro type (it decreae in ), the utility delivered ex-pot to each type, u(w 0 ) c(; e 0 ()), varie a well (the e ect of on thi utility i ambiguou), implying that ome type of A enjoy ex-pot more than their reervation utility u, while other le. Finally, it i traightforward to ee that when A i rik neutral, the optimal e ort pro le i determined by c e (; e 0 ()) = y 0 (e 0 ()), and that any wage pro le fw 0 ()g 2[;] atifying R w 0()f()d = 3 By the boundedne of y() and the continuity of the relevant function, an optimal contract exit. Moreover, it i unique up to a et of zero meaure. 0

11 u + R c(; e 0())f()d and w 0 () 0 for all 2 [; ] i optimal. 3.2 The Pure Moral Hazard Benchmark When an audit reveal both the e ort level exerted by A and hi type, 4 P o er a contract o nw n; fe (); w ()g 2[;] where for each type, (i) w n i the alary, paid if no audit i performed, (ii) e () i the e ort required, and (iii) w () i the wage paid if an audit reveal that A exerted at leat e ort e (). Since c e > 0, type of A exert either e ort e () or no e ort. To implement e (), the contract mut thu atify the incentive compatibility condition ru(w ()) + ( r) u(w n) c (; e ()) ( r) u(w n ), for any 2 [; ]. The optimal contract then olve the problem max w n 0;fe()0;w()0g 2[;] Z Z [y (e()) rw()] f()d ( r) w n (4).t. ru(w()) c(; e()) 0 for all 2 [; ] (5) [ru (w()) c (; e ())] f()d + ( r) u (w n ) u (6) The next propoition, whoe proof i in appendix A, elicit the condition that determine the optimal contract when A i rik avere, and the e ect of r on the value of thi contract. 5 To focu the expoition, we aumed here that the contraint w n 0 doe not bind and then conider the generic cae when thi contraint may bind for the model with moral hazard and advere election. Propoition 2 Aume that u 00 (w) < 0, for all w. Alo, aume that it i optimal for P to induce all type of A to exert e ort. The optimal contract under pure moral hazard i determined by (5), 4 An alternative peci cation of a model with pure moral hazard i one where the type i obervable ex-pot even without an audit, i.e., with probability. In line with the dicuion from ection 2, we focu in thi paper on modeling ituation where it i infeaible for P to acquire on a regular bai information about all employee, be that their e ort level or their cot type. We therefore choe the modeling peci cation a de ned above. 5 Given propoition 2, the optimal contract can be computed in principle a follow. Firt, (8) and a binding (5) determine the pair (w (); e ()) for any in the et on which w () > w n ; clearly, thi et depend on w n. On the other hand, (8) determine e () for with w () = w n alo a a function of w n. Subtituting thee into the binding contraint from (6) determine w n, and then the ret of the contract.

12 (6) ati ed with equality, and for all 2 [; ], w () w n 0, and = 0 whenever ru(w ()) c(; e ()) > 0 (7) u 0 (w ()) c e(; e ()) = y 0 (e ()) (8) The value of the optimal contract i increaing in r. The participation contraint in (6) i ati ed with equality; otherwie P can reduce the alary w n without a ecting (5). (8) equate again, for each type, the marginal bene t and marginal cot for P of implementing additional e ort. To undertand (7), note that ince A i rik avere, P aim to minimize the wage rik impoed on A, ubject to providing the right incentive. If, contrary to (7), w () < w n, then u 0 (w ()) < u 0 (w n ) and thu the cot for P of delivering additional utility to A i lower when done by mean of increaing w () than by that of w n ; therefore, P can increae w () and decreae w n o that A participation contraint in (6) continue to be ati ed but with a lower expected wage paid. On the other hand, when w () i et higher than w n, the correponding rik i impoed on A o a to create incentive to exert e ort; thi i again done with a minimum variance in wage, and therefore the incentive contraint in (5) bind, a tated by (7). A propoition 2 ugget, the optimal contract under pure moral hazard eentially peci e a alary w n to be paid to all type a long a A i not caught hirking, and a reward w () w n > 0 o ered to ome type when they pa an audit. A we how in appendix A, the e ort pro le e () i decreaing, the wage pro le i generically non-monotonic, while the revenue generated by di erent type, y (e ()) rw () ( r) w n, i decreaing in. In term of ex-pot experienced utility, type with w () > w n enjoy le than their reervation utility, while ome of the remaining type enjoy more. 6 6 To ee thi, note that the expected utility delivered to type, i.e., ru (w ()) + ( r) u (w n ) c (; e ()) equal ( r) u (w n ) for with w () > w n and (5) binding, and i higher than ( r) u (w n ) for the ret. Since on average the utility experienced by A i u, it mut be that ( r) u (w n ) < u. 2

13 The following propoition conider the cae when A i rik neutral. The reult follow immediately from the fact that there exit a wage pro le o nw n; fw ()g 2[;] that implement the full-information e ort pro le fe 0 ()g 2[;], while delivering A the ame ex-ante expected wage a under full information. 7 Therefore, if A i rik neutral, P can attain the ame payo a under full information. Thi payo i independent of the intenity of monitoring. Propoition 3 Aume that u(w) = w, for all w. Then e () = e 0 (), for all 2 [; ]. Moreover, the value of the optimal contract equal that under full information. 3.3 The Model with Moral Hazard and Advere Selection It i traightforward to ee that whenever it i optimal for P to induce ome particular type to exert e ort, it mut be optimal to alo do o for all lower cot type. We conider therefore contract where P induce type 2 [; b] to exert e ort, with the threhold b 2 [; ] optimally choen by P. To implify the expoition, we implicitly aume in mot of the enuing analyi a et of model parameter uch that in the correponding optimal contract, the alary w n i nonnegative. We then preent in propoition 5 the condition de ning the optimal contract when we allow for the nonegativity contraint on w n to potentially bind. 8 Principal Problem By the Revelation Principle, one can think of P problem a that of elect- o nb 2 [; ]; w n ; fe (); w ()g 2[;b] that extract A private information ing an optimal contract from type in the et [; b], induce each type 2 [; b] to exert e ort e (), and the type in (b; ] 7 For intance, the wage chedule de ned by w () = c(; e0()) for all, and r wn = u ati e (5) and (6), r and thu implement fe 0()g 2[;]. Moreover, R [rw() + ( r) wn ] f()d = u + R [c(; e0())] f()d. Thu A expected wage equal that under full information implying that thi contract i optimal ince it value attain the theoretical upper bound, the value under full information. 8 Note here that () from P problem enure that in any incentive compatible contract, w() > 0 for all 2 [; b]. 3

14 to hirk. The optimal contract i thu the olution to the problem max b2[;];w n 0;fe()0;w()0g 2[;b] Z b [y (e()) rw()] f()d ( r) w n (9).t. 2 arg max [ru (w(e)) c (; e (e))], for all 2 [; b] (0) e2[;b] ru (w()) c (; e ()) 0, for all 2 [; b] () max [ru (w(e)) c (; e (e))] 0, for all 2 (b; ] (2) e2[;b] Z b [ru (w()) c (; e ())] f()d + ( r) u (w n ) u. (3) where (0) i the incentive compatibility condition that induce type in [; b] to truthfully reveal themelve, while (2) induce type in (b; ] to hirk rather than exert an e ort level peci ed for one of the type in [; b]. While (0) i a omewhat tandard incentive compatibility condition under advere election, the peci c form of () and (2) di er from other type of incentive compatibility contraint under moral hazard from the literature and are determined by the particular type of monitoring technology (random perfect monitoring) that we examine here. The following lemma implie that we can replace () and (2) with the weaker condition from (4) in the above problem. Lemma 4 Any contract that ati e (0), will atify () and (2) if and only if ru (w(b)) c (b; e (b)) 0, and = 0 whenever b <. (4) Proof. Conider rt the cae when b <. We aume throughout that (0) i ati ed and tart by howing that then (4) implie () and (2). Note rt that ru (w()) c (; e ()) ru (w(b)) c (; e (b)) ru (w(b)) c (b; e (b)) for all 2 [; b], where the rt inequality i implied by (0) and the econd by b. Therefore, (4) implie (). Next, ince whenever b, we have ru (w(e)) 4

15 c (; e (e)) ru (w(e)) c (b; e (e)) for any e 2 [; b], it follow that max ru (w(e)) e2[;b] c (; e (e)) max ru (w(e)) c (b; e (e)) = ru (w(b)) c (b; e (b)), where the equality follow from (0). Thu, e2[;b] (4) implie (2). For the convere, note that () immediately implie ru (w(b)) c (b; e (b)) 0. Auming by contradiction that ru (w(b)) c (b; e (b)) > 0, by the continuity of c () in, there exit " > 0 uch that for all 2 (b; b + "), we have ru (w(b)) c (; e (b)) > 0, which contradict (2). When b =, then (2) i automatically ati ed, while from the above argument, it i clear that () i ati ed if and only if ru (w(b)) c (b; e (b)) 0. To olve for the optimal contract, we employ the tandard Firt-Order Approach. Lemma 5 validate thi approach in the current framework by howing the equivalence between the incentive compatibility of a contract with repect to truthful type revelation, on the one hand, and the rt order condition of A problem in (0), plu the monotonicity of the e ort pro le e(), on the other. It proof, which build on a tandard trategy in the literature, i preented in appendix A2. Lemma 5 A contract induce truthful type revelation for all 2 [; b] if and only if e 0 () 0 a.e. 2 [; b] (5) ru 0 (w()) w 0 () = c e (; e())e 0 () a.e. 2 [; b] (6) To keep the expoition in the main text imple we will ignore the monotonicity contraint in (5) and olve the relaxed problem, a de ned by (9), (3), (4) and (6). We preent the analyi of the optimal contract problem for the cae when the monotonicity contraint from (5) bind in appendix A6. Note alo at thi point that (5) and (6) imply that in any incentive compatible contract w 0 () 0 (and alo that w 0 () < 0 whenever e 0 () < 0). Finally, given our aumption that w n 0 doe not bind, the participation contraint in (3) mut bind at optimum ince otherwie w n can be reduced to increae the value of the contract. We conider therefore in the following that (3) i ati ed with equality. 5

16 Optimal Control Approach We conider rt the cae where A i rik avere. To olve P problem we employ method from optimal control theory. We rt recat the problem in term of induced utilitie u n u (w n ) and u() u (w ()), for 2 [; b]; thee utilitie will replace the repective contingent wage a P choice variable. By denoting the invere utility function h u, de ned on the range of the function u, we have that w n = h(u n ) and w () = h (u ()). 9 Under thee tranformation, (0) become 2 arg max [ru (e) c (; e (e))], and o, under the Firt e2[;b] Order Approach, the incentive compatibility condition in (6) i ru 0 () = c e (; e ()) e 0 (). Given thi, the control variable in the optimal control problem i x() e 0 (), while the tate variable are e() and u(). In addition, to account for the participation condition in (3), we introduce a new tate variable v () Z [ru () c (; e ())] f()d (7) and rewrite the binding contraint in (3) a the tranverality condition v (b) = u n u ( r) u n. The other tranverality condition on v i v () = 0. There are no tranverality condition on the two remaining tate variable, e and u. We olve for the optimal contract in two tep. Firt, for any xed value of u n, we olve an optimal control problem where the deciion variable are b and fx()g 2[;b]. In the econd tep, we maximize the reulting optimal value function with repect to u n, a a tandard tatic optimization 9 Thi tranformation require the additional aumption that for every e ort level there exit a wage uch that the participation contraint of the agent i ati ed (ee Bolton and Dewatripont (2005) pp. 54.) 6

17 problem. The optimal control problem in the rt tep i max b2[;];fx()g 2[;b] Z b [y (e()) rh (u())] f()d (8).t. e 0 () = x() (9) u 0 () = r c e (; e ()) x () (20) v 0 () = [ru () c (; e ())] f() (2) v () = 0; v (b) = u n (22) ru (b) c (b; e (b)) 0 and = 0 if b < (23) Current exitence theorem for olution of optimal control problem do not yield the complete et of propertie of the olution to the above problem required in the enuing analyi. We therefore make aumption 7 preented below in the following. Part (i) of the aumption can alternatively be derived a an implication of ome u cient boundne condition on c ee and c e. 20 Part (ii) enure that the olution for the optimal cuto b i determined by the tandard in the literature (equality) condition from (33). Part (iii) enure that u n i determined optimally by a tandard rt-order condition and that the Dynamic Envelope Theorem ha the tandard form. The upercript b elicit the fact that the repective trajectory i the olution correponding to a given cuto b. De nition 6 We ay a function i C () p if it i continuou and piecewie continuouly di erentiable. Aumption 7 (Exitence and Smoothne) (i) For any xed b, there exit a olution to 20 The two Filipov-Ceari type exitence theorem that could potentially be applied to a ituation where the Hamiltonian i linear in the control variable and the control ha an unbounded upport are preented in ection.c on page 392 in Ceari (983). Theorem.4.vii doe not apply a tated ince none of the growth condition are ati ed. However, thee growth condition are employed in the proof of the theorem to conclude that the value function of the correponding optimization problem i bounded. In our cae, for any xed value of b, the boundedne follow from that fact that the value i lower than that of the relaxed problem where condition (9), (20) and (23) are dropped and r =, i.e., by the value of the contract under full information, which i nite. The theorem can then be applied under the additional aumption that c ee and c e are bounded, which are ued to infer the required propertie on what the theorem in Ceari (983) denote by A 0(t; x) and B(t; x). 7

18 (8)-(23) with the correponding tate variable e b (); u b () 2[;b] being C () p The function b! e b () and b! u b () are C () p (8)-(23) i continuouly di erentiable a a function of u n and of r. function of. (ii) for all 2 [; ]. (iii) The optimal value of problem The Hamiltonian aociated with the problem (8)-(23) i H (e; u; v; x; ; 2 ; 3 ; ) [y (e) rh (u)] f() + x + 2 r c e (; e) x + 3 [ru c (; e)] f() (24) Since thi Hamiltonian i linear in the control variable x, while the domain of x i unbounded, 2 a olution to thi problem necearily involve a o-called ingular control, i.e., it = 0 for all. 22 By the Pontryagin Maximum Principle, 23 there exit C () p function (), 2 () and 3 (), and a calar, uch that the following condition are necearily ati ed at the optimum 2 Recall that we are olving the relaxed problem where we drop the monotonicity condition in (5) and thu the domain i R. If intead we incorporate that condition, the olution may involve a o-called bang-ingular-bang = 0 when x() < 0, and > 0 when x() = 0. See appendix A5 for the See, for intance, page 247 in Bryon and Ho (975) for a dicuion of ingular control on unbounded domain. In regard to that dicuion, note that ince in our problem there are no initial or terminal condition on the tate variable e and u, which are thoe a ected by the control x, the optimal control will not require Dirac function impule at or meant to generate jump to the ingular olution. 23 See Theorem 4.2 on page 8 in Caputo (2005) for a more tandard verion of thi reult, or Theorem on page 78 in Seiertad and Sydaeter (987) for a verion that account for the tate contraint at b in (23). 8

19 olution of problem 0 () = 0 2() = 0 3() = = () + 2 () r c e (; e ()) = = y0 (e) f() 2 () r c ee (; e ()) x () + 3 () c e (; e ()) = rh0 (u ())f() 3 () = 0 (28) () = 0; (b) [ru (b) c (b; e (b))] = c e (b; e (b)) (29) 2 () = 0; 2 (b) [ru (b) c (b; e (b))] = r (b) 3 () 2 R; 3 (b) 2 R (3) 0, with = 0 and b = if ru (b) c (b; e (b)) > 0 (32) In addition, ince b i a choice variable, we have the condition H (e (b) ; u (b) ; v (b) ; x (b) ; (b) ; 2 (b) ; 3 (b) ; b) 0, and = 0 if b < (33) which i the tandard neceary condition for free end-time optimal control problem. 25 Condition (25) equate the marginal cot, (), and marginal bene t, 2 () r c e (; e ()), for P of decreaing the level of e ort required from type. 26 There alo exit a econd-order neceary condition, which in the cae of a ingular control take the form of the o-called generalized Legendre-Clebch condition. 27 A we how in appendix A4, in our problem, thi condition i ati ed if, for intance, c ee 0 along the trajectory of the 24 Note that (29) i redundant given (25) and (30), which i why it i not ued when deriving the optimal contract. 25 See, for intance, Theorem 0.2 on page 266 in Caputo (2005). The interpretation of (33) follow from the fact that the value of the Hamiltonian at capture the total value to P (or virtual urplu) generated by type. 26 See page 89 Caputo (2005) for an interpretation of the cotate variable in dynamic optimization problem. Note that in our cae, () < 0, a that cotate variable capture the bene t of decreaing the tate variable e(), rather than increaing it, ince e 0 () < 0. On the other hand, 2() > 0, a it capture the bene t of decreaing u(). 27 Alo refered to a the Kelley condition; ee, for intance, page 246 in Bryon and Ho (975). 9

20 olution to (25)-(33). 28 Thi additional aumption on c (; ) i only u cient, not neceary for the generalized Legendre-Clebch condition to be ati ed. Lemma 8 tate the u ciency of condition in (25)-(33) for the problem (8)-(23), and the uniquene of the correponding olution. o Lemma 8 (Su ciency and Uniquene) If nb ; fe (); u (); v (); x ()g 2[;b] ati e (25)- (33) with cotate variable f (); 2 (); 3 ()g 2[;b], then it i the unique olution to (8)-(23). Proof of lemma 8. We employ the Arrow Su ciency Theorem (ee, Theorem 3.4 on page 60 in Caputo (2005)) auming rt that b i not a choice variable, but xed. In our cae, the o maximized Hamiltonian evaluated at the cotate function nf (); 2 (); 3 ()g 2[;b] equal [y (e) rh (u)] f() + 3 () [ru c (; e)] f() by (25). Since, a we how later, 3 () > 0, thi maximized Hamiltonian i concave in (e; u; v) and trictly concave in (e; u) by the aumption impoed on y () and c (; ) in ection 2. The Arrow Su ciency Theorem implie that the neceary condition are u cient and the uniquene of the tate variable in the olution. 29 The theorem doe not tate the uniquene of the control, but ince x () = e 0 (), the uniquene of the control follow immediately a well. To account for the fact that b i in fact a choice variable, one can then employ a reult from Seiertad (984) to conclude the claim of the lemma. Since the correponding detail are lightly more technical, they are deferred to appendix A6. To complete the derivation of the neceary condition for the problem in (9)-(3), we denote by V(u n ) the value function of the optimal control problem in (8)-(23), a a function of u n. The 28 Thi condition implie that the marginal cot of e ort c e() increae fater in e ort for higher. It i immediately ati ed if, for intance, c(; e) = c () c 2 (e), with c increaing and c 2 increaing and convex. 29 The Arrow Su ciency Theorem, a tated in Caputo (2005), require trict concavity of the maximized Hamiltonian in (e; u; v) for the uniquene of the olution. However, by following it proof, it i evident that if the maximized Hamiltonian i trictly concave in (e; u) and contant in v, a in our cae, then fe (); u ()g 2[;b] mut be unique. The uniquene of the remaining tate variable fv ()g 2[;b] follow then from it de nition in (7). 20

21 neceary rt order condition for the choice variable u n i then d du n [V(un ) ( r)h(u n )] = 0 (34) Propertie of the Optimal Contract Note rt that u n a ect the value of the contract only through u n, i.e., dv(un ) du @u n. By the Dynamic Envelope Theorem 30 n = 3 (b), and thu dv(un ) du n = ( r) 3 (b). Since (28) implie that 3 () i contant, it follow from (34) that 3 () = h 0 (u n ) = u 0 (w n ) > 0, for all 2 [; b]. Employing thi reult and (25) into the de nition of the Hamiltonian H, we combine (33) with the requirement that b = if ru (b) c (b; e (b)) > 0 from (32) to conclude the following reult. Lemma 9 An optimal contract mut atify y (e (b)) rw (b) u 0 (w n ) [ru (w (b)) c (b; e (b))], and = 0 if b < (35) Intuitively, there are two e ect of P implementing poitive e ort for a type. Firt, it generate an additional marginal revenue to P, y (e ()) rw ()). Second, it deliver an additional net utility to A from an ex-ante point of view, ru (w ())) c (; e ()), and allow reducing the alary w n. While the rt e ect can be negative in an optimal contract, condition (35) require that the um of thee two e ect at b be alway non-negative. On the other hand, when b <, ince the utility ru (w (b)) c (b; e (b)) mut be zero by (4) for incentive compatibility reaon, the marginal revenue generated by type b mut be zero a well (otherwie b would be increaed). The following lemma, whoe proof i in appendix A3, identi e a relationhip between w n and the wage pro le fw ()g 2[;b] in an optimal contract, repreenting the counterpart of (7) here. The lemma follow from (32) and the fact proved in appendix that = R b u 0 (w f()d ()) u 0 (w n). 30 See, for intance, Theorem 9. on page 232 in Caputo (2005). 2

22 Lemma 0 An optimal contract mut atify Z b u 0 (w ()) f()d u 0 (w n 0, and = 0 if ru (w(b)) c (b; e (b)) > 0 (36) ) When ru (w(b)) c (b; e (b)) > 0 (which by (4) can occur only when b = ), at optimum, P can equalize the marginal utility delivered to A through an increae of w n by a mall amount with the expected increae in utility that could be delivered to A by increaing each value w (), for 2 [; b], by the ame amount. Therefore R b u 0 (w f()d ()) u 0 (w n ) = 0. When ru (w(b)) c (b; e (b)) = 0 (which generically occur when b < ), P may not be able to perfectly equalize thee invere marginal utilitie. More preciely, he may not be able to et the wage pro le fw ()g 2[;b] low enough without inducing A to hirk for at leat ome type in [; b]. To preempt hirking, P keep the wage w () high enough and lower w n below the level that would equalize the invere marginal utilitie; therefore R b generically trict. u 0 (w f()d ()) u 0 (w n ), a elicited by (36), with the inequality being An implication of lemma 0 i that unlike the cae from ection 3.2, where the audit alo revealed A type, in the cae with advere election tudied here, the wage w () may be lower than the alary w n for ome type, i.e., A may be penalized when evaluated even if he exerted the level of e ort required for hi type. Thi i neceary a if P were to increae w () whenever w () < w n (and imultaneouly reduce w n to keep A participation contraint binding), aiming to reduce the rik to A, in order to preerve the incentive for truthful type revelation, he would alo need to increae the remaining contingent wage, including thoe higher than w n. On net, thi may ubject A to more rik, thu rendering an increae of w () uboptimal. Lemma, whoe proof i in appendix A4, determine the e ort level e () for each type a o a function of the wage pro le nw n ; fw ()g 2[;b]. Remark 2 i alo proved in appendix A4. 22

23 Lemma An optimal contract mut atify for all 2 [; b] c e (; e ()) u 0 (w ()) f() + c e (; e ()) Z u 0 (w ()) u 0 (w n f()d = y 0 (e ())f() (37) ) Remark 2 We have Z h u 0 (w ()) u 0 (w n ) i f()d > 0 for all 2 (; b). While the moral hazard in the model induce a departure from the e cient outcome by requiring a wage pro le that ubject A to rik, the advere election induce ine ciency in the choice of o the e ort level. For the given wage pro le nw n ; fw ()g 2[;b], the e ort level e () elicited by equation (37) maximize type "virtual urplu" for contracting ituation with random auditing, y(e)f() c (; e) R h i f()d. 3 Unlike the cae of pure moral c(;e) u 0 (w ()) f() u 0 (w ()) u 0 (w n ) hazard tudied in ection 3.2, P cannot implement the e ort that maximize the "ocial urplu" y(e) c(;e) u 0 (w ()). If he did, ome of the type in [; ), for which their own marginal cot of e ort i lower than that of type, would chooe it in place of their precribed e ort level. Intead, P implement the lower 32 level of e ort e () which olve (37) and eentially in a dicrete-type verion of the model would make the type jut below indi erent between hi precribed e ort level and e () while all other type in [; ) trictly prefer their precribed e ort level. The magnitude Z h i of thi downward ditortion i a ected by the poitive factor u 0 (w ()) u 0 (w n) f()d. Thi ditorting factor, which for any i a meaure of the cumulative utility gain delivered by the wage adjutment awarded to type in [; ) when paing an audit, and thu of the information rent that type i paid, i increaing in on [; ), where olve w () = w n, i.e., a long a type receive a reward when they are audited, and i decreaing on (; b]. The maximum ditortion i applied by thi factor to the type whoe wage when paing an audit equal the alary w n. 3 Unlike many other agency model, where both player utilitie are quailinear in tranfer and thu thee tranfer vanih from the expreion of the virtual urplu, what we refer to here lightly improperly a the virtual urplu alo depend on wage. The ame obervation i valid for the expreion which we refer to a the ocial urplu later on. 32 Thi follow from c e > 0, c ee > 0, y 00 < 0 and the reult of remark 2. 23

24 Note that at, (37) become ce(;e()) u 0 (w ()) = y 0 (e ()), implying the familiar no ditortion at the top property; when etting the optimal e ort and wage for type, P doe not need to account for potential deviation from lower-cot type and can implement the e cient e ort level for. Moreover, when b = and ru (w()) c (; e ()) > 0, employing (36) into (37) it follow that c e(;e ()) u 0 (w ()) = y 0 (e ()). In thi cae, and unlike mot other contracting ituation tudied in the literature, 33 the optimal contract with random perfect monitoring alo exhibit no ditortion at the bottom. Note that for type on (; b], the ditorting factor i decreaing in, a over thi range of type, the wage when paing an audit i lower than the alary. When b =, at the ditorting factor i zero. Propoition 3 collect our nding and preent the neceary and u cient condition that elicit the optimal contract in thi model when A i rik avere. 34 Propoition 3 Aume that u 00 (w) < 0, for all w. The olution for the optimal contract under moral hazard and advere election i given by (3) ati ed with equality, (4), (6), and (35)-(37). It deerve mentioning here that the contraint ru (w(b)) c (b; e (b)) 0 from (4) doe not necearily bind in an optimal contract when b =. Thi may occur if, for intance, (i) u i o u ciently high, requiring a high wage pro le nw n ; fw ()g 2[;b], (ii) the marginal output y 0 i low for high level of e ort implying that P optimally chooe to implement low level of e ort, and (iii) the marginal cot of e ort c e i low for low level of e ort, implying that a wage pro le 33 An exception are ome model with multidimenional creening; ee, for intance, Rochet and Stole (2002). 34 Given propoition 3, the optimal contract can be computed in principle a follow. When b <, then ru (w (b)) c (b; e (b)) = 0 by (35), and thu (36) i ati ed o with inequality. Then (37) determine implicitly e () for each 2 [; b], a a function of nw n ; fw ()g 2[;b]. Thi alo give e 0 () a a function of the ame wage pro le. Then, fw ()g 2[;b] i the olution of the di erential equation de ned by (6), with initial condition given by (3), ru (w (b)) c (b; e (b)) = 0 and y (e (b)) rw (b) (we need three condition becaue there are alo the two unknown w n and b). When ru (w (b)) c (b; e (b)) > 0, then b =, while from the fact that (36) i ati ed with equality, w n i determined a a function of fw ()g 2[;b]. fe ()g 2[;b] and fw ()g 2[;b] are derived then a above. While thi how that the optimal contract can in principle be computed with the condition identi ed in propoition 3, a practical numerical implementation would involve contructing a ytem of di erential equation in e() and 2() and their derivative, with the two equation obtained from the econd derivative of the equality in (25) with repect to (ee the computation of d 2 in appendix A4) and (27), and two initial condition on 2() given by (30). 24

25 choen o a atify A participation contraint and to minimize hi wage rik 35 i u cient to alo incentivize A to exert e ort. Eentially, when P need A for a job that require low and inexpenive e ort, he o er a contract that ati e A participation contraint with a mooth wage pro le acro all contingencie, which alo provide A with u cient incentive to exert e ort. The following corollary tate the intuitive fact that the revenue generated by di erent type of agent i decreaing in the value of the type. It proof i in appendix A6. Corollary 4 The revenue generated by type, y (e ()) rw () ( r) w n, i decreaing in. The next propoition conider the general cae where the contraint w n 0 may bind. 36 Propoition 5 Conider the cae where the contraint w n 0 may bind. The olution for the optimal contract under moral hazard and advere election i then given by (3), (4), (6), (35)- (37) with 3 replacing u 0 (w n ), and the following additional condition 3 u 0 (w n ) 3 0, and = 0 if Z b [ru (w ()) c (; e ())] f()d > u and w n = 0 (38) 0, and = 0 if w n > 0 (39) Proof. By (28), 3 () i again a contant, which we denote by 3. When we allow for the contraint w n 0 to potentially bind, the participation contraint in (3) may not necearily bind in the optimal contract. Thu, in (22), we have v (b) u n, and therefore in (3), 3 0, and 3 = 0 whenever v (b) > u n. Since v (b) > u n can optimally occur in a olution only when the contraint u n 0 bind, thi immediately implie (38). Next, (34) become d du n [V(u n ) ( r)h(u n )] 0 35 Thi implie that the utility u i delivered to A not only through a high value of w n, but alo through high value for the wage fw ()g 2[;b] o to reduce the dicrepancy between the wage with an audit and the alary w n. 36 Propoition 5 ugget two additional potential cae when determining the optimal contract beide that conidered in propoition 3. Firt, if R b [ru (w()) c (; e ())] f()d = u and wn = 0, then 3 i undetermined, but w n = 0 provide the additional condition for computing the contract. Second, if R b [ru (w()) c (; e ())] f()d > u and w n = 0, then (3) i no longer ati ed with equality, but 3 = 0 and w n = 0 provide the additional condition. 25