The Value of Informativeness for Contracting

Transcription

1 The Value of Informativene for Contracting Pierre Chaigneau HEC Montreal Alex Edman LBS, Wharton, NBER, CEPR, and ECGI Daniel Gottlieb Wharton June 27, 2014 Abtract The informativene principle demontrate qualitative benefit to increaing ignal preciion. However, it i difficult to quantify thee benefit and compare them againt the cot of monitoring ince we typically cannot olve for the optimal contract and analyze how it change with preciion. We conider a tandard agency model with rik-neutrality and limited liability a in Inne 1990, where the optimal contract i a call option. The direct effect of reducing ignal volatility i a fall in the value of the option and thu the agent expected wage, benefiting the principal. The indirect effect i a change in the agent effort incentive. If the original option i deeply out-of-the-money, the agent can only beat the trike price if he exert effort and there i a high noie realization. Thu, a fall in volatility weaken effort incentive. A the agency problem become weaker, the gain from increaed preciion fall toward zero. Thee reult potentially jutify pay-for-luck and the abence of relative performance evaluation. Separately, increae in informativene lead to at-the-money option being optimal. Keyword: Contract theory, principal-agent model, executive compenation, limited liability, pay-for-luck, relative performance evaluation, option, informativene principle. JEL Claification: D86, J33 pierre.chaigneau@hec.ca, aedman@london.edu, dgott@wharton.upenn.edu. We thank eminar participant at Wharton and at the Rik Theory Society 2014 meeting for helpful comment, and Shiying Dong for excellent reearch aitance. 1

2 A major reult in contract theory i the informativene principle Holmtrom 1979, Shavell 1979, Gjedal 1982, Groman and Hart 1983, Kim It argue that the principal hould maximize the preciion of the performance meaure ued to evaluate the agent. Greater preciion allow the principal to ue a cheaper contract to implement at leat the ame effort level. However, in practice, increaing informativene i cotly. Inveting in a uperior monitoring technology involve direct cot. Engaging in relative performance evaluation RPE involve the indirect cot of forgoing the benefit of pay-for-luck documented by prior reearch e.g. Oyer 2004, Raith 2008, Axelon and Baliga 2009, and Gopalan, Milbourn, and Song Potentially for thi reaon, numerou violation of RPE have been found in practice. Aggarwal and Samwick 1999 and Murphy 1999 how that CEO pay i determined by abolute, rather than relative performance. Jenter and Kanaan 2013 imilarly find an abence of RPE in CEO firing deciion. Whether thee violation are an efficient repone to the indirect cot of RPE i unclear. Bertrand and Mullainathan 2001 find that CEO are paid for poitive exogenou hock, particularly in firm with weak governance, conitent with the view that they are inefficient. Indeed, Bebchuk and Fried 2004 argue that the abence of RPE i a key piece of evidence that CEO compenation i not determined by efficient contracting with hareholder, and intead reult from rent extraction by CEO. The informativene principle argue that there are qualitative benefit to increaing ignal preciion. However, for a principal to decide whether to invet in greater preciion, he mut quantify thee benefit in particular, relate them to the underlying parameter of the contracting problem o that he can compare them againt the cot of preciion. Similarly, to evaluate whether the general abence of RPE i efficient, it i ueful to undertand under which etting the benefit of informativene are mallet, and compare them againt the cae in which RPE i particularly abent in reality. Such quantification i difficult under the general framework in which the informativene principle wa derived. A i well-known e.g. Groman and Hart 1983, in a 1 Shavell 1979 how that additional information on the agent effort ha poitive value. Gjedal 1982 and Groman and Hart 1983 how that if the information tructure A i ufficient for the information tructure B in the ene of Blackwell, then A i aociated with a lower agency cot than B. Holmtrom 1979 how that any ignal which i informative about the agent action will be included in the contract. Kim 1995 how that the information tructure A i more efficient than B if the cumulative ditribution function of the likelihood ratio under A i a mean-preerving pread of the one under B. 2

3 general etting it i not poible to olve for the optimal contract. We cannot analyze preciely how the contract change in repone to increaed informativene, and thu quantify the cot aving from contract redeign. Thi paper addree thi open quetion. We conider the tandard etting of rik neutrality and limited liability, which allow u to take an optimal contracting approach. Thee retriction lead to optimal contract that we oberve in practice a hown by Inne 1990, the agent ha a call option. A fall in the trike price increae the option delta and thu the agent effort incentive, but alo augment the value of the option and thu hi expected wage. Thu, the trike price i the minimum poible to atify the agent incentive contraint. We tart by conidering general ignal ditribution. We how that an increae in informativene in the ene of econd-order tochatic dominance ha two effect, each of which ha a clear economic interpretation. Firt, ignoring the incentive contraint, a fall in volatility directly reduce the value of the option and thu the agent expected wage. Second, the increae in preciion change the agent incentive. The heart of the paper analyze thi incentive effect and how how it direction depend on the model underlying parameter. The agent effort incentive tem from the difference in value between two option the le valuable option that he receive when he hirk option-when-hirking, and the more valuable option that he receive when he work and improve the ignal ditribution option-when-working. Change in ignal preciion affect the value of thee option differentially. If the option atifie increaing difference, i.e. effort and preciion are complement an increae in preciion augment the enitivity of the option value to effort, then a rie in informativene augment effort incentive. The principal can thu increae the trike price of the option, i.e. reduce it delta, without violating the agent incentive contraint. Thi trike price increae further reduce the expected wage, and reinforce the firt, direct effect. In contrat, if the option atifie decreaing difference, i.e. effort and preciion are ubtitute, an increae in informativene weaken effort incentive, offetting the firt effect. In the limit, it fully offet it, rendering the total benefit of preciion zero. The key reult from the general model i that we derive a imple condition, which hold for any ignal ditribution and i eay to verify, that govern whether the option atifie increaing or decreaing difference and thu whether a rie in informativene raie or lower effort incentive. We then focu the model on ditribution with a location and cale parameter, uch 3

4 a the Normal and logitic ditribution. The exitence of a cale parameter the volatility of the ditribution allow u to fully characterize change in preciion by change in thi volatility parameter. In turn, we can ue the concept of the option vega enitivity to volatility to analyze how change in preciion affect the value of the option-when-hirking compared to the option-when-working, and thu the agent incentive. The exitence of a location parameter allow u to quantify the vega of each option by comparing thi location parameter to the option trike price; the trike price in turn depend on the model underlying parameter e.g. the everity of the agency problem. In um, for ditribution with a location and cale parameter, the condition that determine whether informativene trengthen or weaken effort incentive implifie to a threhold condition for the trike price. Firt, conider the cae in which the cot of effort i high, i.e. the moral hazard problem i evere. The option will have a low trike price, o that it delta i ufficiently large to induce working. Since the trike price i low, the option-when-working will be deeply in-the-money, and the option-when-hirking i cloer to at-the-money. The vega of an option i highet when it i at-the-money. Thu, the vega of the option-whenhirking i greater, and an increae in informativene reduce it value fater than the option-when-working. Overall, the fall in volatility increae the agent incentive. Intuitively, when volatility i high, the agent effort incentive are weak becaue, even if he hirked, he would till earn a high wage if he received a poitive hock. The agent i not worried about the fact that, if he hirk and receive a negative hock, the ignal will be very low, becaue hi payoff can be no lower than zero due to limited liability. We next conider a low cot of effort, which lead to the trike price being high. Then, the option-when-hirking will be deeply out-of-the-money, and the option-whenworking will be cloer to at-the-money. Thu, the vega of the latter option i greater, and it value fall with informativene fater than the option-when-hirking, lowering incentive. Intuitively, when the trike price i high, the agent will only receive a poitive wage if he exert effort and receive a ufficiently poitive hock. When volatility fall, uch hock are le likely, and o the agent may not get paid even if he doe work. Thu, hi effort incentive decline. In um, the effect of informativene on effort incentive depend on whether the initial trike price of the option i above or below a threhold. Thu, when incentive are trong weak to begin with, e.g. for CEO rank-and-file worker, an increae in informativene further increae reduce incentive, amplifying lowering the gain 4

5 from informativene. For the Normal ditribution, the gain from informativene are monotonically increaing in the cot of effort, and thu the everity of the agency problem. In contrat, an analyi focuing only on the direct effect of informativene on the value of the option and ignoring the incentive contraint, would ugget that the gain from informativene are highet when the option i at-the-money i.e. a moderate initial trike price and a moderate agency problem. In addition to tudying whether a firm hould endogenouly chooe to increae informativene, our analyi alo invetigate the impact of exogenou change in informativene. An exogenou increae in volatility ee Gormley, Mata, and Milbourn 2013 and DeAngeli, Grullon, and Michenaud 2013 for natural experiment will increae reduce the effort incentive of agent with out-of-the-money in-the-money option. Thu, if firm recontract in repone to thee exogenou hock, firm with in-the-money option hould increae their CEO incentive relative to firm with outof-the-money option, either by granting additional option, or reducing the trike price of new grant or exiting option. 2 For tractability, the analyi feature a binary effort level. In the continuou-effort analog, in order to implement a given effort level, the contract mut enure that the agent will not deviate to a lightly lower or a lightly higher effort level i.e. incentive contraint will be local. Thi ituation reemble a binary model in which the low effort level i very cloe to the implemented high effort level. In thi cae, the threhold for the initial trike price that determine whether informativene increae or decreae effort i the mean value of the ignal. If the initial trike price i above below thi threhold, increae in informativene lower raie the trike price toward the threhold. Thu, improvement in informativene e.g. increae in tock market efficiency move the trike price cloer to the mean value of the ignal, and thu lead to at-the-money ATM option being optimal. Bebchuk and Fried 2004 argue that the almot univeral practice of granting ATM option i uboptimal and that out-of-the-money option are more effective becaue the agent only get paid if performance i very high ee alo Rappaport Such an argument ignore the incentive effect: out-of-the-money option have lower delta and thu provide fewer incentive. The analyi alo ugget that accounting or taxation conideration that 2 Acharya, John, and Sundaram 2000 alo tudy the repricing of tock option theoretically, although in repone to change in the mean rather than volatility of the ignal. Brenner, Sundaram, and Yermack 2000 analyze repricing empirically. the 5

6 favor ATM option need not induce uboptimal contracting. A recent paper by Dittmann, Maug, and Spalt 2013 alo conider the incentive contraint when aeing the benefit of a pecific form of increaed informativene indexing tock and option and imilarly how that indexation may weaken incentive. They ue a quite different etting from our, which reflect the different aim of each paper. Their primary goal i to calibrate real-life contract, and o their model incorporate rik averion to allow them to input rik averion parameter into the calibration. However, under rik averion, it i very difficult to olve for the optimal contract. They therefore retrict the contract to compriing alary, tock, and option, and hold tock contant when changing the contract to retore the agent incentive upon indexation. They acknowledge that the actual aving from indexation will be different if the principal ue an initially optimal contract and repond optimally to change in incentive. In contrat, our primary goal i theoretical. We incorporate rik neutrality and limited liability, allowing u to take an optimal contracting approach. In addition, our model allow the analyi of reduction in volatility through other mean than indexation, for example inveting in a uperior monitoring technology. Other explanation for pay-for-luck have been propoed in the literature, partially reviewed by Edman and Gabaix Oyer 2004 how that pay-for-luck may be optimal if the value of worker outide option vary with economic condition and if re-contracting i cotly. Raith 2008 how that it may be preferable to bae compenation on meaure of output rather than input when the agent ha private information on the production technology. Axelon and Baliga 2009 argue that, for contract to be renegotiation-proof, the manager mut have private information that caue him to have a different view from the board on the value of hi long-term pay. Indutry performance i an example of uch information, and o it may be efficient not to filter it out. Gopalan, Milbourn, and Song 2010 how that tying the CEO pay to indutry performance induce him to chooe the firm indutry expoure correctly. Thi paper proceed a follow. Section 1 preent the model. Section 2 how that the optimal contract take the form of a call option. Section 3 derive the gain from a reduction in the variance of the performance meaure. Section 4 conclude. Appendix A contain all proof not in the main text. 6

7 1 The Model We conider a tandard principal-agent model with rik neutrality and limited liability, imilar to Inne The timing i a follow. At time t = 1, the principal firm offer a compenation contract W to the agent worker. At t = 0, the agent chooe hi effort level e {0, e}. Effort of e = 0 i of zero cot to the agent, and e = e cot him C > 0. We will ometime refer to e = e a high effort or working, and e = 0 a low effort or hirking. At t = 1, the agent contribution to firm value output q i realized. A in the literature on performance meaurement e.g. Baker 1992, output i generally not contractible, ince it i difficult to meaure an employee contribution independently of hi colleague. Intead, contract can depend on a performance meaure ignal = q + η, where η i a mean-zero random variable that i uncorrelated with effort: E [η e] = 0. For example, η may be a market or indutry hock, the contribution of other worker, or meaurement error. We aume that output q i not contractible and the contract depend on a eparate ignal, o that we can change ignal preciion without affecting output volatility. However, the model allow for the cae in which output i contractible, which correpond to the degenerate ditribution concentrated at η = 0 i.e. ignal i perfectly informative about output. Conditional on effort e, the ignal i continuouly ditributed according to the probability denity function PDF f θ e with full upport on [, ], where the bound and may or may not be finite. Let F θ e denote the cumulative ditribution function CDF of. A high ignal i good new about effort in the ene of the trict monotone likelihood ratio property MLRP. Formally, for all θ and for all ignal 1 and 0 with 1 > 0, f θ 1 ē f θ 1 0 > f θ 0 ē f θ 0 0. Strict MLRP implie that the ditribution of performance i ordered according to trict firt-order tochatic dominance FOSD : F θ 0 > F θ e for all and all θ. The real-valued parameter θ, which lie in an interval Θ, capture the informativene or preciion of the ignal, and order the ditribution in term of econd-order 7

8 tochatic dominance. Formally, the mean of the ignal i independent of θ, and θ θ = t F θ e d t F θ e d, 1 for all t [, ], where the bound may or may not be finite. Thu, increae in θ generate more precie ignal ditribution in the ene of mean-preerving pread. Our analyi olve for the optimal contract for each given level of preciion θ. Thi approach applie to etting in which the principal cannot influence the preciion of the ignal but it i affected by exogenou force uch a technological change; the analyi derive empirical prediction on how thee change affect the form of the optimal contract and the expected wage. In addition, our approach alo applie to etting in which the principal can chooe the level of preciion θ at a cot κ θ, where κ i increaing and convex. Under the interpretation that η arie from meaurement error, removing the hock correpond to an improvement in the monitoring technology, in which cae κ θ refer to the cot of uch an improvement. For example, Cornelli, Kominek, and Ljungqvit 2013 how that board of director engage in extenive and thu cotly monitoring to gather oft information on the CEO competence, trategic choice, and effort. Under the interpretation that η i a market or indutry hock, increaing preciion θ correpond to relative performance evaluation RPE, in which cae the cot κ θ tem from two ource. Firt, it can arie from the literal cot of implementing RPE. While the actual calculation of indutry performance, given a peer group, i relatively cotle, the determination of the peer group may involve the hiring of compenation conultant. Second, the cot can alo repreent the lo of the benefit of pay-for-luck highlighted by prior work, e.g. Oyer 2004, Raith 2008, Axelon and Baliga 2009, and Gopalan, Milbourn, and Song The dicount rate i normalized to zero. Given a contract W and a level of effort e, the agent expected wage i E [W e] = W f θ e d. The agent i rik-neutral and o maximize hi expected wage, le the cot of effort. He i protected by limited liability and ha a reervation utility of zero. The principal i alo rik-neutral and chooe a contract W, an effort level e, and a preciion θ that 8

9 maximize expected output E [q], le the expected wage E [W ] and cot of preciion κ θ. Following Inne 1990, we make two aumption on the et of feaible contract. Firt, the agent i protected by limited liability, o that W 0 for all. Second, pay-performance enitivity lie between 0 and 1: W + ɛ W and + ɛ W + ɛ W for all, ɛ. Thee contraint mut be atified if the agent can freely borrow to artificially increae output and the principal can freely detroy output. If the contraint on the left did not hold, the agent would artificially increae output, thereby increaing the ignal and thu hi payoff. If the contraint on the right did not hold, the principal would exercie her control right to burn output, thereby reducing the ignal and increaing her payoff. Thee contraint can be expreed a 1 W + ɛ W ɛ 0 for all ɛ. It thu follow that W i Lipchitz continuou and, therefore, differentiable almot everywhere. Hence, with no lo of generality, we can aume that the contract W i a cadlag function atifying 0 W 1 at all point of differentiability. 3 In the firt bet, effort i verifiable. There i no incentive contraint and only a participation contraint. If the principal wihe to induce high effort, thi contraint i given by: E[W e] C 0. 2 To atify 2, the principal pay an expected wage E [W e] that equal the agent cot of effort C. Thu, if E [q e] E [q 0] > C, 3 high effort i optimal for the principal. We aume 3 throughout, ele even under the firt-bet, the principal would not want to induce effort. In the econd bet, the agent effort i unverifiable and o the contract mut atify 3 A cadlag function i everywhere right-continuou and ha left limit everywhere. 9

10 an incentive contraint. The agent will exert effort if and only if: E[W e] E [W 0] C. 4 Following tandard argument, thi incentive contraint will bind. In contrat, the participation contraint will be lack if the principal wihe to implement high effort. 4 We thu ignore it in the analyi that follow. We define X θ implicitly by X θ X θ [f θ e f θ 0] d = C. 5 We will how in Lemma 1 that X θ exit and i unique. The intuition behind 5 i that, if the agent i given a call option on, X θ i the trike price uch that working increae the value of the agent option by an amount equal to the cot of effort, o that the incentive contraint i atified with equality. We make the following aumption to enure that e = e i econd-bet optimal: E [q e] E [q 0] X θ X θ f θ ed 0. 6 The firt term, E [q e] E [q 0], i the benefit to the principal of inducing e = e. The econd term i the cot of the contract required to do o. If 6 did not hold, the principal would allow the agent to hirk, in which cae the problem would be trivial and the contract would involve a contant wage. Note that 6 implie E [q e] E [q 0] C + X θ X θ f θ 0d, which i tronger than 3, the condition that guarantee that high effort i optimal in the firt-bet. The additional term X θ X θ f θ 0d arie becaue the agent will earn rent from hirking a he may enjoy a very poitive hock and generate a high ignal; thu, he mut be offered rent for working to atify hi incentive contraint. 4 If the agent hirk, hi wage cannot fall below zero no matter how low the ignal i, due to limited liability. Thu, the expected wage upon hirking, E [W 0], i poitive. To atify the incentive contraint 4, we mut have E[W e] C E [W 0] 0, and o the participation contraint 2 i automatically atified. 10

11 The principal problem i to chooe a contract W and informativene θ to minimize the um of the expected wage and the cot of preciion, ubject to the agent incentive and limited liability contraint, plu the contraint on the lope of the contract. She chooe a cadlag function W and an informativene parameter θ to minimize E [W ē] + κ θ 7 ubject to E [W e] E [W 0] + C, 8 0 W, and 9 0 W 1 at all point of differentiability of W. 10 Our contracting problem i the dual to the one in Inne In hi model, the agent entrepreneur chooe a financing contract ubject to hi own incentive contraint and the participation contraint of the principal invetor. In our model, the principal firm chooe an employment contract ubject to the incentive and participation contraint of the agent worker. In both model, the optimal contract ha the ame form; the only difference i which party capture the rent. Since Inne tudie a financing etting, the optimal contract for the principal i debt. Thu, the agent ha equity, which i a call option on the firm aet; here, we will how that the agent receive a call option on the ignal. Another difference i that Inne feature a continuou action et. Hi focu wa to derive the form of the optimal contract and thu he wihe to do o in the mot general etting. Our goal i different: given that the optimal contract i a call option, we tudy how change in informativene affect the agent incentive and thu the trike price. We thu pecialize to a binary effort level. With a continuou effort level, a change in informativene θ may alter the optimal effort level. It i well known that olving for the optimal effort level in addition to the cheapet contract that induce a given effort level i extremely complex ee, e.g., Groman and Hart 1983, and thu many paper focu on the implementation of a given effort level e.g. Dittmann and Maug 2007, Dittmann, Maug, and Spalt 2010, Edman and Gabaix 2011 how 5 Indeed, Inne 1990 doe not olve for the optimal effort level or tudy how it i affected by the parameter of the etting, but how that an optimum exit. 11

12 that, if the benefit of effort are multiplicative in firm ize and the firm i ufficiently large, it i alway optimal for the principal to implement the highet effort level and o the optimal effort level i indeed fixed. We thu conider a binary effort etting where high effort i optimal. 2 The Optimal Contract Thi ection olve for the optimal contract for a given level of informativene θ. The analyi i imilar to Inne Our main reult will come in Section 3, which analyze the gain from increaing informativene θ. Let W θ denote the optimal contract that implement high effort for a given informativene level θ. Lemma 1 etablihe that W θ i a call option on, where the trike price X θ i choen to atify the incentive contraint 5 with equality: Lemma 1 Optimal contract For a given θ, there exit a unique optimal contract, characterized by e = e, and W θ = max{0, X θ }, 11 where X θ i determined by the unique olution of 5. The etting i lightly different from Inne 1990, ince the principal i contracting on a ignal rather than output. We how that the Inne 1990 reult of the optimality of a call option extend to thi cae, and the intuition i the ame. The abolute value of the likelihood ratio i highet in the tail of the ditribution of, o the ignal i mot informative about effort in the tail. The left tail cannot be ued for incentive purpoe due to the limited liability contraint, and o incentive are concentrated in the right tail. Thi maximize the likelihood that poitive payment are received by a working agent. With an upper-bound on the lope, the optimal contract involve call option on with the maximum feaible lope, i.e. W = 1. Lemma 2 below how that the trike price fall with the cot of effort. Lemma 2 Let X θ be the trike price in the optimal contract for a given θ. Then, X θ i trictly decreaing in the cot of effort C. 12

13 Proof. The Appendix how that the expreion 5, which implicitly determine X θ, can be rewritten a Applying the implicit function theorem yield: X θ [F θ 0 F θ e] d = C. 12 dx θ dc = 1 F X θ 0 F X θ e < By trict FOSD, the denominator in equation 13 i poitive effort improve the ditribution of the ignal. The higher the cot of effort C, the higher the agent reward mut be for improving the ditribution of the ignal, to encourage him to induce effort. Lowering the trike price raie the delta of the option the enitivity of the option to the value of and thu the agent incentive. 3 The Value of Informativene Thi ection calculate the gain from increaing informativene. Section 3.1 conider general ignal ditribution and provide a condition under which increae in informativene raie the agent effort incentive, which hold for any ditribution that atifie MLRP o that the optimal contract i a call option. Section 3.2 focue on ymmetric, unbounded ditribution with a location and cale parameter and relate thi condition and thu the effect of informativene on incentive to the initial trike price and thu the everity of the agency problem. Section 3.3 graphically illutrate the benefit of informativene for the Normal ditribution. It alo prove analytically that the benefit from informativene are monotonically increaing in the cot of effort, and thu monotonically decreaing in the initial trike price. 3.1 General Ditribution The total effect of increaing informativene on the expected wage can be decompoed a follow: d dθ E [W e] = E [W e] + E [W e] dx θ. 14 } θ {{} X θ dθ }{{} direct effect incentive effect 13

14 The firt component i the direct effect, E [W e]. Holding contant the trike θ price, an increae in ignal preciion change the value of the option; we will later prove that thi effect i negative. Thi reduction in the cot of compenation i the benefit of informativene highlighted by Bebchuk and Fried 2004 in their argument that the lack of RPE i inefficient. In the Holmtrom 1979 etting of a rik-avere agent, an increae in informativene reduce the rik borne by the agent and thu allow the principal to lower the expected wage without violating the agent participation contraint. In our etting of rik neutrality and limited liability, an increae in preciion reduce the value of the option. The econd component i the incentive effect, X θ E [W e] dx θ, which arie becaue the increae in preciion caue the trike price to rie by dx θ dθ to maintain incentive compatibility. dθ X θ E [W e] i negative any increae in the trike price reduce the value of the option and thu the cot of compenation but the ign of dx θ i unclear. dθ We thu eek to derive condition under which an increae in preciion raie or lower the optimal trike price. The following definition will be ueful: Definition 1 Let Θ, E R 2. A function g θ, e : Θ E R atifie increaing difference if, for all θ L < θ H and e L e H, g θ H, e H g θ H, e L g θ L, e H g θ L, e L. 15 It atifie decreaing difference if g atifie increaing difference. θ and e are complement if g θ, e atifie increaing difference, ubtitute if g θ, e atifie decreaing difference, and neutral if g θ, e atifie both increaing and decreaing difference. The increaing difference condition 15 mean that the incremental gain i.e., increae in the value of g from effort, g θ, e H g θ, e L, i increaing in θ. That i, effort and informativene are complement in term of their effect on g. Converely, decreaing difference mean that the incremental gain from effort i decreaing in θ. Thu, effort and informativene are ubtitute. Indeed, increaing difference i the mot common definition of complementarity, wherea decreaing difference i the mot common definition of ubtitutability. 6 In our etting, if g i differentiable, increaing 6 There i a very large literature uing thee concept for undertanding outcome of game e.g. Bulow, Geanakoplo, and Klemperer 1985 and Milgrom and Robert

15 difference i equivalent to the ingle-croing condition: g g θ, ē θ, 0 d 0. θ θ We are concerned with how change in preciion affect the incentive contraint 4. The agent incentive tem from the fact that exerting effort increae the value of hi option. If he work, hi option i worth E [W e]; we refer to thi a an optionwhen-working. If he hirk, he receive an option-when-hirking worth E [W 0]. Hi effort incentive are given by the difference in the value of thee option, i.e. E [W e] E [W 0]. 16 Since a change in preciion θ affect the option-when-working and the option-whenhirking to different degree, it affect the agent effort incentive 16. When preciion and effort are complement, i.e. E [W e] atifie increaing difference, increae in preciion augment the agent effort incentive: {E [W e] E [W 0]} > 0, 17 θ We thu wih to undertand the condition under which E [W e] atifie increaing difference. We do o by uing integration by part ee Appendix A to rewrite the agent expected wage a E [W e] = E [ e] X θ + Xθ F θ e d. 18 The third term, X θ F θ e d, the area under the CDF between and X θ. It can be interpreted economically a the value of a put option with a trike price of X: Pr < X θ e E [X θ < X θ, e] = Xθ X θ f ed = Xθ F θ e d, where the lat equality follow from integration by part. Under thi interpretation, equation 18 i the put-call parity equation. The agent call option equal the expected value of the ignal, minu the trike price, plu the value of a put option. Let π X θ, e X F θ e d denote the value of a put option with a trike price of X. 15

16 By econd-order tochatic dominance equation 1, the value of the put option i decreaing in the preciion of the ignal π X θ θ θ, e 0. 7 To tudy whether E [W e] atifie increaing difference, we examine each of the three term on the right-hand ide RHS of 18 in turn. While E [ e] depend on e, it i independent of θ ince change in θ repreent mean-preerving pread. In addition, X θ depend on θ but not e. Thu, θ and e are neutral in their effect on both of thee term, and non-neutral only in their effect on the third term X θ Thi obervation lead to the following Lemma: F θ e d. Lemma 3 The agent expected pay E [W e] atifie increaing difference if and only if the area under the CDF, X θ F θ e d, atifie increaing difference. The uefulne of Lemma 3 lie in the fact that, while the value of the call option contain everal term ee 18, the area under the CDF X θ F θ e d i a ingle term, and o it i relatively eay to verify whether it atifie increaing difference. While it may eem intuitive that the value of the call option atifie increaing difference if and only if the value of the put option atifie increaing difference, the value of Lemma 3 i that we can check whether expected pay atifie increaing difference by tudying a ingle term X θ F θ e d not that thi term can be interpreted a the value of a put option. The condition in Lemma 3 i imple to check and general: it hold for all ignal ditribution that atify MLRP. We thu apply the definition of ubtitute and complement in Definition 1 to the area under the CDF, X θ F θ e d. Thi application allow u to determine the effect of informativene on the trike price X θ, which i ummarized in Propoition 1. Propoition 1 The optimal trike price X θ i increaing in informativene θ dx θ dθ > 0 if informativene and effort are complement at the trike price X θ, decreaing in informativene dx θ dθ are neutral at X θ. < 0 if they are ubtitute at X θ, and contant dx θ dθ = 0 if they When preciion and effort are complement, exerting effort augment the value of the call option by a greater amount when preciion i high. A a reult, the agent 7 In the Black-Schole model, we have a trict inequality. Thi i becaue the Black-Schole model aume a lognormal ditribution for tock return, o an increae in preciion which correpond to a decreae in volatility affect the whole ditribution. However, in our etting with a general ditribution, a change in θ may only affect the part of the ditribution above the trike price X θ, where the put option ha zero payoff, and o it value doe not change. 16

17 marginal benefit from effort, E [W e] E [W 0], i increaing in informativene. Thi looen the incentive contraint and allow the principal to increae the trike price thu reducing the expected wage while till inducing effort. Thu, in addition to the direct benefit of informativene it reduce the expected wage, holding contant the trike price X θ, the principal further benefit from the incentive effect of greater informativene it allow the trike price X θ to increae, further reducing the expected wage. Propoition 1 in turn lead to Lemma 4 below. Lemma 4 Partial and total effect of informativene on expected wage: d E [W e] dθ > E [W e] θ Proof. From equation 14, we have: d dθ E [W e] = E [W e] + E [W e] dx θ } θ {{} X θ dθ }{{} d E [W e] and dθ direct effect = π X θ, e θ incentive effect if and only if dx θ dθ > [ 1 π ] X θ, e dxθ X θ dθ = π X θ, e [1 F θ X θ e] dx θ θ dθ. E [W e] are both negative, and the former i more negative θ > 0, i.e. effort and preciion are i.e. it abolute value i higher if and only if dx θ dθ complement. The direct effect, E [W e], i negative. An increae in preciion decreae the θ 0 and thu the expected wage. Turning to the incentive value of the put option π X θ θ effect, higher preciion augment the trike price by dx θ dθ principal to pay an additional dx θ dθ with probability 1 F θ X θ e., which in turn require the dollar whenever the price exceed X θ, which occur The ign of dx θ in turn depend on whether informativene and effort are ubtitute or complement. When they are complement, then dx θ dθ > 0. The trike price dθ increae, further reducing the expected wage and reinforcing the direct effect. When they are ubtitute, then dx θ < 0 and the effect go in oppoite direction. While the dθ value of the option decreae with preciion, the agent require a lower trike price to induce effort, which partially offet the benefit to the principal. Even when dx θ dθ < 0 and the incentive effect work in the oppoite direction to the 17

18 direct effect, it can never outweigh it. The total effect d E [W e] i alway weakly dθ negative, i.e. increaing preciion weakly reduce the expected wage. Thi reult arie from revealed preference. If reducing preciion reduced the expected wage, the principal would have added in randomne to the contract, and o the initial contract would not have been optimal. Even though the incentive effect cannot outweigh the direct effect, it i till important to conider a it affect the optimal level of preciion θ that the principal hould chooe, ince increaing preciion i cotly. 8 Indeed, it i poible that the incentive effect exactly offet the direct effect, and o that the total gain from informativene equal exactly zero: ee Appendix B for an example. 3.2 Ditribution with a Location and Scale Parameter Section 3.1 how that, with general ditribution, the effect of preciion on incentive depend on whether effort and preciion are complement or ubtitute; it alo give a condition that determine which cenario we are in. We now add more tructure to the ignal ditribution which allow u to relate whether we have complement or ubtitute to the underlying parameter of the agency problem. We can thu determine from model primitive whether change in preciion increae or decreae the agent incentive. We conider the cae in which the ignal ha a ymmetric ditribution with unbounded upport and location and cale parameter, i.e., their ditribution and denity function can repectively be written a F e = G e and f e = 1 g e. All uch ditribution can be fully characterized by their mean e and tandard deviation. Example include the Normal, logitic, Cauchy, and Laplace ditribution. Since the volatility of a ignal i the invere of it preciion, we have = 1 θ, where θ i the ame informativene / preciion parameter we had earlier. The effect of preciion on incentive depend on how change in preciion affect the value of the option-when-working compared to the option-when-hirking. The introduction of a cale parameter i ueful becaue we can fully parameterize change in preciion by change in. We can thu examine how change in volatility affect the value of thee two option and thu the agent effort incentive uing the familiar 8 Unfortunately, it i not poible to olve for the optimal level of preciion θ in cloed form. In term of comparative tatic, θ i decreaing in the marginal cot of informativene κ θ, but the relationhip with the parameter of the agency problem C i ambiguou. 18

19 concept of the option vega : the enitivity of it value to volatility. Specifically, comparing the vega of the two option will allow u to ae the effect on the agent overall incentive. The introduction of a location parameter i ueful a it allow u to relate the vega of each option to the ditance between the option initial trike price and the location parameter. The initial trike price in turn depend on the underlying parameter of the agency problem. Thu, we will be able to relate the incentive effect to the underlying parameter of the etting. The aumption of ymmetry and unbounded upport are ueful for tractability. Appendix D extend the reult to aymmetric ditribution and to ditribution with a bounded upport uch a the uniform ditribution. With a location and a cale parameter, the ignal can now be written a: = q + ε, 20 where ε ha a mean of 0 and volatility of 1, and parameterize the volatility of the ignal. Since the location of the ditribution can be altered without changing it cale, we can et E [q] = e without lo of generality. All of the reult in Section 3.1 continue to hold, except we will now denote the trike price by X rather than X θ. Propoition 2 below derive a precie condition under which Lemma 3 and Propoition 1 hold, i.e. the put option atifie increaing difference, and o the optimal trike price i increaing in informativene. Propoition 2 Effect of volatility on trike price. dx e. Thi in turn hold if and only if 2 d > 0 if and only if X > ˆX ˆX ˆX f θ e f θ 0 > C. 21 Propoition 2 tate that the incentive effect i poitive, i.e. an increae in preciion augment incentive and thu the trike price, if and only if X i below a threhold ˆX e 2. To undertand the intuition, equation 17 now become {E [W e] E [W 0]} < Since preciion can be parameterized by volatility, we are intereted in how change 19

20 in volatility increae the agent incentive to exert effort i.e. whether effort and volatility are complement or ubtitute and how thi depend on the initial trike price of the option. To highlight the dependence of the option value on the trike price, let Y e, X denote the value of an option where the mean value of the underlying variable i e and the trike price i X. We thu have Y e, X = E [W e] Y 0, X = E [W 0]. The left-hand ide LHS of inequality 22 repreent the effect of change in on incentive. Thi i equal to the vega of the option worth Y e, X minu the vega of the option worth Y 0, X. The vega of an option i alway poitive, highet for an ATM option ee Claim 1 in Appendix C, and decline when the option move either in-the-money or out-of-the-money. Thu, the vega of the option worth Y e, X i highet at X = e, and o if the option ha a trike price of X = e, then it i inthe-money by e. The vega of the option worth Y 0, X i highet at X = 0, and o if 2 2 the option ha a trike price of X = e, then it i out-of-the-money by e. Overall, at 2 2 a trike price of X = e, both option are equally away-from-the-money and have the 2 ame vega ee Claim 2 in Appendix C. Thu, increae in reduce the value of the option-when-working and option-when-hirking equally. The incentive to exert effort, Y e, 2 e Y 0, e 2, are unchanged, and o the trike price X doe not need to change. We thu have dx = 0 for X = X. d Now conider X < X. Then, Y 0, X i cloer to being ATM than Y e, X, and o it ha a higher vega. The intuition i a follow. Volatility increae the value of an option becaue the option holder benefit from it aymmetric payoff: hi downide rik i limited, but he benefit from the upide gain. Since the trike price i low, if the agent work and receive an option worth Y e, X, the expected ignal e i very far from the kink X, and thu the agent benefit little from the aymmetry. Thu, when volatility increae, a working agent gain from any upide but alo loe from any downide, and o Y e, X rie little with. In contrat, if the agent hirk and receive Y 0, X, the expected ignal 0 i cloe to the kink X. Thu, when volatility increae, a hirking agent gain from any upide but doe not loe from any downide. Thu, Y 0, X rie ignificantly with. In um, an increae in reduce the agent effort incentive, and o a fall in X i needed to retore incentive compatibility, ince 20

21 uch a fall increae the value of Y e, X more than Y 0, X. In imple language, when volatility rie and X < X, the agent think: I m not going to bother working hard, becaue even if I do, I might be unlucky and o profit will be low. I might a well hirk, becaue even if I get unlucky and profit become very low, that doen t matter, becaue I can t get paid le than zero no matter how low profit get. Finally, conider X > X. Then, ince Y e, X i cloer to being ATM than Y 0, X, it ha a higher vega. Since Y e, X i cloe to the kink, when volatility increae, an agent who work benefit from the upide potential and i protected from the downide rik. Thu, Y e, X rie ignificantly with. In contrat, if the agent hirk and receive Y 0, X, the expected ignal 0 i well below the kink. Thu, when volatility increae, the agent doe not bear the downide rik, but i unlikely to benefit from the upide potential either: even if noie i poitive, the option will till be out-ofthe-money. Thu, Y 0, X rie little with. In um, an increae in augment the agent effort incentive, and o a rie in X i poible without violating the incentive contraint. In imple language, when volatility rie and X > X, the agent think: If volatility were low, I wouldn t bother working becaue the target X i o high that I wouldn t meet it, even if I did work. But, now that volatility i high, I will work becaue if I do, and I get lucky, I ll meet the target. In the language of the general model of Section 3.1, when C i mall the agency problem i weak, then X > X and o we have decreaing difference. Since informativene and incentive are ubtitute, an increae in informativene reduce incentive and o require the trike price to fall in repone. When C i large the agency problem i trong, then X < X and o we have increaing difference. informativene can be accompanied by an increae in the trike price. An increae in Propoition 2 implie that, in all cae, improvement in informativene draw the trike price X toward X = ē. In the current dicrete model, there are two effort level, 2 ē and 0. In a continuou-effort analog, where the principal wihe to implement effort of ē, the contract mut enure induce the agent to exert effort of ē rather than ē + ε or ē ε, i.e. the incentive contraint mut be local. In our dicrete model, a local incentive contraint reemble the cae in which the implemented high effort level ē i very cloe to the low effort level 0. If ē 0, then X 0. Moreover, ince the contract induce the agent to exert effort of ē, the mean value of the ignal i ē and o an ATM option will have a trike price of ē 0. Thu, if the initial trike price i higher lower than X 0, improvement in informativene e.g. increae in tock 21

22 market efficiency will lower raie the optimal trike price toward 0, i.e. bring the option cloer to ATM. Indeed, Appendix F ketche a continuou effort model which how that increae in informativene bring the option cloer to ATM. Bebchuk and Fried 2004 argue that the almot univeral practice of granting ATM option i inefficient and that out-of-the-money option would be cheaper for the firm. Similarly, Rappaport 1999 advocate out-of-the-money option a they reward the agent only for exceptional performance. However, uch view ignore the incentive effect: out-ofthe-money option have lower delta and thu may provide the agent with inufficient incentive. Murphy 2002 note that in-the-money option would provide the tronget incentive, but that the tax code dicourage uch option. One interpretation i that the tax code lead to firm chooing ATM option when in-the-money option may be more efficient. Our analyi indeed ugget that increae in informativene lead to option optimally being cloe to ATM. 9 In addition, Propoition 2 ugget that exogenou fall in will have different effect on the incentive of agent depending on the moneyne of their option. In particular, it will reduce increae the incentive of agent with out-of-the-money inthe-money option. Thu, firm may wih to reduce the trike price of out-of-themoney option to retore incentive. Option repricing i documented empirically by Brenner, Sundaram, and Yermack 2000, although they do not tudy if it i prompted by fall in volatility. Acharya, John, and Sundaram 2000 alo tudy the repricing of option theoretically, although in repone to change in the mean rather than volatility of the ignal. Finally, note that the above analyi take an optimal contracting approach, o the lope of the contract i the maximum poible without violating the contraint 10. We thu have W = 1: the agent i the reidual claimant of any increae in the ignal a long a X θ. Thu, the principal change X θ to enure that the incentive contraint bind. An alternative approach i to retrict the contract to compriing ATM option, e.g. for accounting or tax reaon, and intead meet the incentive contraint by 9 Hall and Murphy 2000 retrict the contract to conit of option, rather than taking an optimal contracting approach, and calibrate the optimal trike price depending on the CEO rik averion, the proportion of hi wealth in tock, and the proportion of hi wealth in option. They how that, in mot cae, the range of optimal trike price include the current tock price, i.e. correpond to an ATM option. Dittmann and Yu 2011 feature a rik-taking a well a effort deciion, and retrict the contract to coniting of fixed alary, tock, and option. They how that in-the-money option are typically optimal. 22

23 varying the lope of the contract. Indeed, Murphy 1999 and Bebchuk and Fried 2004 document that ATM option are almot univerally granted. Appendix E demontrate an analogou reult for thi cae. With ATM option, we have X = e X = e and o 2 effort and preciion are ubtitute. An increae in informativene require the number of option granted to increae, to maintain incentive compatibility. An increae in the number of option granted augment the expected wage, jut like a decreae in the trike price. A a reult, the total effect of informativene on expected pay i le than the direct effect. Thu, the reult of the core model, where X > X, extend to the cae of ATM option. 3.3 Normal Ditribution We now demontrate graphically the importance of conidering the incentive contraint when evaluating the effect of informativene on incentive, i.e. tudying the total rather than direct effect. We need to aume a pecific ditribution to enable u to calculate thee derivative, and o we conider the cae of the Normal ditribution. Let ϕ and Φ denote the PDF and CDF of the tandard Normal ditribution, repectively. Then, the total effect de [W e] d E [W e] de[w e] d and direct effect E[W e] are given analytically by: [ ] X e X e ϕ X e ϕ X = ϕ 1 Φ Φ X Φ X e 23 Xθ e = ϕ. 24 Thee expreion are derived in Appendix A. In Figure 1, we illutrate thee effect numerically for a range of value of X. Note that the exogenou parameter that i changing i C, but ince X i trictly decreaing in C equation 13, there i a one-toone correpondence between C and X. Thu, we draw the graph with X rather than C on the x-axi, a i tandard for graph of option value. de[w e] = d To undertand Figure 1, recall from 14 that the total effect i given by E[W e] E[W e] dx + θ E[W e] X θ. The direct effect,, tend to zero a the trike price d approache either or. The vega of an option i greatet when the option i ATM, i.e. X = 1. An ATM option benefit mot from the aymmetry in an option payoff: a high noie realization lead to a large increae in the option payoff, but a low noie realization ha no effect a the agent will not exercie the option. 23