Incentives, Project Choice and Dynamic Multitasking * (Job Market Paper)

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1 Incentves, Project Choce and Dynamc Multtaskng * Job Market Paper) Martn Szydlowsk January 14, 2013 Abstract I study the optmal choce of nvestment projects n a contnuous-tme moral hazard model wth multtaskng. Whle n the frst best, projects are nvarably chosen by the net present value NPV) crteron, moral hazard ntroduces a cutoff for project selecton whch depends on both a project s NPV as well as ts rsk-return rato. The cutoff shfts dynamcally dependng on the past hstory of shocks, the current frm sze, and the agent s contnuaton value. When the rato of contnuaton value to frm sze s large, nvestment projects are chosen more effcently, and project choce depends more on the NPV and less on the rsk-return rato. The optmal contract can be mplemented wth an equty stake, bonus payments, as well as a personal account. Interestngly, when the contract features equty only, the project selecton crteron resembles a hurdle rate. 1 Introducton In the neoclasscal nvestment framework, frms operate a sngle technology and contnue the same actvty on a dfferent scale as they grow. Wthn ths standard paradgm, the lterature has studed how agency problems affect the optmal level of nvestment. However, the paradgm not take nto account that frms are crucally dependent on choosng the rght projects. Wth agency, executng dfferent projects makes t necessary to dynamcally change manageral ncentves. The queston s then how to ncentvze the manager to select the optmal portfolo of projects and how the cost of provdng ncentves dstorts the project selecton over tme. * I am ndebted to Jeffrey Ely, Mchael Fshman, and Bruno Strulovc for ther gudance. Ths paper has benefted from helpful dscussons wth Bruno Bas, Jance Eberly, Arvnd Krshnamurthy, Alessandro Pavan, Mark Satterthwate, and Yuly Sannkov. I would also lke to thank semnar partcpants at the Canadan Economc Theory Conference, the North Amercan Summer Meetng of the Econometrc Socety, the Internatonal Conference on Game at Stony Brook, and the Transatlantc Doctoral Conference. Northwestern Unversty, Department of Economcs, 302 Arthur Andersen Hall, 2001 Sherdan Road Evanston, IL Emal: martn.szydlowsk@u.northwestern.edu Webste: JEL Classfcaton: D86, G11, G31, G32, M12, M52 1

2 In ths paper, I characterze a frm s optmal project choce n a contnuous-tme moral hazard model. I show that even though frm and manager are rsk-neutral, project choce s not determned by the NPV crteron alone, but nstead by a project-specfc markup over NPV, whch changes dynamcally dependng on the frm s past payoffs. Ths markup s drven entrely by the cost of ncentves. In my model, the frm hres a manager and has access to a fxed portfolo of projects. Each project yelds a rsky payoff stream, whch s characterzed by the project s rsk-return profle. The manager s effort s requred for projects to be proftable, but may be unobserved, whch s the source of the agency frcton. When the manager s effort allocaton s observed, the frm s project selecton follows the NPV crteron, snce the frm s rsk-neutral. That s, whenever a project s average payoff s hgher than the cost of effort, the project s chosen rrespectve of ts rsk. Ths crteron s statc, and the portfolo of chosen projects never changes. Wth moral hazard, there s both over- and undernvestment n projects relatve to the NPV crteron. 1 Whle undernvestment s drven by the cost of ncentves, overnvestment s caused by the prncpal s nablty to punsh the manager n the presence of a lmted lablty constrant. 2 Intutvely, snce the prncpal cannot demand money from the manager, the only opton after suffcently bad performance s to fre hm and lqudate the frm. 3 Because the manager may shrk, the prncpal must ncentvze each chosen project by makng the contract dependent on the project s output. Ths ncreases the lkelhood that a path of bad outcomes leads to lqudaton, and s the source of the cost of ncentves. Undernvestment occurs precsely because a project s postve NPV may not compensate for the ncrease n lqudaton rsk. Overnvestment occurs because the prncpal seeks a less neffcent punshment scheme. Assgnng projects to the manager ncreases hs effort cost and serves as an alternatve to frng, even when the NPV of the assgned projects s negatve. Snce a project s markup reflects the cost of ncentves, t s hgher for projects wth a hgh rskreturn rato. Intutvely, a hgh rato means that t s dffcult to detect shrkng, and therefore the project s dffcult to ncentvze. The dynamcs of the markup are determned by how the cost of ncentves changes wth the frms past payoffs, and how far the contract s from the lqudaton boundary. When the past performance s suffcently good, the cost of ncentves decreases and the frm chooses projects whch are more dffcult to ncentvze. If the projects rsk-return rato s superlnear, ths translates nto takng more rsky and more proftable projects. When lqudaton becomes suffcently unlkely, the frm s project selecton approaches the frst best. All negatve 1 Throughout the paper, I use the term undernvestment whenever a project wth postve NPV s not taken, and overnvestment whenever one wth negatve NPV s taken. Snce only postve-npv projects are taken n the frst best, undernvestment means equvalently that a project whch s taken n the frst best s not taken under moral hazard. The analog holds for overnvestment. 2 Gven rsk neutralty, ths s necessary to ensure that the optmal contract s not trval. See Secton 3 for detals. 3 It s possble to study a varant of the model n whch the manager s replaced after bad performance, and the frm s subject to hrng costs. The ntuton I outlned and the qualtatve results n my model would be unchanged n ths case. 2

3 NPV proejcts are phased out, and all postve NPV projects are taken regardless of ther rsk. My result that frms close to lqudaton forgo rsky projects s supported by Rauh 2009), who shows emprcally that penson funds wth weak credt ratngs, whch may be nterpreted as a proxy for the fund s bankruptcy probablty, choose safer nvestments, whle fnancally sound ones do the opposte. My fndng also dffers from the semnal rsk shftng result n Jensen and Mecklng 1976). In that paper, the possblty of lqudaton leads frms to take on excessve rsk, whle n my work, t deters the frm from takng rsky projects. In addton to choosng projects, the frm n my settng has a captal stock and operates a neoclasscal nvestment technology. Ths allows me to make predctons on how the optmal project choce changes wth the frm s captal stock. Holdng past performance constant, smaller frms choose ther projects more effcently, whle larger ones forgo postve NPV projects whch are suffcently rsky, and may overnvest n projects. By ncorporatng captal, my model nests DeMarzo et al. 2012), who study captal nvestment n a contnuous-tme moral hazard framework, and fnd that the agency frcton leads to too lttle nvestment. The dstncton between projects and captal nvestment enables me to address a set of qualtatvely dfferent questons. Whle DeMarzo et al. 2012) are study the level of nvestment at any gven pont n tme, I make predctons on whch knds of projects are chosen. In addton, my framework also yelds nsghts nto frms use of hurdle rates, the exercse of real optons under agency, and manageral ncentve contracts. When the prncpal only observes the sum of all project outputs, project choce resembles a hurdle rate. In partcular, the NPV of each chosen project s above the same threshold, whch depends on the project wth the lowest NPV currently chosen. Ths hurdle rate allocaton s not effcent and the contract carres excessve rsk. Consequently, my model suggests that hurdle rates, whch are wdely observed n practce, arse when the frm s unable to condton the contract on ndvdual projects, or unable to fnd ncentve schemes whch condton on ths nformaton. 4 Commonly, devatons from the NPV crteron are explaned by real optons. 5 In the real optons framework, postve NPV projects are not taken because they carry an opton value of watng, and negatve NPV projects are taken f they ental the opton to start addtonal projects n the future. Whle real optons rely on ether rreversbltes or fxed costs, my model can generate devatons from the NPV crteron n the absence of both, and solely as a consequence of moral hazard. Snce fxed costs of startng or stoppng projects are relevant n practce, I ntroduce them nto the contract n Secton 6. I show that my model serves as an approxmaton to a real optons framework under agency wth small fxed costs. The approxmaton result has a precse nterpretaton. We can compute each project s margnal beneft accordng to the crteron n my model, and choose projects accordngly. The loss n value from ths scheme s neglgble, and wrong 4 I wll explctly study the relaton between hurdle rates and manageral compensaton structures observed n practce n Secton 5. 5 See Dxt et al. 1994) for further references to ths lterature. 3

4 projects are almost never chosen, provded that the fxed costs are small. Thus, my model can serve as a gudelne for choosng projects n the presence of fxed costs. In Secton 5, I derve an mplementaton of the optmal contract whch features an equty stake, a fxed wage, and bonus payments. Ths mplementaton ratonalzes the majorty of manageral contracts found n realty, and s due to the ntroducton of multple projects. As Murphy 1999) documents, most CEO s contracts consst of equty, wage and a bonus, and the latter s a lnear weghted functon of the CEO s performance across dfferent categores. Ths s exactly the case n my model. In contrast, an mplementaton whch conssts of equty only can mplement the hurdle rate allocaton, but fals to mplement the second best. Thus, hurdle rates arse n frms whose manageral contracts put too much emphass on equty. My mplementaton also ratonalzes the manager buyng and sellng equty at ex-ante determned transfer prces. The optmal equty share n my model s not statc, as n DeMarzo and Sannkov 2006) or DeMarzo et al. 2012), and has to be adjusted when the project selecton changes. These equty transfers may dstort ncentves, snce f the manager expects to be strpped of shares n the future, he may be less lkely to put n effort. The transfer prces are desgned to exactly offset ths ncentve effect. The manager s contnuaton value n the mplementaton equals the sum of the frm s cash balances and the value of a personal account. 6 Keepng the account value constant mples that frms wth hgher cash buffer, relatve to frm sze, choose ther projects more effcently. The paper proceeds as follows. Secton 2 provdes an overvew of related lterature. Secton 3 ntroduces the model, and llustrates basc results on the ncentve scheme and the prncpal s value functon. Secton 4 s the core of the paper and dscusses the optmal project selecton scheme both under output- and project-based ncentves. It also consders extensons of the orgnal settng n whch the agent can steal nstead of exertng effort, and the frm can allocate funds between projects. The mplementaton outlned n the paragraphs above s derved n Secton 5. Fnally, Secton 6 provdes a dscusson how my setup relates to the real optons framework whle Secton 7 concludes. 2 Related Lterature The present model s related to three strands of lterature. The technques employed to characterze the dynamc contract stem from the lterature on contnuous tme contractng, most notably Schattler and Sung 1993) and Sannkov 2008). Recent contrbutons n ths lterature whch share certan features wth my setup nclude Bas et al. 2010), who study nvestment and downszng of frm sze as a way to ncentvze accdent preventon, He 2009), n whose model the agent s effort 6 The use of cash balances s smlar to DeMarzo et al. 2012), whle the exstence of a personal account s novel to my model, and due to the changng nature of the manager s optmal pay-performance senstvty. 4

5 drectly affects the evoluton of frm sze, and Fong 2007) who studes a bnary effort decson wth two agents. The closest paper to mne s DeMarzo et al. 2012), whch s the frst to study frm nvestment n a contnuous-tme agency framework. My model features multple projects wth varyng rsk-return profles n addton to a contnuous nvestment varable, and nests DeMarzo et al. 2012). My man focus however s on the optmal choce of projects, nstead of the level of captal nvestment, and my paper s amed to be complementary to the analyss of DeMarzo et al. The two models also dffer n terms of mplementaton. Due to the multtask structure, my mplementaton uses transfer prces for equty, bonus payments and a personal account n addton to cash, whle the mplementaton n DeMarzo et al. 2012) reles on cash and equty only. Also, the optmal equty share n my settng changes over tme, and an mplementaton wth constant equty share cannot be optmal. The problem of multtaskng has receved sgnfcant attenton snce the semnal artcle of Holmstrom and Mlgrom 1991). 7 Due to the complex nature of the problem, dynamc studes of multtaskng are rare. The most recent ones nclude Manso 2006), who studes the trade off between two tasks nterpreted as exploraton and explotaton, and Mquel-Florensa 2007), who answers under whether two tasks should be executed sequentally or n parallel, dependng on the strength of the externaltes between them. In a contnuous tme setup, Hartman-Glaser et al. 2010) consder a multtaskng model where an underwrter ssues a mortgage backed securty, and may shrk n selectng the mortgages, whch wll default wth dfferent rates. They fnd that bundlng the mortgages s optmal, whch s remnscent of a smlar statc result by Laux 2001), and the underwrter wll ether exert effort n all mortgages or none. Fnally, my model s related to the lterature on optmal nvestment. The real optons lterature 8 offers a complementary vew on the ssue of project choce, n whch both fxed costs and an opton value of watng drve devatons from the NPV crteron. Although the real optons framework has been extended to ncorporate agency frctons, see p.e. Grenader and Wang 2005), Grenader and Malenko 2010) and Morellec and Schürhoff 2010), studes are mostly lmted to the choce of a sngle project. Ths s because takng one project wll affect the value of other projects, va rreversbltes or fxed costs, whch makes t dffcult to characterze the the optmal choce of multple projects. In my model, the externalty between projects s well behaved, whch allows for the characterzaton of an entre project portfolo. The captal budgetng lterature, see Harrs and Ravv 1996) and Harrs and Ravv 1998), studes the choce of projects when a dvson agent has prvate nformaton about project qualty and has an ncentve to msreport. In Harrs and Ravv 1996) both over-and under-nvestment relatve to the NPV crteron can occur, dependng on whether the project s of low or hgh qualty, and the 7 For a recent contrbuton and further references, see Bond and Gomes 2009). 8 See e.g. Dxt et al. 1994) for a comprehensve overvew. 5

6 optmal contract can be mplemented by allocatng a fxed budget to the agent. In a smlar setup, Berkovtch and Israel 2004) derve an mplementaton whch takes the form of an nternal rate of return, whch s smlar to my result on the hurdle rate. Fnally, Malenko 2011) consders a dynamc verson of the problem, and derves the captal budgetng mechansm n contnuous tme. Snce n the captal budgetng lterature, projects only have an undmensonal qualty assocated wth them nstead of rsk and return, t s dffcult to compare my results. If the average project payoff n my framework s nterpreted as qualty, and the relaton between payoff and the SN rato s postve and suffcently large, then my model wll mply that there are too many low qualty projects and too few hgh qualty projects n the frm s portfolo, n lne wth the above. Another related area s delegated portfolo management as found n Cadenllas et al. 2007), He and Xong 2008), Ou-Yang 2003) and Makarov and Plantn 2010). The key dfference between my model and the portfolo choce framework, s that, very smlar to the real optons lterature, project choce s a bnary decson. Ths allows me to characterze selecton crtera as well as the delay n project mplementaton stemmng from the agency frcton. 3 Model Setup 3.1 Projects and Investment Technology I study a frm whch has access to a fxed number of projects and hres an agent. The shareholders of the frm act as the prncpal. Tme s contnuous, ndexed by t R +, and the horzon s nfnte. Each project yelds rsky payoffs, and the agent decdes n whch projects to exert effort at any gven tme. A project s average payoff s postve only f the agent works, and otherwse t s zero. Formally, for any project {1,..., N}, the cumulatve output x t s gven by dx t = µ a t dt + σ db t. 1) The Brownan Moton B t to captures the dosyncratc nose n the project s payoff, whle the bnary varable a t {0, 1} denotes the agent s decson to work or shrk. µ s the project s average proftablty when the agent exerts effort, and σ measures the amount of nose. For two projects and j, the Brownan Motons B t and B jt are mutually ndependent, so that the path of each project s payoffs s determned by the agent s effort and the nose n that project alone. 9 I denote the agent s allocaton of effort among projects wth the vector a t = a t ) N =1, and the set of all possble allocatons wth A = {0, 1} N. The event a t = 1 shall be nterpreted as project beng assgned to the agent, or alternatvely project beng mplemented at tme t. 9 Formally, B t B js for all tmes t, s 0. Ths helps ensure that n the optmal contract, any dependence between projects s drven by the agency frcton, and not by assumptons on the frm s technology. 6

7 The frm has a captal stock and operates a neoclasscal nvestment technology. 10 A hgher captal stock ncreases the payoff of each project lnearly. Gven manageral effort allocaton a t and captal stock π t, the ncremental payoff of the prncpal s 11 π t a t dx t. Thus, total payoffs depend on both the captal stock, whch I also nterpret as frm sze, and the number of mplemented projects. I denote nvestment as a fracton of captal as I t. The captal stock ncreases wth nvestment, deprecates over tme at rate δ > 0, and follows the law of moton dπ t = I t δ) π t dt. 2) Investment s subject to an adjustment cost, for whch I assume the functonal form π t κ I t ). The functon κ.) s ncreasng, convex, and satsfes κ 0) = 0. Unlke project choce, nvestment n captal s not subject to agency. 3.2 Utlty Functons and the Contract Space The vector of project-specfc Brownan Motons B t := B 1t,..., B Nt ) s defned on a complete probablty space Ω, F, P) wth fltraton F t, whch satsfes the usual condtons. 12 Whle each project s output can be fully observed by the prncpal and contracted upon, effort s unobservable. The prncpal commts to a contract, whch conssts of a cumulatve consumpton process c = {c t R + : t 0}, a prescrbed effort process 13 a = {a t A : t 0}, and a frng tme τ. Effort and consumpton are progressvely measurable wth respect to F t, and τ s a stoppng tme. Both prncpal and agent are rsk-neutral. The agent s effort cost s lnear, symmetrc n effort for each project, and gven by h a t. Ths specfcaton, together wth the mutual ndependence of the nose n project outputs, ensures that any dependence between projects along the path of the optmal contract s drven by the agency frcton, nstead of assumptons on technology or 10 Ths allows my paper to nest the model of DeMarzo et al. 2012). All assumptons n the current paragraph are analogous to that paper. All qualtatve results n my paper contnue to hold when nvestment s omtted. In ths case, the scaled HJB Equaton 8) s the frm s actual HJB equaton, and all terms nvolvng I t dsappear. 11 Throughout the paper, I abbrevate the sum over projects N =1 as where no confuson can arse. 12 See Karatzas and Shreve 1991), p The dscreteness of the effort process poses a potental problem. If the agent s effort a t s not of bounded varaton on an nterval of tme, the agent s contnuaton value process may not be suffcently well behaved to guarantee a unque strong soluton. The problem can be solved ether by assumng a small postve swtchng cost whch s ncurred by the prncpal whenever the effort changes, so that t wll never be optmal to change the project allocaton more than once on a suffcently small nterval of tme, or by consderng an ε-optmal strategy whch leaves the project allocaton constant on such nterval. The model wth swtchng costs s studed n Secton 6, and the exstence of ε-optmal strateges s proven n Proposton 17) n the Appendx. 7

8 preferences. The agent s dscounted lfetme utlty W 0 s gven by [ˆ τ W 0 = E e γt dc t π t h ) ] a t dt F 0. 3) 0 The agent s protected by a lmted lablty constrant. That s, for any tme t, the ncrement n hs consumpton payments dc t may not be negatve. 14 The prncpal receves the payoffs from each project, pays the agent s compensaton and bears the adjustment costs from nvestment n captal. Her expected dscounted payoff s gven by [ˆ ) ) ] τ J 0 = E e rt π t µ a t π t κ I t ) dt dc t W 0, π 0. 4) 0 I assume that prncpal and agent have dfferent dscount factors, and that the agent s less patent,.e. r < γ. As noted n DeMarzo and Sannkov 2006), ths prevents the prncpal from postponng the agent s consumpton forever. I also mpose an upper bound Ī on relatve nvestment, so that I t Ī < r + δ, to ensure that the frm s value functon s bounded.15 Fnally, when the frm s shut down, the prncpal can recover a fracton l > 0 of current captal, and her payoff s lπ t. The agent receves an outsde payoff of zero n ths case. 3.3 Incentve Compatblty To make the dynamc contractng problem tractable, I show that at any gven pont n tme, the entre hstory of the contract can be summarzed by a two state varables, the captal stock and the agent s expected contnuaton utlty. 16 I focus on the contnuaton value n ths secton. In the next secton, I establsh that the state varables can be transformed nto a sngle one. For any ncentve compatble contract a, c, τ), ths contnuaton utlty s gven by [ˆ τ W t = E e γs t) dc s π t h t a t ds ) {a s, c s } s t, F t ] Lemma 1 below uses the martngale representaton theorem 17 to derve a law of moton for the contnuaton value W t, whch follows a dffuson process wth respect to the multdmensonal 14 Precsely, we have dc t = c t lm t t c t 0 almost surely for all t R +. Ths assumpton rules out trval contracts n whch t s costless to ncentvze effort by demandng arbtrarly hgh payments from the agent when a low path of output realzes for one of the projects. Qualtatvely, the results n my paper reman unchanged f I assume dc t cdt for some c > If I = r + δ the prncpal s value or equvalently shareholders value) of the frm mght be nfnte, snce the frm could grow at a fast enough rate to negate any dscountng. 16 Usng the agent s contnuaton value as a state varable s a common technque n dynamc contracts. See e.g. Spear and Srvastava 1987) for an llustraton. 17 See Karatzas and Shreve 1991), Theorem 4.15, p. 182 for the statement and Sannkov 2008) for ts applcaton to contracts n contnuous tme.. 8

9 Brownan Moton B t. Intutvely, gven any project selecton rule a and consumpton schedule c, the only source of uncertanty n the model s the vector of Brownan nose terms B t, and therefore at each pont n tme the agent s contnuaton value must be a functon the past realzatons of nose. The lemma also states an ncentve compatblty condton whch ensures that the agent exerts effort. Lemma 1. For any progressvely measurable effort process a and consumpton process c, there exsts } a collecton of progressvely measurable and square ntegrable stochastc processes {ψ t ) N =1 : 0 t τ, such that dw t = γw t + π t h a t ) The contract s ncentve compatble IC) f and only f whenever a t = 1. dt dc t + π t ψ t db t. 5) ψ t σ µ h 6) The parameter ψ t measures the senstvty of the agent s contnuaton value wth respect to the nose n project s output. Snce the prncpal can control both consumpton and effort, she s able to determne how much the contnuaton value responds to project outputs, and we can thnk of ψ t beng chosen drectly n the optmal contract. When the output of project features an unexpected jump by db t, the agent s contnuaton value changes by ψ t db t. To see how ths mpacts hs decson to exert effort, consder a devaton for a short perod of tme dt, durng whch the agent s shrkng n project. Wthout exertng effort, hs utlty rses by π t hdt. Because the prncpal would not know that the agent s shrkng, she expects that db t = πt σ dx t µ dt), whle the true process s dx t = σ db t. Hence, the prncpal s expectaton about the realzaton of nose falls short by π t µ σ dt, and by the representaton 5), the agent loses ψ π t µ σ dt n contnuaton utlty. To nduce effort, ths loss must be larger than π t h, whch leads to Equaton 6). Lemma 1 also llustrates why the sgnal to nose rato SN = µ σ s mportant for provdng ncentves. When the agent shrks, he affects the prncpal s belefs about the realzaton of db t. When the project s relatvely safe, and the rato s large, observng a shortfall n output by µ dt whle the agent s workng s a very unlkely event, and corresponds to a large negatve realzaton of the Brownan nose. Thus, t s easy to detect shrkng and the agent s contnuaton value does not have to react much to output to provde ncentves. 18 Snce σ µ s the rsk-return rato of the project, an alternatve nterpretaton s that shrkng s dffcult to detect for projects wth hgh rsk-return rato, and easy to detect for projects wth low rsk-return rato. Analogously to the dscrete tme contractng lterature, Equaton 5) should be nterpreted as a 18 Formally, n Equaton 5), the shortfall n output s equvalent to a very large negatve realzaton of db t, whch for gven ψ t mples that W t falls by a relatvely large amount, whle the opposte s true for when σ s large. 9

10 promse keepng constrant. Gven a contnuaton value W t, hgher consumpton dc t mples that ceters parbus, the agent s promsed value at the end of a small nterval of tme W t+dt wll be smaller, whle demandng more effort mples that the prncpal has to promse more utlty to the agent n the future. 3.4 The Optmal Contract Wth { the result of Lemma } 1, the optmal contract can be expressed as a choce of processes ψ t ) N =1, c t, I t : 0 t τ, and a frng tme τ. The prncpal seeks to maxmze the frm value 4), subject to the promse keepng constrant 5), the law of moton for frm sze 2), and the ncentve compatblty condton 6). Formally, we have JW 0, π 0 ) = max s.t. dw t = {ψ t,c t,τ,i t } E γw t + π t h dπ t = π t I t δ) dt ψ t σ µ h f a = 1. [ˆ ) ] τ e rt π t µ a t dt dc t π t κ I t ) dt + e rτ γπ t F 0 0 a t ) dt dc t + π t ψ t db t The prncpal s problem depends on two varables, the ntal contnuaton value W 0 and the ntal captal stock π 0. To smplfy the analyss, I show that the prncpal s value functon can be expressed as a functon of a sngle state varable, whch s the agent s scaled contnuaton value w t = W t π t. Intutvely, all components on the rght hand sde of the prncpal s value functon J W, π) are multples of π t, except for the payout to the agent dc t. 7) Snce ths payout s a choce varable, the prncpal can smply maxmze over the payout relatve to captal dct π t. 19 Smlarly, the laws of moton for the frm sze and the agent s contnuaton value are also lnear n π t. In partcular, dvdng Equaton 5) by π t mples dw t π t = γ W t π t + h a t ) dt dc t + ψ t db t, whch determnes the law of moton for the scaled value w t. Ths suggests that the prncpal s value functon s lnear n π 0, and has the functonal form J W 0, π 0 ) = π 0 j w 0 ) for some functon j.) to be determned. I verfy ths ntuton n the proof of Proposton 2 n Secton A.1 of the Appendx. 19 Formally, the prncpal optmzes over d c t = dc t π t, and the term dc t s replaced by π td c t. Wth a slght abuse of notaton, I keep denotng the scaled payouts as dc t. 10

11 4 Propertes of the Optmal Contract 4.1 Shape of the Value Functon In ths secton, I show that the prncpal s value functon s the soluton to a verson of the Hamlton- Jacob-Bellman HJB) equaton, wth the scaled contnuaton value w t as the only state varable. Ths soluton s used n the followng sectons to characterze the choce of projects, nvestment n captal, and the payout and frng polces of the frm. 20 Proposton 2. Let n t = a t denote the number of projects taken at tme t. The HJB equaton rj w) = sup a,i wth the boundary condtons µ a κ I) + j w) γ I + δ) w + hn) 8) +j w) 1 ψ 2 a + I δ) j w) 2 j 0) = l j w) = 1 j w) = 0 has a unque twce contnuously dfferentable soluton on the nterval [0, w], and equals the prncpal s optmal value functon. The regon 0, w) s parttoned nto regons on whch a partcular project selecton a s optmal. The value functon s strctly concave on 0, w), and three tmes contnuously dfferentable on any subset of 0, w) wth nonempty nteror on whch project choce s constant. The thrd dervatve j w) exhbts a jump whenever project selecton changes. Fgure 1 llustrates the shape of the value functon, whch has the followng features. When w t hts zero, the agent s fred, and the prncpal receves the scrap value of lπ t. Ths s reflected n the boundary condton j 0) = l. Intutvely, because the agent s protected by lmted lablty, the worst future path of consumpton the prncpal can pay to hm nvolves zero consumpton forever. 21 Exertng effort n the future would mply a negatve contnuaton value, as can be seen from Equaton 5). By shrkng forever the agent can guarantee hmself a contnuaton payoff of at least zero. Thus, once w t = 0, any ncentve compatble contract nvolves shrkng forever and no consumpton payments. The prncpal s expected value n ths case s zero, and t s optmal for her to fre the agent n order to receve the scrap value of lπ t. 20 The shape of the value functon and the termnaton and payout polces are analogous to DeMarzo and Sannkov 2006) and DeMarzo et al. 2012), and are explaned ntutvely below for the convenence of the reader. Proposton 2 confrms that the characterzaton s robust to allowng for multple projects, whch ntroduces addtonal challenges n establshng the results. See Appendx A.2 for detals. 21 Formally, dc s = 0 for all s t. 11

12 The exstence of the frng boundary also explans why the prncpal s value functon s concave, and why t may be ncreasng n the agent s contnuaton value. Introducng more rsk n the contract, exemplfed by hgher ψ t, makes the agent s value react more strongly to the nose n output, and ncreases the lkelhood of httng the lqudaton boundary due to a sequence of bad outcomes along the path of the contract. Ths termnaton s neffcent, and therefore an ncrease n rsk must lower the prncpal s utlty. A hgher contnuaton value for the agent mples ether less effort or hgher expected consumpton payments n the future, both of whch lower the prncpal s payoff. At the same tme, a hgher w t mples a lower lkelhood of frng, whch s benefcal for the prncpal. When w t s hgh, the frst effect domnates and the prncpal s value s decreasng n w t. When the agent s value s low, and termnaton s suffcently lkely, the reducton n frng probablty domnates, and the prncpal s value s ncreasng n w t. 22 When w t = w, the agent s pad a dscrete amount, and dc t > 0. Whenever the prncpal pays the agent, her value changes by 1 j w t )) dc t. 23 Therefore, dc t > 0 whenever j w t ) 1, and dc t = 0 otherwse. Snce the prncpal s value s concave n w, and the agent s value jumps downwards whenever dc t > 0 by Equaton 5), the pont at whch the agent s pad s unque, whch leads to the boundary condton j w) = 1. The second condton j w) = 0 s the super contact condton, and guarantees that the payment threshold w s chosen optmally Project Choce Wthout moral hazard, a project s chosen at all ponts n tme f the average payoff s hgher than the effort cost,.e. µ h, and never chosen otherwse. Hence, project choce follows the NPV crteron, and s ndependent of the agent s contnuaton value W t or frm sze π t. Wth moral hazard, the choce of projects s determned from the HJB Equaton 8). The equaton mples that at any pont n tme, the prncpal s problem s separable n each project once the scaled contnuaton value s taken nto account. Therefore, the margnal beneft of each project can be determned separately, and s gven by b w t ) = µ + j w t ) h + j w t ) 1 2 ψ2. 9) A project s executed whenever the margnal beneft s postve. Whether ths s the case depends on the project s average payoff µ, the cost of compensatng the agent for effort j w t ) h, and the 22 Note that ths regme s not renegotaton proof as the prncpal s value functon s ncreasng n the agent s value. If renegotaton s allowed, the prncpal may agree to promse the agent a hgher value snce ths would be mutually benefcal. The overall value for the prncpal s lower compared to the case wth commtment, snce the agent s ncentves are dmnshed f he antcpates the contract to be renegotated. Thus, the prncpal wll always commt f she has the ablty. See also DeMarzo and Sannkov 2006) for a dscusson of renegotaton n a related settng. 23 The frst term s the drect loss n cash pad to the agent, whle the second term measures how the change n the agent s contnuaton value affects the prncpal. 24 Snce j w) = 1, the prncpal s ndfferent between payng and not payng the agent at w. If for example j w) < 0, t would be optmal for the prncpal wat untl w t reaches some w > w, and then pay the agent, snce at ths pont 1 j w )) dc t > 0. The optmal payment threshold w s the one at whch ths s not proftable. 12

13 jw). w w Fgure 1: Shape of jw) and contnuaton regons cost of provdng ncentves j w t ) 1 2 ψ2. Whle the margnal beneft of project may depend on both past and current choces regardng other projects, any potental dependence s summarzed by the dynamcs of w t and the value functon j w). To see how project choce and NPV relate, we can rewrte Equaton 9) as b w) = rnpv + j w) + 1 ) h j w) h2 SN 2. 10) The margnal beneft of mplementng a project depends postvely on both the net present value NPV = µ h r and the project s sgnal to nose rato SN = µ σ. 25 Whle a hgher NPV mples hgher expected payoffs from the project, the sgnal to nose rato works though the agent s ncentves. The hgher the rato, the easer t s to detect shrkng, and the agent s ncentves can be weaker wthout ceasng to motvate effort. By the representaton n Equaton 5), ths s equvalent to a lower volatlty of w t, whch reduces the lkelhood of httng the boundary at whch the agent s fred. The term j w) + 1) h measures the ncrease n socal value from movng the contnuaton value away from the lqudaton boundary, and s always postve. Settng b w) = 0, we can derve the mnmal NPV whch the frm requres to mplement a project, NPV SN, w) = 1 r j w) + 1 ) h 1 2r j w) h2 SN 2. Consequently, all projects wth hgher than the mnmal NPV are mplemented, whle all others are not. The threshold s a functon of the current scaled contnuaton value, as well as the project s sgnal to nose rato. Fgure 2 llustrates the non-lnear relatonshp between NPV and SN, and 25 For an alternatve nterpretaton, note that ψ = σ µ h, where σ µ can be understood as the project s rsk-return rato. 13

14 b w) > 0 NPV SN. Fgure 2: NPV vs. SN boundary outlnes the set of projects whch are chosen when w t = w. In the optmal contract, both over- and undernvestment can occur. Undernvestment s due to the cost of ncentves, and occurs whenever a project s NPV s not suffcent to compensate for the ncrease n termnaton probablty, whle overnvestment s caused by the agent s lmted lablty constrant. 26 performance. The prncpal cannot demand monetary compensaton from the agent after bad Instead, her only means of punshment s to termnate all projects once w t hts zero, whch also precludes her from gettng any payments n the future besdes the scrap value lπ t. To avert termnaton, the prncpal can allocate more projects to the agent, whch ncreases hs dsutlty of effort, and serves as a less neffcent punshment devce. By Equaton 5), ths ncreases the average growth rate of the contnuaton value, and therefore lowers the probablty of httng the frng boundary n the future. Ths effect may compensate for the negatve NPV of a project, and the frm benefts from lowerng the future expected termnaton probablty, n exchange for current losses. However, the projects assgned as a punshment cannot be too dffcult to ncentvze. Otherwse the prncpal would have to ncrease the volatlty of the agent s contnuaton value, whch makes frng more lkely and undermnes the ntended effect. In accordance wth Fgure 2, we therefore observe undernvestment n projects whch are relatvely dffcult to ncentvze, and have a low sgnal to nose rato, and overnvestment n projects where shrkng s easy to detect Standard explanatons for overnvestment nclude empre buldng as n Jensen 1986), and prvate benefts to the agent. In my model the effects are entrely drven by the agency frcton. 27 A secondary effect n favor of overnvestment s that the agent s pad only when w t reaches w. When the contnuaton value s low, the probablty that w t hts zero before reachng w s hgh, and thus n expectaton, the prncpal only has to compensate the agent for a fracton of the ncurred effort cost. Ths explans why j w) > 1 for w < w. 14

15 .. w small.. w medum.. w large NPV SN. Fgure 3: Project Cutoffs as a functon of w 4.3 Project Portfolo Dynamcs The choce of projects evolves over tme as w t changes, and each project s margnal beneft functon s non-monotonc n w. A hgher contnuaton value decreases the lkelhood of frng. Ths ncreases the probablty that the agent collects payments, and thus the cost of compensatng hm for effort j w) h. It also lowers the cost of ncentvzng an addtonal project. For w t suffcently hgh, ths mples that the cost of ncentves s decreasng n the agent s value. It may be ncreasng when w t s low however. In ths case, the prncpal s value s ncreasng n w t, and for hgher w she stands to lose more f a sequence of bad realzatons of output drves the contract nto termnaton, whch outweghs the decrease n termnaton probablty. When the contnuaton value approaches the payout boundary w, the cost of ncentves becomes neglgble and the relatve mportance of a project s SN rato vanshes, whle the nstantaneous cost of compensatng for effort s exactly h. 28 Thus, the margnal beneft functon at w s b w) = µ h, and the project allocaton converges to the NPV crteron. Any negatve NPV projects wth low ψ, whch have been taken at a lower contnuaton value are phased out, whle any projects wth postve NPV and hgh ψ are taken. Ths result s llustrated n Fgure 3. The red functon shows the project boundary for small w, whle the black and blue functons represent the boundares for successvely larger w. As w w, the break-even NPV lne n the fgure approaches the x-axs. When the contnuaton value grows, and the cost of ncentves declnes, t s ntutve to thnk that the frm s optmal portfolo shfts towards hgh-npv, low-sn projects, whle projects wth negatve 28 Formally, ths follows from the boundary condtons n Proposton 2. 15

16 NPV are sorted out. Proposton 3 detals n whch sense ths ntuton holds. Whenever the cost of ncentves s decreasng n w, the margnal beneft of any project whch s suffcently dffcult to ncentvze s ncreasng. Thus, f at a certan cutoff a new project s taken, ts rsk-return rato must be relatvely hgh compared to other projects. The opposte holds for projects whch are relatvely easy to ncentvze, whch see ther margnal beneft decrease wth w. If a project s removed, t must be among these. Proposton 3. If j w) 0, b w) < 0 for all projects. If j w) > 0, there exsts a cutoff ψ w) such that b w) > 0 f and only f ψ > ψ w). If then b w) > 0. ψ 2 > 1 ψj 2 a j, n j If the relatonshp between rsk and return s lnear for every project, the result n Proposton 3 can be sharpened, and for w suffcently large, an ncrease mples that the frm adds projects whch have both hgher rsk and hgher return. Corollary 4. Suppose the σ = Kµ for all and K > 0. Then for all w, the projects wth the hghest return µ are chosen, and b w) > 0 for all whenever j w) 0. Fnally, I characterze the externaltes between projects nduced by the agency problem. In the frst best, project choce s statc and projects are chosen ndependently of each other. Under moral hazard, takng w as gven, b w) s stll ndependent of b j w) for j. In ths sense, the choce of projects s ndependent condtonal on the scaled contnuaton value. Ths s surprsng n the lght of the lterature on statc multtaskng under moral hazard. For nstance, Laux 2001) shows that n a settng wth a rsk neutral prncpal and agent, and lmted lablty, bundlng projects ncreases the prncpal s payoff, snce t allows to extract more of the agent s rents by loosenng the lmted lablty constrant. In my settng, there s no such frst order effect of project choce on payoff, snce the value functon s twce contnuously dfferentable. 29 Intutvely, there s no hysteress effect as n the real optons lterature and the frm can freely swtch between projects, whch rules out frst-order externaltes. There s, however, a second order effect. Choosng a project generates an externalty not on the current payoff of other projects, but on the rate at whch ther value changes wth w. Proposton 5. At any threshold ŵ where a project s added or removed, j + ŵ) < j ŵ). 29 If I were to ntroduce a fxed cost wth project mplementaton, the value functon would not be C 2 at the cutoffs and hence there would be a frst order externalty between projects. Implementng another projects causes a dscrete jump n j, and hence n b for all projects. See Secton 6. 16

17 4.4 Project Choce and Investment My setup nests the framework of DeMarzo et al. 2012), whch deals wth optmal nvestment n captal. In ths secton, I compare how moral hazard dstorts the choce of captal and projects. In the optmal contract, optmal nvestment I w) s strctly below the frst best level of nvestment, whch s consstent wth DeMarzo et al. 2012). The frst order condton for nvestment n Equaton 8) mples that κ I w)) = j w) j w) w. Snce κ s convex and the rght hand sde s ncreasng by the concavty of j w), I w) s ncreasng. At w, nvestment s stll below the frst best level, whch can be seen from pluggng the boundary condtons nto Equaton 8), and comparng the resultng expresson to the frst best payoff j fb w) = max I µ h) + r I + δ w, where µ h) + = max {µ h, 0}. Ths explans why for all w, I w) s lower than the optmal nvestment wthout agency. In my settng, a low scaled contnuaton value mples both less effcent nvestment and portfolo choce. However, t does not mply undernvestment n both projects and I w), and t s possble that low I s coupled wth overnvestment n projects. 4.5 Output- vs. Project-Based Incentves In Secton 4.2, I have shown that n the optmal contract the prncpal adjusts the pay-performance senstvty ψ t separately for each project. In ths secton, I show that the project selecton crteron resembles a hurdle rate when the prncpal s forced to condton the contract on the total output, dx t = dx t, and thus has to bundle all projects. In ths case, all projects chosen at a partcular tme have an NPV whch les above a reference level, whch s determned by the cost of ncentves for the entre portfolo of chosen projects. Repeatng the argument from Lemma 1 shows that the agent s contnuaton value satsfes dw t = γw t + h ) a t dt dc t + ψ t π t σ db t. 11) Unlke n Equaton 5), the vector Brownan Moton B t enters wth a sngle factor ψ t, because the prncpal s forced to condton on the total output. The agent s IC constrant becomes a t = 1 ψ t h µ. 12) At any pont n tme, the frm chooses a certan portfolo of projects, and by Equaton 12), the 17

18 project wth the lowest NPV determnes ψ, the rsk exposure requred to ncentvze all projects n the portfolo. To see why ths s the case, consder a varant of the ntuton outlned n Secton 3.3. If the agent shrks on project for a small perod of tme dt, he saves hdt n effort cost, but at the same tme forgoes µ dt n payoff, and by the representaton 11), suffers a reducton n contnuaton value by µ ψ. The devaton s not proftable when ψ h µ, and snce the prncpal can choose only one ψ, t must be hgh enough to ncentvze effort on all mplemented projects. Snce the prncpal s value functon s concave, we have h ψ t = max. :a t =1 µ Therefore, gven a portfolo and a value for ψ, each project n the portfolo must satsfy NPV h ψ 1 ψ, otherwse t would not be proftable. Therefore, the portfolo appears to have been chosen usng a hurdle rate or mnmum NPV crteron. Snce the hurdle depends on both the scaled contnuaton value w as well as the current project selecton, t shfts non-monotoncally as w changes, but t wll converge to the NPV crteron when w approaches w. Under project based ncentves, the total rsk n the contract s gven by ψ2 = σ2 whle output based ncentves rase t to ) σ2 max h :a =1. Hence, for any effort profle a wth µ at least two projects mplemented, the agent s rsk exposure s strctly hgher, as long as h µ h µ for some mplemented projects and j. Therefore, condtonng the ncentve contract on total output alone cannot be effcent, snce t ntroduces unnecessary rsk n the contract. 30 h µ ) 2, 4.6 Allocaton of Funds n Projects Suppose that each perod, the prncpal can dstrbute kπ resources among the projects to ncrease the effectveness of manageral effort, where k 0, 1). The resource allocaton satsfes π kπ, 13) and s complementary to the agent s effort, so that dx t = π t µ a t dt + σ π t db t. 30 Ths ntuton can be verfed by wrtng out the prncpal s HJB equaton wth output based ncentves, and comparng t to Equaton 8). 18

19 Total nstantaneous cash flow hence follows dx t = π t π t µ a t dt + σ db t ), where π t = π t π t s the fracton of resources allocated to project. In the frst best, the frm engages n an extreme form of wnner pckng, snce only the project wth the hghest NPV receves all the funds. Wth agency however, project fundng not only acts to ncrease the cash flow, but also serves to change ncentves. Gven a fundng allocaton π, the rsk exposure requred to motvate effort s gven by ψ h σ µ 1 π, and project fundng serves to lower the requred pay-performance senstvty, because t mproves the sgnal to nose rato of the project output dx t, whch makes shrkng easer to detect. In ths sense, fundng has an added beneft next to mprovng the effcency of the agent s effort. prncpal s scaled HJB Equaton 8) now changes to rj w) = sup a,i, π +j w) 1 2 π µ a κ I) + j w) γ I + δ) w + hn) h σ ) ) 1 2 a + I δ) j w) λ π k, µ π The where λ s the Lagrange multpler assocated wth resource constrant 13). mplemented, ts captal allocaton solves the FOC 31 Gven project s µ λ j w) h2 SN 2 1 π 3 = 0, whch mples that j w) h 2 ) 1 3 π =. SN 2 λ µ ) Hence, project fundng s decreasng n the project s SN rato, and low rsk projects receve lower fundng compared to hgh rsk projects, snce for hgh rsk projects, the margnal value of lowerng the cost of ncentves s hgher. The lnk between return and fundng remans postve, and hgher payoff projects receve relatvely more funds. As the followng Lemma shows, project fundng ncreases n w only for projects wth suffcently hgh NPV. Ths s ntutve, snce as w rses, the costs of exposng the agent to rsk declne, and therefore the motve to dstort funds away from hgh payoff and towards hgh rsk projects dmnshes as well. Lemma 6. π w) s postve whenever µ λ > λ w) j w) j w) h and negatve otherwse. Moreover, 31 Note that λ > µ for all mplemented projects, snce otherwse π, whch volates 13). 19

20 λ w) j w). Proof. We have π w = 1 j w) h 2 ) 2 3 SN 2 λ w) µ ) 3 h 2 SN 2 j w) µ λ w)) + λ w) j w) λ w) µ ) 2 whch s postve whenever the condton holds. The result on λ w) can be obtaned by pluggng the above expresson nto 13), and solvng for λ w). 4.7 Project Dynamcs wth Stealng In ths secton, I assume the agent can steal the project output nstead of shrkng, and there s no nvestment n captal. The prncpal chooses between ether not executng a project, n whch case there s no output to steal, or executng the project and deterrng the agent from stealng. 32 Ths setup smplfes the dynamcs of the project selecton, because n equlbrum the prncpal does not have to compensate the agent for effort. Instead, she trades off the project s average payoff aganst the costs of provdng ncentves. The agent receves a prvate beneft of ϕµ per unt of tme when he steals, where ϕ 0, 1). Stealng entals a loss of socal value gven by µ 1 ϕ) > 0. The HJB Equaton 8) becomes rj w) = max a A and each project s margnal beneft functon smplfes to µ a + j w) γw + j w) 1 ψ 2 a, 14) 2 b w) = µ + j w) 1 2 ψ2. The dynamcs of the project selecton crteron are drven by the cost of ncentves alone, and are descrbed n the proposton below. Proposton 7. When the agent can steal the project output and the prncpal cannot nvest n captal, then there exsts a unque threshold w 3 0 such that j w) < 0 for all w < w 3 and j w) > 0 for all w > w 3. Left of w 3, no projects are added as w ncreases, and rght of w 3, no projects are removed as w ncreases. The Proposton mples j w), and wth t the cost of provdng ncentves, s hghest for ntermedate values of w. The prncpal s wllngness to tolerate addtonal rsk n the contract depends on the lkelhood of termnaton, and the potental gans and losses from hgh or low output. When 32 Thus, I rule out the case where the prncpal executes the project, but allows the agent to steal. Ths may be justfed wth ether legal or reputatonal concerns of the frm. Moral hazard frameworks wth stealng nclude DeMarzo and Sannkov 2006) and DeMarzo et al. 2012). 20

21 w s low, the prncpal s expected payoff s close to the termnaton payoff. The prncpal loses lttle f a bad outcome occurs, and gans much f a good outcome propels the contract nto a regon where termnaton s unlkely. Therefore, whle she s stll rsk averse, her tolerance for addtonal volatlty s relatvely hgh. As w ncreases, termnaton becomes less lkely, but at the same tme the prncpal s value, and wth t the loss from a bad outcome, ncreases. For w < w 3, ths effect domnates, and j w) s decreasng. If w s suffcently hgh, the frst effect domnates, and j w) ncreases towards zero. In lne wth ths ntuton, when w s low, the prncpal takes on more projects than when w s at ntermedate values, and thus gamble by ncreasng the volatlty n the contract. The case wth stealng and no nvestment also smplfes the dynamcs of the hurdle rate, and the assocated project selecton when the contract condtons on total output, as n Secton 4.5. Proposton 8. In the case of output based ncentves, let µ w) be the hurdle rate when the agent s contnuaton value s w, and ψ w) = max :a =1 h µ be the pay performance senstvty assocated wth the optmal project portfolo at w. When the agent can steal and the prncpal cannot nvest, the result n Proposton 7 holds. For w < w 3, µ w) s ncreasng on any nterval where project choce stays constant, and jumps up when project choce changes, whle ψ w) jumps down when project choce changes. The opposte holds for w > w 3. For w < w 3, the cost of ncentvzng the project portfolo s ncreasng, and so s the hurdle rate µ w), as long as the project selecton remans constant. When the selecton changes, t must be that a project s removed, and ψ w) jumps downwards, whch mples a lower cost of ncentves for the new portfolo, and therefore a lower hurdle rate. 5 Implementaton In ths secton, I dscuss how the optmal contract can be mplemented. I consder two setups, dependng on whether the frm can or cannot ssue equty on ndvdual projects. In the frst case, the frm holds a cash balance and assgns an equty share n every actve project to the agent. The shares are vested, meanng that the agent may lose shares n current projects, or gan shares n new ones dependng on hs performance. When shares are ssued only on the frm, and not the ndvdual projects, equty s not suffcent to mplement the optmal contract. The ntuton for ths s analogous to Secton 4.5, and I show that the hurdle rate allocaton from that secton s mplemented. To acheve the second best, the mplementaton must feature a measure of the agent s performance n the ndvdual projects, whch n my model shares many features of bonus contracts observed n practce. 21