Dynamic Moral Hazard, Persistent Private Information, and Limited Liability

Transcription

1 Dynamic Moral Hazard, Persisen Privae Informaion, and Limied Liabiliy Suvi Vasama y November 13, 212 Job Marke Paper Absrac We analyze a coninuous-ime principal-agen model wih sochasic oupu ha is correlaed over ime. The oupu is unobservable by he principal. The correlaion has he consequence ha boh he curren oupu and fuure prospecs are agen s privae informaion. Thus, he agen s privae informaion is persisen. We idenify necessary and su cien condiions for incenive compaibiliy in such environmens. We derive he renegoiaion-proof equilibria of he game. We show ha in he early sages of he relaionship, he rm is liquidaed ine cienly early compared o he rs-bes. Renegoiaion-proofness imposes ha he liquidaion decision is sequenially raional o he principal. The paymens o he agen are delayed unil he rs-bes is reached. Once he rm has accumulaed su cien nancial slack, he agen sars o receive income and he rm is operaed e cienly. JEL Classi caion: D82, D86 Keywords: Principal-agen model, limied liabiliy, coninuous ime, persisen privae informaion Aalo Universiy School of Economics, P.O. Box 2121, FI-76 Aalo, Finland, suvi.vasama@aalo. y I am especially graeful o Juuso Välimäki for numerous conversaions and encouragemen while advising me wih his projec. I would also like o hank Dirk Bergemann, Eduardo Faingold, Johannes Hörner, Thomas Marioi, Pauli Muro, Larry Samuelson, Bruno Srulovici, Juuso Toikka and Hannu Variainen for helpful commens and conversaions regarding his projec. 1

2 1 Inroducion Mos conracual relaionships evolve over ime. The world in which he principal and he agen inerac is dynamic, and economic environmen changes over ime. In paricular, pas reurns predic fuure oucomes. Therefore, he agen has superior informaion no only abou he curren performance, bu also abou fuure prospecs. We examine a dynamic principal-agen model in which boh players are riskneural. The principal has access o unlimied funding bu he agen is proeced by limied liabiliy. Tha is, he conrac canno impose negaive paymens on him. Wih his formulaion we absrac away from he consumpion-smoohing moive ha is an essenial driving force in models wih risk-averse agen. 1 Insead, we wan o concenrae on he e ecs of ren exracion and e ciency on he opimal conrac. While consumpion smoohing is essenial for applicaions of insurance and public policy, we believe ha ren exracion and e ciency beer capure he moives ha drive he dynamics in applicaions of rm heory or corporae nance. In paricular, we concenrae on rm liquidaion ha can be inerpreed as bankrupcy or insolvency. Our saring poin is a dynamic cash- ow diversion model. The principal owns a rm ha is operaed by he agen. The cash- ow is unobservable by he principal and mus be repored by he agen. I is sochasic and serially correlaed over ime. Wih such correlaion we wan o capure he e ec of changing economic environmen. We assume ha he agen has he opporuniy o lie abou he rue cash- ow realizaion. 2 Such a lie reduces he principal s payo, and allows he agen o enjoy privae bene s. Furhermore, since he cash- ow is correlaed over ime, also he principal s expecaion abou he fuure value becomes more pessimisic. In our model his divergence in he expecaions is no correced over ime even if he agen repors ruhfully in he fuure. Tha is, he agen has persisen privae informaion. To induce ruhful reporing by he agen, he principal has o compensae he agen for no divering he cash- ow. As sandard in he lieraure, he principal has wo insrumens o provide he agen incenives. She can eiher deliver him 1 For dynamic conracing wih risk aversion and quasi-linear uiliy see, for example, Pavan, Segal and Toikka (211), Garre and Pavan (211, 212), and Baaglini and Lamba (212). We discuss he relaed lieraure in more deail in he nex subsecion. 2 Alernaively, we could assume ha he agen can exer cosly, unobservable e or o increase he rm oucome. The principal would hen have o compensae he agen for aking he desired level of e or. 2

3 nonnegaive income or liquidae he rm. We assume ha such liquidaion is irreversible. If he rm is liquidaed, is asses can be sold in re sales. This liquidaion value generaes an ouside opion o he principal. Unlike mos previous cash- ow diversion models he rm value is persisen in our framework. 3 This persisence has a direc consequence on he opimal liquidaion decision. Since cash- ows are persisen, rm value is sochasic whereas he value of he ouside opion is consan. Wih low pas performance, he value of he rm decreases relaive o he liquidaion value. If he value falls oo low, i is opimal o liquidae he rm. Moreover, noice ha if he rm is operaed for a furher insan, a posiive shock may occur. This posiive shock increases he rm value in a persisen manner. If a negaive shock occurs, he rm is liquidaed. Thus, liquidaion has opion value. Nex, we require ha he conrac be renegoiaion-proof. Tha is, we allow he principal o o er an alernaive conrac o he agen a any poin in ime. The agen has he opporuniy o eiher accep or rejec he alernaive conrac. If he agen rejecs, he original conrac remains valid. If he agen acceps, he conrac is replaced. As a consequence, a renegoiaion-proof conrac will ful ll he following requiremen. For no public hisory, here is a conrac ha boh (i) is superior o he principal, and (ii) he agen would accep. 4 To see how renegoiaion works in our framework, noice rs ha he players never have an incenive o renegoiae he paymens. The liquidaion decision is, in urn, raher vulnerable agains renegoiaions. In a framework wih independen cash- ows, liquidaion would never be renegoiaion-proof. 5 Since he rm value is una eced by pas losses, a furher operaion of he rm remains pro able. Thus, he player would always wan o renegoiae he conrac o avoid he ine cien liquidaion. Wih correlaed cash- ows he rm value falls wih low performance. If he value falls oo low, he rm is liquidaed even in he rs-bes conrac. Moreover, he principal has o compensae he agen for his informaion ren whenever he rm is operaed. Wih his cos he ouside opion is relaively more valuable o he principal. Thus, i is opimal for her o liquidae he rm ine cienly early. Indeed, i urns ou ha he renegoiaion-proof conracs in our seing are such ha he liquidaion decision is sequenially raional o he principal. To undersand he opimal conrac in our framework, i is helpful o discuss wo benchmark cases. The rs-bes of he problem can be hough as a sandard 3 A noable excepion is DeMarzo and Sannikov (211) who examine a model wih uncerainy abou fundamenals. 4 Similar ideas have been applied in earlier adverse selecion models, see Baaglini (25) and Maesri (212). 5 See e.g. DeMarzo and Sannikov (26), DeMarzo and Fishman (27), or Biais, Marioi, Planin and Roche (27) for a framework wih independen cash- ows. 3

4 opion pricing problem. (See Dixi an Pindyck (1994).) The rm is operaed unil he cash- ow reaches a cerain hreshold. A ha hreshold, he ouside opion becomes relaively more valuable and he rm is liquidaed. Nex, we move o he case in which he cash- ow is unobservable by he principal. We rs derive a benchmark case in which he players are unable o wrie a long-erm conrac. The principal hen hires he agen insananeously from a spo marke. In he absence of long-erm conracs he agen mus be insananeously rewarded for his informaion ren. The principal liquidaes he rm if he rm value falls oo low such ha he ouside opion becomes more valuable. Because of he agen s informaion ren, he rm is less valuable o he principal han in he rs-bes. Thus, he rm is liquidaed ine cienly early. Finally, we consider he case in which he principal is able o propose he agen long-erm conracs. To undersand he dynamics of he opimal conrac, i is helpful o hink of he model wihou limied liabiliy. Then he principal could sell he rm o he agen who operaes i e cienly. In our framework limied liabiliy prevens his rs-bes soluion. Of course, he issue here is ha he agen has no wealh o buy he rm from he principal. However, wih good pas performance he rm has generaed cash- ows and he agen has earned money as compensaion for his informaion ren. Now he agen can use his money o buy a larger share of he rm from he principal. Equivalenly, we can say ha he paymens o he agen are delayed in he beginning of he relaionship. The opimal conrac accumulaes he agen s coninuaion value in he beginning of he relaionship. Once he rs-bes is reached, he agen sars o receive income. As sandard in he principal-agen framework, he rm can be operaed more e cienly if he agen owns a larger share of i. The inuiion behind his resul is sraighforward. If he agen owns a larger share of he rm, he receives a larger share of he surplus. Therefore, he cares more abou he e ciency, which relaxes his incenive compaibiliy consrain. 1.1 Relaed Lieraure This paper conribues o he fas growing lieraure on dynamic cash- ow diversion models. In heir pah-breaking, paper DeMarzo and Sannikov (26) develop a dynamic principal-agen model ha can easily be solved by coninuous-ime mehods. 6 They show how he opimal conrac can be implemened by sandard securiies. Examining he agency problem inside he rm conribues o explaining he opimal capial srucure. 6 DeMarzo, Fishman, He and Wang (21) exend he model o allow for invesmens. 4

5 In DeMarzo and Sannikov, he rm can cover losses by raising funds on a credi line. Once he credi line is exhaused, he rm is liquidaed. Afer good performance, he rm can accumulae funds o pay o he credi line and o collec he necessary funds o proec iself agains bankrupcy. A rm wih higher pas performance is beer proeced agains larger losses and is herefore more valuable. The capial srucure in he model implies ha he rm value is persisen over ime. Thus he rm will be liquidaed in a renegoiaion-proof equilibrium when he nancier canno pro from coninuaion. However, since cash- ow is independenly disribued, he rm is sill able o generae posiive expeced pro s. This observaion suggess ha resrucuring he rm s liabiliies and shareholder s equiy is in general pro able. In ha case, a liquidaion decision is necessarily ine cien and would be renegoiaed. DeMarzo and Fishman (27) examine a discree-ime version model and develop conceps for de ning renegoiaion in dynamic capial srucure models. Tchisyi (26) exends he model by assuming ha he cash- ow follows a wo sage Markov process. He shows ha he conrac is opimally implemened by using a credi line wih ineres rae ha is increasing in rm value. Biais, Marioi, Planin and Roche (27) examine a relaed model which hey implemen using cash-reserves. They show ha he discree-ime model converges o is coninuous-ime limi. Biais, Marioi, Roche and Villeneuve (21) consider a model wih sochasic losses ha follow a Poisson process. Williams (211) examines a principal-agen model wih persisen privae informaion and a risk-averse agen. He shows how o characerize he opimal conrac using di eren sae variables. Srulovici (212) exends he model o allow for renegoiaion, and develops new mehods for comparing payo s across saes in sochasic games. Our paper is also relaed o a recen model by DeMarzo and Sannikov (211) who examine a model wih persisen, unobservable shocks on fundamenals. They show ha in he opimal conrac, he rm collecs cash-reserves o proec iself agains fuure losses. Once he rm has accumulaed he necessary reserves, he rm is run e cienly. In ha case he rm is only liquidaed if he value of he fundamenals falls oo low. DeMarzo and Sannikov (211) di ers from our framework in wo imporan dimensions. If he agen akes an ine cien acion in heir model, boh he principal s insananeous value and her belief abou he fundamenals are a eced. Therefore, he players beliefs abou he fundamenals diverge. However, if he agen akes he e cien acion in he fuure, his divergence is correced auomaically. In our model, he divergence of expecaions is persisen. Tha is, if he agen lied abou he evoluion of he underlying Brownian moion, he expecaions di er in he fuure even if he agen repors ruhfully. This persisence 5

6 creaes an addiional challenge for modelling. Secondly, in DeMarzo and Sannikov he players ouside opions depend on he curren value. Workers skills and he rm value afer managerial urnover depend on he pas performance. In our model he res value of he rm is consan. This feaure re ecs he fac ha he value of rm asses 7 is relaively sable over ime. Such model is especially suied o examine quesions relaed o corporae nance and capial srucure heory. Pavan, Segal and Toikka (212) examine incenive compaibiliy and revenue maximizaion in a general environmens wih quasi-linear uiliy. In heir model persisen shocks imply consan disorions over ime. This resul follows from he fac ha he impulse responses are bounded in he model. 8 This consrain has he implicaion ha he principal canno le he agen s value become arbirary negaive o save he cos of inducing ruhful revelaion in he fuure. Thus here is no gain for he principal o pospone disorions for he fuure. Since he agen is risk-neural in our model, here is no moive for consumpion smoohing. Thus, delaying paymens o he agen is cosless. Therefore, he principal opimally delays paymens o he agen unil he rs-bes is reached. Garre and Pavan (211, 212) exend he model o allow for he agen s risk-aversion and apply he mehods o examine managerial urnover. They do no consider liquidaion. The res of he paper is organized as follows. In Secion 2 we de ne he general seing of he model. Secion 3 derives he rs-bes of he problem. Secion 4 presens he opimal conrac. The Secion 4.1 derives he opimal conrac for he case ha he players are unable o engage in a long-erm relaionship. The principal hen hires he agen from a spo marke. In Secion 4.3 we show ha he principal can improve her payo by o ering a long-erm conrac o he agen. This resul holds even if we allow he players o renegoiae he conrac. The Secion 4.4 discusses full commimen. The Secion 5 concludes. 2 Seing We examine a coninuous-ime game wih wo players: a principal and an agen. Boh players are risk-neural, and discoun he fuure a a common rae r >. The principal has unlimied access o funding, bu he agen is proeced by limied liabiliy. Therefore, he conrac canno impose negaive paymens on him. 7 Wih rm asses we refer here o machines and oher asses ha can be sold in re sales in circumsances of nancial disress. 8 See Thomas and Worral (199) and Wang (1995) for relaed resuls wih iid informaion srucure. 6

7 The agen runs a rm ha generaes a sochasic oupu a each poin in ime. The rm needs an iniial invesmen I o ge sared. The agen has no iniial wealh such ha his invesmen has o be covered by he principal. If he rm is liquidaed is asse can be sold in re sales for he value B. We assume ha his liquidaion irreversible. A sochasic process x = fx ; F ; < 1g de ned on he probabiliy space (; F; P ) drives he oupu. In oher words, he rm generaes a cash- ow level fx g a each poin in ime. The cash- ow is sochasic and changes according o dx = dz, (1) where fz s : s g is a sandard Brownian moion. Thus a each poin in ime, here is a persisen, exogenous shock on he rm s oupu ha is normally disribued wih zero mean and volailiy. The iniial value of he cash- ow is x = x, and is unknown o he principal. The rue level of he cash- ow is unobservable o he principal and is repored by he agen. The agen can diver cash- ow for his privae bene s by lying abou he realizaion of he underlying Brownian moion. Following he sandards in he lieraure, we assume ha such diversion is ine cien. Formally, he agen s lying sraegy l = fl s : s g is a sochasic process progressively measurable wih respec o F. Given he sraegy chosen by he agen, he principal receives a cash- ow level x l s ds, ^x a each as long as he rm is operaed. We assume ha he agen s lying sraegy is absoluely coninuous wih respec o ime. Whenever he cash- ow is unobservable o he principal, ^x deermines he principal s expecaion abou he fuure value of he rm. Since ^x follows a maringale from he poin of view of he principal, ^x corresponds o he principal s expecaion abou he fuure value of he cash- ows. We concenrae on equilibria in which he agen repors ruhfully. 9 Then a equilibrium, he principal s expecaion and he rue value coincide. O he equilibrium pah, he principal s expecaion di ers from he agen s. From he principal s poin of view he uncerainy in he economy is driven by he repored Brownian moion f^z s : s g. Given her expecaions, his process evolves according o d^x = d^z (2) = dz l d. 9 Since divering cash- ows is ine cien, all equilibria of he game necessarily have his feaure. 7

8 If he agen lies abou he underlying Brownian moion, he principal s expecaion abou he cash- ow di ers from he rue level. Then here is a gap x ^x. We say ha he agen is divering cash- ows and enjoying privae bene s. For each dollar divered, he agen obains a privae bene 2 [; 1]. Thus, if he agen lied abou he shocks in he pas, he now enjoys a privae bene of (x ^x ): Since he cash- ow is persisen, also he agen s privae bene is persisen. Diversion is ine cien such ha divering a dollar is relaed wih a social cos of 1. Following he cash- ow repor, he principal delivers he agen a nonnegaive ow of income i = I(^x s : s ). Limied liabiliy imposes ha he ow income canno ake negaive values such ha i for all. In he beginning, he paricipans agree on a conrac = (i; ) ha deermines he income process for he agen, and a sopping ime a which he rm is liquidaed. The public hisory of he game consis of he pas repors, pas liquidaion decisions and pas paymens. The lraion ^F conrols he arrival of public informaion. The privae hisory of he agen consiss of he public hisory and of he pas realizaions of he cash- ow which are capured by he lraion F. The principal forms her expecaion condiional on he public hisory. She receives he cash- ow from he agen and delivers him a ow income. If she erminaes he projec, she receives he erminaion value. This erminaion payo can be inerpreed as he res value of he asses of he rm ha can be sold in re sales. Formally, he principal s expeced payo is given by ^E e r(s ) (^x s i s )ds + e r( ) B. The agen delivers he cash- ow o he principal and receives a paymen ow. The expeced uiliy of he agen can be wrien as E e r(s ) [i s + (x s ^x s )] ds. 8

9 To simplify he expressions, we normalize he agen s ouside opion o zero. Scaling he ouside opion o implies ha he agen has no opion value in he model. As will be seen laer, he case in which he cash- ow follows a maringale and he agen has ouside opion is special in he sense ha he incenive compaibiliy consrains ake a paricularly simple form. 3 Firs-Bes In his secion we derive he opimal soluion o he problem when he principal and he agen joinly maximize heir pro s. In his case, he problem reduces o an opion problem ha can be solved using sandard mehods. In his version of he model he payo relevan hisory is rivially summarized by he curren value of he cash- ow process. Thus, we can use x as a sae variable. A each poin in ime he players face he following decision problem. They can run he rm for one furher insan o gain rens from he cash- ow ha he rm generaes. Alernaively, hey can liquidae he rm and sell is asses in re sales. When comparing he wo alernaives, he players will ake ino accoun ha he liquidaion is irreversible, and has herefore opion value. If hey run he rm for a furher insan, a posiive shock may occur such ha he cash- ow reaches a higher level. If he shock is negaive, he players always have he opion o inerrup he cash- ows, and liquidae he rm. The players maximize hey join payo ha is a sum of he principal s and he agen s payo. We de ne he social value of he rm as s (x ), f(x ) + v (x ). The social value consiss of he value of he cash- ows ha he players receive as long as he rm is operaed and he liquidaion value ha is realized when he rm is liquidaed. The opimal decision amouns o choosing an opimal sopping ime a which he rm is liquidaed and is asses sold in he re sales. The sopping ime is chosen such ha i maximizes he social value of he rm s (x) = sup E e r(s ) x s ds + e r( ) B The players run he rm unil he cash- ow his a hreshold level k F B. A his poin, his ouside opion becomes more valuable, and he rm is liquidaed. Le, inf : x k F B denoe he sopping ime a which he cash- ows are inerruped and he ouside opion realized. 9

10 A ime, he players compare he res value of he rm and he ineremporal value ha hey would have received if hey had operaed he rm furher. Formally, his rade o can be wrien as B s (x ). (3) The players operae he projec unil he expeced pro reaches he ouside opion. Thus, he rm is liquidae a he poin of ime a which (3) equals. As long as he rm is operaed, he principal earns an insananeous payo of rs (x ) d. (4) This payo consiss of wo pars. Firs, as long as he rm is operaed, i produces cash- ows. Second, liquidaion has opion value. Thus, when shocks are realized, he value of he rm changes. Applying Iô s lemma when he cash- ow evolves according o (1), we obain x s xx (x ) (5) By combining (4) and (5), we nd ha he rm earns an insananeous payo of x d rs (x) s xx (x ) d (6) in he coninuaion region. Nex, he players can eiher coninue operaing he rm or erminae he cash- ows and realize he ouside opion. Naurally, hey opimally choose he he opion relaed wih he higher expeced pro. Combining (3) and (6), we can describe he players problem of choosing he opimal sopping hreshold in heir Hamilon-Jacobi-Bellman equaion = sup B s (x ) ; x rs (x ) s xx (x ) (7) The boundary condiions for he problem can be derived using he sandard arbirage argumens. Coninuiy of he value funcion requires ha he liquidaion value of he rm maches wih he value of coninuing he projec a he sopping poin. Furhermore, di ereniabiliy of he value funcion requires ha he rs derivaives mach a he boundary. A he boundary, he sandard value maching and smooh pasing condiions s(k F B ) = B and s x (k F B ) = mus hold. By solving (7), we can wrie he rs-bes value of he projec as a funcion of he iniial cash- ow s (x) = x r k F B r e (x kf B) + Be (x kf B), (8) 1

11 where = p 2r. The opimal hreshold is k F B = rb p 2r. (9) The social value of he rm consiss of hree erms. The rs erm in he righ hand side of (8) corresponds o he ineremporal value of he cash- ow. The second erm conains he expeced loss from inerruping he cash- ow. The las erm describes he expeced pro from liquidaion. From (9) we can see ha he rm is liquidaed a a level a which he value lies below he liquidaion value. This follows from he opion value naure of he liquidaion decision. By waiing an insan furher here is he opporuniy ha posiive shocks drive he cash- ow a a higher level. Liquidaion, in urn, is irreversible. We can also see from (9) ha if he volailiy is higher, he opimal sopping hreshold is lower. A higher cash- ow volailiy implies ha a possible posiive shock can be higher. This implies ha a higher level can be achieved afer a shorer ime period. 4 Opimal Conrac In his secion, we de ne he opimal conrac. Firs, we de ne he opimal conracing problem. An incenive compaible conrac ( ; x) maximizes he principal s pro f ( ; x) = max E e r(s ) (x s i s ( ; x s ))ds + e r( ) B (1) subjec o delivering he value v ( ; x) = E e r(s ) i s ( ; x s )ds o he agen for he ruhful sraegy l = for all, ha induces he ow payo ^x = x o he principal, and a lower pro (11) v ( ; x) ^v ( ; x) (12) = E e r(s ) [(x s ^x s ) + i s ( ; ^x s )] ds for any alernaive repor sraegy. By varying he principal s iniial value, we can examine di eren divisions of relaive bargaining power. We focus on he case 11

12 in which he principal holds all he bargaining power and can make he agen a ake i or leave i o er. We also discuss brie y he case in which here are many poenial nanciers. In ha case he agen is able o exrac all he rens, whereas he principal breaks even such ha f = I Incenive Compaibiliy In his secion, we derive necessary and su cien condiions for incenive compaibiliy in our framework. To examine incenive compaibiliy, we mus deermine how he agen s coninuaion value changes in response o ucuaion of he oupu. Thus we need a racable represenaion of he agen s coninuaion value. This represenaion should capure how he agen mus be rewarded for posiive oupu realizaions o be willing o reveal upward movemens of he underlying Brownian moion. Similarly, following downward movemens a punishmen mus follow. Of course, since he agen is proeced by limied liabiliy, he punishmens are bounded from below. For he derivaion, we conjecure ha he coninuaion value of he agen, W is coninuous. We verify in he Secion 4.4 below ha his conjecure is indeed correc a he equilibrium. In derivaion, we rely on mehods similar o DeMarzo and Sannikov (26, 211) and Sannikov (27, 28). Tha is, we examine how he agen s coninuaion value changes following he ucuaions of he Brownian moion. Then we look for condiions for incenive compaibiliy. The basic di erence o he earlier papers is ha he cash- ow is persisen over ime. Hence, one would expec also he agen s paymens o be described by a persisen process. As usual in he lieraure, he agency problem arises, because he principal is unable o discover he rue source of a negaive shock. The shock may have been caused exogenously by he naure, or may be he consequence of an ine cien acion chosen by he agen. To preven he agen from aking he ine cien acion, he principal has o develop an incenive scheme ha compensaes he agen for no doing so. Unlike he previous models, he shocks are persisen in our framework. The persisence creaes an addiional challenge for modelling. Our soluion concep builds on he following idea. We decompose he agen s coninuaion value in wo pars. The rs par describes how he value changes if he agen repors a shock. This par is relevan for incenive compaibiliy. The second par capures he e ec of delaying paymens. This par conrols for he ineremporal allocaion of he agen s income. We show ha delaying paymens o he agen corresponds o 1 As usual in he principal-agen lieraure, he opimal conrac is more e cien if he agen holds all he bargaining power. 12

13 public saving of he agen s value. As sandard in he principal-agen framework, incenive compaibiliy is una eced if he paymens o he agen are backloaded. In his secion we examine how he agen s value changes following he ucuaions of he Brownian moion. When rying o deermine he agen s payo, we face he following challenge. Boh he agen s sraegy and he uncerainy in he economy are deermined by f^z s : s g. The payo s, in urn, are deermined by he level f^x s : s g of he process. Thus, if he agen lies abou he underlying Brownian moion, here are wo consequences. Firs, he principal s expecaion changes. Tha is, o he equilibrium pah, he players have di eren expecaions abou he fuure value of he rm. In our model his divergence is no correced even if he agen repors ruhfully in he fuure. Secondly, here is a persisen change in privae bene s. To nd a racable represenaion of he agen s coninuaion value, we mus be able o compare he agen s deviaion payo and his promised value if he repors ruhfully. Roughly speaking, we compare he agen s promised ow paymen wih his privae bene. Boh he privae bene and he promised paymen change in a persisen manner in response o he ucuaions of he Brownian moion. Now recall ha wih limied liabiliy, he principal has wo insrumens o induce ruhful reporing. She can eiher liquidae he rm or provide he agen wih posiive paymens. In oher words, in he coninuaion region he agen has o be compensaed for his informaion ren. Of course, his compensaion can eiher be delivered o he agen hrough immediae income, or in erms of deferred paymens. Noe ha limied liabiliy has he consequence ha i is impossible o coninue operaing he rm and provide he agen wih incenives o repor ruhfully, unless he receives a posiive payo for his repor. Therefore, i is helpful o le he projec be liquidaed a he poin a which he agen s coninuaion value his his ouside opion. Formally, we de ne a sopping ime = inff : ^W = g, (13) where ^W denoes he principal s assessmen of he agen s coninuaion value, or he promised value of he agen. A equilibrium, W = ^W, and he principal correcly anicipaes he agen s coninuaion value. To derive incenive compaibiliy condiions in he coninuaion region, we need o examine how he agen s value evolves in response o he ucuaions of he Brownian moion. Towards his end, we rewrie he agen s value o consider incenive compaibiliy across cash- ow levels raher han across ime. Tha is, we rewrie he agen s value o compensae him for a level of cash- ow ha was reached by he pah of repored Brownian moion f^z s : s g. The income process fi : g describes when he agen acually receives paymens. As we will see laer, his process depends on he enire pah of he cash- 13

14 ow process. Now le process fw : g describe he ow value ha he agen is promised for his repors. This process depends on he pah of he repored Brownian moion. Now he agen s lifeime income has o equal o his promised paymens. A =, he promise keeping consrain requires ha ^W = ^E e r i d = ^E e r w d. Of course, his reward can be paid ou o he agen eiher now or in some fuure period. Thus, he process w is jus a book-keeping procedure ha describes how much he agen was punished or rewarded for pas ucuaions of he cash- ows. Noice ha if he agen divers cash- ows, his privae bene unil ime is ^ e rs (x s ^x s )ds. We can make wo immediae observaions abou he agen s abiliy o diver cash- ows. Firs, he curren value of he cash- ow x is a su cien saisic for he fuure privae bene s. Tha is, he agen s privae bene is adaped o he lraion generaed by x. This propery has he consequence ha incenive compaibiliy in fuure periods only depends on he curren value of x. Conversely, incenive compaibiliy in fuure periods guaranees ha he agen has no incenive o lie abou he cash- ows if he lied abou he cash- ow in he pas. Secondly, he rue cash- ow level x follows a maringale. The reward process w s describes he corresponding compensaion when he agen repors ruhfully. Then his uiliy from ruhful reporing can be wrien as ^ e rs w s ds. We impose some resricions on w o be able o deermine he condiions ha are necessary and su cien for incenive compaibiliy. Firs, since he cash- ow is x measurable, we wan w o be ^x measurable. Secondly, since x follows a maringale, we also wan w o follow a maringale. 11 Now we can use he maringale represenaion heorem o derive he sensiiviy of w o he ucuaions of he repored Brownian moion. This sensiiviy is conrolled by he principal. A he projec has generaed cash- ows in he pas, and he agen has been rewarded. Moreover, he cash- ow is a curren level x which implies a cerain expeced value of fuure rewards and privae bene s. The agen s coninuaion 11 If x follow a more general di usion, also w mus be de ned more generally. In he case in which he agen s ouside opion, his process coincides wih he single crossing condiion for he agen s value. See Garre and Pavan (211) for relaed resuls in discree ime. 14

15 value consiss of wo pars: he pas promises ha were no paid ou as income and of fuure expeced rewards and privae bene s. Formally, W = e r(s ) (w s i s )ds + E e r(s ) (w s + (x s ^x s )ds. (14) Since he agen canno save privaely, he rs par on he righ-hand side of (14) is known a period. Tha par can be inerpreed as deferred paymens o he agen. We de ne m, e r(s ) (w s i s ) ds. Since he agen has limied liabiliy, he deferred paymens play an imporan role when deermining he opimal conrac. If he rm has generaed cash- ows in he pas, he agen s coninuaion value has increased. A higher coninuaion value makes i possible o punish he agen harder. Equivalenly, we can say ha he agen s share of he rm increases. If he agen owns a higher share of he rm, he cares more abou he ucuaions of he oupu. This relaxes he incenive consrain. The following lemma provides a represenaion of he agen s coninuaion value Lemma 1 For each, he agen s coninuaion value can be wrien as W = m + E e r(s ) (w s + (x s ^x s ))ds. (15) The rewards or punishmens are described by a sochasic process fw s g s evolves over ime according o ha w = w + s d^z s. (16) The sensiiviy process f s g s describes how he agen s ow payo evolves in response o he ucuaions of he repored Brownian moion. I is deermined by he hisory of he pas repored shocks F ^z = f^z s : s g. The deferred paymens are colleced in m ha consiss of pas promises minus he income ha he agen received from he principal m = e r(s ) (w s i s )ds. Proof. See Appendix. Nex we look for condiions ha guaranee ha he agen repors cash- ows ruhfully. Here he incenive compaibiliy condiions mus be deermined such 15

16 ha he agen s payo is nowhere higher if he lies abou he underlying Brownian moion. Noe ha whenever cash- ow increases, he agen s privae bene increases. The marginal increase for each addiional uni is. For he agen o repor an upward shock ruhfully, his compensaion has o increase by a leas. Secondly, limied liabiliy imposes ha he paymens o he agen are nonnegaive. This consrain implies ha he punishmens ha he principal can impose o he agen are limied. However, by delaying paymens o he agen, he principal can increase he agen s wealh o allow for harder punishmens. Noe ha he principal s assessmen of he agen s value is ^W = m + w r. Here we can immediaely observe ha a higher value of m allows w o ake lower values. Tha is, he principal has he opporuniy o punish he agen harder. Also, a comparison wih (13) indicaes ha a higher coninuaion value induces a laer sopping ime. Finally, he iniial value of w has o be high enough for he agen o be willing o repor x ruhfully. Noice ha he agen will never se he iniial value oo low o rigger an immediae liquidaion. Le k, inff^x : ^x x and ^W = g, and noice ha for ^x = k, we have ha w =. A =, he agen can repor he iniial value ruhfully and obain he corresponding iniial promise of w. If he agen divers, he can repor o earn an iniial privae bene of (x k ). Following he sandard argumen, we can conclude ha he agen repors x ruhfully if only if w (x k ). The necessary and su cien condiions for incenive compaibiliy can be summarized in he following proposiion Proposiion 1 For, a necessary and su cien condiions for incenive compaibiliy can be wrien as (17) and Moreover, he agen s limied liabiliy requires ha w (x k ) (18) i (19) 16

17 for all and, herefore, W, where he agen s coninuaion value W is as de ned in (15). Proof. See Appendix. 4.2 Spo Conracs In his secion, we consider a version of he model in which he players are unable o engage in a long-erm relaionship. 12 In his case he principal hires he agen insananeously from a spo marke o operae he rm. Wih spo conracs he agen is rewarded wih insananeous income. The level of he income depends on he realizaion of he cash- ow. To deermine he opimal spo conrac, i is imporan o idenify he payo relevan hisory. For he agen, he payo relevan hisory consiss of he curren value of he cash- ow process. The principal condiions her sraegy on her expecaion abou he curren value. For any incenive compaible conrac, he expecaion coincides wih he agen s repor, i.e., ^E[x ] = x. The inabiliy o engage in a long-erm relaionship resrics he principal s abiliy o condiion he paymens o he agen on he hisory. To see why his is he case, suppose ha he principal promises he agen a higher paymen in some fuure period for a higher repor oday. However, in he absence of long-erm conracs he principal has no incenive o keep her promise. This implies ha he agen has no incenives o repor ruhfully in he rs place. Wih spo conracs, he opimal conrac akes he following simple form. A each poin in ime, he agen delivers a cash- ow repor ^x condiional on he curren value of he cash- ow. Following he repor, he principal delivers he agen income i(^x ). The principal hires he agen in he marke coninuously unil her expecaion of he fuure value of he rm falls oo low. If he expecaion falls oo low, he ouside opion becomes more valuable o he principal and he rm is liquidaed. Formally, le, inff : ^x k S g (2) denoe he value of he cash- ow a which he rm is liquidaed. We assume for now ha he liquidaion decision is a hreshold policy, and show laer ha his propery holds a he equilibrium. 12 Technically speaking, we can drop he conrac from he argumens in he opimal conracing problem (1)-(12). 17

18 The principal s problem can now be wrien as f(x) = sup E e r(s ) (x s i s )ds + e r( ) B subjec o he incenive compaibiliy condiion (17), and he limied liabiliy consrain (19). Similar o he rs-bes case, he principal faces he rade o beween leing he agen operae he rm furher, or liquidaing i for is res value. If he rm is liquidaed he principal earns he value of he ouside opion, bu loses he coninuaion value of he fuure cash- ows. Tha is, B f(x ). (21) In he case ha he rm coninues o operae, he principal earns an insananeous payo equal o her discoun rae. This payo consiss of he cash- ow minus he compensaion for he agen, and of he change of value funcion ha resuls from he changes in cash- ow repor. The laer e ec can be calculaed by using Iô s lemma, aking ino accoun he evoluion of he repor process in (2). Thus he agen s value in he coninuaion region sais es (x i )d rf(x )d f xx(x )d. (22) Combining (21) and (22), we can wrie he principal s Hamilon-Jacobi-Bellman equaion as = sup B f(x ); x i rf(x ) f xx(x ). (23) From (23) we can see immediaely ha he principal s paymen is linear in he agen s income i. Thus, o maximize pro s, he principal mus provide he agen he lowes possible consumpion ha is compaible wih ruhful reporing. To deermine he opimal paymen schedule, noe ha he agen never riggers an early liquidaion. Then he mos ha he agen can diver is (x k S )d. For he agen o repor ruhfully, he mus be compensaed a leas for his informaion ren. Thus, we nd ha he agen earns a ow income of i = (x k S ): 18

19 Nex we will derive he opimal liquidaion hreshold k S from he principal s opimizaion problem. Subsiuing he opimal consumpion level in (23), we obain = sup B f(x ); (1 ) x + k S rf(x ) f xx (x ). (24) The boundary condiions are similar o he rs-bes case. A he liquidaion resul, he principal s value funcion has o mach wih he liquidaion value, i.e., f(k S ) = B. Moreover, for he sopping decision o be opimal, he rs derivaives have o mach a he boundary. This yields us he smooh pasing condiion f x (k S ) =. Nex we reurn o he agen s problem. As long as he principal les him o operae he rm, he earns an insananeous payo equal o his discoun rae. A equilibrium, his payo mus equal o he informaion ren ha he agen is able o exrac above he principal s opimal sopping hreshold. Thus we can wrie he agen s Hamilon-Jacobi-Bellman equaion as 13 rv (x ) = (x k S ) (25) Since he agen does no sop, his only boundary condiion is he value maching condiion v(k S ) =. The opimal insananeous conrac can be summarized in he following proposiion Proposiion 2 The opimal spo conrac is of he following form. For each, he agen receives a ow income i = (x k S ) from he principal. The principal earns a ow pro of x i = (1 ) x + k S. The rm is liquidaed a, when he cash- ow level reaches he value k S = rb (1 ) p 2r. (26) The principal receives he liquidaion value B. The agen receives his ouside opion. 13 Noe ha since he agen s ouside opion is normalized o and he receives no paymen for he cash- ow below he hreshold, v xx (x) =. This illusraes he propery ha he agen has no opion value in he model. 19

20 Proof. See Appendix. The equilibrium value funcions can be calculaed explicily from he ordinary di erenial equaions (24) and (25) respecively. From he rs, we obain ha he principal earns he value f(x ) = (1 ) x + k S r k S r e (x ks) + Be (x ks ) By rewriing (25), we nd ha he agen receives he value (27) v (x ) = (x k S ). (28) r From (28) we see ha he agen receives a share of he cash- ow ha exceeds he liquidaion hreshold. The inuiion for he resul is as follows. The agen always has he opion o diver cash- ows as long as he does no rigger an early liquidaion. To preven he agen from divering cash- ows, he principal has o compensae he agen for his informaion ren above he hreshold. Below he hreshold, he erminaion hrea serves as a punishmen mechanism. In ha region, he principal is able o exrac all he rens from he agen. Since he ouside opion is and divering cash- ows and enjoying privae bene s has posiive value, he agen never divers cash- ows o rigger an early liquidaion. The principal s pro in (27) consiss of hree pars. Firs, she receives he ne value of he cash- ow afer he agency cos. Above he hreshold, his value consiss of he cash- ow ne of he agency cos. Below he hreshold, he whole value is exraced. The second erm denoes he gain or loss from sopping, and he las erm equals o he res value of he rm ha is received when he rm is liquidaed. Like in he rs-bes case, he opimal sopping hreshold (26) depends on he volailiy of he cash- ow. Again, his dependence arises from he opion value of sopping. However, he value of coninuaion is now lower han in he rsbes case. The principal has o give up a share of he value of a posiive shock. Oherwise, he agen would conceal he shock from he principal. Therefore, he opimal sopping hreshold is higher han in he rs-bes case. Thus, he cash- ows are inerruped earlier han is socially opimal. Because of he early sopping, he social value of he rm is lower han in he rs-bes case. The ine ciency only vanishes in he limi as!. Figure 1 shows he opimal value as a funcion of he curren level of he cash- ow for low and high incenive cos respecively. Wih higher agency cos, he agen is able o exrac more rens from he principal. Moreover, cash- ows are erminaed a higher level such ha he e ciency loss is more severe. 2

21 1 Low Incenive Cos 1 High Incenive Cos Firs-bes 6 Firs-bes Profi 4 Principal's Profi Profi 2 Agen's Profi Cash-flow 4 Agen's Profi 2 Principal's Profi Cash-flow Figure 1: For L = :3, H = :9, B = 5, = 5, and r = 1% Full Incenive Compaibiliy In his secion we discuss some properies of he agen s coninuaion value in more deail. We show ha a equilibrium, he agen earns he same payo if he divers cash- ows han if he repors ruhfully. Since he agen s value only depends on he rue value of he cash- ow, and is independen of his repor, he is indi eren beween his sraegies. This indi erence holds for all hisories. Indi erence guaranees ha ruhelling is always opimal o he agen. In our model, his resul holds boh on and o he equilibrium pah. Thus he agen always has he righ incenives o repor ruhfully. The agen s rue coninuaion value for some arbirary sraegy k S ^x x is given by W = i r + r (x ^x ). (29) Firs noice ha if he agen delivers he repor ^x, he receives a paymen i = (^x k S ). Suppose ha he agen divered he amoun x bene (x ^x ). ^x. Then he earns he privae 21

22 Subsiuing hese values in (29), we nd ha he agen s coninuaion value sais es W = i r + r (x ^x ) = (x k S ). r Tha is, he agen s value only depends on he rue value of he cash- ow and no on his repor. 4.3 Renegoiaion-Proof Conracs In his secion we derive he opimal conrac in he case in which he principal and he agen are able o engage in a long-erm relaionship. However, his conrac can be renegoiaed and replaced by a new conrac a any poin in ime. As discussed earlier, a renegoiaion-proof conrac has o ful ll he following requiremen. A conrac is renegoiaion-proof if afer no public hisory, here exiss a conrac ha (i) is superior o he principal, and (ii) he agen would accep. To be more concree, we allow he principal o propose an alernaive conrac o he agen a any poin in ime. The agen can eiher accep or rejec he o er. If he agen acceps, he conrac is replaced. If he agen rejecs, he old conrac remains valid. We resric he aenion on conracs ha are incenive compaible. Moreover, a equilibrium he principal only proposes conracs ha are no iself replaced by an alernaive conrac following any public hisory. To undersand he dynamics of he opimal conrac, i is helpful o hink of he problem wihou limied liabiliy. If he agen had wealh, he principal would opimally sell him he rm for is expeced value. Wih limied liabiliy such rsbes soluion is infeasible. However, wih good pas performance, he agen has accumulaed wealh. In he opimal conrac, he agen uses his wealh o buy a larger share of he rm. If he agen holds a larger share of he rm, his value is more sensiive owards he ucuaions of he oupu. Equivalenly, we can hink of he principal rewarding he agen by promising him a higher coninuaion value. In he opimal conrac he paymens o he agen are delayed unil he rs-bes is reached. Once he rs-bes is achieved, he agen sars o receive income and he rm coninues operaion unil i is liquidaed in he rs-bes. Inuiively speaking, he rs-bes value canno be achieved because he principal is no able o punish he agen hard enough. This relies on he fac ha he agen does no have money. Now by delaying paymens o he agen, he principal 22

23 makes he agen wealhier. A wealhier agen can, of course, be punished harder. Once he agen has accumulaed enough wealh, he rs-bes can be reached. Formally, he opimal conrac can be described using wo sae variables. The curren level of he cash- ow process re ecs he fuure pro abiliy of he rm. Since he acual level is unobservable o he principal, she uses he agen s repor o form her expecaions. Furhermore, following he sandard in he lieraure we use he agen s coninuaion value as a sae variable. In he model he coninuaion value of he agen describes he disance beween he curren level of he cash- ow and he liquidaion hreshold. Increasing he coninuaion value implies an earlier sopping ime. A laer liquidaion increases he oal value of he rm unil o he poin a which rs-bes is achieved. Of course, since he principal holds all he bargaining power, she only increases he coninuaion value so much ha he incenive consrain jus binds. Now noice ha if x k S, i follows from he analysis in Secion 4.3 ha he principal always prefers o operae he rm insead of liquidaing i. Thus he principal canno commi o liquidae he rm for higher cash- ow levels. She would always have he incenive o propose an alernaive conrac ha prevens he liquidaion. Of course, he agen is able o anicipae his. Thus, for any conrac he agen can choose repor sraegies such ha ^x = k S for all x k S. Then he earns an expeced payo of W S = (x k S ). r Secondly, recall ha he principal gains by increasing he agen s coninuaion value because increasing he agen s wealh allows her o punish him harder. Now his is, of course, only pro able if he principal can gain by punishing he agen harder. As we will see laer, his is he case unil he rs-bes is reached. Nex, we wan o deermine he agen s coninuaion value ha is needed o implemen he rs-bes. Reasoning along he same lines han in he Secion 4.3, we can conclude ha implemening he rs-bes requires ha he agen is compensaed for an informaion ren of (x k F B )d. We also know ha he principal will no be willing o leave he agen a higher value for his informaion ren. The smalles coninuaion value needed o compensae he agen for ha informaion ren is W F B = (x k F B ). r 23

24 Since he principal holds all he bargaining power, she exracs all he rens from he agen. Thus she increases he agen s coninuaion value in such a way ha he agen is jus indi eren beween immeadiae reward and deferred paymens. To nd he dynamics of he opimal conrac, we rs derive he law of moion of he agen s coninuaion value. Here we have o ake ino accoun wo e ecs. Firs, he coninuaion value changes following he ucuaions of he oupu. This follows from he fac ha he agen is punished or reward for he ucuaions of he repored Brownian moion. Secondly, when he paymens are deferred, he agen mus be compensaed for he discouning. Thus, when he paymens o he agen are delayed, hey earn an ineres rae r. Nex we move o he principal s problem. There are wo facs o prove. Firs, we mus show ha he incenive consrains bind. Tha is, he principal always chooses he lowes possible reward or punishmen ha guaranees ha he agen repors ruhfully. Secondly, we show ha he paymens are delayed unil he rs-bes is reached. Firs-bes is reached when he agen s coninuaion value his a he same ime han he cash- ow level his he e cien liquidaion hreshold. Remark 1 Like in models wih uncorrelaed oupu, he rs-bes conrac is no unique. Since he players discoun he fuure using a common discoun rae, he ineremporal allocaion of he wealh becomes irrelevan once he rs-bes is reached. The only resricion we mus impose on he se of conracs is ha W < 1 for all. Tha is, he principal canno delay he paymens o he agen forever Evoluion of he Coninuaion Value Noice ha he principal has wo ways of rewarding agen: She can eiher deliver he agen immediae paymens, or increase his coninuaion value. The reward process fw : g capures he reward or punishmen ha he agen receives for his repor. The income process fi : g describes when and how much income he agen receives from hen principal. Inuiively speaking, he income process describes when he reward is acually paid ou o he agen. Recall from Secion 4.2 ha he principal s assessmen of he agen s coninuaion value can be wrien as ^W = m + w r, where he erm m capures he deferred paymens. The second erm describes he agen s curren expeced compensaion ha depends on he curren repored cash- ow level. 24

25 Di ereniaing wih respec o, we nd ha he agen s coninuaion value evolves according o d ^W = dm + dw r. (3) If he principal delays paymens o he agen, his coninuaion value increases. If he value is delayed, he agen mus be compensaed for discouning. Therefore, m evolves according o dm = (rm + w i ) d By subsiuing in (3), we nd ha he coninuaion value evolves according o d ^W = (r ^W i )d + d^z where = =r, and r ^W = rm +w. Following he sandards in he lieraure, we can use he agen s coninuaion value as a sae variable o deermine he opimal income process chosen by he principal. The nex lemma shows ha he incenive compaibiliy is no a eced if he paymens o he agen are delayed Lemma 2 If he condiions described in he Proposiion 1 hold, and he agen s coninuaion value evolves according o he agen has no incenive o lie abou he cash- ow. d ^W = (r ^W i )d + d^z, (31) As we will see in he nex subsecion, principal s opimaliy requires ha he incenive consrains binds. Reasoning along he same lines han in Secion 4.3, we know ha o implemen he rs-bes, he agen s coninuaion value has o reach he value W F B = (x k F B ). r Moreover, since he principal has all he bargaining power, he agen will earn he same iniial value han in he spo conracs case. Then, from he analysis of he previous secions, we know ha w r = (x k S ). r Tha is, o susain rs-bes, he principal needs o accumulae paymens unil m F B = (ks k F B ). r 25