The Irrelevance of Control Rights in Agency Models under Risk Neutrality 1

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1 The Irrelevnce of Control Rights in Agency Models under Risk Neutrlity Susheng Wng 2 Novemer 2 Third Revision Astrct: The lloction of control rights is key issue in incomplete contrcts. In this pper, we dd n exit option, the right of erly termintion, to the stndrd gency model for employment contrcts. We ddress two questions: () Who should hve this right? (2) Wht is the effect of its inclusion in contrct? Under risk neutrlity, we find tht () it does not mtter for investments nd economic efficiency to whom this right elongs, lthough it my ffect contrctul terms nd income shring, nd (2) lthough the lloction of this right does not mtter, its inclusion cn hve either positive or negtive effect on economic efficiency. Our nlysis provides enchmrk for more generl nlyses on control rights under risk version. Keywords: Informtion Reveltion, Control Rights, Agency Model, Risk Neutrlity JEL Clssifiction: D23, D2, D8 Acknowledgement: The refereeing process resulted in mjor improvement to the pper nd the input from the two nonymous referees is grtefully cknowledged. All errors remin our own. 2 Deprtment of Economics, Hong Kong University of Science nd Technology, Cler Wter By, Hong Kong. Emil: s.wng@ust.hk.

2 . Introduction As pointed out y Mskin nd Tirole (999), one of the three key ssumptions in the incomplete contrct literture is the risk-neutrlity ssumption. Agency models with risk neutrlity re populr in pplictions ecuse of the existence of n optiml liner contrct. In world where informtion is reveled ex post, control rights re n importnt prt of the model. In this study, we ddress two questions: How does the inclusion of control rights ffect investments nd economic efficiency? How does the lloction of control rights ffect investments nd economic efficiency? We provide simple nswers to these questions. Following the pioneering works y Grossmn nd Hrt (986), Hrt nd Moore (99) nd Hrt (995), control rights hve ecome hot topic. In the world of incomplete contrcts, control rights re generlly defined s the ownership of residul rights. In our principl-gent model in prticulr, control right is defined s the right of erly exit. If the exit option is not exercised, the contrctul greement remins vlid; if the exit option is exercised, the contrctul greement ecomes invlid nd new contrct my e negotited ex post. In equilirium, the optiml contrct is renegotition-proof nd hence vlid throughout the coopertion period. Rel world contrcts, such s employment contrcts, typiclly contin sttements on erly termintion. In prctice, there re lrge vritions in exit options mong employment contrcts. In some contrcts, employers hve the control rights of exit. For exmple, in professionl sports, employers impose trnsfer fees to prevent plyers from leving without the employers consent. However, in most lor contrcts, employees hve the control rights of exit. According to Meccheri (28, p.5-6), penlties on mnger-grde stff for quitting jos re rre in lrge Western corportions. In most cses, n employer s erly termintion of employment is conditionl, while employees hve the control rights of exit, mening tht they cn quit the jo nytime without penlty. For exmple, mny lrge US firms commit to the soclled zero-firing policy tht prevents them from lying off employees without employees consent. In Jpn, it is trdition for employers to provide lifetime employment. The oserved vritions in ssignment of termintion rights my e ecuse termintion rights re inconsequentil so tht they pper in mny different forms, which is consistent with our irrelevnce result; it my lso e ecuse the trding prties re risk verse so tht termintion rights re relevnt nd need to e specified conditionl on the circumstnces. In the ltter cse, our irrelevnce result provides enchmrk for the understnding of such dependence. In some cses, the right of erly exit seems prticulrly crucil in contrctul reltionship. For exmple, in the reltionship etween venture cpitlist nd n entrepreneur in venture cpitl investment, the right to replce the entrepreneur ex post y professionl mnger is considered y reserchers to e n importnt incentive mechnism. It is elieved tht venture cpitlists prefer to finnce risky firms y convertiles ecuse convertiles offer Pge 2 of 23

3 them this right (Shlmn, 99). In university hirings, tenure-trck contrcts re populr ecuse such contrcts gives deprtment the right to grnt lifetime jo security ex post. The key here is the right to exercise the option ex post fter the principl hs otined sufficient knowledge out the gent s performnce nd ility. A contrct etween n employer nd n employee typiclly includes cluse for erly termintion. Different countries nd cultures my hve very different prctices regrding erly termintion of contrct. In some cses, n employee cn quit nd the employer cn fire worker nytime without penlty, while in others, the unilterl erly termintion of contrct incurs severe finncil penlty. We wonder how much the right of erly termintion in prticulr nd the lloction of control rights in generl ctully mtter in principl-gent reltionship. We hve n gency model with doule morl hzrd. In mny rel-world principl-gent reltionships, esides the gent, the principl pplies effort nd investment to ffect outcomes. For exmple, the demnd for product is ffected not only y downstrem firm s (the gent) sles effort ut lso y n upstrem firm s (the principl) mnufcturing inputs tht determine the qulity of the product. The reltionship etween frnchiser nd frnchisee is nother exmple. In her empiricl nlysis, Lfontine (992) concludes tht doule morl-hzrd rgument on frnchising est explins the dt. In university, university uthorities try est to provide good reserch environment y securing reserch funding sources, hiring str reserchers, reducing teching duty, nd providing sound reserch rewrding system. Erly theoreticl studies on gency models with doule morl-hzrd include Romno (994), Bhttchryy nd Lfontine (995), nd Kim nd Wng (998). Our model environment is such tht certin informtion nd options ecome ville over time. The issue of control rights is prticulrly relevnt to such setting. We focus on the right of erly termintion, y which n employer cn dismiss n employee or n employee cn quit without penlty. Such n option will ensure ex-post efficiency with Cosin rgining nd my sometimes led to ex-nte efficiency. Ex-nte efficiency, which is typiclly referred to s economic efficiency, is the key in our investigtion. As we know from Kim nd Wng (998), this gency model cnnot chieve the first est (under doule morl hzrd) even if oth prties re risk neutrl. Our question is: Cn control rights in this gency model improve efficiency if the control rights re llocted ppropritely? In our model, the two prties re symmetric in the sense tht one is the principl (with full rgining power ex nte) nd the other is the gent. We will derive the solutions for different lloctions of control rights. With symmetry nd risk neutrlity, we show tht the lloction of control rights is irrelevnt for investments nd economic efficiency, lthough it does ffect the terms in contrct. This result hs not een known efore. Also, we find tht, lthough the lloction of the right does not ffect efficiency, the inclusion of the right in contrct cn enhnce or reduce efficiency. In prmetric cse, we derive conditions under which the inclusion of the right in contrct cn enhnce or reduce economic efficiency. In prticu- Pge 3 of 23

4 lr, lthough the control rights on jo security do not ffect investments nd economic efficiency, they do ffect the structure of the contrct nd the income shring scheme in the contrct s well s the ility threshold eyond which the gent is to leve the jo voluntrily or forcefully. The rest of the pper is orgnized s follows. Section 2 lys out the sic model. Section 3 presents the optiml solutions under different circumstnces. Section 4 discusses the results. Section 5 concludes with few remrks. All the derivtions nd proofs re in the ppendix. 2. The Model Consider n gency model in which principl hires n gent to undertke project. The project lsts two periods. At the eginning of the project, ech prty invests effort, from the gent nd from the principl. The costs of effort re privte nd re respectively c ( ) nd C ( ). The efforts nd re not verifile. The two effort investments generte composite effort h (, ), which ffects the distriution of output. 3 The output is lso dependent on the gent s type, which is represented y n index q Î +. This index is unknown to oth prties t time (ex nte), nd it ecomes known to oth prties t time (ex post). We ssume tht the disinformtion out q is symmetric, i.e., the two prties hve common distriution function F ( q) out q ex nte, with density function fq ( ). Given effort inputs (, ) nd the gent s ility q, the density function of output is f[ x, q, h(, )]. 4 As in the stndrd gency model, output is contrctile. An income shring rule is represented y function sx ( ) from the output spce to, where sx ( ) is the pyment to the gent when output is x. The set of shring rules is = { s: s is Leesgue integrle }. In ddition to n income-shring rule, we llow for n ex-post option in contrct. Unlike the income rights defined y the income shring rule, this option is control right, which cn e given either to the principl or the gent. The ility q of the inside gent is rndom ex nte. When q is reveled ex post, the principl considers to replce the inside gent y n outside gent from the lor mrket. Depending on who hs the right, monetry trnsfer etween the principl nd the inside gent from one direction or the other my e mde. The 3 Here, efforts nd my or my not e oservle ex nte or ex post. Their oservility is not n issue here since the equilirium efforts nd re known to oth prties t ny moment in time. Since the efforts re not verifile, no mtter whether they re oservle or not, they re not enforcele y court. 4 For exmple, in specil cse when x = qh(, ) + e, where e is rndom shock, the output distriution function is Fx ( x) = Fe [ x-qh(, )], where F e is the distriution of e. This Fe [ x-qh(, )] is specil cse of our output distriution function Fx [, q, h (, )]. Pge 4 of 23

5 outside gent with ility q is ville ex post for fixed mrket wge rte s, where q nd s re known constnts. Here we ssume tht the firm is competitive in the lor mrket so tht it tkes the mrket wge rte s given. When the outside gent with ility q is hired ex post, the output density function for x is f [ x, q, (, )]. h Here, we ssume different density function f to llow for inputs y the outside gent. 5 Hence, contrct in our model is of the form: contrct = { shring rule sx ( ), right to dismiss or quit }. The contrctle option of exit is unique feture in our model. The control right in our model refers to the control of this exit option. We re concerned with two issues: () Who should hve this right? (2) Wht is the effect of the inclusion of this right in contrct? Our model is different from multi-gent model. An outside gent in our model gives vlue to the exit option rther thn competes directly with the inside gent. The outside gent serves s second instrument in controlling incentives, in ddition to the output shring rule. The ility of worker hs een found to e crucil fctor in gency models in the literture. Bker nd Holmström (995) find tht wge growth nd promotion re not closely relted. Insted, they find tht ility is wht drives oth wge growth nd promotion. They find strong evidence in support of the existence of internl lor mrkets, ut they find tht externl lor mrkets hve limited influence on internl lor mrkets. Gions nd Wldmn (999) construct selection model sed on ility. Their model explins few key empiricl findings, such s positive seril correltion in wge growth nd positive correltion etween wge growth nd promotions. Also, Lzer nd Oyer (24) find empiricl evidence indicting n importnt impct of externl lor mrkets on internl lor mrkets. These studies justify the inclusion of n employee s ility nd the existence of n outside gent s key fctors in our model. The timing of the events is illustrted in the following figure. Contrcting, θ rndom, Renegotition, θ relized Revenue Distriuted 2 Ex Ante Ex Post End Figure. The Timing of Events 5 For exmple, f [ x, q, (, )] h cn e f[ x, q, h( +, )], where is the outside gent s optiml effort. Also, to represent the cse where the inside gent s investment does not hve lrge impct on output fter he or she is dismissed, f [ x, q, (, )] h cn e f[ x, q, h(, )], where h cn e different from h. However, for simplicity, we do not explicitly model the outside gent s optiml decision. Pge 5 of 23

6 . At t =, the principl offers contrct to the gent. If the contrct is ccepted, the gent nd the principl simultneously invest nd nd incur costs c ( ) nd C ( ), respectively. 2. At t =, uncertinty out q is resolved. The principl considers the option to replce the inside gent with n outside gent. Suppose tht n outside gent with q is ville for mrket wge rge s. Depending on who hs the control right, the principl my rgin with the inside gent for new contrct or simply fire him without hving to py ny compenstion. 3. At t =2, the project is finished nd pyments re mde sed on the existing contrct. In prctice, when one prty proposes erly termintion of the reltionship (either the employee expresses n intention to leve or the employer requests the employee to leve), the other prty often offers to negotite. Whoever hs the control right of exit hs n dvntge in renegotition. To mke our model more relistic, we llow contrct to e renegotited ex post when new informtion is ville, i.e. when q ecomes known. By tht time, the investments (, ) re sunk ut the inside gent is replcele. As new informtion rrives, the two prties my wnt to tke dvntge of the opportunity to improve ex-post efficiency nd shre the gin from renegotition. Given ( q,, ), the expected revenue t t = is The expected welfre gin t t = from renegotition is r (,, q) º xf[ x, q, h (, )] dx. () g (,, q) º r (,, q )-r (,, q) - s. (2) We dopt Cosin rgining, y which renegotition yields ex-post efficiency. Hence, given the gent s type q with investments (, ), there will e renegotition if nd only if gq (,, )>. Define the threshold q (, ) y eqution: r (,, q )= r (,, q ) + s. (3) Then, if nd only if q < q will the inside gent e replced y n outside gent, regrdless of who hs the right of control. Without renegotition, the expected revenue t t = is R (, ) º r (,, q) df( q). With renegotition, the expected revenue t t = is q (, ) R (, ) º r (,, ) -s df ( ) + r (,, ) df ( ). [ q ] q q q (4) q(, ) Hence, the expected gin t t = from renegotition is q(, ) G (, ) º g (,, q) df( q), (5) Pge 6 of 23

7 where R (, )= R (, ) + G (, ). (6) Stndrd ssumptions re imposed on the functions: R(, ) >, R(, ), R(, ) >, R(, ), c ( ) >, c ( ) >, c ( ) >, C ( ) >, C ( ) >, C ( ) >. (7) 3. The Solutions 3.. Employment without n Exit Option As enchmrk, we first consider the stndrd gency model in which erly termintion is not n option. In this cse, contrcts re of the form: In this cse, the gent s ex-nte pyoff is nd the principl s ex-nte pyoff is contrct = { shring rule sx ( )}. U(,, s) s( x) f[ x, q, h(, )] f( q) dxdq, º V( s,, ) º [ x-sx ( )] f[ x, q, h (, )] f( q) dxdq = R (, )-Us (,, ). Suppose tht fter the contrct is signed, the two prties ply Nsh gme to determine efforts/investments nd. Then, the two first-order conditions (FOCs) re U( s,, ) = c ( ), V( s,, ) = C ( ), nd the principl s second-est prolem is P º mx R (, ) -Us (,, ) -C ( ) sî, Î, Î s.t. IC : U (,, s) = c ( ), IC : V(,, s)= C ( ), IR : U (,, s) ³ u + c( ), where u is the gent s reservtion utility, IC stnds for n incentive comptiility condition, nd IR stnds for n individul rtionlity condition. When oth efforts re contrctile, the principl s first-est prolem is P º mx R (, ) -Us (,, ) -C ( ) sî, Î, Î s.t. U(,, s) ³ u + c( ). Pge 7 of 23

8 Lemm. For the first-est prolem, the first-est efforts (, ) R (, ) = c ( ), R (, ) = C ( ), re determined y nd fixed contrct is optiml: s ( x)= u + c( ). Proposition. For the second-est prolem, the second-est efforts (, ) y P º mx R (, ) -c ( ) -C ( ) -u Î, Î h (, ) s.t. R (, ) = c ( ) + C ( ), h (, ) re determined (8) nd liner contrct is optiml: s ( x)= x+, c ( ) where = R (, ) nd = u + c( )- R(, ). We hve < < nd the first-est outcome is not chievle. Bhttchryy nd Lfontine (995) were the first to provide this result for specil output process with distriution function of the form F( x- h). Kim nd Wng (998) provided the sme result with generl distriution function F( x, h ). Here, the optimlity of liner contrct is due to risk neutrlity nd the filure to chieve the first est is due to the principl s inility to commit ex nte to the efficient level of effort. Remrk. One dvntge of liner contrct is tht the vlidity of the first-order pproch (FOA), i.e., U(,, s )< c ( ) nd (,, V )< ( ) s C for ll (, ), cn e esily gurnteed y the simple conditions in (7). Hence, we hve no need to discuss the vlidity of the FOA. Remrk 2. We hve implicitly ssumed tht n outside gent is not ville ex nte. To justify this, we my think of the inside gent s the est ville person for the jo out of pool of jo pplicnts Employment with n Exit Option: Agent in Control When erly termintion is llowed, the ownership of the control rights over this option ecomes n issue. In this section, the inside gent hs the control right, which mens tht he or she cn quit nytime without penlty. In this cse, the contrct is of the form: contrct = { shring rule sx ( ), right to quit y the gent }. Pge 8 of 23

9 At time, q is known. The principl considers the option of replcing the gent y n outside gent from the mrket. Given q nd s(), if the principl wnts to replce the inside gent, the principl hs to rgin with the gent. Suppose tht n outside gent with ility q cn e hired for fixed wge s. By (3), the principl will replce the inside gent ex post if nd only if q < q. With the distriution of rgining powers ( d, - d) nd the net gin gq (,, ) from renegotition, where d Î (, ), the Nsh rgining solution implies tht the ex-post pyoffs of the gent nd principl re, respectively, u( qd, ) º s( x) f[ x, q, h(, )] dx+ dg(,, q), v( qd, ) º [ x- s( x)] f[ x, q, h(, )] dx+ (-d) g(,, q)= r(,, q) -s -u( qd, ), where rq (,, ) nd gq (,, ) re defined in () nd (2), respectively. Note tht since oth prties enefit ex post from renegotition when q < q, the initil contrct is ndoned voluntrily. Then, the ex-nte pyoffs of the gent nd the principl re, respectively, q(, ) q (, ) U( s,,, d) º u( q, d) f( q) dq+ sx ( ) f[ x, q, h (, )] f( q) dxdq = sx ( ) f[ x, q, h (, )] f( q) dxdq+ dg (, ), q(, ) V(,, s, d) º v( q, d) f( q) dq+ [ x-s( x)] f[ x, q, h(, )] f( q) dxdq q (, ) = R (, )-Us (,,, d), where G (, ) nd R (, ) re defined in (5) nd (6), respectively. Then, the principl s second-est prolem is P º mx R (, ) -Us (,,, d) -C ( ) sî, Î, Î s.t. IC : U (,, s, d)= c ( ), IC : V(,, s, d)= C ( ), IR : U (,, s, d) ³ u + c( ). When oth efforts re contrctile, the principl s first-est prolem is P º mx R (, ) -Us (,,, d) -C ( ) sî, Î, Î s.t. U(,, s, d) ³ u + c( ). (9) Lemm 2. For the first-est prolem, the first-est efforts (, ) R (, )= c ( ), R (, )= C ( ), re determined y nd fixed contrct is optiml: s ( x)= u c( ) dg(, + - ). Pge 9 of 23

10 Proposition 2. For the second-est prolem, the second-est efforts y nd liner contrct is optiml: P º mx R (, ) -c ( ) -C ( ) -u Î, Î h (, ) s.t. R (, ) = c ( ) + C ( ), h (, ) (, ) re determined () ( ) (, c -dg ) where = R (, ) outcome is not chievle. s ( x)= x+, nd = u + c( )-R(, )- dg(, ). The first-est 3.3. Employment with n Exit Option: Principl in Control In this section, the principl hs the control right of exit, which mens tht he or she cn dismiss the gent nytime without penlty. In this cse, the contrct is of the form: contrct = { shring rule sx ( ), right to terminte y the principl }. At time, q is known. The principl considers the option of replcing the inside gent y n outside gent. Given q nd s(), the principl cn simply dismiss the gent since the principl hs the control right. Suppose tht n outside gent with ility q cn e hired for fixed wge s. The net gin from exercising the control right is gq (,, ). By (3), the principl will replce the gent ex post if nd only if q < q. If the inside gent is not fired, the wge pyment will still e sed on the initil contrct. But if the inside gent is fired, the ex-post pyoffs of the principl nd gent re, respectively, u( q) º, v( q) º r(,, q )- s. Then, the ex-nte pyoffs of the gent nd the principl re, respectively, U(,, s) º s( x) f[ x, q, h(, )] f( q) dxdq, q(, ) q(, ) V(,, s) º v( qfq ) ( ) dq+ [ x-s( x)] f[ x, q, h(, )] fq ( ) dxdq q (, ) = R (, )-Us (,, ). Pge of 23

11 Hence, the principl s second-est prolem is P º mx R (, ) -Us (,, ) -C ( ) sî, Î, Î s.t. IC : U (,, s) = c ( ), IC : V (,, s)= C ( ), IR : U (,, s) ³ u + c( ). () When oth efforts re contrctile, the principl s first-est prolem is P º mx R (, ) -Us (,, ) -C ( ) sî, Î, Î s.t. U(,, s) ³ u + c( ). Lemm 3. For the first-est prolem, the first-est efforts (, ) R (, )= c ( ), R (, )= C ( ), re determined y nd fixed contrct is optiml: u + c( ) s ( x)=. q(, -Fé ) ù ê ë ú û Proposition 3. For the second-est prolem, the second-est efforts y P º mx R (, ) -c ( ) -C ( ) -u Î, Î nd liner shring rule is optiml: h (, ) s.t. R (, ) = c ( ) + C ( ), h (, ) (, ) re determined (2) s ( x)= x+, c ( ) u + c( )-D(, ) where = (, D, ) = nd [ q(, -F )] first-est outcome is not chievle. D (, ) º r (,, q) df( q). The q(, ) 4. Discussion 4.. Alloction of Control Rights One immedite question is: who should hve the control right? By Propositions 2 nd 3, the nswer is surprisingly simple. We cn see tht prolems () nd (2) re exctly the sme, which implies the following irrelevnce result. Pge of 23

12 Proposition 4 (Irrelevnce). Under risk neutrlity, control rights re irrelevnt in gency prolems. More specificlly, investments (optiml efforts) nd economic efficiency (socil welfre) re independent of the lloction of control rights, lthough contrctul terms nd income shring s well s the threshold for erly termintion re dependent on the lloction of control rights. This irrelevnce result requires the ssumptions of risk neutrlity nd Cosin rgining. Cosin rgining ensures no loss of socil welfre when the prties renegotite. With risk neutrlity, the Nsh rgining solution lloctes the enefit of renegotition proportionlly to the two prties so tht there is no distortion in ggregte pyoff. This explins why the irrelevnce result holds under risk neutrlity nd Cosin rgining. From Propositions 3, we further find tht, under risk neutrlity, the inclusion of control rights nd the lloction of control rights will not ffect the existence of n optiml liner contrct. Proposition 5 (Liner Contrct). Under risk neutrlity, the inclusion of control rights nd the lloction of control rights will not ffect the existence of n optiml liner shring rule. The existence of n optiml liner income shring rule under risk neutrlity is well known. We extend this result to models with control rights. We show tht the existence of n optiml liner contrct s well s the independence of the existence on the lloction of control rights Inclusion of Control Rights A further question is: how will the inclusion of control rights in contrct ffect economic efficiency? Tht is, lthough the lloction of control rights does not mtter, the inclusion of control rights my. The nswer to this question is quite complicted. Under the first est, we know tht the inclusion of control rights, or the inclusion of more choices in contrct generlly, will lwys enhnce economic efficiency. However, under the second est, the inclusion of control rights llows more choices ut lso cuses more opportunistic ehviors. We find tht, under the second est, the inclusion of control rights cn sometimes reduce efficiency nd t other times improve efficiency. To determine the roles of different fctors ffecting the nswer to this question, we consider prmetric cse in which few key fctors re represented y prmeters. For convenience of nottion, let gent e the inside gent nd gent 2 e the principl. Consider the following prmetric cse: g 2 g 2 h (, )= m 2 + m2, x( q, h)= Ah q, c ( )=, C ( )=, u=, (3) Pge 2 of 23

13 where m i ³ represents the weight of gent i s investment, A is rndom ex post with EA ( )=, nd q is rndom ex nte nd tkes two possile vlues q L ³ nd qh > q L with proility r nd - r, respectively. Suppose q < q < q, 6 nd denote q º rq + - r q, q º rq + -r q. L ( ) ( ) L H H H ρ Φ(θ ) θ L θ θ ~ θ θ H Figure 2. The Distriution of Aility Then, the second-est solution for the gency model without control rights is g m = q, g g m g 2 m 3 g m2 + g m ç èg g2 g2m + gm2 = q, g g m q æm m m m ö P = + -, 2 ç ø nd the second-est solution for the gency model with control rights is Let We cll g m = q, g g m g 2 m 3 g m2 + g m ç èg g2 g2m + gm2 = q, g g m q æm m m m ö P = + - -rs. 2 ç ø p m m,. 2 º p2 º g g2 p i gent i s productivity. Consider condition on the productivity: 6 This mens q gm + gm s ³ gg q - q gm 2 + gm 2 L, which is minor restriction on the prmeters. Pge 3 of 23

14 + - pp =, (4) + 2 p p2 c p p2 where 2rs s c º =. q - q æ q -q ö L ( q - ql) ç r + q çè 2 ø Eqution (4) defines n elliptic curve on re shown in Figure The curve nd the comprison of P ginst P p 2 c p p2 p + p2 = c p + p 2 < > p c Figure 3. Effects of the Inclusion of Control Rights in Contrct The figure implies the following result. Proposition 6. Given the prmetric functions in (3), we hve the following results: () When either p or p 2 is sufficiently lrge, the inclusion of control rights enhnces efficiency. When oth p nd p 2 re sufficiently smll, the inclusion of control rights reduces efficiency. () When q is lrge or when s is smll, the inclusion of control rights tends to enhnce efficiency. (c) Although the two prties positions re symmetric ex nte, their productivities p nd p 2 re eqully relevnt to the impct of the inclusion of control rights on efficiency. Control rights llow the holder to renegotite the terms in contrct. As indicted in (2), the expected welfre gin from renegotition is gq (,, ). Hence, the two prties incentives (the choices of nd ) re not only ffected y the income-shring rule ut lso y this welfre gin. Specificlly, t t =, the income-shring rule determines n lloction of rq (,, ), while the Nsh rgining solution determines n lloction of gq (,, ). The ltter lloction cuses n dditionl considertion (the so-clled opportunistic ehvior) y the prties when they mke choices of nd ex nte. Tht is, the possiility of renegotition my cuse the Pge 4 of 23

15 prties to engge in opportunistic ehviors when mking their initil investments. This opportunistic ehvior, stemming from the right to renegotite for etter terms ex post, my enhnce or reduce the incentive to invest. Proposition 6() indictes tht, if t lest one prty is sufficiently productive, the enefit from this opportunistic ehvior will outweigh its negtive effect. Control rights cn lso provide protection ginst risks. Both prties re uncertin out the inside gent s ility (even the inside gent is uncertin out his or her own ility ex nte). The inclusion of the termintion right in the contrct llows one of the prties to renegotite for replcement y n outside gent when more cple or lower cost outside gent is ville. This possiility of replcement reduces the risk of hving n inside gent whose ility is uncertin. This enefit is shred y oth prties through Nsh rgining, which is uilt into our optiml renegotition-proof contrct. This explins Proposition 6(). Proposition 6(c) emphsizes the equl relevnce of p nd p 2 even though the two prties positions re symmetric. However, this equl relevnce is for overll economic efficiency; income shring nd contrctul terms re still in fvor of the holder of the control right. 5. Concluding Remrks As is well known, risk neutrlity in income ensures the existence of n optiml liner contrct. We show tht risk neutrlity in income lso leds to the irrelevnce of the lloction of control rights to investments nd economic efficiency. We hve ssumed Cosin rgining. Without this ssumption, sed on Hrt nd Moore (28) nd Hrt (29), the expected welfre gin from renegotition ecomes (- l) g (,, q) for some constnt l Î (, ). With this chnge, we need to replce R (, ) in Proposition 2 y R (, )- lg (, ), ut other results remins the sme, where P in Proposition 6 refers to tht in Proposition 3. Since R (, ) in Proposition 2 is reduced to R (, ) - lg (, ) while no chnge in Proposition 3, the solution with the gent in control is less efficient thn tht with the principl in control. The explntion is simple: when the gent is in control, the principl hs to negotite with the gent when the principl wnts to replce the gent, which results in some welfre loss; when the principl is in control, the principl does not need to renegotite with the gent when the principl wnts to replce the gent. This result suggests tht, when renegotition is likely to cuse grievnce of either or oth sides, control rights should e given to the principl, otherwise they cn e given to either side. Although in this pper we discuss specific right the right of erly termintion other kinds of control rights cn e similrly nlyzed. Also, lthough we cll q the gent s ility, it cn certinly e used to represent other kinds of informtion tht re only reveled over time. Pge 5 of 23

16 Finlly, our model serves s enchmrk for generl models on control rights. It my help us understnd how risk ttitudes determine the optiml lloction of control rights in more generl model. For exmple, s we know, if the principl is less risk verse thn the gent, the principl should shre more risks; the question is: should the principl lso e llocted more control rights? With risk neutrlity, output risk does not mtter. But with risk version, it does. If oth the gent nd the principl re risk verse, then on the one hnd, the version ginst the risk of the externl lor mrket will strengthen the principl s preference for the right of exit; on the other hnd, the version ginst the risk of losing the jo will strengthen the gent s preference for the right of exit. As in the current model, n optiml contrct hs to lnce the incentives nd the risks. Intuitively, giving the right to the gent improves the incentives for the gent, while giving the right to the principl improves the principl s risk position. Hence, the nswer to the question of who should hve the right of exit will depend on trdeoff etween the gent s incentives nd the principl s risk ttitude. Appendix This ppendix contins ll the proofs nd derivtions. A.. Proof of Lemm 2 Since the IR condition must e inding, the first-est prolem ecomes P º mx R (, ) -c ( ) -C ( ) -u sî, Î, Î s.t. U(,, s, d) = u + c( ). This prolem cn e solved in two steps. We first find the optiml efforts (, ) y mximizing the ojective function without the constrint. Then, given the efforts, we consider fixed contrct sx ( )= tht stisfies the constrint. We find f[ x, q, h(, )] f( q) dxdq+ dg(, )= u + c( ), which immeditely determines. A.2. Proof of Proposition 2 From (3), we find q r (,, q )- r (,, q) q r (,, q )-r (,, q) r (,, q) r (,, q) =, =. q q Since r (,, q)= h xf [ x, q, h(, )] dx, r (,, q)= h xf [ x, q, h(, )] dx, h h Pge 6 of 23

17 we hve q (, ) q(, ) =. h (, ) h (, ) (5) We lso find q G (, )= é q r(,, q) r(,, q) s ù ê - - fq ( ) + [ r(,, q) -r(,, q) ] fq ( ) dq ë úû q { - } = h xf [ x, q, h(, )] dx xf [ x, q, h(, )] dx f( q) dq, h h q { - } G(, )= h xf[ x, q, h (, )] dx xf[ x, q, h (, )] dx f( q) dq, h h implying We lso hve Thus, (6) nd (6) imply G(, ) G(, ) =. h (, ) h (, ) R(, ) R(, ) =. h (, ) h (, ) R (, ) R(, ) =. h (, ) h (, ) (6) Therefore, the following two equtions imply ech other: h(, ) h(, ) R(, )= C ( ) + c ( ), R(, )= c ( ) + C ( ). h (, ) h (, ) Since U (,, s, d)= h s( x) f [ x, q, h(, )] f( q) dxdq+ dg (, ), h U( s,,, d)= h sx ( ) f[ x, q, h (, )] f( q) dxdq+ dg(, ), h y (6), we hve Thus, h h h U( s,,, d)= U( s,,, d) - dg(, ) + dg(, )= U( s,,, d). h h h h (,,, )= (, ) (,,, )= h h V s d R -U s d R(, ) - U(,, s, d)= V(,, s, d). h h h Therefore, the two IC conditions in prolem (9) imply Pge 7 of 23

18 c ( )= U (,, s, d)= R (, ) -V (,, s, d) h = (, ) (,,, )= h R - V sd R(, )- C ( ). h h Also, since the IR condition in prolem (9) must e inding, prolem (9) ecomes Pº mx R (, ) -c ( ) -C ( ) -u sî, Î, Î h (, ) s.t. R (, ) = c ( ) + C ( ), h (, ) U (,, s, d)= c ( ), U(,, s, d)= u + c( ), This prolem cn e solved in two steps. We first find the optiml efforts (, ) Pº mx R (, ) -c ( ) -C ( ) -u Î, Î h (, ) s.t. R (, ) = c ( ) + C ( ). h (, ) (7) (8) from (9) And then we find contrct tht stisfies the following two conditions: Conditions (2) nd (2) cn e written s U (,, s, d )= c ( ), (2) Us (,,, d )= u+ c ( ), (2) c ( )= h s( x) f [ x, q, h(, )] f( q) dxdq+ dg (, ), h u+ c ( )= sx ( ) f[ x, q, h (, )] f( q) dxdq+ dg (, ). Given efforts (, ) from prolem (9), consider liner contrct of the form sx ( )= x+. Then, conditions (2) nd (2) ecome c ( )= R(, ) + dg(, ), u + c( )= R(, ) + + dg(, ), implying ( ) (, c -dg ) =, = ( ) (, u c dg ) R(, ). R (, ) Tht is, there exists liner contrct stisfying (2) nd (2). Finlly, y compring the IC condition in prolem (9) with condition R (, )= c ( ) for the first est in Lemm 2, we immeditely know tht (, ) of (9) cnnot chieve the first est. Pge 8 of 23

19 A.3. Proof of Lemm 3 Since the IR condition must e inding, the first-est prolem ecomes P º mx R (, ) -c ( ) -C ( ) -u sî, Î, Î s.t. U(,, s) = u + c( ). This prolem cn e solved in two steps. We first find the optiml efforts (, ) y mximizing the ojective function without the constrint. Then, given the efforts, we consider fixed contrct sx ( )=, the simplest contrct, tht stisfies the constrint. We find or f x h dxd u + c q [, q, (, )] f( q) q = ( ), where q f q q ( ) d = u c( ), + q º q(, ). This eqution determines. A.4. Proof of Proposition 3 We hve q U (,, )= ( ) [,, (, )] ( ) ( ) ( ) [, s h sx f, (, )], h x h dxd sx f x h dx q q f q q-f q q U(,, )= ( ) [,, (, )] ( ) ( q ) ( ) [, s h sx f, (, )]. h xq h f q dxdq-f q sx f xq h dx q (22) By (5), Thus, h U( s,, )= U( s,, ). h h (,, )= (, ) (,, )= h h V s R -U s R(, )- U( s,, )= V( s,, ). h h h Therefore, the two IC conditions in prolem () imply h ( )= (,, )= (, ) (,, )= (, ) (,, )= h c U s R -V s R - V s R(, )- C ( ). (23) h h Also, since the IR condition in prolem () must e inding, prolem () ecomes Pge 9 of 23

20 P º mx R (, ) -c ( ) -C ( ) -u sî, Î, Î h (, ) s.t. R (, ) = c ( ) + C ( ), h (, ) U (,, s)= c ( ), Us (,, )= u+ c ( ), This prolem cn e solved in two steps. We first find the optiml efforts Î, Î P º mx R (, ) -c ( ) -C ( ) -u h (, ) s.t. R (, ) = c ( ) + C ( ). h (, ) And then we find contrct tht stisfies the following four conditions: (, ) from (24) (25) Conditions (26) nd (27) re U (,, s)= c ( ), (26) U(,, s)= u + c( ), (27) c ( )= s( x) f [,, (, )] ( ) h x q h f q dxdq q u + c( )= s( x) f[ x, q, h(, )] f( q) dxdq. q Given efforts (, ) from prolem (25), consider liner contrct of the form sx ( )= x+. Then, conditions (26) nd (27) ecome c ( )= D (, ), u c( )= D(, ) [ ( q + + -F )], where q º q(, ) nd D (, ) º r (,, q) df( q). q(, ) Tht is, there exists liner contrct tht stisfies (26) nd (27) with ( ) ( ) (, c u + c -D ) = (, =. D, ) ( -F q ) Finlly, y compring the IC condition in prolem (25) with condition R (, )= c ( ) for the first est in Lemm 3, we immeditely know tht (, ) of (25) cnnot chieve the first est. Pge 2 of 23

21 A.5. The Exmple We find s = q q -, h (, ) r (,, q)= qh (, ), R (, )= q h (, ), q G (, )= h (, ) ( q) dq = r( q ql) h (, )= ré( q ql) h (, ) sù F -, q ë - - û L R (, )= érql + ( - r) qh + r( q-ql) ùh (, ) -rs = q ë û h (, ) -rs, D (, )= h (, ) qf( q) dq = (- r) q h (, ). q(, ) The concvity of R (, ) implies the concvity of R (, ), G (, ) nd D (, ). Consider generl form R(, )= A+ Bh (, ), with some constnts A nd B. For the second-est prolem, the IC condition is H m Bm = g + g. 2 m2 Then, the second-est prolem is g g P º mx A+ Bh(, ) - - Î, Î g g2 s.t. B = +. m m 2 2 (28) The Lgrnge function is g g æg g ö = + (, )- - + l ç + - m m çè ø 2 L A Bh 2 B 2 æ g ö æ g ö g g = Bm l çbm l A. çè ø è ø m ç m 2 The FOCs immeditely imply B m + l B m + l g m g m 2 =, =. Sustituting these into the IC condition yields B l = -. g g + m 2 m2 Pge 2 of 23

22 Then, g m g m = B, = B, g gm2 + g2m g2 g2m + gm2 nd g g B æm m m m ö P = + ( + )- ( ) - ( ) = ç, ø 2 2 A B m 2 2 m2 A 2 ç èg g2 g2m + gm2 nd B gm 2 gm 2 m + m + 2 gg 2 gm 2 + gm 2 h (, )= =. Therefore, for the gency model without renegotition, with A =nd B = q, we hve g m = q, g g m g 2 m g m = q, g g m g m 2 q æm m m m ö P = ç ø ç èg g2 g2m + gm2 For the gency models with renegotition, with A= - rs nd B = q, we hve g m = q, g g m g 2 m g m = q, g g m g m 2 q æm m m m ö P = + - -rs. 2 ç ø ç èg g2 g2m + gm2 References Bker, G., Holmstrom, B. (995). Internl Lor Mrkets: Too Mny Theories, Too Few Fcts. Americn Economic Review, 85(2), Ppers nd Proceedings, Bhttchryy, S., Lfontine, F. (995). Doule-Sided Morl Hzrd nd the Nture of Shre Contrcts. Rnd Journl of Economics, 26(4), Dsgupt, S., To, Z. (998). Contrctul Incompleteness nd the Optimlity of Equity Joint Ventures. Journl of Economic Behvior nd Orgniztion, 37(4), Pge 22 of 23

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