The Informativeness Principle Under Limited Liability

Transcription

1 The Informativene Principle Under Limited Liability Pierre Chaigneau HEC Montreal Alex Edman LBS, Wharton, NBER, CEPR, and ECGI Daniel Gottlieb Wharton Augut 7, 4 Abtract Thi paper how that the informativene principle doe not automatically extend to etting with limited liability. Even if a ignal i informative about e ort, it may have no value for contracting. An agent with limited liability i paid zero for certain output realization. Thu, even if thee output realization are accompanied by an unfavorable ignal, the payment cannot fall further and o the principal cannot make ue of the ignal. Similarly, a principal with limited liability may be unable to increae payment after a favorable ignal. We derive neceary and u cient condition for ignal to have poitive value. Under bilateral limited liability and a monotone likelihood ratio, the value of information i non-monotonic in output, and the principal i willing to pay more for information at intermediate output level. Keyword: Informativene principle, contract theory, principal-agent model, limited liability, pay-for-luck, relative performance evaluation, option. JEL Claification: D86, J33 pierre.chaigneau@hec.ca, aedman@london.edu, dgott@wharton.upenn.edu. thank the Dorinda and Mark Winkelman Ditinguihed Scholar Award. Gottlieb

2 The informativene principle (Holmtrom (979), Shavell (979), Gjedal (98), Groman and Hart (983), Kim (995)) i believed to be one of the mot robut reult in contract theory. The textbook of Bolton and Dewatripont (5, Section 4.7) tate that the moral hazard literature ha produced very few general reult for example, even the intuitive reult that a lower cot of e ort increae the optimal e ort level need not hold but the informativene principle i one of the few reult that i general. They write: among the main general prediction of the model i the informativene principle, which ay that the incentive contract hould be baed on all variable that provide information about the agent action. Formally, a ignal i valuable for the principal i.e. an optimal contract i a function of if and only if output q i not a u cient tatitic for e ort e given (q; ). The informativene principle wa derived in etting without contraint on the contract. However, contracting contraint do exit in reality. Perhap the mot important one i limited liability, which applie to almot all contract between employee and rm. Entrepreneur raiing nancing alo enjoy limited liability, becaue their equity cannot fall below zero. Thi paper tudie whether the informativene principle continue to hold under thi important contraint. The textbook of Tirole (6, p3) conjectured that it doe: he write: added rik i bad when the limited liability contraint i binding, but we how that thi i not alway the cae. Section conider a model in which both the agent and principal are rik-neutral and the only contracting contraint i that the agent i protected by limited liability. We how that, if output q i a u cient tatitic for e ort e given (q; ), then it remain the cae that the ignal ha zero value. However, the revere i no longer true: the optimal contract may be independent of even if output i not a u cient tatitic. Some ignal have zero value even though they are informative. The intuition i a follow. Without contracting contraint, the principal can alway make ue of an informative ignal by adjuting the agent wage accordingly. If To etablih the exitence of an optimal mechanim, Holmtrom (979) and much of the enuing literature aumed that haring rule have a bounded total variation. A he tate, thi retriction i natural from a pragmatic point of view a well, ince the agent wealth put a lower bound, and the principal wealth... an upper bound [on payment]. When deriving the informativene principle, however, he implicitly aumed that the olution wa interior. He conjectured that the informativene principle may not hold under contracting contraint ( if, for adminitrative reaon, one ha retricted attention a priori to a limited cla of contract (e.g., linear price function or intruction-like tepfunction), then informativene may not be u cient for improvement within thi cla ) but doe not formally tudy the implication of limited liability for the informativene principle.

3 low output i accompanied by a low ignal, the agent i paid even lower than with a high ignal. However, with limited liability, the agent i already being paid zero under low output. The principal cannot punih the agent further upon a low ignal, and o the ignal i of no value even though it i informative about e ort. Signal are only ueful if they can be ued to reduce the wage, which under limited liability require the wage to be trictly poitive to begin with. In turn, the agent i only paid a poitive wage for the output level that maximize the likelihood ratio. Thu, ignal are only ueful if they are informative about e ort at thi output level. Section conider additional contracting contraint. In Section., the principal, a well a the agent, i protected by limited liability. Thu, the contract can no longer involve the principal paying only in the maximum likelihood ratio tate, a the required payment would exceed her pledgeable income. A in Inne (99), the optimal contract i a live-or-die contract where the agent receive nothing if output fall below a threhold q, and the entire output if it exceed the threhold. Section. impoe an additional monotonicity contraint which require payment to each party to be non-decreaing in output. The optimal contract i an option on rm output: the agent receive nothing if output fall below q, and the reidual q q if output exceed it. In both cae, the contract depend on the ignal if and only if it i informative about e ort when q = q, i.e. at the center of the ditribution, becaue then it can be ued to allow the threhold q to vary with the ignal. Signal that are informative about e ort at the tail of the output ditribution are of no value. The intuition for q < q i a above: the agent i already being paid zero, and cannot be paid le upon a low ignal without violating limited liability. Turning to q > q, the principal cannot receive any le upon a high ignal. Without the monotonicity contraint, the principal receive zero and cannot receive le without violating limited liability; with the monotonicity contraint, the principal receive q and cannot receive le without violating monotonicity. Appendix B in the Online Appendix how that the reult continue to hold under a continuum of e ort level, holding xed the target e ort level. The reult have a number of economic implication. Firt, the preence of limited liability require u to re ne our notion of informativene. When only the agent ha limited liability, a ignal i valuable if and only if it i informative about e ort for output level that maximize the likelihood ratio. When the principal alo ha limited liability, a ignal i valuable if and only if it i informative about e ort at the center of the ditribution. In particular, a ignal could be informative about e ort almot every- 3

4 where, and till have no value in contracting. Second, the common practice of paying agent for luck, i.e. not ltering out indutry hock, i not necearily uboptimal. If a rm u er a catatrophe, the manager i typically paid zero, regardle of whether it wa down to bad luck (e.g. indutry performance wa alo poor) or hirking. In reality, intance of pay for luck typically concern very good or very bad outcome for example, Bertrand and Mullainathan () conider how CEO pay varie with pike and trough in the oil price but additional ignal are only valuable for moderate outcome. Third, the value of information i non-monotonic in output. Our reult ugget that the principal hould only invet in ignal (e.g. through cotly monitoring) at moderate output realization. Limited Liability on Agent We conider a model with a principal ( rm) and an agent (worker). Both partie are rik-neutral and the agent i protected by limited liability. The agent exert unobervable e ort of e f; g, where e = ( low e ort ) cot the agent, and e = ( high e ort ) cot C >. E ort improve the probability ditribution of output q fq ; q ; :::; q Q g in the ene of rt-order tochatic dominance. It alo a ect the probability ditribution of an additional ignal f ; ; :::; S g. Let q; (p q; ) denote the joint probability of (q; ) conditional on high (low) e ort. Both output and the ignal are contractible. We refer to each realization of an output/ignal pair (q; ) a a tate. The principal o er a vector of payment fw q; g to the agent conditional on the tate. She ha full bargaining power. The agent accept the contract if it ati e hi individual rationality contraint ( IR ): q; w q; C : () q; The agent exert high e ort if the following incentive compatibility contraint ( IC ) i ati ed: ( q; p q; ) w q; C: () q; 4

5 Finally, the agent limited liability contraint ( LL ) implie: w q; 8q; : (3) The IC () and LL (3) imply that the IR () i automatically ati ed, and o we ignore it in the analyi that follow. We aume that the cot of e ort C i u ciently low that the principal wihe to implement high e ort, ele the optimal contract trivially involve a zero wage. The principal problem i to nd the contract fw q; g that minimize the expected wage: ubject to IC (). min H;w H; + H; w H; + L; w L; + L; w L; (4) w q; Lemma below tate that a ignal i valuable if and only if it i informative about e ort in the tate where the wage i trictly poitive. Thu, the informativene principle that the contract doe not depend on if and only if the likelihood ratio i independent of continue to hold under limited liability when conidering tate in which the wage i trictly poitive. (All proof are in Appendix A.) Lemma Let (w q; ) be an optimal contract for implementing e = with w q; > and w q; > for ome q,, and. Then, w q; = w q; if and only if q; p q; = q; p q;. Lemma tate that the wage i trictly poitive only in tate that maximize the likelihood ratio. Lemma Let the vectorn of payment o (w q; ) be an optimal contract for implementing e =. If q; q p q; < max ; (q ; ), then w q; =. p q ; Combining thee reult, a ignal i valuable if and only if it i informative about e ort in tate with the highet likelihood ratio. Thi reult i tated in Propoition below: Propoition A ignal ha poitive value if and only if, for all (q; ) arg max (q ; ) there exit ^ uch that q; p q; 6= q;^ p q;^ : q ; p q ;, 5

6 When the agent exhibit limited liability, a ignal ha poitive value if and only if it a ect the likelihood ratio at the output level at which the likelihood ratio i maximized without the ignal, a only then i the wage aociated with thi output level poitive. In thi cae, the principal can improve on the contract by making the wage at thi output level contingent upon the ignal increae it at the ignal where (q; ) ha the highet likelihood ratio and decreae it to zero at other ignal realization. In contrat, a ignal i not ueful if it change the likelihood ratio only for output level at which the likelihood ratio i not maximized. Since the wage i zero to begin with, the principal cannot decreae it upon a low ignal. In um, if output q i a u cient tatitic for e ort e given (q; ), the ignal ha zero value to the principal. However, even if q i not a u cient tatitic, till ha zero value if it i informative about e ort only for output level at which the likelihood ratio i not maximized. The preence of limited liability require u to re ne our notion of informativene what matter i whether ignal are informative about e ort in tate with the maximum likelihood ratio, rather than in general. We conclude thi ection with two example. The rt one illutrate the reult from Propoition : Example Conider a etting with binary output and binary ignal with the following conditional probabilitie: = 3 = 3 Marginal 3 e = e = Likelihood Ratio q = q H q = q L q = q H q = q L q = q H q = q L Note that q i not a u cient tatitic for e given (q; ) ince the likelihood ratio conditional on q = q L i not contant: p L; L; = =6 = = 3 ; p L; L; = =6 =4 = 3 : By Lemma, the optimal contract entail paying only in tate (q H ; ) and (q H ; ) where the likelihood ratio i maximized at Since the likelihood ratio i equal in both tate, any hare of payment acro thee tate generate the ame payo to the principal a long a they add up to 4 5 (which i the amount required to atify incentive 6

7 compatibility). One olution i w H; = w H; =, w 5 L; = w L; =. The wage doe not depend on, becaue it i only informative at output level for which the wage i already zero. Since an optimal contract i not a function of the ignal realization, the ignal ha zero value. The econd example how that the key driver of our reult i limited liability and not rik neutrality. With rik averion, paying di erent amount in tate with equal likelihood ratio not only fail to improve incentive, but alo force the principal to compenate the agent for the additional rik. Thu, rik averion remove the multiplicity of optimal contract from the previou example. Example Let the agent utility be u (w; e) = p w following conditional probabilitie: e = e = e; e f; g. Conider the = q = q H q = q L q = q H q = q L = Marginal Let +. The likelihood ratio conditional on q = q L are not contant whenever 6= : p L; = L; ; p L; = L; : In Appendix A, we how that the unique olution i w H; = w H; = 4 and w L; = w L; =. Thu, the optimal contract i again independent of the ignal even though the ignal i informative about e ort. Additional Contracting Contraint When the only contraint i limited liability on the agent, generically the wage i poitive in a ingle tate only. In a tradition initiated by Inne (99), we now introduce additional contracting contraint that yield more realitic contract. For tractability, we retain the aumption of binary e ort level; Appendix B tudie the continuou e ort cae. 7

8 Although the reult can be eaily replicated for the dicrete output cae of Section, we expand the model to a continuum of output ince it impli e notation. Formally, output i now ditributed over an interval q [; q], where q may be +. e f; g and the ignal f ; :::; S g are peci ed a before. E ort Since the model combine dicrete and continuou variable, it i convenient to pecify the ditribution in term of the conditional f(qje; ) and the marginal ^^e Pr ( = ^je = ^e). The joint ditribution of (q; ) conditional on e ort i determined by their product. Conditional on e ort e and ignal, output q i ditributed according to the probability denity function ( PDF ): f (qje; ) The marginal ditribution of q i: ( (q) if e = p (q) if e = : f (qje = ) = (q) ; and f (qje = ) = p (q) : (5) Let LR (q) (q) p (q) (6) denote the likelihood ratio aociated with ignal at output q, and w (q) denote the agent payment conditional on output q and ignal. The IC i:. Bilateral Limited Liability w (q) [ (q) p (q)] dq C: (7) In thi ubection, we aume that both partie are ubject to limited liability: w (q) q: (8) A in Section, IC and the agent LL guarantee that IR hold. Although the aumption of a dicrete ignal pace i unimportant for our reult, it allow u to avoid unneceary meaurability iue. Since i dicrete and q i continuou, we ue the notation (q) rather than q; a in the previou ection. 8

9 The principal o er a contract that minimize the expected wage: min fw (q)g w (q) (q) dq; ubject to (7) and (8). Recall that in Section, where only the agent i ubject to limited liability, the principal pay only in the maximum likelihood ratio tate. Thi require her to make a very large payment, which would violate her LL. 3 Thu, he mut pread the payment out acro more tate. A in Inne (99), the olution involve paying the minimum amount poible zero when the likelihood ratio i below a certain threhold, and the maximum amount poible the whole output when it exceed that threhold. The threhold level i choen o that IC hold with equality. larget one: 4 If more than one uch threhold exit, the optimal contract involve the up ( ^ : Z LR (q)>^ A we how in Appendix A, exit. w (q) [ (q) Lemma 3 The optimal contract i w (q) = LR(q)> q. p (q)] dq = C ) : (9) In the remainder of thi ection, we aume that the ditribution of output conditional on (q; ) atify the monotone likelihood ratio property ( MLRP ): i increaing in q 8. (q) p (q) Then, Lemma 3 implie that there exit unique threhold q ; :::; q S, uch that the principal keep (pay) the entire output when it i below (above) that threhold. For a given, the threhold level olve LR (q ) = and Z q>q q [ (q) p (q)] dq = C: () 3 Intuitively, with a continuum of output, the principal wihe to concentrate the payment a much a poible in a neighborhood around the maximum likelihood ratio tate. Without limited liability on the principal, exitence of an optimal contract i typically an iue. The contract cannot involve the principal paying only in the tate with the highet likelihood ratio (a with dicrete output) ince thi i a et of meaure zero, o it mut involve her paying in a neighborhood around that tate. Without limited liability, the principal can generically improve on the contract by concentrating the payment in a maller neighborhood, in which cae an optimal contract fail to exit. Accordingly, Section aumed dicrete output. 4 In general, there may be an interval of optimal threhold which happen with probability zero. Under full upport and continuity of the PDF, the optimal threhold i unique. 9

10 In general, thee threhold will be contingent on the ignal, and o the contract i contingent upon both output and the ignal. Propoition give a neceary and u cient condition under which the threhold are independent of the ignal (q = LR () 8 ). In thi cae, there exit an optimal contract that i contingent only upon output and the ignal ha zero value. Propoition Suppoe the conditional ditribution atify MLRP. The optimal contract i independent of the ignal if and only if, 8 and (q ) p (q ) = (q ) p (q ) for q given by R q q q [f (qje = ) f (qje = )] dq = C: Thu, any ignal that doe not a ect the likelihood ratio at the threhold output q doe not in uence the optimal contract and ha zero value. Information about e ort at output level in the tail of the ditribution i not valuable. The ignal ha no value for q < q becaue the principal i already paying the lowet amount poible (zero). It ha no value for q > q becaue the principal i already paying the full output. A ignal may a ect the likelihood ratio almot everywhere and till have zero value: all that matter i the likelihood ratio at the threhold q, which ha zero meaure. A a reult, ignal are only ueful for moderate output realization. It may not be optimal to invetigate whether very good or very bad performance wa down to luck (e.g. indutry condition) or e ort. For example, if the rm u er a catatrophe, the manager i red and paid zero. Invetigating whether the catatrophe wa due to hirking or bad luck i not ueful, ince the agent will be paid zero in either cae. In practice, ituation in which executive are rewarded for luck typically involve extreme realization; our model ugget that thee contract might indeed be optimal.. Monotonicity Contraint We now introduce a monotonicity contraint a in Inne (99): w (q + ) w (q) ; 8 > : () Inne (99) juti e thee contraint on two ground. If the contraint on the right did not hold, the agent could borrow on hi own account to arti cially increae output,

11 raiing hi payo. If the contraint on the left did not hold, the principal would exercie her control right to burn output, raiing her payo. For implicity, we aume that the likelihood ratio i unbounded from above: lim LR (q) = +; 8: () q!q Thi aumption i not needed for our main reult. However, it allow u to rule out corner olution, thereby enuring that the threhold in the optimal contract are lower than q: The optimal contract i then given by Lemma 4: Lemma 4 Suppoe the conditional ditribution atify MLRP. The optimal contract i w (q) = max fq; z g ; where (z ; :::; z S ) atify the IC (7) with equality and R q z (q) dq R q z p (q) dq = R q z (q) dq R q p z (q) dq 8; ; Lemma 4 yield a tandard option contract, where the threhold z i the trike price. 5 If output exceed z, the agent receive the reidual q z, rather than the entire output a in Section.. In general, z may depend on the realized ignal : Propoition 3 give the condition under which the trike price i independent of the ignal (i.e. z = z 8 ), and o the ignal ha no value for the contract. Propoition 3 Suppoe the conditional ditribution atify MLRP. The optimal contract i independent of the ignal if and only if, 8 ; ; Pr (q > z ; je = ) Pr (q > z ; je = ) = Pr (q > z ; je = ) Pr (q > z ; je = ) ; where z i given by R q z (q z ) [f (qje = ) f (qje = )] dq = C: Propoition 3 how that the intuition behind Section. continue to apply when a monotonicity contraint i introduced. Regardle of whether the monotonicity contraint i impoed, the agent receive zero for low output level and the maximum 5 Inne (99) conider a nancing model, where the contract tipulate the payment from the agent to the principal. The optimal contract i debt, and o the agent ha equity an option on output. Here, we conider a contracting model, where the contract tipulate the payment from the principal to the agent, in line with the literature on the informativene principle.

12 poible for high output level without monotonicity he receive the maximum poible without violating the principal limited liability contraint; with monotonicity he receive the maximum poible without violating the monotonicity contraint. The principal only degree of freedom i on the threhold z that eparate low from high output level. Thu, the ignal i only valuable if it lead to the principal optimally etting di erent trike price i.e. it a ect the likelihood ratio of the event that the agent option i in-the-money. 3 Concluion Thi paper tudie whether the informativene principle hold under limited liability, an important contraint in mot contracting environment. The limited liability (and, where impoed, monotonicity) contraint bind at almot all output level under the optimal contract. A a reult, ignal may be informative about e ort at almot all output and till have zero value ince the wage already lie at the boundary of the contracting pace, the principal cannot ue the ignal to modify the wage. When the agent limited liability i the only contractual contraint, a ignal i valuable if and only if it i informative about e ort at the tate with the highet likelihood ratio, which ha zero meaure. With limited liability on the principal or monotonicity contraint, a ignal i valuable if and only if it i informative about e ort at a ingle intermediate output, which alo ha zero meaure. Thu, the principal willingne to invet in ignal i greatet for intermediate output realization.

13 Reference [] Bertrand, Marianne, and Sendhil Mullainathan (): Are CEO rewarded for luck? The one without principal are. Quarterly Journal of Economic 6, [] Bolton, Patrick, and Mathia Dewatripont (5): Contract Theory (MIT Pre, Cambridge). [3] Gjedal, Froytein (98): Information and incentive: the agency information problem. Review of Economic Studie 49, [4] Groman, Sanford J., and Oliver D. Hart (983): An analyi of the principalagent problem. Econometrica 5, [5] Holmtrom, Bengt (979): Moral hazard and obervability. Bell Journal of Economic, [6] Inne, Robert D. (99): Limited liability and incentive contracting with ex-ante action choice. Journal of Economic Theory 5, [7] Kim, Son Ku (995): E ciency of an information ytem in an agency model. Econometrica 63, 89. [8] Poblete, Joaquín and Daniel Spulber (): The form of incentive contract: Agency with moral hazard, rik neutrality, and limited liability. Rand Journal of Economic 43, [9] Shavell, Steven (979): Rik haring and incentive in the principal and agent relationhip. Bell Journal of Economic, [] Tirole, Jean (6): The Theory of Corporate Finance (Princeton Univerity Pre, Princeton). 3

14 A Proof Proof of Lemma Fix a vector of wage that atify IC, and conider the following perturbation: w q; = w q; + ; and wq; q; p = w q; : q; q; p q; Thi perturbation keep the incremental bene t from e ort contant and therefore preerve IC. LL continue to hold for > if w q; >, and for < if w q; >. The expected wage (4) increae by: q; q; p q; q; q; p q; : (3) If the original contract entail w q; = w q; > (i.e., a trictly poitive wage for output q that doe not depend on whether the ignal i or ), then uch a perturbation would atify both IC and LL. Thu, for thi contract to be optimal, uch a perturbation cannot reduce the expected wage. The term in (3) mut be non-poitive 8 : which yield q; p q; = q; p q;. Proof of Lemma q; q; p q; = q; ; q; p q; Conider the following perturbation, which, a before, keep the incremental bene t from e ort contant, thereby preerving IC: w q; = w q; + LL continue to hold for > if w q ; ; and wq q; p ; = w q ; q; q ; p : q ; >, and for < if w q; >. The expected wage (4) increae by: q; q ; q; p q; q ; p : (4) q ; n o q Let (q; ) arg max ; (q ; ) denote a tate with the highet likelihood ratio p q ; 4

15 and conider a tate (q ; ) that doe not have the highet likelihood ratio: q ; p q ; < q; p q; : (5) From (5), the term inide the parenthee in (4) i trictly negative. Thu, the principal can reduce the expected wage by electing >, which i feaible without violating LL when w q ; tate that do not maximize the likelihood ratio. Proof of Example >. A a reult, the olution entail zero payment in all It can be hown that the principal would like to implement high e ort. Writing in util, the principal program become: min (u i;j ) u H; + ( ) u H; + u L; + ( ) u L; ubject to u H; + u H; + ( ) u H; ( ) u H; + u L; + + u L; + ( ) u L; (IR) u L; ; (IC) u i;j ; 8i; j: (LL) A in the core model, IC and LL imply IR, and o IR can be ignored. It i traightforward to verify that > implie u L; = wherea + > implie u L; = : Subtituting back in the program, the principal olve: ubject to min u H;j u H; + u H; + ( ) u H; ( ) u H; : The unique olution i u H; = u H; =. Converting to dollar unit, we obtain w H; = w H; = 4 and w L; = w L; =. Proof of Lemma 3 5

16 The principal program i ubject to min fw (q)g w (q) (q) dq w (q) q w (q) [ (q) p (q)] dq C: While it i poible to obtain a proof along the ame line of Section, we follow a more direct approach here. The in nite-dimenional Lagrangian i: L = w (q) (q) dq ( w (q) [ (q) p (q)] dq C ) : The rt-order condition are: ( ) ( q w (q) = if (q) [ (q) p (q)] > < ) ; (6) a well a IC, which mut bind: Z LR (q) q [ (q) p (q)] dq = C: (7) Rearranging (6), it follow that w (q) = q if LR (q) > ; and w (q) = if LR (q) <. It remain to be veri ed that the Lagrange multiplier exit. Note that the LHS of (7) converge to zero a % + and to E [qje = ] E [qje = ] a &. By aumption, C i mall enough that the principal wihe to implement high e ort. In particular, high e ort mut be optimal in the rt-bet when e ort i obervable: E [qje = ] C > E [qje = ] : Thu, by the Intermediate Value Theorem, there exit a Lagrange multiplier (; +) for which (7) hold. 6

17 Proof of Propoition The threhold q are independent of if the likelihood ratio LR () evaluated at q are the ame for any, where q i the unique threhold that olve the IC in (). For all ignal and uch that the likelihood ratio (6) evaluated at q are the ame, we have: LR (q ) = LR (q ) () (q ) p (q ) = (q ) p (q ) : (8) Uing the de nition of the marginal ditribution (5), when (8) hold, we can write the IC in () a q q [f (qje = ) f (qje = )] dq = C: (9) where the threhold q i independent of the realization of. Finally, we how that q i unique. Any contract uch that w(q; ) > for q < ^q, where ^q i implicitly de ned by f(^q je = ; ) = f(^q je = ; ) and i unique due to MLRP, i dominated. Indeed, etting w(q; ) = for q < ^q would not a ect payment on other interval, would increae incentive relative to the initial contract, and would reduce the expected wage. Therefore, retricting attention to contract decribed in Lemma 3 uch that w(q; ) > only if q ^q, the expreion under the integral ign in (9) i poitive, which implie that q i unique. Proof of Lemma 4 By the monotonicity contraint (), w () i Lipchitz continuou and, therefore, di erentiable almot everywhere. Hence, without lo of generality, we can aume that w (q) i a cadlag function atifying w (q) at all point of di erentiability. 6 We will adopt a two-tep approach. Firt, we olve for the optimal contract for a xed minimum wage w () = Z. A we will how, the olution involve a contract where the agent receive a xed minimum wage of Z if output i below a threhold, and i the reidual claimant above the threhold. Then, we will how that the minimum i zero, o the olution i a tandard option contract. Formally, pick contant Z and conider the following relaxed program: min w () w (q) (q) dq () 6 A cadlag function i everywhere right-continuou and ha left limit everywhere. 7

18 ubject to w (q) [ (q) p (q)] dq C; _w (q) ; and w () = Z xed. Note that monotonicity _w (q) implie that Z i both neceary and u cient for the agent LL to hold. We will ignore the principal LL and verify that it i ati ed later. Introduce the auxiliary variable y (q) _w (q) and et up the Hamiltonian: H (w; y; ; ; q) f w (q) + [w [ (q) p (q)] C] + (q) y g ; where w are tate variable, y are control variable, are co-tate variable, and i a (tate-independent) Lagrange multiplier. The neceary optimality condition are: y (q) arg max y (q) y ) y (q) = ( ) if (q) ( < > ) = _ ) (q) [ (q) p (q)] = _ (q) ; () and the tranverality condition (q) = : Condition () yield: _ (q) > () > p (q) (q) () LR (q) > ; where LR (q) (q) i the likelihood ratio, which we aumed to be increaing. p(q) Thu, the LHS of the lat inequality above i decreaing in q while the RHS i contant. Hence, there exit a threhold q [; q] uch that _ (q) > for q < q and _ (q) < for q > q. (Note that if q = or q = q, one of thee interval vanihe). Therefore, i bell-haped, with a unique maximum at q and (at mot) two local minima one at and and another at q. We claim that q < q, i.e. i increaing 8 q if q = q. Then, the tranverality condition (q) = and () imply that w (q) i contant (w (q) = w () = Z 8 q), which violate IC. Thu, it cannot be a olution. 8

19 There are two cae to conider. Firt, (). In thi cae, we mut have (q) > 8 q (; q), ince the only candidate for global minima are and q and () = (q). Second, () <, and o there exit a threhold q (; q) uch that (q) < if q < q and (q) > if q > q. We can combine both cae by letting q [; q) denote the threhold below which (q) <. The olution i then w (q) = Z + max fq q ; g (3) for ome q [; q). Thi conclude the rt part of the proof. We now how that the olution entail Z = 8. We ubtitute the agent wage from (3) into the principal objective function to yield: w (q) (q) dq = Z + (q q ) (q) dq ; (4) q and into the IC contraint to yield: q (q q ) [ (q) p (q)] dq C: (5) Monotonicity i automatically ati ed by (3). A before, the agent LL hold if and only if Z. Since Z increae the objective function in (4) but doe not a ect the IC contraint in (5), the olution entail Z = 8. Thu, the olution of the relaxed program i where the threhold q w (q) = max fq q ; g ; (6) are uch that the IC contraint hold with equality: q (q q ) [ (q) p (q)] dq = C: We nally mut verify that the principal LL i ati ed. Thi i true ince w (q) = max fq q ; g q 8 q. Thu, we have etablihed that the optimal contract i an option where the trike price fq g are uch that the IC hold with equality. Next, we characterize the trike price. Subtituting the contract derived in (6) into the objective function (4), the latter 9

20 become fq min g =;:::;S q (q q ) (q) dq: (7) Denoting the Lagrange multiplier aociated with the incentive contraint by, the neceary rt-order condition with repect to q q where i independent of. i (q) dq + [ (q) p (q)] dq = (8) q R q p (q) dq q R q q (q) dq = < ; (9) We mut verify that the principal LL hold in the optimal contract. Thi i the cae if the optimal q i nonnegative 8. For q, uing (q) = for q <, we can rewrite the principal objective function (7) a q (q q ) (q) dq = (q q ) (q) dq (3) For q, uing (q) = p (q) = for q <, R q (q) dq = R q p (q) dq =, and P = P, we can rewrite the LHS of the incentive contraint in (5) a Thu, for q q [ (q) p (q)] dq q =, the Lagrangian L i: [ (q) p (q)] dq q [ (q) p (q)] dq (3) L(q ) = (q q ) (q) dq q [ (q) p (q)] dq: (3) It rt derivative with repect to q an optimum for any. i R q (q) dq <, and o q < cannot be A the neceary rt-order condition for an optimum only applie to interior olu-

21 tion, we need to rule out corner olution by etablihing that q < q 8. Firt, note that an option with q q i equivalent to an option with q = q. Thu, we need to check that lim q!q L (q ) >. We have lim L (q q!q ) = lim y!q f (y) (q y) + [ (y) p (y)] (q y)g ; (33) where the expreion in bracket ha the ame ign a p(y) (y). Since aumption! y!q, we indeed have lim q!q L (q ) > if >. Finally, we () implie p(y) (y) etablih that > by contradiction. If, the LHS of (8) i trictly negative 8. Thi implie that the optimum i q in (5). Proof of Propoition 3 If the olution entail z = z, IC become: = q 8, which violate the incentive contraint z (q z ) [f (qje = ) f (qje = )] dq = C; which i not a function of the ignal ditribution (for xed marginal ditribution f(qje = ) and f(qje = )). The condition from Lemma 4 i R q R q z (q) dq p z (q) dq = R q z (q)dq R q p z (q) dq ) Pr (q > z ; je = ) Pr (q > z ; je = ) = Pr (q > z ; je = ) Pr (q > z ; je = ) ; which tate that the likelihood ratio of the event that the agent option i in-themoney i not a function of the ignal realization.