DEPARTMENT OF ECONOMICS UNIVERSITY OF CRETE

DEPARTMENT OF ECONOMICS UNIVERSITY OF CRETE BE.NE.TeC. Workng Paper Seres Workng Paper: 007-06 Endogenous Strategc Manageral Incentve Contracts Constantne Manasaks, Evangelos Mtrokostas, Emmanuel Petraks Busness Economcs & NEw TeChnologes Laboratory www.soc.uoc.gr/benetec

Endogenous Strategc Manageral Incentve Contracts Constantne Manasaks Evangelos Mtrokostas Emmanuel Petraks Abstract Ths paper studes the endogenous structure of ncentve contracts that frms owners offer to ther managers, when these contracts are lnear combnatons ether of own profts and own revenues, or of own profts and compettor s profts or, fnally, of own profts and own market share. In equlbrum, each owner has a domnant strategy to reward hs manager wth a contract combnng own profts and compettor s profts. Contrary to the receved lterature, the case where there s no ex-ante commtment over any type of contract that each owner offers to hs manager s also examned. In equlbrum, each type of contract s an owner s best response to the competng owner s choce. JEL classfcaton: D43; L Keywords: Olgopoly; Manageral delegaton; Endogenous contracts. The authors wsh to thank Cham Fershtman, Nkolaos Georgantzs, Mchael Kopel and partcpants at the XXII Jornadas de Economa Industral - JEI006 meetng - at Barcelona and the ASSET Meetngs 006 at Lsbon for ther helpful comments and suggestons. Full responsblty for all shortcomngs s ours. Correspondng author. Department of Economcs, Unversty of Crete, Unv. Campus at Gallos, Rethymnon 7400, Greece, Tel: +3083077409, Fax: +3083077406; e-mal: manasaks@stud.soc.uoc.gr. Department of Economcs, Unversty of Crete; e-mal: mtrokostase@erms.soc.uoc.gr. Department of Economcs, Unversty of Crete; e-mal: petraks@erms.soc.uoc.gr.

Introducton Orthodox economc theory treats frmsaseconomcagentswhosemanob- jectve s to maxmze profts. However, already n the 950s, Baumol (958) suggested a sales-maxmzaton model of frms objectve functon as a realstc alternatve to the proft-maxmzaton one. More recently, Fershtman and Judd (987) argue that a proper analyss of the frm s objectve functon should be undertaken under the prsm of the separaton of ownershp and management. They further argue that such an analyss should ncorporate the structure of the ncentves that owners offer to managers n order to motvate them. The purpose of ths paper s precsely to study the endogenous emergence of the structure of contracts that frms owners offer to ther managers so as to motvate them, and how ths n turn nfluences market performance and socetal outcomes. The strategc use of manageral ncentve contracts has been ntroduced n the lterature by the path-breakng papers of Vckers (985), Fershtman (985), Fershtman and Judd (987) and Sklvas (987). In ths lne of research, each frm s owner offers an ncentve contract to hs manager n order to drect hm to a more aggressve behavor n the market, so as to force the competng manager to reduce output. In partcular, n the above seres of papers, n two-staged olgopoly models, n the frst stage of the game, proft-maxmzng owners choose compensaton schemes for ther managers that are lnear combnatons of own profts andownrevenues. Inthesec- ond stage, managers, knowng compensaton schemes, compete n quanttes. Each frm s owner, when determnng hs manager s ncentves, has an opportunty to become a Stackelberg leader, provded that the rval owners do Manageral theores of the frm and agency theory have emphaszed that the aforementoned separaton leads to neffcences due to asymmetrc nformaton and dfferng objectves of managers and owners (e.g., Wllamson, 964; Jensen and Mecklng, 976; Fama and Jensen, 983). Theroleofnon-proft ncentves nto manageral compensaton practces s evdent from emprcal studes n the agency tradton, such as those of Jensen and Murphy (990) and Lambert et al. (99). These studes reveal that executve bonuses and salares are assocated wth both frm sze and proft level, wth the sze correlatons beng the stronger of the two.

not delegate output decsons to managers. In equlbrum, all owners act n the same way at the game s contract stage and frms end up n a prsoners dlemma stuaton. 3 Regardng the structure of manageral ncentve contracts, recent emprcal evdence has hghlghted new perspectves. Gbbons and Murphy (990), along wth Joh (999) and Aggarwal and Samwck (999) support that n most cases, frms owners, when constructng the ncentve contracts for ther managers, take nto account compettors profts along wth ther own profts. Mller and Pazgal (00; 00; 005) formalze ths argument n what they call Relatve Performance contracts. In a two-staged game, frstly, owners offer to ther managers contracts that are lnear combnatons of own profts and compettors profts. In the second stage, managers compete n an olgopolstc ndustry. Equlbrum analyss suggests that when frms owners manpulate ther managers ncentves, they drect the quantty-settng (prce-settng) managers to a more (less) aggressve behavor, compared to the strct own-proft maxmzng behavor that owners would exercse. From another perspectve, Peck (988) mentons that market share s hghly ranked n managers objectves. In a survey for corporate objectves among 000 Amercan and 03 Japanese top managers, Peck (988) documents that ncreasng market share s ranked thrd n the Amercan and second n the Japanese sample. Jansen et al. (007) and Rtz (005) formalze the case of Market Share contracts. In a duopolstc market, n the frst stage of a two-staged game, each owner offers to hs manager a contract that s a lnear combnaton of own profts and own market share. In the second stage, managers compete ether n quanttes or n prces. Ther man result s that for the case of Cournot (Bertrand) competton, quanttes (prces) set from managers compensated wth Market Share contracts are hgher than those set by strct proft-maxmzng owners. The lterature, so far, seems to seek an explanaton for the emergence of the varous types of manageral ncentve contracts. 4 The core substance 3 If managers compete n prces, Fershtman and Judd (987) and Sklvas (987) fnd that owners encourage ther managers to rase prces and keep sales low. 4 A crucal assumpton of the relevant lterature s that delegaton s observable. Katz (99) argues that unobservable contracts have no commtment value at all. Fershtman 3

of the present paper s an attempt to study the endogenous structure of ncentve contracts that frms owners offer to ther managers, when these contracts are lnear combnatons ether of own profts and own revenues (Profts-Revenues contracts), or of own profts and compettor s profts (Relatve Performance contracts) or, fnally, of own profts and own market share (Market Share contracts). Our envsaged product market s a homogenous Cournot duopoly. The examnatons are restrcted to frms wth equally effcent producton technologes, reflected n equal (constant) margnal producton costs. In ths envronment, we consder a three-staged game wth observable actons: n the frst stage, each frm sowner commtstoone typeofcontracttocompen- sate hs manager. In the second stage of the game, gven that the types of contracts have become common knowledge and can not be reset, each owner sets the weght (manageral ncentve parameter) between own profts and ether own revenues, or compettor s profts, or, fnally, own market share. In the thrd stage of the game, managers compete alàcournot. The analyss suggests that each frm s owner domnant strategy s to reward hs manager wth a Relatve Performance type of contract. Intutvely, although the strategc use of manageral ncentve contracts leads to a prsoners dlemma stuaton always, ths stuaton s the least severe n case of the Unversal Relatve Performance Contracts, mplyng that ths confguraton s the least compettve and frms profts are the hghest. A Relatve Performance contract makes a manager s behavor less susceptble to strategc manpulaton by rvals. Less scope for strategc manpulaton gves to the rval owner less reason to provde ncentves for aggressve behavor to hs manager and thus, competton remans softer. The analyss also suggests that competton becomes fercer the more Profts-Revenue managers are actve, as each one of them makes hs rval relatvely more aggressve. Thus, competton under the Unversal Profts-Revenue confguraton turns out to and Judd (987) support that even f contracts are not observable, they wll become common knowledge when the game s beng repeated for several perods. More recently, Kockesen and Ok (004) argue that to the extent that renegotaton s costly and/or lmted, n a general class of economc settngs, strategc aspects of delegaton may play an mportant role n contract desgn, even f the contracts are completely unobservable. 4

be the most ferce. Then, the analyss s extended towards the case where there s no ex-ante commtment overthetypeofcontractthateachowner offers to hs manager, mplyng that each owner can ndependently shft from one type of contract to another. In ths envronment, the followng two-staged game wth observable actons s studed: n the frst stage, each owner chooses the type of contract to reward hs manager and sets the certan manageral ncentve parameter. In the second stage, gven that the types of contracts and the parameters that owners have chosen are common knowledge and can not be reset, managers compete n quanttes. Interestngly enough, t s found that the magntude of each ndvdual owner s ncentve to devate from one type of contract to another s zero. Thus, each type of contract, namely, a Profts-Revenues contract, a Relatve Performance contract, a Market Share contract s owner s best response to owner j s choce. The ntutve explanaton behnd ths fndngsbasedonthecondtonsthatmustbefulflled n equlbrum: frstly, snce producton decsons are taken by managers, n equlbrum, ther reacton curves must be ntersected. Secondly, the fact that frm s (frm j s) owner offers an ncentve contract to hs manager, as a strategc tool n order to become Stackelberg leader aganst frm j (frm ), standard textbook analyss of the Stackelberg model mples that, n equlbrum, there must be tangency between frm s (frm j s) soproft curveand manager j s (manager s)reactoncurve. Thetwostagedgamescharac- terzed by multplcty of equlbra. By employng focal pont analyss, t s found that each owner wll fnally choose to compensate hs manager wth a Relatve Performance contract. Wth respect to the socetal effects of the alternatve manageral ncentve contracts, t s found that ther use ncreases total welfare, compared to the benchmark case where t s the owners of the frmsthattaketheproducton decsons. The strategc use of ncentve contracts by owners ncreases consumers surplus and decreases total ndustry profts. The latter effect s found to be completely compensated by the former. Moreover, the Unversal Profts-Revenues Contracts equlbrum leads to the hghest total welfare, snce t s the most compettve. 5

Our paper contrbutes to the lterature n two ways. Frstly, by nvestgatng the endogenous structure of ncentve contracts that owners offer to ther managers, ths s, to the best of our knowledge, the frst paper that compares market, as well as socetal, outcomes of alternatve types of manageral ncentve contracts. Secondly, ths paper s among the frst to extend the lterature on strategc manageral delegaton by consderng the case n whch there s no ex-ante commtment over any type of contract that each owner offers to hs manager The rest of the paper s organzed as follows: Secton presents the model. In Secton 3, the dfferent subgames are solved and n Secton 4, the subgame Nash perfect equlbrum of manageral contracts s derved. In Secton 5, a welfare analyss s carred out and n Secton 6, the case where there s no exante commtment over the types of contracts that owners offer to managers s examned. Secton 7 offerssomeconcludngremarks. The Model We consder a homogenous good ndustry where two frms, denoted by, j =,,6= j compete n quanttes. The (nverse) demand functon for the fnal good s lnear, and s gven by: P (Q) =A Q () where, A>0 and Q = q + q s the aggregate output. Frms are endowed wth constant returns to scale technologes and have constant unt cost c = c j = c. Thus,frm s profts are gven by: Π =(A q q j c) q () In ths ndustry, each frmhasanownerandamanager. FollowngFershtman and Judd (987), owner, s a decson maker whose objectve s to maxmze the profts of the frm. Ths could be the actual owner, a board of drectors, or a chef executve offcer. Manager refers to an agent that the owner hres to make real tme operatng decsons. 6

Each frm s owner has the opportunty to compensate hs manager by offerng to hm a take t or leave t ncentve contract. 5 Each owner can choose one among three alternatve types of ncentve contracts to compensate hs manager. The frst s the Profts-Revenues (PR) type of contract. Followng Fershtman and Judd (987) and Sklvas (987), under ths type of contract, the ncentve structure takes a partcular form: the rsk-neutral manager s pad at the margn, n proporton to a lnear combnaton of own profts and own revenues. More formally, frm s manager wll be gven ncentve to maxmze: U PR = a PR Π +( a PR )R (3) where Π and R are frm s profts and revenues respectvely. 6 a PR s the manageral ncentve parameter that s chosen optmally by frm s owner so as to maxmze hs profts. There s no restrcton on a PR, allowng even negatve values. If a PR <, frm s manager moves away from strct proft-maxmzaton towards ncludng consderaton of sales and thus, he becomes a more aggressve seller n the market. If a PR =, manager s behavor concdes wth owner s objectve for strct proft-maxmzaton. The second type of contract s the Relatve Performance (RP) one. Followng Mller and Pazgal (00; 00; 005), under ths type of contract, frm s owner compensates hs manager puttng unt weght on own profts and a weght a RP on rval s profts. Thus, manager s utlty functon takes the form: 7 5 Although n real lfe the terms of manageral contracts can be determned va ownersmanagers negotatons, t s a regular assumpton n the strategc delegaton lterature that the market for managers s perfectly compettve and the frms owners have all the power durng negotatons,.e., they offer to ther managers take t or leave t ncentve contracts (see Vckers, 985; Fershtman and Judd, 987; Sklvas, 987; and Mller and Pazgal, 00; 00; 005). 6 U RR wll not be the manager s reward n general. Snce the manager s reward s lnear n profts and sales, he s pad A +B U RP for some constants A, B,wthB > 0. Sncehe s rsk-neutral, he acts so as to maxmze U RP and the values of A and B are rrelevant. 7 In Mller and Pazgal (00), owner compensates hs manager puttng weght of ( a RP )onownprofts and a weght a RP on the dfference between own profts and the profts of the rval frm, mplyng that U RP =( a RP )Π + a RP (Π Π j ). Ths s equvalent to eq. (4). 7

U RP = Π a RP Π j (4) When the objectve functon s wrtten n ths manner, t becomes apparent that f a RP > 0, manager puts negatve weght on the rval frm s performance. If a RP =0,manager s behavor concdes wth owner s nterest for own proft-maxmzaton. Followng Mller and Pazgal (00), n ordertoavodthestuatonwhereamanagersmoreconcernedwthhsrval than the performance of hs own frm, t s further assumed that a RP. The thrd type of contract s the Market Share (M ) one. Followng Jansen et al. (007) along wth Rtz (005), under ths type of contract, frm s owner compensates hs manager wth a contract consttuted by a lnear combnaton of own profts and own market share. In ths case, manager s utlty functon takes the form: U M = Π + a M q (5) q + q j where the manageral ncentve parameter, a M, s a non-negatve number chosen optmally by owner n order to maxmze hs profts. In order to examne whch types of manageral ncentve contracts preval n equlbrum, consder a three-staged game wth the followng tmng: n the frst stage, each frm s owner commts to one of the three types of contracts D, D = PR, RP, M for rewardng hs manager. Then, n the second stage of the game, gven that the types of contracts have become common knowledge and can not be reset, each owner sets the correspondng manageral ncentve parameter a D. In the thrd stage of the game, managers compete 8

alàcournot. 8,9 The equlbrum concept employed s the subgame perfect equlbrum. 3 Equlbrum manageral ncentve contracts In ths part of the paper, the alternatve strateges of frms owners, regardng the contracts that they wll choose to reward ther managers, are nvestgated. The payoffs ofthedfferent subgames are presented n the followng Payoff Matrx. The representaton of the game n the strategc form OWNER PR RP M PR Π PR, Π PR Π (pr rp) pr, rp, Π Π (pr m), Π (pr m) OWNER RP Π (rp pr), Π (rp pr) Π RP, Π RP Π (rp m), Π (rp m) M Π (m pr), Π (m pr) Π (m rp), Π (m rp) Π M, Π M 3. Unversal Profts-Revenues Contracts Under the present canddate equlbrum confguraton, n the frst stage of the game, frms owners smultaneously and non-cooperatvely commt to 8 At ths pont, t s useful to bear n mnd two alternatve nterpretatons of the game. Accordng to the frst one, followng Fershtman and Judd (987) and Sklvas (987), an owner hres a manager and drects hm through an approprate ncentve contract. The alternatve nterpretaton s the one presented by Mller and Pazgal (00). In the latter, the problem faced by the owner of each frm s to choose the best type of manager among those that are avalable, whle each manager s commtted to behavng n a certan manner by vrtue of hs personalty type. More specfcally, n Mller and Pazgal (00), potental managers take on a contnuum of atttudes toward relatve performance whch s captured by ther type, ϕ. However, the dfference between Fershtman and Judd (987) and Mller and Pazgal (00) s only semantc, snce owners have all the barganng power (by assumpton) when settng the contracts. 9 As a benchmark, the no-delegaton case where producton decsons are taken by frms owners, s consdered. In ths case, the reacton functon s q C(qC j )=(A c qc j )/ whle equlbrum output, profts and total welfare are q C =(A c)/3, Π C =(q C ) and TW C =4(A c) /9 respectvely. Dagrammatcally, the benchmark Cournot case s at pont E C, n Fugure. 9

Profts-Revenues contracts to compensate ther managers. Thus, n the thrd stage of the game, manager chooses q n order to maxmze hs utlty gven by eq. (3), takng as gven the output of hs rval, q j,alongwth the manageral contracts and the manageral ncentve parameters set n the prevous stages. The frst order condton of eq. (3) provdes manager s reacton functon: q PR (qj PR )= A apr c qj PR (6) When owner compensates hs manager wth a Profts-Revenues contract, the slope of manager s reacton curve s dqpr = dqc dqj PR =,.e. t s dqj C equal to the reacton curve s slope n the benchmark case, where t s owner who decdes for the level of producton. From eq. (6), manager, when choosng the output level of hs frm, consders a PR c as the margnal cost of producton. Owner, by choosng an ncentve parameter a PR <, the aforementoned margnal cost, a PR c, s lower than that consdered under owner s strct proft-maxmzng behavor, c. Thus, owner makes hs manager more aggressve,.e. he drects hm to produce output at a level hgher than that produced under a strct proft-maxmzng behavor. Dagrammatcally (Fgure ), as owner sets a PR 0 he nduces an outward and parallel shft n hs manager s best response curve, from RC C (benchmark Cournot) to RC PR. Solvng the system of the reacton functons, output levels are: q PR (a PR,a PR j )= A + c(apr j 3 a PR ) so as to max- In the second stage of the game, each owner chooses a PR mze profts gven by: (7) Π PR (a PR,a PR j )= A + c(a PR j a PR ) A + c(a PR j + a PR 3) 9 (8) The frst order condton of eq. functon: (8) provdes owner s best response 0

a PR (a PR j )= (A + c) a PR j 6 (9) 4c Solvng the system of the best response functons, the (canddate) equlbrum manageral ncentve parameters are: are: A +6c a PR = (0) 5c Pluggng a PR n eq. (7) and (8), frm s equlbrum output and profts (A c) q PR = 5 () (A Π PR c) = 5 () 3. Unversal Relatve Performance Contracts Under the present canddate equlbrum confguraton, n the frst stage of the game, frms owners smultaneously and non-cooperatvely commt to Relatve Performance contracts. Thus, n the thrd stage of the game, manager chooses q n order to maxmze hs utlty gven by eq. (4), takng as gven the output of hs rval, q j, along wth the manageral contracts and the manageral ncentve parameters set n the prevous stages. The frst order condton of eq. (4) provdes manager s reacton functon: q RP (qj RP )= A c ( + arp )qj RP contract, the slope of manager s reacton curve s dqrp dqj RP dq C dq C j (3) When owner compensates hs manager wth a Relatve Performance = (+arp ) < = dqpr dqj PR,sncea RP > 0. From eq. (3), manager, when choosng ) as the competng man- By choosng an the output level of hs frm, consders ( + a RP ager s best response to the level of output that he sets. ncentve parameter a RP > 0, the compettor s best response consdered by the manager, ( + a RP ), s lower than that consdered under a strct own proft-maxmzng behavor, qj C. Thus, owner makes hs manager more

aggressve,.e. he drects hm to produce a level of output, hgher than that produced under a strct own proft-maxmzng behavor. Dagrammatcally (Fgure ), as owner sets a RP, he changes the slope of hs manager s reacton curve, compared wth q C (qj C ), fromrc C to RC RP. Solvng the system of the reacton functons, output levels are: q RP (a RP,a RP j )= (A c)(a RP +) + a RP j a RP 3+a RP a RP j so as to max- In the second stage of the game, each owner chooses a RP mze profts gven by: Π RP (a RP,a RP j )= (A c) a RP 3+a RP The frst order condton of eq. functon: + a RP + a RP j a RP a RP j a RP j (4) (5) (5) provdes owner s best response a RP (a RP j )= arp j +3a RP j (6) Now, explotng the problem s symmetry, t s assumed that a RP = a RP j = a RP. Hence, the symmetrc equlbrum manageral ncentve parameter s: a RP = 3 (7) Pluggng a RP are: n eq. (4) and (5), frm s equlbrum output and profts q RP 3(A c) = 8 Π RP 3(A c) = 3 (8) (9) 3.3 Unversal Market Share Contracts Under the present canddate equlbrum confguraton, n the frst stage of the game, frms owners smultaneously and non-cooperatvely commt to Market Share contracts. Thus, n the thrd stage of the game, manager

chooses q n order to maxmze hs utlty gven by eq. (5), takng as gven the output of hs rval, q j, along wth the manageral contracts and the manageral ncentve parameters set n the prevous stages. Wth respect to the frst-order condtons of eq. (5), for, j, 6= j, the equlbrum quanttes of the thrd stage of the game satsfy the followng system of equatons: A c q M q M + a M A c q M q M + a M q M = RC M (q M +qm ) (a M, a M, q M, q M )=0 q M = RC M (q M +qm ) (a M, a M, q M, q M )=0 (0) Ths s a system of analytcally non-solvable equatons. Jansen et al. (007) prove that for any feasble par (a M, a M ) of weghts, the reacton curves are smooth and concave, ntersectng once, say at q M (a M,a M ),q M (a M,a M ). Then, snce " the Jacoban determnant: R # R q J F = M q M R R = (qm +qm ) h 3(q M +qm ) 3 +(q M +3qM )a M [(q M +q M )(q M +3qM )+a M ]a M > 0 (q M q M q M +qm ) the Implct Functon Theorem mples that the functons q M = q M (a M,a M ) and q M = q M (a M,a M ) have contnuous partal dervatves wth respect to the weghts a M and a M. 0 Pluggng q M and q M n eq. (), frm s profts n the second stage of the game are gven by: Π M (a M,a M )= A q M (a M,a M ) q M j (a M,a M ) c q M (a M,a M ) () Then, the frst-order condtons for frms and, for an equlbrum n the second stage of the game, are formulated: Π M (am,am ) a M Π M (am,am ) a M =0 qm a M =0 qm a M A c q M q M A c q M q M q M q M a M q M q M a M =0 =0 () Moreover, 0 Jansen et al. (007) prove that the second-order condtons for the second stage of the game do hold. 3

q M a M = J F det " R a M R a M R q M R q M # q M a RP = J F det " R q M R q M R a M R a M # q M a M = J F det " R a M R a M R q M R q M # q M a M = J F det " R Pluggng qm, qm a M, qm a M and qm a M n eq. () and solvng the system of a M eq. (0) and (), manager s equlbrum manageral ncentve parameter and frm s equlbrum output are: q M R q M R a M R a M # Pluggng q M a M = q M = 0 (A c) 49 4+ (A c) 4 n eq. (), frm s equlbrum profts are: Π M (A c) = 0 + (3) (4) (5) 3.4 Coexstence of Profts-Revenues and Relatve Performance Contracts Consder the asymmetrc canddate equlbrum confguraton (pr rp)where frm s owner compensates hs manager wth a Profts-Revenues contract whle frm s owner rewards hs manager wth a Relatve Performance one. Under the present canddate equlbrum, n the thrd stage of the game, manager () chooses q (q ) n order to maxmze hs utlty gven by eq. (3) [(4)]. The correspondng reacton curves for managers and are gven by eq. (6) and eq. (3) respectvely. Solvng the system of the reacton functons, output levels are: The formulas for qm, qm a M, qm a M a M and qm a M are gven n Appendx A. Of course, due to the symmetrc ndustry structure, the reverse confguraton where owner compensates hs manager under a Relatve Performance contract whle owner rewards hs manager under a Profts-Revenues one, s as well (mplctly) proposed as a canddate confguraton. 4

q pr (a pr,a rp )= (A + c)( apr ) 3+a rp q rp (a pr,a rp A ( + arp )+c(a pr a pr a rp )= 3+a rp ) (6) (7) In the second stage of the game, owners set the ncentve parameters so as to maxmze profts gven by: Π pr (a pr,a rp (A + c capr )[A + c [(a pr ) a rp + a pr ]] )= (3 + a rp ) (8) Π rp (a pr,a rp [A ( + arp )+c(a rp a rp a rp )] [A + c [(a pr +)a rp a pr ]] )= (3 + a rp ) (9) The correspondng best response functons for owners and are: a pr (a rp )= A (arp ) + c(3a rp +5) 4c(a rp +) (30) a rp (a pr )= A + c (apr ) A + c ( 3a pr (3) ) Solvng the system of the best response functons, (canddate) equlbrum manageral ncentve parameters are: a (pr rp) = a (pr rp) = (3) Note that a (pr rp) = a (pr rp) =. Thsmplesthatwhenowner compensates hs manager wth a Profts-Revenues contract, whle owner j compensates hs manager wth a Relatve Performance contract, owner j s best response to owner samanagerwhosconcernedwthfrm s profts as much as he s concerned wth hs frm s profts. Thsmplesthatownerj s best response to owner s a manager whose behavor reaches ts maxmum level of aggressveness,.e. a (pr rp) =. At the same tme, owner s best response to owner j s a manager wth a strct proft-maxmzng behavor. 5

Pluggng a (pr rp) and a (pr rp) n eq. (6), (7), (8) and (9), equlbrum outputs and profts are: (A c) q (pr rp) = 4 (A c) q (pr rp) = (33) (A Π (pr rp) c) = 6 (A Π (pr rp) c) = 8 (34) Note that, q (pr rp) > q (pr rp) and Π (pr rp) > Π (pr rp). Intutvely, when owner compensates hs manager wth a Profts-Revenues contract, the slope of hs manager s reacton curve s gven by dqpr dq j =. When owner compensates hs manager wth a Relatve Performance contract, the correspondng slope s dqrp +a dq j = RP. Snce a RP > 0 always holds, t s straghtforward that dqrp dq dq j > PR dq j,.e. the reacton curve of the manager who s compensated under the Relatve Performance contract s steeper than that correspondng to the Profts-Revenues contract. In the present confguraton of contracts, the pont where manager s reacton curve ntersects manager s reacton curve mples that the manager sets output at a level hgher than the competng manager does,.e. q (pr rp) >q (pr rp) whch results n Π (pr rp) > Π (pr rp). 3.5 Coexstence of Profts-Revenues and Market Share Contracts Consder the asymmetrc canddate equlbrum confguraton (pr m)where frm s owner compensates hs manager wth a Profts-Revenues contract whle frm s owner rewards hs manager wth a Market Share one. Under the present canddate equlbrum, n the thrd stage of the game, manager () choosesq (q ) n order to maxmze hs utlty gven by eq. (3) [(5)]. The equlbrum quanttes of the thrd stage of the game satsfy the followng system of equatons: 6

A a pr c q pr q m = RC (pr m) (a pr, a m, q pr A c q m q pr + a m, q m )=0 q pr = RC (pr m) (q pr +qm ) (a pr, a m, q pr, q m )=0 (35) Ths s a system of analytcally non-solvable equatons. Snce the Jacoban determnant: " R # R q J F = pr q m =3+ (qm +3qpr )a m > 0 (q pr +qm ) 3 R q pr R q m the Implct Functon Theorem mples that the functons q (pr m) = q (pr m) (a pr a m ) and q (pr m) = q (pr m) (a pr, a m ) have contnuous partal dervatves wth respect to the weghts a pr and a m. Pluggng q (pr m) and q (pr m) n eq. (), the frst-order condtons, for an equlbrum of the second stage of the game, are formulated. Then, followng the procedure appled n the case of Unversal Market Share Contracts, equlbrum manageral ncentve parameters and outputs are: 3, a (pr m) = 4A 3c 7(A c) c a (pr m) = 0 7 (A c) 8 (36) q (pr m) = 3 7 (A c) 4 q (pr m) = 5 7 (A c) (37) Pluggng q (pr m) and q (pr m) n eq. (), equlbrum profts for frms and are: Π (pr m) = 8(A c) 53 + 3 7 Π (pr m) = 7 5 (A c) 8 (38) 3 The formulas for qpr a pr, qm a pr, qpr a m and qm a m are gven n Appendx A. 7

3.6 Coexstence of Market Share and Relatve Performance Contracts Consder the asymmetrc canddate equlbrum confguraton (m rp)where frm s owner compensates hs manager wth a Market Share contract whle frm s owner rewards hs manager wth a Relatve Performance one. Under the present canddate equlbrum, n the thrd stage of the game, manager () choosesq (q ) n order to maxmze hs utlty gven by eq. (5) [(4)]. The equlbrum quanttes of the thrd stage of the game satsfy the followng system of equatons: q m A c q m q rp + a rp A c +(a rp )q m q rp = RC (m rp) (a m, a rp, q m, q rp )=0 (q m +qrp ) = RC (m rp) (a m, a rp, q m, q rp )=0 (39) Ths s a system of analytcally non-solvable equatons. Snce the Jacoban determnant: " R # R q J F = m q rp =3+a rp + am [( a rp )q m )] +qrp (3+arp > 0 (q m+qrp ) 3 R q m R q rp the Implct Functon Theorem mples that the functons q (m rp) = q (m rp) (a m, a rp ) and q (m rp) = q (m rp) (a m, a rp ) have contnuous partal dervatves wth respect to the weghts a rp and a m. Pluggng q (m rp) and q (m rp) n eq. (), the frst-order condtons, for an equlbrum of the second stage of the game, are formulated. Then, followng the procedure appled n the case of Unversal Market Share Contracts, equlbrum manageral ncentve parameters and outputs are: 4 a (m rp) =0 a (m rp) = (40) (A c) q (m rp) = 4 (A c) q (m rp) = (4) Pluggng q (rp m) and q (rp m) n eq. (), equlbrum profts for frms 4 The formulas for qm a m, qrp a m, qm a rp and qrp a rp are gven n Appendx A3. 8

and are: (A Π (m rp) c) = 6 (A Π (m rp) c) = 8 (4) Thefactthata (m rp) =0whle a (m rp) =, mples that when owner j compensates hs manager wth a Relatve Performance contract, whle owner compensates hs manager wth a Market Share contract, owner j s best response to owner s a manager who s concerned wth frm s profts as much as he s concerned wth hs frm s profts. Subsequently, owner j s best response to owner s a manager whose behavor reaches ts maxmum level of aggressveness,.e. a (m rp) =. At the same tme, owner s best response to owner j s a manager wth a strct proft-maxmzng behavor. Ths s the reason why, q (m rp) >q (m rp) whch results n Π (m rp) > Π (m rp). 4 Equlbrum manageral ncentve contracts In ths part of the paper, frms owners strateges n the frst stage of the game are nvestgated. In order to answer the queston wth respect of the optmal structure of contracts that owners choose to compensate ther managers, frms profts under the dfferent subgames are compared. The results are summarzed n the followng Proposton: Proposton Under Cournot competton and ex-ante commtment over the type of contract that each frm s owner chooses, each owner s domnant strategy s to reward hs manager wth a Relatve Performance type of contract. Ths Proposton establshes a strategc ratonale for commttng to Relatve Performance contracts. The rest of ths secton s devoted for gvng the ntuton behnd ths result n detal. Comparng equlbrum output and profts under the three symmetrc subgames,.e. the symmetrc confguratons of manageral contracts, wth those obtaned n the benchmark case where producton decsons are taken by owners, the followng Corollary summarzes the fndngs: 9

Corollary () q PR () q RP () q M >q C and Π PR < Π C. >q C and Π RP < Π C >q C and Π M < Π C. The reason why the use of a manageral ncentve contract ncreases output s the followng: owner, by usng an ncentve contract strategcally, drects hs manager to a more aggressve behavor n order to force the competng manager to reduce output. Because each owner acts n the same way at the game s contract stage, frms end up n a prsoners dlemma stuaton. Naturally, the ncreased market supply, n comparson to the no-delegaton case, leads to lower profts. These fndngs are n the sprt of the well-known result n the ndustral organzaton lterature that frms competng n quanttes have no ncentve to engage n Stackelberg warfare. Then, equlbrum output and profts under the three symmetrc confguratons of manageral contracts are ranked. The followng Corollary summarzes the fndngs:. Corollary () q RP <q M <q PR. () Π RR > Π M > Π PR. Intutvely, the strategc use of manageral ncentve contracts leads to a prsoners dlemma stuaton always. Ths stuaton s the least severe n case of the Unversal Relatve Performance Contracts canddate equlbrum. Moreover, t turns out that the Unversal Market Share Contracts canddate equlbrum s more (less) compettve than the Unversal Relatve Performance (Profts-Revenues) Contracts one. Ths mples that the former equlbrum, when compared wth the later, s characterzed by hgher output and lower profts. Consderng the asymmetrc equlbra as well, the results are the followng: Corollary 3 q (pr rp) = q (m rp) >q (pr m) >q (pr m) >q (pr rp) = q (m rp) > q C. 0

Wth respect to q (pr rp) = q (m rp) and q (pr rp) = q (m rp),recallthat when owner compensates hs manager ether wth a Profts-Revenues contract or wth a Market Share contract and owner j compensates hs manager wth a Relatve Performance contract, owner j s best response to owner s a manager whose behavor reaches ts maxmum level of aggressveness. At thesametme,owner s best response to owner j s a manager wth a strct proft-maxmzng behavor. Ths s the reason why the subgames (pr rp) and (m rp) lead to dentcal frm-level quanttes and profts. Note also that q S hq (pr m), q (pr m), S = PR,M,RP, mplyng that n the symmetrc confguratons of manageral ncentve contracts, each manager sets output at a level lower (hgher) than the level set by the manager compensated wth thecontract thatntensfes the competton relatvely less (more), snce there s no asymmetry that an owner can explot. Corollary 3 leads to the followng rankng regardng the equlbrum profts of the dfferent subgames: Corollary 4 Π C Π (m rp). > Π (pr rp) = Π (m rp) > Π (pr m) > Π (pr m) > Π (pr rp) = As for the equlbrum profts, Π S RP. Ã Regardng the ndustry-level equlbrum output Q = Ã X profts T Π = Π!, the results are the followng: = hπ (pr m), Π (pr m), S = PR, M,! X q and total Corollary 5 () Q PR >Q (pr m) >Q M >Q RP >Q (pr rp) = Q (m rp) > Q C. () T Π C >TΠ RP >TΠ (pr rp) = T Π (m rp) >TΠ M > T Π (pr m) >TΠ PR. The results stated n Corollary 5 reveal the compettveness of the alternatve confguratons of manageral ncentve contracts. Recall that although the strategc use of manageral ncentve contracts leads to a prsoners dlemma stuaton always, ths stuaton s the least severe n case of the =

Unversal Relatve Performance Contracts, mplyng that ths confguraton s the least compettve and frms profts are the hghest. The ntuton behnd ths result goes as follows: a Relatve Performance contract makes a manager s behavor less susceptble to strategc manpulaton by rvals. Less scope for strategc manpulaton gves to the rval owner less reason to provde ncentves for aggressve behavor to hs manager and thus, competton remans softer. Note also that competton becomes fercer the more Profts-Revenue managers are actve, as each one of them makes hs rval relatvely more aggressve. Thus, competton under the Unversal Profts-Revenue confguraton turns out to be the most ferce. 5 Welfare analyss In ths Secton we perform a welfare analyss. More specfcally, total welfare under the alternatve confguratons of contracts s compared wth total welfare n the benchmark case where producton decsons are taken by frms owners. Let total welfare TW be defned as the sum of frms profts and consumers surplus. TW D = T Π D +CS D,D= PR, RP, M, (pr rp), (pr m), (m rp) (43) wth T Π D and CS D = Q D beng the overall ndustry profts and consumers surplus respectvely. Usng equlbrum results, total welfare for each confguraton of contracts s gven n Appendx B. The results are summarzed n the followng Proposton: Proposton Under Cournot competton: () The use of manageral ncentve contracts ncreases total welfare,.e., TW D >TW C. () The Unversal Profts-Revenues Contracts (Unversal Relatve Performance Contracts) equlbrum leads to the hghest (lowest) total welfare,

.e., SW PR >SW (pr m) >SW M >Q RP >SW (pr rp) = SW (m rp). As far as the frst part of the Proposton s concerned, t has already been clear that, compared wth the no-delegaton case, the strategc use of an ncentve contract by an owner, n order to become Stackelberg leader, dstorts hs manager s ncentves away from proft-maxmzaton, ncreasng output and consumers surplus. Ths tends to ncrease total welfare. Because each owner acts n the same way, n equlbrum, the ncreased market supply leads to lower ndustry profts than those obtaned under the benchmark nodelegaton case. The reduced total ndustry profts tend to decrease total welfare, but ths effect s completely compensated by the hgher consumers surplus. It s found that the postve effect of ncreased consumers surplus on total welfare domnates the negatve effect of decreased profts whch results n TW D >TW C. Regardng the second part of the Proposton, t has already been mentoned that competton becomes fercer the more Profts-Revenue managers are actve, as each one of them makes hs rval relatvely more aggressve. The fercer the competton, the hgher the consumers surplus and the lower the frms profts. Note also that snce t s the consumers surplus effect that domnates on total welfare, the rankng of total welfare s dentcal wth the rankng of the ndustry-level equlbrum total output, stated n Corollary 5 (). 6 Equlbrum manageral ncentve contracts under no commtment So far analyss was grounded on the assumpton that each frm s owner commts to a partcular type of contract n the frst stage of the game. In ths part of the paper, the case where there s no ex-ante commtment over the types of contracts that owners offer to managers s nvestgated. In order to examne whch types of manageral ncentve contracts preval n equlbrum, we consder a two-staged game wth the followng tmng: n the frst stage, each frm s owner chooses one type of contract to reward hs manager and 3

sets the correspondng manageral ncentve parameter a D, D = PR, RP, M. The crucal, yet (due to the symmetrc ndustry) reasonable assumpton here s that the precse contract (the type of contract and the manageral ncentve parameter) that owner sets s not observable by the rval owner, before contract-settng s everywhere completed. Thus, each owner can ndependently shft from one type of contract to another. In the second stage of the game, managers compete alàcournot. To nvestgate the condtons under whch, a canddate equlbrum confguraton of strateges, n the types of contracts (one for each owner) at the frst stage, s proposed. Subsequently, t s checked whether or not t survves all possble devatons. Ifyes, the proposed equlbrum s a Subgame Perfect Nash Equlbrum. Consder the symmetrc confguraton where both frms owners choose to compensate ther managers wth Profts-Revenues contracts. Fgure offers the vsualzaton of ths canddate equlbrum (pont E PR ). Note that n equlbrum, two condtons must be fulflled: frstly, snce producton decsons are taken by managers, ther reacton curves RC PR and RC PR must be ntersected. Secondly, the fact that frm s (frm s) owner offers an ncentve contract to hs manager, as a strategc tool n order to become Stackelberg leader aganst frm (frm ), mples that, n equlbrum, there must be tangency between frm s (frm s) soproft curveπ (Π )and manager s (manager s) reacton curve RC PR (RC PR ). Unversal Profts-Revenues Contracts s an equlbrum confguraton only f no owner has an ncentve to unlaterally devate by offerngtohsmanager ether a Relatve Performance contract or a Market Share one. Therefore, twopossbledevatonshavetobechecked. Regardng the frst devaton, suppose that owner stcks to the Profts- Revenues contract, belevng that owner wll offer to hs manager the same type of contract. In the frst stage of the game, owner sets the parameter a PR = that corresponds to the Unversal Profts-Revenues Contracts 3 equlbrum. Owner devates towards compensatng hs manager wth a Relatve Performance contract. Thus, owner uses hs reacton functon a (rp pr) (a ) wth a = a PR = to optmally adjust the parameter a 3 for hs manager s contract. Thus, owner sets: 4

a PR d a PR d = The devant owner s profts wll be Π PR d and a = a PR = 3. = Π(rp pr) (a,a ),wherea = Π PR (A c) d = 5 Interestngly enough Π PR d = ΠPR, mplyng that the magntude of owner s ncentve to devate from the Unversal Profts-Revenues Contracts equlbrum towards compensatng hs manager wth a Relatve Performance contract, s zero. 5

Fgure : Equlbrum under the benchmark Cournot, Unversal Profts-Revenues Contracts and Unversal Relatve Performance Contracts. Dagrammatcally, when owner devates from the Unversal Profts- Revenues Contracts canddate equlbrum towards compensatng hs manager wth a Relatve Performance contract, owner drects hs manager to the Relatve Performance reacton curve RC RP and, by readjustng the manageral ncentve parameter, optmally readjusts ts slope untl RC RP.Compared wth RC PR, manager under the RC RP has now become a more aggressve seller, because gven q he now produces more. However, the manager of the devant owner sets output at a level equal to the one he would set under the canddate equlbrum (E PR ). Only at pont E PR,bothofthe aforementoned equlbrum condtons hold. More specfcally, only at pont 6

E PR the reacton curve RC RP s ntersected wth RC PR and also, the frm s (frm s) soproft curveπ (Π )stangenttothemanager s (manager s) reacton curve RC PR (RC PR ). Note that nether pont C nor pont C can emerge n the subgame perfect equlbrum because the slope of the frm s soproft curveπ s non-zero,.e. ponts C and C do not belong n the manager s reacton curve RC RP. In the same lnes, examnng the second devaton,.e., the case where owner devates from the Unversal Profts-Revenues Contracts canddate equlbrum towards compensatng hs manager wth a Market Share contract, one fnds that the magntude of owner s ncentve to devate from the Unversal Profts-Revenues Contracts canddate equlbrum towards compensatng hs manager wth a Market Share contract, s zero. Wth respect to the subgame equlbrum of the game, the results are summarzed n the followng Proposton: Proposton 3 Under Cournot competton and no ex-ante commtment over thetypeofcontractthateachfrm s owner chooses, each type of contract, namely, a Profts-Revenues contract, a Relatve Performance contract, a Market Share contract, s owner s best response to owner j s choce. Ths result s drven by the fact that n equlbrum, two condtons must be fulflled: frstly, managers reacton curves must be ntersected and secondly, each frm s soproft curve must be tangent to the competng manager s reacton curve. The above condtons do hold for all the dfferent types of contracts that owners can offer to ther managers, regardless the functonal forms of cost and demand. Therefore, assumng no ex-ante commtment over any type of contract that owners wll offer to managers, even a contract of dfferent type could be an owner s best response to the competng owner s choce. Note that the two-staged game s characterzed by multplcty of equlbra. Subsequently, the followng queston s addressed: whch types of the alternatve manageral ncentve contracts wll fnally emerge n equlbrum? Usng the results for equlbrum profts from Secton 3, by employng focal pont analyss, t s found that rval owners would realze that t s n ther 7

mutual nterest to move towards the equlbrum that ensures them the hghest profts. Therefore t s expected that each owner wll fnally choose to compensate hs manager wth a Relatve Performance contract. 7 Concluson In ths paper we have nvestgated the endogenous structure of ncentve contracts that frms owners offer to ther managers, when these contracts can be ether of a Profts-Revenues type, or of a Relatve Performance, or, fnally, of a Market Share one. The analyss suggested that each frm s owner domnant strategy s to reward hs manager wth a Relatve Performance type of contract. Intutvely, a Relatve Performance contract makes a manager s behavor less susceptble to strategc manpulaton by rvals. Less scope for strategc manpulaton gves to the rval owner less reason to provde ncentves for aggressve behavor to hs manager and thus, competton remans softer, mplyng hgher profts. Consderng the case n whch there s no ex-ante commtment over any type of contract that each frm s owner offerstohsmanager,theanalyssrevealedthateachtypeofcontractsa best response to the competng owner s choce. Ths result s drven by the fact that n equlbrum, two condtons must be fulflled: frstly, managers reacton curves must be ntersected and secondly, each frm s soproft curve must be tangent to the competng manager s reacton curve. By employng focal pont analyss, t was found that each owner wll fnally choose to compensate hs manager wth a Relatve Performance contract. Regardng the socetal consequences of the alternatve delegaton schemes, the use of manageral ncentve contracts s found to ncrease total welfare. Moreover, the Unversal Profts-Revenues Contracts equlbrum leads to the hghest total welfare, snce t s the most compettve. Our results have been derved n the context of a duopolstc market wth a lnear demand and cost model. We are of the opnon that the duopolstc market reveals all essental dfferences between the alternatve types of contracts. We are also aware of the lmtatons of our analyss n assumng specfc functonal forms. However, the propertes of the equlbrum cond- 8

tons that drve our results that allows us to argue that these results wll also hold under general demand and cost functons. The use of more general forms would jeopardze the clarty of our fndngs, wthout sgnfcantly changng ther qualtatve character. Appendx Appendx A: Unversal Market Share Contracts q M a M q M a M q M a M q M a M q M h ( q M +qm ) 3 +a M qm = (q M +qm ) h 3(q M +qm ) 4 +(q M +qm )(q M +3qM )a M +[(qm +qm )(3qM +qm )+am ]am h q = M ( q M +qm ) 3 +a M qm am qm (q M +qm ) h 3(q M +qm ) 4 +(q M +qm )(q M +3qM )a M +[(qm +qm )(3qM +qm )+am ]am h q = M ( q M +qm ) 3 +a M qm am qm (q M +qm ) h 3(q M +qm ) 4 +(q M +qm )(q M +3qM )a M +[(qm +qm )(3qM +qm )+am ]am q M h (q M +q M ) 3 +a M qm = (q M +qm ) h 3(q M +qm ) 4 +(q M +qm )(q M +3qM )a M +[(qm +qm )(3qM +qm )+am ]am Appendx A: Coexstence of Profts-Revenues and Market Share Contracts q pr a pr q m a pr q pr a m q m a m = q pr (q pr +qm ) 3(q pr +qm ) 3 +(3q pr +qm )a m q pr (q pr +qm ) 3(q pr +qm ) 3 +(3q pr +qm )a m h ( q pr +qm ) 3 (q pr qm )a m 3(q pr +qm ) 3 +(3q pr +qm )a m = = c = c + am qpr ( qpr +qm ) 3+ (3qpr +qm ) am ( qpr +qm ) 3 Appendx A3: Coexstence of Market Share and Relatve Performance Contracts q m a m q rp a m = (q m+qrp ) = 3+a rp (q m+qrp ) q rp + am [ qm ( a 3+a rp q rp ( a rp rp ) +qrp ( 3+arp ( qm +qrp ) 3 ) + am [ qm ( a rp )] ) +qrp ( 3+arp ( qm +qrp ) 3 9 )]

q m a rp q rp a rp = = 3+a rp 3+a rp q m + (qrp qm ) am ( qm +qrp ) 3 + am [ qm ( a q m rp ) +qrp ( 3+arp ( qm +qrp ) 3 am qrp (q m+qrp ) 3 + am [ qm ( a rp )] ) +qrp ( 3+arp ( qm +qrp ) 3 Appendx B: Total Welfare )] TW PR = (A c) 5 TW RP = 5(A c) 3 TW M = 7(A c) 9 3 TW (pr rp) = 5(A c) 3 TW (pr m) = 6(A c) 3 + 5 7 TW (m rp) = 5(A c) 3 References Aggarwal, R.K., Samwck, A.A., 999. Executve compensaton, strategc competton, and relatve performance evaluaton: theory and evdence. Journal of Fnance 54, 999-043. Baumol, W., 958. On the theory of olgopoly. Economca 5, 87-98. Fama, E.F., Jencen, M.C., 983. Separaton of ownershp and control. Journal of Law and Economcs 6, 30-35. Fershtman, C., 985. Manageral ncentves as a strategc varable n duopolstc envronment. Internatonal Journal of Industral Organzaton 3, 45-53. Fershtman, C., Judd, K.L., 987. Equlbrum ncentves n olgopoly. Amercan Economc Revew 77, 97-940. Gbbons, R., Murphy, K.J., 990. Relatve performance evaluaton for chef executve offcers. Industral and Labor Relatons Revew 43, 30S-5S. Jansen, Thjs, Ler, van Are, Wtteloostujn, van Arjen, 007. A note on strategc delegaton: the market share case. Internatonal Journal of Industral Organzaton, forthcomng. 30

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