S-shaped Incentive Schemes and Pay Caps

Transcription

1 S-haped Incenve Scheme and Pay Cap Tony Haao Cu Carlon School o Managemen Unvery o Mnneoa Jagmohan S. Rau The Wharon School Unvery o Pennylvana Mengze Sh Roman School o Managemen Unvery o Torono March 0 ll auhor have conrbued equally o he paper. Tony Haao Cu an Proeor o Markeng a he Carlon School o Managemen, Unvery o Mnneoa. Jagmohan S. Rau Joeph J. rey Proeor, Proeor o Markeng, a he Wharon School o Bune, Unvery o Pennylvana. Mengze Sh ocae Proeor o Markeng a he Roman School o Managemen, Unvery o Torono. Ynghao Zhang provded uperb reearch aance. mal correpondence: cu@umn.edu, rau@wharon.upenn.edu, and mh@roman.uorono.ca. I

2 S-haped Incenve Scheme and Pay Cap brac S-haped ncenve cheme and pay cap are arly common n pracce. Th paper demonrae he opmaly o -haped ncenve cheme and pay cap by ncorporang alepeople averon o pay nequy no he andard agency model. Our analy how ha alepeople dere or pay arne ncreae he convey o he opmal ncenve cheme a mall ale bu ncreae he concavy a large ale. Conequenly, he opmal compenaon plan -haped. Wh averon o pay nequy, he opmal ncenve cheme alway conan an upper bound or oal paymen. or praccal mplemenaon, we propoe a capped quoa plan o appromae he opmal -haped cheme. Our numercal analy ndcae ha he capped quoa plan ha an average non-opmaly o le han % n paramerc pace uded. The numercal analy alo eplore he ource o non-opmaly and he relaonhp beween marke characerc and he opmal ze o pay cap. Keyword: agency heory; pay arne; ale orce compenaon; pay cap II

3 . Inroducon S-haped ale ncenve cheme are common n pracce. ypcal -haped ncenve cheme conan varyng common rae. or eample, a ale ncenve cheme may nclude a alary o $0,000, no common or he r $ mllon ale, a common rae o 5% or addonal ale aer he $ mllon mark, and hen a lower common rae o % or any ale above $ mllon See old lne BCD n gure or an lluraon. The $ mllon mark oen nerpreed a a ale quoa. pecal cae o he -haped ncenve cheme a pay cap, whch mpoe a mamum amoun o oal ale common o be earned by a aleperon. lluraed n gure, an ncenve cheme can conan a common cap o $50,000 See lne BC n gure. On reachng he pay cap, ale common rae eecvely become zero. lernavely one could nerpre h a a cap o $70,000 or he oal pay. ccordng o 005 Hew Sale Compenaon Survey, 43% o he compane ued pay cap or alepeople and 47% ued pay cap or ale manager. More recenly, 009 Incenve Pracce Sude by ZS ocae how ha -haped ale ncenve cheme were ued n 65% o 85% o ale eam n pharmaceucal ndury, wh -haped common rae more common han pay cap. Some bonu conrac can alo be nerpreed a -haped ncenve cheme. or eample, an ncenve cheme may nclude a bonu o $0,000 or reachng $ mllon ale, an era bonu o $0,000 or reachng he $ mllon ale arge, bu no addonal bonu or reachng any new mleone. Such paern were requenly oberved n ale conrac lluraed n Oyer 000. lhough -haped ncenve cheme are commonly oberved, o he be o our knowledge, prevou reearch ha no nvegaed he underlyng reaon or uch ncenve cheme. Th paper nended o brdge h gap by developng a heory bul on he preme ha ale agen are concerned abou pay arne. arne an mporan ue n ale managemen becaue he perceved pay arne can aec alepeople ob aacon Smh 00, whch, n urn, may aec perormance and loyaly n -haped ncenve cheme alo called a regreve ncenve cheme or a deceleraor. Many proeonal por league ncludng B mpoe alary cap or ndvdual ahlee and eam Yang, Sh and Goldarb 009.

4 Ramawam and Sngh 003, Card e al. 00. Th parcularly mporan n ndure where a aleperon relaonhp wh her clen crcal n mananng accoun ably. gure : S-haped Sale Incenve Scheme $70,000 Toal Pay C % Mamum D 5% $0,000 B 0 $ M $ M Sale Our model ocue on arne aocaed wh pay equy beween deren alepeople,.e. he nra-rm compenaon nequy" dened n Joeph and Kalwan 998. ollowng ehr and Schmd 999, Cu, Rau, and Zhang 007, and Ho and Su 009, we nclude he duly or unarne, more peccally, he averon o pay nequy compared o peer alepeople, a an addve erm n he alepeople uly uncon. Unlke he convenonal approach yped by Bau, Lal, Srnvaan, and Saeln 985 here aer reerred o a BLSS ha conder each ndvdual aleperon eparaely, h paper ude a group o alepeople mulaneouly n order o accoun or arne concern. Our analy ugge ha averon o pay nequy ncreae he convey o opmal ncenve cheme or mall value o ale oucome. When ale volume low, a aleperon mo lkely o eperence duly reulng rom pay nequy. he averon o pay nequy creae era ncenve or he aleperon, he rm provde le moneary ncenve hrough ale common. In conra, when ale volume large, arne concern ncreae he concavy o he opmal ncenve cheme becaue he rm wan o conan he epeced pay nequy acro alepeople. Combnng he above wo reul,

5 n he uaon where BLSS would recommend a lnear or conve orm o compenaon cheme, accounng or arne concern lead o an -haped opmal ncenve cheme. We alo demonrae ha ndeed opmal or he rm o place an upper bound on he oal common pay o he alepeople. In her emnal paper, BLSS deny hree deren hape o opmal compenaon cheme: concave, conve, or lnear correpondng o aleperon rk olerance beng lower han one, hgher han one, or equal o one. In her reul, he common rae alway change monooncally a he ale ncreae and hu he -haped compenaon cheme canno be opmal n a BLSS cone n pe o a very verale peccaon o uly and ale repone uncon. Rau and Srnvaan 996 here aer reerred o a RS udy he opmal degn o quoa-baed compenaon cheme or a ale orce wh deren errory poenal. RS ocu on he conve cae o BLSS opmal cheme and demonrae he near opmaly o quoa plan. In oher ude relaed o ale orce compenaon, a mple lnear common rucure oen ued. Such eample nclude he compenaon cheme ncorporang cuomer aacon Hauer, Smeer, and Werneel 994; Kalra, Sh, and Srnvaan 003, compenaon cheme or ndependen ale agen Calderaro and Coughlan 009, monorng Joeph and Thevaranan 998, mul-produc Lal and Srnvaan 993, normaon aymmery Lal and Saeln 986, and delegaon e.g., Bhardwa 00; Dong, Yao and Cu 0. Smple lnear common rucure are alo common n lab-baed emprcal ude e.g., Ghoh and John 000. Our paper conrbue o he emergng leraure on arne and mpac on rm managemen raege. Srnvaan 98 how ha an equal common rae polcy can be opmal when a aleperon eek a ar-ncome and chooe h eor o mach he equable earnng. However, Srnvaan 98 doe no ncorporae he arne concern eplcly no agen uly uncon o udy how uch concern could aec he opmal ale common rucure. ehr, leander and Schmd 007 how epermenally ha when ome agen are ar-mnded, bonu conrac provde beer ncenve han conrac wh upron wage paymen. In an epermenal udy, Güh e al. 00 conder heerogeneou agen who care abou arne boh beween he prncpal and hemelve, and among he agen. They aume ha agen producve are deermnc bu unequal, and hu he 3

6 prncpal can ner her eor rom oupu. The auhor how ha n uch cae, a prncpal wll oer le aymmerc compenaon plan o he heerogeneou agen when agen know each oher compenaon plan, han when hey do no have uch normaon. Our udy conder homogeneou agen wh ochac ale producon uncon uch ha he agen eor canno be compleely nerred rom her oupu. The model by Derau and Sappngon 007 mo cloely relaed o our paper. Derau and Sappngon 007 ncorporae concern o nequy no an advere elecon model and how ha agen averon o e-po nequy no conranng or he prncpal he agen are dencal e ane. Our analy however ugge ha alepeople concern or arne wll aec he compenaon plan oered by he prncpal even when alepeople are dencal e-ane. nally, Lm 00 how ha when conean care abou oucome relave o oher conean, a cone hould have a hgher proporon o wnner. Our reearch alo add o he growng leraure on ncorporang behavoral heore no quanave model o beer underand how hee aec rm raegc markeng decon e.g., maldo and Jan 005; Bradlow, Hu and Ho 004a, 004b; Chen and Cu 00; Chen, Iyer and Pazgal 00; Cu, Rau and Zhang 007; enberg, Krhna and Zhang 00; Harde, Johnon and ader 993; Ho, Lm, and Camerer 006; Ho and Zhang 008; Kalra and Sh 00; Lm, hearne and Ham 009; Orhun 009; Syam, Krhnamurhy and He 008; ec.. The re o h paper organzed a ollow. In he ne econ, we oulne an agency model ha ncorporae arne concern. We provde he man analycal reul on opmal compenaon cheme n Secon 3. In Secon 4 we decrbe he algorhm and he reul o numercal ude on quoa plan wh pay cap. nally, we conclude wh man ndng rom our analy and drecon or uure reearch.. Model Conder a rm ha employ dencal alepeople,e., wh he ame ale producvy. ach o hee alepeople mached wh one ndependen errory. We ue he ame nde or alepeople 4

7 and errore, where =,,. Sale oucome n a errory depend on he ellng eor o he aleperon agned o he errory, bu ndependen o he eor o oher alepeople. We denoe ale n errory by, and aleperon eor by. rom he rm perpecve, all errore are he ame e ane. Speccally, ale n all errore ollow he ame cumulave drbuon, where =,,. ollowng he andard prncpal-agen model, we aume he rm or ale manager canno oberve a aleperon eor. Snce ale oucome are ochac, he rm canno perecly ner a aleperon eor rom he aleperon ale oucome eher. We le denoe he compenaon plan or aleperon. Snce ale are he only obervable oucome, and errore are ndependen, we conder a ale-baed compenaon uncon =. We aume ha > 0; ha, a aleperon compenaon ncreae wh h ale volume. Th aumpon mple ha ale volume and compenaon ollow he ame rank order. More peccally, he probably o a aleperon pay beng he hghe whn a group he ame a ha o h ale beng he hghe. nally, we aume ha alway pove; n oher word, we only adm hoe compenaon plan no nvolvng any pobly o moneary lo or alepeople. Saleperon Decon ach aleperon decde he ellng eor o mamze h epeced value compued over he drbuon o all alepeople ale, =,,,, a ollow: Ma U ma c.. ndcaed n quaon, aleperon uly depend on ncome and pay arne. More peccally, U aleperon uly o h compenaon pay and he coecen o duly rom pay nequy. We ollow he andard aumpon ha uly uncon U monooncally ncreang bu he margnal uly U monooncally decreang wh compenaon. We urher aume a aleperon duly rom pay nequy eparable rom he uly rom pay, and a lnear uncon o he pay nequy caled by a arne parameer. Thee aumpon are commonly ued n he leraure 5

8 on arne Charne and Rabn 00; ehr and Schmd 999; Ho and Su 009. quaon alo ndcae ha aleperon ncur duly c rom eor. gan we ollow he andard aumpon ha duly uncon c monooncally ncreang and conve wh he eor level; ha, c > 0 and c 0. mployee concern or pay nequy are well documened n he leraure. kerlo and Yellen 990 revew a l o alernave reaon or why employee may dlke pay nequy; rangng rom eleeem, recprocy, o ealouy. Our model n quaon aume ha, when evaluang pay nequy, a aleperon compare h pay wh he hghe pay n he group. We aume ha e-po each aleperon wll know he ale and pay acheved by he op perormer whn he reerence group. Mo organzaon have a polcy o ecrecy wh regard o wage and alare. However, vrually all he ale orce ollow he pracce o recognzng op perormer and n he proce dcloe how well hey perormed, and oen publcze her compenaon. Cone are oen organzed a boh company and dvon level. ane, however, when makng decon on her ellng eor, alepeople orm epecaon abou he ancpaed pay nequy baed on her bele on ale oucome. ng reearch ndcae ha perceved arne o pay deermned by an ndvdual comparon o h pay o oher n he reerence group, and he ypcal reerence group nclude oher wh mlar kll Carroll and To 977, Kahneman, Knech, and Thaler 000. In our model, we conder alepeople wh dencal kll and errore, and hence dencal ale eecvene. Whn he reerence group o ale repreenave, we aume ha a aleperon conder he hghe pay n he group a he reerence pon. Th conen wh reul n Marn 98. When echncan n a acory were aked wheher hey lked o know he hghe, average, or lowe pay o echncan or comparon o her own wage, nerengly, mo o hem waned o know he hghe pay among all echncan. Snce our model ocue on arne aocaed wh pay nequy beween deren alepeople whn he ame rm, he arne all n he doman o nra-rm compenaon nequy" dened n Joeph and Kalwan 998 and peer-nduced arne dened n Ho and Su 009. In our model, e- 6

9 ane all he alepeople are dencal n her able and errore are mlar o one anoher and eor n he equlbrum; however, e-po hey may generae deren amoun o ale, and hereore be compenaed derenly, reulng n deren eor-reward rao. However, unlke n a ale cone where one aleperon pay depend on oher alepeople perormance Kalra and Sh 00, n our model oher alepeople perormance may aec he ocal aleperon only hrough he duly or pay nequy, bu no pay. rm Decon The rm decde compenaon cheme S =,,, o mamze epeced pro rom all alepeople. We ormulae he rm problem a ollow: ma S.. U ma c R, =,, arg ma U ma c, =,, 4.. In quaon, he rm pro conrbuon rom aleperon equal o pro rom ale mnu compenaon, where he rm pro margn. quaon 3 he parcpaon conran or ndvdual raonaly condon or each o he alepeople. Speccally, n quaon 3, each aleperon ne urplu uly mnu co o eor hould be a lea a large a an oude amoun. oe ha R a uncon o eor, and hu model pay-eor equy a dcued n Ramawam 0 and Sngh 003, and mlar n pr o he equable pay n Srnvaan naly Snce a aleperon perceved pay nequy relave o wha oher ge, we need o analyze conrac or all alepeople mulaneouly. In he equlbrum analy, we ocu on a ocal aleperon, 7

10 aumng he remanng - alepeople are under equlbrum conrac; ha, =, whch dencal or all. a reul, all hee - alepeople epend = amoun o eor a uggeed by IC condon 4. Beore conducng he ormal analy on opmal compenaon cheme, ueul o underand how a margnal ncreae n compenaon o he ocal aleperon would aec he pay nequy eperenced by he ocal aleperon and anoher non-ocal aleperon k. We dcu h margnal eec n Secon Margnal ec o Sale on Pay Inequy rom aleperon perpecve, nce he remanng - alepeople all epend equlbrum eor, n each o hee errore ale ollow an dencal and ndependen drbuon,.. The probably ha aleperon acheve he hghe ale. oe ha probably decreae wh. aleperon le lkely o be he op aleperon when compared o a larger reerence group. ow conder he non-ocal aleperon k. H epeced pay nequy meaured by he gap wh he op-perormng aleperon: ma.. k k k k d [ ][ ma ma ] d k k 5 In quaon 5, we eparae he epeced pay nequy o aleperon k no wo par: when aleperon ha he hghe ale n he group, he r par, - k, he pay nequy o aleperon k; oherwe, he econd par, ma k, he pay nequy o aleperon k. In quaon 5, gven aleperon ale, aleperon k k epec a pay nequy 8

11 Pay_Inequy_k = [ ][ ma ma ] k k I we ubue k wh, we oban aleperon epeced pay nequy a ollow: Pay_Inequy_= [ ][ ma ma ]. ow we can dcu he margnal eec on epeced pay nequy due o a mall ncreae n aleperon pay a ale,. r, wh probably k k, aleperon acheve he hghe ale n he group and doe no eperence pay nequy, bu he remanng - alepeople eperence pay nequy. In h cae, an ncreae n wll urher ncreae he pay nequy eperenced by hoe - alepeople bu wll no change aleperon own pay nequy. Second, wh probably -, aleperon doe no acheve op ale and hence eperence pay nequy, whch can be reduced wh an ncreae o. The remanng - alepeople may eperence pay nequy bu he gap ndependen on aleperon ale. Thereore, a very mall ncreae o wll no change hee - alepeople pay nequy. Overall, wh a un ncreae o, he pay nequy eperenced by each o he oher - alepeople epeced o ncreae by bu aleperon own pay nequy epeced o decreae by, wh a oal ncreae o -. ; The above analy yeld wo ueul mplcaon. r, a a very low level o ale, aleperon mo lkely o eperence pay nequy. a reul, he aleperon wll have an era ncenve o work harder becaue he ncremenal ale can no only ncreae compenaon bu alo reduce pay nequy. Conequenly, a maller common rae wll be ucen o nduce a dered level o eor. Second, a a very hgh level o ale, aleperon mo lkely o be he op ale agen and unlkely o eperence pay nequy. In h cae, any era eor and ncremenal ale wll be dermenal becaue o he ncreaed pay nequy eperenced by he remanng - alepeople. Conequenly, a rm may wan o lower he common rae o conan he eor. 9

12 0 3.. Opmal Compenaon Scheme We now analyze he opmal compenaon cheme dened by he opmzaon problem n quaon, 3, and 4. To characerze he oluon, we ollow he Lagrangean approach adoped n BLSS. L + ma 0.. R c U + ma '.. c U 6 where and are Lagrangean mulpler. Snce we conduc margnal analy wh a ocal aleperon, ueul o wre he above Lagrangean no ollowng eplc orm. d L + ma 0.. R c d U + ma '.. e c d U where pay nequy gven by quaon 5. In a ymmerc equlbrum, opmal compenaon a ale hould ay ollowng neceary condon: 0,,,...,,, L Takng advanage o he ymmery, he above condon can be wren a - L U ' U ' =0. ollowng he analy n BLSS, we can rewre he above condon no

13 U ' [ ][ ] 7 When β = 0, quaon 7 reduced o he amlar orm n BLSS. In order o derve more eplc oluon, pecc aumpon are requred on alepeople uly uncon and he drbuon or ale uncon. ollowng BLSS, o begn wh, we udy hree uly peccaon wh deren rae o change o rk olerance; more peccally, we conder a power uncon, 0< < whch ha he rae o change o rk olerance equal o / - δ and larger han, a log uncon log whch ha he rae o change o rk olerance equal o, and an eponenal uly uncon a whch ha he conan rk averon and hence he rae o change o rk olerance equal o 0. Second, ollowng BLSS, we udy ale repone uncon wh / = a-b a>0 and b>0. Such drbuon nclude Gamma, Bnomal, and ormal drbuon commonly aumed n he leraure. or eample, n Gamma drbuon, he ale condonal on eor level characerzed by he probably drbuon uncon q q q q, where he mean ale equal q g g g e o g h k and varance equal o g q. Under Gamma drbuon, we have g g ' kq h k. h k Subung uly uncon and Gamma ale repone uncon no quaon 7, we can derve he opmal compenaon plan. We ue he ubcrp p, l, and e o denoe power uncon, log uncon, and eponenal uncon. 3 Cae. Power uly uncon : 3 The dervaon o +B alo apple o normal and bnomal drbuon.

14 ] ][ [ p k h k h kq k h k h kq ] ][ [ b a b a ] [ B B 8 Cae. Log uly uncon log: ] [ l B B 9 Cae 3. ponenal uly uncon wh conan rk averon a - : ] [ e B B L 0 When β = 0, he opmal compenaon cheme a hown n quaon 8, 9, and 0 reduced o he correpondng BLSS oluon. The opmal compenaon uncon n 8, 9, and 0 are reduced o conve, lnear, and concave uncon o ale, repecvely. ollowng BLSS, we reer o a he alary parameer and B a he common rae parameer. Snce no a lnear uncon, he acual alary and common rae wll be nonlnear uncon o and B. Dependng on uly peccaon, rom quaon 8, 9, and 0 we can derve he correpondng opmal amoun o alary a,, and L. Snce he value o depend on parameer β and, we canno drecly ner rom h reul how he alary may vary wh he parameer. e we udy he propere o opmal compenaon cheme n quaon 8, 9, and 0. We ummarze he propere n hree lemma leadng o he r propoon. LMM : 0 and B > 0.

15 Proo: See ppend or dealed proo. 4 Lemma conrm ha he alary parameer pove wh an upper lm o /β, and common rae parameer B pove. oe ha Lemma hold or all hree ype o uly uncon uded. Snce we are prmarly nereed n he hape o he compenaon cheme, we ne eamne how he lope o he compenaon cheme change a ncreae. Th ummarzed n he ollowng lemma. d d LMM : or any > 0, we have 0 or mall and 0 or large. urher, he d d opmal compenaon upper-bounded. The upper bound equal o,, and L or power uly uncon, log-uly uncon, and eponenal uly uncon, repecvely. Lemma ndcae he opmaly o mpong an upper lm on ale compenaon. ale ncreae, he opmal compenaon alo ncreae. However, when he ale volume ucenly large, he margnal compenaon ncreae converge o zero. I we appromae he general compenaon uncon by a pecewe lnear uncon BLSS, page 77, Lemma ugge ha he common rae wll evenually converge o zero. Th eecvely creae an upper lm on he oal ale compenaon. The preence o an upper lm alo mple a concave compenaon uncon when he ale are ucenly large. Lemma provde he eplc epreon or he opmal upper lm on he ale compenaon, and how ha he upper lm depend on hree parameer: β,, and. More peccally, he upper lm more rngen.e., a maller mamum pay when he arne parameer β large, a larger reerence group, and/or a maller uly parameer. r, when alepeople are more concerned abou pay arne, he upper lm on compenaon hould be lower. Th becaue a low upper lm can reduce he epeced pay nequy and ncreae he ancpaed pay arne. The magnude o he arne 4 Unle oherwe noed, all proo are provded n ppend. 3

16 parameer β may depend on he vbly o ale comparon. Salepeople can become more concerned wh arne when oher alepeople perormance are more vble. Such vbly can be naurally hgher n he ndure wh ewer clen. Vbly can alo be aclaed by ceran company pracce. or eample, announcemen made by he compane o recognze her op alepeople, comparon ale manager ue o creae compeon among he alepeople n cone Sh and Yang 009. Second, when alepeople ace a larger reerence group, he opmal compenaon cheme hould have a lower upper lm. Th becaue he unepecedly large compenaon or one aleperon can caue pay nequy or he remanng - alepeople. larger mple a greaer aggregae pay nequy and eelng o unarne. Thu, a lower upper lm more benecal n rercng epeced pay nequy among a larger group o alepeople. In pracce, may depend on how he alepeople are organzed and how hey orm he reerence group. or eample, all alepeople whn he ame drc are een a one reerence group, hen a larger drc ha a bgger. lernavely, only he alepeople whn he ame cohor o recru are condered n he reerence group, hen hould depend on he number o new recru and he urnover rae. nally, a larger ndcae hgher margnal uly or compenaon and a hgher rk olerance. I alepeople are more movaed by he addonal pay, a hgher upper lm can nduce more ellng eor and revenue. Thu, a larger h more aenon o ale and le aenon o he arne concern. a reul, he upper lm can be hgher. LMM 3: or he cae o a power uly uncon and a log uly uncon, conve or mall. or he cae o eponenal uly uncon, conve or mall and only >. Lemma 3 ndcae ha alepeople arne concern ncreae he convey o he opmal compenaon cheme a low level o ale. Recall ha n he abence o arne concern, a n BLSS, he opmal compenaon cheme lnear when alepeople ehb a log uly uncon. Inerengly, he lnear urn o conve a mall ale when β > 0. The nuon or h reul eplaned earler n he margnal analy. When he ale are low, a aleperon more lkely o uer pay nequy. Th creae addonal ncenve or he aleperon o work harder o reduce he epeced pay nequy. a reul, he rm can provde le moneary ncenve hrough ale common a mall 4

17 ale. Snce he ncenve o reduce pay nequy ronger a maller ale, he requred common rae hould ncreae wh ale level. Thu, arne concern urn an oherwe lnear compenaon curve no a conve hape a mall level o ale. When arne concern ucenly rong o ha >, even an oherwe concave compenaon cheme under eponenal uly uncon can become conve. Overall, or maller ale oucome, Lemma 3 how ha alepeople averon o pay nequy have a pove eec on her ncenve o work harder. e we combne he reul rom lemma -3 and ummarze our man heorecal reul n Propoon. PROPOSITIO :.. Under a power uly uncon, when alepeople eperence duly rom pay nequy,.e., > 0, he opmal compenaon plan an ncreang and conve uncon or mall, and an ncreang and concave uncon or large wh an upper lm on he compenaon equal o... Under a log uly uncon, when alepeople eperence duly rom pay nequy,.e., > 0, he opmal compenaon plan an ncreang and conve uncon or mall, and an ncreang and concave uncon or large wh an upper lm on he compenaon equal o..3. Under an eponenal uly uncon, when alepeople eperence duly rom pay nequy,.e., > 0, or mall he opmal compenaon plan an ncreang and conve uncon β>, and or large an ncreang and concave uncon wh an upper lm on he compenaon a L. Propoon ndcae ha, when alepeople are ucenly avere o pay nequy, he opmal compenaon cheme hould be a conve uncon o ale or mall and a concave uncon o ale or large. Thu, Propoon provde one poble eplanaon or why we commonly oberve compane ung -haped compenaon cheme. Propoon alo ndcae an upper lm or he compenaon cheme. dcued earler, he opmal upper lm hould be lower wh larger value o β and. e we compare our reul wh BLSS o how how he concern or pay arne wll aler he rucure o he opmal compenaon cheme. The reul can be be eplaned wh he cae o log 5

18 uly uncon lluraed n gure. In h cae, BLSS would recommend a lnear compenaon uncon whch equvalen o a alary and a conan common rae, a depced by he hn old lne. Wh arne concern > 0, he opmal compenaon cheme become he -haped; he hck old lne n gure. There are hree noceable change: rom lnear o convey or maller value o ale, rom lnear o concavy or larger value o ale, and a eeper curve a he nermedae range o ale. r, a mall ale, a aleperon mo lkely o uer rom eperencng pay nequy. The averon o ancpaed pay nequy creae an addonal ncenve or he aleperon o work hard. a reul, a le common-nduced ncenve needed a lower ale. Second, a large ale, a aleperon very lkely o be he op aleperon. ny era ale wll ncreae he pay nequy eperenced by he remanng alepeople. Snce uch duly coly o he rm, opmal or he rm o mnmze he common rae a large ale. Thrd, gven he lower ale common rae a boh mall and large ale, he rm hould provde hgher common rae a he nermedae ale n order o acheve he dered level o ale. We could appromae he old -haped compenaon uncon by a pecewe lnear uncon. The common rae rucure n gure reemble he common rucure n gure. The reul aocaed wh wo oher uly peccaon, he power uncon and he eponenal uncon, are qualavely mlar. The arne concern alway ncreae he convey a mall ale and he concavy a large ale. Under he power uly uncon, compared o BLSS, arne concern urn an oherwe conve ale compenaon uncon no an -haped uncon. Under he eponenal uly uncon, ucenly large arne concern urn an oherwe concave uncon no an -haped uncon. Thu ar we have heorecally eablhed he opmaly o -haped compenaon cheme. However, he compenaon uncon a a oluon o quaon 7 may be poenally comple or a rm o communcae o alepeople. BLSS ugge n pracce rm ypcally ue quoa-baed common plan o appromae he opmal uncon. Such an appromaon lead o mnmal lo o opmaly, a demonraed by RS. ollowng he ame approach, n he ne econ, we nvegae eay-o-mplemen -haped compenaon plan lke hoe n gure. We ue numercal analy o 6

19 Sale Compenaon eamne he non-opmaly aocaed wh our appromaon. Snce he preence o an upper lm a dncve eaure o our reul n Propoon, we wll alo nvegae he opmal ze o pay cap and relaonhp wh a number o marke characerc. gure : arne ec n Opmal Pay Srucure and Pecewe Lnear ppromaon Common Rae BLSS-opmal pay S-haped Compenaon Scheme 0 ~ Sale Low-ale area: arne concern add era ncenve or ocal aleperon o work harder o avod eperencng negave uly rom pay nequy. Inermedae-ale area: rm gve ou hgher common rae o nduce eor lo due o pay cap. Hgh-ale area: rm ha ncenve o cap he pay o conan alepeople pay nequy negave eernaly. Opmal BLSS Compenaon Scheme Opmal S-haped Sale Compenaon Scheme ppromaed Compenaon Scheme wh Sale Cap 4. ppromang S-Shaped Compenaon Scheme wh Capped Quoa Plan: umercal naly To mplemen he opmal -haped compenaon cheme n a large ale orce ha con o mulple group o alepeople and errore, we conder a capped quoa plan commonly ued n pracce. 5 Th ale rucure reemble one oen ued n a mul-dvon rm where he ale envronmen repone 5 The pay cap n a capped quoa plan no he ame a he upper lm or he ale compenaon n he opmal plan gven by Propoon. we demonrae laer n he numercal ude, he opmal pay cap n he capped quoa plan are ypcally maller han he upper lm or he opmal ale compenaon plan. 7

20 uncon and organzaonal culure are oen mlar whn each dvon, bu can be que deren acro dvon. 6 n Secon 3, we aume alepeople whn each group o be dencal and ue only her peer whn he dvon a he reerence group o orm arne udgmen. capped quoa plan con o a alary, a common rae R, ale quoa q, and cap-nducng ale q or ale agen group ee gure 3. The rm mplemen he ame compenaon plan whn each group, bu allow ome derence beween group o accoun or her deren ale envronmen. ollowng RS, we aume he bae alary and common rae R o be dencal or all group, bu he ale quoa q and cap-nducng ale q are pecc o ale dvon. The group-pecc ale quoa and ale cap depend on he group ale repone uncon and her alepeople rk preerence and enve o pay nequy. We ormulae he capped quoa plan or ale group a ollow: q q R q q q q R q In quaon, each aleperon n ale group receve only a alary beore reachng ale quoa q. er reachng he quoa, each aleperon earn a ale common a rae R or he ncremenal ale above q. The aleperon wll no receve any addonal ale common aer reachng he cap-nducng ale q, cappng oal amoun o common a q - q R. The capped quoa plan gven by deren rom he opmal -haped compenaon cheme gven by Propoon n wo way. r, he capped quoa plan a pece-we lnear appromaon o a general -haped uncon. Thu, he capped quoa plan doe no perecly he curvaure o opmal - haped uncon. ollowng RS, we dene uch lo o opmaly a hape-nduced non-opmaly. Second, he capped quoa plan mpoe he ame alary and common rae R or all group. Th rercon on and R can ubanally reduce he compley o compenaon degn n a mul-dvon ale orce, bu come a a co. gan ollowng RS, we dene heerogeney-nduced non-opmaly a he lo o opmaly arng rom he conran on equal alary and common rae acro group. 6 We ue he erm group and dvon nerchangeably. 8

21 gure 3: Capped Quoa Plan Pay Cap Toal Pay : Bae Salary R: Common Rae 0 q q Sale Opmal S-haped Sale Compenaon Scheme Bac Quoa Plan wh Pay Cap Our numercal ude have wo man obecve. We nend o nvegae he perormance o capped quoa plan. Speccally, n he pr o RS, we eamne how well a capped quoa plan appromae he opmal -haped compenaon cheme on opmaly. We alo eamne wheher he ource o ub-opmaly due o a lower ale eor and hereore lower epeced ale or due o overcompenaon. We alo eplore he deermnan o pay cap. Whle deermnan o quoa and common rae have been uded eenvely n he leraure, pay cap have no been ormally eamned. To deny he key drver, we eplore a e o envronmenal varable uch a he ze o he reerence group, aleperon characerc, and marke-errory characerc. 4.. umercal naly Mehod To reman comparable wh prevou numercal analye conduced n RS, we ar by aumng ha ale volume ollow a gamma drbuon wh ollowng probably drbuon uncon: q q g q g e q g q. Gamma drbuon enure pove ale wh a lnear 9

22 epeced ale uncon, [ ] g h k where h denoe he bae ale level whou era eor and k denoe he aleperon average margnal producvy o eor. Sale uncerany n gamma drbuon capured by he coecen o varaon gven by / q, a larger q ndcang maller uncerany. Second, he uly uncon o a aleperon n group gven by [ ] d where 0 < <, d > 0, and >. Speccally, each aleperon' uly rom pay ollow a power uly uncon. The larger he parameer, he le rk-avere he aleperon. Snce >, a aleperon duly or eor a conve uncon o. RS provde more deal on how o nerpre hee parameer. The numercal analy cover a paramercal pace dened by deren value o parameer,,, R 0, d,, h, k, q. ollowng RS, we e pro margn η equal o one. Table ummarze he numercal value o he parameer ued, grouped no hree caegore: aleperon characerc, errory characerc, and aleperon-errory characerc. In addon o all he parameer ncluded n RS, our analy conan wo new parameer: arne concern β and number o alepeople n each group. Pleae noe ha he parameer capure he ze o reerence group. We nclude hree level o arne concern; = 0 no arne concern, = 0.00 mld concern or arne, and = 0.0 rong concern or arne. We alo nclude hree level o reerence group ze; =, 3, and 5. or he re o parameer, o provde comparably wh prevou reearch, we ue he ame level o parameer value a n RS. ll ogeher, he parameer value n Table provde a oal o 3 =768 combnaon. 7 To avod unneceary repeon, we reer o RS or more normaon on nerpreng each o hee parameer and he elecon o her repecve value or numercal analy. We ue numercal opmzaon echnque o analyze boh he opmal compenaon cheme gven by quaon 8 and he opmal quoa plan wh pay cap gven by quaon. In order o oban he analycal reul requred o olve he opmal compenaon cheme, we eend he opmzaon analy 7 The cae o = 0 ame a n RS and no ued n he analy o pay cap. 0

23 n ppend. Baed on hee analycal reul, we numercally olve he opmal -haped compenaon cheme or any gven e o,,, R 0, d,, h, k, q, ollowng a wo-age numercal mehod mlar wh Bau and Kalyanaram 990. We brely ummarze he wo-age numercal procedure below. Saleperon Characerc Terrory Characerc Saleperon-Terrory Characerc Table : Parameer Value Parameer Symbol Value. arne Concern 0; 0.00; 0.0. Rk veron /3; ¾ 3. Mnmum Uly R 0 30; umber o Reerence Salepeople ; 3; 5 5. Duly Mulpler d ;.5 6. Duly lacy 4; 5 7. Bae Sale h 0; Cerany n Sale Q ; 0 9. Sellng ecvene k 500; 3000 Sage. nd he upper bound o eor level, m, ha he rm can pobly nduce. Speccally, we e = 0 and hen calculae B = BM and = m uch ha boh IR condon quaon 3 and IC condon quaon 4 are bndng. Due o he compley o he compenaon plan quaon 8, we are unable o oban a cloed orm epreon or m. Hence, we compue BM and m ung a dchoomc earch algorhm. Sage. nd opmal and B wh he ollowng wo-ep proce.. Varyng rom 0 o m ncremenally n mall ep, or each, we olve or and B ung he ame procedure a n Bau and Kalyanaram 990 and calculae he correpondng pro or he rm.. er earchng or all poble value o, we nd he opmal ha yeld he hghe pro. The aocaed and B are he parameer or he opmal compenaon plan. Ung a mlar mehod, we numercally olve he opmal quoa plan wh a pay cap or any gven paramercal e o,,, R 0, d,, h, k, q. The procedure brely ummarzed below.

24 Sage. Wh varyng rom 0 o M and R rom 0 o RM, or each ncremenal ep, we nd opmal ale quoa and cap-nducng ale or each par o and R or all combnaon. Here M deermned baed on he arng alare under -haped compenaon plan and RM e o be one becaue he common rae hould no eceed one. We hen calculae he rm oal pro by addng up he pro under all combnaon. or a parcular combnaon, we olve or q and q ung he ollowng wo-ep proce.. Wh varyng rom o, or each we olve or q and q ung an ehauve earch algorhm wh a reaonable ncremenal earchng ep and calculae he correpondng pro or he rm. Here and are e baed on he level o ale eor under he -haped plan.. er earchng or all poble, we nd he opmal ha yeld he hghe pro. The aocaed q and q are he ale quoa and cap-nducng ale or he quoa plan wh a pay cap. Sage. er earchng or all poble par o and R, we nd he opmal par and R ha yeld he hghe oal pro. Th par o and R, along wh all he aocaed q and q, are he parameer requred o dene he quoa plan wh pay cap. oe ha he opmal -haped compenaon cheme mamze a rm epeced pro rom each e o parameer value. aurally h wll mamze he oal epeced pro rom all 768 e o parameer value alo. However, n earchng or he opmal capped quoa plan, we mamze he oal pro rom all 768 e o parameer value. we conran alary and common rae o be equal, he pro rom he opmal capped quoa plan wll be lower han he pro rom opmal alored -haped cheme. 4.. Reul We r eamne how well he capped quoa plan perorm n comparon o he opmal -haped compenaon cheme when alepeople are concerned wh pay equy. La, we udy he deermnan or opmal pay cap.

25 on-opmaly o he Capped Quoa Plan ollowng RS, we meaure he perormance by he magnude o non-opmaly he percenage lo o opmaly a dened by S Cap 00. Here reer o he rm epeced pro under op S S he opmal -haped compenaon cheme and reer o he epeced pro under he capped quoa Cap plan. op nclude boh hape-nduced and heerogeney-nduced non-opmaly dened by RS. Snce and are wo new parameer unque o our model o pay equy, n Table we how he level o nonopmaly o he capped quoa plan or each combnaon o and. or each e o and, here are 8 combnaon o parameer value. In each cell o Table, we preen he average, large, and malle non-opmaly value compued over hee 8 cenaro. Smlar o ndng n RS, Table how he average non-opmaly, combnng hape-nduced and heerogeney-nduced non-opmaly, o be mall, and no more han % n all cenaro. Th ugge ha he capped quoa plan perorm que well n appromang he opmal -haped compenaon cheme alored o each ndvdual aleperon. Source o on-opmaly To urher underand ource o non-opmaly aocaed wh he capped quoa plan, we eplore he eec on nermedae oucome uch a ale volume.e., he revenue or he rm and he alepeople alare co o compenaon. Table 3 how he change n epeced ale volume a well a change n alepeople epeced alare under he capped quoa plan v. he opmal -haped compenaon plan. I he change n ale volume pove, ugge ha on average he epeced ale volume under he capped quoa plan lower han ha under he opmal -haped compenaon plan. Smlarly, a pove change n alary ugge ha a aleperon earn a lower alary under he capped quoa plan han under he opmal -haped compenaon plan. 3

26 Baed on Table 3, we nd ha when arne concern weak.e., =0.00 or when he reerence group mall.e., =0.0 and =, on average epeced ale volume are lower under he capped quoa plan han under he -haped plan. In hee cae, he average epeced alary alo lower under he capped quoa plan. Th ugge ha when he arne concern weak or when he reerence group mall, he non-opmaly n he rm epeced pro can are rom lower ale volume. The reducon n ale volume moly due o he pay cap, a emprcally demonraed n Mra and ar 00. Tha, when arne concern weak or when he reerence group mall, he hape-nduced non-opmaly rom pay cap domnae and lead o reduced ale volume. Inerengly, when he arne concern are rong and reerence group ze large,.e., =0.0 and =3 or =5, on average boh he epeced ale volume and epeced alare under capped quoa plan are hgher han hoe under -haped compenaon plan! Thu, when alepeople have rong arne concern and he ze o he reerence group large, he non-opmaly may reul rom hgher alare o alepeople raher han change n ale volume. Snce he alary rerced o be ame or all alepeople, he rm reduce ncenve and ncreae alary o manage he epeced pay nequy. Tha, when arne concern are rong and reerence group ze large, he heerogeney-nduced non-opmaly domnae and lead o hgher epeced alare. Table : on-opmaly n Pro o he Capped Quoa Plan = 3 = 3 = 5 = %, 7.3%, 0.9%.90%, 6.90%, 0.44%.79%, 6.3%, 0.34% = %, 5.97%, 0.33%.37%, 6.8%, 0.00% 0.5%, 7.4%, 0.00%. ll non-opmaly comparon are relave o he opmal -haped compenaon cheme.. Mean are compued over 8 aleperon-errory combnaon. 3. r percenage repreen average non-opmaly, econd and hrd percenage repecvely repreen he large and malle non-opmaly over 8 aleperon-errory combnaon. 4

27 Table 3: Change n peced Sale Volume and Salepeople Salary = 3 4 = 3 = 5 = %, 7.34%, -6.34%.97%, 9.7%, -8.5%.86%, 8.%, -6.0%.78%, 9.5%, -7.7%.74%, 8.%, -6.98%.6%, 6.7%, -0.3% = %, 8.83%, -5.59% 0.56%, 3.4%, -8.8% -0.%, 8.04%, -7.76% -3.63%, 8.%, -3.4% -.7%, 5.73%, -9.8% -9.%,.5%, -36.6%. ll change are relave o he reul n opmal -haped compenaon cheme.. Mean are compued over 8 aleperon-errory combnaon. 3. The number on op ndcae change n ale volume and he number a boom ndcae change n alepeople alary. 4. r percenage repreen average change, econd and hrd percenage repreen he large and malle change, repecvely, over 8 aleperon-errory combnaon. Deermnan o Opmal Pay Cap In order o udy how opmal pay cap are aeced by aleperon-pecc, errory-pecc, and aleperon and errory-pecc parameer led n Table, we conduc a regreon analy on 768 mulaed reul, ung he opmal pay cap a he dependen varable and he ollowng parameer a ndependen varable: ze o reerence group, alepeople arne concern, her rk averon, bae ale h, ellng eecvene k, and cerany n ale q. The dealed regreon reul are repored n ppend 3 and he qualave reul ummarzed n Table 4. Table 4: Summary o Regreon Reul or Pay Cap Parameer ec on Pay-Cap h k q 5

28 ec o Sze o Reerence Group on Pay Cap. Our reul ugge ha he opmal pay cap wll be lower when he reerence group become larger. When ar-mnded alepeople compare her pay wh he reerence pay,.e., he hghe pay n he reerence group, he opmal ale cap hould decreae wh he ze o reerence group. Wh a larger reerence group, a aleperon le lkely o become he op aleperon and hence more lkely o eperence duly rom he comparon wh he op achever. Moreover, a hgher pay or he op aleperon would lead o a greaer amoun o aggregae duly when more alepeople uer rom he eperence o pay nequy. Snce he duly o pay nequy ha a negave mpac on he rm pro, he rm hould e a lower pay cap n order o conan he epeced pay nequy. ec o arne Concern on Pay Cap. Regreon reul how ha he opmal pay cap wll be lower when alepeople are more concerned wh pay nequy. When alepeople are more ar-mnded, he rm hould e a lower pay cap o reran he ale derence among alepeople o reduce he epeced duly rom pay derence. Wh a lower pay cap, alepeople who are oherwe equal, wll no be rewarded or very lucky realzaon o demand. ec o Rk veron on Pay Cap. Our regreon reul ugge ha he opmal pay cap hould be lower when alepeople are more rk avere. maller mple lower margnal value or era pay o he alepeople. a reul, era ale common wll be le eecve n movang he alepeople. Seng a lower pay cap, whle reducng he epeced pay nequy, wll no gncanly depre he ellng eor rom alepeople. Conequenly he opmal pay cap lower. ec o Bae Sale h, Sellng ecvene k, and Cerany n Sale q on Pay Cap. The regreon reul how ha, when he bae ale larger larger h, he pay cap hould be hgher. When ale eor more eecve n generang ale larger k, he pay cap hould be greaer. Snce ellng eor are very eecve, a rm wan o e a hgh pay cap n order no o depre he producve ellng eor. When he uncerany n ale hgh maller q, he rm hould e a hgher pay cap. larger 6

29 ale uncerany make more dcul o nduce ellng eor. a reul, a hgher pay cap requred o avod dampenng alepeople movaon. 5. Concluon and uure Reearch Th paper demonrae ha alepeople concern or pay nequy can aec he degn o opmal compenaon cheme. Speccally, concern or pay nequy can lead o opmal compenaon cheme o be -haped, and alo ugge he ue o pay cap. In addon o provdng a heorecal ba or - haped compenaon cheme and pay cap, h paper ugge he pobly o mplemenng -haped compenaon cheme hrough a mple capped quoa plan. umercal analye llurae near opmal perormance o capped quoa plan whn he paramerc cenaro uded. Reul rom our numercal ude urher ugge ha rm hould mpoe lower pay cap when agen are more ar-mnded, more rk avere, and/or are a par o a larger reerence group. Pay cap hould be hgher when he errore have larger bae demand, are more reponve o ellng eor, and/or ace more ochac demand. Overall, when mpong a pay cap, a rm need o balance he bene rom reduced pay nequy wh he poenal co due o dmnhed ellng eor. We vualze everal poble drecon or uure reearch. r, uure reearch may conder alernave way o ormulae reerence pon o model nequy. Th paper aume ha a aleperon compare h pay wh he hghe pay among equvalen peer. Whle here ome emprcal uppor or our ormulaon, here are alernave way o ormulae reerence pon; ncludng he average pay whn he reerence group, he arge e by he manager, or au quo. Second, uure reearch may udy alepeople reacon o pay nequy n oher way. or eample, n addon o eperencng duly a modeled n h paper, alepeople may hrk o nenonally punh he prncpal kerlo and Yellen 988,990; Rabn 993. Thrd, uure reearch may nvegae he eecvene o oher ype o compenaon cheme n dealng wh ar-mnded agen uch a bonu conrac or ru conrac wh upron wage paymen uded n ehr, leander and Schmd

30 Reerence kerlo, G.., J. L. Yellen arne and unemploymen. mer. conom. Rev , The ar wage-eor hypohe and unemploymen. Quar. J. conom maldo, W., S. Jan Conpcuou conumpon and ophcaed hnkng. Managemen Sc Bau,. K., G. Kalyanaram On The Telave Perormance o Lnear veru onlnear Compenaon Plan. In l J. Re. n Markeng Bau,. K., R. Lal, V. Srnvaan, R. Saeln Saleorce Compenaon Plan: n gency Theorec Perpecve. Markeng Sc Bhardwa, P. 00. Delegang prcng decon. Markeng Sc Bradlow,. T., Y. Hu, T. Ho. 004a. learnng-baed model or mpung mng level n paral conon prole. J. Markeng Re ,,. 004b. Modelng behavoral regulare o conumer learnng n conon analy. J. Markeng Re Calderaro, abo and nne T. Coughlan Sped-up channel: The role o p n herarchcal ellng organzaon. Markeng Sc. 6, 3-5. Card, D, Ma,., More,., and. Saez. 00. Inequaly a work: The eec o peer alare on ob aacon. BR Work Paper Carroll, S. J. and H. L. To Organzaonal Behavor. Chcago: S. Clar Pre. Chen, Y., T. H. Cu. 00. The bene o unorm prce or branded varan. Workng Paper., G. Iyer,. Pazgal. 00. Lmed memory, caegorzaon, and compeon. Markeng Sc Cu, T. H., J. S. Rau, Z. J. Zhang arne and Channel Coordnaon. Managemen Sc Derau, R., D.. M. Sappngon quy and advere elecon. J. conom. Managemen Sraegy Dong, Y., O. Yao, T. H. Cu. 0. When acquon pol reenon: Drec ellng v. Delegaon under CRM. Workng Paper. ehr,., K. leander, K. M. Schmd arne and conrac degn. conomerca , K. M. Schmd heory o arne, compeon and co-operaon. Quar. J. conom enberg,. M.,. Krhna, Z. J. Zhang. 00. Do we care wha oher ge? behavor approach o argeed promoon. J. Markeng Re Ghoh, M., G. John permenal evdence or agency model o aleorce compenaon. Markeng Sc Güh, W., M. Köngen, J. Kovac,. Zala-Mezo. 00. arne whn rm: The cae o one prncpal and mulple agen. Schmalenbach Bune Revew Hew ale compenaon urvey Harde, B. G. S.,. J. Johnon, P. S. ader Modelng lo averon and reerence dependence eec on brand choce. Markeng Sc Hauer, J. R., D. I. Smeer, B. Wernerel Cuomer aacon ncenve. Markeng Sc Ho, T.-H.,. Lm, C.. Camerer Modelng he pychology o conumer and rm behavor wh behavoral economc. J. Markeng Re

31 Ho, T.-H., X. Su Peer-nduced arne n Game. mer. conom. Rev Ho, T., J. Zhang Degnng prcng conrac or boundedly raonal cuomer: Doe he ramng o he ed ee maer? Managemen Sc Joeph, K., M. U. Kalwan The Role o Bonu Pay n Saleorce Compenaon Plan. Indural Markeng Managemen Joeph, K. and. Thevaranan Monorng and ncenve n ale organzaon: n agencyheorecal perpecve. Markeng Sc. 7, Kahneman, D., J. L. Knech, and R. H. Thaler arne a a Conran on Pro Seekng: nlemen n he Marke. In Choce, Value, and rame, eded by D. Kahneman and. Tverky, Kalra,., M. Sh. 00. Degnng opmal ale cone: heorecal perpecve. Markeng Sc ,. 00. Conumer value-mamzng weepake & cone: heorecal and epermenal nvegaon. J. Markeng Re. 47, ,, K. Srnvaan Saleorce compenaon cheme and conumer nerence. Managemen Sc Lal, R., V. Srnvaan Compenaon Plan or Sngle- and Mul-Produc Saleorce: n pplcaon o he Holmrom-Mlgrom Model. Managemen Sc Lal, R., R. Saeln Saleorce Compenaon Plan n nvronmen wh ymmerc Inormaon. Markeng Sc Lm,. 00. Socal Lo veron and Opmal Cone Degn. J. Markeng Re , M. J. hearne, S. H. Ham Degnng Sale Cone: Doe he Prze Srucure Maer? J. Markeng Re Marn, J. 98. Relave deprvaon: heory o drbuve nuce or an era o hrnkng reource. In Reearch n organzaonal behavor: n annual ere o analycal eay and crcal revew, vol. 3, ed. Larry L. Cummng and Barry M. Saw. Greenwch, CT: JI Pre. Mra, S. and H. ar. 00. rucural model o ale-orce compenaon dynamc: emaon and eld mplemenaon. Workng paper. Sanord Unvery. Orhun,. Y Opmal produc lne degn when conumer ehb choce e-dependen preerence. Markeng Sc Oyer, P heory o ale quoa wh lmed lably and ren harng. J. Labor conom Rabn, M Incorporang arne no game heory and economc. mer. conom. Rev Rau, J. S., V. Srnvaan Quoa-baed compenaon plan or mul-errory heerogeneou aleorce. Managemen Sc Ramawam, S.., J. Sngh neceden and Conequence o Mer Pay arne or Indural Salepeople. J. Markeng Sh, Mengze and Yupn Yang 009. eedback conrol durng ale cone. Workng paper. Unvery o Torono. Smh, J. C. 00. Pay growh, arne and ob aacon: Implcaon or nomnal and real wage rgdy. Workng Paper. Srnvaan, V. 98. n nvegaon o he equal common rae polcy or a mul-produc aleorce. Managemen Sc

32 Syam,., P. Krhnamurhy, J. D. He Tha' wha I hough I waned? Mwanng and regre or a andard good n a ma-cuomzed world. Markeng Sc Yang, Yupn, Mengze Sh, and v Goldarb 009, mang he Value o Brand llance n Proeonal Team Spor, Markeng Sc

33 3 ppend In h append we prove Lemma, Lemma, and Lemma 3. Propoon can be drecly nerred rom hee hree lemma. In he proo, ollowng BLSS 985, we analyze hree alernave uly peccaon correpondng o deren rae o change o rk olerance: Cae : Power uncon wh 0<< wh rae o change o rk olerance =/-δ >. In h cae, ] [ p B B Cae : Log uncon log wh rae o change o rk olerance =. In h cae, ] [ l B B Cae 3: ponenal uncon wh conan rk averon a - wh rae o change o rk olerance=0. In h cae, ] [ e B B L 3 Proo or Lemma : and B > 0. Cae : Power uly uncon I < 0 or, hen 0 < 0, whch conradc wh he aumed power uly uncon. rom quaon, we have 0 0 ] [ p B B 4 and p B B ] [ ] [ B B 5

34 To be conen wh BLSS, we dene he bae alary parameer a. In order o enure, we have. rom he aumpon >0, we have 0. Togeher wh 4 and 5, he condon o 0 mple ha. We hereore have. To prove B > 0, ake he r order dervave o wh repec o : p B B d d p B [ ] 6 or any B 0, we would have, whch conradc wh aumpon >0. Cae : Log uly uncon The opmal compenaon uncon under log-uly uncon,, can be regarded a a pecal cae o p wh = 0. ll he ep o he proo n Cae hold wh = 0. or nance, quaon 6 become d l d B B B [ ] 7 gan, or any B 0, we would have, whch conradc wh aumpon >0. Cae 3: ponenal uly uncon Snce he opmal compenaon uncon ha he propery ha L, he proo or h e l cae agan very mlar wh Cae. r, e 0 L B B [ ] 0 L and e L B B [ ] L 3

35 The condon o 0 mple ha. We hereore have. To prove B > 0, ake he r order dervave o e wh repec o : d d e l, 8 d d l d whch ha he ame gn a l gven n 7. Thu, he dervave n 8 wll alo be negave wh d B 0, becaue he r erm n 8 alway pove Q..D. Proo or Lemma. or any > 0, we have or mall and or large. urher, he opmal compenaon ha he upper bound equal o [ ],, and or power uly uncon, log-uly uncon, and eponenal uly uncon, repecvely. Cae : Power uly uncon rom quaon 6 we have: d p d 0 B 0 0 p 9 d p d B 0 p B B 0 umpon >0 and quaon 5 ogeher mply ha p or any 0. oe ha or =0, we have p whch conen wh he reul n BLSS. Cae : Log uly uncon n he proo or Lemma, we can regard he opmal compenaon uncon under log-uly uncon, l, a pecal cae o p wh = 0. gan, all he ep o he proo n Cae hold wh = 0. The 33

36 34 r-order dervave 7 become B/-β >0 when = 0, and converge o 0 when goe o nne. The upper bound mpled o /β-. Cae 3: ponenal uly uncon The proo or h cae ollow proo or Cae. Speccally, l l e l l e d d d d d d d d oe ha n above wo equaon, he r erm are alway ne. Speccally, l 0 he alary erm equal o /-β and l he pay upper bound a /β-, or he cae o log-uly uncon. Q..D. Proo or Lemma 3. or he cae o power uly uncon and log uly uncon, conve or mall. or he cae o eponenal uly uncon, conve or mall and only β>. Cae : Power uly uncon rom quaon 6 we can derve he econd-order dervave o wh repec o a below ' d d p p p B B B p 3 B B p, where B B and B. We hu have 3 ' B B d d p p p p >0 baed on Lemma and Lemma. Cae : Log uly uncon