Optimal incentive contracts under loss aversion and inequity aversion

Transcription

1 Fuzzy Optim Decis Mking Optiml incentive contrcts under loss version nd inequity version Chi Zhou 1 Jin Peng 2 Zhibing Liu 2 Binwei Dong 3 Springer Science+Business Medi, LLC, prt of Springer Nture 2018 Abstrct This pper studies model of principl-gent problem under loss version nd inequity version. The model nlyzes how loss version nd inequity version ffect the wge structure in optiml contrct design. The results demonstrte tht the presence of loss version would led to set of rising wge levels nd tht rnge of wge levels is wider if principl is more loss verse. In ddition, the principl s profit decreses in the principl s degree of loss version nd in the risk neutrl gent s degree of inequity version. Nevertheless, the wge growth of risk verse gent will be reduced. Furthermore, the incentive mechnism of non-contrctible effort will cuse higher wge growth thn the one of contrctible effort. The increse of relized profit level or the decrese of loss version level would led to too equitble lloctions for the risk neutrl gent. Under this incentive mechnism, n increse in the risk verse gent s concern for equity will be convergence towrds liner shring rules, while the principl who hs more sensitive to the loss my offer much lower wge level. B Chi Zhou czhou@tjut.edu.cn B Jin Peng pengjin01@tsinghu.org.cn Zhibing Liu lzb @hgnu.edu.cn Binwei Dong bwdong@tju.edu.cn 1 School of Mngement, Tinjin University of Technology, Tinjin , Chin 2 Institute of Uncertin Systems, Hunggng Norml University, Hunggng , Hubei, Chin 3 Institute of Systems Engineering, Tinjin University, Tinjin , Chin

2 C. Zhou et l. Keywords Uncertinty theory Contrct theory Morl hzrd Loss version Inequity version Subdifferentil 1 Introduction Principl-gent problems exist in ll wlks of socil life, such s economics, sociology nd mrketing. Once one prty uthorizes nother one to perform some tsks, principl-gent problem ppers between the two prticipnts. The former is clled principl, nd the ltter is gent. Generlly speking, there exists bounded rtionlity behvior nd other-regrding concerns in principl-gent problems. For exmple, person s well-being depends not only on his own profit, but lso on how his own profit compres to reference level. Recently, there hve been lot of interesting studies in bounded rtionlity behvior nd theoreticl frmeworks hve been developed to model other-regrding concerns tht include reciprocity, envy, nd concerns bout firness. However, very few reserches in existing literture study how contrct-theoretic predictions chnge when loss version nd inequity version re tken into ccount. We introduce loss version nd inequity version into the Englmier nd Wmbch (2010)) setting where principl hires n gent who, by his choice of effort, determines the uncertinty distribution of profits. The principl perceives the potentil profit in gins nd losses reltive to the reference level. Menwhile, the principl s behviors exhibit loss version, which mens losses loom lrger thn gins of the sme mgnitude (Mez nd Webb 2007), the gent s behviors often exhibit inequity version, which mens tht the gent cres bout how his wge compres to the firm s profits nd tht principl fers quits nd reduced effort when the wge pid is unfir (Dur nd Glzer 2008). Survey evidence lso shows tht 69% of the firms mngers tht were interviewed offer forml py structures becuse they cn crete internl equity. Most of those mngers (78%) view internl equity s n importnt fctor in keeping high morle nd hrmony in their firms. Also, 49% of the mngers view it s importnt in job performnce (Rey-Biel 2008). In ddition, the perceived gin or loss of the owner nd the perceived equity of the contrctor re vitl for projects success in Chin s construction mrket. For instnce, the owner worries bout getting ripped off by the contrctor which is dvntges in informtion nd professionl fter signing the contrct. The perceived of inequity of the contrctor, which cused by the owners writing lot of shift of responsibility cluses in contrct, hs influenced on coopertion. In order to study this problem, we construct n optiml contrct model in which the principl is ssumed to be loss verse nd the gent is ssumed to be inequity verse. The model cn be considered from perspective of contrctible or non-contrctible effort. To this end, we suppose the principl utility tht consists of instntneous utility nd gin-loss utility. Since the gin-loss utility function is nonsmooth, subdifferentil is introduced to study optiml contrct for the loss-verse principl. Furthermore, the gent s utility function includes n equitble lloctions-regrding concern. By deriving the Lgrngin nd the first order necessry conditions, the structures of wge level re discussed in optiml contrct model under loss version nd inequity version.

3 Optiml incentive contrcts under loss version The min findings of our pper re s follows. First, the optiml contrct under loss version exhibits set of wge schemes if effort is contrctible nd the gent is risk neutrl with income, nd the rnge of wge levels is wider when the principl is more loss verse. For the cse of contrctible effort nd risk neutrl gent, the principl s profit decreses in the principl s degree of loss version, nd lso decreses in the gent s degree of inequity version. Second, the wge growth will be reduced if effort is contrctible nd the gent is risk verse with income. Third, the optiml contrct under loss version is strictly incresing with high wge growth if effort is not contrctible nd the gent is risk neutrl with income. In ddition, the optiml contrct specifies more equitble distribution of overll profit s the relized profit level increses or the loss version level decreses for the risk neutrl cse. Finl, the optiml contrct under loss version is lso strictly incresing if effort is not contrctible nd the gent is risk verse with respect to income. Furthermore, n increse in the gent s concern for equity will cuse the wge levels converge to liner shring rules. An increse in the principl s loss version levels will decrese the wge level. Recent studies hve extensively nlyzed other-regrding concerns in the field of contrct theory. Fehr nd Schmidt (1999) introduced n importnt type of inequity version, which ssumed tht individuls hve desire to be fir under inequity version. Fehr nd Schmidt (2004) studied two-tsk principl-gent experiment in which only one tsk is contrctible. This behvior is consistent with theories of firness. Rey- Biel (2008) studied optiml contrcts in simple model where employees re verse to inequity, s modeled by Fehr nd Schmidt (1999). Dur nd Glzer (2008) estblished principl-gent model to study profit-mximizing contrcts when worker envies his employer. Envy tightens the worker s prticiption constrint nd so clls for higher py nd softer effort requirement. Gouksin nd Wn (2010) investigted n interply between workers in orgniztions under the ssumption tht workers exhibit behviorl bises: envy, jelousy, or dmirtion towrd the other coworkers compenstions. They ssumed workers cre bout their reltive position, nd studied the impct of this ssumption on their efforts nd on their optiml incentive contrcts. Rsch et l. (2012) discussed the strtegic behvior nd mechnism design under inequity version in brgining situtions with two-sided symmetric informtion. Englmier nd Wmbch (2010) nlyzed the clssic morl hzrd problem with the dditionl ssumption tht gents re inequity verse. And on this bsis Cto (2013) considered the principl-gent problems under inequity version. The results suggest tht the first-order pproch is restricted in the presence of inequity version. Expected utility theory, which ws formulted by Neumnn nd Morgenstern (1947), reigned for severl decdes s the dominnt normtive model of decision mking under uncertinty. It ws ssumed tht ll resonble people would wish to obey the xioms of the theory. These xioms re widely believed to represent the essence of rtionl behvior under uncertinty. However, it is well known tht mny people behve in boundedly rtionl wys tht systemticlly violte these xioms. To describe nd interpret such behvior, Khnemn nd Tversky (1979) hd proposed n lterntive descriptive model of economic behvior tht they clled prospect theory. The most prominent feture of prospect theory is loss version, tht is, people cre much more bout losses reltive to their reference level thn bout gins. A few recent reserches hve delt with the problem of incorporting loss version into con-

4 C. Zhou et l. trct theory. Mez nd Webb (2007) incorported loss version into principl-gent theory to explin the tendency for good performnce to be rewrded nd for poor performnce to go unpunished. Crbjl nd Ely (2012) studied optiml contrct design between monopolist nd continuum of potentil buyers under symmetric informtion nd reference-dependent preferences. In ddition, Deng et l. (2013) proposed principl-gent model, ssuming the retiler s degree of loss version to be symmetric informtion. Although the prospect theory is not yet common in the context of optiml contrct models, there re mny fields where it (especilly, loss version) hs been successfully pplied, for exmple, sset pricing (Popescu nd Wu 2007), mcroeconomic growth models (Foellmi et l. 2011), job serch models (Zhou et l. 2017) nd supply chin pricing decisions (Zhng et l. 2014). Following the models of Englmier nd Wmbch (2010) nd Cto (2013), we investigte the wge structure under loss version nd inequity version by designing optiml contrct. Moreover, we study the effects of loss version nd inequity version on wge level. The existing literture depicted the uncertin informtion in principl-gent problems by rndom vrible. However, describing the uncertinty s rndomness is not dequtely resonble. Therefore, new pproch bsed on the experts subjective judgement, clled uncertinty theory (Liu 2007), ws proposed. From then on, uncertinty theory hs plyed the importnt role of mthemticl model to del with humn uncertinty. Mny reserchers pply uncertinty theory to the principl-gent problems (Wu et l. 2014; Chen et l. 2018). For instnce, Mu et l. (2013) estblished n uncertin contrct model to study principl-gent reltionship between one enterprise nd one rurl migrnt worker. In ddition, Zhou et l. (2017b) studied dynmic recruitment problem with enterprise performnce in the uncertin environment nd proposed n optiml serch strtegy for the recruitment firm to serch or employ job pplicnt. If the principl hs historicl dt of the potentil profit per unit effort, then the profit my be rtionlly chrcterized s rndom vrible. A model cn be built to mximize the principls expected net profit, nd its deterministic equivlent model cn be obtined nturlly due to the probbility mesures specilty. Similrly, the optiml solution of the model under stochstic environment cn be obtined. However, in relity, there is no historicl dt for most of the cses. The effort the gent exerts not only directly ffects the profit, but lso indirectly ffects the profit. Moreover, the potentil profit per unit effort is often not esy to mesure directly. Actully, for the potentil profit without historicl dt, the principl cn invite some domin experts to evlute the belief degree, which represents the strength with which the principl nd the gent believe the indetermincy quntity tht ech profit my occur. And then, the uncertinty distributions of the unit outputs cn be estimted by uncertin sttistics, which is methodology for collecting nd interpreting experts experimentl dt by uncertinty theory. The reminder of this rticle is orgnized s follows. In Sect. 2, we present the principl-gent model. Section 3 discusses the optiml contrct model with contrctible effort. Section 4 studies the similr model with non-contrctible effort. Moreover, we nlyze the structure of optiml contrcts for the situtions where effort is contrctible nd non-contrctible. The conclusions re summrized in Sect. 5.

5 Optiml incentive contrcts under loss version 2 The model Consider principl-gent problem in which principl (he) hires n gent (she) to work for him. The principl is ssumed to be loss verse nd the gent is llowed to be inequity verse. He uthorizes her to perform tsk, nd then she exerts effort to perform the tsk with cost. However, the potentil profit per unit effort is unknown for the two prticipnts, becuse it cnnot be predicted exctly in dvnce. Assume tht the profit x is continuously distributed in compct intervl, b] with uncertinty distribution F(x e) when the gent exerts effort level e, nd let = df(x e)/dx. The wge w(x) is pid to the gent fter obtining the profit. As the principl is loss verse, his expected net profit U P = (1 α)(x w(x)) + αv(x 2w(x)) ] df(x e), (1) where the proportion prmeter α 0, 1] nd v ( x 2w(x) ) = { x 2w(x), if x 2w(x) 0 φ(x 2w(x)), if x 2w(x) <0, (2) with the loss version prmeter φ>1. In Eq. (1), the component x w(x) is the principl s instntneous utility, nd the component v ( x 2w(x) ) is the gin-loss utility function which is determined by the discrepncy between net profit x w(x) nd reference point w(x). Therefore, the reference point ffects principl s behvior vi the mgnitude of the perceived gin or loss reltive to the reference point. Figure 1 shows tht the gin-loss utility function is piecewise-liner pproximtion of Khnemn nd Tversky (1979) s kinked power utility function which weighs losses hevier thn gins. The core of prospect theory for our purpose is the notion of loss version nd the existence of reference point. Hence, to cpture the symmetry between gins nd losses, this simple piecewise-liner function entils the essentil feture of loss version (Foellmi et l. 2011; Su2009). Fig. 1 An exmple of kinked gin loss utility function v(z) + = = φ

6 C. Zhou et l. Empiricl evidence suggests tht the lrger the perceived gin/loss is, the lrger the corresponding bsolute impct on profit is. In ddition, the ssumption bout the proportion prmeter α cn derive the following remrk. And to some extent the proportion prmeter represents the principl s degree of loss version. For convenience, both loss version prmeter φ nd proportion prmeter α re known s the loss version levels. Remrk 1 If the prmeter α = 0, the gin-loss utility hs no effect on the expected net profit. This implies tht the proposed model is virtully degenerted into the originl model of Englmier nd Wmbch (2010). If the prmeter α = 1, the profit utility is fully equl to the gin-loss utility. Moreover, ssume tht the gent s utility function hs three prts: her benefit u(w(x)) from her wge, n equitble lloctions-regrding concern βg(x 2w(x)) nd her cost c(e) from exerting effort. Therefore, the gent s expected utility is expressed s follows. U A = u(w(x)) c(e) βg(x 2w(x)) (3) with G (x 2w(x)) > 0, if x w(x) >w(x), (3b) G (x 2w(x)) < 0, if x w(x) <w(x), G ( ) >0, G(0) = 0, G (0) = 0, (3c) where β > 0 represents the degree of inequity version of the gent, u (w) > 0, u (w) 0, c (e) >0 nd c (e) 0. This model is in line with Fehr nd Schmidt (1999) s model. The convexity of G( ) implies n version towrds lotteries over different levels of inequity. The reservtion utility level U mens tht the principl hs to obey the gent s prticiption constrint U A > U. Similr to the monotone ( fe(x e) likelihood rtio property, we lso ssume x > 0. This implies tht higher the reliztion of profit the more likely it is tht high effort is exerted. As shown by Whitt (1980), the monotone likelihood rtio property lso implies tht F e (x e) <0 (uncertin dominnce condition). Furthermore, the gent cn secretly boost or destroy profits. This ensures tht the slope of the optiml contrct is bounded between 0 w (x) 1. ) 3 Optiml contrct with contrctible effort In this section, the optiml contrct model is investigted where the principl cn contrct on effort, i.e., there is no morl hzrd problem. In this problem, the principl mximizes his expected profit net of wge pyments nd hs to obey only the gent s prticiption constrint (PC). Therefore, the principl s decision problem cn be written s follows, mx U P = e,w(x) s.t. U A = (1 α)(x w(x)) + αv(x 2w(x)) ] df(x e) u(w(x)) βg(x 2w(x)) ] df(x e) c(e) U. (PC)

7 Optiml incentive contrcts under loss version To exmine the effect of loss version nd inequity version on optiml contrct, we first ssume tht the gent is risk neutrl with respect to income, i.e., u(w(x)) = kw(x), k > 0. Since v(x 2w(x)) is piecewise nd continuous utility function which weights losses in profit hevier thn gins, it is pprent tht v(z) is nonsmooth t z = x 2w(x) = 0. To study the effects of the symmetry between gins nd losses on the profit, the subdifferentil of v(z) t z = 0 is defined s the set 1,φ], where 1 nd φ re the one-sided limits 1 = lim z 0 + Tht is, v(z) z nd φ = lim z 0 v(z), respectively. z v(x 2w(x)) = 1,φ]. (4) By introducing loss version levels, the contrct structure is chnged s follows. Proposition 1 If effort is contrctible nd the gent is risk neutrl with income, the optiml contrct under loss version is w(x) = x g ( 1 + α λk 2λβ ), g where g( ) represents the inverse function of G ( ). Proof The principl s problem is given by ( 1 α + 2φα λk 2λβ )], (5) mx U P = e,w(x) s.t. U A = (1 α)(x w(x)) + αv(x 2w(x)) ] dx u(w(x)) βg(x 2w(x)) ] dx c(e) U. (PC) The Lgrngin for the principl s problem is L = (1 α)(x w(x)) + αv(x 2w(x)) ] dx λ U u(w(x)) βg(x 2w(x)) ] ] dx + c(e), where λ is the lgrnge multiplier. Then, the first order condition is s follows. L 2w(x)) = (1 α) + α v(x + λu (w(x)) w(x) w(x) + 2λβ G (x 2w(x)) = 0. Dividing by nd rerrnging yields v(x 2w(x)) α 1 + α + λu (w(x)) + 2λβG (x 2w(x)) = 0. w(x)

8 = 2 v(x 2w(x)) = 2φ, 2]. Further- By Eq. (4), we hve more, v(x 2w(x)) w(x) C. Zhou et l. 1 + α 2φα, 1 α] + λu (w(x)) + 2λβG (x 2w(x)) = 0. Thus, it follows from u (w(x)) = k tht ] 1 + α λk 1 α + 2φα λk, = G (x 2w(x)). 2λβ 2λβ Since G ( ) is monotonous due to G ( ) >0, it is cler tht g ( 1 + α λk 2λβ ), g Hence, we hve w(x) = x ( )] 1 α + 2φα λk = x 2w(x). 2λβ g ( ) 1+α λk 2λβ, g ( )] 1 α+2φα λk 2λβ. Proposition 1 shows tht optiml contrct under loss version exhibits set of wge schemes. For higher loss version level α or φ, the rnge of wge schemes should be wider. However, n increse in β will increse the wge level. If φ pproches one, which corresponds to the cse without loss version, the wge level will only be constnt. The results of Proposition 1 re illustrted with n exmple in Fig. 2. We first consider tht the inverse function g(x) = 3x, α = 2 1, β = 2 1, k = 1, nd φ = 1.5. In ddition, we lso ssume the inverse function g(x) = x 3 nd the loss version prmeter φ cn be chosen in the set {1.5, 2, 2.5}. FromFig.2, we find tht optiml contrct under loss version is not unique wge level, but set of wge schemes. Menwhile, n increse in the principl s loss version will negtively enlrge rnge of wge schemes, nd while n increse in the gent s inequity version will rise the levels of wge schemes. This implies tht optiml contrct under loss version mkes wge pyments more flexible, nd provides the principl more choices. When the principl is more loss verse, he is more likely to lower the gent s wge level. And s the gent s inequity version is incresing, she wnts higher wge level to improve the fir-feeling. By tking the prtil derivtive of Eq. (1) with respect to α, we simply derive the following proposition. Proposition 2 With contrctible effort nd risk neutrl gent, the principl s profit decreses in the principl s degree of loss version, α, nd decreses in the gent s degree of inequity version, β. To enrich our model, we introduce risk version for the gent. Proposition 3 If effort is contrctible nd the gent is risk verse with income, the slope of optiml contrct under loss version is 0 <w (x) < 1 2.

9 Optiml incentive contrcts under loss version () w(x)=x/ w(x)=x/ w(x) x (b) w(x)=x/2 1/2 w(x)=x/2 27/16 w(x)=x/ w(x) x Fig. 2 Optiml contrct with contrctible effort nd risk neutrl gent. Cse g(x) = 3x, b cse g(x) = x 3 Proof From Proposition 1, the Lgrngin nd the first order condition cn be rewritten s 1 + α 2φα, 1 α] + λu (w(x)) + 2λβG (x 2w(x)) = 0. Tking the derivtive of this with respect to x yields u (w(x))w (x) + 2βG ( )(1 2w (x)) = 0.

10 C. Zhou et l. Moreover, we hve w (x) = 2βG ( ) 4βG ( ) u (w(x)). Note tht w (x) >0 due to G ( ) >0 nd u (w(x)) < 0. Furthermore, w (x) = u (w(x)) 8βG ( ) 2u (w(x)), u where (w(x)) 8βG ( ) 2u (w(x)) < 0. Therefore, 0 <w (x) < 2 1 holds. In the presence of inequity version we lwys observe some profit shring, even if it is not necessry for incentive resons or when profits re not good performnce mesure. While the presence of loss version will led to rnge of wge levels. From Proposition 3, we hve tht the presence of risk version will reduce the wge growth. 4 Optiml contrct with non-contrctible effort In this section, we ssume tht effort cnnot be contrcted. Thus the problem becomes morl hzrd problem. The principl lso mximizes his expected profit net of wge pyments nd hs to obey the gent s prticiption constrint (PC) nd incentive comptibility constrint (IC). This incentive comptibility constrint hs the form e rg mx U A = ẽ u(w(x)) βg(x 2w(x)) ] df(x ẽ) c(ẽ) (IC) nd indictes tht the effort level is optiml for the gent. Then, the principl s problem of finding n optiml contrct cn be formulted s mx U P = w(x) s.t. U A = b e rg mx U A = ẽ (1 α)(x w(x)) + αv(x 2w(x)) ] df(x e) u(w(x)) βg(x 2w(x)) ] df(x e) c(e) U, (PC) u(w(x)) βg(x 2w(x)) ] df(x ẽ) c(ẽ). (IC) In order to solve this problem, we replce the bove mximiztion problem by its first order condition u(w(x)) βg(x 2w(x)) ] dfe (x e) c (e) = 0. (IC ) The cse of risk neutrl gent is lso considered first, i.e., u(w(x)) = kw(x). The stndrd contrcting model (with non-inequity verse gent nd non-loss verse principl) shows tht the principl offers wge contrct with slope one, mking the

11 Optiml incentive contrcts under loss version gent residul climnt of ll ccruing profits. As the gent is risk neutrl with respect to income, nd effort is not contrctible, the optiml contrct is strictly incresing with slope between 1/2 nd 1. However, the solution in the cse with risk neutrl but loss verse nd inequity verse gent looks different. Proposition 4 If effort is not contrctible nd the gent is risk neutrl with income, the optiml contrct under loss version is strictly incresing with slope where to x. w (x) = 1 μ 2 + ( ( ) fe (x e) λ + μ f e(x e) 1 + α, 1 + 2φα α] ) 2 4βG, (6) (x 2w(x)) ( ) fe (x e) represents the derivtive of the likelihood rtio f e (x e) with respect Proof The optimiztion problem is given by mx U P = w(x) s.t. U A = b (1 α)(x w(x)) + αv(x 2w(x)) ] dx u(w(x)) βg(x 2w(x)) ] dx c(e) U, f e (x e) u(w(x)) βg(x 2w(x)) ] dx c (e) = 0. (IC ) And the lgrngin tkes the form (PC) L = (1 α)(x w(x)) + αv(x 2w(x)) ] dx λ U u(w(x)) βg(x 2w(x)) ] ] dx + c(e) μ 0 f e (x e) u(w(x)) βg(x 2w(x)) ] ] dx + c (e), (7) where λ nd μ re the lgrnge multipliers. Therefore, the first order condition cn be divided by nd rewritten s λ + μ f ] e(x e) k + 2βG (x 2w(x)) ] α 2φα, 1 α] =0. (8) By tking the derivtive of Eq. (8) with respect to x, μ 2βG (x 2w(x))(1 2w (x)) + ( ( ) fe (x e) λ + μ f e(x e) ) k + 2βG (x 2w(x)) ] = 0. (9)

12 C. Zhou et l. ( ) Since fe (x e) > 0, it holds tht where ll terms but k+2βg ( (x 2w(x)) λ+μ fe(x e) Thus we get w (x) = 1 μ 2 + ( ( ) fe (x e) λ + μ f e(x e) k+2βg ( (x 2w(x)) λ+μ fe(x e) ) k + 2βG (x 2w(x)) 4βG, (10) (x 2w(x)) ) re obviously positive. To ensure tht ) is lso positive, we consider the bove first order condition. λ + μ f ] e(x e) k + 2βG (x 2w(x)) ] =1 + α, 1 α + 2φα]. (11) k + 2βG (x 2w(x)) λ + μ f e(x e) Substituting Eq. (12) into Eq. (10) yields w (x) = 1 μ 2 + ( ( ) fe (x e) λ + μ f e(x e) 1 + α, 1 α + 2φα] = ( ) 2. (12) λ + μ f e(x e) 1 + α, 1 + 2φα α] ) 2 4βG. (x 2w(x)) ( ) Since the both terms fe (x e) nd G (x 2w(x)) re strictly positive, the optiml contrct in Proposition 4 is strictly incresing with slope between 1/2 nd 1. And if effort is not contrctible nd the gent is risk neutrl with income, the slope of optiml contrct under loss version exhibits rnge of vrition. An increse in the loss version level α or φ will expnd the scope of the slope, while higher concern for equity β will reduce the scope of the slope. To further chrcterize the optiml contrct, we provide property by nlyzing how the contrct vries with x. Proposition 5 If gent is risk neutrl in income, the optiml contrct specifies more equitble distribution of overll profit s the relized profit level increses or the loss version level decreses. Proof Note tht the first order condition cn be rewritten s ] G (x 2w(x)) = α, 1 α + 2φα] 2β λ + μ f k. (13) e(x e)

13 Optiml incentive contrcts under loss version If x increses, then f e(x e) increses due to the monotone likelihood rtio property. Therefore, the right hnd side of this first order condition decreses. Thus the bsolute vlue of G ( ) decreses, in turn implying less inequity. If loss version prmeter φ or proportion prmeter α decreses, this lso implies less inequity. As noted bove in the stndrd model with risk neutrl gent, it is optiml to implement the full informtion effort level. Under inequity version this is not possible s the slope of the wge contrct cn provide incentives nd insure ginst vrying degrees of inequity. If the principl gve higher powered incentives nd perceived loss version, these would led to too inequitble lloctions for which the gent would hve to be compensted up front. Now we llow for risk version in the gent s preferences. In this cse, the structure of the wge contrct is lso given s follows. Proposition 6 If effort is not contrctible nd the gent is risk verse with respect to income, the optiml contrct under loss version is strictly incresing. Proof As in the proof of Proposition 4, the principl s problem, the Lgrngin nd the first order condition re derived. Thus the first order condition is λ + μ f ] e(x e) u (w(x)) + 2βG (x 2w(x)) ] =1 + α, 1 α + 2φα]. Differentiting this eqution with respect to x yields ] λ + μ f e(x e) u (w(x))w (x) + 2βG (x 2w(x))(1 2w (x)) ] ( ) + μ fe (x e) u (w(x)) + 2βG (x 2w(x)) ] = 0. Therefore, the bove eqution cn be rerrnged to w (x) = 2βG (x 2w(x)) 4βG (x 2w(x)) u (w(x)) ( ) + μ fe(x e) 2βG (x 2w(x))+u (w(x)) λ+μ fe(x e) 4βG (x 2w(x)) u (w(x)), ( ) where G ( ) >0, u (w(x)) 0 nd fe (x e) > 0. As ll terms re strictly positive, it holds tht w (x) >0. Furthermore, we investigte how the gent s concern for equity nd the principl s concern for loss ffect the wge contrct. It implies tht the concerns for equity nd loss become the min driving force for the structure of the wge contrct. Corollry 1 If the gent s concern for equity β increses, the optiml contrct converges to w(x) = 2 1 x. If the principl s loss version prmeter φ nd proportion prmeter α increse, the optiml contrct is strictly decresing if w(x) > 2 1 x.

14 C. Zhou et l. Proof The first order condition from the proof of Proposition 4 cn be rewritten s 1 + α, 1 α + 2φα] λ + μ f e(x e) = u (w(x)) + 2βG (x 2w(x)). Recll tht G (x 2w(x)) > 0, if x w(x) >w(x), nd G (x 2w(x)) < 0, if x w(x) <w(x). The first order condition cn be hold in ny point t finite intervl. The left hnd side is constnt with given x.forx 2w(x) >0, i.e., w(x) < 2 1 x,we hve G (x 2w(x)) > 0. An increse in β will increse G (x 2w(x)) nd decrese u (w(x)). Thus w(x) is incresing in x.forx 2w(x) <0, i.e., w(x) > 2 1 x, we obtin G (x 2w(x)) < 0. Therefore, n increse in β will lso increse G (x 2w(x)) nd increse u (w(x)). Thus w(x) is decresing in x. Note tht the prticiption constrint is u(w(x)) βg(x 2w(x)) ] dx c(e) U. Dividing the bove expression by β, we obtin ] u(w(x)) G(x 2w(x)) dx c(e) β β U β. When β, G(x 2w(x))] dx 0. Thus this inequity cn be stisfied if G( ) = 0, which implies tht w(x) = 2 1 x. If the loss version level α or φ increses, the left hnd side is lso incresing. Therefore, the term u (w(x)) of the right hnd side is lso incresing if w(x) > 2 1 x. Since u (w(x)) is decresing in w(x), it holds tht w(x) is strictly decresing. The results of this corollry imply tht n increse in inequity version will led to liner shring rules. Furthermore, the principl who hs more sensitive to the loss will offer much lower wge level. From Fig. 3, the rnge of the wge levels is provided to illustrte the bove results when the function G (x) = x 3, u (w) = 2 w 2, (0 <w< 1), α = 2 1, β = 2 1, nd the loss version index φ cn be chosen in the set {1.5, 2, 2.5}. Moreover, Fig. 4 shows how the proportion prmeter ffects the rnge of wge levels, when the loss version index φ = 1.5 nd the proportion prmeter α cn be chosen in the set {0.5, 0.8}. Figures 3 nd 4 present the numericl computtion of these effects on the rnge of wge levels, when loss version levels φ nd α increse. This implies tht the rnge of wge levels is negtively enlrging s loss version prmeter φ is incresing, nd the rnge of wge levels is overll descending with the increse of loss version proportion prmeter α. Tht is, the more loss verse the principl is, the lower wge level is offered. In prticulr, the proportion prmeter α positively ffects

15 Optiml incentive contrcts under loss version φ=1 φ=1.5 φ=2 φ= w(x) Fig. 3 The effect of φ on the rnge of wge levels x α=0.5 α= w(x) x Fig. 4 The effect of α on the rnge of wge levels the proportion of gin-loss utility. While the proportion of gin-loss utility is going up, the principl is lowering the rnge of wge levels in order to ensure profitbility. Towrds better understnding of the optiml contrct model under loss version nd inequity version, we mke comprison between the proposed model nd the originl models s follows. The originl models only ssumed tht the gent is inequity verse, nd the principl is neither risk verse nor inequity verse (Englmier nd Wmbch 2010; Cto 2013), nd exmined the effects of inequity version on the structure of optiml contrcts. Different from them, this proposed model ssumes tht the principl is loss verse nd the gent is inequity verse, nd nlyzes the effects

16 C. Zhou et l. () the originl models the proposed model 2 w(x) x (b) w(x) the originl models the proposed model x Fig. 5 Comprison between the proposed model nd the originl models. Cse of contrctible effort, b cse of non-contrctible effort of loss version nd inequity version on the optiml contrcts. From perspective of contrctible or non-contrctible effort, we further discuss the wge structure nd the incentive mechnism. In fct, the optiml contrct model under loss version nd inequity version is nlyzed by the pplying the subdifferentil, since the gin-loss utility function is nonsmooth in the proposed model. For the cse of contrctible effort, Fig. 5 presents the numericl computtion of this comprison between the proposed model nd the originl models. In this numericl exmple, the inverse function is g(x) = 3x, β = 2 1, k = 1, φ = 1.5, the proportion prmeter α = 2 1 in the proposed model nd the proportion prmeter α = 0in the originl models. For the cse of non-contrctible effort, Fig. 5b lso presents the

17 Optiml incentive contrcts under loss version numericl computtion of tht comprison. The numericl exmple considers tht the function G (x) = x 3, u (w) = 2 w 2, (0 <w<1), β = 2 1, φ = 1.5, nd the proportion prmeter α cn be chosen in the set {0, 0.5}. The results show tht the presence of loss version would led to set of rising wge levels, nd the wge level in the proposed model could be lower thn tht in the originl models. In ddition, the incentive mechnism of non-contrctible effort cse cuses higher wge growth thn the one of contrctible effort cse. 5 Conclusion The min contribution in this pper is presented in incorporting loss version nd inequity version into the optiml contrct model. Therefore, we develop model of morl hzrd problem in the presence of loss version nd inequity version, in which the principl is loss verse nd the gent is inequity verse. We further derive the implictions for wge structure from optiml contrct model with contrctible nd non-contrctible effort. With contrctible effort nd risk neutrl gent, the results demonstrte tht the optiml contrct under loss version exhibits set of wge levels, nd the rnge of wge level is wider when the principl is more loss verse. Furthermore, the principl s profit decreses in the principl s degree of loss version, nd decreses in the gent s degree of inequity version. If the gent is risk verse with income, the wge growth will be reduced. With non-contrctible effort nd risk neutrl gent, the optiml contrct under loss version is strictly incresing with high wge growth. Moreover, s the relized profit level increses or the loss version level decreses, there leds to too equitble lloctions. If the gent is risk verse with income, the wge structure will not chnge. Furthermore, the optiml contrct converges to liner shring rules s the gent s concern for equity increses. And the optiml contrct will be strictly decresing s the principl s loss version levels increse. In the future work, we cn incorporte competition into contrct design in the presence of loss version. For instnce, principl will offer wge contrct to motivte gents nd induce the competition mong gents. In ddition, it is lso interesting to study dynmic decision in the optiml incentive contrcts under loss version nd inequity version. Acknowledgements This work is supported by the Ntionl Nturl Science Foundtion of Chin (Nos nd ), Humnity nd Socil Science Youth Foundtion of Ministry of Eduction of Chin (No. 17YJC630232), nd the Chin Postdoctorl Science Foundtion (No. 2017M610160). References Crbjl, J. C., & Ely, J. (2012). Optiml contrcts for loss verse consumers. Technicl report, University of Queenslnd, School of Economics. Cto, S. (2013). The first-order pproch to the principl-gent problems under inequlity version. Opertions Reserch Letters, 41(5), Chen, Z., Ln, Y., & Zho, R. (2018). Impcts of risk ttitude nd outside option on compenstion contrcts under different informtion structures. Fuzzy Optimiztion nd Decision Mking, 17(1),

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