Paying Gig Workers. Patrick Kampkötter University of Tübingen Dirk Sliwka University of Cologne, CESifo and IZA. Preliminary version

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1 Sebastia Butschek Uiversity of Cologe Payig Gig Workers Patrick Kampkötter Uiversity of Tübige Dirk Sliwka Uiversity of Cologe, CESifo ad IZA Prelimiary versio November 7, 06 Abstract We study the compesatio of gig workers i a atural field experimet. To derive testable predictios, this paper presets a formal model capturig a cetral feature of olie freelace work: gig workers ability to choose both how much to work ad how big a effort to make. We aalyse the set-up i a pricipal-aget model, showig that the optimal cotract icludes a sales-based commissio ad uses a gig-based piece rate to isure a risk-averse aget. This piece rate is icreasig i her risk aversio, itrisic motivatio ad ability. We the predict the effects of itroducig a gig piece rate while reducig the commissio rate. The effects o the agets choices of quatity ad quality are heterogeeous i their risk aversio, itrisic motivatio ad ability. Key Words: Icetives, Risk Aversio, Itrisic Motivatio, Sales Compesatio, Multitaskig, Field Experimet JEL Classificatio: M5, J33, D3 sebastia.butschek@wiso.ui-koel.de patrick.kampkoetter@ui-tuebige.de dirk.sliwka@ui-koel.de.

2 Itroductio The rise of the o-demad ecoomy has see a proliferatio of firms that rely o a workforce composed of freelacers rather tha regular employees. May of these firms operate virtual platforms o which freelacers are matched with customers. There is cosiderable variatio betwee platforms i terms of the work parameters freelacers get to set themselves. A aspect emphasised by may platforms, however, is that freelacers set their ow schedules, decidig how much ad whe to work. Aother characteristic shared by may platforms is that compesatio is purely output-based. I other words, gig workers compesatio is ofte a direct fuctio of the success of the gig. As a cosequece, gig workers typically face substatially higher icome ucertaity tha regular employees while potetially ejoyig greater flexibility i their work arragemets. I this project we study compesatio cotracts for freelacers i the o-demad ecoomy. To do so, we first describe a formal model that allows us to aalyse differet potetial drivers of freelacers quatity ad quality choices. To empirically test the most importat hypotheses derived from the theory, we will coduct a field experimet o a olie platform i a secod step. I the curret versio of this paper we preset the aalysis of the formal model. The field experimet will be carried out i collaboratio with a olie platform ru by a retail firm. That platform acts as a itermediary betwee cliets 3 ad gig workers, who provide remote shoppig advice. Their service may result i the olie sale of physical goods, which is hadled by the platform. Gig workers decide how much they wat to work: they set the quatity of slots they make available for cliet cosultatios. Their efforts also determie the quality achieved: the usefuless of their advice affects the sales to each cliet. At the outset, gig workers compesatio is commissio- Whether or ot cotractually defiig the status of o-demad workers as that of a idepedet cotractor rather tha a employee is legally valid give the relevat jurisdictio s labour law is beig questioed by legal scholars - see, e.g., Prassl 07. A recet article i The Ecoomist discusses the example of Uber s drivers The Ecoomist Prit Editio, November We will use the words "customer" ad "cliet" iterchageably i this paper.

3 based, payig them a fractio of et sales to the cliets they advised. This is the status quo agaist which we will test a itervetio that chages the compesatio scheme. The goal of the theoretical aalysis preseted here is to provide ituitio about the mechaisms that might be at play, to iform the desig of the optimal compesatio policy for gig workers, ad to geerate testable predictios for the field experimet. We first aalyse a pricipal-aget model that captures essetial features of the relatioship betwee a platform ad a gig worker. That is, we cosider a pricipal desigig a cotract to motivate a aget who ot oly determies the quality of her work o all gigs but also how much labour iput to provide how may gigs to offer. We first characterise the optimal cotract ad show that optimal pay makes use of a sales-based commissio but also isures a risk-averse aget through a gig-based piece rate order bous. That is, the pricipal optimally icludes a paymet i the cotract that depeds oly o the umber of gigs. This is i cotrast to purely commissio-based pay optimal for a risk-eutral aget. The key idea here is that a pure commissio rate iduces icetives to provide too little labour iput whe the aget is risk averse as the payoff from each gig is ucertai. A order bous, however, provides stroger icetives to icrease labour supply. Secod, we aalyse the heterogeeity i agets reactios to the cotract with respect to risk aversio, ability ad itrisic motivatio. Whe modellig itrisic motivatio we allow for both coscietiousess 4 ad task ejoymet as potetial drivers of a desire to work ad to do a good job. We show that quality is icreasig i a aget s itrisic motivatio ad that a itrisically motivated aget s quality choice respods less strogly to the commissio rate. Fially, we study a specific applicatio of our model that allows us to derive predictios for the field experimet. We cosider a pricipal that iitially pays her aget a pure commissio without a order bous. The above result suggests that if the aget is somewhat risk averse the platform should 4 We use coscietiousess i the sese of the Big Five persoality trait, measurig the extet to which a aget is drive by a sese of duty whe performig a task. 3

4 chage its compesatio policy ad itroduce a order bous. We formally aalyse a hypothetical experimet where - i expectatio - the itroductio of a order bous is paid for by the reductio i the commissio rate. That is, order bous ad commissio are calibrated o a populatio of agets i such a way that the average aget s pay per order remais costat if agets do ot adjust quality. We the show that such a move from a pure commissio to a combiatio of a commissio ad a order bous leads to a icrease i average quatity which will be more proouced for more risk averse agets ad less proouced for more able ad more itrisically motivated agets. The shift i compesatio will reduce quality but this decrease will be less proouced the higher the aget s itrisic motivatio. Fially, for a shift of ay give size, profits will icrease if ad oly if the aget is suffi cietly risk averse. The Model Our framework builds o a multi-taskig model i the spirit of Holmström ad Milgrom 99. Cosider a aget who works for a pricipal, providig a service to customers. The aget chooses the umber of cliet orders to fulfil [0; ] ad the average service quality q [0; q]. The aget has potetially imperfectly kow ability a N m, σ a with m > 0. She has covex costs of effort c q, where c qq > 0, c qqq 0, c > 0 ad c q > 0 such that the margial cost of q icreases whe goes up - providig a give level of quality o more orders requires more effort. We also assume that the margial average cost of quality per order is weakly decreasig i the umber of orders: q c q, 0 which is equivalet to c q q, c q q,. followig coditios that guaratee iteral solutios: Moreover, we impose the c q 0, 0, c q, 0 0 q, lim q q c q q,, lim c q, q ad 4

5 c c qq + c q > q,. 5 Note that, i additio to the direct disutility of effort, the cost fuctio ca accommodate several behavioural compoets. For example, the aget may be itrisically motivated for the task ad thus, for istace, may have egative margial costs of effort up to a poit or may have psychological costs of ot providig a appropriate quality give the quatity she has chose. We will later o cosider a specific example to explore this possibility. Whe the aget fulfils orders, she geerates a level of sales S a + q + i i where i N 0, σ. The aget has a outside optio that yields a reservatio value w A > 0 with certaity. We allow for the possibility that the aget is risk averse with costat absolute risk aversio, where her Arrow- Pratt measure of absolute risk aversio is r. Both total sales S ad the umber of orders worked are verifiable ad we cosider liear cotracts that pay a wage w α + β + γ S, where β 0 is a order bous, i.e., a order-based piece rate that does ot deped o quality, ad γ [0, ] is a commissio rate. 3 Aalysis 3. Characterizig Optimal Cotracts The aget s objective fuctio is EU [ α + β + γ a + q + ] i c q,. 5 The first four coditios are Iada-type coditios that also allow for egative margial costs at small levels of effort. The last coditio guaratees the cocavity of the aget s objective fuctio. i 5

6 The variace of the aget s compesatio is V [ α + β + γ a + q + ] i γ σ a + σ. i As the aget exhibits costat absolute risk aversio see, for istace, Milgrom ad Roberts 99, Wolfstetter 00 for details, she maximises max α + β + γ m + q c q,,q rγ σ a + σ with first order coditios 6 β + γ m + q c q, rγ σ a + σ 0 IC ad γ c q q, 0. IC To determie the optimal cotract the pricipal maximises her expected profits max γ m + q β α α,β,γ,,q subject to the icetive compatibility costraits IC ad IC ad the aget s participatio costrait α + β + γ m + q c q, rγ σ a + σ wa. As the aget s participatio costrait must be bidig otherwise profits could be icreased by reducig α without violatig the icetive compatibility costraits we ca substitute α from the bidig participatio costrait. The pricipal thus maximises m + q c q, rγ σ a + σ subject to the icetive compatibility costraits. We first characterise the optimal cotract uder risk eutrality r 0: 6 See the Appedix for a proof of the cocavity of the objective fuctio. 6

7 Propositio If the aget is risk eutral r 0 the optimal cotract ever etails a strictly positive order bous, i.e. β 0, ad the commissio rate is γ. Proof: See Appedix. Ituitively, the order bous β provides icetives oly for quatity while the commissio rate γ provides udistorted icetives for both quality ad quatity: uder a commissio rate the aget s pay is a liear trasformatio of the pricipal s profits. Itroducig a order bous distorts the aget s decisio favourig quatity see, for istace, Feltham ad Xie 994 ad Schedler 008 for aalyses of performace measure cogruece i multitaskig models. However, this picture may chage whe the aget is risk averse. The reaso is that a commissio cotract imposes icome risk o the aget. A aget s risk aversio lowers her margial retur to quatity as each additioal order comes with a icome risk while risk aversio does ot affect the returs to extra quality provisio. I other words, risk aversio distorts the icetive effects of the commissio, iterferig with its ability to provide appropriate quatity icetives to risk averse agets. Here, a order bous may become effective as it geerates icetives to provide quatity without imposig risk o the aget. Ideed, we ca show that uder risk aversio the optimal cotract etails both a lower commissio rate ad a strictly positive order bous: Propositio If the aget is risk averse, the optimal cotract always icludes a order bous β > 0 ad a commissio rate γ <. Proof: See Appedix. 3. Itrisic Motivatio We ow impose more structure o the cost fuctio i order to study comparative statics with respect to behavioural determiats of the aget s effort 7

8 reactio. Cosider the followig specific cost fuctio: κ c q, q η τ q q + ν with η, κ, η [0, [ ad τ [ 0, q ]. If τ η 0, the aget is purely extrisically motivated. I this case, the margial cost of fulfillig aother order icreasig as well the margial cost of providig more quality per order icreasig q is strictly icreasig. Moreover, the cost of providig a quality level q o each order is liearly icreasig i the umber of order i.e, for simplicity we igore potetial learig effects here. If, however, η > 0, the cost fuctio captures two behavioural motives for doig more ad better work. To see the role the parameters τ ad η play cosider the itrisic beefit from completig a order η τ q q. The parameter η measures the aget s overall itrisic motivatio to complete orders, τ measures task ejoymet ad q is a level of quality that is optimal from a welfare perspective. 7 A aget with a higher η has a stroger icetive to choose a quality level that is close to the ormatively optimal level. For simplicity we assume that q is equal to the first-best quality. 8 If τ 0 the η τ q q < 0; oce a aget has decided to complete a order, the itrisic motivatio to provide higher quality puts a burde o the aget. Such a pealty for ot doig a good job may be icurred by a idividual that does ot ejoy the task but is drive by a sese of duty, a feelig of obligatio, or by coscietiousess. If, however, τ > 0, her itrisic motivatio may give pleasure to the aget. I fact, if τ q the η τ q q > 0 for all q 0, q. I this case the aget ejoys workig for o a cliet order while of course still tradig off the fu of workig with the effort costs captured i the first term i. 7 Note that we have a settig i mid where the aget picks a set of items that is set to customers via mail. The customers ca the decide which items to keep ad ejoy free returs of the items they do ot wat. As the firm icurs costs without earig aythig o all retured items its ideal aget oly selects items the customer wats to keep. I other words, i our settig the firm s objective fuctio is closely aliged with the customer s iterests. 8 We thus have that, q arg max,q m + q κ q η τ q q + ν which yields quality level q. κ 8

9 We first characterise a aget s reactio to a cotract with a commissio rate γ [0, ] ad a order bous β 0. We substitute the margial costs ito the icetive compatibility coditios IC ad IC to obtai the followig result: Propositio 3 The aget chooses quality level ad quatity q γ + ηq κ + η ν + rγ σ β + γm + γ + ηq a κ + η η q τ rγ σ. The aget s choice of quality is icreasig i her itrisic motivatio η while the effect of the commissio rate γ o her choice of quality q is decreasig i η. The aget s choice of quatity is decreasig i her risk aversio r ad icreasig i mea ability m. Her choice of is icreasig i η if ad oly if task ejoymet τ is suffi cietly strog. Proof: See Appedix. If the aget is itrisically motivated she makes a greater effort to provide quality; at the same time her quality provisio is less resposive to the commissio rate. The itrisic desire to do a good job may lead to a reductio i quatity if task ejoymet τ is small, i.e., if the aget does ot much ejoy the task per se but is evertheless itrisically compelled to provide quality, e.g., by her coscietiousess. Such a aget aticipates that she will ivest more effort every time she completes a order, receivig lower utility tha a purely selfish aget. She ratioally fulfils fewer orders, workig harder o each idividual order. Istead a aget who is itrisically motivated ad ejoys the task suffi cietly high τ both completes a higher umber of orders ad provides higher quality o them. 9

10 Fially, whe we further simplify the model by assumig that there is o ucertaity about the aget s talet σ a 0 we ca derive closed-form solutios for the optimal cotract parameters: Propositio 4 Whe σ a 0 the optimal commissio rate is γ ad the optimal order bous is β rσ + rσ κ + η, cm + η m + q + rσ κ + η + + rσ κ + η I the optimal cotract the commissio rate is strictly decreasig i the aget s risk aversio r ad itrisic motivatio η. The optimal order bous is strictly icreasig i the aget s risk aversio r, itrisic motivatio η ad mea ability m. Proof: See Appedix. 3.3 Applicatio: Derivig Experimetal Predictios We ow tur to a applicatio desiged to yield testable predictios for the field experimet we will coduct. Cosider a pricipal that iitially pays her aget a pure commissio without a order bous. Our results from 3. suggest that if the aget is somewhat risk averse the pricipal should reduce the commissio rate while itroducig a order bous. We aalyse a particular chage i cotract the pricipal may experimet with: itroducig a order bous that is paid for by the simultaeous reductio of the commissio rate. To this ed, cosider a shift from a pure commissio rate γ 0 ]0, ] to a lower commissio rate γ < γ 0 combied with a order bous β > 0; the relative size of γ 0 γ ad β is calibrated o a populatio of agets i such a way that the average aget s pay per order remais costat if agets do ot adjust quality. We ow aalyse the heterogeeous effects of such a 0

11 itervetio o expected quatity, quality ad profits. For this purpose we assume that agets kow their ability a whe choosig their efforts ad thus a perso i is characterized by a vector a i, r i, η i, τ i. Moreover, we assume that the persoality traits are ucorrelated. 9 First ote that a shift that keeps the paymet per order costat at prior quality i the populatio of agets will imply that As E [γ 0 a i + q i0 ] β + γ m + E [q i0 ] β γ 0 γ m + E [ γ0 + ηq κ + η i [ γ + η E [ q i ] E i q γ 0 + η i q ] E κ + η i κ + η i it is clear that there will be a loss i quality ad [ ] E γ γ 0 κ+η η i i ]. η i γ γ 0 κ + η i > 0 [ ] γ γ 0 κ + η i such that the loss i quality is the smaller, the more itrisically motivated a aget is higher η i. We ca moreover show that quatity icreases heterogeeously ad that profits icrease for a certai type of aget: Propositio 5 Cosider a shift from a pure commissio rate γ 0 ]0, ] to a lower commissio rate γ < γ 0 combied with a order bous β > 0. i Such a shift reduces expected quality E [ q i ] < 0. The effect is the smaller, the more itrisically motivated a aget is i.e., E[ q i η i ] η > 0. i ii The shift icreases expected quatity E [ i ] > 0; the effect is the larger, the more risk averse the aget i.e., E[ i r i ] r i > 0, the less able the aget i.e., E[ i a i ] a i < 0, ad the less itrisically motivated the aget is i.e., E[ i η i ] η < 0. i 9 Note that oly the proof of claim iii i Propositio 5 will hige o the assumptio that the traits are ucorrelated. The predictios for the effects o quality ad quatity also arise whe the traits are correlated.

12 iii The shift icreases expected profits if ad oly if the aget is suffi cietly risk averse. Proof: See Appedix. 4 Coclusio A importat feature of freelace work is the freedom to set oe s ow schedule: a freelacer i our settig, for example, ca chage the umber of jobs from oe day to the ext. I this paper we ivestigate the cosequeces of such worker flexibility for a firm tryig to determie the optimal pay structure for its workforce. We formally aalyse a pricipal-aget model that icorporates the aget s choice of quatity worked ad accommodates heterogeeity i risk aversio, ability ad itrisic motivatio. We show that the optimal cotract for a risk-averse aget i this settig combies a sales-based commissio rate with a order-based piece rate order bous. Moreover, the optimal order bous is icreasig i the aget s risk aversio, ability ad itrisic motivatio. Based o this model we derive predictios that we ca test i a atural field experimet we will coduct i collaboratio with a olie platform. For this purpose we study the effects of a move from a pure commissio rate to a combiatio of a order bous ad a lower commissio rate set i such a way that at prior quality levels expected paymets per gig remai costat. The model predictios that we will test are the followig: first, the itervetio leads to a icrease i average quatity ad this icrease is more proouced for more risk averse agets. Secod, the shift i compesatio reduces quality but to a lesser extet the higher the aget s itrisic motivatio. Fially, for a shift of ay give size, profits will icrease if ad oly if the aget is suffi cietly risk averse. Extesios of our model provide a framework for studyig other questios beyod the predictios we test i the field experimet. These iclude the dyamics of employee ad employer learig i.e. by studyig the effecs of chages i σ a - the ucertaity about a aget s ability as well as the

13 selectio ad sortig of workers ito ad out of freelace jobs depedig o idividual characteristics ad o the meu of cotracts offered. 3

14 5 Appedix Cocavity of the aget s objective fuctio The objective fuctio is strictly cocave if c q, rγ σ a < 0 c qq q, < 0 c q, rγ σ a cqq q, γ c q q, 0 the latter is equivalet to c q, + rγ σ a cqq q, γ c q q, 0 which always holds if for all γ, q, i if γ c q q, 0 c q, c qq q, γ c q q, c q, c qq q, γ c q q, which always holds whe γ if c q, c qq q, + c q q, ii if γ c q q, < 0 c q, c qq q, γ c q q, γ c q q, c q, c qq q, which always holds because of the covexity of the cost fuctio. Proof of Propositio : The Lagragea becomes L m + q c q, λ β + γ m + q c q, λ γ c q q, 4

15 ad L β λ 0 L γ λ m + q λ 0 L m + q c q, + λ c q, λ γ c q q, 0 L q c q q, λ γ c q q, + λ c qq q, 0. Thus λ 0 ad, i tur, λ 0 such that m + q c q, 0 c q q, 0 From the icetive compatibility costraits we must thus have that γ c q q, 0 which implies γ. Ad which implies that β 0. β + γ m + q c q, 0 Proof of Propositio : From the Lagragea we obtai L m + q c q, rγ σ a + σ λ β + γ m + q c q, rγ σ a + σ λ γ c q q, 5

16 L β λ L γ rγ σ a + σ λ m + q rγ σ a + σ λ 3 L m + q c q, rγ σ a + σ +λ c q, + rγ σ a λ γ c q q, 4 L q c q q, λ γ c q q, + λ c qq q,. 5 Settig through 5 equal to zero, we have λ 0 from ad cosequetly λ rγ σ a + σ from 3. Substitutig these ad simplifyig, the remaiig two coditios become m + q c q, rγ σ a rγ σ a + σ cq q, 0, 6 c q q, rγ σ a + σ cqq q, 0. 7 Usig IC we ca substitute c q q, γ ito 7 to obtai γ rγ σ a + σ cqq q, 0 γ + r σ a + σ c qq q, < γ + r σ a + σ c qq q, 8 whe r > 0. Moreover, from IC ad 6 we have that m+q c q, rγ σ a rγ σ a + σ cq q, β+γ m + q c q, rγ σ a + σ m + q + rγ σ rγ σ a + σ cq q, β + γ m + q β γ m + q rγ σ a + σ cq q, + rγ σ 6

17 If we substitute γ β +r σ a + σ c qqq, m + q + r σ a + σ c qq q, r + r σ a + σ c qq q, +r + r σ a + σ c qq q, σ a + σ cq q, σ β r σ a + σ cqq q, m + q c q q, + r σ a + σ c qq q, + rσ 9 + r σ a + σ c qq q, This will be strictly positive if r > 0 ad σ a + σ c qq q, m + q c q q, + r σ a + σ c qq q, +σ > 0 will always be the case for all m > 0 if c qq q, q c q q,. Because of the assumptio that c q q, c q q, this coditio holds if c qq q, q c q q,. 0 Note that due to c qqq 0 the margial costs of quality are weakly covex ad thus c q, 0 c q, q + c qq, q 0 q 7

18 which implies coditio 0. c qq, q q c q, q c q, 0 }{{} 0 q, Proof of Propositio 3: Coditios IC ad IC become κ β + γ m + q q η τ q q ν rγ σ a + σ 0 such that from IC ad γ κq + η q q 0. γ κq + η q q 0 with q γ + ηq κ + η q γ κ + η ad q η q κ + η γ + ηq κ + η κq γ κ + η γ κ + η > 0 We compute by rearragig IC ad simplifyig to obtai κ β + γ m + q q η τ q q ν rγ σ a + σ 0 κ β + γ m + q q η τ q q ν rγ σ a rγ σ 0 8

19 β + γ m + q κ q η β + γ m + γ+ηq κ+η τ q q rγ σ ν + rγ σ a κ γ+ηq κ+η η τ γ+ηq κ+η q rγ σ ν + rγ σ a β + γm + γ γ+ηq κ+η κ γ+ηq + η κ+η τ η γ+ηq κ+η γ+ηq κ+η q + q rγ σ β + γm + γ γ+ηq κ+η κ γ+ηq + η κ+η τ η ν + rγ σ a γ+ηq + η γ+ηq κ+η ν + rγ σ a β + γm + γ+ηq κ+η γ+ηq κ+η η q τ rγ σ ν + rγ σ a β + γm + γ+ηq κ+η η q τ rγ σ ν + rγ σ. a κ+η q η q rγ σ We the have τ η ν + rγ σ a > 0, r r { ν + rγ σ a β + γm + γ + ηq κ + η η q τ γ σ a ν + rγ σ a γ σ a ν + rγ σ a β + γm + γ + ηq κ + η η q τ β + γm + γ+ηq κ+η η q τ rγ σ ν + rγ σ a rγ σ rγ σ } γ σ ν + rγ σ a γ σ ν + rγ σ a γ σ a ν + rγ σ a γ σ ν + rγ σ a < 0 9

20 η ad γ + ηq q κ + η γ + ηq ν + rγ σ a 4 κ + η q τ ν + rγ σ γ + ηq c + η q γ a κ + η q τ γ + ηq c + η q γ + ηq γ q κ + η ν + rγ σ a κ + η + τ γ + ηq cq + γ + ηq ηq γ + ηq γ q κ + cη + η ν + rγ σ a κ + η + τ cq γ + cηq + γηq + η q γ ηq γ κ q cηq η q ν + rγ σ a κ + η + τ cq γ γ κ q ν + rγ σ a κ + η + τ τ ν + rγ σ a γ cq κ + η. η is strictly egative if τ 0 ad for τ q it is equal to q κ + η γ cq ν + rγ σ a κ + η κ q + cηq + η q γ + cq γ κ q ν + rγ σ a κ + η cηq + η q + cq γ γ ν + rγ σ a κ + η η κ + η + γ γ κ ν + rγ σ a κ + η > 0 Moreover, is strictly icreasig i τ which completes the proof. Proof of Propositio 4: The optimal values of γ ad β are obtaied by 0

21 substitutig c qq ad c q ito expressios 8 ad 9: γ + r σ a + σ c qq q, β r σ a + σ cqq q, m + q c q q, rσ + + r σ a + σ c qq q, + r σ a + σ c qq q, The cost fuctio is give by κ c q, q η τ q q + ν. Substitutig ad settig σ a 0, c q κq + η q q c qq κ + η c q κq + η q q γ + rσ κ + η which is strictly decreasig i r ad i η. Substitutig c qq ad c q ito β ad settig σ a 0 we obtai β rσ κ + η m + q cq + η q q rσ + rσ + κ + η + rσ κ + η rσ cm + η m + q rσ + rσ κ + η + rσ + rσ κ + η cm + η m + q + rσ κ + η + + rσ κ + η For the comparative statics ote that β ca be rearraged to obtai. β σ cm + η m + q r + σ κ + η + r + σ κ + η,

22 β η which is strictly icreasig i r ad i m. Fially, m + q rσ + rσ κ + η cm + η m + q rσ + rσ κ + η m + q rσ + rσ κ + η m + q rσ cm rσ η m + q + rσ κ + η rσ rσ + rσ κ + η m + q + rσ κ + η + rσ > 0. κ + η rσ + rσ κ + η 3 rσ + rσ κ + η 3 Proof of Propositio 5: Claim i directly follows from the cosideratios i the text. Claim ii: Cosider ν i ν β + γ a i + γ + η i q κ + η i γ 0 a i + γ 0 + η i q κ + η i β + γ γ 0 a i + γ + η i q κ + η i η i q τ η i q τ γ 0 + η i q κ + η i r iγ σ r iγ 0σ + r i γ 0 γ σ ν ν β + γ γ 0 a i + γ γ 0 + γ γ 0 η i q κ + η i β + γ γ 0 a i + γ γ 0 γ + γ 0 + γ γ 0 η i q ν β γ 0 γ κ + η i a i + γ + γ 0 + η i q κ + η i + r i γ 0 γ σ + r i γ 0 γ σ + r i γ 0 γ σ

23 [ ] with β γ 0 γ m + E γ0 +η i q κ+η i i ν ν γ 0 γ m + E γ 0 γ m a i + E [ γ0 + η i q κ + η i ] γ 0 γ [ γ0 + η i q ] γ + γ 0 + η i q κ + η i κ + η i a i + γ + γ 0 + η i q κ + η i + r i γ 0 γ σ Now cosider the effect of the treatmet o quatity i the populatio, which is give by E [ i ] ν [ γ0 + η γ 0 γ E i q γ + γ 0 + η i q ] + κ + η i κ + η i E [r i] γ 0 γ σ [ γ0 γ γ 0 γ E κ + η i ν γ 0 γ ν [ E κ + η i Now we ca cosider the partial derivatives ad ] + E [r i] γ 0 γ σ > 0 ] + E [r i] γ 0 γ σ > 0 E [ i r i ] γ 0 γ σ > 0, r i c E [ i a i ] γ 0 γ < 0 a i ν E [ i η i ] γ 0 γ 4q κ + η i γ + γ 0 + η i q η i ν 4 κ + η i γ 0 γ cq γ γ 0 ν κ + η i γ 0 γ γ γ 0 ν κ + η i < 0. Claim iii: Compare profits geerated by a aget i before ad after the + r i γ 0 γ σ 3

24 shift. Iitial profits are Profits after the shift are Π i0 γ 0 i0 a i + q i0 such that the chage i profits is Π i γ i a i + q i β i Π i γ a i + q i β i γ 0 i0 a i + q i0 substitutig it ν Π i γ γ 0 ν β t + γ t a i + γ t +η i q a i + γ + η i q κ + η i κ+η i γ 0 a i + γ 0 + η i q κ + η i q η τ i i r iγ t σ, t {0, }, β β + γ ν a i + γ + η i q κ + η i η i q τ i r iγ 0σ First, ote that Π i is a liear fuctio of r i. Takig the first derivative with respect to r i we obtai Π i r i γ σ c a i + γ + η i q [ γ 0 γ 0 β ν κ + η i a i + γ 0 + η i q κ + η i [ ] Substitutig β γ 0 γ m + E γ0 +η i q κ+η i q τ i η i r iγ σ a i + γ 0 + η i q κ + η i γ σ γ 0 ν γ 0σ a i + γ + η i q γ γ κ + η i [ Π i σ γ 0 γ r i c 0 a i + γ 0 + η i q γ κ + η γ a i + γ + η i q i κ + η [ i +γ γ0 + η γ 0 γ m + E i q ]] κ + η i ] + βγ. a i + γ 0 + η i q κ + η i 4

25 Hece, [ E [ Π i r i ] σ r i c [ > σ c as required. [ γ γ0 + η 0 γ 0 m + E i q ] [ γ γ + η κ + η γ m + E i q ] i κ + η [ i +γ γ0 + η γ 0 γ m + E i q ]] κ + η [ i γ γ0 + η 0 γ 0 m + E i q ] [ γ γ0 + η κ + η γ m + E i q ] i κ + η [ i +γ γ0 + η γ 0 γ m + E i q ]] κ + η [ ] i σ m + E γ0 +η i q κ+η i [ γ c 0 γ 0 γ γ + γ γ 0 γ ] [ m + E γ0 +η i q σ γ c 0 γ γ0 > 0, κ+η i ] 5

26 Refereces Feltham, G. ad J. Xie 994. Performace measure cogruity ad diversity i multi-task Pricipal/Aget relatios. The Accoutig Review 69, Holmström, B. ad P. Milgrom 99. Multitask pricipal-aget aalyses: Icetive cotracts, asset owership ad job desig. Joural of Law, Ecoomics ad Orgaizatio 7, 4 5. Milgrom, P. ad J. Roberts 99. Ecoomics, Orgaizatio ad Maagemet. Eglewood Cliffs: Pretice-Hall. Prassl, J. 07. Humas as a Service. Oxford Uiversity Press. Schedler, W Whe is it foolish to reward for a while beefitig from b? Joural of Labor Ecoomics 6 4, Wolfstetter, E. 00. Topics i Microecoomics. Cambridge Uiversity Press. 6