Job assignments as a screening device

Transcription

1 Forthcoming in International Journal of Indutrial Organization Job aignment a a creening device Juan D. Carrillo Columbia Buine School ECARES and CEPR Abtract We tudy intra-firm competition for promotion when the ability of the competing candidate i imperfectly oberved. We how that firm hould offer the job that require the highet degree of involvement to the candidate whoe ability i known with leat certainty typically, junior and outider becaue thee individual have the tronget career-concern incentive to perform efficiently. Alo, when firm have to delegate the election of the creening procedure to their inider candidate, then the proportion of internal promotion relative to external hiring i exceively high. Keyword: Screening, job allocation, delegation, favoritim, career concern. JEL Claification: D81, L22. I thank Iabelle Broca, Micael Catanheira, Deni Gromb, Jacque Lawarrée, Canice Prendergat, Loïc Sadoulet, the participant at everal eminar, the Editor Simon Anderon and two anonymou referee for their comment. Juan D. Carrillo, Columbia Buine School, Uri Hall 812, 3022 Broadway, New York, NY 10027, Ph: , FAX: , <jc2095@columbia.edu>, web:

2 1 Introduction Selection and promotion of the mot able employee i a key determinant of the economic performance of firm. It i therefore not urpriing to oberve that mot organization devote a ubtantial amount of reource to creen candidate, through an implicit or explicit ytem of relative performance evaluation. The creening procedure can be roughly divided in two tep: firt, each employee ability or capacity to perform in the firm i accurately teted againt that of hi peer, then the mot uitable candidate i elected for the job or promoted. 1 The mechanim behind thee evaluation and election procee may look trivial. For intance, uppoe that the manager i forced to delegate the deign of the creening procedure to one of the job-candidate. One might think that uch delegation i relatively le cotly when the candidate in charge of the creening i on average better than hi competitor. Given hi a priori dominance, he might want to deign a proce a efficient a poible i.e one that reveal the maximum amount of information to the manager in order to prove hi uperiority. In fact, the oppoite i true: relying on the employee choice of the creening proce i more cotly the higher the expected quality of the individual to whom thi deciion i delegated. Thi wa firt pointed out by Baliga and Sjötröm 2001, and it contitute the tarting point of the paper. Conider the following ituation. Two candidate compete for one poition in an organization. Their talent or fit between their ability and the need of the organization i partly unknown both to the organization and to the individual themelve. More preciely, we aume that candidate differ both in their expected ability and in the variance of thi etimation: the ability of the employee already working in the firm the enior or incumbent i le uncertain than that of a recent hire or an individual from the job market the junior or outider. We further aume that there i a fixed number of creening tak that erve to imperfectly evaluate the true capacity of the two candidate. Each tak can be performed by one and only one of them. Given that the rik-neutral manager want to appoint the candidate with highet ability among the two, he will ue thee tak to elicit a much information a poible from the two individual. How i thi achieved? In Propoition 1 we how that the manager hould plit the tak between the two candidate. Furthermore, the candidate whoe ability i more uncertain the junior/outider perform more tak than hi rival, but the optimal allocation doe not depend on the candidate prior expected abilitie. One may at firt be urpried that the choice of the manager i not influenced by the relative poition of the candidate. The intuition i imply that the manager maximize the total quantity of information that the creening proce convey about the two 1 Although we will focu on the relationhip between a manager and two job candidate, there are many other ituation that could be analyzed with a imilar model: a political party that chooe between two potential leader for the upcoming electoral campaign, a firm that chooe a ubcontractor for the activitie outide it core buine, etc. 1

3 individual. In thi model, the value of each ignal depend on the preciion of the information which i affected by the prior knowledge about the ability of that candidate but not on it expected value. Uing the manager optimal allocation a a benchmark, we can then characterize the inefficiencie that arie when the allocation of creening tak ha to be delegated to one of the candidate. Contrary to the manager optimal rule, the optimal choice of the candidate will depend on the relative expected abilitie of the two competitor Propoition 2. An individual with a lower expected ability than hi rival want to maximize the probability of leapfrogging the opponent, which i achieved by maximizing the amount of information revealed during the creening proce. It i true that information i more likely to confirm hi inferiority than to reveal a uperiority. However, it i alo hi bet and only chance to be ex-pot conidered more able than hi opponent. Uing a ymmetric reaoning, one can ee that a candidate who i on average better than the rival want to minimize the chance of being leapfrogged by the opponent. We how that minimization of information i obtained by concentrating all the available tak on the hand of the incumbent, whoe ability i already relatively well known. Again, the incentive of the candidate ahead and the candidate behind do not depend on the magnitude of the difference in their expected abilitie. Overall, there i a conflict of interet between the manager who want to elect the mot able candidate for the job and the candidate who want to maximize their probability of being elected. Depite thi tenion, the manager can alway avoid the cot of delegation; it uffice to allow the leat able candidate to chooe hi preferred creening procedure. Our theory alo predict that when the mot able candidate i in charge of allocating the creening tak, then it may be optimal to bia the competition againt him Corollary 1. After having determined the inefficiencie of delegating the creening proce, we move on to analyze the uitability of favoring ome type of employee over other. To thi purpoe, we aume that there are two qualitatively different creening tak, each candidate ha to perform one and only one of them, and the manager decide which one i performed by each of them. A before, both type of tak allow the manager to imperfectly infer the ability of the candidate. Furthermore, creening tak are productive, in the ene that they affect directly the utility of the manager, and one of them i more productive than the other. Lat, performance in each tak depend tochatically not only on the ability of the candidate who undertake it, but alo on hi unobervable effort. Ability i, by aumption, mot valuable in the mot productive tak. Then, abent effort conideration, the manager optimal rule i trivial: he firt allocate the candidate with highet expected prior ability to the mot productive creening tak and then keep for the job the one with highet expected poterior ability given the oberved performance. Including the poibility of effort modifie thi concluion. Once individual are allocated to tak, they do not 2

4 internalize the poible effect of their effort in the outcome of the creening activity. Hence, they will never exert effort to affect the performance in the creening job thi concluion i reminicent of the well-known literature on career concern, ee Holmtröm Intead, their incentive to work hard are excluively due to the fact that a high performance ignal a high ability, and therefore increae the chance of being elected for the job. We how that, depite the difference with the tandard career concern paper candidate compete for a fixed price and they do not fully recoup via wage the benefit of their effort, the uual reult that the individual with mot uncertain ability ha the tronget interet in putting effort to bia the perception of hi ability hold in our etting. Since the manager recoup part of the benefit generated by the employee effort, he will take into account their different implicit incentive when deciding the allocation rule. In particular, we how in Propoition 3 that the manager may find it optimal to favor the election of the outider/junior for the mot valuable tak even if hi expected ability i lower than that of the incumbent/enior. To um up, thi paper analyze the efficiency loe when the creening procedure ha to be delegated. It alo argue that favoring candidate whoe ability i relatively low and relatively unknown may ometime be optimal for the organization. The model draw other general implication for iue related to the optimal hiring policie in organization. Before preenting the model, we would like to briefly mention ome paper directly related to our. A previouly mentioned, Baliga and Sjötröm 2001 already pointed out the idea that above average agent want to uppre information and below average agent want to generate it. Their paper analyze the following ituation. Two agent have private information about the quality of a project developed by one of them. The project i correlated with the ability of the owner. The principal want to implement the project only if it i good, and then promote one of the agent. Naturally, agent do not alway have incentive to reveal their information truthfully, becaue it affect the promotion deciion of their bo. Their paper differ from our in objective, modelling and reult. Their goal i to characterize the optimal contract offered by the principal. The ingredient of the model are complete contract, aymmetric information and poibility to provide or hide information via the implementation or not of the project. One of the central reult i that, in the optimal renegotiation-proof contract, the deciion whether to implement the project mut be delegated to it owner. By contrat, our paper take delegation for granted. The goal of our Section 2 i to analyze the cot of delegating the creening proce. To thi purpoe, we build a model with incomplete but ymmetric information, with a fixed number of creening tak, and the only tool we conider i the allocation of thee tak in particular it i not poible to uppre information. Our main contribution i to characterize the optimal allocation of tak from the viewpoint of the different actor in our economy. Thi allow u to determine the cot of 3

5 delegation, which doe not depend on the difference in prior expected abilitie. From thi, we alo conclude that if the choice of tak allocation ha to be delegated to a relatively able individual, it i then optimal to bia the competition againt him. 2 Precott and Vicher 1980 were among the firt author to analyze creening a a determinant of firm performance. However, our work alo focue on different iue than their. For example, in our paper candidate are in competition, o our key variable i relative ability rather than abolute ability. Alo, we fix the number of creening tak and determine how to efficiently allocate them between different candidate, rather than optimize over the number of tet period for each one of them. Section 3 of our model can be een a an extenion of the eminal paper by Holmtröm 1999 on career concern. Contrary to the work in thi literature, in our paper candidate compete for a fixed price. Thi implie that the principal partly recoup the benefit generated by the employee effort. A a reult, he ue the dicretion in the allocation of tak to favor the individual with higher implicit incentive to exert effort, that i the junior or outider. Prendergat and Stole 1996 already analyzed a ituation in which different type of agent have different incentive to ditort information: junior agent overreact to new in order to ignal that they are talented while enior agent underreact to avoid ignaling that their previou behavior wa wrong. Ditortion are therefore of different and maybe complementary nature than in our etting, and rely on private information and a ignaling motive. Meyer 1991 i a claical paper on dynamic incentive and the optimal deign of tournament when the principal can extract a poitive but limited amount of information from the performance of hi ubordinate. Lat, Carmichael 1988 i a claical reference on optimal incentive for efficient hiring and promotion when the agent within the organization have an input into deciion like in academia for example. 2 Screening with the number of tak We conider the deciion problem of a rik-neutral manager from now referred to a principal who ha to determine the value of different job-candidate from now referred to a agent in order to keep the mot able one. More preciely, the principal ha to chooe between two agent a {i, o}: an incumbent i who ha been already working in the firm and an outider o who i new to the firm. The model can alo be interpreted a the choice between a enior agent i who ha been in the firm for a long time and a recent hire or junior one o. The two agent differ excluively in their ability, θ i and θ o, to perform tak, or the fit between their talent and the need of the organization. All the actor have imperfect but ymmetric information about 2 Note that in Baliga and Sjötröm 2001, the contet may alo be biaed in favor of the ex-pot leat able agent. However, uch trategy i optimal in their paper becaue it provide incentive for truth-telling given the aymmetry of information, and in our paper becaue it provide incentive for optimal allocation of tak. 4

6 the unidimenional ability parameter of both agent. Formally, they all know that abilitie are drawn from the following ditribution: θ a N m a, σ 2 a a {i, o} 1 Agent differ in two repect. Firt, they have different expected abilitie m a. For our analyi we will conider all the poible pair m i, m o. Second, the preciion of the etimate of the agent abilitie σ 2 a are alo different. For the ret of the paper, we will aume that the ability of the incumbent/enior i i known with more accuracy than the ability of the outider/junior o: Aumption σ 2 i < σ2 o. We introduce thi aumption becaue, in our view, the longer the previou labor relation between the principal and the agent, the more accurate i the information that the principal ha collected about the capacity of the agent. In other word, the principal ha a better knowledge about how good the matching or fit between firm and incumbent or firm and enior employee i than between firm and outider or firm and junior employee. Note that, given the principal rik-neutrality, thi doe not give any a priori advantage to incumbent and enior agent. 3 The agent elected by the principal to work in the firm receive an exogenou wage b > 0. 4 The other agent receive zero, which i the outide wage or opportunity cot not modelled in the paper. For implicity, the performance of the agent elected for the job i equal to hi ability. The principal cannot commit to a choice rule for the election between the agent. He will therefore retain the agent with highet expected ability conditional on the information available at the election tage 5 ee Corollary 1 for a dicuion of what happen when we relax thi aumption. In order to increae the knowledge about the ability of the agent, the principal can aign them ome creening tak. In thi ection, we conider the following three pecific characteritic of the creening proce: i A fixed number n of creening tak need to be plit between the two competitor. All creening tak are identical and each of them can be performed by one and only one agent. The cot of delegation are determined by comparing the optimal allocation from the principal perpective with the optimal allocation from the agent perpective. ii Screening tak are unproductive, i.e. performance of the agent in thee activitie ha no intrinic value for the principal. 6 3 The reader not convinced by thi interpretation may imply retain that the variance in the ability of agent i different and exogenouly given. 4 Thi i made for implicity. A we dicu later on, including more ophiticated contract do not necearily alter the inight of the paper, although ome incentive problem might be alleviated ee Remark 1. 5 Our reult immediately extend to the cae of poitive correlation between ability and performance. All that matter i that the higher the agent ability, the higher the likelihood of hi being elected. 6 However, our qualitative reult would till hold if creening tak affected the payoff of the organization. 5

7 iii Performance on a creening tak i a tochatic function of the ability of the agent to whom it i aigned. Formally, if agent a undertake tak k {1,..., n}, hi performance x k a i: x k a = θ a + ε k a where ε k a i.i.d. N 0, σ 2 ε. 2 The outcome x k a i publicly oberved. Thi information i valuable even if the principal doe not derive any direct utility from the outcome of the tak and the agent are not rewarded for their performance. In fact, each piece of new give ome information about the agent ability. Therefore, it influence the deciion of the principal to keep one agent or the other, and hence the payoff of the three actor in thi economy. In the cae of the Normal ditribution, computing the poterior belief about the agent ability given hi performance in the creening tak i particularly imple. For intance, if agent a realize tak 1 to with performance {x 1 a, x 2 a,..., x a} the poterior ditribution of hi ability become: 7 θ a {x k a} k=1 N λ a m a + 1 λ a x k a, σa 2 λ a k=1 σ 2 ε where λ a = σε 2 + σa 2. 3 In word, the poterior ditribution i alo Normal. It mean i a weighted average of the prior mean and the um of the performance. It variance i determinitic and decreaing in, the number of tak realized. Note that, in order to update the belief about the agent ability, the total performance in creening tak that we denote by Xa k=1 xk a i a ufficient tatitic. From 2 and the independence of the noie term, we can ee that: N m a, σε 2 + σa 2 4 X a Suppoe that the incumbent realize creening tak with total performance Xi, and the outider the remaining n one with total performance Xo n. Denote by w {i, o} the winner, that i the agent ex-pot elected by the principal after oberving the performance in the n tak. Recall that the on-the-job productivity of the winner i equal to or poitively correlated with hi imperfectly known ability and that the principal cannot commit on the election rule. Given hi rik-neutrality, the principal will therefore chooe for the job the agent with highet expected poterior ability conditional on the outcome of the creening tak. Formally: { } w = arg max E[θ i Xi ], E[θ o Xo n ] {i,o} Our objective in thi ection i to determine the cot of delegation. We thu compare the firt-bet cenario in which the principal i reponible for the allocation of creening tak between the two agent to the econd-bet cenario in which either the incumbent/enior or the 7 In order to avoid integer problem, we will treat and n a real number in the optimization problem below. 5 6

8 outider/junior chooe how to plit the tak. In both cae, the winner i determined by the principal according to hi ex-pot optimal election rule given by equation 5. From the preference and payoff decribed above, we can notice that the principal and the agent have conflicting goal: the former want to maximize the probability of electing the mot able agent while the latter want to maximize their own probability of being elected. Becaue of thi tenion, the different individual have different preference over the revelation of information, and therefore different deire concerning the optimal plitting rule of the creening activitie. The timing of our game i ummarized a follow. allocation of creening tak ignal election of winner by principal agent wage production and principal payoff, n X i, Xn o w {i, o} b if a = w 0 if a w θ w b time Figure 1. Timing in the number of creening tak game. Note that 5 implicitly aume that the principal keep one and only one agent. Nothing would change if we included the poibility of not retaining any agent. Alo, keeping exactly one agent i formally equivalent to retaining both agent and promoting one and only one of them to a more valuable job a for example in Baliga and Sjötröm, Principal optimal allocation of creening tak Think of the following ituation. The principal i the manager of a mall oftware company who need to hire or promote a oftware expert. To evaluate the capacity of the candidate, the manager ak them to develop ome imple computer program. The incumbent and the outider realize and n of thee program a creening tak, repectively. The ex-ante welfare of the principal i the expected ability of the winner net of the fixed remuneration b. The expected ability of the winner i the um of two factor. Firt, Vi the expected ability of the incumbent conditional on being elected i.e. on being the one with highet expected poterior ability weighted by hi ex-ante probability of being elected. Second, Vo n the expected ability of the outider conditional on being elected, weighted by hi ex-ante probability of being elected. Formally: [ ] [ ] V i = E V n o = E θ i E[θi Xi ] > E[θ o Xo n ] [ θ o E[θi Xi ] < E[θ o Xo n ] Pr E[θ i Xi ] > E[θ o Xo n ], ] [ ] Pr E[θ i Xi ] < E[θ o Xo n ]. Define, W V i + V n o 7

9 and denote by p the number of tak optimally delegated to the incumbent from the principal viewpoint n p are then optimally delegated to the outider. We have: p = arg max W 6 At thi tage, we can characterize the optimal choice of the principal. Propoition 1 When the principal decide how to allocate the creening tak, then p [0, n/2, where i independent of m i and m o, / σ 2 i W ign m i 0, and lim σ σi 2 i 2 = n/2. 8 Moreover, σ2 o W m o = ign m i m o and ign 2 W f m i 2 W f m o = ign m i m o. Proof. Uing integration by part and traightforward although tediou algebra we get that: 9 where f = W = m i Φ σ 4 i σ 2 ε + σ2 i [ mi m o + m o 1 Φ f + n σ4 o σ 2 ε +n σ2 o ] mi m o + f ϕ f mi m o, 7 f and ϕ and Φ are repectively the denity and c.d.f. of the tandard Normal ditribution. From 7 and given that a property of the Normal ditribution i ϕ x = x ϕx, we can eaily check that: = ϕ > 0, and therefore p = arg max o that < n/2. 10 W m i W m o W f mi m o f f. Simple computation how that f < 0. Given σ 2 i < σ2 o, then f n/2 < 0, Differentiating f ; σi 2 we obtain that / σ2 i f ; σi 2/ σ2 i > 0. Alo, W = Φ, = 1 Φ, f n/2 = 0 when σi 2 = σ2 o. Finally, alo from 7, = 2 Φ 1 and mi m o f W m i mi m o f m o 2 W f m i 2 W f m o = 2 m i m o f f ϕ mi m o f mi m o f. The principal i intereted in determining how able i the mot able agent. Therefore, he want to ue the creening proce to extract a much information a poible from both agent. Given that the informational content of each tak decreae with the number of tak previouly realized, then maximal information i dicloed when creening tak are divided between the two agent. Propoition 1 provide an analytical characterization of thi optimal allocation. It how that a the ex-ante difference between the variance of abilitie decreae σi 2 σo, 2 then the difference between the number of tak performed by each agent alo decreae n. The mot intereting characteritic of Propoition 1 i that the prior expected abilitie of the two agent will not affect the allocation of tak by the principal. One could think that if one σ 4 i 8 More preciely, = arg max + n σ4 σ ε 2 o. + σ2 σ i ε 2+n σ2 o 9 See any advanced book in Statitic dealing with the Normal ditribution, or Carrillo and Mariotti 2001, Lemma 1 for a imilar proof in a context of electoral competition between political partie. 10 Note that if f 0 > 0 for which n σi 2 > σε 2 i a ufficient condition then the olution for i interior. That i, it i optimal for the principal that both agent realize ome creening tak. 8

10 agent i on average much better than the other, then the principal hould concentrate hi effort in teting thi agent, jut to make ure that he i a good a he look. Thi intuition i incorrect. In fact, the principal focue excluively on the amount of information collected with the creening proce. The maximization of total knowledge i obtained by equating the marginal quantity of learning about each agent, which depend on the accuracy of the previou information their ex-ante variance in ability but not on the type of information their ex-ante expected ability. However, thi i not to ay that the principal i not affected by the prior expected abilitie of the agent. Firt and trivially, hi welfare increae with the agent ex-ante expected ability W m i > 0 and W m o > 0. More interetingly, keeping the um of expected abilitie contant, the principal i wore-off the cloer the expected abilitie are formally, W m i W m o > 0 if m i > m o. The idea i imply that the principal will only keep one agent. Hence, the higher the difference in their prior expected abilitie, the higher the poterior expected ability of the mot able agent. Lat, even though the relative benefit of creening one agent or the other do not depend on m i, m o, the marginal benefit of plitting the creening tak are higher the cloer the prior expected abilitie formally, 2 W f m i 2 W f m o > 0 if m i < m o. If the prior abilitie are ufficiently different, the agent with highet prior expected ability i mot likely to keep hi advantage after the revelation of information. The evaluation tak are then going to revere the prior ranking only with a mall probability, which make creening relatively le valuable. Propoition 1 ha alo ome intereting implication. It predict that a an agent pend more and more time within the organization, it become le and le important to tet him before deciding about hi promotion. We hould therefore rarely oberve enior employee being crutinized. At the ame time, it alo ay that optimal teting depend excluively on the relative prior knowledge about the competitor. Therefore, being rigorouly teted hould not be interpreted by agent a a negative indicator of their perceived capacity. In other word, according to our theory and other thing being equal, there hould be no correlation between which of the agent competing for the poition i teted more exhautively and which one i finally promoted. 2.2 Agent optimal allocation of creening tak The idea according to which the principal i the ultimate reponible for the election of the agent mot uitable for the job eem quite natural. If poible, he would alo prefer to decide how to allocate efficiently the creening tak between agent. However, it i not unuual to oberve a delegation of thi choice to one of the agent. There are everal reaon for the neceity of delegation. The implet one i work overload. The principal perform everal other tak within the organization. Therefore, even if deigning the creening proce i important, the opportunity cot of pending time on thi problem might be too high beide, it i a priori unclear how 9

11 cotly delegation i, and the purpoe of the ection i preciely to evaluate it. A econd reaon for delegation i uperior information of agent. For intance, uppoe that there are N potential creening tak, it i only poible to evaluate the performance in a ubet n of them, and creening i time-conuming, o that only n tak can be undertaken before the election i made. If only agent know the n tak where performance can be evaluated, then for N ufficiently large the principal will be better-off by delegating the allocation choice rather than making a blind gue. Thi idea i related to Aghion and Tirole 1997, where the principal may find it optimal to delegate authority to the agent even if the interet of the two partie are not perfectly congruent. 11 Denote by a the number of tak undertaken by the incumbent/enior if agent a chooe the allocation, in which cae n a i the number of tak undertaken by the outider/junior. Recall that, abent commitment device, the agent elected by the principal after the creening proce i the one with highet expected poterior ability. Hence, from the perpective of agent a {i, o}, the optimal number of tak that the incumbent/enior hould undertake i: i = arg max o = arg max [ ] Pr E[θ i Xi ] > E[θ o Xo n ], 8 [ ] Pr E[θ i Xi ] < E[θ o Xo n ]. 9 Given 8 and 9, we can now characterize the optimal choice from the agent viewpoint, which alo inform u about the inefficiencie of delegation. Propoition 2 The optimal allocation of tak from the perpective of the incumbent/enior i and the outider/junior o i: i = n if m i > m o and i = if m i < m o. o = if m i > m o and o = n if m i < m o. Proof. Given 3, 4 and 8, the probability of keeping an incumbent who realize tak i: Pr [ λ i m i + 1 λ i X i > λ n o m o + 1 λn o n and that of keeping the outider i 1 Φ mi m o f ] Xo n = Φ = Φ mo m i f mi m o f. Hence, i = arg min if m i > m o and i = arg max f if m i < m o. Converely, o = arg max f if m i > m o and o = arg min f if m i < m o. Recall that f < 0. Given σi 2 < σo, 2 then f0 > fn. Therefore, n = arg min f and, a before, = arg max f. 11 In Baliga and Sjötröm 2001, delegation i part of the renegotiation-proof contract offered by the principal given the aymmetry of information with the agent. f 10

12 Suppoe that the agent with highet prior expected ability decide the allocation of tak. Thi agent want to minimize the probability of being leapfrogged by hi rival. In order to keep hi advantage with the greatet poible probability, he will deign a creening procedure that convey the minimum amount of information. Converely, the agent with lowet prior expected ability will maximize the information revealed during the creening proce in order to overcome hi prior handicap with greatet probability. Thi reult i reminicent of the work by Baliga and Sjötröm 2001 and may at firt be urpriing. In fact, we tend to think that the higher the ability of an agent, the higher hi willingne to dicloe information. However, in our etting the oppoite i true: the agent with highet expected ability can only loe from the revelation of information, while the agent with lowet expected ability can only win from it dicloure. An intereting quetion i to undertand how can agent maximize rep. minimize the amount of information revealed. One obviou alternative i to maximize rep. minimize the number of creening tak, but thi poibility i ruled out in our etting. The only degree of freedom i the rule for plitting tak. Since the informational content of each tak decreae with the number of tak previouly realized, then by concentrating all the tak on the hand of the agent whoe ability i bet known the incumbent/enior, the amount of information revealed i minimized. Thi poibility will be choen by the agent with highet expected ability when the creening procedure i delegated to him. Converely, maximal information i dicloed when creening tak are plit among the two agent, with < n/2 tak allocated to the incumbent. Thi alternative will be optimally elected by the agent whoe expected ability i lowet. Overall, the combination of Propoition 1 and 2 ugget that depite the conflict of interet between the principal intereted in electing the agent with highet ability and the agent intereted in maximizing their probability of being elected, delegation of creening will not be cotly a long a the agent with lowet expected ability bear thi reponibility. Thi reult hold independently of the magnitude of the difference between the expected abilitie of the two agent m i m o, and alo independently of the number of creening tak available n. 12 At thi point we can offer ome implication for creening and turnover in organization. Conider a firm where the incumbent agent control the allocation of the creening job independently of their relative expected ability. Firt, if we adopt a dynamic perpective, our model ugget that it may not alway be deirable to hire the bet available applicant. Naturally, ability affect productivity and therefore i valuable. However, it alo block the quality of the future creening procedure. In other word, the organization might find it optimal not to retain an agent who i 12 In Baliga and Sjötröm 2001, the principal cannot avoid all the cot of delegation becaue, in their paper, the only way to generate information i by implementing a project. Therefore, the principal trade-off the cot of the project and the benefit of information wherea the agent with lowet expected ability only maximize the revelation of information. 11

13 lightly better than the average outide applicant becaue the gain of hi higher expected ability will be offet by the cot of an inefficient future election rule. Second, there i an intereting relation between turnover rate and efficient creening. On the one hand, low firing and turnover implie that the agent working in the organization mut be better than the outide pool of applicant. On the other hand, it may jut be the conequence of an inefficient creening rule et by incumbent who are jut above average. Third, conider a chain of command in which a enior agent decide whether to perform a job himelf or leave it to hi junior colleague. If the enior i, on average, relatively good then he will perform all the tak to avoid that the principal learn about the ability of the junior member. Thi reult can be een a complementary to the chain of command and tranmiion of information argument of Friebel and Raith Remark 1. If we could make the payoff of the agent promoted a function of hi job productivity an incentive contract of the type b w E[θ w ] and the incumbent were delegated the allocation of tak, then he would chooe o a to maximize Vi. Even in that cae, the incumbent objective would not be aligned with thoe of the principal, o the inefficiency highlighted earlier when the incumbent i relatively able would till be preent. 13 Note however that b fixed eem a reaonable aumption in tatutory job, where agent compete for a promotion but pay cale are fixed. Remark 2. The reader might view our modelling of creening a being exceively implitic. Firt, one hould treat the total number of creening tak a an optimization variable. 14 Although thi would quantitatively change our reult, our baic inight would only be reinforced: the marginal value for the principal of a new tak would till be poitive and greatet when allocated to the agent with mot uncertain ability, wherea the agent with highet ability would till prefer to concentrate all tak on the hand of the agent with leat uncertain ability or, even better, to minimize the total number of tak. Second, allowing both agent to perform the ame tak could ometime facilitate the principal choice. Yet, if there are multiple way to undertake each tak, there i no fundamental difference between both agent completing the ame tak or each agent completing a different one. Third, and more interetingly, if tak were allocated equentially i.e., tak k + 1 were allocated after oberving agent a performance x k a in tak k, the principal optimal trategy would till be to allocate each tak to the agent whoe ability i mot uncertain. However, contrary to our current reult, the interet of the principal and the agent with lowet ability would no longer be perfectly aligned, ince that individual would not necearily remain 13 The principal can obtain an alignment of interet by making the incumbent payoff contingent on the productivity of the agent appointed for the job. However, a mechanim where an agent i fired and till get a payoff from the productivity of hi competitor retained for the job eem both awkward and difficult to enforce. 14 Our model implicitly aume that the cot for the firm of organizing the creening proce i nil up to n tak and infinite afterward. Obviouly, optimizing over one variable the fraction of tak undertaken by each agent intead of two variable the number of tak undertaken by each agent coniderably implifie the technical apect of the model. 12

14 behind during the whole creening proce. 2.3 Principal welfare and optimal bia in the election rule Up to now, we have aumed that the tak allocation can be delegated to any agent. However, if we interpret our model a the competition for a poition between an incumbent and an outider, delegation to the latter doe not make much ene, a he i not even working in the firm. If the principal i forced to delegate the creening procedure and the incumbent i the only candidate for thi activity, Propoition 1 and 2 ugget that thi agent will chooe an allocation that dicloe too little information whenever he i relatively good. However, thi inefficiency will be alleviated if the principal can commit on the election procedure. Corollary 1 If the highet ability agent deign the creening proce, it may be optimal for the principal to commit to bia the deciion rule in favor of the lowet ability agent. Proof. Suppoe that m i = m o +, with > 0. By Propoition 2, information i minimized i = n. Information can be maximized if the principal commit ex-ante to the election rule w = arg max {i,o} {E[θ i Xi ], E[θ o Xo n ] + 2 } intead of 5. A 0, the cot of thi commitment goe to zero wherea the benefit of i = intead of i = n i trictly poitive. The agent who chooe the creening rule will either minimize or maximize revelation of information depending on whether he i on average more or le able than hi rival. Biaing the contet in favor of the leat able agent i formally equivalent to putting ahead the worker who i behind. If the difference between the expected abilitie of the agent i mall, the cot of committing to thi ex-pot inefficient deciion rule i mall, wherea the benefit of inducing the agent to maximize rather than minimize the amount of information revealed can be ubtantial. 15 Thi reult ha an intereting implication: organization that bia the promotion deciion againt their own employee and intead rely mainly on the outide labor market, are likely to perform better than organization in which inider worker are favored. It i alo counterintuitive, a in practice we frequently oberve the oppoite bia. 16 From the analyi of Section 2.1 and 2.2 we can alo determine whether the principal and the agent benefit from an accurate creening proce and a highly volatile ability of agent. Corollary 2 Independently of which individual chooe the allocation of the creening tak, the welfare of both the principal and the agent with lowet expected ability increae in the variance of 15 See Meyer 1991 for a comprehenive theoretical analyi of optimal deign of cutoff rule in contet. 16 One hould be careful when interpreting thi reult, a there may be other reaon mainly baed on incentive iue for the optimality of the oppoite bia. 13

15 the agent abilitie σi 2 and σo 2 and decreae in the variance of the noie σε. 2 Converely, the welfare of the agent with highet expected ability decreae in σi 2 and σo, 2 and increae in σε. 2 Proof. Immediate if we note that f σ 2 i > 0, f σ 2 o > 0 and f σ 2 ε < 0. The principal and the lowet ability want to maximize information revelation, wherea the highet ability agent want to minimize it. The amount of learning i greater when performance provide an accurate ignal about the capacity of the individual σ 2 ε mall and when the ability of the agent i very volatile σ 2 a high, becaue then the prior knowledge about their capacity i carce. Overall, the fact that the principal i going to keep only the bet agent make him endogenouly rik-lover with repect to the agent ability. 3 Screening with the quality of tak Once we have tudied the cot of delegation, we can analyze the iue of favoritim, defined a the poibility of chooing for the bet job the agent with lowet expected ability. We conider an extended verion of the model preented in Section 2. A before, the principal chooe between an incumbent/enior and an outider/junior with different and imperfectly known abilitie. The agent elected for the job or promoted receive an exogenouly fixed wage b while the other one get 0. The agent on-the-job productivity i equal to hi ability. Unlike in Section 2, we aume that there are only two creening tak available n = 2 and that each agent ha to perform one and only one of them = 1. We will generically ue the indexe α, β {h, l} with α β to denote the two tak. Simplifying the number of tak allow u to better focu on new iue, namely thoe related to vertical differentiation of the creening activitie and agent incentive to perform efficiently. The new feature of the creening proce are the following: i Screening tak are valuable, i.e. the performance of the agent in thee activitie enter directly in the utility function of the principal. ii The performance x α a of agent a allocated to creening tak α depend tochatically not only on hi ability θ a a before but alo on hi effort e a exerted which i not oberved by the principal and the type of creening tak α realized: x α a = νθ a, e a ; α 10 where, for all x R: a Pr [x α a < x] / θ a < 0, b Pr [x α a < x] / e a < 0, c Pr [x h a < x] < Pr [x l a < x], d Pr [x h a < x] / θ a < Pr [x l a < x] / θ a < 0. iii Both agent have the ame cot of exerting effort ce, with c > 0 and c > 0. According to thi formalization, performance in the creening tak i higher in a tochatic ene the higher the agent ability and effort part a and b. Furthermore, performance i alo 14

16 higher in tak h which tand for high than in tak l which tand for low part c. Lat, ability i alo relatively more valuable in tak h than in tak l part d. The pecific timing of the game that we analyze i the following. {i, o} {h, l} matching agent/tak effort x α i, xβ o e i, e o performance in creening tak election of winner by principal w J {i, o} agent wage b if a = w J 0 if a w J job productivity principal payoff θ w b time Figure 2. Timing in the quality of creening tak game. In thi game, the principal ha two objective: firt, to maximize the expected return of the um of the agent performance in the creening tak E[x α i + xβ o ] and econd, to keep the mot able one for the job. By ii-d, then abent effort conideration, the principal would allocate the agent with highet expected prior ability to the mot valuable creening tak. Formally, E[θ i ] > E[θ o ] Pr [x h i + x l o < x e a = 0] < Pr [x l i + x h o < x e a = 0]. Then, conditional on the information elicited, the agent with highet expected poterior ability would be retained for the job we denote w J the winner. However, ince different type of individual have different incentive to work hard, thi may no longer be true when effort conideration are taken into account. The objective of thi ection i to analyze the optimal allocation of agent to the different creening tak given thi moral hazard iue. Contrary to the previou ection, we will aume that the principal doe not delegate the allocation choice. 17 Remark 3. We have aumed for ymmetry with the previou ection that agent do not obtain any direct reward from their performance in the creening tak. Given that thi activity i now intrinically valuable, one could think that agent hould get a wage contingent on the type of the creening tak performed, with b h > b l > 0. A long a thee wage are not contingent on the outcome of the creening tak e.g. they are paid before performance i oberved, all the reult hold and ome other inight can be gained with thi extra ingredient. Remark 4. Intead of creening and election, one could eaily reinterpret our etting a a twoperiod, job-allocation model. In thi alternative game, two agent working in a firm are allocated according to their ability and anticipated effort to two job with different productivity h and l 17 Determining the agent preference over the different tak i trivial: if they receive no compenation for their performance in the creening tak, they are indifferent between h and l and if they are remunerated according to the type of creening tak realized ee Remark 3 below, they both prefer tak h independently of their ability. 15

17 and different payoff b h and b l. At the end of the firt period and given their performance x i and x o, there i reallocation between job h and l according to their updated ability. Remark 5. The model i an extenion of the two-period verion of Holmtröm 1999 career concern paper. By aumption, wage are not contingent on current performance, o the agent ha no incentive to put effort in order to increae it. However, ome effort might be incurred in order to bia the principal perception of the agent ability. A in a rat race, thi i perfectly anticipated by the principal and no bia occur in equilibrium. 18 The major difference i that in Holmtröm 1999 and all the ubequent literature there i perfect competition for agent, o in equilibrium individual are paid according to their expected ability θ a and anticipated effort e a. Intead, in our etup there i a fixed reward b for being elected for the job. Given thi payoff dichotomy, agent only recoup part of the benefit generated with their effort. Then, it i a priori unclear whether the tandard inight of the career concern literature alo hold in thi model. The literature on career concern ha demontrated that the pecific functional form of ν may affect the incentive of the different type of agent to exert effort. In the next ubection, we olve our model for the cae in which i the marginal effect of effort in the outcome doe not depend on the ability of the individual formally, 2 Pr [x α a < x] / e a θ a = 0 and ii effort i more valuable the higher the ability of the individual formally, 2 Pr [x α a < x] / e a θ a < Career concern when ability and effort are independent Thi cae which i the one analyzed in the paper by Holmtröm 1999 and almot all the ubequent literature i characterized by a production function additively eparable in effort and ability. The pecific functional form of the performance function ν that we are going to adopt ee 10 i: x α a = ρ α [ θ a + e a + u a ], u a i.i.d. N 0, σ 2 u and ρ h > ρ l > We proceed to the formal analyi of the game. Recall that ability i mot valuable in the mot productive creening tak, only the outcome and not the effort i oberved by the principal, and abilitie determine the agent future job productivity. Hence, the principal will elect for the job the agent with highet expected ability conditional on the performance and anticipated effort of both agent in the creening tak. Denote by ẽ a agent a anticipated effort. We have: w J = arg max {i,o} { } E[θ i x α i, ẽ i ], E[θ o x β o, ẽ o ] 18 Our reult would not be affected if we aumed that the performance of the agent elected for the job depended alo on hi effort. A in the tandard career concern model, the agent in the lat period here, once in the job ha no incentive to exert effort becaue there i no future perception of ability to bia

18 Agent will have incentive to exert effort even if they are not compenated for it ee Remark 5 or if the payoff for the creening activity i unk a in the interpretation given by Remark 3. In fact, for any level of effort anticipated by the principal, exerting effort influence current performance, which in turn affect the perception of ability by the principal, and therefore the probability of being elected for the job. Anticipating the principal election rule given by 12, and for a given allocation α, β of creening tak, the maximization problem of the agent are: [ ] e i = arg max Pr E[θ i x α i e, ẽ i ] > E[θ o x β o e o, ẽ o ] b ce e [ ] e o = arg max e Pr E[θ o x β o e, ẽ o ] > E[θ i x α i e i, ẽ i ] b ce From 11 and uing the ame technique a in the previou ection for updating belief, we get: x E[θ a x α α a e a, ẽ a ] = 1 γ a m a + γ a e a a ẽ a where γ a = σ2 a ρ α σu 2 + σa Given 12, 13 and the fact that x α a e a N ρ α m a + e a, ρ 2 ασa 2 + σu 2, one can how that: [ ] Pr[w J = i] = Pr E[θ i x α i e i, ẽ i ] > E[θ o x β o e o, ẽ o ] = Φ mi m o+γ i e i ẽ i γ oe o ẽ o gσi 2,σ2 o,σu 2 σi where g = 4 + σ4 σi 2 o. 19 Note that 14 doe not depend on how agent were allocated +σ2 u σo+σ 2 u 2 to tak ρ h and ρ l. The reaon i that the principal i able to dicount for the fact that, other thing being equal, the agent in tak h will exhibit a higher performance than the agent in tak l. In equilibrium, expectation mut be fulfilled, that i e a = ẽ a. 14 Uing 14, it i therefore immediate that equilibrium effort ẽ a are uniquely determined by: c γ a ẽ a = gσi 2, σ2 o, σu 2 ϕ mi m o gσi 2, σ2 o, σu 2 b 15 Note that ẽ a i proportional to γ a, and therefore ẽ o > ẽ i. From 14, the job will be offered to the incumbent/enior and to the outider/junior with the following equilibrium probabilitie: mi m o mo m i Pr [ w J = i ] = Φ gσi 2, σ2 o, σu 2 and Pr [ w J = o ] = Φ gσi 2, σ2 o, σu 2. Once we have determined which agent will be elected by the principal for the job, it i poible to analyze by backward induction the optimal allocation of agent to the different creening tak. The high-productive creening tak h will be executed by the agent atifying: } w S = arg max {m i + ẽ i, m o + ẽ o {i,o} Thi lead immediately to our next reult. 19 Thi function i the analogue of f in Section 2 when = 1 and n =

19 Propoition 3 It i optimal for the organization to favor the outider/junior agent in the allocation of creening tak. More preciely, tak h will be performed by agent o when m i m o [0, ẽ o ẽ i. Favoritim i more likely to occur, the higher b and the maller m i m o. Proof. According to 16, w S = o if m o > m i ẽ o ẽ i. By inpection of 15 and given that ẽ γ o > γ i, we obtain that ẽ o > ẽ i. Beide, o ẽ i m i m < 0 and ẽo ẽ i o b > 0. When two agent compete for a promotion, the incentive of agent a to bia the perception of hi ability i proportional to γ a σ 2 a/σ 2 a + σ 2 u. Indeed, when the initial knowledge about hi ability i very imprecie σ 2 a big, the ignal convey an important amount of information. In that cae, the agent ha a trong interet in biaing the principal perception of hi ability and, to thi purpoe, he exert a great amount of effort. Although no bia occur in equilibrium, thi provide an advantage to the agent with mot uncertain ability whenever effort enter directly in the performance function ν : thi agent i ometime elected for tak h even though hi expected ability i maller than that of hi rival. Formally, agent o i favored i.e. elected for the mot valued tak even if he i on average wore than hi opponent a long a m o m i + ẽ i ẽ o, m i. Suppoe that agent obtain a direct payoff from undertaking tak h ee Remark 3 or we interpret our model a a dynamic job-allocation game ee Remark 4. Other thing being equal, our model ha the counterfactual reult that in order to perform efficiently a firm hould aign the job requiring the highet degree of involvement to outider and young employee. To um up, young agent provide more effort than their old peer. Since the principal i partly reidual claimant for the agent effort, he will maximize the return of the firm by biaing the competition in their favor. Note that young worker will benefit from uch advantage only in the hort run if at all a noted by 12, the final election will be efficient and the individual eventually appointed to the job will alway be the one with highet expected poterior ability. Alo, note that if either σ 2 i = σ2 o or σ 2 u = 0, then for all pair m i, m o both agent exert the ame effort e i = e o. Thi prove that the key for the difference in effort between the two agent i their different incentive to bia future perception of ability, and not the exogenou difference in their prior expected abilitie. Lat, from Propoition 3 we can perform ome comparative tatic. Suppoe that the expected ability of the junior agent i maller than that of the enior one m o < m i. From 15 and 16, we can notice that both agent will put more effort if they are in a cloe race than if there i little uncertainty about who will be elected for the job m i m o < 0. Moreover, the increae in effort when the race i cloe will be greatet for the agent whoe ability i the leat known ẽ o ẽ m i m o < i m i m o. Similarly, a the value of the job, b, increae, the young agent i more likely to be favored becaue hi incentive to exert effort increae more than thoe of hi opponent. ẽ a 18