On the incentive effects of job rotation

European Economic Review 98 (2017) 424–441

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European Economic Review journal homepage: www.elsevier.com/locate/euroecorev

On the incentive effects of job rotationR Hendrik Hakenes a,b,∗, Svetlana Katolnik c a b c

Institute for Financial Economics and Statistics, University of Bonn, Adenauerallee 24–42, Bonn D-53113, Germany CEPR, 2nd Floor, 33 Great Sutton Street, London EC1V 0DX, UK Institute of Managerial Accounting, Leibniz Universität Hannover, Königsworther Platz 1, Hannover D-30167, Germany

a r t i c l e

i n f o

Article history: Received 30 June 2016 Accepted 10 July 2017 Available online 20 July 2017 JEL classification: D83 J24 L23 Keywords: Career concerns Job rotation Information Incentives Reputation

a b s t r a c t A new employee may work hard to build his reputation. This effect is greatest when he starts. The longer he is employed at his job, the more the firm will already have learned about his ability. The incentives for an employee to influence the firm’s perceptions of his ability decrease over time. If rotating the employee to a different job leads to new uncertainty about his ability, this generates a fresh impulse for effort. However, job rotation also reduces the employee’s time horizon, thus reducing future rents from reputation. This trade-off leads to a unique optimum. We derive a simple rule of thumb for an optimal rotation time. Our main results still hold for cases of complete but imperfect information transmission. The incentive effects of job rotation also prevail in a setting in which skills are job-specific. We study several extensions: rotations across multiple employees, absence of commitment to a rotation time, different bargaining positions, discounting over time, and learning by doing effects. © 2017 Elsevier B.V. All rights reserved.

1. Introduction In the presence of career concerns, talent uncertainty of employees plays a central role in the provision work incentives. The employees’ wish to develop a positive reputation and receive a pay raise serves as an incentive device: a new employee with uncertain ability wants to work hard to receive a positive assessment. However, an employee who has been at his job for an extended time cannot improve his reputation much more. After some time, there may be a need of a fresh start to create new incentives. Restricting access to past information may therefore be optimal when motivating effort is the goal. If rotating the employee to a different job leads to new uncertainty about the employee’s ability, this provides a new impulse for effort. However, this mechanism comes at the cost of lower incentives because the rotation is anticipated. The relevant

R We would like thank an associate editor and two reviewers for their valuable comments and suggestions. We also would like to thank Matthias Fahn, Dirk Jenter, Ian Jewitt, Matthias Kräkel, Christian Lukas, Sebastian Pfeil, Martin Peitz, Ulrich Schäfer, Jens Robert Schöndube, Barbara Schöndube-Pirchegger, Dirk Sliwka, and Stefan Wielenberg for helpful comments. Participants in the GEABA symposium in Magdeburg, the Colloquium on Personnel Economics in Köln, the Annual Conference of the German Academic Association for Business Research in Leipzig, the VfS Annual Conference in Hamburg, and the Workshop on Accounting and Economics in Odense provided helpful suggestions. Remaining errors are our own. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. ∗ Corresponding author at: Institute for Financial Economics and Statistics, University of Bonn, Adenauerallee 24–42, D-53113 Bonn, Germany. E-mail addresses: [email protected], [email protected] (H. Hakenes), [email protected] (S. Katolnik).

http://dx.doi.org/10.1016/j.euroecorev.2017.07.003 0014-2921/© 2017 Elsevier B.V. All rights reserved.

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aspect of the employee rotation is thus to reduce the amount of information. Other mechanisms may have similar effects, such as rotating the employee’s boss.1 In our model, we concentrate on the incentive aspects of job rotation, abstracting from any technological benefits such as learning on the job or searching for a good employee-job match. Like Cisternas (2017a), we use a continuous-time version of Holmström’s (1999) multi-period reputation formation model: a firm hires an employee with an initially unknown ability, and the employee exerts effort to improve the firm’s expectations of his ability, and therefore his future wages. In early periods, the employee’s type is known only vaguely, so the firm places substantial weight on performance (output) in updating its beliefs about the employee’s talent. As information accumulates, the employee’s type is revealed increasingly more, and new observations regarding output have less impact on perceived ability. Therefore, the incentives to invest in costly effort decrease over time. We extend Holmström’s (1999) work in three ways. First, we focus on intra-firm learning and assume that only the firm and the employee observe the employee’s performance. Second, the firm can use job rotation as a tool to artificially introduce some new uncertainty about ability. The role of job rotation in the model is similar to the stochasticity of the employee’s ability in Holmström’s (1999) model – it prevents the employee’s ability from becoming fully known, and thus keeps the employee’s incentives alive. Third, we endogenize the length of these time intervals to optimally balance the employee’s reputation incentives over time. There is a unique optimal rotation time, determined by an implicit equation. This optimal rotation time follows a simple rule of thumb: it is four times the ratio of output uncertainty to prior type uncertainty. Right after being rotated, the employee works 60 percent more than in the first best optimum. With the optimal choice of rotation time, the employee reaps 65 percent of the first best level of utility.2 To focus on the pure incentive effects, we first analyze a setting in which ability is constant across jobs, and in which job rotation is used to introduce new uncertainty regarding this ability. Below, we consider an environment in which the employee’s productivity can vary across jobs (Section 6). In this case, job re-assignments may be used to improve the quality of the employee-job match and the optimal rotation decision depends on output. Even when productivity is completely match-specific, the forces underlying the finite rotation time are still prevalent, and this may render contracting to a rotation time optimal. We also analyze how the optimal rotation time is influenced by the amount of information that is communicated to the new department (Section 4). We present different extensions that suggest the robustness of our model: if employees work in teams, the rotation time increases (Section 5.1). Commitment to the rotation time is not a necessary assumption (Section 5.2). More ex ante bargaining power with the firm increases the rotation time (Section 5.3). Rotating the employee is optimal if discounting is sufficiently low (Section 5.4). Finally, if the employee becomes more skilled over time, the rotation time depends on the type of on-the-job learning (Section 5.5). Literature. In addition to the fundamental Holmström (1999), the most closely related work is Höffler and Sliwka (2003). Their article shows that installing a new manager means deleting information about employees’ past performance. If employees compete for a promotion, this competition is re-invigorated. However, the effects of this deletion may lead to the wrong employee being promoted. In Höffler and Sliwka (2003), managers are replaced, but rotating the employees can serve the same purpose. A tournament between employees is at the heart of the argument. However, our paper does not require a tournament, as a single employee is sufficient. Replacing the manager, instead of rotating employees, would also work in our model. Under both approaches, artificially introducing uncertainty about ability is key.3 The implicit incentives we study are comparable to other models on career concerns, beginning with Fama (1980); Ricart i Costa (1988) and Gibbons and Murphy (1992). Following the extensions by Dewatripont et al. (1999a, 1999b), some recent work has examined the implications of career concerns for organizational transparency (Arya and Mittendorf, 2011; Koch and Peyrache, 2011; Mukherjee, 20 08a, 20 08b) and task design (Bar-Isaac and Hörner, 2014; Ortega, 2003). Consistent with the basic notion that improved information may reduce the strength of future incentives, this paper investigates the role of the firm’s organizational assignments on learning about types and the incentives provided to employees.4 The general learning mechanism used in this paper is related to the seminal job-matching article of Jovanovic (1979). In his 1 The practice of job rotation is pervasive and increasing. Osterman (1994, 20 0 0) documents that in 1997, 55.5% of large and medium-sized private sector firms in the U.S. implemented job rotation for more than half of their core employees, which is up from 26.6% in 1992. A 1993 U. S. survey by Gittleman et al. (1998) reports adoption rates of 24.2% for firms with 50 or more employees. In a survey of Danish private sector firms, Eriksson and Ortega (2006) show that rotation schemes were implemented for 19.5% of hourly paid workers with variations by firm size and industry. 2 We have set our model in a pure rational framework, and the employee builds a reputation to earn higher wages. Other interpretations are close and yield comparable results. Assume, for example, that a person wants to be likeable in a group. Likeability is influenced by character, but the person can also take an action to seem more likeable. Then, if the person is new to a group, he will put in a high effort, trying to increase his likeability. Over time, the true character is learned by the group, and the effort does not pay off any more. One can also think of a mixed situation, where an employee works hard in order to be liked by his boss. In that case, one would have to put reputation instead of the monetary wage into his utility function. 3 Our theoretical predictions differ in two dimensions. First, in the tournament setting of Höffler and Sliwka (2003), wages have an option-like feature in that the expected increase in compensation depends on an employee’s probability of promotion. Therefore, incentives are determined by the expected heterogeneity of employees’ abilities. This force is not present in our model, as wages proportionally increase with an employee’s posterior reputation. Second, in Höffler and Sliwka (2003), the ability risk (associated with the expected difference in employees’ uncertain talents) influences the utility by means of an incentive effect and an assignment quality effect. In our model, the ability risk (associated with the employee’s uncertain talent) influences the utility by means of divergent incentive effects. 4 The optimality of coarse information has also appeared in the performance feedback literature (see, for instance, Martinez, 2009; Ederer, 2010; Hansen, 2013), studying how incentives are affected by the release of interim performance information.

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continuous-time setting, contract length and the suitability of an employee for a certain job depend on the employee’s ability as ability is revealed through past performance information. Our paper also relates to other models of dynamic agency that are based on continuous-time contracting. Following the seminal work of Holmström and Milgrom (1987) and the more recent contribution by Sannikov (2008), our approach is most closely connected to models that use continuous-time information to learn about unknown profitability over time. Examples include Faingold and Sannikov (2011) for a model on reputation dynamics, DeMarzo and Sannikov (2017) and He et al. (2017) on optimal long-term contracting with learning, Giat and Subramanian (2013) on imperfect public information and asymmetric beliefs, Cisternas (2017b) on private belief effects and ratcheting, and Bonatti et al. (2017) on the relation between market structure and career concerns. Our approach is similar to Cisternas (2017a), who considers a continuoustime analogue of Holmström’s (1999) model in which skill accumulation is endogenous. In a linear Gaussian setup with additively separable output technology comparable to ours, there is a deterministic equilibrium in which career concern incentives are determined by the sensitivity of beliefs about ability to unexpected output. Since the uncertainty about ability decays as learning accumulates, the firm’s beliefs become increasingly less responsive to new output surprises, which causes the employee’s effort to decrease over time. We prove an equivalent result, but our emphasis is on an endogenous duration of job assignments. Thus, the equilibrium effort in our model represents a finite time analogue of the career concerns result (without strategic skill accumulation) of Cisternas (2017a) for the cases of constant ability and no discounting over time.5 Models of the long-run sustainability of reputation incentives emphasize the importance of mechanisms through which type uncertainty is replenished (Cripps et al., 20 04; 20 07). There are many ways to model such mechanisms. One is simply to allow for variations in productivity over time (similar to Holmström’s (1999) model with changing types), as is undertaken in Benabou and Laroque (1992); Cole et al. (1995); Mailath and Samuelson (2001); Phelan (20 06); Wiseman (20 08) and Board and Meyer-ter-Vehn (2013). Another way is to assume that there are limits to information accumulation by endowing players with finite memory (Liu and Skrzypacz, 2014), by slowing down or reducing the release of information (Jeon, 1996, Dellarocas, 2006, Deb and Bar-Isaac, 2016), by imposing costs for observing history (Liu, 2011), or by adding a third player that precludes complete learning (Ekmekci, 2011, Hörner and Lambert, 2016, Pei, 2016). As with this approach, our model studies the role of organizational structure in re-introducing uncertainty to sustain reputation incentives. Our paper can also be thought of as a step towards understanding the trade-offs between the divergent incentive effects that are created by means of organizational reassignment: frequent rotations often create uncertainty and new incentives, but they also limit the payback horizon, and thus lead to lower incentives over time. 2. The model We introduce a principal-agent framework with continuous time and infinite horizon. Consider a labor market that has many identical firms and many employees, with discount rate ρ < 1 close to 1. There are many different jobs in each firm such that a large number of employees can be hired. Unfilled jobs are always available so that employees have the bargaining power. At the initial date t = 0, a firm owner and an employee (both risk neutral) sign an employment contract. Most importantly, the contract fixes a rotation time T: after T periods, the employee will be assigned to a different job within the firm.6 Aggregate output Yt that is produced up to time t ∈ [0, T] is a linear function of the employee’s constant ability η, 7 his unobservable choice of efforts {aτ }0 ≤ τ ≤ t , and a noise component,



Yt =

t 0

(η + aτ ) dτ + σ Zt ,

(1)

where σ > 0 is a constant and {Zt }t ≥ 0 is a standard Wiener process. The employee’s ability η is not known with certainty: the common prior belief is that η is normally distributed with mean η0 and variance s20 . By performing a job, the firm and the employee symmetrically learn about η by observing the employee’s performance over time. The desire to manipulate the firm’s beliefs influences the employee’s effort. However, effort is costly with a strictly convex quadratic cost per unit of time, C (at ) = c at2 /2, where c > 0 is a constant. The first-best effort choice is thus aFB = 1/c. The incremental news contained in Yt over a time step dt can be summarized by

dYt = (η + at ) dt + σ dZt ,

(2)

which represents the increase in output as a result of the employee’s ability, his effort and the noise term. Once T is reached, the employee is assigned to a new job. We assume that the employee’s old division cannot communicate the previous performance information to the new division. In Section 4, we relax this assumption and and assume imperfect communication. Our approach is in line with the related career concerns literature (see, for instance, Bar-Isaac, 2007), which assumes that an employee’s productive history is only observable until he is relocated. Our model can also be reinterpreted to encompass 5 A further comparison to Holmström (1999) is given in Prat and Jovanovic (2014), who employ a continuous-time framework with binary effort and show that Holmström’s (1999) result can be expanded to include risk aversion. Consistent with our model’s findings, an employee’s equilibrium effort under spot contracting converges to zero as soon as the beliefs about the employee’s type become sufficiently precise. 6 While our analysis focuses on a representative employee, all results carry over to the case in which multiple employees work on different individual jobs and in which these employees are rotated across jobs within the firm, see Section 5.1. 7 The assumption of a constant ability is relaxed in Section 5.5.

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Fig. 1. Model timeline.

multidimensional abilities, allowing the employee to be good at some jobs and bad at others, see Section 6. In such a setting, job rotation is used as a tool to search for the best employee-job match; thus, job rotation does not need to reduce the amount of information processing. Ex post, it may turn out that an employee’s ability is low. Moreover, effort incentives decrease over time. Thus, to sustain reputation incentives through job rotation, a notion of commitment is needed. Like Milgrom and Roberts (1988) and Waldman (2003), we assume that the firm and the employee can credibly commit to an assignment rule. Suppose that the choice of T is publicly verifiable, while wages and outputs are not. In this case, the credibility of commitment can easily be sustained explicitly (e. g., by relying on court enforcement). In Section 5.2, we relax the verifiability assumption and show that credible commitment can also be sustained implicitly by the value of future relationships (i. e., by relying on a relational contract).8 To make our analysis comparable to Holmström (1999), let us assume that explicit incentive contracts, linking wage to output, cannot be written. As in Holmström’s (1999) model, there are sufficiently many homogeneous firms at date t = 0 that are bidding for the employee’s service. The firms anticipate the employee’s optimal rotation time. However, we focus on intra-firm learning and assume that only the firm and the employee observe the employee’s performance, and that wages are determined by bargaining.9 This approach differs from Holmström (1999), where the firm and the market have precisely the same information so that with labor market competition, wages are determined by the employee’s market reputation. For the sake of simplicity, we assume that the employee has the ex interim bargaining power, making a take-it-or-leave-it offer to the firm in each period. However, the choice of the optimal rotation time does not depend on how the surplus in ex interim renegotiation is split between the firm and the employee, provided that the employee’s reputation remains a matter of concern (only the ex ante bargaining power matters, see Section 5.3). In our case, each optimal contract offer that will be accepted by the firm specifies that the employee receives a wage wt equal to his expected marginal product, conditional on the observed output path {Yτ }0 ≤ τ ≤ t until the current date t. Thus, if the firm anticipates that the employee reacts to the path {Yτ }0 ≤ τ ≤ t by exerting effort aˆt , it will pay the employee the wage

wt := Etaˆ [η + at ] = ηˆ t + aˆt ,

(3)

where Etaˆ [·] is the conditional date-t expectation operator and ηˆ t := Etaˆ [η] is the firm’s estimate of the employee’s type. Here, the hat stands for anticipated variables, a variable without a hat stands for the realization. Consequently, if the employee receives wage wt at date t and exerts effort at , he will obtain the instant utility

ut := wt − C (at ) = ηˆ t + aˆt − c at2 /2. The employee’s objective is to maximize the aggregate expected utility. The utility until date T is that the aggregate utility is

U :=

∞ 

ρi T

i=0

 0

T

ρ t ut dt.

(4) T 0

ρ t ut dt, which implies

(5)

At the beginning of the Appendix we prove that, if ρ is close to 1 such that the role of discounting is negligible, maximizing U is equivalent to maximizing the long-run average undiscounted utility,

U¯ :=

1 T



T 0

ut dt =

1 T



T 0

(wt − C (at )) dt.

(6)

We discuss the effect of smaller discount rates in Section 5.4. Each job assignment until the time T of a rotation has exactly the same structure, see Fig. 1.

8 Credible commitment can also be achieved by means of technologically determined constraints, when re-examination of the initial rotation time leads to costly changes in production. This notion is particularly intuitive if we require that the frequency with which changes to the production technology are made is lower than the frequency with which the firm updates beliefs about the employee’s type. 9 Other models of career concerns that incorporate limited observability by the market include Arya and Mittendorf (2011); Jeon (1996); Meyer (1994); Ortega (2003) on team production problems, and Mukherjee (20 08a, 20 08b) on strategic information disclosure. A detailed survey of this topic is provided by Gibbons and Waldman (1999).

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3. Equilibrium 3.1. Bayesian learning and incentives t Consider an employee at date 0 ≤ t ≤ T, after the employee has previously invested an aggregate effort of At := 0 aτ dτ . With the anticipated level Aˆt of aggregate effort, the resulting output Yt , given in (1), is normally distributed with mean

η0 t + Aˆt and with variance

s20 t 2 + σ 2 t.

Using Bayes’ theorem for normally distributed random variables, we obtain the following expressions for the firm’s posterior mean, ηˆ t , and the posterior variance, st2 := Vart [η], of the employee’s ability:

ηˆ t = Etaˆ [η] = E[η|Yt , Aˆt , t] =

σ2 s20 t

+σ

2

η0 +

  

+σ

Yt − Aˆt , t

(7)

1 − qt t

(8)

=:1−qt

1 s−2 0

+σ

2

  

=:qt

st2 = Vart [η] = Var[η|t] =

s20 t s20 t

=

−2 t

σ

s20 2 2 s0 t + 2

σ

=

σ 2.

The employee’s perceived type consists of a weighted average of the prior η0 and of an estimate of ability derived from output Yt . The sum of both weights is 1 in each case; thus, the weight on the prior, denoted qt , increases if the new information is very noisy (large σ 2 ). By contrast, if the prior is very noisy (large s20 ), then more weight 1 − qt is placed on the output measure. The firm does not observe the employee’s actions, so its beliefs are based on aggregate anticipated effort, Aˆt . The employee also knows his actual efforts such that his beliefs are determined by (7), in which At and not Aˆt is included, and by (8). In equilibrium, the firm’s Bayesian updating is based on a correct anticipation, aˆt = at∗ for all t ∈ [0, T], where the asterisk is used to denote equilibrium values. However, the potential belief divergence allows the employee to manipulate the firm’s beliefs. More information always reduces uncertainty: the longer the employee works at a job, the smaller is the variance s20 t 2 + σ 2 t of the output measure relative to time t. In other words, the more time passes, the more obvious the employee’s true ability η becomes, and the less noise st2 there is. In the extreme case, if tenure tends to infinity, t → ∞, then the employee’s expected ability converges to his real ability, ηˆ t → η, and its variance converges to zero, st2 → 0, which thus implies that the variance of the employee’s wage decreases over time and tends to zero, as t → ∞. To determine the equilibrium effort, consider an infinitesimal time interval, [t , t + dt ], where the employee chooses to deviate from the anticipated effort aˆt in favor of an alternative effort at > aˆt . With preferences as in (6), and given the information at time t, the employee’s aggregate future expected utility (continuation value, cf. Sannikov (2008)) is

Vt := Eta =

Eta

= Eta



t

T

 T



(wτ − C (aτ )) dτ = Eta (ηˆ τ + aˆτ − C (aτ )) dτ t

 T  t

qτ

 T  t

qτ

η0 + ( 1 − q τ ) η0 +

1 − qτ

τ

Yτ − Aˆ τ

 τ 0

τ

a2 + aˆτ − c τ 2



dτ

(dYv − aˆv dv ) + aˆτ − c

a2τ 2





dτ .

(9)

An increase in the current effort and in the costs of effort has two effects. The first effect captures an instantaneous performance improvement (see (2)): additional effort at − aˆt at time dt increases short-term performance dYt by (at − aˆt ) dt. The second effect is associated with the persistent manipulation of beliefs (see (7)): the firm that expects the employee to exert effort level aˆt < at will mistakenly attribute the higher output dYt − aˆt dt to a higher ability value ηˆ t . This belief manipulation is beneficial for the employee, since he will be rewarded for higher ability in the future. The employee’s problem thus consists of maximizing Vt in (9), depending on two offsetting influences: the benefit from distorting the firm’s ability beliefs, which is captured by the employee in the form of additional wages wτ for all dates τ ∈ [t, T], and the loss associated with the employee’s current disutility of effort, which is determined by the cost function C(at ). t t The firm updates its ability beliefs based on the component Yt − Aˆt = 0 dYτ − aˆτ dτ = 0 (η + aτ − aˆτ ) dτ + σ Zt of the output history which is not explained by anticipated effort. A unit increase in the current unexpected effort at − aˆt for an instant dt results in the same dt unit increase in Yt − Aˆt . Accordingly, if Yt − Aˆt rises by one unit, then the firm’s posterior ηˆ t of the employee’s ability will rise by (1 − qτ )/τ units. Hence, we can interpret (1 − qτ )/τ as the sensitivity of beliefs to unexpected output (cf. Cisternas, 2017a).10 An increase in Yt − Aˆt increases ηˆ t at all future dates until rotation. Thus, the employee considers not only the initial reaction of the firm’s beliefs to output changes but also the endurance of these 10 The sensitivity (1 − qt )/t is deterministic because the variance st2 of the posterior ability beliefs decreases deterministically over time (see (8)). Wages are linear in the firm’s posterior ability beliefs ηˆ t , so the employee’s effort in our model is deterministic as well, like in Holmström (1999) and Cisternas (2017a).

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429

Fig. 2. Employee’s equilibrium effort at∗ as a function of time t. Note: Here and in the rest of the paper, parameters are σ = 1, s0 = 1, c = 1, and η0 = 0.

beliefs. Accordingly, the value of future wage benefits is obtained by multiplying the output shift (at − aˆt ) dt by the marginal benefit (1 − qτ )/τ , and by aggregating it for all dates τ ∈ [t, T].11 Using (7), we can write the employee’s problem as

max (at − aˆt ) dt

{at }0≤t≤T

= max (at − aˆt ) dt {at }0≤t≤T





T

1 − qτ

τ

t T

t

s20

dτ − C (at ) dt

s20 τ + σ 2

dτ − c

at2 dt. 2

(10)

The first-order condition yields the employee’s optimal effort at∗ . This effort does not depend on the firm’s anticipated effort aˆt , or on the employee’s true type, or his beliefs about it. It is summarized in the following proposition. Proposition 1. The employee’s equilibrium effort is

at∗ =





s2 T + σ 2 1 log 02 . c s0 t + σ 2

(11)

Fig. 2 outlines three basic characteristics of the employee’s equilibrium effort, at∗ . First, a comparison of the two panels reveals that the greater the prior type uncertainty s20 , the more the employee must prove himself and the more effort he spends in equilibrium. This contrasts with the uncertainty σ 2 of the performance measurement: the larger the σ 2 , the less is learned about the employee’s type, and the smaller are the incentives to invest in increasing visibility. Second, the equilibrium effort increases when the rotation time is extended because the larger the payback period T − t, the longer the employee can benefit from a wage increase based on increased effort. Thus, for small values of t, effort is high. As time reaches t = T , the employee does not expend any effort at all. Third, the functions are convex in t. If the second effect were the only prevalent time effect, then the effort should be linear in T − t. However, it is not, which means that the additional information that is aggregated over time plays an important role. Initially, for small values of t, beliefs about the employee’s type are imprecise. Hence, the employee’s incentives to increase beliefs about his type are great. For larger values of t, the employee’s reputation is firmly established, and there is thus only limited room for manipulating beliefs. This is the standard learning effect, which implies that returns to effort decrease as t increases because the sensitivity (1 − qt )/t of beliefs to unexpected output is lower when the ability is known more precisely. 3.2. Timing of rotation We thus arrive at our main question: what is the optimal rotation time T∗ ? On the equilibrium path, the firm infers the employee’s ability η from the true optimal level of effort, aˆt = at∗ for all dates t ∈ [0, T]. Since the firm and the employee share the same information sets, (Yt − Aˆt )/t has mean η0 , and the firm’s posterior belief regarding η also has mean η0 . The firm’s profits are zero, so the expected equilibrium wage corresponds to the employee’s perceived mean productivity, E[wt∗ ] = η0 + at∗ . Incorporating the equilibrium effort at∗ of (11), the employee’s expected date-t utility is given by

ut∗ = E[wt∗ ] − C (at∗ ) = η0 + at∗ − c at∗ 2 /2 = η0 +







s2 T + σ 2 s20 T + σ 2 1 1 log 02 − log c 2c s0 t + σ 2 s20 t + σ 2

2

.

(12)

11 Although (1 − qτ )/τ is not the actual marginal rent in connection with effort, as there is no room for belief manipulation in equilibrium, it is used to establish incentive compatibility.

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Fig. 3. Employee’s expected average utility U¯ as a function of the rotation time T.

Using this expression in (6), we can write the employee’s date-0 maximization problem as

max U¯ = max E T

1  T  T

T

1 = max T T T

0



0

 

a∗ 2 1 T (wt∗ − C (at∗ )) dt = max η0 + at∗ − c t dt T

T





0

2



s2 T + σ 2 s20 T + σ 2 1 1 η0 + log 02 − log c 2c s0 t + σ 2 s20 t + σ 2

σ2

η0 +

= max

T

2 c s20 T

log

s2 T 0

σ2

+1

2

2 dt

.

(13)

The employee’s wages are strictly increasing and concave in T, and the effort costs are strictly increasing and convex. The objective function (13) is thus strictly concave in T so that there is a unique optimum for the rotation time. The first-order condition is

0=

∂ U¯ ∂ = ∂T ⎡ ⎢

=⎣

T 0

σ 2 log



wt∗ dt/T − ∂T

s20 T ∗

σ

2

+1

∂



c s20 T ∗ 2

−

T 0

C (at∗ ) dt/T ∂T

s20 T ∗

σ +1

−1 ⎤ ⎡

2

c T∗

⎥ ⎢ ⎦−⎣

σ 2 log



s20 T ∗

σ +1



2

s20 T ∗

σ +1 2

c s20 T ∗ 2

−1  −

s20 T ∗

σ +1 2

c T∗

−1 +

σ 2 log



s20 T ∗

σ +1 2

2 c s20 T ∗ 2

2 ⎤ ⎥ ⎦. (14)

Combining the first terms in squared brackets, canceling out the second, and factoring out identical expressions, (14) can be simplified to

0= =



s2 T ∗ 1 log 0 2 + 1 ∗ cT σ 2 s20

T

∗

σ2

−

 s2 T ∗ 0

σ2

 s2 T ∗ 0



+1

σ s2 T ∗ 2

+ 1 log

0

σ2

−1

−



+1 .

σ2 2 s20

T∗

log

s2 T ∗ 0

σ

2

+1



(15)

We obtain the following result. Proposition 2. The optimal rotation time is

T ∗ = T0

σ2 s20

,

(16)

where T0 ≈ 3.922 is the unique solution of 2 T0 − (T0 + 1 ) log[T0 + 1] = 0, σ 2 is the noise in the output function, and s20 is the initial type uncertainty. Fig. 3 shows that the employee’s expected average utility U¯ is non-monotonic over the rotation time T. Specifically, U¯ is influenced by the employee’s prior estimation η0 of ability, the cost factor c, and the variance ratio σ 2 /s20 . The variance ratio provides a simple rule of thumb for the optimal rotation time, T ∗ ≈ 4 · σ 2 /s20 . Intuitively, the better the employee’s type is already known ex ante compared to the noise σ 2 of performance measurement, thus the smaller s20 is, the lower employee rotation is needed. To see this, consider the two marginal effects on the employee’s utility U¯ (see (14)): at the optimum, an increase in the output noise σ 2 decreases the marginal costs more than the benefits, so it becomes optimal to increase

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T∗ . With increasing noise s20 of ability, the marginal costs increase more than the benefits; thus, T∗ decreases. Why? Even though a larger type uncertainty s20 increases the variance s20 t 2 + σ 2 t of the output measure, and thus implies a higher posterior variance st2 of the employee’s ability, it also increases the weight 1 − qt placed on output in updating beliefs about the employee’s type, and with it the employee’s incentives. Therefore, if the employee’s type is virtually unknown (large s20 ), then he has an incentive to work hard, particularly at the beginning, until more information is revealed. The employee can then only achieve optimal workload by cutting down the time T until rotation, i. e., by being rotated more frequently. If the employee’s type is virtually known (small s20 ), then his incentives to work are small and suboptimal. Consequently, he must be incentivized to expend more effort. In fact, he also wants to be incentivized more, because he ultimately reaps all the benefits himself. This can be achieved by choosing a long rotation time, so the incentives to increase beliefs about his type are higher. In the extreme case, if the type uncertainty tends to zero, s20 → 0, the optimal rotation time tends to infinity, T∗ → ∞. Consequently, in our model, the benefits of job rotation apply only if there is a measure of career uncertainty. As an immediate corollary from Proposition 2, the effort is larger than the first best action at the beginning, at t = 0, and then becomes smaller than the first best action (even zero) towards the rotation time T. This can be seen by substituting T∗ into at∗ from (11), which yields a∗0 ≈ 1.594/c. In other words, right after being rotated, the employee works about 60 percent more than he should. There is a second corollary. Substituting the optimal rotation time T∗ into the employee’s target function (13) yields

U¯ ∗ = η0 +

σ2 2 c s20

T∗

log

s2 T ∗ 0

σ

2

+1

2

= η0 +

log[T0 + 1]2 0.324 = η0 + . 2 c T0 c

(17)

The utility depends on the average ability η0 plus a term that is inversely proportional to the cost factor c. By choosing the optimal rotation time T∗ , the employee can gain 0.324/c, independent of the variances s20 and σ 2 . Remember that the first-best effort choice is aFB = 1/c, leading to a utility of U¯ FB = η0 + 0.5/c. Hence, the optimal choice of rotation time yields 0.324/0.5 = 64.8 percent of the first-best utility level. If one deviates from the optimum T∗ , this gain decreases depending on the fraction σ 2 /s20 , and approaches zero as a limit.

4. Partial information transfer When the employee rotates within the firm, typically some of the information will be passed to the employee’s new division, but not all: there are job reference letters, there may be rumors, the superiors of the employee may have communicated in advance, and so on. Thus, instead of our former assumption that the employee’s new division does not receive any information from the employee’s previous job, let us now assume that the information is communicated to the new division, but with some noise. We especially want to analyze how the level of noise influences the optimal rotation time. Assume again that the employee has ability η. The prior at date 0 has mean η0 and variance s20 . Until the first rotation, the employee produces an output YT . We assume that the old division writes a reference letter for the employee based on YT ; it communicates YT plus a noise term  that has zero mean and variance ν 2 . Hence, with ν 2 → 0, we capture the case in which the reference letter is perfectly informative. The baseline model obtains if we let ν 2 → ∞, in which case the reference letter does not contain any information. Potentially, the rotation time will not be constant over time. We therefore consider the steady state; that is, a phase in which the additional information is cancelled out exactly by the additional noise in each job. Let t = 0 denote the rotation time in the steady state. Also, assume that the information in the reference letter follows a normal distribution with mean μr and variance σr2 . These variables are endogenous to the model. We now show how σr2 depends on ν 2 . At each rotation time T, the employee leaves his old job, takes his reference letter, and starts a new job. The new division’s updated prior belief of the employee’s ability is normally distributed with mean and variance

η0,new := E0 [η | σr2 ] =

s20,new := Var0 [η

σr2

s20 + σr2

| σr2 ] =

s20 σr2 s20

η0 +

+ σr2

.

s20

s20 + σr2

μr ,

(18)

(19)

If the reference letter is extremely precise, σ r → 0, then the prior variance goes to zero, s20,new → 0, and there is thus no learning. In contrast, if the reference letter is extremely noisy, σ r → ∞, we are back to the baseline model, η0, new → η0 and s20,new → s20 .

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Now the employee starts to work. This adds another signal Yt with mean η0,new t + Aˆt and with variance s20,new t 2 + σ 2 t to the division’s information. At date t, after observation of output Yt , the division’s posterior ability beliefs are characterized by

σ2

ηˆ t,new := Etaˆ [η | σr2 ] =

s20,new t



+σ



=:qt,new

2 and by st,new := Vart [η

2

s20,new t



 s20,new σ 2

| σr2 ] =

+σ

s20,new t

(Yt − Aˆt )

s20,new t

η0,new +

2

+σ



=:1−qt,new

=

(20)

t

2



1 − qt,new t

σ 2.

(21)

At the date of rotation, t = T , the posterior variance is given by

s2T,new =

1 − qT,new T

s20 σr2 σ 2

σ2 =

s20 σr2 T + (s20 + σr2 ) σ 2

.

(22)

Because the noise term has variance ν 2 , we have σr2 = s2T,new + ν 2 , and thus s2T,new = σr2 − ν 2 . Plugging this into (22), we can eliminate s2T and solve for σr2 ,

ν

σ = 2 r

2



ν2 + 4

ν+



s20 σ 2 s20

T +σ

(23)

2

as the steady state value of the variance of reference letters. In each new job, information is added because the employee’s performance is observed, and information is lost because of the noise in the reference letter. The more time the employee remains at his job, the more precise the reference letter becomes. If T → ∞, then σr2 ≈ ν 2 asymptotically. From the employee’s perspective, an additional effort at − aˆt at time instant dt will increase future wages through two channels. First, the employee will immediately influence his current division’s beliefs about his type, and hence his future wages until the next rotation time T. Like in (10), the expected aggregate wage benefits are determined by the sensitivity (1 − qt,new )/t of beliefs to unexpected output. These benefits are given by (at − aˆt ) dt times



T

1 − qτ ,new

τ

t

dτ = dt



s20,new 2 s0,new +

T

τ

t

σ2

dτ = log

s2

0,new T s20,new t

+ σ2

+ σ2

.

(24)

Second, because of the additional effort, the reference letter will also be marginally better. Specifically, YT − Aˆ T increases by (at − aˆt ) dt; hence, the current division’s final assessment ηˆ T,new increases by (at − aˆt ) (1 − qT,new )/T . The reference letter for the next division is improved by the same amount, and the new prior η0, new goes up by that amount times the factor s20 /(s20 + σr2 ) (see (18)). In (20), this is multiplied with qt, new . Aggregated over the period [T, 2 T], the expected impact on the employee’s wage is (at − aˆt ) dt times

s2 1 − qT,new · 2 0 2 T s 0 + σr = =

s20,new s20,new

T +σ

·

2

s20,new s20,new

·

T + σ2



2T

q(τ −T ),new dτ =

T



s20

s20

+σ

2 r

s20

s20

+σ

2 r

T



T 0

qτ ,new dτ

σ2 dτ τ + σ2 s2 T + σ 2

0,new log . σ2

s20,new

0

·

s2 1 − qT,new · 2 0 2 T s 0 + σr

σ2 s20,new

(25)

Now at date T, the new division writes another reference letter for the next division; thus, the employee expects that the additional effort at − aˆt at date dt also influences his wage in the next job, but this time with a squared factor,

1 − q

T,new

T

·

2 

s20

s20

+σ

2 r

0

T

qτ ,new dτ .

(26)

This is repeated, a geometric series emerges. The expected aggregate additional wage benefits through all future reference letters are (at − aˆt ) dt times

φ

1−φ

·

σ2 s20,new

log

s2

0,new

T + σ2

, 2

σ

with

φ=

s20,new 2 s0,new T +

σ

2

·

s20

s20

+ σr2

.

(27)

Note that for large values of ν 2 , the variance of the reference letter σr2 is also large and, consequently, φ is small. This means that if the reference letter is very noisy, it will only marginally affect the employee’s efforts. At each date t, the employee’s equilibrium effort at∗ is given by the first-order condition of the maximization of





max (at − aˆt ) dt (24 ) + (27 ) − c at2 /2 dt

{at }0≤t≤T

⇔





at∗ = (24 ) + (27 ) /c.

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433

Fig. 4. Effect of information noise ν on effort at∗ .

Fig. 5. Effect of information noise ν on utility U¯ and rotation time T∗ .

Given at∗ , the expected instantaneous utility is ut∗ = η0 + at∗ − c at∗ 2 /2. The employee maximizes the expected average utility, T ∗ maxT U¯ = u dt/T . The explicit solution of the first-order condition, dU¯ /dT = 0, is too complex, but the expression for U¯ 0

t

is in closed form. Fig. 4 illustrates the equilibrium effort at∗ for two numerical examples: ν = 1 (gray curve), which is the model with information transmission, and ν → ∞ (black curve), which corresponds to our baseline model. If the reference letter is extremely noisy, ν → ∞, the effort is exactly as in Fig. 2. For smaller values of ν , and thus a more informative reference letter, the optimal effort is smaller at the beginning and larger at the end of the employee’s assignment. Intuitively, as the employee wants to influence his reference letter, he works even right before the rotation. However, right after the rotation, it is harder for him to manipulate the new division’s beliefs; hence, he works less. Fig. 5 shows the employee’s expected average utility U¯ (left panel) and the optimal rotation time T∗ (right panel) under varying levels of the information noise ν . More information transmission between jobs leads to a higher average utility. The rotation time still matters: for both T → 0 and T → ∞, the average utility converges to zero. If the reference letter is very noisy, ν → ∞, the model converges to our baseline model; hence, the optimal rotation time converges to T ∗ = 3.922 σ 2 /s20 . For intermediate values of ν , the optimal T∗ increases. If the reference letter is very precise, the optimal T∗ decreases and converges to zero as ν → 0. Intuitively, to restore incentives, the employee needs to rotate very often such that his true type is never revealed with too much precision. Hence, there is a ν (here, ν ≈ 1.1) for which T∗ becomes maximal. In the limit of a very informative reference letter, ν → 0, the first-best utility level can be reached if the employee rotates fast. 5. Extensions 5.1. Multiple employees Job rotation schemes are typically promoted within large firms in which multiple employees perform a wide range of tasks and in which these employees are periodically reallocated across different jobs.12 Depending on the application, it is 12

See Gittleman et al. (1998) and Papageorgiou (2016) in support of a positive association between firm size and the implementation of job rotation.

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Fig. 6. Effect of team size n on effort at∗ and rotation time T∗ .

possible that employees perform different individual jobs and routinely rotate these jobs among themselves. Furthermore, a job could be performed by more than one employee, and rotations could take place within teams. Let us discuss a few concrete cases. First, take our basic model but assume that the firm has an infinite number of jobs and n > 1 employees. Then, the rotation time will be exactly the same as in the baseline model. Second, assume there are n > 1 employees and m ≥ n jobs. Then, an employee can only rotate m − 1 times, and his abilities will be revealed gradually so that his effort converges to zero. Third, assume that there are n > 1 employees and m ≥ n jobs and suppose that for each period τ , one of the employees (or one of the jobs) is replaced. Then, the rotation time cannot be longer than m τ /n. With multiple employees jointly working on one task, the rotation problem changes to some degree, especially if the firm cannot observe an individual employee’s output. In line with the literature on reputation in teams (see Jeon, 1996; Arya and Mittendorf, 2011; Chalioti, 2016), consider a production technology that is additive in individual inputs,

Yt (n ) =

 0

t

(η1 + . . . + ηn + a1τ + . . . + anτ ) dτ + σ Zt ,

(28)

where the superscript denotes the employee (team member) i = {1, . . . , n}. Assume that the prior belief is that abilities ηi are independently and normally distributed across employees, with mean η0i and variance s20 . The posterior mean ηˆ ti is revised based on the measure of team output that is imputed as i’s ability, which now depends on the anticipated efforts and on the posterior beliefs about team members’ abilities. When at each date t ∈ [0, T], employee i is paid his conditionally expected marginal product, his effort will be determined by the sensitivity of beliefs to unexpected output, (1 − qτ (n ))/τ for all τ ∈ [t, T], where

1 − qt (n ) :=

s20 t

s20 t

+ (n − 1 )

s20 t

+σ

2

=

s20 t 2 n s0 t +

σ2

.

(29)

A larger team has a similar effect to that of an increase in the output uncertainty σ 2 : the added uncertainty (n − 1 ) s20 with regard to team members’ abilities reduces the weight (1 − qτ (n ))/τ on the output measure (and increases the weight qτ (n) on the prior reputation η0i ) in updating ηˆ ti , and thereby reduces i’s incentives to influence this measure. Similar to the baseline model, the returns to effort are greater if the production environment is more certain (smaller σ 2 ), if the own ability is less known (larger noise s20 contained in s20 t), and if the team members’ abilities are better known (smaller added noise (n − 1 ) s20 ). It is possible to solve explicitly for the equilibrium effort. As shown in Fig. 6, effort decreases with the number n of employees at a decreasing rate. It decreases more strongly when the duration T − t until rotation is extended. These effects occur because the noise s20 of each additional team member has a greater influence on the learning process when the team is small and when the payback period is large. The rotation time increases with the number of employees, as each employee’s reputation incentives decrease, ∂ at∗ /∂ n < 0 and ∂ T∗ /∂ n > 0.13 5.2. No commitment to the rotation time T We have assumed that the firm and the employee can commit to a rotation time T∗ . In the absence of commitment, there are a number of new aspects. First, there is a time inconsistency problem. Ex ante, the employee prefers a rotation time of T∗ , but once some time has passed, the effort level decreases. Shortly before T∗ , for example, the effort level is

13

Due to the symmetric structure of employees’ variances s20 , the equilibrium (at∗ , T ∗ ) is symmetric as well. This assumption simplifies the pre2

2

sentation, but it is not crucial for the analysis. If, instead, si0 = s0 for all i = {1, . . . , n}, then we can write 1 − qti (n ) = si0 t/( (s10 + . . . + sn0 2 ) t + σ 2 ) =  2 2 si0 t /(t ni=1 si0 + σ 2 ). Substituting this into employee i’s objective function allows us to solve for the equilibrium effort.

H. Hakenes, S. Katolnik / European Economic Review 98 (2017) 424–441

435

Fig. 7. Rotation time T∗ as a function of employee’s ex ante bargaining power b0 .

close to zero. Consequently, ex interim, the employee would like to rotate earlier. In the absence of any commitment, the equilibrium rotation time would be T = 0. Because of Proposition 1, the equilibrium effort level would be a∗ = 0. To get rid of these problems, we have assumed that the rotation time T∗ is contractible. More precisely, we have assumed that the rotation time is verifiable by an outsider (for example, a court), but wages and outputs are not. However, this assumption can be relaxed. For exposition, assume that the firm is large and has a continuum of employees. Let us also assume that rotations are observable to other employees, but wages and outputs are not.14 Nothing is verifiable. The firm can then implicitly commit to a rotation time T∗ , using a relational contract.15 In our case, a relational contract means that if the firm agrees to shorten the rotation time of some employee, all the other employees will believe that the firm no longer keeps up its implicit commitment. They will believe that the firm returns to the equilibrium with T = 0 and will choose a∗ = 0 accordingly. Because the benefit only lasts for an infinitesimal time span, but the downside of zero effort lasts forever, reducing T∗ is never optimal for the firm, for no possible parameter constellation. In other words, for a commitment to some rotation time T∗ , we only need to assume the observability of rotation, not its verifiability.16 5.3. Different bargaining positions In the baseline model, all the bargaining power in the negotiation process is allocated to the employee. To show that our results are robust to alternative solutions of the bargaining problem, let us assume that bt ∈ [0, 1] is a measure of the employee’s relative date-t bargaining position vis-à-vis the firm. As discussed above, it is convenient to differentiate between the employee’s ex ante control rights b0 (determined by the initial labor market competition) and his ex interim bargaining power bt . Changing the ex interim bargaining game has a negative impact on incentives because the employee bears the T full costs of his effort but gains only an aggregate future benefit of t bτ (1 − qτ )/τ dτ from improved performance and the associated belief manipulation. However, as long as reputation concerns create an implicit incentive for the employee, bt = 0 for all t ∈ [0, T], the rotation time T∗ will only be affected by b0 . Allowing for different initial bargaining positions b0 , bargaining over T can be modeled as an alternating-offer game between the firm and the employee in which a cooperation strategy is based on the asymmetric Nash solution (Binmore et al., 1986). Formally, the Nash solution maximizes a bargaining power-adjusted product of the firm’s and the employee’s expected average utilities, providing each party with a share of the surplus that is proportional to their bargaining powers. At each date t, the firm’s expected profits are given by (1 − bt ) (ηˆ t + aˆt ) = (1 − bt ) (η0 + at∗ ). Because increasing T raises the employee’s optimal efforts, the firm will always choose the largest possible rotation time. By contrast, as the employee adjusts his efforts to the loss of ex interim bargaining power, his optimal T will remain unchanged relative to the baseline model. Hence, as shown in Fig. 7, making the firm ex ante stronger at the bargaining table, i. e., decreasing b0 , implies that the employee will be rotated less frequently, ∂ T∗ /∂ b0 < 0. 5.4. Time discounting To assess the robustness of our results to the inclusion of discounting, assume that the discount rate ρ can take any value between zero and one. Our baseline model corresponds to the special case in which ρ is near one. When the employee is 14 The observability of assignments (and non-observability of outputs and wages) is a frequent assumption in the internal labor market literature (see, for instance, Greenwald, 1986; Ricart I Costa, 1988). 15 Bull (1987) and MacLeod and Malcomson (1989) have set the foundations for this literature, and Gibbons (1998) gives an early overview, but the literature is still thriving. 16 We have to admit that the model is rigged in our favor. We have a continuous-time model, and we have no discounting. In typical models with relational contracts, it is not possible to reach the full-commitment solution. However, if the discount factor approaches one, one can come close.

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Fig. 8. Rotation time T∗ as a function of the discount factor ρ .

impatient and discounts future benefits, ρ < 1, the positive impact of current effort on future wages will diminish, and this will depresses the employee’s incentives. Hence, as shown in Fig. 8, the optimal rotation time increases when the employee discounts the future more steeply, ∂ T∗ /∂ ρ < 0. Moreover, decreasing the discount rate leads to a critical value, say ρ , such that for ρ ≤ ρ , rotating the employee is never optimal, T∗ → ∞. The main conclusions of our model thus hold for cases in which time discounting is relatively low.

5.5. Learning by doing There are multiple ways in which the employee can learn over time. He may get better at a job the longer he performs that specific job in the same division (see Arrow, 1962, for this “traditional” learning argument). He may perform a job and acquire abilities that he can also use in other divisions (see Lindbeck and Snower, 20 0 0, for the relation between job rotation and intertask learning). Alternatively, the employee may even become more productive as he works in more divisions (see Aoki, 1986; Gibbons and Waldman, 2004, for a discussion of this type of argument). None of these effects are covered in the baseline model, but they can easily be accommodated, resulting in some intuitive comparative statics. If the employee becomes more productive as long as he works in the same division (on the same task), then the optimal rotation time will lengthen (larger T∗ ). If the employee just learns and the knowledge he acquires can be applied in any division, then the rotation time will not be influenced. If it is beneficial to have worked in many divisions, then the employee will rotate faster, and T∗ will decrease.

6. Job-specific skills In our baseline model, the employee’s ability is the same for each job. Job rotation is used for sustaining the employee’s reputation incentives over time. In this section, we consider the case of job-specific, multidimensional skills. For example, the same employee can be a bad sales person but work great in the back office. In reality, both aspects are prevalent: the ability may be different for different employees, but the quality of the employee-job fit may also differ for each employee. To make our point, we adjust the baseline model as little as possible. Specifically, we now assume that η measures the quality of the match between the employee and a job. We discuss two different assumptions. First, we consider a situation in which the firm must set a rotation time. This could be the case, for example, in a consultancy firm that has a fixed schedule for promotion. After T years, the employee is either promoted or fired (in which case he may find an equivalent job). In such a situation, different abilities may be required after each promotion: η is drawn anew. With this (fairly specific) set of assumptions, the results of our baseline model remain identical. Second, more naturally, we assume that the firm can choose between a fixed rotation (or promotion) time T, or no commitment at all. In that case, one must slightly adjust the model to avoid degeneration of the equilibrium. (Optimally, the employee would try to find a match with η → ∞, and then stay on this job forever. However, because η = ∞ is not possible, there is always an even better match, and the employee would have to keep on searching.) For example, we could assume a positive cost for each rotation. We could assume that the employee discounts future returns. Another possibility is to assume that η is drawn from a distribution with bounded support. For exposition, let us assume a discount factor ρ < 1. As a benchmark, we consider the case without commitment to a rotation time. By working at a job, and by choosing an effort at , the employee creates an output Yt and, accordingly, earns a wage of wt . If the output is low (i. e., low compared to the expected output, given the effort), the employee will infer that the match quality is poor, and he will rotate. Let Y∗ (T) denote the critical output level at time T such that the employee decides to rotate. Eq. (1) gives the output level for a given action choice and realization of the random component Zt . After rotation, the employee works at a new job with a new

H. Hakenes, S. Katolnik / European Economic Review 98 (2017) 424–441

match quality, which provides some utility level U, equal to the ex ante utility. This utility is

⎛

 U=

∞ 0

⎝



Y
(t )





⎞

wt − C (at ) fZ (Z ) dZ fη (η ) dη⎠ ρ t dt +

⎛



∞ 0

⎝



Y ≥Y ∗

437

⎞

U fZ (Z ) dZ fη (η ) dη⎠ ρ t dt.

(30)

(t )

Remember that wt depends on the output, and thus on the random component Zt . The critical Y∗ (T) is exactly the employee’s indifference point: the continuation value from staying at the current job is equal to the ex-ante utility U of a new job. Importantly, the wages wt before a rotation are not influenced by the choice of the rotation time. Now assume that the employee can commit to a rotation time. Denote this rotation time T∗∗ . For comparability, let us assume that the rotation time can depend on output, T∗∗ (Y). Let Y∗∗ (T) denote the inverse of this function. The employee commits to a rotation whenever the output Y at date T is smaller or equal to Y∗∗ (T). Importantly, with commitment, the employee may end up in a situation where he must rotate, although at that point, he would rather stay. He may also find himself in a situation where he must stay, although he would prefer to rotate. On the other hand, by choosing some Y∗∗ (T), the employee can actively shape the remaining time on the job, and thus his incentives to exert effort, and with it the firm’s expectations about his effort – and his future wages. For a thought-experiment, assume that the employee is at the indifference point Y∗ (T). Time T has passed, and the output has fallen short of expectations such that the employee is just indifferent between rotating and staying. Now, from an exante perspective, if the employee would have committed to a rotation time T∗∗ (Y), would he have chosen a smaller or a larger T? Here, the two fundamental effects of our baseline model kick in. On the one hand, with a fresh start after rotation, the employee would have had new incentives to work, and thus would have earned a higher wage. Because of this effect alone, he would have chosen a shorter rotation time. On the other hand, we know that for a too short rotation time T, the incentives to work converge to zero. This effect alone implies a longer rotation time. Like in the baseline model, by isolating these two effects, we obtain an optimal rotation time. Summing up, if T is small, then in the region near the indifference point Y∗ (T), the employee would like to commit to a later rotation time. If T is already large, then in the region near the indifference point Y∗ (T), the employee would like to commit to an earlier rotation time. There are thus two possible scenarios. If the match turns out to be good, then the output may never even come near to the indifference point Y∗ (T), and the employee will stay on the job forever. If the match turns out to be bad, the employee will rotate at some point. Then, the possibility to choose the rotation policy ex ante will lead to contracting of rotation times. Let T denote the critical time for which Y∗ (T) and Y∗∗ (T) intersect. This T represents an “optimal” rotation time in the sense that, if commitment is possible, the rotation time will move towards T . The employee would not, however, commit exactly to the rotation time T – instead, the rotation time would still depend on the output Y. Because T is determined by analogous rationales as T∗ of Proposition 2, it will follow similar comparative statics. Consequently, the incentive effects of job rotation prevail, no matter whether the type of the employee or the quality of the match (or both) are unknown. 7. Discussion To evaluate the applicability of our analysis, it is useful to compare it with the predictions of alternative theories of job rotation frequency. We review four such theories. First, earlier work has advocated frequent rotations as a means of learning about the productivity of employees and/or of different jobs (Meyer, 1994; Ortega, 2001). In these models, the firm learns relatively more with rotation because of the larger productivity uncertainty after the job change. Both this approach and ours predict the same qualitative result that a greater value of learning (i. e., a lower ratio of output uncertainty to prior productivity uncertainty) renders frequent rotations beneficial, but our main contribution lies in linking these learning effects to the incentives of employees. Whereas frequent rotations lead to higher incentives to invest in reputation because of the replenishing uncertainty, they also lead to lower incentives due to a shorter time horizon over which to amortize such investments. Second, job rotation is also justified on the grounds of improving the quality of the employee-job match (Li and Tian, 2013; Papageorgiou, 2016). According to that view, job rotation is an attempt to move employees into jobs in which they will be the most productive so as to overcome a mismatch loss.17 Like in our model, the adoption of rotation is affected by the arrival of performance information over time, with the information being relevant to the job at hand and for the employee at hand. However, whereas our model explains the practice of job rotation based on learning about the employee’s ability and based on the employee’s incentives, the alternative approach, which focuses on the matching motives only, does not. Moreover, under the job matching approach, job reassignments occur only in circumstances in which performance information turns out to be unfavorable. Thus, as shown in Section 6, the optimal rotation time will depend on output. With match-specific abilities, our analysis identifies that the incentive effects of job rotation may still prevail. However, this approach cannot account for regular, periodic job transfers.

17 Self-selection into job rotation can also be a means of signaling high productivity, which may allow firms to better match pay to an employee’s true skills (Arya and Mittendorf, 2006a).

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Third, job rotation may be a useful device for alleviating the ratchet effect (Arya and Mittendorf, 2004; Ickes and Samuelson, 1987; Milgrom and Roberts, 1992). By assigning an employee to different jobs, the employee knows that his current job performance would not be relevant to the prediction of future job performance, so he has no incentives to conceal the productivity of his current job. In these models, job rotation will only arise if the employee can extract some informational rent about this productivity, which the firm can learn with rotation.18 In contrast, our model shows that job rotation is effective in voluntarily limiting the amount of performance information available to the firm.19 Under both approaches, frequent rotations are beneficial when the initial productivity uncertainty, and thus the extent of learning are large. However, whereas in the above-mentioned models this result arises because there is a productivity disadvantage of job rotation that derives from job-specific knowledge, our work presents a complementary approach that does not need to rely on the typical technological explanations. Fourth, it is often argued that job rotation can help mitigate corruption (Choi and Thum, 2003) and prevent employees’ influence activities (Eguchi, 2005).20 By ensuring that employees do not occupy jobs for a long time, job rotation plays a role in limiting employees’ opportunities for diverting profits to their own benefit. Frequent rotations can then be optimal in situations in which explicit contracts fail to motivate risk-averse employees effectively, and thus in situations in which performance measurement is characterized by substantial uncertainty (Eguchi, 2005). Likewise, in our model, a high uncertainty of performance measurement implies that career concerns do not function well in inducing high effort. However, this leads to exactly the opposite implication: it is then optimal to rotate an employee less frequently as a means of increasing his efforts over time. Furthermore, if we reinterpret job reassignments as promotions or demotions (see Milgrom and Roberts, 1988), allowing the employee to affect the firm’s assignment decisions through means other than productive work may actually encourage the employee’s influence activities. Even though the motives for job rotation can vary across firms, our analysis highlights a novel rationale for the use of job rotation that applies to the many employment settings in which work incentives are created by employees’ career concerns. In contrast to alternative theories of job rotation, the determination of the optimal rotation time in our model relies on both learning and incentives, and does not rely on technical motives. However, because our forces for job rotation are complementary to alternative theories, distinguishing empirically among these theories may be difficult and requires further study. It may be worth noting, however, that our theoretical predictions are consistent with the empirical studies by Campion et al. (1994) and Eriksson and Ortega (2006), who emphasize that job rotation is less used for reducing the boredom of long-tenured employees with limited future prospects.21 Our results can also be interpreted consistently with the recent evidence, providing insights into the impact of organizational change on information, reputation incentives and performance (Hertzberg et al., 2010; Jenter et al., 2017; Mühlheußer et al., 2016).22 Within a more general context, our model yields insights into the emerging trends in workplace reorganization, including the increased dissolution of longterm contracts, rising inter-firm job change, limited employment duration, use of flexible staffing, and adoption of triangular employment relations with many firms and temporary or contingent contracts.23 8. Conclusion In our model, the employee has incentives to work to build his reputation. Over time, his true type is gradually revealed, and incentives are diluted. Introducing new uncertainty about ability would create new incentives, but it would shorten the employee’s time horizon right at the start. In this paper, we ask the following questions. If one could restrict access to past information, what would be the optimal frequency of re-introducing uncertainty? How is this frequency affected by the quality of information transmission? What frequency is optimal if the employee works in a team? And how does the frequency depend on the employee’s discount factor? In the paper, we treat job rotation as one strategy to introduce new uncertainty, at least partially. Promoting the employee, replacing the boss, or keeping incomplete records on the employee’s performance would serve similar purposes. Many other aspects are relevant for job rotation, boss replacement or record keeping, but our model focuses on the incentive effects. 18 Prescott and Townsend (2006) identify similar informational effects and incentive-related benefits of job rotation. Arya and Mittendorf (2006b) consider job rotation as a tool for smoothing information across employees. 19 A related idea is presented in DeVaro and Gürtler (2016), where commitment to ongoing job rotation prevents strategic shirking of employees who try to favorably influence their future job assignments. In their model, the firm voluntarily forgoes the opportunity to learn about employees’ abilities, which leads to less efficient future job assignments but improves the employees’ incentives in early periods. 20 Additional incentive-based approaches consider job rotation as a means of incentivizing labor-saving technical change (Carmichael and MacLeod, 1993) and of limiting routine and monotonous work (Cosgel and Miceli, 1999). 21 Ference et al. (1977) have used this motivational argument in the context of “plateaued” employees. 22 Hertzberg et al. (2010) find evidence that the routine use of job rotation can improve loan officers’ reporting incentives driven by their reputation concerns. In the literature on employee turnover, Mühlheußer et al. (2016) show that managerial replacement can restore employees’ incentives to demonstrate their skills. Jenter et al. (2017) look at the stock price reaction of firms in which the CEO suffered a sudden death. Suicides are not included, so sudden deaths are exogenous and can be used to exclude endogeneity problems. The stock price is used to proxy the CEO’s performance. The deaths of short-tenured CEOs cause value losses, whereas the deaths of long-tenured CEOs cause value gains. This is in line with our theory, if the CEO is interpreted as the employee of shareholders. When the CEO has only just started his job, his incentives are high, so the death should result in a value loss. When the CEO has been in the firm for a long time, incentives are low, so the death should result in a value gain. 23 Several studies have generated evidence regarding workplace restructuring and the implementation of flexible staffing arrangements, primarily referring to the US labor market (see, for example, Kalleberg, 20 0 0; Gramm and Schnell, 2001; Houseman, 2001).

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439

Our general incentive mechanism can also be applied beyond the particular setup of the model. Another instance in which agents may choose to introduce uncertainty by narrowing down career horizons affects firm-bank relationships, where a borrower’s refinancing terms can depend on his past performance information that is revealed over time. Or, alternatively, limiting the duration of buyer-seller connections may be optimal to smooth incentives of firms to sustain their reputation for quality, as in Strausz (2009). Further research could analyze the influences of other production and information technologies, such as of a multitask learning setup. As Dewatripont et al. (20 0 0) note, it is likely that multitasking reduces an employee’s aggregate efforts because it weakens the link between effort and realizations of current output. Related to Cisternas (2017a), multitasking can also affect the allocation of effort between directly productive work and strategic human capital acquisition. In the latter case, effort affects output only indirectly through its effect on the rate of change of the employee’s skills. An investment in effort has, then, a delayed impact on output, and the value of this investment depends on the horizon length. These circumstances may lead back to optimal rotation decisions. Appendix A We first prove that maximizing the long-run average undiscounted utility U¯ in (6) corresponds to maximizing the longrun utility with discounting U in (5) when the discount factor ρ approaches 1. The long-run utility with discounting is given by

U :=



1 1 − ρT

T 0

ρ t ut dt.

(A.1)

Maximizing U with respect to T yields the first-order condition, ∂ U/∂ T = 0,

ρ T log ρ ( 1 − ρ T )2



T 0



1 1 − ρT

ρ t ut dt +

T 0

ρ t ∂T ut dt + ρ T uT = 0,

(A.2)

where ∂ T ut denotes the partial derivative of ut with respect to T. Factoring out 1 − ρ T ,

ρ T log ρ 1 − ρT



T 0

ρ t ut dt +



T 0

ρ t ∂T ut dt + ρ T uT = 0.

(A.3)

Because of L’Hôpital’s rule, ρ T log ρ /(1 − ρ T ) becomes −1/T for ρ → 1. Hence, the whole expression becomes

−



1 T

T

0



ut dt +

T 0

∂T ut dt + uT = 0.

(A.4)

Maximizing the long-run average undiscounted utility U¯ in (6) gives the first-order condition, ∂ U¯ /∂ T = 0,

−

1 T2



0

T

ut dt +

1 T



T 0

∂T ut dt + uT = 0,

(A.5)

which is equivalent to (A.4). This proof is a special case of Dutta (1991). Proof of Proposition 1. The employee’s problem is to choose efforts at for all dates t ∈ [0, T] that maximize his continuation value Vt of (9). To characterize the employee’s incentive compatibility condition, consider deviations from anticipated effort aˆt for all t ∈ [0, T]. If the employee deviates only at time instant [t , t + dt ] by exerting at − aˆt > 0, Yt − Aˆt increases by (at − aˆt ) dt. According to (7), the firm’s conditional mean ηˆ t increases by (at − aˆt ) dt (1 − qτ )/τ for all τ ∈ [t, T], increasing Vt by  (at − aˆt ) dt tT (1 − qτ )/τ dτ . Thus, given the history {Yτ }0 ≤ τ ≤ t of performance, a necessary condition for at to be optimal is that it maximizes

max (at − aˆt ) dt log

{at }0≤t≤T

s2 T + σ 2

0 s20 t

+σ

2

−c

at2 dt, 2

(A.6)

which is equivalent to (10). The first-order condition for this maximization problem,

log

s2 T + σ 2

0

s20 t + σ 2

dt − c at∗ dt = 0,

(A.7)

yields (11) in Proposition 1. As at∗ is independent of the firm’s anticipated aˆt , we have determined the unique equilibrium.  Proof of Proposition 2. On the equilibrium path, the firm correctly anticipates the employee’s efforts, aˆt = at∗ for all dates t ∈ [0, T]. Hence, at each t ∈ [0, T], the employee’s expected equilibrium wage is E[wt∗ ] = η0 + at∗ . Using this in (6), and inserting the equilibrium effort at∗ of Proposition 1, implies that T∗ must maximize U¯ in (13). The corresponding first-order condition, ∂ U¯ /∂ T = 0, is given in (15). Substituting T0 := s20 T ∗ /σ 2 in (15) yields

2 T0 − (T0 + 1 ) log[T0 + 1] = 0,

(A.8)

440

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with the numerical approximation T0 ≈ 3.922. The optimal rotation time is thus T ∗ = T0 · σ 2 /s20 ≈ 3.922 · σ 2 /s20 . The secondorder condition for maximization, ∂ 2U¯ /∂ T 2 < 0, requires that

T02 − T0 (3 T0 + 2 ) log[T0 + 1] + (T0 + 1 )2 log[T0 + 1]2 < 0, which is fulfilled at T0 ≈ 3.922.

(A.9)



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