Matthias Kräkel
email: m.kraekel@uni-bonn.de.

Anja Schöttner
email: anja.schoettner@uni-bonn.de.

Technology Choice, Relative Performance Pay, and Worker Heterogeneity

Keywords: complementarities, heterogeneous workers, production technology, relative performance pay JEL Classi…cation: D82, D86, J33, M52

We identify a new problem that may arise when heterogeneous workers are motivated by relative performance pay: If workers'abilities and the production technology are complements, the …rm may prefer not to adopt a more advanced technology even though this technology would costlessly increase each worker's productivity. Due to the complementarity between ability and technology, under technology adoption the productivity of a more able worker increases more strongly than the productivity of a less able colleague. As a consequence, both workers'motivation to exert e¤ort is reduced. We show that this adverse incentive e¤ect is dominant and, consequently, keeps the …rm from introducing a better production technology if talent uncertainty is su¢ ciently high and/or monitoring of workers is su¢ciently precise.

A c c e p t e d M a n u s c r i p t 1 Introduction

A fundamental incentive problem in organizations arises from the fact that a …rm often has only coarse information on its workers'e¤ort. In particular, performance signals are often only ordinal and/or unveri…able. In the …rst case, the …rm only observes an ordinal ranking of worker performance. In the latter case, performance is observable by the …rm but not by a third party. In such situations, incentive schemes like individual bonuses or piece rates are not feasible because they require individual performance signals or they are subject to potential employer opportunism. If worker performance is unveri…able, ex-post the …rm can save labor costs by wrongly claiming that workers have performed poorly. Since workers anticipate such opportunistic behavior, incentives would be completely erased.

However, when only ordinal and/or unveri…able performance signals are available, the …rm can still rely on relative incentive schemes that distribute a …xed amount of money among the workers according to their relative performance [START_REF] Malcomson | Work incentives, hierarchy, and internal labor markets[END_REF][START_REF] Malcomson | Rank-order contracts for a principal with many agents[END_REF]. 1 In practice, we can observe diverse variants of such incentive schemes, e.g., bonus pools [START_REF] Kanemoto | Firm reputation and self-enforcing labor contracts[END_REF]Rajan andReichelstein 2006, 2009;[START_REF] Budde | Bonus pools, limited liability, and tournaments. Governance and the E¢ ciency of Economic Systems[END_REF], job-promotion tournaments [START_REF] Baker | The internal economics of the …rm: Evidence from personnel data[END_REF][START_REF] Treble | The internal economics of the …rm: Further evidence from personnel data[END_REF], sales contests [START_REF] Kalra | Designing optimal sales contests: A theoretical perspective[END_REF][START_REF] Murphy | Sales sontest e¤ectiveness: An examination of sales contest design preferences of …eld sales forces[END_REF][START_REF] Lim | Designing sales contests: Does the prize structure matter[END_REF], and forced-distribution systems [START_REF] Murphy | Performance measurement and appraisal: Motivating managers to identify and reward performance[END_REF][START_REF] Thomas | The rise and fall of Enron[END_REF]. Under each variant, the …rm commits to pay a certain collective amount of money to the workers. Such a commitment is credible because a third party can verify whether the entire amount has been paid out by the …rm. Since the …rm is forced to pay out the total amount of money, it has no incentive to misrepresent the workers'performance. This important self-commitment property assures worker incentives.

In this paper, we point out that the use of relative performance pay can be highly problematic if the …rm can choose between di¤erent production 1 Another well-known solution to the unveri…ability problem are relational (or selfenforcing) contracts (see, e.g., [START_REF] Bull | The existence of self-enforcing implicit contracts[END_REF][START_REF] Baker | Relational contracts and the theory of the …rm[END_REF]. For a relational contract to be feasible, the …rm's loss from reneging must be su¢ ciently large, e.g., the employeremployee relationship needs to be sustained with su¢ ciently high probability in the future and the associated future pro…t must not be discounted heavily.

2

A c c e p t e d M a n u s c r i p t technologies. We characterize situations in which the …rm foregoes to install a new technology although this technology would increase each worker's productivity and is costlessly available. When choosing the technology, the …rm faces the following trade-o¤: On the one hand, a more advanced technology enhances each worker's productivity (productivity e¤ect). On the other hand, if worker ability and …rm technology are complements and workers di¤er in their abilities, the new technology increases the productivity of a more able worker more strongly than the productivity of a less able worker. Thus, the outcome of worker competition for bonus shares is less responsive to changes in e¤ort and, consequently, both workers exert less e¤ort (adverse incentive e¤ect). If the adverse incentive e¤ect dominates the productivity e¤ect, the …rm will not adopt the advanced technology.

In a next step, we use a parameterized example to highlight the impact of worker heterogeneity on technology choice. We show that, the higher the degree of worker heterogeneity and the higher the uncertainty about workers' ex-ante unknown talents, the more likely the …rm is to choose the less productive technology. In particular, we compare two labor market situations that di¤er in the expected ability of the workers. We demonstrate that the …rm may adopt the more advanced technology only in the situation with lower expected worker ability. Such a scenario occurs if talent uncertainty in the situation with higher average ability is su¢ ciently high compared to the situation with lower average ability. Furthermore, if workers' equilibrium e¤orts are rather small under either technology due to imprecise performance measurement or steep marginal e¤ort costs, the adverse incentive e¤ect of technology adoption is not severe. As a result, if the …rm's monitoring technology is imprecise, the …rm is more inclined to invest in a better production technology. Hence, if worker ability and production technology are complements in the …rm's production function, monitoring technology and production technology are substitutes.

Theoretic contributions to moral hazard in principal-agent relationships typically consider either limited liability or risk aversion of workers as contractual frictions (e.g., La¤ont and Martimort 2002, chapter 4). Until Section 4, the paper focuses on the case where workers are risk neutral and protected A c c e p t e d M a n u s c r i p t by limited liability. To check the robustness of our …ndings, Section 5 turns to the case of risk averse workers. There, we analyze both the case of limitedly liable workers and of unlimited liability. Whereas limited liability of risk-averse workers leads to the same two opposite e¤ects as the case of risk neutral workers -a positive productivity e¤ect and an adverse incentive e¤ect, switching to unlimited liability adds a third e¤ect: If a better technology decreases e¤ort due to more uneven worker competition, e¤ort costs will be reduced as well. Under unlimited liability, this cost reduction directly bene…ts the …rm, which can then lower expected wage payments since the workers' participation constraint can always be made binding. Hence, the …rm is in favor of introducing the better technology under unlimited liability rather than under limited one.

The theoretical setting with ability and technology being complements …ts well with the situation observed in the last decades where …rms intensely invested in information technologies (IT). Initially, investment in IT was used to save labor and to substitute capital for low-ability work. However, nowadays IT and workers' abilities are mainly seen as complements (see, among many others, [START_REF] Applegate | Information technology and tomorrow's manager[END_REF][START_REF] Berndt | High-tech capital formation and labor composition in U.S. manufacturing industries: An exploratory analysis[END_REF][START_REF] Hitt | The role of information technology in modern production: Complement or substitute to other inputs? Wharton School[END_REF][START_REF] Bresnahan | Information technology, workplace organization, and the demand for skilled labor: Firmlevel evidence[END_REF]. IT is used by high-ability workers for improving time to market in research and development and improving service to key customers, for example. In other words, rather complex IT is used by …rms for intensively exploiting the potential of their high-ability workers, hence making them more productive.

Besides the literature cited above, our paper is related to the work on rank-order tournaments starting with the seminal articles by [START_REF] Lazear | Rank-order tournaments as optimum labor contracts[END_REF], [START_REF] Green | A comparison of tournaments and contracts[END_REF] and [START_REF] Nalebu¤ | Prizes and incentives: Towards a general theory of compensation and competition[END_REF]. Subsequent papers pointed to speci…c disadvantages of tournaments. Two major problems of tournaments have been emphasized in the literature. First, workers can improve their relative positions in the ranking by investing in counterproductive e¤ort or sabotage [START_REF] Lazear | Pay equality and industrial politics[END_REF][START_REF] Konrad | Sabotage in rent-seeking contests[END_REF][START_REF] Chen | Sabotage in promotion tournaments[END_REF][START_REF] Münster | Selection tournaments, sabotage, and participation[END_REF][START_REF] Amegashie | Sabotaging potential rivals[END_REF][START_REF] Gürtler | On sabotage in collective tournaments[END_REF]. Second, similar to cartels in market competition, tournament participants can collectively gain by a stable collusion that minimizes e¤ort costs [START_REF] Ishiguro | Collusion and discrimination in organizations[END_REF][START_REF] Chen | Fighting collusion in tournaments[END_REF] A c c e p t e d M a n u s c r i p t [START_REF] Sutter | Communication, cooperation and collusion in team tournaments -an experimental study[END_REF]. In this paper, we identify a further problem of bonus pools or tournaments -an adverse e¤ect on technology choice given that worker ability and production technology are complements.

The remainder of the paper is organized as follows. In the next section, we introduce the model setup. Section 3 solves the workers'problem of e¤ort choice under a given bonus-pool incentive scheme. Section 4 focuses on the …rm's problems of designing the optimal bonus-pool contract and choosing the optimal production technology. Section 5 extends the analysis to the case of risk averse workers. Section 6 concludes.

The Model

We consider a …rm that employs two workers. All parties are risk neutral.2 Workers are protected by limited liability so that all payments to them must be non-negative. Worker i's (i = 1; 2) contribution to …rm pro…t (or, for short, output) is h(e i ; a i ; ) + " i + . Here, e i 0 denotes worker i's e¤ort choice, a i is the worker's exogenously given ability, and characterizes the …rm's production technology. The random variable " i describes idiosyncratic noise with " 1 and " 2 being identically and independently distributed with density f (") and cumulative distribution function (cdf) F ("). Hence, the density of the composed random variable " 2 " 1 is a symmetric convolution around zero. We denote this convolution by g( ) and the corresponding cdf by G( ) and assume that g( ) is single-peaked at zero and that G( ) is twice di¤erentiable. The random variable measures common noise, which in ‡uences both workers in the same way (e.g., the economic situation of the …rm). 3 Neither e¤ort e i nor a worker's contribution h(e i ; a i ; ) + " i + is observable by the …rm. Instead, the …rm observes a noisy and unveri…able A c c e p t e d M a n u s c r i p t binary signal s with:4 s = ( s 1 if h(e 1 ; a 1 ; ) + " 1 + > h(e 2 ; a 2 ; ) + " 2 + s 2 otherwise (1)

According to (1) the …rm has only access to ordinal information concerning worker performance. Signal realization s = s i indicates that worker i has performed better than worker j (i; j = 1; 2; i 6 = j). 5The technology parameter 2 f L ; H g is chosen by the …rm. It can either use a more advanced technology = H or a less advanced one = L < H . For simplicity, we assume that technology adoption is free. 6 The function h(e i ; a i ; ) is strictly increasing in e¤ort, ability, and the technology parameter, i.e., @h @e i ; @h @a i ; @h @ > 0. In particular, this means that, holding e¤ort constant, a better technology increases the output of each worker. Furthermore, output is concave in e¤ort, i.e., @ 2 h @e 2 i 0. The marginal productivity of e¤ort increases with a better technology, i.e., @ 2 h @e i @ 0. However, we do not impose a restriction on the sign of @ 2 h @a i @ , implying that technology and ability can be substitutes or complements. In case of substitutes ( @ 2 h @a i @ 0), the marginal productivity of ability decreases under the better technology. Put di¤erently, productivity di¤erences due to distinct abilities are evened out because the advanced technology increases the productivity of less able workers more strongly. For example, this happens if the new technology makes the production task easier for workers of lower ability, so that they Page 7 of 27 A c c e p t e d M a n u s c r i p t can keep up with more capable colleagues. Such a situation might occur if the …rm adopts an easier-to-handle computer operation system, like switching from MS-DOS to MS-Windows. By contrast, if technology and ability are complements ( @ 2 h @a i @ > 0), more able types bene…t more from the advanced technology, e.g., if the new technology is complex and di¢ cult to handle (as replacing typewriters with personal computers). Finally, to simplify the analysis, we assume that the marginal productivity of e¤ort does not interact with ability, i.e., @ 2 h @e i @a i = 0. 7 A worker's ability can be either high or low, a i 2 fa L ; a H g, where a H > a L 0. The probability that a worker is of high ability is denoted by p 2 (0; 1) and is common knowledge. After accepting the contract o¤ered by the …rm and entering into the employment relationship, each worker becomes familiar with the task to be conducted in this particular …rm, and can thus assess how good he will be at it. Consequently, every worker learns his own ability. Moreover, each worker also observes the type of his colleague, whereas the …rm never observes workers'abilities. This assumption captures the fact that employees who work closely together usually possess better information about one another's talents than the …rm. For simplicity, an agent's reservation utility is zero. Worker i's costs of e¤ort are c(e i ) with c 0 (e i ) ; c 00 (e i ) > 0 for all e i > 0 and c(0) = 0. To guarantee interior solutions, we further impose the restriction that @h @e i (0; a i ; ) > c 0 (0). The relative performance signal (1) renders individual pay-for-performance schemes infeasible, but the …rm can employ a relative incentive scheme such as a …xed bonus pool B 0 to provide its workers with e¤ort incentives. In order to induce appropriate incentives, the …rm announces to pay B to the better performing worker 7 The assumption @ 2 h @ei@ai = 0 implies that we focus on the analysis of "unfair contests" in the sense of O' Kee¤e et al. (1984) and [START_REF] Schotter | Asymmetric tournaments, equal opportunity laws, and a¢ rmative action: Some experimental results[END_REF], who di¤erentiate between "unfair" and "uneven" contests as two alternative ways of modeling heterogeneous players. In unfair contests, players exerting the same e¤ort level have di¤erent winning probabilities. Technically, e¤ort and ability enter the production function additively, leading to symmetric equilibria. However, in uneven contests e¤ort and ability are multiplicatively connected (either in the production or the cost function), thus yielding asymmetric equilibria. [START_REF] Kräkel | Technology choice and incentives under relative performance schemes[END_REF] show that the main results derived in the present paper also extend to uneven contests.

A c c e p t e d M a n u s c r i p t and ( 1) B to his co-worker. Limited liability requires that 0 1. Timing is as follows. At the …rst stage, the …rm makes the technological choice 2 f L ; H g, which is publicly observable. Thereafter, it o¤ers two randomly chosen workers a bonus-pool contract (B; ) specifying size and distribution of the bonus pool.8 Given that workers accept, they enter the …rm and observe abilities. In stage 3, workers simultaneously choose their e¤ort levels. Then, outputs and the signal (1) are realized. Finally, based on the realization of s, the …rm shares B between the workers according to the prespeci…ed sharing rule .

Workers'E¤ort Choices

In this section, we derive workers'equilibrium e¤ort levels given the …rm's technological choice and the bonus-pool contract (B; ). These equilibrium e¤orts characterize the incentive constraints for the …rm's optimization problem at the contracting stage 2. When workers choose e¤ort, they know the technology parameter . Thus, given the e¤ort choice e 2 of worker 2, worker 1 chooses e¤ort e 1 to maximize (1 ) B [1 prob{h(e 1 ; a 1 ; ) + " 1 + > h(e 2 ; a 2 ; ) + " 2 + }] + B prob{h(e 1 ; a 1 ; ) + " 1 + > h(e 2 ; a 2 ; ) + " 2 + } c(e 1 ) = (1 ) B + (2 1) B G (h(e 1 ; a 1 ; ) h(e 2 ; a 2 ; )) c(e 1 ):

Similarly, worker 2 solves max e 2

(1

) B + (2 1) B [1 G (h(e 1 ; a 1 ; ) h(e 2 ; a 2 ; ))] c(e 2 ):
We assume that the functional forms are such that worker i's objective function is concave in e i for all e j (i; j = 1; 2; i 6 = j). 9 Thus, the equilibrium e¤ort A c c e p t e d M a n u s c r i p t levels (e 1 ; e 2 ) are characterized by the two …rst-order conditions g(h(e 1 ; a 1 ; ) h(e 2 ; a 2 ; )) @h @e 1 (e 1 ; ) (2 1) B c 0 (e 1 ) = 0;

(2) g(h(e 1 ; a 1 ; ) h(e 2 ; a 2 ; )) @h @e 2 (e 2 ; ) (2 1) B c 0 (e 2 ) = 0: (3)

Note that @h @e i is independent of a i because of our assumption @ 2 h @e i @a i = 0. From the …rst-order conditions it follows that c 0 (e 1 )= @h @e 1 (e 1 ; ) = c 0 (e 2 )= @h @e 2 (e 2 ; ). Thus, since c 0 (e i )= @h @e i is strictly increasing in e i , the equilibrium is unique and symmetric, e 1 = e 2 =: e . Hence, equilibrium e¤ort e (a 1 ; a 2 ; B; ; ) is implicitly given by g(h(e ; a 1 ; ) h(e ; a 2 ; )) @h @e 1 (e ; ) (2 1) B c 0 (e ) = 0: (4) Implicit di¤erentiation of equation ( 4) leads to our …rst proposition.

Proposition 1 If a 1 6 = a 2 and @ 2 h @a i @ > 0, then e (a 1 ; a 2 ; B; ; ) can be decreasing in the technology parameter . In particular, de d < 0 if @ 2 h @e i @ = 0.

Proof. Let a 1 6 = a 2 . 10 Applying the implicit function theorem to (4) gives sign de d = sign g 0 (h(e ; a 1 ; ) h(e ; a 2 ; )) @h @ (e ; a 1 ; ) @h @ (e ; a 2 ; ) @h @e 1 (e ; ) +g (h(e ; a 1 ; ) h(e ; a 2 ; )) @ 2 h @e 1 @ (e ; ) :

(5)

Because @ 2 h @e 1 @ (e ; ) 0, the second term on the right-hand side of ( 5) is nonnegative. Thus, we can obtain de d < 0 only if the …rst term on the right-hand strict concavity is ensured if [g 0 (h(e i ; a i ; ) h(e j ; a j ; )) ( @h @ei (e i ; )) 2 + g(h(e i ; a i ; ) h(e j ; a j ; )) @ 2 h @e 2 i (e i ; )] ( 21) B c 00 (e i ) < 0 for all e i ; e j ; a i ; a j . This condition is satis…ed if the density g( ) is su¢ ciently ‡at (i.e., the variance of " 2 " 1 is su¢ ciently large) and/or c( ) is su¢ ciently convex. 10 For a 1 = a 2 , equation (4) boils down to g(0) @h @e1 (e ; ) (2 1) B c 0 (e ) = 0. Here, de =d 0 due to @ 2 h @e1@ (e ; ) 0.

9

A c c e p t e d M a n u s c r i p t side is negative. This is the case if and only if @ 2 h @a i @ > 0: Then, we either have g 0 (h(e ; a 1 ; ) h(e ; a 2 ; )) < 0 and @h @ (e ; a 1 ; ) @h @ (e ; a 2 ; ) > 0 (if a 1 > a 2 ) or g 0 (h(e ; a 1 ; ) h(e ; a 2 ; )) > 0 and @h @ (e ; a 1 ; ) @h @ (e ; a 2 ; ) < 0 (if a 1 < a 2 ) because @h @a i > 0 and g( ) is single-peaked at zero. Proposition 1 shows that, if ability and technology are complements and workers are heterogeneous, adopting an enhanced production technology may have an adverse e¤ect on e¤ort, i.e., decrease workers' equilibrium e¤ort choices. This is the case whenever the counteracting positive e¤ect on the marginal productivity of e¤ort, @ 2 h @e i @ , is not too strong and, in particular, if such an e¤ect does not exist, @ 2 h @e i @ = 0. The intuition for this …nding can be best seen by inspection of (4): Since g ( ) is single-peaked at zero, equilibrium e¤orts will be lower the higher jh(e ; a 1 ; ) h(e ; a 2 ; )j. As technology and ability are complements, a better technology makes an initially asymmetric competition with a 1 6 = a 2 even more asymmetric (i.e., jh(e ; a 1 ; ) h(e ; a 2 ; )j increases), which further weakens both workers'incentives. By contrast, if ability and technology are substitutes or workers are homogeneous, equilibrium e¤ort always increases with a superior technology.

The Firm' s Decisions

We now consider the stage where the …rm decides on the optimal bonus-pool contract (B; ), given the technology parameter . Let P H := p 2 (P L := (1 p) 2 ) denote the probability of a homogeneous match with two highability (low-ability) workers and P HL := p(1 p) the probability of a heterogeneous match. Anticipating workers'equilibrium behavior e (a 1 ; a 2 ; B; ; ) as incentive constraint, the …rm chooses B and in order to maximize expected output net of wage costs, i.e., Thereby, the …rm has to take into account the limited liability constraint 2 [0; 1] and the workers' participation constraints. Recall that, before signing the contract, both workers are identical because they are characterized by symmetric ability uncertainty at this point. Thus, using that 1 G(x) = G( x) by symmetry of g(x), each worker faces the same participation constraint Ef( 21) B[G(h(e (a 1 ; a 2 ; B; ; ); a 1 ; ) h(e (a 1 ; a 2 ; B; ; ); a 2 ; ))] + (1 ) B c(e (a 1 ; a 2 ; B; ; ))g 0:

P
Here, the expectation operator refers to the di¤erent possible realizations of the abilities a 1 and a 2 . In order to solve the …rm's problem, …rst note that we can ignore the participation constraint: Under any ability match, each worker can ensure himself a non-negative expected utility and, hence, his reservation value, by entering the competition for bonus shares and choosing zero e¤ort. Thus, it is rational for him to accept any feasible bonus-pool contract with nonnegative B and 0 1. Moreover, the …rm optimally chooses = 1 to maximize workers' e¤ort for any given bonus pool B (see (4)). Therefore, we now de…ne e (a 1 ; a 2 ; B; ) := e (a 1 ; a 2 ; B; 1; ). The following proposition summarizes our …ndings for the optimal bonus-pool contract: 11

The proposition shows that the optimal bonus-pool contract is a winner-11 For brevity we skip 2E ["] + 2E [ ] in the …rm's objective function. Furthermore, we assume that the functional forms are such that the …rm's objective function is strictly concave in B. For example, this is the case if (i) h(e i ; a i ; ) is linear in e i and (ii) c 000 > 0. Then, by (i), g(h(e ; a 1 ; ) h(e ; a 2 ; )) @h @e1 (e ; ) is independent of e . Thus, from (ii) and (4) it follows that e is strictly concave in B. Consequently, h(e ; a i ; ) is also strictly concave in B because h(e i ; a i ; ) is increasing and strictly concave in e i .

A c c e p t e d M a n u s c r i p t

takes-all contest with the best performing worker obtaining the entire bonus pool. The optimal size of this bonus pool trades o¤ marginal incentives via implemented e¤orts e (a 1 ; a 2 ; B ( ) ; ) and marginal labor costs. 12Now we turn to the …rst stage, where the …rm chooses the production technology 2 f L ; H g. The …rm's pro…t under the optimal bonus-pool contract with pool size B = B ( ) is given by Although the …rm faces a binary decision problem, di¤erentiation of the objective function with respect to is helpful for deriving our results on the optimal technology choice. Applying the envelope theorem, the impact of technology on …rm pro…t is given by The partial derivatives of h with respect to re ‡ect the direct e¤ect of a marginal technology improvement on output for a given worker match. This e¤ect is always positive by the assumption that @h @ > 0. The remaining A c c e p t e d M a n u s c r i p t terms characterize the impact of an enhanced technology on workers'e¤ort choices and, consequently, output. By the proof of Proposition 1, in the two homogeneous matches where workers are either both of low or both of high ability, equilibrium e¤ort is increasing in the technology parameter , i.e., @e @ (a k ; a k ; B ; ) 0 for k = L; H. However, if workers are heterogeneous, equilibrium e¤ort may be decreasing, i.e., @e @ (a H ; a L ; B ; ) < 0 is possible. As outlined in the discussion of Proposition 1, @e @ (a H ; a L ; B ; ) < 0 particularly holds if technology and ability are complements and e¤ort and technology are independent. In such a situation, a better technology H > L exacerbates the problem of asymmetric worker competition and leads to a negative incentive e¤ect. If this negative incentive e¤ect dominates the direct positive impact of technology on output for all 2 [ L ; H ], then the …rm will optimally choose the less productive technology L . By contrast, if the probability of a heterogeneous worker match is rather low, then the adverse incentive e¤ect will not be decisive for the …rm's adoption decision. Formally, from ( 7) we obtain that @ (B ( ); ) @ 0 for all if p approaches either 0 or 1. The following proposition summarizes our …ndings:

(B ( ); ) = P H 2h(e (
@ (B ( ); ) @ = 2P H @h @e (e (
Proposition 3 If, for all 2 [ L ; H ], X i2fH;Lg P i @h @e
(e (a i ; a i ; B ; ); a i ; ) @e @ (a i ; a i ; B ; ) + @h @ (e (a i ; a i ; B ; ); a i ; ) < P HL X i2fH;Lg @h @e (e (a H ; a L ; B ; ) ; a i ; ) @e @ (a H ; a L ; B ; ) + @h @ (e (a H ; a L ; B ; ) ; a i ; )

then the …rm will prefer technology L to H . However, if p approaches either 0 or 1, the …rm prefers H to L . We now present an example where the …rm indeed prefers the inferior technology. Furthermore, the example allows to identify further determinants that prevent the adoption of a superior technology. We assume that the production function is h (e i ; a i ; ) = e i + a i .

(8)

Thus, e¤ort and technology are independent while ability and technology

13

A c c e p t e d M a n u s c r i p t are complements. 13 Furthermore, let " 2 " 1 be normally distributed with " 2 " 1 N (0; 2 ) and e¤ort costs be given by the exponential function c (e i ) = exp(ce i ) 1 with c > 0. 14 At the e¤ort stage, each worker's equilibrium strategy, as given by ( 4), can now be written as g((a 1 a 2 ) )( 21)B = c exp(ce )

, e = 1 c " ln (2 1) B c p 2 (a 1 a 2 ) 2 2 2 2 # : (9)
Obviously, e decreases in ja 1 a 2 j, as Proposition 1 predicts for the speci…c production function ( 8). Moreover, if a 1 6 = a 2 , then equilibrium e¤ort is smaller under H than under L for any given bonus-pool contract (B; ).

Using that = 1 under the optimal contract, at stage 2, the …rm's objective function ( 6) is given by

(B) = 2 c ln B c p 2 2p(1 p) (a H a L ) 2 2 c 2 +2 (pa H + (1 p) a L ) B;
which yields the optimal bonus pool B = 2 c . Inserting into the …rm's objective function leads to

(B ) = 2 c ln 2 c 2 p 2 1 2p(1 p) (a H a L ) 2 2 c 2 +2 (pa H + (1 p) a L ) :
13 All the following results continue to hold for a more general production function where e¤ort and technology are complements, h(e i ; a i ; ) = (1 + k )e i + a i ; as long as this complementarity is not too strong, i.e., the exogenously given constant k 0 is su¢ ciently small.

14 An exponential function allows for su¢ ciently steep cost increases to guarantee existence of a pure-strategy equilibrium in the bonus-pool game between the workers. Such cost function has also be used by [START_REF] Tadelis | The market for reputations as an incentive mechanism[END_REF], [START_REF] Kräkel | Optimal risk taking in an uneven tournament game with risk averse players[END_REF][START_REF] Kräkel | Risk taking in asymmetric tournaments[END_REF][START_REF] Kräkel | Optimal risk taking in an uneven tournament game with risk averse players[END_REF].

14

A c c e p t e d M a n u s c r i p t Hence, the …rm will prefer = L to = H if and only if

(B ; L ) > (B ; H ) , p(1 p) (a H a L ) 2 c 2 ( H + L ) > pa H + (1 p) a L , V ar [a] c 2 ( H + L ) > E [a] (10)
with V ar [a] denoting the variance and E [a] the mean of unknown worker ability from the …rm's perspective. Condition (10) can now be nicely interpreted. The left-hand side characterizes the detrimental incentive e¤ect of a better technology, whereas the right-hand side measures the positive direct impact on expected output. The condition will be satis…ed if c and 2 are rather small. Intuitively, the …rm should adopt the better technology if workers'e¤ort choices are not very responsive to incentives. Then, equilibrium e¤orts are rather small under either technology and, consequently, the detrimental incentive e¤ect of a technology improvement is negligible. E¤ort responsiveness is low when the marginal e¤ort cost function c 0 (e i ) is steep and/or the winner of the contest for B is determined by luck rather than e¤ort, i.e., if the variance of the random variable " 2 " 1 is large.

Furthermore, condition (10) holds for large values of a H a L , i.e., for a su¢ ciently high degree of worker heterogeneity. In that case, the negative incentive e¤ect of a more advanced technology is particularly strong (compare ( 9)). This …nding will be reinforced if technology itself has a signi…cant in ‡uence on output and, hence, the outcome of worker competition, i.e., if H + L is large. This also means that the …rm should not adopt the better technology if output is particularly responsive to ability (i.e., @h @a i = is large).

The impact of p on technology choice can be illustrated by a numerical example. Plotting the two pro…t functions (B ; H ) (solid graph) and (B ; L ) (dashed graph) as functions of p (using H = 2, L = 1, c = 1, = 0:2, a H = 0:2 and a L = 0:1) yields Figure 1. If p is rather small or rather large, the …rm faces a homogeneous worker match with a very high probability. In that case, the adverse incentive e¤ect due to worker heterogeneity is negligible and the …rm prefers the better technology (compare Proposition 3). However, for intermediate values of p the adverse incentive e¤ect becomes crucial so that the …rm optimally chooses L . Finally, we can compare the technology choices of a …rm in two hypothetical situations I and II that are characterized by di¤erent ability distributions in the labor market. Let V ar [a s ] = p s (1 p s ) (a Hs a Ls ) 2 denote the variance and E [a s ] = p s a Hs + (1 p s ) a Ls the mean of workers'unknown ability in situation s (s = I; II) with E [a I ] > E [a II ]. That is, situation s = I o¤ers, on average, a better worker pool than situation s = II. Then, condition (10) states that the …rm may prefer the advanced technology only in s = II but not in situation s = I. This is the case if V ar [a I ] V ar [a II ] is su¢ ciently large. In other words, although ability and technology are complements, an improved labor market (in terms of worker ability) may not foster the adoption of better technologies if the improvement is accompanied by higher talent uncertainty.

Page 17 of 27 In this section, we extend our analysis to the case of risk-averse workers. We now assume that each worker has the utility function

U (e i ) = u (I i ) c (e i ) ;
where u ( ) is monotonically increasing and concave with u(0) = 0. I i denotes worker i's monetary income, which consists of his share in the bonus pool in our context. Furthermore, c ( ) has the same properties as the cost-of-e¤ort function in Section 2 and each worker's reservation utility is still normalized to zero. Our aim is to show that, if workers are risk averse, the adverse incentive e¤ect of a superior technology may also prevent the adoption of this technology. From Proposition 1 we know that an adverse incentive e¤ect of technology adoption is, in particular, present if e¤ort and technology are independent and if technology and ability are complements. Hence, to simplify our analysis, we focus on production function ( 8), which exhibits these characteristics. In what follows, we will consider both limited and unlimited liability of workers.

In analogy to the analysis of Section 3, the two workers maximize their respective expected utilities u (( 1) B) + [u ( B) u ((1 ) B)] G (h(e 1 ; a 1 ; ) h(e 2 ; a 2 ; )) c(e 1 ) and u ((1 ) B) + [u ( B) u ((1 ) B)] [1 G (h(e 1 ; a 1 ; ) h(e 2 ; a 2 ; ))] c(e 2 ):

The two …rst-order conditions yield g(h(e 1 ; a 1 ; ) h(e 2 ; a 2 ; )) [u ( B) u ((1 ) B)] = c 0 (e 1 ) @h @e 1 (e 1 ; ) = c 0 (e 2 ) @h @e 2 (e 2 ; )

;

showing that if a pure-strategy equilibrium exists, it must be unique and symmetric, e = e 1 = e 2 , with g(h(e ; a 1 ; ) h(e ; a 2 ; )) @h @e 1 where H( ) denotes the inverse function of c 0 (e i ).

At the contract stage, the …rm chooses an optimal bonus-pool contract in order to maximize15

(B) = P H 2[H ([u ( B) u ((1 ) B)] g (0)) + a H ]+ (11)
P L 2[H ([u ( B) u ((1 ) B)] g (0)) + a L ]+ 2P HL [2H ([u ( B) u ((1 ) B)] g ((a H a L ) )) + (a H + a L ) ] B subject to the workers'participation constraint Ef[u ( B) u ((1 ) B)]
[G(h(e (a 1 ; a 2 ; B; ; ); a 1 ; ) h(e (a 1 ; a 2 ; B; ; ); a 2 ; ))] +u ((1 ) B) c(e (a 1 ; a 2 ; B; ; ))g 0

, u ((1 ) B) + [u ( B) u ((1 ) B)] EfG((a 1 a 2 ) )g
Efc(H ([u ( B) u ((1 ) B)] g ((a 1 a 2 ) )))g 0:

First, we analyze the case of limited liability with 2 [0; 1]. Since u (0) = 0 and hence u ((1 )B) 0, we can apply the same argument as in Section 3, which allows us to ignore the participation constraint. Consequently, again a winner-takes-all contest is optimal for the …rm to induce maximal incentives for a given bonus pool. Expected net pro…ts can therefore be written as

(B) = P H 2[H (u (B) g (0)) + a H ] + P L 2[H (u (B) g (0)) + a L ]+ 2P HL [2H (u (B) g ((a H a L ) )) + (a H + a L ) ] B:
Let B ( ) denote the optimal size of the bonus pool. Then, applying the envelope theorem at the …rst stage of the game, where the …rm chooses A c c e p t e d M a n u s c r i p t reduces equilibrium e¤ort due to more uneven worker competition, the workers'e¤ort costs also decrease. Under unlimited liability, the …rm gains from this reduction in e¤ort costs: It can lower each worker's expected utility from bonus payments by the same amount, making the participation constraint again just binding. At the same time, incentives can be held constant, i.e., a lower B is accompanied by a larger . By contrast, under limited liability, the …rm does not bene…t from reduced e¤ort costs because it cannot lower workers' expected utility from bonus payments without further decreasing incentives ( cannot be larger than 1).

Conclusion

The previous analysis has shown that a …rm that uses relative performance pay to provide its workforce with e¤ort incentives may refrain from implementing an advanced production technology, even if the adoption of this technology is free. A necessary condition for the …rm to prefer an inferior technology is that a worker's ability and the production technology are complementary, i.e., a better technology raises the productivity of more able workers more strongly. Then, under an enhanced technology, competition among heterogeneous workers becomes more uneven. As a consequence, workers are discouraged from exerting e¤ort. If this adverse incentive e¤ect is su¢ ciently strong, it outweighs the advantageous e¤ect of an increased productivity under the new technology.

This main result leads to several empirical predictions. First, the adverse incentive e¤ect is the stronger the more responsive the workers'e¤ort choice is to incentives. In particular, this means that …rms which are able to assess workers'performances quite precisely (i.e., 2 is low) are less inclined to adopt a superior production technology than …rms with a less accurate monitoring technology. Thus, production and monitoring technologies are substitutes.

Second, higher talent uncertainty among workers exacerbates the adverse incentive e¤ect of a new technology. Presuming that talent uncertainty decreases as workers stay longer with the …rm and are promoted along the …rm's hierarchy, our analysis suggests that a …rm bene…ts more from introducing A c c e p t e d M a n u s c r i p t new technologies on higher layers. Thus, taking into account costs for technology adoption, new technologies (e.g., computer systems) should …rst be implemented on higher hierarchy levels, while adoption on lower levels takes place as technology costs decrease.

Third, we have focused on a situation where the …rm can only use relative performance pay due to a lack of cardinal and veri…able performance signals and because relational contracts are not feasible. However, in practice, the …rm may prefer relative performance pay even if individual incentive schemes (e.g., piece rates) are, in principle, available. Important reasons may be the elimination of common shocks (see the random variable above) or lower measurement costs. Our analysis implies that, given the feasibility of different forms of incentive contracts, a …rm may want to revise its incentive scheme after the adoption of a new production technology. For example, before the availability of a new production technology, the …rm might prefer relative performance pay to individual incentive contracts because the former exhibits lower costs for measuring employee performance. However, after technology adoption, it might be worthwhile for the …rm to invest in a monitoring technology that allows it to apply individual performance pay. Then, the …rm avoids the adverse incentive e¤ect that would occur under a relative incentive scheme. In general, our analysis identi…es a new comparative advantage of individual incentive pay if (i) worker ability and the production technology are complements and (ii) the adoption of advanced technologies is crucial for …rm success.

Fourth, our paper points out that the problem of not adopting a more productive technology under relative performance pay is less severe if workers are unlimitedly liable. The reason is that, in this case, the …rm can lower the total wage sum when e¤ort incentives are diminished. Hence, labormarket regulations that imply positive worker rents (e.g., minimum wages) may impede the fast di¤usion of a new technology in industry compared to a situation where workers receive compensation equal to their outside options.

Finally, one may wonder whether welfare could strictly increase if a welfare-maximizing third party and not the …rm chooses the production tech-A c c e p t e d M a n u s c r i p t nology in stage 1. 18 In general, adoption of the more productive technology should become more likely in this situation since the third party trades o¤ the adverse incentive e¤ect against all positive technology e¤ects -including the impact on worker rents. In particular, there are cases where a third party has a strong interest in introducing the better technology: [START_REF] Gürtler | Optimal tournament contracts for heterogeneous workers[END_REF] show that under limited liability a …rm sometimes prefers to implement more than …rst-best e¤ort in order to exploit worker competition to reduce rents. In such a situation, the …rm may forego the better technology to retain incentives. By contrast, the third party may adopt the superior technology to improve productivity and, at the same time, lower e¤ort towards the e¢ cient level.

A c c e p t e d M a n u s c r i p t

Figure 1 :

 1 Figure 1: Firm pro…ts as a function of p:

  Under production technology (8), the workers' equilibrium strategy boils down to e = H ([u ( B) u ((1) B)] g ((a 1 a 2 ) )) ;

  H 2h(e (a H ; a H ; B; ; ); a H ; ) + P L 2h(e (a L ; a L ; B; ; ); a L ; ) +2P HL [h(e (a H ; a L ; B; ; ); a H ; ) + h(e (a H ; a L ; B; ; ); a L ; )]

	+2E["] + 2E[ ] B:
	10

  H ; a H ; B; ); a H ; ) + P L 2h(e (a L ; a L ; B; ); a L ; ) +2P HL [h(e (a H ; a L ; B; ); a H ; ) + h(e (a H ; a

	Proposition 2 The …rm chooses the bonus-pool contract (B; ) = (B ( ) ; 1)
	with
	B ( ) = arg max B 0	P H 2h(e (a

L ; B; ); a L ; )] B:

  a H ; a H ; B ; ); a H ; ) + P L 2h(e (a L ; a L ; B ; ); a L ; )+ 2P HL [h(e (a H ; a L ; B ; ); a H ; ) + h(e (a H ; a L ; B ; ); a L ; )] B ( ):

We discuss the case of risk-averse workers in Section 5.

A similar noise structure can be found in[START_REF] Lazear | Rank-order tournaments as optimum labor contracts[END_REF] and[START_REF] Green | A comparison of tournaments and contracts[END_REF]. The assumption of a unimodal distribution is also common in tournament models; see, e.g.,[START_REF] Dixit | Strategic behavior in contests[END_REF],[START_REF] Drago | A collective tournament[END_REF],[START_REF] Hvide | Tournament rewards and risk taking[END_REF][START_REF] Chen | Sabotage in promotion tournaments[END_REF].

Binary-signal models are often used in the literature on moral hazard; see, among many others,[START_REF] Demougin | Contractual design with correlated information under limited liability[END_REF] and[START_REF] Demougin | Monitoring versus incentives[END_REF].

The assumption of rather poor information (in particular, ordinal or unveri…able information) on worker performance is quite realistic for lower hierarchy levels. For example, several …rms apply holistic work organization where missing occupational barriers make the collection of cardinal and veri…able information prohibitively costly or even impossible. Instead, …rms rely on relative performance evaluation by workers' superiors. See[START_REF] Ichniowski | The e¤ects of human resource management practices on productivity: A study of steel …nishing lines[END_REF] and[START_REF] Lindbeck | Multitask learning and the reorganization of work: From tayloristic to holistic organization[END_REF] on holistic organizations. Relative performance evaluation is also an integral part of the traditional personnel policy of Japanese …rms; see, e.g.,Endo (

1994).6 In practice, technology adoption is typically costly, where the adoption of a more advanced production technology is more expensive to the …rm. Thus, introducing costs for technology adoption would only reinforce our result that the …rm may prefer the less productive technology.

Supplementing the bonus-pool contract, the …rm could also o¤er a …xed payment to each worker. However, we do not consider this possibility because it does not lead to a superior contract from the …rm's point of view.

As already noted byLazear and Rosen (1981, see p. 845, fn. 2) agents' objective functions are not necessarily concave under a relative performance scheme. In our model,

Our …nding is similar to Proposition 3 in[START_REF] Budde | Bonus pools, limited liability, and tournaments. Governance and the E¢ ciency of Economic Systems[END_REF]. However, Budde considers homogeneous workers and allows for more than two contestants.

Again, we neglect 2E ["] + 2E [ ], because both terms are irrelevant for the …rm's contract-design problem. Furthermore, recall that g (x) = g ( x) due to symmetry.

Note that under u 0 each worker would receive a strictly positive rent because of his zero reservation utility.

We thank two anonymous referees for valuable comments. Financial support by the Deutsche Forschungsgemeinschaft (DFG), grant SFB/TR 15, is gratefully acknowledged.

4P HL H 0 (u (B ) g ((a H a L ) )) u (B ) g 0 ((a H a L ) ) (a H a L ) :

The …rst expression on the right-hand side of ( 12) is positive and measures the direct e¤ect of a better technology on expected …rm output. The second expression is negative since g 0 ((a H a L ) ) < 0 due to g ( )'s single peakedness, indicating the detrimental incentive e¤ect. The …rm prefers the less productive technology if the …rst e¤ect is dominated by the latter one. Thus, the reasoning is quite similar to our argumentation in the case of risk-neutral workers based on condition (7).

If workers are of unlimited liability, payments to workers can be negative so that we can drop the restriction 2 [0; 1]. Consequently, we cannot ignore the participation constraint any longer. To simplify notation, we de…ne

with v := u 1 being monotonically increasing and strictly convex. Thus, 11) and the new de…nitions, the …rm's contractdesign problem can be rewritten as follows:

2 due to the symmetry of g ( ).

A c c e p t e d M a n u s c r i p t

Let (u ; u ) = (u ( ) ; u ( )) denote the solution to the contract-design problem, leading to pro…ts = (u ; u ). At the …rst stage, the Lagrangian to the …rm's problem with solution (u ; u ) reads as

with 0 as the Lagrange multiplier for the participation constraint. Clearly, the participation constraint is binding at the optimum. Otherwise, the …rm could decrease B and, at the same time, appropriately increase so that u u = u ( B) u (( 1) B) remains constant. In other words, by reducing the size of the bonus pool and increasing the share of the better performing worker the …rm could reduce its labor costs while holding incentives constant. The …rm would proceed in this way and reduce u (( 1) B) until the participation constraint becomes binding. Thus, in general, we have > 0. Furthermore, in the optimum, u is negative and hence > 1. 17 In order to analyze the impact of technology choice on optimal …rm pro…ts , we have to apply the envelope theorem to the Lagrangian L (u ; u ; ):

Here, we de…ne a := a H a L and D 12) and ( 13) shows that, under unlimited liability, the direct productivity e¤ect on output and the detrimental incentive e¤ect work again into opposite directions (…rst line of ( 13)). However, now we have an additional positive e¤ect (second line of ( 13)) that mitigates the adverse incentive e¤ect compared to the case of limited liability. This additional e¤ect makes the introduction of a better technology more likely under unlimited than under limited liability. When a better technology